Big grains eat small ones — LSW t^(1/3) coarsening
A two-phase mixture is never truly at equilibrium while many small particles remain: their curved interfaces store surface energy. Over time the dispersed phase coarsens — large droplets grow while small ones dissolve and vanish — so the system reduces its total interfacial area. This simulation shows that classic process, tracking the size distribution and the famous t^(1/3) growth of the mean radius.
c_eq(r) = c_inf · exp(2γΩ / rRT) ≈ c_inf (1 + l_c/r) — Gibbs-Thomson solubility.
dr/dt ∝ (D/r)(1/r* − 1/r) — diffusion-limited growth vs critical radius r*.
<r>³ − <r₀>³ = K·t ⇒ <r> ~ t^(1/3) — LSW scaling law.
The same physics that makes large salt crystals grow at the expense of fine ones also ruins old ice cream: water recrystallises into ever-larger ice grains, giving that gritty texture. Ostwald ripening is why so many microstructures slowly coarsen — and why fast quenching is used to lock in fine, strong ones.
Ostwald ripening is a coarsening process in a two-phase mixture where larger particles, droplets or grains grow over time at the expense of smaller ones. The total amount of the dispersed phase stays roughly constant while the number of particles drops and the average size increases, because the system lowers its total interfacial (surface) energy.
The Gibbs-Thomson effect makes the local equilibrium solubility at a curved interface higher for smaller radii. Small particles therefore sit in a solution that is supersaturated relative to large particles. Solute diffuses down this concentration gradient from small particles to large ones, so small particles shrink and dissolve while large ones grow.
Lifshitz, Slyozov and Wagner (LSW) theory predicts that for diffusion-limited coarsening the cube of the mean radius grows linearly with time: <r>³ − <r₀>³ = K·t, so the mean radius increases as <r> ~ t^(1/3). On a log-log plot of mean radius versus time the slope approaches 1/3.
It plots the mean particle radius against simulated time on logarithmic axes. A reference line of slope 1/3 is drawn so you can compare. As the field coarsens the measured curve settles toward the same slope, demonstrating the LSW power law emerging from many individual particle interactions.
Yes, to a good approximation. Mass (volume) of the dispersed phase is conserved during ripening; it is only redistributed from small particles to large ones. The simulation conserves total area: whatever a shrinking particle loses is transferred to growing neighbours, so the area histogram shifts to larger sizes without creating or destroying material.
At any instant there is a critical radius equal to the current mean radius. Particles larger than critical grow, particles smaller than critical shrink. As the mean radius itself increases with time, the critical radius drifts upward, so particles that were once growing can eventually fall below the moving threshold and start to dissolve.
Temperature controls the diffusion coefficient, which sets the rate constant K. Higher temperature means faster diffusion and quicker coarsening, but it does not change the t^(1/3) exponent. In the simulation the temperature or diffusion slider rescales how fast the mean radius climbs while keeping the same scaling law.
It governs grain coarsening in alloys, precipitate growth that weakens age-hardened metals, sintering of ceramics, recrystallisation of ice cream and the growth of large salt or sugar crystals over time. It also drives the destabilisation of emulsions and foams and the growth of nanoparticles during synthesis.
In Ostwald ripening particles never touch; material moves by diffusion of dissolved solute through the matrix. In coalescence two particles physically merge on contact. Both coarsen the microstructure, but they give different size-distribution shapes and different growth exponents, so distinguishing them helps identify the dominant mechanism.
Because the driving force is interfacial curvature. As particles grow, curvatures decrease and concentration differences between particles shrink, so diffusive fluxes weaken. The t^(1/3) law captures this slowdown: equal absolute growth takes ever longer as the structure coarsens, which is why microstructures can remain metastable for long times.