Category:Interface pinning

Interface pinning^[1] is used to determine the melting point from a molecular-dynamics simulation of the interface between a liquid and a solid phase. The typical behavior of such a simulation is to freeze or melt, while the interface is pinned with a bias potential. This potential applies an energy penalty for deviations from the desired two-phase system. It is preferred simulating above the melting point because the bias potential prevents melting better than freezing.

The Steinhardt-Nelson^[2] order parameter ${\displaystyle Q_6}$ discriminates between the solid and the liquid phase. With the bias potential

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

penalizes differences between the order parameter for the current configuration ${\displaystyle Q_6({\mathbf{R}})}$ and the one for the desired interface ${\displaystyle A}$. ${\displaystyle \kappa}$ 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 ${\displaystyle \langle Q_6\rangle}$ in equilibrium and the desired order parameter ${\displaystyle A}$. This difference relates to the the chemical potentials of the solid ${\displaystyle \mu_\text{solid}}$ and the liquid ${\displaystyle \mu_\text{liquid}}$ phase

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

where ${\displaystyle N}$ is the number of atoms in the simulation.

Computing the forces requires a differentiable ${\displaystyle Q_6(\mathbf{R})}$. In the VASP implementation a smooth fading function ${\displaystyle w(r)}$ is used to weight each pair of atoms at distance ${\displaystyle r}$ for the calculation of the ${\displaystyle Q_6(\mathbf{R},w)}$ order parameter. This fading function is given as

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

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

How to

Interface pinning uses the ${\displaystyle Np_zT}$ ensemble where the barostat only acts along the ${\displaystyle z}$ direction. This ensemble 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 ${\displaystyle x}$-${\displaystyle y}$ 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 ${\displaystyle n}$.

OFIELD_Q6_FAR

Defines the far-fading distance ${\displaystyle f}$.

OFIELD_KAPPA

Defines the coupling strength ${\displaystyle \kappa}$ of the bias potential.

OFIELD_A

Defines the desired value of the order parameter ${\displaystyle A}$.

The following example INCAR file calculates the interface pinning in sodium^[1]:

TEBEG = 400                   # temperature in K
POTIM = 4                     # timestep in fs
IBRION = 0                    # run molecular dynamics
ISIF = 3                      # use Parrinello-Rahman barostat for the lattice
MDALGO = 3                    # use Langevin thermostat
LANGEVIN_GAMMA_L = 3.0        # friction coefficient for the lattice degree of freedoms (DoF)
LANGEVIN_GAMMA = 1.0          # friction coefficient for atomic DoFs for each species
PMASS = 100                   # mass for lattice DoFs
LATTICE_CONSTRAINTS = F F T   # fix x-y plane, release z lattice dynamics
OFIELD_Q6_NEAR = 3.22         # near fading distance for function w(r) in Angstrom
OFIELD_Q6_FAR = 4.384         # far fading distance for function w(r) 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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