75 lines
2.9 KiB
Markdown
75 lines
2.9 KiB
Markdown
# The Interface Pinning method for studying solid-liquid transitions
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In this example we will use the interface pinnig method to study a solid-liquid transition.
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This is done by adding a harmonic potential to the Hamiltonian
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that bias the system towards two-phase configurations:
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U_bias = 0.5*K*(Q-a)^2
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The bias field couple to an order-parameter of crystallinity Q. The implementation use long-range order:
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Q=|rho_k|,
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where rho_k is the collective density field of the wave-vector k.
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For future reference we note that the structure factor S(k) is given by the variance of the collective density field:
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S(k)=|rho_k|^2.
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### About the method
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It is recommended to get familiar with the interface pinning method by reading:
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[Ulf R. Pedersen, JCP 139, 104102 (2013)](http://dx.doi.org/10.1063/1.4818747)
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A detailed bibliography is provided at
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<http://urp.dk/interface_pinning.htm>
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and a brief introduction can be found at YouTube:
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[](http://www.youtube.com/watch?v=F_79JZNdyoQ)
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### Implimentation in LAMMPS
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For this example we will be using the rhok fix.
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fix [name] [groupID] rhok [nx] [ny] [nz] [K] [a]
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This fix include a harmonic bias potential U_bias=0.5*K*(|rho_k|-a)^2 to the force calculation.
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The elements of the wave-vector k is given by the nx, ny and nz input:
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k_x = (2 pi / L_x) * n_x, k_y = (2 pi / L_y) * n_y and k_z = (2 pi / L_z) * n_z.
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We will use a k vector that correspond to a Bragg peak.
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## Example: the Lennard-Jones (LJ) model
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We will use the interface pinning method to study melting of the LJ model
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at temperature 0.8 and pressure 2.185. This is a coexistence state-point, and the method
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can be used to show this. The present directory contains the input files that we will use:
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in.crystal
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in.setup
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in.pinning
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1. First we will determine the density of the crystal with the LAMMPS input file in.crystal.
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From the output we get that the average density after equilibration is 0.9731.
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We need this density to ensure hydrostatic pressure when in a two-phase simulation.
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2. Next, we setup a two-phase configuration using in.setup.
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3. Finally, we run a two-phase simulation with the bias-field applied using in.pinning.
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The last column in the output show |rho_k|. We note that after a equilibration period
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the value fluctuates around the anchor point (a) -- showing that this is indeed a coexistence
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state point.
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The reference [JCP 139, 104102 (2013)](http://dx.doi.org/10.1063/1.4818747) gives details on using the method to find coexistence state points,
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and the reference [JCP 142, 044104 (2015)](http://dx.doi.org/10.1063/1.4818747) show how the crystal growth rate can be computed from fluctuations.
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That method have been experienced to be most effective in the slightly super-heated regime above the melting temperature.
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## Contact
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Ulf R. Pedersen;
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<http://www.urp.dk>;
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ulf AT urp.dk
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