Surface heterogeneity random

Crystallites grow from nuclei. In homogeneous nucleation, nuclei are created by random statistical thermodynamic fluctuations of the local structure of the polymer. In heterogeneous nucleation, nuclei are created on surfaces of randomly distributed microscopic insoluble particles present in the melt. Since thermodynamic fluctuations are very small in magnitude while impurities are almost always present in significant amounts in a polymer, heterogeneous nucleation normally dominates. However, homogeneous nucleation can also become important at low values of Tx [186,187]. 

Recent analyses of contact angle hysteresis has treated surface heterogeneity in terms of random fluctuations see the excellent review by deGennes The approach is to treat both surface roughness and variations in surface composition as weak fluctuations, i.e. deviations from the ideally smooth surface, du/dy (Fig. 11) and from the difference in solid/liquid and solid/vapor surface energies, — yj - Both fluctuations are considered to be equivalent in perturbing the contact line and are analyzed in terms of their effect on the elastic line energy. 

This weak fluctuations or random noise approach to surface heterogeneity is in some ways more realistic than the earlier model of Johnson and Dettre in that the latter is highly anisotropic (in terms of roughness) and discontinuous (in composition). Real surfaces generally have a very random roughness and a high dispersion in surface composition. Thus the weak fluctuation approach focuses attention on local condi-... 

The other extreme model of surface topography, "random topography, assumes that sites characterized by different values of e are distributed on an energetically heterogeneous surface completely at random. The function s 5- now takes the form [37]... 

Equation (81) suggests that the surface heterogeneity is induced solely by the MPSD. This approach is more fundamentally sound than the approach of energy distribution in Eq. (55), since the distribution of energy sites, F(e), may not follow any particular type of distribution function. On the other hand, the microporous structure of the activated carbon results from the random processes of carbonization and activation and therefore may possibly be characterized by some distribution functions. Hence, assuming a function f r) is more reasonable than arbitrarily imposing a function F e). In the MPSD approach, F e) can be derived from /(r) according to the relationship... 

For a block copolymer system where the differences in surface energy between the two components is less, wetting constraints can lead to surface heterogeneity (62). For the same composition, block copolymers, segregate more effectively to the surface than graft copoljuners, which in turn are more surface active than random copolymers, though problems with microphase separation are reduced for random copolymers (61). Other chain architectures have been investigated and also been shown to he surface active (67). 

Contemporary theory on the multi-site occupancy adsorption on heterogeneous solid surfaces is rather scanty. The reason is that such description involves a new degree of complexity and only a few papers have been published on this subject [260,261]. Rudzihski and co-workers [262-265] accepted the Everett s equation for the multisite occupancy adsorption from liquid mixtures [266] on solid surfaces as a starting expression for description of adsorption on heterogeneous surfaces with random and patchwise topography. The short review on the subject is presented in reference [267]. 

Rudzinski, W., and Jagiello, J., A simple approach to the 2D mobile adsorption of gases on heterogeneous solid surfaces exhibiting random surface topography. Adsorpt. Sci. Technoh. 6(1). 35-51 (1989). 

The discussion given so far applies to XR from surfaces covered by homogenous thin films. The interpretation of reflectivity data from heterogeneous monomolecular layers is more difficult. Consider, e.g., a surface with randomly distributed domains of lipid and protein or domains of two lipids with different reflectivity, say, R q = r qj)f and Ri qz) = I f2( 2)P> where r, q are complex functions describing the amplitudes of the reflectivity, cf. eq. (20). 

Here, we mention only the precursory results obtained by Nicholson and Silvester [103] for adsorption on the surface with random and patchwise surface topographies. This work led to the conclusion that the smooth sigmoid isotherms are not necessarily associated with surface heterogeneity nor are stepped isotherms indicative of homogeneous surfaces. Nicolson and... 

Equation (14), although derived from the approximate random walk theory, is rigorously correct and applies to heterogeneous surfaces containing wide variations in properties and to perfectly uniform surfaces. It can also be used as the starting point for the random walk treatment of diffusion controlled mass transfer similar to that which takes place in the stationary phase in GC and LC columns. 

FIG. 9 Changes of the monolayer film critical temperature with the concentration of impurities obtained from the Monte Carlo simulations (open circles) and resulting from the mean field theory (solid line). (Reprinted from A. Patrykiejew. Monte Carlo studies of adsorption. II Localized monolayers on randomly heterogeneous surfaces. Thin Solid Films, 205 189-196, with permision from Elsevier Science.)... 

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