The order parameter in binary mixtures
Phase transitions in a binary mixture are investigated by means of the collective variables method with a reference system. It is shown that the system is described with two sets of collective variables: η~k and ξ~k. The CV connected with the order parameter is η~k₌₀ in the case of the...
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| Cite this: | The order parameter in binary mixtures / O.V. Patsahan // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 235-241. — Бібліогр.: 10 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1202632025-02-09T20:57:40Z The order parameter in binary mixtures Параметр порядку у бінарних сумішах Patsahan, O.V. Phase transitions in a binary mixture are investigated by means of the collective variables method with a reference system. It is shown that the system is described with two sets of collective variables: η~k and ξ~k. The CV connected with the order parameter is η~k₌₀ in the case of the gas-liquid critical point as well as in the case of the mixing-demixing phase transition. The particular form of η0 for each of these phenomena can be determined by means of the relations between the microscopic parameters,temperature, density and concentration of the system under consideration. Фазові переходи у бінарній суміші досліджуються з допомогою методу колективних змінних з виділеною системою відліку. Показано, що система описується двома наборами колективних змінних: η~k і ξ~k . Колективною змінною,зв’язаною з параметром порядку є змінна η~k₌₀ як у випадку критичної точки газ-рідина, так і у випадку фазового переходу змішування-незмішування. Конкретна форма η0 для кожного з цих явищ визначається співвідношенням між температурою, густиною, концентрацією і мікроскопічними параметрами системи, яка розглядається. 1999 The order parameter in binary mixtures / O.V. Patsahan // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 235-241. — Бібліогр.: 10 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.235 PACS: 05.70.Fh, 05.70.Jk, 64.10.+h https://nasplib.isofts.kiev.ua/handle/123456789/120263 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Phase transitions in a binary mixture are investigated by means of the
collective variables method with a reference system. It is shown that
the system is described with two sets of collective variables: η~k and ξ~k.
The CV connected with the order parameter is η~k₌₀ in the case of the
gas-liquid critical point as well as in the case of the mixing-demixing
phase transition. The particular form of η0 for each of these phenomena
can be determined by means of the relations between the microscopic
parameters,temperature, density and concentration of the system under
consideration. |
| author |
Patsahan, O.V. |
| spellingShingle |
Patsahan, O.V. The order parameter in binary mixtures Condensed Matter Physics |
| author_facet |
Patsahan, O.V. |
| author_sort |
Patsahan, O.V. |
| title |
The order parameter in binary mixtures |
| title_short |
The order parameter in binary mixtures |
| title_full |
The order parameter in binary mixtures |
| title_fullStr |
The order parameter in binary mixtures |
| title_full_unstemmed |
The order parameter in binary mixtures |
| title_sort |
order parameter in binary mixtures |
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Інститут фізики конденсованих систем НАН України |
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1999 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120263 |
| citation_txt |
The order parameter in binary mixtures / O.V. Patsahan // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 235-241. — Бібліогр.: 10 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT patsahanov theorderparameterinbinarymixtures AT patsahanov parametrporâdkuubínarnihsumíšah AT patsahanov orderparameterinbinarymixtures |
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2025-11-30T16:57:05Z |
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2025-11-30T16:57:05Z |
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1850235237685002240 |
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Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 235–241
The order parameter in binary mixtures
O.V.Patsahan
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 290011 Lviv, Ukraine
Received August 31, 1998, in final form November 30, 1998
Phase transitions in a binary mixture are investigated by means of the
collective variables method with a reference system. It is shown that
the system is described with two sets of collective variables: η~k and ξ~k.
The CV connected with the order parameter is η~k=0
in the case of the
gas-liquid critical point as well as in the case of the mixing-demixing
phase transition. The particular form of η0 for each of these phenomena
can be determined by means of the relations between the microscopic
parameters,temperature, density and concentration of the system under
consideration.
Key words: binary mixture, collective variables, phase transition, order
parameter
PACS: 05.70.Fh, 05.70.Jk, 64.10.+h
1. Introduction
The choice of the order parameter in multicomponent fluid mixtures is a serious
problem because the character of the phase transition can change continuously from
the pure gas-liquid transition to a mixing-demixing one. The question of the physical
nature of the order parameter in binary fluid mixtures has been considered until
recently from the point of view of both the phenomenological theory [1,2] and the
microscopic approach [3]-[6]. Nowadays the commonly accepted idea is that both the
gas-liquid and mixing-demixing phase transitions are accompanied by total density
fluctuations as well as by relative density fluctuations. The only symmetrical mixture
exhibits a complete distinction between these two processes [6]. However, such an
“ideal” system is not likely to occur in reality. In real mixtures the contribution from
each type of the fluctuation processes changes along the critical curve. The evaluation
of such contributions at each critical curve point is essential to the definition of the
order parameter and to the understanding of the phase transition character in the
mixture. In our approach the question of the physical nature of the order parameter
seems to have a consistent and clear solution.
c© O.V.Patsahan 235
O.V.Patsahan
In this paper we propose a microscopic approach to the study of phase transitions
in multicomponent fluids. It is based on the method of collective variables (CV)
with a distinguished reference system (RS) [7,8]. The point is that the statistical
description of the phase transition is to be performed in the appropriate phase space
specific to a certain physical model. Among the independent variables of this space
there should be those connected with order parameters. This phase space forms a set
of CV . Each of them is a mode of density fluctuations corresponding to the specific
feature of the model under consideration. In particular, for a magnetic system the CV
are variables connected with spin density fluctuation modes, for a one-component
fluid – with particle density fluctuation modes. What is the content of the CV for a
multicomponent system? We will answer this question below.
2. Functional representation of the grand partition function of
a binary mixture
Let us consider a classical two-component system of interacting particles con-
sisting of Na particles of species a and Nb particles of species b. The volume of the
system is V , the system temperature is T .
Let us assume that the interaction in the system has a pairwise additive charac-
ter. The interaction potential between a γ particle at ~ri and a δ particle at ~rj may
be expressed as a sum of two terms:
Uγδ(rij) = ψγδ(rij) + φγδ(rij),
where ψγδ(r) is a potential of a short-range repulsion that will be chosen as an
interaction between two hard spheres σγγ and σδδ. φγδ(r) is an attractive part of
the potential which dominates at large distances. An arbitrary positive function
belonging to the L2 class can be chosen as the potential φγδ(r).
Further consideration of the problem is done in the extended phase space: in
the phase space of the Cartesian coordinates of the particles and in the CV phase
space. An interaction connected with repulsion (potential ψγδ(r)) is considered in
the space of the Cartesian coordinates of the particles. We call this two-component
hard spheres system a reference system (RS). The interaction connected with an
attraction (potential φγδ(r)) is considered in the CV space. The phase space overflow
is cancelled by introducing the transition Jacobian. The contribution of the short-
range forces to the long-range interaction screening is ensured by averaging this
Jacobian over the RS.
Then a grand partition function in the CV representation with a RS can be
written as
Ξ = Ξ0Ξ1,
where Ξ0 is the grand partition function of the RS. The thermodynamic and struc-
tural properties of the RS are assumed to be known. We assume that in the region of
temperatures, concentrations and densities we are interested in, the thermodynamic
236
The order parameter in binary mixtures
functions of the RS remain analytic. Ξ1 has the following form:
Ξ1 =
∫
(dρ) (dc) exp
[
βµ+
1 ρ0 + βµ−
1 c0 −
β
2V
∑
~k
[Ṽ (k)ρ~kρ−~k
+ W̃ (k)c~kc−~k
+ Ũ(k)ρ~kc−~k
]
]
J(ρ, c). (2.1)
Here the following notations are introduced:
ρ~k and c~k are the CV connected with total density fluctuation modes and relative
density (or concentration) fluctuation modes in the binary system.
Functions µ+
1 and µ−
1 have the form:
µ+
1 =
√
2
2
(µa
1 + µb
1), µ−
1 =
√
2
2
(µa
1 − µb
1) (2.2)
and are determined from the equations
∂ ln Ξ1
∂βµ+
1
= 〈N〉, (2.3)
∂ ln Ξ1
∂βµ−
1
= 〈Na〉 − 〈Nb〉; (2.4)
Ṽ (k) =
(
φ̃aa(k) + φ̃bb(k) + 2φ̃ab(k)
)
/2,
W̃ (k) =
(
φ̃aa(k) + φ̃bb(k)− 2φ̃ab(k)
)
/2,
Ũ(k) =
(
φ̃aa(k)− φ̃bb(k)
)
/2, (2.5)
J(ρ, c) =
∫
(dω) (dγ) exp
[
i2π
∑
veck
(ωkρk + γkck)
]
J(ω, γ), (2.6)
J(ω, γ) = exp
[
∑
n>1
∑
in>0
(−i2π)n
n!
∑
~k1...~kn
M (in)
n (0, . . . , 0)
× γ~k1 . . . γ~kinω~kin+1
. . . ω~kn
]
. (2.7)
Index in is used to indicate the number of variables γ~k in the cumulant expansion
(2.6). CumulantsM
(in)
n are expressed as linear combination of the partial cumulants
Mγ1...γn and are presented for γ1, . . . , γn = a, b and n 6 4 in [6] (see Appendix B in
[6]).
Formulas (2.1)–(2.7) are the initial formulas in our study of phase transitions in
binary fluids.
237
O.V.Patsahan
3. The order parameter in a binary mixture
Let us consider the functional integral in Gaussian approximation (2.1)–(2.7).
This approximation, also known as the random-phase approximation, yields the
correct qualitative picture of the phenomena under consideration. After integration
over variables γk and ωk, Ξ1 can be rewritten as
ΞG
1 =
1
2π
∏
~k
′ 1
π
1
√
∆(k)
∫
(dρ) (dc) exp
[
ρ0(βµ
+
1 + ℵ1/∆)
+ c0(βµ
−
1 + ℵ2/∆)− (M
(0)
1 ℵ1 +M
(1)
1 ℵ2)
− 1
2
∑
~k
[ρ~kρ−~k
A11(k) + c~kc−~k
A22(k) + 2ρ~kc−~k
A12(k)]
]
, (3.1)
where
ℵ1 =M
(2)
2 M
(0)
1 −M
(1)
2 M
(1)
1 , ℵ2 =M
(0)
2 M
(0)
1 −M
(1)
2 M
(0)
1 ,
A11(k) = −1
2
( β
V
Ṽ (k) +
M
(2)
2
∆
)
,
A22(k) = −1
2
( β
V
W̃ (k) +
M
(0)
2
∆
)
,
A12(k) = −1
2
( β
V
Ũ(k)− M
(1)
2
∆
)
, (3.2)
∆ =M
(0)
2 M
(2)
2 − (M
(1)
2 )2.
In order to determine the phase space of the CV connected with the order parameters
we introduce independent collective excitations by diagonalizing the square form in
(3.1) by means of the orthogonal transformation:
ρ~k = A(k)η~k +B(k)ξ~k ,
c~k = C(k)η~k +D(k)ξ~k . (3.3)
The explicit forms for coefficients A(k), B(k), C(k) and D(k) are given in Ap-
pendix A.
As a result, we have
ΞG
1 =
1
2π
∏
~k
′ 1
π
1
√
∆(k)
∫
(dη) (dξ) exp
[
η0(AM1 + CM2) + ξ0(BM1 +DM2)
− (M
(0)
1 ℵ1 +M
(1)
1 ℵ2)/∆(0)− 1
2
∑
~k
(ε11(k)η~kη−~k
+ ε22(k)ξ~kξ−~k
)
]
, (3.4)
where
εii(k) = −(A11(k) + A22(k)∓
√
(A11(k)−A22(k))2 + 4A2
12(k)). (3.5)
238
The order parameter in binary mixtures
One of the quantities (3.5) (or both) tends to zero as the critical temperature
is approached at a certain wave vector ~k∗. Thus the CV η~k∗ (or ξ~k∗) can be iden-
tified as the order parameter, where the wave vector ~k∗ should correspond to the
minimum of one of the functions ε11(k) or ε22(k) (or both). These functions depend
on temperature, the attractive potentials φ̃γδ(k) and the characteristics of the RS.
The RS enters into (3.5) via the cumulants Mγδ(k). Mγδ(k) can be expressed by
the Fourier transforms of the direct correlation functions Cγδ(k) using the Ornstein-
Zernike equations for a mixture.
Figure 1. Density-concentration projection of the mean field critical line of the
model binary mixture at α = 1.0, q = 0.9 and r = 0.6 (α = σaa/σbb, q =
−φ̃bb(0)/|φ̃aa(0)|, r = −φ̃bb(0)/|φ̃ab(0)|).
Coefficients ε11(k) and ε22(k) are studied both as wave vector functions at dif-
ferent values of temperature T , density and concentration including the gas-liquid
and mixing-demixing critical points [9] and as temperature functions at ~k = 0 [10].
The results show that branch ε11(k) is the first to reach zero no matter whether
the system approaches the gas-liquid or gas-gas demixing critical point. Moreover,
ε11(k) and ε22(k) have the minima at ~k = 0 [9]. Hence we can draw the following
conclusions:
1. Branch ε11(k) is always critical.
2. Because ε11(k) has the minimum at ~k = 0, the CV connected with the order
parameter is η0 in the case of the gas-liquid critical point as well as in the case
of the mixing-demixing phase transition. The particular form of η 0 for each
239
O.V.Patsahan
of these phenomena can be determined by means of the relations between the
microscopic parameters, temperature, density and concentration of the system,
e.g. by means of coefficients A(0), B(0), C(0) and D(0).
Thus, the proposed approach enables us, on microscopic grounds, to define the
order parameter at each point along a critical curve and to understand the character
of the phase transition in the binary mixture. Figure 1 shows the (η, c) projection
of the (T, η, c) critical surface of the model binary mixture (η is the packing density
of the mixture, c is the concentration of species b). The arrows show the direction
of the strong fluctuations (order parameter) along the critical curve.
Based on the Gaussian distribution (3.1)–(3.2) we have determined the critical
branch and, correspondingly, the CV η0 connected with the order parameter. The
purpose of our further study will be calculating the binary mixture behaviour in the
vicinity of its critical points.
Appendix A
The coefficients A(k), B(k), C(k) and D(k) have the forms:
A =
√
2|A12|[4A2
12 + (A11 − A22)
2 − (A11 −A22)
√
(A11 −A22)2 + 4A2
12]
−1,
B =
√
2|A12|[4A2
12 + (A11 − A22)
2 + (A11 − A22)
√
(A11 − A22)2 + 4A2
12]
−1,
C = −
√
2
2
|A12|
(A12)
[A11 − A22 −
√
(A11 − A22)2 + 4A2
12]
× [4A2
12 + (A11 − A22)
2 − (A11 −A22)
√
(A11 −A22)2 + 4A2
12]
−1,
D = −
√
2
2
|A12|
(A12)
[A11 − A22 +
√
(A11 −A22)2 + 4A2
12]
× [4A2
12 + (A11 − A22)
2 + (A11 − A22)
√
(A11 − A22)2 + 4A2
12]
−1.
References
1. Anisimov M.A., Gorodetskii E.E., Kulikov V.D., Sengers J.V. // Phys. Rev. E, 1995,
vol. 51, No. 2, p. 1199–1215.
2. Anisimov M.A., Gorodetskii E.E., Kulikov V.D., Povodyrev A.A., Sengers J.V. //
Physica A, 1995, vol. 220, p. 277–324.
3. Chen X.S., Forstmann F. // J. Chem. Phys., 1992, vol. 97, No. 5, p. 3696–3703.
4. Parola A., Reatto L. // J. Phys.: Condens. Matter, 1993, vol. 5, p. B165–B172.
5. Parola A., Reatto L. // Advan. in Phys. 1995, vol. 44, No. 3, p. 211–298.
6. Yukhnovskii I.R., Patsahan O.V. // J. Stat. Phys., 1995, vol. 81, No. 3/4, p. 647–672.
7. Yukhnovskii I.R., Holovko M.F. Statistical Theory of Classical Equilibrium Systems.
Naukova Dumka, Kiev, 1980.
8. Patsagan O.V., Yukhnovskii I.R. // Theor. Math. Phys., 1990, vol. 83, No. 1, p. 387–
397.
240
The order parameter in binary mixtures
9. Patsahan O.V., The phase transitions in binary systems. I. Random phase approxi-
mation. Preprint Inst. Cond. Matter Phys. Acad. Sci. Ukraine, ICMP–92–2U, Lviv,
1992 (in Ukrainian).
10. Patsahan O.V. // Ukr. Fiz. Zh., 1996, vol. 41, No. 9, p. 877–884 (in Ukrainian).
Параметр порядку у бінарних сумішах
О.В.Пацаган
Інститут фізики конденсованих систем НАН Укpаїни,
290011 Львів, вул. Свєнціцького, 1
Отримано 31 серпня 1998 р., в остаточному вигляді –
30 листопада 1998 р.
Фазові переходи у бінарній суміші досліджуються з допомогою ме-
тоду колективних змінних з виділеною системою відліку. Показано,
що система описується двома наборами колективних змінних: η~k і
ξ~k . Колективною змінною,зв’язаною з параметром порядку є змінна
η~k=0
як у випадку критичної точки газ-рідина, так і у випадку фазового
переходу змішування-незмішування. Конкретна форма η0 для кож-
ного з цих явищ визначається співвідношенням між температурою,
густиною, концентрацією і мікроскопічними параметрами системи,
яка розглядається.
Ключові слова: бінарна суміш, колективна змінна, фазовий
перехід, параметр порядку
PACS: 05.70.Fh, 05.70.Jk, 64.10.+h
241
242
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