Category:Interface pinning: Difference between revisions

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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 $Q_6$ discriminates between the solid and the liquid phase. With the bias potential

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

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

where $N$ is the number of atoms in the simulation.

Computing the forces requires a differentiable $Q_6(\mathbf{R})$ . In the VASP implementation a smooth fading function $w(r)$ is used to weight each pair of atoms at distance $r$ for the calculation of the $Q_6(\mathbf{R},w)$ 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 $n$ and $f$ are the near- and far-fading distances, respectively. The radial distribution function $g(r)$ of the crystal phase yields a good choice for the fading range. To prevent spurious stress, $g(r)$ should be small where the derivative of $w(r)$ is large. Set the near fading distance $n$ to the distance where $g(r)$ goes below 1 after the first peak. Set the far fading distance $f$ to the distance where $g(r)$ goes above 1 again before the second peak.

How to

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

This category currently contains no pages or media.

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

[steinhardt:prb:83-2] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28, 784 (1983).

[1]

[2]

@@ Line 1: / Line 1: @@
-All tags and articles that deal with Interface Pinning calculations are members of this category.
+'''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.
+<!-- == Theory == -->
+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.
-== Theoretical Background ==
+The Steinhardt-Nelson{{cite|steinhardt:prb:83}} order parameter <math>Q_6</math> discriminates between the solid and the liquid phase.
+With the bias potential
-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.
+:<math>U_\text{bias}(\mathbf{R}) = \frac\kappa2 \left(Q_6(\mathbf{R}) - A\right)^2 </math>
-The Steinhardt-Nelson Q 6 {\displaystyle Q_{6}} order parameter is used for discriminating the solid from the liquid phase and the bias potential is given by
+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>.
+<math>\kappa</math> is an adjustable parameter determining the strength of the pinning.
-U bias ( R ) = κ 2 ( Q 6 ( R ) − a ) 2 {\displaystyle U_{{\textrm {bias}}}({\mathbf {R}})={\frac \kappa 2}\left(Q_{6}({\mathbf {R}})-a\right)^{2}}
+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>\langle Q_6\rangle</math> in equilibrium and the desired order parameter <math>A</math>.
+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
-where Q 6 ( R ) {\displaystyle Q_{6}({{\mathbf {R}}})} is the Steinhardt-Nelson Q 6 {\displaystyle Q_{6}} orientational order parameter for the current configuration R {\displaystyle {\mathbf {R}}} and a {\displaystyle a} is the desired value of the order parameter close to the order parameter of the initial two phase configuration.
+:<math>
+N(\mu_\text{solid} - \mu_\text{liquid}) =
+\kappa (Q_{6,\text{solid}} - Q_{6,\text{liquid}})(\langle Q_6 \rangle - A)
+</math>
-With the bias potential enabled, the system can equilibrate while staying in the two phase configuration. From the difference of the average order parameter ⟨ Q 6 ⟩ {\displaystyle \langle Q_{6}\rangle } in equilibrium and the desired order parameter a {\displaystyle a} one can directly compute the difference of the chemical potential of the solid and the liquid phase:
+where <math>N</math> is the number of atoms in the simulation.
-N ( μ solid − μ liquid ) = κ ( Q 6 solid − Q 6 liquid ) ( ⟨ Q 6 ⟩ − a ) {\displaystyle N(\mu _{{\textrm {solid}}}-\mu _{{\textrm {liquid}}})=\kappa (Q_{{6{\textrm {solid}}}}-Q_{{6{\textrm {liquid}}}})(\langle Q_{6}\rangle -a)}
+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
-where N {\displaystyle N} is the number of atoms in the simulation.
+:<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.
 == How to ==
-A comprehensive documentation on interface pinning calculations is given in {{TAG|Interface pinning calculations}}.
+'''Interface pinning''' uses the <math>Np_zT</math> ensemble where the barostat only acts along the <math>z</math> 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 <math>x</math>-<math>y</math> plane perpendicular to the action of the barostat.
+Set the following tags for the '''interface pinning''' method:
+;{{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>.
+The following example {{TAG|INCAR}} file calculates the interface pinning in sodium{{cite|pedersen:prb:13}}:
+ {{TAGBL|TEBEG}} = 400                   # temperature in K
+ {{TAGBL|POTIM}} = 4                     # timestep in fs
+ {{TAGBL|IBRION}} = 0                    # run molecular dynamics
+ {{TAGBL|ISIF}} = 3                      # use Parrinello-Rahman barostat for the lattice
+ {{TAGBL|MDALGO}} = 3                    # use Langevin thermostat
+ {{TAGBL|LANGEVIN_GAMMA_L}} = 3.0        # friction coefficient for the lattice degree of freedoms (DoF)
+ {{TAGBL|LANGEVIN_GAMMA}} = 1.0          # friction coefficient for atomic DoFs for each species
+ {{TAGBL|PMASS}} = 100                   # mass for lattice DoFs
+ {{TAGBL|LATTICE_CONSTRAINTS}} = F F T   # fix x-y plane, release z lattice dynamics
+ {{TAGBL|OFIELD_Q6_NEAR}} = 3.22         # near fading distance for function w(r) in Angstrom
+ {{TAGBL|OFIELD_Q6_FAR}} = 4.384         # far fading distance for function w(r) in Angstrom
+ {{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
+== References ==
+<references/>
+<noinclude>
 ----
-[[The_VASP_Manual|Contents]]
-[[Category:VASP|Interface Pinning]][[Category:Molecular Dynamics|Molecular Dynamics]]
+[[Category:VASP|Interface pinning]][[Category:Molecular dynamics]]