Interfacial correlation functions

Ethylene/vinyl acetate copolymer Interfacial correlation function Complex modulus Storage modulus Loss modulus... 

The proof of this result is based on exact bounds on the asymptotic behavior of the interfacial correlation functions, obtainable from the Bogoliubov inequality, and uses a reductio ad absurdum a self-maintained interface is assumed to exist and it is then shown that this assumption leads to a contradiction. The key step in the demonstration consists in deriving and making use of the asymptotic behavior at large separations of the direct correlation function of Ornstein-Zernike, c(r, r ), defined in terms of the more familiar pair correlation function ft(r, r ) which measures the probability of having a molecule at point r given that there is one at point r ... 

III. INTERFACIAL CORRELATION FUNCTIONS AN EXACTLY SOLUBLE MODEL... 

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

Finally, we return to the case of antisymmetric surfaces, i.e. a situation where one surface of the thin film favors the A-rich phase and the other the B-rich phase (Fig. Id). Simulations were recently carried out [266] in order to test the predictions Eq. (127) on the anomalous interfacial broadening (Sect. 2.5). Figure 24 demonstrates that this phenomenon can indeed be readily observed. Using the interfacial tension a that has been independently measured [215], o= 0.015, and the correlation length 3.6 lattice spacings from a direct study of the bulk correlation function, one can evaluate Eq. (127) quantitatively, using... 

Pair correlation functions can also be used to show differences in structure between the bulk PEM and interfacial regions. Figure 13 shows the difference in the water network in the aqueous domain of bulk membrane of Nafion to those adsorbed on to a catalyst surface through the Oh o Oh o POF at all the water con-... 

Correlation function Elongational strain Strain rate Craze strain Elastic strain Shear strain Interfacial length Extension ratio... 

Another correlation function that has received much attention in simulations of liquids is the center of mass velocity. (The dynamical variable A in equation (15) is set to the center of mass velocity). The integral of this correlation function is proportional to the diffusion constant of the liquid. For an interfacial system, this diffusion constant is in general anisotropic and location-dependent the diffusion rate along the direction normal to the interface, D, is different from the one... 

The decay of the orientational correlation function is highly nonexponential and one needs at least four exponentials to fit it. The average orientational correlation time, t, is slower by about a factor of 20 than that of its bulk value. The orientational correlation function for the interfacial water molecules will, of course, decay in the very long time (of the order of tens of nanoseconds), either because of "evaporation" of the interfacial water molecules or rotation of the micelle. 

Fundamental frequency of the resonator Correlation function for surface roughness Root mean square height of a roughness Wave vector of shear waves in quartz, (Uy pq//rq Correlation length of surface roughness Thickness of the liquid film Thickness of interfacial layer Molecular dynamics Pressure in a liquid Quartz crystal microbalance Hydrodynamic roughness factor Electrochemical roughness factor Coordinates (normal and lateral)... 

It is therefore entirely feasible to have the observed intensity I q) converted first into the correlation function by Fourier inversion, and then determine the interfacial boundary area S from the initial slope of the latter. Since the Porod law is applicable in the limit of q -> oo, it is natural that the corresponding information is contained in T (r) in the limit of r - 0. In practice, however, performing the Fourier transform on observed intensity data and obtaining the correlation function for small r with sufficient accuracy are difficult tasks. 

Microemuisions exist for values of the parameter y [in and see Eqs. (33) and (34)] less than 1 and greater than —1, with more negative values associated with more structure. As can be seen, the correlation function is an exponentially decaying oscillatory function of the separation r. On the other hand, for values of y > 1, the Fourier transform is simply a sum of two monotonically decaying exponential functions, and the liquid is unstructured. It is this difference in bulk behavior that proves crucial to the interfacial wetting behavior. 

A Monte Carlo simulation [102] of a system with short-range forces confirmed these notions. The correlation function clearly exhibited exponentially damped oscillations. From the ratio of the wavelength and correlation length, the value of y characterizing the system could be obtained from Eq. (35), and it was found that 1 > y > 0, indicating that the microemulsion was structured but weakly so. Within the mean-field calculation, however, this is still strong enough that the middle phase should not wet the oil/water interface. However, measurement of all three interfacial tensions within the simulation revealed that Antonow s rule was obeyed, so that the interface was indeed wetted by the middle phase, an effect clearly attributable to the fluctuations included in the simulation. 

Deviations from spherical shape can be modeled as a growth into prolate or oblate shapes. The area enclosed volume ratio constraint implies a constraint in the possible values of the two semiaxes describing the particle size. Halle [62] calculated the correlation functions for the combined particle tumbling and surface diffusion of prolate and oblate particles. His results have, for example, been applied to microemulsion systems [48,58], focusing on the ratio yVpr(0)// 5,sph(0). where the subscripts pr and sph refer to prolate and spherical particles, respectively. For a given ratio of interfacial area to enclosed volume, which specifies the radius R of the sphere, y, pr(0)//. sph(0) is a function of the prolate axial ratio, Diat, and R. Knowing Djat and R from other experiments, it is possible to determine the aggregate axial ratio from the relaxation experiment. 

The Guinier, Debye-Bueche, Invariant and Porod analyses are all based on the assumption of well defined phases with sharp interfacial boundaries. In addition, the Guinier approach is based on the assumption that the length distribution function (23.15), or probability Poo(r) that a randomly placed rod (length, r) can have both ends in the same scattering particle (phase) is zero beyond a well defined limit. For example, for monodisperse spheres, diameter D, Poo = 0, for r > D. In the Debye-Bueche model, Poo has no cut off and approaches zero via an exponential correlation function only in the limit r oo [45,46]. 

SAXS is a more precise tool to quantitatively evaluate the two-phase structure of PU by providing the data of interdomain spacing, domain size, and interfacial thickness [40]. Figure 7.28 illustrated typical SAXS intensity profiles (/(S)S vs. 20) for PU/C20A nanocomposites. The one-dimensional correlation function F(Z) that is related to the electron density distribution within specimens is expressed as follows ... 

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