Depletion effects in smectic phases of hard rod--hard sphere mixtures
Depletion effects in smectic phases of hard rod–hard sphere mixtures
Yuri Mart́ınez-Ratón∗
Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas,
Escuela Politécnica Superior, Universidad Carlos III de Madrid,
Avenida de la Universidad 30, E-28911 Leganés, Madrid, Spain.
Giorgio Cinacchi†
Dipartimento di Chimica, Università di Pisa Via Risorgimento 35, I–56126, Pisa, ITALY
Enrique Velasco‡
Departamento de F́ısica Teórica de la Materia Condensada and Instituto de Ciencia
de Materiales Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain.
Luis Mederos§
Instituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cient́ıficas, E-28049 Cantoblanco, Madrid, Spain.
(Dated: November 3, 2018)
It is known that when hard spheres are added to a pure system of hard rods the stability of
the smectic phase may be greatly enhanced, and that this effect can be rationalised in terms of
depletion forces. In the present paper we first study the effect of orientational order on depletion
forces in this particular binary system, comparing our results with those obtained adopting the usual
approximation of considering the rods parallel and their orientations frozen. We consider mixtures
with rods of different aspect ratios and spheres of different diameters, and we treat them within
Onsager theory. Our results indicate that depletion effects, and consequently smectic stability,
decrease significantly as a result of orientational disorder in the smectic phase when compared with
corresponding data based on the frozen–orientation approximation. These results are discussed in
terms of the τ parameter, which has been proposed as a convenient measure of depletion strength.
We present closed expressions for τ , and show that it is intimately connected with the depletion
potential. We then analyse the effect of particle geometry by comparing results pertaining to
systems of parallel rods of different shapes (spherocylinders, cylinders and parallelepipeds). We
finally provide results based on the Zwanzig approximation of a Fundamental–Measure density–
functional theory applied to mixtures of parallelepipeds and cubes of different sizes. In this case, we
show that the τ parameter exhibits a linear asymptotic behaviour in the limit of large values of the
hard–rod aspect ratio, in conformity with Onsager theory, as well as in the limit of large values of
the ratio of rod breadth to cube side length, d, in contrast to Onsager approximation, which predicts
τ ∼ d3. Based on both this result and the Percus–Yevick approximation for the direct correlation
function for a hard sphere binary mixture in the same limit of infinite asymmetry, we speculate that,
for spherocylinders and spheres, the τ parameter should be of order unity as d tends to infinity.
I. INTRODUCTION
In recent years experimental mixtures that closely re-
semble a hard rod–hard sphere system have been studied;
typically the rods are represented by tobacco mosaic or
fd–virus particles, while the spheres are represented by
polystyrene latex particles or globular proteins [1, 2, 3].
The phase diagrams found include the isotropic phase, as
well as phases with liquid-crystalline symmetry. Among
these, nematic (N) and smectic (Sm) phases have been
found. In addition, bulk demixing transitions, as well
as microsegregated phases of various symmetries, have
been observed[2]. Intensive theoretical effort has been
∗Electronic address: yuri@math.uc3m.es
†Electronic address: g.cinacchi@sns.it
‡Electronic address: enrique.velasco@uam.es
§Electronic address: l.mederos@icmm.csic.es
devoted to the understanding of these systems. A useful
concept in this effort is that of a depletion force. Struc-
tural and thermodynamic stability of systems consisting
of hard particles can be understood solely in terms of
entropic effects which, in mixtures, can be reinterpreted
as an attractive depletion force. Some studies have been
done recently in an attempt to quantify attractive deple-
tion interactions between solute particles, of anisotropic
shape, mediated by solvent particles that can be isotropic
or possess themselves liquid-crystalline order [4, 5, 6].
One of the microsegregated phases that has been ob-
served is the lamellar phase. This phase, having smectic
symmetry, consists of alternate pure layers of rods and
spheres. The lamellar phase can be greatly stabilised
with respect to the corresponding smectic phase in the
pure–rod fluid, as predicted in Ref. [7], and shown in
Refs. [8, 9, 10, 11, 12, 13]. In the lamellar phase, deple-
tion forces result from the fact that it is entropically more
favourable for spheres to occupy the interstitial space,
which creates a large effectively attractive force between
http://arxiv.org/abs/0704.1140v1
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mailto:g.cinacchi@sns.it
mailto:enrique.velasco@uam.es
mailto:l.mederos@icmm.csic.es
adjacent layers of rods; enhanced smectic–phase stability
ensues.
Thus far, theoretical studies of the lamellar phase have
mostly used, among others, the following approxima-
tions: parallel rodlike particles with frozen orientational
order, specific particle geometry, Onsager theory [14] and
neglect of particle flexibility. Cinacchi et al.[15] stud-
ied the hard-spherocylinder (HSPC)–hard-sphere (HS)
mixture using Onsager theory and free particle orienta-
tions, but presented phase diagrams including isotropic,
nematic and the lamellar phase, without specifically ad-
dressing any stability issue. In this paper we lift stepwise
the first three of these restrictions and assess their impact
on the enhancement of smectic stability in the mixture
with respect to the pure rod system.
Density–functional theory is a convenient theoretical
tool to study the structure and phase behaviour of soft–
condensed matter. Onsager theory is a most success-
ful version of density–functional theory, intended for
bodies interacting through hard potentials. Some of
the shortcomings of Onsager theory are improved by
Fundamental–Measure theory [16] (FMT), which intro-
duces the fundamental particle measures in the theory
and should therefore be quite general. Our theoretical
analysis is based on both these density–functional ap-
proximations. Specifically, we first use the HSPC model
within Onsager theory to include the orientational de-
grees of freedom of the rods by means of an orientational
distribution function and the associated order parame-
ter. The results will be discussed in terms of a parame-
ter, τ , which is directly related to the depletion strength
in smectic phases [10]. Then, we discuss the effect of
particle geometry by considering various shapes for the
hard particles in the frozen-orientation approximation.
Finally, we consider a FMT approximation for mixtures
of hard parallelepipeds (HPAR) and cubes (HC), and
analyse it in the context of the Zwanzig approximation
[17]; the latter consists of considering particle orienta-
tions to be restricted to three mutually perpendicular
axes. Recent advances on the application of FMT to
freely rotating particles have focused on anisotropic par-
ticles with infinitely narrow breadths (needles)[18]. Since
one of our aims is to investigate how the depletion mech-
anism changes as the breadth ratio of the particles is
varied, these recent developments are not appropriate in
our context, and the use of the Zwanzig approximation
seems to be justified for lack of a better approach. FMT
provides a different theoretical viewpoint with respect to
which the predictions of Onsager theory can be assessed.
In fact, the main purpose of this part of the work is to
assess the impact of a proper inclusion in the theory of
pair correlations on the depletion mechanism in smectic
phases. In the framework of FMT, we additionally study
in detail the asymptotic limit of τ with respect to dif-
ferent parameters relating to the aspect ratio of the rod
and the relative sizes of HPARs and HCs. With a view
to completing this study, and based on some expressions
used to calculate the τ parameter, which we evaluate for
hard spheres, we speculate that, for HSPC–HS mixtures,
τ ∼ 1 as the ratio of rod breadth to sphere diameter
tends to infinity.
In the following section the general procedure used to
locate the nematic–smectic spinodal line is illustrated;
for the sake of clarity, this is illustrated in the context
of Onsager theory for a binary mixture of hard rods and
HS’s. In Section III the τ parameter, which is used to
quantify the stability of the smectic phase when HS’s
are added, is defined, and a closed expression, valid in
the context of Onsager theory and parallel rods, is pre-
sented. The results are presented in Section IV, which
has three subsections. The first is devoted to the effect
of free particle orientations, the second addresses the ef-
fect of particle shape and the third contains the results
for mixtures of HPAR’s and HC’s, analysed by means of
FMT in the Zwanzig approximation. This last subsection
is in turn divided into two parts. The first is devoted to
the results obtained by applying Onsager theory to the
Zwanzig HPAR–HC system, while in the second we de-
scribe the results from FMT, together with a detailed
discussion on the asymptotic limits of the τ parameter.
The conclusions are presented in Section V. The Ap-
pendices collect expressions for the Fourier transforms
of the overlap functions for the various particle geome-
tries explored in the paper, together with a few details
on the numerical minimisation of the functional in the
context of the frozen–orientation approximation. An ex-
act expression for τ in terms of the correlation functions,
valid for a general mixture of freely rotating particles,
is also included. Explicit expressions for these functions
in the Zwanzig–Onsager and Zwanzig–FMT approaches
for HPAR–HC and HS binary mixtures are provided.
Finally, the asymptotic behaviour of the τ parameter,
within the framework of FMT, and for the HPAR–HC
mixture, is described.
II. ONSAGER THEORY FOR HARD
ROD–HARD SPHERE MIXTURES AND
STABILITY ANALYSIS
The first theory we employ to describe hard rod–hard
sphere mixtures is Onsager theory, first proposed by
Onsager[14] for a pure system of hard rods and exten-
sively used in theoretical approaches to orientational or-
dering in hard–rod fluids. It can be regarded as a trun-
cated virial expansion, up to second order in density, of
the excess part of the free energy. Thus, it contains the
orientational–dependent second–virial coefficient exactly.
It is a density–functional theory, since the free energy de-
pends functionally on the orientational distribution func-
tion. The extension of Onsager theory to mixtures is
straightforward; the reader is referred to Ref.[15] for a
detailed account on its implementation for the present
model fluids.
We consider a two-component mixture of hard rods
and hard spheres labelled with 1 and 2, respectively. The
density functional in the Onsager approximation, assum-
ing uniaxial smectic symmetry and taking z as the coor-
dinate along the smectic layer normal, is written as
ρi(z)
ρi(z)
− ρ1(z)Sr(Q)
i,j=1
dz′ρi(z)ρj(z
′)fij(z − z
, (1)
where Φ ≡ βF/V is the Helmholtz free-energy density
per unit thermal energy, ds the smectic layer spacing,
Sr(Q) is the orientational-entropy density of the hard
rods, Q their orientational order parameter, ρ1(z) and
ρ2(z) are local densities for the two components, and
fij(z) are angular averages of the overlap functions be-
tween species i and j, already integrated in the xy plane.
The procedure used to calculate these averages has been
described in detail elsewhere for the case of mixtures of
spherocylinders [15]. In the case where one adopts the
popular approximation of considering that particles pos-
sess perfect orientational order (Q = 1), an approxima-
tion used in all previous theoretical analyses of the hard-
rod–hard-sphere mixture [7, 8, 9, 10, 13], the functions
fij(z) can usually be written exactly, depending on the
particle geometry. Note that f11 and f12 both depend on
Q, but certainly not f22. In the nematic phase, ρi(z) is
a constant.
Since we are interested in searching for the nematic-
smectic spinodal, we consider the following perturbations
on the constant nematic densities:
ρi(z) = ρxi (1 + λi cos kz) , i = 1, 2 (2)
where ρ is the total mean density, x2 ≡ x, x1 = 1−x are
the molar fractions of the two components, and k is the
smectic wave number. Note that λi can be positive or
negative; in the case that λ1λ2 < 0, smectic layers rich
in one of the components will alternately appear, giving
rise to a microsegregated smectic phase. The free energy
difference between smectic and nematic states, to second
order in the λi’s, is:
i + ρ
i,j=1
xixjλiλj f̃ij(k;Q)
where f̃ij(k) are the following cosine Fourier transforms:
f̃ij(k) =
dtfij(t) cos kt, i, j = 1, 2. (4)
The Hessian matrix, ∂2(β∆F/N)/∂λi∂λj , necessary to
study the stability of the nematic phase against smectic
fluctuations, is
1 + ρx1f̃11(k;Q)
] ρx1x2
f̃12(k;Q)
ρx1x2
f̃12(k;Q)
1 + ρx2f̃22(k)
The spinodal line is then obtained by solving the equa-
tions
∆(k0, ρ0, Q0) ≡
1 + ρ0x1f̃11(k0;Q0)
1 + ρ0x2f̃22(k0)
−ρ20x1x2
f̃12(k0;Q0)
ρ0,k0,Q0
ρ0,k0,Q0
= 0 (6)
for ρ0, k0 and Q0, the values of the density, wave-vector
and order parameter of the unstable nematic at the spin-
odal, respectively; note that these quantities will depend
on the value of the composition x. The second equation
ensures that instability will occur at some particular k
vector for the first time, as the density is decreased from
above down to the value at the spinodal. For the pure
case (x = 0) the instability equations reduce to
1 + ρ0f̃11(k0;Q0) = 0, (7)
∂f̃11
ρ0,k0,Q0
ρ0,k0,Q0
= 0. (8)
These are the equations defining a spinodal to an ordered
phase in a pure system: 1 − ρ0c̃(k0) = 0, where c̃(k) is
the Fourier transform of the direct correlation function
(in Onsager theory c̃(k) = −f̃(k)).
III. THE τ PARAMETER
The stability of the nematic phase against smectic-type
fluctuations can be quantified in different ways. In this
paper we use the parameter τ proposed by Dogic et al.[10]
in their analysis of the hard-rod-hard-sphere mixture:
τ = lim
. (9)
In this expression η is the total packing fraction of the
mixture, η2 the partial packing fraction of spheres, and
the derivative is evaluated at the nematic-smectic spin-
odal line. Other definitions are possible; for example, one
could use the spinodal line p(η2), with p the pressure, in-
stead of η(η2). Here we adhere to the τ parameter as a
measure of smectic stability. It turns out, as shown be-
low, that the τ parameter can be directly related to the
depletion potential, so that τ contains basic information
on depletion forces in the system.
We now obtain a compact expression for τ in the frame-
work of Onsager theory, considering the case of perfect
orientational order (Q = 1). The generalization to free
orientational order and a general density functional is
given in the Appendix C. Our strategy consists of ob-
taining the nematic–smectic spinodal line η = η(η2) per-
turbatively in the packing fraction η2 (i.e. for small val-
ues of η2), and then extracting the value of τ from the
first-order term. We begin by first expanding Eqns. (6)
at small x. We assume the expansions
ρ = ρ0 + ρ
0x+ ..., k = k0 + k
0x+ ..., (10)
where now ρ0 and k0 are the density and wave-vector, at
the x = 0 spinodal, and ρ′0 and k
0 are the derivatives of ρ
and k with respect to the molar fraction x, at the x = 0
spinodal, respectively. Inserting these equations into the
first two of Eqns. (6) we obtain, at zeroth order in x:
1 + ρ0f̃11(k0) = 0, f̃
11(k0) = 0. (11)
To first order in x we get
= 1− ρ20f̃
12(k0), k
′ = −2
f̃12(k0)f̃
12(k0)
f̃11(k0)f̃
11(k0)
. (12)
Now, differentiating the equations η = η1 + η2 = ρ(1 −
x)v1 + η2 and η2 = ρxv2, and taking the limit x→ 0, we
obtain the relation
τ = lim
where v1, v2 are the volumes of the two particle species,
rods and spheres respectively. This equation, along with
Eqn. (11), gives η0 = −v1/f̃11(k0), and
τ = 1− η20
f̃212(k0)
f̃12(k0)
f̃11(k0)
. (14)
In fact, this expression remains valid for binary mixtures
composed of any convex bodies, but can only be used in
the approximation of frozen particle orientations (Q = 1)
and in the framework of the Onsager approximation. The
τ parameter is calculated by solving Eqns. (11) and then
evaluating Eqn. (14).
We now proceed to show the direct relation existing
between the parameter τ and the depletion interaction
between two parallel rods mediated by the solvent parti-
cles (hard spheres), at least in the limit where the density
of the solvent particles ρ2 and their diameter, relative to
the breadth of the rods, are very small. These are the
conditions under which the Asakura–Oosawa approxima-
tion [19] is valid, and the depletion potential becomes
Vdep(r, ρ2) = −ρ2 [f12 ∗ f12] (r) (the asterisk standing for
convolution). Now
∂Ṽdep(k0)
= −f̃212(k0), (15)
and Eqn. (14) gives
τ = 1 +
∂Ṽdep
f̃211(k0)
(this equation can also be obtained from a mean–field
perturbative treatment of the fluid, by considering par-
allel hard rods that interact via a depletion potential in
mean–field approximation). Eqn. (16) relates the τ co-
efficient with the depletion potential, and shows that τ
contains basic information on depletion forces. In turn,
it provides a condition under which τ may be negative,
and thus directly links the depletion potential to the en-
hancement of smectic–phase stability in fluid mixtures of
hard rods and spheres.
IV. RESULTS
A. Mixtures of HSPC and HS: frozen versus free
orientations
All previous theoretical analyses of the lamellar phase
in the hard-rod–hard-sphere mixtures [7, 10, 13] have
relied on the approximation of considering perfect ori-
entational order. In this section we assess the impact
of this severe approximation on the formation of smec-
tic phases in the HSPC/HS mixture. We have investi-
gated a number of HSPC/HS mixtures using the orig-
inal Onsager theory, changing the HSPC aspect ratio
κ1 ≡ (L1 + D1)/D1 and the HSPC breadth relative to
the HS diameter, d ≡ D1/D2. Fig. 1 shows the spin-
odal line η–η2 of the nematic–smectic transition, for the
cases of frozen and free orientations, and for κ1 = 8 and
various values of the scaled HS diameter. We note that,
in all cases shown, the smectic phase is stabilised with
respect to the nematic phase (negative slope at η2 = 0)
on adding HS to the pure HSPC fluid; this is always the
case when d ≥ 1 (spheres smaller than the cylinder di-
ameter). The effect is amplified when both the HSPC
aspect ratio length (κ1) is increased and the sphere di-
ameter is decreased (d increases). However, the effect is
less pronounced when the HSPC are free to orient their
main axes. The τ parameter, shown in Fig. (2), reflects
this behaviour, which points to a less strong depletion
effect due to orientational fluctuations.
B. Other hard rod–hard sphere mixtures: effect of
particle shape
To investigate the effect of the shape of the rods on the
depletion effect and, in turn, on the stability of the smec-
tic phase, Vesely[13] has recently considered parallel-
rod models consisting of linear overlapping hard-sphere
chains, hard ellipsoids and hard spheroellipsoids. Here
we have studied, still in the approximation of frozen ori-
entations, two additional types of hard particles: cylin-
ders (HCYL) and parallelepipeds with a square trans-
verse shape (HPAR). To minimize the effect of the dif-
ferent particle geometries, the particles were chosen to
have the same volume (see later). In all cases the minor-
ity component continues to consist of hard spheres. The
functions fij(z), and also their Fourier transforms, can
be calculated analytically in these cases. The relevant
expressions can be found in the Appendix A.
0 0.005 0.010 0.015 0.020
d -1=1/3
0 0.005 0.010 0.015 0.020
d -1=1/3
FIG. 1: Nematic–smectic spinodal line η(η2), according to
Onsager theory, for the case κ1 = 8, and d
−1 = 1, 1/2 and 1/3
(indicated as labels). (a) Frozen–orientation approximation;
(b) free orientations. Note that horizontal and vertical scales
in both graphs are the same.
Results for the τ parameter in the case of HCYL/HS
and HPAR/HS mixtures, along with the previously con-
sidered HSPC/HS mixtures, are shown in Figs. 3(a)-(c).
The τ parameter is plotted against the aspect ratio of the
rods κ1 in Fig. 3(a) for different values of the scaled in-
verse HS diameter d = D1/D2 (D1 being the side length
of the parallelepiped in the case of HPAR); Fig. 3(b) is
a zoom of (a) in the region of small κ1. Finally, the τ
parameter against d for different values κ1 is plotted in
Fig. 3(c). We can see from Fig. 3(a) that τ tends to
decrease linearly, becoming negative for sufficiently large
aspect ratios. This effect is more and more pronounced
as the diameter of the HS is decreased. An interesting
feature is that, when the diameter of HS is greater than
that of the rod, depletion is larger, and therefore smec-
tic stability is enhanced more substantially, as one goes
0 1 2 3 4 5 6
κ 1=21
FIG. 2: Values of τ as a function of inverse HS diameter ratio
d, for various values of HSPC length–to–breadth ratio κ1.
Lines: Frozen–orientation approximation (Q = 0); from top
to bottom: κ1 = 6, 8, 11 and 21. Symbols: free orientations.
Squares: κ1 = 5; open circles: κ1 = 8; filled circles: κ1 = 11,
and triangles: κ1 = 21.
from HPAR to HCYL or HSPC (note that these two are
very similar). The opposite behaviour (i.e. depletion in
HPAR is enhanced with respect to HCYL and HSPC)
results when the HS diameter is equal or less than that
of the HSPC [Fig. 3(a)]. For short rods τ exhibits oscil-
latory behaviour [see Fig. 3(b)]. Finally, fixing κ1 and
increasing d, we find that the τ parameter decreases as
a cubic power-law in d and that its value for HPAR is
much less than that for HSPC or HCYL.
The linear and cubic dependences of τ with respect to
κ1 and d, respectively, can be elucidated from Eqn. (14)
using the fact that the volume ratio can be expressed as
v1/v2 = κ1d
3 for the HCYL case (the other cases being
similar), while the squared ratio of Fourier transforms of
the overlap functions tends asymptotically, for large κ1
and d, to a constant. Simple calculations give, in this
limit,
τHC = τHSPC = −
3 (17)
with k∗0 = k0L1 the reduced wave number of the N–Sm
spinodal instability in the one–component system.
To study the effect of a varying particle shape on the
phase behaviour of the rod-sphere mixtures, we have per-
formed a full minimization of the Helmholtz free-energy
functional with respect to smectic-like density profiles of
the rods [ρ1(z)] and spheres [ρ2(z)]. The minimization
was carried out with respect to the Fourier amplitudes
of the truncated Fourier expansion of the density pro-
files and smectic wave number (details are provided in
Appendix B). The choice of a Fourier expansion is justi-
fied by the fact that the HS equilibrium density profiles
0 10 20 30 40 50 60
2 3 4 5 6 7 8
0 2 4 6 8 10
κ1=10
FIG. 3: (a) The τ parameter vs. κ1 for different values of
the parameter d (indicated in the graph). (b) A zoom of
(a) for small κ1’s. (c) τ vs. d for different values of κ1 as
indicated in the figure. (c) τ parameter as a function of d for
different values of κ1, as indicated. Results for HSPC, HCYL,
and HPAR are plotted with solid, dashed, and dotted lines,
respectively.
obtained with this minimization differ very significantly
from those obtained via the usual single–amplitude ex-
ponential parametrization, as shown in Fig. 4. We can
see that the parametrization produces too sharp density
peaks in the HS profile, which enhances the depletion
effect.
0 0.2 0.4 0.6 0.8 1
FIG. 4: Equilibrium density profiles of the equimolar HSPC–
HS mixture (different components are labelled in the figure)
resulting from minimization with respect to the Fourier am-
plitudes (solid lines) and with respect to a single–amplitude
exponential parametrization (dashed lines). The HSPC has
an aspect ratio κ1 = 10 and its breadth coincides with the HS
diameter, taken as unity. The pressure in reduced units was
chosen to be 0.4.
Since we would like to compare the results from the
density–functional minimization of mixtures composed of
rods of different geometries (HSPC, HCYL and HPAR),
a sensible criterion is required to choose the sizes of the
particles so as to make sure that these differences arise
from the shape and not from different sizes and volumes.
For this purpose we have used three different criteria:
(i) all particles have the same diameter and the same
length, (ii) all particles have the same length and the
same volume, and (iii) all particles have the same aspect
ratio and the same volume. Comparing the equilibrium
density profiles resulting from use of these three criteria,
we have concluded that differences between particles are
minimised using the third criterion (these results are not
shown here).
We have calculated, using the free, i.e. Fourier–based,
minimization, the phase diagrams of binary mixtures of
rods (HSPC, HCYL and HPAR) and HS. The HSPC and
HS diameters were taken to be equal and adopted as unit
of length. The aspect ratios of all the rods were set to
10, all particle volumes being the same. The following
relations result:
DHC =
1− (3κHSPC)
]−1/3
DHSPC,
DHP =
DHC, Lµ = κHSPCDµ, (18)
with µ=HSPC, HC or HP. The resulting phase diagrams
are shown in Fig. 5. The first interesting feature of
all phase diagrams is the presence of a tricritical point,
where the nature of the N–Sm transition changes from
0 0.2 0.4 0.6 0.8 1
HCYLHPAR
FIG. 5: Phase diagrams in the reduced pressure–composition
plane of three different mixtures composed of rods of different
geometries (HSPC, HCYL, and HPAR) with HS. Different
lines corresponding to each mixture are labelled in the figure.
Also indicated are the stability regions of the nematic and the
smectic phases. Solid lines represent the binodals of the N–
Sm coexistence, while the continuous N–Sm transitions have
been indicated by a dashed line.
second to first order[20]. For pressures higher than that
of the tricritical point, the demixing instability region
becomes wider. The width of this demixing window in-
creases for mixtures going from HSPC to HCYL and
then HPAR. This scenario clearly demonstrates that in-
creasing differences in shape between spheres and rods
enhance the depletion in smectics, resulting in a wider
demixing gap.
C. Zwanzig model: Onsager theory versus FMT
In order to check whether the effects presented above
are robust with respect to the theory and approxima-
tions used, we have analysed a mixture of HPAR and HC
using Fundamental–Measure theory in the Zwanzig ap-
proximation. Zwanzig [21] introduced a model that con-
siderably reduces the complexities associated with min-
imising a density-functional that depends on particle ori-
entations. The approximation consists of restricting the
possible orientations of a rod to lie along the three or-
thogonal axes x, y and z, treating a pure system as a
three-component mixture (one component for each orien-
tation). The approximation is very useful and has been
used quite substantially. In our FMT context, the re-
sulting theory treats spatial correlations very accurately
so that it should better describe the onset of smectic or-
der in the nematic fluid. We have used the HPAR-HC
system and the Zwanzig approximation because in this
combination a FMT can be rigorously formulated [17].
The Zwanzig approximation should not be terribly lim-
iting, since the nematic fluid is well oriented. In order to
elucidate the effect of restricted orientations separately
from that inherent to the proper inclusion of higher–order
correlations, a study of the HPAR-HC mixture using a
Zwanzig–Onsager theory, together with that on mixtures
of parallel HPAR and HC, is necessary.
1. Zwanzig model in the Onsager formulation
In Appendix D we have written expressions for the
Fourier transforms of the correlations functions c̃ij(k) as
obtained from the Zwanzig–Onsager approach. These
expressions are necessary to calculate the τ parameter
using Eqn. (C.10). The results obtained are plotted in
Fig. 6. In the same figure the parallel case is also plotted
for the sake of comparison. The usual cubic power–law
behaviour of τ with respect to d is obtained; this is typical
of any Onsager approach, as discussed in Sec. IVB. The
Zwanzig approach gives a lower value for τ as compared
to the parallel case. This result is similar to that already
obtained using the freely rotating Onsager approach for
HSPC. For high values of κ1, both approaches collapse
into a single line, which is due to the high value of the
order parameter Q of a fluid composed of long particles.
0 2 4 6 8 10
FIG. 6: τ parameter as a function of d for three different
values of κ1, as indicated. Solid and dashed lines correspond
to the Zwanzig and parallel approaches, respectively.
2. Zwanzig model in the FMT formulation
Details on this approximation can be found in Ref.
[17]. Here we only give a brief sketch of the theory. We
continue to use the notation introduced in the previous
section for the dimensions of the particles, i.e. the length
of the HPAR is L1, the side length is D1 (these particles
are assumed to have a square section), and D2 is the side
length of the cubes. Since the unit vector Ω̂ only has
three possible orientations, the one–particle distribution
functions ρs(r, Ω̂) can be expressed as
ρs(r, Ω̂) =
ρsµ(r)δ(Ω̂ − êµ) (19)
where êµ, µ = 1, 2, 3, are unit vectors along the three
perpendicular directions xyz, respectively, and ρsµ(r) are
the local density of species s parallel to the µ-axis. The
excess part of the free-energy density, in reduced thermal
units, is obtained in [17], and has the form
Φex(r; {ρs}) = −n0 ln(1− n3) +
n1 · n2
1− n3
n2xn2yn2z
(1− n3)2
with the functions {nα} (α = {0, 1x, 1y, 1z, 2x, 2y, 2z, 3})
being weighted densities obtained as
nα(r) =
dr′ρsµ(r
′)ω(α)sµ (r− r
′), (21)
where ω
sµ are characteristic functions whose spatial in-
tegrals give the fundamental measures of the particles
(edge length, surface and volume). The ideal part of
the free energy density in reduced thermal units for this
model is
Φid(z) =
ρsµ(z) [log ρsµ(z)− 1] , (22)
so that the total free energy per unit volume and unit
thermal energy can be calculated as
dz [Φid(z) + Φexc(z)] . (23)
Density and order–parameter profiles can be defined; in
particular, the density profile is
ρs(z) =
ρsµ(z). (24)
The nematic–smectic spinodal line can be obtained in
this theory using the same kind of arguments used for
the Onsager theory (Sec. II). Note, however, that the
expression (14) for the τ parameter is not valid in this
context, and we need to use the more general formula
(C.10), valid for any mixture of freely rotating particles.
In the Appendix C we describe in detail how this formula
is obtained. An alternative way to calculate this param-
eter is to implement a numerical differentiation scheme
on the spinodal line η(η2).
The nematic–smectic spinodal lines for the case κ1 = 8
and various values of the scaled cube side length d−1 =
D2/D1 are shown in Fig. 7, whereas the corresponding τ
parameters for different values of κ1 are shown in Fig. 8.
As was the case in the mixtures analysed using Onsager
theory, τ decreases as the size of the cubes is diminished,
0 0.005 0.01 0.015 0.02 0.025
d -1=0.8
FIG. 7: Nematic–smectic spinodal line η–η2, as obtained from
Fundamental–Measure theory for the HPAR-HC mixture, for
κ1 = 8 and various values of d
−1, as indicated in the graph.
indicating a stronger depletion effect and a corresponding
enhancement of the smectic–phase stability. An interest-
ing feature of these results is that they appear to exhibit
linear behaviour for large values of d (i.e. as the size of
the cubes becomes smaller). Fig. 8 shows this by means
of least–square fits to linear functions. The coefficients of
the fit are functions of the aspect ratio κ1, and, in turn,
appear to be linear in κ1, as shown in Fig. 9.
0 2 4 6 8 10
τ κ1=6
FIG. 8: τ parameter as a function of d for various values of
the parallelepiped length–to–breadth κ1 = L1/D1 (indicated
in the graph). The data are indicated by symbols; linear least-
square fits τ = α(κ1) + β(κ1)d are included as lines for each
value of κ1.
To explain the linear behaviour obtained for the τ pa-
rameter as a function of d, we have analytically calcu-
lated the asymptotic behaviour of τ in the régime d≫ 1
for a mixture of parallel parallelepipeds and cubes in the
6 8 10 12 14 16 18 20
FIG. 9: Symbols: coefficients α (circles) and β (squares) as a
function of κ1. Solid lines: linear fits α(κ1) = α1 + α2κ1 and
β(κ1) = β1 + β2κ1, with β2 = −0.038.
FMT formulation. In Fig. 10 we compare the τ param-
eter, for various values of κ1, obtained from the Zwanzig
and parallel FMT approaches. An interesting feature is
that depletion is enhanced in the Zwanzig approach, a
result opposite to that from the Onsager approach (Fig.
0 2 4 6 8 10
κ1 =6
FIG. 10: The τ parameter as a function of d calculated from
the Zwanzig (solid line) and parallel (dashed line) FMT ap-
proaches. Values of κ1 are indicated as labels.
Details on the calculation of the asymptotic behaviour
of τ with respect to d have been relegated to Appendix
F. From the general expression of τ (Eqn. C.10), partic-
ularized for the parallel case (Q = 1, which makes the
third terms in the numerator and denominator to van-
ish), the linear behaviour can be explained if the sum of
the two terms in the numerator are of order d−2. Since
the volume ratio is v1/v2 = κ1d
3, we obtain the asymp-
totic behaviour τ ∼ κ1d = L1/D2. This result can also
be obtained using the Zwanzig model (Figs. 8 and 9),
which shows that the third term of the numerator in Eqn.
(C.10) is also of order ∼ d−2. The coefficient β(κ1) of the
linear fits of τ with respect to κ1 (Fig. 9) depends lin-
early on κ1 and has a slope β2 = −0.038 (see caption of
Fig. 9), which is in the range of values predicted by the
asymptotic limit ofW (κ1) ≡ τ/(κ1d) as a function of κ1,
shown in Fig. (11).
The function (∂∆/∂ρ2)
[Eqn. (F.1)] is of order d−2
only within the FMT formulation, the Onsager approach
giving a term ∼ d0. This, in turn, shows the importance
of properly taking account of pair correlations between
particles in order to adequately describe depletion inter-
actions. The asymptotic limit coincides with the adhe-
sive limit because, when the mixture is highly asymmet-
ric, the attractive depletion potential between two big
particles becomes narrower and deeper, tending in the
limit to a Dirac delta function at contact. This result
is confirmed by our calculations in Appendix F, which
show that the inverse Fourier transform of one of the
terms of the function (∂∆/∂ρ2)
is a Dirac delta func-
tion at contact. We have repeated the same analysis
in Appendix F for the case of a HS binary mixture in
the FMT formulation (PY approach). We have obtained
that (∂∆/∂ρ2)
∼ O(d−3). Also, a Dirac delta func-
tion at contact is present in the inverse Fourier trans-
form of (∂∆/∂ρ2)
. The difference in the limit of in-
finite size asymmetry between the HPAR-HC and HS
cases is due to the difference in particle shapes: when
two big particles are closer enough the parallel sides of
two parallelepipeds exclude many more small particles
from their interstitial space than the curved surface of
two big hard spheres. The discussion above forces us to
speculate about the possible asymptotic behaviour of the
τ parameter for the HSPC-HS mixture in the limit d≫ 1:
since v1/v2 ∼ d
3, τ ∼ O(d0) [Eqns. (C.10) and (F.2)];
this is a consequence of the caps of the spherocylinders
being spherical. This argument does not apply to the
behaviour of τ in the limit of large κ1 for fixed d, which
continues to be linear.
V. CONCLUSIONS AND PERSPECTIVES
The first result from this study on depletion effects in
smectic phases is that, within Onsager theory, the in-
clusion of orientational degrees of freedom significantly
reduces the enhanced stability of the smectic phase over
the nematic phase as compared to that obtained from
the frozen-orientation approximation. A second conclu-
sion that can be drawn is that when the surface curvature
of the rods is very different from that of the other species
(e.g. the case of HPAR and HS), depletion is enhanced
and the smectic phase stabilizes at lower packing frac-
tions as HS become more abundant. This effect is also
reflected in the fact that the N–Sm demixing gap in the
phase diagram widens as one goes from HSPC to HP
particles through the HC geometry. Finally, the inclu-
sion of more realistic pair correlations between particles,
in the manner of FMT theory, changes the asymptotic
behaviour of the τ parameter (a convenient measure of
depletion in smectics) for large values of the ratio d, i.e.
the asymmetry between the breadth of the rod and the
sphere diameter; in particular, for HPAR, τ increases lin-
early with d, while for HSPC we speculate that τ goes
to a finite limit. The limiting case of large aspect ratios
and finite d is similarly captured by both Onsager and
Fundamental–Measure theories.
In this work we have studied the depletion mecha-
nism along the N–Sm spinodal line for different mixtures
by explicit calculation of the τ parameter evaluated on
this line in the pure fluid. An interesting task could
be to extend this study to equilibrium smectic phases,
consisting of well-developed density peaks, by defining
some quantity evaluated at the smectic density profile.
In this case depletion-based mechanisms may add inter-
esting phenomenology, such as strong microsegregation
between different species, which could be directly quan-
tified using some suitably defined parameter. Work along
this direction is currently in progress.
Acknowledgments
We are grateful to the referee for his/her comments,
suggestions and careful reading of the manuscript. We
thank MIUR (Italy) and Ministerio de Educación y
Ciencia (Spain) for financial support under the 2005
binational integrated program. We gratefully ac-
knowledge financial support from Ministerio de Edu-
cación y Ciencia under grants Nos. FIS2005-05243-
C02-01, FIS2005-05243-C02-02, FIS2004-05035-C03-02,
BFM2003-0180 and from Comunidad Autónoma de
Madrid (S-0505/ESP-0299).
YMR was supported by a Ramón y Cajal research con-
tract from the Ministerio de Educación y Ciencia.
VI. APPENDIX
Appendix A. Overlap functions: case of frozen
orientations
When particle orientations are parallel, the functions
fij(z) and their Fourier transforms can be written exactly
for a large class of particle shapes and their mixtures. We
consider in turn each of the particle geometries analysed
in the paper.
In the case of mixtures of HSPC of length L1 and
breadth D1, and HS of diameter D2, the Fourier trans-
forms are
f̃11(k) =
{sin [k(D1 + L1)]
− D1k cos [k(D1 + L1)]− sin (kL1)} ,
f̃12(k) =
{sin [(D12 + L1)k/2]
D12 cos [(D12 + L1)k/2]− sin (kL1/2)
f̃22(k) =
[sin (kD2)− kD2 cos (kD2)] , (A.1)
where D12 = D1 +D2.
For mixtures of HCYL of length L1 and breadth D1,
and HS of diameter D2, the appropriate Fourier trans-
forms:
f̃11(k) =
2πD21
sin (kL1) ,
f̃12(k) = 2π
k(L1 +D2)
k(L1 +D2)
k(L1 +D2)
− sin
− sin
+ sin
f̃22(k) =
[sin (kD2)− kD2 cos (kD2)] (A.2)
where J1(x) and H1(x) are the Bessel and Struve func-
tions of first order, respectively.
For mixtures of HPAR with length and square side
length L1, D1, respectively, and HS of diameter D2, the
Fourier transforms are
f̃11(k) =
sin (kL1) ,
f̃12(k) = 8
sin [k(L1 +D2)/2]
k(L1 +D2)
− sin
k(L1 +D2)
− sin
+ sin
f̃22(k) =
[sin (kD2)− kD2 cos (kD2)] . (A.3)
Appendix B. Minimisation of Onsager functional in
the frozen-orientation approximation
We assume a Fourier expansion for the density profiles
ρi(z):
ρi(z) ≡ ρxiψi(z) = ρxi
s(i)n cosnkz
, (B.1)
with k the smectic wave number, and s
n the n-th Fourier
amplitude for species i. Introducing this expression into
the free-energy funtional, Eqn. (1), we obtain
Φ = ρ
log ρ− 1 +
xi lnxi +
dzψi (z) lnψi (z)
+ ρW(x, k, {s
, (B.2)
Here we have expressed the excess part in terms of the
function
W(x, k, {s(i)n }) =
f̃ij(0) +
s(i)n s
n f̃ij(kn)
, (B.3)
where kn = nk and the Fourier transforms f̃ij(k) were
defined in Section II (since we are assuming perfect ori-
entational order, we set Q = 1). From this, the pressure
βP = ρ+ ρ2W(x, k, {s(i)n }) (B.4)
To calculate phase equilibria in the mixture it is more
convenient to work in conditions of constant pressure.
Then, we minimize the Gibbs free energy per particle
and unit thermal energy,
(Φ + βP ) (B.5)
with respect to the smectic wave number and the co-
efficients {s
n }. The corresponding derivatives can be
written down explicitely but they have to be solved nu-
merically using an iterative method. In practice, the
Fourier expansions have been truncated; they include ca.
40 terms so as to satisfy a stringent convergence criterion
in the iterative procedure (the number of Fourier ampli-
tudes was chosen to guarantee an absolute error less than
10−7 in the density profiles).
Appendix C. τ parameter: general case
The strategy is to obtain the nematic–smectic spinodal
line perturbatively in the composition x, since we only
need to know the spinodal in the neighbourhood of x = 0
to obtain the τ parameter. The equations to solve are
three: two defining the spinodal, the third giving the
equilibrium state of the nematic fluid on the spinodal
[Eqns. (6)]. These equations can be written as
∆(k, ρ1, ρ2, Q) = [1− ρ1c̃11(k, ρ1, ρ2, Q)]
× [1− ρ2c̃22(k, ρ1, ρ2, Q)]− ρ1ρ2c̃
12(k, ρ1, ρ2, Q) = 0
(C.1)
∂∆(k, ρ1, ρ2, Q)
= 0 (C.2)
∂Φ(ρ1, ρ2, Q)
= 0, (C.3)
where c̃ij(k, ρ1, ρ2, Q) are the Fourier transforms of the
direct correlation functions between species i and j. The
relationship between infinitesimal changes in the vari-
ables {k, ρ1, ρ2, Q} along the spinodal can be calculated
from (C.1) as
dρ1 +
dρ2 +
dk = 0.
(C.4)
Similarly, the relation between the changes in the order
parameter Q and densities {ρ1, ρ2} can be obtained from
(C.3) as
∂Q∂ρ1
dρ1 +
∂Q∂ρ2
dρ2 +
dQ = 0.
(C.5)
Defining the coefficient υ ≡ dρ1/dρ2, we obtain from
Eqns. (C.4) and (C.5):
υ = −
∂2Φ/∂Q∂ρ2
(∂2Φ/∂Q2)
∂2Φ/∂Q∂ρ1
(∂2Φ/∂Q2)
, (C.6)
where the condition given in (C.2) of the absolute mini-
mum of ∆(k, ρ1, ρ2, Q) with respect to the wave number
k was used. Note that this expression is valid for any
composition x of the mixture. Now, evaluating all the
derivatives at ρ2 = 0 we obtain, from Eqn. (C.1)
= −ρ0
∂c̃11
, (C.7)
= −ρ0
∂c̃11
, (C.8)
= −ρ0
∂c̃11
, (C.9)
where the superscripts mean that all derivatives are eval-
uated at x = 0, k = k0, and Q = Q0, values correspond-
ing to the one-component fluid spinodal [note that, from
Eqn. (C.1), we get 1 − ρ0c̃11 = 0 at ρ2 = 0, which has
been used to obtain the derivatives above].
Now, using the fact that, for the one-component fluid,
1−ρ0c̃
11 = 0, and that the τ parameter can be expressed
through the coefficient υ evaluated at x = 0 as τ = 1 +
(v1/v2)υ
(0), we finally obtain
τ = 1−
∂c̃11
∂c̃11
∂2Φ/∂ρ2∂Q
(∂2Φ/∂Q2)
∂c̃11
∂c̃11
∂2Φ/∂ρ1∂Q
(∂2Φ/∂Q2)
(C.10)
which is a general expression for the τ parameter. In
the Onsager approximation, we have c̃ij(k, ρ1, ρ2, Q) =
f̃ij(k,Q) and the derivatives of c̃ij with respect to the
densities ρi vanish, while for parallel rods (Q = 1) the
derivative with respect to Q also vanishes, and we obtain
expression (14).
For a fluid mixture of particles without orientational
degrees of freedom, such as a mixture of parallel paral-
lelepipeds and cubes or a mixture of hard spheres, expres-
sion (C.10) for τ can be reinterpreted as follows. From
Eqn. (C.10), the τ parameter can be re-expressed as
τ = 1−
∂S−1ef
∂S−1ef
, (C.11)
where S−1ef (k, ρ1, ρ2) is the inverse structure factor of an
effective one-component fluid of particles, labelled as 1,
with interactions between them being mediated by parti-
cles labelled as 2. This structure factor can be calculated
by evaluating the second functional derivative with re-
spect to ρ1(r) of the semi-grand canonical free-energy
functional
Υ[ρ1] ≡ F [ρ1, ρ2]− µ2
drρ2(r), (C.12)
at the bulk densities[22]. Here F [ρ1, ρ2] is the Helmholtz
free-energy functional of the binary mixture and the den-
sity profile ρ2(r) in Eqn. (C.12) is calculated from the
condition of fixed chemical potential of species 2:
δF [ρ1, ρ2]
δρ2(r)
= µ2. (C.13)
The result for this effective structure factor is
S−1eff (k, ρ1, ρ2) = 1− ρ1c̃11(k, ρ1, ρ2)
ρ1ρ2c̃
12(k, ρ1, ρ2)
1− ρ2c̃22(k, ρ1, ρ2)
. (C.14)
After the inclusion of its first derivative with respect to
ρ1 and ρ2, evaluated at ρ2 = 0 in Eqn. (C.11), we exactly
obtain Eqn. (C.10) without the third terms in both nu-
merator and denominator (as they vanish for fluids with-
out orientational degrees of freedom).
Appendix D. Correlation functions for HPAR-HC
mixture in the Zwanzig–Onsager approach
In the Zwanzig approximation the rods point along one
of the Cartesian axes x, y or z. For the binary HP-HC
mixture in the Zwanzig approach the correlation func-
tions evaluated at k = (0, 0, k) can be calculated as
− c̃11(k) = 2x
f̃1x,1x(k) + f̃1x,1y(k)
+ x2‖f̃1z,1z(k)
+ 4x⊥x‖f̃1x,1z(k), (D.1)
−c̃12(k) = 2x⊥f̃1x,2(k) + x‖f̃1z,2(k), (D.2)
−c̃22(k) = f̃22(k), (D.3)
where the subindexes 1µ (µ = x, y, z) label particle
1 (rods), which point along the direction µ, while the
subindex 2 labels cubes. x⊥ and x‖ are respectively the
fraction of rods perpendicular and parallel to the nematic
director (which is taken to be parallel to the z axis).
These variables are functions of the nematic order pa-
rameter Q [i.e. x⊥ = (1 − Q)/3, and x‖ = (1 + 2Q)/3],
the equilibrium values of which should be calculated from
the extremum condition of the free-energy density with
respect to Q. This condition reads
1 + 2Q
− 2ρ1 (L1 −D1)
(D.4)
The expressions for the Fourier-transformed overlap func-
tions are
f̃1x,1y(k) = 2(L1 +D1)
2 sin(kD1)
f̃1x,1x(k) = 8L1D1
sin(kD1)
f̃1x,1z(k) = 4D1(L1 +D1)
sin [k(L1 +D1)/2]
f̃1z,1z(k) = 8D
sin(kL1)
f̃1x,2(k) = 2(L1 +D2)(D1 +D2)
sin [k(D1 +D2)/2]
f̃22(k) = 8D
sin(kD2)
f̃1z,2(k) = 2(D1 +D2)
2 sin [k(L1 +D2)/2]
, (D.5)
Note that, due to the discrete axial symmetry of the par-
ticles, some overlap functions can be expressed in terms of
others; for instance, f̃1x,1x = f̃1y,1y, and f̃1z,1x = f̃1z,1y.
Appendix E. Correlation functions for HPAR-HC
and HS mixtures in the FMT approach
To obtain the Fourier transforms of the correlation
functions for HP-HC mixtures [17], and for HS mixtures
[16], we have used the FMT functional, which has the
same structure in both systems, namely:
− c̃ij(k) = χ
(0)f̃ij(k) + χ
R̃ij(k) + χ
S̃ij(k)
+ χ(3)Ṽij(k), (E.1)
where f̃ij , R̃ij , S̃ij and Ṽij are Fourier transforms of the
overlap function, and of the mean radius, surface and
volume of the overlap region between particles i and j,
respectively. For a HPAR-HC mixture the mean radius
and the surface area of the overlap region are vectors
oriented along the directions parallel to the edge lengths
(mean radius) and perpendicular to the sides (surface
area) of the parallelepipeds. For the HS mixture they are
scalars, as are the quantities χ(1) and χ(2), which for the
HPAR-HC mixture are vectors. Finally, the correlation
functions depend on the wave vector k or on its mod-
ule k in the HPAR-HC and HS mixtures, respectively.
The expressions for χ(i) corresponding to the HPAR-HC
mixture are
χ(0) =
, χ(1) =
(1− η)2
, (E.2)
(2) =
(1− η)2
(1− η)3
, (E.3)
χ(3) =
(1− η)2
2ξ1ξ2
(1− η)3
6ξ2xξ2yξ2z
(1 − η)4
(E.4)
with ρ =
i ρi and η =
i ρivi the total
density and packing fraction, respectively, ζ2 =
(ξ2yξ2z , ξ2zξ2x, ξ2xξ2y), and
ξ1x = ξ1y =
ρiDi, ξ1z = ρ1L1 + ρ2D2, (E.5)
ξ2x = ξ2y = ρ1L1D1 + ρ2D
2, ξ2z =
i , (E.6)
with L1 and D1 the length and breadth of the paral-
lelepiped, while D2 is the edge-length of the cube.For
the HS mixture we have
χ(0) =
, χ(1) =
(1− η)2
, (E.7)
χ(2) =
(1 − η)2
(1 − η)3
, (E.8)
χ(3) =
(1 − η)2
2ξ1ξ2
(1− η)3
(1− η)4
, (E.9)
where
ρiDi, ξ2 = π
i , (E.10)
and Di (i = 1, 2) the particle diameters.
To calculate the τ parameter we only need expressions
for c̃11 and c̃12. The Fourier transforms of the geometric
measures of overlapping bodies for the HPAR-HC fluid
with a wave number k = (0, 0, k) (smectic symmetry) are
f̃11(k) = 8D
sin(kL1)
, (E.11)
R̃11(k) = 4D
sin(kL1)
sin(kL1)
sin(kL1/2)
, (E.12)
S̃11(k) = 2D
sin(kL1/2)
sin(kL1/2)
sin(kL1)
, (E.13)
Ṽ11(q) = 4D
sin(kL1/2)
(E.14)
The expressions for particles 1 and 2 are
f̃12(k) = 2D
sin(kL12/2)
(E.15)
R̃12(k) = 2D12
sin(kL12/2)
,D1D2
sin(kL12/2)
sin(kL1/2) sin(kD2/2)
, (E.16)
S̃12(q) = 2D1D2
sin(kL1/2) sin(kD2/2)
sin(kL1/2) sin(kD2/2)
sin(kL12/2)
(E.17)
Ṽ12(q) = 4(D1D2)
2 sin(kL1/2) sin(kD2/2)
, (E.18)
where D12 = D1 +D2 and L12 = L1 +D2.
For HS mixtures these expressions have the form
f̃ij(k) =
sin(kDij/2)
cos(kDij/2)
, (E.19)
R̃ij(k) =
sin(kDi/2) sin(kDj/2)
cos(kDij/2)
, (E.20)
S̃ij(k) =
(4π)2
sin(kDi/2) sin(kDj/2)
sin(kDij/2)
, (E.21)
Ṽij(k) =
(4π)2
sin(kDi/2) sin(kDj/2)
sin(kDij/2)
+ cos(kDi/2) cos(kDj/2)] , (E.22)
for i, j = 1, 2.
Appendix F. Asymptotic behaviour of τ
The asymptotic behaviour of (∂∆/∂ρ2)
for d =
D1/D2 ≫ 1 can be calculated from (C.8) and (E.1),
particularizing the latter equation to HPAR-HC and HS
mixtures. After some algebra, we arrive at
= −ρ0
∂c̃11
y(1 + y)
2(1 + y)2T 21 (k
0) + 2yT1(k
0) + y
2T 21 (k
,(F.1)
where κ1 = L1/D1, T0(x) = cosx, and T1(x) = sinx/x,
while y = η0/(1 − η0), and k
0 = k0L1. The expression
corresponding to the HS fluid reads
y(1 + y)
2y2(1 + 3y)TV (k
0) + Tδ(k
+ 2yTf(k
0) + y(1 + 6y)TS(k
0)} , (F.2)
where we have defined the dimensionless quantities
TV (k
0) = 6v
1 Ṽ (k
0), Tδ(k
0) = 3s
1 δ̃(k
0), Tf (k
3v−11 f̃(k
0), and TS(k
0) = 6(v1s1)
−1S̃(k∗0), with s1 and v1
the surface area and volume of particle 1, while δ̃(q) is the
Fourier transform of the Dirac delta function δ(D1−|r|).
The presence of a delta function indicates that, in the
limit d → ∞, the one-component sticky-sphere limit of
the fluid is obtained. We arrive at the same conclusion, in
the same limit, for the HPAR/HC mixture, by compar-
ing Eqns. (F.1) and (F.2). Each term in the right-hand
side of Eqn. (F.1), from left to right, has the same mean-
ing as in the HS mixture: they are related to the Fourier
transform of the volume, adhesiveness, overlap function,
and surface area of the overlap region between two parti-
cles, respectively. For example, the inverse Fourier trans-
form of T0(k
∗), with the smectic symmetry q = (0, 0, q),
results in a term proportional to δ(L1 − |z|)/2, which
reflects the stickiness of parallelepipeds along the z di-
rection. A complete effective density functional for the
infinite asymmetric limit was worked out for hard-cube
mixtures in Ref.[23]. The direct correlation function re-
sulting from this functional has a Dirac delta function
located at contact of the sides of the cubes.
However, there is an important difference between the
HP-HC and HS mixtures, which is related to the square
and cubic power dependence of the expressions (F.1)
and (F.2) with respect to the asymmetric parameter
d = D1/D2. This result is related to the fact that the
planar geometry of the sides of parallelepipeds enhances
the depletion interaction between two big particles. We
should compare these results with those obtained using
the Onsager approach, where c̃ij(k, ρ1, ρ2) = f̃ij(k) and
then (∂∆/∂ρ2)
∼ O(d0). Thus, we can conclude that
depletion interactions in this limit cannot be properly
described by the Onsager approximation.
Finally, to obtain the asymptotic behaviour of the τ
parameter for the HP-HC mixture in the limit d → ∞
, we use Eqns. (C.10), (F.1) and (E.1), particularized
for HP, together with the spinodal instability condition
1− ρ0c̃
11 = 0, to obtain explicitly
τ ∼ κ1dW (κ1), (F.3)
W (κ1) = −
2(1 + y)2T 21 (k
0/2) + [T0(k
0) + 2yT1(k
+ y2T 21 (k
3 + y−1
+ 2(3 + y)T1(k
3 + y + 6(1 + y)3
T 21 (k
(F.4)
In Fig. (11) the function W (κ1) is plotted. As can be
seen from the figure, (i) W varies very little with respect
to κ1, and (ii) depletion is maximum in the Onsager limit
(κ1 → ∞). Note that, on taking the latter limit, the
condition d = D1/D2 ≫ 1 must be fulfilled.
It is well known that, when the PY approximation is
used, the condition 1−ρ0c̃11(k, ρ0) > 0 is always fulfilled
for all η and q in the physical parameter region. Because
of the absence of a fluid-solid spinodal in this approxi-
mation, we cannot calculate the τ parameter to estimate
the effect of depletion at the freezing transition.
0 0.2 0.4 0.6 0.8 1
-0.04
-0.038
-0.036
-0.034
-0.032
FIG. 11: The function W (κ1).
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