Critical exponents roughness

The Hurst exponent H characterizes the roughness of the saturated interface. As for the local slope p, it is literally the approximate average slope of the local structure. Its value is related to the intersection of C2(r, t) with the vertical axis. For normal scaling, it does not depend on time. - If p does depend on time, the scaling becomes anomalous. As one can see, to be able to characterize a growing interface, one needs the height-height correlation function C2(r, t), and more specifically, the critical exponents FI and p. 

Table 26.1 Critical exponents a, P, y, v, and q for boiling. The correlation length gives roughly the sizes of droplets. Source JM Yeomans, Statistical Mechanics of Phase Transitions, Clarendon Press, New York (1992). 

The direct comparison of results obtained by various methods is difficult. At the same time, the experimental data qualitatively meet the current theories of adsorption and theoretical predictions regarding dependence of the layer thickness on the adsorbed amount, molecular mass, nature of a solvent and adsorption conditions. Corresponding dependencies in scaling form may differ by the critical exponents. It is also worth noting that the adsorption behavior of macromolecules is very specific for individual system polymer-solvent-adsorbent. It was shown that adsorbance strongly depends on the structure of adsorbent surface (physical roughness, its characteristics, the existence of... 

To this end, let us use eqn. (66) of Chap. 3, permitting one to estimate roughly the critical distance, Rcr, at which there occurs a transition from one exponent to the other. By substituting into this formula the values 7d = 2 eV (the depth of a trap for elr in water-alkaline matrices), [Pg.191]

The dependence of the accuracy of determining equilibrium composition by means of methods which do not require stoichiometric analysis, on the accuracy of the parameters c i = 1, 2,..JV is best seen from the relationship (5.142). The exponent on the right-hand side of this relationship includes a difference of two large numbers. In the case of most compounds, the absolute value of the parameter C lies in the interval (20 150) at above-critical temperatures. Thus, if it is required that the value of the equilibrium molar fraction be determined with an accuracy of p %, the value of the thermodynamic parameter c must be determined to an accuracy of roughly lOOp %. The reason consists in the fact, that the relation c 8 Ci i = 1, 2,. ..,iV follows from the equation (5.142). For the sake of simplicity, the influence of errors of determining Lagrangian multipliers was neglected. 

The experimentally observed power law exponent of about 3-5 has been given some theoretical justification by Bueche (1952). Below the critical value, Mw.c< the molecular weight-viscosity relationship is roughly linear, the greater dependence above this point usually being ascribed to entanglement effects. 

The mean-field nature of our entanglement model is perhaps its weakest point. It has long been recognized that mean-field theories are rather rough approximations near the critical points for systems with d <4 dimensions. For example, the experimentally measured exponent for the liquid-gas critical point is (J= 0.33 and not S = 0.5. Mean-field theories generally predict more abrupt transitions than are actually observed. The discrepency between mean-field theories and experiments stems from the absence of fluctuation phenomena which are often more important near the transition. 

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