Critical Fluctuations and Spinodal Decomposition

This appearance of a peak which grows in intensity, initially at a fixed position and then shifting to lower scattering angles, can in fact be considered 

We consider here the concentration fluctuations in the homogeneous phase and also the manner, in which these are reflected in measured scattering functions. 

g stands for the free energy density of the mixture 

The equation can be further simplified, if a linearization approximation is used. Clearly the state with a uniform concentration, 

This is a useful result. It relates the Gibbs free energy of a given fluctuation state to two parameters only, namely the integral or mean values of 6(j)) and (V ( )2. 

We see that the situation has now become more involved. As we shall learn in the next section, rather than follows from an experimental determination of the spinodal. 

This appearance of a peak which grows in intensity, initially at a fixed position and then shifting to lower scattering angles, can in fact be considered as indicative of a spinodal decomposition. One can say that the peak reflects the occmrence of wave-hke modulations of the local blend composition, with a dominance of particular wavelengths. Furthermore, the intensity increase indicates a continuous amplitude growth. This, indeed, is exactly the process sketched at the bottom of Fig. 4.13. 

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Critical Fluctuations and Spinodal Decomposition 135 For the gradient term we obtain in similar manner... 

The interfacial tension between demixed liquid phases is one of the important physical properties of polymer solutions and polymer mixtures. In particular, the interfacial tension near critical point is closely related to concentration fluctuations associated with critical opalescence and spinodal decomposition. Recently, some theoretical treatments have been presented for the interfacial tension of demixed polymer solutions and polymer mixtures. For poly-... 

Finally, we will briefly discuss the properties of polymer blends under shear flow. In small molecule mixtures, shear flow is known to produce an anisotropy of critical fluctuations and anisotropic spinodal decomposition [244, 245], In polymer mixtures, the shear has the additional effect of orienting and stretching the coils, thus making the single-chain structure factor anisotropic. In the framework of the Rouse model these effects have been incorporated into the RPA description of polymer blends [246, 247]. Assuming a velocity field v = yyex, where x, y, z are cartesian coordinates, y the shear rate, and ex is a unit vector in x direction, the single chain structure factor becomes [246, 247]... 

Demixing, Fig. 2 (a) Characteristic lengths (a) and (b) nucleation barrier AF plotted versus concentration cb. Full curves show the predictions of the Cahn-Hillard mean-field theory of nucleation and spinodal decomposition for the critical wavelength and the correlation length of concentration fluctuations in a metastable... 

Within the spinodal line, any small fluctuation in composition will lead to a lowering of the free energy under these conditions phase separation will proceed immediately via a mechanism of amplification of random composition fluctuations called spinodal decomposition (Binder 1991). In the metastable part of the phase diagram a small composition fluctuation actually raises the free energy and, in order to begin the phase-separation process, a droplet of the minority phase, of a size greater than a critical size, has to be nucleated. Thus this mechanism of phase separation is known as nucleation and growth. 

When a" > 0, the amplification factor is negative, and density fluctuations are damped. When a" < 0, density fluctuations exceeding a critical wavelength grow, and spinodal decomposition occurs. This critical wavelength is given by (90). The amplification factor exhibits a sharp maximum at a wavenumber, B, which can be obtained from (97) by differentiation. 

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