# Isotropic-nematic phase separation in asymmetrical rod-plate mixtures

**ABSTRACT** Recent experiments on mixtures of rodlike and platelike colloidal particles have uncovered the phase behavior of strongly asymmetrical rod-plate mixtures. In these mixtures, in which the excluded volume of the platelets is much larger than that of the rods, an extended isotropic (I)–plate-rich nematic (N−)–rod-rich nematic (N+) triphasic equilibrium was found. In this paper, we present a theoretical underpinning for the observed phase behavior starting from the Onsager theory in which higher virial terms are incorporated by rescaling the second virial term using an extension of the Carnahan–Starling excess free energy for hard spheres (Parsons’ method). We find good qualitative agreement between our results and the low concentration part of the experimental phase diagram. © 2001 American Institute of Physics.

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**ABSTRACT:**It is well known that the increase of the spatial dimensionality enhances the fluid-fluid demixing of a binary mixture of hard hyperspheres, i.e. the demixing occurs for lower mixture size asymmetry as compared to the three-dimensional case. However, according to simulations, in the latter dimension the fluid-fluid demixing is metastable with respect to the fluid-solid transition. According to the results obtained from approximations to the equation of state of hard hyperspheres in higher dimensions, the fluid-fluid demixing might becomes stable for high enough dimension. However, this conclusion is rather speculative since none of the above works have taken into account the stability of the crystalline phase (nor by a minimization of a given density functional, neither spinodal calculations or MC simulations). Of course, the lack of results is justified by the difficulty for performing density functional calculations or simulations in high dimensions and, in particular, for highly asymmetric binary mixtures. In the present work, we will take advantage of a well tested theoretical tool, namely the fundamental measure density functional theory for parallel hard hypercubes (in the continuum and in the hypercubic lattice). With this, we have calculated the fluid-fluid and fluid-solid spinodals for different spatial dimensions. We have obtained, no matter of the dimensionality, the mixture size asymmetry nor the polydispersity (included as a bimodal distribution function centered around the asymmetric edge-lengths), that the fluid-fluid critical point is always located above the fluid-solid spinodal. In conclusion, these results point to the existence of demixing between at least one solid phase rich in large particles and one fluid phase rich in small ones, preempting a fluid-fluid demixing, independently of the spatial dimension or the polydispersity.Journal of Statistical Mechanics Theory and Experiment 07/2011; · 1.87 Impact Factor - SourceAvailable from: Francesco Sciortino[Show abstract] [Hide abstract]

**ABSTRACT:**Concentrated solutions of short blunt-ended DNA duplexes, down to 6 base pairs, are known to order into the nematic liquid crystal phase. This self-assembly is due to the stacking interactions between the duplex terminals that promotes their aggregation into poly-disperse chains with a significant persistence length. Experiments show that liquid crystals phases form above a critical volume fraction depending on the duplex length. We introduce and investigate via numerical simulations, a coarse-grained model of DNA double-helical duplexes. Each duplex is represented as an hard quasi-cylinder whose bases are decorated with two identical reactive sites. The stacking interaction between terminal sites is modeled via a short-range square-well potential. We compare the numerical results with predictions based on a free energy functional and find satisfactory quantitative matching of the isotropic-nematic phase boundary and of the system structure. Comparison of numerical and theoretical results with experimental findings confirm that the DNA duplexes self-assembly can be properly modeled via equilibrium polymerization of cylindrical particles and enables us to estimate the stacking energy.08/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**Concentrated solutions of short blunt-ended DNA duplexes, down to 6 base pairs, are known to order into the nematic liquid crystal phase. This self-assembly is due to the stacking interactions between the duplex terminals that promotes their aggregation into poly-disperse chains with a significant persistence length. Experiments show that liquid crystals phases form above a critical volume fraction depending on the duplex length. We introduce and investigate via numerical simulations, a coarse-grained model of DNA double-helical duplexes. Each duplex is represented as an hard quasi-cylinder whose bases are decorated with two identical reactive sites. The stacking interaction between terminal sites is modeled via a short-range square-well potential. We compare the numerical results with predictions based on a free energy functional and find satisfactory quantitative matching of the isotropic-nematic phase boundary and of the system structure. Comparison of numerical and theoretical results with experimental findings confirm that the DNA duplexes self-assembly can be properly modeled via equilibrium polymerization of cylindrical particles and enables us to estimate the stacking energy.Macromolecules 01/2012; 45(2):1090-1106. · 5.93 Impact Factor

Page 1

Isotropic-nematic phase separation in asymmetrical rod-plate mixtures

H. H. Wensink, G. J. Vroege,a)and H. N. W. Lekkerkerker

Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, Utrecht University,

Padualaan 8, 3584 CH Utrecht, The Netherlands

?Received 4 June 2001; accepted 25 July 2001?

Recent experiments on mixtures of rodlike and platelike colloidal particles have uncovered the

phase behavior of strongly asymmetrical rod-plate mixtures. In these mixtures, in which the

excluded volume of the platelets is much larger than that of the rods, an extended isotropic

(I)–plate-rich nematic (N?)–rod-rich nematic (N?) triphasic equilibrium was found. In this paper,

we present a theoretical underpinning for the observed phase behavior starting from the Onsager

theory in which higher virial terms are incorporated by rescaling the second virial term using an

extension of the Carnahan–Starling excess free energy for hard spheres ?Parsons’ method?. We find

good qualitative agreement between our results and the low concentration part of the experimental

phase diagram. © 2001 American Institute of Physics. ?DOI: 10.1063/1.1403686?

I. INTRODUCTION

Dispersions of hard rod- or platelike colloidal particles

have been known to exhibit a spontaneous transition from an

isotropic (I) phase, in which the particles are randomly ori-

entated to an orientationally ordered uniaxial nematic (N)

phase.1–6The physical basis for understanding the phase be-

havior of anisometrical particles has been described in the

classic work of Onsager.7In his paper, Onsager formulated

the statistical mechanics of the problem by means of a virial

expansion of the free energy. He showed that the transition

can be explained on the basis of repulsive interactions be-

tween the particles as embodied in the second virial term.

The phase behavior of mixtures of rodlike and platelike

particles is intrinsically richer than that of rods and platelets

separately. Depending on the concentration and composition,

several liquid crystal phases may be formed. An issue which

is currently subject to debate is the stability of the so-called

biaxial nematic phase. In this liquid crystalline phase, both

rods and plates are orientationally ordered but in mutually

perpendicular directions. For symmetrical mixtures, i.e., in

case the excluded volumes of all particle pairs are set equal,

the biaxial phase appears to be a commonly established fea-

ture in many theoretical studies.8–13Under certain condi-

tions, the biaxial phase may, however, be unstable with re-

spect to demixing into the separate uniaxial nematic phases,

one containing predominantly rods ?the N?-phase? and plate-

lets ?the N?-phase?.14,15In case of asymmetrical mixtures,

i.e., mixtures for which the excluded volume of the platelets

is much larger than that of the rods, there is strong experi-

mental evidence of this demixing transition. Recently, van

der Kooij and one of us16,17studied the phase behavior of

mixtures of hard boehmite rods ?aspect-ratio 10? and gibbsite

platelets ?aspect-ratio 15?. No evidence of a biaxial phase

was found. The absence of a biaxial phase in these mixtures

is consistent with the observation that the nematic phases are

either strongly enriched in rods or in platelets. Since biaxial

order is preferably found in well-balanced nematic phases

?i.e., with approximately equal mole fractions of rods and

plates12? the biaxial phase is unlikely to appear in these

asymmetrical mixtures.

In this paper, we use the Onsager formalism to calculate

the phase diagram of asymmetrical mixtures of cylindrical

rods and platelets with equal aspect-ratios and equal dimen-

sions, so that the excluded volume of the platelets is much

larger than that of the rods. In view of the aforementioned

experimental results, we focus on the coexistence between

the isotropic and both uniaxial nematic phases. This means

that we do not take into consideration the biaxial nematic

phase.

We find qualitative agreement between our diagram and

the experimental one,16,17at least in the low concentration

part of the diagram. In both cases the phase behavior is ini-

tially dominated by coexistence between an isotropic phase

and a plate-rich nematic (N?) phase and, subsequently, by

an extended (I–N?–N?) triphasic area. This indicates that a

triphasic equilibrium at which the two uniaxial nematic

phases coexist with an isotropic phase may be expected in a

broad range of rod-plate compositions.

II. THEORY

Onsager already pointed out that the second virial ap-

proach, although valid for infinitely thin needles, cannot be

justified for thin disks. The reason for this is that disks, being

two-dimensional objects, have a nonzero probability of inter-

section and thus a finite excluded volume even at zero thick-

ness. The relative importance of three-body interactions in

terms of the ratio B3/B2

has been estimated by Onsager7at O(1). More accurate pre-

dictions were obtained from computer simulations,18giving

B3/B2

this value does not differ much from the B3/B2

hard spheres. Even for slender rods with aspect ratios smaller

than roughly 100, the results from the second virial approxi-

mation are no longer quantitatively correct. For example, for

2?with B3the third virial coefficient?

2?0.51 for disks with aspect ratio L/D?0.1. Note that

2?0.625 for

a?Fax: ?31-302533870. Electronic mail: G.J.Vroege@chem.uu.nl

JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 1515 OCTOBER 2001

73190021-9606/2001/115(15)/7319/11/$18.00© 2001 American Institute of Physics

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Page 2

spherocylinders with aspect ratio 10 the third to second virial

coefficient ratio was predicted19at B3/B2

clearly of the same order of magnitude as the values men-

tioned previously.

Hence, in view of these results one must conclude that

Onsager’s approach of truncating the free energy after the

second virial coefficient cannot be justified quantitatively for

mixtures of rods and platelets with an aspect ratio of roughly

10, which is a typical value for the experimental systems

under consideration.

In order to make quantitative comparison with experi-

ments possible we somehow have to account for the effect of

higher-order correlations between particles. In the present

study we use Parsons’ approach20to incorporate higher virial

terms into the Onsager free energy, albeit in an approximate

manner. The approach, based on the so-called decoupling

approximation in which orientational and translational de-

grees of freedom are treated separately, comprises a rescaled

form of the Carnahan–Starling free energy for hard spheres

to describe the ?excluded volume? interactions between the

anisometrical particles. The well-known Carnahan–Starling

?CS? equation of state has proven to be very accurate for

hard-sphere fluids at volume fractions up to the freezing frac-

tion (??0.5), giving results that are almost indistinguish-

able from simulation predictions.

The success of the approach relies on the incorporation

of higher-order interactions, albeit in an approximate man-

ner. Lee21has shown that Parsons’approach gave an accurate

description of the isotropic to nematic transition in a system

of hard ellipsoidal particles and spherocylinders as compared

to MC-simulations. To give an example, the I–N transition

density for hard spherocylinders with aspect-ratio L/D?5

was calculated at a volume traction ??0.400 which practi-

cally coincides with the value predicted from Monte Carlo

simulations.22Recently, Camp and others13,15showed that

the approach could also be applied rather successfully to

mixtures of rod- and platelike ellipsoids. It was shown to

give improved agreement with computer simulations over

the Onsager theory. The agreement with simulations was

shown to be within 10%.

In this section we present an analytical theory based on

the approximate Gaussian trial orientation distribution func-

tion ?ODF? as formulated by Odijk et al.,23which is a sim-

plified version of the trial ODF used by Onsager.7With the

Gaussian ODF, we retain a tractable theory since it enables

us to minimize the free energy analytically. On the other

hand, the Gaussian ODF has proven to be very reliable in

describing highly ordered nematic phases despite the fact

that it is not a solution of the stationary condition as obtained

from formal minimization of the free energy functional.24

For bidisperse systems of rods with different lengths, the

Gaussian ODF successfully explained generic features like

the fractionation effect, the widened biphasic gap23and,

somewhat later, the existence of triphasic and nematic–

nematic equilibria.25A recent analysis by van Roij24based

on elaborate numerical calculations of the exact high density

ODF essentially confirmed all conclusions of Ref. 25, thus

emphasizing the virtues of the Gaussian approximation.

2?0.313, which is

In this paragraph, we will first give a description of the

Onsager formalism for mixtures of rods and platelets. After

that, the Parsons approach will be explained in more detail.

A. Onsager theory

We consider a binary mixture of hard rods and hard

platelets in a macroscopic volume V. The particles involved

are characterized by four parameters: the length Lrand the

diameter Drof the rods ?with Lr?Dr? and the diameter Dp

and length ?thickness? Lpof the platelets ?with Dp?Lp?. The

generic shape of the particles is depicted in Fig. 1. The de-

tails of the exact shape of the particles are found to be irrel-

evant for the general argument, provided that the particles

are sufficiently anisometrical, i.e., Lr/Dr?1 and Dp/Lp

?1. When we study mixtures of rods and plates we have to

realize that, in the uniaxial nematic phases, one type of par-

ticle may align along a nematic director ?say the z-axis?

while the other type tends to orient its axis randomly in a

plane perpendicular to the director ?the xy plane?. Hence-

forth, for the sake of definiteness, we will use subscript 1 to

refer to the species oriented along the z axis and subscript 2

to the species oriented in the xy plane. The composition

variable, x?N2/(N1?N2), thus represents the mole fraction

of the species with preferred perpendicular orientation.

Hence, x represents the mole fraction of platelets in case of

I–N?coexistence and the fraction of rods in case of I–N?

coexistence.

Onsager described the nematic phase of a dilute solution

of anisometrical particles in terms of the orientational distri-

bution function ?ODF?, fj(?), representing the distribution

of the angles between the axis of the particle and the nematic

director. The ODF must be normalized according to

?fj(?)d??1, where ? is the solid angle of the particle’s

normal vector. In the isotropic state, all orientations are

equally probable which implies fiso?1/4?. In the present

study, we use Gaussian trial ODFs with variational param-

eters ?jto describe the angular distribution of the particles in

the nematic phase. Hence, we write for particles oriented

along the director

FIG. 1. Generic shape and orientation axes of the rod- and platelike par-

ticles.

7320J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Wensink, Vroege, and Lekkerkerker

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Page 3

f1?????

?1

4?exp??1

?1

2?1?2?

2?1?????2?

0????

2

4?exp??1

?

2????

,

?1?

and for particles oriented perpendicular to the nematic direc-

tor

f2?????

?2

?2??3exp??1

2?2?

?

2???

2?

0????.

?2?

Note that f1(?) is peaked around the nematic director ??

?0 and ???? whereas f2(?) attains its maximum value in

the plane perpendicular to the nematic director (???/2).

The formation of an isotropic state ?with fjconstant? or a

nematic state ?with fja peaked distribution? is caused by a

competition between orientational entropy ?favoring the iso-

tropic state? and excluded volume entropy ?favoring the

nematic state?. Onsager7defined ?jas a measure for the

?negative of the? orientational entropy

?j?? fj???ln?4?fj????d?,

j?1,2,

?3?

which has its minimum (?j?0) in the isotropic state but

increases as the orientational entropy decreases. An impor-

tant implication of using Gaussian trial ODFs is that it en-

ables us to calculate ?janalytically. For the nematic state we

obtain after substituting Eqs. ?1?, ?2? and straightforward in-

tegration

?1?ln?1?1,

2?ln?2?ln2

?4?

?2?1

??1?.

?5?

In the second virial approximation, the interactions between

hard particles may be expressed as an excluded volume en-

tropy depending on the excluded volume between two par-

ticles. Onsager gives us the following expression for the ex-

cluded volume between two platelets ?i.e., circular disks?, a

platelet and a ?cylindrical? rod and two rods as a function of

the angle ? between the particles’ axes

vexcl

pp?????

2Dp

3?sin???O?Dp

2Lp?,

vexcl

rp?????

4LrDp

2?cos???O?LrDpDr?,

?6?

vexcl

rr????2Lr

2Dr?sin???O?LrDr

2?.

Note that we restrict ourselves to the leading order contribu-

tions, which is justified for sufficiently anisometrical par-

ticles. Using the isotropic averages ??sin???iso??/4 and

??cos???iso?1/2 we obtain the average excluded volume be-

tween two randomly oriented particles in the isotropic phase

vexcl,iso

pp

??2

8Dp

3,

vexcl,iso

rp

??

8LrDp

2,

?7?

vexcl,iso

rr

??

2Lr

2Dr.

A measure for the average excluded-volume interaction be-

tween two particles is given by the average of its angular

dependence, defined as follows:

?jk???

vexcl,iso

vexcl

jk

jk???

fj???fk????d? d??

?8?

which implies ?jk?1 for the isotropic phase. For the nematic

phase, we must substitute Eq. ?6?. Hence, for two rods or

platelets orientated either along or perpendicular to the nem-

atic director this integral reads

??? ?sin???,????fj???fj????d? d??,

?jj?4

j?1,2.

?9?

Similarly, for a rod and a plate with mutually perpendicular

orientations we have

?12?2?? ?cos???,????f1???f2????d? d??.

?10?

Unlike Eq. ?3?, the excluded volume integrals cannot be cal-

culated straightforwardly since the integrands depend on the

interparticle angle ?(?,??). However, we can make head-

way by performing an asymptotic expansion of the integrals,

valid for small angles ? and ???/2???, implying that the

particles’ axes in general marginally deviate from the nem-

atic director or the xy plane. Obviously, the asymptotic ex-

pansion is only justified for large ?j, i.e., the Gaussian

ODFs must be sharply peaked around their maximum values.

Retaining only the leading order terms of the asymptotic

expansions, we readily obtain for ?11?in case of orientation

along the director?

?11?

4

???1

?O??1

?3/2?,

?11?

which was already found by Odijk et al.23The excluded vol-

ume integral for a rod and a plate with mutual perpendicular

orientations ?12requires a bit more effort ?see Appendix A

for details?. The result is as follows:

??

?12??81

?1?

1

?2??O??1

?3/2,?2

?3/2?.

?12?

Remarkably, the leading order term is equal to the one ob-

tained for rods with two different lengths.23Finally, for the

averaged excluded volume between two particles preferen-

tially oriented in the xy plane we can write

?22??22,0?1?F??2??,

where the leading order term ?22,0is simply the average ex-

cluded volume between two particles j randomly orientated

in the xy plane (?????0) relative to vexcl,iso

?22,0??4/????

00

?13?

22

?

d??

?1?

?

d? sin??

8

?2.

?14?

The ?2-depending correction term F is rather difficult to

obtain. It reads

7321J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Phase separation in rod-plate mixtures

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Page 4

F??2??

where K?ln4?1

will elaborate on its derivation in Appendix A. It is important

to note that F scales as ?2

scale as ??1/2. So, for large ?j, F leads to a small contri-

bution compared to ?11and ?12.

The total Helmholtz free energy F of the rod-plate mix-

ture ?in units kBT per particle? can now be expressed in

terms of the parameters ?jand ?jk

1

?2?

1

2ln?2?K??O??2

?2ln?2?,

?15?

2?E?3

2and ?E?0.577 Euler’s constant. We

?1ln?2whereas ?11and ?12both

?F

N?cst?lnc?1??1?x?ln?1?x??x lnx

??1?x??1?x?2

?c??1?x?2?11?2x?1?x?q12?12?x2q22?22?,

where ??(kBT)?1?kBand T have their usual meaning of

Boltzmann’s constant and absolute temperature?. The con-

stant cst includes terms independent of the mole fraction and

concentration. Furthermore, c is the total number density of

particles rendered dimensionless by relating it to vexcl,iso

?16?

11

via

c?1

2vexcl,iso

11

N1?N2

V

??

?

4Lr

?2

16Dp

2Dr

N

V

N?-phase

3N

V

N?-phase

.

?17?

The last term in the free energy Eq. ?16? can be identified as

a ?dimensionless? second virial coefficient B˜2multiplied by

the dimensionless concentration c. Note that cB˜2constitutes

the excess part of the free energy which accounts for the

interactions between the hard particles. Using the expres-

sions for ?jand ?jkin Eq. ?16? and minimizing with respect

to ?1and ?2yields

?1

1/2?2??1/2??1?x??21/2xq12h?Q??c,

?18?

?2

1/2??25/2??1/2?1?x?q12g?Q??H?x,?2??c,

with the definitions

?19?

Q??2/?1,

h?Q??Q1/2g?Q???

?20?

Q

Q?1?

1/2

,

?21?

and

q12?vexcl,iso

12

/vexcl,iso

11

,

q22?vexcl,iso

22

/vexcl,iso

11

.

?22?

Furthermore, H is the contribution arising from F(?2)

8

?2xq22?2

H?x,?2??

?1/2?1?2K?ln?2?,

?23?

which is small for large ?2. To simplify matters, we set F

?and H? equal to 0 for the moment, so that ?22??22,0

?8/?2. Henceforth, this will be referred to as the zeroth

order problem, denoted by subscripts 0. Within this approxi-

mation, it is possible to combine both minimization equa-

tions in order to obtain an expression only involving the ratio

of both ?’s. Taking the ratio of Eqs. ?18? and ?19? gives

Q0

1/2?

23/2?1?x?q12g?Q0?

?1?x??21/2xq12h?Q0?,

?24?

which is an implicit equation for Q0only containing the

mole fraction x and not the concentration c, so Q0

?Q0(x). A similar equation was obtained by Odijk et al. for

bidisperse rods. Fortunately, Eq. ?24? can be solved analyti-

cally, unlike the one obtained for bidisperse rods. After rear-

ranging terms, Eq. ?24? can be rewritten as a simple qua-

dratic equation in Q0, which is easily solvable. In practice, it

is convenient to rewrite the excluded volume terms ?jkin

terms of Q0,x and c using the minimization equations ?18?,

?19? and then substitute Q0(x) as found from Eq. ?24?. The

free energy of the uniaxial nematic phase is then written

explicitly in terms of the composition and dimensionless

concentration of the phase.

To locate phase transitions, we must know the osmotic

pressure and chemical potential of both species. These are

easily calculated as derivatives of the free energy. In the

nematic phase ?denoted by subscript n? we get in dimension-

less notation

??

N1,N2,T

? ˜n??1

2vexcl,iso

11

?F

?V?

??3?x?c?

8

?2q22x2c2,

?25?

? ˜1,n???

?F

?N1?

N2,V,T

?lnc?ln?1?x???1?2c??1?x??11?xq12?12?,

? ˜2,n???

N1,V,T

?26?

?F

?N2?

?lnc?lnx??2?2c??1?x?q12?12?xq22?22?.

?27?

Similarly, we obtain for the isotropic phase ?denoted by sub-

script i? using the isotropic values ?j?0 and ?jk?1

? ˜i?c?c2B˜2,i,

?28?

? ˜1,i?lnc?ln?1?x??2c??1?x??xq12?,

?29?

? ˜2,i?lnc?lnx?2c??1?x?q12?xq22?,

?30?

where B˜2,i??(1?x)2?2x(1?x)q12?x2q22? is the dimen-

sionless second virial coefficient for the isotropic phase.

When we use Eqs. ?15? and ?23?, the implicit equation

for the nematic phase becomes much more difficult. In fact,

it can be shown that Q will be dependent on the concentra-

tion as well, which considerably complicates our further

analysis. We will therefore not attempt to solve the full equa-

tion. To make headway, we will account for F(?2) in a per-

turbative way. Since F is only a small contribution, param-

eters ?j,?jand ?jkobtained from the minimization of the

free energy will only marginally differ from the ones ob-

tained for F?0. Hence, F can be considered as a perturba-

7322J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Wensink, Vroege, and Lekkerkerker

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Page 5

tion to the parameters obtained from the zeroth order prob-

lem. The approach will be discussed in Appendix B.

As mentioned in the introductory part, although Onsag-

er’s second virial approximation works well for sufficiently

elongated rods it does not give quantitative results for plate-

like particles. Therefore, in case of rod-plate mixtures, many-

body interactions involving platelets will undoubtedly play a

role in the regime where nematic phases appear. In order to

make quantitative progress, we have to somehow account for

the effect of higher virial terms. A method which has proved

to be useful up to now is to rescale the excess part of the

Onsager free energy using a modified form of the Carnahan–

Starling excess free energy for hard spheres. This approach,

known as the Parsons approach, will be discussed in the next

section.

B. The Parsons approach

The Parsons approach involves an expression of the ex-

cess free energy in terms of the semi-empirical Carnahan–

Starling free energy for hard spheres26

fCS????

?FCS

N

ex

???4?3??

?1???2,

?31?

where ? is the volume fraction of hard spheres. For a one-

component system of hard anisometrical particles this free

energy is multiplied by the prefactor ??vexcl??/8v0with v0

the particle volume and ??vexcl?? the average excluded vol-

ume. Note that ??vexcl??/8v0?1, in the case of hard spheres.

For binary mixtures of anisometrical particles, the prefactor

may be rewritten as ??v ¯excl??/v ¯0in terms of the following

mole fraction averages13

??v ¯excl???vexcl,iso

11

??1?x?2?11?2x?1?x?q12?12?x2q22?22?

?vexcl,iso

11

B˜2,

v ¯0??1?x?v0,1?xv0,2,

?32?

???1?x??1?x?2,

where ? is the total volume fraction of particles. Quantita-

tively, the results from this approach compare reasonably

well with computer simulations as was shown by Camp

et al.13,15Having established this, we can write the excess

free energy within the Parsons approach as follows:

?Fex

N

?c??1?x?2?11?2x?1?x?q12?12?x2q22?22?f˜CS???

?cB˜2f˜CS???,

where f˜CS(?)?fCS(?)/4?. Replacing the excess free en-

ergy cB˜2in Eq. ?16? by Eq. ?33? gives us the Onsager–

Parsons free energy for a binary mixture of hard rods and

platelets. Minimization with respect to ?jnow yields ?for the

zeroth order problem, F?0?

?1

?33?

1/2?2??1/2??1?x??21/2xq12h?Q0??cf˜CS???,

?34?

?2

1/2?25/2??1/2q12?1?x?g?Q0?cf˜CS???.

Note that the implicit equation in Q0(??2/?1) is left un-

changed since the concentration dependent part cf˜CS(?)

cancels. The total volume fraction ? of rods and platelets is

related to the dimensionless concentration c and the mole

fraction x via

??c,x??2c??1?x?

vex,iso

v0,1

11

?x

v0,2

vex,iso

11 ?.

?35?

It is now straightforward to recalculate the osmotic pressure

and chemical potentials by substituting Eq. ?35? and Q0(x)

?from Eq. ?24?? in the Onsager–Parsons free energy and tak-

ing the adequate derivatives. However, these expressions are

rather intricate since additional derivatives of f˜CSwith re-

spect to c and x are now involved. Within the Parsons ap-

proach, the osmotic pressure of the nematic phase ?denoted

by n? reads

P?cn???2?xn?cn?

? ˜n

8

?2cn

2xn

2q22f˜CS??1?? ln f˜CS

? lnc?,

?36?

and the chemical potentials

P?? ˜1,n?2 ln f˜CS???2?x??cf˜CSx2q22

? ˜1,n

8

?2?

?? ln f˜CS

??1?x?,

?37?

? ˜2,n

P?? ˜2,n?ln f˜CS???2?x??cf˜CSx2q22

8

?2?

? ln f˜CS

?x

.

Recall that these expressions only hold for the zeroth order

problem ??22??22,0?. When using the full expression for ?22

Eq. ?13?, additional terms arising from the perturbation

theory must be included ?see Appendix B?. These contribu-

tions are omitted here for the sake of simplicity, but can be

obtained straightforwardly. For the isotropic phase ?denoted

by i? we obtain

2f˜CSB˜2,i?1?ci

? ˜i

P?ci?ci

? ln f˜CS

?ci?,

?38?

? ˜1,i

P?lnci?ln?1?xi?

?cif˜CS?B˜2,i

? ln f˜CS

??1?xi??2?1?xi??2xiq12?,

?39?

? ˜2,i

P?lnci?lnxi

?cif˜CS?B˜2,i

? ln f˜CS

?xi

?2?1?xi?q12?2xiq22?.

III. PHASE DIAGRAMS

We can now construct the phase diagrams by imposing

the standard conditions of equal osmotic pressure and chemi-

cal potentials in the coexisting phases, using the expressions

7323J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Phase separation in rod-plate mixtures

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Page 6

from the Onsager–Parsons free energy and adding the per-

turbative contributions. The phase diagrams were con-

structed as follows. The I–N?and I–N?coexistence curves

were calculated separately using (q11,q22)?(qpp,qrr) and

(q11,q22)?(qrr,qpp), respectively, and solving the coexist-

ence equations iteratively. A triphasic equilibrium was lo-

cated at the intersection point ?triple osmotic pressure? of the

two isotropic branches. At the triple pressure, an isotropic

phase coexists with two nematic phases, each with different

mole fraction and number density. The location of the triple

point ?in terms of x and c? could subsequently be used as a

starting point for the calculation of the N?–N?coexistence

branches.

To give a graphic representation of the results, a ? ˜-x

diagram is shown in Fig. 2 and a diagram in terms of volume

fractions in Fig. 3. The volume fraction representation may

be more convenient from an experimental standpoint. The

volume fractions are obtained straightforwardly from Eq.

?35?. The tie lines are given by horizontal lines in Fig. 2

because of equality of osmotic pressure. In the volume frac-

tion representation these tie lines may be shown to form

straight lines.27In the latter case, we can also draw dilution

lines, i.e., straight lines radiating from the origin, along

which the overall mole fraction x of the parent system re-

mains constant. Obviously, in the ? ˜-x representation these

dilution lines run vertically.

To facilitate comparison with experimental results we

have matched the dimensions of the particles under consid-

eration to the average size of the gibbsite particles used in

experiment.17Accordingly, we have chosen equal aspect ra-

tios for the rods and platelets, i.e., Lr/Dr?Dp/Lp?15. Fur-

thermore, the long and short dimensions of the particles were

chosen to be equal, so that Lr?Dpand Dr?Lp. The ratios

of the excluded volumes are then given by

qrr?1,

qrp?15/4,

qpp?15?/4,

?40?

indicating that the isotropic excluded volume of two platelets

is almost 12 times larger than the excluded volume of two

rods. The resulting mixture is thus strongly asymmetrical

(vexcl

Several features are notable from Figs. 2 and 3. First,

there is a reentrant transition at mole fractions xpbetween

0.44 and 0.58. Experimentally, this would imply that a dilute

sample containing 50% platelets undergoes numerous phase

transitions upon concentrating, going from the isotropic state

I to I?N?, N?, I?N?,I?N??N?and finally N??N?.

Asimilar reentrant transition was found in binary mixtures of

rods with different lengths.28,23Furthermore, the triphasic

equilibrium, represented by a triple line in the ? ˜-x represen-

tation, clearly manifests itself in the volume fraction repre-

sentation as a triphasic triangle which covers a fair part of

the phase diagram. Accordingly, a large range of composi-

tions will pass through the three phase area. It appears that a

very small mole fraction of platelets in the isotropic phase

already leads to a three phase equilibrium upon increasing

the overall concentration.

We also investigate the phase behavior of a mixture with

reduced asymmetry, i.e., the differences in excluded volume

are chosen to be less extreme. For that, we consider a mix-

ture with aspect ratio Lr/Dr?Dp/Lp?50 and size ratio Lr

?2Dp, Dr?2Lp. So, again, both species are equally aniso-

metrical but the size of the platelets is now reduced to half

the size of the rods. This leads to the following excluded

volume ratios

rr?vexcl

rp?vexcl

pp).

qrr?1,

qrp?3.13,

qpp?4.91.

?41?

Note that qppis reduced by a factor 2.4 compared to Eq.

?40?. Yet, the excluded volume of two ?randomly oriented?

platelets is still about five times larger than that of two rods.

The corresponding phase diagrams are shown in Figs. 4

and 5.

Again, we observe a reentrant transition around xp

?0.5, albeit less prominent than in Fig. 2. A remarkable

difference with Fig. 2, however, is the presence of an azeo-

tropic point in the plate-rich part ?the ‘tail’ part? of the ? ˜-x

diagram ?see Fig. 4?b??. The azeotropic point is located at

mole fraction xp?0.718. Accordingly, a mixture containing

71.8% platelets ?with Lr?2Dp? does not fractionate during

FIG. 2. Phase diagram as calculated from the perturbation theory in the

osmotic pressure-composition (? ˜-x) plane for a mixture of rods and plates

(Lr/Dr?Dp/Lp?15) with equal dimensions, i.e., Lr?Dpand Dr?Lp.

Note the reentrant phenomenon near xplate?0.44.

FIG. 3. Phase diagram in the ?rod??platerepresentation corresponding to

Fig. 2. Thick lines indicate phase boundaries; thin lines represent tie lines

connecting coexisting phases.

7324J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Wensink, Vroege, and Lekkerkerker

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Page 7

phase separation, i.e., the coexisting I and N?daughter

phases have the same mole fraction as the parent sample.

An azeotropic point always corresponds to an extremum

in the osmotic pressure as a function of the mole fraction. In

this case, there is a minimum in the osmotic pressure at xp

?0.718, as can be seen from Fig. 4?b?. By recalculating the

phase diagram for various ratios Lr/Dpon the interval 1

?Lr/Dp?2 one can show that the azeotropic point ?and

hence the minimum in ? ˜? shifts toward higher mole frac-

tions as the ratio Lr/Dpis lowered. Obviously, noting its

absence in Fig. 3, the azeotropic point must leave the scene

at some point on this interval. This, so-called ‘‘critical’’ ratio

can be determined by calculating, for instance, the initial

fractionation, which we define as xp

mally close to 1, as a function of the ratio Lr/Dp. At the

critical ratio, the azeotropic point is located at xp?1 and the

initial fractionation is thus equal to zero. Note that the initial

fractionation is positive in case of an azeotropic point since

the coexistence pressure decreases initially at xp?1. In case

of no azeotropic point, the coexistence pressure increases

upon lowering xpand the initial fractionation is negative.

The critical ratio Lr/Dpwas found to be 1.54, independent

of the aspect ratio Lr/Dr?Dp/Lp. Hence, in case of equal

aspect ratios, we may expect an azeotropic point only if the

rods are sufficiently larger than the platelets ?i.e., Lr

?1.54Dp?.

I?xp

Nfor xpinfinitesi-

IV. DISCUSSION

We have constructed the phase diagram of a rod-plate

mixture with strongly asymmetric excluded volumes (vex

?vex

approach, we used Gaussian ODFs with adjustable param-

eters ?jto describe the distribution of angles in the uniaxial

nematic phases. To account for the effect of higher virial

terms, we have rescaled the excess free energy according to

Parsons’ method.

In the present calculations, we have set up a perturbation

theory to account for the ln?2/?2-term Eq. ?15? in the

asymptotic expansion of the excluded volume integral ?22.

To check the validity of this theory, we may compare it with

a full numerical approach, in which the minimization equa-

tions are solved numerically along with the coexistence

equations. As an example, we have depicted some I–N?

binodals obtained from several approaches in Fig. 6. The

differences between the approaches seem to become more

pronounced at low aspect ratios. First, agreement between

the perturbation theory and the full numerical solution is

very good. In fact, it remains surprisingly good even at low

aspect ratios ?below 15? whereas at high ratios (?20) the

curves almost become indistinguishable. Second, the zeroth

order approximation clearly starts to deviate from the other

curves as the aspect ratio is lowered. Crucially, the zeroth

order approximation ?i.e., retaining only the leading order

constant in ?22? completely breaks down at low aspect ratios

?say, Lr/Dr?18? giving unphysical binodals. This break-

down is not encountered within the perturbation theory or the

numerical approach. Therefore, we conclude that the second

order contribution F to ?22is an essential ingredient in our

calculations, since it enables us to calculate the phase behav-

ior of mixtures with aspect ratios comparable to that of ex-

perimental systems ?i.e., Lr/Drroughly between 10 and 15?.

plate

rod) using Onsager’s second virial approximation. In our

FIG. 4. ?a? Phase diagram as calculated from the perturbation theory in the

? ˜-x plane for Lr/Dr?Dp/Lp?50. The platelets are half the size of the

rods ?Lr?2Dpand Dr?2Lp?. A reentrant phenomenon near xplate?0.5 is

evident. ?b? Magnification of the ‘tail part’ of the diagram. Note the azeo-

tropic point at xaz?0.718.

FIG. 5. Phase diagram in the ?rod??platerepresentation corresponding to

Fig. 4?a?. Thick lines indicate phase boundaries; thin lines indicate represent

tie lines connecting coexisting phases. A particular dilution line correspond-

ing to the azeotropic point is drawn. Note that the associated tie line coin-

cides with the azeotropic dilution line.

7325J. Chem. Phys., Vol. 115, No. 15, 15 October 2001 Phase separation in rod-plate mixtures

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Page 8

Let us now compare our calculated phase diagram with

the experimental one as constructed in Refs. 16 and 17. We

have depicted the experimental diagram in Fig. 7. This dia-

gram can be compared directly with our volume fraction rep-

resentation in Fig. 3. From a qualitative point of view, both

diagrams agree very well. The general topology of both dia-

grams is very similar, at least in the low concentration part. A

notable feature is that the phase behavior is largely domi-

nated by coexistence between the isotropic phase and the

plate-rich nematic (N?) phase whereas coexistence between

I and the rod-rich nematic (N?) phase is only evident in a

small area close to the rod axis. This is a manifestation of the

asymmetrical nature of the mixture, i.e., the excluded vol-

ume of the platelets is much larger than the excluded volume

of the rods. Note that the fractionation between the I and N?

phases is also much more pronounced, particularly at high

osmotic pressures, than for I–N?coexistence. Finally, an

I–N?–N?triphasic area covering a significant part of the

phase diagram is present in both theory and experiment. A

generic feature that does not seem to appear in the experi-

mental diagram is the reentrant transition. However, a de-

tailed investigation of the lower part of the experimental

phase diagram is probably required to detect this phenom-

enon.

The high concentration part of the experimental phase

diagram is essentially different from the calculated one be-

cause additional high density liquid crystal phases ?with par-

tial positional order, such as the columnar (C) phase and the

not yet identified17rod-rich phase X? come into play which

are not taken into account theoretically. Most importantly,

the theoretically predicted N?–N?demixing beyond the

triphasic area is not observed experimentally. The absence of

this equilibrium is probably caused by a combination of

polydispersity and the presence of additional transitions from

nematic to high-density liquid crystal phases ?C and X?. As a

result, the N?–N?demixing is disrupted by several multi-

phase equilibria involving more than three phases, in particu-

lar, the remarkable five-phase equilibrium ?see Fig. 7?. This

observation is very striking since it seems to contradict the

Gibbs phase rule, which states that for an effective two-

component system of hard particles only bi- and triphasic

equilibria can be expected. A possible explanation for these

phenomena is therefore the polydispersity of the colloidal

species, i.e., the particles differ in size and shape. Since both

species have a fairly high polydispersity ?around 30%? the

resulting mixture effectively contains almost infinitely many

components, which may lead to coexistence between arbi-

trarily many phases. Hence, the challenge raised for further

theoretical research is not only to investigate the stability of

high-density liquid crystal phases ?e.g., columnar, smectic,

etc.? but also to address the effect of polydispersity on the

phase behavior of rod-plate mixtures.

APPENDIX A

In this Appendix, we give approximate analytical results

for ?12and ?22by performing asymptotic expansions of the

integrals.

Excluded volume integral for particles with polar and

equatorial orientation: ?12

Inserting the Gaussian ODFs Eqs. ?1?, ?2? into ?12, Eq.

?10? yields

?12??1?

??/2

??/2

?exp??1

Here, ? is the polar angle ?i.e., the angle between the particle

axis and the z-axis? and ???/2??? is the meridional angle

between the particle vector and its projection onto the xy

plane. Furthermore, ? is the azimuthal angle between the

projections of the particle vectors onto the xy plane. Recall

that the ODFs are sharply peaked around ??0 and ??0.

Hence, using

?2

?2??3?

2??1?2??2?2??d? d?cos??d?sin??. ?A1?

?/2?

?/2?

0

2?

?cos??

?cos????cos? sin??sin? cos? cos??,

?A2?

and substituting the asymptotic expressions, we can approxi-

mate

FIG. 6. I–N?binodals for Lr/Dr?Dp/Lp?18 and Lr?Dp, Dr?Lpas

calculated from several approaches; zeroth order approximation ?dashed

line?, perturbation theory ?dotted line? and full numerical approach ?solid

line?.

FIG. 7. Experimental phase diagram for mixtures of colloidal boehmite rods

(Lr/Dr?10) and gibbsite platelets (Dp/Lp?15). Picture taken from Ref.

17. The phase behavior becomes considerably complicated beyond the

triphasic area due to the presence of additional N–C and N–X transitions.

7326J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Wensink, Vroege, and Lekkerkerker

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Page 9

?12?4?1?

?exp??1

?2

?2??3?

2??1?2??2?2??d?? d? d?.

0

?/2?

0

?/2?

0

2?

???? cos??

?A3?

In order to get rid of the absolute value sign in the integrand

we must split the integral into parts. Noting that

??? cos??0 if ???,

??? cos??0 if ??? and if ??G????G, ?A4?

??? cos??0 if ??? and if ?????G

?with cos?G???/?? we may split Eq. ?A3? into three sepa-

rate integrals

?12?4?1?

0

?exp??1

?4?

0

??G

?d?? d? d??4?

0

?exp??1

?2

?2??3??

2??1?2??2?2??d?? d? d?

?/2?

?/2?

2??1?2??2?2??d?? d? d??,

?12?4?1?

?/2?

0

?/2?

??

?

???? cos??

?/2?

?

? exp??1

2??1?2??2?2??

?/2?

?G

?

?

? cos?

?A5?

?2

?2??3?I1?I2?I3?.

Note that we can extend the integrations in I1with respect to

? and ? to infinity since the exponent decays rapidly to zero

for large ?1and ?2. The integral is then easily calculated

and yields I1?2?(?1?2)?1. For the second one, it is con-

venient to reverse the order of integration. Hence, we write

I2??4?

?/200

?exp??1

??4?

?/20

??1?exp??1

??

?/2?

2??1?2??2?2??? d?? d? d?

??

2?2?2cos2???? d? d?.

?? cos ?

?

exp??1

2?1?2??2

?1

?A6?

The integral can be worked out straightforwardly and yields

2?(?1?2)?1????1/(?1??2)??1?. For the third integral

we use the relation ??G

I3?4?

0

?

?exp??1

?? cos?d?????2??2to write

?/2?

?/2??2??2

2??1?2??2?2??? d? d?,

?A7?

substituting y?1

2?1(?2??2) yields

3/2?

0

I3?25/2?1

?/2

exp??1

2??1??2??2?d?

??

0

?

y1/2exp??y?dy?2?/??1

3??1??2?.

?A8?

Finally, substituting all contributions back into Eq. ?A5? thus

yields for ?12

?12??8

?1?

where the second term in the expansion can be shown to be

of the order ?j

??

1

1

?2??O??1

?3/2,?2

?3/2?,

?A9?

?3/2.

Excluded volume integral for particles with equatorial

orientation: ?22

Inserting the Gaussian ODFs Eqs. ?1?, ?2? into ?22, Eq.

?9? yields

?2??2?

??/2

??/20

?exp???2

which we can approximate, similar to ?12, as

?22?16

?

00

?exp???2

Here, K(?,??) is the azimuthally integrated kernel

K??,?????

0

?22?4

?

?2

?/2?

2??2???2??d? d?sin??d?sin???,

?/2?

2?

?sin??

?A10?

?2

?2??2?

??

?

K??,???

2??2???2??d? d??.

?A11?

2?

?sin??d???

0

2??1?cos2? d?. ?A12?

We can expand K around ??0 since the particle vectors, on

average, only marginally deviate from the xy plane. Here, ?

is the angle between the two particle vectors u(?) and

u?(?,?),

u??

cos?

0

sin??,

u???

cos?? cos?

cos?? sin?

sin???.

Taking the square of the inner product of the two vectors and

substituting the asymptotic expressions ?up to second order

in ?? yields

cos2???cos? cos?? cos??sin? sin???2

???1?

1

2?2??1?

1

2??2?cos??????2.

Using cylindrical coordinates ???R sin? and ???R cos??

and expanding up to second order in R gives

cos2???1?R2?cos2??2R2cos? sin? cos??O?R4?

?A13?

and

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Page 10

?sin????1?cos2?

??1??1?R2?cos2?

??1?R2cos? sin?cos?

The kernel K now reads29

K?R,????

0

??

0

1??1?R2?cos2??¯?.

?A14?

2??1??1?R2?cos2? d?

2?R2cos? sin? cos?

?1??1?R2?cos2?d??¯ .

?A15?

The second integral is zero, since the integrand is an odd

periodic function ?with period 2??. Likewise, all higher con-

tributions depending on odd powers of cos? are zero. The

first integral in Eq. ?A15? can be rewritten as the complete

elliptic integral of the second kind E(?) via

?

0

2??1??1?R2?cos2? d??4?

0

?/2?1?? sin2? d?

?4E???,

?A16?

where ??1?R2. This integral can be expanded around R

?0 up to second order30

2?ln?

2?ln4?lnR?1

We now have the following expansion for the azimuthally

integrated kernel:

E????1?1

4

R??1

2?R2?

3

16?ln?

4

R??13

12?R4? . . . .

?1?1

2?R2?O?R4?.

?A17?

K??,????4?2?ln4?

1

2ln??2???2??

1

2???2???2?

?O???4ln???

?A18?

valid for small ? and ??. Substituting this into Eq. ?A11?

gives ?in terms of the cylindrical coordinates (?,R)?

?2??2?

00

?exp???2

which can be solved straightforwardly, yielding

?2?1?1

2

?2

?22?32

?

?2

??

2??1?1

2?ln4?lnR?1

2??

2R2?R d? dR,

?A19?

?22?

8

ln?2

?

ln?2&??

1

2?E?1

?2

?

?O??2

?2ln?2?,

?A20?

where ?E?0.5772156649... denotes Euler’s constant. Note

that the ln?2/?2term is the leading order ?2depending term

in this expansion.

The next step is to work out the higher order integrals

contributing to the kernel Eq. ?A15?. It can be shown that

these integrals give additional O(?2

After tedious derivations we obtain the following final ex-

pression for ?22

?1) contributions to ?22.

?22?

8

?2?1?1

2

ln?2

?2

?

ln4?

1

2?E?3/2

?2

??O??2

?2ln?2?

?

8

?2?1?F??2???O??2

?2ln?2?,

?A21?

which now contains all contributions up to order O(?2

?1).

APPENDIX B

Let ?1,0and ?2,0be the solutions of the minimization

Eqs. ?18? and ?19? for the zeroth order problem ?F?H?0?,

?1,0

1/2?2??1/2??1?x??21/2xq12h?Q0??c,

?B1?

?2,0

1/2?25/2??1/2q12?1?x?g?Q0?c.

When F is nonzero and H is given by Eq. ?23? these solu-

tions will only be marginally affected, since H is a small

contribution. Hence, we can write

?1??1

0???1,

?2??2

0???2,

?B2?

where ??1and ??2are the perturbations, such that ??1/?1

and ??2/?2

are small parameters. We can linearize

Q(??2/?1) with respect to these perturbation parameters

?Q0?1???2

?2

and, accordingly,

g?Q???1?Q0??1/2?1?1

2

?1??1/2?1?1

0

0

Q?

?2

?1

0???2

0???1

0???1

?1

0?,

?B3?

Q0

Q0?1?

Q0?1?

??2

?2

0???1

?1

0??,

0??.

?B4?

h?Q???1?Q0

2

1

??2

?2

0???1

?1

Similarly, we get

?1

1/2??1,0

1/2?1???1

2?1

0? and ?2

1/2??2,0

1/2?1???2

2?2

0?.

?B5?

To find explicit expressions for ??1and ??2we must sub-

stitute the above expressions into the minimization equations

?with H nonzero?. All zeroth order terms must cancel out, by

construction, thus leaving an inhomogeneous set of linear

equations in ??1and ??2

x??2??2?1?x??1?Q0??Q0x???1,

?B6?

??2??

Q0

1?Q0???2?Q0??1??2?2,0

where the term depending on H is the inhomogeneous term

ensuring nonzero solutions for ??1and ??2. After solving

the linear set we obtain explicit expressions for ??1and ??2

?not shown here? which can be rewritten as functions of

Q0(x), x and c, using Eq. ?B1?. It can be shown that the

perturbations scale with concentration as ??j?Kj(x)c

?c lnc, where Kj(x) are functions of the mole fraction only.

Hence, the perturbation F leads to an additional c lnc con-

tribution ?up to leading order? to the usual c2-dependence of

?1,0and ?2,0Eq. ?B1?.

1/2cH?x,?2,0?,

7328J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Wensink, Vroege, and Lekkerkerker

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Page 11

The final step is to linearize the expressions ?jand ?jk.

Substituting Eq. ?B2? in Eqs. ?3?, ?5? and expanding up to

first order in ??1and ??2thus yields for the orientational

entropy

?1??1,0???1?ln?1,0?1???1

?1,0,

??1??

?B7?

?2??2,0???2?1

2?ln?2,0?ln2

??2

2?2,0.

?B8?

Similarly, we get for the excluded volume entropy

?11??11,0???11?4??1/2??1,0

and

?1/2?

??1

2?1,0

3/2?,

?B9?

?12??12,0???12

?23/2??1/2??

??

Obviously, the expression for ?22Eq. ?13? remains un-

changed, since F constitutes the direct perturbation. Note

that the leading order terms in Eqs. ?B7?, ?B8? and ?B9?,

?B10? correspond to the expressions for ?jand ?jkgiven in

Sec. IIA. It is now straightforward to reexpress the free en-

ergy Eq. ?16? of the nematic phase by simply adding pertur-

bation terms depending on ??jand ??jk. Hence, we obtain

the following perturbative contributions ?denoted by ?? to the

osmotic pressure and chemical potentials of the nematic

phase which must be added to the corresponding expressions

given in Sec. IIA:

?? ˜n??c?Q0?1?x???1

1

?1,0?

1

?2,0?

1/2

h?Q0???1

2?1,0

3/2

?g?Q0???2

2?2,0

3/2 ??.

?B10?

?2,0?1

2x??2

?2,0?

?1

2?1?x??Q0

2cxW?x,c,Q0?,

?B11?

?? ˜1,n?

x

2??1

?2,0???2

?2,0??Q0

??1

?2,0,

?B12?

?? ˜2,n??1

2Q0

2??1

?2,0?W?x,c,Q0?,

?B13?

with

W?x,c,Q0??

1

4?c

xq22

2?1?x?2?1?Q0??ln?2,0?2K?.

q12

?B14?

Note that the terms depending on W are the direct perturba-

tions ?arising from ??22?.

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?0

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2?d?R4cos2?/?1?(1?R2)cos2?

and

?0

2?d?R4cos2?/?1?(1

7329 J. Chem. Phys., Vol. 115, No. 15, 15 October 2001Phase separation in rod-plate mixtures

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