Stages of spinodal decomposition early

The results for the glass crystallization of PET annealed at 80 °C as before are shown in Fig. 8. In the early stage of spinodal decomposition up to 20 min, the characteristic wavelength A remains constant at a value of 15 nm, which agrees with the theoretical expectation that only the amplitude of density fluctuations increases whilst keeping a constant characteristic wavelength. In the late stage from 20 to 100 min it increases up to 21 nm just before crystallization. Such a time dependence of A in nm can be represented by... 

The characteristic length scale in the early stage of spinodal decomposition will correspond approximately to this wavelength.8... 

K.B. Rundman and J.E. Hilliard. Early stages of spinodal decomposition in an aluminum-zinc alloy. Acta Metall, 15(6) 1025-1033, 1967. 

The validity of the linear theory observed for the early stage of spinodal decomposition is chiefly related to the large size of the chain molecules. As shown above, characteristic quantities as the time t or the wavelength Am(0) of the fastest growing fluctuation are proportional to Ro and Rg, respectively. Furthermore, the Landau-Ginzburg criterion (cf. condition 2)) ensures that the mean-field regime is sufficiently extended. 

Results are shown in Figs. 12 and 13. All blend specimens were set iso-thermally above LCST and kept there for a maximum of 5 min. As will be seen, this corresponds only in some cases to an early stage of spinodal decomposition depending on temperature. The diffusion coefficients governing the dynamics of phase dissolution below LCST are in the order of 10"14 cm2 s"1. Figure 12 reflects the influence of the mobility coefficient on the phase dissolution. As can be seen, the apparent diffusion coefficient increases with increasing temperature of phase dissolution which expresses primarily the temperature dependence of the mobility coefficient. Furthermore, it becomes evident that the mobility obeys an Arrhenius-type equation. Similar results have been reported for phase dis-... 

In the early stage of spinodal decomposition, varies with phase separation time, therefore it cannot be scaled by a single length parameter (t). It is necessary to normalize the structure function by the invariant function, i.e.. 

Generally, the spinodal decomposition of polymer mixtures is classified into three stages, each of which is called early, intermediate, and late stage, respectively [50]. In the early stage of spinodal decomposition, whose dynamics can be well described by the linearized theory [74], the amplitude of the fluctuations exponentially increases with time without any variation in the wavelength of the fluctuations. The phase separation up to 5000 Monte Carlo (MC) steps in Fig. 9c corresponds to this stage. With increasing amplitude the linear approximation... 

Interpretation of the phase separation fluorescence results requires that accurate values of the equilibrium binodal compositions for a particular temperature be available. These are necessary in order to calculate the volume fraction of the rich phase in the two component system. This volume fraction is assumed to be constant during the early stages of spinodal decomposition. 

Two sets of experiments on phase separation kinetics have been performed. The first series of experiments was designed to demonstrate the feasibility of using excimer fluorescence to test Cahn s kinetic treatment of the early stages of spinodal decomposition. 

In the early stage of spinodal decomposition, remains close to the average concentration 4>o of the initial solution. If we neglect the terms higher than first order in concentration fluctuation u defined by... 

The early stage of spinodal decomposition corresponds to the special case of eq 2.16 in which the non-linear sum term is negelcted, i.e.,... 

Equation 2.20 predicts for the early stage of spinodal decomposition that S k, t) at any fixed k smaller than k increases exponentially, where k has been defined in Section 2.4.2, and that S k, t) grows with time most rapidly atk = where fcm is given by eq 2.11. The S k, t) vs. k relation may show a maximum, but the position of the maximum depends on So(k) and 5t, i.e., it is not always found at k = km and moves with time. Finally, substitution of eq 2.20 leads to... 

Snyder et al. [23] were among the earlier workers who studied the early stage of spinodal decomposition of polymer blends, with the distinct aim of testing the Cahn-Hilliard theory. They measured /total as a function of time for three PS/PVME blends and found that ln(/total ) increased linearly with time t over a certain range of early time. If R k) defined by eq 2.8 has a sharp maximum at fc = Fni> eq 2.7 may be approximated by u x,t) e.xp (//(fcni)t), so that exp (2/ (fcm)t). Therefore, the initial slope of ln(/totai ) vs. t can be equated to R(km)- Snyder et al. used this idea to analyze their data (the same idea had already been used by Nishi et al. [11]). However, we have to note that the peak of R k) given by eq 2.8 is not so sharp as to justify the approximation used. 

An optimized bi-continuous periodic structure occurs at the early stage of spinodal decomposition. The small domains coalescence with each other at the later stage, in order to minimize the total interfacial area and thus the total free energy of the system. The structural evolution at the later stage is called Ostwald ripening (Ostwald 1896). According to the Porod law. 

What kinds of different morphological features occur at the early stages of spinodal decomposition and nucleation ... 

In the following, we restrict our attention to the early stages of spinodal decomposition. In the analysis of experiments one often uses the Landau-de Gennes functional (Eq. 96) which results in the Cahn-HilUard-Cook theory (105] for the early stages of phase separation. This treatment predicts that Fourier modes of the composition independently evolve and increase exponentially in time with a wavevector-dependent rate, 4>A(q, t) exp[it(q)fj. Therefore, it is beneficial to expand the spatial dependence of the composition in our dynamic SCF or EP calculations in a Fourier basis of plane waves. As the linearized theory suggests a decoupling of the Fourier modes at early stages, we can describe our system by a rather small number of Fourier modes. 

Figure 1.4.40 also indicates that the interdomain distance during the early stages of spinodal decomposition is around 1 p,m. Relevant binodal compositions for this system at the operating conditions were mentioned in Section 1.4.2.2. 

It is evident from these figures that coarsening occurs even at the early stages of spinodal decomposition in the reactive system. Figure 1.5.4 shows their corresponding composition profiles at indicated phase separation time values. 

For deep quench systems, and for the early stage of spinodal decomposition, we get the following solution to Eq. (A.IO) ... 

Reduction of Eq. (A. 10) to binary and one-dimensional systems for composition-dependent M and D is found elsewhere (De Fontaine, 1967 Caneba and Soong, 1985). The solution procedure involves expressing both diffusivity and mobility in truncated polynomial form up to the quartic term. Results of all calculations in binary polymer/solvent systems indicate that the interdomain distance and domain size progressively increase with time. Thus, interdomain distances developed in the early stages of spinodal decomposition are the smallest achievable in the system. 

The Eq. (A.9) describes early-stage as well as late-stage behavior of a spinodal decomposition process in any coordinate system. We assume that Ky independent of concentration. For simplicity, let us consider the case of early stages of spinodal decomposition. Also, both diffusion and mobility coefficients are assumed to be constant at the average composition of the system. We get... 

The scale of phase separation for polymer blends at the early stage of spinodal decomposition is very small and is of the order of a few hundreds or thousands of angstroms and is typical of the morphology found in many blended systems. 

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