Correlation functions interfaces

Lamellar morphology variables in semicrystalline polymers can be estimated from the correlation and interface distribution fiinctions using a two-phase model. The analysis of a correlation function by the two-phase model has been demonstrated in detail before [30,11] The thicknesses of the two constituent phases (crystal and amorphous) can be extracted by several approaches described by Strobl and Schneider [32]. For example, one approach is based on the following relationship ... 

Having determined the effect of the diffusive interfaces on the structure parameters, we now turn to the calculation of H and K in microemulsions. In the case of oil-water symmetry three-point correlation functions vanish and = 0. In order to calculate K from (77) and (83) we need the exphcit expressions for the four-point correlation functions. In the Gaussian approximation... 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

Much more detailed information about the microscopic structure of water at interfaces is provided by the pair correlation function which gives the joint probability of finding an atom of type/r at a position ri, and an atom of type v at a position T2, relative to the probability one would expect from a uniform (ideal gas) distribution. In a bulk homogeneous liquid, gfn, is a function of the radial distance ri2 = Iri - T2I only, but at the interface one must also specify the location zi, zj of the two atoms relative to the surface. We expect the water pair correlation function to give us information about the water structure near the metal, as influenced both by the interaction potential and the surface corrugation, and to reduce to the bulk correlation Inunction when both zi and Z2 are far enough from the surface. 

Spohr provides a detailed discussion of the water pair correlation function at the water/Pt(100) interface." His results are shown in Fig. 3 for the oxygen-oxygen, oxygen-hydrogen and hydrogen-hydrogen pair correlations when one of the reference atoms is in either the first or the second layer, but otherwise a complete averaging over the locations and orientations of the other atom has been performed. The pair correlations... 

Much less attention has been paid to the dynamic properties of water at the solution/metal interface (or other interfaces). Typical dynamic properties that are of interest include the diffusion constant of water molecules and several types of time correlation functions. In general, the time correlation function for a dynamic variable of interest A(t) is defined as... 

EVB-based MD simulahons, as well as conhnuum dielectric approaches, involve empirical correlations between the structure of acid-functionalized interfaces in PEMs and proton distributions and mobilities in aqueous domains. The results remain inconclusive with respect to the role of packing... 

For the model system considered in Eikerling et al., i the chemical composition and water content are fixed. Only minimal hydration could be considered. A more recently begun work aims explicitly at the understanding of structural correlations and dynamics at acid-functionalized interfaces between polymer and water in PEMs. It directly addresses the question of... 

A different analysis of the scattering pattern uses the Debye correlation function (14), derived for a random two-phase structure with sharp interfaces ... 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

In the case of a liquid/vapour interface it is straightforward to reduce the above procedure of approximating the inhomogeneous direct correlation functions Cayivi, F2) given by Equations (19)-(20) and (23)-(26) to the non-linear interpolation between two phases, vapour and liquid. 

For a planar interface the interpolation (19)-(20) of the inhomogeneous site-site direct correlation function in terms of the homogeneous ones simplifies the SS-LMBW equation (18) to the form... 

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