On the composition of nuclei in binary systems

Abstract

It is shown that some past thermodynamic treatments of the so-call ’’classical’’ nucleus for nucleation in a liquid–liquid miscibility gap have neglected certain important terms. These occur in the chemical potential and are associated with the dependence of interfacial tension on composition. A careful thermodynamic development is presented which yields the more complete result. In 1959 Cahn and Hilliard (CH) presented a thermodynamic theory for the nucleus based on the use of ’’gradient’’ terms in the interfacial transition zone in place of the conventional interfacial tension. The properties of their nucleus differed appreciably from those of what they termed the ’’classical’’ nucleus in connection with which more conventional interfacial concepts are used. For example, at the spinodal, the CH nucleus has no free energy of formation and is of infinite size with a composition infinitesimally different from that of the parent solution while the ’’classical’’ nucleus requires finite free energy, has finite size, and a composition close to that of one branch of the coexistence curve. There are many other differences. However, when the ’’classical’’ nucleus is subjected to the complete thermodynamic treatment it assumes all of the features of the CH nucleus, and proves to be essentially equivalent. Since the CH method and the classical have certain complementary features, one or another is the most useful depending on the problem involved. Both Heady and Cahn (HC) and Sundquist and Oriani (SO) have performed experiments which indicate that the conventional theory of nucleation fails for supercoolings within a miscibility gap. Although the analysis in this paper indicates that their estimates of the thermodynamic properties of the nucleus, from experimental data, via the so-called ’’tangent rule’’ may be incomplete, the more precise free energy barrier is found to be even smaller than their estimate, thus adding strong support to their claim that the conventional theory fails.