The Interface Pinning method for studying solid-liquid transitions

In this example we will use the interface pinnig method to study a solid-liquid transition. This is done by adding a harmonic potential to the Hamiltonian that bias the system towards two-phase configurations:

U_bias = 0.5*K*(Q-a)^2

The bias field couple to an order-parameter of crystallinity Q. The implementation use long-range order:

Q=|rho_k|,

where rho_k is the collective density field of the wave-vector k. For future reference we note that the structure factor S(k) is given by the variance of the collective density field:

S(k)=|rho_k|^2.

About the method

It is recommended to get familiar with the interface pinning method by reading:

[Ulf R. Pedersen, JCP 139, 104102 (2013)](http://dx.doi.org/10.1063/1.4818747)

A detailed bibliography is provided at

<http://urp.dk/interface_pinning.htm>

and a brief introduction can be found at YouTube:

[![Interface Pinning](http://img.youtube.com/vi/F_79JZNdyoQ/0.jpg)](http://www.youtube.com/watch?v=F_79JZNdyoQ)

Implimentation in LAMMPS

For this example we will be using the rhok fix.

fix [name] [groupID] rhok [nx] [ny] [nz] [K] [a]

This fix include a harmonic bias potential U_bias=0.5*K*(|rho_k|-a)^2 to the force calculation. The elements of the wave-vector k is given by the nx, ny and nz input:

k_x = (2 pi / L_x) * n_x, k_y = (2 pi / L_y) * n_y and k_z = (2 pi / L_z) * n_z.

We will use a k vector that correspond to a Bragg peak.

Example: the Lennard-Jones (LJ) model

We will use the interface pinning method to study melting of the LJ model at temperature 0.8 and pressure 2.185. This is a coexistence state-point, and the method can be used to show this. The present directory contains the input files that we will use:

in.crystal
in.setup
in.pinning

1. First we will determine the density of the crystal with the LAMMPS input file in.crystal. From the output we get that the average density after equilibration is 0.9731. We need this density to ensure hydrostatic pressure when in a two-phase simulation.

2. Next, we setup a two-phase configuration using in.setup.

3. Finally, we run a two-phase simulation with the bias-field applied using in.pinning. The last column in the output show |rho_k|. We note that after a equilibration period the value fluctuates around the anchor point (a) -- showing that this is indeed a coexistence state point.

The reference JCP 139, 104102 (2013) gives details on using the method to find coexistence state points, and the reference JCP 142, 044104 (2015) show how the crystal growth rate can be computed from fluctuations. That method have been experienced to be most effective in the slightly super-heated regime above the melting temperature.

1. Contact
   
   Ulf R. Pedersen; http://www.urp.dk; ulf AT urp.dk