Category:Interface pinning: Difference between revisions

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'''Interface pinning''' is a method for finding melting points from a [[:Category:Molecular dynamics|molecular-dynamics]] simulation of a system where the liquid and the solid phase are in contact.  
Use '''interface pinning''' to determine the melting point from a [[:Category: Molecular dynamics|molecular-dynamics]] simulation of the interface of a liquid and a solid phase.
<!-- == Theory == -->
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.


== Theory ==
The Steinhardt-Nelson order parameter <math>Q_6</math> discriminates between the solid and the liquid phase.
To prevent melting or freezing at constant pressure and constant temperature, a bias potential applies penalty energy for deviations from the desired two-phase system.
With the bias potential


The Steinhardt-Nelson order parameter <math>Q_6</math> is used for discriminating the solid from the liquid phase and the bias potential is given by
:<math>U_\text{bias}(\mathbf{R}) = \frac\kappa2 \left(Q_6(\mathbf{R}) - Q_{6,\text{pinned}}\right)^2 </math>


:<math>U_\textrm{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>Q_6({\mathbf{R}})</math> and the one for the desired interface <math>Q_{6,\text{pinned}}</math>.
<math>\kappa</math> is an adjustable parameter determining the strength of the pinning.


where <math>Q_6({\mathbf{R}})</math> is the <math>Q_6</math> order parameter for the current configuration <math>\mathbf{R}</math> and <math>a</math> is the desired value of the order parameter close to the order parameter of the initial two-phase configuration.
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>Q_{6,\text{pinned}}</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


With the bias potential enabled, the system can equilibrate while staying in the two-phase configuration. From the difference between the average order parameter <math>\langle Q_6 \rangle</math> in equilibrium and the desired order
:<math>
parameter <math>a</math> one can directly compute the difference of the chemical potential of the solid and the liquid phase:
N(\mu_\text{solid} - \mu_\text{liquid}) =  
 
\kappa (Q_{6,\text{solid}} - Q_{6,\text{liquid}})(\langle Q_6 \rangle - Q_{6,\text{pinned}})
:<math> N(\mu_\textrm{solid} - \mu_\textrm{liquid}) =\kappa (Q_{6 \textrm{solid}} - Q_{6 \textrm{liquid}}) (\langle Q_6 \rangle - a) </math>
</math>


where <math>N</math> is the number of atoms in the simulation.
where <math>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.
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) -->
<math>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>w(r)</math> to weight each pair of atoms at distance <math>r</math> for the calculation of the <math>Q_6</math> order parameter
We use a smooth fading function <math>w(r)</math> to weight each pair of atoms at distance <math>r</math> for the calculation of the <math>Q_6</math> order parameter


:<math> w(r) = \left\{ \begin{array}{cl} 1  &\textrm{for} \,\, r\leq n \\
:<math> w(r) = \left\{ \begin{array}{cl} 1  &\textrm{for} \,\, r\leq n \\
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                       0  &\textrm{for} \,\,f\leq r \end{array}\right. </math>
                       0  &\textrm{for} \,\,f\leq r \end{array}\right. </math>


Here <math>n</math> and <math>f</math> are the near- and far-fading distances given in the {{TAG|INCAR}} file respectively. A good choice for the fading range can be made from the radial distribution function <math>g(r)</math> of the crystal phase. We recommend to use the distance where <math>g(r)</math> goes below 1 after the first peak as the near fading distance <math>n</math> and the distance where <math>g(r)</math> goes above 1 again before the second peak as the far fading distance <math>f</math>. <math>g(r)</math> should be low where the fading function has a high derivative to prevent spurious stress.
<!-- 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 given in the {{TAG|INCAR}} file 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 ==
== How to ==


The '''interface pinning''' method uses the <math>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 '''interface pinning''' method uses the <math>Np_zT</math> ensemble where the barostat only acts in 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:
The following variables need to be set for the '''interface pinning''' method:

Revision as of 08:40, 7 April 2022

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

penalizes differences between the order parameter for the current configuration and the one for the desired interface . 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 in equilibrium and the desired order parameter . This difference relates to the the chemical potentials of the solid and the liquid phase

where is the number of atoms in the simulation.

Computing the forces requires a differentiable . We use a smooth fading function to weight each pair of atoms at distance for the calculation of the order parameter


Here and are the near- and far-fading distances given in the INCAR file respectively. The radial distribution function of the crystal phase yields a good choice for the fading range. To prevent spurious stress, should be small where the derivative of is large. Set the near fading distance to the distance where goes below 1 after the first peak. Set the far fading distance to the distance where goes above 1 again before the second peak.

How to

The interface pinning method uses the ensemble where the barostat only acts in 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 .
  • OFIELD_Q6_FAR: This tag defines the far-fading distance .
  • OFIELD_KAPPA: This tag defines the coupling strength of the bias potential.
  • OFIELD_A: This tag defines the desired value of the order parameter .

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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