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##

Interface pinning

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 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 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.

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 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

For more details on the interface pinning method see Ref. [162].

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