Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures
The statistical model of the water solution of radioactive elements and the porous clayey matrix is proposed. The generalized transport equations for the description of diffusion, sorption, radiative processes and chemical reactions are obtained taking into account the electromagnetic processes.
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1181962017-05-30T03:04:22Z Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures Tokarchuk, M.V. Hlushak, P.A. Krip, I.M. Shymchuk, T.V. The statistical model of the water solution of radioactive elements and the porous clayey matrix is proposed. The generalized transport equations for the description of diffusion, sorption, radiative processes and chemical reactions are obtained taking into account the electromagnetic processes. Запропоновано стастичну модель "водний розчин радiоактивних елементiв - пориста глиниста матриця". Отримано узагальненi рiвняння переносу для опису дифузiї, сорбцiї, радiоактивних процесiв та хiмiчних реакцiй з врахуванням електромагнiтних процесiв. 2007 Article Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures / M.V. Tokarchuk, P.A. Hlushak, I.M. Krip, T.V. Shymchuk // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 179-188. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 05.60.+w, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi DOI:10.5488/CMP.10.2.179 http://dspace.nbuv.gov.ua/handle/123456789/118196 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The statistical model of the water solution of radioactive elements and the porous clayey matrix is proposed.
The generalized transport equations for the description of diffusion, sorption, radiative processes and chemical
reactions are obtained taking into account the electromagnetic processes. |
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Tokarchuk, M.V. Hlushak, P.A. Krip, I.M. Shymchuk, T.V. |
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Tokarchuk, M.V. Hlushak, P.A. Krip, I.M. Shymchuk, T.V. Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures Condensed Matter Physics |
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Tokarchuk, M.V. Hlushak, P.A. Krip, I.M. Shymchuk, T.V. |
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Tokarchuk, M.V. |
| title |
Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures |
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Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures |
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Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures |
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Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures |
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Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures |
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reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures |
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Інститут фізики конденсованих систем НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118196 |
| citation_txt |
Reactionary - electrodiffusion equations of transport processes of electrolyte solutions of radioelements through porous clayey structures / M.V. Tokarchuk, P.A. Hlushak, I.M. Krip, T.V. Shymchuk // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 179-188. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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2025-07-08T13:32:30Z |
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2025-07-08T13:32:30Z |
| _version_ |
1837085807975858176 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 2(50), pp. 179–188
Reactionary – electrodiffusion equations of transport
processes of electrolyte solutions of radioelements
through porous clayey structures
M.V.Tokarchuk1, P.A.Hlushak1, I.M.Krip2, T.V.Shymchuk2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii
Street, 79011 Lviv, Ukraine
2 National University “Lvivska Polytekhnika”, 12 Bandera Str., 79013 Lviv, Ukraine
Received March 27, 2007 in final form May 23, 2007
The statistical model of the water solution of radioactive elements and the porous clayey matrix is proposed.
The generalized transport equations for the description of diffusion, sorption, radiative processes and chemical
reactions are obtained taking into account the electromagnetic processes.
Key words: diffusion, sorption, electromagnetic processes, porous clayey matrix, Chornobyl catastrophe
PACS: 05.60.+w, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi
1. Introduction
The experimental and theoretical studies of migration processes of radionuclides in soils [1,2]
which were intensively carried out prior to Chornobyl catastrophe, have shown that the diffusion
processes are the basis mechanisms of the radionuclide transport. These processes become more
complicated because the soil has such features as heterogeneity, porosity, capability of adsorbing
ions, etc. Therefore, the study of the mechanisms of adsorption, desorption, radionuclide diffusion
in soil are of considerable interest from the viewpoint of their use in designing the radiation-
absorbing barriers. These barriers can be constructed as obstacles on the way of subsoil waters
which contain isotopes of caesium, strontium, europium, plutonium, americium, etc. Such a com-
position of isotopes in subsoil waters is characteristic of Chornobyl zone. Bentonite clay is of special
interest while constructing such barriers. The phenomenological approach [2] with the application
of the modified diffusion equation (the so-called Fick equation) is used for a general theoretical
description of radionuclide diffusion in a soil. Additional parameters, which characterize the prop-
erties of a soil, are present in the equation of radionuclide diffusion. The soil is considered to
be a continuous medium with stationary values of diffusion constants of radionuclides which are
determined experimentally. Probably, such an approach cannot describe complicated migration
processes of radionuclides including specific interactions with ions, molecules, and colloid particles
in the soils. The mathematical modelling of radionuclide transport in the upper soil layers [3–10] is
performed on the basis of macroscopic diffusion and hydrodynamic equations in which the trans-
port coefficients, namely, diffusion coefficient and adsorption-desorption constants are the unknown
parameters which should be primarily determined experimentally. Apparently, such an approach
is incapable of describing the complex processes of radionuclide migration taking into account the
specific interactions between ions, molecules, colloidal particles. Radioactive elements are capable
of creating different forms of aquated ions, molecules, double and mixed complexes, mononuclear
and polynuclear hydrolysates, colloidal particles and products of the radiolysis processes [11]. This
is a specific feature of water solutions that contain radioactive elements 235,238U,238,239Pu, 241Am,
90Sr, 134,137Cs, 242,244Cm and others. From the analysis of radionuclide forms in water and other
solutions it follows that the radionuclides basically create the composite complexes in three forms:
c© M.V.Tokarchuk, P.A.Hlushak, I.M.Krip, T.V.Shymchuk 179
M.V.Tokarchuk et al.
cationic, anionic and neutral (colloidal particles, polymers), which follows from the analysis of
radionuclide forms in water and other solutions.
In particular, the sorption of plutonium in a “subsoil water – natural soil” system is investigated
in [12] using a radiometric method (through α – radiation of 239Pu) depending on the concentration
of plutonium, its oxidative state and the composition of the soil. Subsoil water of carbonate-
bicarbonate type with pH=7 contains ions Pu4+, PuO2+
2 , PuO2+
2 in this system. It is shown
that the soil, which contains the clay minerals of kaolinite and montmorillonite groups 30–35%,
carbonates 20%, quartz 3–5%, minerals of micas and hydromicas group up to 30%, is a good
adsorbent. However, it has not been found out reliablly which exactly mechanism decreases the
quantity of Pu in a solution since the slightly soluble compounds Pu(OH)4 and hydroxocarbonates
Pu(OH)2CO3 at the given values of pH have been discovered. It is revealed that the composition
of the soil has the greatest effect upon the interphase distribution of plutonium. Both the surface
soil which contains aluminosilicates, plant remains, and the soil out of clayey minerals, micas
and carbonates show the high enough and better properties with respect to Pu(V), Pu(VI). The
presence of cations is typical of these soils. However, the results of desorption have shown that
only a small part of Pu(V) and Pu(VI) caused by the ionic mechanism is sorbed. The relation of
different forms of plutonium in subsoil waters depends on concentration of Pu, HCO−
3 , CO2−
3 , H+.
Interphase distribution of plutonium will be defined both by taking into account the composition
and properties of subsoil waters and the physical and chemical features of soils. The existence
of such forms should be taken into account at the description of the processes of radionuclide
migration in soils and underground waters. Statistical models in which consideration is conducted
at a microscopic level, satisfy the same requirements. Here in order to study the migration of
radionuclides it is necessary to use the statistical approaches which are based on the equitable
account of the radiative particles and ions, water molecules and colloidal particles. Their presence
due to an interaction can strongly effect the transport of radionuclides. It is important to note the
results of works [13,14], in which the processes of an adsorption of uranyl, caesium and strontium
on a surface of silicates and aluminosilicates are explored using the quantum-chemical calculations
(an ab-initio method). It is shown that the surfaces adsorb the compounds UO2+
2 , Cs+, Sr2+.
The detailed information on distribution of charges and distances between atoms of an adsor-
bate and sorbent is obtained, and the forces of bonds are calculated. These results are extremely
important in constructing the statistical models of such systems. The detailed analysis of possible
physicochemical processes which can occur at penetration of aqueous solutions through the layer
of bentonite clay has been fulfilled. The obtained results give grounds to claim that the clayey
porous matrix in an isoelectric state is characterized by coagulative compact structure. The ion-
ization of functional groups causes the deformation of the porous associate skeleton. Thus, the
density of porous clayey matrix due to the action of two factors is not a stationary value during
the process of electrolyte infiltration. The first factor is the Coulomb repulsion of structural par-
ticles, the second one is the magnification of water sorption due to the swelling of clayey matrix.
The effect of these two factors increases at the absence of strong chemical interaction when the
ion sorption by a porous surface is defined by their charge condition. The superfine clay particles
get a negative charge in the water medium. Therefore, ions H+, Fe3+ , Cu2+, [Fe(CN)6]
4− ap-
pear to be counterions for clayey structure. The dissociation degree of available acidic groups is
defined by the level p of the disperse encirclement. Ions abstain with different force by the surface
of porous associates at electrolytes entering the disperse clayey medium. The adsorption of ions
that form nonsoluble compounds will be stronger than the adsorption of ions which are capable of
forming the water-soluble salts. The expression which describes the ion exchange at the formation
of nonsoluble compounds can be presented in the form:
RH+ + Xn+Yp− ⇒ RX + RH+Yp−.
In the case when the water-soluble salts are formed, the expression for the description of ion
exchange process is as follows:
R−H+ + A+B− ⇔ R−A+ + H+B−,
180
Reactionary – electrodiffusion equations of transport processes of electrolyte solutions
where R are the particles of the porous associate matrix which plays the role of an ion exchanger,
X, Y , A, B, H are the ions which take part in the exchange. Therefore, with sufficient confidence
it can be asserted that the entering of the water-soluble compounds causes structural changes of a
porous matrix due to a sorption of polyvalent ions. The basic effect of this process is the formation
of compact associates which decrease the charge of the surface. This decrease of the charge should
be taken into account while constructing the transport equations on the border of electrolyte
and the surface of the porous medium. These examinations are also actual at calculations within
the framework of the molecular hydrodynamics of electro-osmotic processes in montmorillonite
clay [15].
This paper proposes a statistical model of “the water solution of radioelements – the porous
clayey matrix” system. The generalized reactionary-diffuse transport equations for the description
of the diffuse, radiative, sorbtion processes and chemical reactions has been obtained.
2. Statistical model of “the water solution of radioelements – the porous
clayey matrix” system. The generalized reactionary-diffu sion transport
equations
For theoretical estimations of superficial processes it is important to establish a set of pa-
rameters which can be defined experimentally. Superficial processes occur at interaction of water
solutions of radioactive elements with the surface and porous space of clayey structures which are
modified by ferrocyanide of iron and copper. The electrochemical impedance is one of the inter-
esting standard approaches of theoretical estimations and experimental measurements in case of
interaction of electrolytes with electroconductive systems [16]. It is important to consider the possi-
bility of using the electrochemical impedance methods for the research of electrochemical reactions,
diffusion and adsorption processes in superficial area “the electrolyte – the porous matrix”. The
experimental characteristics of electrochemical impedance can be connected with the reactionary-
diffusion equations, which take into account the processes of diffusion of ions, molecules, and their
adsorptions. Thus, one of the important problems is the calculation of the transport processes of
water solution particles in a porous space of the clayey matrix where the adsorption processes can
really take place. The electrolyte conductivity and the ion diffusion in pores can be given by the
expression:
σE = φ3/2σ, DE = φ3/2D,
where φ is the porosity of clayey matrix. The change of ion concentration in porous space of the
clayey matrix in the semi-phenomenological theory of electrochemical processes can be described
within the framework of the generalized diffusion model, in which the driving force is the difference
of potentials ΦE of electrolyte and Φs of clayey matrix U = Φs − ΦE:
φ
∂
∂t
C = D∇2C +
(1 − t+)
F
~∇ ·~jE − KadC − λC, (2.1)
where
~jE = −σ~∇ΦE +
σRBT
F
(1 − t+)
1
C̃
~∇C (2.2)
is the electric current of solution anions (Cs+, Sr2+, UO2+
2 , PuO2+
2 ), C is the ion concentration,
t+ is the transfer number of ions, D is their diffusion constant, RB is the gas constant, F is the
Faraday constant, Kad is the constant of ion adsorption on the pore surface of the clayey matrix, λ
is the decay constant of the corresponding radionuclide. In the solid phase of a porous matrix the
change of the filled density Θ by radionuclide ions of solution can be described by the equation:
∂
∂t
Θ = Ds
~∇ ·
[
(1 +
d ln γ
d ln Θ
)~∇Θ
]
− λΘ, (2.3)
where Ds is the diffusion constant of ions in the solid phase of matrix, γ is the ion activity coefficient.
Thus, the potential Φs of the porous matrix is determined by the electric current:
~js = −σs
~∇ · Φs, (2.4)
181
M.V.Tokarchuk et al.
where σs is the electrical conductivity of the porous matrix. The essential shortcoming of this
approach is the absence of the dynamics of the solvent molecules, of the polarization effects on
“the electrolyte – the porous matrix” interface, of the physical processes connected with α − β −
γ− radiation of radionuclides, namely of the microradiolysis processes, of the oxidation-reduction
reaction.
The statistical theory of diffusion processes for water solutions of radioelements in soils and in
subsoil waters was developed in work [17]. Here the generalized diffusion equations are obtained
taking into account the spontaneous radioactive decays and the decays under the action of neutrons.
They can be modified with the inclusion of processes of an adsorption and a desorption of solution
ions on the soil particles as well as of electromagnetic processes:
~∇ · ~B(~r; t) = 0, ~∇ · ~D(~r; t) = φ
∑
α
Zαeδnα(~r; t),
~∇× ~E(~r; t) =
∂
∂t
~B(~r; t), ~∇× ~H(~r; t) =
∂
∂t
~D(~r; t) +~j(~r; t), (2.5)
~j(~r; t) = φ
∑
α
Zαe(δ ~Jα
d (~r; t) + δ ~Jα
E (~r; t) + δnα(~r; t)~v(~r; t)),
where ~B(~r; t), ~E(~r; t) are the intensity vectors of the average values of magnetic and electric fields
and ~H(~r; t), ~D(~r; t) are the induction vectors that correspond to them. φ = Vf/V is the porosity
of clay matrix, Vf is the volume which occupies the water solution of electrolyte in a porous space
of the matrix, V is the complete volume, Zα is the valency of ions, δnα(~r; t) is the average value
fluctuations of ion densities of sort α (namely Cs +, Sr2+, UO2+
2 , PuO2+
2 , etc.) of the water solution
of electrolytes. ~j(~r; t) is the density of the complete average ionic current which is expressed through
diffusive δ ~Jα
d (~r; t), ionic δ ~Jα
E (~r; t) and convective δnα(~r; t)~v(~r; t) currents. ~v(~r; t) is the average
velocity of the water solution in porous matrix. δnα(~r; t) and δ ~Jα
d (~r; t), δ ~Jα
E (~r; t), δnα(~r; t)~vf (~r; t)
are interlinked by transport equations for each phase. They are obtained using the method of
nonequilibrium statistical operator similar to [17] and are as follows:
∂
∂t
δnα(~rl; t) = −
∂
∂~rl
(
δ ~Jα
d (~rl; t) + δ ~Jα
E (~rl; t) + δnα(~rl; t)~v(~rl; t)
)
−
∑
β
Aαa(~rl; t) δna(~rl; t) − Aα(~rl; t)δn
α(~rl; t), (2.6)
where
Aαβ(~rl; t) =
∫
∞
0
J(~rl;E; t)σαβ(E) dE + Lαβλβ , (2.7)
Aα(~rl; t) =
∫
∞
0
J(~rl;E; t)σα(E) dE + Lαλα,
~Jα
d (~rl; t) =
∑
as
1
Vs
∫
Vs
d~r′s
∫ t
−∞
eε(t′−t)Dαa(~rl, ~r
′
s; t, t
′)
×
∂
∂~r′s
δna(~r′s; t, t
′) +
∑
a,b
∑
k,s
1
Vs
∫
Vs
d~r′s
1
Vk
∫
Vk
d~r′k
t
∫
−∞
eε(t′−t)
× Kα,ab(~rl, ~r
′
s, ~r
′
k; t, t′)δna(~r′s; t
′)δnb(~r′k; t′)dt′,
δ ~Jα
E (~r; t) =
∑
γa
∑
sk
1
Vs
∫
Vs
d~r′s
1
Vk
∫
Vk
d~r′k
t
∫
−∞
eε(t′−t)Dαa(~rl, ~r
′
k; t, t′)Φaβ
nn(~r′k, ~r′s)Zβe ~E(~r′s; t). (2.8)
In these formulas Φaβ
nn(~r′k, ~r′s) = 〈n̂a(~r′k)n̂β(~r′s)〉0 is the equilibrium pair distribution function of
particles, 〈. . .〉0 denotes the averaging with the grand canonical Gibbs distribution. n̂α(~r′k) =
182
Reactionary – electrodiffusion equations of transport processes of electrolyte solutions
Nα
∑
i=1
δ(~ri −~r′k) is the microscopic density of particle number of α sort in the k phase. α indicates the
sorts of ions of water molecules and of clayey matrix particles. Dαβ(~rl, ~r
′
s; t, t
′) are the generalized
diffusion coefficients as functions of coordinates and time, which describe dissipative processes;
Kα,ab(~rl, ~r
′
s;~rk”; t, t′) are the generalized coefficients of reactions which describe the processes of
adsorption, desorption and possible chemical reactions of the hydrolysis, radiolysis, complexations.
The last two terms in (2.6) describe the change of radionuclide density with time, caused by neutron
currents and spontaneous decays. The first summand describes the origination of radionuclides of
α sort from all other nucleuses of β sort owing to (n, γ), as well as (n, f) reactions when the
corresponding radionuclide of α sort belongs to decay products. The second summand describes
the radionuclide decay of α sort under the action of neutrons and the spontaneous radioactive
decay. The functions Aαβ(~rl; t), Aα(~rl; t) are the velocities of corresponding reactions. J(~rl;E; t) is
the spectrum of the neutron density current at the point ~rl at the moment of time t. σαβ(E) is the
microscopic cross-section of radionuclide formation of α sort at the capture of neutrons with energy
E by nucleus of β sort. Lαβ is the probability of radionuclide formation of α sort at the radioactive
nucleus decay of β sort. λα, λβ are the nucleus decay constants of α, β sorts. The set of equations
(2.6)–(2.8) is nonlinear. This set takes into account the diffusion, adsorption, desorption, ionic
conduction, electromagnetic as well as radiative processes with the consideration of the porosity
of a clayey matrix. Each of the currents (2.7), (2.8) gives the contribution to (2.5), while δ ~Jα
E (~rl; t)
has the field contribution which is connected with the generalized ionic conductivity of electrolyte
solution in the clayey matrix.
The generalized reactionary transport equations for ions, electrons and water molecules, prod-
ucts of radiolysis are obtained for the description of the reactionary phenomena in a system of
the aqueous solution of radioelement-porous clayey structures taking into account the electromag-
netic processes. These equations are convenient for more severe comprehension of complexity of
the physicochemical processes and the statistical substantiation of the electrochemical impedance
methods. They agree with the averaged Maxwell equations for electromagnetic processes:
~∇ · ~Bl(~r, t) = 0,
~∇ · ~Dl(~r, t) =
N
∑
a=1
Zaenl
a(~r, t) + enl
e(~r, t) +
∑
f
~df · ~∇nl
f (~r, t) + enl
β(~r, t),
~∇× ~El(~r, t) = −
∂
∂t
~Bl(~r, t),
~∇× ~Hl(~r, t) =
∂
∂t
~Dl(~r, t) +~jl
i(~r, t) +~jl
d(~r, t) +~jl
e(~r, t) +~jl
β(~r, t), (2.9)
where ~Bl(~r, t), ~El(~r, t), ~Dl(~r, t), ~Hl(~r, t) are, respectively, the intensities and the inductions of elec-
trical and magnetic fields of water solution. These intensities and the inductions are created by
ions of a sort with density nl
a(~r, t); by electrons with density nl
e(~r, t) which arise from the radiolysis
of water solution by β− electrons; by electrons with the density nl
β(~r, t), which are radiated by
radionuclides in solution volume and on the surface of clayey matrix; as well as by water molecules
and by radiolysis products H2O2, HO2 with particle density nl
f (~r, t) of f sort and with electrical
dipole moment ~df . Za is the ion valency of specie a. ~jl
i(~r, t)~j
l
e(~r, t)~j
l
β(~r, t), ~jl
d(~r, t) are respectively
the average currents of ion charge, of low-energy electrons and β-electrons, as well as of the dipole
moment of the polarized molecules. The expressions for currents can be obtained using the method
of nonequilibrium statistical operator [17]. In approximation of the constant transport coefficients
they have the following structure for ions:
~jl
i(~r, t) =
∑
a
Zae
(
−
∑
ξ
Dlξ
aa
~∇ · nξ
a(~r, t) −
∑
ξ
∑
b
Dlξ
ab
~∇ · nξ
b(~r, t) −
∑
f
Daf
~∇ · nl
f (~r, t)
−
∑
ξ
Dlξ
ae
~∇ · nξ
e(~r, t) +
∑
ξ
Dlξ
aβ
~∇ · nξ
β(~r, t) + nl
a(~r, t)~vl(~r, t)
)
+
∑
ab
σab
~El(~r, t)
183
M.V.Tokarchuk et al.
+
∑
af
1
mf
~σaf · ~∇ ~El(~r, t) +
∑
a
σae
~El(~r, t) +
∑
a
σaβ
~El(~r, t) −
∑
a
Ka
adna(~Sω, t)Θs(~Sω, t)
+
∑
a
Ka
desn
l
a(~r, t)Θs(~Sω, t), (2.10)
where the index ξ indicates: l is the water solution, s is the porous matrix. Ka
ad, Ka
des are the
constants of adsorption and desorption for ions of a sort. na(~Sω, t) is the ion density of a sort
adsorbed on the surface ~Sω of the interface of the systems the water solution of radioactive elements
and the porous clay matrix. Θs(~Sω, t) is the adsorption site density on the surface of a porous
matrix. The ionic current of charges satisfies the conservation law:
∂
∂t
ρi(~r, t) = −~∇ ·~ji(~r, t), (2.11)
where ρi(~r, t) =
∑
a
Zaena(~r, t) is the complete density of ion charge. σab = ZaeDabZbe is the partial
electrical ion conductivity of the a and b sorts, Dab is the interdiffusion coefficient. ~σaf = ZaeDaf
~df
is the partial electrical conductivity of ions of the a sort and molecules of the f sort, Daf is the
interdiffusion coefficient of ions and molecules. ~vl(~r, t) is the average particle velocity in electrolyte.
The current for molecules of the solvent has the following form:
~jl
d(~r, t) =
∑
f
1
mf
~df · ~∇
(
− Dff
~∇ · nl
f (~r, t) −
∑
ξ
∑
b
Dlξ
fb
~∇ · nξ
b(~r, t)
−
∑
ξ
Dlξ
fe
~∇ · nξ
e(~r, t) −
∑
ξ
Dlξ
fβ
~∇ · nξ
β(~r, t) + nl
f (~r, t)~vl(~r, t)
)
+
∑
bf
1
mf
~∇ · ~σfb
~El(~r, t) +
∑
f
1
mf
~∇·
↔
σ ff ·~∇
1
mf
~El(~r, t)
+
∑
f
1
mf
~∇ · ~σfe
~El(~r, t) +
∑
f
1
mf
~∇ · ~σfβ
~El(~r, t), (2.12)
where
↔
σ ff= ~dfDff
~df is the conductivity of dipole molecules of solution, Dff is the diffusion
constant of molecules. In the current (2.12) the processes of adsorption and desorption of solution
molecules are not taken into account. The electrical current for electrons has the structure similar
to the ions:
~jl
e(~r, t) = −e
∑
ξ
∑
b
Dlξ
eb
~∇ · nξ
b(~r, t) − e
∑
f
Def
~∇ · nl
f (~r, t)
− e
∑
ξ
Dlξ
ee
~∇ · nξ
e(~r, t) − e
∑
ξ
Dlξ
eβ
~∇ · nξ
β(~r, t) + enl
e(~r, t)~vl(~r, t)
+
∑
b
σeb
~El(~r, t) +
∑
f
1
mf
~σef · ~∇ ~El(~r, t) + σl
ee
~Es(~r, t) + σl
eβ
~Es(~r, t), (2.13)
where σl
ee, σl
eβ are the electrical conductivities of electrons and β-electrons in electrolyte. The
current of high-energy β-electrons has the same structure as ~jl
e(~r, t). Thus it is formally necessary
to exchange the index e for β. However, by considering that the β-electrons have got a consid-
erable energy, their correlations with molecules of the solution will be negligible in comparison
with dynamic correlations with electrons and ions both in the solution and in the porous matrix.
Therefore, for ~jl
β(~r, t) it is possible to write the expression:
~jl
β(~r, t) = −e
∑
ξ
∑
b
Dlξ
βb
~∇ · nξ
b(~r, t) −e
∑
ξ
Dlξ
βe
~∇ · nξ
e(~r, t) − e
∑
ξ
Dlξ
ββ
~∇ · nξ
β(~r, t) + enl
e(~r, t)~vl(~r, t)
+
∑
b
σβb
~E
(
l~r, t) + σl
βe
~E(
s~r, t) + σl
ββ
~E(
s~r, t), (2.14)
184
Reactionary – electrodiffusion equations of transport processes of electrolyte solutions
~jl
e(~r, t), ~jl
β(~r, t) as well as the ion current satisfy the conservation laws for the charge.
The similar equation systems for average electromagnetic fields and currents are obtained for
the “porous clayey matrix” subsystem:
~∇ · ~B(
s~r, t) = 0,
~∇ · ~D(
s~r, t) =
N
∑
a=1
Zaens
a(~r, t) + ens
e(~r, t) + ens
β(~r, t),
~∇× ~E(
s~r, t) = −
∂
∂t
~B(
s~r, t),
~∇× ~H(
s~r, t) =
∂
∂t
~D(
s~r, t) +~js
i (~r, t) +~js
e(~r, t) +~js
β(~r, t), (2.15)
where the currents of electrons and ions in a porous matrix are presented by expressions:
~js
i (~r, t) =
∑
a
Zae
(
−
∑
ξ
Dsξ
aa
~∇ · nξ
a(~r, t) −
∑
ξ
∑
b
Dsξ
ab
~∇ · nξ
b(~r, t)
−
∑
f
Dsl
af
~∇ · nl
f (~r, t) −
∑
ξ
Dsξ
ae
~∇ · nξ
e(~r, t) −
∑
ξ
Dsξ
aβ
~∇ · nξ
β(~r, t)
+ ns
a(~r, t)~vs(~r, t)
)
+
∑
ab
σab
~Es(~r, t) +
∑
a
σae
~Es(~r, t) +
∑
a
σaβ
~Es(~r, t), (2.16)
~js
e(~r, t) = −e
∑
ξ
∑
b
Dsξ
eb
~∇ · nξ
b(~r, t) − e
∑
f
Def
~∇ · nl
f (~r, t)
− e
∑
ξ
Dsξ
ee
~∇ · nξ
e(~r, t) − e
∑
ξ
Dsξ
eβ
~∇ · nξ
β(~r, t)
+ ens
e(~r, t)~vs(~r, t) +
∑
b
σeb
~Es(~r, t) + σs
ee
~Es(~r, t) + σs
eβ
~Es(~r, t), (2.17)
~js
β(~r, t) = −e
∑
ξ
∑
b
Dsξ
βb
~∇ · nξ
b(~r, t) − e
∑
ξ
Dsξ
βe
~∇ · nξ
e(~r, t)
− e
∑
ξ
Dsξ
ββ
~∇ · nξ
β(~r, t) + ens
β(~r, t)~vs(~r, t)
+
∑
b
σβb
~El(~r, t) + σs
βe
~Es(~r, t) + σs
ββ
~Es(~r, t). (2.18)
The porous space of matrix is filled with particles of the water solution, namely with water
molecules, with the radiolysis products which evolved from α−β−γ – irradiations of the solution.
In this case it is necessary to supplement the set of equations (2.16)–(2.18) by the current densities
of solution molecules in porous space of clayey matrix:
~js
d(~r, t) =
∑
f
1
mf
~df · ~∇
(
− D̃ff
~∇ · ns
f (~r, t) −
∑
ξ
∑
b
D̃lξ
fb
~∇ · nξ
b(~r, t)
−
∑
ξ
D̃lξ
fe
~∇ · nξ
e(~r, t) −
∑
ξ
D̃lξ
fβ
~∇ · nξ
β(~r, t) + ns
f (~r, t)~vl(~r, t)
)
+
∑
bf
1
mf
~∇ · ~̃σfb
~Es(~r, t) +
∑
f
1
mf
~∇ ·
↔̃
σ ff ·~∇
1
mf
~Es(~r, t)
+
∑
f
1
mf
~∇ · ~̃σfe
~Es(~r, t) +
∑
f
1
mf
~∇ · ~̃σfβ
~Es(~r, t), (2.19)
where D̃ff , D̃lξ
fb, D̃
lξ
fβ ,
↔̃
σ ff , ~̃σfe are the interdiffusion coefficients and conductivities of ions and
molecules in porous space of the clayey matrix. Thus, the combined equations (2.6)–(2.19) con-
185
M.V.Tokarchuk et al.
tain the coefficients of interdiffusion and self-diffusion as well as the conductivities of ions, elec-
trons, molecules, the products of radiolysis in solution and porous space of the clayey matrix.
The intensities and inductions of electrical and magnetic fields are connected by expressions
~Bξ = µ0
~Hξ
~Dξ = ε0εξ
~Eξ, where εξ is the generalized dielectric function of the corresponding
subsystem. The both systems of equations for the electrolyte solution and the porous matrix are
interconnected by the interphase partial diffusion constants Dξ′ξ
αβ , (ξ, ξ′ = l, s, α, β = a, f, e) and
by the limiting conditions on the boundary “the electrolyte solution – the porous matrix”:
~n · ( ~Bs − ~Bl) = 0, ~n · ( ~Ds − ~Dl) = Q(~Sω, t),
~n × ( ~Es − ~El) = 0, ~n × ( ~Hs − ~Hl) = Q(~Sω)~vs(~Sω, t), (2.20)
where Q(~Sω, t) is the complete surface electrical charge on the boundary “the electrolyte – the
porous matrix” which satisfies the conservation laws:
∂
∂t
Q(~Sω, t) = ~n ·~ji(~Sω, t), ~vs(~Sω, t) = ~vl(~Sω, t),
Q(~Sω, t) =
∑
a
Zaena(~Sω, t). (2.21)
The unit vector ~n is directed perpendicularly to the interface “the electrolyte solution – the porous
matrix”. The density of adsorbed ions na(~Sω, t) on the surface ~Sω satisfies the reactionary-diffuse
equations taking into account the processes of radioactive decays of nuclides Cs+, Sr2+, UO2+
2 ,
PuO2+
2 :
∂
∂t
na(~Sω, t) = ∇~Sω
Dsurf
a ∇~Sω
na(~Sω, t) + DT∇
2T (~r; t) + Ka
adna(~r, t)Θs(~Sω, t)
− Ka
desn
l
a(~Sω, t)Θs(~Sω, t) −
∑
β
Aaβ(~Sω, t) nβ(~Sω, t) − Aa(~Sω, t)n(~Sω, t), (2.22)
where DT is the constant of thermal diffusion of radionuclides adsorbed on the clayey matrix
surface, T (~r; t) is the local temperature, Dsurf
a is the diffusion of sort a particles on the clayey matrix
surface. The temperature effects on the surface of clayey matrix are generated by radioactive decays
of radionuclides. The presented system of the equations (2.6)–(2.22) describes the complicated
physicochemical processes of diffusion, adsorption, desorption, radioactive decay, which include
the surface phenomena on the boundary “the water radionuclides solution - the porous clayey
matrix”. The particles of clayey matrix will effect the mechanical characteristics (turgescency,
dissolubility) since the clayey matrix can exude by water solution. Therefore, it is necessary to add
to (2.6)–(2.22) the conservation laws for particle momenta of the solution and the clayey matrix,
which have the following form in frequency representation:
−iωρl~vl = ~∇·
↔
τ l +
∑
j
Zje(n
0
l ~el + nlE
0
s ), −iωρs~vs = ~∇·
↔
τ s, (2.23)
where
↔
τ l,
↔
τ s is the stress tensors of a solution and a matrix, and ρl, ρs are their complete densities,
respectively. The sum of the right part describes the electrical force which operates on a redundant
charge, without taking into account the Lorentz force. For stress tensors we have the corresponding
expressions:
↔
τ l= Kl
~∇ · ~vl − iωη
(
∇~vl + ∇~vT
l −
2
3
~∇ · ~vl
↔
I
)
,
↔
τ s= Ks
~∇ · ~vs + G
(
∇~vs + ∇~vT
s −
2
3
~∇ · ~vs
↔
I
)
, (2.24)
where
↔
I is the individual tensor, Kl, η are the coefficients of volumetric and shift viscosity of a
solution and Ks, G are the modules of volumetric and shift elasticity of the porous clayey matrix.
186
Reactionary – electrodiffusion equations of transport processes of electrolyte solutions
The limiting conditions for stress tensors on the boundary of the phases are as follows:
~n · (
↔
τ l −
↔
τ s) = Q0~es, ~vl − ~vs = 0. (2.25)
Thus, we have obtained the systems of transport particle equations (2.5)–(2.25) of the solution
of radioactive element electrolytes through porous clayey matrix with the surface active centres of
absorption Θs(~Sω, t). They are centres of modifying the surface of carbonate-containing bentonite
clayey by ferrocyanides of iron and copper [18,19]. Thus the composite physicochemical processes
of diffusion, absorption, desorption, radioactive decay, mechanical tensions with inclusion of surface
phenomena on the boundary “the water solution of radionuclides - the porous clayey matrix” were
taken into account.
This study was supported by the STCU under the project No. 1706.
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187
M.V.Tokarchuk et al.
Реакцiйно-електродифузiйнi рiвняння процесiв переносу
розчинiв електролiтiв радiоактивних елементiв крiзь пористi
глинистi структури
М.В.Токарчук1, П.А.Глушак1, I.М.Крiп2, Т.В.Шимчук2
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1
2 Нацiональний унiверситет “Львiвська полiтехнiка”, 79013 Львiв, вул. С.Бандери, 12
Отримано 27 березня 2007 р., в остаточному варiантi – 23 травня 2007 р.
Запропоновано стастичну модель “водний розчин радiоактивних елементiв – пориста глиниста ма-
триця”. Отримано узагальненi рiвняння переносу для опису дифузiї, сорбцiї, радiоактивних процесiв
та хiмiчних реакцiй з врахуванням електромагнiтних процесiв.
Ключовi слова: дифузiя, сорбцiя, електромагнiтнi процеси, пориста глиниста матриця,
Чорнобильська катастрофа
PACS: 05.60.+w, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi
188
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