ABSTRACT
The information one can extract from liquid reflectivity experiments is the electron density profile in the direction perpendicular to the interface, the thickness and density of layers at the interface, and the interface roughness. When we study a liquid surface or interface, the measured reflectivity will differ from the theoretical Fresnel reflectivity. There are three main causes for this behavior. There will be an inherent roughness due to the finite size of the atoms. It is not possible to obtain an interface or surface flatter. The broadening of the profile is due to the following:
• Finite size of atoms • Capillary waves • Other roughness contributions In order to determine the structure of our interface, we need to separate the capillary roughness from other sources of roughness that might be present, which contribute to the surface diffuse scattering. Roughness is defined as a statistical deviation of the local surface from the mean surface. Therefore, statistical parameters and functions such as the root mean square deviation from the mean surface (the rms roughness), correlation lengths, correlation functions or power spectral densities can be introduced. For fluctuating surfaces, these values depend on space and time [5]. For a liquid in an equilibrium state, these statistical parameters exclusively depend on the physical properties of the liquid and its surroundings. In the simplest case, the surface tension, the viscosity, temperature, and gravity are sufficient to describe a liquid surface or interface. As liquid surfaces are thermally excited at temperatures T π 0, the local surface z(x, y) is not static but fluctuating. Energy density: e(r, t) Temperature field: T(r, t) with the position vector r and the time t. In equilibrium, the sum of all forces is zero. In a simple approach, only the forces acting on volume elements r ∂v/∂t, resulting from pressure gradients —p and friction –hDv (with the viscosity h) are taken into account. From this follows r h∂∂ = -— +v vt p D ,(9.7)
which is a simplified form of the Navier-Stokes equation. To solve Eq. (9.7) in the case of capillary waves, J. Jäckle used the fact that small deviations of the local surface from the mean surfaces can be approximated by a sum of plane waves. For incompressible and source-free liquids, the additional condition — ◊ v =0 applies. This condition can be written as follows: ik v z vx z' + ∂∂ = 0 (9.8) The whole problem can be decoupled in vx and vy. One boundary is simply given by v = 0 deep in bulk liquid or at a solid-liquid interface. At the surface, the situation is more complicate and the stress tensor s has to be included into the calculation with the surface tension g, gravity g and the vertical surface displacement uz(x, y) (see Fig. 9.3).
