# How to estimate latent heat of polymorphic phase transitions

1. Jul 20, 2014

### g_mogni

Hello,

I know that the latent heat of a first-order isothermal and isobaric phase transition is in general given by the difference in enthalpies between the two phases, but my question is: do you know if there is there a way to estimate its value "from first principles" for the case of a first-order polymorphic solid-solid structural phase transition (say the transition from the cubic-diamond and beta-tin phases of crystalline silicon)? By first principles I mean not resorting to classical thermodynamics, but considering for example the difference in interatomic forces between the two phases. Any insight on this would be greatly appreciated!

Gabriele

Last edited: Jul 20, 2014
2. Jul 21, 2014

### jpmit

Hi Gabriele,

The latent heat L = TΔS , where ΔS is the difference in entropy between the two phases. At constant volume, ΔS = -ΔU/T where ΔU is the difference in internal energy between the two phases (this can be derived from the thermodynamic differential dU = PdV - TdS). So to compute the latent heat, you need to compute ΔU.

Now, how to compute ΔU? You need to compute the ensemble average - in your case in the NPT ensemble - of the Hamiltonian for both phases, and subtract one from the other. The usual approach to doing this is Monte-Carlo simulation (Metropolis method), see e.g. the books 'Computer Simulation of Liquids' by Allen and Tildesley and 'Understanding Molecular Simulation' by Frenkel and Smit for a good description of this. These books are really for classical force fields, the formalism is basically identical for quantum systems as well but the practical methods for actually computing ensemble averages would be different (more difficult) in that case.

ps: I am not quite sure what you mean by "considering for example the difference in interatomic forces between the two phases". The interatomic forces are what they are (or, more likely, what your simplified model of the forces say they are). It probably wouldn't be sensible to compute two internal energies with different force field descriptions. If you did, it would be non-trivial to relate the two internal energies, and hence compute an accurate ΔU value.

Last edited: Jul 21, 2014
3. Jul 21, 2014

### g_mogni

Hi there, thanks for your help. The issue here is that I would like to be able to estimate the latent heat independently from the formula L=TΔS. The reason is that I would like to be able to calculate the transition pressure for a first-order polymorphic phase transition of crystalline silicon, and for that I need to come up with independent estimates for each of L and the TΔS terms (such that at the transition pressure the two terms are equal). I already have a method for calculating the entropy from phonon frequencies. In the normal case of a isothermal and isobaric phase transition under hydrostatic stress, the latent heat would normally be given by the difference in enthalpies of the two phases ΔU+pΔV, such that the equation L=TΔS (which is valid at the transition pressure) reduces to the equivalence of the gibbs free energies G=U+pV-TS of the two phases, i.e. the conventional way of finding the transition pressure. The problem is that the expression ΔU+pΔV for the latent heat is only valid under hydrostatic stress. For non-hydrostatic stress, which is the case I would like to study, there isn't an obvious expression for the enthalpy. Hence why I was wondering if maybe there was a way of estimating the latent heat based on microscopic statistical mechanics considerations on the difference between the structures of the two phases, and not on classical thermodynamics. Hope I have been clear enough.