Nov., 1959
POTENTIAL AROUNDA CHARGED COLLOIDAL SPHERE
1869
POTENTIAL AROUND A CHARGED COLLOIDAL SPHERE BY THOMAS J. DOUGHERTY AND IRVIN M. KRIEGER Department of Chemistrv and. Chemical Engineering, Case Institute of Technology, Cleveland, Ohio Received April 17, 1969
An approximation to the solution of the Poisson-Boltzmann equation in spherical symmetry was obtained by integrating a closely-related differential equation. The solution obtained reduces under proper limiting conditions to the familiar Dehye-Hiickel and flat plate approximations. The calculated otentials and surface charge densities compare favorably with those obtained by Hoskin by numerical integration of the soisson-Boltzmann equation.
Introduction Most lyophobic colloids are stabilized by the electrical repulsion between particles resulting from the surface charge on the particles. Since electrical forces determine the stability of the colloidal systems, it follows, afortiori, that they play a dominant role in determining the thermodynamic and kinetic properties of the system. The exact calculation of these potentials presents a very difficult problem in statistical mechanics, but frequently the potentials can be calculated, to a satisfactory approximation, from the Poisson-Boltzmann equation. The derivation of this equation and discussions of the approximations involved in the derivation are available in several places’J so these subjects will not be considered here. Even with this approximation the problem remains formidable, since the Poisson-Boltzmann equation is a non-linear secondorder partial differential equation. Our treatment will be restricted to the case of spherical symmetry, in which case the problem is one-dimensional and we need deal only with an ordinary differential equation. Two well-known approximate solutions of this problem are the Debye-Huckel solution3 for “small potentials” and the flat plate approximation for very large sphere^.^ I n this paper we will develop a method for obtaining an approximate solution t o the Poisson-Boltzmann equation which contains both the DebyeHuckel and the flat plate solutions as special cases. Approximate Solution of the Poisson-Boltzmann Equation The Poisson-Boltzmann equation can be written in the form 4n vZ1c.(i.)= - D Nizie exp (-ziqb/kT) (1) V
we can write equation 2 in terms of dimensionless variables. (4)
Letting N =
Ni, xi =
Ni/N, and
=
i xiXi2,
equation 4 becomes
i
If a is the radius of the colloid particle and we let 7 = KU be the value of p a t the surface of the particle, then the condition for the validity of the flat plate approximation is a, >> 1 / ~or 7 >> 1, since 1 / is ~ essentially the distance from the particle surface beyond which the potential becomes small. We make now the change of variable Y = PA7 4 (6) where a! is a constant to be specified later. With this transformation, equation 5 becomes
+
We now observe that if the variable y multiplying the exponential term in equation 7 could be shifted into the exponent to multiply d, then we would obtain a new equation which would have the same form as the Poisson-Boltzmann equation for the flat plate problem, and hence solutions could be obtained by elementary means. Therefore we investigate next the possibility of commuting the two operations on 4 in the product y exp( - x i 4) which appears in equation 7 to obtain the new equation
where # is the potential, D the dielectric constant of the solvent, V the volume of solvent, E the absolute magnitude of the electronic charge, Ni the number of ions of valence x i , and other symbols Because of the boundary conditions on the potenhave their usual meaning. In the case of spherical tial, 4, y4 and d(yd)/dy must all approach zero as y symmetry, equation 1 becomes approaches infinity. Thus the boundary conditions force the solutions of equations 7 and 8 to agree a t large y. I n addition, they must agree for small 6, as can be shown by expanding the exponentials in Introducing the quantities both equations and retaining only the first power of (1) R. H. Fowler and E. A . Guggenheim, “Statistical Thermob. (We recall that electroneutrality requires that dynamics,” The Macmillan Co., New York, N. Y.,1939, pp. 385xixi = 0). Finally, the commutation is valid a t 391. (2) A. Munster, “Statistische Thermodynamik,” Springer Verlag, Berlin, 1956,pp. 764-790. (3) P. Debye and E. Hiickel, Physik. Z . , 24, 185 (1923); 26, 97 (1924). (4) E. J. W. Verwey and J. Th. G. Overbeek, “Theory of the Stability of Lyophobic Colloids,” Elsevier Publishing Co., Inc., New York, N. Y.,1948,p . 25.
i
the point y = 1; by suitable choice of a we may make this occur a t any desired distance from the particle surface. An appropriate choice is a = l/2, which is in the region where the potential gradient is large. A first integral of equation 8 is
THOMAS J. DOUQHERTY
1870
readily obtained. Multiplying both sides of equat.ion 8 by 2 d(yd)/dy we obtain
AND
IRVIN M. KRTEGER
I(X) =
43
1 (1 ;ln (1
Vol. 63
j: (e2zA + 28-zA - 3)-'/2 dx
-
-
=
+ 2) 1 + [3ezb/(ezb + 2) I*/*) + 2)]'/~)(1+ 3ezA/(eA + 2)]'/2)
[3ezx/(ezA [3erb/(ezb
]'/Q)(
(18)
which integrates to
Returning to our original dimensionless variables and rearranging, we obtain for the final result
In the limit y 4 w , d approaches zero exponentially, so that 4, y+ and dy+/dy all approach zero. These conditions determine C.
where
c
=
-
zi
=
-1
(11)
i
With this value for C, equation 10 becomes
On taking the square root of both sides of equation 12, we are faced with an ambiguity in sign. The sign is chosen which leads to a finite (actually zero) potential a t infinity. Taking the square root and integrating from the surface out to an arbitrary point, we obtain dv4
=
'>"'
-
(13)
Thus the problem has been reduced to quadratures; that is, to evaluating integrals of the form
Case 3.-A third case of considerable interest is the n to one electrolyte where n >> 1. Physically this corresponds to the situation where the ions of valence z are the counterions and the ions of valence -nz are the colloid particles themselves. Here 21 = --nxz, nz1 = z2 = n / ( n 1) and = nz2,giving
+
(20)
This integral cannot be evaluated exactly in closed form, but as long as ZX >> l/n we can disregard the term e-*d in the integrand. With this approximation and neglecting one in comparison to n equation 20 becomes I(X) e
1:
2 [tan-'
- I)-lh dX = (e-A - l ) + I / s - tan-'
(ezh
(e-&-
l)I/z] (21)
This result can also be expressed in the form where X = yd, XO = y060. These integrals can be evaluated either graphically or numerically by standard methods. We will now examine some special cases for which solutions can be obtained in closed form. Case 1 : Symmetrical electrolyte (valencies of all ions having same absolute magnitude) .-For and = this case, x1 = -x2 = z, x1 = x2 = 9,so that I(X) =
4 2 l'(ezh
+ e-zA - 2)-% dX =
In terms of our original dimensionless variables the result is
A form more convenient for computational purposes is
+
(22)
where A , = [exp
(-)r + a
- '"]l
This approximation fails at large distances where the potential becomes very small. If this restriction were ignored then equation 22 would lead us to believe that the potential changes sign periodically as p increases, giving a rather peculiar damped oscillatory behavior. It is interesting to prove that the exact solutions of our approximate form of the Poisson-Boltzmann equation can never oscillate. To prove this we first note that from equation 12 it can be shown that the slope of the potential versus p curve can be zero only when the potential is zero. Now if we can show that the potential is zero only a t p = w the proof will be complete. To prove this we need only prove that
where A I = tanh [ T Z ~ O / ~ ( T a)]. We divide the integral into two parts Case 2 : Two to one electrolyte (valencies of ions of greater charge is numerically twice the valence of ions of opposite charge) .-Here ZI = -2z2 = -22, 2x1 = x2 = 2/3, and = 2 9 , and therewhere X* is sufficiently small to allow us to neglect fore equation 14 takes the form all powers of X higher than the second in the expan-
POTENTIAL AROUNDA CHARGED COLLOIDAL SPHERE
Nov., 1959
1871
sion of the exponentials in the second part of the X i = 1 and zizi=O integral. Recalling that
c i
i
(electroneutrality) and making the suggested expansion we obtain
0.6 --
and thus
t
which is equivhlent to equation 23; this completes the proof. Properties of the Solution We will now study some of the properties of our approximate solutions. We consider in particular the result for a symmetrical electrolyte which can be compared to the exact calculations of Hoskinb who integrated the Poisson-Boltzmann equation numerically for this case. First we note that in the limits do+O and w - 0 3 we obtain the DebyeHuckel and the flat plate solutions, respectively, as we woiild expect. There is, however, an important difference between our solution for very large T ar d the flat plate approximation. The Bat plate solution predicts that the potential approaches zero as e - p whereas our solution predicts a Debye-Huckel form for the potential for large p. An interesting and somewhat unexpected result is that our solution predicts that the DebyeHuckel solution is correct in the limit -0 even if 4o is large. (In fact, OUI result reduces to the Debye-Huckel result when 740 is small.) This result has been observed by Mullere on the basis of more exact calculations than those presented here. It should be noted that it is the presence of the quantity a which enables us to obtain this result. This shows that the inclusion of this quantity improves the approximation a t small values of T . To facilitate comparison with Hoskin's results, we define, following Hoskin, the function C* ( p ; ~ , q h ) = $/+DH where $DH is the value of the potential calculated from the Debye-Huckel formula and q5 is the value of the potential calculated from our solution. The function C ( p ; ~ , 4 0 ) = $IH/+DH is defined similarly where 4~ is the value of the potential calculated by Hoskin. Figures 1 and 2 show comparisons of these two functions for the case of a symmetrical electrolyte (the only case treated by Hoskin) where we have taken a = l/z in equation 17. Table I shows a comparison of the limiting values of these functions as p-t 0. Values of C*( 00 ; T , + ~ ) for a = 0, and 1are presented to show the effect of the parameter a. The choice a = '/z gives generally satisfactory agree) the two to one ment. Values of C * ( ~ ; T , + Ofor are also included in Table electrolyte with a = I to show the effect of valence type. These limiting values for C* have been calculated from the relatively simple formulas C*( 03 ;T,+O) = [4(7+a)/(Tz+o) ltanh [ T X + ~ / ~ ( T a) + ] for the symmetrical (5) N. E. Hoskin, Trans. FaradaU Soc., 49, 1471 (1953). (6) H.Muller, Kolloidcham. Beihefte, 26, 257 (1928).
0.41
~
0.0
~
I
1.0
I
I
2.0
3.0
~
4.0
p-7.
Fig. 1.-Comparison of the potentials calculated from equation 17 (solid lines) with those calculated by Hoskin (open circles). Continued in Fig. 2.
. 0.8
h
t h... 4
G
0.6
0.4
1.o
0.0
2.0
3.0
4.0
p-7.
Fig. 2.-Continuation of Fig. 1. The data were plotted on two graphs to avoid confusion.
electrolyte and C*( 00 ;r,do) = 6 ( ~ + a ) A z / ~ z for & the two to one electrolyte where A2 is defined as in equation 19. The approximation we have made introduces an error i n - h e curvature of the potential versus p curve. The choice a = l/z is made so that this error will be approximately as often negative as positive and, therefore, a t large distances the error will be averaged out to some extent. However, if we were interested only in the region immediately adjacent to the surface of the particle, then we would want the error in the curvature to be small in this region. I n this case, we would choose a! = 0. I n the next section we will calculate the surface potential from the surface charge density. This calculation requires accurate values for the potential gradient a t the particle surface, and so for this calculation we will choose a = 0. Evaluation of the Surface Potential From equation 12 we obtain
* dy
=
- (--?.-)'I2 (.
+
(
xiei
which can be written in the form
'
='y9
-
Y*
(27)
THOMAS J. DOUGHERTY AND IRVINM. KRIEGER
1872
"
TABLE I
Symmetrical electrolyte T. do) C(m; aL1/z a - I
C*(m.
a - o
1 3 5 15
0.9797 ,9797 .9797 .9797 .9797
0.9908 .9850 , 9831 .9809 .9757
0.9242 .9242 .9242 ,9242 .9242
0.9645 .9430 .9364 ,9286 .9242
(T
(-)"2
e*,'"=; 7,:~) .-
z+o = 1
W
0.9962 ,9884 .9858 .9821 ,9757
0.9915 ,9861 .9841 ,9814
0.8795 ,8431 .8331 ,8219 8157
0.9664 ,9466 .9395 .9299
0.7524 .6833 .6652 .6453 ,6346 0.5343 ,4433 .4224 .4006 .3894
xie-zi+a
-
1)'"
(29)
one trolyte elec-
7,$0)
- +o/7 -
The requirement that the normal component of the electric induction be continuous a t the surface yields
T W O ~ O
T
=
Gip= r
COMPARISON OF VALUESOF C*( ; r, 40) A N D C( m ; 7 , +o) FOR VARIOUS VALUESOF THE PARAMETERS 7 AND 260
Vol. 63
(30)
DK~T
where B is the surface charge density. The corresponding expressions for the Debye-Huckel approximation are
z+o = 2
1 3 5 15 m
0.9797 ,9555 .9458 ,9327 .9242
Equations 29 and 30 can be used to calculate the surface potential if the surface charge density is known. This calculation can be performed either graphically or by some appropriate numerical technique. From equations 29, 30, 31 and 32 we can calculate the ratio of the surface charge density calculated from our approximation to that calculated from the Debye-Huckel approximation for the same surface potential.
z+o = 4
0,7616 ,7616 .7616 .7616 .7616
1 3 5 15 m
,6034 .6034 .6034 ,6034
3 5 15 W
0.8742 .8106 .7928 ,7726 .7616
0.9242 .8469 .8187 .7830 .7616
0.8734 .8160 ,7980 .7752
.6673 .6433 .6173 ,6034
.7477 .6786 .6305 ,6034
.6695 ,6470 .6202
2'0
=8 0.7616 .6034 .5586 .5088 .4820
...
_ -_ - -
.3095 ,2926 ,2754 .2668
...
d'
U
CDH
=
( 1 ;rhbo
For the case of a symmetrical electrolyte, equation 33 becomes
Table I1 shows a comparison of the values of this ratio computed from equation 34 to those obtained by Hoskin. The maximum error, which occurs when x + ~ = 6 and T = 1, is less than 8%. Hoskin's value for z + ~ = 8 and r = 15 (marked with m , . a question mark) we believe to be too large since our solution should become exact when r becomes very large and Hoskin's value does correspond to our result for = m . With this point excluded, (28) we see that our approximation always gives higher where we have set a = 0 for the reasons mentioned values for the surface charge density, (the difa t the end of the previous section. Setting p = r ference in the two values for e+0 = 6 and r = 15 in equation 28 we obtain for the potential gradient is probably meaningless) and thus we have an a t the surface upper limit for this quantity. TABLEI1 1 3 5 15
0.4820 .4820 ,4820 .4820 .4820
0.6526 .5467 ,5218 .4956 ,4820
0.6220 ,5435 ,5218 ,5001
COMPARISON OF THE VALUES OF
,
U/UDH 290
r = l
r = 3 7 = 5 7
=
15
r = m
Hoskin Eq. 33 Hoskin Eq. 33 Hoskin Eq. 33 Hoskin Eq. 33 Hoskin Eq. 33
0,2989 .2340 .2208 .2075 .2009
CALCULATED FROM EQUATION 34 TO THOSECALCULATED B Y HOSKIN
= 1
1.015 1.021 1.027 1.032 1.032 1.035 1.039 1.039
ZdO
= 2
1.063 1.088 1.116 1.131 1.135 1.146 1.160 1.164
Z@O
=
4
1.317 1.407 1.557 1.610 1.641 1.678 1.751 1.763
ZQD 5
6
2.017 2.170 2.686 2.754 2.914 2.949 3.202 3.193
sdo = 8
3.741 3.911 5.311 5.367 5,843 5,852 6.823(?) 6.459
...
...
...
...
...
1.042
1.175
1.813
3.339
6.822
