On the statistical theory of visco-elastic properties of electrolyte solutions
Based on the generalized kinetic equation for single and pair correlation
 distribution functions, analytical expressions for dynamic coefficients of
 shear and bulk viscosity and corresponding elastic modulus are obtained.
 The frequency dispersion of these coefficients and...
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| Опубліковано в: : | Condensed Matter Physics |
|---|---|
| Дата: | 2001 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2001
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| Цитувати: | On the statistical theory of visco-elastic
 properties of electrolyte solutions
 / S. Odinaev, A. Dodarbekov // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 277-282. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860263446330736640 |
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| author | Odinaev, S. Dodarbekov, A. |
| author_facet | Odinaev, S. Dodarbekov, A. |
| citation_txt | On the statistical theory of visco-elastic
 properties of electrolyte solutions
 / S. Odinaev, A. Dodarbekov // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 277-282. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | Based on the generalized kinetic equation for single and pair correlation
distribution functions, analytical expressions for dynamic coefficients of
shear and bulk viscosity and corresponding elastic modulus are obtained.
The frequency dispersion of these coefficients and elastic modulus are
caused mainly by translational and structural relaxation processes. The
asymptotic behaviour of these expressions at low and high frequency is
investigated.
Отримано аналітичні вирази для динамічних коефіцієнтів зсувної
та об’ємної в’язкості та відповідних модулів пружності на основі
розв’язку узагальненого кінетичного рівняння для одно- та двочастинкових кореляційних функцій. Частотна дисперсія цих коефіцієнтів і модулів пружності зумовлена в основному трансляційними та
структурними релаксаційними процесами. Встановлена асимптотична низько- та високочастотна поведінка отриманих виразів.
|
| first_indexed | 2025-12-07T18:57:52Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 277–282
On the statistical theory of visco-elastic
properties of electrolyte solutions
S.Odinaev, A.Dodarbekov
Physical-Technical Institute of the Academy of Sciences of Tadjikistan,
229/1 Ajni Str., 734063 Dushanbe, Tadjikistan
Received August 5, 2000
Based on the generalized kinetic equation for single and pair correlation
distribution functions, analytical expressions for dynamic coefficients of
shear and bulk viscosity and corresponding elastic modulus are obtained.
The frequency dispersion of these coefficients and elastic modulus are
caused mainly by translational and structural relaxation processes. The
asymptotic behaviour of these expressions at low and high frequency is
investigated.
Key words: kinetic equation, translational and structural relaxation, shear
and bulk viscosity, elastic modulus, electrolyte solutions
PACS: 61.20.Qg, 66.10.Ed
Since electrolyte solutions are widely used, we need to know their viscoelastic,
thermoelastic, acoustic and electroconductive properties. A number of theoretical
and experimental works are devoted to the study of viscoelastic properties of solu-
tions. An especially relevant task is taking into account the modification of viscosity
stimulated by the following factors: contributions of the solution structure, hydra-
tion, charge and size of ions causing a modification of the solution structure as well
as the process of restoring the solution equilibrium (relaxation).
In works [1,2], starting from the kinetic equations for single and pair distribution
functions, expressions for static transport coefficients of a binary solution are ob-
tained which are expressed by means of molecular parameters and a pair distribution
function. In [3] the frequency dependence of the transport coefficients is generalized
by Zwanzig in the case of binary mixtures. The main attention is paid to the research
of the contribution of internal degrees of freedom to bulk viscosity and dependence
of transport coefficients on concentration. The viscosity coefficients, obtained in [4]
are presented as integrals of time correlation functions and their determination is a
difficult task.
It is known, that similar to the static case for the frequency dependence, the bulk
viscosity is many times larger than shear viscosity, and its magnitude is determined
by interactions of structural units of a solution. In water and in an aqueous solution
c© S.Odinaev, A.Dodarbekov 277
S.Odinaev, A.Dodarbekov
the bulk viscosity arises mainly because of structural relaxation processes [5]. The
theoretical calculations made by Fisher with co-workers [6] show that the hydration
of ions changes the bulk viscosity four times more than the shear one. The authors
explain by the fact that an essential moment for hydration is the compression of
water under the influence of the electrical field of ions.
The main achievement of the statistical theory of the transport and elastic prop-
erties of electrolyte solutions is the definition of kinetic factors and modules of
elasticity in two limiting cases of slow and fast processes. In the main, the existing
theories start from the offer concerning a single characteristic relaxation time of
kinetic factors which is not enough for the description of the processes of structural
relaxation. Moreover, the knowledge of frequency-dependent factors of transport
and modules of elasticity with considering the contribution of a structural relax-
ation, will permit to investigate theoretically such acoustic properties of solutions as
a variance of a sound velocity, a sound absorption coefficient, as well as spectrum of
collective modes. The present work is devoted to a research of viscoelastic properties
of electrolyte solutions based on the uniform microscopic theory with considering
the contribution of various relaxational processes.
We consider a single-phase electroneutral inhomogeneous system. Na, ma, da and
Nb, mb, db are the numbers, masses and diameters of structural units of a solution
of the sorts a and b, respectively. The particles of solution interact by means of
spherically-symmetrical potential Φab (|~r |), ~rab = ~q2 − ~q1 is a distance between the
particles. Let us assume that the solvent is a neutral medium, offering resistance to
the driven ions of the soluted substance and results in hydration.
In the case of small deviation of the solution from an equilibrium (in linear
approximation) the stress tensor is determined microscopically as follows [7]:
σαβ(q1, t) = −
∑
a
(
k
P aδ
αβ +Kαβ
a
)
+
1
2
∑
a
∑
b
∫ ∂Φab
∂r
·
rαrβ
r
nabd~r, (1)
where
~r = ~rab/dab
is the reduced distance between particles of the solution,
dab =
1
2
(da + db)
and
k
P a(~q1, t) =
1
3
∫ P̃ 2
a
ma
fad~pa
is the kinetic part of pressure,
Kαβ
a (~q1, t) =
1
ma
∫
(
P̃ α
a P̃
β
a −
1
3
P̃ 2
a δ
αβ
)
fad~pa
is the kinetic part of the viscous stress tensor,
nab(~q1, ~q2, t) =
∫
fabd~pad~pb
278
Visco-elastic properties of electrolyte solutions
is the binary density,
P̃ α
a = pαa −mav
a(~q1, t)
is the relative impulse of a particle of the sort a.
Using the kinetic equations for fa and fab, which consider the contribution of
spatial density correlation and velocity correlation [7] in the framework of superpo-
sition approximation for the three-particle distribution function fabc, for the P
k
a and
Kαβ
a one obtains:
∂
k
P a
∂t
+
5
3
k
P a(0)div~v +
2
3
∂
k
S
α
a
∂g21
−
2
3
∑
b
∫ ∂φab
∂rα
Jα
2(a)d~r = 0, (2)
∂Kαβ
a
∂t
+ 2
k
pa(0)
{
∂va
∂qβ1
}
+ 2
∂
k
S
α
a
∂qβ1
− 2
∑
b
∫
{
∂φab
∂rα
Jβ
2 (a)
}
d~r = −
2βa
ma
Kαβ
a . (3)
For a perturbed part of a binary density we shall receive the Smoluchowski
equation. Its solution has a form:
n
′
ab =
t
∫
0
dt1
∞
∫
−∞
Gab(r, r1, t− t1)Fab(~g1, ~r1, t1)d~r1, (4)
where Gab(r, r1, t − t1) is the solution to the homogeneous Smoluchowski equation
which can be written in the radial-symmetrical case as:
Gab =
2(rr1)
(2π)3
[π/ωab(t− t1)]
−1/2
{
exp
[
−
(r − r1)
2
4ωab(t− t1)
]
− exp
[
−
(r + r1)
2
4ωab(t− t1)
]}
,
(5)
Fab = −ϕabdiv~v − nanb
(
rarb −
1
3
r2δαβ
)
∂ ln gab
∂ ln r
{
∂va
∂qβ1
}
− (nanb/dab)
(
eb
βb
−
ea
βa
)
∂ ln g0ab
∂r
ra
r
Ea, (6)
ϕab = 2nanbg
0
ab
{
1 +
1
6
∂ ln g0ab
∂ ln r
−
1
2
+
[(
∂ ln g0ab
∂ lnn
)
t
+ γ
(
∂ ln g0ab
∂ lnT
)
n
]}
,
γ =
1
nCν
(
∂P
∂T
)
n
,
{
∂Mα
∂qβ
}
=
1
2
(
∂Mα
∂qβ
+
∂Mβ
∂qα
−
2
3
δαβdiv ~M
)
, Mα = vα,
k
S
α
a etc.,
ωab =
kT
d2ab
(
βa + βb
βaβb
)
,
βa, βb are the friction coefficients, g0
ab(r) is the equilibrium radial distribution func-
tion, na and nb are the number density of particles of sorts a and b, respectively,
k
S
α
a
is the kinetic part of heat of sort a.
279
S.Odinaev, A.Dodarbekov
Substituting (4) and also solutions (2) and (3) in case of independent flows into
(1), performing the Fourier-transformation on time, we get for the dynamic modules
of elasticity and viscosity coefficients:
K(ω) = Ks
+
(
ρ
N0
µ
)2
∑
a
∑
b
2π
3
CaCbd
3
abω
∞
∫
0
drr2
∂Φab
∂r
r
∫
0
Gab
2 (r, r1, ω)
∗
ϕab r1dr1, (7)
µ(ω) =
∑
a
k
Pa (0)(wτa)
2
1 + (ωτφ)2
+
(
ρ
N0
M
)2
∑
a
∑
b
2π
15
CaCbd
3
abω
∫
∞
0
drr2
φab
∂r
∫ r
0
Gab
2
∂g0ab
∂r1
r21dr1 , (8)
ηv(ω) =
(
ρ
N0
M
)2
∑
a
∑
b
2π
3
CaCbd
3
ab
∞
∫
0
drr3
∂φab
∂r
r
∫
0
Gab
1
∗
ϕab r1dr1, (9)
ηs(ω) =
∑
a
k
P a (0)τa
(1 + (ωτa)2)
+
(
ρ
N0
M
)2
∑
a
∑
b
2π
15
CaCbd
3
ab
∞
∫
0
drr2
∂Φab
∂r
r
∫
0
Gab
1
˙∂g0ab
∂r1
r1dr1, (10)
where
k
P a (0) = nakT0,
Cab
1,2(r, r1, ω) = ±
τab
2
(
2
ωτab
)
[
(sinϕ1 ∓ cosϕ1)e
−ϕ1 − (sinϕ2 ∓ cosϕ2)e
−ϕ2
]
,
τa = ω−1
a =
ma
2βa
, τab = ω−1
ab =
d2ab
kT
·
βaβb
βa + βb
,
∗
ϕ= ϕab/nanb, ϕ1,2 = (9ωτab/2)
1/2(r ∓ r1),
Ks is an adiabatic module of elasticity; ρ is the density of the solution,N 0 is the Avo-
gadro constant, M is molar mass of the solution, Ca and Cb are the concentrations
of particles of sorts a and b, respectively.
Expressions (7)–(10) describe a dynamic behaviour of viscoelastic properties of
electrolyte solutions. The frequency dependence of modules of elasticity and factors
of viscosity is caused, mainly, by the process of structural relaxation. This depen-
dence is determined by the behaviour of functions Gab
1 (r, r1, w) and Gab
2 (r, r1, w).
The translational and structural relaxations in solutions of electrolytes according to
(7)–(10) play unequal role. The relaxation of shear viscosity and shear module of
elasticity is both translational and structural, while the relaxation of bulk viscosity
and bulk module of elasticity is structural only.
The realization of numeric calculations of the obtained expression (7)–(10) de-
mands a concrete choice of the potential of interparticle interaction and the radial
equilibrium distribution function which is considered as known because of (8).
280
Visco-elastic properties of electrolyte solutions
Let us consider some limiting cases of the obtained results. At low frequencies,
when wτ ≪ 1, the bulk module of elasticity tends to its adiabatic value K s, and
shear module of elasticity tends to zero under the law ω3/2, and the bulk and shear
viscosity coefficients tend to their static values as linear functions of ω 1/2, which
completely agree with the results of works [9,10], obtained using the method of
molecular dynamics for liquids.
In a high-frequency regime, when ωτ ≫ 1, the modules of elasticity do not
depend on frequency and coincide in the form with high-frequency modules of elas-
ticity obtained by Zwanzig for liquids [11], and the factors of viscosity tend to zero
proportionally to ω−1.
Thus, conducted asymptotic evaluations of the obtained results for the solutions
of electrolytes completely correspond to general conclusions of the statistical theo-
ry of viscoelastic properties of liquids. At slow processes, the expressions (7)–(10)
describe viscous properties, and in the case of very fast processes, the same expres-
sions describe the elastic properties of solutions. In a high-frequency region, the
liquid behaves as amorphical solid state and besides the bulk module of elasticity,
the shear module of elasticity occurs which ensures the possibility of distribution of
both longitudinal and transversal acoustic modes in the solutions of electrolytes.
References
1. Bearman R.J., Kirkwood J.G. // Chem. Phys., 1958, vol. 28, No. 1, p. 136.
2. Naghizadeh J. // Chem. Phys., 1963, vol. 39, No. 12, p. 3406.
3. Haus J.W. // Chem. Phys., 1974, vol. 60, No. 7, p. 2438.
4. Mo K.S., Gubbins K.E. // Mol. Phys., 1976, vol. 31, No. 3, p. 825.
5. Erdey-Grous T. Transport Phenomena in Aqueous Solutions. Moskow, Mir, 1976.
6. Fisher I.Z., Zajtseva A.M. // Dokl. Acad. Nauk SSSR, 1964, vol. 154, p. 1175.
7. Odinaev S., Adchamov A.A. Molecular Theory of Structural Relaxation and Transport
Phenomena in Liquids. Dushanbe, Donish, 1998.
8. Yukhnovskii I.R., Holovko M.F. Statistical Theory of Classical Equilibrium Systems.
Kyiv, Naukova Dumka, 1980.
9. Lagarkov A.N., Sergeev V.M. // Usp. Fiz. Nauk, 1978, vol. 125, No. 3, p. 409.
10. Evans D.J., Henly C.J., Hess Z. Non-Newtonian phenomena in simple liquids. – In
collection “Fizika za rubezhom”, Moskow, Mir, 1986, p. 7–28.
11. Zwanzig R., Montain R.D. // Chem. Phys., 1965, vol. 43, No. 12, p. 4464.
281
S.Odinaev, A.Dodarbekov
До статистичної теорії в’язкопружних властивостей
розчинів електролітів
С.Одінаєв, А.Додарбеков
Фізико-технічний інститут Академії наук Республіки Таджикистан,
Республіка Таджикистан, 734063 Душанбе, вул. Айні 229/1
Отримано 30 серпня 2000 р.
Отримано аналітичні вирази для динамічних коефіцієнтів зсувної
та об’ємної в’язкості та відповідних модулів пружності на основі
розв’язку узагальненого кінетичного рівняння для одно- та двочас-
тинкових кореляційних функцій. Частотна дисперсія цих коефіцієн-
тів і модулів пружності зумовлена в основному трансляційними та
структурними релаксаційними процесами. Встановлена асимпто-
тична низько- та високочастотна поведінка отриманих виразів.
Ключові слова: кінетичне рівняння, трансляційна та структурна
релаксація, зсувна та об’ємна в’язкість, модулі пружності, розчини
електролітів
PACS: 61.20.Qg, 66.10.Ed
282
|
| id | nasplib_isofts_kiev_ua-123456789-120433 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:57:52Z |
| publishDate | 2001 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Odinaev, S. Dodarbekov, A. 2017-06-12T07:37:37Z 2017-06-12T07:37:37Z 2001 On the statistical theory of visco-elastic
 properties of electrolyte solutions
 / S. Odinaev, A. Dodarbekov // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 277-282. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 61.20.Qg, 66.10.Ed DOI:10.5488/CMP.4.2.277 https://nasplib.isofts.kiev.ua/handle/123456789/120433 Based on the generalized kinetic equation for single and pair correlation
 distribution functions, analytical expressions for dynamic coefficients of
 shear and bulk viscosity and corresponding elastic modulus are obtained.
 The frequency dispersion of these coefficients and elastic modulus are
 caused mainly by translational and structural relaxation processes. The
 asymptotic behaviour of these expressions at low and high frequency is
 investigated. Отримано аналітичні вирази для динамічних коефіцієнтів зсувної
 та об’ємної в’язкості та відповідних модулів пружності на основі
 розв’язку узагальненого кінетичного рівняння для одно- та двочастинкових кореляційних функцій. Частотна дисперсія цих коефіцієнтів і модулів пружності зумовлена в основному трансляційними та
 структурними релаксаційними процесами. Встановлена асимптотична низько- та високочастотна поведінка отриманих виразів. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the statistical theory of visco-elastic properties of electrolyte solutions До статистичної теорії в’язкопружних властивостей розчинів електролітів Article published earlier |
| spellingShingle | On the statistical theory of visco-elastic properties of electrolyte solutions Odinaev, S. Dodarbekov, A. |
| title | On the statistical theory of visco-elastic properties of electrolyte solutions |
| title_alt | До статистичної теорії в’язкопружних властивостей розчинів електролітів |
| title_full | On the statistical theory of visco-elastic properties of electrolyte solutions |
| title_fullStr | On the statistical theory of visco-elastic properties of electrolyte solutions |
| title_full_unstemmed | On the statistical theory of visco-elastic properties of electrolyte solutions |
| title_short | On the statistical theory of visco-elastic properties of electrolyte solutions |
| title_sort | on the statistical theory of visco-elastic properties of electrolyte solutions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120433 |
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