Coarsening process

For fi/fj in the range 0 to 1 this result is negative. Wy= -AA is therefore positive, and this is what drives the coarsening process. 

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 coarsening process (growth of dominant wavelength) takes place during the approach to equilibrium as well. We carried out simulations verifying this. We started the process with 2 dimensional sine wave initial conditions, and with no incident flux, so that the surface relaxed towards a plane. After a short transient (during which the... 

Other terms in the current, which we did not include, could make the current zero for a given slope nia. This stabilizes the slope of the growing structures around Wo, and explains the phenomena of selected slope. Although the selected (or "magic") slope has been observed in many experiments, it is not necessarily present in every case and is not believed to be important in the coarsening process. 

At this point we are able to compare the coarsening process of the Monte-Carlo simulation with that of the continuum equation, (5). We rescale the surfaces obtained in the simulation to the dimensionless variables Fl(X,T), and compare the time evolution of the free energy associated with the rescaled surface of the simulation (with different parameter values) with the free energy of the continuum equation. As we expect, the free energy (Fig. 4) decreases in time. But it turns out that the... 

An another way of describing the coarsening process is to study the time dependence of the characteristic feature separation n (the lateral size of the mounds, defined as the first zero crossing of the correlation function (/j(0)fe(f))). In most cases fc scales as a power of time,... 

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... 

Aspects of nucleation, growth of a single crystal, volume growth, and coarsening processes are discussed in greater depth in Chapter 4. It will be seen that although some of these processes can be quantified well, it is not possible yet to quantitatively predict the kinetics of many of these processes. 

In a two-phase composite material of isolated spherical particles embedded in a matrix, there is a driving force to transport material from particles enclosed by isotropic surfaces of larger constant mean curvature to particles of smaller constant mean curvature. This coarsening process and the motion of internal interfaces due to curvature are treated in Chapter 15. 

For positive values of the control parameter , stationary, spatially periodic solutions y/s(x) = y/x(x I 2n/q) of (53) may be found with and without forcing. However, in the case of a vanishing forcing amplitude (a = 0) in (53), this equation has a i//-symmetry and one has a pitchfork bifurcation from the trivial solution l/r = 0 to finite amplitude periodic solutions as indicated in Fig. 19. In the unforced case, however, periodic solutions of (53) are unstable for any wave number q against infinitesimal perturbations that induce coarsening processes [114, 121],... 

In Figs. 27 and 28 we also show the dynamics of the characteristic length scales and snapshots for a driving amplitude slightly below the critical one. One can see that at t 103 there is competition between the influence of the forcing and the coarsening process, which finally wins. 

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. 

Two effects attributed to long-chain molecules cause features of the coarsening processes taking place in phase-separating polymer systems ... 

Ternary solutions of immiscible polymers in a low-molecular solvent display wide miscibility gaps. Consequently, they invariably involve demixing above a critical concentration of total polymer by spinodal decomposition and subsequent coarsening processes. When solvent evaporation progresses the enhanced viscosity will slow down the rate of phase separation to a level at which no further phase changes can be observed. 

Conflicting results have been found for the explicit time evolution of the correlation length during isothermal phase separation. A 1/3-power law in the growth of patterns, which is characteristic for the hydrodynamically controlled Lifshitz Slyozov process, was confirmed in Ref. [99] while an exponential increase over a certain period of time was established in Ref. [21]. Nevertheless, it is evidenced that in blends comprising liquid-crystalline polymers spinodal decomposition and subsequent coarsening processes take a course similarly to isotropic liquid mixtures. 

We observed that the phase separation in the 107. aqueous UPC solutions is the spinodal decomposition. The linearized theory of Cahn-Hilliard predicts many of the qualitative features of the SD in the present system, but is not adequate in the quantitative comparison. The phenomenon of SD is non-linear in nature dominated by the late stage of SD. The growth mechanism has been identified to be the coarsening process driven by surface tension. The kinetics of phase separation at the 107. aqueous HPC resembles the behavior of off-critical mixtures. 

Of greater interest to us is the late stage, where the spinodal pattern is described by regions of rather uniform composition separated by interfaces whose width A is much less than the domain size a (see Fig. 9-4d). In this stage, phase separation proceeds by a coarsening process in which the domains get larger while their composition and interfacial width... 

A second diffusive coarsening process is diffusion and coalescence of droplets. That is, the droplets move around by Brownian motion, collide, and occassionally coalesce into larger droplets. This process follows the same slow-diffusion law as evaporation-condensation, namely a t (Vicsek 1989 White and Wiltzius 1995). 

Figure 9.6 Coarsening process during phase separation of a polymer-rich phase (dark) from a solvent-rich phase (light). In the earliest period, image (b), the polymer-rich phase is the majority phase but eventually, in image (d), the majority phase is the solvent-rich one. The elastic stresses in the polymer-rich phase prevent it from...

Another peculiar phenomenon is double phase separation in which each of the two phases formed during spinodal decomposition of a mixture of A and B becomes unstable to a second phase separation, in which droplets of B-rich phase appear in the A-rich domain and droplets of A-rich phase appear within the B-rich domains (Tanaka 1994b). This phenomenon is thought to occur when the capillary coarsening process (in which the domain size grows as a oc t) outruns the diffusion process and the A-rich domains are left with a small excess concentration of B over that allowed at bulk equilibrium. This excess of B cannot diffuse to the interface with the A-rich phase as fast as that interface moves away by capillary coarsening. The excess B therefore nucleates into droplets of B-rich phase within the coarsened A domains. The converse occurs in the A-rich domains. 

Coarsening processes produce microstruotural changes without causing shrinkage... 

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