Temperature Dependence and Foam Diagrams

Talanquer and co-workers [195, 196] have considered the nucleation of bubbles in binary mixtures. Intuitively [197], one would expect that the nucleation rate increased with increasing temperature (as it does, e.g., in a compressible one-component polymer solution), but they have found the inverse behavior for certain parameters [195,196], i.e., a decrease of the nucleation rate with increasing temperature at fixed composition and pressure. Intriguingly, there is no qualitative difference between the structure of critical bubbles with normal and inverse nucleation rate behavior [196]. In Fig. 18 (a) we present the temperature dependence of the nucleation barrier at low pressure pa /ksT = 0.001 and high pressure pa /ksT = 0.16. 

The former corresponds effectively to a one-component compressible polymer solution, while the character of a compressible binary mixture becomes more apparent at higher pressures in the vicinity of the triple line. The composition is held constant, and the temperature is varied. From Fig. 8 (b) we conclude that the composition of the coexisting phases remains almost constant in the temperature interval 0.75 kiTje 0.82 for both pressures. At low pressure, the nucleation barrier decreases monotonously with temperature as expected. At higher pressure, however, the nucleation barrier exhibits a non-monotonous dependence on temperature AG exhibits both a maximum and a minimum upon increasing temperature at fixed molar fraction. The inset compares the radial density distributions of the critical bubbles and planar interfaces at ksT/e = 0.7573. In both cases the solvent density at the center of the bubble is higher than at coexistence and there is an enrichment of solvent at the interface of the bubble. However, there are no qualitative differences in the structure, in agreement with the observation of Talanquer and co-workers [196] for binary Lennard-Jones mixtures. 

To obtain further information, we plot the interfacial tension y and the excess of solvent at the interface between coexisting phases in the bulk in panel (b). At low pressure, y decreases with temperature and there is almost no excess of solvent at the interface. At high pressure, also the interfacial tension exhibits a maximum as a function of temperature and there is a pronounced interfacial excess. In our specific model, this non-monotonous temperature dependence of the interfacial tension can 

It is instructive, however, to set our finding also in context of the phase diagram. In Fig. 19 we plot binodals and spinodals as a function of composition x and temperature T at constant pa /ksT = 0.16. For the composition x = 0.68 the spinodal of the polymer-rich liquid is located at A Fspin/e = 0.675239, i.e., just 2% below the triple temperature. In the SCF calculations the nucleation barrier vanishes at Fspin, 

The binodals of the liquid-vapor phase coexistence as a function of molar fraction and temperature resemble the binodals of a one-component system as a function of density and temperature At high temperature, there is a critical point. Upon decreasing temperature the polymer-rich phase becomes more concentrated in polymer, while the solvent concentration increases in the vapor phase. The spinodal of the polymer liquid, however, exhibits a non-monotonous temperature dependence of the composition. This dependence is parallel to the non-monotonous behavior of the nucleation barrier as we increase temperature. In fact, at the pressure considered, and even more so at lower pressures (cf. Fig. 20), there exists an extended temperature region, where the polymer-fraction at the spinodal of the liquid decreases upon increasing temperature. 

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