Mass transfer through interfaces

Gibbs-Thomson condition

Gibbs-Thomson condition for a curved interface

Gibbs-Thomson condition

For a curved interface at equilibrium:

(585)\[\mu_{in}^{eq}(\phi_{in})=\mu_{out}^{eq}(\phi_{out})=\mu_{i}^{eq}\]

where the index \(i\) is for “interface”. The pressure difference is given by the Laplace’s law:

(586)\[p_{in}(\phi_{in})-p_{out}(\phi_{out})=\sigma\kappa\]

With its definition:

With its def (Eq. ([eq:Def_p-c_local])), Eq. ([eq:Equilib_p_Curved]) writes

(587)\[\phi_{in}\mu_{in}^{eq}-f(\phi_{in})-\phi_{out}\mu_{out}^{eq}+f(\phi_{out})=\sigma\kappa\]

Using common tangent construction Eq. (387) for \(f_{dw}(\phi_{out})-f_{dw}(\phi_{in})\):

(588)\[\mu_{i}^{eq}(\phi_{in}-\phi_{out})+\mu^{eq}(\phi_{out}-\phi_{in})=\sigma\kappa\]

and finally:

(589)\[\boxed{\mu_{i}^{eq}=\mu^{eq}+\frac{\sigma\kappa}{(\phi_{out}-\phi_{in})}}\]

This is the Gibbs-Thomson condition which means that the equilibrium chemical potential of a curved interface is equal to the equilibrium chemical potential for a flat interface corrected by the curvature and the surface tension.

Ostwald ripening

Consequence of Gibbs-Thomson condition with \(\boldsymbol{j}_{CH}\): Ostwald ripening

Ostwald ripening

The Gibbs-Thomson condition writes:

(590)\[\mu_{i}=\mu^{eq}+\frac{1}{(\phi_{out}-\phi_{in})}\frac{2\sigma}{R}\]

If we consider two droplets of radius \(R_{1}\) and \(R_{2}\) with \(R_{1}<R_{2}\), Eq. () implies:

(591)\[\mu_{i}(R_{2})<\mu_{i}(R_{1})\]

There exists a flux from \(\mu_{i}(R_{1})\) to \(\mu_{i}(R_{2})\) because \(\boldsymbol{j}_{CH}\propto-\boldsymbol{\nabla}\mu_{\phi}(\boldsymbol{x},t)\) (see Fig. Fig-Ostawald-Ripening). Consequence: the smallest droplet disappears and the biggest one grows. This is the Ostwald ripening.

Fig. 62 Principle of Ostwald ripening

Droplet growth