Solid nuclei and liquid droplets: A parallel treatment for 3 phase systems

Abstract

For solid phase self assembly into crystals or large diameter polymers, the presence of a liquid-liquid demixing transition has been known to have an accelerating effect on the nucleation process. We present a novel approach to the description of accelerated nucleation in which the formation of solid phase aggregates and liquid-like aggregates compete as parallel pathways to formation of dense phases. The central idea is that the small aggregates that would ultimately form the liquid phase are sufficiently labile to sample the configurations that would form the solid, so that the growing cluster begins as a liquid, and switches into growth as a solid when the aggregates have equal free energies. This can accelerate the reaction even when the liquid-demixed state is thermodynamically unfavorable. The rate-limiting barrier is therefore the energy at which there is a transition between liquid and solid, and the effective nucleus size is then concentration independent, even though for both nucleated demixing and nucleated crystallization, the nucleus size does depend on concentration. These ideas can be expressed in a chemical potential formalism that has been successfully used in nucleation of sickle hemoglobin, but not to our knowledge previously employed in describing LLD processes. The method is illustrated by considering existing data on Lysozyme.

Keywords: kinetics; liquid-liquid demixing; nucleation; protein assembly.

Figure 1

Energetic barrier to association, as a function of aggregate size. The energy is the chemical potential of an aggregate of the given size, relative to the monomers. The net cost in energy for small aggregates creates an effective barrier. The aggregate size at which the barrier peaks is the nucleus. The energy turning point is achieved when the cost of removing an additional monomer from solution and adding it to the aggregate is less than the energetic gain from the joining the aggregate. The curves can become lower either because the cost is less (high concentration of monomer) or because the gain is greater (strong intermolecular contacts for example). As the height decreases, the size of the nucleus also gets smaller until ultimately the barrier vanishes and the monomers are immediately unstable relative to any size aggregate. A dashed line drawn at ΔG = kT. A barrier of that height becomes insignificant against thermal fluctuations, and is the cloud point for LLD.

Figure 2

Three possible types of interaction between solid nucleation (solid curve) and a coexisting liquid aggregates (dashed lines). The liquid curve may become isoenergetic with the solid curve before the nucleus (a), and then the transformation of liquid droplets into solid nuclei produces no increment in rate. When the liquid curve crosses the solid nucleus curve at higher aggregates than the nucleus as in (b), a growing droplet that transforms into a solid will not need to surmount as high a barrier as the solid does. This can happen even when the liquid phase is still unstable. The point at which the curves cross then functions as the nucleus. A third possibility (c) is that the liquid barrier is enclosed in the solid curve, meaning that liquid droplet nucleation controls the reaction. In the case shown, the droplets would transform to solids at a point at which they had become more stable than monomers. In all three cases, however, the final structures are equivalent.

Figure 3

Log (base 10) of nucleation rate (number mL⁻¹ min⁻¹) of lysozyme crystals as a function of log (base 10) of supersaturation, defined as the activity of the monomers divided by the activity at solubility. Data is from Bhamidi et al.10 for 4% NaCl, pH 4.5, at 16°C, but the supersaturation relation allows us to treat the case of 13°C by using the solubility at that point. The solid line is the best linear fit to the upper region. The dashed line extrapolates the fit. In the theory for simple nucleation processes, linear behavior in this plot is not expected, in contrast to the straight line behavior expected when a crossing process acts as a nucleus. At lower supersaturation, as shown in Figure 4(a), the crossover pathway and the straightforward nucleation pathway have similar rates, suggesting that both pathways might be active, and that the net rate would exceed the extrapolated value from the high concentration, similar to what is observed.

Figure 4

Free energy barriers deduced for lysozyme at 13°C. The liquid clusters grow almost linearly, while the solid clusters display a clear maximum. Panel (a) is for a supersaturation of 14, which corresponds to c = 1.17 mM at 13°C. Panel (b) is for supersaturation of 29 which corresponds to c = 2.55 mM at 13°C. In panel (b), the intersection of the free energies effectively lowers the barrier for crystal formation by 1.7 kT (or 0.57 kcal/mol). This would speed up the rate by 5.5 times. In panel (a), the intersection is almost at the peak of the crystal nucleation barrier, and the acceleration of crystal formation is only around 22%, making both nucleation and growth via liquid intermediate almost equally likely.