## Sunday, September 20, 2015

### Surface tension and the non-polar solvation entropy

I made a stupid sign mistake in one of my video lectures and have spent part of the weekend sorting things out in my mind.  So here is a note to self before while it is still fresh in my mind.

The non-polar free energy of solvation can be written as
$$\Delta G_{\text{np-solv}} = \gamma_{\text{np}} SASA$$
where where $SASA$ is the solvent accessible surface area.  The argument is that one of the main contributions to $\Delta G_{\text{np-solv}}$ is the energy required to make the molecular cavity in the solvent, which, for macroscopic objects, is a function of the surface tension of the liquid $\gamma$ and the surface area of the cavity.
$$\Delta G_{\text{np-solv}} \propto \gamma SASA$$
$\gamma$ is positive for water so $\Delta G_{\text{np-solv}}$ is positive in water.

So far, so good.  But this simple picture fails when considering the solvation entropy
$$\Delta S_{\text{np-solv}} = - \left( \frac{\partial \gamma_{\text{np}}}{\partial T} \right) SASA$$
For water the bulk surface tension decreases with increasing temperature as you would expect, which suggests that $\Delta S_{\text{np-solv}}$ is positive when in fact it is observed to be negative.

So if $\gamma_{\text{np}}$ has anything to do with $\gamma$ this would imply that the surface tension associated with molecular-sized cavities increase with temperature.  It is not clear why that would be so and this, in part, has led Graziano to argue that, effectively, $\gamma_{\text{np}}$ has nothing to do with $\gamma$ but is a strictly empirical parameter.

A little more detail that ultimately doesn't shed any more light
$\gamma_{\text{np}}$ is also positive but not equal to $\gamma$. One reason is that $\Delta G_{\text{np-solv}}$ also contains contributions from repulsion and dispersion interactions with the solute.  However, if one computes $\Delta G_{\text{np-solv}}$ from hard sphere simulations the corresponding $\gamma_{\text{np}}$ values still does not match the $\gamma$ value for bulk water.

Tolman has argued that the surface tension depends on the curvature of the surface and suggested the following approximation
$$\gamma (R) = \gamma \left( 1 - \frac{2\delta}{R} \right)$$
where $R$ is the cavity radius and $\delta$ is a parameter called the Tolman length.  When $R < 2\delta$
$$\frac{\partial \gamma (R)}{\partial T} = \left( \frac{\partial \gamma}{\partial T} \right) \left( 1 - \frac{2\delta}{R} \right)$$
will indeed be positive, but only when $\Delta G_{\text{np-solv}}$ is negative.  What is observed is positive $\Delta G_{\text{np-solv}}$ and  $\Delta S_{\text{np-solv}}$.

Ashbaugh has pointed out that a  temperature-dependent $\delta$ solves this problem but Graziano fired back that since there is no analytical form for $\delta$, $\frac{\partial \delta}{\partial T}$ is just another temperature dependent parameter, and you might as well use $\frac{\partial \gamma_{\text{np}}}{\partial T}$ as a parameter (I am paraphrasing here).