Coarse-Grained Modeling of a Deformable Nematic Vesicle
CCoarse-Grained Modeling of a Deformable Nematic Vesicle
Jun Geng ( 耿 君 ), Jonathan V. Selinger, and Robin L. B. Selinger Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA (Dated: December 18, 2011)We develop a coarse-grained particle-based model to simulate membranes with nematic liquid-crystal order. The coarse-grained particles form vesicles which, at low temperature, have orienta-tional order in the local tangent plane. As the strength of coupling between the nematic directorand the vesicle curvature increases, the vesicles show a morphology transition from spherical toprolate and finally to a tube. We also observe the shape and defect arrangement around the tips ofthe prolate vesicle.
PACS numbers: 82.70.Uv, 61.30.Jf, 87.16.A-
Over the past twenty-five years, a major theme of re-search in condensed-matter physics has been the com-plex interaction of geometry with orientational order andtopological defects. Both theoretical and experimentalstudies have investigated order and defects on surfacesof fixed shape , such as colloidal particles or droplets [1–7]. These studies have shown, for example, that a ne-matic phase on a spherical surface will form four defectsof topological charge +1/2 each, and these defects maybe exploited to develop colloidal particles that will self-assemble into tetrahedral lattices for photonic applica-tions [8]. Inspired by this potential application, many au-thors studied how to control the arrangement of the fourhalf-charged defects in a nematic phase. Many effectshave been considered, including elastic anisotropy [3], ex-ternal field [4], and curvature of the colloidal particles [6].Further research has investigated orientational orderand defects on deformable vesicles , which serve as simpleanalogues for biological membranes [9–12]. These stud-ies show that defects in the orientational order will de-form fluid vesicles into non-spherical shapes. For exam-ple, some vesicles have tilt order, which can be modeledas XY order in the local tangent plane; these vesicleswill exhibit two defects of topological charge +1 each,and can deform into prolate or oblate shapes [11]. Othervesicles composed of T-shaped lipids or surfactants withrod-like heads may have nematic order in the local tan-gent plane [13]. Theoretical studies have predicted thatthe four half-charged defects will induce these vesicles todeform into tetrahedra [9].So far, theoretical studies of orientational order in de-formable vesicles have considered systems that are ideal-ized in several ways: at zero temperature, with no elasticanisotropy, with only certain couplings between orien-tational order and curvature, and with shapes that areslight perturbations on perfect spheres. A key question iswhether vesicle shape would be qualitatively different ifany of these simplifying assumptions were relaxed. Com-puter simulation provides a useful approach to this is-sue. For example, simulations can investigate problemswhere the geometry is not a perfect sphere but rathera more complex disordered shape, with bumps of pos- itive and negative Gaussian curvature. One simulationmethod uses a triangulated-surface model with tangent-plane orientational order; this method has indeed showncomplicated tube and inward tubulate shapes [12]. How-ever, a disadvantage of the triangulated-surface model isthat the connectivity of the surface is fixed, unlike exper-imental systems in which molecules can detach from andrejoin the vesicles.In this article, we develop an alternative method tosimulate orientational order in deformable nematic vesi-cles, using a coarse-grained particle-based model. Thismethod allows the particles to be assembled into a mem-brane with orientational order in the local tangent plane.The membrane spontaneously selects its own shape,which may be flat, spherical, or more complex. Further-more, the interaction of the coarse-grained particles canbe correlated with molecular features. Using this model,we calculate the arrangement of topological defects andthe shape of the vesicle as a function of the interactionparameters. In particular, we find a morphology transi-tion from spherical to prolate and finally to a tube as thecoupling between nematic order and curvature increases.To develop an appropriate simulation approach, wegeneralize a coarse-grained membrane simulation model without tangent-plane order due to Ju Li and collabo-rators [14–17]; see also Ref. [18]. In their approach, abilayer membrane is represented by a single layer of in-teracting coarse-grained point particles, each of whichcarries a polar vector degree of freedom ˆn representingthe preferred layer normal direction. The interaction po-tential favors association of particles with ˆn vectors ly-ing parallel and side-by-side. In simulations, the particlesself-assemble into pancake-shaped single-layer aggregatesshowing liquid-like self-diffusion and membrane elastic-ity. When another term is added to the potential favor-ing a slight splay between the ˆn vectors of neighboringparticles, particles spontaneously coalesce into sphericalshells. Each particle represents not a single molecule buta large patch of membrane containing many molecules,and the surrounding solvent is implicit. This approach ismore coarse-grained compared to other implicit-solventlipid models in the literature [19] as it does not sepa- a r X i v : . [ c ond - m a t . s o f t ] D ec FIG. 1. Each coarse-grained particle has a vector ˆn , whichaligns along the local membrane normal, and a vector ˆc , whichhas nematic alignment within the local tangent plane. rately resolve hydrophilic head/hydrophobic tail compo-nents across the width of a lipid bilayer.To simulate a membrane with tangent-plane order, forinstance formed by T-shaped lipids, we generalize thepreceding model by defining a coarse-grained point par-ticle with two vector degrees of freedom, as shown inFig. 1. The vector ˆn again defines the preferred layernormal direction, and a new vector ˆc represents the localnematic director orientation in the membrane’s tangentplane. Both ˆn and ˆc are unit vectors and are alwaysperpendicular to each other. Particles interact with eachother via an anisotropic Lennard-Jones-type pairwise po-tential with a distance cut-off: u ij ( ˆn i , ˆn j , ˆc i , ˆc j , x ij ) = (1) u R ( x ij ) + [1 + α [ a ( ˆn i , ˆn j , ˆc i , ˆc j , ˆx ij ) − u A ( x ij ) . In this expression, the repulsive and attractive parts ofthe potential are given by u R ( r ) = ε [( R cut − r ) / ( R cut − r min )] for r < R cut , u A ( r ) = − ε [( R cut − r ) / ( R cut − r min )] for r < R cut , u R ( r ) = u A ( r ) = 0 for r ≥ R cut . (2)Note that the repulsive and attractive terms have expo-nents 8 and 4, respectively, in contrast with the expo-nents 12 and 6 for the Lennard-Jones potential. Thesereduced exponents soften the potential and enhance thefluidity of the membrane. The coefficient α controlsthe strength of the anisotropic orientational interactions,which are defined by the function: a ( ˆn i , ˆn j , ˆc i , ˆc j , ˆx ij ) = (3)1 − [1 − ( ˆn i · ˆn j ) − β ] − [( ˆn i · ˆx ij ) − γ ] − [( ˆn j · ˆx ij ) − γ ] + 2 η [( ˆc i · ˆc j ) − . In this function, β = sin ( θ ) and γ = sin ( θ / θ represents the preferred angle between the ˆn vectors oftwo neighboring particles. As shown by Li and cowork-ers [14], if θ = 0, particles tend to self-assemble into flatmembranes with the ˆn vectors along the membrane nor-mal. If θ (cid:54) = 0, then the ˆn vectors of neighboring particlesalign with a separation angle θ ; i.e. they favor a splay,and particles self-assemble into vesicles with spontaneouscurvature. The novel aspect of our potential is the η term, whichfavors parallel or antiparallel alignment of the ˆc vectorsof neighboring particles, giving rise to nematic liquid-crystal order in the plane of the membrane. By varyingthe parameter η at fixed temperature, we can vary thestrength of the nematic order, and hence the Frank elasticconstants of the liquid crystal and the strength of thecoupling between curvature and orientational order.We perform a series of simulations with about 10,000particles over a range of η from 0.2 to 0.5, at tempera-ture k B T ≈ . ε . Other parameters are d = 1 . (cid:15) = 1 . α = 3 . r min = 2 / d , R cut = 2 . d , and θ = 0 . d using therandom sequential absorption method [20]. The value of θ was selected to set the model membrane’s spontaneouscurvature to match the vesicle’s initial radius. In the ini-tial state, ˆn vectors are aligned radially and ˆc vectorsare randomly oriented in the local tangent plane. Thetemperature is increased and maintained by a Langevinthermostat. The ˆc nematic director field quickly ordersto form a population of + / − half-charge defects; this tex-ture coarsens via defect pair-annihilation until four posi-tive half-charge defects remain. The vesicle then relaxestoward its equilibrium shape.Varying the coupling parameter η gives rise to a se-quence of shape transitions. Snapshots of the front, side,and top views of the simulation results are shown inFig. 2. Black dots indicate the locations of the four posi-tive half-charged defects in the tangent plane (except forthe largest η = 0 . ˆc vectors are shown as line segments in the localtangent plane. The ˆn vectors are not shown. The coarse-grained particles are semi-transparent so that defects arevisible on both the near and far sides of the vesicles.From the snapshots in Fig. 2, we can see that changing η changes the nematic director field and defect configura-tion as well as vesicle shape. For the smallest η = 0 .
2, thefour defects align themselves approximately on a greatcircle. This great circle is consistent with the contin-uum prediction of Ref. [3] for nematic vesicles with astrong elastic anisotropy, so it may indicate that this po-tential gives a substantial difference between the effec-tive Frank constants. As η increases to 0.25, the defectsmove to form approximately a regular tetrahedron. Forthese small values of η , the vesicle shape is fairly close tospherical, indicating that the coupling between directordistortions and curvature is not yet enough to substan-tially distort the sphere. As η increases further to 0.30,the defects shift further, with two defects moving to eachend of the vesicle. At the same time, the vesicle elon-gates substantially between these two poles, forming aprolate shape. For η ≥ .
35, the nematic order becomesstronger, so that the defect cores cost more energy. Wethen observe holes nucleating at the centers of the defects, η Front Side Top0.20.250.30.350.40.450.5FIG. 2. (Color online) Simulated vesicles for several values ofthe parameter η . Color indicates distance from the vesicle’scenter of mass, and black dots indicate particles in or near adefect core. Particles are semi-transparent so that defects onboth near/far sides are visible. and the vesicle coexists with a particle gas. The defectconfiguration also changes from an elongated tetrahedron( η =0.3–0.4) to rectangular ( η = 0 . η = 0 .
5, each pair of defects fuses to forma large pore at each end of the vesicle, thus eliminatingthe defects and transforming the vesicle into a tube. Themiddle of the tube is still swollen, not a perfect cylinder,because of the model membrane’s spontaneous curvature.
FrontTopFIG. 3. (Color online) Enlarged front and top views of thevesicle for η = 0 .
3, with line segments showing the local ne-matic orientation ˆc in relation to the overall shape. Particlesare opaque; only defects on the near side are visible. To show the relationship between the local nematicorientational ˆc and the overall shape more clearly, Fig. 3shows front and top views of the vesicle with η = 0 .
3. Inthese views the coarse-grained particles are opaque, sothat only two defects (on the near side) are visible, andthe defects can easily be identified as topological charge1/2. From the front view, we see that the local nematicorientation ˆc is aligned along the long axis of the prolatevesicle, allowing the ˆc vectors of neighboring particles tobe nearly parallel in three dimensions (3D); transversealignment would have much higher 3D interaction en-ergy. From the top view, we see that the local nematicorientation ˆc is aligned perpendicular to the separationbetween the two defects. Hence, the director distortionbetween the defects is almost entirely bend rather thansplay. Furthermore, the top view of the vesicle is notcircular but extended along the average nematic direc-tor, perpendicular to the separation between the defects.Thus, the overall vesicle has a biaxial, potato-like shape.It is remarkable that the half-charged defects are ar-ranged in pairs, with a pair at each end of the vesicle, inspite of the usual repulsion between defects. We specu-late that this arrangement occurs because the defects areattracted to the regions of high positive Gaussian curva-ture at each end of the vesicle. This attraction to thecurved regions competes with the repulsion between de-fects to favor an optimum separation between the defects,which depends on the coupling parameter η . A similarpairing of defects has been seen in analytic calculationsby Kralj [6] for the optimum positions of nematic defectson colloidal particles with a fixed ellipsoidal shape. Herewe see that the pairing occurs even when the shape is notfixed but is free to deform.Our simulations can be compared with the predictionsof Park et al. [9], who performed analytic calculations ofthe shapes of deformable membranes with nematic andgeneral n -atic order (note n = 2 for a nematic phase).Their theory predicts that a nematic vesicle should de-form into a shape with the symmetry of a regular tetrahe-dron, with a half-charged defect at each vertex. By con-trast, our simulations never show the tetrahedral defor-mation, but only the extended biaxial, potato-like shape.We speculate that this discrepancy occurs because theirtheory is idealized in two ways. First, their free-energyfunctional considers only intrinsic coupling of directorvariations with curvature through a covariant derivative,explicitly neglecting other couplings allowed by symme-try [21]. The interaction potential in our simulation in-cludes an extrinsic coupling of the nematic director to the3D curvature direction. In previous work we showed thatthe extrinsic coupling greatly changes the director fieldon surfaces with fixed curvature [22]; here we see that italso affects shapes of deformable membranes. This effectmay explain why the vesicle becomes extended along itslong axis and why its ends become extended in a biax-ial way; both distortions reduce the interaction energyof ˆc vectors in 3D. Second, their free-energy functionalmakes the approximation of a single Frank elastic con-stant, while our interparticle potential presumably givesdifferent effective Frank constants. Shin et al. [3] showedthat anisotropy of Frank constants changes the arrange-ment of defects; here it also affects the membrane shape.The nucleation of pores in a bilayer membrane dependson processes at length scales well below the particle spac-ing in our coarse-grained model [23]. While our coarse-grained simulation results demonstrate qualitatively thatpores spontaneously nucleate at topological defects, moredetailed theoretical analysis and molecular-scale simula-tions are needed to understand the process in detail. An-other question is whether total enclosed volume is con-served during vesicle shape relaxation. The formation ofpores in our simulation suggests that fluid may leak into or out of the vesicle as it evolves in shape, so we have notimposed a condition of fixed volume.In conclusion, we have developed a coarse-grainedparticle-based model for simulating membranes with ori-entational order, and we have used it to study vesiclesin the nematic liquid-crystal phase. The simulation re-sults show surprisingly complex vesicle shapes and defectconfigurations, which arise from features in the interpar-ticle potential. 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