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# Ostwald Ripening

## Introduction

Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.

## Thermodynamics

Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, $\mu^{\mathrm{E}} = \mu^{\mathrm{I}}$. For an ideal gas,

$\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,$

where $\mathrm{V}_m$ is the molar volume. Since $\mu^{\mathrm{E}} = \mu^{\mathrm{I}}$,

$\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}$

and $\mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}}$. From the ideal gas law, $\mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}}$ and assuming $\mathrm{V}_m^{\mathrm{I}}$ to be independent of $p^{\mathrm{I}}$,

$\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.$

Also, $\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},$

where $\gamma$ is the interfacial tension. This can be easily derived from $\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0$ for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size.

$S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}},$

where $\alpha = 2 \gamma \mathrm{V}_m /R T$ defines a characteristic length scale. For most systems, $\alpha \approx 10^{-7}$ cm (Kabalnov, 1992).

## Kinetics

While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on a short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets.

When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as

$\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,$

with D the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of $\tau = r^3/\omega$.

Lifshitz-Slezov theory assumes that the rate-limiting step is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with $S_M$, $S_E$ the solubilities and $R_M$, $R_E$ the diffusion resistances in the membrane and external phases, respectively, then

$\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).$

Here, $R_M = 1/4 \pi r D_E$ and $R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}$, with $\delta$ the membrane thickness and $C$ the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to $r^2$ instead of $r^3$. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999).

Experiments verify that under certain conditions, $r^3$ grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of $\omega$. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).

In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. $r_{\mathrm{L}} \rightarrow \infty$), ripening stops when the Laplace pressure $\Delta p_{\mathrm{L}}$ in the small droplets is equal to the difference in osmotic pressure, yielding

$\Delta c = \frac{2 \gamma}{\mathrm{R T} r},$

with $\delta c$ the concentration difference between droplets (Norde, 2003).

If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically,

$\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma.$

When $\mathrm{d} \Delta p_{\mathrm{L}} = 0$ ripening stops, therefore $\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}$ and for spheres $2 \mathrm{d} \log(r) = \mathrm{d} Area$, so $\gamma = 2 K,$ where K is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high K, and therefore can be used to inhibit ripening.

## Applications

### Ice Cream

After warming up, during recrystallization when temperatures decrease again, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture after melting and refreezing. Ostwald ripening of ice crystals (Clarke, 2003).

### Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters

Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.

When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece et. al. showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy (Source includes illustrative movie).

### Geology

Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl et. al. studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).

They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum.