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

In document Electrostatic fluctuation and double layer interactions between surfaces (Page 69-72)

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

(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

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