Self Assembly of Janus Ellipsoids
SSelf Assembly of Janus Ellipsoids
Ya Liu, Wei Li, Toni Perez, and J. D. Gunton
Department of Physics, Lehigh University, Bethlehem, PA 18015
Genevieve Brett
Department of Physics, Skidmore College, Saratoga Springs, NY 12866
We propose a primitive model of Janus ellipsoids that represent particles with an ellipsoidal coreand two semi-surfaces coded with dissimilar properties, for example, hydrophobicity and hydrophilic-ity, respectively. We investigate the effects of the aspect ratio on the self-assembly morphology anddynamical aggregation processes using Monte Carlo simulations. We also discuss certain differencesbetween our results and those of earlier results for Janus spheres. In particular, we find that thesize and structure of the aggregate can be controlled by the aspect ratio.
INTRODUCTION
Colloidal particles with anisotropic properties interactthrough an energy that depends not only on their spatialseparation but also on their relative orientations. Thisis a relatively new field that has been receiving consid-erable attention in recent literature [1–6]. Various site-specific techniques such as template-assisted fabricationand physical vapor deposition have been developed tosynthesize patchy colloidal particles of different shapes,patterns and functionalities [7–9]. The self-assembly ofthose building blocks into a desired mesoscopic structureand function are considered as a bottom-up strategy toobtain new bulk materials that have potential applica-tions in broad fields including drug delivery, photoniccrystals, biomaterials and electronics. The effects ofanisotropy have been classified by Glotzer and Solomonusing the concept of an anisotropy dimension includingpatchiness, aspect ratio, faceting etc [1, 10].Patchiness on spheres with different number, size andarrangements have been successfully fabricated in col-loids experiments and reveal interesting properties andcrystal structures [3–5, 11]. One particular example isthe Janus sphere with two dissimilar semi-surfaces, whichhas been extensively studied by experiments and theories[3, 12–14]. Examples of dissimilar, coded surfaces in-clude hydrophobic/hydrophilic, charged/uncharged, andmetallic/polymer surfaces. A variety of stable structuresand unusual phase behaviors have been found under dif-ferent chemical conditions of the solution by both exper-iments and computer simulations. Without losing thegenerality of the dissimilarity of two surfaces, a primitivetwo-patch Kern-Frenkel model has been used to modelJanus spheres [12, 15]. In Monte Carlo simulations, thissimplified model reproduces the main experimental fea-tures including self-assembly morphology and sheds lighton potential applications in engineering and theoreticalstudies of reentrant phase diagrams [14, 16].As suggested by Glotzer and Solomon, it’s natural toextend the study to explore the role of aspect ratio, inwhich patches are arranged on an anisotropic core such as spheroids. The anisotropy dimension is related to theaspect ratio and recent studies using ground-state energycalculation reveal many interesting structures such as he-lix [17, 18]. However, how the aspect ratio affects the self-assembly morphology is still not clear. In experiments,ellipsoidal colloids can be engineered with high monodis-persity using techniques such as deforming the sphericalsilica by ion fluence [19, 20], which makes the patched el-lipsoidal surface possible if one can combine this withthe template-assisted fabrication technique. Recently,unpublished work shows that Janus football-like ellipsoidhas been fabricated [21]. Suspensions of Janus ellipsoidalparticles provide a good candidate to study the effect ofaspect ratio on self-assembly and new feature of colloidalphase transformation.In this report, we first propose a theoretical modelof Janus ellipsoids with hard-core repulsion and quasi-square-well attraction. The properties of the model arealso discussed. In the second part, we use Monte Carlosimulations to study the effect of aspect ratio of ellipsoidson the self-assembly morphology and dynamical proper-ties. In the last section we present a brief conclusion.
MODEL
In our simulation study, a primitive model is presentedfor Janus spheroidal particles, with the lengths of theprinciple axes denoted by a (cid:54) = b = c . Depending on theaspect ratio, defined as (cid:15) = ab , the ellipsoid is charaterizedas oblate if (cid:15) < (cid:15) > r = ( x , y , z )whose axial orientations are given by u T = ( u , u , u ), v T = ( v , v , v ), w T = ( w , w , w ). The equation forthis ellipsoid has the explicit form[ u · ( r − r )] a + [ v · ( r − r )] + [ w · ( r − r )] b = 1 (1)In terms of a matrix representation, Eq. 1 can be rewrit-ten as X T AX = 0, where X T = ( x, y, z,
1) and A is a a r X i v : . [ c ond - m a t . s o f t ] A ug × A are listed as follows:1) A = u a + v b + w c , A = u u a + v v b + w w c A = u u a + v v b + w c , A = − x A − y A − z A A = u a + v b + w c , A = u u a + v v b + w w c A = − x A − y A − z A A = u a + v b + w c , A = − x A − y A − z A A = − − x A − y A − z A .Here A is normalized so that a point in the interior of theellipsoid X satisfies X T AX < A ) < U ij = U f ( r ij , u i , u j ). Asillustrated for two oblate ellipsoids in Fig. 1, attractivepatches are coded by red and the orientation is chosen tocoincide with the principle axis u in the body-fixed frameof reference . The orientational interaction is defined as: u j u i r ij i FIG. 1: Plot of two interacting ellipsoids labeled as i and j, inwhich attractive and hardcore repulsive surfaces are coded byred and blue, respectively. r ij is center-to-center displacementpointing from j to i. u i and u j is the patchy orientation. θ i is the patch angle of ith ellipsoid. f ( r ij , u i , u j ) = (cid:40) u i · ˆ r ij ≤ cos δ, u j · ˆ r ij ≥ − cos δ δ = π corresponds to Janus parti-cle. An attractive interaction exists if two red patchesface each other. The standard square-well potential hasbeen used for Janus spheres; however, the determina-tion of the accurate spatial relation between ellipsoids iscomputationally time-consuming. We thus introduce aquasi-square-well potential defined as U = (cid:40) ∞ if particles overlap − U H ( σ ij + 0 . σ − r ij ) otherwise (3) where U is the well depth, H ( x ) denotes the Heavi-side function, σ represents the length of the longer axis(max(2a,2b)). r ij is the center-to-center distance be-tween ellipsoids: r ij = | r ij | with r ij = r i − r j . σ ij = 2 b [1 − χ r ij · u i + ˆ r ij · u j ) χ u i · u j + (ˆ r ij · u i − ˆ r ij · u j ) − χ u i · u j )] − / (4)with χ = (cid:15) − (cid:15) +1 , ˆ r ij = r ij r ij [22]. σ ij is introduced as anapproximation to characterize the spatial relation, suchthat there is no overlapping interaction if r ij ≥ σ ij , pro-vided that the ellipsoids are represented by a Gaussianfunction: exp( − r · γ − · r ) with γ = a uu + b ( vv + ww )[22]. This approximation has been widely used to studyanisotropic particles such as liquid crystals and granularmaterial [23]. We note that in our study, this approx-imation is only applied to the potential but not to thegeometric overlapping which will be determined using aprecise method. Therefore, in our case H ( σ ij +0 . σ − r ij )represents a quasi-square-well potential with width 0 . σ .Specific cases under the condition u i = u j are illustratedin Fig. 2 for the aspect ratio (cid:15) = 0.1, 0.5 and 0.9 fromleft to right. Two particles will interact if their red shellstouch each other. The hard-core repulsion is provided FIG. 2: Plot of quasi-square-well potential under the condi-tion u i = u j for (cid:15) = 0.1, 0.5 and 0.9 from left to right. The redand blue surfaces represent the attraction range and hardcorerepulsion associated with each ellipsoid, respectively. Viewangle of each figure is tuned for better visualization. by the following geometric relation. Given two ellipsoids A : X T AX = 0 and B : X T BX = 0, one introduces thecharacteristic polynomial F ( λ ) = det( A − λ B ). A and B are normalized so that the interiors of A and B satisfy X T AX < X T BX <
0. The roots of the charac-teristic equation F ( λ ) = 0 have two positive real valuesand the rest characterizes the geometric relation betweenellipsoids [24, 25] such that1. A and B are separate if and only if F ( λ ) = 0 has twodistinct negative roots;2. A and B touch each other externally if and only if F ( λ ) = 0 has a negative double root.3. Otherwise, A and B overlap.Sturm sequence methods are applied to numerically de-cide if two roots are distinguishable. The primitive modelwe propose recovers the Kern-Frenkel Janus sphere modelwhen (cid:15) = 1, since under this condition, σ ij = σ and con-sequently, the attraction is simplified to H (1 . σ − r ij ).The introduced quasi-square-well potential has an advan-tage that there is no ambiguity when defining the con-nectivity of aggregates during the self-assembly process,and could be easily generalized to more realistic models.Our model is different from the Kern-Frenkel modelwhen (cid:15) (cid:54) = 1 since then the magnitude U is a func-tion of both the separation and orientation due to theanisotropic ellipsoidal core. The aspect ratio (cid:15) affectsthe shape of the interacting potentials. To characterizethe effective interaction between particles, we calculatethe second viral coefficient B : B = 12 V π ) (cid:90) [1 − e − βU ] d r d r d u d u (5)The results of Monte Carlo integrations of B /B hs as afunction of temperature for different (cid:15) are shown in Fig.3, where without specification, (cid:15) = 0 . , . B hs = πσ stands for the sec-ond viral coefficient for hard spheres with diameter equalto σ . The numerical error is less than 1%. The solid curverepresents the theoretical prediction for a Janus sphere: B /B hs = 1 − ( δ − e βU −
1) with the interactionrange δ = 1 . σ in the study [15]. As (cid:15) increases to 1, B approaches the value for the Janus sphere from above. (cid:1) (cid:2) (cid:3)(cid:2)(cid:4)(cid:5)(cid:3)(cid:2)(cid:5)(cid:5)(cid:3)(cid:6)(cid:4)(cid:5)(cid:3)(cid:6)(cid:5)(cid:5)(cid:3)(cid:4)(cid:5)(cid:5) (cid:7) (cid:1) (cid:8)-(cid:10)-(cid:11) (cid:5)(cid:5)(cid:12)(cid:2) (cid:5)(cid:12)(cid:13) (cid:5)(cid:12)(cid:14) (cid:5)(cid:12)(cid:15) (cid:6) FIG. 3: Plot of B vs. k B T /U for the aspect ratio (cid:15) =0.1 (green down triangle), 0.5 (red triangle) and 0.9 (bluecircle) from top to bottom. The solid curve is the theoreticalprediction for a Janus sphere, i.e. (cid:15) = 1. SIMULATION RESULTS
Standard Monte Carlo (MC) simulation in the NVTensemble has been applied to study the self-assemblypathway of these interacting Janus particles. We investi-gate a system in a 30 × ×
30 box with periodic bound- ary conditions. The particle aspect ratios range from0.1 to 0.9 with number density ρ = 0 . βU = 3 with β = k B T , where k B isthe Boltzmann constant. The system is initialized as arandomly-distributed noninteracting gas of monomers. Arandom translation followed by a random rotation is car-ried out for each monomer. In particular, the rotation isperformed using the method of quaternion parameters.More than ten independent runs for each aspect ratiohave been carried out up to 5 × Monte Carlo steps(MCS) and an ensemble average is taken by averagingover all runs. We monitor the evolution of − E/U (shownin Fig. 4), the negative average potential scaled by theattraction strength, which characterizes the number ofinteracting neighbors of each particle. Since the initialsystem has no interaction, − E/U starts from 0. It growsquickly and the dynamics slows down while approachingequilibrium. The system for (cid:15) = 0 . (cid:15) = 0 . . × MCS. This differ-ent growth tendency for the different aspect ratios is dueto the distinct aggregation mechanism that dominates atdifferent times. As we will show in the later section, theaggregation at the early stage is dominated by monomerdiffusion and interactions with small, formed oligomers.Depending on the aspect ratio, the system is composedof monomers, small oligomers, micelles and vesicles. Af-terwards, the aggregation dynamics is mainly the diffu-sion and collision of those small clusters that have muchsmaller diffusion constants than monomers. The value of (cid:15) affects the dynamics, as shown in Fig. 4, such that theaggregation process is relatively faster but reaches a lessstable structure when (cid:15) is larger. (cid:2) (cid:5) (cid:6) (cid:6)4(cid:6)(cid:8)(cid:6)4(cid:8)(cid:8)(cid:8)(cid:6) (cid:9)(cid:10)(cid:11) (cid:6) (cid:8)(cid:12)6(cid:6)(cid:14) (cid:15)(cid:12)6(cid:6)(cid:14) (cid:16)(cid:12)6(cid:6)(cid:14) (cid:17)(cid:12)6(cid:6)(cid:14) (cid:18)(cid:12)6(cid:6)(cid:14) FIG. 4: Plot of − E/U vs. MCS. The curves from top tobottom correspond to the aspect ratio (cid:15) = 0.1 (green), 0.5(red), and 0.9 (blue) To illustrate the difference in aggregation morphology,we take snapshots of the cluster growth as shown in Fig. (cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:6)(cid:7)(cid:4)(cid:3)(cid:7)(cid:4)(cid:8)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:7)(cid:1)(cid:1)(cid:9)(cid:2)(cid:4)(cid:5)(cid:8) (cid:5) (cid:6)(cid:7)(cid:5)(cid:3)(cid:8)(cid:1)(cid:10)(cid:2)(cid:3)(cid:4)(cid:1)
FIG. 5: Plot of cluster formation for (cid:15) = 0 . , . .
5. The cluster is defined such that two monomers areconnected if they interact; there is no ambiguity with re-spect to this interaction in our model. From left to rightin Fig. 5, we show typical structures for (cid:15) = 0.1, 0.5,and 0.9 that are collected from all simulations. Smalloligomers such as trimer, tetramer, pentamer, hexamer,and heptamer are similar and monomers form usual poly-gons, except that for (cid:15) = 0 . . σ . When (cid:15) < .
5, two ellipsoids have the possibility to interacteven when they are separated by the third one. As theoligomer grows, it starts forming a micelle (single layer)and a vesicle (double layer) as shown for clusters 13 and39 ( (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 .
1, therelevant clusters (8 and 17) display two and three layers,respectively. Those structures have been found in thecase of Janus spheres and have an effect on breaking thethermal correlations between particles. They thus affectthe system phase behavior. Eventually for (cid:15) = 0 .
9, clus-ters with different structure are formed, including twosmall oligomers joining together (46), as well as morecomplicated chains (72). For (cid:15) = 0 .
5, it’s possible to form a triple-layer compact cluster (51) and a dumbbellcluster(62). For (cid:15) = 0 .
1, due to the relation between theattractive range and aspect ratio, multiple-layer struc-tures have been observed in the simulations (30, 52).As is well known, Monte Carlo simulation doesn’t pro-vide an accurate description of the kinetics of the sys-tem; however, it still reveals some useful features. Weinvestigate the time evolution of number of clusters asillustrated in Fig. 6. The number of clusters for larger (cid:15) drops faster until reaching about 10 MCS. After that,the tendency reverses. The observation is consistent withthe energy evolution and B . At the early stage, the dy-namics is dominated by monomer motion and the sys-tem at (cid:15) = 0 . (cid:15) = 0 . (cid:1) (cid:2) (cid:3) (cid:4) (cid:5)(cid:6) m (cid:8) (cid:9) m (cid:10) (cid:11) (cid:2) (cid:12) (cid:13) (cid:5)(cid:6) (cid:17)(cid:10)(cid:18) + 6(cid:5)1+(cid:20) 2(cid:5)1+(cid:20) 3(cid:5)1+(cid:20) 4(cid:5)1+(cid:20) 5(cid:5)1+(cid:20) m FIG. 6: Plot of number of cluster vs. MCS for (cid:15) = 0.9 (bluecircle), 0.5 (red up triangle) and 0.1 (green down triangle).Error bars come from the statistical variance of independentruns. The inset shows the evolution of the number of clustersdefined through energy (blue) and distance (red) for (cid:15) = 0 . We have calculated the distribution of cluster size,which illustrates the effect of aspect ratio on the systemapproaching equilibrium, as shown in Fig. 7. For larger (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:2) (cid:6)(cid:7) (cid:4) (cid:2) (cid:8) (cid:9) (cid:1)(cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:2)(cid:1)(cid:4)(cid:1)(cid:2)(cid:1)(cid:5)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:7) (cid:10)(cid:11)(cid:7)(cid:3)(cid:4)(cid:12)(cid:5)s(cid:14)(cid:2)(cid:15)(cid:12) (cid:1) (cid:3)(cid:1) (cid:4)(cid:1) (cid:5)(cid:1) (cid:8)(cid:1) (cid:9)(cid:1)(cid:1) (cid:9)(cid:3)(cid:1) (cid:9)(cid:4)(cid:1) (cid:1)(cid:2)(cid:3)(cid:3)(cid:4)(cid:5)(cid:4)
FIG. 7: Plot of distribution of cluster size for (cid:15) = 0.1 (greendown triangle), 0.5 (red triangle) and 0.9 (blue circle). Insetsfrom top to bottom are configurations of the largest clusterfound in the simulations for (cid:15) = 0.1, 0.5 and 0.9, respectively.The number associated with each configuration is the clustersize. (cid:15) , the distribution has a broader range with a lower peak,which is consistent with the morphology shown in Fig. 5.Due to the complex structure formation such as a chain, (cid:15) = 0 . (cid:15) = 0 .
1, the distribu-tion is narrow and has a higher peak, which indicates thatthe clusters are more uniform. The largest cluster with size 55 has a similar shape as the oligomer (52) shownin Fig. 5; the structure of this cluster is like a blob withseveral layers. The structure is more stable in terms ofits energy and prevents further aggregation to form morecomplex forms. The case of an aspect ratio 0.5 (74) isintermediate between these two extremes of 0.1 and 0.9and the resulting cluster reveals two vesicles forming to-gether through a bridge-like structure. Next, in order toinvestigate in detail how ellipsoids organize in the cluster,we consider the correlation between patch orientations: u i · u j of two bonded ellipsoids (for which there exists apatchy attraction). The distribution P ( u i · u j ) is illus-trated in Fig. 8. (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:2) (cid:6)(cid:7) (cid:4) (cid:2) (cid:8) (cid:9) (cid:10)(cid:10)5(cid:12)(cid:10)5(cid:13) (cid:7) (cid:12) s(cid:15)s(cid:7) (cid:13) -(cid:12) -(cid:10)5(cid:17) (cid:10) (cid:10)5(cid:17) (cid:12) FIG. 8: Plot of distribution of u i · u j between two bondedellipsoids for (cid:15) = 0.1 (green), 0.5 (red) and 0.9 (blue). Typicalconfigurations corresponding to each peak are shown. For (cid:15) = 0 . P ( u i , u j ) develops two peaks; typical con-figurations corresponding to this correlation are shown inthe figure. These two peaks correspond to two ellipsoidsfacing toward ( u i · u j ∼ −
1) and opposite to ( u i · u j ∼ (cid:15) decreases, one peak ( u i · u j ∼ − (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 0 . MCS and is composedof a large number of monomers, micelles, vesicles andlarge complex clusters at 5 × MCS.For (cid:15) = 0 .
5, the configuration shows a somewhat dif-
FIG. 9: Snapshots of the system for (cid:15) = 0 . , . t = 10 and 5 × MCS. ferent feature (middle panels of Fig. 9) . The systemforms micelles and vesicles at 10 MCS and finally in-cludes monomers, large numbers of micelles and vesicles,and their combined aggregates. For (cid:15) = 0 .
1, the configu-ration show that clusters forms slowly and only includessmall oligomers and a large number of micelles and vesi-cles at 5 × . CONCLUSION
In the article, we propose a primitive model to studythe self-assembly of Janus ellipsoids. The interactionsbetween ellipsoids include a hard-core repulsion and aquasi-square-well attraction, where the latter exists whenthe patchy surfaces of two interacting ellipsoids orientin designated directions. The anisotropy in our modelcomes from two aspects: the patch interaction and theanisotropic core, which are controlled by a patchy angle δ and an aspect ratio (cid:15) , respectively. We particularly fo-cus on the Janus ellipsoids in which δ = π and (cid:15) rangesfrom 0.1 to 0.9, and address the effects of aspect ratio onthe self-assembly morphology and dynamical properties.Our model could be easily extended to consider morerealistic systems. For example, the typical interactionbetween colloids is in the range (0 . σ ∼ . σ ), which however is computationally expensive to simulate. Janusellipsoids with the interacting range comparable to 0 . σ can be realized by nanoparticles using the technique suchas induced phase separation [26]. Our results show thatfor larger aspect ratio, the self-assembly process is rel-atively faster and the morphology is more complicated,including chain structures as well as micelles and vesicles.The structures for smaller aspect ratio are more uniformand multiple-layer vesicles dominate. In our Monte Carlosimulation, for (cid:15) = 0 . B -scaling in the calculation of the phase diagram. This B scaling has been successfully applied, for example, forpatchy spheres [27]. In addition, it is important see howour results depend on the range of the interaction, e.g.0.2 σ , as is the case for colloidal interactions. We also findfor small aspect ratio that ellipsoids tend to form vesi-cles, which suggests a potential application for particleencapsulation. Perhaps the most important result of thestudy for materials engineering is the fact that the sizeand structure of the aggregates can be controlled by theaspect ratio, which should be an interesting result froma design viewpoint. ACKNOWLEDGEMENTS
This work was supported by grants from the MathersFoundation and the National Science Foundation (GrantDMR-0702890). One of us (GB) was supported by theNSF REU Site Grant in Physics at Lehigh University.Simulation work was supported in part by the NationalScience Foundation through TeraGrid resources providedby Pittsburgh Supercomputing Center. We thank Wen-ping Wang at HongKong University for providing the el-lipsoid code. [1] S. C. Glotzer and M. J. Solomon, Nature Materials ,557 (2007).[2] W. K. Kegel and H. n. w. Lekkerkerker, Nature Materials , 5 (2011).[3] Q. Chen, J. K.Whitmer, S. Jiang, S. C. Bae, E. Luijten,and S. Granick, Science , 199 (2011).[4] A. B. Pawar and I. Kretzschmar, Langmuir , 9057(2009).[5] Q. Chen, S. C. Bae, and S. Granick, Nature , 381(2011).[6] F. Romano and F. Sciortino, Nature Materials , 171(2011).[7] A. B. Pawar and I. Kretzschmar, Macromol. Rapid Com-mun. , 150 (2010).[8] S. Jiang, Q. Chen, M. Tripathy, E. Luijten, K. S.Schweizer, and S. Granick, Adv. Mater. , 1060 (2010). [9] V. N. Manoharan, M.T.Elsesser, and D.J.Pine, Science , 483 (2003).[10] Z. Zhang and S. C. Glotzer, Nano Lett. , 1407 (2004).[11] L. Hong, S. Jiang, and S. Granick, Langmuir , 9495(2006).[12] F. Sciortino, A. Giacometti, and G. Pastore, Phys. Rev.Lett. , 237801 (2009).[13] W. L. Miller and A. Cacciuto, Phys. Rev. E , 021404(2009).[14] F. Sciortino, A. Giacometti, and G. Pastore, Phys. Chem.Chem. Phys. , 11869 (2010).[15] N. Kern and D. Frenkel, J. Chem. Phys. , 9882(2003).[16] A. Reinhardt, A. J. Williamson, J. P. K. Doye, J. Car-rete, L. M. Varela, and A. A. Louis, J. Chem. Phys. ,104905 (2011).[17] S. N. Fejer and D. J. Wales, Phys. Rev. Lett. , 086106(2007).[18] S. N. Fejer, D. Chakrabraty, and D. J. Wales, Soft Matter , 3553 (2011).[19] T. van Dillen, A. van Blaaderen, and A. Polman, Mate-rials Today , 40 (2004).[20] C. C. Ho, A. Keller, J. A. Odell, and R. H. Ottewill,Colliod and Polymer Science , 469 (1993).[21] A. J. DeConinck, R. F. Shepherd, A. R. Cote, S. Granick,K. S. Schweizer, and J. A. Lewis, poster (MRL 2009).[22] B. J. Berne and P. Pechukas, J. Chem. Phys. , 4213(1971).[23] C. M. Care and D. J. Cleaver, Rep. Prog. Phys. , 2665(2005).[24] W. Wang, J. Wang, and M.-S. Kim, Computer AidedGeometric Design , 531 (2001).[25] Y.-K. Choi, J.-W. Chang, W. Wang, M.-S. Kim,and G. Elber, IEEE Trans.Visualization and ComputerGraphics , 311 (2009).[26] R. K. Shah, J.-W. Kim, and D. A. Weitz, Adv. Mater. , 1949 (2009).[27] H. Liu, S. K. Kumar, F. Sciortino, and G. T. Evans, J.Chem. Phys.130