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Phase field calculus, curvature-dependent energies, and vesicle membranesQiang Dua

a Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

First published on: 06 July 2010

To cite this Article Du, Qiang(2011) 'Phase field calculus, curvature-dependent energies, and vesicle membranes',Philosophical Magazine, 91: 1, 165 — 181, First published on: 06 July 2010 (iFirst)To link to this Article: DOI: 10.1080/14786435.2010.492768URL: http://dx.doi.org/10.1080/14786435.2010.492768

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Philosophical MagazineVol. 91, No. 1, 1 January 2011, 165–181

Phase field calculus, curvature-dependent energies, and

vesicle membranes

Qiang Du*

Department of Mathematics, Pennsylvania State University,University Park, PA 16802, USA

(Received 2 January 2010; final version received 7 May 2010)

Lipid bilayer vesicles are membranes formed spontaneously in aqueousenvironments under suitable conditions. They have been extensively studiedin literature as a simple model of cell membranes. In this work, we discusssome recent developments in the phase field modeling and simulations ofvesicles. We describe how the equilibrium and dynamic properties of thevesicles can be effectively described by the phase field calculus which is anatural translation of the energy variations with respect to the vesiclegeometry to the variations of the corresponding diffuse interfacial energywith respect to the phase field functions. We also illustrate how phase fieldtechniques can be generalized to model a multi-component vesicle, a vesicleinteracting with a surrounding fluid and the adhesion of a vesicle to asubstrate.

Keywords: diffuse interface; phase field; curvature; vesicle deformation;vesicle fluid interaction; vesicle substrate adhesion

1. Overview

Interfaces and defects are ubiquitous in nature and they may take on different formsin different materials and can be studied on different spatial and time scales. One ofthe popular approaches used to study interfaces and defects on the continuum level isthe general Landau formalism [1] where the physical state is described by one (ormore) smoothly defined order parameter(s) which takes on nearly constant valuesexcept in a thin layer or small neighborhood surrounding the interfaces and defects,together with a governing free energy E¼E( ) which incorporates the relevantphysics. Such a formalism dates back to the work of van der Walls in 1893 on thegas–liquid interface, and over the years it has been successfully applied to studymany problems that involve various interfaces, defects and phase transitions. Theformalism is now commonly called a phase field or diffuse interface approach (twoterms that we use interchangeably in this paper). Examples of this formalism includecelebrated work like the Cahn–Hilliard diffuse interface theory of phase transition[2,3], and the Ginzburg–Landau model of superconductivity [4,5], as well as morerecent high-order extensions such as the phase field crystal models [6] and phase field

*Email: [email protected]

ISSN 1478–6435 print/ISSN 1478–6443 online

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models of fluid vesicle membranes [7]. The latter is the focus of this review. We beginwith a brief introduction to the background of vesicle membrane modeling and thegeometric and analytic ingredients in the phase field calculus. We then discuss thephase field modeling and simulations of single-component equilibrium vesicles,followed by the extension to multi-component vesicles. Afterwards, we present phasefield models for vesicle–fluid hydrodynamic interactions and vesicle–substrateinteractions. Issues related to the topological transformations of vesicles and theGaussian curvature contributions are also explored. There is surely more work onthe phase field modeling of vesicle membrane yet to be carried out, in order toprovide more quantitative validations of the models and to develop more effectiveand predictive simulation tools for practical applications; some of these importantissues and related questions are discussed in the final section of this paper.

In this exposition, while we attempt to include the significant contributions fromdifferent groups, it should be said that the works discussed here are certainly morebiased towards those we are more closely involved in. Indeed, even though thesubject of phase field modeling and simulation of vesicle membranes is relativelynew, it has been attracting more and more attention in recent years and this reviewprovides at best a partial account. We refer to [7–36] for more details on relevantstudies and additional references.

2. Phase field calculus and diffuse interface modeling

A well known example of the diffuse interface (phase field) approach is the diffuseinterface description of material interfaces developed by Cahn and Hilliard [2] forphase transition problems. We refer to [3,37–39] for related reviews.

In a typical diffuse interface description of a material interface, a smoothlydefined phase field function (or order parameter) is used which takes on two nearlyconstant values, say �1 for illustration, away from the interface on its two differentsides, and exhibits a smooth but more significant variation within a thin diffusetransition layer containing the interface. The thickness of the transition layer istypically characterized by a small positive parameter �, and the surface � is recoveredas a particular level set of , say { (x)¼ 0}, see Figure 1.

To construct a diffuse interface or phase field model of the interface, a keyquestion to be answered would be: how can one use such a function to representthe geometric features and physical properties associated with the interface �? Let us

Figure 1. An interface � with a thin diffuse interface layer, a typical one-dimensional phasefield profile, and a phase field simulation of thin film microstructures on a substrate[40]. Reprinted from Journal of Computational Physics, Vol. 208, Issue 1, P. Yu, S.Y. Hu,L.Q. Chen and Q. Du, An iterative-perturbation scheme for treating inhomogeneouselasticity in phase-field models, p. 34, Copyright 2005, with permission from Elsevier.http://www.sciencedirect.com/science/journal/00219991.

166 Q. Du


consider the geometric features for the moment, and in the simplest case, they may

involve the enclosed volume of the interface and possibly the surface area. For the

phase field descriptions of such quantities, the answers are well known [2]. For

example, as �! 0, the functional

E�ð Þ ¼

Z�

�

2jr j2 þ

1

4�ðj j2 � 1Þ2

� �d� ð1Þ

approaches a constant multiple of the surface area of � (in the sense of �-convergence)

[41–43]. Contributions of other energies may also be incorporated, such as the aniso-

tropic interfacial energy and the bulk misfit elastic energy [44,45]. It is then of both

theoretical and practical interest to study the structures of phase field functions which

minimize the various forms of the energy functionals under different constraints,

and/or their dynamic behavior as they evolve in time via some forms of a gradient

system (that is, steepest-descent dynamics) of the energy or through the interaction

with other external systems. The equilibrium phase field function is determined by

the Euler–Lagrange equation of (1) and in the unconstrained case, it leads to

�E�ð Þ ¼ �D þ1

�2ðj j2 � 1Þ ¼ 0 , ð2Þ

where �E�( ) denotes the variation of the energy with respect to the function .Meanwhile, well-known representatives of the associated dynamics include the

Allen–Cahn (Ginzburg–Landau) equation,

� t ¼ D �1

�2ðj j2 � 1Þ ð3Þ

for a time relaxation parameter �, and the Cahn–Hilliard equation,

� t ¼ �D D �1

�2ðj j2 � 1Þ

� �, ð4Þ

as well as hydrodynamic systems of coupled phase-field Navier–Stokes equations

[20,46]. In applications, these diffuse interface (phase field) models represent various

different dynamics; for example, the Allen–Cahn equation can be used to describe

the order–disorder kinetics or so-called Model-A dynamics of a non-conserved

variable, while the Cahn–Hilliard equation can describe the kinetics of spinodal

decomposition and it is also called Model-B dynamics of a conserved variable [47].There has been much work in the last 50 years related to the various extensions of

diffuse interface theory and their applications. We again refer to the reviews cited

earlier for surveys of many of these studies.Recently, the phase field diffuse interface framework has also been extended to

model interfaces governed by curvature-dependent energies, such as the lipid bilayer

vesicles which are basic models of cell membranes. Cellular membranes change

conformation in striking ways during many important biological processes such as

movement, division, and vesicle trafficking [48,49]. A crucial feature describing such

changes is given by the membrane curvature which is now viewed as an active means

to create membrane domains and to help in inducing biological signaling [49]. Given

the fluid-like behavior of vesicle membranes, an important energetic term entering a

Philosophical Magazine 167


continuum mechanics description of a vesicle surface is the bending elastic energy,

formulated by by Canham, Evans and Helfrich [50–53] as

EH ¼

Z�

b

2ðH� c0Þ

2þ cKþ �

� �ds, ð5Þ

where � is the surface representation of the vesicle, � denotes the surface tension, b

and c are the bending rigidities, which are usually on the order of some tens of kBT

for phospholipid bilayers, H and K are the mean and Gaussian curvatures of the

surface �, and c0 is the spontaneous curvature (which may also be modeled by the

area difference elasticity) [18,54].As a simple illustration, let us take the case c0¼ c¼ 0 in (5) and consider only the

contribution of the mean curvature energy term:

Eb ¼

Z�

H2 ds: ð6Þ

Rather than using a coarse-grained MD approach [55–58] or taking an explicit

sharp interface geometric description [53,59–62], following the Landau formalism,

we developed a phase-field/diffuse-interface bending energy model in [7] by

exploring the intrinsic connections between the geometric calculus and the

corresponding phase field calculus. We note that the Ginzburg–Landau formalism

has been used to study the properties of lipid bilayers in many contexts; see for

example [63]. Our focus here is on its application to the curvature-dependent

energies.Our starting point is the well-known connection between the energy functional E�

in (1) and the interfacial area/surface tension in the sharp interface limit, that is, as

�! 0. Moreover, we also know that geometrically, the variation of interfacial area is

related to the mean curvature of the surface [41–43,64]. Thus, motivated by the

following diagram:

Eð Þ ¼

Z�

�

2jr j2þ

1

4�ðj j2�1Þ2

� �d� ��!

�-convergence

�!0Surface area B¼

Z�

d�

variation#of variation#of �

�E

� ¼��D þ

1

�ðj j2�1Þ ��!

at f ¼0g!�

�!0Mean curvature

�B

�� ~n¼H

Figure 2. A bilayer vesicle assembled by amphiphilic molecules, and a computed equilibriumvesicle configuration in the shape of a discocyte [7]. Reprinted from Journal of ComputationalPhysics, Vol. 198, Issue 2, Q. Du, C. Liu and X. Wang, A phase field approach in the numericalstudy of the elastic bending energy for vesicle membranes, p. 450, Copyright 2004, withpermission from Elsevier. http://www.sciencedirect.com/science/journal/00219991.

168 Q. Du


that reveals the connection between the geometric and the phase field descriptions, it

is plausible that the functional

Eb�ð Þ ¼

b

2�

Z�

�D þ1

� ð1� 2Þ

� �2

dx ð7Þ

is consistent with (6) as �! 0 (up to a constant multiple 4ffiffiffi2p=3 [17,65], which is

neglected here for simplicity). We note that the Gaussian curvature term in (5) is

related to the Euler number, whose diffuse interface formulation [66] will be

discussed later.The variational problem of minimizing the phase field bending energy (7) was

first described in [7], which is another variant of the Ginzburg–Landau formalism.

Based on the analysis in [17,65,67], we also have the following diagram in connection

with the sharp interface limit:

Eb�ð Þ ¼

1

�

Z�

�D þ1

� ð1� 2Þ

� �2

d� ��!�-conv.

�!0Bending energy Eb¼

Z�

H2 d�

variation#of variation#of �

�Eb�ð Þ ¼ �4 f �1

�ð3 2�1Þ� f

where f¼4 � 1�2ð 2�1Þ�

��!at f ¼0g!�

�!0

Willmore stress

�Eb

�� ~n¼�D�H�2HðH2�K Þ:

One may find detailed calculations on the sharp interface limit of the variational

derivatives of the phase field bending energy functional in [67]. The diagram is

another successful illustration of the consistency between the geometric calculus and

the phase field calculus which puts the phase field bending energy functional (7) on a

solid mathematical foundation.Given the natural derivation of the phase field energy functional (7), it is no

surprise that it has been used in other areas of study as well. For example, a two-

dimensional version of the functional has been previously adopted for image analysis

applications [68] which traced it back to a conjecture of DeGiorgi [69]. We note also

that there have also been a number of studies on phase field or Ginzburg–Landau

type models for nearly flat bilayers where the energy functionals can be viewed as a

combination of the energy (1) with a linearization of (7) [63,70].

3. Equilibrium configurations of single-component vesicles

Based on the bending elasticity model, the equilibrium shape of a single-component

vesicle can be obtained by minimizing (5) subject to a fixed surface area and enclosed

volume [53,71]. Numerical simulations based on minimizing the phase field bending

elastic energy (7) have been given in [7,19], subject to a given surface area imposed by

a prescribed value of B( ) and a given volume imposed by a prescribed value of the

integral of . In the phase field simulations, the interfacial width was taken to be

sufficiently small so that the consistency between the computed results based on the

phase field models and that based on the sharp interface descriptions can be

numerically confirmed. The equilibrium shapes are obtained through gradient flow



dynamics associated with the energy (7) subject to both volume and surface areaconstraints.

A number of well-known equilibrium configurations have been obtained alongwith a detailed energy bifurcation diagram, see Figure 3 for plots of the discocytesand stomacytes. Among the various other computational findings, one that is ofparticular interest is the phase field simulation which reproduces the non-axisymmetric equilibrium shapes such as the one shown in Figure 4, which hasbeen previously studied in [72–74]. Also shown in the figure is an equilibrium vesicleshape with a high genus, which is easily obtained in the phase field setting since theimplicit surface representation allows the simulation of interfaces with differenttopological structures.

The effect of the spontaneous curvature [75] has been examined within the phasefield context as well in [18]. In particular, with c0 being spatially inhomogeneous, anexample with a nonzero spontaneous curvature near two opposite ends of the vesicleis also shown in Figure 3. The phase field simulations of the equilibrium shapesprovide further evidence for the effectiveness of the diffuse interface modelingapproach.

Since [7], there have been other studies that were based on the formulation (7) formodeling vesicle bending elastic energy. For example, it has been used in [10] tostudy both the equilibrium and dynamic shapes of vesicles, much in the same spirit as[7] and [18], but with a local surface area conservation constraint rather than theglobal constraint adopted by [7]. It is known that for the computation of equilibriumvesicle shapes, the local and global area constraints are in fact equivalent based onthe Euler–Lagrange equations for both formulations.

4. Multi-component vesicles and vesicles with free edges

In recent experimental studies, multi-component vesicles with different lipidmolecule compositions (and phases) have been shown to display complex

Figure 4. A pair of phase field functions can simultaneously determine both the vesicle surfaceand the two different phases; comparisons of some phase field simulations of a two-component vesicle in [36] with the experimental observations in [76]. With kind permissionfrom Springer Science+Business Media: Journal of Mathematical Biology, Modelling andsimulations of multi-component lipid membranes and open membranes via diffuse interfaceapproaches, Vol. 56, Issue 3, 2008, p. 347, X. Wang and Q. Du.

Figure 3. Equilibrium shapes of vesicles: discocytes, stomacytes, non-axisymmetric torus, avesicle with inhomogeneous spontaneous curvature and a high-genus surface.

170 Q. Du


morphology involving micro-domains [76,77]. There is strong evidence suggestingthat phase segregation and interaction contribute critically to the membranesignaling, trafficking, and sorting processes [78]. Equilibrium shapes of a multi-component vesicle are often described by minimizing an energy that includes theelastic bending energy of the membrane and the line tension energy at the interfacebetween the components or a mixing energy [8,14,28,30,32,79–81]. The mixingenergy was often given by a natural Ginzburg–Landau free energy expansion withrespect to a local composition fluctuation profile, which is like a phase field variable.The strong segregation limit of such a mixing energy is consistent with theformulation given by the line energy. In the last few years, there have also been newstudies on the use of phase field models to simulate multi-component vesicles and thephase separation process [24,26,33,35,36].

In the phase field model presented in [36] for a two-component vesicle, a pair ofphase field functions (�, ) was introduced so that one of them would identify thevesicle surface while the other would signify the different phases, which is in the samespirit as the level set description of an object with co-dimension 2 [82]. Theequilibrium configurations can be computed by minimizing the total energy subjectto a prescribed total volume and the areas (total compositions) of individualcomponents. A number of simulations were performed in [36] to match the reportedexperimental findings [76] and to provide numerical evidence for the equilibriumproperties of such configurations.

Rather than taking the phase field descriptions of both the vesicle and theindividual phases, phase field or Ginzburg–Landau type models have been developedin [8,14,30,32,57,81,83] and also [24,26,33,35] based on a sharp-interface geometricdescription of the vesicle but with a diffuse interface (phase field) representation ofthe individual phases. Many interesting results have been obtained. For example, anumber of three-dimensional equilibrium patterns were provided in [30] for aconstant bending rigidity but a composition-dependent spontaneous curvature.Working with a near planar geometry, [26] examined the morphological evolutionand stability of lipid membranes initialized in a variety of compositional andgeometric configurations, and it was also observed that rigid topographical surfacepatterns have a strong effect on the phase separation and compositional evolution inthese systems. Based on two-dimensional simulations, it was found in [33] that due tothe differences between the spontaneous curvatures and the bending rigidities, theformation of buds, asymmetric vesicle shapes and vesicle fission can take place intwo-component vesicles. There are also theoretical analyses available; for instance,

Figure 5. Comparisons of simulated open vesicles [36] with that observed in experiments in[85] and a sequence of equilibrium shapes showing the closing of the vesicle with increasingline tension. With kind permission from Springer Science+Business Media: Journal ofMathematical Biology, Modelling and simulations of multi-component lipid membranes andopen membranes via diffuse interface approaches, Vol. 56, Issue 3, 2008, p. 347, X. Wang andQ. Du.



careful asymptotic analysis in the sharp interface limit of such types of models hasbeen performed in [24].

It was noted that the phase field method developed in [36] for two-componentclosed vesicles can be readily applied to model and simulate a single-component openvesicle which has been an experimentally and analytically studied subject [84–89].A key idea is to include a ghost component which seals the open vesicle, but does notcontribute to the total energy. In [36], taking advantage of the effective handlingof topological changes by the phase field approach, a number of simulated equilib-rium shapes of open vesicles were provided along with comparisons with someexperimentally observed shapes.

5. Dynamics of vesicles

The study of the vesicle dynamics has been of much theoretical and practical interest[58,80,90–98]. Naturally, the study of the corresponding phase field models has alsoattracted the most attention of the community, see [9,10,15,16,20,25,29,31] on workrelated to mostly single-component vesicles and [26,33,35] for the dynamics of two-component vesicles. As the latter has been discussed in the previous section, we focuson the single-component case here.

The hydrodynamic model developed and analyzed in [16,20] is of a phase-fieldNavier–Stokes type, similar to that developed in [46,99–101], but with the phase fieldsurface tension energy replaced by the phase field bending elasticity energy:

ut þ u � ru ¼ �4uþ rpþ �Eb�ð Þr ,

t þ u � r ¼ ��A2�Eb� ð Þ,

r � u ¼ 0:

8><>: ð8Þ

For convenience, here �Eb�ð Þ denotes the variation of the phase field bending elasticenergy described earlier, u is the fluid velocity, p is the pressure, and � is a dampingparameter which is zero for the pure transport corresponding to a no-slip boundarycondition or it can be taken to be a small positive number with A2

¼ I for a non-conserved Allen–Cahn type dynamics or A2

¼�D for a conversed Cahn–Hilliardtype dynamics. The system of Equations (8) can be derived either from the principleof virtual work or from the least action principle [16,20]. In the case �4 0, thedamping term can help stabilize the numerical simulations [20]. To preserve thevolume and surface area constraints, in [16,20], both Lagrange multiplier and penaltyformulations have been suggested.

An immediate consequence of the model derivation given in [16,20] is that thesystem (8) enjoys the energy dissipation law given by

d

dt

1

2

Z�

juj2 d�þ E�ð Þ

� �¼ �

Z�

�jruj2 þ �jA�Eb� ð Þj2

� �d�, ð9Þ

which, for � ¼ 0, is the phase field analog of the energy dissipation law in the sharpinterface limit:

d

dt

1

2

Z�

juj2 d�þ

Z�

kH2 d�

� �¼ ��

Z�

jruj2 d�:

172 Q. Du


The consistency of the two energy laws has been rigorously established in [20].

Moreover, as �! 0, it was shown that the term �Eb�ð Þr approaches the Willmore

stress [20]. Preliminary numerical simulations of initially resting vesicles and

subsequently transported in a background fluid were presented in [20]; see

Figure 6 for an illustration. Notice that, while topological changes have taken

place, the total volume and the surface area have remained constant, and so has the

Euler number which accounts for the energy contribution from the change in the

Gaussian curvature. More numerical studies remain to be carried out, especially in

complex flow situations and with a dynamic change in the contribution of the

Gaussian curvature energy.Phase field models of vesicle dynamics have also been studied in [10–13] based on

the basic dynamic equations in [10] which is a gradient flow of the phase field energy

functional (7). Interesting phenomena involving pearling instabilities and polymer-

induced tubulations have been studied. We note that such a gradient dynamic system

can also be treated as a special case of the system (8) with u¼ 0 and A2¼�D:

t ¼ D �4 f �1

�ð3�2 � 1Þ� f

� �with f ¼ 4��

1

�2ð�2 � 1Þ�,

which is the H�1 gradient flow, more commonly called the conserved Cahn–Hilliard

type dynamics, of the energy functional (7). Following (9), it has a specialized energy

dissipation law:

d

dtE�ð Þ ¼ ��

Z�

jr�Eb�ð Þj2 d� ¼ ��

Z�

jð�DÞ�1=2 tj2 d�:

A hydrodynamic phase field model has also been used in [9,31,102] with a different

phase field formulation. Though named an advected-field approach, the work in

[102] appears to be the earliest attempt to develop a phase field type hydrodynamic

model for vesicles. While there is no explicit formulation of the phase field bending

elastic energy there, the vesicle motion was represented by the advection of the phase

field function (with additional damping) and a method was proposed in the two-

dimensional setting to compute the stress due to the variation of the bending energy.

Such a stress calculation is more involved than the Willmore stress formula

associated with the phase field elastic bending energy (7). Later, in [29], a

thermodynamic phase field model was derived for the hydrodynamics of vesicles

under the assumption of local membrane incompressibility. An extensive list of

simulations have been performed in [9,31,102] to study the various motions of the

Figure 6. An elliptical shaped vesicle is flushed by a fluid jet [20]. Reprinted from Physica D:Nonlinear Phenomena, Vol. 238, Issues 9–10, Q. Du, C. Liu, R. Ryham andX.Wang,Energeticvariational approaches in modeling vesicle and fluid interactions, p. 923, Copyright 2009, withpermission from Elsevier. http://www.sciencedirect.com/science/journal/01672789.



vesicles in a flow environment, including the tank-treading, tumbling, vacillating andbreathing modes. More recently, the work of [25] offered a very general derivation ofthe so-called thermodynamically consistent phase field models that covered both theapproaches used in [16,20] and [9,29,31].

To model the interactions of an immersed interface in a fluid environment subjectto fluctuation forces, a number of approaches have been developed in recent years,such as the discrete particle based Brownian dynamics [97], Stokesian dynamics [103]and dissipative particle dynamics [104], interfacial Langevin equations [105], and thestochastic immersed boundary methods [106]. Within the diffuse interface frame-work, we derived a stochastic implicit interface model (SIIM) for an immersedinterface in an incompressible viscous fluid subject to fluctuating forces [15,107]. Forexample, for an incompressible viscous fluid in the low Reynolds number regime, ifwe let u be the fluid velocity, p the pressure, and the phase field function for theimmersed interface, then one particular form of the SIIM is given by

ut � �4uþ rp ¼ � ð�Eb� Þr Þ þ Fþ _f,

r � u ¼ 0,

t þ ð � uÞ � r ¼ 0:

8><>:

Here, is a spatial averaging kernel which is often used in immersed boundarymethods, with * denoting the spatial convolution, F is a deterministic body force and_f is a stochastic force whose particular form obeys the fluctuation dissipationtheorem. The deterministic version of the coupled equations is consistent with theleast action principle [99]. The energy functional E¼E( ) can take on a diffuseinterface form like (1) or (7) or can take on the form used in level-set methods likethose given in [108–111].

6. Vesicle substrate adhesion

Adhesion is a key mechanism for the survival of many cells and organisms and itoccurs ubiquitously in nature. It is now established that stem cells differentiate intovarious cell types, including neurons, myoblasts, and osteoblasts depending on thestiffness of the elastic substrate that they adhere to [112]. While the cytoskeletalnetworkwithin the cell provides the keymechanism for the adhesion to a substrate andsenses the stiffness of the substrates [113,114], investigations have also revealed theimportant role played by cell membranes [115]. Recently, we also extended thephase field bending energy model to account for the interaction of the vesicles withboth flat and curved substrates. For a single-component vesicle interacting with asubstrate �s through an adhesion potential W¼W(x), which is a function of thedistance between a point x on the vesicle to the substrate, the total phase field energythen becomes

Eað Þ ¼

Z�

� D �1

�2ð 2 � 1Þ

� �2

�WðxÞð�2 � 1Þ2

�þ G

1�

2x3

" #d�, ð10Þ

where G is the gravity constant. A number of different choices of the adhesion poten-tial W¼W(x) has been used, including a Gaussian potential and a Leonard-Jones

174 Q. Du


type potential. The phase field formulation is based on sharp interface versions used in

many existing studies of vesicle–substrate adhesions, see for example [116–118]. In the

limit, when the bending effect becomes negligible, the overall vesicle shape approaches

the limiting profile given by the Laplace–Young equation for a soap film with a

nonzero contact angle which gives unbounded bending energy. With small but

nonzero bending, a boundary layer develops near the contact point where the

curvature changes rapidly [119].To validate the model, in [120], the results of phase field simulations were

compared with those obtained by the sharp interface computation. The effect of the

substrate curvature on bounding and unbounding transitions has been discussed,

similar to the study in [117]. We also used a Gaussian adhesion potential coupled

with the gravitational effect to study the interplay between gravity and the adhesion,

which is a subject previously examined in [121]. The phase field numerical

simulations presented in Figure 7 show that gravity pushes down the vesicles to

the bottom substrate while stronger adhesion leads to more visible protrusions.In other related work, we studied the effect of adhesion on the phase separation

process [122] by extending a phase field formulation for a two-component vesicle

[8,32] to include the effect of an adhesion potential between the vesicle and a

substrate. The total energy is of the form

Eð�Þ ¼

Z�

bð�Þ

2

H� c0ð�Þ

2þ �l

�

2

��r��2 þ 1

4�ð�2 � 1Þ2

� �� wð�ÞPðxÞ

� dx,

with �¼ �(x) being the phase field function, � the interfacial width parameter, �l aline tension constant, and P¼P(x) an adhesion potential as a function of the

distance from x2� to the substrate, so that the values of b(�), c0(�) and w(�) at

�¼�1 give, respectively, the values of the bending moduli, spontaneous curvatures

and the adhesion coefficients corresponding to the two different phases. Though the

formulation remains valid in general, to simplify the computation, we only

conducted numerical simulations in the case that both the vesicle membrane and

the substrate are axisymmetric in three dimensions. Not only were a number of

typical profiles found for equilibrium two-component axis-symmetric vesicles subject

to adhesion, numerical experiments have also been conducted in [122] to support the

observation of [123] that the adhesion may promote the phase separation of a multi-

component membrane.

Figure 7. A vesicle adhered to a cylindrical substrate: the three-dimensional view (left)and different cross-sections with and without the gravitational effect and subject tostronger and weaker adhesion (right) [120]. Reprinted from Journal of ComputationalPhysics, Vol. 228, Issue 20, J. Zhang, S. Das and Q. Du, A phase field model forvesicle substrate adhesion, p. 7837, Copyright 2009, with permission from Elsevier.http://www.sciencedirect.com/science/journal/00219991.



7. Gaussian curvature, Euler number and topological transformation

In the Helfrich formulation of the bending elastic energy (5), there is a contribution

coming from the Gaussian curvature which is viewed as a very important factor in a

number of biological processes such as vesicle fusion [124].For constant Gaussian bending modulus, it is well known that the resulting term

is a topological quantity related to the Euler number of the vesicle surface. It is

perhaps an intriguing statement that the phase field method is not only capable of

dealing with complex topological changes of the interfaces given the implicit surface

representation, but it is also possible to effectively retrieve topological information

from the phase field formulations. This is largely based on the framework of phase

field calculus: one can derive a counterpart of the Gauss–Bonnet theorem for

surfaces in terms of phase field functions. A perhaps pioneering work in this regard is

given in [66] where integral formulae have been presented to show that one can

reliably post-process the phase field solutions to recover the so-called phase field

Euler number, an approximation to the conventional Euler number of the interface

under consideration.To give an example, we consider a typical phase field function ¼ (x) in three-

dimensional space with �c¼ {x |�c5 (x)5 c} containing level surfaces of similar

topology for a suitable parameter c. Define

MðxÞij ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2cjr jp rirj �

rjr j2 � r

2jr j4ri rj

� �and Ge ¼

Z�c

�ðMðxÞÞdx,

ð11Þ

with �(M ) being the trace of (Adj(M )) which is the adjoint matrix of M.

Computationally, it was shown that Ge gives a good approximation to the

conventional Euler number in the sharp interface limit. A numerical example is

given in Figure 8. The notable changes of computed phase field Euler numbers occur

while the vesicles are going through topological transformations. The fractional

number values of the Euler number also signify the formation of self-intersecting

surface singularities during the merging process even though the phase field function

remains smooth during the whole simulation. Recently, we also considered simplified

algorithms to retrieve topological information directly from the implicit diffuse

interface description [21].

Figure 8. A phase field simulation of a few spherical like vesicles merge into a single vesiclethrough topological transformation, and the corresponding phase field Euler numbercomputed on-the-fly [66]. Reprinted from Journal of Computational Physics, Vol. 212,Issue 2, Q. Du, C. Liu, and X. Wang, Simulating the deformation of vesicle membranes underelastic bending energy in three dimensions, p. 757, Copyright 2005, with permission fromElsevier. http://www.sciencedirect.com/science/journal/00219991.

176 Q. Du


8. Conclusions

In this brief review, we have discussed how the ideas of phase field calculus can beused as an analog of the standard geometric calculus to develop reliable phase field

models for interfaces governed by curvature-dependent energies. A particular

motivation is the study of the bending elastic energy model for vesicle membranes.Our focus here has been on the most basic modeling issues and its several applica-

tions. There are obviously many important issues that we have not addressed here.First of all, in applications of the diffuse interface framework, the studies are

mostly carried out through numerical simulations [3,39]. Since a diffuse interface

model often gives a single set of equations with regular/smooth solutions in the

computational domain, the initial code development is relatively easier, in compar-ison with its sharp-interface counterpart. Yet, the solutions may display large

gradients in narrow interfacial regions which raise tremendous challenges to

numerical analysts in differentiating and balancing both the modeling and numericalerrors [125]. The effectiveness of phase field simulations rely on efficient numerical

methods that can resolve the phase field function and other related physical

variables. For much of the phase field simulations we have performed, we haveutilized both finite element methods and the Fourier spectral methods. The former

can work for more general geometries and with more general boundary conditions,while the latter is particularly effective for cubic domains with periodic boundary

conditions and for simulations that approach the sharp interface limit [126]. We refer

to [22,23,36,126] for the implementation and theoretical analysis of these numericalmethods, along with their adaptive implementations. With the Moving Mesh Fourier

Spectral methods [127,128], the implementation can offer anisotropic adaptivity

which also significantly improves the computational efficiency and stability. For thephase field bending elasticity models, it has also been demonstrated that, with

isotropic adaptive finite elements based on local mesh refinement and coarsening

governed by rigorous residual-based a posteriori error estimators, the computationalcost of a typical three-dimensional phase field simulation can be proportional to

O(��2) where � is the interfacial width parameter. In comparison, the computational

cost on a uniform mesh would be at least of order O(��3) [22]. We refer to[22,127,128] for more detailed discussions and extensive lists of references on the

adaptive methods for phase field simulations. A challenging issue requiring muchinvestigation is the development of efficient anisotropic three-dimensional finite

element methods for phase field simulations. Meanwhile, quantitative validation of

phase field simulations is obviously another important issue that needs to beseriously addressed.

Furthermore, we note that the successful applications of the phase field approach

can be expedited with the development of reliable and adaptive methods for not onlythe numerical solutions but also the information processing from the large-scale

simulations such as the retrieval of the hidden topological structures and other

important statistics. The phase field Euler number provides an example of the factthat it is sometimes constructive to take an alternative view: while the diffuse

interface approach is known to be insensitive to topological changes of the

underlying interface, topological features of the interface can give importantinformation about the underlying physical or biological processes which should not



be missed. In the phase field community, there have been studies of the topologicalstructures simulated by phase field methods, and commonly they are done byanalyzing the reconstructed images of implicitly captured interfaces [129]. The studyin [66] shows that it is possible to perform some of these tasks directly from phasefield simulations through a numerical integration. Indeed, in the application of phasefield modeling to multi-physics, multi-scale studies of many physical and biologicalprocesses, extracting statistics on the interfaces is often a more important task thansimply reconstructing their detailed geometric structures. Such a trend reflects thenature of integrated, adaptive and intelligent scientific computation [130].

Finally, on the application of phase field models to the study of vesicles, thereis also additional work such as [27] which incorporated the effect of electromag-netic interactions. While we have discussed phase field models for both the vesicledynamics and the vesicle–substrate adhesions, we have not combined them for thestudy of the dynamic vesicle–substrate interactions in a moving fluid environment.This is certainly a very interesting and challenging topic for future research. Thereare naturally many other interesting issues related to vesicles where the phase fieldformulation can become useful, such as the study of vesicle fusion due to virus entry[131] and vesicle-mediated particle interactions [55,132]. Many ideas presented herecan also find applications to other related problems. Moreover, the popularity andthe amazing unreasonable effectiveness (borrowing the words of [133]) of the phasefield methods in various applications also signify the need for further mathematicalanalysis and computational studies.

Acknowledgements

This research is supported in part by NSF-DMS 0712744 and NIH NCI-125707. I would liketo thank a number of people, including Roy Nicolaides and Morton Gurtin of CarnegieMellon who introduced the Cahn–Hilliard theory to me more than two decades ago; MaxGunzburger of Florida State who brought to my attention the Ginzburg–Landau modelswhich led to many joint works on the subject, together with other collaborators; and Long-Qing Chen who shared with me his knowledge and insight on the various aspects of phase fieldmodeling in materials sciences at Penn State. On the phase field modeling and simulations ofvesicles, I am particularly grateful to Chun Liu and Xiangqiang Wang for many stimulatingdiscussions and collaborations that resulted in a number of works discussed here. I also thankthe contributions made by other collaborators that include Rolf Ryham, Jian Zhang, SovaDas, Liyong Zhu, Manlin Li and Yanxiang Zhao. In addition, I thank the authors of [76] and[85] who permitted the use of their experimental pictures in [21] for comparison purposes;some of them are again shown in this paper.

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Philosophical Magazine Phase field calculus, curvature ... · Philosophical Magazine Vol. 91, No. 1, 1 January 2011, 165–181 Phase field calculus, curvature-dependent energies,