# Reviev in Computational Chemistry vol 19 - Lipkowitz K.B.

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--2 T”

r/2 \r KD' KDr

-2KDrdr

2k2 R2

-ö:;ã [(A1 (R)a1e-KDa1 )2 + (A2(R)a2e-KDa2 )2]

1 /1 2

R/2

\r3 KDr4 KDr5

2KDrdr

e

[330]

00

00

270 The Poisson-Boltzmann Equation

The integrals can be evaluated to give

16a1a2eKD(a1+a2) A1(R)A2(R)(1 + kdR) ^r bF(R) =-LLBZ2--r2-e |331]

As observed for two interacting cylinders, for large kdR, the coefficients are independent of separation distance and approximately independent of the sphere radii.331 Force curves predicted by Eq. [331] are in good agreement with those obtained by Ledbetter, Croxton and McQuarrie who solved the PB equation numerically.331 Finally, we integrate the force according to Eq. [281] and, as for the planar and cylindrical cases, analytical evaluation of the integral requires that we restrict ourselves to separations of at least two to three Debye lengths. Using Eq. [204], the leading terms in the interaction potential between two charged spheres are found to be

8aia?e kdh0

bVpB (R)=otr

2^2 +(1+É)%((1 - 22)S2- (2|Ú)ò,)

+ é+É91) ((1 - S1)e, - (?>2)

|332]

where e and g vary as exp(-kdH0). As done previously for two interacting cylinders, we have introduced the ( parameters of Eq. [206] (and in the definition of a0 as well) that are useful for taking the KDa ! 0 limit. This potential in the low-charge-density (DH) limit is

PVdh(R) = La^aCa^R ^1&2 + s 1S2 + s2S1)e-KDH° |333]

Equations [332] and [333] represent the leading terms in an asymptotic expansion of the interaction potential and as such become increasingly inaccurate as the separation distance between the spheres decreases. To evaluate the potential in the opposite limit in which the spheres are almost touching, we combine Eqs. [281] and [329] and change the order of integration to write

V (R) = 2p

1 1

0 , R

P(R*, p, 0)dR*

p dp |334]

Note that the term in brackets represents the interaction potential per unit area as determined by the pressure due to two charged rings of radius p and width dp at the dividing plane. The inner integral in Eq. [334] can be evaluated by following the generalization of Derjaguin’s method328 by Hogg, Healy, and Fuerstenau.112 For values of p less than the radius of the smaller sphere (= amin), this pressure and the resulting interaction potential can be approximated by that due to two opposing and parallel planar rings if the spheres are

Analytical Solutions to the Poisson-Boltzmann Equation 271

sufficiently close; what “sufficiently close’’ means will be determined below. Thus, in place of the bracketed term we can insert the interaction potential per unit area between two planes with charge densities aj and a2 separated by a distance

H(R, p)= R — af — p2 — a2 — p2 for p < am This allows us to write Eq. [334] as

[335]

Vhhf(R) = 2ÿ

Vpl[a1, a2, H(R, p)]p dp

[336]

where the planar interaction potential Vpl is given by Eq. [110]. Equation [335] may be inverted analytically for [p(H)]2

[p(H)]

2 (2a1 + H0 — H)(2a2 + H0 — H)(2ax + 2àã + H0 — H)(H — H0)

4(a1 + a2 + H0 — H)

which lets us write Eq. [336] as

[337]

Hm

Hn

Vpl(a1, S2, H)j

RH

(a2 — a2)

(R — H)3

dH,

Hm

R

[338]

Clearly, Eq. [338] represents an approximation to the actual interaction energy since (1) the rings become progressively less parallel as p and hence H increases, (2) this ignores contributions from that part of the larger sphere beyond the smaller sphere radius as well as those from the backsides of both spheres, and (3) there are pressure contributions on the dividing plane beyond amin not considered at all. These concerns are minimized if the system meets two conditions:332 (1) the closest spacing between spheres is much less than the smaller radius, and (2) the thickness of both spherical double layers is small:

H0

-< 1 < KDan

amin

[339]

a

0

The first condition guarantees that most field lines contributing to the interaction are parallel, satisfying point (1), while the second condition implies that the interaction falls off quickly with curvature, satisfying point (2). The second

272 The Poisson-Boltzmann Equation

condition also allows us to extend the upper integration limit from Hmax to infinity since there are then no significant contributions beyond this value, thus rendering point (3) moot. Although we apply the Derjaguin approximation only to spheres in this chapter, it is equally applicable to any two curved surfaces provided é1 and a2 are interpreted as the radii of curvature of the surfaces at their points of closest approach.332 We noted earlier that Sparnaay has obtained expressions for the interaction for two parallel and for two crossed cylinders within the Derjaguin approximation for low and moderate surface

potentials.314

Consider the interaction potential according to Eq. [338] for two spheres of equal radius a:

p t2a

V(H0 + 2a)= 2

Vpl(sb s2, H*)(2a - H*)dH*, H* = H - H0 [340]

Let us first compare the potential between two identical spheres to that between two planes. We see below that the distance dependence of the leading term in the potential between two spheres is the same as that between two planes, exp(-kdH), so the ratio of potentials is approximately

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