Equivalence of the two forms of the Gibbs adsorption equation

633

Appendices

APPENDIX 1A: EQUIVALENCE OF THE TWO FORMS OF THE GIBBS ADSORPTION EQUATION Following Ref. [1] here we derive Eq. (1.70) from Eq. (1.68). We consider a solution of various species (i = 1,2,...,N), both amphiphilic and non-amphiphilic. As before we will use index “1” to denote the surfactant ions, whose adsorption determines the sign of the surface electric charge and potential. A substitution of ai¥ from Eq. (1.71) into Eq. (1.68) yields

-

N æ N ~ö ds ~ = å Gi d ln ais + çç å zi Gi ÷÷dF s kT i =1 ø è i =1

(1A.1)

Since the solution as a whole is electroneutral, one can write [2-4] N

~

å zi Gi = 0

(1A.2)

i =1

From Eqs. (1.69) and (1A.2) one obtains the following expression for the surface electric charge density rs : N N r r~s º s = å zi Gi = -å zi Li Z1e i =1 i =1

(1A.3)

Further, in view of Eqs. (1.69), (1.71) and (1A.2) one can transform Eq. (1A.1) to read -

N N ö æ N ds = å Gi d ln ais + å L i d ln ai¥ - çç å zi L i ÷÷dF s kT i =1 i =1 ø è i =1

(1A.4)

With the help of Eqs. (1.49), (1.69) and (1A.3) one can bring Eq. (1A.4) in the form N N ds ~ = å Gi d ln a is + å L i da i¥ + r~ s dF s kT i =1 i =1

(1A.5)

where ¥

L ~ L i º i = ò [exp(- z i F) - 1]dz ai¥ 0

(1A.6)

On the other hand, integrating Eq. (1.57), along with Eq. (1A.6), one can deduce Fº

¥

2

N 1 æ dF ö ~ dz = a i¥ L i ÷ å 2 òç k c 0 è dx ø i =1

Differentiating Eq. (1A.7) one obtains

(1A.7)

634

Appendices

N N ~ ~ dF º å L i dai¥ + å ai¥dL i i =1

(1A.8)

i =1

where “d” denotes a variation of the respective thermodynamic parameter corresponding to a small variation in the composition of the solution. Further, with the help of Eqs. (1A.6) and (1.55) one obtains ¥

N

N ~ a d L = å i¥ i ò å ai¥ zi exp(- zi F)dFdz i =1

0 i =1

¥

2 = - ò r~dFdz = 2 kc 0 =-

¥

d 2F ò dz 2 dFdz 0 ¥

(1A.9) 2

2 æ dF ö 1 æ dF ö ÷ dF s - 2 ò d ç ÷ dx 2 ç k c è dz ø z =0 k c 0 è dz ø

Then combining Eqs. (1A.7), (1A.9) and (1.59) one obtains N

~

å ai¥dL i = r~sdF s - dF

(1A.10)

i =1

A substitution of Eq. (1A.10) into Eq. (1A.8) yields N

~ 2dF º å L i da i¥ + r sdF s

(1A.11)

i =1

Next, the substitution of Eq. (1A.11) into the Gibbs adsorption equation (1A.5) leads to æs ö N - dç + 2 F ÷ = å Gi d ln ais è kT ø i =1

(T = const.)

(1A.12)

Comparing the definition of F, Eq. (1A.7), with Eq. (1.61) one finds that 2F = -sd/kT. The substitution of the latter result into Eq. (1A.12), along with Eq. (1.19), finally gives the sought for Eq. (1.70). REFERENCES: APPENDIX 1A 1. P.A. Kralchevsky, K.D. Danov, G. Broze, A. Mehreteab, Langmuir 15 (1999) 2351. 2. S. Hachisu, J. Colloid Interface Sci. 33 (1970) 445. 3. D.G. Hall, in: D.M. Bloor, E. Wyn-Jones (Eds.) The Structure, Dynamics and Equilibrium Properties of Colloidal Systems, Kluwer, Dordrecht, 1990; p. 857. 4. D.G. Hall, Colloids Surf. A, 90 (1994) 285.

635

Appendices

APPENDIX 8A: DERIVATION OF EQUATION (8.31)

Following Ref. [1] we consider the configuration of two floating particles depicted in Fig. 8.2, where the meaning of the notation is explained. For small particles, (qRk)2