Interface pinning calculations

Interface Pinning is a method for finding melting points from an MD simulation of a system where the liquid and the solid phase are in contact. To prevent melting or freezing at constant pressure and constant temperature, a bias potential applies a penalty energy for deviations from the desired two phase system.

The Steinhardt-Nelson $Q_6$ order parameter is used for discriminating the solid from the liquid phase and the bias potential is given by

$U_\textrm{bias}(\mathbf{R}) = \frac\kappa2 \left(Q_6(\mathbf{R}) - a\right)^2$

where $Q_6({\mathbf{R}})$ is the Steinhardt-Nelson $Q_6$ orientational order parameter for the current configuration $\mathbf{R}$ and $a$ is the desired value of the order parameter close to the order parameter of the initial two phase configuration.

With the bias potential enabled, the system can equilibrate while staying in the two phase configuration. From the difference of the average order parameter $\langle Q_6 \rangle$ in equilibrium and the desired order parameter $a$ one can directly compute the difference of the chemical potential of the solid and the liquid phase:

$N(\mu_\textrm{solid} - \mu_\textrm{liquid}) =\kappa (Q_{6 \textrm{solid}} - Q_{6 \textrm{liquid}}) (\langle Q_6 \rangle - a)$

where $N$ is the number of atoms in the simulation.

It is preferable to simulate in the super heated regime, as it is easier for the bias potential to prevent a system from melting than to prevent a system from freezing.

$Q_6(\mathbf{R})$ needs to be continuous for computing the forces on the atoms originating from the bias potential. We use a smooth fading function $w(r)$ to weight each pair of atoms at distance $r$ for the calculation of the $Q_6$ order parameter:

${\displaystyle w(r) = \left\{\begin{array}{cl} 1 & \textrm{for} r\leq n \frac{ (f^2 - r^2)^2(f^2 - 3n^2 + 2r^2)}{(f^2 - n^2)^3} & \textrm{for} n<r<f 0 & \textrm{for $f\leq r$} \end{array} \right.}$

where $n$ and $f$ are the near and far fading distances given in the INCAR file respectively. A good choice for the fading range can be made from the radial distribution function $g(r)$ of the crystal phase. We recommend to use the distance where $g(r)$ goes below 1 after the first peak as the near fading distance $n$ and the distance where $g(r)$ goes above 1 again before the second peak as the far fading distance $f$ . $g(r)$ should be low where the fading function has a high derivative to prevent spurious stress.

The interface pinning method uses the $Np_zT$ ensemble where the barostat only acts on the direction of the lattice that is perpendicular to the solid liquid interface. We recommend to use a Langevin thermostat and a Parrinello-Rahman barostat with lattice constraints as demonstrated in the listing below assuming a solid liquid interface perpendicular to the $z$ direction. The listing shows the section of the INCAR file relevant for interface pinning that was used to determine the triple point of sodium:

TEBEG = 400              # temperature in K
POTIM = 4                # timestep in fs
IBRION = 0               # do MD
ISIF = 3                 # use Parrinello-Rahman barostat for the lattice
MDALGO = 3               # use Langevin thermostat
LANGEVIN_GAMMA = 1.0     # friction coef. for atomic DoFs for each species
LANGEVIN_GAMMA_L = 3.0   # friction coef. for the lattice DoFs
PMASS = 100               # mass for lattice DoFs
LATTICE_CONSTRAINTS = F F T   # fix x&y, release z lattice dynamics
OFIELD_Q6_NEAR = 3.22  # fading distances for computing a continuous Q6
OFIELD_Q6_FAR = 4.384  # in A
OFIELD_KAPPA = 500     # strength of bias potential in eV/(unit of Q)^2
OFIELD_A = 0.15        # desired value of the Q6 order parameter
%TODO: ref

For more details on the interface pinning method see reference ^[1].