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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 order parameter is used for discriminating the solid from the liquid phase and the bias potential is given by
where is the Steinhardt-Nelson orientational order parameter for the current configuration and 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 in equilibrium and the desired order parameter one can directly compute the difference of the chemical potential of the solid and the liquid phase:
where 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.
needs to be continuous for computing the forces on the atoms originating from the bias potential. We use a smooth fading function to weight each pair of atoms at distance for the calculation of the order parameter:
where and 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 of the crystal phase. We recommend to use the distance where goes below 1 after the first peak as the near fading distance and the distance where goes above 1 again before the second peak as the far fading distance . should be low where the fading function has a high derivative to prevent spurious stress.
The interface pinning method uses the 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 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 Angstrom OFIELD_KAPPA = 500 # strength of bias potential in eV/(unit of Q)^2 OFIELD_A = 0.15 # desired value of the Q6 order parameter
For more details on the interface pinning method see reference .