** log(h) [A]**

** 63 4.2 Pressure between dipolar surfaces**

One can calculate the interaction free energy (and hence the pressure by differentiation) using the various approximations of the preceding

chapter. First concentrate on dipoles constrained to point perpendicular to the surfaces (hence the parallel components of the susceptibility are zero). The HNC results will be compared with the following approximations:

(a) Ideal dipoles, second order perturbation theory. This is just Eqs (3.5.2) and (3.5.4), but doesn't include the asymptotic part of the correlation

function Eq. (3.5.3).

**(b)** Zwitterions, second order perturbation theory. This uses the reflection
coefficient (3.3.15), together with the approximation **G°zz(k) = G°zz(0), **and the
expansion of the logarithm in the free energy expression (3.3.12) through to
terms of order **(G°zz(0) ^**

**(c)** Zwitterions, perturbation theory, including the asymptotic part of the
correlation function. Same as (b) but includes the term linear in k ( Eq. (3.4.4)
using Eq. (3.4.6c)). This approximation is the analogue of Eqs (3.5.2-4),

generalised to zwitterions.

**(d)** Zwitterions, infinite order perturbation theory. **G°zz(k) ** as in (c)
inserted into the reflection coefficient (3.3.15) and the full logarithm

expression (3.3.12) is used for the free energy.

**(e)** Simplified mean field perturbation approximation. The **G°zz(k) **is
obtained from **x(0) **(using **G°zz(0) ** from table 4.1 in Eq. (3.4.7)) inserted into
Eq. (3.4.3) using the zwitterion potentials (3.4.6). Then the reflection

coefficient and free energy are determined as in (d).

All of these approximations are identical in the asymptotic limit of large separations. However the accuracy at smaller separations is expected to improve in going from a to e, as we now explicitly show. As before, the HNC is considered to be a benchmark against which to test the other approximations. This is reasonable since the hypemetted chain closure is

64 known to work well in general for systems with long range interactions (see Part I and references therein). In addition, HNC results for the susceptibility of these particular two dimensional dipolar systems have already been

favourably compared with simulation data (§4.1).

Figures 4.2 show the pressure between surfaces of perpendicular dipoles at an area per dipole of 75Ä2 (chosen to reflect the surface density of polar lipid bilayers) and for moments of p=l and 5 eÄ. Since there are no images here, the attraction arises from the correlations between dipoles on opposite surfaces, and asymptotically decays as h'5. For the 1Ä zwitterions,

o

we see that the ideal dipolar approximation (a) is already very good at 10A separation (always measured between the zwitterion midpoints) and that the simplified mean field perturbation approximation (e) is excellent over nearly the whole regime shown. For the system with the higher moment, |i= 5 eÄ, the ideal dipole is within an order of magnitude of the HNC at 10Ä, and is

o

virtually coincident with it by about 30A separation. The approximations progressively improve, and again (e) gives virtually exact agreement with the HNC even at very small separations.

Figures 4.3 show the effect of images on the pressure due to the
surface dipoles alone. It is desirable to highlight these novel surface effects
and so the zero frequency Lifshitz term has not yet been added (but see fig. 4.4
below). In the attractive (small separation) regime, the difference between
the various approximations is most apparent. However, by the time the
image repulsion dominates, all the approximations agree, and now the
pressure goes asymptotically as * hr4.* Figure 4.4 shows the total pressure for
the 5Ä zwitterions in the HNC approximation. Below about 15Ä the pressure
for the cases with and without dielectric discontinuities are quite similar.
Here it is obviously dominated by the direct correlational attraction between
dipoles on opposite surfaces in both cases. Beyond that separation, the

pressure for the case with images is essentially given by the zero frequency van der Waals attraction. This decays as h'3 and dominates the more rapidly

6

log h [Ä]

**Figure **4.2a The attractive pressure between surfaces of perpendicular
dipoles (|i=lÄ) versus separation (log scales). The HNC results for

monovalent zwitterions ( ) are compared to the ideal dipole approximation ( a ,...) and the simplified mean field perturbation

approximation (e, ). The area per diopole is 75Ä2, £,=^=80 (no images) and T=300K. The susceptibility used in the approximations is taken from the HNC results in Table 4.1.

log h [A]

**Figure **4.2b Same as preceding figure but for p=5Ä. The HNC zwitterion
results ( ---) are compared to four approximate formulae (given in the
text): a ( ...); b ( --- • • ) (would be coincident with approximation
c); d ( ); and e ( ).

log h [A]

**Figure **4.3a The attractive pressure between surfaces of perpendicular
dipoles (p=5A) versus separation (log scales). The HNC results for

monovalent zwitterions ( ) are compared are compared to five
approximate formulae: a ( ... ); b ( * * ); c ( * * **) ’,**

d ( ---); and e ( ---- --- ). The area per diopole is 75Ä2, e,=tfO,

**6o=2, w=3Ä ** and T=300K The susceptibility used in the approximations is

taken from the HNC results in Table 4.1b. The zero frequency van der Waals attraction is not included.

log h [A]

**Figure **4.3b Same system as above but in the repulsive regime (the zero
frequency van der Waals attraction is not included). The ideal dipole

prediction ( a ,...) is compared to the HNC ( ) and the simplified mean field perturbation approximation (e, ), the latter two virtually coincident.

**log(h) [Ä]**

**Figure **4.4 HNC results for the attractive pressure, -P, as a function of

o

separation, h, between two surfaces with perpendicular 5A zwitterions in the presence of dielectric discontinuities (ex = 80, e2 = 2 and w = 3 A). The inverse density is 75 A2 and T=300K. The total pressure (---),

^dipolar + ^VdW’ compared to the static VdW interaction, PVdW, al°ne (--- ). As a reference, we have also included the pressure curve (---) for the system in absence of discontinuities [same as in fig. 4.2b].

**65**