Long-Range Interactions Macroscopic Theory

Note that the case of small contact angles treated here means that S 0, which only occurs when y / — ysv 0. This is the same expression for the contact angle as Eq. (4.2), for small contact angles, when cos  

1 - (0 /2). Although the macroscopic derivation of Young s equation is simpler, the approach described here, which self-consistently solves for the profile, can be generalized to explicitly include the interactions between the fluid and vapor and the substrate as described below.  

We now consider the case of complete wetting (spreading power 5 0) for the case where there are van der Waals interactions, which tend to thicken the film. For a finite amount of fluid spreading on an infinite solid substrate, the equilibrium film profile will therefore not be a monolayer, but a pancake with a maximum thickness that is determined by the balance of the surface tensions and the van der Waals energies (see Fig. 4.3). 

For simplicity, imagine a two-dimensional drop (with a one-dimensional interface between the fluid drop and the vapor). We assume that the substrate area is larger or equal to the size of the drop and that it therefore does not 

The variation of g with respect to the profile h(x) is given by the Euler-Lagrange equation  

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Before discussing this new method it is useful to recall briefly the methods which we have already discussed. Note, first of all that calculations of the dielectric tensor must be based, as is known, upon a microscopic theory Such a theory for ionic crystals was first developed by Born and Ewald (2) for the infrared spectral region. The application of this approach for the region of exciton resonances has also been demonstrated in (3). In an approach identical to that of Born and Ewald (2) the mechanical excitons (see Section 2.2) are taken as states of zeroth-approximation. In the calculation of these states the Coulomb interaction between charges has to be taken into consideration without the contribution of the long-range macroscopic part of the longitudinal electric field. If this procedure can be carried out, then the Maxwell total macroscopic fields E and H can be taken as perturbations. In the first order of perturbation theory, we find... 

On the nanoscale the shape of such droplets can deviate significantly from the macroscopic shape of a spherical cap due to the finite range of the intermolecular interactions involved. As described later, one can actually relate the macroscopic equilibrium contact angle, the character of wetting transitions, and the shape of nanodroplets to the intermolecular interactions by using, for example, classical density functional theory (DFT). But the long range of intermolecular interactions also affects the motion of droplets. In particular it can lead to lateral interactions between droplets and structures, which are absent on the macroscopic scale. 

The purpose of this Chapter is to describe the dielectric properties of liquid crystals, and relate them to the relevant molecular properties. In order to do this, account must be taken of the orientational order of liquid crystal molecules, their number density and any interactions between molecules which influence molecular properties. Dielectric properties measure the response of a charge-free system to an applied electric field, and are a probe of molecular polarizability and dipole moment. Interactions between dipoles are of long range, and cannot be discounted in the molecular interpretation of the dielectric properties of condensed fluids, and so the theories for these properties are more complicated than for magnetic or optical properties. The dielectric behavior of liquid crystals reflects the collective response of mesogens as well as their molecular properties, and there is a coupling between the macroscopic polarization and the molecular response through the internal electric field. Consequently, the molecular description of the dielectric properties of liquid crystals phases requires the specification of the internal electric field in anisotropic media which is difficult. 

Micro.scopic theories lead to an expression for ( fl(fVo,t b, ), which is, generally, not a true macroscopic quantity and contains many microscopic details insignificant in the long-wave macroscopic limit. In particular, this enables the pair interaction potential in the model Ilaniiltoiiian to be replaced by the short-range 8 function. 

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