Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane”
Diffusion of ions under inhomogeneous conditions in the two-phase system “initial electrolyte solution-membrane” is investigated. Expressions for the friction constant, self- and interdiffusion coefficients are derived by means of a nonequilibrium statistical operator for an inhomogeneous case. Th...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” / R.I. Zhelem, M.V. Tokarchuk, I.P. Omelyan, E.M. Sovyak // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 53-62. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1199122017-06-11T03:03:12Z Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” Zhelem, R.I. Tokarchuk, M.V. Omelyan, I.P. Sovyak, E.M. Diffusion of ions under inhomogeneous conditions in the two-phase system “initial electrolyte solution-membrane” is investigated. Expressions for the friction constant, self- and interdiffusion coefficients are derived by means of a nonequilibrium statistical operator for an inhomogeneous case. They are applied to the system, thus the behaviour of the position-dependent selfdiffusion coefficient for NaCl near a reverse osmosis membrane is evaluated. Pозглядається дифузiя iонiв у неодноpiдних умовах двофазної системи вихiдний pозчин електpолiту – мембpана. З допомогою методу неpiвноважного статистичного опеpатоpа отpимано виpази для константи теpтя, коефiцiєнтiв самодифузiї та взаємної дифузiї в неодноpiдному випадку. Їх застосовано до даної системи, і pозpаховано поведiнку коефiцiєнта самодифузiї для NaCl, залежного вiд вiддалi до звоpотноосмотичної мембpани. 1999 Article Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” / R.I. Zhelem, M.V. Tokarchuk, I.P. Omelyan, E.M. Sovyak // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 53-62. — Бібліогр.: 16 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.53 PACS: 05.60.+w, 73.40, 66.10.Cb, 82.65.Fr http://dspace.nbuv.gov.ua/handle/123456789/119912 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
Diffusion of ions under inhomogeneous conditions in the two-phase system
“initial electrolyte solution-membrane” is investigated. Expressions for the
friction constant, self- and interdiffusion coefficients are derived by means
of a nonequilibrium statistical operator for an inhomogeneous case. They
are applied to the system, thus the behaviour of the position-dependent
selfdiffusion coefficient for NaCl near a reverse osmosis membrane is evaluated. |
| format |
Article |
| author |
Zhelem, R.I. Tokarchuk, M.V. Omelyan, I.P. Sovyak, E.M. |
| spellingShingle |
Zhelem, R.I. Tokarchuk, M.V. Omelyan, I.P. Sovyak, E.M. Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” Condensed Matter Physics |
| author_facet |
Zhelem, R.I. Tokarchuk, M.V. Omelyan, I.P. Sovyak, E.M. |
| author_sort |
Zhelem, R.I. |
| title |
Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” |
| title_short |
Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” |
| title_full |
Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” |
| title_fullStr |
Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” |
| title_full_unstemmed |
Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” |
| title_sort |
calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119912 |
| citation_txt |
Calculation of distribution functions and diffusion coefficients for ions in the system “initial electrolyte solution – membrane” / R.I. Zhelem, M.V. Tokarchuk, I.P. Omelyan, E.M. Sovyak // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 53-62. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT zhelemri calculationofdistributionfunctionsanddiffusioncoefficientsforionsinthesysteminitialelectrolytesolutionmembrane AT tokarchukmv calculationofdistributionfunctionsanddiffusioncoefficientsforionsinthesysteminitialelectrolytesolutionmembrane AT omelyanip calculationofdistributionfunctionsanddiffusioncoefficientsforionsinthesysteminitialelectrolytesolutionmembrane AT sovyakem calculationofdistributionfunctionsanddiffusioncoefficientsforionsinthesysteminitialelectrolytesolutionmembrane |
| first_indexed |
2025-07-08T16:53:50Z |
| last_indexed |
2025-07-08T16:53:50Z |
| _version_ |
1837098475520524288 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 53–62
Calculation of distribution functions
and diffusion coefficients for ions in the
system “initial electrolyte solution –
membrane”
R.I.Zhelem, M.V.Tokarchuk, I.P.Omelyan, E.M.Sovyak
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 290011 Lviv, Ukraine
Received July 14, 1998
Diffusion of ions under inhomogeneous conditions in the two-phase system
“initial electrolyte solution-membrane” is investigated. Expressions for the
friction constant, self- and interdiffusion coefficients are derived by means
of a nonequilibrium statistical operator for an inhomogeneous case. They
are applied to the system, thus the behaviour of the position-dependent
selfdiffusion coefficient for NaCl near a reverse osmosis membrane is eval-
uated.
Key words: diffusion in liquids, transport processes, distribution function,
osmosis, membrane
PACS: 05.60.+w, 73.40, 66.10.Cb, 82.65.Fr
1. Introduction
Reverse osmosis occupies a significant place among membrane processes for a
mixture or electrolyte solution separation into components [1–11]. In contrast to
other separation methods, reverse osmosis across a membrane occurs at a molec-
ular level. In this process a membrane plays the role of a specific selective barrier
and puts through itself primarily certain components from the solution to be sep-
arated. The selectivity of reverse osmosis considerably depends on an interaction
of ions and solution molecules with the molecules and components (as membranes
are composite materials) of the membrane on its surface and inside. In this connec-
tion, in order to describe, forecast and improve the separation in modern devices
it is important to work out a nonequilibrium statistical theory for reverse osmosis
coupled with computer simulation on the basis of an equivalent account of both
solution particles and membrane structure. Spatial inhomogeneity of the system
should be considered too. In [12], an analytical calculation of generalized diffusion
c© R.I.Zhelem, M.V.Tokarchuk, I.P.Omelyan, E.M.Sovyak 53
R.I.Zhelem et al.
coefficients was performed for ions and molecules in the three-phase system “initial
electrolyte solution-membrane” filtrate. It turned out that all diffusion coefficients
could be expressed through equilibrium singlet and pair distribution functions.
Their evaluation is a complex problem of statistical mechanics. So [13] was aimed
at obtaining qualitative distribution functions for the beginning of the separation
process when the dissolved ions are located on one side of the reverse osmosis
membrane only. Then, it suffices to consider the two-phase system “initial elec-
trolyte solution-membrane”. In addition, the description is simplified by assuming
the solvent and the membrane to be continuous diffusive media with appropriate
dielectric constants ε1 > ε2, whereas ions are represented by charged hard spheres.
Here we are based on the same model. First, we complete the mathematics on
inhomogeneous self- and interdiffusion coefficients and arrive at the expression for a
friction constant. In the next chapter we apply the result to the model of the reverse
osmosis membrane and, using the known distribution functions, evaluate position-
dependent self-diffusion coefficients for the two-phase system “initial electrolyte
solution-membrane”.
2. Inhomogeneous diffusion coefficients for ions in a two-ph a-
se system “initial electrolyte solution-membrane”
In papers [11,12], generalized diffusion equations were obtained for the solution
transport through reverse osmosis membranes by means of the Zubarev nonequilib-
rium statistical operator. We will consider Markovian processes for ionic diffusion
when the flux of k-species particles in phase l is described by the equation:
ĵk(rl; t) = −
∑
k′
∑
l′
∫
V
l′
dr′
l D
kk′(rl, rl′)
∂
∂rl′
δnk′(rl′; t), (1)
where
Dkk′(rl, rl′) = (2)
=
∑
k′′
∑
l′′
∫
V
l′′
drl′′
∞
∫
0
dt〈(1−P0) ĵ
k(rl)T0(t)(1− P0) ĵ
k′′(rl′′)〉0
×
[
Φ̃−1
nn(rl′′ , rl′)
]
k′′k′
are normalized inhomogeneous diffusion coefficients for the particles of species k,
k′ in phases l, l′. δnk′(rl′; t) = 〈n̂k′(rl′)〉
t − 〈n̂k′(rl′)〉0 is a deviation of the density
profile from its equilibrium value 〈n̂k′(rl′)〉0 =
Nk
′
l′
V
l′
fk′
1 (rl′), which is a moving force
of diffusion. The indices l, l′ stand for the phases: 1–initial electrolyte solution, 2–
membrane, 3–filtrate. It is naturally to assume that
∑
l′′ concerns the neighbouring
phases only, as the correlation between the initial solution and filtrate is negligible.
In equation (2), P0 is a Mori projection operator and T0(t) is an evolution operator
defined in [13], which takes into account the projection.
54
Diffusion coefficients for an inhomogeneous system
To make use of these formulas we focus our attention on the inverse functions
[
Φ̃−1
nn(rl, rl′)
]
kk′
to be determined by the relations:
∑
k′
∑
l′
∫
drl′Φ
kk′
nn (rl, rl′)
[
Φ̃−1
nn(rl′, rl′′)
]
k′k′′
= δkk′′δ(rl − rl′′) , (3)
where
Φkk′
nn (rl, rl′) = 〈δn̂k(rl) δn̂
k′(rl′)〉0
are connected with the total correlation function hkk′
2 (rl, rl′). To find an explicit
form for
[
Φ̃−1
nn(rl, rl′)
]
kk′
it is necessary to consider the details of the model for the
membrane, solution and initial diffusion conditions.
The initial solution is supposed to be a two-species system of charged hard
spheres of equal valency Z+ = |Z−| = Z and size σ in a continuous medium with
dielectric constant ε1. The membrane is modelled by a hard wall with dielectric
constant ε2. Here ε1 > ε2. Then ions located in the vicinity of the membrane suffer
intensive repulsion caused by electrostatic images, i.e. negative adsorption, which
is confirmed by equilibrium density profiles of the system (see the next chapter).
The phenomenon produces a potential barrier which does not allow the salt ions
to penetrate the membrane phase. Consequently, initial conditions for equilibrium
concentrations of ions in various phases are to be the following:
〈n̂k(r1)〉0 6= 0, 〈n̂k(r2)〉0 = 0, 〈n̂k(r3)〉0 = 0.
They show an absence of ions in the membrane and filtrate phases at the beginning
of a diffusion process. It means that
Φkk′
nn (rl, rl′) =
{
≡ 0, l 6= l′
6= 0, l = l′ = 1
.
So, having in mind that all nonzero quantities concern the initial electrolyte solu-
tion only, we omit the both phase indices l, l′ and the sum over them in (1)–(3).
It should be noted that in doing so the important membrane influence (negative
adsorption) is not excluded, namely, it affects hkk′
2 (rl, rl′) and, according to (3),
Φkk′
nn (r, r
′) as well. To determine the latter we make use of the relationship:
Φkk′
nn (r, r
′) =
δ〈n̂k(r)〉0
δ ln zk′(r′)
, (4)
where zk
′
(r′) is an inhomogeneous activity of a k′-species particle. Then, equation
(3) l = l′ = l′′ = 1 can be represented as
∑
k′′
∫
dr′′
δ〈n̂k(r)〉0
δ ln zk′′(r′′)
δ ln zk
′′
(r′′)
δ〈n̂k′(r′)〉0
= δkk′δ(r − r′) , (5)
55
R.I.Zhelem et al.
this is a trivial Ornstein-Zernike equation, while the unknown function
[
Φ−1
nn(rl, rl′)
]
kk′
is expressed in terms of direct correlations ckk
′
2 (r, r′) [14,15]:
δ ln zk(r)
δ〈n̂k′(r′)〉0
=
[
Φ−1
nn(rl, rl′)
]
kk′
=
δkk′δ(r − r′)
〈n̂k′(r′)〉0
− ckk
′
2 (r, r′). (6)
The solution obtained is exact, however, ckk
′
2 (r, r′) serves as a basic function in the
equilibrium theory for calculating structural distribution functions. It is usually
approximated (PY, HNC approximations, their modifications [14,15]) to have the
total correlation function hkk′
2 (r, r′) with the help of the Ornstein-Zernike equation.
We are going to perform a calculation of inhomogeneous diffusion coefficients
for ions (2) in the Gaussian approximation for the time correlation function 〈(1−
P0) ĵ
k(rl)T0(t)(1− P0) ĵ
k′′(rl′′)〉0, then, in accordance with [12] one arrives at
Dkk′(r, r′) =
∞
∫
0
dτλkk′
0 (r, r′) exp
{
−
λkk′
2 (r, r′)
2!
τ 2
}
= λkk′
0 (r, r′)
√
π
2λ
kk′
2 (r, r′)
,
(7)
where λkk′
0 (r, r′) and λkk′
2 (r, r′) are obtained in [13] and have the form:
λkk′
0 (r, r′) =
∑
k′′
∫
dr′′〈(1− P0) ĵ
k(rl)T0(t)(1− P0) ĵ
k′′(rl′′)〉0
×
(
δk′′k′δ(r
′′ − r′)
〈n̂k′′(r′′)〉0
− ck
′′k′
2 (r′′, r′)
)
(8)
=
kBT
mk
(
δkk′δ(r − r′)− fk
1 (r) · c
kk′
2 (r′, r′)
)
,
λkk′
2 (rl, rl′) =
∑
l′′
∑
k′′
∫
V
l′′
drl′′
(
〈 ĵk(rl)
¨̂
jk
′′
(rl′′)〉0 +
Nk
Vl
Nk′′
Vl′′
(kBT )
2
mkmk′′
(9)
×
∂
∂rl
∂
∂rl′′
{
fk
1 (rl)f
k′′
1 (rl′′)
[
Φ̃−1
nn(rl, rl′′)
]
kk′′
})[
Φ̃−1
nn(rl′′, rl′)
]
k′′k′
,
λ
kk′
2 (r, r′) = λkk′
2 (r, r′)
/
λkk′
0 (r, r′). (10)
Inserting (7) and (8) in (1), we get the flux
ĵk(r; t) = Dkk(r)
∂
∂r
δnk(r; t) +
∑
k′
∫
dr′D
kk′
(r, r′)
∂
∂r′
δnk′(r′; t) , (11)
where
Dkk(r) = Dk(r) =
kBT
mk
√
π
2λ
kk
2 (r)
(12)
is an inhomogeneous self-diffusion coefficient for a k-species particle with respect
to the membrane surface in the initial electrolyte solution phase, while
D
kk′
(r, r′) =
kBT
mk
fk
1 (r)c
kk′
2 (r, r′)
√
π
2λkk′
2 (r)
(13)
56
Diffusion coefficients for an inhomogeneous system
is that of interdiffusion for k, k′ -species particles.
At last, on the basis of the explicit form for inverse functions (6) and definition
(10), λ
kk
2 (r) has the following form:
λ
kk
2 (r) = (14)
=
1
3mk
[
∂2
∂r2
Uk(r) +
∑
k′
Nk′
V1
∫
∂2
∂r2
Ukk′(r, r
′) fk′
1 (r′)gkk
′
2 (r, r′) dr′
]
+
5
3mk
[
∂2
∂r2
Uk(r) +
∑
k′
Nk′
V1
∫
fk′
1 (r′)
( ∂2
∂r2
Ukk′(r, r
′) gkk
′
2 (r, r′)
+
∂
∂r
Ukk′(r, r
′)
∂
∂r
gkk
′
2 (r, r′)
)
dr′
]
,
where gkk
′
2 (r, r′) are pair correlation functions, fk
1 (r) – density profiles, Uk(r) and
Ukk′(r, r
′) are interaction potentials defined in the next chapter. The quantity
λ
kk
2 (r) is an analogue of the friction constant γ2 for particle diffusion in a medium.
One can be convinced of the fact that considering a bulk case, when Uk(r) ≡ 0,
fk′
1 (r′) ≡ 1, gkk
′
2 (r, r′) = gkk
′
2 (|r − r′|), namely, the classical result of S.Rice is
recovered [16]. An advantage of expression (14) is in the appearance of a position
dependent friction constant for an inhomogeneous system. It should also be noted
that λ
kk
2 (r) describes self-diffusion between k-species particles but equally treats
other species which form a combined medium for the given particles. It is clear that
equation (14) may be applied to any inhomogeneous system with its microscopic
characteristics.
3. Numerical results and conclusions
As mentioned above, equilibrium theories do not provide an exact behaviour
of the direct correlation function ckk
′
2 (r, r′), especially at small distances. So we
restrict ourselves to the estimate of self-diffusion coefficients for the salt NaCl in
the vicinity of the reverse osmosis membrane within the described model. To do it
on the basis of equations (12), (14) one needs the density profiles, pair correlation
functions and interaction potentials. The problem was solved in [13]. Here we only
remind the necessary results.
The singlet potential Uk(r) consists of a trivial Gibbs part
ϕw(zi) =
{
∞, |zi| < σ/2
0, |zi| > σ/2
, (15)
and an electrostatic interaction with its own image:
Uw
im(zi) =
ε1 − ε2
ε1 + ε2
Z2
i e
2
2zi
, (16)
where zi and Zi are the distances from the membrane and the ionic charge, respec-
tively. In the case of pair interaction there are three different potentials, namely,
57
R.I.Zhelem et al.
0.0 1.0 2.0 3.0 4.0 z/σ0.4
0.6
0.8
1.0
1.2
1.4
1.6
f1(z)
�
�
�
�
△
△
△
△
∗
∗
∗ ∗
∗- 4%
�- 15%
△- 27%
Figure 1. Density profiles for Na+ and Cl−
ions in water near the reverse osmosis mem-
brane at various mass concentrations
the hard-sphere one
ϕ(rij) =
{
∞, |rij| < σ
0, |rij| > σ
, (17)
the Coulomb potential for charges
Zi, Zj
Ub
C(rij) =
ZiZje
2
rij
(18)
and the interaction of the i-th ion
with the j-th ion’s image
Ub
im(r
′
ij) =
ZiZje
2
r′ij
ε1 − ε2
ε1 + ε2
, (19)
where r′ij is a distance between them.
The inhomogeneous distribution
near the planar wall can be de-
scribed by the singlet fk
1 (z1) and pair
fkk′
2 (z1, z2, r12) densities which are to
be found. In our case of a symmetric
1:1 electrolyte the following relations
are valid:
f+
1 (z1) = f−
1 (z1) = f(z1) , (20)
f++
2 (z1, z2, r12) = f−−
2 (z1, z2, r12) , f+−
2 (z1, z2, r12) = f−+
2 (z1, z2, r12) ,
where “+” denotes positive particles, while “–” – negative ones. The functions are
coupled by the first equation of the BBGKY chain [13]:
df(z1)
dz1
+
dU1(z1)
dz1
f(z1) (21)
+ ρ
∫
V
∂
∂z1
ϕ(r12)
[
f++
2 (z1, z2, r12) + f+−
2 (z1, z2, r12)
]
dr2
+ ρ
∫
V
∂
∂z1
U++
C (r12)
[
f++
2 (z1, z2, r12)− f+−
2 (z1, z2, r12)
]
dr2
+ ρ
∫
V
∂
∂z1
U++
im (r′12)
[
f++
2 (z1, z2, r12) + f+−
2 (z1, z2, r12)
]
dr2 = 0.
It is closed by the superposition approximation
f
++
+−
2 (z1, z2, r12) = f(z1)F
0
hs(r12) exp{±G(z1, z2, r12)}, (22)
58
Diffusion coefficients for an inhomogeneous system
0.0 1.0 2.0 3.0 4.0 z/σ0.8
1.0
1.2
1.4
D(z)/Dbulk
∗
∗
∗
�
�
�
△
△
△
∗- 4%
�- 15%
△- 27%
Figure 2. Self-diffusion coefficients for Na+
and Cl− ions in water near the reverse os-
mosis membrane at various mass concen-
trations
wherein F 0
hs(r12) is a pair distri-
bution function for hard spheres,
G(z1, z2, r12) is responsible for the in-
homogeneous electrostatic contribu-
tion. The latter, along with U1(z1), is
discussed in detail in [13]. Equation
(21) was solved for an aqueous NaCl
solution (ε1 = 81) near a frequently
used polymeric membrane with ε2 =
4 at temperature 250C. The diame-
ters σ for both Na and Cl ions were
assumed to be 3.9Å. Numerical cal-
culations for three mass concentra-
tions (4%-sea water, 27%-the high-
est concentration at 250C) are de-
picted in figure 1. In all the cases
negative adsorption is observed. The
contact value increases with the rise
of concentration, while the density
maximum shifts closer to the mem-
brane. It can be explained by the
fact that structural ordering is more
intensive for larger concentrations in
comparison with the Coulomb inter-
action. However, all the solutions achieve the bulk properties almost simultane-
ously, namely, at z ≈ 4σ − 5σ.
Now let us focus on the diffusion coefficients in accordance with equation (12).
It is easy to see that electrolyte symmetry due to Coulomb interactions makes
some quantities entering (14) equivalent. Thus, the set of equalities (20) may be
extended by those for a single particle interaction
U+
1 (z1) = U−
1 (z1) = U1(z1), (23)
which is used in (21) and for a pair potential
U++(r, r0) = U−−(r, r0), (24)
U+−(r, r0) = U−+(r, r0).
Instead of the inhomogeneous pair correlation function g2, the same bulk one is
accepted to simplify the calculations. For qualitative estimates the masses of Na
and Cl ions are assumed to be equal. Then their diffusion coefficients coincide, too:
D++(r) = D−−(r) = D(r). (25)
In contrast to the evaluation of distribution functions, Coulomb interaction is
excluded from pair potentials for the friction constant λ
kk
2 (r) (14), because the
59
R.I.Zhelem et al.
effect of particle dimensions is dominant especially at high densities typical of a
liquid. However, electrostatics has an effect through the distribution function (21)
which is calculated with the total potential (hard spheres plus Coulomb). With all
the assumptions we arrive at the following:
D(z) =
Dbulk
√
A(z) + B(z) + C(z)
, (26)
where
A(z) = 3/4
∆U(z)
πN/V σ3g2(σ)
, (27)
B(z) = 1/2
1
∫
Max(−z,−1)
f1(z + z′) dz′, (28)
C(z) = 5/4
(
−
1
∫
Max(−z,−1)
f1(z + z′) dz′ + f1(z + 1) +H(z − 1)f1(z − 1)
)
, (29)
g2(σ) is a contact value for the bulk hard sphere pair correlation function, H(x) is
the Heaviside unit function, Dbulk is a bulk diffusion coefficient attained at high z,
when A(z) → 0, B(z) → 1, C(z) → 0. It exceeds the experimental value. This is
not surprising, as the Markovian approximation was used to obtain (26), as well
as other simplifications were made to treat the problem analytically. Thus, we put
Dbulk = 1 and focus our attention on the relative behaviour of D(z).
In figure 2, qualitative self-diffusion coefficients for Na and Cl ions in the system
“initial solution–membrane” are represented, as computed from (26). Intensive
repulsion near the membrane (negative adsorption) brings about an overall rise
of the coefficient at small distances where the effect of the term A(z) is the most
significant. Far from the membrane it monotonically tends to zero, while all the
typical behaviour of the self-diffusion coefficient is determined by B(z) and C(z).
B(z) is responsible for the bulk value Dbulk at z → ∞. Refraction of the plot for
the largest concentration at z ≈ 1.5σ is caused by the hard sphere potential which
has a step-like form and abruptly decreases the singlet density from a contact value
to zero. So, expression (26) adequately describes the position dependent diffusion
coefficient for ions near the dielectric membrane surface that operates on the basis
of a negative adsorption phenomenon.
References
1. Lakshminarayanaiah N. Transport Phenomena in Membranes. N.-Y. and London,
Academic Press, 1967, 517 p.
2. Dytnerskii Yu.I. Membrane Processes for Liquid Mixture Separation. Moscow,
Khimiya, 1975, 229 p. (in Russian).
60
Diffusion coefficients for an inhomogeneous system
3. Kozhevnikov N.E., Leonov A.I. Reverse Osmosis and Ultrafiltration. Reviews on Cer-
tain Productions of Chemical Industry. Moscow, Institute of experimental chemistry,
1973, No. 35, 21 p. (in Russian).
4. Lonsdale H.K. Theory and practice of reverse osmosis and ultrafiltration. In the book:
Technological Processes Using Membranes, Moscow, Mir, 1976, p. 130–159 (in Rus-
sian).
5. Rade C.E. Principles of reverse osmosis. In the book: Technological Processes Using
Membranes, Moscow, Mir, 1976, p. 115–130 (in Russian).
6. Dytnerskii Yu.I. Reverse Osmosis and Ultrafiltration.Moscow, Khimiya, 1978, 349 p.
(in Russian).
7. Sourirajan S. Reverse Osmosis. London, Logos, 1979, 578 p.
8. Nikolayev N.I. Diffusion in Membranes. Moscow, Khimiya, 1980, 231 p. (in Russian).
9. Dukhin S.S., Churayev N.V., Shilov V.N., Starov V.M. Problems of reverse osmosis
simulation. // Uspekhi Khimii, 1988, vol. 57, No 6, p. 1010–1030 (in Russian).
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sian).
11. Kuryliak I.J., Tokarchuk M.V. Statistical theory of electrolyte solutions transport
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13. Yukhnovskii I.R., Zhelem R.I., Omelyan I.P., Sovyak E.N., Tokarchuk M.V. Diffusion
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edition, 1990, 556 p.
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61
R.I.Zhelem et al.
Pозpахунок функцiй pозподiлу та коефiцiєнтiв
дифузiї iонiв у системi “вихiдний pозчин
електpолiту – мембpана”
P.I.Желем, М.В.Токаpчук, I.П.Омелян, Є.М.Сов’як
Інститут фізики конденсованих систем НАН Укpаїни,
290011 Львів, вул. Свєнціцького, 1
Отpимано 14 липня 1998 р.
Pозглядається дифузiя iонiв у неодноpiдних умовах двофазної систе-
ми вихiдний pозчин електpолiту – мембpана. З допомогою методу
неpiвноважного статистичного опеpатоpа отpимано виpази для кон-
станти теpтя, коефiцiєнтiв самодифузiї та взаємної дифузiї в неод-
ноpiдному випадку. Їх застосовано до даної системи, і pозpаховано
поведiнку коефiцiєнта самодифузiї для NaCl, залежного вiд вiддалi до
звоpотноосмотичної мембpани.
Ключові слова: дифузiя в piдинах, пpоцеси пеpеносу, функцiя
pозподiлу, осмос, мембpана
PACS: 05.60.+w, 73.40, 66.10.Cb, 82.65.Fr
62
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