Phase changes in matter generally occur by building up from small nuclei of the new phase. Scattering experiments and computer simulations reveal the characteristic size of the smallest of these nuclei.
Rain and snowfall, petroleum refining and the purification of pharmaceuticals by crystallization are examples of natural and industrial processes that involve a phase transition. Understanding the molecular mechanisms underlying these ubiquitous events is therefore of widespread interest. Writing in Journal of Physical Chemistry B, Albert C. Pan and colleagues1 suggest a novel way of measuring the characteristic size of the smallest building-block of a new phase that can grow spontaneously, and show how this quantity changes across thermodynamic conditions ranging from metastability to instability.
In phase transitions such as crystallization and boiling, the two phases involved have different densities (ice floats on water) and entropies (heat must be added to water to make it boil). Such transitions are known as first-order transitions, and they require that the starting, or mother phase, have a higher free energy (energy available to do work) than the new phase at the same temperature and pressure. In this situation, the mother phase — for example, liquid water below freezing — will give way to the new phase, ice. The mother phase is said to be metastable with respect to the new phase, and the degree of departure of the system from equilibrium conditions is referred to as its supersaturation.
First-order phase transitions occur through a nucleation mechanism2, in which a molecular aggregate of the new phase begins to form within the mother phase. The lower free energy of the new phase favours nucleus formation, but the process is hindered by the energetic cost of forming an interface between the nucleus and the mother phase. The driving force for nucleus formation is proportional to the volume of the nucleus, whereas the cost of forming an interface is proportional to its surface area. The former will thus balance the latter only for a sufficiently large nucleus, and nuclei larger than a critical size grow spontaneously, whereas those smaller than this threshold dissolve back into the mother phase. This picture of a nucleus as a localized and rare fluctuation underlies classical nucleation theory2, and is an adequate model at small supersaturations. Under these conditions, the nucleation time is long relative to the structural relaxation time, and the nucleation process can be treated by equilibrium (thermo- dynamic) considerations involving the work needed to form the critical nucleus.
Pan et al.1 use structure factors that they obtain in neutron-scattering experiments on phase-separating polyolefin blends to determine the length scale associated with the critical nucleus, a procedure developed originally by co-author Nitash Balsara and his group3,4. The structure factor, S(q), is the intensity distribution, as a function of the scattering vector, q, of probing neutrons that have been deflected off molecules in the mixture5. This scattering vector is given by q=(4π/λ)sin(θ/2), where θ is the scattering angle and l the wavelength of the incident neutrons. The growth of structures with a characteristic size l is associated with an increase in scattering, and so the structure factor, at q∼1/l.
Structure-factor curves obtained at different times during a nucleation process merge at a critical scattering vector qc (Fig. 1). In other words, for scattering vectors larger than qc, the scattering intensity is independent of time, and structures of characteristic size smaller than 1/qc do not grow in time. The merging of curves at qc is thus a signature of the size of the critical nucleus3,4. The fact that the characteristic size of critical nuclei can be determined from the structure factor is surprising: it suggests that the fluctuations that give rise to these nuclei resemble ordinary concentration fluctuations more than the localized, high-intensity fluctuations envisioned in classical nucleation theory.
The authors also calculate the structure factor through computer simulation of a model phase-separating binary mixture. The measured and computed structure factors are qualitatively very similar. Furthermore, the critical nucleus size obtained from the computed structure factor agrees with that obtained independently by calculating the free-energy cost of forming nuclei of different sizes, the maximum of such a free-energy curve corresponding to the critical nucleus. The results provide a direct test of the relationship between the critical nucleus size and the intensity distribution of scattered radiation, and so point to a new way of measuring the size of a critical nucleus. As this quantity is difficult to obtain experimentally, this is an important development. The structure factor by itself, however, does not yield direct information on the number of molecules in the critical nucleus.
At large supersaturations, the interface between nucleus and mother phase becomes progressively diffuse. In this situation, theory focuses on the free-energy cost of establishing a non-sharp interface6,7. According to the classical theory, the size of the critical nucleus decreases steadily with increasing supersaturation, produced using different values of temperature or pressure, and remains finite at the ‘spinodal’ point where the mother phase becomes thermodynamically unstable. In contrast to phase transitions that originate from a metastable phase, transitions from an unstable mother phase (‘spinodal decomposition’6) occur spontaneously, bypassing the need for critical nuclei to form. Theories that approximate interfacial thermodynamics by neglecting density or concentration fluctuations predict a sharp transition from nucleation to spinodal decomposition and a diverging critical nucleus at the spinodal6. Scaling arguments, in contrast, predict a smooth transition from metastability to instability8. In agreement with recent experiments on concentrated protein solutions9, Pan et al.1 find that the size of the critical nucleus decreases smoothly with supersaturation and remains finite at the spinodal. They see no evidence of a diverging nucleus size.
An important open question is the geom-etry of the critical nucleus. Computer simulations10 indicate that, at large supersaturations, nuclei send out offshoots to become ramified, fractal objects. But the definition of the critical nucleus size used by Pan et al. in their simulations assumes a compact object, and it would be interesting to account for the shape of the critical nucleus in future calculations.
Although a qualitative picture of the transition from nucleation to spinodal decomposition exists7, this has yet to result in a predictive theory of nucleation at large supersaturations. Because of the ambiguity associated with the very notion of a critical nucleus under these conditions7, such a theory will probably require kinetic arguments (based on the rates of growth and decay of embryonic nuclei)11, rather than thermodynamic arguments based on the free-energy cost of forming a critical nucleus. The work of Pan et al. provides microscopic insight on the smallest building-block of a new phase, an important ingredient for future theoretical descriptions.
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Applied Physics A (2007)