Category:Interface pinning

Use interface pinning to determine the melting point from a molecular-dynamics simulation of the interface of a liquid and a solid phase. Because the typical behavior of such a simulation is to freeze or melt, the interface is pinned with a bias potential. This potential applies an energy penalty for deviations from the desired two-phase system. Prefer simulating above the melting point because the bias potential prevents melting better than freezing.

The Steinhardt-Nelson order parameter [math]\displaystyle{ Q_6 }[/math] discriminates between the solid and the liquid phase. With the bias potential

[math]\displaystyle{ U_\text{bias}(\mathbf{R}) = \frac\kappa2 \left(Q_6(\mathbf{R}) - A\right)^2 }[/math]

penalizes differences between the order parameter for the current configuration [math]\displaystyle{ Q_6({\mathbf{R}}) }[/math] and the one for the desired interface [math]\displaystyle{ A }[/math]. [math]\displaystyle{ \kappa }[/math] is an adjustable parameter determining the strength of the pinning.

Under the action of the bias potential, the system equilibrates to the desired two-phase configuration. An important observable is the difference between the average order parameter [math]\displaystyle{ \langle Q_6 \rangle }[/math] in equilibrium and the desired order parameter [math]\displaystyle{ A }[/math]. This difference relates to the the chemical potentials of the solid [math]\displaystyle{ \mu_\text{solid} }[/math] and the liquid [math]\displaystyle{ \mu_\text{liquid} }[/math] phase

[math]\displaystyle{ N(\mu_\text{solid} - \mu_\text{liquid}) = \kappa (Q_{6,\text{solid}} - Q_{6,\text{liquid}})(\langle Q_6 \rangle - A) }[/math]

where [math]\displaystyle{ N }[/math] is the number of atoms in the simulation.

Computing the forces requires a differentiable [math]\displaystyle{ Q_6(\mathbf{R}) }[/math]. We use a smooth fading function [math]\displaystyle{ w(r) }[/math] to weight each pair of atoms at distance [math]\displaystyle{ r }[/math] for the calculation of the [math]\displaystyle{ Q_6 }[/math] order parameter

[math]\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\lt r\lt f \\ 0 &\textrm{for} \,\,f\leq r \end{array}\right. }[/math]

Here [math]\displaystyle{ n }[/math] and [math]\displaystyle{ f }[/math] are the near- and far-fading distances given in the INCAR file respectively. The radial distribution function [math]\displaystyle{ g(r) }[/math] of the crystal phase yields a good choice for the fading range. To prevent spurious stress, [math]\displaystyle{ g(r) }[/math] should be small where the derivative of [math]\displaystyle{ w(r) }[/math] is large. Set the near fading distance [math]\displaystyle{ n }[/math] to the distance where [math]\displaystyle{ g(r) }[/math] goes below 1 after the first peak. Set the far fading distance [math]\displaystyle{ f }[/math] to the distance where [math]\displaystyle{ g(r) }[/math] goes above 1 again before the second peak.

How to

Interface pinning uses the [math]\displaystyle{ Np_zT }[/math] ensemble where the barostat only acts along the [math]\displaystyle{ z }[/math] direction. This uses a Langevin thermostat and a Parrinello-Rahman barostat with lattice constraints in the remaining two dimensions. The solid-liquid interface must be in the [math]\displaystyle{ x-y }[/math] plane perpendicular to the action of the barostat.

Set the following tags for the interface pinning method:

OFIELD_Q6_NEAR

Defines the near-fading distance [math]\displaystyle{ n }[/math].

OFIELD_Q6_FAR

Defines the far-fading distance [math]\displaystyle{ f }[/math].

OFIELD_KAPPA

Defines the coupling strength [math]\displaystyle{ \kappa }[/math] of the bias potential.

OFIELD_A

Defines the desired value of the order parameter [math]\displaystyle{ A }[/math].

The following is a sample INCAR file for interface pinning of sodium^[1]:

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

References

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