Theory of conductance of colloidal electrolytes in univalent salt

The Journal of Physical Chemistry, Vol. 83, No. 12, 7979

Conductance of Colloidal Electrolytes

1663

Theory of Conductance of Colloidal Electrolytes in Univalent Salt Solutions Dirk Stigter Western Regional Research Center, SEA, USDA, Albany, California 94710 (Received September 20, 1978: Revised Manuscrlpt Received February 20, 1979) Publication costs assisted by the US. Department of Agriculture

The conductance theory in this paper is a sequel of the recent treatment of the electrophoresis of highly charged colloidal cylinders with a Gouy-Chapman type electrical double layer, allowing for the same hydrodynamic and electrostatic interactions: electrophoretic hindrance and relaxation effect. The approach accounts in detail for the transport of the small ions near the colloid particle. The interaction between colloid particles is neglected. The theory is developed for randomly oriented, long cylinders (no end effects). For spherical colloids only the final expressions are given. In both cases the colloid conductance is expressed in terms of interaction integrals. These integrals have been evaluated by extending the computer programs written earlier for the electrophoresis problem. Conductance and electrophoresis depend on the same parameters: the reduced surface potential and the reduced radius of the colloid, and the reduced mobilities of the small ions. Numerical results are presented that span the range of practical interest of these parameters for colloid spheres and cylinders in univalent salt solutions, including new data on electrophoresis.

Introduction Several aspects of the electrical conductance of colloidal solutions are treated in the literature.lB2 The results are too fragmentary or too limited in scope for the evaluation of the kinetic charge of the colloid particles. In general, the theory of transport in solutions is conceptually and mathematically complex. Much effort has been devoted to the proper interpretation of the migration of charged solutes in an external electric field. In strong electrolyte solutions3 conductance and transference data have served to test ionic distribution functions (Poisson-Boltzmann) and the assumptions of continuum hydrodynamics (Navier-Stokes) in the molecular domain. In colloidal electrolyte solutions112 the interpretation of electrical transport experiments relies also on the use of ionic distributions and of continuum hydrodynamics. As argued earlier14 the relative errors in the Poisson-Boltzmann equation are expected to be less in the colloidal range than for small ions. The same is true for the Navier-Stokes equation (compare ref 3, p 125). This provides a suitable, albeit approximate, physical basis from which to proceed in colloidal solutions. On the mathematical side the electronic computer has added a new dimension to theoretical chemistry in recent years. Earlier approximations in mathematical analysis can now be tested and, if desired, be replaced by more exact numerical results. The present study rests on earlier work where computer techniques have been applied in the theory of electrophoresis of highly charged colloidal spheres4b5and cylinders617surrounded by a Gouy-Chapman ionic atmosphere. The theory is now extended to the conductance of colloidal solutions where we account for the same two interaction effects that enter into the treatment of electrophoresis: (a) the electrophoretic hindrance or retardation, due to the extra velocity of the counter charge imparted to it by the liquid flow around the central colloid particle, and (b) the relaxation effect, arising from the distortion of the equilibrium distribution of the small ions around the colloidal ion in motion. This approach means that the electrophoresis and conductance problems are solved by the same approximation since both results depend on the same hydrodynamic and electrostatic perturbations caused by the colloid ion and the small ions moving in the external field.

The treatment takes into account the interactions between the colloid and the small ions in its vicinity. In experimental practice colloidal solutions may be diluted to eliminate interactions between the colloidal particles, except in salt-free solutions. In the latter case one expects the hydrodynamic and electric interactions between colloid particles to be long range, that is, of the same range as the interparticle distance. Hence it is difficult to eliminate such interaction by extrapolating to infinite dilution of colloid. This difficulty is well illustrated by experimental data such as Eisenberg’s conductance measurements of aqueous poly(methacry1ate) solutions.8 For this reason we exclude salt-free colloidal solutions. We consider only dilute colloidal solutions which contain excess univalent salt. The present paper contains the theory and numerical results. In the following paperg applications are made to detergent micelles, to poly(methacrylates), and to DNA. In the next section we first establish the relation between the conductance and the ionic mobilities in colloidal solutions. In subsequent sections the treatment is specialized to highly charged, randomly oriented, long cylinders and to colloidal spheres. The theory is developed for positively charged colloids with a negative Gouy-Chapman ionic atmosphere. Consistent charge reversal on colloid and small ions make the results applicable to anionic colloids. Relation between Colloidal Conductance and Ionic Mobilities As in statistical theorylO we take the view that colloidal solutions are not homogeneous. The starting point is the fact that colloid ions are much larger than the small ions and carry a much larger electric charge. The colloid ions and their ionic atmospheres form islands in an otherwise uniform salt solution. We shall derive the contribution of a single “colloid island” to the conductance of the solution. Consider an aqueous solution of a univalent salt, molarity c3, and a colloidal electrolyte, molar equivalent concentration c2 based on titration or fixed charge. (The subscript 1is reserved for water as the main component,) Formally we can write the specific conductance K , of the solution as a sum of contributions of the colloid and of the salt

This article not sublect to U S . Copyright. Published 1979 by the American Chemical Soclety

1604


~ O O O K ,= C&

+ ~3h3

(1)

where h2 is the equivalent conductance of the colloid component and h3that of the salt. Such a division of K, is not unique since it depends on the allocation of the various interaction effects between colloid and small ions to either the salt or the colloid. It is convenient to take for h3the equivalent conductance of the salt in the uniform salt solution far from the colloid particles, outside their double layers. This choice for h3also fixes the colloid conductance and relegates to A2 all the kinetic interactions in the transport process: electrophoretic hindrance and relaxation effect of the small ions in the vicinity of the colloid ion. The usual treatment of the conductance of nonhomogeneous medial1 is based on the analogy with the electrostatics of nonhomogeneous dielectrics, that is, on solutions of Laplace’s equation for dielectric mixtures (compare ref 3, p 312). In colloidal electrolyte solutions, however, the situation is more complex because the current distribution near a colloid particle does not satisfy the Laplace equation. Fortunately, in electrophoretic theory one evaluates the distribution of the small ions, of the electric field, and of the liquid velocity around the colloid ion. These distribution functions are major ingredients of the conductance theory. In the remainder of this section we prepare a framework for the use of such electrophoretic information. Suppose that the colloid-salt solution is contained in a rectangular volume V of length 1, and with an external field of average strength X perpendicular to the cross section with area A = V/1. With eq 1 one obtains for the total current I through the solution I = K,XA = (czhz + ~3h3)XA/1000 (2) When the colloid concentration is sufficiently low the colloid particles and their double layers form isolated inhomogeneities in the otherwise uniform salt solution. Therefore, on a colloidal scale the current density i in the solution is not uniform. Nevertheless, under steady state conditions the total current

I= SidA A

(3)

must be the same through all cross sections. For the purpose of extrapolation the surface integral in eq 3 is converted into a volume integral by multiplication with (1/1)Sldl I = 11 S vi d V

(4)

Comparison with eq 2 gives

We now take the limit of eq 5 at, infinite dilution. For a (relatively large) volume u of salt solution that contains one single colloid particle with n native charges eq 5 yields nhz/N,, =

lk

du - C3h3V/1000

(6)

where N,, is Avogadro’s number. In eq 6 the term in h2may be derived from experiments, as shown in the following paper.g The remaining terms are expressed in the charge transport by individual ions. The colloid ion has electrophoretic mobility U/X and carries a kinetic charge Ze ( n native charges and n - Z counterions inside the shear surface). So the contribution

Dirk Stigter

of the colloid ion to the integral is ZeU/X. We recall that in the definition of h2we have taken for AS the equivalent conductance of the bulk salt solution, far from the colloid particles. In this region the velocities of the small ions are

u+ = eX/f+ and u- = -eX/f(7) where f+ and f- are friction factors that include the interactions between the small ions in the bulk salt solution. So in eq 6 we have

where F is the Faraday. The local current density everywhere in the solution outside the colloid particle is connected with the local velocities of the small ions i = e(u+u+- v-u-) (9) where v+ and v- are the local concentrations (number densities) of the small ions. Collecting results in eq 6 one obtains

(10) The term ZeU/X in eq 10 represents the current carried by the colloid ion, So the integration in the last term of eq 10 is over the volume u of solution outside the shear surface of the colloid particle. Without the colloid particle and its double layer the last two terms of eq 10 cancel exactly. In the presence of the colloid the result is not so simple. In the vicinity of the colloid ion u+ and u- are not given by eq 7 and, furthermore, the local concentrations u+ and u- differ from the bulk values because of interactions with the highly charged, moving colloid ion. These interactions can be derived in the case of colloidal spheres and long cylinders from the electrophoresis theory of such particle^.^-^ The next section gives the treatment for colloidal cylinders. Conductance of Colloidal Cylinders We adopt the model and the assumptions of electrophoresis theory:6i7 positively charged, long cylinders with random orientation and surrounded by a negative, Gouy-type ionic atmosphere; no end effects; uniform viscosity and uniform dielectric constant in the solution outside the shear surface of the cylinder; no conductance of the cylinder inside its shear surface; allowance for high surface potentials, and polarization of the double layer in the external electric field (relaxation effect); solution obeying Ohm’s law. The notation of ref 6 and 7 is used. For randomly oriented cylinders the conductance is obtained by averaging the interactions in the principal 0rientations.I We first deal with the cylinder oriented perpendicularly to the external field. The local ion concentrations near the cylinder at rest (no external field) are related to the electrostatic potential y = e$/kT in the double layer by Boltzmann’s law

In this equation and others below, the upper sign refers to the positive co-ions and the lower sign to the negative counterions, In the external field X the double layer is polarized. The relative perturbations of the ion concentrations are expressed by the functions 6+



v, = v,0(1

+ 6,)

(12)

The distortion of the equilibrium potential y by the external field is measured by the dimensionless perturbation potential g. The total electrostatic potential A is given by eA/kT = y + g (13) The local ion velocities near the cylinder are a superposition of the local liquid velocity u, an electric ion migration proportional to -VA, and a diffusion velocity. Instead of eq 7 we haye u* = u 7 (e/fdVA - ( k T / f d V ( l nv d (14) The local ion currents are, with eq 13 and 14 v+u+= vau - (kT/fd[fv*V(y + 8) + Vv+l

V * U ~= Y*OU

- (kT/fh)vaOV(fg

+ )6,

(16)

represents the deformation of the applied field X by the insulating cylinder with radius xo = KU, if it were uncharged. For reasons of symmetry we need in eq 17 only the differential operator in the field direction 1 sin8 a -v cos 29- a - -K ax x a29 and application in eq 17 yields

v,u* = v*"u +

\

+ L'[ ds)

d cos2 29 -(fH dx

(25)

where E is a computed function of y and x. We now return to the integral in eq 10. For u we take a cylindrical volume of solution between the shear surface at r = a and an outer radius r = b that encloses the entire double layer, and with the length L of the cylinder. The terms in f l inside the brackets in eq 24 yield the conductance of all small ions in u with the bulk mobilities of eq 7. This contribution cancels the second term in eq 10, except for the counterions:

Ze2/f- (26)

+

f*

where L+ and L- are computed in the electrophoresis of the cylinder in a transverse field reported in ref 6. The average liquid velocity a t distance x = Kr from the randomly oriented cylinder is following eq 22 of ref 7

f&K

where K is the reciprocal Debye length, 29 is the angle between the radial vector r and the direction of the applied field X, H a n d D, are dimensionless functions of the radial variable x = Kr. The function H = x x:/x (18)

-v,O

+ f*

( u ) = (DkTX/l2rqe)(y

Substituting for g and 6* from eq 3-7, 3-8, and 3-40-3-42 in ref 6, eq 16 becomes eX V*U+ = V*'U -v+OV((fH - D*) COS 29) (17)

+

cylinder parallel with the field, to produce the result for the randomly oriented cylinder. Now, in parallel orientation there is no relaxation effect and, hence, the counterpart of L, vanishes. So the average ion transport function for random cylinders is eX (24) (v+u+)= V*'(U) -vho(fl - 2/L;t(x)]

(15)

We consider solutions that follow Ohm's law. Therefore, in eq 15 only the terms proportional to X need to be retained. Moreover, terms that depend only on the radial distance r from the central axis of the cylinder do not contribute to the integral in eq 10 and are dropped also. The remainder of eq 15 is reduced to

1665

- D,)

+ sin2 8 f H X- D+

Evaluation of the integral in eq 10 requires the average with respect to 29

or, using eq 18 and 20, for the perpendicular orientation

The friction factors of the small ions are expressed in terms of their equivalent conductances in the bulk salt solution 1 l/f+= X,/Fe and - = L / F e (27)

f-

With eq 24, 26, and 27 the colloid conductance in eq 10 becomes ZFU FL b nA2 = -+ ZX(v+O - v?)(u)Zrr dr -

X

+ xs

In eq 28 one recognizes the separate contributions to nA2 of the colloid ion, the 2 counterions, and the interaction integrals. These integral terms would vanish if there were no interaction between the small ions and the colloid ion. They are the counterparts of the well-known interactions in electrophoresis. The first integral is the electrophoretic hindrance which slows down the counter charge with the liquid velocity u around the moving cylinder. The second integral is the relaxation effect. In both terms the integrand vanishes outside the double layer so that one integrates essentially from the shear surface of the cylinder, a t r = a, to r = ~ 0 . For computations and for practical applications it is convenient to write eq 28 as follows nA2 = ZXcOu + ZX-

and with eq 3-52 of ref 6 for D+ eX ( v a u a ) l = ~ I O ( U ) L + -vko(f1

f*

-L(x)J

(23)

The dimensionless functions L+ and L- are intermediate products of the electrophoresis computation. In eq 23 the functions ( u ) and L,(x) depend on the orientation of the cylinder. As derived in ref 7 these functions for the perpendicular orientation must be mixed in a 2:l ratio with the corresponding functions for the

+ ZXeh+ ZX+rel+ ZX-,l

For randomly oriented cylinders6J FU DkTF hcoii = X = -(YO 127r17e + JXoE dx)

(29)

(30)

With the relation for the cylinder 27rLv*O -eTY

--

ZK'

-

2xo(d~/dx)o

one obtains from the integrals in eq 28 for the electrophoretic hindrance

1666


F Xeh

1

1;

= %2xo(dy/dx)o

(eY - e-Y)(u)x dx

(32)

for the relaxation contribution of the co-ions

and of the counterions

In general, Atoll in eq 30 is evaluated by an iteration method using a computer, as explained e l ~ e w h e r e . The ~?~ functions ( u ) / X in eq 32 and L, in eq 33 and 34 are byproducts of this computation so that an extension of the program in the final iteration cycle yields h e h and XfIe1.

Dirk Stigter

values for yo = 0 have been obtained from Henry's linearized theory.12 Note that the influence of the counterions on f is larger than that of the co-ions, more so at higher surface potential of the colloid. The electrophoretic hindrance term is presented in Table I1 as the ratio &.h/Xcoll. It is clear that in eq 29 for the conductance the interaction z X e h offsets a substantial fraction of the term ZXcoll and cannot be neglected. For cylinders and for spheres the ratio Xeh/Xcoll approaches the same limit for large xo. This limit is a function of yo and can be derived from the theory of the flat double layer where one finds for xo = 2 cash (y0/2) - 2 Xeh _ -1 (42) Xcoll Yo sinh (Yam

+

Conductance of Colloidal Spheres The treatment for colloidal spheres proceeds along the same lines as for cylinders and yields very similar results. In eq 29 we have for spheres with reduced radius xo = K U , kinetic charge Ze and native charge ne

The electrophore.tic hindrance ratio is very insensitive to the mobility of the small ions. Table I1 shows that increasing m+ or m- by a factor of 2 changes h e h / h c o l l by less than 2%, even for yo = 5. Results for the relaxation effect are given in Tables I11 and IV as the ratios X+Iel/A+ and Using the low potential approximations for y and L,, one can show that for yo 0 both relaxation terms converge to the same analytic form (apart from a sign difference). For spheres this limit is

The average liquid velocity at distance x from the sphere is

-A*rel-

-

A,

exo -f e-x(x - x2/x5) dx 6(1 + x O ) l ,

yo = 0 (43)

and for cylinders Instead of eq 32-34 we now have

The quantities 4, L+, and L- in eq 35-38 represent dimensionless functions of x which are computed for the electrophoresis of the sphere according to ref 4. Numerical Results a n d Limitin'g Cases The interaction terms in eq 29 have been evaluated for spheres and for cylinders with various parameter values, in conjunction with the computation of the electrophoretic mobility U/X of the colloid. All results in this section are presented in dimensionless form, that is, independent of temperature, viscosity, or dielectric constant of the solution. The factor that connects the experimental conductance Atoll with the reduced mobility U' of the colloid XcOu = A J / X = (DlzTF/67rve)U' (39) has in aqueous systems at 25 "C the value DkTF/67rae = 12.86 cm2 0-l equiv-l

(40)

The same factor connects the conductance A+ of the co-ions and A- of the counterions with the dimensionless friction parameters m+ and m- used in the computation m, = (DkTF/67rae)(l/h,) (41) In water at 25 "C m, = 0.184 corresponds to A, = 70, close to the values in KC1 solutions. Table I gives results for the electrophoretic mobility in terms of the factor f in U' = fyo,computed from eq 35 and 30. The data for mL = 0.184 and yo = 1-3 are taken from Table I of ref 6, the remaining data are new. The limiting

where KOand K1 are modified Bessel functions. Comparison of Tables I11 and IV shows that for higher potentials the relaxation effect is much stronger for the counterions than for the coions. Moreover, the effects for the two ion species are not coupled to any significant extent, that is, is nearly independent of m-, and XFre1 is essentially independent of m+. Additional results (not shown here), computed with an approximate double layer potential, suggest that the relaxation effect for both ions vanishes for xo = m, for spheres as well as cylinders. As observed above, the electrophoretic ratio in Table I1 changes very little with m+ or m-. The variation of the other conductance functions with m+and m- is shown in Figures 1 and 2 for some representative cases. Figure 1 confirms the approximately linear relation between the colloid mobility and m* already reported by Wiersema et a1.5 for spheres. A similar linear relation is found for cylinders in Figure 1,and for the relaxation ratios in Figure 2 for both spheres and cylinders. This means that, in practical applications, with m+and m.. different from the present values, the various conductance functions may be obtained by linear superposition of the corrections from the recorded values in Tables I11 and IV. For most small ions m+ or m- is between 0.184 and 0.368, that is, A+ is between 70 and 35 at 25 "C in water. In this range accurate interpolations can be made in the tables. The accuracy of the tabulated values depends on the convergence of the various functions in the iterative computation, Most runs were programmed to produce results with less than 0.1% error in Xcoll, that is, in the f values of Table I. The results for Xeh/Xcoll in Table I1 are equally accurate. The uncertainty in the relaxation functions of Tables I11 and IV is usually somewhat larger, in particular for small xo and large yo, where possible

The Journal of Physical Chemistty, Vol. 83, No.


12, 1979 1667

TABLE I: Electrophoretic Mobility Factor f i n U' = f y , and in U/X = f D t / 6 n q as a Function of Reduced Radius x, = Ka, Reduced Surface Potential y o = e f / k T of Colloid, and Reduced Friction Factors m , = DhTF/Gnqeh+ - of Small Ions x , = 0.1

yo

0.2

0.5

Colloidal Spheres 1 2

5

10

15

m

1.154 1.130 1.043 0.914 0.769 0.633

1.245 1.225 1.143 1.012 0.852 0.694

1.306 1.282 1.209 1.086 0.927 0.761

1.5 1.5 1.5 1.5 1.5 1.5

0.901 0.754 0.620

1.000 0.839 0.682

1.075 0.915 0.749

1.5 1.5 1.5

0.872 0.714 0.578

0.968 0.790 0.628

1.044 0.863 0.693

1.5 1.5 1.5

m, = 0.184, m_ = 0.184 0 1 2 3 4 5

1.000 0.997 0.987 0.972 0.953 0.928

1.001 0.996 0.979 0.950 0.909 0.860

1.008 0.997 0.963 0.907 0.837 0.760

3 4 5

0.965 0.944 0.918

0.940 0.895 0.842

0.895 0.821 0.741

3 4 5

0.962 0.935 0.903

0.933 0.881 0.819

0.882 0.780 0.710

1.025 1.008 0.956 0.876 0.783 0.689

1.062 1.040 0.969 0.864 0.748 0.639

m+= 0.368, m- = 0.184 0.862 0.766 0.671

0.850 0.732 0.623

m+ = 0.184, m- = 0.368 0.845 0.738 0.637

0.829 0.700 0.589

Long Colloidal Cylinders with Random Orientation

yo

x, = 0.1

0.2

0.5

10

m

1.223 1.198 1.129 1.032 0.933 0.846

1.312 1.290 1.227 1.130 1.018 0.914

1.5 1.5 1.5 1.5 1.5 1.5

0.803

1.022 0.923 0.838

1.121 1.009 0.906

1.5 1.5 1.5

= 0.184, m_ = 0.368 0.909 0.930 0.843 0.850 0.789 0.788

1.004 0.899 0.815

1.100 0.978 0.873

1.5 1.5

1

m, 0 1 2 3 4 5

1.001 0.995 0.973 0.940 0.902 0.862

1.006 0.997 0.970 0.931 0.888 0.844

1.024 1.011 0.974 0.924 0.871 0.823

3 4 5

0.931 0.890 0.848

0.922 0.876 0.831

0.914 0.859 0.811

3 4 5

0.926 0.880 0.835

0.916 0.866 0.819

0.907 0.849 0.799

= 0.184,

2

m- = 0.184

1.057 1.039 0.991 0.928 0.866 0.812

1.113 1.091 1.031 0.952 0.877 0.813

m+ = 0.368, m. 0.918 0.855 0.801

m,

5

= 0.184 0.942 0.866

1.5

0.9

m - = 0.184

-0--

08

CYLINDERS 0.184

ml

0.8

9

f

07

% ..

0.184

m-

0.7

_- A A A-

m-

=

0.184

SPHERES 06

m+ 0.6 0

=

0.184

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

m+ or rn-

Figure 1. Change of mobility factor fof colloid as a function of the mobility m- of the counterions (solid lines) and of the mobility mt of the co-ions (broken lines).

05

02

0 4

06

08

IO

m + or m -

divergence of the iterative computation was the limiting factor. Discussion We first consider the theoretical limit of the conductance for xo = 0. For a very small radius a of colloid spheres one

Figure 2. Change of the relaxation effect of the counterions, -A-,& as a function of the mobility m- of the counterions (solid lines) and of the mobility m + of the co-ions (broken lines).

might expect a linkage with the conductance theory of ordinary salt solutions. However, this is where the present theory is incorrect because it neglects the Brownian motion

The Journal of Physlcal Chemistry, Vol. 83, No. 12, 1979

1668

Dirk Stigter

TABLE 11: Ratio of Electrophoretic Hindrance of Small Ions and Colloid Conductance, h , ~ / h , , ~ , as a Function of Reauced Radius x, = Ka, Reduced Surface Potential y o = e.t/kT of Colloid, and Reduced Friction Factors m = DIzTF/Gnqeh, of Small Ions Colloidal Spheres y" x, = 0.1

0.2

0.5

2

1

5

10

15

m

-0.407 -0.420 -0.459 -0.519 -0.595 -0.676

-0.445 -0.457 -0.493 -0.547 -0.612 -0.683

-0.460 -0.472 -0.505 -0.555 -0.616 -0.681

-0.500 -0.510 -0.538 -0.577 -0.619 -0.661

0 1 2 3 4 5

-0.043 -0.045 -0.049 -0.055 -0.065 -0.082

-0.079 -0.085 -0.093 -0.108 -0.133 -0.174

-0.166 -0.172 -0.192 -0.228 -0.286 -0.368

5

-0.082

-0,175

-0,369

m, = 0.184, m- = 0.184 -0.247 -0.327 -0.257 -0.340 -0.288 -0.378 -0.342 -0.441 -0.419 -0.524 -0.515 -0.617 m+= 0.368, m- = 0.184 -0.515 -0.616

-0.676

-0.683

-0.681

-0.661

5

-0.083

-0.177

-0.376

m+= 0.184, m_ = 0.368 -0,526 -0.630 -0.688

-0.699

-0.693

-0.661

Long Colloidal Cylinders with Random Orientation x, = 0.1

yo

0.2

0.5

1

5

10

m

-0.448 -0.461 - 0.496 - 0.547 -0.607 -0.665

-0.469 - 0.481 -0.513 -0.561 -0.615 -0.670

-0.500 -0.510 - 0.538 -0.571 -0.619 - 0.661

-0.664

- 0.669

-0,661

-0.669

-0.675

- 0.661

2

m+= 0.184, m_ = 0.184 0 1 2 3 4 5

-0.187 -0.195 -0.210 -0.237 -0.278 -0.335

- 0.234

-0.264 - 0.302 -0.356 - 0.426

-0.305 -0.316 - 0.347 - 0.398 -0.463 - 0.538

5

-0.336

-0.426

-0.538

5

-0.338

- 0.429

-0.541

- 0.242

- 0.405 - 0.418

-0.359 -0.372 - 0.408 -0.463 - 0.530 - 0.600

-0.455 -0.511 -0.576 - 0.641

m+ = 0.368, m_ = 0.184 -0.600

- 0.639

m+ = 0.184, m_ = 0.368 -0.604

- 0.644

TABLE 111: Relaxation Effect on Conductance of Co-ions, h + d / h + as , a Function of Reduced Radius x, = Ka, Reduced Surface Potential y o = e c / k T of Colloid, and Reduced Friction Factors m,- = DkTF/6nveh+ of Small Ions x, = 0.1

Colloidal Spheres 1 2 m+= 0.184, m_ = 0.184

0.2

0.5

0.166 0.130 0.099 0.073 0.050 0.023

0.165 0,126 0.094 0.065 0.052 0.025

0.161 0.112 0.076 0.050 0.033 0.021

0.151 0.096 0.059 0.038 0.024 0.015

0.132 0.077 0.044 0.025 0.016

3 4 5

0.002 -0.045 - 0.080

- 0.026

- 0.008

- 0.001

-0.045

-0.016

- 0.004

0.002 0.000

3

0.073 0.050 0.024

yo

4 5

0.005

0.065 0.053 0.028

0.010 m+= 0.368, m., = 0.184 0.008 0.009 0.011

m+= 0.184, m- = 0.368

0.053 0.035 0.024

0.039 0.026 0.016

0.026 0.016 0.010

5

10

15

0.094 0.049 0.025 0.013 0.007 0.004

0.064 0.031 0.015 0.007 0.004 0.002

0.046 0.023 0.010 0.005 0.002 0.001

0.004 0.001 0.000

0.001 0.000 0.000

0.000 0.000 0.000

0.014 0.008 0.004

0.007 0.004 0.002

0.005 0.003 0.001

Long Colloidal Cylinders with Random Orientation yo

xo = 0.1

0.2

0.5

m,

= 0.184,

1

2

5

10

0.111 0.062 0.034 0.019 0.011 0.007

0.073 0.037 0.018 0.010 0.006 0.003

0.047 0.022 0.010 0.005 0.003 0.002

m- = 0.184

1 2 3 4 5

0.165 0.120 0.086 0.062 0.044 0.032

0.162 0.113 0.071 0.053 0.039 0.025

0.152 0.098 0.062 0.040 0.026 0.016

3 4 5

0.034 0.017 0.008

0.031 0.016 0.008

m, = 0.368, m- = 0.184 0.024 0.018 0.013 0.009 0.007 0.005

0.011 0.006 0.003

0.004 0.002 0.001

0.001 0.000 0.000

3 4 5

0.062 0.045 0.033

0.054 0.038 0.026

m, = 0.184, m- = 0.364 0.040 0.030 0.026 0.018 0.017 0.011

0.020 0.012 0.007

0.010 0.006 0.003

0.005 0.003 0.002

0

0.136 0.081 0.048 0.029 0.018 0.011



1669

, as Function of Reduced Radius x, = K a , Reduced TABLE IV: Relaxation Effect o n Conductance of Counterions, Surface Potential y o = e f / k T of Colloid, and Reduced Friction Factors m -, = DkTF/Gnveh+ - of Small Ions Colloidal Spheres

x, = 0.1

yo

0.2

0.5

1

2

5

10

15

m+= 0.184, m_ = -0.151 -0.226 -0.328 -0.458 -0.599 -0.730

0.184 -0.132 - 0.216 -0.335 - 0.481 - 0.632 -0.764

-0.094 -0.171 -0.287 - 0.438 -0.598 -0.737

-0.064 -0.123 -0.220 -0.357 -0.514 - 0.664

-0.048 -0.096 -0.177 - 0.299 - 0.447 -0.597

m+= 0.368, m_ = -0.457 -0.522 -0.596 - 0.644 - 0.722

0.184 - 0.479 - 0.629 - 0.760

-0.437 -0.596 -0.735

- 0.356

- 0.297

-0.513 - 0.661

- 0.445

- 0.506

- 0.416 - 0.587

-0.351 -0.517' -0.669

1 2 3 4 5

-0.166 -0.207 -0.250 -0.300 -0.352 -0.410

-0.165 -0.213 - 0.268 -0.335 - 0.414 -0.500

-0.161 -0.224 -0.306 -0.407 -0.525 - 0.648

3 4 5

-0.300 -0,350 -0.410

-0.334 -0.410 -0.500

- 0.407

3 4 5

-0.393 -0.475 -0.560

-0.430 - 0.535 -0.650

- 0.501

0

m,

0.184, m_ = 0.368 -0.545 -0.557 - 0.721 -0.701 - 0.846 -0.828

=

-0.640 -0.780

-0.676 - 0.802

-0.733

- 0.597

Long Colloidal Cylinders with Random Orientation yo

x, = 0.1

0.2

0 1 2 3 4 5

-0.165 -0.220 -0.285 -0.354 - 0.424 -0.491

-0.162 -0.224 -0.298 -0.378 - 0.457 -0.523

3 4 5

-0.352 -0.421 -0.485

-0.376 -0.452 -0.520

3 4 5

-0.418 -0.502 -0.570

- 0.438

-0.525 -0.593

0.5 m, = 0.184, -0.152 -0.225 -0.313 -0.405 - 0.489 -0.555

m+= 0.368, m_ = 0.184 -0.410 - 0.497 -0.486 - 0.563 -0.552

- 0.403

m+= 0.184, m_ = 0.368 -0,461 -0.459 - 0.548 -0.551 -0.610 -0.610

of the central colloid particle. Computations by Wiersema4 suggest that this approximation does not cause large errors in the colloidal size range. It was already pointed out in the Introduction that for small K , that is, a t low salt concentration, the interactions between the colloid particles are long range and cannot be eliminated or neglected. In summary, the present theory is expected to fail for small X O = Ka. The influence of uncharged colloid particles on the conductance of salt solutions also deserves some comment. According to the present theory the interactions per kinetic colloid charge, the terms Aeh and A*rel, remain finite for yo = 0. This means that in eq 29 for uncharged colloids, Z = 0, the colloid conductance nA2 must vanish. In other words, the presence of an uncharged, nonconducting colloid particle does not change the conductance of a salt solution except for a dilution effect. This conclusion differs from results in the literature. According to Rayleigh and others2t3J a volume fraction p of uncharged colloid particles reduces the solution conductance by a factor 1- p/2, apart from the trivial dilution factor 1 - p. For colloid particles with volume v, the factor 1 - p/2 would correspond to a nonvanishing contribution per particle equal to the conductive transport by half the colloid volume of solution: nA2 = -(A+

CN*"uc + A-)-- 1000 2

1 m- = 0.184 -0.136 -0.214 -0.310 -0.410 - 0.499 - 0.565

for Z = 0

(45)

where c is the molarity of the salt solution. In general the effect of the colloid charge on the conductance should be superimposed on that of the uncharged colloid (compare eq 18). So the discrepancy of eq 45 and

2

5

10

-0.111

- 0.073

- 0.188 - 0.288

-0.135 - 0.226 -0.335 -0.439 -0.524

-0.047 -0.092 -0.164 -0.262 -0.368 -0.466

-0.396 - 0.490 -0.558 -0.395 - 0.488 -0.558

- 0.334

-0.521

-0.261 -0.369 -0.463

- 0.443

-0.37,9 -0.487 -0.562

-0.303 -0.417 -0.508

- 0.438

-0.538 - 0.601

TABLE V: Values of R from Eq 46 for m , = m _ = 0.184

x, = N,

0.1

0.2

0.5

1

2

5

0 0 0 0 0 0 0 1 48.9 15.7 4.06 1.58 0.64 0.18 2 201 64.8 17.1 6.86 2.87 0.86 3 460 150 41.8 17.5 7.62 2.39 4 827 281 83 36.3 16.3 5.58 5 1331 478 150 67.6 31.4 11.3

10

0 0.063 0.31 0.95 2.31 4.94

the result nA2 = 0 from eq 29 for 2 = 0 is significant also for charged colloids. To give an impression of the practical importance of this discrepancy we take the ratio R of the relaxation effect for charged colloids, Z(A+rel+ per particle, and the right-hand side of eq 45. For colloid spheres in a salt solution with A+ = A- this ratio is

- R = - Z(A+rel + A-reJ -2000 A+ + AcIv,,v,

\

where 0 is the numerical correction factor in the charge-potential relation given as P = I/I(DH) in Table 21 of ref 13. Table V shows values of R obtained from the relaxation data of Tables I11 and IV for m+ = m- = 0.184. It appears that R >> 1except for high xo and low potentials. So under most conditions, where R >> 1, the exact treatment of the influence of uncharged particles on the conductance is only of academic interest. There are a t least two reasons to prefer the present theory leading to nA2 = 0 for Z = 0.

1670


First, eq 45 has been derived by extrapolating results for a collection of uncharged particles of volume fraction p and interacting through long-range electrostatic perturbations. It is not obvious that these theories2y3J1are valid in the limit p = 0, the case of a single colloid particle. Secondly, all theories2i3J1assume a uniform medium, that is, in eq 14 the last term is neglected. Detailed study shows that the functions D+ and D- in eq 17 incorporate a part of the perturbation potential g. This part of g does not vanish for 2 = 0 but is compensated exactly by an ionic diffusion effect deriving from 6, and 6- (compare eq 16). In fact, the perturbations of the electric field and of the ion concentrations are intimately connected in the relaxation effect. Separate treatment of the electric field perturbation might lead into wrong interpretations. The relaxation effect will be discussed further in the following paper.

Dirk Stigter

References and Notes (1) H. R. Kruyt, Ed., “Colloid Science”, Elsevier, New York, 1949. (2) S. S. Dukhin and B. V. Derjaguin, “Electrokinetic Phenomena”, Vol. 7 of “Surface and Colloid Science”, E. Matijevic, Ed., Wiley, New York, 1974. (3) R. A. Robinson aml R. H. Stokes, “Electrolyte Solutions”, Butterworths, London, 1959. (4) P. H. Wiersema, Thesis, Utrecht, 1964. (5) P. H. Wiersema, A. L. Loeb, and J. Th. G. Overbeek, J . Colloid Interface Sci., 22, 78 (1966). (6) D. Stigter, J. Phys. Chem., 82, 1424 (1978). (7) D. Stigter, J . Phys. Chem., 82, 1428 (1978). (8) H. Eisenberg, J . Polym. Sci., 30, 47 (1958). (9) D. Stigter, J . Phys. Chem., following paper in this issue. 10) D. Stigter, Biopolymers, 16, 1435 (1977). 34, 481 (1892). 11) J. W. Rayleigh, Phil. Mag. (9, 12) D. C. Henry, f r o c . R . Soc. London, Ser. A , 133, 106 (1931). 13) A. L. Loeb, J. Th. G. Overbeek, and P. H. Wiersema, “The Electrical Double Layer Around a Spherical Colloid Particle”, M.I.T. Press, Cambridge, 1961. (14) J. A. Schellman and D. Stigter, Biopolymers, 16, 1424 (1977).

Kinetic Charge of Colloidal Electrolytes from Conductance and Electrophoresis. Detergent Micelles, Poly(methacrylates), and DNA in Univalent Salt Solutions Dirk Stigter Western Regional Research Center, SEA, USDA, Albany, California 94710 (Received September 20, 1978; Revised Manuscript Received February 20, 1979) Publication costs assisted by the U.S. Department of Agriculture

The specific conductance of colloid-salt solutions is used to derive the limiting equivalent conductance A2 of the colloid component (infinite dilution of colloid). The method is employed for detergent micelles in NaCl solutions, for partially neutralized poly(methacry1ic acid) (PMA) in alkali bromide solutions, and for DNA in various chloride solutions. The derivation of A2 accounts for the expulsion of salt from the ionic atmosphere around the charged colloid. This salt correction depends on the specific conductance data available. It is different in the three systems under study, and may be as high as 30% of A*. All interactions are treated for the Gouy-Chapman double layer model. A2 is interpreted in terms of the kinetic charge of the colloid, as suggested in the preceding paper. The results are compared with the electrophoretic charge. Differences between the two sets of kinetic charges average 8.3% for the micelles and 9.3% for PMA. This is within the errors of the experiments and the theoretical models. In view of other transport theories it is noteworthy that large relaxation effects are found in all systems under study. Furthermore, the relaxation effect is much larger for the counterions than for the colloid, due to the diffusion in the polarized ionic atmosphere. This shows that the law action = reaction is insufficient to correlate the relaxation effects on the colloid and on its counter charge.

Introduction The main purpose of this paper is a comparison of two different calculations of the charge of colloid particles: (a) from the electrophoretic l potential and (b) from the conductance of the solution and the mobilities of the charged constituents. An outstanding reason is that the comparison of the two separate results for the kinetic charge of the colloid ion provides a valuable test of the theoretical model and the underlying assumptions. Furthermore, the kinetic charge is an important characteristic of the immediate ionic environment of biocolloids such as DNA which codetermines their physicochemical properties and biological performance. Theories of electrophoresis and of conductance are available for highly charged spheres and long cylinders with a Gouy-Chapman electrical double l a ~ e r . l - ~The theories are now applied to evaluate the kinetic charge of micelles of sodium dodecyl sulfate (NaDS), of alkali poly(methacry1ates) (PMA), and of double helical DNA. The diameter of NaDS micelles and of the DNA helix

make such particles very suitable for testing transport theories. The particles are large enough to be of colloidal size, that is, one applies the laws of continuum theory (Navier-Stokes, Poisson-Boltzmann) with some confidence, following arguments given in ref 20, p 125 and ref 22, p 1424. On the other hand, the particles are sufficiently small to make meaningful connections between structural parameters and transport measurements. For instance, a comprehensive analysis of experimental data of NaDS micelles has shown that the hydrodynamic shear surface of the micelles coincides within 1A with the surface enveloping the hydrated sulfate groups of the micellized detergent ions! Such information combined with accurate data for the kinetic charge is a valuable key to the ionic charge distribution near the colloid surface. It is unfortunate that for charged cylinders an accurate interpretation of viscosity and diffusion data is still impossible because there is no theory of the relaxation effect in these experiments. So for PMA and DNA the kinetic diameter is, within limits, an adjustable parameter. The

This article not subject to US. Copyright. Published 1979 by the American Chemical Society

Theory of conductance of colloidal electrolytes in univalent salt

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