Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 144, No. 2
PHYSICS
E. A. STRELTSOVA
DETERMINATION OF THE ELECTROPHORETIC VELOCITY OF AN ION BY THE METHOD OF DISTRIBUTION FUNCTIONS
(Presented by Academician N. N. Bogolyubov, 27 XII 1961)
In the study of the electrical conductivity of solutions of strong electrolytes, an essential role is played by the calculation of the electrophoretic force acting on an ion under the influence of an external electric field, and of the additional electrophoretic velocity of the ion caused by this force.
The qualitative aspect of the phenomenon of electrophoresis was studied in detail by the authors of works ((^{2-4})); however, for quantitative estimates these authors adopted hypotheses that make it difficult, and sometimes simply impossible, to obtain formulas of approximation higher than the first. In later investigations other authors ((^{5-7})) give formulas for the electrophoretic velocity in the form of power series, continuing, however, to adhere to the principle of a phenomenological approach to the phenomenon and thereby casting doubt on the accuracy of the results obtained.
In the present work an attempt is made at a quantitative study of the phenomenon of electrophoresis, based on the application of the statistical method of distribution functions, due to N. N. Bogolyubov ((^{1})). In ((^{9,10})) the motion was investigated of a system of (N) charged particles, connected by Coulomb interaction, in a viscous liquid under the action of an external field with potential (U_a(t, q_a)); (q{q^\alpha}), (\alpha = 1, 2, 3), are Cartesian coordinates determining the position of an ion in a macroscopic volume (V). Let the velocity of the considered (i)-th ion of the (a)-th class be (\mathbf{v}{ai}). This velocity is decreased, owing to the action of ions of the opposite sign moving in the opposite direction and carrying along particles of the liquid, by an amount proportional to some function (f = |q_i - q_j|) is the distance between the ions,}(1/r_{ij})), where (r_{ij
[
\mathbf{v}{ai}
-
\sum}
\mathbf{v}{bj} f\right).}!\left(\frac{1}{r_{ij}
]
(f_{ab}(1/r_{ij})) turns out to be a tensor, the components of which we determine by making use of the expressions known in hydrodynamics for the velocity of liquid particles carried along by a solid sphere of radius (a), moving through a viscous liquid ((^{11})). Taking the ion radius (a) to be considerably smaller than the distance between the ions (r_{ij}), in the expressions for the velocity we shall discard terms of order (a^3/r^3). In Cartesian coordinates the components of the tensor have the form
[
f_{ab}^{\alpha\beta}
=
f_{ab}^{\beta\alpha}
=
\frac{3b}{4|q_a - q_b|}
\left[
\delta_{\alpha\beta}
+
\frac{(q_a^\alpha - q_b^\alpha)(q_a^\beta - q_b^\beta)}
{|q_a - q_b|^2}
\right];
]
(b) is the radius of an ion of the (b)-th class, (\delta_{\alpha\beta}=1) for (\alpha=\beta), (\delta_{\alpha\beta}=0) for (\alpha\ne\beta).
The resistance force acting on an ion is
[
\mathbf{P}{ai}
=
-\lambda
\left(
\mathbf{v}{ai}
-
\sum}
\mathbf{v}{bj} f\right)}!\left(\frac{1}{r_{ij}
\right),
]
where (\lambda_a = 6\pi\eta a); (\eta) is the coefficient of viscosity of the liquid.
In the first approximation we shall assume that this force has a Stokesian nature, i.e., that (\mathbf v_{ai}=-\mathbf P^0_{ai}/\lambda_{ai}); then
[
\mathbf v_{ai}
=
-\frac{\mathbf P^0_{ai}}{\lambda_{ai}}
+
\sum_{\substack{1\le i\ne j\le N_b\ 1\le b\le m}}
-\frac{\mathbf P^0_{bj}}{\lambda_{bj}}
\, f_{ab}!\left(\frac{1}{r_{ij}}\right).
]
The potential energy of the system is
[
U
=
\sum' \Phi_{ab'}!\left(\left|q_{ai}-q_{b'j}\right|\right)
+
\sum_{\substack{1\le i\le N_a\ 1\le a\le m}}
U_a(t,q_{ai}),
]
where (\sum') denotes summation over all distinct pairs of particles;
[
\Phi_{ab'}!\left(\left|q_{ai}-q_{b'j}\right|\right)
=
\frac{e_a e_{b'}}{B\left|q_{ai}-q_{b'j}\right|}
]
is the Coulomb potential of binary interaction; (e_a) is the charge of an ion of the (a)-th class; (B) is the dielectric constant of the solution. The electric-field strength depends only on time, so that
[
\frac{\partial U_a(t,q_{ai})}{\partial q_{ai}^{\alpha}}
=
-e_{ai}E^{\alpha}(t).
]
The component of the force along one of the coordinate axes is
[
P_{ai}^{0\alpha}
=
\sum'
\frac{\partial \Phi_{ab'}!\left(\left|q_{ai}-q_{b'j}\right|\right)}
{\partial q_{ai}^{\alpha}}
-
e_{ai}E^{\alpha}(t).
]
In view of the identity of particles of one species, the velocity component along this axis will have the form
[
v_{ai}^{\alpha}
=
\frac{e_a}{\lambda_a}E^{\alpha}(t)
-
\frac{1}{\lambda_a}
\sum_{1\le b'\le m}
\frac{\partial \varphi_{ab'}!\left(\left|q-q'\right|\right)}
{\partial q^{\alpha}}
\left(N_{b'}-\delta_{ab'}\right)
+
]
[
+
\sum_{\substack{1\le \beta\le 3\ 1\le b'\le m}}
E^{\beta}(t)\,
\frac{e_{b'}}{\lambda_{b'}}
\, f_{ab'}^{\alpha\beta}!\left(\frac{1}{r}\right)
\left(N_{b'}-\delta_{ab'}\right)
-
]
[
\sum_{\substack{1\le \beta\le 3\ 1\le \beta'\le m}}
\frac{1}{\lambda_{b'}}
\, f_{ab'}^{\alpha\beta}!\left(\frac{1}{r}\right)
\left(N_{b'}-\delta_{ab'}\right)
\sum'{1\le b''\le m}
\frac{\partial \varphi}!\left(\left|q'-q''\right|\right)
{\partial q'^{\beta}}
\left(N_{b''}-\delta_{b'b''}\right).
]
Let us average the velocity over all particles:
[
\overline{v_{ai}^{\alpha}}
=
\int\cdots\int v_{ai}^{\alpha}D_N\cdots dq\cdots;
]
(D_N(q_1\ldots q_N)) is the Gibbs distribution function of the probabilities of the positions of all particles.
Carrying out the averaging operation and the limiting passage (N\to\infty), (V\to\infty), (N/V=n), we obtain an expression for the averaged velocity consisting of 9 terms, 5 of which depend on the distribution functions (g_{ab}(t,q_a,q_b)), determined in ((9,10)). Of these 9 terms, 7 vanish either because of the condition of electrical neutrality of the system or because of the properties of the functions (g_{ab}(t,q_a,q_b)). Two terms remain:
[
\overline{v_{ai}^{\alpha}}
=
\frac{e_a E^{\alpha}(t)}{\lambda_a}
+
]
[
+
\frac{1}{8\pi\eta V}
\sum_{\substack{1\le \beta\le 3\ 1\le b'\le m}}
E^{\beta}(t)
\iint
e_{b'}n_{b'}
\left[
\frac{\delta_{\alpha\beta}}{\left|q-q'\right|}
+
\frac{(q^{\alpha}-q'^{\alpha})(q^{\beta}-q'^{\beta})}{\left|q-q'\right|^3}
\right]
g_{ab'}(t,q,q')\,dq\,dq'.
]
If, in solving the problem, one confines oneself to finding terms linear with respect to (\bar E), then for (g_{ab'}(t,q,q')) one may take only its equilibrium value, found in (8). Substituting this value into the second term of the expression for the velocity, after lengthy, though not complicated, transformations we obtain:
[
\begin{aligned}
\overline{v}{ai}={}&
\frac{e_a E^\alpha(t)}{\lambda_a}
-\frac{e_a E^\alpha(t)\chi}{6\pi\eta}\,e^{-\chi a}
+\frac{e_a^2 E^\alpha(t)\sum_b e_b^3 n_b}{3\eta(B\theta)^2}\operatorname{Ei}(2\chi a)
\
&+\frac{2\pi e_a E^\alpha(t)\sum_b e_b^2(e_a+e_b)n_b \sum} e_{b'}^3 n_{b'}
{3\eta\chi^2(B\theta)^3}
\left{ e^{-\chi a}\operatorname{Ei}(\chi a)
-2\operatorname{Ei}(2\chi a)+e^{\chi a}\operatorname{Ei}(3\chi a)-\ln 3\cdot e^{-\chi a}\right}
\
&+\frac{\pi e_a E^\alpha(t)\left(\sum_b e_b^3 n_b\right)^2}
{3\eta\chi^2(B\theta)^3}
\left{
\frac{2}{3}e^{-2\chi a}
+\left[\ln 3(2+\chi a)-\frac{4}{3}\right]e^{-\chi a}
\right.
\
&\left.\qquad\qquad
+\left[(\chi a-1)e^{\chi a}-1\right]\operatorname{Ei}(3\chi a)
+4\operatorname{Ei}(2\chi a)-(\chi a+2)e^{-\chi a}\operatorname{Ei}(\chi a)
\right},
\
\chi^2&=\frac{4\pi\sum_s e_s^2 n_s}{B\theta},\qquad \theta=kT.
\end{aligned}
]
The formula contains tabulated exponential integral functions
[
\operatorname{Ei}(y)=\int_y^\infty \frac{e^{-t}}{t}\,dt
=-0.5772-\ln y+y-\frac{y^2}{2\cdot 2!}+\cdots
]
and is suitable for direct calculations.
The first term of the formula represents the ion velocity due to the action of the external field; the second is the known expression for the electrophoretic velocity, found in (2–4) under the assumption that the “ionic atmosphere” preserves spherical symmetry; the remaining terms give the correction due to deformation of the ionic atmosphere, which in fact occurs during the motion of the ion in the external electric field.
The author expresses deep gratitude to Academician N. N. Bogolyubov for discussion of the results.
Kyiv Technological Institute
of Light Industry
Received
27 XII 1961
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