1. SOP TRANSACTIONS ON THEORETICAL PHYSICS
ISSN(Print): 2372-2487 ISSN(Online): 2372-2495
DOI: 10.15764/TPHY.2014.03008
Volume 1, Number 3, September 2014
SOP TRANSACTIONS ON THEORETICAL PHYSICS
Phase Diagram of Colloids Immersed in
Binary Liquid Mixtures
M. Badia1, K .Elhasnaoui3*, A. Maarouf3, T.El hafi3, M. Benhamou2,3
1 Royale Air School, Mechanical Dept, DFST, BEFRA, P.O.Box 40002, Menara, Marrakech
2 ENSAM ,Moulay Ismail University P.O.Box 25290, Al Mansour, Meknes
3 LPPPC, Sciences Faculty Ben M’sik, P.O.Box 7955, Casablanca, (Morocco)
*Corresponding author: elhasnaouikhalid@gmail.com, abdelwahad.maarouf@gmail.com
Abstract:
We consider an assembly of spherical colloids of radius R immersed in a A — B binary liquid
mixture close to its consolute point Tc It is assumed that particles prefer to be surrounded by
one of the two components (A). Fluctuations of composition imply a reversible flocculation of
these particles in the non-preferred B-rich phase, which originates from a universal long-range
attractive Casmir potential with a well established decay r 2xy (the critical exponent 2xy is about
1.03). The aim is a quantitative study of the phase diagram shape of this colloid System,
using the random-phase approximation with a hard-sphere reference. The phase diagram
is drawn in the h t plane, where h is the packing fraction and t = (T Tc)/Tc the reduced
shift of temperature T from the consolute point Tc. The problem is governed by three relevant
parameters, which are the packing fraction h, the temperature shift t and an energy strength
˜u(attraction energy between two particles separated by a distance that is equal to the hard-
sphere diameter s per kBT unit). This energy is given by an explicit universal function depending
only on the ratio of the real particle diameter b = 2R to s. We first determine the coordinates
(h⇤, t⇤) , of the critical point, where t⇤ is the temperature shift at the critical temperature t⇤. To
a given value of the r 2xy energy strength ˜u corresponds a value of the temperature shift t⇤.
All possible points (t⇤, ˜u) constitute a continuous critical line, along which the colloid System
undergoes a phase transition from liquid state to gas state. We show that this curve is described
by a universal equation, i.e. ˜u ⇠ (t⇤)gt
where gt is the critical exponent characterizing the
behavior near the consolute point, of the compressibility of binary liquid mixture without colloids.
Second, we determine the exact from of the spinodal curve in the (h/h⇤, t/t⇤) plane. We show
that this curve is universal. Third, we determine exactly the state equation of colloid fluid and put
it on a universal form. We finish by drawing the complete phase diagram involving coexistence
and spinodal curves in the (h/h⇤, t/t⇤) plane.
Keywords:
Colloids; Hard Sphere Casimir Potential; Phase Diagram
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2. Phase Diagram of Colloids Immersed in Binary Liquid Mixtures
1. INTRODUCTION
We start by considering a binary mixture made of two incompatible liquids A and B. At some critical
temperature (consolute point), the mixture phase separates. From a thermo-dynamic point of view,
phase separation is a real phase transition, which is characterized by the appearance of macrodomains
alternatively rich in A and B liquids. The size of these domains is the thermal correlation length x,
which becomes divergent when the consolute point is reached, i.e. x ⇠ a|T Tc| nt . Here, a is a
typical microscopic length and nt a critical exponent. Phase separation is thus accompanied by long-
wavelength fluctuations extending over spatial distances of order of x. These fluctuations are responsible
for phenomena investigated below.
The real System we consider here consists of an assembly of inert colloidal particles immersed in
a binary liquid mixture near its consolute temperature Tc. Colloids are particles of mesoscopic size,
which are the subject of numerous theoretical and experimental studies, because of their abundant
industrial applications. To simplify, colloids will be assumed to be spherical of the same diameter b = 2R
(monodisperse System). Thus, we are in the presence of a critical binary liquid mixture in contact with
spherical surfaces.
The physics of phase separation of binary liquid mixtures in contact with interacting surfaces is a very
rich problem. For example, near the consolute point, colloids have tendency to be surrounded by one
component (A) of the mixture [1]. Colloids clothed by A-liquid located in the B-rich sida reversibly
flocculate duo to the presence of long-wavelengtli fluctuations of composition. Quantitatively, this
flocculation results from a universal long-range attractive Casimir potential with a well established decay
r 2xy [2], where the critical exponent 2xy is about 1.03 (see below). This problem was discussed in a
short note by Fisher and de Gennes [3]. Very recently, in a series of experiments [4, 5], one has considered
silica beads with diameter about 0.1µm immersed in lutidine (A)-water (B) binary mixture. Near the
consolute point, it was found that the silica colloids exhibit a sharply defined reversible flocculation. Those
colloids prepared using the St?ber method are known to adsorb on lutidine preferentially [4]. Aggregation
takes place in the water rich-side of the phase diagram.
The purpose of this work is the determination of the phase diagram shape of an assembly of spherical
colloids of the same diameter b = 2R immersed in a binary liquid mixture close to the consolute point.
These colloidal particles attract each other by an attractive Casimir potential, whose form will be
precised below. Here, the effective medium (solvent) is the non-preferred phase (B). Use will be made
of the random-phase approximation (RPA) [6] with a hard-sphere reference. The phase diagram will
be drawn in the h t plane, where h is the packing fraction and t = (T Tc)/Tc the reduced shift of
temperature T from the consolute temperatureTc. As we will see below, three relevant parameters govern
the physics of the problem, namely, the packing fraction h, the temperature shift t and an energy strength
˜u, which is the dimensionless attraction energy between two particles separated by a (minimal) distance a
that is equal to the hard-sphere diameter. As shown below, the energy strength ˜u is an explicit universal
function depending only on the ratio of the real particle diameter b = 2R to s .
Our findings are the following. Within the framework of RPA, we first determine analytically the
coordinates (h ⇤, t⇤) of the critical point, which is the top of the coexistence curve of colloid System.
Here, t* is the temperature shift at the critical temperature T⇤ that should not be confused with consolute
point Tc. To a given value of the energy strength `u (or equivalently of the ratio b/s ) corresponds a value
of the temperature shift t*. All possible points (t⇤, ˜u) constitute a critical line (C), along which the
colloid system undergoes a phase transition from a liquid state to a gas state. We show that this curve is
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3. SOP TRANSACTIONS ON THEORETICAL PHYSICS
described by a universal equation, that ˜u ⇠ (t⇤)gt
, where gt is the critical exponent defining the behavior
near the consolute point of the compressibility relative to the binary mixture without colloids. Second,
we determine the exact form of the spinodal curve in the (h/h*, t/t*)-plane. We show that this curve is
universal. Finally, we determine the parametric equations defining the coexistence curve and draw the
complete phase diagram in the (h/h⇤, t/t⇤) plane.
The presentation of the paper is planned as follows. Sect. 2 is devoted to a succinct recall of the
attractive Casimir potential expression. We present in Sect. 3 the RPA formulation of the problem. In
Sect. 4, we investigate the exact shape of the phase diagram within the framework of RPA. We draw our
conclusions in Sect. 5.
2. THE CASIMIR INTERACTION POTENTIAL
We first start by recalling some useful backgrounds. Consider a binary mixture made of two incompa-
tible liquids A and B. We denote by fA and fB their respective compositions. For incompressible mixture,
these compositions are related by the incompressibility condition, i.e. fA +fB = 1
To study the phase behavior near the consolute point, one introduces an order parameter y, which is
the composition fluctuation of one species, say A. It is defined as the distance between two points along
the coexistence curve, that is
y =
fA fB
fA +fB
· (1)
The critical behavior is usually investigated through the connected two-point correlation function
defined as usual by [7–9]
G r r0
=< y (r)y r0
> < y (r) >< y r0
> · (2)
The notation < . > indicates the thermal average over ail possible configurations of the order parameter.
The Fourier transform of the correlation function is directly proportional to the scattering intensity, which
can be measured in a light of neutron scattering experiment.
We recall that the two-point correlation function behaves at small distances compared to the thermal
correlation length x as [7–9]
G r r0
'
B
|r r0|2xy
, r r0
x · (3)
The characteristic length x becomes singular when the consolute point is approached, i.e.
x ⇠ a|t| nt
, t =
T Tc
Tc
, (4)
where a is a typical microscopic length, t the reduced shift of temperature from the consolute point Tc
and nt a thermal critical exponent. In relationship (3), By is some known amplitude [2] and xy the scaling
dimension of field y, which is related to the traditional thermal critical exponent ht by [7–9]
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4. Phase Diagram of Colloids Immersed in Binary Liquid Mixtures
xy =
1
2
(d 2+ht) (5)
where d is the space dimensionality. The three-dimensional best value of exponent xy was found to be
xy ' 0.515 [2] . Of course, the correlation function vanishes (exponentially) for distances much larger
than x.
Now, consider an assembly of N spherical colloids immersed in a binary liquid mixture close to its
consolute point. We assume that these colloids have the same radius R. Thus, the polydispersity effects
are neglected. Recently, it was established that fluctuations of composition imply a long-range attractive
Casimir force [2] between particles located in the non-preferred phase, say (B). Each colloid is surrounded
by the preferred phase (A).
Intuitively, the Casimir interaction potential Uc (r r0) between two particles separated by a distance
|r r0| must be proportional to the two-point correlation function. As a matter of fact, one has shown the
result [2]
Uc (r r0)
kBT
= Ey R2xy
G r r0
(6)
where R is the colloid radius and Ey some known universal amplitude [2].
Since the values of the two-point correlation function are appreciable only for distances much smaller
than the length x, thon, the expected Casimir potential reads
Uc
kBT
' Dy
✓
b
r
◆2xy
, (r < x). (7)
Here, b = 2R is the colloid diameter and Dy a universal amplitude. From its expansion to first order in
e = 4 d(4 is the critical dimension of the System) extracted from Ref. [2], we find for this amplitude
the three-dimensional value Dy ' 6.64. The result (7) was established using the so-called small-sphere
expansion combined with conformal invariance [2].
The above expression for the Casimir effective potential call several remarks. First, this potential has a
universal character, independently on the chemical details of system. Second, the value of exponent xy ,
at d = 3 implies a decay r 1.03of the Casimir potential for large separations. In fact, this power law is
conform with the decay r 1as found by de Gennes using an approximate theory [10]. We note that the
above form of the Casimir potential is the same for two widely separated colloidal particles immersed in
a one-component fluid near the liquid-vapor critical point. Finally, this result was extended to helium
at lambda temperature [2], whose scaling dimension xe (of composite field y2 is xe = d 1/nt. At
three-dimensions, one has xe ' 1.51 [2]. This corresponds to a slow decay r 3.02 of the Casimir potential
with an amplitude of order of unity.
In principle, we have all ingredients to investigate the phase diagram of colloids immersed in binary
liquid mixtures close to their consolute point, using the established form (7) of the effective interaction
potential. This is precisely the aim of the following section.
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5. SOP TRANSACTIONS ON THEORETICAL PHYSICS
3. THE RPA FORMULATION
To determine the shape of the phase diagram, use will be made of the RPA method [6, 11]. It is known
that this approach is reliable only for low particle density. To use it, we will need a reference potential.
We choose the simpler one, which is the hard sphere-potential
U0 (r) =
(
•, r s
0, r s
(8)
where s is the hard-sphere diameter s b. By reference System, we mean a colloid System at the
same temperature T and bulk density r but in which the interaction potential is U0 (r).
Then, the effective potential governing the physics of the colloid System is the sum
U (r) = U0 (r)+Uc (r), (9)
where Uc (r) is the long-range Casimir potential (s < r < x) defined by Eq. (7). Beyond the thermal
Figure 1. Reduced interaction potential U (r)/kBTversus the reduced distance r/s.
correlation length, that is for r x, the interaction potential goes exponentially to zero. The above
decomposition means that, there is a competition between the Casimir potential favoring the flocculation
of colloids and the hard-sphere potential favoring rather their dispersion. The shape of potential U (r) is
depicted in Figure 1. The spirit of the RPA consists of writing the direct correlation function c(r) of a
monodisperse colloidal dispersion as
c(r) = co (r) bUc (r) (10)
Here, b = 1/kBT and co (r) is the direct correlation function of reference system. The Fourier transform
of the function c(r) is defined by
˜c(q) =
Z
dr c(r) exp(iq.r) (11)
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6. Phase Diagram of Colloids Immersed in Binary Liquid Mixtures
where q is the wave-vector. This transform is related to the structure factor S (q) through the standard
Ornstein-Zernike relation
S(q) = 1 r ˜c(q) (12)
Fourier transforming relationship (10) and according to the above equality, one finds the following
expression for the structure factor within the framework of RPA
S(q) = S0(q)
⇥
1+brS0(q) ˜Uc(q)
⇤ 1
(13)
Here, eUc (q) is the Fourier transform of Casimir potential, which is directly proportional to the scattering
intensity from a binary liquid mixture without colloids. We will need the hard-sphere direct correlation
function expression at vanishing wave-vector. It is a function of the packing fraction h = (p/6)rs3
given by [6]
r ˜c0 =
8h +2h2
(1 h)4
(14)
(Here, the parameter h should not be confused with the thermal critical exponent ht defining the critical
behavior of compressibility of the A — B mixture without colloids).
We are now interested in the quantitative nature of the spinodal decomposition, which drives the System
from small colloid density state (gas) to high colloid density one (liquid). The desired phase diagram is
generally drawn in the (h,T) plane.
4. THE PHASE DIAGRAM
4.1 The Critical Point
The top of the spinodal curve is the critical point where the System undergoes a phase separation. Its
coordinates (h⇤,T⇤) can be determined solving the coupled equations
1 ˜c(0) = 0, (15)
∂
∂h
[1 ˜c(0)] = 0. (16)
Explicitly, we find
1+4h +4h2 4h3 +h4
(1 h)4
+br eUc (0) = 0 (17)
8+20h 4h2
(1 h)5
+b
6
ps3
eUc (0) = 0. (18)
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7. SOP TRANSACTIONS ON THEORETICAL PHYSICS
By subtraction, we obtain the critical packing fraction
h⇤ ' 0.128. (19)
Incidentally, note that, within the framework of RPA, the critical packing fraction is independent on the
form of interaction potential.
To get the critical temperature T*, we will need the expression of eUc (0) appearing in System (17)-(18).
We first set
u = kBT Dy
✓
b
s
◆2xy
, (20)
which is the contact energy between two particles, i.e. energy at minimum separation r = s Expression
of quantity eUc (0) can be obtained integrating the potential given by relation (7) over the r-variable running
from s to x, to find
b ˜Uc (0) ' 4p ˜u(2 ht)s3
✓
x
s
◆ht 2
, (21)
with the notation ˜u = u/kBT. We have used the scaling relations: d 2xy = gt/nt and gt = nt (2 ht)
[7–9].
Near the consolute point Tc, we have the scaling law x ⇠ s t nt , with an amplitude of order unity.
This implies that the quantity b eUc (0) defined by Eq. (21) scales as
b ˜Uc (0) ' 4p ˜u(2 ht)s3
t gt
, (22)
where t is the temperature shift defined above. Therefore, the contribution of the Casimir interaction
scales as the isotherm compressibility (or susceptibility).
We are know all ingredients to determine the critical temperature T*. We first set
t⇤
=
T Tc
Tc
, (23)
which is the corresponding temperature shift. In fact, to each value of the dimensionless energy strength
˜u corresponds a value of the critical temperature parameter t*. The choice of ˜u is equivalent to give the
ratio of the particle diameter b to the hard-sphere one s. We have then a critical line along which the
colloidal System undergoes a phase transition, which will be drawn in the (t, - ˜u)-plane. The analytical
shape of this line can be deduced from Eqs. (17) and (22). We find
˜u ' A⇤
(t⇤
) gt
, (24)
with the amplitude
A⇤ ' 0.138. (25)
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8. Phase Diagram of Colloids Immersed in Binary Liquid Mixtures
The critical line is thus governed by the thermal critical exponent gt whose best three-dimensional value
is gt ' 1.2411 [8]. We have used the best value of the critical exponent ht at d = 3, which is ht ' 0.031
[8]. The shape of critical line (C) of Eq. (24) is depicted in Figure 2.
Figure 2. Critical line (C) in the (t , ˜u )-plane.
This curve separates two domains I and II. The former corresponds to high energy strength, where
colloids have tendency to flocculate (liquid state). Low energy strength domain II corresponds rather to a
dispersion of colloids (gas state).
4.2 The Spinodal Curve
The spinodal curve is given by Eq. (17) emanating from the divergence condition of the compressibility
of colloid System. Without details, we give the equation defining this curve in the (h,t) plane
t = t⇤
[f (h)/ f (h)] 1/gt
, (26)
With
f (h) =
1+4h +4h2 4h3 +h4
h (1 h)4
. (27)
Spinodal curve of Eq. (26) calls the following important remark. First, note that this can be rewritten
on the following form, in terms of reduced variables h/h⇤ and t/t⇤
t
t⇤ = g(h/h⇤) (28)
Where
g h h⇤ = f (h) (h⇤). (29)
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9. SOP TRANSACTIONS ON THEORETICAL PHYSICS
Second, the function g(x) is universal, that is independent on the chemical nature of the problem, on one
hand, and the geometric form of colloids (through their radius R), on the other hand. The non-universality
is completely contained in critical temperature parameter t*. The spinodal curve of Eq. (28) is shown in
Figure 3.
Figure 3. Spinodal curve in the (h/h*, t/t*)-plane.
Now, it remains us to determine the state equation and coexistence curve of colloid System.
4.3 The State Equation
Such an equation can be derived from the relationship giving the isotherm compressibility kT of colloid
fluid, i.e.
rkT = ∂r ∂ P (30)
where P is the pressure and r the particle density. Within the framework of RPA, the isotherm
compressibility is given by
kBTr kT = [1 r ˜c(0)] 1
(31)
where ˜c(0) is the Fourier transform of the direct correlation function at vanishing wavevector. Combin-
ing Eqs. (30) and (31) and after integration over the r variable, we find the following equation for the
pressure
P = P0 +
r
2
eUc (0) (32)
where Po is the hard-sphere pressure that is given by the following h-depending function [6]
P0
r kBT
=
1+h +h2
(1 h)3
. (33)
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10. Phase Diagram of Colloids Immersed in Binary Liquid Mixtures
Relationship (32) tells us that the attractive Casimir interaction yields a reduction of the pressure.
Introduce now the dimensionless energy parameter
w =
6
p s3
·
eUc (0)
kBT
. (34)
At the critical point, we find from Eq. (18) that the critical value w* of this parameter is a pure number,
i.e.
w⇤
=
8+20h⇤ +8(h⇤)2
(1 h⇤)5
' 21.205 (35)
We have use the fact that h⇤ ' 0.128. Now, set
ep ⌘ P
rkBT (36)
that is a dimensionless pressure. Combing Eqs. (22), (24) and (32) to (35) to obtain the universal state
equation
˜p= j ( ˜h, ˜t) (37)
with the notation ˜h ⌘ h/h⇤, t ⌘ t/t⇤. The two-factor universal scaling function j is found to be
j ( ˜h, ˜t)=
1+h⇤. ˜h +(h⇤)2
. ˜h2
(1 h⇤. ˜h)3
w⇤h⇤
2
· ˜h ·(˜t) gt
(38)
All non-universality is entirely contained in the critical temperature shift t* (through˜t). By universality,
we mean that this state equation within the RPA is independent on the chemical nature of A and B-
components and the radius R of colloids.
4.4 The Coexistence Curve
To determine the shape of such a curve, we will need the chemical potential expression. The starting
point is the classical Gibbs-Duhem relation, i.e.
✓
∂P
∂r
◆
T
= r
✓
∂µ
∂r
◆
T
, (39)
where P is the pressure given by relationship (32). Integration of Eq. (39) over the r-variable yields
µ
kBT
=
µ0
kBT
w⇤
·h(
t
t⇤
) gt
(40)
with
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11. SOP TRANSACTIONS ON THEORETICAL PHYSICS
µ0
kBT
=
1+h +h2
(1 h)5
+
3
2(1 h)3
+ln(
h
1 h
) (41)
which defines the hard-sphere chemical potential µ0. Universal relationship (40) shows a chemical
potential reduction due to the attractive interaction between colloidal particles.
The analytical form of the coexistence curve can be determined through the equality of pressures and
chemical potentials at coexistence, i.e.
P(hI) = P(hII), (42)
µ (hI) = µ (hII), (43)
where P is the pressure given by Eqs. (36) to (38) and µ the chemical potential defined by Eqs. (40)
and (41). There hI and hII stand for the packing fractions of the two phases at coexistence. We show that
the above equalities yield the relationships
h
✓
hI
h⇤
,
hII
h⇤
◆
= 0, (44)
t
t⇤
= k
✓
hI
h⇤
,
hII
h⇤
◆
, (45)
where h (x, y) and k(x, y) are known two-factor universal scaling functions. These parametric equations
give the coexistence curve.
The complete phase diagram involving coexistence and spinodal curves is shown in Figure 4.
Figure 4. Complete phase diagram shape in the (h/h*, t/t*)-plane.
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12. Phase Diagram of Colloids Immersed in Binary Liquid Mixtures
5. CONCLUSIONS
We considered an assembly of spherical colloids of the same radius R immersed in a binary liquid
mixture close to its consolute point Tc. We assumed that particles prefer to be surrounded only by one
component (preferred phase). Long-wavelength fluctuations of composition imply a reversible flocculation
of colloids in the other side of the phase diagram (non-preferred phase). This flocculation originates from
a universal long-range attractive Casimir potential with a well established decay r 1.03. The aim was a
quantitative study of the phase diagram structure of these colloids, within the framework of RPA with a
hard-sphere reference. The phase diagram is drawn in the h — t plane, where h is the packing fraction
and t = (T — Tc) /Tc the reduced shift of temperature T from the consolute point Tc. Three relevant
parameters govern the physics of the problem, which are the packing fraction h, the temperature shift t
and a dimensionless energy strength ˜u, which is the attraction energy between two particles separated by
a (minimal) distance that is equal to the hard-sphere diameter s , per kBT unit. This energy is an explicit
universal function depending only on the ratio of the real particle diameter b = 2R to s. We have first
determined the coordinates (h*, t*) of the critical point, where h* is the critical packing fraction and t*
the temperature shift at the critical temperature T*. To a given value of the energy strength ˜u corresponds
a value of the temperature shift t*. All possible points (t*, ˜u) constitute a critical line (C), along which
the colloid System undergoes a phase transition from a liquid state to a gas state. We have shown that,
such a curve can be described by a universal equation, i.e. ˜u ⇠ (t⇤)gt
, where gt is the critical exponent
defining the behavior near the consolute point of the compressibility of the binary liquid mixture without
colloids. Second, we have determined the exact form of the spinodal curve in the (h/h*, t/t*)-plane.
We have shown that this curve is universal. Also, we determined the parametric equations defining the
coexistence curve and drawn the complete phase diagram in the (h/h*, t/t*)-plane.
We emphasize that the present analysis can be extended, in a straightforward way, to helium at lamda-
point, substituting the critical exponent xy ' 0.515 by xe ' 1.51 and the amplitude Dy ' 6.64 appearing
in Eq. (7) by De ' 1.
Extension of the present studies to charged colloidal particles immersed in critical binary liquid mixtures
is in progress.
ACKNOWLEDGMENTS
We are much indebted to Professors M. Daoud and J.-L. Bretonnet, for helpful discussions.
References
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[4] D. Beysens and D. Est`eve, “Adsorption phenomena at the surface of silica spheres in a binary liquid
mixture,” Physical Review Letters, vol. 54, no. 19, p. 2123, 1985.
119

13. SOP TRANSACTIONS ON THEORETICAL PHYSICS
[5] M. L. Broide, Y. Garrabos, and D. Beysens, “Nonfractal colloidal aggregation,” Physical Review E,
vol. 47, no. 5, p. 3768, 1993.
[6] J. Hansen and I. McDonald, “Theory of Simple Liquids,” 1976.
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SOP welcomes authors to contribute their research outcomes under the following rules:
Although glad to publish all original and new research achievements, SOP can’t bear any
misbehavior: plagiarism, forgery or manipulation of experimental data.
As an international publisher, SOP highly values different cultures and adopts cautious attitude
towards religion, politics, race, war and ethics.
SOP helps to propagate scientific results but shares no responsibility of any legal risks or harmful
effects caused by article along with the authors.
SOP maintains the strictest peer review, but holds a neutral attitude for all the published articles.
SOP is an open platform, waiting for senior experts serving on the editorial boards to advance the
progress of research together.