el hasnaoui khalid

African Journal Of Mathematical Physics Volume 8(2010)101-114

Casimir force in confined biomembranes

K. El Hasnaoui, Y. Madmoune, H. Kaidi,M. Chahid, and M. Benhamou

Laboratoire de Physique des Polymeres et Phenomenes CritiquesFaculte des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco

[email protected]

abstract

We reexamine the computation of the Casimir force between two parallel interactingplates delimitating a liquid with an immersed biomembrane. We denote by D their sepa-ration, which is assumed to be much smaller than the bulk roughness, in order to ensurethe membrane confinement. This repulsive force originates from the thermal undulationsof the membrane. To this end, we first introduce a field theory, where the field is not-ing else but the height-function. The field model depends on two parameters, namelythe membrane bending rigidity constant, κ, and some elastic constant, µ ∼ D−4. Wefirst compute the static Casimir force (per unit area), Π, and find that the latter decayswith separation D as : Π ∼ D−3, with a known amplitude scaling as κ−1. Therefore,the force has significant values only for those biomembranes of small enough κ. Second,we consider a biomembrane (at temperature T ) that is initially in a flat state away fromthermal equilibrium, and we are interested in how the dynamic force, Π (t), grows in time.To do calculations, use is made of a non-dissipative Langevin equation (with noise) thatis solved by the time height-field. We first show that the membrane roughness, L⊥ (t),increases with time as : L⊥ (t) ∼ t1/4 (t < τ), with the final time τ ∼ D4 (required timeover which the final equilibrium state is reached). Also, we find that the force increases intime according to : Π (t) ∼ t1/2 (t < τ). The discussion is extended to the real situationwhere the biomembrane is subject to hydrodynamic interactions caused by the surround-ing liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh) and Πh (t) ∼ t2/3 (t < τh),with the new final time τh ∼ D3. Consequently, the hydrodynamic interactions lead tosubstantial changes of the dynamic properties of the confined membrane, because bothroughness and induced force grow more rapidly. Finally, the study may be extended, ina straightforward way, to bilayer surfactants confined to the same geometry.Key words: Biomembranes - Confinement - Casimir force - Dynamics.

I. INTRODUCTION

The cell membranes are of great importance to life, because they separate the cell from the surroundingenvironment and act as a selective barrier for the import and export of materials. More details concern-ing the structural organization and basic functions of biomembranes can be found in Refs. [1− 7]. Itis well-recognized by the scientific community that the cell membranes essentially present as a phospho-lipid bilayer combined with a variety of proteins and cholesterol (mosaic fluid model). In particular, thefunction of the cholesterol molecules is to ensure the bilayer fluidity. A phospholipid is an amphiphile

0c⃝ a GNPHE publication 2010, [email protected]

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K. El Hasnaoui et al. African Journal Of Mathematical Physics Volume 8(2010)101-114

molecule possessing a hydrophilic polar head attached to two hydrophobic (fatty acyl) chains. The phos-pholipids move freely on the membrane surface. On the other hand, the thickness of a bilayer membraneis of the order of 50 Angstroms. These two properties allow to consider it as a two-dimensional fluidmembrane. The fluid membranes, self-assembled from surfactant solutions, may have a variety of shapesand topologies [8], which have been explained in terms of bending energy [9, 10].In real situations, the biomembranes are not trapped in liquids of infinite extent, but they rather con-fined to geometrical boundaries, such as white and red globules or liposomes (as drugs transport agents[11− 14]) in blood vessels. For simplicity, we consider the situation where the biomembrane is confinedin a liquid domain that is finite in one spatial direction. We denote by D its size in this direction. Fora tube, D being the diameter, and for a liquid domain delimitated by two parallel plates, this size issimply the separation between walls. Naturally, the length D must be compared to the bulk roughness,L0⊥, which is the typical size of humps caused by the thermal fluctuations of the membrane. The latter

depends on the nature of lipid molecules forming the bilayer. The biomembrane is confined only whenD is much smaller than the bulk roughness L0

⊥. This condition is similar to that usually encountered inconfined polymers context [15].The membrane undulations give rise to repulsive effective interactions between the confining geometricalboundaries. The induced force we term Casimir force is naturally a function of the size D, and mustdecays as this scale is increased. In this paper, we are interested in how this force decays with distance.To simplify calculations, we assume that the membrane is confined to two parallel plates that are a finitedistance D < L0

⊥ apart.The word ”Casimir” is inspired from the traditional Casimir effect. Such an effect, predicted, for thefirst time, by Hendrick Casimir in 1948 [16], is one of the fundamental discoveries in the last century.According to Casimir, the vacuum quantum fluctuations of a confined electromagnetic field generate anattractive force between two parallel uncharged conducting plates. The Casimir effect has been confirmedin more recent experiments by Lamoreaux [17] and by Mohideen and Roy [18]. Thereafter, Fisher andde Gennes [19], in a short note, remarked that the Casimir effect also appears in the context of criticalsystems, such as fluids, simple liquid mixtures, polymer blends, liquid 4He, or liquid-crystals, confinedto restricted geometries or in the presence of immersed colloidal particles. For these systems, the criticalfluctuations of the order parameter play the role of the vacuum quantum fluctuations, and then, theylead to long-ranged forces between the confining walls or between immersed colloids [20, 21].To compute the Casimir force between the confining walls, we first elaborate a more general field theorythat takes into account the primitive interactions experienced by the confined membrane. As we shall seebelow, in confinement regime, the field model depends only on two parameters that are the membranebending rigidity constant and a coupling constant containing all infirmation concerning the interactionpotential exerting by the walls. In addition, the last parameter is a known function of the separationD. With the help of the constructed free energy, we first computed the static Casimir force (per unitarea), Π. The exact calculations show that the latter decays with separation D according to a power

law, that is Π ∼ κ−1 (kBT )2D−3, with a known amplitude. Here, kBT denotes the thermal energy, and

κ the membrane bending rigidity constant. Of course, this force increases with temperature, and hassignificant values only for those biomembranes of small enough κ. The second problem we examined isthe computation of the dynamic Casimir force, Π (t). More precisely, we considered a biomembrane attemperature T that is initially in a flat state away from the thermal equilibrium, and we were interested inhow the expected force grows in time, before the final state is reached. Using a scaling argument, we firstshowed that the membrane roughness, L⊥ (t), grows with time as : L⊥ (t) ∼ t1/4 (t < τ), with the finaltime τ ∼ D4. The latter can be interpreted as the required time over which the final equilibrium stateis reached. Second, using a non-dissipative Langevin equation, we found that the force increases in timeaccording to the power law : Π (t) ∼ t1/2 (t < τ). Third, the discussion is extended to the real situationwhere the biomembrane is subject to hydrodynamic interactions caused by the flow of the surroundingliquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh) and Π (t) ∼ t2/3 (t < τh), with the new finaltime τh ∼ D3. Consequently, the hydrodynamic interactions give rise to drastic changes of the dynamicproperties of the confined membrane, since both roughness and induced force grow more rapidly.This paper is organized as follows. In Sec. II, we present the field model allowing the determination ofthe Casimir force from a static and dynamic point of view. The Sec. III and Sec. IV are devoted to thecomputation of the static and dynamic induced forces, respectively. We draw our conclusions in the lastsection. Some technical details are presented in Appendix.

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II. THEORETICAL FORMULATION

Consider a fluctuating fluid membrane that is confined to two interacting parallel walls 1 and 2. Wedenote by D their finite separation. Naturally, the separation D must be compared to the bulk membraneroughness, L0

⊥, when the system is unconfined (free membrane). The membrane is confined only whenthe condition L << L0

⊥ is fulfilled. For the opposite condition, that is L >> L0⊥, we expect finite size

corrections.We assume that these walls are located at z = −D/2 and z = D/2, respectively. Here, z means theperpendicular distance. For simplicity, we suppose that the two surfaces are physically equivalent. Wedesign by V (z) the interaction potential exerted by one wall on the fluid membrane, in the absence ofthe other. Usually, V (z) is the sum of a repulsive and an attractive potentials. A typical example isprovided by the following potential [22]

V (z) = Vh (z) + VvdW (z) , (2.0)

where

Vh (z) = Ahe−z/λh (2.0)

represents the repulsive hydration potential due to the water molecules inserted between hydrophiliclipid heads [22]. The amplitude Ah and the potential-range λh are of the order of : Ah ≃ 0.2 J/m2 andλh ≃ 0.2−0.3 nm. In fact, the amplitude Ah is Ah = Ph×λh, with the hydration pressure Ph ≃ 108−109

Pa. There, VvdW (z) accounts for the van der Waals potential between one wall and biomembrane, whichare a distance z apart. Its form is as follows

VvdW (z) = − H

12π

[1

z2− 2

(z + δ)2 +

1

(z + 2δ)2

], (2.0)

with the Hamaker constant H ≃ 10−22 − 10−21 J, and δ ≃ 4 nm denotes the membrane thickness. Forlarge distance z, this implies

VvdW (z) ∼Wδ2

z4. (2.0)

Generally, in addition to the distance z, the interaction potential V (z) depends on certain length-scales,(ξ1, ..., ξn), which are the interactions ranges. The fluid membrane then experiences the following totalpotential

U (z) = V

(D

2− z

)+ V

(D

2+ z

), −D

2≤ z ≤ D

2. (2.0)

In the Monge representation, a point on the membrane can be described by the three-dimensional positionvector r = (x, y, z = h (x, y)), where h (x, y) ∈ [−D/2, D/2] is the height-field. The latter then fluctuatesaround the mid-plane located at z = 0.The Statistical Mechanics for the description of such a (tensionless) fluid membrane is based on thestandard Canham-Helfrich Hamiltonian [9, 23]

H [h] =

∫dxdy

[κ2(∆h)

2+W (h)

], (2.0)

with the membrane bending rigidity constant κ. The latter is comparable to the thermal energy kBT ,where T is the absolute temperature and kB is the Boltzmann’s constant. There, W (h) is the interactionpotential per unit area, that is

W (h) =U (h)

L2, (2.0)

where the potential U (h) is defined in Eq. (2), and L is the lateral linear size of the biomembrane.Let us discuss the pair-potential W (h).

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Firstly, Eq. (2) suggests that this total potential is an even function of the perpendicular distance h, thatis

W (−h) = W (h) . (2.0)

In particular, we have W (−D/2) = W (D/2).Secondly, when they exist, the zeros h0’s of the potential function U (h) are such that

V

(D

2− h0

)= −V

(D

2+ h0

). (2.0)

This equality indicates that, if h0 is a zero of the potential function, then, −h0 is a zero too. The numberof zeros is then an even number. In addition, the zero h0’s are different from 0, in all cases. Indeed, thequantity V (D/2) does not vanish, since it represents the potential created by one wall at the middle ofthe film. We emphasize that, when the potential processes no zero, it is either repulsive or attractive.When this same potential vanishes at some points, then, it is either repulsive of attractive between twoconsecutive zeros.Thirdly, we first note that, from relation (2), we deduce that the first derivative of the potential function,with respect to distance h, is an odd function, that is W ′ (−h) = −W ′ (h). Applying this relation tothe midpoint h = 0 yields : W ′ (0) = 0. Therefore, the potential W exhibits an extremum at h = 0,whatever the form of the function V (h). We find that this extremum is a maximum, if V ′′ (D/2) < 0,and a minimum, if V ′′ (D/2) > 0. The potential U presents an horizontal tangent at h = 0, if only ifV ′′ (D/2) = 0. On the other hand, the general condition giving the extrema hm is

dV

dh

∣∣∣∣h=D

2 −hm

=dV

dh

∣∣∣∣h=D

2 +hm

. (2.0)

Since the first derivative W ′ (h) is an odd function of distance h, it must have an odd number of extremumpoints. The point h = hm is a maximum, if

d2V

dh2

∣∣∣∣h=D

2 −hm

< − d2V

dh2

∣∣∣∣h=D

2 +hm

, (2.0)

and a minimum, if

d2V

dh2

∣∣∣∣h=D

2 −hm

> − d2V

dh2

∣∣∣∣h=D

2 +hm

. (2.0)

At point h = hm, we have an horizontal tangent, if

d2V

dh2

∣∣∣∣h=D

2 −hm

= − d2V

dh2

∣∣∣∣h=D

2 +hm

. (2.0)

The above deductions depends, of course, on the form of the interaction potential V (h).Fourthly, a simple dimensional analysis shows that the total interaction potential can be rewritten on thefollowing scaling form

W (h)

kBT=

1

D2Φ

(h

D,ξ1D, ...,

ξnD

), (2.0)

where (ξ1, ..., ξn) are the ranges of various interactions experienced by the membrane, and Φ (x1, ..., xn+1)is a (n+ 1)-factor scaling-function.Finally, we note that the pair-potential W (h) cannot be singular at h = 0. It is rather an analyticfunction in the h variable. Therefore, at fixed ratios ξi/D, an expansion of the scaling-function Φ, aroundthe value h = 0, yields

W (h)

kBT=

γ

2

h2

D4+O

(h4

). (2.0)

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We restrict ourselves to the class of potentials that exhibit a minimum at the mid-plane h = 0. Thisassumption implies that the coefficient γ is positive definite, i.e. γ > 0. Of course, such a coefficientdepends on the ratios of the scale-lengths ξi to the separation D.In confinement regime where the distance h is small enough, we can approximate the total interactionpotential by its quadratic part. In these conditions, the Canham-Helfrich Hamiltonian becomes

H0 [h] =1

2

∫dxdy

[κ (∆h)

2+ µh2

], (2.0)

with the elastic constant

µ = γkBT

D4. (2.0)

The prefactor γ will be computed below. The above expression for the elastic constant µ gives an ideaon its dependance on the film thickness D. In addition, we state that this coefficient may be regarded asa Lagrange multiplier that fixes the value of the membrane roughness.Thanks to the above Hamiltonian, we calculate the mean-expectation value of the physical quantities,like the height-correlation function (propagator or Green function), defined by

G (x− x′, y − y′) = ⟨h (x, y)h (x′, y′)⟩ − ⟨h (x, y)⟩ ⟨h (x′, y′)⟩ . (2.0)

The latter solves the linear differential equation(κ∆2 + µ

)G (x− x′, y − y′) = δ (x− x′) δ (y − y′) , (2.0)

where δ (x) denotes the one-dimensional Dirac function, and ∆ = ∂2/∂x2 + ∂2/∂y2 represents the two-dimensional Laplacian operator. We have used the notations : κ = κ/kBT and µ = µ/kBT , to mean thereduced membrane elastic constants.From the propagator, we deduce the expression of the membrane roughness

L2⊥ =

⟨h2

⟩− ⟨h⟩2 = G (0, 0) . (2.0)

Such a quantity measures the fluctuations of the height-function (fluctuations amplitude) around theequilibrium plane located at z = 0. We show in Appendix that the membrane roughness is exactly givenby

L2⊥ =

D2

12, (2.0)

provided that one is in the confinement-regime, i.e. D << L0⊥. Notice that the above equality indicates

that the roughness is independent on the geometrical properties of the membrane (through κ). Weemphasize that this relation can be recovered using the argument that each point of the membrane hasequal probability to be found anywhere between the walls [24].The elastic constant µ may be calculated using the known relation

L2⊥ =

1

8

kBT√µκ

. (2.0)

This gives

µ =9

4

(kBT )2

κD4. (2.0)

This formula clearly shows that this elastic constant decays with separation D as D−4. The term µh2/2then describes a confinement potential that ensures the localization of the membrane around the mid-plane. Integral over the hole plane R2 of this term represents the loss entropy due to the confinement ofthe membrane. The value (19) of the elastic constant is compatible with the constraint (17).Therefore, the elaborated model is based on the Hamiltonian (13), with a quadratic confinement poten-tial. We can say that the presence of the walls simply leads to a confinement of the membrane in a region

105


of the infinite space of perpendicular size L⊥.We define now another length-scale that is the in-plane correlation length, L∥. The latter mea-sures the correlations extent along the parallel directions to the walls. More precisely, the prop-agator G (x− x′, y − y′) fails exponentially beyond L∥, that is for distances d such that d =√

(x− x′)2+ (y − y′)

2> L∥.

From the standard relation

L2⊥ =

kBT

16κL2∥ , (2.0)

we deduce

L∥ =2√3

(κ

kBT

)1/2

D . (2.0)

In contrary to L⊥, the length-scale L∥ depends on the geometrical characteristics of the membrane(through κ).The next steps consist in the computation of the Casimir force at and out equilibrium.

III. STATIC CASIMIR FORCE

When viewed under the microscope, the membranes of vesicles present thermally excited shape fluctu-ations. Generally, objects such as interfaces, membranes or polymers undergo such fluctuations, in orderto increase their configurational entropy. For bilayer biomembranes and surfactants, the consequence ofthese undulations is that, they give rise to an induced force called Casimir force.To compute the desired force, we start from the partition function constructed with the Hamiltoniandefined in Eq. (13). This partition function is the following functional integral

Z =

∫Dh exp

−H0 [h]

kBT

, (3.0)

where integration is performed over all height-field configurations. The associated free energy is suchthat : F = −kBT lnZ, which is, of course, a function of the separation D. If we denote by Σ = L2 thecommon area of plates, the Casimir force (per unit area) is minus the first derivative of the free energy(per unit area) with respect to the film-thickness D, that is

Π = − 1

Σ

∂F∂D

. (3.0)

This force per unit area is called disjoining pressure. In fact, Π is the required pressure to maintain thetwo plates at some distance D apart. In term of the partition function, the disjoining pressure rewrites

Π

kBT=

1

Σ

∂ lnZ

∂D=

1

Σ

∂µ

∂D

∂ lnZ

∂µ. (3.0)

Using definition (19) together with Eqs. (23) and (24) yields

Π = −1

2

∂µ

∂DL2⊥ . (3.0)

Explicitly, we obtain the desired formula

Π =3

8

(kBT )2

κD3. (3.0)

From this relation, we extract the expression of the disjoining potential (per unit area) [25]

Vd (D) = −∫ D

∞Π(D′) dD′ =

3

16

(kBT )2

κD2. (3.0)

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The above expression of the Casimir force (per unit area) calls the following remarks.Firstly, this force decays with distance more slowly in comparison to the Coulombian one that decreasesrather as D−2.Secondly, this same force depends on the nature of lipids forming the bilayer (through κ). In this sense,contrarily to the Casimir effect in Quantum Field Theory [16] and in Critical Phenomena [20], the presentforce is not universal. Incidentally, if this force is multiplied by κ, then, it will become a universal quantity.Thirdly, at fixed temperature and distance, the force amplitude has significant values only for those bilayermembrane of small bending rigidity constant.Fourthly, as it should be, such a force increases with increasing temperature. Indeed, at high temperature,the membrane undulations are strong enough.Finally, the numerical prefactor 3/8 (Helfrich’s cH -amplitude [9]) is close to the value obtained usingMonte Carlo simulation [26].In Fig. 1, we superpose the variations of the reduced static Casimir force Π/kBT upon separation D, fortwo lipid systems, namely SOPC and DAPC [27], at temperature T = 18C. The respective membranebending rigidity constants are : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J. These values correspondto the renormalized bending rigidity constants : κ = 23.9 and κ = 12.2. The used methods for themeasurement of these rigidity constants were entropic tension and micropipet [27]. These curves reflectthe discussion made above.

FIG. 1. Reduced static Casimir force, Π/kBT , versus separation D, for two lipid systems that are SOPC (solidline) and DAPC (dashed line), of respective membrane bending rigidity constants : κ = 0.96 × 10−19 J andκ = 0.49× 10−19 J, at temperature T = 18C. The reduced force and separation are expressed in arbitrary units.

IV. DYNAMIC CASIMIR FORCE

To study the dynamic phenomena, the main physical quantity to consider is the time height-field,h (r, t), where r = (x, y) ∈ R2 denotes the position vector and t the time. The latter represents the timeobservation of the system before it reaches its final equilibrium state. We recall that the time heightfunction h (r, t) solves a non-dissipative Langevin equation (with noise) [28]

∂h (r, t)

∂t= −Γ

δH0 [h]

δh (r, t)+ ν (r, t) , (4.0)

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where Γ > 0 is a kinetic coefficient. The latter has the dimension : [Γ] = L40T

−10 , where L0 is some length

and T0 the time unit. Here, ν (r, t) is a Gaussian random force with mean zero and variance

⟨ν (r, t) ν (r′, t′)⟩ = 2Γδ2 (r − r′) δ (t− t′) , (4.0)

and H0 is the static Hamiltonian (divided by kBT ), defined in Eq. (13).The bare time correlation function, whose Fourier transform is the dynamic structure factor, is definedby the expectation mean-value over noise ν

G (r − r′, t− t′) = ⟨h (r, t)h (r′, t′)⟩ν − ⟨h (r, t)⟩ν ⟨h (r′, t′)⟩ν , t > t′ . (4.0)

The dynamic equation (28) shows that the time height function h is a functional of noise ν, and wewrite : h = h [ν]. Instead of solving the Langevin equation for h [ν] and then averaging over the noisedistribution P [ν], the correlation and response functions can be directly computed by means of a suitablefield-theory, of action [28− 31]

A[h, h

]=

∫dt

∫d2r

h∂th+ Γh

δH0

δh− hΓh

, (4.0)

so that, for an arbitrary observable, O [ζ], one has

⟨O⟩ν =

∫[dν]O [φ [ν]]P [ν] =

∫DhDhOe−A[h,h]∫DhDhe−A[h,h]

, (4.0)

where h (r, t) is an auxiliary field, coupled to an external field h (r, t). The correlation and responsefunctions can be computed replacing the static Hamiltonian H0 appearing in Eq. (13), by a new one :H0 [h, J ] = H0 [h]−

∫d2rJh. Consequently, for a given observable O, we have

δ ⟨O⟩JδJ (r, t)

∣∣∣∣J=0

= Γ⟨h (r, t)O

⟩. (4.0)

The notation ⟨ . ⟩J means the average taken with respect to the action A[h, h, J

]associated with the

Hamiltonian H0 [h, J ]. In view of the structure of equality (33), h is called response field. Now, if O = h,we get the response of the order parameter field to the external perturbation J

R (r − r′, t− t′) =δ ⟨h (r′, t′)⟩J

δJ (r, t)

∣∣∣∣J=0

= Γ⟨h (r, t)h (r′, t′)

⟩J=0

. (4.0)

The causality implies that the response function vanishes for t < t′. In fact, this function can be related tothe time-dependent (connected) correlation function using the fluctuation-dissipation theorem, accordingto which

Γ⟨h (r, t)h (r′, t′)

⟩= −θ (t− t′) ∂t ⟨h (r, t)h (r′, t′)⟩c . (4.0)

The above important formula shows that the time correlation function C (r − r′, t− t′) =⟨φ (r, t)φ (r′, t′)⟩c may be determined by the knowledge of the response function. In particular, weshow that

L2⊥ (t) =

⟨h2 (r, t)

⟩c= −2Γ

∫ t

−∞dt′

⟨h (r, t′)h (r, t′)

⟩. (4.0)

The limit t → −∞ gives the natural value L2⊥ (−∞) = 0, since, as assumed, the initial state corresponds

to a completely flat interface.Consider now a membrane at temperature T that is initially in a flat state away from thermal equilibrium.At a later time t, the membrane possesses a certain roughness, L⊥ (t). Of course, the latter is time-dependent, and we are interested in how it increases in time.

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We point out that the thermal fluctuations give rise to some roughness that is characterized by theappearance of anisotropic humps. Therefore, a segment of linear size L effectuates excursions of size [32]

L⊥ = BLζ . (4.0)

Such a relation defines the roughness exponent ζ. Notice that L is of the order of the in-plane correlationlength, L∥. From relation (20), we deduce the exponent ζ and the amplitude B. Their respective values

are : ζ = 1 and B ∼ (kBT/κ)1/2

.In order to determine the growth of roughness L⊥ in time, the key is to consider the excess free energy(per unit area) due to the confinement, ∆F . Such an excess is related to the fact that the confiningmembrane suffers a loss of entropy. Formula (27) tells us how ∆F must decay with separation. Theresult reads [32]

∆F ∼ kBT/L2max ∼ kBT (B/L⊥)

2/ζ, (4.0)

where Lmax represents the wavelength above which all shape fluctuations are not accessible by the confinedmembrane. The repulsive fluctuation-induced interaction leads to the disjoining pressure

Π = −∂∆F

∂L⊥∼ L

−(1+2/ζ)⊥ . (4.0)

In addition, a care analysis of the Langevin equation (28) shows that

∂L⊥

∂t∼ −Γ

∂∆F

∂L⊥= Γ×Π ∼ ΓL

−(1+2/ζ)⊥ . (4.0)

We emphasize that this scaling form agrees with Monte Carlo predictions [32, 33]. Solving this first-orderdifferential equation yields [34]

L⊥ (t) ∼ Γθ⊥tθ⊥ , θ⊥ =ζ

2 + 2ζ=

1

4. (4.0)

This implies the following scaling form for the linear size

L (t) ∼ Γθ∥tθ∥ , θ∥ =1

2 + 2ζ=

1

4. (4.0)

Let us comment about the obtained result (39).Firstly, as it should be, the roughness increases with time (the exponent θ⊥ is positive definite). Inaddition, the exponent θ⊥ is universal, independently on the membrane bending rigidity constant κ.Secondly, we note that, in Eq. (39), we have ignored some non-universal amplitude that scales as κ−1/4.This means that the time roughness is significant only for those biomembranes of small bending rigidityconstant.Fourthly, this time roughness can be interpreted as the perpendicular size of holes and valleys at time t.Fifthly, the roughness increases until a fine time, τ . The latter can be interpreted as the time over whichthe system reaches its final equilibrium state. This characteristic time then scales as

τ ∼ Γ−1L1/θ⊥⊥ , (4.0)

where we have ignored some non-universal amplitude that scales as κ. Here, L⊥ ∼ D is the finalroughness. Explicitly, we have

τ ∼ Γ−1D4 . (4.0)

As it should be, the final time increases with increasing film thickness D.On the other hand, we can rewrite the behavior (39) as

L⊥ (t)

L⊥ (τ)=

(t

τ

)θ⊥

. (4.0)

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This equality means that the roughness ratio, as a function of the reduced time, is universal.Now, to compute the dynamic Casimir force, we start from a formula analog to that defined in Eq. (24),that is

Π (t)

kBT=

1

Σ

∂ ln Z

∂D=

1

Σ

∂µ

∂D

∂ ln Z

∂µ, (4.0)

with the new partition function

Z =

∫DhDhe−A[h,h] . (4.0)

A simple algebra taking into account the basic relation (35a) gives

Π (t)

kBT= −1

2

∂µ

∂DL2⊥ (t) , (4.0)

which is very similar to the static relation defined in Eq. (25), but with a time-dependent membraneroughness, L⊥ (t).Combining formulae (43) and (46) leads to the desired expression for the time Casimir force (per unitarea)

Π (t)

Π (τ)=

(t

τ

)θf

, (4.0)

where Π (τ) is the final static Casimir force, relation (25). The force exponent, θf , is such that

θf = 2θ⊥ =ζ

1 + ζ=

1

2. (4.0)

The induced force then grows with time as t1/2 until it reaches its final value Π (τ). At fixed time andseparation D, the force amplitude depends, of course, on κ, and decreases in this parameter accordingto κ−3/2. Also, we note that the above equality means that the force ratio as a function of the reducedtime is universal.In Fig. 2, we draw the reduced dynamic Casimir force, Π (t) /Π(τ), upon the renormalized time t/τ .

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FIG. 2. Reduced dynamic Casimir force, Π (t) /Π(τ), upon the renormalized time t/τ .

Finally, consider again a membrane which is initially flat but is now coupled to overdamped surfacewaves. This real situation corresponds to a confined membrane subject to hydrodynamic interactions.The roughness now grows as [35]

L⊥ (t) ∼ tθ⊥ , θ⊥ =ζ

1 + 2ζ=

1

3. (4.0)

Therefore, the roughness increases with time more rapidly than that relative to biomembranes free fromhydrodynamic interactions.In this case, the dynamic Casimir force is such that

Πh (t)

Π (τh)=

(t

τh

)θf

, (4.0)

where Π (τh) is the final static Casimir force, relation (25). The new force exponent is

θf = 2θ⊥ =2ζ

1 + 2ζ=

2

3. (4.0)

There, τh ∼ D3 accounts for the new time-scale over which the confined membrane reaches its finalequilibrium state. Therefore, the dynamic Casimir force decays with time as t2/3, that is more rapidlythan that where the hydrodynamic interactions are ignored, which scales rather as t1/2. As we saidbefore, this drastic change can be attributed to the overdamped surface waves that develop larger andlarger humps.We depict, in Fig. 3, the variation of the reduced dynamic force (with hydrodynamic interactions),Πh (t) /Π(τh), upon the renormalized time t/τh.

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FIG. 3. Reduced dynamic Casimir force (with hydrodynamic interactions), Πh (t) /Π(τ), upon the renormalizedtime t/τh.

V. CONCLUSIONS

In this work, we have reexamined the computation of the Casimir force between two parallel wallsdelimitating a fluctuating fluid membrane that is immersed in some liquid. This force is caused by thethermal fluctuations of the membrane. We have studied the problem from both static and dynamic pointof view.We were first interested in the time variation of the roughening, L⊥ (t), starting with a membrane that isinially in a flat state, at a certain temperature. Of course, this length grows with time, and we found that: L⊥ (t) ∼ tθ⊥ (θ⊥ = 1/4), provided that the hydrodynamic interactions are ignored. For real systems,however, these interactions are important, and we have shown that the roughness increases more rapidly

as : L⊥ (t) ∼ tθ⊥ (θ⊥ = 1/3). The dynamic process is then stopped at a final τ (or τh) that representsthe required time over which the biomembrane reaches its final equilibrium state. The final time behavesas : τ ∼ D4 (or τh ∼ D3), with D the film thickness.Now, assume that the system is explored at scales of the order of the wavelength q−1, where q =(4π/λ) sin (θ/2) is the wave vector modulus, with λ the wavelength of the incident radiation and θ thescattering-angle. In these conditions, the relaxation rate, τ (q), scales with q as : τ−1 (q) ∼ q1/θ⊥ = q4

or(τ−1h (q) ∼ q1/θ⊥ = q3

). Physically speaking, the relaxation rate characterizes the local growth of the

height fluctuations.Afterwards, the question was addressed to the computation of the Casimir force, Π. At equilibrium,using an appropriate field theory, we found that this force decays with separation D as : Π ∼ D−3, witha known amplitude scaling as κ−1, where κ is the membrane bending rigidity constant. Such a force isthen very small in comparison with the Coulombian one. In addition, this force disappears when the

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temperature of the medium is sufficiently lowered.The dynamic Casimir force, Π (t), was computed using a non-dissipative Langevin equation (with noise),solved by the time height-field. We have shown that : Π (t) ∼ tθf (θf = 2θ⊥ = 1/2). When the hydro-dynamic interactions effects are important, we found that the dynamic force increases more rapidly as :

Πh (t) ∼ tθf(θf = 2θ⊥ = 2/3

).

Notice that we have ignored some details such as the role of inclusions (proteins, cholesterol, glycolipids,other macromolecules) and chemical mismatch on the force expression. It is well-established that thesedetails simply lead to an additive renormalization of the bending rigidity constant. Indeed, we writeκeffective = κ + δκ, where κ is the bending rigidity constant of the membrane free from inclusions, andδκ is the contribution of the incorporated entities. Generally, the shift δκ is a function of the inclusionconcentration and compositions of species of different chemical nature (various phospholipids formingthe bilayer). Hence, to take into account the presence of inclusions and chemical mismatch, it would besufficient to replace κ by κeffective, in the above established relations.As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants,although the two systems are not of the same structure and composition. One of the differences is themagnitude order of the bending rigidity constant.

APPENDIX

To show formula (17), we start from the partition function that we rewrite on the following form

Z =

∫Dh exp

−H [h]

kBT

=

D/2∫−D/2

dzΦ(z) . (5.0)

Also, it is easy to see that the membrane mean-roughness is given by

L2⊥ =

D/2∫−D/2

dzz2Φ(z)

D/2∫−D/2

dzΦ(z)

. (5.0)

The restricted partition function is

Φ (z) =

∫Dhδ [z − h (x0, y0)] exp

−H [h]

kBT

. (5.0)

Here, H [h] is the original Hamiltonian defined in Eq. (3). Of course, this definition is independent on thechosen point (x0, y0), because of the translation symmetry along the parallel directions to plates. Noticethat the above function is not singular, whatever the value of the perpendicular distance.

Since we are interested in the confinement-regime, that is when the separation D is much smaller thanthe membrane mean-roughness L0

⊥(z ∼ h << L0

⊥), we can replace the function Φ par its value at z = 0,

denoted Φ0. In this limit, Eq. (A.2) gives the desired result.This ends the proof of the expected formula.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions,during the ”First International Workshop On Soft-Condensed Matter Physics and Biological Systems”,14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbahfor fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University ofGrenoble) for their kinds of hospitalities during his regular visits.

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REFERENCES

1 M.S. Bretscher and S. Munro, Science 261, 12801281 (1993).2 J. Dai and M.P. Sheetz, Meth. Cell Biol. 55, 157171 (1998).3 M. Edidin, Curr. Opin. Struc. Biol. 7, 528532 (1997).4 C.R. Hackenbrock, Trends Biochem. Sci. 6, 151154 (1981).5 C. Tanford, The Hydrophobic Effect, 2d ed., Wiley, 1980.6 D.E. Vance and J. Vance, eds., Biochemistry of Lipids, Lipoproteins, and Membranes, Elsevier, 1996.7 F. Zhang, G.M. Lee, and K. Jacobson, BioEssays 15, 579588 (1993).8 S. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA,1994.

9 W. Helfrich, Z. Naturforsch. 28c, 693 (1973).10 For a recent review, see U. Seifert, Advances in Physics 46, 13 (1997).11 H. Ringsdorf and B. Schmidt, How to Bridge the Gap Between Membrane, Biology and Polymers Science, P.M.

Bungay et al., eds, Synthetic Membranes : Science, Engineering and Applications, p. 701, D. Reiidel PulishingCompagny, 1986.

12 D.D. Lasic, American Scientist 80, 250 (1992).13 V.P. Torchilin, Effect of Polymers Attached to the Lipid Head Groups on Properties of Liposomes, D.D. Lasic

and Y. Barenholz, eds, Handbook of Nonmedical Applications of Liposomes, Volume 1, p. 263, RCC Press, BocaRaton, 1996.

14 R. Joannic, L. Auvray, and D.D. Lasic, Phys. Rev. Lett. 78, 3402 (1997).15 P.-G. de Gennes, Scaling Concept in Polymer Physics, Cornell University Press, 1979.16 H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap B 51, 793 (1948).17 S.K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997).18 U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998).19 M.E. Fisher and P.-G. de Gennes, C. R. Acad. Sci. (Paris) Ser. B 287, 207 (1978); P.-G. de Gennes, C. R.

Acad. Sci. (Paris) II 292, 701 (1981).20 M. Krech, The Casimir Effect in Critical Systems, World Scientific, Singapore, 1994.21 More recent references can be found in : F. Schlesener, A. Hanke, and S. Dietrich, J. Stat. Phys. 110, 981

(2003); M. Benhamou, M. El Yaznasni, H. Ridouane, and E.-K. Hachem, Braz. J. Phys. 36, 1 (2006).22 R. Lipowsky, Handbook of Biological Physics, R. Lipowsky and E. Sackmann, eds, Volume 1, p. 521, Elsevier,

1995.23 P.B. Canham, J. Theoret. Biol. 26, 61 (1970).24 O. Farago, Phys. Rev. E 78, 051919 (2008).25 We recover the power law Π ∼ D−3 that is known in literature (see, for instance, Ref. [8]), but the corresponding

amplitude depends on the used model.26 G. Gompper and D.M. Kroll, Europhys. Lett. 9, 59 (1989).27 U. Seifert and R. Lipowsky, in Structure and Dynamics of Membranes, Handbook of Biological Physics, R.

Lipowsky and E. Sackmann, eds, Elsevier, North-Holland, 1995.28 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989.29 H.K. Jansen, Z. Phys. B 23, 377 (1976).30 R. Bausch, H.K. Jansen, and H. Wagner, Z. Phys. B 24, 113 (1976).31 F. Langouche, D. Roekaerts, and E. Tirapegui, Physica A 95, 252 (1979).32 R. Lipowsky, in Random Fluctuations and Growth, H.E. Stanley and N. Ostrowsky, eds, p. 227-245, Kluwer

Academic Publishers, Dordrecht 1988.33 R. Lipowsky, J. Phys. A 18, L-585 (1985).34 R. Lipowsky, Physica Scripta T 29, 259 (1989).35 F. Brochard and J.F. Lennon, J. Phys. (Paris) 36, 1035 (1975).

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Brownian dynamics of nanoparticles in contactwith a confined biomembrane

Y. Madmoune, K. El Hasnaoui, A. Bendouch, H. Kaidi,M. Chahid, and M. Benhamou

Laboratoire de Physique des Polymeres et Phenomenes CritiquesFaculte des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco

[email protected]

abstract

The system we consider is a fluid membrane confined to two parallel reflecting wallsthat are separated by a finite distance, L, assumed to be small in comparison to the bulkroughness. The attractive membrane is surrounded by small colloidal particles (nanopar-ticles). The purpose is the study of Brownian dynamics of these particles, under a changeof a suitable parameter, such as temperature, T , or colloid-membrane interaction strength,w. The Brownian dynamics is investigated through the knowledge of the time particledensity, which solves the Smoluchowski equation. Solving this equation around the mid-plane, where the essential of phenomenon occurs, we obtain the exact form of the localparticle density, as a function of the perpendicular distance and time. In the derivedexpression, appears some time-scale, τ , which scales as τ ∼ L3/w. This scale-time canbe interpreted as the required time over which the colloidal suspension reaches their finalequilibrium state. Also, τ can be regarded as the time-interval over which the particlesare trapped in holes and valleys.Key words: Biomembranes - Nanoparticles - Confinement - Brownian dynamics.

I. INTRODUCTION

The biomembranes play a crucial role in life. Indeed, they separate the cell from the surroundingenvironment, and act as a selective barrier for the import and export of materials. These biological ma-terials are complex systems, but they possess a natural structural organization, where each componenthas a specific function [1− 7]. Nowadays, the scientific community recognizes that the cell membranesessentially present as a phospholipid bilayer combined with a variety of proteins and cholesterol. Forexample, the function of the cholesterol molecules is to ensure the bilayer fluidity. A phospholipid is com-monly defined as an amphiphile molecule that is composed of a hydrophilic polar head attached to twohydrophobic (fatty acyl) chains. We note that the phospholipids move freely on the membrane surface.On the other hand, the thickness of the bilayer is of the order of 5 nanometers. This two facts allow toconsider this bilayer as a two-dimensional fluid membrane. Experiment shows that the fluid membranes,self-assembled from surfactant solutions, may have a variety of shapes and topologies [8]. These formshave been theoretically explained in terms of bending energy [9, 10].Usually, the biomembranes are not immersed in liquids of infinite extent, but they are rather confined togeometrical boundaries. Typical examples are provided by white and red globules or liposomes, as drugs


91

khalid

Highlight

Y. Madmoune et al. African Journal Of Mathematical Physics Volume 8(2010)91-100

transport agents [11− 14], in blood vessels.To obtain quantitative results, we consider the situation where the biomembrane is trapped in a liquiddelimitated by two parallel reflecting walls, which are a finite distance, L, apart. By finite distance, wemean that the separation L is much smaller than the bulk membrane mean-roughness, ξ0⊥ (film geome-try). The latter can be regarded as the typical size of humps caused by the thermal fluctuations of themembrane. Of course, the scale ξ0⊥ depends on the nature of lipid molecules forming the bilayer. Thecondition L < ξ0⊥ then ensures the confinement of the biomembrane. Such a condition is similar to thatusually encountered in confined polymers context [15].In real situations, the biomembranes are not pure, but they are in the presence of various entities, likeproteins, small and macro-ions, or more complex structures [16]. For example, the membrane suspensionsused in detergency and cosmetics are usually in contact with numerous additives (macromolecules andcolloids), in order to improve their efficiency and to control their viscoelastic properties [17]. A simpleway to study this influence consists in regarding these entities as small spherical colloids. This assumptionhas a physical sense only when one is concerned with those phenomena occurring at scales greater thanthe characteristic size of neighboring entities (diameter of particles, gyration radius of macromolecules,etc.).The organization of nanoparticles around a fluctuating fluid membrane, embedded in a liquid of infiniteextent, is the subject of very recent theoretical works [18− 20]. The question was addressed to statisticalproperties of the particles mediated by the membrane undulations. In particular, it was found that theseundulations give rise to an aggregation of the beads in the vicinity of the fluid membrane. Such an aggre-gation is caused by the appearance of mutual attractive forces due to their contact with this membrane.Also, the attention has been paid to the investigation of the phase transition [18] that drives the colloidsfrom a dispersed phase (gas) to a dense one (liquid).In a very recent work [21], one has studied the Brownian dynamics of nanoparticles of very low density,which are in contact with an interacting fluid membrane. The colloids and membrane were assumed tobe trapped in a liquid of infinite extent. More precisely, the problem to solve was how these particles arepushed by the external potential towards the interface. The Brownian dynamics was studied through thetime evolution of the particle density, when some suitable parameter, such as temperature, pressure, ormembrane environment, is changed from an initial value to a final one.We recall that the Brownian motion governs various time-dependent phenomena ranging from suspen-sions [22− 27] to polymer solutions [28]. This motion can be investigated using two approaches, namelythe Smoluchowski equation solved by the distribution function and Langevin equation. Although thetwo theoretical formulations are different, but they are physically equivalent. The Smoluchowski equa-tion that is a generalization of the usual diffusion equation, has a clear relevance to thermodynamics ofirreversible processes. The Langevin equation, however, has no direct relationship to thermodynamics,but it provides a successful tool for the description of wider classes of stochastic processes.In this paper, the purpose is to extend the study to Brownian dynamics of colloidal particles in contactwith an attractive fluid membrane, where the host liquid is delimitated by two impenetrable walls. Moreprecisely, the question is how this dynamics can be affected by confinement. As we shall see below, thisconfinement induces drastic changes of the statistical properties of beads.To study the Brownian dynamics of a colloidal dispersion around a confined interacting fluid membrane,use is made of a theoretical approach based on the Smoluchowski equation. In fact, this equation de-scribes the evolution of the particle density in time, and involves a known mean-force external potentialexperienced by the particles [19]. To simplify, the immersed particles are assumed to be point-like and ofvery low-density. The first assumption remains valid as long as we are concerned with strong membraneundulations, while the second means that the mutual interactions between particles can be ignored. Thus,the only remaining interaction is an external potential originating from the statistical fluctuations of thismembrane. In the distance-range of interest, that is around the fluid membrane, we determine the exactform of the local particle density. The latter depends on the perpendicular distance z from the mid-planelocated at z = 0, time t, and some characteristic time-scale τ ∼ L3/w, where L is the separation betweenthe confining walls and w is the colloid-membrane interaction strength. We emphasize that the time-scaleτ can be regarded as the required time over which the beads are trapped in the new holes and valleys.This paper is organized as follows. In Sec. II, we present the essential of field theory allowing the calcu-lation of a basic quantity that is the mean-force potential due to the membrane undulations. Browniandynamics study is the aim of Sec. III. Some concluding remarks are drawn in the last section.

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II. THEORETICAL FORMULATION

Start with a fluctuating fluid membrane, free from particles, which is confined to two parallel reflectingwalls 1 and 2. These are separated by a finite distance L that is assumed to be much smaller thanthe bulk membrane roughness, ξ0⊥, when the system is unconfined (free membrane). The membrane isconfined only when the condition L < ξ0⊥ is fulfilled.

In the Monge representation, a point on the membrane can be described by the three-dimensional positionvector r = (x, y, z = h (x, y)), where h (x, y) ∈ [−L/2, L/2] is the height-field. The latter then fluctuatesaround the mid-plane located at z = 0.The Statistical Mechanics for the description of such a (tensionless) fluid membrane is based on thestandard Canham-Helfrich Hamiltonian [9, 29]

H0 [h] =1

2

∫dxdy

[κ (∆h)

2+ µh2

], (2.0)

with the elastic constant [30]

µ =9

4

(kBT )2

κL4. (2.0)

Here, κ is the bending rigidity constant that is directly proportional to the thermal energy kBT . Infact, the term µh2/2 describes a confinement potential that ensures the localization of the membranearound the mid-plane. Integral over the hole plane R2 of this term represents the loss entropy due to theconfinement of the membrane. We recall that formula (2) of the elastic constant is compatible with theroughness expression, i.e.

ξ2⊥ =⟨h2

⟩− ⟨h⟩2 =

1

8

kBT√µκ

=L2

12, (2.0)

provided that one is in the confinement-regime where L << ξ0⊥. Such a quantity measures the fluctuationsof the height-function (fluctuations amplitude) around the equilibrium plane located at z = 0. We recallthat the result (3) was recently derived in Ref. [30].Now, consider an assembly of N colloidal particles moving around a fluctuating fluid membrane. Tosimplify calculations, the particles are assumed to be point-like. In fact, this assumption makes senseonly if the particle size is much smaller than the membrane roughness ξ⊥ = L/2

√3. Typically, the

considered particles have diameter of a few tens of nanometers, while the roughness is of the order of 1micrometer. In addition, we suppose that there is no direct colloid–colloid interaction. This assumptionis valid only when the colloidal dispersion is of very low-density. We recall that, in this paper, we areconcerned with the influence of the membrane undulations on the particles movement. Of course, theprimitive mutual interactions between nanoparticles should be taken into account when one is interestedin their phase transition (colloidal aggregation) near an attractive fluid interface [19].The total Hamiltonian describing physics of colloids and membrane reads [19]

H [h] = H0 [h] +Hcm [h] , (2.0)

where H0 [h] is the bare Hamiltonian defined in Eq. (1). In the above definition, Hcm accounts forthe colloid-interface interaction, which is generally a complicated function of particles positions andconfigurations of the interface. For the sake of simplify, we suppose that the interaction Hcm is of contacttype, and depends only on the relative perpendicular distances between particles and surface. Then, theproposed form is

Hcm [h]

kBT= −w

2

N∑i=1

δ [zi − h (ρi)] . (2.0)

Here, δ is the one-dimensional Dirac function. The discrete sum is performed over all particles positions,ri = (ρi, zi), 1 ≤ i ≤ N . In the above definition, w > 0 represents the surface coupling constant that

93


measures the colloid-membrane interaction strength. In fact, w plays the role of an extrapolation lengthas usually encountered in Surface Critical Phenomena [31− 33]. We note that the interaction magnitudew may be influenced by a change of temperature or membrane environment. In this model, we supposethat the attractive interface is penetrable and the colloids can accommodate on its both sides.As shown in Ref. [19], the membrane undulations give rise to one, two and more bodies interactionsbetween beads. The exact calculations of these effective interactions were achieved taking advantageof the so-called cumulant method traditionally used in Statistical Field Theory [34, 35]. In this work,we focus our attention on those colloidal suspensions of very low-density (ideal gases), only. In theseconditions, the mutual interactions between particles can be neglected. Therefore, the only remaininginteraction is the attractive one-body interaction, U (z), which describes the direct potential betweencolloids and interface. Its expression was found to be [19]

U (z) = U0 exp

−6z2

L2

, (2.0)

with the negative amplitude

U0 = −√

3

2π

w

LkBT . (2.0)

The quantity |U0| is the potential depth. Originally [19], the above relations explicitly incorporate the

roughness ξ⊥ that we replaced by its expression : ξ⊥ = L/2√3.

Let us discuss the above expression of the external potential felt by the nanoparticles.Firstly, in addition to the perpendicular distance z, the interaction potential naturally depends on theseparation L between the reflecting walls, and the surface coupling constant w.Secondly, this one-body potential exhibits a minimum at the mid-plane z = 0. In addition, it is symmetricaround this point.Thirdly, remark that the potential depth |U0| depends on three kinds of parameters, which are theabsolute temperature T , the surface coupling constant w, and the film thickness L. For instance, if T andw are fixed, the potential depth is inversely proportional to separation L. This means that the externalpotential experienced by beads has a significant magnitude only for those membranes confined to verynarrow geometries. If T and L are now fixed to some values, the potential depth linearly increases withincreasing surface coupling constant w.Fourthly, we emphasize that |U0| must be small in comparison with the thermal energy kBT . This implies

that the coupling constant w is bounded from above, i.e. w < w∗ =√2π/3× L ≃ 1. 4472× L.

Finally, as it should be, in the absence of the colloid-membrane interaction (w = 0), the one-body potentialvanishes.The (reduced) external potential experienced by the nanoparticles, U (z) /kBT , is depicted in Fig. 1,upon the renormalized perpendicular distance z/L, for two values of the surface coupling constant w :w1 = 0.5 × L and w2 = 0.9 × L. As it should be, the curve drawn with parameter w2 is below thatassociated with w1 < w2.

94


FIG. 1. Reduced mean-force potential versus the renormalized perpendicular distance z/L, for two values ofthe surface coupling constant w : w1 = 0.5× L and w2 = 0.9× L.

The above expression of the one-body potential is the principal ingredient for the Brownian dynamicsstudy of very low-density particles, which are located near a soft membrane. But, in order to facilitatecalculations and get exact results, the above expression for the external potential must be simplified.Since the essential of phenomenon occurs in the interval |z| < ξ⊥ = L/2

√3, such a potential reduces to

[21]

U (z) ≃ U0 +W (z) , z < L⊥ , (2.0)

with the harmonic potential

W (z) =1

2kz2, (2.0)

where the elastic constant k is as follows

k = −U0

ξ2⊥= 12

√3

2π

w

L3kBT > 0 . (2.0)

The above equality shows that the elastic constant k scales with separation L as : k ∼ L−3.We note that the potential depth |U0| has as effect to renormalize the density amplitude [21].

III. TIME EVOLUTION OF THE PARTICLE DENSITY

Consider, now, an assembly of colloidal particles moving around a fluctuating fluid membrane. Undera sudden change of a suitable parameter, such as temperature, pressure or membrane environment, the

95


system is out equilibrium. We assume that the system is subject to Brownian dynamics by changingthe colloid-membrane interaction strength w, but the temperature and film thickness remain fixed. Thischange may be caused by affecting the membrane environment. The problem can be studied throughthe local particle density, n (z, t). The latter represents the number of colloids per unit volume, atdistance z and at time t. More precisely, we are interested in how this density evolves in time before thecolloidal suspension reaches its final equilibrium state. For simplicity, we will neglect mutual interactionsbetween nanoparticles. This hypothesis makes sense at least for small particle densities. Hence, theonly interaction experienced by the beads is an external potential caused by the membrane undulations.Within the harmonic approximation, this potential is defined in Eqs. (9) and (10).To determine the time evolution of the local particle density, use will be made of the Smoluchowskiequation [27, 28], which is a linear partial differential equation, of first order and second order, withrespect to time and perpendicular distance, respectively.Before writing and solving this equation, we shall need some backgrounds. We first recall the expressionof the equilibrium particle density

neq (z) = A exp

−U (z)

kBT

, (3.0)

where A is a normalization constant and U (z) is the one-body potential due to the membrane undulations.If this potential is approximated by its harmonic form, the above definition becomes

neq (z) = n0 exp

−W (z)

kBT

, (3.0)

where n0 is now the value of the particle density at the mid-plane z = 0.When the colloidal dispersion is out of equilibrium, in addition to distance, the density depends ontime. This means that the nanoparticles execute Brownian dynamic but in the presence of the harmonicpotential W (z). In order to compute this local density, we first recall that the Brownian diffusion iscorrectly described by the Fick’s law. The latter stipulates that the flux of matter, j (z, t), is directlyproportional to the spatial gradient of density, that is

j = −D∂n

∂z− 1

ζ

∂W

∂z, (3.0)

with

D =kBT

ζ(3.0)

the diffusion constant and ζ the friction coefficient, of which the inverse 1/ζ is the mobility. If we designby a the particle radius and by ηs the solvent viscosity, the friction coefficient ζ can be calculated fromhydrodynamics [36] : ζ = 6πηsa. Equality (14) states that the diffusion constant characterizing thethermal motion is related to the quantity ζ, which expresses the response to an external field. Such anequality is a consequence of the well-known dissipation-fluctuation theorem [27, 28].On the other hand, relation (13) must be combined with the local conservation law of matter

∂n

∂t+

∂j

∂z= 0 . (3.0)

Combining Eqs. (13) and (15) yields the Smoluchowski equation

∂n

∂t=

1

ζ

∂

∂z

(kBT

∂n

∂z+ n

∂W

∂z

)(3.0)

solved by the local particle density n (z, t). At equilibrium, that is ∂n/∂t = 0, the above equation reducesto : kBT∂n/∂z + n∂W/∂z = 0, whose solution is neq (z) = n0 exp exp −W (z) /kBT, which is nothingelse but the density defined in Eq. (12).Replacing the harmonic potential W (z) by its explicit form (9) gives

∂n

∂t= D

∂2n

∂z2+

k

ζz∂n

∂z+

k

ζn . (3.0)

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This new Smoluchowski equation must be supplemented by two boundary conditions, which are

n (z, t = 0) = ni (z) , n (z, t = ∞) = nf (z) . (3.0)

If the temperature T and separation L are fixed, the inial and final equilibrium particle densities, ni (z)and nf (z), are completely determined by the initial and final surface coupling constants wi and wf ,respectively. Therefore, the dynamic is caused by a change of the membrane environment.Taking advantage of those mathematical techniques used in Ref. [21], we show that the solution to theSmoluchowski equation (17) is given by

n (z, t) = nf (z) +[2πDτf

(1− e−2t/τf

)]−1/2∫ ∞

−∞dy exp

−

(zet/τf − y

)22Dτf

(e2t/τf − 1

) [ni (y)− nf (y)] ,

(3.0)where the initial and final equilibrium particle densities are given by

ni (z) = ni0 exp

− kiz

2

2kBT

= ni

0 exp

− z2

2Dτi

, (3.0)

nf (z) = nf0 exp

− kfz

2

2kBT

= nf

0 exp

− z2

2Dτf

, (3.0)

with the time-scales τi and τf

τi =ζ

ki=

1

12

√2π

3

D−1

wiL3 , τf =

ζ

kf=

1

12

√2π

3

D−1

wfL3 , (3.0)

where wi and wf > wi are the initial and final surface coupling constants. This means that the colloid-membrane interaction is suddenly increased from wi to wf . The above relations suggest that the timesτi and τf depend on the colloid-membrane interaction and film thickness. In particular, the time-scaleτf may be interpreted as the time beyond which the colloidal system reaches its final equilibrium state.Then, a weak colloid-membrane interaction necessitates a great time before the colloidal system tends toits final state. In fact, τf has another physical meaning, and can be regarded as the required time overwhich the particles are trapped in new holes and valleys.Now, after integration over the y variable, in Eq. (19), we obtain a closer form for the time particledensity

n (z, t) = ni0

[1 + η

(e−2t/τf − 1

)]−1/2

exp

− 1

1 + η(e−2t/τf − 1

) z2

2Dτi

, (3.0)

with the reduced time-shift

η =τi − τf

τi=

wf − wi

wf> 0 . (3.0)

The quantity η then represents the relative shift of the colloid-membrane interaction strengths wi andwf . The density amplitude, in Eq. (23), was obtained using the matter conservation law

∫ +∞

−∞ni (z) dz =

∫ +∞

−∞nf (z) dz ≡ Γ . (3.0)

Here, Γ represents the adsorbance, which is defined as the total number of colloids (per unit area) locatednear membrane. Combining Eqs. (20), (21) and (25) yields the relationship√

2πDτf nf0 =

√2πDτi n

i0 = Γ . (3.0)

Let us comment the density expression (23).First, we note that, it is easy to see that the solution (23) satisfies the two boundary conditions (18).

97


The initial and final equilibrium states are defined in Eqs. (20) and (21).Second, when it is reduced by ni

0, the time particle density depends on three dimensionless factors, namelythe renormalized distance z/

√2Dτi, the time-ratio t/τf and the time-shift η = (τi − τf ) /τi. Therefore,

all microscopic details (colloid-membrane interaction) are entirely contained in τi and τf .Finally, we emphasize that the time particle density curve exhibits a maximum at z = 0, and it issymmetric around this point, whatever be the values of t/τf and η.We depict, in Fig. 2, the reduced time particle density n (z, t) /ni

0 versus the renormalized distancez/

√2Dτi, choosing three values of the time-ratio t/τf : 0, 0.5, and ∞. The former corresponds to the

initial state and the second to the final one. These curves are drawn with parameter η = 0.5. This valuemeans that the final surface coupling constant wf is two times more important than the initial one wi,that is wf = 2wi.

FIG. 2. Reduced local particle density, n (z, t) /ni0, versus the renormalized distance z/

√2Dτi, with three values

of the time-ratio t/τf : 0 (dashed line), 0.5 (solid line), ∞ (line in dots). These curves are drawn choosing thevalue η = 0.5 (wf = 2wi).

IV. CONCLUSIONS

This work is dedicated to the Brownian dynamics study of small colloidal particles in contact with anattractive penetrable fluid membrane. The host liquid was assumed to be delimitated by two parallelreflecting walls, which are a finite distance apart. The membrane surrounded by beads is confined onlyif the film thickness is much smaller than the bulk membrane mean-roughness.Physics was discussed in terms of three relevant parameters, which are the absolute temperature, T , theseparation between walls, L, and colloid-membrane interaction strength, w. In our study, we have fixedthe temperature and film thickness to some values, and varied the surface coupling constant. This canbe experimentally achieved modifying the membrane environment.For the present study, we have started from three hypothesizes : (1) the particles are point-like, (2) theyare of very low-density (in order to forget their mutual interactions), and (3) strongly interact with themembrane.To achieve the investigation of the Brownian dynamics, use was made of a theoretical formalism basedon the Smoluchowski equation. The latter is solved by the time particle density we were interested in.We have exactly computed this physical quantity, around the mid-plane of the film, where the essential ofphenomenon occurs. Within this distance-domain, the mean-force external potential was approximated

98


by an harmonic one. This means that we were in the presence of Brownian particles moving in anharmonic potential, that is, in addition to the usual diffusion, these experience small oscillations with afrequency ν scaling as ν ∼

√w/L3. Hence, the harmonic approximation used for the mean-force potential

is largely justified for those fluid interfaces of small enough colloid-membrane interaction strength.As we have shown, the time particle density depends on a time-scale, τ , scaling as τ ∼ L3/w. We haveinterpreted this scale-time as the required time over which the nanoparticles reach their final equilibriumstate. Also, τ can be regarded as the time-interval where the particles are trapped in holes and valleysof size slightly smaller than the film thickness L.We note that, when the separation L is much greater than the bulk roughness ξ0⊥, in addition to theabove evoked parameters, the physical phenomenon depends on the specific membrane characteristic (viathe bending rigidity constant κ). In this case, the Brownian dynamics can be caused by a change of theparameter κ. This situation has less physical interest, since, in this case, finite size effects contribute tothe leading behavior only by exponentially small corrections.As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants,although the two systems are not of the same structure and composition.

ACKNOWLEDGMENTS


REFERENCES

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Bungay et al., eds, Synthetic Membranes : Science, Engineering and Applications, p. 701, D. Reiidel PulishingCompagny, 1986.

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(2006).21 A. Bendouch, M. Benhamou, and H. Kaidi, Elec. J. Theor. Phys. 5, 215 (2008).22 A. Einstein, Ann. Physik. 17, 549 (1905) ; 19, 371 (1906); see, also, Investigation on the Theory of the Brownian

Movement, E.P. Dutton and Copy Inc, New York, 1926.23 N. Wax, Noise and Stochastic Processes, Dover Publishing Co., New York, 1954.24 M. Lax, Rev. Mod. Phys. 32, 25 (1960) ; 38, 541 (1966).25 R. Kubo, Rep. Prog. Phys. 29, 255 (1966).26 C.W. Gardiner, Handbook of Stochastic Methods, 3d ed., Springer, 2004.27 H. Risken and T. Frank, The Focker-Planck Equation : Methods of Solutions and Applications, 2d ed., Springer,

1989.28 M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press Oxford, 1986.29 P.B. Canham, J. Theoret. Biol. 26, 61 (1970).30 K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou, Induced forces in confined biomem-

branes, submitted for publication.31 K. Binder, in : Phase Transitions and Critical Phenomena, Vol. 8, edited by C. Domb and J.L. Lebowitz,

Academic Press, London, 1983.32 H.W. Diehl, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 10,

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100

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Physica A 389 (2010) 3465–3475

Contents lists available at ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

Statistical mechanics of bilayer membranes in troubled aqueous mediaM. Benhamou ∗, K. Elhasnaoui, H. Kaidi, M. ChahidLaboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco

a r t i c l e i n f o

Article history:Received 30 March 2010Available online 14 April 2010

Keywords:Bilayer membranesVesiclesImpuritiesEquilibriumStatistical mechanics

a b s t r a c t

We consider a bilayer membrane surrounded by small impurities, assumed to be attractiveor repulsive. The purpose is a quantitative study of the effects of these impurities onthe statistical properties of the supported membrane. Using the replica trick combinedwith a variational method, we compute the membrane mean-roughness and the heightcorrelation function for almost-flat membranes, as functions of the primitive elasticconstants of themembrane and some parameter that is proportional to the volume fractionof impurities and their interaction strength. As results, the attractive impurities increasethe shape fluctuations due to the membrane undulations, while repulsive ones suppressthese fluctuations. Second, we compute the equilibrium diameter of (spherical) vesiclessurrounded by small random particles starting from the curvature equation. Third, thestudy is extended to a lamellar phase composed of two parallel fluid membranes, whichare separated by a finite distance. This lamellar phase undergoes an unbinding transition.We demonstrate that the attractive impurities increase the unbinding critical temperature,while repulsive ones decrease this temperature. Finally, we say that the presence of smallimpurities in an aqueous medium may be a mechanism to suppress or to produce anunbinding transition, even the temperature and polarizability of the aqueous medium arefixed, in lamellar phases formed by parallel lipid bilayers.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Usually, the aqueous media supporting biological membranes are assumed to be homogeneous. Actually, any realsystem inevitably contains impurities. Underwell-controlled conditions, the particles can be removed from the surroundingmedium. But, if these entities are present, it is also interesting to study their effect on the statistical properties of thebiomembranes, such as fluctuations’ spectrum and dynamical behavior. In general, random inhomogeneities tend todisorder the system. It is important to make a distinction between annealed and quenched disorders. The former is usedwhen impurities and host constituents (phospholipids) are in equilibrium [1]. This means that their respective mobilitiesare comparable. If it is not the case, that is host constituents and impurities are out of equilibrium, the disorder is ratherquenched [1]. When the Statistical Mechanics is used, the latter consists to trace over all membrane undulations, beforeperforming the summation over the impurities’ disorder. Although the quenched disorder is harder to analyze, it remainsmore realistic than its annealed counterpart. Indeed, the thermal and the noise averaging have very different roles.In this paper, the physical systemwe consider is a fluidmembrane (flat or closed) trapped in a troubled aqueousmedium.

The latter is impregnated by a weak amount of impurities that may be attractive or repulsive regarding the membrane. Theaim is to show how these entities can modify the statistical properties of the fluid membrane. These properties will bestudied through the fluctuations’ amplitude. To model the system, we suppose that the impurities act as a random externalpotential with a Gaussian distribution (uncorrelated disorder). In addition, we suppose that the disorder is quenched. To

∗ Corresponding author.E-mail address: [email protected] (M. Benhamou).

0378-4371/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2010.03.049

khalid

Highlight


3466 M. Benhamou et al. / Physica A 389 (2010) 3465–3475

do calculations, the replica theory [2,3] that is based on a mathematical analytical continuation, usually encountered inQuantumMechanics [4] and Critical Phenomena will be made use of [5,6].Our finding are as follows. First, using the above evoked theory, we compute the mean-roughness of an almost-flat

membrane, as a function of the primitive parameters of the pure membrane (free from impurities) and a certain parameterdepending on the volume fraction of impurities and their interaction strength. The main conclusion is that, attractiveimpurities increase the shape fluctuations due to the thermal undulations, while repulsive ones tend to suppress thesefluctuations and then lead to a strongmembrane confinement. Second, we analyze the impurities effects on the equilibriumshape of closed vesicles solving the curvature equation (for spherical vesicles).We show that the vesicle ismore stable in thepresence of repulsive impurities, in comparison with attractive ones. Thereafter, the study is extended to lamellar phasesformed by two parallel fluid membranes. We demonstrate that the presence of random impurities drastically affects thephysical properties of the lamellar phase, in particular, the unbinding transition driving the system from a bind state to astate where the two membranes are completely separated.This paper is organized as follows. In Section 2, we describe the fundamentals of the used model. Section 3 deals with

the computation of the fluctuations amplitude of a single almost-flat fluid membrane, surrounded by attractive or repulsiveimpurities. We compute, in Section 4, the equilibrium diameter of a closed vesicle in the presence of impurities. Extensionof study to lamellar phases is the aim of Section 5. Finally, some concluding remarks are drawn in the last section.

2. Effective field theory

Consider a fluctuating fluidmembrane embedded in a three-dimensional liquid surrounded by very small impurities. Forthe sake of simplicity, we suppose that the impurities are point like. Within the framework of the Monge representation,a point on the membrane can be described by the position-vector (r, z = h (r)), where r = (x, y) ∈ R2 is the transversevector and h (r) is the height function.The Statistical Mechanics of fluid membranes free from impurities is based on the Canham–Helfrich Hamiltonian [7]

H0 [h] =∫d2r

[κ2(∆h)2 +

µ

2h2], (1)

with κ the membrane bending modulus. The confinement energy (per unit area) µh2/2 (µ > 0) is responsible for thelocalization of themembrane in some region of the Euclidean space, where it fluctuates around an equilibrium plane locatedat h = 0. Therefore, the height h takes either positive and negative values. For simplicity, the membrane is assumed to betensionless. In fact, this assumption does not change conclusions made below.To model the impurities effects on the statistical properties of the system, we suppose that these tend to reinforce the

membrane confinement, if they are repulsive, or to render this membrane more free, if the particles are rather attractive.These tendencies can be explained assuming that the confinement parameter is local in space and making the substitution

µ→ µ+ V (r) , (2)

in the above Hamiltonian, where the variation V (r) can be regarded as a random external potential. To simplify, thecorresponding probability distribution is supposed to be Gaussian (uncorrelated disorder), that is

V (r) = 0, V (r) V (r ′) = −vδ2(r − r ′

). (3)

Here,−v is a positive constant proportional to both concentration of impurities and strength of their interaction potential,and δ2 (r) denotes the two-dimensional Dirac distribution.Therefore, the new effective Canham–Helfrich Hamiltonian reads

H [h] =∫d2r

[κ

2(∆h)2 +

12(µ+ V (r)) h2

], (4)

for attractive impurities. For repulsive ones, V (r)must be replaced by iV (r), with i2 = −1. Since the disorder distributionis Gaussian, all its odd moments vanish, but the even ones do not. This implies that all physical quantities, calculated withthe pure imaginary potential iV , are entire numbers.Since the impurities and membranes are not in equilibrium, the disorder is rather quenched, that is we have to average

not the partition function, Z , but its logarithm, ln Z . The latter defines the free energy.For a given (quenched) configuration of impurities, the partition function is

Z =∫

Dhe−A[h], (5)

with the action

A [h] =H [h]kBT

=

∫d2r

[κ

2(∆h)2 +

12

(µ+ V (r)

)h2]. (6)


M. Benhamou et al. / Physica A 389 (2010) 3465–3475 3467

We have used the notations

κ =κ

kBT, µ =

µ

kBT, V =

VkBT

. (7)

In term of the reduced impurities potential V , the second disorder law in Eq. (3) becomes: V (r) V (r ′) = −vδ(r − r ′

), with

v = v/ (kBT )2. A simple dimensional analysis shows that: [κ] = L0, [µ] = L−4 and [v] = L−6, where L is some length thatmay be the membrane thickness.The main quantity to consider is the average of the logarithm of the partition function, ln Z , over disorder. To compute

such a quantity, we use the replica trick [2]. This method consists to formally write

ln Z = limn→0

Zn − 1n

. (8)

After performing the average over disorder, we get

Zn =∫

Dh1 . . .Dhne−A[h1,...,hn], (9)

with the n-replicated effective action

A [h1, . . . , hn] =∫d2r

[κ

2

n∑α=1

(∆hα)2 +µ

2

n∑α=1

h2α +v

8

(n∑α=1

h2α

)(n∑

β=1

h2β

)], (10)

where Greek indices denote replicas. The last term introduces an additional coupling constant v < 0, which is directlyresponsible for the effective interaction between replicas due to the presence of impurities. The above action describesattractive impurities. For repulsive ones, the coupling v must be replaced by −v. The additive quartic term in the aboveaction means that the presence of impurities is accompanied by an increase of entropy, when these are attractive, and byan entropy loss, if they are rather repulsive.The following paragraph is devoted to a quantitative determination of the fluctuations spectrum in the presence of

impurities.

3. Single almost-flat membrane

We start by considering the attractive impurities problem. Of course, the functional integral (9) cannot be exactlycomputed. One way is the use of a variational method. To this end, we consider a bilayer membrane in the presence ofthe following bare action

Aη [h1, . . . , hn] =∫d2r

[κ

2

n∑α=1

(∆hα)2 +η

2

n∑α=1

h2α

], (11)

with the new confinement coupling η > 0. With this action, the partition function, Zη , is exact. We have

Zη = (Z0)n , (12)

with

Z0 =∫

Dh exp−

∫d2r

[κ

2(∆h)2 +

η

2h2]. (13)

The latter accounts for the usual partition function of a bilayer membrane free from impurities.Introduce the mean-value of a functional X [h1, . . . , hn], calculated with the bare action Aη [h1, . . . , hn],

〈X〉0 =1Zη

∫Dh1 . . .DhnX [h1, . . . , hn] exp

−Aη [h1, . . . , hn]

. (14)

In term of this mean-value, the averaged partition function may write

Zn = Zη⟨exp

−(A− Aη

)⟩0 . (15)

Using the standard inequality⟨eX⟩0 ≥ e

〈X〉0 , (16)

we get

Zn ≥ Zη exp−⟨A− Aη

⟩0

. (17)



This implies that

− ln Z = limn→0

1− Zn

n≤ limn→0

1− Zη exp−⟨A− Aη

⟩0

n

. (18)

On the other hand, we have

⟨A− Aη

⟩0 =

∫d2r

[µ− η

2

n∑α=1

⟨h2α⟩0 +

v

8

n∑α=1

n∑β=1

⟨h2αh

2β

⟩0

]. (19)

It is easy to see thatn∑α=1

⟨h2α⟩0 = nσ

2,

n∑α=1

n∑β=1

⟨h2αh

2β

⟩0= n (n+ 2)

(σ 2)2, (19a)

with the squared membrane roughness

σ 2 (η) =181√κ η. (20)

With these considerations,

− ln Z ≤ − ln Z0 +Σ[µ− η

2σ 2 +

v

4

(σ 2)2]

, (21)

whereΣ is the area of the reference plane.Notice that the right-hand side of this inequality is η-dependent function. Therefore, a good approximate value of−ln Z

is

−[ln Z

]MF = minη

[− ln Z0 +Σ

(µ− η

2σ 2 +

v

4

(σ 2)2)]

. (22)

(The subscriptMF is formean-field theory). The variational parameter η is such that

∂

∂η

[− ln Z0 +Σ

(µ− η

2σ 2 +

v

4

(σ 2)2)]

= 0. (23)

Combining the relationship

σ 2 = −2Σ

∂

∂ηln Z0 (24)

and Eq. (20) yields the minimum value of η, which satisfies the following implicit relation

η = µ+ vσ 2. (25)

The squared membrane roughness, σ 2, is then given by the parametric equationsσ 2 =181√κ η,

η = µ+ vσ 2.

(26)

Eliminating η between these equations gives the implicit relation

σ 2 =18

1√κ(µ+ vσ 2

) . (27)

Introduce the usual squared roughness of a membrane free from impurities, σ 20 = 1/8√κ µ. Then, we have

σ 2 = σ 201√

1+ vσ 2/µ. (28)

The latter may be rewritten, in terms of dimensionless variables σ 2/σ 20 , κ , andw = −64κ v(σ 20)3> 0, as

σ 2

σ 20=

1√1− wσ 2/σ 20

, (attractive impurities) . (29)



Table 1Some values of the couplingw and their corresponding ratios σ 2/σ 20 , for attractive impurities.

w σ 2/σ 20

0.1 1.05750.15 1.09370.2 1.13780.25 1.19390.3 1.2715w∗

√3

Notice that the coupling constant w > 0 is directly proportional to the volume fraction of impurities and their interactionstrength.Let us comment about the obtained result.Firstly, this makes sense only when σ 2/σ 20 < 1/w. This condition is fulfilled for very weak disorders (w is small enough).Secondly, since w is positive definite, we have: σ 2 > σ 20 . A comparison between this inequality and that just evoked

above implies that 0 < w < 1.Thirdly, if we set x = σ 2/σ 20 , the above relation can be rewritten as

w =x2 − 1x3

. (30)

Therefore, w is a direct function of x (Fig. 1). This formula may be experimentally used to estimate the impurities couplingw, knowing the experimental value of the ratio σ 2/σ 20 . If w is fixed to some value, the x-variable then solves the followingthird-degree algebraic equation

wx3 − x2 + 1 = 0. (31)

The existence of its roots then depends on the value of the renormalized coupling w. We show that there exists a typicalvaluew∗ = 2/3

√3 of the couplingw, such that:

(1) Forw > w∗ (strong disorder), we have only one root that is negative ;(2) For w = w∗ (medium disorder), we have one negative root and a positive unique root, which is x = σ 2/σ 20 =

√3.

Then, themaximal value of themembrane roughness is σ ∗ = 31/4σ0. This constraint means that the lipid bilayer cannotsupport undulations of perpendicular size (hunt size) greater than σ ∗. This is realized if only if the impurities volumefraction φ is below some value φ∗, directly proportional to the thresholdw∗;

(3) Forw < w∗ (weak disorder), we have one negative root and two positive ones. Only one positive root is acceptable (thesmallest one). At this particular root, the free energy −kBT

[ln Z

]MF is minimal. We show that the absolute minimum

satisfies the inequality

− 4wx6 + 3x4 − 1 > 0. (32)

In this case, the ratio σ 2/σ 20 can be obtained by numerically solving Eq. (31). We report in Table 1 some values of theimpurities couplingw and the corresponding dimensionless squared membrane roughness σ 2/σ 20 . It is easy to see that,the ratio σ 2/σ 20 increases with increasing couplingw, provided thatw is in the intervalw < w∗.

Now, for repulsive impurities, the fundamental relationship (29) is replaced by

σ 2

σ 20=

1√1− wσ 2/σ 20

, (repulsive impurities) , (33)

or equivalently

− wx3 − x2 + 1 = 0, (34)

with the stability condition

4wx6 + 3x4 − 1 > 0. (35)

The equality (34) can be transformed into the following direct function (Fig. 2)

w =x2 − 1x3

, (36)

with x < 1 andw < 0.As it should be, the membrane roughness is reduced by the presence of repulsive impurities, that is σ 2 < σ 20 , and in

addition, σ 2 decreaseswith increasing impurities coupling−w. As amatter of fact, the effect of these particles is to reinforce



Fig. 1. Impurities’ couplingw versus the ratio σ 2/σ 20 , for attractive impurities.

Fig. 2. Impurities’ couplingw versus the ratio σ 2/σ 20 , for repulsive impurities.

Table 2Some values of the couplingw and their corresponding ratios σ 2/σ 20 , for attractive impurities.

w σ 2/σ 20

−0.1 0.9554−0.15 0.9364−0.2 0.9190−0.25 0.9032−0.3 0.8885−0.4 0.8622

the confinement of the considered fluidmembrane. Therefore, a bilayermembrane collapses,when it is trapped in a troubledaqueous media with repulsive impurities. In this case, it is easy to see that the algebraic equation has only one positive root.In Table 2, we give some values of the couplingw and the corresponding ratio σ 2/σ 20 .Another interesting physical quantity is the (connected) height correlation function: G

(r − r ′

)=

⟨h (r) h

(r ′)⟩−

〈h (r)〉⟨h(r ′)⟩. The latter measures the fluctuations of the height function h around its mean-value 〈h〉. Here, 〈.〉 denotes

the thermal expectation mean-value, which must not be confused with average over disorder.



To compute the correlation functions, we start from the generating functional

Z [J] =∫

Dh exp−A [h]+

∫d2r J (r) h (r)

, (37)

with the action A [h] = H [h] /kBT , whereH [h] is the effective Canham–Helfrich Hamiltonian defined in Eq. (4). Here, J isan auxiliary source coupled to the field h. Functional derivatives of Z [J] with respect to source J give all height correlationfunctions, in particular, the propagator

G(r − r ′

)=

δ2ln Z [J]δJ (r) δJ (r ′)

∣∣∣∣∣J=0

. (38)

This definition indicates that the main object to compute is the averaged connected generating functional ln Z [J].To compute the functional ln Z [J], as before, we use the replica method based on the limit

ln Z [J] = limn→0

Zn [J]− 1n

, (39)

with

Zn [J] =∫

Dh1 . . .Dhn exp

−A [h1, . . . , hn]+

n∑i=1

∫d2r J (r) hi (r)

, (40)

where A [h1, . . . , hn] is the action, relation (10).Using the standard cumulant method [5,6] based on the approximate formula

〈exp X〉0 = exp〈X〉0 + (1/2!)

(⟨X2⟩0 − 〈X〉

20

)+ · · ·

, (41)

we find that

Zn [J] ' Zη exp−⟨A− Aη

⟩0 + · · ·

exp

n2

∫d2r

∫d2r ′J (r)G0

(r − r ′

)J(r ′), (42)

with the usual bare propagator

G0(r − r ′

)=

∫d2q

(2π)2eiq.(r−r

′)

κq4 + η. (43)

In Eq. (42), we have ignored high-order terms in J that do not contribute to the propagator. After performing the limit n = 0,we find that the averaged connected generating functional reads

ln Z [J] ' − ln Z0 +Σ(µ− η

2σ 2 +

v

4

(σ 2)2)+12

∫d2r

∫d2r ′J (r)G0

(r − r ′

)J(r ′). (44)

By simple functional derivation, we obtain the expected propagator

G(r − r ′

)= G0

(r − r ′

)=

∫d2q

(2π)2eiq.(r−r

′)

κq4 + η. (45)

Then, the expected propagator identifies with the bare one. Here, the variational parameter η satisfies the implicit equation(25).We then recover the relationship: σ 2 = G (0), where σ 2 is the squared membrane roughness computed above.The next step consists to extend the study to a closed vesicle surrounded by attractive or repulsive impurities.

4. Single vesicle

We start by recalling some basic backgrounds dealt with the equilibrium shape of (spherical) vesicles, whichmay studiedusing Differential Geometry techniques.The vesicle is essentially formed by two adjacent monolayers (inner and outer) that are formed by amphiphile lipid

molecules. These permanently diffuse with the molecules of the surrounded aqueous medium. Such a diffusion thenprovokes thermal fluctuations (undulations) of the membrane. This means that the latter experiences fluctuations aroundan equilibrium plane.Consider a biomembrane of arbitrary topology. A point of this membrane (surface) can be described by two local

coordinates (u1, u2). At each point of the surface, there exists two particular curvatures (minimal and maximal), called



principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2. The quantities R1 and R2 are the principal curvature radii. With thehelp of the principal curvatures, one constructs two invariants that are the mean-curvature

C =12(C1 + C2) , (46)

and the Gauss curvature

K = C1C2. (47)

The principal curvatures C1 and C2 are nothing else but the eigenvalues of the curvature tensor [8].To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely

accepted one is the fluid mosaic model proposed by Singer and Nicolson in 1972 [9]. This model consists to regard the cellmembrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a fluid, while theproteins and other amphiphile molecules (cholesterol, sugar molecules, . . . ) are simply embedded in the lipid bilayer. Wenote that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayerbiomembranes were first studied, in 1973, by Helfrich [7]. The author recognized that the lipid bilayer could be regarded assmectic-A liquid crystals at room temperature, and proposed the following curvature free energy (without impurities)

F =κ

2

∫(2C)2 dA+ κG

∫KdA+

∫γ dA+ p

∫dV . (48)

where dA denotes the area element, and V is the volume enclosedwithin the lipid bilayer. In the above definition, κ accountsfor the bending rigidity constant, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure differencebetween the outer and inner sides of the vesicle. The spontaneous curvature is ignored. The first-order variation gives theshape equation of lipid vesicles [10]

p− 2γ C + 4κC(C2 − K

)+ κ∇2 (2C) = 0, (49)

with the surface Laplace–Bertolami operator

∇2=1√g∂

∂ui

(√gg ij

∂

∂uj

), (50)

where g ij is the metric tensor on the surface and g = det(g ij). For open or tension-line vesicles, local differential equation

(49) must be supplemented by additional boundary conditions we do not write [11]. The above equation have known threeanalytic solutions corresponding to sphere [10],

√2-torus [12–15] and biconcave disk [16].

For spherical vesicles, the solution to the above curvature equation is exact, and we find that the equilibrium radius is

R0 =2γp. (51)

Now, assume that the vesicle is trapped in a troubled aqueous medium. Usually, to take into account the presence ofimpurities, a low-order coupling between their volume fraction, φ, and the mean-curvature C is added to the above freeenergy F , that is

F → F ± kBT∫φCdA. (52)

The positive sign is for attractive impurities and the negative sign for repulsive ones. The new free energy is then

F =κ

2

∫(2C ± 2C0)2 dA+ κG

∫KdA+

∫γ dA+ p

∫dV (53)

with the notation

C0 = kBTφ

4κ, γ = γ − 2κC20 . (54)

Then, the impurities generate an extra spontaneous curvature. This means that these give arise to an asymmetry of thevesiclewhen itsmembrane is crossed. In addition, these impurities additively renormalize the interfacial tension coefficient.Minimizing this new curvature free energy yields

p− 2γ C + κ (2C ± C0)(2C2 ∓ C0C − 2K

)+ κ∇2 (2C) = 0. (55)

For a spherical vesicle, the mean curvature is a constant, and we have K = C2. In this case, the mean-curvature C is a rootof a polynomial of degree 2. This means that we have an exact solution we do not write. In particular, for very small volumefractions (φ 1), we find that the vesicle equilibrium radius writes

RR0= 1±

pkBT8γ 2

φ + O(φ2), (56)



or ∣∣∣∣R− R0R0

∣∣∣∣ ' pkBT8γ 2 φ. (57)

The signs (+) and (−) describe attractive and repulsive impurities, respectively. Here, R0 = 2γ /p denotes the equilibriumradius for a spherical vesicle free from impurities.The above results call the following remarks.Firstly, as it should be, the equilibrium radius of the vesicle increaseswith increasing volume fraction of impurities, when

these are attractive. This means that the vesicle is swollen, but it radius remains close to the unperturbed one R0, as long asthe volume fraction of impurities is very small. For repulsive impurities, however, the vesicle is collapsed.Secondly, formula (57) makes sense only when the volume fraction φ is below some threshold φ∗ = 8γ 2/kBTp. Since the

latter must be small in comparison with unity, the vesicle bears interfacial tensions of coefficients that do not exceed sometypical value γ ∗ =

√kBTp/8.

Finally, at fixed volume fraction φ < φ∗, this formula may be used to estimate the experimental value of the ratio p/γ 2,by a simple measurement of the equilibrium radii R (with impurities) and R0 (without impurities).The following paragraph will be devoted to the study of lamellar phases made of two parallel lipid bilayers.

5. Lamellar phases

The natural question is the extension of the study to the case when we have more than one membrane immersed in aaqueous media, which is impregnated by small random impurities.In our analysis, we start from a lamellar phase composed of two parallel (neutral) fluid membranes free from impurities.

The effects of these entities on physics will be discussed below. The cohesion between these bilayer membranes is ensuredby long-ranged attractive van derWaals forces [17], which are balanced, at short membrane separation, by strong repulsioncoming from hydration forces [18] and by steric shape fluctuations ones resulting from the membrane undulations [7].For two parallel bilayer membranes separated by a finite distance l, the total interaction energy per unit area is

V (l) = VH (l)+ VW (l)+ VS (l) . (58)

The first part

VH (l) = AHe−l/λH (59)

represents the hydration potential (per unit area) that acts at small separations of the order of 1 nm. The correspondingamplitude AH and potential-range λH are about AH ' 0.2 J/m2 and λH ' 0.3 nm. The second part

VW (l) = −W12π

[1l2−

2

(l+ δ)2+

1

(l+ 2δ)2

](60)

is the attractive van der Waals potential (per unit area) that originates from polarizabilities of lipid molecules and watermolecules. Here, W accounts for the Hamaker constant that is in the range W ' 10−22 − 10−21 J, and δ for the bilayerthickness. The latter is of the order of δ ' 4 mm. The last part is the steric shape fluctuations potential (per unit area) [7]

Vs (l) = cH(kBT )2

κ l2, (61)

with kB the Boltzmann’s constant, T the absolute temperature, and κ the common bending rigidity constant of the twomembranes. But in the case of two bilayers of different bending rigidity constants κ1 and κ2, we have κ = κ1κ2/ (κ1 + κ2).There, the coefficient cH is a known numerical coefficient [7].We note that the lamellar phase remains stable at the minimum of the potential, provided that the potential depth is

comparable to the thermal energy kBT . This depends, in particular, on the value of amplitudeW of the direct van der Waalsenergy. The Hamaker constantW may be varied changing the polarizability of the aqueous medium.In a pioneered theoretical paper, Lipowsky and Leibler [19] have shown that there exists a certain threshold Wc

beyond which the van der Waals attractive interactions are sufficient to bind the membranes together, while below thischaracteristic amplitude, the membrane undulations dominate the attractive forces, and then, the membranes separatecompletely. According to the authors,Wc is in the intervalWc ' (6.3− 0.61)× 10−21 J when the bending rigidity constantis in the range κ ' (1− 20) × 10−19 J. We note that the typical value Wc corresponds to some temperature, Tc , calledunbinding critical temperature [19,20]. In particular, it was found [19] that, when the critical amplitude is approached fromabove, the mean-separation between the two membranes, 〈l〉0, diverges according to

〈l〉0 = ξ0⊥

∼ (Tc − T )−ψ , T → T−c , (62)

with ψ a critical exponent whose value is [19]: ψ ' 1.00 ± 0.03. The latter was computed using field-theoreticalRenormalization-Group.



Now, assume that the lamellar phase is trapped in a troubled aqueous medium. As first implication, the impuritiesdrasticallymodify the unbinding transition phenomenon, in particular, the critical temperature. For simplicity, suppose thatthe two adjacent fluid membranes are physically identical. Then, we have to consider two distinct physical situations: Theimpurities attract or push the twomembranes. As we have shown above, for attractive impurities, themembrane roughnessξ⊥ is important, and then, the steric shape fluctuation energy dominates. Therefore, we expect that the unbinding transitionoccurs at a low critical temperature T ∗c we will determine below. For repulsive impurities, however, we have an oppositetendency, that is the unbinding transition takes place at a high temperature greater than Tc (absence of impurities). Thiscan be explained by the fact that the membrane roughness is less important, and then, the direct van der Waals energydominates. In any case, the mean-separation 〈l〉 scales as

〈l〉 = ξ⊥ ∼(T ∗c − T

)−ψ, T → T−c . (63)

The new unbinding critical temperature T ∗c can be estimated as follows. Using formulae (29) and (33), we show that, atfirst order in the impurities couplingw,

ξ⊥

ξ 0⊥

' 1+w

4+ O

(w2), (attractive impurities) , (64)

ξ⊥

ξ 0⊥

' 1−w

4+ O

(w2), (repulsive impurities) . (65)

Combing Eqs. (62) to (65), we find that the difference between the two critical temperatures Tc and T ∗c (without and withimpurities) is as follows

Tc − T ∗c ∼ w + O(w2), (attractive impurities) , (66)

Tc − T ∗c ∼ −w + O(w2), (repulsive impurities) . (67)

The prefactors in the above behaviors remain unknown. The temperature shift is then proportional to the volume fractionof impurities and their interaction strength through the couplingw. Then, we are in a situation similar to finite size scaling.For charged membranes forming the lamellar phase, it was found [19] that the mean-separation between two adjacent

bilayers scales as [19]

〈l〉0 = ξ0⊥

∼ (χ − χc)−ψ , (68)

in the vicinity of χc , with χ the ionic concentration of the aqueousmedium and χc its critical value. For instance, for DPPC inCaCl2 solutions,χc is in the interval [21]:χc ' 84−10mM. In the presence of impurities, we state that themean-separationbehaves as

〈l〉 = ξ⊥ ∼(χ − χ∗c

)−ψ, (69)

with the same exponent ψ . In this case, we have

χ∗c − χc ∼ w + O(w2), (attractive impurities) , (70)

χ∗c − χc ∼ −w + O(w2), (repulsive impurities) . (71)

We note that the same comments may by done in this case.

6. Conclusions

In this paper,we presented a large scope about the effects of impurities on the statistical properties of fluidmembranes. Infact, these drastically affect the living systems behavior. As an example, we can quote a very recent experimental study [22],where the authors have undertaken a series of comparative experiments, in order to explore the effect of impurities in theform of proteins and lipids on the crystallization of membrane proteins in vapor diffusion.For the present study, the impurities were assumed to be attractive or repulsive. Using the replica trick combined with a

variational method, we have computed themembranemean-roughness, as a function of the parameters associatedwith thepuremembrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength.The main conclusion is that, attractive impurities increase the shape fluctuations due to the membrane undulations, butrepulsive ones tend to suppress these fluctuations.Also, we have computed the equilibrium diameter of (spherical) vesicles surrounded by small random particles solving

the curvature equation.Thereafter, we extended discussion to lamellar phase formed by two parallel fluid membranes that are a finite distance

apart. This lamellar phase may undergo an unbinding transition. We have shown that, attractive impurities increase theunbinding critical temperature, while repulsive ones tend to decrease this temperature.



We point out that, the incorporation of a small amount of impurities in an aqueous medium may be a mechanism tosuppress or to produce an unbinding transition (even the temperature and polarizability of the aqueous medium are fixed)within lamellar phases composed of fluid membranes.Finally, the present study may be extended to bilayer membranes of arbitrary topology. The essential conclusion is that,

attractive impurities tend to swell the membrane, while in the presence of repulsive ones, this membrane is collapsed. Also,the used method can be applied to multilayers constituted by several parallel lamellar phases.

Acknowledgements

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ‘‘FirstInternational Workshop On Soft-Condensed Matter Physics and Biological Systems’’, 14–17 November 2006, Marrakech,Morocco. One of us (M.B.) would like to thank Professor C. Misbah for fruitful correspondences, and the Laboratoire deSpectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his regular visits.

References

[1] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996.[2] V.J. Emery, Phys. Rev. B 11 (1975) 239.[3] S.F. Edvards, P.W. Anderson, J. Phys. (France) 5 (1975) 965.[4] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 3rd edition, Pergamin Press, 1991.[5] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989.[6] C. Itzykson, J.-M. Drouffe, Statistical Field Theory: 1 and 2, Cambridge University Press, 1989.[7] W. Helfrich, Z. Natureforsch 28c (1973) 693.[8] K. Wolfgang, Differential Geometry: Curves – Surfaces – Manifolds, American Mathematical Society, 2005.[9] S.J. Singer, G.L. Nicolson, Science 175 (1972) 720.[10] O.-Y. Zhong-Can, W. Helfrich, Phys. Rev. Lett. 59 (1987) 2486.[11] Z.C. Tu, Z.C. Ou-Yang, Phys. Rev. E 68 (2003) 061915.[12] O.-Y. Zhong-Can, Phys. Rev. A 41 (1990) 4517.[13] Z. Lin, R.M. Hill, H.T. Davis, L.E. Scriven, Y. Talmon, Langmuir 10 (1994) 1008.[14] M. Mutz, D. Bensimon, Phys. Rev. A 43 (1991) 4525.[15] A.S. Rudolph, B.R. Ratna, B. Kahn, Nature 352 (1991) 52.[16] H. Naito, M. Okuda, O.Y. Zhong-Can, Phys. Rev. E 48 (1993) 2304.[17] J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, London, 1991.[18] R.P. Rand, V.A. Parsegian, Biochim. Biophys. Acta 988 (1989) 351.[19] R. Lipowsky, S. Leibler, Phys. Rev. Lett. 56 (1986) 2541.[20] R. Lipowsky, E. Sackmann (Eds.), An Extensive List of References on the Subject can be Found in: Structure and Dynamics of Membranes: Generic and

Specific Interactions, vol. 1B, Elsevier, 1995.[21] L.J. Lis, W.T. Lis, V.A. Parsegian, R.P. Rand, Biochemistry 20 (1981) 1771.[22] A. Christopher, et al., Acta Cryst. D 65 (2009) 1062–1073.

An extended study of the phase separation between phospholipids and grafted polymers on a

bilayer biomembrane

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Phys. Scr. 83 (2011) 065801 (6pp) doi:10.1088/0031-8949/83/06/065801

An extended study of the phase separationbetween phospholipids and graftedpolymers on a bilayer biomembrane

M Benhamou, I Joudar, H Kaidi, K Elhasnaoui, H Ridouaneand H Qamar

Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences, Ben M’sik,PO Box 7955, Casablanca, Morocco

E-mail: [email protected]

Received 4 April 2010Accepted for publication 9 May 2011Published 26 May 2011Online at stacks.iop.org/PhysScr/83/065801

AbstractWe re-examine here the phase separation between phospholipids and adsorbed polymer chainson a fluid membrane with a change in some suitable parameter (temperature). Our purpose isto quantify the significant effects of the solvent quality and of the polydispersity of adsorbedloops formed by grafted polymer chains on the segregation phenomenon. To this end, weelaborate on a theoretical model that allows us to derive the expression for the mixing freeenergy. From this, we extract the phase diagram shape in the composition–temperature plane.Our main conclusion is that the polymer chain condensation is very sensitive to the solventquality and to the polydispersity of loops of adsorbed chains.

PACS numbers: 87.16.Dg, 47.57.Ng, 64.60.F

1. Introduction

Biological membranes are of great importance to life, becausethey separate the cell from the outside environment andseparate the compartments inside the cell in order to protectimportant processes and specific events.

Nowadays, it is largely recognized that biologicalmembranes are present as a lipid bilayer composed of twoadjacent leaflets [1, 2], which are formed by amphiphilemolecules possessing hydrophilic polar-heads pointingoutward and hydrophobic fatty acyl chains that form the core.The majority of lipid molecules are phospholipids. Thesehave a polar-head group and two non-polar hydrocarbon tails,whose length is of the order of 5 nm.

Also, the cell membranes incorporate another type oflipid, cholesterol [1, 2]. The cholesterol molecules haveseveral functions in the membrane. For example, they giverigidity or stability to the cell membrane and preventcrystallization of hydrocarbons. The biomembranes alsocontain glycolipids (sugars), which are lipid molecules thatmicroaggregate in the membrane, and may be protectiveand act as insulators. Certain kinds of molecules are

bounded by sphingolipids such as cholera and tetanustoxins. Sphingolipids and cholesterol favor the aggregationof proteins in microdomains called rafts. In fact, these playthe role of a platform for the attachment of proteins while themembranes are moved around the cell and also during signaltransduction.

Proteins (long macromolecules) are another principalcomponent of cell membranes. Transmembrane proteins areamphipathic and are formed by hydrophobic and hydrophilicregions having the same orientation as other lipid molecules.These proteins are also called integral proteins. Their functionis to transport substances, such as ions and macromolecules,across the membrane. There exist other types of proteinsthat may be attached to the cytoplasm surface by fatty acylchains or to the external cell surface by oligosaccharides.These are termed peripheral membrane proteins. They havemany functions; in particular, they protect the membranesurface, regulate cell signaling and participate in manyother important cellular events. In addition, some peripheralmembrane proteins (those having basic residues) tend to bindelectrostatically to negatively charged membranes, such as theinner leaflet of the plasma membrane.

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Phys. Scr. 83 (2011) 065801 M Benhamou et al

We note that the majority of macromolecules forming thebilayer are simply anchored on the membrane and form a softbranched polymer brush [3, 4].

The study of grafted polymers on soft interfaces wasmotivated by the fact that they have potential applicationsin biological materials, such as liposomes [5–8]. These softmaterials were discovered by A Bangham. Currently, theyare major tools in biology, biochemistry and medicine (asdrugs transport agents). Liposomes are artificial vesicles ofspherical shape that can be produced from natural nontoxicphospholipids and cholesterol. But the lipid bilayers have ashort lifetime, because of the weak stability of the vesiclesand their extermination by white blood cells. To have stablevesicles in time, one useful method consists of protectingthem with a coat of flexible polymer chains (coat size ofthe order of 50 nm [1]), which prevents the adhesion ofmarker proteins [6, 7]. In fact, these polymer chains stabilizethe liposomes, due to the excluded volume forces betweenmonomers [8]. Liposomes can also be synthesized fromA–B diblock-copolymers immersed in a selective solvent thatprefers to be contacted by the polymer A. The hydrophobicB-polymer chains then aggregate and form a thin bilayer,while the hydrophilic polymer chains A float in the solvent.These copolymer-based liposomes have properties that areslightly different from those of the lipid ones [9] (highresistance, high rigidity and weak permeability to water).Depending on the choice of copolymers, these liposomes areresistant to detergents [10].

The grafting of polymers onto lipid membranes wasconsidered in a very recent paper [11]. More precisely, thepurpose was to investigate of the phase separation betweenphospholipids and anchored polymers. As assumptions, theaqueous medium was assumed to be a good solvent, and inaddition, the polymer chains were anchored to the interfaceonly by one extremity, a big amphiphile molecule. The latteris chemically different from the phospholipid molecules. Inthis paper, however, we assume that the polymer chainsmay be adsorbed on the membrane by many monomersthat are directly attached to some polar-heads of the hostlipid molecules (mobile anchors), and then organize inpolydisperse loops. The surrounding liquid may be a goodsolvent or a theta solvent. As we shall see below, theloops’ polydispersity and solvent quality drastically affectthe phase behavior of the system. Under a change in asuitable parameter, such as temperature, the phospholipidsand adsorbed polymer chains phase separate into macroscopicdomains alternately rich in the two components. For thedetermination of the phase diagram shape, we elaborate ona theoretical model that takes into account both the loops’polydispersity and the solvent quality.

This paper is organized as follows. In section 2,we derive the expression for the mixing free energy ofthe phospholipid–anchor mixture. To investigate the phasediagram shape is the aim of section 3. Finally, someconcluding remarks are given in section 4.

2. Mixing free energy

Polymers can adhere to biomembranes in several ways:(i) by lipid anchors; that is, the polymer is covalently

bound to the polar-head of a lipid molecule (for manyproteins, the lipid anchors are glycosylphosphatidylinositolunits) [12, 13]; (ii) by hydrophobic side-groups of thepolymers which are integrated into the bilayer [14, 15];(iii) by membrane spanning hydrophobic domains of thepolymer (membrane-bound proteins, for example); and (iv) bya strong adsorption that drives the polymer from a desorbedstate to an adsorbed one [16]. For a single polymer chain, theadsorption (on solid substrates) can be theoretically studiedusing a scaling argument [17], for instance. In addition tothe polymerization degree of the polymer chain, N , andthe excluded volume parameter, v, the study was based onan additional parameter, δ, which is the energy (per kBTunit) required to adsorb one monomer on the surface. For astrong adsorption, δ−1 defines the adsorbed-layer thickness.An adsorption transition takes place at some typical value δ∗

of δ scaling as [17]: δ∗∼ R−1

F ∼ aN−νF (νF = 3/5), whereRF is the Flory radius of the polymer chain. For dilute andsemi-dilute polymer solutions, the adsorption phenomenondepends, in addition to the parameter δ, on the polymerconcentration.

In this paper, we consider situation (iv), where eachmonomer has the same probability of being adsorbed on themembrane surface [18]1. More precisely, a given monomeris susceptible to becoming linked to a polar-head of aphospholipid molecule (anchor). As a result, the polymerchains form polydisperse loops (with eventually one or twotails floating in the aqueous medium). In fact, this assumptionconforms to what was considered in [19]. The case whereno loops are present (adsorption only by one extremity) wasconsidered in [11]. Contrary to the adsorption on the solidsurface, the anchored polymer chains are mobile on the hostmembrane and may undergo the aggregation transition thatwe are interested in. To be more general, the fluid membraneis assumed to be in contact with a good solvent or a thetasolvent.

The purpose is to write a general expression for themixing free energy. The latter will allow the determinationof the phase diagram related to the aggregation of anchoredpolymer chains.

First, we start with the free energy (per unit area) of thepolymer layer (for solid substrates), which is given by [20]

F0

kBT'

1

b2

∫ N

1

([b2S (n)]β + [−b2S′ (n)] ln

[−

S′ (n)

S1

])dn.

(1)For the configuration study, Guiselin [21] considered that eachloop can be viewed as two linear strands (two half-loop). Here,S(n) is the number of strands having more than n monomersper unit area, and b represents the monomer size. The numberN denotes the length of the longest strand. Hereafter, we shalluse the notation S1 = 8/a, which denotes the total number ofgrafted chains per unit area, with 8 being the volume fractionof anchors and a their area. If the mixture is assumed tobe incompressible, then 1 − 8 is the volume fraction of thephospholipid molecules.

Let us come back to the free energy expression (1);note that the first term of the right-hand side represents the

1 The adsorption of an adequate polymer on a fluid membrane made of twophospholipids of different chemical nature may be a mechanism of phaseseparation between these two unlike components as was pointed out.

2


contribution of the (two- or three-body) repulsive interactionsbetween monomers belonging to the polymer layer. Thesecond term is simply the entropy contribution describingall possible rearrangements of the grafted chains in thepolymer layer. There, the exponent β depends on the solventquality [20]. When the grafting is accomplished in a dilutesolution with a good solvent or a theta solvent, the values ofβ are β = 11/6 (blob model) or β = 2, respectively.

For polydisperse polymer layers, the distribution S(n)

is known in the literature [20]. This can be obtained byminimizing the above free energy with respect to thisdistribution.

Without presenting details, we simply sketch the generalresult that [20]

S(n)∼8

an1/(β−1). (2)

For usual solvents, we have

S(n)∼8

an6/5(good solvents), (3)

S(n)∼8

an(theta solvents). (4)

We shall rewrite the distribution S(n) as

S(n) =8

af (n), (5)

with f (n) ∼ 1/n1/(β−1), for polydisperse polymer layers, andf (n) = 1 (for all n), for polymer brushes.

Therefore, F0 can be approximated by

F0

kBT= a−1u8β + η(N )8, (6)

where the linear term in 8 describes the contribution ofentropy to the free energy, where the coefficient η(N ) is asfollows (see the appendix):

η(N ) = a−1

[β + ln(β − 1)]

(1 − N 1/(1−β)

)+

β

β − 1N 1/(1−β) ln N

. (7)

In the asymptotic limit N → ∞, this coefficient goes to

η(N ) → a−1 [β + ln(β − 1)], (8)

provided that β > 1 (note that β = 11/6, for good solvents,and β = 2, for theta ones). This asymptotic limit is alwayspositive definite and inversely proportional to the polar-headarea a. In fact, the positive sign of the coefficient η(N ) agreeswith the entropy loss due to the polymer chains grafting. Aswe shall see below, this linear contribution does not changethe phase diagram in the composition-critical parameter plane.Here, the coupling constant u is as follows:

u =

(b2

a

)β−1

Nζ(N ), (9)

with

ζ (N ) =1

N

∫ N

1[ f (n)]β dn. (10)

Explicitly, we have

ζ (N ) = 1 (polymer brushes), (11)

ζ (N )∼ Nβ/(1−β) (polydisperse systems). (12)

Note that ζ(N ) < 1, since, in all cases, β > 1. Therefore,the polydispersity of loops decreases the repulsive interactionenergy.

Now, we have all the ingredients for the determinationof the expression for the mixing free energy. To this end, weimagine that the interface is present as a two-dimensional (2D)Flory–Huggins lattice [17, 22], where each site is occupiedby an anchor or by a phospholipid molecule. Hence, we canregard the volume fraction of anchors 8 as the probability thata given site is occupied by an anchor. Therefore, 1 − 8 is theoccupation probability of phospholipids.

Before the determination of the desired mixing freeenergy (per site), we note that the latter is the sum of threecontributions, which are the mixing entropy (per site), thevolume free energy (per site) and the interaction energy (persite) coming from the membrane undulations. Actually, theinduced attractive forces due to the membrane undulationsare responsible for the condensation of anchors. These forcesbalance the repulsive ones between monomers along theconnected polymer chains. We then write

F

kBT= 8 ln 8 + (1 − 8) ln(1 − 8) + χ8(1 − 8)

+ u8β + η(N ) 8. (13)

We note that, for polymer chains anchored by one extremity,a big amphiphile molecule (the polymer brush case), the firstcontribution of entropy, 8 ln 8, should be divided by somefactor q , which represents the ratio of the anchor area to thearea of polar-heads of the host phospholipids. In the presentcase, we have q = 1. In equality (11), χ accounts for the Floryinteraction parameter

χ = χ0 −1

A2

(C −

D

kBT

), (14)

where the positive coefficients C and D are such that (indimension 2) [23]

D = −π

∫∞

σ

rU (r) dr, C =π

2σ 2 (covolume). (15)

Here, U (r) is the pair potential induced by the membraneundulations ([11]; [24] and references therein)

U (r) =

∞, r < σ,

−AH

(σ

r

)4, r > σ,

(16)

where σ is the hard disc diameter, which is proportional tothe square root of the anchor area a. There, the potentialamplitude AH plays the role of the Hamaker constant. It wasfound that the latter decays with the bending rigidity constantaccording to ([11]; [24] and reference therein)

AH ∼ κ−2. (17)

3


But this amplitude is also sensitive to temperature. We remarkthat the above mixing free energy is not symmetric under thechange 8 → 1 − 8.

Some straightforward algebra gives the followingexpression for the attraction parameter D:

D = AHπ

2σ 2∼

κ−2

σ 2. (18)

Therefore, those membranes with a small bending modulusinduce significant attraction between anchors.

In formula (12), χ0 > 0 is the Flory interactionparameter describing the chemical segregation betweenamphiphile molecules that are phospholipids and anchors.This segregation parameter is usually written as

χ0 = α0 +γ0

T, (19)

where the coefficients α0 and γ0 depend on the chemicalnature of the various species. Also, the total interactionparameter χ can be written as

χ = α +γ

T, (20)

with the new coefficients

α = α0 −C

A2, γ = γ0 +

D

kB A2. (21)

These coefficients then depend on the chemical nature ofthe unlike components and on the membrane’s characteristicsthrough its bending modulus κ .

If we admit that the coupling constant u has a slightdependence on temperature, we will draw the phase diagramin the plane of variables (8, χ). Indeed, all of the temperaturedependence is contained in the Flory interaction parameter χ .

3. Phase diagram

With the help of the above mixing free energy, we candetermine the shape of the phase diagram in the (8, χ)-planethat is associated with the aggregation process that drives theanchors from a dispersed phase (gas) to a dense one (liquid).We focus only on the spinodal curve along which the thermalcompressibility diverges. The spinodal curve equation can beobtained by equating to zero the second derivative of themixing free energy with respect to the anchor volume fraction8; that is, ∂2 F/∂82

= 0. Then, we obtain the followingexpression for the critical Flory interaction parameter:

χ(8) =1

2

(1

8(1 − 8)+ β(β − 1) u8β−2

). (22)

Above this critical interaction parameter appear two phases:one is homogeneous and the other is separated. Of course, thelinear term in 8 appearing in equality (11) does not contributeto the critical parameter expression.

We remark that, in usual solvents, this critical interactionparameter is increased due to the presence of (two-or three-body) repulsive interactions between monomersbelonging to the polymer layer. This means that these

interactions widen the compatibility domain, and then, theseparation transition appears at low temperature.

Now, to see the influence of the solvent quality,we rewrite the interaction parameter u as u = u0ζ(N ) ∼

u0 Nβ/(1−β) < u0, where u0 is the interaction parameterrelative to a monodisperse system. Thus, the polydispersityof loops has a tendency to reduce the compatibility domain incomparison with the monodisperse case.

The critical volume fraction, 8c, can be obtained byminimizing the critical parameter χ(8) with respect to the8-variable. We then obtain

1

(1 − 8c)2 −

1

82c

+ β(β − 1) (β − 2) u8β−3c = 0. (23)

For good solvents (β = 11/6), we have

1

(1 − 8c)2 −

1

82c

−55

216u8−7/6

c = 0. (24)

Therefore, the critical volume fraction is the abscissa of theintersection point of the curve of the equation (2x − 1)/

x5/6(1 − x)2 and the horizontal straight line of the equationy = (55/216)u. Note that this critical volume fraction isunique, and in addition, it must be greater than the value1/2 (for mathematical compatibility). The coordinates ofthe critical point are (8c, χc), where 8c solves the aboveequation and χc = χ(8c). The latter can be determined bycombining equations (19) and (21). For theta solvents (β = 2),the coordinates of the critical point are exact,

8c =12 , χc = 2 + u, (25)

where the interaction parameter u scales as

u =b2

aN−1. (26)

The above relation clearly shows that the polymer chains’condensation rapidly takes place only when the characteristicmass N is high enough. The same tendency is also seen in thecase of good solvents.

It is straightforward to show that the critical fractionand the critical parameter are shifted to lower values in thecase of polydisperse systems, whatever be the quality of thesurrounding solvent.

In figure 1, we present the spinodal curve formonodisperse (with no loops) and polydisperse (with loops)systems, with a fixed parameter N . We have chosen the goodsolvents situation. For theta solvents, the same tendency isseen.

We present in figure 2 the spinodal curve for apolydisperse system (with loops), at various values of theparameter N . As expected the critical parameter is shifted tohigher values when we augmented the typical polymerizationdegree.

Finally, we compare, in figure 3, the spinodal curves for apolydisperse system (with loops) for the case of theta solventsand those for the case of good solvents, at fixed parametersb and N . All the curves in the figure reflect our discussionsabove.

4


0,0 0,2 0,4 0,6 0,8 1,00

20

40

60

80

Φ

χ

Figure 1. Spinodal curves (in a good solvent) for monodisperse(dashed line) and polydisperse (solid line) systems, when N = 100,with the parameter b2

= 0.5a.

0,0 0,2 0,4 0,6 0,8 1,00

4

8

12

16

20

Φ

χ

Figure 2. Spinodal curves for a polydisperse system, when N = 50(solid line), 100 (dashed line) and 150 (dotted line), with theparameter b2

= 0.5a . We assumed that the surrounding liquid is agood solvent. For theta solvents, the tendency is the same.

4. Discussion and conclusions

This paper is devoted to the thermodynamic study of theaggregation of the polymer chains adsorbed on a soft surface.Such an aggregation is caused by competition betweenthe chemical segregation between phospholipids and graftedpolymer chains, their volumic interactions and the membraneundulations. More precisely, we addressed the question ofhow these polymer chains can be driven from a dispersedphase (gas) to a dense one (liquid), under a change in asuitable parameter, e.g. the absolute temperature.

To be more general, we achieved the study in a unifiedway; that is, we have considered more realistic physicalsituations: the solvent quality (good or theta solvents) and thepolydispersity of the adsorbed loops formed by the grafted

0,0 0,2 0,4 0,6 0,8 1,00

4

8

12

16

20

Φ

χ

Figure 3. Superposition of spinodal curves for a polydispersesystem for the case of theta solvents (solid line) on those for thecase of good solvents (dashed line). For these curves, we choseN = 100 and b2

= 0.5a.

polymer chains. The latter were taken into account throughthe well-known form of the chains’ length distribution.

In doing so, we first computed the expression for themixing free energy by adopting the Flory–Huggins latticeimage usually encountered in polymer physics [17, 22].Such an expression shows that there is competition betweenfour contributions: entropy, chemical mismatch betweenunlike species, interaction energy induced by the membraneundulations and the interaction energy between monomersbelonging to the grafted layer. Such a competition governs thephases succession.

We emphasize that the present work and a previouswork [11] differ from another previous work that wasconcerned with the same problem, but in which the substratewas assumed to be a rigid surface [3]. Therefore, themembrane undulations were neglected. As we have seen,these undulations increase the segregation parameter χ byan additive term, χm , scaling as κ−2. This means thatthe phase separation is accentuated due to the presence ofthermal fluctuations. Compared with the previous work [11],which was concerned with soft brushes with monodisperseend-grafted polymers, the present work is more general, sinceit takes into account both the polydispersity of loops formingthe adsorbed polymer chains and the solvent quality. Thus, thepresent study was achieved in a unified way. As we have seen,these details drastically affect the phase diagram architecture.

Now, let us discuss further the influence of solvent qualityon the critical phase behavior. We recall that the solventquality appears in the free energy (11) through the repulsionparameter u, defined in equation (7). We find, in the N → ∞

limit, that ug ∼ N 1/5uθ , where the subscripts g and θ standfor good and theta solvents, respectively. This implies that thegood solvent plays the role of a stabilizer regarding the phaseseparation.

Finally, this work must be considered as a naturalextension of a study reported previously [11], which wasconcerned with monodisperse end-grafted polymer chains

5


trapped in a good solvent. Therefore, the present workpresents a wide perspective on the phenomenon of segregationbetween the host phospholipids and grafted polymer chains onbilayer membranes.

Acknowledgments

We are indebted to Professors T Bickel, J-F Joanny and CMarques for helpful discussions during the First InternationalWorkshop on Soft-Condensed Matter Physics and BiologicalSystems (14–17 November 2006, Marrakech, Morocco). MBthanks Professor C Misbah for fruitful correspondence andthe Laboratoire de Spectroscopie Physique (Joseph FourierUniversity of Grenoble) for their kind hospitality during hisvisit. We are grateful to the referee for critical remarks anduseful suggestions that helped us to improve the scientificcontent of this paper.

Appendix

The aim is to determine the coefficient η(N ) appearing informula (6). We start with the entropy contribution to the freeenergy

Fentropic

kBT'

1

b2

∫ N

1

([−b2S′(n)

]ln

[−

S′(n)

S1

])dn, (A.1)

where the loop-size distribution is defined in equation (2).Then, the above expression can be rewritten as

Fentropic

kBT= η(N ) 8, (A.2)

with

η(N ) = a−1 1

β − 1

∫ N

1

(nβ/(1−β) ln

[1

β − 1nβ/(1−β)

])dn.

(A.3)Straightforward algebra gives

η(N ) = a−1

[β + ln(β − 1)]

(1 − N 1/(1−β)

)+

β

β − 1N 1/(1−β) ln N

. (A.4)

This concludes the determination of the coefficient η(N ).

References

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[2] Benhamou M 2008 Lipids Insights 1 2[3] Nicolas A and Fourcade B 2003 Eur. Phys. J. E 10 355[4] Aubouy M et al 2000 Phys. Rev. Lett. 84 4858

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http://dx.doi.org/10.1140/epje/i2001-10111-x


Statistics of a single D-manifold restricted to two parallelbiomembranes or a tubular vesicle

M. Benhamou∗ , K. Elhasnaoui, H. Kaidi, M. Chahid

Laboratoire de Physique des Polymeres et Phenomenes CritiquesFaculte des Sciences Ben M’sik, P.O. Box 7955, Casablanca, Morocco

∗ [email protected]

abstract

The purpose is an extensive conformational study of a single polymer immersed in anaqueous medium (good solvent) delimitated by bilayer membranes. To be more general,we assume that the polymer is of arbitrary topology we term D-polymeric fractal or D-manifold, where D is the spectral dimension (for instance, D = 1, for linear polymers, and4/3, for branched ones). The main quantity to consider is the parallel extension of theconfined polymer. To make explicit calculations, we suppose that the polymer is restrictedto a tubular vesicle or two parallel biomembranes. We first show that, for the firstgeometry, the polymer is confined only when the tubular vesicle is in equilibrium state.For the second geometry, the confinement is possible if only if the two parallel membranesare in their binding state, that is below the unbinding or adhesion temperature. In anycase, the parallel gyration radius of the confined polymer is computed using an extendedFlory-de Gennes theory. As result, this radius strongly depends on the polymer topology(through the spectral dimension D) and on the membranes sizes, which are the equilibriumdiameter (function of bending modulus, pressure difference between inner and outer sidesof the membrane, and interfacial tension coefficient), for the first geometry, and themean-separation (function of temperature and interaction strength between the adjacentmembranes), for the second one. Finally, we give the expression of the confinement freeenergy, as a function of the polymer size, and discuss the effects of external pressure orlateral tension on the radius expression for two confining parallel membranes.

Key words: D-Polymeric fractals, Biomembranes, Vesicles, Confinement.

I. INTRODUCTION

The polymer confinement finds many applications in various domains, such as biological functions,filtration, gel permeation chromatography, heterogeneous catalysis, and oil recuperation.The physics of polymer confinement is a rich and exciting problem. Recently, much attention has beenpaid to the structure and dynamics of polymer chains confined to two surfaces, or inside cylindrical pores[1− 7]. Thereafter, the study has been extended to more complex polymers, called D-polymeric fractalsor D-manifolds [8] that are restricted to the same geometries. Here, D is the spectral dimension thatmeasures the degree of the connectivity of monomers inside the polymer [9]. For instance, this intrinsicdimension is 1 for linear polymers and 4/3 for branched ones. The D-manifolds may be polymerized (or


55

khalid

Highlight

M. Benhamou et al. African Journal Of Mathematical Physics Volume 10(2011)55-64

crumpled) vesicles [10]. The polymer confinement between two surfaces and in cylinders with sinusoidalundulations was also investigated [11, 12].The confining geometries may be soft-bodies, as bilayer biomembranes and surfactants, and spherical andtubular vesicles. A particular question has been addressed to polymer confinement in surfactant bilayersof a lyotropic lamellar phase [13], where the authors reported on small-angle X-ray scattering and free-fracture electron microscopy studies of a nonionic surfactant/water/polyelectrolyte system in the lamellarphase region. The fundamental remark is that, the polymer molecules cause both local deformation andsoftening of the bilayer.In the same context, it was experimentally demonstrated [14] that the polymer confinement may induce anematic transition of microemulsion droplets. More precisely, the authors showed that upon confinement,spherical droplets deform to prolate ellipsoid droplets. The origin of such a structural transition may beattributed to a loss of the conformational entropy of polymer chains due to the confinement.In other experiments [15− 17], a new phase has been observed adding an neutral hydrosoluble polymer(PVP) in the lyotropic lamellar phase (CPCI/hexanol/water). Also, one has studied the effect of a neutralwater-soluble polymer on the lamellar phase of a zwitterionic surfactant system [18].The polymer confinement is also relevant for living systems and governs many biological processes. Asexample, we can quote membrane nanotubes that play a major role in intercellular traffic, in particularfor lipid and proteins exchange between various compartments in eukaryotic cells [19− 22]. The traffic ofmacromolecules and vesicles in nanotubes is ensured by molecular motors [23]. Also, these are responsiblefor the extraction of nanotubes [24]. The formation of tubular membrane tethers or spicules [25− 33]results from the action of localized forces that are perpendicular to the membrane. These forces mayoriginate from the polymerization of fibers [34], as actin [35], tubulin [36], or sickle hemoglobin [37].Inspired by biological processes, as macromolecules and vesicles transport, we aim at a conformationalstudy of confined polymers in aqueous media delimitated by biomembranes. More precisely, the purposeis to see how these biomembranes can modify the conformational properties of the constrained polymer.To this end, we choose two geometries : a tubular vesicle (Geometry I), and two parallel biomembranes(Geometry II). To be more general, we consider polymers or arbitrary connectivity called D-polymericfractals or D-manifolds [38]. Linear and branched polymers, and polymerized vesicles constitute typicalexamples. We note that this conformational study in necessary for the description of dynamic propertiesof polymers restricted to these geometries.As we shall below, the polymer confinement is entirely controlled by the confining biomembranes state.Indeed, for Geometry I, the polymer may be confined only when the confining tubular vesicle is atequilibrium. For Geometry II, the confinement is possible if only if the two parallel membranes are inthe binding state.This paper is organized as follows. In Sec. II, we briefly recall the conformational study of unconfinedD-polymeric fractals in good solvent. Sec. III deals with the conformational study of polymers confinedinside a tubular vesicle. In Sec. IV, we extend the study to D-polymeric fractals confined to two parallelbiomembranes. Some concluding remarks are drawn in the last section.

II. UNCONFINED POLYMERIC FRACTAL

Consider a single polymeric fractal of arbitrary topology (linear polymers, branched polymers, polymernetworks, ...). We assume that the considered polymer is trapped in a good solvent. We denote by

RF ∼ aM1/dF (1)

its gyration (or Flory) radius, where dF is the Hausdorff fractal dimension, M is the molecular-weight(total mass) of the considered polymer, and a denotes the monomer size. The mass M is related to thelinear dimension N by : M = ND, where D is the spectral dimension [9]. The latter is defined as theHausdorff dimension corresponding to the maximal extension of the fractal.Naturally, the Hausdorff dimension depends on the Euclidean dimensionality d, the spectral dimensionD and the solvent quality. When the polymer is ideal (without excluded volume forces), its Hausdorffdimension, d0F , is a known simpler function of D [9]

d0F =2D

2−D. (2)

56


For linear polymers (D = 1), d0F = 2 [2], for ideal branched ones (D = 4/3), d0F = 4 [9], and for crumpledmembranes (D = 2), d0F = ∞.Because of the positivity of the Hausdorff dimension, the above expression makes sense only for D < 2.Indeed, this condition is fulfilled for any complex polymer with spectral dimension in the interval 1 ≤D < 2 [9].A polymeric fractal in good solvent is swollen, because of the presence of the excluded volume forces. Thepolymer size increases with increasing total massM according to the power law (1). The first implicationof the polymer swelling is that, the actual Hausdorff dimension dF is quite different from the Gaussianone, defined in Eq. (2). However, there exists a special value of the Euclidean dimensionality called uppercritical dimension duc [39, 40], beyond which the polymeric fractal becomes ideal. This upper dimensionis naturally a D-dependent function, which can be determined [39] using a criterion of Ginzburg type,usually encountered in critical phenomena [41, 42]. According to Ref. [39], duc is given by

duc =4D

2−D. (3)

For instance, the upper critical dimension is 4 for linear polymers [2], and 8 for branched ones [39].We emphasize that, in general, the Hausdorff fractal dimension dF cannot be exactly computed. Manytechniques have been used to determine its approximate value, in particular, the Flory-de Gennes (FD)theory [2]. Using a generalized FD approach, it was found that the fractal dimension is given by [39]

dF = Dd+ 2

D + 2, (4)

below the critical dimension, and it equals the Gaussian fractal dimension d0F described above. Fordimension 3, we have

dF (3) =5D

D + 2. (5)

For instance, for linear polymers, dF (3) = 5/3, and dF (3) = 2, for branched ones (animals).In the following paragraph, we shall focus our attention on the conformational study of a single polymericfractal confined to a long tubular vesicle.

III. CONFINED POLYMERIC FRACTAL IN GEOMETRY I

A. Useful backgrounds

Before studying the conformation of a single polymeric fractal, we recall some basic backgroundsconcerning the equilibrium shape of tubular vesicles. This can be done using Differential Geometrymachineries.The tubular vesicle is essentially formed by two adjacent leaflets (inner and outer) that are composedof amphiphile lipid molecules. These permanently diffuse with the molecules of the surrounded aqueousmedium. Such a diffusion then provokes thermal fluctuations (undulations) of the membrane. This meansthat the latter experiences fluctuations around an equilibrium plane we are interested in.Consider a biomembrane of arbitrary topology. A point of this membrane can be described by twolocal coordinates (u1, u2). From surfaces theory point of view, at each point, there exists two particularcurvatures (minimal and maximal), called principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2. Thequantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, oneconstructs two invariants that are the mean-curvature

C =1

2(C1 + C2) , (6)

and the Gauss curvature

K = C1C2 . (7)

57


We recall that C1 and C2 are nothing else but the eigenvalues of the curvature tensor [43].To comprehend the geometrical and physical properties of the biomembranes, one needs a good model.The widely accepted one is the fluid mosaic model proposed by Singer and Nicholson in 1972 [44]. Thismodel consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely inthe membrane surface like a fluid, while the proteins and other amphiphile molecules (cholesterol, sugarmolecules, ...) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranescrucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes werefirst studied, in 1973, by Helfrich [45]. The author recognized that the lipid bilayer could be regarded assmectic-A liquid crystals at room temperature, and proposed the following curvature free energy

F =κ

2

∫(2C + 2C0)

2dA+ κG

∫KdA+

∫γdA+ p

∫dV . (8)

where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the abovedefinition, κ accounts for the bending rigidity constant, C0 for the spontaneous curvature, κG for theGaussian curvature, γ for the surface tension, and p for the pressure difference between the outer andinner sides of the vesicles. The first order variation gives the shape equation of lipid vesicles [46]

p− 2γC + κ (2C + C0)(2C2 − C0C − 2K

)+ κ∇2 (2C) = 0 , (9)

with the surface Laplace-Bertlami operator

∇2 =1√g

∂

∂ui

(√ggij

∂

∂uj

), (10)

where gij is the metric tensor on the surface and g = det(gij

). For open or tension-line vesicles, local

differential equation (9) must be supplemented by additional boundary conditions we do not write [47].

The above equation have known three analytic solutions corresponding to sphere [46],√2-torus [48− 51]

and biconcave disk [52].For cylindrical (or tubular) vesicles, one of the principal curvature is zero, and we have

C = − 1

R, K = 0 , (11)

where R is the radius of the cylinder. If we ignore the boundary conditions (assumption valid for verylong tubes), the uniform solution to equation (9) is

H = 2

(4κ

p

)1/3

, (12)

where H is the equilibrium diameter. We have neglected the surface tension and spontaneous curvaturecontributions, in order to have a simplified expression for the equilibrium diameter.The above relation makes sense as long as the pressure difference is smaller than a critical value pc thatscales as [53] : pc ∼ κ/R3. The latter is in the range 1 to 2 Pa. The meaning of the critical pressure isthat, beyond pc, the vesicle is unstable. This implies that the equilibrium diameter must be greater than

the critical one Hc = 2 (4κ/pc)1/3

.In what follows, we shall use the idea that consists to regard the tubular vesicle as a rigid cylinder ofeffective diameter H that depends on the characteristics of the bilayer through the parameters κ and p.

B. Parallel extension to the cylinder-axis

Consider now a polymeric fractal of arbitrary topology confined inside a tubular vesicle of equilibriumdiameter H. The host solvent is assumed to be a good solvent. The polymer is confined if only if itsthree-dimensional Flory radius RF3 is much larger than the diameter H, that is H << RF3. At fixedparameters κ and p, this condition implies that the polymer mass M must be greater than some typical

value M∗ that scales as M∗ ∼ (κ/p)5D/3(D+2)

. Now, if the parameter M and p are fixed, the polymer is

58


confined is only if κ << κ∗ ∼ pM3(D+2)/5D. This behavior clearly indicates that the confinement is morefavorable when the tubular vesicle is of small bending rigidity constant. We note that the confinementcondition depends on the nature of the polymeric fractal through its spectral dimension D. Also, thesolvent quality is another factor that influences this confinement.The confinement of polymers of arbitrary topology inside a rigid tube is largely investigated in Ref. [8].In this paper, to achieve the conformational study of a confined polymer, we use a generalized FD theorybased on the following free energy [8]

F

kBT=R2

∥

R20

+ vM2

R∥H2, (13)

where R∥ denotes the polymer parallel extension to the tube-axis, and v is the excluded volume parameter.

There, R0 ∼ aM1/d0F is the ideal radius and R∥H2 represents the volume occupied by the fractal.

Minimizing the above free energies with respect to R∥ yields the desired result

R∥ ∼ aM (D+2)/3D

(a−3κ

p

)−2/9

. (14)

We have used relationship (12).Let us comment about the obtained result.Firstly, notice that, in any case, the parallel radius naturally depends on polymer and tubular vesiclecharacteristics, through M and parameters (κ, p), respectively.Secondly, at fixed polymer mass M , the parallel extension is important for those tubular vesicles of smallbending modulus.Finally, the above behavior is valid as long as the parallel extension R∥ remains below the maximal

extension of the polymer, that is we must have R∥ < aM1/D (maximal extension). This gives a minimaltube diameter

Hmin ∼ aM (D−1)/2D . (15)

Therefore, the confinement of the polymeric fractal implies that the tube diameter is in the intervalHmin < H < RF3. For instance, for linear polymers, Hmin ∼ a, and Hmin ∼ aM1/8, for branched ones.The minimum pore size for linear polymers is then independent on the molecular-weight. This meansthat the linear polymers find their way even through a very small pore. It is not the case for branchedpolymers, for which the minimum pore size increases with the total mass. As pointed out in Ref. [8],it should be possible to construct a porous medium that may separate a mixture of branched and linearmolecules on a basis of their connectivity. Indeed, one can choose an appropriate minimum pore sizethrough which linear polymers can pass, whereas branched polymers cannot. Now, combining relations(12) and (15) yields a minimal value for the bending rigidity constant (at fixed pressure difference)

κmin ∼ a3pM3(D−1)/2D . (16)

Thus, a tubular membrane confines a polymeric fractal only if its bending modulus is in the intervalκmin < κ < κ∗, where the typical value κ∗ is that given above.The following paragraph is devoted to the confinement of a polymer of arbitrary topology to two parallelfluctuating fluid membranes.

IV. CONFINED POLYMERIC FRACTAL IN GEOMETRY II

A. Basic backgrounds

Consider a lamellar phase formed by two parallel bilayer membranes. The cohesion between theselipid bilayers is provided by long-ranged van der Waals forces [54]. But these attractive interactions arebalanced, at short membrane separation, by strong repulsion coming from hydration forces for unchargedbilayers [55]. In addition, for bilayers carrying electric charges, the attractive interaction is reduced by

59


the presence of electrostatic forces [54, 56, 57].For two parallel bilayer membranes that are a finite distance l apart, the total interaction energy (perunit area) is the following sum

V (l) = VH (l) + VE (l) + VW (l) . (17)

The first part represents the hydration energy. The hydration forces that act at small separation of theorder of 1 nm, have been discovered for multilayers under external stress [55]. The adopted form for thehydration energy is an empirical exponential decay

VH (l) = AHe−l/λH . (18)

The typical values of amplitude AH and potential-range λH are AH ≃ 0.2 J/m2 and λH ≃ 0.3 nm. Theelectrostatic energy between two charged membranes also decays exponentially, that is

VE (l) = AEe−l/λE , (19)

provided that the separation l is greater than the Debye-Huckel length. The potential amplitude andits range depend on ionic concentration in the aqueous solution, χ, and the surface charge density atmembranes, σ0. More precisely, λE and AE scale as : λE ∼ χ−1/2 and AE ∼ σ2

0χ−1/2. The last part

is the attractive van der Waals energy that results from polarizabilities of lipid molecules and watermolecules. This interaction energy has the standard form

VW (l) = − W

12π

[1

l2− 2

(l + δ)2 +

1

(l + 2δ)2

]. (20)

The Hamaker constant is in the rangeW ≃ 10−22−10−21 J. In the above expression, the bilayer thicknessδ is of the order of δ ≃ 4 mm.In principle, one must add a steric interaction energy that originates from membranes undulations.According to Helfrich [45], this energy (per unit area) is

Vs (l) = cH(kBT )

2

κl2, (21)

with kB the Boltzmann’s constant, T the absolute temperature, and κ the common bending rigidity con-stant of the two membranes. But, for two bilayers of different bending rigidity constants κ1 and κ2, wehave κ = κ1κ2/ (κ1 + κ2). The precise value of coefficient cH remains an open debate. Indeed, Helfrichgave for this coefficient the value cH ≃ 0.23 [45], but computer simulations predict smaller prefactors,namely, cH ≃ 0.16 [58], cH ≃ 0.1 [59], cH ≃ 0.07 [60] and cH ≃ 0.08 [61]. Therefore, the steric interactionenergy is significant only for those membranes of small bending modulus. Of course, this energy vanishesfor rigid interfaces (κ→ ∞).We note that the lamellar phase remains stable at the minimum of the potential, provided that thepotential-depth is comparable to the thermal energy kBT . This depends, in particular, on the value ofW -amplitude.From a theoretical point of view, Lipowsky and Leibler [62] had predicted a phase transition that drivesthe system from a state where the membranes are bound to a state where they completely separated.Such a phase transition is first-order if the steric repulsive energy is taken into account. But, if this energyis ignored (for relatively rigid membranes), the transition is rather second-order. We restrict ourselves tosecond-order phase transitions, only. The authors have shown that there exists a certain thresholdWc be-yond which the van der attractive interactions are sufficient to bind the membranes together, while belowthis characteristic amplitude, the membrane undulations dominate the attractive forces, and then, themembranes separate completely. In fact, the critical value Wc depends on the parameter of the problem,which are temperature T , and parameters AH , λH , δ and κ. For room temperatures and AH ≃ 0.2 J/m2,λH ≃ 0.3 nm, and δ ≃ 4 nm, one has Wc ≃ (6.3− 0.61) × 10−21 J, when the bending rigidity constantis in the range κ ≃ (1− 20) × 10−19 J. For instance, for egg lecithin, one has [63] κ ≃ (1− 2) × 10−19

J, and the corresponding threshold Wc is in the interval Wc ≃ (6.3− 4.1) × 10−21 J. We note that thetypical value Wc corresponds to some temperature, Tc, called unbinding critical temperature [62, 64].

60


Let us first consider uncharged membranes, and notice that the Hamaker constant may be varied chang-ing the polarizability of the aqueous medium. It was found [62] that, when the critical amplitude isapproached from above, the mean-separation between the two membranes diverges according to

H ∼ (Tc − T )−ψ ∼ (W −Wc)

−ψ,

(W →W+

c or T → T−c

). (22)

Here, ψ is a critical exponent whose value is [62]

ψ ≃ 1.00± 0.03 . (23)

Such an exponent was computed using field-theoretical Renormalization-Group.From an experimental point of view, critical fluctuations in membranes were considered in some experi-ment [65], and in particular, the mean-separation was measured.We have now all ingredients to study the conformation of a single polymeric fractal of arbitrary topology,which is confined to two parallel fluid membranes forming a lamellar phase.

B. Polymer parallel extension

First, we note that the polymer is confined only when its three-dimensional gyration radius RF3 ∼aM (D+2)/5D is much greater than the mean-separation H ∼ (Tc − T )

−ψ, that is H << RF3. This con-

dition implies that the polymer confinement is possible only when temperature T is below some valueT ∗ = Tc − aM−(D+2)/5Dψ. This temperature is then smaller than the critical unbinding one Tc, and theshift Tc − T ∗ essentially depends on the polymer mass M and its spectral dimension D.The first implication of the polymer confinement is that, its behavior becomes two-dimensional. Thismeans that the polymer can be regarded as a two-dimensional polymeric fractal formed by blobs (pan-cakes) of size H. To determine the parallel extension of the polymer, R∥, as before, we start from a FDfree energy

F

kBT=R2

∥

R20

+ vM2

R2∥H

, (24)

where v denotes the excluded volume parameter. There, R0 ∼ aM1/d0F is the ideal radius and R2∥H

represents the volume occupied by the fractal. Minimization of the above free energy with respect to R∥gives

R∥ ∼ aM (D+2)/4D (W −Wc)ψ/4 ∼ aM (D+2)/4D (Tc − T )

ψ/4, (T << T ∗) . (25)

We have used Eq. (22).The above result calls the following remarks.First, the expression of the polymer parallel extension combines two critical phenomena, one is relatedto the long-mass limit of the polymeric fractal and the other to the vicinity of the unbinding transitionof the membranes.Second, in this formula, naturally appears the fractal dimension (D + 2) /4D of a two-dimensional poly-meric fractal.Third, the parallel radius becomes more and more smaller as the unbinding transition is reached. Inother word, this radius is important only when the two adjacent membranes are strongly bound.Finally, the parallel radius expression may be used to determine the unbinding critical exponent ψ, inX-ray experiment, for instance.

V. CONCLUDING REMARKS

In this paper, we have two objectives, namely the conformational study of a polymeric fractal insidea tubular vesicle or between two parallel membranes forming an equilibrium lamellar phase. For theformer, the length scale is the equilibrium diameter that depends on the characteristics of the membrane,which are the bending modulus and the pressure difference between inner and outer aqueous media. For

61


the second geometry, the length scale is the mean-separation between the adjacent membranes.The main quantity to consider was the parallel extension of the confined polymeric fractal. Such aquantity was computed within the framework of an extended Flory-de Gennes theory. Notice that thesame result may be recovered using scaling argument or blob model [8].Another physical quantity to consider is the confinement free energy, denoted ∆F . It is the reducedfree energy of the polymer, measured from the state with H = ∞. Here, H is the tube diameter or themean-separation. The confinement free energy must have the scaling form : ∆F = kBTf (RF3/H), wherethe unknown scaling function f (x) has the following features : f (x) ∼ 0, for x << 1 and f (x) ∼ xm,for x >> 1. The exponent m can be obtained using the fact that ∆F must be extensive as a function

of the total mass M . This gives m = dF (3) = 5D/ (D + 2). Therefore, ∆F = kBTM (H/a)−5D/(D+2)

,

with H ∼ (κ/p)1/3

, for tubular vesicles, and H ∼ (W −Wc)−ψ

, for lamellar phases. Notice that ∆Fmay be measured by comparison of the two concentrations in the pore (tube or slab) and in the bulksolution. We have Cpore/Cbulk = α exp (−∆F/kBT ), where α is a certain coefficient depending on theratio RF3/H.We emphasize that for charged membranes forming the lamellar phase, it was demonstrated [62] that the

mean-separation between two adjacent bilayers scales as : H ∼ (χ− χc)−ψ

, as χ → χ+c , where χ is the

ionic concentration of the aqueous medium and χc is its critical value. Of course, the latter depends onthe nature of the lipid system. For instance, for DPPC in CaCl2 solutions, experimental measurements[66] showed that χc is in the range χc ≃ 84 − 10 mM. In this case, the parallel radius of the polymer

scales as : R∥ ∼ aM (D+2)/4D (χ− χc)ψ/4

.We said above that, for T > Tc, the shape fluctuations drive the membranes forming lamellar phase aparteven in the presence of the direct attractive forces. In this case, the system recovers its bound state bya simple application of an external pressure or a lateral tension.In the presence of an external pressure P , it was found [64] that the mean-separation H scales as :H ∼ ξ⊥ ∼ P−1/3 (ξ⊥ being the membrane mean-roughness). Such a behavior agrees with MC data [67].In this case, the parallel extension of the polymer obeys the scaling law : R∥ ∼ aM (D+2)/4DP 1/12. As itshould be, this extension increases with increasing external pressure.The role of a lateral tension is to suppress the bending undulations and the fluctuation-induced repulsion.In fact, the latter becomes short-ranged, and the long-ranged van der Waals attraction then dominates[64]. For this case, it was found [64] that the mean-separation behaves as : H ∼ ξ⊥ ∼ Σ−1/2, whereΣ represents the lateral tension. Then, the resulting parallel extension is : R∥ ∼ aM (D+2)/4DΣ1/8. Asexpected, the parallel radius of the polymer increases as the lateral tension is augmented.For tubular vesicles, we have assumed that they are tensionless. If it is not the case, and if the pressuredifference between inner and outer aqueous media can be neglected, the equilibrium diameter is : H =

2 (2κ/γ)1/2

, which is solution to Eq. (9). Here, γ is the interfacial tension coefficient. In this case, the

parallel extension of the polymer is : R∥ ∼ aM (D+2)/3D(a2γ/κ

)1/3. Naturally, this extension increases

with increasing interfacial tension coefficient. The minimal diameter corresponds to a typical valueγmin ∼ a−2κM (1−D)/D of γ (κ is fixed). Therefore, the polymer is confined if only if γ < γmin.Finally, further questions in relation with the subject are under consideration.

ACKNOWLEDGMENTS


62


REFERENCES

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64

1

Fluctuation spectra of supported membranes via long-flexible polymers

T. El Hafi 1, M. Benhamou 1,2, K. Elhasnaoui 1, H. Kaidi 1,3

1 LPPPC, Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, Morocco

2 ENSAM, Moulay Ismail University, P.O. Box 25290, Al Mansour, Meknes, Morocco 3 CRMEF, P.O. Box 255, Meknes, Morocco

([email protected])

Abstract. We consider a fluid membrane that is supported to a solid substrate via long-flexible polymers (as proteins or lipopolymers). In this paper, we aim at a quantitative investigations of the associated fluctuation spectra. The study is achieved through the knowledge of the height correlation function between points on the membrane. First, we compute exactly such a function for all pairs of points of the membrane, versus the parameters of the problem, which are the elastic constants of the membrane (bending modulus and interfacial tension) and the molecular-weight of the connected polymer chains. Second, from this correlation function, we extract the exact expression of the fluctuations amplitude and discuss its dependence on molecular-weight. Third, we write exact scaling laws for both height correlation function and fluctuations amplitude. Finally, we emphafsize that the presence of the connected polymers may be a new mechanism to bind membranes in the vicinity of substrates for specific interests. Keywords : Supported membranes; Tethers; Polymers; Correlation function; Fluctuation spectra.

1. Introduction Lipid-bilayer membranes supported on solid substrates are widely used as cell-surface models connecting biological and artificial materials. Usually, the supported biomembranes can be placed either directly on solids or on ultrathin polymer supports that play the role of the extracellular matrix [1]. The main goal is to control, organize and study the properties and function of membranes and membrane-associated proteins.

Supported lipid-membranes have other practical interests [2]. Indeed, they permit the biofunctionalization of inorganic solids and polymeric materials, and provide a natural environment for the immobilization of proteins (such as

khalid

Highlight

2

hormone receptors and antibodies). They allow the preparation of ultrathin, high-electric-resistance layers on conductors and the incorporation or receptors into these insulating layers for the design of biosensors. In addition, they are used in Electrochemistry for the detection of protein activity by measuring the resistance and capacitance changes [3-6].

Currently, supported membranes can be assemled as follows : (i) the inner monolayer of the lipid-bilayer fixed to the substrate by covalent chemical bonds or by ion bridges, (ii) freely supported lipid-protein bilayers separated from the substrate by ultrathin water layers (about ten nanometers), and (iii) bilayer membranes linked to the substrate by grafted or adsobed polymers, as lipopolymers [7]. The fixation of these lipopolymers on flat surfaces was studied using Surface Plasmon Spectroscopy. The film thickness of the adsorbed lipopolymer is about 12 to 20Å.

Statistical Mechanics of supported membranes has been the subject of many theoretical investigations [8]. The majority of these works considered tethers are springs. In this paper, however, the tethers are long-flexible polymer chains. We shall assume that the two extremities of each polymer chain on the inner monolayer of the supported bilayer membrane and wall remain parallel each to other.

The purpose is the study of the fluctuation spectra of supported membrane pairs. This is done through the computation of the height-height correlation function. We have computed this function exactly, from which we extracted an important quantity that is the fluctuations amplitude, even when the polymer chains are polydisperse. Such an amplitude is naturally space-dependent function.

The remaining of presentation proceeds as follows. Section 2 deals with the description of the used extended Hamiltonian. In Section 3, we present the exact computation of the correlation function of the supported membrane. Section 4 is dedicated to the fluctuation spectra and discussion. Some concluding remarks are drawn in the last section. Finally, some technical details are relegated in Appendices A and B.

3

2. Extended Canham-Helfrich Hamiltonian Start with a fluctuating fluid membrane that is embedded in a three-dimensional homogeneous aqueous medium, delimitated by a plan rigid substrate (semi-infinite geometry). We assume that the membrane is linked to the solid surface by polymer chains. We denote by the position-vector of the second extremity of chain on the supported interface. For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is perpendicular to the fluid membrane and substrate. To be more general, the positions of anchoring extremities are assumed to be distributed at random.

Within the framework of the Monge representation, a point on the membrane

can be described by , = ℎ, where = , is the two-dimensional

transverse vector and ℎ is the height-function (perpendicular distance to the surface).

The Statistical Mechanics of the supported fluid membrane is based on an extended Hamiltonian [8] ℎ = ℎ + ∑ ℎ , (1)

with the bare Canham-Helfrich Hamiltonian [9,10] ℎ = ! "∆ℎ + $∇ℎ , (2)

where " is the membrane bending modulus, $ is the interfacial tension coefficient. The second part in the r.h.s of Hamiltonian expression (1) accounts for the elastic contribution due to the presence of the connected hydrocarbon chains (springs), which is not local. There, denotes the elastic constant of polymer chain . The generalized Hamiltonian (1) describes the so-called model D in literature [8].

It will be convenient to rewrite the total Hamiltonian (1) as follows ℎ = ! "∆ℎ + $∇ℎ + &ℎ , (3)

with the local function ' = ∑ ( − . (4)

4

Here, ( denotes the two-dimensional Dirac-function. Then, the term &ℎ/2 plays the role of a local confinement potential that maintains the position of the fluid membrane around some typical distance, ℎ+. 3. Exact height-correlation function With the help of the above total Hamiltonian, we compute the expectation mean-value of various functionals of the height-field ℎ, in particular the two-point correlation function ,, ′ = ⟨ℎℎ′⟩ − ⟨ℎ⟩⟨ℎ′⟩ . (5) Such a function measures the height-fluctuations. As we shall see below, this correlation function allows the determination of the local membrane-roughness. Notice that the presence of tethers breaks the translation symmetry in the parallel directions to the supported membrane.

Now, the aim is the computation of the expected correlation function. To this end, we start from the following second-order differential equation "∆1 − $2Δ1 + '2 ,, ′ = ( − ′ , (6)

with the two-dimensional Laplacian operator Δ1 = 4/4 + 4/4. We

have introduced the rescaling parameters : " = "/56, $2 = $/56, 7 =/56, where 6 is the absolute temperature and 5 is the Boltzmann’ constant. There, '2 = '/56. Differential equation (6) is directly obtained from Hamiltonian (1) by simple functional derivatives.

To compute the height-correlation function, the first step consists to transform differential equation (6) into the following integral equation solved by the propagator ,, ,, ′ = , − ′ − ! ′′, − ′′'2′′,′′ − ′ , (7) or equivalently, ,, ′ = , − ′ − ∑ , − , − ′ , (7a)

5

where , − ′ is the (bare) propagator of a non-supported fluid membrane = 0,

, − ′ = ⟨ℎℎ′⟩ − ⟨ℎ⟩⟨ℎ′⟩ = 89:;<9 =>?;;.A;;BA;;CDE:FGHE:9 . (8)

The notation ⟨… ⟩ means the thermal average, which is performed with the Canham-Helfrich Hamiltonian defined in Eq. (2). The general expression for this propagator we do not recall is known in literature [11-13]. The particular value of the propagator ,0 = JK defines the mean-roughness of the membrane, JK. The latter and the in-plane correlation length (beyond which the

two-point correlation function exponentially fails), J∥, are related by : JK =J∥/16".

To solve integral equation (7a), we set = N, where the subscript 1 ≤ P ≤

is arbitrary. It is straightforward to show that ∑ QN , − ′ = ,N − ′ , (9)

with the matrix elements QN = (N + 7, − N . (10)

The inversion of Eq. (7a) gives , − ′ = ∑ RSNN ,N − ′ , (11)

with the matrix coefficients RN = QN = (N + 7N, − N . (12)

Matrix R is then the transpose of matrix Q.

Combining now relationships (7a) and (11) yields the expression for the height-correlation function ,, ′ = , − ′ − ∑ , − TN,N − ′,N , (13)

6

with coefficients TN = 7 + 7NRSN . (14)

These define a squared matrix, T. In Appendix A, we show that this matrix is symmetric and positive definite.

The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under

translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function , does. The translation symmetry breaking can be directly seen on integral equation (7).

Secondly, since matrix T is symmetric, the propagator is invariant permutting the points and ′.

Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements TN 's.

Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations.

Sixthly, the expected propagator is completely determined by the knowledge of matrix T and bare propagator ,.

Finally, as it should be, at infinity, that is → ∞ and ′ → ∞, ,, ′ goes to zero. 4. Fluctuation spectra and discussion The local membrane mean-roughness, JK, is simply the value of propagator ,, ′ for = ′. From expression (13), we obtain the following exact result JK = JK − ∑ , − TN, − N,N . (15)

Matrix elements TN’s depend on all distances between tethers W − NW’s and

their elastic constants ’s. Before discussing the above finding, we rewrite it on the following scaling

form

7

JK = JKX1 − ∑ Y − TZNY − N,N , (16)

with the dimensionless quantities TZN = JKTN , (17)

Y − = , − /JK . (18)

The latter is small than unity and Y0 = 1. Notice that coefficients TZN’s

depend only on dimensionless elastic constants 7JK’s and renormalized bare

propagators Y − N’s.

Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases :

(i) One tether:

Let be the vector-position. In this case, matrix TZ is a scalar quantity, and we have TZ = 7JK/[1 + 7JK\ . (19)

Combing this equality with formula (13) yields

JK = JK]1 − 7 _a9G7 _a9 Y − . (20)

(ii) Two tethers: We denote by and the position-vectors of the two tethers. In this case, we

find that the matrix elements of TZ are as follows

TZ = _a9b [7 + 77JK\ , TZ = _a9

b [7 + 77JK\ , (21a)

TZ = TZ = − _aFb 77Y − , (21b)

with

8

c = 1 + 7JK + 7JK + 77JKd − 77JKdY − . (21c) Then, the associated local roughness is

JK = JK e1 − TZY − − TZY − −2TZY − Y − f/ . (22)

(iii) The tethers have the same length and their extremities coincide at one point ≡ : This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by JK = JKh1 − iY − , (23) with the constant i = ∑ TZN .,N (24)

Such a constant is nothing else but the sum of all coefficients of matrix TZ. A comparison between relations (20) and (23) suggests that a -watermelon tether

is equivalent to one tether of effective elastic constant jZ, such that i = jZJK/[1 + jZJK\ . (25)

Such a equality relates the new elastic constant jZ to that of one tether, 7. We show in appendix B that the quantity i is exactly given by i = 7JK/[1 + 7JK\ . (26)

By comparison, we have jZ = 7 . (27)

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, JK, remains above the bulk one, JK, at

9

any point of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane.

Thirdly, for any number of tethers, when the considered point is far from the positions of the tethers extremities ’s, that is | − | ≫ J∥, the local

roughness JK fails exponentially to the bulk one, JK. Finally, let us assume that the tethers are long-flexible polymer chains (in

dilute solution) of common polymerization degree, m. In this case, the associated elastic constants are identical and scale as noS~qSmS, where q represents the monomer size. For infinitely long-polymer chains, that is m → ∞,

matrix TZ goes to JK/nor, where r is the matrix unity. In this limit, to first order in mS, the general result (16) becomes JK = JK s1 − JK/no ∑ Y − + tmSu . (28)

The above equality can be seen as a Taylor series in JK/no-variable. 5. Concluding remarks We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains.

The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations.

According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce

Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points.

In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere.

10

For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…).

Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration.

11

Appendix A The aim is to demonstrate that matrix T, defined in Eq. (14), is symmetric and positive definite. To this end, we introduce two matrices, jv and ,, of

respective elements 7(N and , − N, with indices 1 ≤ x ≤ and 1 ≤ P ≤ , and (N = 1, if x = P, and (N = 0, otherwise.

It is easy to see that the matrix B expresses as follows R = r + ,jv , (A.1) where r is the matrix unity. On the other hand, the matrix T can be rewritten as T = jvRS = jvr + ,jvS . (A.2) Such a form clearly suggests that Ty = T, since both matrices jvS and , are symmetric. Then, matrix T is symmetric.

To show that T is positive definite, we consider its inverse matrix TS =RjvS. Explicitly, we have TS = jvS + , . (A.3) The associated matrix elements are TSN = 7S(N + , − N . (A.4)

To prove that matrix T is positive definite, it will be sufficient to show that z ≡ ∑ TSNN,N > 0 , (A.5)

for any set of integer variables , … , , and z = 0, if only if , … , =0, … ,0. Start from the Fourier integral representation of the bare propagator

, − N = 89:;<9 =>?;;.|A;;BA;;~DE:FGHE:9 . (A.6)

We show that

12

z = ∑ + 89:;<9|∑ =>?;;.A;;∑ ~ =?;;.A;;

DE:FGHE:9 . (A.7)

Here, denotes the complex conjugate of , and > 0 is its squared module. Since the elastic constants ", $2 of the fluid membrane are positive, z > 0. The latter vanishes, if only if = 0 x = 1, … , . Matrix TS is then positive definite. Therefore, T is a positive definite matrix.

This ends the proof of the desired mathematical properties of matrix T.

Appendix B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑ TNTSN = ( .,N (B.1)

According to Eq. (A.4), we have ∑ TN[7S(N + JK\ = ( .,N (B.2)

A double summation over indices x and gives ∑ TZN = i = 7JK/[1 + 7JK\ .,N (B.3)

We have Eq. (17).

This ends the proof of formula (25).

13

References [1] More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). [2] E. Sackmann, Science 271, 43 (1996). [3] S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). [4] P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). [5] V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). [6] L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). [7] P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). [8] R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. [9] P.B. Canham, J. Theoret. Biol. 26, 61 (1970). [10] W. Helfrich, Z. Natureforsch 28c, 693 (1973). [11] T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). [12] H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). [13] A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

Statistical Theory and Method Abstracts – Zentralblatt Databasec© 2012 FIZ Karlsruhe

ZMATH 1197.76166Madmoune, Y.; El Hasnaoui, K.; Bendouch, A.; Kaidi, H.; Chahid, M.; Benhamou, M.Brownian dynamics of nanoparticles in contact with a confined biomembrane.Afr. J. Math. Phys. 8, No. 1, 91-100, Article No. 1010, electronic only (2010).Summary: The system we consider is a fluid membrane confined to two parallel reflecting walls that areseparated by a finite distance, L, assumed to be small in comparison to the bulk roughness. The attractivemembrane is surrounded by small colloidal particles (nanoparticles). The purpose is the study of Browniandynamics of these particles, under a change of a suitable parameter, such as temperature, T , or colloid-membrane interaction strength, w. The Brownian dynamics is investigated through the knowledge of thetime particle density, which solves the Smoluchowski equation. Solving this equation around the mid-plane,where the essential of phenomenon occurs, we obtain the exact form of the local particle density, as a functionof the perpendicular distance and time. In the derived expression, appears some time-scale, τ , which scalesas τ ∼ L3/w. This scale-time can be interpreted as the required time over which the colloidal suspensionreaches their final equilibrium state. Also, τ can be regarded as the time-interval over which the particlesare trapped in holes and valleys.

Classification: 76Z05 82D99 92C30 60K35Keywords: biomembranes; nanoparticles; confinement; Brownian dynamicshttp://www.fsr.ac.ma/GNPHE/ajmpVolume8N1-2010/ajmp1010.pdf

– 1 – November 13, 2012

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Baku, Azerbaijan| 5

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 3. May, 2013 T. El Hafi, M. Benhamou, K. Elhasnaoui, H. Kaidi. Fluctuation spectra of supported membranes via long-flexible

polymers. International Journal of Academic Research Part A; 2013; 5(3), 5-10. DOI: 10.7813/2075-4124.2013/5-3/A.1

FLUCTUATION SPECTRA OF SUPPORTED MEMBRANES

VIA LONG-FLEXIBLE POLYMERS

T. El Hafi1, M. Benhamou1,2, K. Elhasnaoui1, H. Kaidi1,3

1LPPPC, Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca,

2ENSAM, Moulay Ismail University, P.O. Box 25290, Al Mansour, Meknes, 3CRMEF, P.O. Box 255, Meknes (MOROCCO)

[email protected]

DOI: 10.7813/2075-4124.2013/5-3/A.1 ABSTRACT We consider a fluid membrane that is supported to a solid substrate via long-flexible polymers (as proteins

or lipopolymers). In this paper, we aim at a quantitative investigations of the associated fluctuation spectra. The study is achieved through the knowledge of the height correlation function between points on the membrane. First, we compute exactly such a function for all pairs of points of the membrane, versus the parameters of the problem, which are the elastic constants of the membrane (bending modulus and interfacial tension) and the molecular-weight of the connected polymer chains. Second, from this correlation function, we extract the exact expression of the fluctuations amplitude and discuss its dependence on molecular-weight. Third, we write exact scaling laws for both height correlation function and fluctuations amplitude. Finally, we emphafsize that the presence of the connected polymers may be a new mechanism to bind membranes in the vicinity of substrates for specific interests.

Key words: Supported membranes; Tethers; Polymers; Correlation function; Fluctuation spectra 1. INTRODUCTION Lipid-bilayer membranes supported on solid substrates are widely used as cell-surface models connecting

biological and artificial materials. Usually, the supported biomembranes can be placed either directly on solids or on ultrathin polymer supports that play the role of the extracellular matrix [1]. The main goal is to control, organize and study the properties and function of membranes and membrane-associated proteins.

Supported lipid-membranes have other practical interests [2]. Indeed, they permit the biofunctionalization of inorganic solids and polymeric materials, and provide a natural environment for the immobilization of proteins (such as hormone receptors and antibodies). They allow the preparation of ultrathin, high-electric-resistance layers on conductors and the incorporation or receptors into these insulating layers for the design of biosensors. In addition, they are used in Electrochemistry for the detection of protein activity by measuring the resistance and capacitance changes [3-6].

Currently, supported membranes can be assemled as follows : (i) the inner monolayer of the lipid-bilayer fixed to the substrate by covalent chemical bonds or by ion bridges, (ii) freely supported lipid-protein bilayers separated from the substrate by ultrathin water layers (about ten nanometers), and (iii) bilayer membranes linked to the substrate by grafted or adsobed polymers, as lipopolymers [7]. The fixation of these lipopolymers on flat surfaces was studied using Surface Plasmon Spectroscopy. The film thickness of the adsorbed lipopolymer is about 12 to 20Å.

Statistical Mechanics of supported membranes has been the subject of many theoretical investigations [8]. The majority of these works considered tethers are springs. In this paper, however, the tethers are long-flexible polymer chains. We shall assume that the two extremities of each polymer chain on the inner monolayer of the supported bilayer membrane and wall remain parallel each to other.

The purpose is the study of the fluctuation spectra of supported membrane pairs. This is done through the computation of the height-height correlation function. We have computed this function exactly, from which we extracted an important quantity that is the fluctuations amplitude, even when the polymer chains are polydisperse. Such an amplitude is naturally space-dependent function.

The remaining of presentation proceeds as follows. Section 2 deals with the description of the used extended Hamiltonian. In Section 3, we present the exact computation of the correlation function of the supported membrane. Section 4 is dedicated to the fluctuation spectra and discussion. Some concluding remarks are drawn in the last section. Finally, some technical details are relegated in Appendices A and B.

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6 | PART A. NATURAL AND APPLIED SCIENCES

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 3. May, 2013 2. EXTENDED CANHAM-HELFRICH HAMILTONIAN Start with a fluctuating fluid membrane that is embedded in a three-dimensional homogeneous aqueous

medium, delimitated by a plan rigid substrate (semi-infinite geometry). We assume that the membrane is linked to the solid surface by 푛 polymer chains. We denote by 푟 the position-vector of the second extremity of chain 푖 on the supported interface. For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is perpendicular to the fluid membrane and substrate. To be more general, the positions of anchoring extremities are assumed to be distributed at random.

Within the framework of the Monge representation, a point on the membrane can be described by 푟, 푧 =ℎ(푟) , where 푟 = (푥, 푦) is the two-dimensional transverse vector and ℎ(푟) is the height-function (perpendicular distance to the surface).

The Statistical Mechanics of the supported fluid membrane is based on an extended Hamiltonian [8] 퐻[ℎ] = 퐻 [ℎ] + ∑ 푘 ℎ (푟 ), (1) with the bare Canham-Helfrich Hamiltonian [9,10]

퐻 [ℎ] = ∫ 푑 푟 [휅(∆ℎ) + 휎(∇ℎ) ], (2) where 휅 is the membrane bending modulus, 휎 is the interfacial tension coefficient. The second part in the

r.h.s of Hamiltonian expression (1) accounts for the elastic contribution due to the presence of the connected hydrocarbon chains (springs), which is not local. There, 푘 denotes the elastic constant of polymer chain 푖. The generalized Hamiltonian (1) describes the so-called model D in literature [8].

It will be convenient to rewrite the total Hamiltonian (1) as follows 퐻[ℎ] = ∫ 푑 푟 [휅(∆ℎ) + 휎(∇ℎ) + 휚ℎ ], (3) with the local function 휌(푟) = ∑ 푘 훿 (푟 − 푟 ). (4) Here, 훿 (푟) denotes the two-dimensional Dirac-function. Then, the term 휚ℎ /2 plays the role of a local

confinement potential that maintains the position of the fluid membrane around some typical distance, ℎ . 3. EXACT HEIGHT-CORRELATION FUNCTION With the help of the above total Hamiltonian, we compute the expectation mean-value of various functionals

of the height-field ℎ, in particular the two-point correlation function 퐺(푟, 푟′) = ⟨ℎ(푟)ℎ(푟′)⟩ − ⟨ℎ(푟)⟩⟨ℎ(푟′)⟩. (5) Such a function measures the height-fluctuations. As we shall see below, this correlation function allows the

determination of the local membrane-roughness. Notice that the presence of tethers breaks the translation symmetry in the parallel directions to the supported membrane.

Now, the aim is the computation of the expected correlation function. To this end, we start from the following second-order differential equation

휅∆ − 휎Δ + 휌(푟) 퐺(푟, 푟′) = 훿 (푟 − 푟′), (6) with the two-dimensional Laplacian operator Δ = (휕 /휕푥 ) + (휕 /휕푦 ). We have introduced the rescaling

parameters : 휅 = 휅/푘 푇, 휎 = 휎/푘 푇, 푘 = 푘 /푘 푇, where 푇 is the absolute temperature and 푘 is the Boltzmann’ constant. There, 휌 = 휌/푘 푇. Differential equation (6) is directly obtained from Hamiltonian (1) by simple functional derivatives.

To compute the height-correlation function, the first step consists to transform differential equation (6) into the following integral equation solved by the propagator 퐺,

퐺(푟, 푟′) = 퐺 (푟 − 푟′) − ∫ 푑 푟′′퐺 (푟 − 푟′′)휌(푟′′)퐺(푟′′ − 푟′), (7) or equivalently, 퐺(푟, 푟′) = 퐺 (푟 − 푟′) − ∑ 푘 퐺 (푟 − 푟 )퐺(푟 − 푟′), (7a) where 퐺 (푟 − 푟′) is the (bare) propagator of a non-supported fluid membrane (푘 = 0),

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INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 3. May, 2013

퐺 (푟 − 푟′) = ⟨ℎ(푟)ℎ(푟′)⟩ − ⟨ℎ(푟)⟩ ⟨ℎ(푟′)⟩ = ∫ ( )

.( ′). (8)

The notation ⟨… ⟩ means the thermal average, which is performed with the Canham-Helfrich Hamiltonian 퐻

defined in Eq. (2). The general expression for this propagator we do not recall is known in literature [11-13]. The particular value of the propagator 퐺 (0) = (휉 ) defines the mean-roughness of the membrane, 휉 . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), 휉∥ , are related by : (휉 ) = 휉∥ /16휅.

To solve integral equation (7a), we set 푟 = 푟 , where the subscript 1 ≤ 훽 ≤ 푛 is arbitrary. It is straightforward to show that

∑ 퐴 퐺(푟 − 푟′) = 퐺 푟 − 푟′ , (9) with the matrix elements

퐴 = 훿 + 푘 퐺 푟 − 푟 . (10) The inversion of Eq. (7a) gives 퐺(푟 − 푟′) = ∑ [퐵 ] 퐺 푟 − 푟′ , (11) with the matrix coefficients

퐵 = 퐴 = 훿 + 푘 퐺 푟 − 푟 . (12) Matrix 퐵 is then the transpose of matrix 퐴. Combining now relationships (7a) and (11) yields the expression for the height-correlation function 퐺(푟, 푟′) = 퐺 (푟 − 푟′) − ∑ 퐺 (푟 − 푟 )푀 퐺 푟 − 푟′, , (13) with coefficients

푀 = 푘 + 푘 [퐵 ] . (14) These define a squared matrix, 푀. In Appendix A, we show that this matrix is symmetric and positive

definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel

directions to the fluid membrane and substrate, but the bare correlation function 퐺 does. The translation symmetry breaking can be directly seen on integral equation (7).

Secondly, since matrix 푀 is symmetric, the propagator is invariant permutting the points 푟 and 푟′. Thirdly, all dependence of this propagator in elastic constants 푘 's is entirely contained in matrix elements

푀 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence

of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix 푀 and bare

propagator 퐺 . Finally, as it should be, at infinity, that is 푟 → ∞ and 푟′ → ∞, 퐺(푟, 푟′) goes to zero. 4. FLUCTUATION SPECTRA AND DISCUSSION The local membrane mean-roughness, 휉 (푟), is simply the value of propagator 퐺(푟, 푟′) for 푟 = 푟′. From

expression (13), we obtain the following exact result

휉 (푟) = (휉 ) − ∑ 퐺 (푟 − 푟 )푀 퐺 푟 − 푟, . (15) Matrix elements 푀 ’s depend on all distances between tethers 푟 − 푟 ’s and their elastic constants 푘 ’s. Before discussing the above finding, we rewrite it on the following scaling form

휉 (푟) = 휉 1 − ∑ 푔 (푟 − 푟 )푀 푔 푟 − 푟, , (16)

with the dimensionless quantities


INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 3. May, 2013

푀 = (휉 ) 푀 , (17)

푔 (푟 − 푟 ) = 퐺 (푟 − 푟 )/(휉 ) . (18) The latter is small than unity and 푔 (0) = 1. Notice that coefficients 푀 ’s depend only on dimensionless

elastic constants 푘 (휉 ) ’s and renormalized bare propagators 푔 푟 − 푟 ’s. Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Let 푟 be the vector-position. In this case, matrix 푀 is a scalar quantity, and we have

푀 = 푘(휉 ) / 1 + 푘(휉 ) . (19) Combing this equality with formula (13) yields

휉 (푟) = 휉 1 − ( )( ) 푔 (푟 − 푟 ) . (20)

(ii) Two tethers: We denote by 푟 and 푟 the position-vectors of the two tethers. In this case, we find that the matrix elements

of 푀 are as follows

푀 = 푘 + 푘 푘 (휉 ) , 푀 = 푘 + 푘 푘 (휉 ) , (21a)

푀 = 푀 = − 푘 푘 푔 (푟 − 푟 ), (21b)

with

퐷 = 1 + 푘 (휉 ) + 푘 (휉 ) + 푘 푘 (휉 ) − 푘 푘 (휉 ) 푔 (푟 − 푟 ). (21c) Then, the associated local roughness is

휉 (푟) = 휉 1 − 푀 푔 (푟 − 푟 ) − 푀 푔 (푟 − 푟 )

−2푀 푔 (푟 − 푟 )푔 (푟 − 푟 )

/

. (22)

(iii) The 푛 tethers have the same length and their extremities coincide at one point 푟 (푟 ≡ 푟 ) : This case corresponds to a 푛-watermelon tether, and we find that the associated local roughness is given by

휉 (푟) = 휉 1 − 퐶푔 (푟 − 푟 ), (23) with the constant 퐶 = ∑ 푀 ., (24)

Such a constant is nothing else but the sum of all coefficients of matrix 푀. A comparison between relations

(20) and (23) suggests that a 푛-watermelon tether is equivalent to one tether of effective elastic constant 퐾, such that

퐶 = 퐾(휉 ) / 1 + 퐾(휉 ) . (25) Such a equality relates the new elastic constant 퐾 to that of one tether, 푘. We show in appendix B that the

quantity 퐶 is exactly given by 퐶 = 푛푘(휉 ) / 1 + 푛푘(휉 ) . (26) By comparison, we have

퐾 = 푛푘. (27) Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local

roughness, 휉 (푟), remains above the bulk one, 휉 , at any point 푟 of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane.

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INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 3. May, 2013 Thirdly, for any number of tethers, when the considered point 푟 is far from the positions of the tethers

extremities 푟 ’s, that is |푟 − 푟 | ≫ 휉∥ , the local roughness 휉 (푟) fails exponentially to the bulk one, 휉 . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common

polymerization degree, 푁. In this case, the associated elastic constants are identical and scale as 푅 ~푎 푁 , where 푎 represents the monomer size. For infinitely long-polymer chains, that is 푁 → ∞, matrix 푀 goes to (휉 /푅 ) 퐼, where 퐼 is the matrix unity. In this limit, to first order in 푁 , the general result (16) becomes

휉 (푟) = 휉 1 − (휉 /푅 ) ∑ 푔 (푟 − 푟 ) + 푂(푁 ) . (28)

The above equality can be seen as a Taylor series in 휉 /푅 -variable. 5. CONCLUDING REMARKS We recall that the purpose of this paper is the investigation of correlations between points belonging to a

supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the

membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations.

According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce

Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points.

In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere.

For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…).

Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration.

REFERENCES

1. More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). 2. E. Sackmann, Science 271, 43 (1996). http://dx.doi.org/10.1126/science.271.5245.43. 3. S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). http://dx.doi.org/10.1021/

la034197v. 4. P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004).

http://dx.doi.org/10.1016/j.bbamem.2003.12.009. 5. V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631

(2006). http://dx.doi.org/10.1021/bc050328n. 6. L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007).

http://dx.doi.org/10.1016/j.elecom.2006.10.045. 7. P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). http://dx.doi.org/

10.1002/1616-5195(200211)2:8<387::AID-MABI387>3.0.CO;2-5. 8. R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. http://dx.doi.org/

10.1140/epje/i2006-10084-2. 9. P.B. Canham, J. Theoret. Biol. 26, 61 (1970). http://dx.doi.org/10.1016/S0022-5193(70)80032-7. 10. W. Helfrich, Z. Natureforsch 28c, 693 (1973). 11. T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). http://dx.doi.org/10.1103/

PhysRevE.70.051404. 12. H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). http://dx.doi.org/10.1209/

epl/i2004-10305-4. 13. A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1

(2006).


INTERNATIONAL JOURNAL of ACADEMIC RESEARCH Vol. 5. No. 3. May, 2013 APPENDIX A The aim is to demonstrate that matrix 푀, defined in Eq. (14), is symmetric and positive definite. To this end,

we introduce two matrices, 퐾 and 퐺 , of respective elements 푘 훿 and 퐺 푟 − 푟 , with indices 1 ≤ 훼 ≤ 푛 and 1 ≤ 훽 ≤ 푛, and 훿 = 1, if 훼 = 훽, and 훿 = 0, otherwise.

It is easy to see that the matrix B expresses as follows

퐵 = 퐼 + 퐺 퐾, (A.1) where 퐼 is the matrix unity. On the other hand, the matrix 푀 can be rewritten as

푀 = 퐾퐵 = 퐾(퐼 + 퐺 퐾) . (A.2) Such a form clearly suggests that 푀 = 푀, since both matrices 퐾 and 퐺 are symmetric. Then, matrix 푀

is symmetric. To show that 푀 is positive definite, we consider its inverse matrix 푀 = 퐵퐾 . Explicitly, we have

푀 = 퐾 + 퐺 . (A.3) The associated matrix elements are

[푀 ] = 푘 훿 + 퐺 푟 − 푟 . (A.4)

To prove that matrix 푀 is positive definite, it will be sufficient to show that

퐹 ≡ ∑ 푥 [푀 ] 푥, > 0, (A.5) for any set of integer variables (푥 , … , 푥 ), and 퐹 = 0, if only if (푥 , … , 푥 ) = (0, … ,0). Start from the Fourier

integral representation of the bare propagator

퐺 푟 − 푟 = ∫ ( )

.

. (A.6) We show that

퐹 = ∑ 푘 푥 + ∫ ( )

∑ . ∑ . . (A.7)

Here, 푧 denotes the complex conjugate of 푧, and 푧푧 > 0 is its squared module. Since the elastic constants

(휅, 휎) of the fluid membrane are positive, 퐹 > 0. The latter vanishes, if only if 푥 = 0 (훼 = 1, … , 푛). Matrix 푀 is then positive definite. Therefore, 푀 is a positive definite matrix.

This ends the proof of the desired mathematical properties of matrix 푀. APPENDIX B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity

∑ 푀 [푀 ] = 훿 ., (B.1) According to Eq. (A.4), we have

∑ 푀 푘 훿 + (휉 ) = 훿 ., (B.2)

A double summation over indices 훼 and 훾 gives

∑ 푀 = 퐶 = 푛푘(휉 ) / 1 + 푛푘(휉 ) ., (B.3) We have Eq. (17). This ends the proof of formula (25).

The Quantum Casimir Effect May Be a Universal ForceOrganizing the Bilayer Structure of the Cell Membrane

Piotr H. Pawlowski • Piotr Zielenkiewicz

Received: 1 November 2012 / Accepted: 29 March 2013 / Published online: 24 April 2013

The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract A mathematic–physical model of the interac-

tion between cell membrane bilayer leaflets is proposed

based on the Casimir effect in dielectrics. This model

explains why the layers of a lipid membrane gently slide

one past another rather than penetrate each other. The

presented model reveals the dependence of variations in the

free energy of the system on the membrane thickness. This

function is characterized by the two close minima corre-

sponding to the different levels of interdigitation of the

lipids from neighbor layers. The energy barrier of the

compressing transition between the predicted minima is

estimated to be 5.7 kT/lipid, and the return energy is esti-

mated to be 3.1 kT/lipid. The proposed model enables

estimation of the value of the membrane elastic thickness

modulus of compressibility, which is 1.7 9 109 N/m2, and

the value of the interlayer friction coefficient, which is

1.9 9 108 Ns/m3.

Keywords Cell membrane Lipid bilayer Casimir

effect

Introduction

Casimir-Polder (Casimir 1948; Casimir and Polder 1948)

forces are universal physical forces arising from a quan-

tized field. They act even between two uncharged metallic

plates in a vacuum, placed a few micrometers apart,

without any external electromagnetic field. The idea that

Casimir forces may play an important role in different

biomembrane systems is quite new. Based on the funda-

mental work of Lifshitz (1956) dealing with retarded van

der Waals forces between macroscopic bodies, first it was

applied to the formation of cellular ‘‘rouleaux’’ (Bradonjic

et al. 2009) and confined biomembranes (El Hasnaoui et al.

2010). In a natural manner, it supplements the theory of

nonretarded van der Waals interactions in a lipid–water

system (Parsegian and Ninham 1970), sometimes offering

an interesting counterproposal.

Thus, forces related to the zero-point energy of quantum

fluctuations may play an important role in biology, and the

analysis of these forces offers a new view into biological

phenomena at the cellular level. Herein, we propose a

simple second-quantized explanation for the fact that cell

membrane bilayer leaflets slip (Otter and Shkulipa 2007)

past one another, rather than penetrate each other. This

membrane feature is of great importance as it determines

the anisotropy of the membrane’s rheological properties.

The relative freedom of movement of molecules along

membrane leaflets and the relative restriction of displace-

ment in the transverse direction account for the great lateral

fluidity and the small perpendicular compressibility of a

membrane (Evans and Hochmuth 1978). These dual

mechanical properties, both fluid-like and solid-like, have

an impact on the structure and function of the proteins

embedded in the lipid bilayer matrix (Andersen and Koe-

ppe 2007). This impact finally determines the status of the

cell membrane as an active barrier, a natural organizer and

an important participant in all processes of life.

The proposed model of the interaction of the cell

membrane bilayer leaflets considers the membrane interior

to be a three-layer dielectric sandwich. The free energy of

P. H. Pawlowski (&) P. Zielenkiewicz

Institute of Biochemistry and Biophysics, Polish Academy of

Sciences, PAS, Pawinskiego 5a, 02-106 Warszawa, Poland

e-mail: [email protected]

P. Zielenkiewicz

Plant Molecular Biology Laboratory, Warsaw University,

Warsaw, Poland

123

J Membrane Biol (2013) 246:383–389

DOI 10.1007/s00232-013-9544-9

khalid

Highlight

the system is related to the electromagnetic field excitations

in the ground state. The free energy depends on the varying

thickness of the central layer, where lipid chains penetrate

the opposite layers and where the density of lipid chains,

and the dielectric permittivity, varies in space. The theo-

retical values of the two energy minima, the membrane

elastic thickness modulus of compressibility and the

interlayer friction coefficient, were set according to this

model.

The Model

The cell membrane interior was considered to be a three-

layer dielectric sandwich, which consists of parallel slabs

as in Fig. 1. The two lateral peripheral regions of this

system are fully occupied by all the lipids of the local

leaflet. The remaining central region contains the hydro-

carbon tails that penetrate from the neighboring layers. The

dielectric constant, ec, of this central layer differs from the

dielectric constant, ep, of the other parts of the membrane.

We assumed that the length of the lipids in each leaflet may

vary within a narrow range, L ± dL/2, where L represents

the average lipid length and dL represents the width of the

length distribution. The distribution of the lipid lengths was

assumed to be uniform. Thus, the perpendicular cross

section of a membrane bilayer matrix resembles ‘‘the two

overlapping combs with some broken teeth.’’

The thickness of the central region, dc, falls to a mini-

mum value equal to dL in the configuration in which the

total membrane thickness, d, equals 2L (Fig. 2a). This

configuration we called ‘‘configuration MTCR’’ (minimal

thickness of the central region). To clarify further consid-

erations, configuration MTCR was treated as the reference

configuration. In this configuration, dc may increase both

with an increase (Fig. 2b) and with a decrease (Fig. 2c) in

the total membrane thickness. Thus, the possible variation

in the thickness of the central region in configuration

MTCR is unidirectional.

For convenience, we introduced the variable x to

describe the difference between the actual total membrane

thickness and the thickness of the membrane in configu-

ration MTCR; that is, x = d-2L. Then to formalize the

description of the central region, one may write

dc = dL ? |x|, where |x| denotes the absolute value.

The dielectric constant, ec, was assumed to vary in space

within the central region due to variations in the lipid

density. Variations along the direction perpendicular to the

membrane plane were postulated. At first approximation, ec

was simply characterized by the spatial average heci. In

general, when the total membrane thickness differs from

the thickness of the reference configuration by x, the

average heci changes as described by the following

formula:

heci ¼1þ ep 1

1þ jxj=dL

1þjxj=dL

x\0

ep x ¼ 0

1þ ep 1

11þx=dL x [ 0

8><

>:ð1Þ

For details, see Appendix.

The plot of Eq. 1 indicates (Fig. 3) that the average

heci increases with the compression of the membrane

thickness (x \ 0) and decreases with membrane thickness

extension (x [ 0). For the membrane in the configuration

MTCR (x = 0), heci strictly equals ep.

Quantum electrodynamic considerations reveal that two

dielectric plates separated by a dielectric medium may be

attracted as a result of the decrease in the zero-point energy

of the quantum electromagnetic field excitations (Srivast-

ava et al. 1985). In the case of the cell membrane, by

approximating the dielectric constant of the central region

with the average value heci, the free energy, F, of the field

per unit area may be described by the following equation

(Bradonjic et al. 2009):

F ¼ p2

720

ep heciep þ heci

2 hcffiffiffiffiffiffiffiffiheci

pdLð Þ3 1þ jxj=dLð Þ3

ð2Þ

where h is the reduced Planck constant and c is the speed of

light in a vacuum; the dependence of heci on x is described

by Eq. 1.

ep ec ep

LL δ

Fig. 1 Three-layer dielectric sandwich model of a cell membrane.

Only the longest and shortest lipids are shown. L average lipid length,

dL width of the lipid length range, ec dielectric constant of the central

region, ep dielectric constant of the peripheral region

384 P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

123

Results

Equations 1 and 2 show that, for ideally flat leaflets

(dL = 0), Casimir forces couple membrane layers at zero

distance (x = 0), with the free energy tending to negative

infinity. More reasonable calculations (dL = 5 A, ep = 2),

taking into account the heterogeneity of the lipid lengths

(Blaurock 1982) and specifying the dielectric constant of

the peripheral regions of the membrane (Huang and Levitt

1977) were also performed. These calculations indicate

(Fig. 4) that there are two finite minima of free energy,

both for the membrane in configurations close to the

MTCR. The first (lower) minimum (F = -4.68 9 10-2 J/

m2) is for leaflets being slightly separated (x = 4.0 A), and

the second one (F = -2.54 9 10-2 J/m2) is for leaflets

being moderately pushed together (x = -2.9 A).

According to the model, in both cases, the thickness of the

central region of the membrane (dc = 9.0 and dc = 7.9 A)

is larger than the assumed thickness in the configuration

MTCR (dc = 5 A).

Discussion

We assumed that the central region of the membrane may

be treated as a kind of leaflet interface. One interesting

question is how the interface is stabilized to avoid leaflet

penetration and sticking. The proposed model may offer a

simple explanation based on the Casimir effect in dielec-

trics. At reasonable values of the applied parameters (see

‘‘Results’’ section), the model predicts two free energy

minima. For the membrane organized in the lower energy

minimum (x = 4 A), its leaflets are separated by 4 A

d = 2L

dc = L

MTCR

a

d > 2L

dc > b

d < 2L

dc > cδ Lδ Lδ

Fig. 2 Unidirectional variations in the thickness of the central region

of the membrane. Only the longest and shortest lipids are shown.

a Configuration MTCR. Total membrane thickness d = 2L. Central

region thickness dc = dL. b Bilayer leaflets farther apart than in

configuration MTCR. Total membrane thickness d [ 2L. Central

region thickness dc [ dL. c Bilayer leaflets closer together than in

configuration MTCR. Total membrane thickness d \ 2L. Central

region thickness dc [ dL. L average lipid length, dL width of the lipid

length range, MTCR minimal thickness of the central region

100-10

1

p

2 p - 1

x/ Lδ

ε

ε

cε

Fig. 3 The space average of the dielectric constant in the central

region, heci, as a function of the ratio x/dL. x the difference between

the total membrane thickness and the membrane thickness in

configuration MTCR, dL width of the lipid length range, ep dielectric

constant in peripheral regions of the membrane

-0,07

-0,05

-0,03

-0,01

0,01

0,03

0,05

-50 -30 -10 10 30 50 70 90

F [J

/m2 ]

x [A]

Fig. 4 The free energy, F, of the field per unit area as a function of a

change, x, in the membrane thickness relative to the thickness of the

MTCR membrane

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect 385

123

greater than in configuration MTCR. At the assumed het-

erogeneity of the lipid lengths (dL = 5 A), this result means

that 20 % of lipids (1-x/dL) penetrate into the neighboring

leaflet (Fig. 5a) but by no more than 1 A (dL-x).

According to the results presented in Fig. 4, transition

into the second (higher) minimum (x = –2.9 A) requires

approximately 5.7 kT/lipid (calculated for T = 300 K and

area per lipid a = 50 A2). In this minimum energy con-

figuration, the membrane leaflets are in closer contact. All

lipids penetrate the opposite layer but no deeper than 7.9 A

(Fig. 5b). The return to the other minimum requires 3.1 kT/

lipid.

The revealed characteristic energies are several times

higher than those estimated assuming the average tension c(0.003 mN/m) of living cells (Blanchard and Rauch 2012).

The result ca kT indicates that the Casimir effect may

be an important contributor to the membrane dynamics,

along with the hydrophobic effect.

To study the mechanical properties of the presented

model membrane, the elastic thickness modulus of

compressibility, k, was numerically estimated using a dif-

ferential second derivative of the free energy at the first

minimum (k = d F00). Assuming a bilayer thickness of

d = 5 nm (Kuchel and Ralston 1998), k = 1.7 9 109 N/

m2; i.e., this estimated value of k is of the same order as

that measured using volume dilatometry of lipid bilayers

(Srinivasan et al. 1974). This result indicates that the

central region of the cell membrane resembles an

‘‘incompressible’’ core that does not allow lipid interleaflet

penetration.

The results of the presented model also enable estima-

tion of the value of the interlayer friction coefficient,

f (friction force per unit area and unit velocity). Approxi-

mating the end of the lipid penetrating the neighbor leaflet

with a hemisphere of a radius r that experiences the action

of the Stokes force from the liquid passing with the

velocity 2v, one may obtain f = 6pgr/a, where g is the

membrane shear viscosity. Taking the typical value of

viscosity g = 0.1 Ns/m2 estimated from the diffusion

coefficient of membrane-spanning proteins in phospholipid

bilayers (Waugh 1982) and assuming that the membrane is

in the first minimum with r = 0.5 A, f = 1.9 9 108 Ns/

m3, which is within the range of reported experimental

values (Shkulipa et al. 2005; Otter and Shkulipa 2007). For

the second minimum, one may expect an eightfold higher

value because of a deeper penetration.

According to the model, a membrane bilayer in a basic

state (lower minimum) should possess relatively small in-

terleaflet friction. The probability that, due to thermal

fluctuations, some regions of the membrane reach second

minimum is relatively small.

At first glance, one may worry about some physical and

mathematical problems with the proposed approach. It is

obvious that some points require additional discussion,

especially phenomena at a physical level neglected in our

model. First of all, is it justified to assume that the lipids of

opposite leaflets interpenetrate each other at all? Steric

interactions seem to be the dominant suppressor of inter-

digitation. However, a spontaneous or induced interdigi-

tated phase of bilayers consisting of double-tail lipids was

confirmed in computer simulation and differential scanning

calorimetry experiments (Kranenburg 2004; Kranenburg

et al. 2004; Mavromoustakos et al. 2011). Moreover, a

simple estimation below shows that the possible steric

effect is not energetically dominant, as one may expect. Let

us assume that during interdigitation the part p of lipids is

compressed and their length decreases by an assumed

certain value, k. Let us also assume that each of two lipid

acyl chains contains the number b of C–C bonds charac-

terized by a certain bond stiffness, g. Then, elastic defor-

mation energy per lipid molecule may be calculated as

pgk2/b. For typical conditions, p = 1/10 (meaning that

50 % of penetrating lipids in the lower minimum are

4.50.5-4.5

100%

-0.5

n 1 n 2

a

ξ [A]

3.951.05-3.95

100%

-1.05

n 1 n 2

b

ξ [A]

Fig. 5 The area densities n1 and n2 of lipids belonging to a given

leaflet as a function of a distance, n, from the membrane midplane.

The range occupied by the longest and shortest lipids is shown above.

a For the first minimum (lower) of free energy, x = 4 A. b For the

second minimum (higher) of free energy, x = -2.9 A


123

compressed), g = 100 N/m, k = 0.5 A´

and b = 15; this

energy equals 0.8 kBT. These are only 7 % of the predicted

value of the energy barrier in a lower minimum and, as

such, may be neglected at first approximation. The next

question is, how much does the derived shape of the free

energy, F (Fig. 4), depend on the specific way that the

interpenetration occurs? It was assumed that the distribu-

tion of lipid lengths was uniform and, in this way, inter-

digitation varied linearly with a distance from the

membrane midplane (Figs. 5, 6). What will change if we

assume the more spectacular variation? In extreme cases,

when a single-point distribution (dL = 0) is assumed, one

infinite minimum of energy at midplane will be obtained

and the energy will increase with distance, like –1/|x|3.

Thus, narrowing the distribution lowers minima and

approaches them together. It should be stressed that the

parameters heci, ep and dL in Eq. 2 may also effectively

describe a more diverse system.

Applied formula for the free energy per unit area, Eq. 2,

assuming ep ? ? and heci ? ?, gives the famous Casi-

mir result for the energy of attraction between ideal mir-

rors. This energy is a result of the change in the zero-point

energy of an empty quantum vacuum. From the other hand,

it is well known that real cell membranes are under the

permanent influence of electrostatic interactions. Assuming

a natural transmembrane electric potential V = 100 MV, a

membrane thickness dm = 10-8, an effective membrane

dielectric constant em = 2 and a vacuum permittivity

e0 = 10-11 [F/m], it is easy to estimate the area density of

the energy of an electric field, 0.5e0em (V2/dm). It is equal to

10-5 J/m2 . This result is three orders less than the energy

of the considered Casimir effect.

It is necessary to underline that the discussed formula is

correct only within certain assumed constraints. One of

them, zero temperature approximation (Landau and Lif-

shitz 1960) may be justified for considered conditions

(kBT hc/dc). The second assumption, i.e., the same

dielectric constant at all frequencies, is formally valid for

ideal dielectrics or in the case of large separations. How-

ever, for values of constant permittivity not so far from

unity (vacuum value), possible error for small distances is

disregarded. Moreover, the bulk term in energy is disre-

garded, which may be dominant for large separations. As

the ratio of energies of attraction of water slabs and lipid

slabs (both estimated using Eq. 2) is as small as 1:10, for

the sake of simplicity, the influence of water outside the

membrane was not considered. Anisotropy in the dielectric

constant is also beyond the scope of this article, and the

model for the dielectric constant in the region of inter-

digitation is very simple, based on a linear superposition of

dielectric constants in terms of the effective densities of

tails.

We realize that real membranes, especially cell mem-

branes, are obviously complex, heterogeneous, nonideal

dielectrics with complicated frequency responses and real

conductivity. They are certainly not perfectly plain and

smooth plates. Despite the above simplifications, the gen-

eral predictions of our model, i.e., the magnitude of the free

energy and the existence of two energy minima, seem to be

still reasonable and wait for more detailed further investi-

gation and confirmation. A lateral Casimir effect for cor-

rugated planes or nonretarded local pairwise van der Waals

forces might provide a better description of the physics

there and provide a more accurate description. For exam-

ple, the last one can replace an inverse-cube law, describ-

ing variation in energy with distance, by an inverse-square

law. Using this method, a simple estimation of energy of

so-called hydrophobic bonding, at a Hamaker function

equal to 7 9 10-21 J and a distance of 50 A´

(Parsegian and

Ninham 1970), gives the value of the density of free energy

close to 10-5 J/m2. This quantity is three orders less than

the energy of lipid–lipid interactions and two orders less

than the energy of water–water interactions estimated in

our model. We think that future numerical brute-force

simulations might make an important contribution toward a

better understanding of the mentioned discrepancy.

In light of these findings, it is evident that the Casimir

effect may play an important role in many biological

phenomena and may be a universal force that organizes the

tensegrity structure of biological systems. Some scientists

might expect spectacular ‘‘levitation’’ forces between the

leaflets, but it appears that there is instead a ‘‘quantum

trap’’ preventing membrane leaflet interdigitation and col-

lapse as well as maintaining a significant gap between

leaflets, which leaves molecules a plane with freedom of

0 L +x/2δL/2 +x/2-L-x /2δL/2-x /2

−δL/2-x /2

n p

n 1 n 2

−δL/2+x /2

ξ

Fig. 6 The area densities n1 and n2 of lipids belonging to a given

leaflet as a function of the distance, n, from the membrane midplane.

The range occupied by the longest and shortest lipids is shown above.

np total number of lipids belonging to a given leaflet per unit surface

area, L average lipid length, dL width of the lipid length range,

x = d - 2L, where d is the total membrane thickness


123

movement. Even without taking into account interlayer

lipid collisions, hydrophobic interactions and the stabiliz-

ing role of proteins, Casimir forces may prevent lipid chain

mishmash or molecular escape. This hypothesis appears to

be fruitful and is worth experimental verification.

Acknowledgements We thank Prof. Bogdan Lesyng and Dr. Marek

Kalinowski from the University of Warsaw for inspiration and valu-

able discussion.

Open Access This article is distributed under the terms of the

Creative Commons Attribution License which permits any use, dis-

tribution, and reproduction in any medium, provided the original

author(s) and the source are credited.

Appendix

The Average Value of the Dielectric Constant

in the Central Region of a Lipid Bilayer

Let variable n represent the distance from the membrane

midplane; then, the central region may be defined as the

membrane layer within the range dL=2 xj j=2 n dL=2þ xj j=2. Here, dL represents the width of the

distribution of lipid lengths, and x = d - 2L, where d is

the total membrane thickness and L is the average lipid

length. In general, x may be positive (thickness expansion),

negative (thickness compression) or zero (configuration

MTCR) and falls within the range x [ dL - 2L. Assuming

that the dielectric constant ec in the central region varies

locally with n, ec depends on the number of lipids, n, that

are passing through the unit area of plane n = constant.

This dependence may be described as

ec ¼ 1þ ðep 1Þ n

np

ð3Þ

where ep is the dielectric constant of the peripheral region

of the membrane fully occupied only by the lipids

belonging to a given leaflet and np is the total number of

lipids belonging to a given leaflet per unit surface area.

The area density n of lipids at distance n can be

described by the equation

n ¼ n1 þ n2 ð4Þ

where n1 and n2 are area densities of lipids belonging to a

given leaflet.

For a uniform distribution of lipid lengths, the constituent

densities n1 and n2 (Fig. 6) can be described as follows:

n1 ¼np L x=2 n\ dL=2 x=2np

21 nþx=2

dL=2

dL=2 x=2 n dL=2 x=2

0 dL=2 x=2\n Lþ x=2

8><

>:

ð5Þ

n2 ¼0 L x=2 n\ dL=2þ x=2np

21þ nx=2

dL=2

dL=2þ x=2 n dL=2þ x=2

np dL=2þ x=2\n Lþ x=2

8><

>:

ð6Þ

Independently of the sign of x, the central region

consists of three layers:

dL=2 xj j=2 n\ dL=2þ xj j=2

dL=2þ xj j=2 n dL=2 xj j=2

dL=2 xj j=2\n dL=2þ xj j=2

8<

:ð7Þ

Calculating the sum in Eq. 4, with the help of Eqs. 5, 6

in respective layers, depending on the sign of x, one can

obtain:

n ¼

np þ np

21þ nx=2

dL=2

dL=2 xj j=2 n\ dL=2þ xj j=2

np 1 xdL

dL=2þ xj j=2 n dL=2 xj j=2 x\0

np þ np

21 nþx=2

dL=2

dL=2 xj j=2\n dL=2þ xj j=2

8>>><

>>>:

ð8Þn ¼ np dL=2 n dL=2 x ¼ 0 ð9Þ

n ¼

np

21 nþx=2

dL=2

dL=2 x=2 n\ dL=2þ x=2

np 1 xdL

dL=2þ x=2 n dL=2 x=2 x [ 0

np

21þ nx=2

dL=2

dL=2 x=2\n dL=2þ x=2

8>>><

>>>:

ð10Þ

The space average of n in the range dL=2xj j=2 n dL=2þ xj j=2 is calculated as

hni ¼

RdL=2þ xj j=2

dL=2 xj j=2

ndn

dLþ xj j ð11Þ

and according to Eqs. 8, 9 and 10 equals

hni ¼np 1þ xj j=dL

1þ xj j=dL

x\0

np x ¼ 0

np1

1þx=dL x [ 0

8><

>:ð12Þ

Then, the space average of the dielectric constant ec

according to Eqs. 3 and 12 can be described as

heci ¼1þ ep 1

1þ jxj=dL

1þjxj=dL

x\0

ep x ¼ 0

1þ ep 1

11þx=dL x [ 0

8><

>:ð13Þ

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123

Colloidal aggregation in critical crosslinked polymer blends

M. Benhamou∗, F. Elhajjaji, K. Elhasnaoui, and A. Derouiche

Polymer Physics and Critical Phenomena Laboratory

Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, Morocco

We consider a low-density assembly of small colloidal particles that are

immersed in a critical crosslinked polymer blend made of two polymers of

different chemical nature. We assume that, near the spinodal temperature at

which the mixture exhibits a microphase separation, the particles preferen-

tially adsorb one of the two polymers. The consequence is that, the beads

aggregate in the non-preferred phase. The aim is an extensive study of the

thermodynamical properties of the colloidal aggregation, under a change of

a suitable parameter such as temperature. This phase transition drives the

colloids from a dispersed phase (gas) to a dense one (liquid). To this end,

we elaborate a new field theory that allows the determination of the effective

free energy of the mobile colloids, as a functional of their density. From this

effective free energy, we draw the phase diagram in particle volume fraction-

temperature plane, and investigate all critical properties of the colloidal floc-

culation, from a static and dynamic point of view. Finally, the discussion is

extended to impregnated crosslinked polymer blends in solution.

I. INTRODUCTION

The aggregation phenomenon within the traditional colloidal solutions are due to the

van der Waals attractive forces, which generally originate from the fact that the particles

possess a dipolar moment [1]. The same phenomenon may produces within the so-called

critical systems with immersed colloidal particles, such as a fluid near the liquid-gas critical

∗Author for correspondence; E-mail : [email protected]

1

khalid

Highlight

point, a mixture of simple liquids (or polymers) near the consolute point, liquid 4He near

the λ-transition or liquid-crystals. For these systems, the critical fluctuations of the order

parameter generate long-range attractive forces between colloids, termed critical Casimir

forces [2]. The word Casimir is attributed to the well-known Casimir effect [3], according

to which two perfectly conducting parallel metals attract each other, due to the vacuum

quantum fluctuations. The latter play the role of the critical fluctuations for physical systems

exhibiting a critical point.

Physics of the colloidal aggregation in critical two-component liquid mixtures is a very

exciting and rich problem, and has received a great deal of attention from a theoretical

and experimental point of view. Theoretically, the critical Casimir effect has been studied

by several methods, such as conformal invariance in dimension 2 [4 − 8], field-theoretical

Renormalization-Group (RG) in dimension 3 [2, 9− 15], and Monte Carlo simulation [16, 17].

Recently, the study was successfully extended to polymer blends [18].

Experimentally, the critical Casimir force was measured in a series of experiments

[19− 27]. Among these, we can quote certain experiments [19− 23] concerning the behavior

of silica beads of small diameter, immersed in a binary liquid mixture made of lutidine and

water. In fact, the silica beads have tendency to adsorb rather lutidine, when one is in

the vicinity of the consolute point of the mixture free from particles. This is the critical

adsorption [28− 41]. The experimental observation shows that, near criticality, the colloids

undergo a reversible aggregation in the water-rich side.

It was then natural to consider more complex systems such as crosslinked polymer blends

(CPBs) confined to two parallel walls [42] or with immersed colloids [43]. CPBs are made

of two chemically incompatible polymers that are densely crosslinked in their one-phase

region, by γ-irradiation [44], for instance. When the temperature is lowered, below some

characteristic temperature depending on the reticulation dose, the CPB phase separates into

mesoscopic phases alternatively rich in unlike polymers. This is the microphase separation

(MPS). We note that the first theory of MPS in CPBs was introduced by de Gennes [45],

followed by several extended works [46].

The incorporation of a small amount of particles in a CPB may have some relevance for

industry. Indeed, these particles can reinforce the mechanical properties of the considered

2

CPB. But, in most practical realizations, the immersed particles are instable and flocculate

under some special physical conditions. It is then useful to study their stability conditions.

In fact, the presence of colloids influences the primitive segregation interactions between

unlike connected polymer chains and their chemical potentials, and the internal rigidity of

the polymer network.

In this paper, we consider a low-density assembly of colloidal particles, immersed in a

critical CPB made of two polymers A and B, of different chemical nature. It is assumed that,

near the spinodal temperature at which the CPB undergoes a MPS, the colloids preferentially

adsorb one polymer, saying A. Thus, we are in the presence of a critical adsorption. The

consequence is that, the particles aggregate in the non-preferred B-rich phase. For instance,

the system may be a hydrogenated polyolefin-deuterated polyolefin mixture incorporating

silicon particles, considered in some recent experiment by Wendlandt and coworkers [47].

The induced force responsible for the colloidal aggregation was computed in Ref. [43], at

the spinodal point. The main result is that, the induced force decays as r−3 exp (−r/ξ∗),

where r is the interparticle distance and ξ∗ ∼ an1/2 is the microdomains size (mesh size)

[45]. Here, a denotes the monomer size and n is the mean number of monomers per strand

(section of chains between consecutive crosslinks).

The purpose is a study of the thermodynamical properties of the colloidal aggregation,

from a static and dynamic point of view. This colloidal aggregation is a phase transition

driving the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we

first elaborate a field theory governing the flocculation phenomenon. Combining this field

theory with the usual cumulant method [48, 49], we derive the expression of the effective free

energy of the moving colloids. The latter is naturally a functional of the volume fraction

of particles. We assume that the colloidal system is disordered. The problem is then the

specification of the disorder nature. To be more realist, we choose the most difficult one,

that is a quenched disorder [50]. This means that one must average the logarithm of the

partition function over the disorder. The opposite case is an annealed disorder, where the

partition function itself is averaged over the external disorder variables. This disorder is

relatively simple, and assumes that the colloids and polymers are at equilibrium. In fact,

such an assumption is not reliable, since, in principle, the diffusion time of colloids is different

3

from that of polymers. To do explicit calculations, we suppose that the disorder is rather

quenched and follows a Gaussian distribution. After averaging over disorder, we find that the

effective free energy of colloids is similar to that relatively to the traditional Flory-Huggins

theory usually encountered in Polymer Physics [51, 52]. The obtained effective free energy

of the dispersed colloids allows the determination of the phase diagram shape in the particle

volume fraction-temperature plane and all critical properties of the aggregation transition.

Finally, the discussion is extended to troubled CPBs in solution.

The remaining of presentation proceeds as follows. We derive, in Sec. II, the expression

of the effective free energy of the immersed colloids. The phase diagram is investigated

in Sec. II. We discuss, in Sec. III, the static and dynamic scattering properties. Some

concluding remarks are drawn in the last section.

II. EFFECTIVE FREE ENERGY

Consider M spherical colloidal particles immersed in a CPB. For simplicity, these par-

ticles are assumed to be small and of the same radius. Thus, we are concerned with a

monodisperse system. The fundamental assumption is that, near the bulk spinodal tem-

perature, the particles preferentially attract the polymer A. This means that one is in the

presence of a critical adsorption, where the colloids are surrounded by the preferred polymer

A. The consequence is that, the particles located in the non-preferred B-rich microdomains

aggregate. In fact, this aggregation is due to a long-ranged attractive force experienced by

the colloids, which was computed in Ref. [43].

To investigate the aggregation phenomenon, we introduce an order parameter or compo-

sition fluctuation, ϕ, which is simply the difference of compositions, ΦA and ΦB, of the two

polymers, that is ϕ = ΦA − ΦB. The order parameter ϕ (r) is then a scalar field depending

on the position vector, r, of the representative point of the mixture.

To investigate the physical properties of the troubled CPB, we must precise the form

of the Hamiltonian, which is a generalization of that introduced by de Gennes for studying

MPS in pure CPBs [45]. In the absence of colloids, the de Gennes’ (bare) Hamiltonian writes

[45]

4

Hb [ϕ]

kBT= a−d

∫ddr

t

2ϕ2 +

a2

2(∇ϕ)2 +

C

2P 2

. (1)

The latter results from a competition between the usual phase separation and the elastic

properties of the polymer network. In the above definition, the integration is performed

over the d-dimensional infinite euclidean space Rd. In the above definition, T stands for the

absolute temperature, kB for the Boltzmann’s constant, and a for the monomer size. The

notation

t =1

2(χc − χ) < 0 (1a)

denotes the distance from the critical temperature, where χ is the standard Flory interaction

parameter, which is inversely proportional to the absolute temperature T , and χc = 2/N is

its critical value if the system is uncrosslinked. Here, N denotes the common polymerization

degree of the polymer chains before they are crosslinked. Therefore, we are concerned with

a monodisperse system. The first two terms in relation (1) constitute the expansion of the

traditional Flory-Huggins free energy [51, 52] around the critical composition. The gradient

term accounts for the interfacial energy between A-rich and B-rich phases, and the third

one describes the gel elasticity. The elastic contribution in this relation was first introduced

by de Gennes by analogy with the polarization of a dielectric medium. The internal rigidity

constant, C, was related to the average number n of monomers per strand by [45]

C ∼ a−2n−2 . (1b)

The quantity 1/n was interpreted as the reticulation dose [44]. In equality (1), the vector

P denotes the average displacement between the centres of masses of A and B strands; it

is the analog of the polarization in the dielectric problem. The monomers A and B play

the role of the positive and negative charges of the dielectric medium, respectively. The

charge fluctuations correspond to the fluctuations of composition. The quantities P and ϕ

are not independent each other, but related by the Maxwell’s relation: divP= −ϕ (r). Such

a relation enables us to transform the gel elasticity contribution as follows

C

2

∫

V

ddrP 2 → −C

2

∫

V

ddrϕ∆−1r ϕ , (1c)

where ∆−1r is simply the inverse Laplacian operator, defined by

5

∆−1r ϕ (r) = −

∫ddr′K (r, r′)ϕ (r′) , (1d)

with the Green function K (r, r′), such that

−∆rK (r, r′) = δd (r − r′) . (1e)

It was found [45] that the system phase separates at the spinodal temperature, that is

for

ts = −2(Ca2

)1/2∼ D , (1f)

where D = n−1 is the reticulation dose. This relation then defines a critical line in the (C, t)-

plane along which the CPB undergoes a MPS. For t > ts, the CPB is in the disordered phase,

and for t < ts, it is rather in the ordered one. We recall that the microdomains size or mesh

size is as follows [45]: ξ∗ = a (Ca2)−1/4

∼ an1/2.

For a CPB with immersed point-like colloids, the proposed Hamiltonian is

H [ϕ]

kBT=Hb [ϕ]

kBT+ t0

M∑

i=1

ϕ2 (ri) + C0

M∑

i=1

P 2 (ri) . (2)

Here, M denotes the total number of particles dispersed in the medium. The two last terms

in the right-hand side describe the contribution of colloids to the Hamiltonian, where the

positive quantities t0 and C0 measure the perturbation of the temperature parameter t and

rigidity constant C due to the presence of the moving particles. For shapely particles, t0 and

C0 express the interactions between the mixture and their surface. A simple dimensional

analysis shows that: [t0] = L0 and [C0] = L−2, where L is some length.

It will be convenient to rewrite the above deformed Hamiltonian on the following form

H [ϕ]

kBT=Hb [ϕ]

kBT+

∫ddrρ (r)O (r) , (3)

with the local particle density

ρ (r) =N∑

i=1

δd (r − ri) (3a)

and local field operator

O (r) = t0ϕ2 (r) + C0P

2 (r) . (3b)

6

Now, to determine the free energy of a CPB, F , where the particles positions, (r1, ..., rM),

are fixed in space, we shall need the expression of the partition function, Z. The latter is

defined by the following functional integral

Z (r1, ..., rM) =1

M !

∫Dϕe−Ab[ϕ]+

∫ddrρ(r)O(r) , (4)

with the bare action Ab [ϕ] = Hb [ϕ] /kBT . The above functional integral is performed over

all possible configurations of ϕ-field. The quantity M ! accounts for the usual symmetry

factor.

It will be useful to introduce the bulk expectation mean-value

〈X〉b = Z−1b ×

∫DϕX [ϕ] e−Ab[ϕ] , (5)

with the partition function of the free CPB

Zb =

∫Dϕe−Ab[ϕ] . (5a)

With these considerations, the partition rewrites as

Z (r1, ..., rM) =ZbM !

⟨exp

∫ddrρ (r)O (r)

⟩

b

. (6)

We note that the bulk mean-value in the above expression can be computed using the

cumulant method usually encountered in Statistical Field Theory [48, 49], based on the ap-

proximative formula

⟨eX⟩b= e〈X〉b+(1/2!)[〈X

2〉b−〈X〉2

b]+... . (7)

Then, we have

Z (r1, ..., rM) =ZbM !

exp

∫ddrρ (r) 〈O (r)〉b +

1

2

∫ddr

∫ddr′ρ (r) ρ (r′) 〈O (r)O (r′)〉b,c + ...

,

(8)

where

〈O (r)O (r′)〉b,c = 〈O (r)O (r′)〉b − 〈O (r)〉b 〈O (r′)〉b (9)

accounts for the connected two-point correlation function constructed with the field operator

O (r). In relation (8), only one and two-body interactions are taken into account. High order

terms describing the three-body interactions and more are then ignored.

7

To compute the desired free energy, we start from the standard formula F = −kBT lnZ

and find

F [ρ]

kBT=

FbkBT

+Fint [ρ]

kBT− S/kB , (10)

where Fb = −kBT lnZb represents the free energy of the host mixture, and Fint is the

contribution of the effective interactions that can be written as

Fint [ρ]

kBT= −

∫ddrρ (r) 〈O (r)〉b,c −

1

2

∫ddr

∫ddr′ρ (r) ρ (r′) 〈O (r)O (r′)〉b,c + ... . (11)

Then, this formula is a combination of a one-body potential, 〈O (r)〉b, and a two-body one,

〈O (r)O (r′)〉b,c. We have

〈O (r)〉b,c = t0⟨ϕ2 (r)

⟩b+ C0

⟨P 2 (r)

⟩b, (12)

〈O (r)O (r′)〉b,c = t20⟨ϕ2 (r)ϕ2 (r′)

⟩b,c

+ 2t0C0⟨ϕ2 (r)P 2 (r′)

⟩b,c

+ C20

⟨P 2 (r)P 2 (r′)

⟩b,c

.

(13)

In equality (10), S accounts for the entropy of particles, and it is essentially the logarithm

of the symmetry factor M !, to which the Stirling’s formula will be applied.

Now, the aim is the computation of the effective free energy of the trapped colloids. To

do calculations, we first assume that their positions in space are disordered at random, with

a quenched disorder. The latter is more realistic than the annealed one, since the monomers

and colloids are not in equilibrium. We denote by ρ0 = M/Ω the number density of particles,

where Ω is the total volume occupied by the troubled CPB. According to the known central

limit theorem, the random density fluctuations around the mean-value ρ0 can be assumed

to be governed by a Gaussian (or bimodal) distribution, that is

P [ρ (r)] = P0 exp

−

1

2ρ20[ρ (r)− ρ0]

2

. (14)

Here, the difference ρ (r)− ρ0 is the density fluctuation and P0 is a normalization constant.

Then, the first and second moments of this distribution are as follows

ρ (r) = ρ0 , (15)

ρ (r) ρ (r′) = ρ20 + ρ0δd (r − r′) . (16)

8

Therefore, we are concerned with a non-correlated disorder.

For a quenched disorder, we have to average not the partition functionZ but its logarithm

lnZ = −F [ρ] /kBT , where F [ρ] is the free energy of expression (11). Then, we have

F [ρ]

kBT=

FbkBT

+Fint [ρ]

kBT− S/kB , (17)

with

Fint [ρ]

kBT≃ −

∫ddrρ (r) 〈O (r)〉b,c −

1

2

∫ddr

∫ddrρ (r) ρ (r′) 〈O (r)O (r′)〉b,c

= −ρ0

∫ddr 〈O (r)〉b,c −

ρ0Ω

2

∫ddr

⟨O (r)2

⟩b,c−

ρ202

∫ddr

∫ddr′ 〈O (r)O (r′)〉b,c . (18)

To determine the architecture of the phase diagram of the colloidal system, we need the

effective free energy (per site), ∆F , that can be obtained from vF [ρ]/Ω by ignoring the

constant Fb and terms proportional to density ρ0. Here, v denotes the volume of colloids.

In order to precise the form of ∆F , we first introduce the dimensionless particle density

(volume fraction of colloids) Ψ = ρ0 × v. After symmetrization (Ψ→ 1−Ψ), the entropy

(per site) writes [51, 52]

vS

kBΩ= −Ψ lnΨ− (1−Ψ) ln (1−Ψ) . (19)

With these considerations, the effective free energy (per site) is given by

∆F

kBT= Ψ lnΨ + (1−Ψ) ln (1−Ψ) + uΨ(1−Ψ) , (20)

with the colloid interaction parameter

u =1

2v

∫ddr 〈O (0)O (r)〉b,c > 0 (21)

that essentially presents as the spatial integral of the two-point correlation function of the

field operator O.

The derived effective free energy calls the following remarks.

Firstly, its expression clearly shows its analogy with that relatively to the standard Flory-

Huggins theory (FH) [51, 52] describing the free polymer blends; u then plays the role of the

Flory interaction parameter. Therefore, the present theory of free energy (20) is a lattice

model, where a given site is occupied by one particle, with a probability Ψ. Thus, 1−Ψ is

9

the probability to have an empty site. In fact, if Ψ is the composition of the dense phase

(liquid phase), that of the dispersed one is 1−Ψ.

Secondly, formula (20) indicates that the critical fluctuations of composition give arise

to an effective interaction energy between colloids that is responsible for their aggregation.

Thirdly, we note that the dimensionless parameter u is temperature-dependent and al-

ways positive definite. This means that, above some critical value, u∗, defined later, one

assists to a coexistence between dilute (gas) and dense (liquid) phases.

Finally, we emphasize that the interaction parameter u is a linear combination of three

bulk correlation functions (integrated over space coordinates), that is

u =1

2v

(t20X00 + 2t0C0X01 + C2

0X11

), (22)

with the coefficients

X00 =

∫ddr

⟨ϕ2 (0)ϕ2 (r)

⟩b,c

, X01 = X10 =

∫ddr

⟨ϕ2 (0)P 2 (r)

⟩b,c

, (22a)

X11 =

∫ddr

⟨P 2 (0)P 2 (r)

⟩b,c

. (22b)

Replacing P 2 by −ϕ∆−1r ϕ and using the Wick theorem [48, 49] yields

X00 = 2

∫ddr [〈ϕ (0)ϕ (r)〉b]

2 , (22c)

X01 = X10 = 2

∫ddr 〈ϕ (0)ϕ (r)〉b

⟨ϕ (0)∆−1

r ϕ (r)⟩b, (22d)

X11 = 2

∫ddr 〈ϕ (0)ϕ (r)〉b

⟨∆−1r ϕ (0)∆−1

r ϕ (r)⟩b. (22e)

Explicitly, we have

X00 (T,C) = 2sda2d

∫ ∞

0

qd−1dq

(a2q2 + t+ C/q2)2, (22f)

X01 (T,C) = X10 (T,C) = 2sda2d

∫ ∞

0

qd−3dq

(a2q2 + t+ C/q2)2, (22g)

X11 (T,C) = 2sda2d

∫ ∞

0

qd−5dq

(a2q2 + t+ C/q2)2. (22h)

10

Here, sd = 2πd/2Γ (d/2) denotes the area of the unit sphere embedded in the d-dimensional

euclidean space Rd, where Γ (z) is the Euler gamma function [53]. In mean-field theory, the

critical behavior emerges in the d→ 4 limit, and we find, near the spinodal point (t→ t+s ),

the following expressions for the coefficients Xij’s

X00 = 2π3a7ξ∗−3

(t− ts)3/2

, t→ t+s , (23a)

X01 = X10 = π3a7ξ∗−1

(t− ts)3/2

, t→ t+s , (23b)

X11 = π3a7ξ∗

(t− ts)3/2

, t→ t+s . (23c)

Therefore, the colloid interaction parameter scales as

u = u0

(ts − t

ts

)−3/2, t→ t+s , (24)

with the positive amplitude

u0 =a4π3

25/2v

(2t20 + 2t0C0 + C2

0

). (24a)

We have used the notation

C0 = C0ξ∗ 2 . (24b)

The central result (24) call two remarks.

Firstly, as it should be, the colloid interaction parameter becomes more and more im-

portant as the spinodal temperature (at which the CPB undergoes a MPS) is reached (from

above), and diverges at this same temperature.

Secondly, all dependance on the perturbation parameters (t0, C0), due to the presence

of fillers, is entirely contained in the positive amplitude u0 defined in Eq. (24a). This

dependence is quadratic in these parameters. In addition, this amplitude increases with

increasing variables t0 and C0.

In Fig. 1, we draw the reduced colloid interaction parameter, u/u0, upon the reduced

(positive) temperature shift ∆t = (ts − t) /ts. Remark that this curve is universal, indepen-

dently on the physical and geometrical properties of colloids and their interactions strength

with the host CPB.

11

The constructed effective free energy allows the determination of the phase diagram shape

in the composition-temperature plane. This is precisely the aim of the following paragraph.

III. PHASE DIAGRAM

Now, the equilibrium volume fraction of particles solves the equation ∂∆F/∂Ψ = 0.

Explicitly, we have

u =1

2Ψ− 1ln

(Ψ

1−Ψ

). (25)

This equation defines the coexistence curve in the (Ψ, u)-plane, along which the dispersed

and dense phases coexist. On the other hand, the spinodal curve, along which the isotherm

compressibility diverges, can be obtained solving ∂2∆F/∂Ψ2 = 0. Then, the spinodal curve

equation is

u =1

2Ψ (1−Ψ). (26)

Finally, the location of the critical point, K, can be determined solving the two simultaneous

equations ∂2∆F/∂Ψ2 = 0 and ∂3∆F/∂Ψ3 = 0. We find that its coordinates (Ψ∗, u∗) are

such as

Ψ∗ =1

2, u∗ = 2 . (27)

The critical value u∗ = 2 corresponds to a well-defined critical temperature, T ∗, which

satisfies the implicit equation

1

2v

[t20X00 (T

∗, C) + 2t0C0X01 (T∗, C) + C2

0X11 (T∗, C)

]= 2 . (28)

Combining the dominant parts of the coefficients X00, X01 and X11 we computed above with

equality (28) yields the critical temperature, T ∗, at which the colloids flocculate

t∗ = ts

[1−

(u02

)2/3]. (29)

We have ts < t∗ < 0, provided that u0 < 2. Notice that t∗ = (χc − χ∗) /2, with χc = 2/N

and χ∗ ∼ T ∗ −1. Here, the amplitude u0 is defined in equality (24). The above relationship

12

then defines a critical surface in the (t0, C0, t∗)-space on which the colloidal system presents

a phase transition.

Therefore, at T = T ∗ and Ψ∗ = 0.5, the colloidal system exhibits a second order phase

transition from a dispersed phase to a dense one, at the critical temperature T ∗, which is

very close to the spinodal temperature Ts, since the perturbation parameters t0 and C0 are

assumed to be small enough.

Come back to Eq. (25) defining the coexistence curve and try to rewrite it in the vicinity

of the critical point. Close to this point, we obtain

u = u∗ +8

3(Ψ−Ψ∗)2 , Ψ→ Ψ∗ , (30)

or equivalently,

Ψ = Ψ∗ ±1

2

√3

2(u− u∗)1/2 , u→ u∗ . (30a)

Notice that u− u∗ ∼ T ∗ − T .

The phase diagram of the colloidal system is drawn in Fig. 2, where the solid and dashed

lines represent the binodal and spinodal, respectively. These lines meet at the critical point

K = (Ψ∗, u∗). The region between the two curves is a metastable domain.

We note that the phase diagram architecture we investigated above has been determined

using the FH approach that is a mean-field theory (MFT), which underestimates the fluc-

tuations of the composition that are strong near criticality. To go beyond this theory, and

in order to get a correct critical phase behavior, use will be made, in the next section, of the

scaling approach that is a direct consequence of the RG theory [48, 49].

IV. SCATTERING PROPERTIES

A. Static properties

Now, consider another main physical quantity that is the structure factor, S (q), which

informs us on the critical properties of the colloidal system. This can be measured in light-

scattering and neutron-scattering experiments. The structure factor is nothing else but the

Fourier transform of the two-point correlation functionG (r) = 〈Ψ(0)Ψ (r)〉−〈Ψ(0)〉〈Ψ(r)〉,

which measures the fluctuations of the particle composition Ψ. We write

13

S (q) =

∫ddq

(2π)deiq.rG (r) , (31)

where q denotes the wave-vector, whose amplitude is given by |q| = (4π/λ) sin (θ/2), where

λ is the wavelength of the incident radiation and θ is the scattering-angle. In particular, the

zero-scattering-angle limit (q → 0) of the structure factor defines the thermal compressibility

of the colloidal system, κT , that is

S (0) = kBTκT . (32)

This compressibility diverges as the critical temperature T ∗ is approached, and we have

κT ∼ |T − T ∗|−γt , (32a)

where γt is a critical exponent whose value depends only the space dimensionality d and not

on the chemical details of the system.

Other critical exponents can be defined. For example, the order parameter ψ = Ψ−Ψ∗

behaves, around T ∗, as

ψ ∼

0 , T ≥ T ∗ ,

(T ∗ − T )βt , T ≤ T ∗ .(33)

with a second critical exponent βt, in the limit ∆µ = µ − µc → 0, where µ is the chemical

potential and µc is its critical value [48, 49]. In term of the colloid interaction parameter u

and composition Ψ, we have

u− u∗ ∼ (Ψ−Ψ∗)1/βt (33a)

that is the equation of the coexistence curve close to the critical point. At T = T ∗, the order

parameter behaves as ψ ∼ |∆µ|1/δt , with a critical exponent δt.

On the other hand, the size of domains rich in colloids or the thermal correlation length

measuring the correlations extent, ξt, diverges at criticality according to

ξt ∼ σ |T − T ∗|−νt , (34)

with the critical exponent νt and an atomic-scale σ that may be the particle diameter.

Also, the specific heat diverges at the critical point with some critical exponent denoted

14

αt. Finally, we define the last critical exponent, ηt, which characterizes the small-distance

behavior of the correlation function, that is

G (r) ∼ r2−d−ηt , r << ξt . (35)

Therefore, the critical phase behavior is completely determined by the knowledge of the

thermal critical exponents (αt, βt, γt, δt, νt, ηt). To compute these exponents, we first remark

that the order parameter ψ plays the role of the magnetization of a magnetic material

exhibiting a paramagnetic-ferromagnetic transition, and ∆µ, the role of an external magnetic

field. Also, the free energy defined in Eq. (20) resembles the Bragg-Williams one usually

encountered in magnetic materials [48, 49]. The conclusion is that, the present second order

phase transition belongs to the same universality class as the paramagnetic-ferromagnetic

one. Thus, the three-dimensional values of the various critical exponents are those of Ising

model we do not recall [48, 49]. Naturally, these good values must be compared to those

obtained within the framework of MFT used in the last section, which are α0t = 0, β0t = 1/2,

γ0t = 1, δ0t = 3, ν0t = 1/2, and η0t = 0. The difference comes from the fact that MFT

underestimates the strong fluctuations of composition of colloids near the critical point.

We recall that the real critical exponents have been computed using the RG-techniques.

In particular, the realistic coexistence curve equation is that defined in Eq. (33a), with a

non-trivial critical exponent βt, different from the mean-field one, β0t = 1/2.

Finally, the structure factor obeys the scaling law

S (q) = qηt−2f (qξt) , (36)

with the scaling-function f (x).

B. Dynamic properties

Assume that the temperature of the system is suddenly lowered from an initial value to

a final one very close to the critical temperature T ∗. Then, the system is out of equilibrium

and the colloid volume fraction relaxes from an initial equilibrium value Ψi to a final one

Ψf . We are interested in the time order parameter ψ (r, t) = Ψ (r, t) − Ψ∗, where r ∈ Rd

15

denotes the position-vector and t the time. The latter represents the time observation of

the system before it reaches its final equilibrium state.

We recall that the time composition fluctuation ψ (r, t) solves a generalized non-

dissipative Langevin equation [49] we do not write. Applying the stochastic RG-techniques

yields the scaling law for the dynamic structure factor. In particular, the latter depends on

the following time-scale

R (t) ∼ Γ1/zt1/z , (37)

with the critical dynamic exponent

z = 4− ηt = 3.469± 0.001 . (37a)

We have used the best value for the critical exponent ηt, that is ηt = 0.031± 0.001 [48, 49].

There, Γ accounts for the kinetic coefficient. The length R (t) can be interpreted as the size

of domains rich in colloids at time t. For MFT, z0 = 4.

The final equilibrium state is reached at a characteristic time, τ , which scales as

τ ∼ Γξzt , (38)

where ξt is the thermal correlation length or the size of domains at the final temperature

(the subscript t is for thermal that must not be confused with time). The above behavior

can be obtained equating R (t) to ξt.

Finally, we write the scaling law for the dynamic structure factor

S (q, t) = qηt−2g (qξt, qR (t)) , (39)

where g (x, y) is a two-factor scaling-function. Of course, the time t must not be confused

with the temperature parameter appearing in Eqs (1), (1a), (23a-c) and (24).

V. DISCUSSION AND CONCLUSIONS

In this work, we were interested in the reversible aggregation of colloids immersed in

a critical CPB. We assumed that the colloids prefer to be surrounded by one of the two

polymers, close to the spinodal point of the free mixture. The aggregation phenomenon

16

results from the existence of long-ranged attractive forces. The question we asked was the

investigation of the phase diagram and critical properties of the immersed colloids. To do

calculations, use was made, first, of MFT, and second, the scaling theory, in order to get a

correct leading phase behavior. In addition, it was assumed that the positions of particles

are disordered, and the disorder is quenched and follows a Gaussian distribution. We first

determined the expression of the effective free energy of colloids, and showed that the latter

is of Flory-Huggins type. From this effective free energy, we drawn the complete shape of

the phase diagram. Second, the scaling theory revealed that the critical exponents have

non classical values in comparison to those relatively to MFT. Third, the discussion was

extended to dynamic properties of the colloidal flocculation.

The essential result is that, the colloidal aggregation, which is a phase transition driving

the system from dilute to dense phases, occurs when one is close to the spinodal point of

the crosslinked polymer mixture free from fillers.

For the description of the colloidal aggregation, we assumed that the colloids are spheric

and of the same diameter. This means that we were concerned only with monodisperse

systems. Actually, the particles are never geometrically identical. They may have different

shapes and sizes. Nevertheless, the present approach remains valid for very small particles,

whatever is their shape.

An interesting problem that we have not undertaken is the effect of an added (good) sol-

vent on the colloidal flocculation. One interesting way is to use the blob model [54, 55]. Quan-

titatively, all results dealt with the aggregation phenomenon are similar to that relatively

to impregnated CPBs in the molten state (no solvent is present) replacing the monomer

size a by the screening length, ξ ∼ aΦ−3/4 [52, 56], the polymerization degree N by the

number of blobs per connected chain, Z (Φ) ∼ NΦ5/4 [52, 56], the segregation parameter

χ by the effective one, χ ∼ χΦ0.26 [55], and the mesh size, ξ∗ ∼ an1/2, by its homologous

ξ∗∼ an1/2Φ−1/8. For theta-solvents, ξ ∼ aΦ−1 [52, 56], Z (Φ) ∼ NΦ2 [52, 56], χ ∼ χΦ0.87

[57], and ξ∗∼ ξ∗ ∼ an1/2. We would have the same discussions. But the presence of the

solvent drastically affects the critical phase behavior. In particular, the critical tempera-

ture, T∗, at which the aggregation transition occurs, depends, in addition to the parameters

(N,C, t0, C0) relatively to the molten state, on the monomer volume fraction Φ. For good

17

solvents, for instance, the swelling of strands due to the excluded volume forces alter the

aggregation phenomenon. It is well-known that a good solvent plays a stabilizer role.

For the description of the colloidal aggregation, we have not taken into account the

primitive interactions between the immersed colloids. This assumption makes sense as long

as the colloid density is very small. Generally, we show that these primitive interactions

contribute to the colloid interaction parameter u by an extra term, up, which presents as

[58]: up = −b0 + a0/kBT > 0, where b0 is the covolume and a0 = −2π∫∞r0

r2U (r), with the

primitive pair-potential U (r). Here, r0 is the unique zero of U (r), that is U (r0) = 0. In this

case, the total colloid interaction parameter is then u + up. The presence of the primitive

interactions naturally induces a change of the phase behavior. In particular, the location

of the critical flocculation temperature is shifted to its higher values. In other word, the

critical fluctuations of the host CPB have tendency to accentuate the colloidal aggregation.

All conclusions concerning the scattering properties remain the same.

Finally, this work must be regarded as a natural extension of a published work dealt

with the colloidal aggregation in uncrosslinked polymer blends with immersed colloids [59].

ACKNOWLEDGMENTS

We are much indebted to Professors Daoud for stimulating correspondences. One of

us (M.B) would like to thank the Laboratoire Leon Brillouin (Saclay) for their kinds of

hospitality during his regular visits.

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FIGURE CAPTIONS

Fig. 1 : Variation of the reduced colloid interaction parameter, u/u0, upon the temperature

shift ∆t = (ts − t) /ts.

Fig. 2 : Phase diagram shape, where the solid line is the binodal and the dashed one

is the spinodal. The two curves meet at the ”liquid-gas” critical point, K, of coordinates

(Ψc = 0.5, u∗ = 2).

23

0,0 0,2 0,4 0,6 0,8 1,00

5

10

15

20

∆t

u/u0

Figure 1

0,00 0,25 0,50 0,75 1,00

2

3

4

5

6

7

.

.

.K

Ψ

u

Figure 2

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