A. Schmitt, R. Varoqui, and J. P. Meullenet
1514 (55) L. Pauling, “Nature of the Chemical Bond”, 3rd ed,Cornell University Press, Ithaca, N.Y., 1960. (56) See paragraph at end of text regarding supplementary material. (57) V. G. Dashevsky and A. I. Kitaigorodsky, Theor., Eksp. Khlm., 3 ,
43 (1967). (58) F. J. Adrian, J . Chem. Phys., 28, 608 (1958). (59) G. W. Whehrd, “Resonance in Organic Chemistry”, Wlley, New York, N.Y., 1955, p 132.
Ionic and Electrical Conductances in Polyelectrolyte Solutions A. Schmitt,’ R. Varoqui, and J. P. Meullenet CNRS, Centre de Recherches sur les Macromol6cules. 6, rue Boussingauh, Strasbourg, France (Received June 28, 1976; Revised Manuscript Recelvsd May 18, 1977) Publication costs asslstsd by Centre de Recherches sur les Macromolecules
A phenomenological theory of electrical transport in polyelectrolyte-plus-salt solutions is developed. The treatment is based on linear force-flux relationships by using binary frictional coefficients within the framework of linear nonequilibrium thermodynamics. Correlations between conductances and reduced self-diffusion coefficients of counterions and coions are established without taking recourse to an explicit model, and former theories on the subject are generalized. In particular, it is shown that the “intrinsic conductivity” of a polyelectrolyte in excess-of-salt solutions may be related in a simple way to the electrophoretic mobility of the polyion, the transport parameters of small ions in pure salt solutions, and to the lowering of the self-diffusion coefficient of small ions in the presence of the polyelectrolyte. A comparison with recent experimental results displays semiquantitative agreement.
I. Introduction The study of the correlation between electrophoretic and self-diffusion ionic mobilities in polyelectrolyte, salt-free, solutions started with the pioneering work of Huizenga, Grieger, and Wal1,l who proposed a simple equation to interrelate their experimental data. Manning2was the first to establish the HGW equation on theoretical grounds and to discuss its physical significance by considering a model polyelectrolyte solution. His result was later generalized by Schmitt and V a r ~ q u iusing , ~ a phenomenological approach. When a low-molecular-weight salt is present, a great variety of experiments, ranging from the salt-free to the excess-of-saltconditions, is in principle available. However, not many extensive results have been published to enable the same kind of interrelations to be made. It is the purpose of the present paper to establish general equations interrelating ionic mobilities, and to propose some guidelines for a more systematic experimental investigation. A comparison is made with the recent work of Devore and Manning4 who extended the results of the Manning theory. 11. Theory (1)The Polyelectrolyte Solution. Let us consider an aqueous solution containing a monodisperse polyelectrolyte and a low-molecular-weightsalt of concentrations c, and c,, respectively. The valences ( 2 ) of the mutual counterion (l),coion (2), and polyelectrolyte (3) are zl,z2, and 2QS,respectively. The conditions of electroneutrality are
counterions might have to be considered as part of the polyion. The effective valence of the polyion is then 23 (23 I zgS),and the number of free counterions per polyelectrolyte molecule v, so that the fraction f of free counterions per polyelectrolyte molecule is
f = ~ / V S= Z ~ / Z ? . S
(3) Finally, we express the stoichiometric fraction, x , of counterions belonging to the polyelectrolyte
x = V&p/(V,Cp +
VlC,)
(4)
x ranges from 0 (excess-of-salt limit) to 1 (salt-free so-
lution). (2) Correlation of Ionic Conductances. (a) Basic Equations. When a uniform and constant electric field is applied to a homogeneous ionic solution, the algebraic ionic diffusion fluxes, Ji,are proportional to the applied field strength, e, in the linear approximation6
Ji = q u i
= qui€
(i = 1,2, 3)
(5)
u; and u; are the mean algebraic velocity and mobility of ionic species i, with respect to the solvent. Iff differs from unity, we suppose that the bound ions move with the polyion, so that the stoichiometric counterion flux, Jls,is expressed as
J,, = J1+
JJ(V, -
v)
(6)
With eq 3,4, and 6, the relation between measured and effective counterion mobilities is u,, = [l- x ( 1 - f ) ]u1 + x(1- f)u3
(7)
Let us finally define the ionic equivalent conductances (positive quantities) by where v1 and v2 are the number of counterions and coions, respectively, per salt molecule and us and v3 are the number of counterions and polyions per polyelectrolyte molecule (v3 = 1). From a thermodynamic point of vie^,^^^ some The Journal of Physical Chemlstry, Vol. 8 1, No. 15, 1977
5 is the Faraday constant.
1515
Electrical Transport in Polyelectrolyte Solutions
(b) Equivalent Conductance of the Coion. Correlations between the various ionic mobilities are easily obtained through the friction coefficient formalism. Under the conditions stated in section I1 2(a), it expresses that the electrical force applied to 1mol of an ion is balanced by the frictional forces with other mobile species. As applied to the coion, the multicomponent equation gives'
- vl)+
= f2l(v2
Z25E
f23(v2
- v3) + f2wv2
( l / f 2 ) ( z 2 s+ u l f 2 l
+
(10) where f i is the total friction coefficient of the coion with its environment, i.e. =
f2
+ f21 +
= f2w
u3f23)
(11)
f23
Let us now make a perturbation approximation and suppose that, for small values of x , f 2 , and fil have the same values as those in a pure salt solution of concentration c,. Thus, the frictional interaction between the coions and the polyion appears only through the f 2 3 coefficient. We denote with superscript u quantities relative to a pure salt solution of concentration c,, so that f2
+
f2"
f23
= f21" +
fiw"
+
f23
IzlI[c,vl(X1,
K - K"
= h2"Pz
This equation will be useful in studying the limiting behavior when an excess of salt is present, as we will now see. (e) Conductances in the Excess-of-Salt Limit. The excess-of-salt limit is characterized by x
