Ostwald ripening

Ostwald ripening is a phenomenon observed in solid solutions and liquid sols that involves the change of an inhomogeneous structure over time, in that small crystals or sol particles first dissolve and then redeposit onto larger crystals or sol particles. Dissolution of small crystals or sol particles and the redeposition of the dissolved species on the surfaces of larger crystals or sol particles was first described by Wilhelm Ostwald in 1896. For colloidal systems, Ostwald ripening is also found in water-in-oil emulsions, while flocculation is found in oil-in-water emulsions.

Mechanism This thermodynamically-driven spontaneous process occurs because larger particles are more energetically favored than smaller particles. This stems from the fact that molecules on the surface of a particle are energetically less stable than the ones in the interior. Consider a cubic crystal of atoms: all the atoms inside are bonded to 6 neighbours and are quite stable, but atoms on the surface are only bonded to 5 neighbors or fewer, which makes these surface atoms less stable. Large particles are more energetically favorable since, continuing with this example, more atoms are bonded to 6 neighbors and fewer atoms are at the unfavorable surface. As the system tries to lower its overall energy, molecules on the surface of a small particle (energetically unfavorable, with only 3 or 4 or 5 bonded neighbors) will tend to detach from the particle and diffuse into the solution. Kelvin's equation describes the relationship between the radius of curvature and the chemical potential between the surface and the inner volume:

Δ μ = 2 σ ν a t r {\displaystyle \Delta \mu ={\frac {2\sigma \nu _{\mathrm {at} }}{r}}}

where μ {\displaystyle \mu } corresponds to the chemical potential, σ {\displaystyle \sigma } to the surface tension, ν a t {\displaystyle \nu _{\mathrm {at} }} to the atomic volume and r {\displaystyle r} to the radius of the particle. The chemical potential of an ideal solution can also be expressed as a function of the solute's concentration if liquid and solid phases are in equilibrium.

μ = k B T log ⁡ ( C e q ) {\displaystyle \mu =k_{\mathrm {B} }T\log(C_{\mathrm {eq} })}

where k B {\displaystyle k_{\mathrm {B} }} corresponds to the Boltzmann constant, T {\displaystyle T} to the temperature and C e q {\displaystyle C_{\mathrm {eq} }} to the solute concentration in a solution in which the solid and the liquid phase are in equilibrium. Combining both expressions the following equation is obtained:

k B T log ⁡ ( C e q C ∞ ) = 2 σ ν a t r → C e q ( r ) = C e q ( ∞ ) e 2 σ ν a t r k B T {\displaystyle k_{\mathrm {B} }T\log \left({\frac {C_{\mathrm {eq} }}{C_{\infty }}}\right)={\frac {2\sigma \nu _{\mathrm {at} }}{r}}\rightarrow C_{\mathrm {eq} }(r)=C_{\mathrm {eq} }(\infty )\mathrm {e} ^{\frac {2\sigma \nu _{\mathrm {at} }}{rk_{\mathrm {B} }T}}}

Thus, the equilibrium concentration, C e q {\displaystyle C_{eq}} , is lower around bigger particles than it is around smaller particles.

C e q ( r ) > C e q ( R ) {\displaystyle C_{\mathrm {eq} }(r)>C_{\mathrm {eq} }(R)}

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