Coarsening exponent

But in the coarsening process of the Monte-Carlo simulations (Fig. 5) the exponent is n = 1/6 for a domain of the parameter space (F, S), while for other regions it is n = 1/4. The region where the value of the coarsening exponent does not equal to the... 

Figure 5. The value of the coarsening exponent n of the Monte-Carlo surfaces and two experimental surfaces as a function of the growth parameters. The points (o) where n = 1/6 coincide with those simulations where the equilibrium part of the free energy did not match that of the continuum equation. The error bars show the parameter range/uncertanity of an Fe/Fe( 100) experiment of Ref. 10 (o measured n = 0.16 0.04) and Ref 12 ( measured n = 0.23 0.02). The estimate of the Ehrlich-Schwoebel barrier is taken from Ref 13 (thin line) and Ref. 14 (thick line).

Experimental results support this parameter-space dependence of the coarsening exponent as well. In case of Fe/Fe( 100) homoepitaxial growth (where there are estimates for the value of the Ehrlich-Schwoebel barrier), at room temperature n = 1/6 has been measured " ( = 0.16 0.02), while at elevated temperature the exponent is 1/4 ( = 0.234 0.02). These results are in excellent agreement with our predictions (Fig. 5). 

Eq. (8). The value of P is, however, slightly greater than the predictions of these models. This difference is presumably caused by nonlocal effects produced by the electric field operating at the growing interfiice [20]. Likewise, the difference in the value of yff leads to z 3, and a coarsening exponent 1/z 0.33. 

The phase separation process at late times t is usually governed by a law of the type R t) oc f, where R t) is the characteristic domain size at time t, and n an exponent which depends on the universality class of the model and on the conservation laws in the dynamics. At the presence of amphiphiles, however, the situation is somewhat complicated by the fact that the amphiphiles aggregate at the interfaces and reduce the interfacial tension during the coarsening process, i.e., the interfacial tension depends on the time. This leads to a pronounced slowing down at late times. In order to quantify this effect, Laradji et al. [217,222] have proposed the scaling ansatz... 

The kinetics of the nonconserved order parameter is determined by local curvature of the phase interface. Lifshitz [137] and Allen and Cahn [138] showed that in the late kinetics, when the order parameter saturates inside the domains, the coarsening is driven by local displacements of the domain walls, which move with the velocity v proportional to the local mean curvature H of the interface. According to the Lifshitz-Cahn-Allen (LCA) theory, typical time t needed to close the domain of size L(t) is t L(t)/v = L(t)/H(t), where H(t) is the characteristic curvature of the system. Thus, under the assumption that H(t) 1 /L(t), the LCA theory predicts the growth law L(t) r1 /2. The late scaling with the growth exponent n = 0.5 has been confirmed for the nonconserved systems in many 2D simulations [139-141]. 

Results of Ref 5 show that integrating these equations generates coarsening the time dependence of the lateral size, Vc, of the mounds scales with a power of time, Vc f, with exponent n = 1/4. This exponent is associated with the leading A term. Similarly, Stroscio et ah found n = 1/6 numerically when only the A term was present. A detailed analytical proof is given by Golubovic. ... 

In case of relaxation to equilibrium, the process is diffusion-dominated and the presence of the A term is verified. For non-equilibrium conditions we have two cases For weakly out of equilibrium (low flux, low Ehrlich-Schwoebel barrier) the A term is still present and dominates the long-time coarsening, characterized by = 1/4. However, for strongly out of equilibrium cases (high flux, high Ehrlich-Schwoebel barrier) the Dt term seems to be dominated by the A term, causing coarsening with exponent n = 1/6. 

From all theoretical and experimental results one may conclude that there is no simple scaling relation over a long period of time. The detailed coarsening mechanisms, which are attributed to the intrinsical non-linearity of the phase separation process, determine the exponent a. The time dependence of a reflects cross-over among different coarsening processes. 

An equation of the form provided in Equation (8) is often used to fit experimental coarsening data. The exponent, n, is used to infer the rate-controlling step for the growth process. This has proven useful for modeling grain growth in a number of metallic alloy... 

During the later stages one observes that the coarsening behavior changes gradually - an apparent temperature-dependent exponent x in the relation qm oc t x is found in the intermediate stage , see Fig. 8a, while the behavior in the transition stage [36] is similar to the behavior of small molecule mixtures... 

The predictions of the coarsening theory that are embodied in Equation 12.3 are as follows. The temporal exponent for the rate of radial growth of the particle is 1/3. Therefore, the radius should vary with It is assumed that the total volume (Vxot) of the coarsening phase is invariant with time after the first-order transition (i.e., phase segregation). Therefore, the number of particles is proportional to According to Equation 12.3 then, the number of particles per unit volume, N, is proportional to... 

The overall rate constant, as opposed to the growth exponent, depends directly on such material parameters as the surface tension, the solubility and diffusion constant of gas molecules in the interstitial liquid, the film thickness and the liquid content (40, 41). Note that the dependence with the liquid fraction is not yet completely known (as measuring the coarsening of wet foams, at constant liquid fraction, is very difficult because of drainage). The studies of sound propagation into foams also turned out to be an indirect method to measure the coarsening (51). 

In classical wet soap foams, the growth exponent is typically A 0.33, whereas in tire liquid crysti, A 0.2 was observ both in tiie nematic and smectic phases. This may be explained by the presence of defects at the surface of the bubbles, thus slowing down the coarsening. Such statement is corroborated with the observation that, in the isotropic phase, the foam rapidly mptures. 

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