Wavelength of spinodal decomposition

The wavelength of spinodal decomposition can be determined from the wavenumber at which the maximum in the scattering occurs. This wavelength. A, is given by... 

Wavelength Takayanagi model parameter Optimal wavelength of spinodal decomposition Chemical potential... 

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... 

Microstructural characteristics of spinodal decomposition are periodicity and alignment. Periodicity arises from wavelengths associated with the fastest-growing initial mode. At later times, the characteristic periodic length increases due to microstructural coarsening. Periodicity can be detected by diffraction experiments. 

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. 

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... 

The theory of spinodal decomposition was first examined by Cahn [164]. It predicts the exponential growth of sinusoidal composition modulations at a fixed wavelength A. The size of each phase may be given by A written as [165],... 

Due to the restriction Eq. (48) which leads to q2a2 o) — 1 < 1. In this limit, the characteristic wavelength kc is much larger than the gyration radii of the polymer coils, i.e. in this regime the behavior of the polymer mixture is qualitatively the same as that of a mixture of small molecules. Using the full q-dependence of A(q) and Scon(q) in Eq. (77) this linearized theory of spinodal decomposition can be extended to deep quenches as well [78]. Here we quote the result for symmetrical mixtures (NA = NB = N, cta = crB = a) only. Then [78]... 

The same fact is true for surface effects on the dynamics of spinodal decomposition in polymer blends [374,375], dynamics of surface enrichment in blends [367, 375, 376], and, last but not least, for surface effects on block copolymers there one may have surface - induced ordering [377, 379] and interesting competition effects between the lamellar ordering (of wavelength X) and film thickness D in thin block copolymer films [380—388]. These phenomena are outside of our consideration here. 

The diffusion coefficient in this polymer-polymer system can be roughly estimated from the rate of spinodal decomposition. To make the estimate, particle size change during spinodal decomposition will be ignored, and average particle size will be used. If the most rapidly growing wavelength from Equation 7 is substituted into Equation 6, then ... 

Figure 7. Time dependence of the wavelength of the fastest growing density fluctuation (Affiax) during a molecular dynamics simulation of isothermal liquid-vapor spinodal decomposition in the three-dimensional Lennard-Jones fluid kT/e = 0.8 p

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