Category:Interface pinning: Difference between revisions

<Interface pinning is a method for finding melting points from a molecular-dynamics simulation of a system where the liquid and the solid phase are in contact.

Theory

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 [math]\displaystyle{ Q_6 }[/math] is used for discriminating the solid from the liquid phase and the bias potential is given by

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

where [math]\displaystyle{ Q_6({\mathbf{R}}) }[/math] is the [math]\displaystyle{ Q_6 }[/math] order parameter for the current configuration [math]\displaystyle{ \mathbf{R} }[/math] and [math]\displaystyle{ a }[/math] 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 [math]\displaystyle{ \langle Q_6 \rangle }[/math] in equilibrium and the desired order parameter [math]\displaystyle{ a }[/math] one can directly compute the difference of the chemical potential of the solid and the liquid phase:

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

where [math]\displaystyle{ N }[/math] 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.

[math]\displaystyle{ Q_6(\mathbf{R}) }[/math] needs to be continuous for computing the forces on the atoms originating from the bias potential. 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. A good choice for the fading range can be made from the radial distribution function [math]\displaystyle{ g(r) }[/math] of the crystal phase. We recommend to use the distance where [math]\displaystyle{ g(r) }[/math] goes below 1 after the first peak as the near fading distance [math]\displaystyle{ n }[/math] and the distance where [math]\displaystyle{ g(r) }[/math] goes above 1 again before the second peak as the far fading distance [math]\displaystyle{ f }[/math]. [math]\displaystyle{ g(r) }[/math] should be low where the fading function has a high derivative to prevent spurious stress.

How to

The interface pinning method uses the [math]\displaystyle{ Np_zT }[/math] ensemble where the barostat only acts on the direction of the lattice that is perpendicular to the solid liquid interface. This uses a Langevin thermostat and a Parrinello-Rahman barostat with lattice constraints in the remaining two dimensions.

The following variables need to be set for the interface pinning method:

• OFIELD_Q6_NEAR: This tag defines the near-fading distance [math]\displaystyle{ n }[/math].
• OFIELD_Q6_FAR: This tag defines the far-fading distance [math]\displaystyle{ f }[/math].
• OFIELD_KAPPA: This tag defines the coupling strength [math]\displaystyle{ \kappa }[/math] of the bias potential.
• OFIELD_A: This tag 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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