A simple and exactly solvable model for a semiflexible polymer chain interacting with a surface

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Physica A 323 (2003) 453–465www.elsevier.com/locate/physa

A simple and exactly solvablemodel for a semi"exible polymer chain

interacting with a surfaceP.K. Mishra∗, S. Kumar, Y. Singh

Department of Physics, Banaras Hindu University, Varanasi 221 005, India

Received 12 December 2002

Abstract

We use the lattice model of directed walks to investigate the conformational as well as theadsorption properties of a semi"exible homopolymer chain immersed in a good solvent in 2Dand 3D. To account for the sti0ness in the chain we have introduced energy barrier for eachbend in the walk and have calculated the persistent length as a function of this energy. For theadsorption on an impenetrable surface perpendicular to the preferred direction of the walk wehave solved the model exactly and have found the critical value of the surface attractions for theadsorption in both 2D and 3D. We have also enumerated all possible walks on square and cubiclattices for the number of steps N6 30 for 2D and N6 20 for 3D and have used ratio methodfor extrapolation. The transition located using this method is in excellent agreement with theresults found from the analytical method. We have compared the results of surface adsorptionfor two di0erent surface orientations. In one of the orientation, surface is considered parallelto the preferred direction and in another it is perpendicular to the preferred direction. Resultsfound in both the cases indicate that for sti0er chains adsorption transition takes place at highertemperature compared to that of "exible chainc© 2003 Elsevier Science B.V. All rights reserved.

PACS: 64.60.−i; 68.35.Rh; 5.50.+q

Keywords: Exact results; Semi"exible polymer; Exact enumeration

∗ Corresponding author.E-mail addresses: [email protected] (P.K. Mishra), [email protected] (S. Kumar),

[email protected] (Y. Singh).

0378-4371/03/$ - see front matter c© 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0378-4371(02)01993-3

mailto:[email protected]



454 P.K. Mishra et al. / Physica A 323 (2003) 453–465

1. Introduction

Biopolymers are known to exhibit under di0erent environments a variety of persistentlengths ranging from being much smaller than the over all length of the polymer, tobeing comparable to the chain length [1]. When the persistent length associated withthe polymer is much smaller than the overall length of the chain, the polymer issaid to be "exible. On the other hand, when the persistent length is comparable tothe chain length, the polymer is said to be rigid. When the persistent length falls inbetween the two extremes, the chain is said to be semi"exible. The conformationalproperties of such chains have attracted considerable attention in recent years becauseof experimental developments in which it has become possible to pull and stretch singlemolecule to measure elastic properties [2]. Such studies reveal a wealth of informationabout the conformational behaviour of semi"exible polymers that are of clear biologicalimportance.An impenetrable surface is known to a0ect the conformational properties of polymers

in a signiEcant way [3,4]. This is due to a subtle competition between the gain ofinternal energy and a corresponding loss of entropy at the surface. Since the "exibilityof the chain a0ects this competition, a semi"exible chain is expected to show di0erentadsorption behaviour compared to that of a "exible polymer chain. A sti0 chain isknown to get adsorbed easily compared to a "exible chain [5].A simple way to account the sti0ness of a semi"exible chain is to constrain the angle

between the successive segments to be Exed. The value of the angle depends on thelocal sti0ness of the chain. This prescription leads to the freely rotating chain model[6]. In the continuum limit the freely rotating chain becomes the so-called worm likechain (WLC) [7]. In these models the persistent length lp is deEned as a characteristiclength for tangent–tangent correlation function 〈t(s)t(s′)〉 � exp(−|s − s′|=lp). Thetangent vector t(s) is deEned as 9r(s)=9t, where r(s) is parametrized in terms of thearc length s of the chain [7].Though the worm like chain model of Kratky and Porod [7] has been used

extensively to study the conformational properties and surface adsorption of asemi"exible chain [8], it cannot mimic exactly the dimensional behaviour of thepolymer chains. In this paper we use the lattice model of directed walk and introducesti0ness in the chain by associating energy with every bend of the walk and calcu-late the bulk and adsorption properties of the chain as a function of sti0ness of thechain.The paper is organized as follows: In Section 2 we describe the lattice model of

directed walk and investigate the bulk properties. We calculate the value of persistentlength as a function of energy associated with the bend. In Section 3 we discuss thesurface adsorption of a semi"exible chain represented by a directed walk on a planeperpendicular to the preferred direction of the walk. We also use the exact enumer-ation technique to locate the adsorption desorption transition and compare the resultwith those found exactly. This allows us to comment upon the accuracy of the methodof exact enumerations.

P.K. Mishra et al. / Physica A 323 (2003) 453–465 455

2. A lattice model for the semi�exible chain

We consider a model of a self-avoiding directed walk on a lattice [9]. Though thedirectedness of a walk amounts to some degree of sti0ness as all directions of thespace is not treated equally, sti0ness in the chain is introduced by associating energybarrier with every turn of the walk. Though the model is very restrictive in the sensethat the bend can be either 90◦ or no bend at all, the model can be solved analyticallyand therefore give the exact values of conformational and adsorption properties of asemi"exible chain. We consider two speciEc cases of directed walks: If the walkeris allowed to take steps along the ±y-axis (in 2D) and only along the +x-axis thewalk is said to be partially directed-self-avoiding walk (PDSAW). On the other hand,if the walker is allowed only along the +y and +x directions then the walk is saidto be fully directed-self-avoiding walk (FDSAW). In the case of 3D, a PDSAW isone in which walker is allowed along the ±y direction but only along the +x and +zdirections while in the FDSAW the walker is allowed to move only along the +x;+yand +z directions.A sti0ness weight k = exp(− �b) where = (kBT )−1 is inverse of the temperature

and �b(¿ 0) is the energy associated with each turn. For k=1 or �b=0 the chain is saidto be "exible and for 0¡k¡ 1 or 0¡�b¡∞ the chain is said to be semi"exible.When �b → ∞ or k → 0, the chain becomes a rigid rod.The partition function of such a chain can be written as

Z(x; k) =N=∞∑N=0

∑all walks of N steps

xN kNb : (1)

Here Nb is the total number of bends in a walk of N steps and x is the step fugacity.

2.1. Conformational properties of the chain in 2D

(i) The case of PDSAW: In 2D, the generating function for PDSAW has twocomponents; one along the directedness (i.e., +x-axis) and other perpendicular to it(i.e., ±y-axis) as shown in Fig. 1. The recursion relation for these two components ofgenerating function are [9]

X = x + x(X + 2kY ) ; (2)

Y = x + x(kX + Y ) : (3)

Solving Eqs. (2) and (3) we get

X =x + (2k − 1)x2

1− 2x + x2 − 2x2k2; (4)

Y =x + (k − 1)x2

1− 2x + x2 − 2x2k2: (5)


xx

k

k

y y

y

x

x

y

y

k

yx

Fig. 1. The diagrammatic representation of the recursion relations Eqs. (2) and (3), for PDSAW. The thickarrows X and Y denote all possible walks with the initial step along the +x and ±y directions, respectively.

The partition function can therefore be written as

Z2Dp:d:(x; k) = X + 2Y =

(4k − 3)x2 + 3x1− 2x + x2 − 2x2k2

: (6)

The critical point for polymerization of an inEnite chain is found from the relation

1− 2x + x2 − 2x2k2 = 0 : (7)

This leads to the critical value of the step fugacity for a given value of k as xc =1=(1 +

√2k).

The sti0ness in the chain increases the value of fugacity for polymerization. Thisdependence is shown in Fig. 2 by long-dashed line in which we plot xc as a functionof �b. We deEne the persistent length as the average distance between two successivebends of the walk, i.e.,

lp = 〈L〉=〈Nb〉 ; (8)

where L= 〈N 〉a; a being the lattice parameter.For the PDSAW in 2D we End

lp =3 + 2

√2

4 + 3√2

[√2 + exp( �b)

]: (9)

The dependency of lp on �b is shown in Fig. 3 by a long-dashed line.The value of lp increases exponentially with the bend energy at a given temperature.(ii) The case of FDSAW: In this case the polymer is directed along the +x and

+y direction; leading to the following recursion relations for the generating functions:

X = x + x(X + kY ) ; (10)

Y = x + x(kX + Y ) : (11)


Fig. 2. The variation of step fugacity xc with �b.

Fig. 3. The variation of lp with bending energy �b.

Solving these equations we get the following value for the partition function Z2Df:d:.

Z2Df:d:(x; k) = X + Y =

2x1− (1 + k)x

: (12)

The critical value of step fugacity is found to be xc = 1=(1 + k). The variation of xcwith �b is shown in Fig. 2 by solid line.


The value of persistent length in this case attains a simple relation, i.e.,

lp = 1 + e �b : (13)

The variation of lp with the bending energy is shown in Fig. 3 by a solid line.

2.2. Conformational properties of the chain in 3D

(i) The case of PDSAW: In the case of PDSAW the polymer chain is directed intwo-directions. The recursion relations for generating functions are [9]

X = x + x(X + 2kY + kZ) ; (14)

Y = x + x(kX + Y + kZ) ; (15)

Z = x + x(kX + 2kY + Z) : (16)

Solving these equations we get the values of X; Y; Z and the partition function as

X = Z =x + (2k − 1)x2

(1 + k − 4k2)x2 − (k + 2)x + 1; (17)

Y =x + (k − 1)x2

(1 + k − 4k2)x2 − (k + 2)x + 1; (18)

Z3Dp:d:(x; k) = X + 2Y + Z =

(6k − 4)x2 + 4x(1 + k − 4k2)x2 − (k + 2)x + 1

: (19)

That is at xc =(k + 2−√

17k)=2(k + 1 − 4k2), the Z3D

p:d: will diverge. In this case

the dependence of the fugacity for polymerization on the sti0ness is more involvedcompared to the case in 2D. The variation of xc with �b is shown in Fig. 2 by thedashed line. For persistent length we End

lp =2A1[exp(2 �b) + exp( �b)− 4]

(1−√17 + 2 exp( �b))A2 + (85 + 21

√17) exp(2 �b)

; (20)

where

A1 =[85 + 19

√17−

(102 + 26

√17)exp( �b) +

(34 + 8

√17)exp(2 �b)

]

and

A2 =[204 + 52

√17−

(272 + 64

√17)exp( �b)

]:

The value of lp as a function of �b is plotted in Fig. 3 by the dashed line.(ii) The case of FDSAW: In this case the polymer is directed along all the three-

directions i.e., along the +x, +y and +z directions. We can write following recursion


relations:

X = x + x(X + kY + kZ) ; (21)

Y = x + x(kX + Y + kZ) ; (22)

Z = x + x(kX + kY + Z) : (23)

The solution of these equations leads to

X = Y = Z =x

1− (1 + 2k)x: (24)

Thus the partition function of the system can be written as

Z3Df:d:(x; k) = X + Y + Z =

3x1− (1 + 2k)x

: (25)

The critical value of the step fugacity is xc=1=(1+2k). The variation of step fugacitywith bending energy �b is shown in Fig. 2 by a dot-dashed line. In this case lp isfound to be

lp = 1 + 12 exp( �b) : (26)

The value of lp as a function of �b is plotted in Fig. 3 by a dot-dashed line.In all the cases discussed above the persistent length shows exponential dependence

on the bending energy.

3. Surface adsorptions

In the case of directed model we have two distinct surfaces; one parallel and theother perpendicular to the directedness of the walk. The adsorption of polymer on asurface parallel to the preferred direction of the walk has been studied in case of 2Dusing the transfer matrix method [9]. The features associated with the adsorption werefound to be same as in the isotropic case except that the critical value of surfaceattraction for adsorption is di0erent. The surface perpendicular to the direction of walkmay give di0erent features as the walk once leaves the surface it cannot return to itdue to restriction on the walk. Here we report the result found analytically for theadsorption of directed semi"exible chain on a surface perpendicular to the directednessof the chain both in two and three dimensions.

3.1. Adsorption of a directed semi3exible chain on a surface perpendicular to thedirectedness of the chain in 2D

(i) The case of PDSAW: In case of 2D, surface is a line represented by x=0. LetS be the component of generating function along the surface and X the componentperpendicular to the surface as shown in Fig. 4. Following the method outlined abovewe can write surface component as:

S = s+ s(s+ kX ) + s2(s+ kX ) + s3(s+ kX ) + · · · ; (27)


Fig. 4. The diagrammatic representation of the recursion relation (27). Each walk of the polymer chain startsfrom O. In this diagram X and S denotes all possible walks with initial step along the +x and along thewall, respectively.

where s = !x, and ! = exp(− �s) being the weight associated with each step alongthe wall.For !=1 Eq. (27) reduces to Eq. (5). The partition function in presence of surface

for PDSAW is found to be

Z2Dsp:d:(k; !; x) = X + 2S; (s¡ 1) : (28)

Combining with Eq. (4) we End

Z2Dsp:d:(k; !; x) =

2sx2(1− 2k) + 2s(1− 2x) + (2sk + 1− s)[x + (2k − 1)x2](1− s)(1− 2x + x2 − 2k2x2)

:

(29)

The critical value of adsorption transition is found from the relation

(1− s)(1− 2x + x2 − 2k2x2) = 0 : (30)

This leads to !c = 1=xc =√2k + 1, which reduces to !c =

√2 + 1 for the "exible

polymer chain. The variation of !c with �b is shown in Fig. 5 by a long dashed line.(ii) The case of FDSAW: It is straight forward to show that for FDSAW the partition

function in the presence of surface is

S = s+ s(s+ kX ) + s2(s+ kX ) + s3(s+ kX ) + · · · : (31)

Using value of X from Eq. (12) we get

Z2Dsf:d:(k; !; x) = X + S =

s(1− (1 + k)x) + x(sk − s+ 1)(1− s)(1− (1 + k)x)

(s¡ 1) (32)


Fig. 5. The exact value of !c for di0erent values of �b. The lines in this Egure correspond to analyticalresults however the dots on the lines correspond to the value obtained from exact-enumeration method andthe cross used to denote the !c value for 3D partially directed case.

which gives adsorption transition point at !c = 1+ k. The variation of !c with �b isshown in Fig. 5 by a solid line.

3.2. Adsorption of a semi3exible directed chain on a surface perpendicular to oneout of the two preferred direction of the chain in 3D

(i) The case of PDSAW: The analysis given above can be generalized in 3D wheresurface dimension is two i.e., x−y plane at z=0. In the case of PDSAW as mentionedabove the choice of the walker is restricted to the +x-axis, ±y-axis and +z-axis. Let Sxand Sy is the component of the total partition function Z3D

sp:d: along the +x and ±y-axes,respectively, however component perpendicular to the wall along +z-axis remains sameas deEned by Eq. (17). We can, therefore write

Sx =s− s2 + 2s2k + Z(2k2s2 + sk − s2k)

1− 2s+ s2 − 2s2k2; (s¡ 1) ; (33)

Sy =s− s2 + s2k + Z(k2s2 + sk − s2k)

1− 2s+ s2 − 2s2k2: (34)

The expression for the partition function in this case found from the relation

Z3Dsp:d:(k; !; x) = Sx + 2Sy + Z : (35)


Substituting the value of Sx, Sy and Z we End

Z3Dsp:d:(k; !; x) =

x(1− x + 2kx)U + s(3− 3s+ 4sk)V(1− 2s+ s2 − 2s2k2)[(1 + k − 4k2)x2 − (k + 2)x + 1]

; (36)

where U and V are

U = 1− 2s+ 3sk + s2 − 3s2k + 2s2k2 ; (37)

V = 1− 2x − kx + x2 + kx2 − 4k2x2 : (38)

The two singularities appearing in Eq. (37) give the critical value of xc =(k + 2−√

17k)=2(1+k−4k2) and !c=2(1+k−4k2)=

(1 +

√2k

)(k + 2−√

17k).

For "exible polymer chain (i.e., k = 1 or �b = 0) it gives xc = (−3 +√17)=4 and

!c = 1:47524 : : : .(ii) The case of FDSAW: For FDSAW, the partition function can be easily be

evaluated. Here we write the Enal form of the partition function as

Z3Dsf:d:(k; !; x) = Sx + Sy + Z =

(2s+ x − 3sx − 3sxk)(1− s− sk)(1− (2k + 1)x)

(s¡ 1) : (39)

The two singularities appearing in Eq. (39) gives the critical value of xc =1=(2k +1),and !c=(2k+1)=(k+1). The variation of !c with �b are given in Fig. 5 for PDSAWand FDSAW by a dashed and dot-dashed lines, respectively.

4. Result from exact enumeration method

Since the analytical approach is limited to very few cases, one often has to resortto numerical methods, such as Monte Carlo simulations or a lattice model using ex-trapolation of exact series expansion (referred to as exact enumeration method). Thelater method has been found to give satisfactory results as it takes into account thecorrections to scaling. To achieve the same accuracy by the Monte Carlo method, achain of about two orders of magnitude larger than in the exact enumeration methodhas to be considered [10].We have enumerated all possible walks of length N6 30 on square lattice and of

length N6 20 on cubic lattice with two di0erent surface orientations. In one of thecase where surface is considered along the preferred direction. However in the anothercase surface is perpendicular to the preferred direction of walks. The canonical partitionfunction is written as

ZN (k; !) =∑Ns

∑Nb

CN (Ns; Nb)!NskNb : (40)

Here Ns is the number of steps on the surfaces. The reduced free energy per monomeris found from the relation

G(k; !) = limN→∞

1N

log ZN (k; !) : (41)

The limit N → ∞ is found by using the ratio method [11] for extrapolation.The transition point for adsorption–desorption is found from the maximum of


Fig. 6. This Egure compares the values of !c for di0erent values of �b for one geometry in which attractivesurface is parallel to preferred direction (i.e., polymer walk is fully directed in the plane of surface) to thatof another with surface perpendicular to the preferred direction of the walk.

92G(k; !)=9�2s (=9〈Ns〉=9�s). The transition points found from this method are shownin Fig. 5 by dots and cross. The results found from this method are in very goodagreement with those found exactly in above sections. This result indicates that as foras locating the adsorption–desorption transition of a long "exible as well as semi"exi-ble chains immersed in a good solvents are concerned the method of exact enumerationgives reliable results. In a polymer chain monomer may be represented either a stepor by a vertex. Former in which bond (step) of chain represents a monomer is re-ferred as step model of the problem while in later where a vertex of chain representsa monomer referred as vertex version of the problem. When surface is perpendicularto preferred direction both models gives the same results. However, in geometricalsituation in which surface is along the preferred direction of walk, critical value ofsurface adsorption is di0erent for both the model in 2D [12,13] and 3D PDSAWs hasbeen discussed in Fig. 6.

5. Conclusion

In spite of the severe restriction imposed on the angle of bending of the chain,the lattice models may provide interesting results for the conformational and surfaceadsorption properties of a semi"exible chain. Introducing directedness in the walk al-lowed us to solve the model exactly in both 2D and 3D. We have calculated the stepfugacity for polymerization of an inEnite chain and the persistent length as a functionof energy associated with the bending. We have been able to obtain the critical value


for adsorption of a directed chain on a surface perpendicular to the preferred directionof the walk analytically in both 2D and 3D. The dependence of this critical value ofsurface attraction on the sti0ness of the chain have been evaluated. Our study alsoshowed that sti0er chains adsorption takes place at higher temperature when comparedto that of "exible chains. We have examined the accuracy of the method of exactenumeration in locating the adsorption–desorption transition and have found that themethod gives values that are in excellent agreement with the exact values.Finally we would like to make a comment on the issue of the weight associated

to each step (and treat it as monomer) and weight associated to each vertex (andtreat it as one monomer). It has been shown that [12,13] critical value of !c = 1:707for step model while it is 1:492066 : : : for the vertex model when surface has beentaken along the preferred direction in 2D. However in the case, when surface has beenconsidered perpendicular to the preferred direction the critical value of !c is found tobe the same for vertex and step model. This is because walk once leaves the surfacenever returns on the surface. Therefore, Ns number of step on the wall remain sameas Ns +1 vertex on the surface. Here !c remains the same. However, when surface isconsidered parallel to the directedness, walker frequently visiting the surface and herethere is di0erence in the critical value of surface adsorption. In fully directed 2D and3D cases either surface is parallel to preferred direction or perpendicular to it does notmake any di0erence to critical value of surface adsorption as walker never returns tothe surface.

Acknowledgements

This work has been Enancially supported by the Council of ScientiEc and Indus-trial Research, New Delhi, India, INSA and Department of Science and Technology,New Delhi, India.

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A simple and exactly solvable model for a semiflexible polymer chain interacting with a surface