The solution of the nonlinear Poisson—Boltzmann equation for thin, spherical double layers

The Solution of the Nonlinear Poisson-Boltzmann Equation for Thin, Spherical Double Layers RAMESH NATARAJAN AND R O B E R T S. S C H E C H T E R 1 Department of Chemical Engineering, The University of Texas, Austin, Texas 78712

Received June 6, 1983; acceptedOctober 11, 1983 A solution to the nonlinear Poisson-Boltzmannequation has been obtained for the region exterior to a spherical, colloidalparticle in a binary, symmetricelectrolytesolution. Using the Debye length as the expansion parameter, uniformly valid asymptoticexpansions for the potential and charge density distributions have been obtained. The results are valid for arbitrary values of the surfacepotential. INTRODUCTION

nomena that include streaming potentials, electrophoresis, electroviscous effects, etc. (6).

The existence o f an electrical, diffuse, double layer has a profound effect on a wide variety of equilibrium and nonequilibrium phenomena in colloid science. The use of the nonlinear Poisson-Boltzmann (PB) equation as a model for the double layer has been subjected to criticism on various grounds (1), but it is accepted that it furnishes a good model in the region exterior to the outer Helmholtz plane. Solutions of the nonlinear PB equations have thus found applicability in explaining the physics of a wide variety of phenomena, some of which are:

The spherical diffuse double layer, because of its importance in the study o f colloidal partides in electrolytes, has attracted considerable theoretical attention over the past 60 years, and the methods of solution of the nonlinear equations may be classified into four categories: (1) Series solutions in powers of a small surface charge or a small surface potential (7, 8). The radius of convergence of the resulting series is small restricting the utility of this solution approach. (2) Numerical methods (9-11) have been used by many investigators with the most complete investigation by Loeb et al. (I1), whose results are given in the form of tables and cover a wide range of values of the applicable parameters and include symmetric and asymmetric electrolytes. Interpolatory formulas (12) have been derived from these tables which facilitate their use. (3) Approximate solutions based on a variational approach (13, 14). (4) Asymptotic expansion methods with a small Debye length as the expansion parameter (15-19). Additional references to work in the Soviet Union using this approach can be obtained in the book by Dukhin and Derju-

(1) The DLVO theory (2, 3) for the empirically proposed Schulze-Hardy rule, which showed the dependence o f the coagulating properties of electrolytic solutions on the valence of their ionic components. (2) The estimation o f ionic radii in solutions (4). (3) The theory of the disjoining pressure due to the overlapping o f diffuse double layers and its application to instability phenomena in thin, liquid films (5). (4) A wide variety of electrokinetic phe1To whom all correspondenceshould be addressed. 50 0021-9797/84 $3,00 Copyright © 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

SPHERICAL DOUBLE LAYERS

guin (20). This promising approach to the solution of the full nonlinear equations has been used in this work to systematically calculate higher-order terms. The solutions obtained in this work are unrestricted with regard to the value of the surface potential but limited to the consideration of binary, symmetric electrolytes for analytic convenience. We have also obtained expressions for the effective surface potential and for the surface charge density in the form of power series expansions in the Debye length. The first few terms in this series have been calculated explicitly and the solution method used here may be used to systematically calculate higher-order terms. The calculation of these higher-order terms is of great interest since they may be used in the construction of Pad~ approximants to the power series, as has been done in this work. The resulting expressions are shown to give accurate estimates of the effective surface potential and the surface charge density (when compared to the numerically obtained results of Loeb et al. (11)) for fairly large values of the Debye length. EQUATIONS AND DISCUSSION

The physical variables are nondimensionalized by using the following reference scales: potential k T / e z , space charge density ezno~, and length a. The small parameter is the nondimensional Debye length given by the formula = (1/a)[ce~kT/2e2z2noo] 1/2. [1] The relevant equations are Poisson's equation relating the potential and the space charge and the diffusion equations for the ionic species in an electric force field. A detailed discussion of these equations has been reported (20). In the spherically symmetric case, the nondimensional equations are [02 + 2r -I O,]~b =

-l/2e-2(p+ -

Or[r2(Orp+ + P~ Orq~)] = 0.

p_),

[2] [31

Some economy of notation has been effected in Eq. [3] by writing the diffusion equa-

51

tions for the individual ionic species in one single equation. The individual equations may be read off by considering either the upper signs or the lower signs including subscripts. This notation is implicit hereafter. The boundary conditions (20) are that the potential on the particle surface is specified and the individual ionic currents to the particle surface vanish. Far from the particle the potential and ionic charge densities decay to the values in the electrically neutral, bulk solution. Hence, a t r = l, q~ = ~b*, [4] Orp +- "q- p +_ O r ~) = O,

[5]

and for r ---, o% ~ O,

[6]

p+--~ 1.

[7]

In this case, Eq. [3] may be integrated to give p_+ = exp[T-qS] [8] and the result substituted in Eq. [2] to yield the familiar form of the nonlinear PB equation [02 + 2r-' 0r]q~ = c-2 sinh ~b.

[9]

For e ~ 1, there is a thin boundary layer of thickness O(E) in which the potential and ionic charge density fields vary sharply. Outside this boundary layer in the outer region the potential is exponentially small and the Debye-Hiickel linearization is valid (18). Asymptotic expansions for the solution are obtained in both regions and the method of matched asymptotic expansions used to generate uniformly valid solutions (21). CONSTRUCTION OF THE SOLUTION

The solutions in the inner and outer regions are denoted by t~in, P-+in, and ~bout, a_+out, respectively. The outer solutions will satisfy the boundary conditions at infinity and the inner solutions the boundary conditions on the particle surface and the two solutions will be required to satisfy matching conditions in their domain of common validity. As noted in the previous section, the DebyeJournal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

52

NATARAJAN

AND SCHECHTER

Hiickel linearization is valid in the outer region so that Eqs. [21 and [3] may be linearized and solved to obtain the well-known solution (4) ~bo,t = A(¢)(e-r/'/r),

and 02 = [1 - 2¢a'1 + E2(3a'l2 - 2a~) - ¢3(4a'13 - 6a'1~2 + 2ag) + O(~4)]02

[ 10]

+ [-ca'; + ~2(3a'la'~ - a~) + d(-6a'12d; P+-out :

1 -T- bout,

[ll]

which satisfies the boundary condition at infinity. The integration constant A is a function of ¢ and must be determined by matching to the inner solution. In order to study the inner region, the boundary layer variable x = (r - 1)/~ ~ O(1) is defined in terms of which Eqs. [2] and [3] may be written as

02 + ~

~

= -v~(m~o - p-~.), [12]

x 4~i~ + (O~o+_i~)(Ox¢~n) = 0.

[1 31

The boundary conditions become, at x = 0, q~in = ~ * ,

[ 14]

OxP++i n "P P±in(Ox([)in) = O.

[15]

+ 3a'~a~ + 3a'la~ - a~) + O(¢4)]0,, [20] where the primes denoted differentiation with respect to s. After substituting Eqs. [ 16]-[20] in Eqs. [ 12] and [13], and O(1) inner equations are obtained as

ats=

02~b0 = --1/2(p+0 -- P--0),

[21]

O2~p+o+ Os[P±o(O/po)] = 0;

[22]

4~o = 4~*,

[231

Om+_o -!--p+O(OsOo) = 0.

[24]

0,

Equations [21 ]-[24] are, as expected, identical to those obtained for a plane diffuse double layer since in this approximation the particle curvature is neglected. The solution possessing the required exponentially decaying behavior for large s is given by 4~o = 4 tanh -1 (pe-~),

The inner expansions are taken in the form

P±O = exp(-T-q~o), (~in = 4 0 ~- E(~I -]- E2~b2

[16]

+emp±m+O(d~+~),

[17]

+

~ma,.(S)

+ O(d"+~).

[181

The straining functions ai(s) are chosen to vanish at s = 0, so that the particle surface is at zero in both the x and s variables. The transformations for the derivatives in Eqs. [12] and [ 13] are then given by 0x = t1 - ~'~ + d ( ~ ; ~ - ~ ) -- E3(atl3 --

[271

o~1 + [(-2a;)(O~o) + ( - a ; + 2)(0s~O)1 --~ --1/2(/9+1 -- P - l ) ,

X = S q- EO/I(S ) q- ¢20/2(S ) • • •

p = tanh (~b*/4).

Proceeding to the calculation of higher order terms, to O(e), the inner equations are

/)±in ~" P+-0 + ~P±l + ~2p+_2

+

[261

where

"~ " " " "~- E m ~ m -~ o ( E m + l ) ,

+ ...

[25]

2a'~a'z + a'3) + 0(~4)10s

Journal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

[19]

[28]

Os2O_+, + 0s[o±l(0s~o) + p±o(0,~l)]

+ [(-2a'0(a~o±o) + (-a'~ + 2)(Oso±o)l + p±o[(-2~'0(0~o) + (-~'i +2)(0,~o)] + [(-2a'0(0~p_+o)(0s¢O)] = 0.

[29]

The arbitrariness in the definition of al(s) may now be utilized in order to eliminate the nonhomogeneous terms in Eqs. [28] and [29]. Hence, we let al(s) be defined by the condition

SPHERICAL DOUBLE LAYERS (-2c¢~)(02~bo) + (-a'~ + 2)(Os~bo) = 0

[30]

53

0~ -- 2(} q-2e-2s\~)a~

upon which Eqs. [28] and [29] reduce to 02(])1 = --I/2(p+l -- fl 1),

Os[Cg~p+~ ++-p+_l(OsqSO) + p_+o(G~b0] = 0,

= 3/20~P1@~ -~ Oli -- 2s

[31]

[32]

and the boundary conditions are at s = 0, q~l = 0,

[33]

Osp+_l + p+_~(Os~O) ++-p+_o(O~4~) = 0.

[34]

[401

with [41 ]

a2(0) = 0.

This may be solved for in the same manner used to obtain a l ( s ) from Eqs. [36] and [37]. The result is

3,

ot2(S) = -

--8(U2-19 2)+'~(U --174)

The solutions of Eqs. [31] to [34] are given by ~bl = p+l = 0.

[35]

Thus, to this order and in fact all higher orders for a suitable choice of the straining function ae(s) which eliminates the nonhomogeneous terms, we will find that the corresponding corrections to the potential and charge density fields will uniformly vanish. Then, the higher-order correction terms are completely determined by the definition of the variable s in Eq. [18] and its use in Eqs. [25] and [26] to determine the O(1) potential and ionic charge density fields. Substituting known quantities into Eq. [30], the condition to be satisfied by al(S) is _2 e - 2S-~

a'l'- 2 [ i + ~ J a ' l

= 2

[36]

+ 2U In (~) + 4 In ( p ) - 2 In2 ( p ) ( 1 -- U4) - -

(1 -- p4) In (1 -

/22

u z) + - -

p2

£u: In (1 - Y) dy ] , [42] 2 y

Xln(1-p2)-2 where

[431

u = pe-L

Also ~i(O)=~

3+

(1-p~)2.

[441

In order to eliminate the inhomogeneous terms in the O(E 3) inner equations, -3(s) has to satisfy the conditions -}- - 2 e - 2 S \

with al(0) = 0.

[371

Since Eq. [36] is a second-order differential equation, the solution will contain two constants of integration, one of which is chosen to satisfy Eq. [37] and the other to eliminate terms that grow exponentially with s, rendering the inner solution unmatchable to the outer solution. Then a l ( s ) = - s - (p2/2)(e-2~ - 1).

[38]

a'ff0) = -(1 - / ~ ) .

[39]

Also The condition to be satisfied in order to eliminate nonhomogeneous terms in the O(E2) inner equations is

a~-2(11

Pp2e~as)OJ3 = ~)2(l/2ol; J - 1 ) - - 2a~ -

4 a ' ~ s + 2 s 2 + a~(3a'~ -

2)

[45]

with a3(0) = 0.

[461

The solution for a'3(s) requires some effort to complete. However, a3(0) may be obtained with less effort by integrating Eq. [451 once, identifying the terms that will lead to exponentially growing terms and picking the constant of integration to eliminate those terms. The result is E

3

Journalof Colloidand InterfaceScience,Vol. 99, No. 1, May 1984

54

NATARAJAN AND SCHECHTER -

3 2

-

(1

p2) In

-

(1

3 12"] (1 -p2)2 - -p2,

11+ ~

x2

p2)

-

J

fl2(x)---'--~+-~

where p2 In (1 - y)

1~ = •Io

dy

In y

X [1 -- EX -~ ff2X2 q- O(153)]. [55]

Similarly, evaluating the inner solution asymptotically for large x gives from Eqs. [25], [50], [53], and [54]

4p 1 + , 7 -

1+6

-2p 4

+(1--/)2P4_____~)l n ( 1 - - p 2 ) + 211J+ O(e 3)} X e-X[1 - ex + ff2x2 -~ O(E3)].

[561

Comparing Eqs. [55] and [56], d(e) is easily determined and after substituting in Eq. [10], we may write 4o.t = Y(e-(r-n/'/r),

s = x + ~&(x) + dfl2(x)

[57]

where

m

m(X) +

[5o1

Substituting Eqs. [38], [42], and [50] into Eq. [18] and expanding for small e, we get p2 ( e-2x _ 1),

fl2(X) = (1 -p2e-2X)fll(x ) - a2(x).

[511 [521

Equations [25], [26], and [50]-[52] then furnish the solution for the potential and charge density fields in the inner region correct to O(e3). The solution for the potential correct to O(e) has been obtained previously (15, 16, 18) while the O(e2) term is a new result. The values of ill(x) and flz(x) as x --* oo are required for matching purposes: ill(x) - 0 x - - -

2

Journal of Colloid and Interface Science,

[54]

4ou~ "~ A ( O e - l / ' e -x

The potential and charge density fields in the inner region are given wholly by Eqs. [25] and [26]. However, Eq. [25] is given in terms of the straining variable s, and in order to obtain the results in terms of the physical variable x, Eq. [18] must be inverted. Hence, let

l(x) = x +

]

In (1 - p2) + 211_.

6 gives

CALCULATION OF POTENTIAL AND CHARGE DENSITY FIELDS

• • • +

p2

[49]

The function a~(0) is a bounded function o f p 2 over the range of values o f p 2, i.e., for 0 p2 < 1. The integrals 11 and 12 may be numerically evaluated or obtained from a table of Debye functions (22) after making suitable transformations to carry 11 and 12 to the standard form.

+

(1 -- p4____~)

The outer solution may now be completed by evaluating the constant A(e) in Eq. [10] by matching to the inner solution. Writing Eq. [ 10] in inner variables and expanding for small

y

i -Tv dy.

3 4 1 +6p2-~p

[48]

and

12= ~ . 2

+

[47]

1r

VoL99, No. 1, May1984

[53]

Y = go(P) + egl(p) + eZg2(p) + O(e 3)

[58]

and where go(P) = 4p,

[591

gl(P) = 2/) 3,

[60]

gz(P) = - P - 6/) 3 + 2P 5

The result in Eqs. [57] and [58] to O(e) has been obtained earlier (18), whereas the O(e 2) term is a new result. It is seen that Eq. [57] differs from the Debye-Hfickel linearized solution only in the form of the coefficient Y. The significance of this coefficient was pointed out by Loeb et al. (11), who defined it as the "effective surface potential," i.e., that potential

SPHERICAL D O U B L E LAYERS

that would give the right potential distribution at large distances from the surface of the sphere if the Debye-Hfickel approximation were to apply everywhere. The usefulness of this concept in the calculation of the free energy of interaction between two charged colloidal particles has been considered by Bell et al. (23). C A L C U L A T I O N OF THE SURFACE C H A R G E

The nondimensionalized surface charge density is given by the equation O" = -- E2(Or~)lr= 1 = - - E(Ox~bo)lx=o

4ep

-- 1 - - p 2 [1 -- ,Oe'l q- ~2(0/t12 -- 0/~) --

~3(OLtl 3 - -

2a'1~ + oz~) + O(e4)]]s=0, [621

where a'l(O), a~(0), and a~(0) are defined in Eqs. [39], [44], and [47], respectively. After substituting into Eq. [62] one may write =

,f,(p)

+

~f~(p) + ~%(p) + ~4f4(p) + O(eS),

[63]

where

4p fi(P) - 1 - p2

[64]

f2(P) = 4p,

[651

55 TABLE I

The Functions gi(P) as Defined in Eqs. [59]-[61] ~b*

p

go

gl

g2

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 12.00 14.00 16.00

0.24492 0.46212 0.63515 0.76159 0.84828 0.90515 0.94138 0.96403 0.97803 0.98661 0.99505 0.99818 0.99933

0.979674 1.848468 2.540595 3.046376 3.393134 3.620593 3.765502 3.856110 3.912104 3.946457 3.980219 3.992711 3.997317

0.029382 0.197372 0.512456 0.883488 1.220824 1.483163 1.668471 1.791836 1.871032 1.920756 1.970475 1.989087 1.995978

-0.049806 -0.306898 -0.708620 -1.081443 -1.342699 -1.500814 -1.590711 -1.641139 -1.669710 -1.686174 -1.701528 -1.706995 -1.708982

results for the effective surface potential Yand the electrophoretic charge density a which have been defined in Eqs. [58] and [63], respectively. It can be seen that these quantities have been expressed in the form of power seres in e, whose first few coefficients are explicitly known as functions of p. These coefficients have been evaluated for a few sample values o f p and the results tabulated in Tables I and II. Evaluation of

Examination of the values of the coefficients f (p) throughf4(p) given in Table II shows that the expansion in Eq. [63] is numerically convergent 2 over a finite range of values of ~ > 0. f4(P) - -4(1 - p 2 ) [--'/2P2 + 1/2P4 This range is dependent on the values of p P and varies from e < 1 at p = 0.25 to e < 20 - 1/2(1 - pZ) In (1 - p2) + 3/4 In p2 at p = 0.98. In this region the series in Eq. × In (1 - p2) _ 3/411 + 3/4/2]. [67] [63] may be summed directly to obtain highly accurate estimates of a. The results in Eq. [63] up to the O(~2) terms It is well known (see Shanks (24) for several have been well known and widely used for examples) that a power series, such as Eq. thin spherical double layers. The O(e 3) and [63], may be summed to its correct value, i.e., O(E4) terms in Eq. [63] have not to our knowl- to the analytic continuation of the sum of the edge been previously derived. corresponding convergent series, even when the series is formally divergent. In order to do f3(p) = - 2 p ( 1 - p2)[1 + I n (1pT- p 2 ) 1 , [66]

N U M E R I C A L RESULTS

In this section, we shall indicate a method that may be used to obtain accurate numerical

2 In the formal computational sense that calculation of additional terms increases the accuracy of the sum. Journal of Colloid and Interface Science, Vol. 99, No. 1, May 1984

56

NATARAJAN AND SCHECHTER TABLE II T h e Functions f ( p ) as Defined in Eqs. [63]-[67] 4,*

p

f,(p)

A(p)

A(p)

f,(p)

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9,00 10.00 12.00

0.24492 0.46212 0.63515 0.76159 0.84828 0.90515 0.94138 0.96403 0.97803 0.98661 0.99505

1.042191 2.350402 4.258559 7.253721 12.100409 20.035750 33.085255 54.579834 90.006022 148.406421 403.426315

0.979675 1.848469 2.540596 3.046377 3.393135 3.620593 3.765502 3.856110 3.912104 3.946457 3.980219

0.014389 0.090799 0.212500 0.317118 0.364880 0.355998 0.311201 0.252204 0.193700 0,142052 0.071954

-0.013526 -0.071409 -0.126775 -0.133182 -0.102318 -0.064331 -0.035418 -0.017818 -0.008419 -0.003804 -0.000710

this various modified summation techniques are available and the choice of the right method may be established theoretically from a knowledge of the analytic behavior of the function being summed or in the absence of this knowledge, empirically, by studying the convergence of a sequence of partial sums. In our case, however, nothing is known about the analytic behavior of a in the complex plane (except on the positive real axis). This, taken along with the fact that only a finite number of terms in the series are known, precludes any rational choice of a method of summation. In this instance, therefore, we are content to sum (Eq. [63]) by calculating the highestorder Pad6 approximant possible from its known terms and comparing the results with the extensive numerical computations of Loeb et al. (11), wherever applicable. The [2/1 ] Pad6 approximant (in the notation of Baker (25)) to a/~ may be directly calculated to give3 _a = f l f 3 + (AJ~ - flJ~)e + (f~ - J~A)e 2

,

A -f4, [681

In Table III, columns 1 and 2, we have compared the prediction of Eq. [68] with those of Loeb et al. (11) (whose results are exact to 3 The surface charge density I as used in Ref. (11) is identical to our a # , the factor of e arising from the different choice of length scales. Journal of Colloid and Interface Science, Vot. 99, No. 1, May 1984

5 significant digits) for E = 2.0 and e = 10.0. For e ~ ~ , the exact value of a/e goes uniformly to the prediction of the Debye-Hfickel theory according to which a/e~b*e

as

e~oo.

[69]

In column 3 of Table III we have compared the predictions made by Eq. [68] for e ~ with that of Eq. [69]. The ability of lowerorder Pad~ approximants to perform an accurate long-range extrapolation for the values of a function from a knowledge of the first few terms of its power series is well known TABLE III Ratio of the Value of the Surface Charge Density Given by Eq [68] to the Exact Values 4a*

~ = 2.0

~ = 10.0

~~

1.00 2.00 3.00 4,00 5,00 6,00 7,00 8.00 9.00 10.00 12.00

0.999 0.998 0.997 0.997 0.997 0.998 1.004 0.999 1.000 1.000 1.000

0.997 0.990 0.982 0.976 0.973 0.973 0.974 0.978 0.984 0.991 0.998

0.995 0.982 0.966 0.950 0.939 0.932 0.929 0.928 0.930 0.933 0.939

N o t e . The exact values for ~ = 2.0 and E = 10.0 are taken from the tables published by Loeb et al. n and for --~ Go from the Debye-Hiickel approximation.

57

SPHERICAL D O U B L E LAYERS

(25) and the results in Table III must be taken as additional evidence of that fact. Considering the circumstances under which Eq. [63] was derived the agreement is remarkable, less than a m a x i m u m of 2.7% at E = 10.0 and less than a m a x i m u m of 7.2% at E ~ oo. The authors are unaware of any other analytical expression that gives an accurate estimate for tr as does Eq. [68] over all values of applicable p and c. The error in Eq. [68] is negligible for small to moderate values of and increases with increasing c to a m a x i m u m error of about 7% at intermediate potentials for e --~ ~ .

TABLE IV Ratio of the Value of the Effective Surface Potential Given by Eq. [70] to the Exact Values ~*

e = 0.5

~ = 1.0

~ = 2.0

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 12.00 14.00

1.000 0.999 0.997 0.995 0.993 0.992 0.990 0.989 0.989 0.988 0.988 0.988

0.999 0.996 0.992 0.986 0.980 0.974 0.969 0.966 0.964 0.962 0.960 0.960

0.998 0.993 0.985 0.972 0.957 0.942 0.929 0.919 0.912 0.907 0.902 0.900

Evaluation of Y

In this case only the first three terms of the expansion are known, so that we m a y form the [ 1/ I ] Pad6 approximant to Ywhich is given by y = gogl + (g2 _ gog2)c

[701

gl -- g2~ Again it is known from the Debye-Hfickel theory that Y--~ ~b*

as

E--~ ~ .

[71]

N o t e . The exact values are taken from the numerical results of Loeb et al. it. For values of ~ > 2.0, Eq. [70] is increasingly inaccurate, giving errors greater than 10%, especially for large values of qS*.

the solution in Ref. (7) and involving exceedingly lengthy calculations and the second a slightly less complicated method based on a variational principle. The results from both these methods have been tabulated in Table I of their paper for e = 0.1, 0.2, and 1.0 and for values of q~* ranging from l to 8. Direct comparison of the results with the predictions of Eq. [70] shows that the latter is to be preferred in this range, especially for large values of ~b* where the methods given in Ref. (23) break down completely. Summarizing, it is found that low-order Pad6 approximants to the series given in Eqs. [58] and [63] for Y and ~, respectively, are helpful in inducing numerical convergence over an extended range o f values ofc. It seems likely that higher-order approximants will give even more improved results; however, this will require the calculation of additional terms in Eqs. [58] and [63], which is a straightforward but algebraically tedious task.

The original series (Eq. [58]) for Yhas very poor convergence properties giving errors of about 4% when compared to the exact values given to 5 significant digits in Table 20 of Ref. (11). F r o m Table IV we can see that the expression given in Eq. [70] is somewhat more accurate giving a m a x i m u m error 1.2% at = 0.5, a m a x i m u m error of 4.0% at ~ = 1.0, and a m a x i m u m error o f 10% at e = 2.0. For values of c > 2.0, the expression (Eq. [70]) is extremely inaccurate for large values of ok* even though it m a y be used circumspectly for low values of ~*. Again, considering the low order of the approximant and limited range of ~ over which the series expansion (Eq. [58]) is useful, the ability of Eq. [70] in inducing convergence over a wide range of c is reACKNOWLEDGMENTS markable. The research was supported by a grant from the National Few analytical approximations to Y are Science Foundation, Grant CPE-8208952. available. However, Bell et aL (23) give two Dr. Schechter holds the Dula and Ernest Cockrell, Sr. methods for estimating IT, the first based on Chair in Engineering.

Journal of Colloid and Interface Science, Vol. 99, No, I, May 1984

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NATARAJAN AND SCHECHTER REFERENCES

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