To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions
We have obtained an equation describing space-time behaviour of the current density component by using kinetic equation for one-particle distribution function for the structural units of the solution with the generalized Vlasov potential. The analytic expression for the complex tensor of electroco...
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| Cite this: | To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions / S. Odinaev, I. Ojimamadov // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 735–740. — Бібліогр.: 8 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1190192025-06-03T16:27:05Z To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions До статистичної теорії дисперсії тензорів електропровідності і діелектричної сприйнятливості розчинів електролітів Odinaev, S. Ojimamadov, I. We have obtained an equation describing space-time behaviour of the current density component by using kinetic equation for one-particle distribution function for the structural units of the solution with the generalized Vlasov potential. The analytic expression for the complex tensor of electroconductivity σ(ω) is given derived from the Fourier-transform and from the comparison with the differential form of the Ohm’s law. This permitted us to obtain the dielectric susceptibility tensor ε(ω) for conducting media. By identifying the longitudal εk and transversal ε⊥ parts one can determine the anisotropy of the dielectric susceptibility for electrolyte solutions. Отримано рівняння, яке описує просторово-часову поведінку компоненти густини струму, використовуючи кінетичне рівняння для одночастинкової функції розподілу структурних компонент розчину з узагальненим потенціалом Власова. Представлено аналітичний вираз для комплексного тензора електропровідності σ(ω) , який виведений з Фур’є-перетворення і з диференціальної форми закону Ома. Це дало змогу отримати тензор діелектричної сприйнятливості ε(ω) для провідного середовища. Виділяючи поздовжню εk і поперечну ε⊥ частини можна визначити анізотропію діелектричної сприйнятливості для електричних розчинів. 2004 Article To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions / S. Odinaev, I. Ojimamadov // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 735–740. — Бібліогр.: 8 назв. — англ. 1607-324X DOI:10.5488/CMP.7.4.735 PACS: 61.20.Qg, 51.10.+y https://nasplib.isofts.kiev.ua/handle/123456789/119019 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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We have obtained an equation describing space-time behaviour of the current density component by using kinetic equation for one-particle distribution function for the structural units of the solution with the generalized Vlasov potential.
The analytic expression for the complex tensor of electroconductivity σ(ω) is given derived from the Fourier-transform and from the comparison with the differential form of the Ohm’s law. This permitted us to obtain the dielectric susceptibility tensor ε(ω) for conducting media. By identifying the longitudal εk and transversal ε⊥ parts one can determine the anisotropy of the dielectric susceptibility for electrolyte solutions. |
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Odinaev, S. Ojimamadov, I. |
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Odinaev, S. Ojimamadov, I. To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions Condensed Matter Physics |
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Odinaev, S. Ojimamadov, I. |
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Odinaev, S. |
| title |
To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions |
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To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions |
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To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions |
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To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions |
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To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions |
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to the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions |
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Інститут фізики конденсованих систем НАН України |
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2004 |
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To the statistic theory of dispersion of tensors of electric conductivity and dielectric susceptibility of electrolyte solutions / S. Odinaev, I. Ojimamadov // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 735–740. — Бібліогр.: 8 назв. — англ. |
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Condensed Matter Physics |
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AT odinaevs tothestatistictheoryofdispersionoftensorsofelectricconductivityanddielectricsusceptibilityofelectrolytesolutions AT ojimamadovi tothestatistictheoryofdispersionoftensorsofelectricconductivityanddielectricsusceptibilityofelectrolytesolutions AT odinaevs dostatističnoíteoríídispersíítenzorívelektroprovídnostíídíelektričnoíspriinâtlivostírozčinívelektrolítív AT ojimamadovi dostatističnoíteoríídispersíítenzorívelektroprovídnostíídíelektričnoíspriinâtlivostírozčinívelektrolítív |
| first_indexed |
2025-12-02T09:05:52Z |
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1850386785655324672 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 4(40), pp. 735–740
To the statistic theory of dispersion of
tensors of electric conductivity and
dielectric susceptibility of electrolyte
solutions
S.Odinaev 1 ∗, I.Ojimamadov 2
1 Physics-Technical Institute, Academy of Sciences,
Republic of Tajikistan, 299/1 Aini Ave, Dushanbe, 734063
2 M.S.Osimi Tajik Technical University,
Akad. Rajabovs Str., 10, Dushanbe 734042, Republic of Tajikistan
Received April 28, 2004
We have obtained an equation describing space-time behaviour of the cur-
rent density component by using kinetic equation for one-particle distri-
bution function for the structural units of the solution with the generalized
Vlasov potential.
The analytic expression for the complex tensor of electroconductivity σ(ω)
is given derived from the Fourier-transform and from the comparison with
the differential form of the Ohm’s law. This permitted us to obtain the di-
electric susceptibility tensor ε(ω) for conducting media.
By identifying the longitudal ε‖ and transversal ε⊥ parts one can determine
the anisotropy of the dielectric susceptibility for electrolyte solutions.
Key words: kinetic equation, distribution functions, frequency dispersion,
dielectric susceptibility tensor, electric conductivity tensor
PACS: 61.20.Qg, 51.10.+y
1. Introduction
Rather high accuracy and easiness of measurement of electroconductivity has
long attracted the efforts of numerous researchers who have gathered enormous
experimental material. However theoretical research of electroconductivity of elec-
trolyte solutions is one of the most complicated and difficult questions of physics of
liquid state. Research of irreversible processes in electrolyte solutions, in particular
of electroconductivity, belongs to Onsager. Generalization on the cases of conduc-
tivity of an alternating current was carried out by Debye and Falkenhagen. Further
∗E-mail: ods@ttu.tajik.net
c© S.Odinaev, I.Ojimamadov 735
S.Odinaev, I.Ojimamadov
development of the theory of electroconductivity, based on the use of nonequilibri-
um multiparticle distribution functions, is presented in works [1–3]. Undoubtedly,
the possibility to determine dynamic coefficients of electroconductivity and dielec-
tric susceptibility is of great interest. Thus, the purpose of the present work is to
determine these parameters as well as the appropriate modules of elasticity in view
of anisotropy depending on constituents, structure of solution and thermodynamic
parameters based on the molecular-kinetic theory.
First of all we accept the kinetic equation for one-particle distribution function
fa(~χa, t) of a type structural units of electrolyte solutions with generalized Vlasov
potential [4]:
∂fa
∂t
+
pα
a
ma
·
∂fa
∂qα
1
+ eaE
α(~q1, t)
∂fa
∂pα
a
+ Fα
a (~q1, t)
∂f 0
a
∂pα
a
= Ic(fa), (1)
where
Fα
a (~q1, t) =
∑
b
∫
{
Fα
ab(l)δ(t − τ) − Ωe−Ω(t−τ)(Fα
ab(l) − Fα
ab(c))
+ e−Ω(t−τ)Fα
ab(el)
}
dτ, (2)
Ic(fa) = βa
∂
∂pα
a
[
p̃α
a
ma
fa + kT (~q1, t) ·
∂fa
∂pα
a
]
, (3)
Fα
ab(`) = nb
∫
∂2Φab
∂r2
·
rαrβ
r2
go
ab(r)
(
Uβ(~q2, t) − Uβ(~q1, t)
)
d~r,
Fα
ab(c) = nb
∫
∂2Kab
∂r2
·
rαrβ
r2
(
Uβ(~q2, t) − Uβ(~q1, t)
)
d~r,
Fα
ab(e`) = nb
(
eb
βb
−
ea
βa
)
·
∫
∂2Φab
∂r2
·
rαrβ
r2
go
ab(r)d~r · Eβ(~q1, t),
Kα
ab(r) =
|~q2−~q1|
∫
−∞
∂Φab(y)
∂y
· Zab(y)dy, ~r = ~q2 − ~q1,
Zab(y) =
[
no
(
∂go
ab(r)
∂no
)
T
+ γTo
(
∂go
ab(r)
∂To
)
n
]
,
γ =
1
noCv
(
∂Po
∂To
)
no
, ~xa = ( ⇀→ qa,
−→p a),
f o
a (pa) = na(2πmkT0)
−3/2 exp(−p2
a/2makT0) – is equilibrium Maxwell distribution
function; ma , ~qa , ~pa – are mass, coordinate and momentum of particles of a type,
respectively, p̃α
a = pα
a −maυ
α(~q1, t) – is relative momentum, ea = zae , eb = zbe , e – is
an elementary charge; za , zb , βa , βb – are valency and friction coefficients of particles
of a and b types, respectively, Eα(~q1, t) – are the components of electric field, Φab(|~r|)
– is the potential energy of interaction between the structural units of solution, and
736
To the statistic theory of dispersion of tensors
g0
ab(r) – is radial distribution function, describing equilibrium structure of solution,
which according to [5], we consider as known, ~Ua(~q, t) – is displacement vector of
particles of a type, Ω = Ωab = τ−1
ab = (kT/d2
ab) · (1/βa + 1/βb) is phenomenological
frequency of structural relaxation, k is Boltzman’s constant, dab = 1/2(da +db) , da ,
db – are diameters of structural units of solutions of a and b types. Right-hand side of
the equation (1), i.e. Ic(fa) is the collision Fokker-Plank term, which is derived in the
approximation of pair interactions. It provides irreversibility of the initial equation
in time, i.e. the possibility to describe the dissipation processes in solutions. For
convenience, let us put the coordinates of particles ~qa = ~q1 , ~qb = ~q2 , ~qc = ~q3
and so on. It should be mentioned that particles of the solution interact by the
potential Φab(|~r|, Ωs), which consists of the sum of the energy of inter-ionic (kations
and anions) Φij , ion-molecular Φis and Φjs , and intermolecular Φss interactions.
Here Ωs = (υs, αs) – are polar angles describing the orientation of the dipole around
the axis, connecting mass centers of interacting particles.
Let us use the definitions of pulse moments of the function fa(~xa, t) according to
[6] and introduce the vector of density of current ~j(~q, t) :
ρa(~q1, t) = eana(~q1, t) = ea
∫
fa(~xa, t)d~pa ,
ρaυ
α
a (~q1, t) = ea
∫
pα
a
ma
fa(~xa, t)d~pa ,
Kαβ
aa (~q1, t) =
∫
p̃α
a p̃β
a
ma
fa(~xa, t)d~pa ,
~j(~q1, t) =
∑
a
~ja(~q1, t) =
∑
a
ρa(~υa − ~υ(~q1, t)), (4)
as well as conditions of electroneutrality
∑
a
eana = e
∑
a
zana = 0,
where ρa , υα
a , Kαβ
aa – are volumetric density of charge, components of average
speed and kinetic part of the tensor of flow of momentum of particles of a type,
respectively.
Using the method of pulse moments of one-particle distribution function fa(~xa, t)
, multiplying equation (1) by (eap
α
a )/ma and integrating by d~pa , taking into account
equations (2)–(4), we derive for the components of the vector of density of current
jα
a (~q1, t) , the following equation:
∂jα
a (~q1, t)
∂t
+ νaj
α
a =
n0
ae
2
a
ma
[
Eα(~q1, t) −
∑
b
1
ea
∫
~e Ω(t−τ)Fα
ab(el)dτ
]
, (5)
where
νa = τ−1
a = βa/ma .
737
S.Odinaev, I.Ojimamadov
Performing Fourier-transformation by time in (5) and solving it in regard to
jα
a (ω, ~q) , for the components of the vector of density of current jα(ω) =
∑
a
jα
a (ω, ~q1),
we receive
jα(ω) =
∑
a
n0
ae
2
a/βa
1 − iωτa
[
δαβ +
∑
b
τcG
αβ
el (0)/ea
1 − iωτc
]
Eβ(ω), (6)
where
Gαβ
el (0) = n0
b
(
eb
βb
−
ea
βa
)
dab
∫
∂2Φab
∂r2
·
r2rβ
r2
g0
ab(r)d~r. (7)
Comparing (6) with the Fourier-image of the differential form of Ohm law, for
complex tensor of electroconductivity σ̃αβ(ω) we have:
σ̃αβ(ω) =
∑
a
nb
ae
2
a/βa
1 − iωτa
[
δαβ +
∑
b
τc · G
αβ
el (0)
ea(1 − iωτc)
]
. (8)
As to each process of transition in hydrodynamic mode, the certain elastic prop-
erties in a high-frequency mode will correspond. Further, according to [6] we shall
introduce the complex tensor of electroelasticity module:
∈̃
αβ
(ω) = −iωσ̃αβ (ω) =∈αβ (ω) − iωσαβ (ω) , (9)
where the real part ∈αβ (ω)− is the dynamic tensor of electroelasticity module, and
imaginary part σαβ (ω)− is the dynamic tensor of electroconductivity.
Substituting (8) into (9), and dividing real and imaginary parts for ∈αβ (ω) and
σαβ (ω) , we receive:
∈αβ (ω) =
∑
a
∈0
a
[
(ωτ̂a)
2δαβ + ω2τ̂aτ̂c(τa + τc)
∑
b
Gαβ
el
ea
]
,
σαβ (ω) =
∑
a
∈0
a
[
τ̂aδ
αβ + τ̂aτ̂c(1 − ω2τaτc)
∑
b
Gαβ
el
ea
]
, (10)
where
τ̂a =
τa
1 + (ωτa)
2 , τ̂c =
τc
1 + (ωτc)
2 ,
∈0
a= n0
ae
2/ma , τ and τc is time of translational and structural relaxation, respecti-
vely.
The expressions in system (10) describe the frequency dispersion of electroelas-
tic modules and coefficients of electroconductivity in a wide interval of variations
of thermodynamic parameters and frequencies of external action. In these expressi-
ons the frequency dependence is caused by contributions of both translational and
structural relaxation. Potential parts of these coefficients contain the integral terms
which are defined by means of energy of interaction of structural units of the solution
Φab(r) and equilibrium radial distribution function g0
ab(r) . At a certain choice of the
model of solution, according to [5], the latter could be considered as known. Debye
738
To the statistic theory of dispersion of tensors
and Falkengagen [7], being restricted by the real part of electroconductivity, investi-
gated frequency dispersion electroconductivity of electrolyte solutions. The second
equation of the system (10) is the generalization of the effect of Debye-Falkenhagen
for electrolyte solutions, the frequency behaviour of which coincides with the results
of general relaxation theories. Formulas of the system (10) make it possible to in-
vestigate the asymptotic behaviour of these expressions both in hydrodynamic and
in high-frequency mode and correspond to the general conclusions of the statistical
theory of electroelastic properties of solutions.
At low frequencies (hydrodynamic mode ω → 0) expressions (10) describe elec-
troconductivity properties of solutions, and at fast processes (high-frequency mode
ω → ∞) – only the elastic properties could be described.
Expressions (10) also permit to investigate dispersion of dielectric susceptibility
of solutions. Following paper [8], it is also possible to determine a frequency dis-
persion of tensor of dielectric susceptibility ε̃αβ (ω) for conducting media, which is
connected to conductivity tensor σ̃αβ (ω) by the ratio
ε̃αβ (ω) = δαβ +
i
ε0ω
σ̃αβ(ω). (11)
The latter enables one to determine longitudinal σ‖, ε‖ and transversal σ⊥, ε⊥
components of these coefficients, as well as based on the Maxwell equations, to
determine the longitudinal ~E‖
(
ω,~k
)
and transversal ~E⊥
(
ω,~k
)
parts of the vector
of electric field ~E
(
ω,~k
)
with regard to its wave vector ~k in electrolyte solutions,
which is the purpose of the further research.
References
1. Ebeling W., Feistel R., Kelbg G., Sanding R. Generalizations of Onsagers semi phe-
nomenological theory of electrolytic conductance. // J. non-equilibr. thermodlyn.,
1978, vol. 3, No. 1, p. 11–28.
2. Sänding R. Theory of linear vectors transport processes in binary isothermal elec-
trolyte solutions. // Z. Phys. Chem. (DDR), 1984, vol. 265, No. 4, p. 663–680.
3. Lessner G. The electric conductivity of stationary and homogenous electrolytes up
to concentration C = 1 mol/L and high electric fields. // Physica A, 1982, vol. 116,
No. 1–2, p. 272–288; Physica A, 1983, vol. 122, No. 3, p. 441–458.
4. Odinaev S., Ojimamadov I. About one kinetic equation with generalized Vlasov’s
potential. – In: Int. conf. PLMMP, abstracts. Kyiv, September 14–19, 2001, p. 35.
5. Yukhnovsky I.R., Holovko M.F. Statistic Theory of Classical Equilibrium Systems.
Kiyv, Naukova Dumka, 1980, p. 372, (in Russian).
6. Odinaev S., Adkhamov A.A. Molecular Theory of Structural Relaxation and Transi-
tion Phenomena in Liquids. Dushanbe, Donish, 1998, p. 230, (in Russian).
7. New issues of contemporary electrochemistry, (ed. J.M.Bokris). Moscow, Inostrannaya
literature, 1962, p. 462, (in Russian).
8. Klimontovich Yu.Ya. Statistic Theory of Non-Equilibrium Phenomena in Plasmas.
Moscow, Moscow State University Publisher, 1964, p. 281, (in Russian).
739
S.Odinaev, I.Ojimamadov
До статистичної теорії дисперсії тензорів
електропровідності і діелектричної
сприйнятливості розчинів електролітів
С.Одінаєв 1 , І.Оджімамадов 2
1 Фізико-технічний інститут, Академія наук,
Душанбе, Республіка Таджикистан
2 Таджицький технічний університет ім. М.С.Осімі,
Душанбе, Республіка Таджикистан
Отримано 28 квітня 2004 р.
Отримано рівняння, яке описує просторово-часову поведінку ком-
поненти густини струму, використовуючи кінетичне рівняння для
одночастинкової функції розподілу структурних компонент розчину
з узагальненим потенціалом Власова.
Представлено аналітичний вираз для комплексного тензора елек-
тропровідності σ(ω) , який виведений з Фур’є-перетворення і з
диференціальної форми закону Ома. Це дало змогу отримати
тензор діелектричної сприйнятливості ε(ω) для провідного сере-
довища. Виділяючи поздовжню ε‖ і поперечну ε⊥ частини можна
визначити анізотропію діелектричної сприйнятливості для елек-
тричних розчинів.
Ключові слова: кінетичне рівняння, функції розподілу, частотна
дисперсія, тензор діелектричної сприйнятливості, тензор
електропровідності
PACS: 61.20.Qg, 51.10.+y
740
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