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| 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.
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| <!-- == Theory == -->
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| Because the typical behavior of such a simulation is to freeze or melt, the interface is ''pinned'' with a bias potential.
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| This potential applies an energy penalty for deviations from the desired two-phase system.
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| Prefer simulating above the melting point because the bias potential prevents melting better than freezing.
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| The Steinhardt-Nelson order parameter <math>Q_6</math> discriminates between the solid and the liquid phase.
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| With the bias potential
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|
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| :<math>U_\text{bias}(\mathbf{R}) = \frac\kappa2 \left(Q_6(\mathbf{R}) - Q_{6,\text{pinned}}\right)^2 </math>
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| 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>.
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| <math>\kappa</math> is an adjustable parameter determining the strength of the pinning.
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| Under the action of the bias potential, the system equilibrates to the desired two-phase configuration.
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| 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>.
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| 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
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|
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| :<math>
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| N(\mu_\text{solid} - \mu_\text{liquid}) =
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| \kappa (Q_{6,\text{solid}} - Q_{6,\text{liquid}})(\langle Q_6 \rangle - Q_{6,\text{pinned}})
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| </math>
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| where <math>N</math> is the number of atoms in the simulation.
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| Computing the forces requires a differentiable <math>Q_6(\mathbf{R})</math>.
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| <!-- PLEASE REPHRASE - I did not understand this part and how it relates to Q_6(R) -->
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| 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
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|
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| :<math> w(r) = \left\{ \begin{array}{cl} 1 &\textrm{for} \,\, r\leq n \\
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| \frac{(f^2 - r^2)^2 (f^2 - 3n^2 + 2r^2)}{(f^2 - n^2)^3} &\textrm{for} \,\, n<r<f \\
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| 0 &\textrm{for} \,\,f\leq r \end{array}\right. </math>
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|
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| <!-- is w(r) equivalent to (1 - t)^2(1 + 2t) with t = (r - n) / (f - n)? -->
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| Here <math>n</math> and <math>f</math> are the near- and far-fading distances given in the {{TAG|INCAR}} file respectively.
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| <!-- END REPHRASE -->
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| The radial distribution function <math>g(r)</math> of the crystal phase yields a good choice for the fading range.
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| To prevent spurious stress, <math>g(r)</math> should be small where the derivative of <math>w(r)</math> is large.
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| Set the near fading distance <math>n</math> to the distance where <math>g(r)</math> goes below 1 after the first peak.
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| Set the far fading distance <math>f</math> to the distance where <math>g(r)</math> goes above 1 again before the second peak.
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|
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| == How to ==
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|
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| 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.
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| The following variables need to be set for the '''interface pinning''' method:
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| *{{TAG|OFIELD_Q6_NEAR}}: This tag defines the near-fading distance <math>n</math>.
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| *{{TAG|OFIELD_Q6_FAR}}: This tag defines the far-fading distance <math>f</math>.
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| *{{TAG|OFIELD_KAPPA}}: This tag defines the coupling strength <math>\kappa</math> of the bias potential.
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| *{{TAG|OFIELD_A}}: This tag defines the desired value of the order parameter <math>a</math>.
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|
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| The following is a sample {{TAG|INCAR}} file for interface pinning of sodium{{cite|pedersen:prb:13}}:
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| {{TAGBL|TEBEG}} = 400 # temperature in K
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| {{TAGBL|POTIM}} = 4 # timestep in fs
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| {{TAGBL|IBRION}} = 0 # do MD
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| {{TAGBL|ISIF}} = 3 # use Parrinello-Rahman barostat for the lattice
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| {{TAGBL|MDALGO}} = 3 # use Langevin thermostat
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| {{TAGBL|LANGEVIN_GAMMA}} = 1.0 # friction coef. for atomic DoFs for each species
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| {{TAGBL|LANGEVIN_GAMMA_L}} = 3.0 # friction coef. for the lattice DoFs
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| {{TAGBL|PMASS}} = 100 # mass for lattice DoFs
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| {{TAGBL|LATTICE_CONSTRAINTS}} = F F T # fix x&y, release z lattice dynamics
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| {{TAGBL|OFIELD_Q6_NEAR}} = 3.22 # fading distances for computing a continuous Q6
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| {{TAGBL|OFIELD_Q6_FAR}} = 4.384 # in Angstrom
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| {{TAGBL|OFIELD_KAPPA}} = 500 # strength of bias potential in eV/(unit of Q)^2
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| {{TAGBL|OFIELD_A}} = 0.15 # desired value of the Q6 order parameter
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| == References ==
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| <references/>
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|
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| <noinclude>
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|
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| ----
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| [[The_VASP_Manual|Contents]]
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|
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| [[Category:VASP|Interface Pinning]][[Category:Molecular Dynamics]]
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