In Ostwald ripening, the higher interfacial chemical potential of small droplets drives diffusion toward larger droplets, increasing the characteristic domain size. The Gibbs--Thomson relation supplies the curvature-dependent surface value, while conserved order-parameter dynamics controls transport through the surrounding phase.
For a three-dimensional droplet with concentration jump , constant mobility , and quasi-static exterior chemical potential,The first formula solves the exterior Laplace equation. The second follows from conservation: the total outward flux removes excess composition at the rate . For a symmetric mixture, and the Gibbs--Thomson relation gives .
For the geometrical diffusion-capacitance analogy approximation , a mass law and constant ice supersaturation , the growth law gives . Integration yields the displayed time. Comparing to a liquid sphere must use equal mass rather than equal radius; evolving supersaturation, ventilation and surface attachment invalidate the constant-coefficient estimate.
For an isolated growing sphere, a quasi-steady exterior diffusion equation, constant surface concentration , and a maintained far-field concentration give . A sharp moving boundary with interior concentration then obeys . The displayed integrated law assumes all concentrations and the diffusion coefficient are constant. It becomes when the seed radius is negligible. This reservoir-fed law differs from curvature-controlled Ostwald ripening in a depleting many-particle system.
When attachment is much slower than exterior diffusion, the surface concentration stays close to the reservoir value. An attachment coefficient with units of velocity gives nearly constant radial growth in the dilute-feed approximation. For a sphere the ratio distinguishes attachment control from transport control. A finite attachment rate leads to a Robin boundary condition rather than a fixed equilibrium surface concentration.
A quasi-static chemical potential outside a fixed-shape droplet solves the same Laplace equation as an electrostatic potential outside an equipotential conductor. If on the droplet and zero on the surrounding reservoirs, its outward diffusive current iswhere is the corresponding electrostatic capacitance in a medium of permittivity . The arbitrary cancels against the capacitance's proportionality to . Thus electrostatic geometry directly determines evaporation rates.
For a spherical droplet at fixed center height above a flat equilibrium reservoir, the diffusion-capacitance analogy and sphere-plane capacitance giveAt large separation, , soThe nearby reservoir increases the total flux and shortens the lifetime. The spherical-shape assumption controls the geometry; the local current density is not uniform over the surface.
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Ostwald ripening is a phenomenon that occurs in solid dispersions, emulsions, and other colloidal systems, where larger particles grow at the expense of smaller ones over time. This process is driven by differences in solubility and chemical potential between particles of different sizes. In a dispersed system, smaller particles tend to have a higher curvature (meaning they have a higher surface area relative to their volume) compared to larger particles.