Mesoscopic description of network-forming clusters of weakly charged colloids
Systems composed of spherical charged particles in solvents containing counterions and inducing effective short-range attraction are studied in the framework of mesoscopic field-theory. We limit ourselves to mean-field approximation (MF) and to weak ordering. We discuss properties of potentials cons...
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Ciach, A. Góźdź, W.T. 2012-04-08T15:48:46Z 2012-04-08T15:48:46Z 2010 Mesoscopic description of network-forming clusters of weakly charged colloids / A. Ciach, W.T. Góźdź // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23603: 1-12. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 61.20.Gy, 64.60.De, 82.70.Dd https://nasplib.isofts.kiev.ua/handle/123456789/32094 Systems composed of spherical charged particles in solvents containing counterions and inducing effective short-range attraction are studied in the framework of mesoscopic field-theory. We limit ourselves to mean-field approximation (MF) and to weak ordering. We discuss properties of potentials consisting of strong short-range attraction and weak long-range repulsion (SALR) in the context of formation of nonuniform distribution of particles on a mesoscopic length scale instead of macroscopic phase separation. In earlier work it was found that spherical, cylindrical and slab-like clusters of particles are formed, and for low enough temperatures the clusters form ordered, periodic bcc, hexagonal and lamellar phases. In addition, a gyroid phase was predicted in which two interwoven regular network-like clusters branching in triple junctions are formed. At properly rescaled density and temperature, the coexistence lines between different ordered phases were found to be universal in MF, with the exception of the gyroid phase. Here the phase diagram is determined for two choices of the SALR potential, one corresponding to a large range of the attractive part of the potential, and the other one to a very small range of attraction. We find that the region of stability of the gyroid phase very weakly depends on the form of the SALR potential within the approximate theory. У межах мезоскопічної польової теорії вивчаються системи, що складаються зі сферичних заряджених частинок у розчинниках, які містять контріони та індукують короткосяжне притягання. Розглянуто наближення середнього поля (СП) і слабке впорядкування, а також властивості потенціалів, що складаються із сильного короткосяжного притягання та слабкого далекосяжного відштовхування (SALR) у контексті формування неоднорідного розподілу частинок на мезоскопічних масштабах довжин замість макроскопічного фазового відокремлення. Було показано, що формуються сферичні, циліндричні та щілиноподібні кластери частинок, і для достатньо низьких температур ці кластери формують впорядковані періодичні об'ємоцентричні, гексагональні та ламеларні фази. Крім того, було передбачено гіроїдну фазу, в якій формуються два переплетені регулярні мережоподібні кластери, що розгалужуються в потрійному з'єднанні. Знайдено, що за відповідного масштабування густини та температури, лінії співіснування між різними впорядкованими фазами є універсальними в наближенні СП, за винятком гіроїдної фази. Визначено фазову діаграму для двох виборів потенціалу SALR: такого, що відповідає великій області притягувальної частини потенціалу, та такого, що відповідає малій області притягання. Виявлено, що область стабільності гіроїдної фази дуже слабо залежить від форми потенціалу SALR у межах наближеної теорії. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Mesoscopic description of network-forming clusters of weakly charged colloids Мезоскопічний опис мережо-формуючих кластерів слабо заряджених колоїдів Article published earlier |
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Mesoscopic description of network-forming clusters of weakly charged colloids |
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Mesoscopic description of network-forming clusters of weakly charged colloids Ciach, A. Góźdź, W.T. |
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Mesoscopic description of network-forming clusters of weakly charged colloids |
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Mesoscopic description of network-forming clusters of weakly charged colloids |
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Mesoscopic description of network-forming clusters of weakly charged colloids |
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Mesoscopic description of network-forming clusters of weakly charged colloids |
| title_sort |
mesoscopic description of network-forming clusters of weakly charged colloids |
| author |
Ciach, A. Góźdź, W.T. |
| author_facet |
Ciach, A. Góźdź, W.T. |
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2010 |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Мезоскопічний опис мережо-формуючих кластерів слабо заряджених колоїдів |
| description |
Systems composed of spherical charged particles in solvents containing counterions and inducing effective short-range attraction are studied in the framework of mesoscopic field-theory. We limit ourselves to mean-field approximation (MF) and to weak ordering. We discuss properties of potentials consisting of strong short-range attraction and weak long-range repulsion (SALR) in the context of formation of nonuniform distribution of particles on a mesoscopic length scale instead of macroscopic phase separation. In earlier work it was found that spherical, cylindrical and slab-like clusters of particles are formed, and for low enough temperatures the clusters form ordered, periodic bcc, hexagonal and lamellar phases. In addition, a gyroid phase was predicted in which two interwoven regular network-like clusters branching in triple junctions are formed. At properly rescaled density and temperature, the coexistence lines between different ordered phases were found to be universal in MF, with the exception of the gyroid phase. Here the phase diagram is determined for two choices of the SALR potential, one corresponding to a large range of the attractive part of the potential, and the other one to a very small range of attraction. We find that the region of stability of the gyroid phase very weakly depends on the form of the SALR potential within the approximate theory.
У межах мезоскопічної польової теорії вивчаються системи, що складаються зі сферичних заряджених частинок у розчинниках, які містять контріони та індукують короткосяжне притягання. Розглянуто наближення середнього поля (СП) і слабке впорядкування, а також властивості потенціалів, що складаються із сильного короткосяжного притягання та слабкого далекосяжного відштовхування (SALR) у контексті формування неоднорідного розподілу частинок на мезоскопічних масштабах довжин замість макроскопічного фазового відокремлення. Було показано, що формуються сферичні, циліндричні та щілиноподібні кластери частинок, і для достатньо низьких температур ці кластери формують впорядковані періодичні об'ємоцентричні, гексагональні та ламеларні фази. Крім того, було передбачено гіроїдну фазу, в якій формуються два переплетені регулярні мережоподібні кластери, що розгалужуються в потрійному з'єднанні. Знайдено, що за відповідного масштабування густини та температури, лінії співіснування між різними впорядкованими фазами є універсальними в наближенні СП, за винятком гіроїдної фази. Визначено фазову діаграму для двох виборів потенціалу SALR: такого, що відповідає великій області притягувальної частини потенціалу, та такого, що відповідає малій області притягання. Виявлено, що область стабільності гіроїдної фази дуже слабо залежить від форми потенціалу SALR у межах наближеної теорії.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/32094 |
| citation_txt |
Mesoscopic description of network-forming clusters of weakly charged colloids / A. Ciach, W.T. Góźdź // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23603: 1-12. — Бібліогр.: 27 назв. — англ. |
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| first_indexed |
2025-11-24T19:09:26Z |
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2025-11-24T19:09:26Z |
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| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 2, 23603: 1–12
http://www.icmp.lviv.ua/journal
Mesoscopic description of network-forming clusters of
weakly charged colloids
A. Ciach, W.T. Góźdź
Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warszawa, Poland
Received February 10, 2010, in final form March 18, 2010
Systems composed of spherical charged particles in solvents containing counterions and inducing effective
short-range attraction are studied in the framework of mesoscopic field-theory. We limit ourselves to mean-
field approximation (MF) and to weak ordering. We discuss properties of potentials consisting of strong short-
range attraction and weak long-range repulsion (SALR) in the context of formation of nonuniform distribution of
particles on a mesoscopic length scale instead of macroscopic phase separation. In earlier work it was found
that spherical, cylindrical and slab-like clusters of particles are formed, and for low enough temperatures the
clusters form ordered, periodic bcc, hexagonal and lamellar phases. In addition, a gyroid phase was predicted
in which two interwoven regular network-like clusters branching in triple junctions are formed. At properly
rescaled density and temperature, the coexistence lines between different ordered phases were found to be
universal in MF, with the exception of the gyroid phase. Here the phase diagram is determined for two choices
of the SALR potential, one corresponding to a large range of the attractive part of the potential, and the other
one to a very small range of attraction. We find that the region of stability of the gyroid phase very weakly
depends on the form of the SALR potential within the approximate theory.
Key words: colloids, clusters, self-assembly, order-disorder phase transitions
PACS: 61.20.Gy, 64.60.De, 82.70.Dd
1. Introduction
Biologically relevant macromolecules, nanoparticles or colloids are typically charged, and there-
fore repel each other. On the other hand, van der Waals and various solvent-mediated interactions
lead to effective attraction between them. The range of attraction is usually shorter than the range
of screened electrostatic repulsion. However, for weakly to moderately charged particles, the magni-
tude of the attraction is significantly larger than the magnitude of repulsion. The resulting potential
is negative for short distances, and positive for larger distances and is known as short-range attrac-
tion long-range repulsion (SALR) potential [1–15]. Short separations of particles corresponding to
the attraction are favorable, whereas the distances close to the maximum of repulsion lead to a
significant increase in energy. Depending on the repulsion barrier and range, as well as on density
ρ and temperature T , the random distribution of particles and macroscopic phase separation can
be less favorable than the formation of clusters of various shape and size. When the average dis-
tance between particles corresponds to strong repulsion, then it is energetically favorable to form
clusters with particles strongly attracting each other, such that the separation between the clusters
is larger than the range of repulsion. When separations between the particles forming the cluster
are mainly within the range of attraction and the separation between clusters is larger than the
range of repulsion, the energy takes low values. This effect may overcompensate the entropy loss
associated with cluster formation. When the density increases such that the average separation
between particles is too small to form spherical clusters with particles attracting each other, and
with average separation between the clusters larger than the range of repulsion, then under the
constraint of a given density, it is favorable to grow the cluster in one dimension, and keep the
distance between elongated clusters larger than the range of repulsion. For still higher densities,
slab-like clusters sufficiently far apart from each other may be created. Such scenario has been
observed both in experiments and in simulations. In experimental systems the clusters have been
c© A. Ciach, W.T. Góźdź 23603-1
http://www.icmp.lviv.ua/journal
A. Ciach, W.T. Góźdź
directly seen by confocal microscopy [3, 7]; for an increasing density, first spherical, then elongated
and finally network-like clusters were observed. In the case of globular protein solutions, cluster
formation is a subject of intensive debate [3, 16]. Simulation [9, 11] and theoretical studies [17, 18]
of particular examples of the SALR potential show that indeed spherical, cylindrical and slab-like
clusters are formed for an increasing density. In addition, for some range of particle density, lower
than the density of crystallization, the clusters form ordered, periodic structures at low enough
temperatures. The results of simulations and recent theoretical prediction [17, 18] indicate that
the sequence of ordered phases is the same regardless of the detailed form of the SALR poten-
tial. In particular, small [11] and very large [9] clusters are spherical, cylindrical and slab-like in
shape for similar volume fractions of particles. Mean-field results [17] indicate that for a suitably
rescaled temperature, the phase diagram is universal when weak ordering of particles occurs, i.e.,
in rescaled variables the phase-coexistence lines for different forms of the SALR potential collapse
onto master curves. There is one exception, however. Between the stability regions of the hexago-
nal phase of cylinders and lamellar phase of slabs, a thermodynamically stable gyroid phase was
found [17], but its stability region is not universal, i.e., it may be different for different forms of
the effective interactions. In this phase particles form two interwoven networks of clusters which
branch in triple junctions. This phase may be related to the branched network of clusters observed
experimentally and in simulations [6, 7, 14]. The prediction of stability of the gyroid phase is based
on mean-field approximation to a mesoscopic field theory, and the results were obtained under
assumption of weak ordering (i.e., strong fluctuations of particles around their average positions,
leading to smooth concentration profiles with small amplitudes). Hence, further evidence is required
to confirm that such a peculiar regular ’gel’ can be a thermodynamically stable phase.
In this work we address an open issue how the stability region of this gyroid phase on the
phase diagram depends on the shape of the SALR potential, and consequently on the size of the
equilibrium clusters at strong dilution. To this end, we consider two forms of the SALR potential,
one leading to very large and the other one to very small clusters, and determine the phase diagrams
for these two versions of the SALR potential using the method developed in [17].
In section 2 we introduce a particular form of the SALR potential, consisting of two Yukawa
terms, and discuss the range of parameters corresponding to phase separation and to clustering. In
section 3 we briefly review the version of the statistical field theory introduced in [17]. This approach
is suitable for a description of collective phenomena leading to ordering on the mesoscopic length
scale. In section 4 we describe the calculations of the phase diagram in the simplest mean-field
approximation in the case of weak ordering, and discuss our results.
2. The interaction potential and its properties
We restrict our attention to colloids, nanoparticles or globular macromolecules that can be
modeled as spherical hard cores with diameter σ, and with additional interactions present for
distances r > σ. We limit ourselves to the interaction potential that has the form
u(r) = uSR(r) + uLR(r) (1)
with
uSR(r) =
{
0 if r > 1,
∞ if 0 < r < 1,
(2)
where r is in σ units. The long-range part uLR(r) is defined for r > 1, and for such separations we
assume that uLR(r) has the form
v(r) = −A1
r
e−z1r +
A2
r
e−z2r, (3)
where zi is the inverse range in σ−1 units. We consider the SR and the LR parts separately and
treat the case with uLR(r) = 0 as a reference system. In microscopic theory we need an extension
23603-2
Mesoscopic description of network-forming clusters of weakly charged colloids
of uLR(r) for r < 1 for calculational reasons. The contribution to the energy of the system coming
from overlapping cores should be avoided, and therefore we assume
uLR(r) =
{
v(r) if r > 1,
0 if r < 1.
(4)
Such an extension is not unique, and other choices made in [19–22] in different context lead to
more accurate results. The simplest possibility of extension of the long-range part of the potential
for r < 1 would be to assume the form (3) for all r > 0. In this case analytical results can be easily
obtained. However, the self-energy would be included, and moreover, there would be no information
on the length scale set by the hard cores. We shall compare the structure factor obtained within
the simplest Gaussian approximation for uLR(r) = v(r) and for uLR(r) given by (4) in the next
section; the form (4) leads to the structure factor that qualitatively agrees with experiments and
simulations, whereas for uLR(r) = v(r) the second peak of the structure factor, associated with
particle-particle correlations within the clusters is missing.
In the mesoscopic theory [17, 18] the key role is played not by the potential alone, but rather
by the potential multiplied by the microscopic correlation function, g(r), calculated for the frozen
mesoscopic density distribution. The form of g(r)uLR(r) is given by (4) for any extension of uLR(r)
for r > 1 for the simplest approximation g(r) ≈ θ(r − 1).
We focus on the case of strong short-range attraction and weak long-range repulsion, and
assume z1 > z2, and A1 > A2. In Fourier representation (4) takes the form
ũLR(k) = 4π
[
A2e
−z2
z2
2 + k2
(
z2
sin k
k
+ cos k
)
− A1e
−z1
z2
1 + k2
(
z1
sin k
k
+ cos k
)
]
. (5)
We use the notation f̃(k) for the Fourier transform of the function f(r). Note that the energy
associated with a plane wave of concentration, ρ∗(z) = Φ cos(kz) + ρ̄∗, is equal to V Φ2ũLR(k)/4,
where V is the volume of the system and we introduced dimensionless density
ρ∗ = ρσ3 (6)
and denoted the space-averaged density by ρ̄∗. When ũLR(k) < 0, a concentration wave with the
wavelength 2π/k leads to a decrease of energy compared to homogeneous distribution of parti-
cles. The minimum of ũLR(k) at k = kb gives the most probable wavelength 2π/kb of the plane
concentration wave, where layers with excess particle density of width π/kb are followed by re-
gions of depleted particle density of the same width. Superpositions of plane waves in different
directions may lead to nonuniform distribution of particles of different symmetries. The energy
associated with the most probable wave of concentration, ũLR(kb) is a convenient energy unit, and
we introduce dimensionless temperature T ∗ by
T ∗ =
kBT
|ũLR(kb)| . (7)
In [17] it was shown that in mean-field (MF) approximation, the spinodal lines are given by a
universal curve in (T ∗, ρ∗) phase diagram, independently of the shape of the SALR potential.
Thus, such units are appropriate for determination of the common features of different systems
that form nonuniform structures on the mesoscopic length scale.
The gas-liquid phase separation may occur when ũLR(0) < 0, and the necessary condition for
the phase separation is
A2
A1
< e−(z1−z2)
(
z1 + 1
z2 + 1
)(
z2
z1
)2
. (8)
The behavior depends on whether ũLR(k) assumes a minimum or a maximum for k = 0. We find
d2ũLR(k)
dk2
|k=0 =
4π
3
[A1e
−z1
(
z3
1 + 3z2
1 + 6z1 + 6
)
z4
1
−
A2e
−z2
(
z3
2 + 3z2
2 + 6z2 + 6
)
z4
2
]
. (9)
23603-3
A. Ciach, W.T. Góźdź
For parameters Ai, zi corresponding to negative values of the above (maximum at k = 0) we
find periodic ordering, because the second extremum, for k > 0, is a minimum. For z1 > 6 and
z2 ≈ 1 this is typically the case if A2/A1 > 10−4. Hence, even very weak repulsion suppresses phase
separation for very short-range of attraction. For longer range of attraction we need a sufficiently
strong repulsion to suppress phase separation in favor of clustering.
The parameters Ai and zi may be state-dependent, and in different systems the dependence
on temperature and density may be different. In particular, for charged colloids in the presence
of counterions, z2 is the inverse screening length which is a function of temperature and density
of the form z2 ∝
√
ρ/T . Here we are not interested in any particular system, but rather try to
determine the shape of the phase diagram in the universal variables. Note that when the potential
depends on temperature and density, the same holds for the energy unit |ũLR(kb)|.
We choose two sets of parameters for the potential given by (3).
A1 = 1, A2 = 0.2, z1 = 1, z2 = 0.5, (10)
A1 = 1, A2 = 0.05, z1 = 3, z2 = 0.5.
The shape of the potential in the cases I and II is quite different. In the case I large clusters
are formed, because kb ≈ 0.6088 whereas in the case II the clusters are rather small, because
kb ≈ 1.7926.
3. Field theory
Consider nonuniform density distributions ρ∗(x). The probability that the density has the given
form ρ∗(x) in the mesoscopic description is given by [17]
p[ρ∗(x)] = Ξ−1e−βΩMF[ρ∗(x)], (11)
where β = 1/(kBT ), kB is the Boltzmann constant, and ΩMF[ρ∗] is the grand-thermodynamic
potential in the system with the density constrained to have the given nonuniform form ρ∗(x),
βΩMF[ρ∗] =
1
2
∫
x
ρ∗(x)βuLR(|x − x
′|)ρ∗(x′) +
∫
x
(
βfh[ρ
∗|x) − µ∗ρ∗(x)
)
, (12)
where µ∗ = βµ/σ3, and we use the notation
∫
x
≡
∫
dx. fh[ρ∗|x) is the free-energy density of the
reference system (with uLR = 0) at x. In general, fh[ρ
∗|x) is a functional of ρ∗(x). Here we limit
ourselves to the local-density approximation, where the free-energy density at the space position
x is a function of the local density at x
fh[ρ
∗|x) = fh(ρ
∗(x)). (13)
We adopt the PY approximation for the free energy of hard spheres at density ρ∗,
βfh(ρ∗) = ρ∗ ln(ρ∗) − ρ∗ + ρ∗
[
3s(2− s)
2(1− s)2
− ln(1 − s)
]
, (14)
where s = ρ∗π/6. Note that in this theory the form of ΩMF[ρ∗(x)] depends on the choice for the
extension of uLR(r) for r < 1 (see the first term in equation (12)). Let us first consider βΩMF for
uniform states ρ∗ = const.
βΩMF(ρ∗)/V = −β∗p∗ =
ρ∗2
2
β∗ũ∗LR(0) + βfh(ρ∗) − µ∗ρ∗, (15)
where we used the thermodynamic relation Ω = −pV , introduced ũ∗LR(k) = ũLR(k)/|ũLR(kb)|,
β∗ = 1/T ∗, and p∗ = p/|ṽ(kb)|.
23603-4
Mesoscopic description of network-forming clusters of weakly charged colloids
Figure 1. The LR part of the interaction potential, uLR(r) given by (4), for two choices of
parameters. Left: A1 = 1, A2 = 0.2, z1 = 1, z2 = 0.5. Right: A1 = 1, A2 = 0.05, z1 = 3,
z2 = 0.5.
Figure 2. Fourier transform of the LR part of the interaction potential, ũLR(k) given by (5), for
two choices of parameters. Left: A1 = 1, A2 = 0.2, z1 = 1, z2 = 0.5. Right: A1 = 1, A2 = 0.05,
z1 = 3, z2 = 0.5.
Minimum condition of ΩMF(ρ∗):
∂ΩMF
∂ρ∗
= 0 (16)
has the explicit form
ρ∗βũLR(0) +
dβfh(ρ∗)
dρ∗
= µ∗ (17)
and
ũLR(0) =
∫
x
uLR(x) = 4π
[
A2e
−z2
z2
2
(
z2 + 1) − A1e
−z1
z2
1
(
z1 + 1)
]
. (18)
Densities that satisfy the above for a given µ∗ and T ∗ correspond to the most probable density of
23603-5
A. Ciach, W.T. Góźdź
the uniform system and are denoted by ρ∗
0. The explicit form of (16) is
µ∗ = ρ∗0β
∗ũ∗LR(0) + ln(ρ∗
0) − ln(1 − s0) +
14s0 − 13s2
0 + 5s3
0
2(1 − s0)3
. (19)
The correlation function for mesoscopic density in the Gaussian approximation is just given by
G̃(k) =
1
C̃(k)
=
ρ∗0(1 − s)4
β∗ũ∗LR(k)ρ∗0(1 − s)4 + (1 + 2s)2
, (20)
where
C̃(k) =
δ2βΩMF
δρ̃∗(k)δρ̃∗(−k)
= β∗ũ∗LR(k) +
d2βfh(ρ
∗)
dρ∗2
. (21)
In figure 3 G̃(k) is shown for the first set of parameters of the SALR potential (see equation (10)),
for the two extensions of ũ∗LR(r) for r < 1. The second maximum for r ≈ 6, present in experimental
systems, occurs when equation (4) is used for ũ∗LR(r).
Figure 3. Correlation function in Fourier representation, G̃(k) (equation (20)), in the diluted
phase (structure factor is given by S(k) = G̃(k)/ρ∗) for A1 = 1, A2 = 0.2, z1 = 1 and z2 = 0.5
and ρ∗ = 0.1, and T ∗ = 1.58. The dashed and solid lines are for uLR = v and for uLR given by
equation (4) respectively.
The boundary of stability of ΩMF is given by
C̃(kb) = 0. (22)
Note that the boundary of stability of the functional (12) significantly depends on the form of
ũ∗LR(r) for r < 1, where it is not uniquely defined. In the context of the mesoscopic theory,
where ũ∗LR(r) in (12) should be replaced by ũ∗LR(r)g(r), the quantitative results depend on the
assumed form of the correlation function g(r) with density constrained to have a fixed form on the
mesoscopic length scale. Here we assume g(r) = θ(r− 1). The explicit expression for the boundary
of stability of the uniform phase is
β∗ =
(1 + 2s)2
ρ∗(1 − s)4
. (23)
The MF line of instability depends on the state dependence of the temperature unit |ũLR(kb)|.
Instability with respect to separation of uniform phases occurs when kb = 0, and equation (23)
represents the spinodal line in this case. Metastable separation occurs provided that the condition
(8) is satisfied, and is given by (23) with ũ∗LR(kb) replaced by ũ∗LR(0).
23603-6
Mesoscopic description of network-forming clusters of weakly charged colloids
4. First order transition to microsegregated phases
For weakly first-order transitions we assume that the density has the form
ρ(x) = ρ∗
0 + η(x). (24)
Next we expand fh(ρ∗) about ρ∗0, and we obtain
β∆ΩMF[ρ∗0, η(x)] = βΩMF[ρ∗0 + η(x)] − βΩMF[ρ∗0]
=
1
2
∫
k
η̃(k)β∗ũ∗LR(k)η̃(−k) +
∞
∑
n=3
An
n!
∫
x
η(x)n , (25)
where
An =
∂nβfh(ρ)
∂ρ∗n
|ρ∗=ρ∗
0
. (26)
The stability requirement for η → ∞ allows us for truncating the expansion at even n > 4. Here
we shall limit ourselves to n = 4. Explicit expressions for A3 and A4 are given in Appendix.
The functional (25) has a well know structure [23–26] considered before in phenomenological
approaches to block copolymers, highly charged colloids and microemulsions. We repeat the same
procedure and assume that η(x) is periodic and can be written in the form
η(x) =
∑
n
Φngph
n (x), (27)
where gph
n (x) represents orthonormal basis functions for the n-th shell that have the symmetry of
the phase ph, and satisfy the normalization condition
∫
Vu
gn(x)2
Vu
= 1. (28)
By
∫
Vu
we denote an integration over the unit cell of the ordered structure. The volume of the unit
cell is denoted by Vu. Φn is the n-th amplitude.
Stable or metastable phases correspond to the global or a local minimum of ∆ΩMF[ρ∗0, η(x)].
At phase coexistence between two phases ∆ΩMF[ρ∗0, η(x)] for these phases takes equal values. At
the transition between the fluid and the ordered phase ∆ΩMF[ρ∗0, η(x)] = 0.
4.1. One-shell approximation
Let us first limit ourselves to the one-shell approximation. For structures possessing different
symmetries the Fourier transform of g1 has the form
g̃ph
1 (k) =
(2π)d
√
2nph
nph
∑
j=1
(
wδ(k − k
j
b) + w∗δ(k + k
j
b)
)
, (29)
where ww∗ = 1 and 2nph is the number of vectors k
j
b in the first shell of the structure ph. For
the first shell |kj
b | = kb. Since uLR(k) assumes a minimum for k = kb, the contribution to the
grand potential from the first shell is lower than the contributions from the remaining shells.
The explicit forms of the functions gph
1 in real-space representation for lamellar, hexagonal, bcc
and gyroid phases are given in Appendix. Within the given symmetry, the amplitude Φ1 of the
order parameter corresponds to the minimum ∆ΩMF
ph of ∆ΩMF[ρ∗0 + Φ1g
ph
1 ] with respect to Φ1. As
already shown by Leibler [24], for A3 6= 0, the ordered phase, coexisting with the fluid, has the bcc
symmetry in MF. For A3 = 0 the transition is to the striped (lamellar) phase [24]. In the considered
system A3(ρ
∗) < 0 for ρ∗ < ρ∗c , A3(ρ
∗) = 0 for ρ∗ = ρ∗c , and A3(ρ
∗) > 0 for ρ∗ > ρ∗c , where ρ∗c
23603-7
A. Ciach, W.T. Góźdź
is the critical density of the gas-liquid separation. In the PY approximation (14) ρ∗c ≈ 0.2457358.
From (25)–(28) we obtain in the one-shell approximation
∆ΩMF =
1
2
C̃(kb)Φ2
1 +
A3κ
ph
3
3!
Φ3
1 +
A4κ
ph
4
4!
Φ4
1 , (30)
where
κph
n =
∫
Vu
gph
1 (x)n
Vu
. (31)
From the extremum condition
∂∆ΩMF
∂Φ1
= C̃(kb)Φ1 +
A3κ
ph
3
2
Φ2
1 +
A4κ
ph
4
3!
Φ3
1 = 0 (32)
we obtain the amplitude
Φ1+,− =
√
3
[
−
√
3A3κ3 ±
√
3(A3κ3)2 − 8A4κ4C̃(kb)
]
2A4κ4
(33)
for each phase characterized by different geometric factors κn given in Appendix. Transition be-
tween the disordered and the bcc phase, as well as the amplitude of the density profiles along
the transition line are given by analytic expressions in [17], where the remaining transition lines
between ordered phases, obtained numerically, are also shown.
4.2. Two-shell approximation
The accuracy of the one-shell approximation depends on the shape of ũLR(k), in particular,
on the ratio ũLR(kb2)/ũLR(kb), where kb2 is the wavenumber corresponding to the second shell.
The approximation is the worse the smaller is the above ratio, since the second shell should not
be disregarded when it leads to comparable energy as the first shell. For a given shape of the
potential, the value of kb2 matters, and it is different for different structures. As discussed in detail
in [17, 27], for the gyroid phase kb2 = kb2/
√
3, whereas for the remaining phases kb2/kb is much
larger, so the value of ũLR(kb2)/ũLR(kb) is much larger too. We thus consider the gyroid phase
in the two-shell approximation, and the remaining phases in the one-shell approximation. In the
two-shell approximation the grand potential assumes the form
∆ΩMF =
1
2
C̃(kb)Φ2
1 + κ3
A3
3!
Φ3
1 + κ4
A4
4!
Φ4
1
+
1
2
C̃(k2b)Φ
2
2 + κ0,3
A3
3!
Φ3
2 + κ0,4
A4
4!
Φ4
2 + κ2,1
A3
2
Φ2
1Φ2 + κ2,2
A4
2!2!
Φ2
1Φ
2
2 , (34)
where the geometric factors
κph
n,m =
∫
Vu
gph
1 (x)ngph
2 (x)m
Vu
. (35)
for the gyroid phase are given in Appendix. The minimum condition with respect to Φi gives
Φ1± = ±
√
− C̃(kb2)Φ2 + κ0,3
A3
2 Φ2
2 + κ0,4
A4
3! Φ3
2
κ2,1
A3
2 + κ2,2
A4
2 Φ2
, (36)
Φ2± =
−κ2,1A3 ±
√
∆
κ2,2A4
, (37)
with
∆ = κ2
2,1A
2
3 − 2κ2,2A4
(
C̃(kb) + κ3,0
A3
2
Φ1 + κ4,0
A4
6
Φ2
1
)
. (38)
23603-8
Mesoscopic description of network-forming clusters of weakly charged colloids
For ρ∗ < ρ∗c and ρ∗ > ρ∗c the minimum of ∆ΩMF corresponds to Φ1+ and Φ2−, and to Φ1− and
Φ2+ respectively. It is important to note that the structure given by two-shell and by one-shell
approximation is significantly different. Namely, in the one-shell approximation a single network
of excess density is present, whereas for Φ1Φ2 < 0 two much thinner networks of excess density
are formed. The phase diagrams calculated for the two potentials with parameters given in equa-
tion (10) (see figures 1–2) are shown in figures 4, 5. Note that despite significantly different shape
Figure 4. Phase diagram obtained in MF approximation with the bcc, hexagonal and lamellar
phases considered in the one-shell and the gyroid phase in the two-shell approximation for
A1 = 1, A2 = 0.2, z1 = 1 and z2 = 0.5. The outer solid lines are the coexistence lines between
the disordered and the bcc crystal phases, the bcc and the hexagonal phases coexist along the
dashed lines, along the dotted lines the hexagonal and the lamellar phases coexist, and inside
the shaded regions the gyroid phase is stable. For densities lower than ρ∗c clusters of particles
form ordered patterns, whereas for ρ∗ > ρ∗c the pattern is formed by bubbles – regions with
depleted particle density.
Figure 5. Phase diagram obtained in MF approximation with the bcc, hexagonal and lamellar
phases considered in one-shell and the gyroid phase in the two-shell approximation for A1 = 1,
A2 = 0.05, z1 = 3 and z2 = 0.5. The outer solid lines are the coexistence lines between the
disordered and the bcc crystal phases, the bcc and the hexagonal phases coexist along the
dashed lines, along the dotted lines the hexagonal and the lamellar phases coexist, and inside
the shaded regions the gyroid phase is stable. For densities lower than ρ∗c clusters of particles
form ordered patterns, whereas for ρ∗ > ρ∗c the pattern is formed by bubbles – regions with
depleted particle density.
of the interaction potential, the extent of the stability region of the gyroid phase on the phase
diagram given in reduced variables (ρ∗, T ∗) is similar. Moreover, the phase diagram in variables
(ρ∗, T ∗) is similar to the phase diagram obtained earlier [17] for the SALR potential characterized
by the parameters A1 = 140 exp(8.4), A2 = 30 exp(1.55), z1 = 8.4 and z2 = 1.55, and corre-
sponding to kb ≈ 1.94026. The ratio of ṽ(b2)/ṽ(kb) for the cases I and II and for the previously
considered potential takes the values 0.9762993733, 0.8973863512, 0.9119505483 respectively. In the
23603-9
A. Ciach, W.T. Góźdź
case of large clusters (case I, π/kb ≈ 5 ) the gyroid phase extents to higher temperatures and oc-
cupies a larger range of densities than in the case of small clusters (case II, π/kb ≈ 1.7 and [17],
π/kb ≈ 1.6). We conclude that in the simple MF approximation, the phase diagram in reduced
variables, although not universal beyond the one-shell approximation, rather weakly depends on
the shape of the effective interaction potential. The region of stability of the gyroid phase increases
when the minimum of the interaction potential in Fourier representation becomes shallower, i.e.,
when ṽ(kb2)/ṽ(kb) increases.
5. Summary
We have considered the SALR potential representing a wide class of biological and soft matter
systems in MF approximation and in the case of weak ordering. We first discussed the properties
of such kind of potential, and next focused on determination of the phase diagram in reduced
variables. Our main goal was to determine the stability region of gyroid phase with Ia3d symmetry
for various shapes of the SALR potential. We have chosen two qualitatively different cases, I and
II, corresponding to very large clusters and to small clusters of colloids, respectively. We have
found that the phase diagrams in reduced units in these two cases are similar. The gyroid phase is
located between the hexagonal and the lamellar phase, and may be related to a branched network of
clusters observed experimentally and in simulations [6, 7, 14].The smaller is the difference between
the energy gain associated with the formation of a planar density wave with the wave number
k ≈ kb and the gain for the most probable wavelength k = kb, the more favorable is the gyroid
phase.
We expect that the free-energy landscape contains many local minima corresponding to a
distorted network, and that the formation of thermodynamically stable regular structure of the
gyroid phase in experimental system may be preempted by a glass-like disordered branched network
that locally resembles the gyroid phase. Confirmation of the existence of the predicted gyroid phase
by experiment or simulation remains a challenge.
Acknowledgement
This work is dedicated to Prof. Ihor Mryglod on the occasion of his 50 birthday. We grate-
fully acknowledge partial support by the Polish Ministry of Science and Higher Education, Grant
No. NN 202 006034.
6. Appendix. Explicit expressions of functions and parameters
Parameters in the functional (25)
A3(ρ) =
12s3 + 20s2 + 5s − 1
ρ2(1 − s)5
, (39)
A4(ρ) =
2 − 12s + 30s2 + 112s3 + 48s4
ρ3(1 − s)6
(40)
geometric factors in equations (30) and (34)
κbcc
3 =
√
4
3
, (41)
κbcc
4 =
15
4
, (42)
23603-10
Mesoscopic description of network-forming clusters of weakly charged colloids
κIa3d
3,0 =
1√
6
, (43)
κIa3d
4,0 =
17
8
, (44)
κIa3d
0,3 =
2√
3
, (45)
κIa3d
0,4 =
15
4
, (46)
κIa3d
2,1 = −
√
3
6
, (47)
κIa3d
2,2 =
1
2
, (48)
κhex
3 =
√
2
3
, (49)
κhex
4 =
5
2
, (50)
κlam
3 = 0 , (51)
κlam
4 =
3
2
, (52)
where κn,0 ≡ κn.
Expressions for the functions g1(x). We use the notation x = (x1, x2, x3).
g`
1(x) =
√
2 cos(kbx1), (53)
ghex
1 (x) =
√
2
3
[
cos(kbx1) + 2 cos
(kbx1
2
)
cos
(
√
3kbx2
2
)
]
, (54)
gbcc
1 (x) =
1√
3
∑
i<j
(
cos
(kb(xi + xj)√
2
)
+ cos
(kb(xi − xj)√
2
)
)
, (55)
gIa3d
1 (x) =
√
8
3
[
cos
(kbx1√
6
)
sin
(kbx2√
6
)
sin
(2kbx3√
6
)
+ cos
(kbx2√
6
)
sin
(kbx3√
6
)
sin
(2kbx1√
6
)
+ cos
(kbx3√
6
)
sin
(kbx1√
6
)
sin
(2kbx2√
6
)
]
. (56)
Expression for the function g2(x) for the gyroid phase
gIa3d
2 (x) =
1√
3
∑
i<j
(
cos
(2kb(xi + xj)√
6
)
+ cos
(2kb(xi − xj)√
6
)
)
. (57)
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Мезоскопiчний опис мережо-формуючих кластерiв слабо
заряджених колоїдiв
А. Цях., В.Т. Гузьдзь
Iнститут фiзичної хiмiї Польської Академiї Наук, Варшава, Республiка Польща
В рамках мезоскопiчної польової теорiї вивчаються системи, що складаються iз сферичних заря-
джених частинок у розчинниках, якi мiстять контрiони та iндукують короткосяжне притягання. Ми
обмежуємося наближенням середнього поля (СП) i слабким впорядкуванням. Ми обговорюємо
властивостi потенцiалiв, що складаються iз сильного короткосяжного притягання i слабкого дале-
косяжного вiдштовхування (SALR) у контекстi формування неоднорiдного розподiлу частинок на
мезоскопiчних масштабах довжин замiсть макроскопiчного фазового вiдокремлення. У попереднiй
роботi показано, що формуються сферичнi, цилiндричнi i щiлиноподiбнi кластери частинок, i для
достатньо низьких температур цi кластери формують впорядкованi перiодичнi об’ємоцентричнi, ге-
ксагональнi i ламеларнi фази. Крiм того, було передбачено гiроїдну фазу, в якiй формуються два
переплетенi регулярнi мережоподiбнi кластери, що розгалужуються в потрiйному з’єднаннi. Було
знайдено, що при вiдповiдному масштабуваннi густини i температури, лiнiї спiвiснування мiж рiзни-
ми впорядкованими фазами є унiверсальними в наближеннi СП, за винятком гiроїдної фази. У цiй
статтi визначається фазова дiаграма для двох виборiв потенцiалу SALR: такого, що вiдповiдає ве-
ликiй областi притягуючої частини потенцiалу, та такого, що вiдповiдає малiй областi притягання.
Ми виявляємо, що область стабiльностi гiроїдної фази дуже слабо залежить вiд форми потенцiалу
SALR у межах наближеної теорiї.
Ключовi слова: колоїди, кластери, самоорганiзацiя, фазовi переходи лад-безлад
23603-12
Introduction
The interaction potential and its properties
Field theory
First order transition to microsegregated phases
One-shell approximation
Two-shell approximation
Summary
Appendix. Explicit expressions of functions and parameters
|