Why the concentration of solute in the phase in equilibrium with particle will increase as the radius of curvature of the particle decrease?
Answers
If the ripening is controlled by diffusion across the continuous phase, then the cube of the diameter increases linearly with time (α=3) and the ripening rate Ω3 can be derived using the Lifshitz and Slyozov theory [2, 3].
(2)Ω3=64γ Diff S VmRT
where S is the molecular solubility, Vm is the molar volume, Diff is the molecular diffusion coefficient of the dispersed phase in the continuous one, γ is the surface tension of the oil-water interface and R is the molar gas constant. The theory also predicts that the size distribution becomes quite narrow and asymptotically self-similar, in agreement with experiments. In principle, Eq. (2) is valid in the limit of very dilute emulsions. In general, it is expected that emulsions with higher volume fractions of disperse phase will have faster absolute growth rates [10-14] than those predicted by Lifshitz and Slyozov model. The theoretical rate of ripening must therefore be corrected by a factor f(ϕ) that reflects the dependence of the coarsening rate on the dispersed phase volume fraction ϕ [10, 14].
Kabalnov [15] and Taylor [16] found that the presence of ionic micelles in the continuous phase had a surprisingly small effect on the rate of ripening, despite the fact that the solubility of the dispersed phase is largely enhanced. It is argued that due to electrostatic repulsion, ionic micelles cannot absorb oil directly from emulsion drops. In the presence of non-ionic surfactants, much larger increases in the rate of ripening might be expected due to larger solubilization capacities and to the absence of electrostatic repulsion between droplets and micelles. Weiss et al. [17
When the dispersed phase is composed of a binary mixture, the growth may be arrested if one component is almost insoluble in the continuous phase, therefore retaining the soluble one, due to the gradual loss of mixing entropy [18]. The osmotic pressure of the trapped species within the droplets can overcome the Laplace pressure differences that drive the coarsening and ‘osmotically stabilize’ the emulsions. Webster and Cates [19] gave rigorous criteria for osmotic stabilization of monodisperse and polydisperse emulsions in the dilute regime. The same authors [20] have also examined the concentrated regime in which the droplets are strongly deformed and therefore possess a high Laplace pressure. These authors conclude that osmotic stabilization of dense emulsions also requires a pressure of trapped molecules in each droplet that is comparable to the Laplace pressure that the droplets would have if they were spherical, as opposed to the much larger pressures actually present in the system.
Mass transfers in emulsions may be driven not only by differences in droplet curvatures but also by differences in their compositions. This is observed when, e.g. two chemically different oils are emulsified separately and the resulting emulsions are mixed. This phenomenon is called composition ripening [21, 22]. Mass transfer from one emulsion to the other is controlled by the entropy of mixing and proceeds until the compositions of the droplets become identical. The most spectacular evidence of composition ripening comes from the so-called ‘reverse recondensation’ which occurs when the two emulsions differ significantly both in their initial size and in their rate of molecular diffusion. If the larger sized emulsion is composed of the faster diffusing oil, then molecular diffusion occurs in the ‘reverse’ direction, i.e. from large to small droplets [23].
4, 5, 17, 24, 25], i.e.26] have examined the coarsening of concentrated alkane-in-water droplets (ϕ≈20-40%) stabilized by a non-ionic poly(ethoxylated) surfactant, following a temperature quench. Since the rate of ripening does not depend on the alkane chain length, they conclude that the transfer of oil from the smaller drops to the larger ones does not occur by diffusion across the continuous phase but rather through the direct contact of the droplets (permeation). A similar conclusion is drawn by Schmitt et al. [27 w/w). The fact that the two types of instability disappear simultaneously strongly suggests that they possess the same microscopic origin: hole nucleation in the thin liquid films. Due to the activated nature of the process, only the largest holes grow spontaneously and produce coalescence events. Although the smaller holes are evanescent, they allow the diffusion of matter between droplets and therefore they contribute to the permeation, increasing and even dominating the overall mass flux and hence ripening rate. Since the ripening is surface-controlled, the square of the diameter should increase linearly with time. Schmitt et al show that the quadratic scaling (α=2) correctly accounts for the evolution of the diameter as a function of time for emulsions stabilized by nonionic poly(ethoxylated) surfactants.