## Ostwald Ripening

### Quick Start

Life for a particle or colloid is unfair. You start with a mixture of smaller and larger particles and instead of them exchanging molecules to become an overall average size, the small particles get rapidly smaller and disappear, while the larger particles get bigger. This is so unintuitive that it causes much puzzlement to those who find lots of big particles in a dispersion they were sure had earlier contained lots of small ones.

The effect is driven by a large surface energy, γ, a large diffusion coefficient, D, a large solubility in the medium, c; and smaller radii r0 at time t0 are affected more acutely than larger ones. As you play with the settings over your desired time-scale, see how quickly the radius at time t, rt, increases at the start, even if the rate soon slows down.

### Ostwald Ripening

γ dyne/cm
D 10-6cm²/s
c mole %
V cc/mole
r0 nm
tmax hr

Ostwald ripening shows the unfairness of physics. The bigger particles grow at the expense of the smaller ones. The driving force is the interfacial tension γ; the higher it is the more energy it requires to create a small radius (high curvature) droplet. The calculation is of rt (actually the average of rt), the radius at time t, which depends on:

• tmax - the maximum time, up to 48hr for the ripening calculation
• r0 - the average particle radius at t=0
• γ - the interfacial energy. A high particle surface-to-solvent interfacial energy increases the driving force
• D - the diffusion coefficient of the solute molecule through the medium
• V - the molar volume of the solute molecule (if you don't know this, just enter the MWt instead). Up to Nov 2022 the formula used V but now uses V².
• RT - the gas constant R and temperature T in °K, assumed to be 300.
• c - the Mole Fraction solubility of the solute in the solvent (water). Solutes with high solubility speed up Ostwald ripening.

rt³-r0³ = 8γDcV^2t/(9RT)

There are four lessons about Ostwald ripening.

1. Small particles ripen more quickly so nanoparticles of soluble solutes may quickly become microparticles
2. A low interfacial energy stops Ostwald ripening (the driving force decreases) so choose a surface modifier (dispersant molecule) that brings the particle's interfacial energy close to zero.
3. D doesn't change much across typical solutes, so the only way to decrease it is to increase the viscosity of the solution.
4. The theory applies to smooth spheres. If you have rough particles then the effective r0 is much lower and so "spikey" parts of the solute will move first.