Entropy-driven demixing in binary hard-core mixtures: From hard spherocylinders towards hard spheres

PHYSICAL REVIEW E NOVEMBER 1997VOLUME 56, NUMBER 5

Entropy-driven demixing in binary hard-core mixtures:From hard spherocylinders towards hard spheres

Marjolein Dijkstra1 and Rene´ van Roij21Centre Europeén de Calcul Atomique et Molećulaire, Ecole Normale Supe´rieure de Lyon,

46 Allee d’Italie, 69364 Lyon Cedex 07, France2Laboratoire de Physique, Ecole Normale Supe´rieure de Lyon, 46 Alleé d’Italie, 69364 Lyon Cedex 07, France

~Received 21 May 1997!

We present a computer simulation study of a binary mixture of hard spherocylinders with different diameters(D1,D2) and the same lengths (L15L25L). We first study a mixture of spherocylinders with lengthsL515D2 andD150, which can be regarded as a mixture of rodlike colloids and ideal needles. We find clearlyan entropy-driven isotropic-isotropic (I -I ) demixing transition in this mixture. In addition, we study a mixtureof spherocylinders with diameter ratioD1 /D250.1 and we investigated theI -I demixing transition as afunction of the lengthL of the particles. We observe a stableI -I demixing for all values ofL in the range of3<L/D2<15, but we could not reach the limitL50, i.e., the hard-sphere mixture with diameter ratio of 0.1.Striking agreement is found forL/D2515 with the results that follow from the second virial theory forinfinitely elongated rods. ForL/D252, we did not find a demixing transition till a total packing fraction ofh50.581, which is higher than the packing fraction at which freezing occurs for a pure system of thick rods.Thus this result and the extrapolation of our finite-L data toL50 gives us a fingerprint that the fluid-fluiddemixing transition in the binary hard-sphere mixture with a diameter ratio of 0.1 is metastable with respect tofreezing or does not exist at all at densities below close packing.@S1063-651X~97!07310-8#

PACS number~s!: 64.70.Md, 64.75.1g, 61.20.Ja

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I. INTRODUCTION

One of the most intriguing aspects of mixtures is the psibility of a spontaneous transition from a mixed to a dmixed state. Such demixing transitions have been obsein binary alloys and also in complex fluids, as, e.g., in owater mixtures. Traditionally, such transitions have beenplained on the basis of relatively unfavorable interactionergies between unlike particles, as, e.g., described byBragg-Williams theory for solutions and the Flory-Huggitheory for polymers. However, an interesting and longstaing question is whether a demixing transition can alsoobserved in a mixture in which potential energy doesplay a role. In a hard-core mixture, for instance, the topotential energy of the system vanishes, as only nonoverping configurations contribute to the configurational integrIn that case only entropic contributions are left to the frenergy, and phase separation can only occur in such an amal fluid if demixing results in an increase of the entropThe main contribution to the entropy of mixing of a binamixture comes from the entropy of mixing of an ideal miture, given bySid(x)52NkB@x lnx1(12x)ln(12x)#, wherex denotes the number fraction of one component andN thenumber of particles. It follows from the convexity of thfunction that phase separation always results in a decreaSid(x). This would imply that one would never observedemixing transition in a hard-core mixture. Also the intuitivnotion that entropy is related to the disorder of the systsuggests that a mixed system should have a higher entthan a demixed one at the same density and energy.

The simplest example of a well-studied mixture is tbinary hard-sphere mixture. The Ornstein-Zernike~OZ!

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equations of this system were theoretically analyzed in 1by Lebowitz and Rowlinson, who concluded that the homgeneous fluid phase is stable with respect to demixing,gardless of the diameter ratio, composition, or pressure@1#.These conclusions, which are based on the Percus-Ye~PY! closure of the OZ equations, were supported by ottheoretical studies@2# and by computer simulations@3–7#,and do agree with our intuitive notion of entropy. Howevein the early 1990s Biben and Hansen provided evidence fspinodal instability if the diameter ratio is more extreme th1:5 @8#. Their conclusion is based on the Rogers-Young csure of the OZ equations, which is supposed to be maccurate than the PY closure. They identified thedepletioneffect—known from colloid-polymer mixtures@10,11#—asthe demixing mechanism. Essentially the same instabwas also reported by Lekkerkerker and Stroobants@9#. Thedepletion mechanism in a binary hard-sphere fluid is baon the gain of free volume for the small spheres due to ctering of the larger spheres. To be more precise, in the cthat two large spheres are far apart from each other the sspheres are excluded from a depletion layer with thickns1/2 around the large spheres, wheres1 is the diameter ofthe small spheres. When the two large spheres are brointo contact, the depletion zones overlap and the volumecessible to the small particles increases. The resulting gaentropy of the small particles is the driving force that makthe large particles cluster. Apart from this entropic picturedepletion effect, there is also the kinetic picture in which t‘‘sea’’ of small particles generates an effective attractiveteraction between two large spheres, if the latter are so ctogether that no small particle fits in between. This unbanced osmotic attraction is then responsible for the liquvapor like demixing into a phase rich in large spheres a

5594 © 1997 The American Physical Society

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56 5595ENTROPY-DRIVEN DEMIXING IN BINARY HARD-CORE . . .

one rich in small spheres@9#. For completeness, we mentiothat the effective potential or depletion force between tlarge spheres due to a solvent of small spheres hascalculated theoretically by Maoet al. @12#, and computed ina simulation by Bibenet al. @13#. Using this effective pairpotential in a Monte Carlo study a tendency for phase seration is found. However, this phase separation is differfrom that found in simple liquids, since it results in a rapgrowth of huge clusters and subsequently a very slow reation of the clusters. We finally also note that the demixtransition in the binary hard-sphere fluid has also been foby Rosenfeld, within a self-consistent density functiontheory @14#.

Experiments, however, suggest that the demixing tration in sufficiently asymmetric binary hard-sphere mixturis strongly coupled to the freezing transition, and thatactual coexistence is that between a crystalline phase ofmarily big spheres and a fluid phase with primarily smspheres@17,18#. These results are supported by recent dsity functional calculations, which show a fluid-solid phaseparation in binary hard-sphere mixtures@15,16#. Whetheror not there is a~metastable! fluid-fluid spinodal in binaryhard-sphere mixtures remains an open question.

Computer simulation of phase separation in a mixturevery dissimilar spheres is difficult because of slow equilibtion. The numerical difficulties are less severe for lattmodels. Indeed, a purely entropic demixing transition hbeen shown by computer simulation of a lattice model obinary mixture of hard parallel cubes, if the diameter ratiothe cubes is sufficiently large@19,20#. Recent density func-tional calculations on this system have shown similar res@21#. Thus these results show that a demixing transitionbe driven by entropic effects alone. It is, however, uncleawhat extent this demixing transition is enhanced by thetice and by the discrete orientations of the cubes. It is tdesirable and interesting to consider an off-lattice systemhard-core particles.

An example of such an off-lattice mixture that has bestudied by computer simulation is the mixture of sphericolloids and rodlike polymers@22#. The colloids are represented by hard spheres with a diameters, and the rodlikeparticles by infinitesimally thin needles of lengthL. A de-mixing transition was found forL/s.0.3. Also, in computersimulations of a colloid-polymer mixture, where the colloiare modeled as hard spheres and the polymers as ideal lchains, a demixing transition was found@23#. However, inboth cases the polymers are ideal and do not interacteach other.

In this paper we present a computer simulation studybinary mixtures of hard spherocylinders with different diaeters (D1,D2) and the same lengths (L15L25L). Withinthe Onsager theory for hard rods, Sear and Jackson shofor D150 and L/D2→` that isotropic-isotropic (I -I ) de-mixing can preempt the isotropic-nematic (I -N) transition@24#. Our simulations of a system ofL/D2515 andD150clearly show anI -I demixing transition and thus confirm thresults of Sear and Jackson. For a finite diameterD1 , vanRoij and Mulder estimated recently that thisI -I demixing inthick-thin mixtures of hard rods is stable with respect toI -N transition, as long asD1 /D2.0.2 @25#. Again in thesecases, the depletion effect is identified as the demix

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mechanism. It is important to realize that these theoretresults only apply to hard rods in the limit of extreme elogation. In the present simulation study we consider theI -Idemixing transition as a function of the lengthL of the par-ticles, for a diameter ratioD1 /D250.1. Since the theoreticaresults are supposed to be exact forL/D2→`, we expect toreproduce the theoretical results closely for sufficienlarge—but finite—values ofL/D2 . For L50 the system re-duces to the binary hard-sphere mixture with diameter raD1 /D250.1. Although we could not reach the limitL50~with D1 /D250.1!, we did find stableI -I demixing for allvalues ofL considered, ranging from 3<L/D2<15. To ourknowledge, this is the first evidence of a fluid-fluid demixintransition driven by entropy alone in an off-lattice systemnonideal hard-core particles. Moreover, extrapolatingfinite-L data toL50 gives us a fingerprint that the fluid-fluidemixing spinodal in the binary hard-sphere system is mstable with respect to freezing.

II. COMPUTER SIMULATIONS

In order to determine the coexistence curve of the tdemixed phases directly, we carried out Gibbs ensemMonte Carlo simulations. In this method, the two coexistiphases are simulated in separate simulation boxes whichexchange volume and particles at a given temperature inder to fulfill the phase equilibrium requirements of equpressures and chemical potentials. During the simulationdinary Monte Carlo steps are performed in both phasesorder to equilibrate both systems internally. However,acceptance ratio for exchanging a large particle is smalldense system of small particles, as a large particle willmost always overlap with one of the small particles. In ordto speed up equilibration, we used collective particle mothat employed a generalization of the configurational-bMonte Carlo scheme of Refs.@26, 27#. In this approach, wefirst choose randomly a large spherocylinder in one boxtry to insert this particle at a random position in the othbox. When the particle overlaps with another large sphecylinder the trial move is immediately rejected. If no suchoverlap is found, the small spherocylinders overlapping wthe large spherocylinder in its new position are removedare then moved to the volume vacated by the large sphcylinder in the first box using a generalization of the Rosebluth sampling. The trial move is then accepted with a proability determined by the ratio of the Rosenbluth weightsthe new and old configurations. For more technical detawe refer the reader to the Appendix. In our simulations,performed four types of trial moves.

~1! Random displacement and rotation of a particle in oof the boxes.

~2! Small particle exchange between the boxes.~3! Large particle exchange between the boxes using c

figurational bias Monte Carlo method.~4! Volume exchange between the boxes.In each simulation, we measure the number fractionx of

the thick spherocylinders in both boxes, given byx5N/(N1M ), whereN and M are, respectively, the number of ththick and thin particles in that box. In addition, the numericvalue of the pressure was determined by virtual voluchanges. This method is based on the fact that the pressu

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5596 56MARJOLEIN DIJKSTRA AND RENEvan ROIJ

minus the volume derivative of the Helmholtz free energwhich can be related to the acceptance ratio of virtual vume changes as described in Ref.@28#.

Most runs consisted of 1042105 cycles per particle pethermodynamic state point. In each cycle, we attempt aplacement and rotation of a particle in one of the boxes,we try to exchange volume and a particle between the boFor the particle exchange we select with equal probabthe box where a particle is removed and the species ofparticle that will be transferred. The system size varies frabout 100 to 200 thick spherocylinders with 400–800 tspherocylinders.

III. RESULTS

A. L 515D2 , D1 /D250 and 0.1

The first set of Gibbs ensemble Monte Carlo simulatiowas performed on a binary mixture of hard spherocylindof lengthsL15L25L515D2 with D150. Here, and in thesequel of this paper, we useD2 as unit length. In Fig. 1, weshow the resulting dimensionless pressurebPbd of the co-existing phases as a function of the number fractionx. Hereb5pL1

2D1/4, d5D2 /D1 , and b51/kBT with kB Boltz-mann’s constant andT the temperature. We observe clearlydemixing transition in the isotropic phase for pressubPbd.7. Phase coexistence is found between an isotrophase of primarily thin spherocylinders (x.0) and an iso-tropic phase with a mixture of thick and thin spherocylinde(0.1<x<0.8). From Table I, we see that the critical pointestimated from the simulation agrees very well with the othat followed from the second virial theory for infinitelelongated rods of Ref.@25#. This good agreement is mainldue to the fact thatx'0.11 corresponds to only a smafraction of particles with a finite aspect ratio, while mostthe particles have diameterD150 and hence satisfyL/D1→`. Another reason for this good agreement is ththis transition occurs at low densities, so that the high

FIG. 1. The reduced pressurebPbd versus the number fractionx for a mixture of hard spherocylinders with lengthsL15L2

515D2 and diameterD150. The open squares denote theI -I de-mixing, the full circles and the stars theI -N demixing. The fulllines are the theoretical binodals taken from Ref.@29#.

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order virial coefficients of the thicker rods are not that important. In the same figure we also plotted the theoretbinodals obtained from Ref.@29#, which is based on previouwork. The deviation of the theoretical binodal and the simlated coexistence data at higher pressures atx.0.2 can beattributed to the third- and higher-order virial coefficientsthe thicker rods, which are neglected in the theory.

In order to investigate if thisI -I demixing transition isstable with respect to the isotropic-nematic transition,also performed Gibbs ensemble simulations starting frombox in which the particles are aligned and a box withisotropic configuration. We traced theI -N coexistence curvestarting from the pure thick system (x51) and graduallyadding more and more thin rods. For a pure system of thspherocylinders, theI -N coexistence is found to be at a presure ofbPbd516.15, which is slightly lower than the valuof bPbd517.141 in Ref.@22# obtained by integration of theGibbs-Duhem equation that describes theL/D dependence~instead of the temperature dependence! of the pressure awhich the two phases coexist. The addition of small particto this pure system of thick spherocylinders leads to anI -Nphase separation at higher pressures. Phase coexistenfound between an isotropic phase and a nematic phase wslight tendency of the large particles to be in the nemaphase and a slight tendency of the small particles to be inisotropic phase. However, the phase coexistence regionarrow. At higher pressures, we find that thisI -N transitionintervenes in theI -I demixing transition, giving rise to anI -I -N triple point, at which two isotropic phases~one withprimarily thin rods and the other with a mixture of thick anthin! coexist with a nematic phase of primarily thick rods. Aeven higher pressures, we find a wide phase coexistencgion between an isotropic and a nematic phase.

The simulation results of Ref.@22# show that a first-orderphase transition occurs from a nematic to a smectic phasea pure system of thick rods (x51) with L/D515 at a den-sity of about r* 50.5, wherer* 5r/rCP and rCP52/(&1)L/D). We only performed simulations up to a densiof r* 50.4484 for a system consisting of thick rods (x51), which corresponds with a pressurebPbd530.16, sowe did not consider the smectic phase in any detail.

Next we performed simulations of a spherocylinder mture with both diameters finite. We setD1 /D250.1 and keptL/D2515. In Fig. 2, we plot again the resulting coexistenpressure versus the number fraction. From Table I, wethat we find again good agreement between the critical pestimated in the simulations and calculated in the theoryRef. @25#. We again investigated if thisI -I demixing transi-tion is stable with respect to theI -N transition. Starting froma pure system of thick spherocylinders, we find a broa

TABLE I. The reduced pressurebPbd and the number fractionof the thick particlesx of the critical points predicted by theor~Ref. @25#! and estimated from the simulations for different diameter ratiosd5D2 /D1 and lengthL515D2 .

Theory Simulationd bPbd x bPbd x

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I -N demixing region at higher pressures, when we add tspherocylinders. Again, we found a triple point at which twisotropic phases coexist with a nematic phase, and a wI -N coexistence region for even higher pressures.

B. Towards hard spheres . . .

We showed above that we have been able to find a puentropy-driven fluid-fluid (I -I ) demixing transition that preempts theI -N transition. The question immediately ariswhy this transition is found in simulations of hardspherocylinder mixtures, while it is still not found in computer simulations of hard-sphere mixtures. One of the rsons is that the fluid-fluid demixing transition fospherocylinder mixtures occurs at much lower packing frtions than for hard-sphere mixtures. To illustrate this,compare three different theoretical estimates for the tpacking fractionh at the critical point of the fluid-fluid de-mixing transition of a hard-sphere mixture with a diameratio of D1 /D250.1 with the corresponding simulated valuin the hard-spherocylinder mixture ofL/D2515. The totalpacking fractionh5h11h2 is the sum of the packing fractions of the thinner and thicker species. For the hard sphwe haveh.0.47 @8#, 0.53@9#, and 0.37@14#, which is to becontrasted withh50.0736 for the spherocylinders. Anothreason why simulations of hard-sphere mixtures are difficis the low number fractionx at which the demixing transitiontakes place. The same three hard-sphere theories give focritical x the valuesx.0.02 @8#, 0.0056@9#, and 0.002@14#,while the spherocylinders showx50.2. A low number frac-tion means that a huge amount of small particles is neefor each large particle. Computer simulations of very asymetric mixtures are difficult as all the computer time will bspent on moving the small particles around, while displament of a large particle is hardly accepted in a dense sysof small particles.

FIG. 2. The reduced pressurebPbd versus the number fractionx for a mixture of hard spherocylinders with lengthsL15L2

515D2 and diameterD150.1D2 . The open squares denote theI -Idemixing, the full circles and the stars theI -N demixing. The fulllines are the theoretical binodals taken from Ref.@29#.

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In order to make a connection with the hard-sphere mixtures, we considered for fixedD1 /D250.1 a number of de-creasing lengthsL/D2510,8,5,3. In Fig. 3, we show the re-sults for theI -I demixing for the different values ofL/D2 .We see that the phase coexistence region becomes smawith decreasingL. A second and third virial theory of ahard-rod mixture with discrete orientations~Zwanzig model@30#! does not reproduce this narrowing of theI -I coexist-ence region, which remains therefore unexplained. The number fraction of the thick spherocylindersx and the reducedpressure of the critical point remain more or less the same fall lengths considered. In Fig. 4, we plot the packing fractionsh2 of the thick spherocylinders versus the packing fractionsh1 of the thin spherocylinders for the estimated critica

FIG. 3. The reduced pressurebPbd versus the number fractionx for a mixture of hard spherocylinders with varying lengthsL1

5L2 and diameterD150.1D2 .

FIG. 4. The packing fractionh1 of the thin spherocylindersversus the packing fractionh2 of the thick spherocylinders for theestimated critical points of theI -I demixing transition in a mixtureof hard spherocylinders with varying lengthsL15L25L and D1

50.1D2 ~stars! and the theoretical predicted critical points for thehard-sphere mixtures withD150.1D2 taken from Refs.@8, 9, 14#.

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TABLE II. The packing fractions of the thin particlesh1 , the thick particlesh2 , and the total packingfraction h5h11h2 , at which the critical point of theI -I demixing transition is estimated from the simultions for a mixture of hard spherocylinders with lengthsL15L25L and diameter ratio ofD1 /D250.1. Wealso tabulated for comparison the packing fractions at which a certain type of phase transition occurs fisotropic phase for a pure fluid of thick spherocylinders (x51) for different values ofL/D2 .

Critical point Pure fluidL/D2 h1 h2 h11h2 h Type of transition

15 0.0028 0.0708 0.0736 0.1777 isotropic-nematic10 0.0042 0.1084 0.1126 isotropic-nematic

8 0.0052 0.1398 0.1450 isotropic-nematic5 0.0079 0.2131 0.2210 0.4532 isotropic-nematic3 0.0138 0.4349 0.4487 0.5115 isotropic-solid2 0.532366 isotropic-solid0 0.47 isotropic-solid

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points of theI -I demixing transition for the various lengthWe also plot the three above mentioned theoretically pdicted critical densities for the hard-sphere mixtures wD1 /D250.1 @8,9,14#. We see that the packing fractionwhich the critical point is estimated increases enormouwhen we decrease the length of the spherocylinders. In TII, we compare the critical packing fractions of the sphecylinders with those at which apure fluid of thick sphero-cylinders will undergo a phase transition. The latter datataken from Ref.@22#. ForL/D252, we found no phase separation below a total packing fraction ofh5h11h250.581. From Table II, we can see that the isotropic-sotransition for a pure system of thick spherocylinders wL/D52 occurs already at a lower packing fraction. Thmight be a fingerprint that the isotropic-isotropic demixitransition becomes metastable with respect to the freetransition, or does not exist at all at densities below clopacking. A possible reason for the apparent absencespinodal instability might be the narrowing of theI -I coex-istence region with decreasing length of the spherocylindas discussed above. Unfortunately, equilibration probleprevented us from increasing the packing fraction even mso that we could not investigate whether a metastaisotropic-isotropic demixing transition exists above tfreezing transition. Also, slow equilibration preventedfrom decreasing the length-to-diameter ratio even morethat the hard-sphere mixture (L50) could not be studieddirectly—as expected.

In summary, we present the results of a computer simtion study of binary mixtures of hard spherocylinders wdifferent diameters (D1,D2) and the same lengths (L15L25L). We find clearly anI -I demixing transition in amixture of spherocylinders with lengthsL15L2515D2 , andD150 or D150.1D2 . Good agreement is found with thresults obtained from a second virial theory for infinitelong rods. To our knowledge, this is the first evidence ofluid-fluid demixing transition driven by entropy alone in aoff-lattice system of nonideal hard-core particles. In additiwe investigated theI -I demixing transition as a function othe lengthL and diameter ratioD1 /D250.1. We find astable I -I demixing for all values ofL in the range of 3<L/D2<15, but we could not reach the limitL50, i.e., thehard-sphere mixture with diameter ratio of 0.1. ForL/D252, we could not observe a demixing transition below a to

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packing fraction ofh50.581, which is above the packinfraction of the freezing transition of a pure mixture of thicrods. Slow equilibration prevented us from increasingpacking fraction even more and from decreasing the lengto-diameter ratio. A similar slowing down of the dynamichas been found in simulations of the hard-sphere mixtureBiben et al. @13#. Thus extrapolating our finite-L data toL50 gives us a fingerprint that the fluid-fluid demixing trasition in the binary hard-sphere mixture with a diameter raof 0.1 is metastable with respect to freezing, or does not eat all at densities below close packing.

ACKNOWLEDGMENT

We thank Thierry Biben for a critical reading of thmanuscript.

APPENDIX

In Gibbs ensemble Monte Carlo simulations, the two cexisting phases are simulated in separate simulation boand exchange of particles in the boxes are essential fortaining equal chemical potentials. However, in a dense stem of thin spherocylinders, random insertion of a thispherocylinder is difficult. To overcome this problem, wremove the thin particles that hinder the insertion of the thparticle and reinsert them into the space vacated by the tparticle. The algorithm for exchanging a thick spherocylder goes as follows.

~1! Choose first one of the boxes with equal probabilit~2! Choose randomly one of the thick spherocylinders a

insert this in the other box at a random position and withrandom orientation.

~3! Check if the thick particle at its new position overlapwith another thick particle. If there is such an overlap thmove will immediately be rejected.

~4! If there is no such overlap, check which of the thparticles have an overlap with the thick particle at its nposition.

~5! The thin particles that have an overlap with the thiparticle at its new position, saym particles, will be insertedin the volume that will become free in the old box of ththick spherocylinder, where we have removed the thick pticle.

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The insertion of them thin particles into the free volumeis not simple since the probability of random insertion ofmparticles in a restricted volume is small. We therefore usemethod similar to the one proposed by Biben for hard-sphmixtures@27#. We will describe the method below. When whave already insertedi 21 small particles, we will attempt toinsert thei th particle.

~1! Choose firstk random orientations for thei th particle,say v i( j ) and (j 51, . . . ,k).

~2! Define for each orientationv i( j ) a rectangular box, centered around the center of the removed thick particThe diameter of each box is given by the absolute va

of the components of the vectorRW 5L2V21L1v i( j )1(D11D2)(1,1,1), whereV2 is the old orientation ofthe removed thick particle, and where the axes ofsimulation box were chosen as basis vectors. We tchoose for each orientationv i( j ) a random positionrW i( j ) for the thin particle in the corresponding box. Wthen check if this thin particle has indeed overlap wthe removed thick particle. If not, we try a newrW i( j )until the overlap condition is satisfied.

~3! We now compute the Boltzmann factor ex@2bu„rW i( j ),v i( j )…# for all these trial insertions bychecking if there is an overlap with any of the other thiand thin particles and thei 21 particles that are alreadinserted. For hard-core particles, this Boltzmann factoeither zero in the case of overlap, or one when there isoverlap with other particles.

~4! One of the trial insertions, say positionrW i with orienta-tion v i , is now selected with a probability

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and orientationv i are one of thek trial insertions. Thei th thin particle will be inserted at this position and thcorresponding partial ‘‘Rosenbluth weight’’wi will bestored, where

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~5! These steps will be repeated until we have inserted amparticles.

~6! For the old configuration, a similar procedure is usedorder to calculate the old Rosenbluth factor. We agtry to add them overlapping particles in the space thatvacated by moving the thick particle from its new potion to its old position. When we have already inserti 21 thin particles, we will attempt to insert thei th par-ticle. In order to do that, we selectk21 trial directionsfor the thin particle. Thekth trial direction will be theoriginal configuration of the thin particle. We store thcorresponding partial ‘‘Rosenbluth weight’’ and we s

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nn

lect the original configuration of the thin particle. Wrepeat this until we have added allm particles.

In order to use this method to transfer a thick particle froone box to the other box in a Gibbs ensemble simulation,impose detailed balance on the Monte Carlo scheme. Timplies that in equilibrium, the rate at which thick particleare transferred from one box to the other equals the revrate.

Let us first consider the partition function of a systemNI thick particles in a box with a fixed configuration ofM Ithin particles and a volumeVI .

Q~NI ,VI ,T!5VI

NI

VNINI!E dsI

NIexp@2bU~G I!#, ~A4!

whereV is the thermal volume@31#, andU(G I) is the exter-nal potential ofNI thick particles and theM I thin particles inbox I interacting with each other by a hard-core potentNote that we used scaled coordinatessNI and that the con-figurational integral of the thin particles is neglected asconfiguration of the thin particles is fixed. However, ththick particles do interact with the thin particles and thusexternal potentialU(G I) depends on the configuration of ththin particles. If we now considerNII5N2NI particles inanother box with a fixed configuration ofM II thin particlesand a volumeVII5V2VI , the partition function of the twoboxes becomes

Q~NI ,NII ,VI ,VII ,T!5VI

NI~V2VI!N2NI

VNNI! ~N2NI!!E dsI

NI

3exp@2bU~G I!#E dsIINII

3exp@2bU~G II !#. ~A5!

Now we consider the case that the two systems canchange particles in such a way that the total number of pticles distributed over the two boxes remains constant,that the volumesVI andVII can change in such a way that thtotal volumeV5VI1VII remains constant. In this case, whave to consider all possible distributions of theN particlesover the two boxes and we have to integrate over the voluVI , which gives for the partition function

Q~N,V,T!5 (NI50

N E0

V

dVI

VINI~V2VI!

N2NI

VNNI! ~N2NI!!E dsI

NI

3exp@2bU~G I!#E dsIIN2NI

3exp@2bU~G II !#. ~A6!

It follows now that the probability to find a configuratiowith NI particles in box I with a volume ofVI and positionsI

NI andN2NI particles in box II with a volumeV2VI with

positionssIIN2NI is given by

ile-

s

l-th

cy

othdat

ickedtoes

nw

teox


N~NI,VI,T,sINI,sII

N2NI!;VI

NI~V2VI!N2NI

NI! ~N2NI!!

3exp@2bU~G I!#exp[2bU~G II !].

~A7!

For the configurational-bias method, we used the detabalance condition@26# to determine the probability of acceptance

N~o!p~o→n!5N~n!p~n→o!, ~A8!

whereN(o) and N(n) are, respectively, the probabilitiethat the system is in the original conformationo and in thenew conformationn. The transition matrixp(o→n) is equalto the probability to generate a trial move fromo to n timesthe probability to accept this trial move, i.e.,p(o→n)5a(o→n)3acc(o→n). In order to satisfy the detailed baance condition one can use the Metropolis scheme withacceptance rule:

acc~o→n!5minS 1,N~n!a~n→o!

N~o!a~o→n! D . ~A9!

Let us now assume that we want to move a thick spheroinder from box I to box II. The overlappingm thin sphero-cylinders in box II are now moved to box I. In order tcompute the acceptance rule, it is convenient to splitparticles in box I in particles that will keep their position anorientation going from the old to the new configuration, sNI thick particles andM I thin particles, and in particles thawill be moved, so one thick particle andm thin particles. Inbox II, we findNII thick particles andM II thin particles thatwill keep their position and orientation and again one thparticle andm thin particles that will be moved. We illustratthis in Fig. 5 for clarity. Note that the number of thick anthin particles in the old configuration of box I is equalNI11 andM I and that the number of thick and thin particlin the old configuration of box II isNII andM II1m. UsingEq. ~A7!, we find for the probability of the new configuratio

d

e

l-

e

y

N~n!5VI

NI~V2VI!N2NI

~NI!! ~N2NI!!exp@2bU„G I~n!…#

3exp@2bU„G II~n!…# ~A10!

while the old configuration has a probability of

N~o!5VI

NI11~V2VI!

N2NI21

~NI11!! ~N2NI21!!exp@2bU„G I~o!…#

3exp@2bU„G II~o!…#. ~A11!

For the ratio of the two probabilities, we find

N~n!

N~o!5

~V2VI!~NI11!

VI~N2NI!

3exp@2bU„G I~n!…#exp@2bU„G II~n!…#

exp@2bU„G I~o!…#exp@2bU„G II~o!…#.

~A12!

The Boltzmann factor for the new configuration can nobe written as follows:

FIG. 5. Schematic picture of the configurational-bias MonCarlo method for moving a thick spherocylinder from box I to bII in a Gibbs ensemble simulation.

r-

factor

exp@2bU„G I~n!…#exp@2bU„G II~n!…#5exp$2b@U~GNI

I !1U~GNI

I ,GM I

I !1U~GNI

I ,GmI !1U~Gm

I !1U~GM I

I !U~GM I

I ,GmI !#%

3exp$2b@U~GNII

II !1U~GmovedII ,GNII

II !1U~GmovedII ,GM II

II !U~GNII

II ,GM II

II !

1U~GM II

II !#%, ~A13!

whereU(GNI

I ) andU(GM I

I ) are, respectively, the external potential of a cluster ofNI thick particles and a cluster ofM I thin

particles interacting with each other by a hard-core potential. The external potential of a cluster ofNI thick particles with acluster ofM I thin particles is denoted byU(GNI

I ,GM I

I ), while U(GmovedII ,GNII

II ) is the external potential of the moved thickpa

ticle in box II with a cluster ofNII thick particles. The superscript denotes in which box the particles are. The Boltzmannfor the old configuration is given by

ert the

ert the

eely onceparticle


exp@2bU„G I~o!…#exp@2bU„G II~o!…#5exp$2b@U~GNI

I !1U~GNI

I ,GM I

I !1U~GmovedI ,GNI

I !1U~GmovedI ,GM I

I !U~GM I

I !#%

3exp$2b@U~GNII

II !1U~GM II

II !1U~GmII !1U~GM II

II ,GmII !

1U~GNII

II ,GmII !1U~GNII

II ,GM II

II !#%. ~A14!

We now find for the ratio of the two probabilities

N~n!

N~o!5

~V2VI!~NI11!

VI~N2NI!

exp$2b@U~GNI

I ,GmI !1U~Gm

I !1U~GM I

I ,GmI !#%

exp$2b@U~GmovedI ,GNI

I !1U~GmovedI ,GM I

I !#%

3exp$2b@U~Gmoved

II ,GNII


II !#%

exp$2b@U~GmII !1U~GM II

II ,GmII !1U~GNII

II ,GmII !#%

. ~A15!

The probability to generate the new configuration starting from the old configuration is given by the probability to insm thin particles in box I in the space vacated by moving the thick particle.

a~o→n!5)i 51

mexp@2bu~rW i ,v i !#

Z$rW i ,v i %

5

P i 51m expH 2bFU~GNI

I ,Gmi

I !1U~GM I

I ,Gmi

I !1(j 51

i 21

U~Gmj

I ,Gmi

I !G JkmW

5exp$2b@U~GNI

I ,GmI !1U~GM I

I ,GmI !1U~Gm

I !#%

kmWn, ~A16!

whereWn5P i 51m wi

n , i.e., the total Rosenbluth weight of them particles in the new configuration.The probability to generate the old configuration starting from the new configuration is given by the probability to ins

m thin particles in box II in the space vacated by the thick particle.

a~n→o!5exp$2b@U~GNII

II ,GmII !1U~GM II

II ,GmII !1U~Gm

II !#%

kmWo~A17!

and we find for the acceptance rule

acc~o→n!5minS 1,~V2VI!~NI11!Wn

VI~N2NI!Wo

exp$2b@U~GmovedII ,GNII


II !#%

exp$2b@U~GmovedI ,GNI

I !1U~GmovedI ,GM I

I !#% D . ~A18!

Note that the number of thin particles does not occur in the acceptance rule, as the thin particles cannot be chosen fra thick particle, that will be moved, is selected. The acceptance rule depends only on the potential energy of the thickwith all the particles that will keep their positions and orientations in boxes I and II.

.

e

tter

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Documents

Entropy-driven demixing in binary hard-core mixtures: From hard spherocylinders towards hard spheres