JOURNAL OF COLLOID SCIENCE 11,617-622 (1956)

CALCULATION OF M E M B R A N E PRESSURE FOR A SOL OF SPHERICAL PARTICLES

R. K Schofield 1

With mathematical appendix by

D. H. Rees 1

Received April 5, 1956

ABSTRACT

Electronic computation has been used to obtain the Poisson-Boltzmann distribu- tion of counter ions between an outer spherical surface, radius R0, where the poten- tial gradient is zero, and an inner concentric surface, radius R, representing a uni- formly charged spherical particle. Here co is the concentration of the counter ions at the outer surface, and cm is their mean concentration in the volume between the two surfaces. The results of the computations are given in a plot of Co/C,, against

R In R0 " For given values of R and R0, co at first increases linearly with the number of

counter ions per particle, but ult imately approaches a limit. Values of the membrane potential, p0, are calculated from the equation Po = NkT(co ~ C)/1000 for 1% and 0.1% sols of particles 10 -6 cm. in radius.

INTRODUCTION

Using D o n n a n ' s t h e o r y to ca lcula te the equ i l ib r ium pressure , p, across a m e m b r a n e s epa ra t i ng pure w a t e r f rom a d i lu te aqueous sol of cha rged spher ica l pa r t i c les one ob ta in s as a first a p p r o x i m a t i o n

N k T (c,, -[- C). [1] pm--- 1 ~

I n th is express ion N is A v o g a d r o ' s number , k B o l t z m a n n ' s cons tan t , T the abso lu te t e m p e r a t u r e , c m the m e a n m o l a r concen t ra t ion of the coun te r ions, and C the mo la r concen t ra t ion of the par t ic les . E q u a t i o n [1] gives an ove re s t ima t e of the m e m b r a n e pressure if t he concen t ra t ion , c, of the coun te r ions a t the charged surfaces of the par t i c les g r ea t l y exceeds the i r concen t ra t ion , Co, a t po in t s fu r thes t a w a y f rom the par t ic les . I t seems to the a u t h o r t h a t the equa t ion

N k T (Co + C) [2] p0=

should p rov ide a b e t t e r a p p r o x i m a t i o n to the m e m b r a n e pressure t h a n Eq. [1].

1 Rothamsted Experimental Station, Department of Statistics, Itarpenden, Hefts., England.

617

618 n.K. SCIIOFIELD

~V[ETttOD

With this thought in mind the author enlisted the help of Dr. P. M. Grundy, who explored means for solving the Poisson-Boltzmann equation applied to the space between an outer spherical surface of radius R0, at which the potential gradient is zero and the ionic concentration is co, and an inner concentric surface of radius R representing a uniformly charged spherical particle at which the ionic concentration is c. Taking the case where all the counter ions have the same valency v, Dr. Grundy introduced a parameter t defined by the equation

4 t~R~c = to 2 R o 2 Co - [3]

in which fl stands for 1000 D ! ~ T ' e being the electron charge and D the

R dielectric constant. For any chosen value of to, t can be related to In R0 by

numerical integration (see Appendix). Integrations were carried out on the E.D.S.A.C. at Cambridge through

the kind cooperation of Dr. J. C. P. Miller. These computations provided part of the data used in preparing Fig. 1. More recently Mr. D. H. Rees

R 0,05 0,1 0.2 0~3 0.4 0.5 0.6 0.8 1.0 Wo I I I I I

=Into 3.0 2.5 2.0 1.5 1.0 0.5

_in ~ I i , t I i r I I I P I I I I ~ I L $ I I L I Ro 3.0 2.5 2.0 1.5 1.0 0.5 _R ~ I I I Ra 0.05 0.1 0.2 0.3 0.4 0,5 0.6 0,8

FIG. 1. Co/C~ p l o t t e d a g a i n s t In R/Ro f o r c o n s t a n t v a l u e s of to a n d ¢ .

1.0

0.9

0.8

i 0.7

i° '6 co

,0.5

0,4

0.3

0.2

0.1

1.0

CALCULATION OF MEMBRANE PRESSURE

T A B L E I

619

For R = 10-6 cm.

R Co Cm R--o to c,n 4, co (~i l l imoles po 0

per liter) (era. of water) counter ions per particle

1 % S o l

0.217 0.69 ~ ~ 0.374 9.4 ~¢ " 0.7 0.100 9.07 0.364 9.2 875 " 0.8 0. 534 4.48 0.278 7.1 127 " 0.9 0.696 3.18 0.220 5.6 77 " 1.0 0.777 2.43 0.178 4.5 56 " 1.1 0.828 1.94 0.147 3.8 43 " 1 . 2 0.862 1.59 0.124 3.2 35 " 1.3 0.887 1.34 0.105 2.7 29 " 1.4 0.905 1.14 0.091 2.4 24 " 1.5 0.919 0.98 0.079 2.1 21

0 . 1 % S o l

0.101 0.95 ~ ~ 0.0428 1.1 " 1.0 0.387 8.18 0.0386 1.0 242 " 1.1 0.656 5.78 0.0319 0.8 115 " 1 . 2 O. 779 4.57 0.0268 0.7 84 " 1.3 0.832 3.77 0. G229 O. 6 67

" 1.4 0.867 3.17 0.0198 0.5 55 " 1.5 0.890 2 . 7 2 0.0172 0.4 47

has used the computer at Ro thamsted to trace the values of t to the limit t - 0 for to = 0.4, 0.5 • • • 1.2. At the same t ime values were obtained for the ratio co/c m, where c m is the mean molar concentration of counter ions in the volume between the concentric spheres of radii R0 and R, respectively.

~:~ESULTS

R Figure 1 is a plot of co/c,~ against In R0" To illustrate its use let us con-

sider its application to a sol in which uniform spherical particles are dis- persed in 97 times their volume of water. Taking (Ro/R) ~ = 98 gives R/Ro = 0.217, so we read off the values of co/cm at which the curves for successive values of to intersect the vertical for R/Ro = 0.217. These values are set out in the third column of the upper par t of Table I. The correspond-

ing values of 4 = In c are given in the fourth column. These can also be co

read roughly from Fig. 1. They are not used in the following calculations, but are included for general interest. A unit value of ¢ is approximately equal to 25 my. difference in electrical potential between the outer sphere and the particle surface.

620 R.K. SCttOFIELD

If we take R = 10 -6 cm., R0 is 4.6 × 10 -6 cm.; from Eq. [3]

4 Co - - to2Ro2v~ fl .

Taking v = 1 and ~ = 1.06 X 10 is cm./millimoles, we obtain

0.178 Co - millimoles/liter.

to ~

The values of Co obtained in this way are entered in the fifth column of Table I, and the corresponding vMues of p0 (Eq. [2]) in centimeters of water are given in the sixth column. The concentration of the particles themselves to the membrane pressure is only 1 ram., since in a solution containing 1 millimole/liter each molecule occupies 1.661 >( 10 -is cm2 and so

C -- 3 X 1.661 4~r(4.6) 3 = 0.00407 millimoles/liter.

The last column shows the number of univalent counter ions balancing the charge on each particle which is given by c , , / C .

A similar calculation was made for a more dilute sol of volume ration 970, i.e., R / R o = 0.101. The results are given in the lower par t of Table I. The value of R was again taken to be 10 -G cm. so R0 = 9.9 X 10 -6 cm. Here C is just ten times smaller than before, but for the same charge per particle Co is not reduced to quite one-tenth. This reflects the fact that , for the same particle charge, c6 /c ,~ is greater for the more dilute sol.

DISCUSSION

It will be seen from both parts of Table I that as the charge on the particles increases the membrane pressure p0, approaches a maximum value. Most of the surface charge densities implied in Table I are low; even 875 unit electron charges on the surface of a particle 10 -6 cm. in radius corre- Sponds to an area of 144 A. 2 per unit charge.

The entries in the last three columns of Table I depend on the arbitrary choice of 10 -6 cm. for R. To construct a table for R = 2 X 10 -6 cm., the values of Co and po in Table I must be divided by 4, while the values of c,~/C must be doubled. The surface densities of charge will be halved.

If R were increased to 10 -5 cm., the maximum membrane pressure for a 1% sol would be only 1 Into. of water. If R were reduced to 10 -7 cm., membrane pressures approaching I00 cm. of water would be given by a 0.i % sol, but the particles would carry only a few unit charges so that co would not greatly exceed C. Thus the treatment developed in this paper is best suited to aqueous sols of spherical particles neither much larger nor much smaller than 10 -8 cm. in radius. For larger particles the pressures

CALCULATION OF MEMBRANE PRESSURE 621

could be increased by using more concentrated sols, but caution is needed in applying the treatment when R/Ro exceeds 0.4.

In conclusion, brief consideration will be given to the effect of a small external electrolyte concentration on the equilibrium membrane pressure. We take the case represented by the fifth row of Table I, but suppose that externally there is 2 X 10 -5 M concentration of uni-univalent electrolyte. The ion product, being 4 X 10 -l° outside, must have the same value inside, the expected values being 0.179 and 0.002 millimole/liter. As before C is 0.004, so the total inside concentration is 0.185. Since the total outside con- centration is 0.040, the concentration difference effective in producing the membrane pressure is 0.145, giving a pressure of 3.6 cm. of water instead of 4.5 cm.

A P P E N D I X

With only one kind of ion present the Poisson-Boltzmann equation takes the form

V2 b _ -41rveNco - ve¢/ [4] 1000D e kT '

where ~ is the electric potential, v the valency of the counter ions, e the electronic charge, N Avogadro's number, D the dielectric constant, k Boltzmann's constant, and T the absolute temperature. The molar concen- tration of the counter ions at the surface of an inner sphere of radius R whose potential is -~b is given by c. In particular, c = Co on the surface of the sphere of radius R - R0 with zero potential.

By making the following substitutions and taking into account the spherical symmetry of the system, Eq. [4] can be reduced to

r 21 d ( d r r 2-~d~b)=2e¢-+°, [5]

where

- - v e ~ _ i n c ¢ - kT co

87r~2N 4 B - 1000DkT - t2R2v~c (by virtue of Eq. [3])

R ?, - -

R0

to 2 = e ¢0

The problem has been to find the first value of r (< i) for which ¢ becomes infinite given the initial values

¢---¢0

022 R.K. SCttOFIELD

de _ 0 dr

a t r -- 1.

The solutions calculated on the E.D.S.A.C. were derived from the numerical integration of an equation of the same form as Eq. [5]. Because of the rapid increase in the values of ~ these results were useful only over a limited range of values of r. In particular, the limiting value of r as ~b --~ could not be determined. In order to determine the whole range of values of r the substitution u --- In r was made. This reduced Eq. [5] to the pair of first-order differential equations

di" i" - i ' t - t 2 -- 1

du

dt

du

with the initial conditions

t

dt t = fo, du to a t u = O.

[6]

The integration of these equations was carried out on the Elliott Bros. "401" digital computer, now located a t Rothamsted Experimental Sta- tion, with the use of a modified form of the Runge K u t t a process (1). An integration interval of 2 -9 was used and a t every 17th step values of u, t,

dt Co and e - i t were printed. When t first became < 2 -6 the program was d u ' c,,~ '

reset automatical ly to print out every step of the integration. The graphs in Fig. 1 for to = 0.4, 0.5 . . . 1.2 were drawn directly from

the printed results. The graphs for to = 1.3, 1.4, 1.5 were drawn from calcu- lations based on E.C.S.A.C. results as far as they were valid. The curves for constant ¢ were determined by s tandard interpolation:

REFERENCE

1. WILKES, M. V.; WHEELER, D. S., A.ND GILL, S., "Programs for an Electronic Digi- tal Computer." Addison Wesley Press, Cambridge, Massachusetts, 1951.