** log(h) [A]**

** 51 the simplified mean field approximation for zwitterions of length a and**

**moment p.=qa interacting only via electrostatic potentials. Then components**
**of the dipole field tensor are**

T „(k ) = ; 2 ( 1 - cos(k a ) ) (1 + A e 2kw) (3.4.6a)

a2 12

f (k) = *—* 2i sin(k a/2) 2 smh(ka/2) A e_2kw (3.4.6b)

e k a 2 12

**l**

t 2(k) = *~* *~* *2* ( 2 - 2e ^ " A e 2kw( 2 - e"kl - e1^ ) ) (3.4.6c)

**e k a **_{l} 12

**The simplified mean field approximation then consists of inserting**
**these into the formula for the polarisation correlation function (3.4.3) and**

A

**taking the reponse function to be given by its value at k=0, T(0). This is**
**related to the susceptibility by Eq. (3.4.3) at k=0, or explicitly**

1 + 4itß *xJQ* / (e^)

(3.4.7)

**Hence an accurate numerical calculation of the susceptibility will give the**

A

**value of T(0) and the pressure in the simplified mean field approximation**
**follows, by replacingf T(k) by T(0) in Eq. (3.4.3).**

**3.4C Low coupling approximation**

**The parallel and perpendicular components of the susceptibility can be**
**estimated in the low coupling regime. The short ranged response function X**
**is not only independent of the long range tail of the interaction potential, but**
**is determined approximately by the system interacting with a truncated**
**potential (see Part I). Therefore, to a first approximation, its value is given by**
**that of an ideal dipolar gas in which all orientations are equally likely**

**T*(0) = x “ (0) = T > ) = | p p 2 ** **(3.4.8)**
**This approximation is analogous to the Debye-Hückel theory of electrolytes.**

**52**

The dipole field tensor T includes short range potentials, and these

A

contribute to T(0). If the sytem is not too highly coupled, then the core

repulsions may be neglected. There then remains an electrostatic term in the

A

zz component. Zwitterions of length a (and fixed moment p) have Tzz(0) =

*—AtzIz-^s.* . Hence for freely orientable dipoles one has the approximations

A o
G >
**A 0 ** **1 ** **o ** **A H**
g "(0) - T PH2 , G°(0) - - .
**57 ** **3 ** **1 + 47tßpp2 ** **e a**
**PH2 **

**11**

**11**

_{(3.4.9a)}

and for perpendicularly constrained dipoles one has

0 A 0 A 0

G (0) = G (0)_{xxv }_{'}_{yyv }_{7} _{G > )}

**1 + 47tßpp2 /** (3.4.9b)

Note that this approximation predicts that the zz component of the susceptibility goes to zero in the limit of point dipoles. This limit is a high coupling limit (since the charge must increase if the dipole moment |i=qa is held fixed) and in actual fact the susceptibility becomes constant at

3.5 A sy m p to tic r e s u lts

**53**

T he long w av elen g th lim it of th e co rrelatio n function can be in se rte d

in to th e reflection coefficient a n d th e in te ra c tio n free energy d eterm in ed

n u m erically from Eq. (3.3.12). H ow ever, since th e co n trib u tio n from long

w av elen g th s becom es in creasin g ly d o m in an t a t la rg e r sep aratio n s, one

could a rg u e t h a t a n asym ptotic expansion in pow ers of se p ara tio n w ould be

m ore co n sisten t. To o b tain th is, one expands th e lo g arith m in Eq. (3.3.12),

a n d collects from each te rm coefficients w ith th e sam e pow er of k. F rom

scaling a rg u m e n ts (change v ariab les, k=q/h), each pow er of k corresponds to

successive te rm s in th e asym ptotic expansion. N ote t h a t in th is derivation

th e n e a re s t im ages, w hich are in d ep e n d en t of sep aratio n , a re included in th e

reference system .

F ir s t for th e case of th e d ip o lar surfaces em bedded in th e c en tral

dielectric m ed iu m (h=d-2w). U sin g th e long w av elen g th lim it of th e

co rrelatio n s Eq. (3.4.4) in th e fo rm u la for th e reflection coefficient Eq. (3.3.11),

one o b tain s th e firs t sev eral te rm s in th e asym ptotic expansion. T he leading

te rm is

z eta function. T his is th e u su a l L ifshitz zero frequency re s u lt for th e

in te ra c tio n b etw een two half-spaces (c/E q . (1.3.6)); i t is in d ep e n d en t of th e

d ip o lar surfaces. T he n e x t o rd er term , w here th e coefficients co n trib u te a

factor of k, is

H ere a n d a fte r we u se th e n o tatio n X= G ^ fO ), *Z=* G°zz(0), W±= l± A12 e '2kw .
T his te rm corresponds to firs t o rd er p e rtu rb a tio n theory, k eeping only th e

te rm s lin e a r in k. A sym ptotically, th is te rm decays as a n in v erse cubic in (3.5.1)

w h ere £3(x)=X xn/n 3 w hich m ay be considered to be a g en eralised R iem an n

-kh -kd

**54**