Interface pinning calculations: Difference between revisions

Revision as of 11:48, 16 October 2024

Interface pinning uses the $Np_zT$ ensemble where the barostat only acts along the $z$ 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 $x$ - $y$ 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 $n$ .
OFIELD_Q6_FAR: Defines the far-fading distance $f$ .
OFIELD_KAPPA: Defines the coupling strength $\kappa$ of the bias potential.
OFIELD_A: Defines the desired value of the order parameter $A$ .

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

↑ U. R. Pedersen, F. Hummel, G. Kresse, G. Kahl, and C. Dellago, Phys. Rev. B 88, 094101 (2013).

[pedersen:prb:13-1] U. R. Pedersen, F. Hummel, G. Kresse, G. Kahl, and C. Dellago, Phys. Rev. B 88, 094101 (2013).

[1]

@@ Line 1: / Line 1: @@
-'''Interface pinning'''{{cite|pedersen:prb:13}} is used to determine the melting point from a [[:Category: Molecular dynamics|molecular-dynamics]] simulation of the interface between a liquid and a solid phase.
+'''Interface pinning''' uses the <math>Np_zT</math> ensemble where the barostat only acts along the <math>z</math> direction.
-<!-- == Theory == -->
+This ensemble uses a Langevin thermostat and a Parrinello-Rahman barostat with lattice constraints in the remaining two dimensions.
-The typical behavior of such a simulation is to freeze or melt, while the interface is ''pinned'' with a bias potential.
+The solid-liquid interface must be in the <math>x</math>-<math>y</math> plane perpendicular to the action of the barostat.
-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{{cite|steinhardt:prb:83}} order parameter <math>Q_6</math> discriminates between the solid and the liquid phase.
+Set the following tags for the '''interface pinning''' method:
-With the bias potential
+;{{TAG|OFIELD_Q6_NEAR}}: Defines the near-fading distance <math>n</math>.
+;{{TAG|OFIELD_Q6_FAR}}: Defines the far-fading distance <math>f</math>.
+;{{TAG|OFIELD_KAPPA}}: Defines the coupling strength <math>\kappa</math> of the bias potential.
+;{{TAG|OFIELD_A}}: Defines the desired value of the order parameter <math>A</math>.
-:<math>U_\text{bias}(\mathbf{R}) = \frac\kappa2 \left(Q_6(\mathbf{R}) - A\right)^2 </math>
+The following example {{TAG|INCAR}} file calculates the interface pinning in sodium{{cite|pedersen:prb:13}}:
+ {{TAGBL|TEBEG}} = 400                   # temperature in K
-penalizes differences between the order parameter for the current configuration <math>Q_6({\mathbf{R}})</math> and the one for the desired interface <math>A</math>.
+ {{TAGBL|POTIM}} = 4                     # timestep in fs
-<math>\kappa</math> is an adjustable parameter determining the strength of the pinning.
+ {{TAGBL|IBRION}} = 0                    # run molecular dynamics
+ {{TAGBL|ISIF}} = 3                      # use Parrinello-Rahman barostat for the lattice
-Under the action of the bias potential, the system equilibrates to the desired two-phase configuration.
+ {{TAGBL|MDALGO}} = 3                    # use Langevin thermostat
-An important observable is the difference between the average order parameter <math>\langle Q_6\rangle</math> in equilibrium and the desired order parameter <math>A</math>.
+ {{TAGBL|LANGEVIN_GAMMA_L}} = 3.0        # friction coefficient for the lattice degree of freedoms (DoF)
-This difference relates to the the chemical potentials of the solid <math>\mu_\text{solid}</math> and the liquid <math>\mu_\text{liquid}</math> phase
+ {{TAGBL|LANGEVIN_GAMMA}} = 1.0          # friction coefficient for atomic DoFs for each species
+  {{TAGBL|PMASS}} = 100                   # mass for lattice DoFs
-:<math>
+ {{TAGBL|LATTICE_CONSTRAINTS}} = F F T   # fix x-y plane, release z lattice dynamics
-N(\mu_\text{solid} - \mu_\text{liquid}) =
+  {{TAGBL|OFIELD_Q6_NEAR}} = 3.22         # near fading distance for function w(r) in Angstrom
-\kappa (Q_{6,\text{solid}} - Q_{6,\text{liquid}})(\langle Q_6 \rangle - A)
+ {{TAGBL|OFIELD_Q6_FAR}} = 4.384         # far fading distance for function w(r) in Angstrom
-</math>
+ {{TAGBL|OFIELD_KAPPA}} = 500            # strength of bias potential in eV/(unit of Q)^2
+ {{TAGBL|OFIELD_A}} = 0.15               # desired value of the Q6 order parameter
-where <math>N</math> is the number of atoms in the simulation.
-Computing the forces requires a differentiable <math>Q_6(\mathbf{R})</math>.
-<!-- PLEASE REPHRASE - I did not understand this part and how it relates to Q_6(R) -->
-In the VASP implementation a smooth fading function <math>w(r)</math> is used to weight each pair of atoms at distance <math>r</math> for the calculation of the <math>Q_6(\mathbf{R},w)</math> order parameter. This fading function is given as
-:<math> 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 \\
-&\textrm{for} \,\,f\leq r \end{array}\right. </math>
-<!-- is w(r) equivalent to (1 - t)^2(1 + 2t) with t = (r - n) / (f - n)? -->
-Here <math>n</math> and <math>f</math> are the near- and far-fading distances, respectively.
-<!-- END REPHRASE -->
-The radial distribution function <math>g(r)</math> of the crystal phase yields a good choice for the fading range.
-To prevent spurious stress, <math>g(r)</math> should be small where the derivative of <math>w(r)</math> is large.
-Set the near fading distance <math>n</math> to the distance where <math>g(r)</math> goes below 1 after the first peak.
-Set the far fading distance <math>f</math> to the distance where <math>g(r)</math> goes above 1 again before the second peak.
 == References ==