Spinodal length scale

Therefore, the locus of the values ( ) with a vanishing second derivative of A delimits the region of the miscibility gap in which spinodal decomposition occurs. This locus is referred to as the spinodal (figure C2.1.10 (bl). The length scale of the concentration fluctuations at the beginning of the separation process is controlled by... 

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

FIGURE 8.9 Spinodal decomposition process modeled using the Cahn-Hilliard equation. Time and length scales are dimensionless. Number of time steps following the quench are (a) 10,000, (b) 40,000, (c) 160,000, and (d) 640,000. Courtesy of Dr. Nigel Clarke (University of Durham, U.K.). 

Fig. 2a), R oc [csp(r) c] However, there is no clear evidence for such a divergent length scale when one carries out quenches into the region of the phase diagram where the spinodal is expected to occur [3]. Rather, there is compelling evidence from both experiments and computer simulations that the transition from nucleation to spinodal decomposition is completely gradual [3] (Fig. 2a). 

In summary, the predictions of analytic PRISM theory [67] for the phase behavior of asymmetric thread polymer Uends display a ly rich dependence on the single chain structural asymn try variables, the interchain attractive potential asymmetries, the ratio of attractive and repulsi interaction potential length scales, a/d, and the thermodynamic state variaUes t) and < ). Moreover, these dependences are intimately coupled, which mathematically arises within the compressible PRISM theory from cross terms between the repulsive (athermal) and attractive potential contributions to the k = 0 direct correlations in the spinodal condition of Eq. (6.6). The nonuniversality and nonadditivity of the consequences of molecular structural and interaction potential asymmetries on phase stability can be viewed as a virtue in the sense that a great variety of phase behaviors are possible by rational chemical structure modification. Finally, the relationship between the analytic thread model predictions and numerical PRISM calculations for more realistic nonzero hard core diameter models remains to be fully established, but preliminary results suggest the thread model predictions are qualitatively reliable for thermal demixing [72,85]. 

The collecti partial structure factors, MM-(k), are easily written down, and in an approximate theory finite length scale spinodal instabilites may be present corresponding to MM (k = k ) = co which is equivalent to [86] ... 

As indicated above the phase separation process can also be studied using optical microscopy as has been done by Verhaegh et al. [96]. They observed spinodal decomposition in a well-defined model colloid-polymer mixture of silica spheres in cyclohexane with dissolved polydimethyl siloxane chains of qr ss 1, and found similar results to those plotted in Fig. 4.16. Their findings agree with the picture sketched above and the characteristic length scale A follows the regimes of (4.41) and (4.42). 

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