fix wallforce

Syntax

fix ID group_ID wallforce eps sigma zw delta rw cleavplane file <filename> 

• eps = (energy units)

• sigma = (distance units)

• zw = position of the walls respect to the cleaving plane

• delta = width of the interpolation region in the m function

• rw = cut-off of the LJ potential

• cleavplane = position of the cleaving plane

• <filename> = external file which contains the position of the walls

Description

On each side of the cleaving plane we place two planes with fixed interaction sites. Each site interacts with surrounding atoms through a wall shaped potential which is constructed from the repulsive core of the LJ potential, defined using a standard Weeks–Chandler–Anderson splitting Hansen and McDonald1:

\[ \begin{equation} \phi(r) = 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^{6}}{r^{6}} \right) + \epsilon \end{equation} \]

Interactions of the atom with the walls are given by

\[ \Phi(\mathbf{r};z_w)=m(\Phi_1, \Phi_2) \]

where \(m\) is a modified minimum function defined in 2 and \(\Phi_1\), \(\Phi_2\) are given by:

\[\begin{split} \begin{align} \Phi_1(\mathbf{r};z_w) & = \sum_{j} \phi \left( \left|\mathbf{r} - \left(\mathbf{r}_j^{(1)} - z_w\mathbf{n}\right) \right| \right) = \sum_{j} \phi \left( \left|\mathbf{r} - \mathbf{r}_j^{(1)} + z_w\mathbf{n} \right| \right) \\ \Phi_2(\mathbf{r};z_w) & = \sum_{j} \phi \left(\left|\mathbf{r} - \left(\mathbf{r}_j^{(2)} + z_w\mathbf{n} \right) \right| \right) = \sum_{j} \phi \left( \left|\mathbf{r} - \mathbf{r}_j^{(2)} - z_w\mathbf{n} \right| \right) \end{align} \end{split}\]

where \(\mathbf{n}\) is the (unit) normal to the cleaving plane. In the cleaving procedure, the strength of the interactions sites is varied during the simulation using the parameter \(z_w\) which varies from an initial value \(z_{w,i}\) to a final value \(z_{w,f}\). There is no a priori prescription for these two values except that the initial value \(z_{w,i}\) must be such that the walls do not interact with the atoms in the system.

Note that the way the wall potential is defined the atoms interacts with only one of the two walls (the one that gives the lowest interactions).

The format of the file is:

N1
x1 y1 z1 
x2 y2 z2
...
xN1 yN1 zN1
N2
x1 y1 z1 
x2 y2 z2
...
xN1 yN1 zN1

where \(N_1\) is the total number of walls on one side of the cleaving plane, whereas \(N_2\) is the total number of walls on opposite side of the cleaving plane. Here is an example:

132
  0.28595722359534864       0.16509748001949254        0.0
  0.85787167078604587        1.1556823601364479        0.0
   1.4297861179767433       0.16509748001949254        0.0
...
   12.296160614599994        10.401141241228032        -0.57191
   11.724246167409296        11.391726121344988        -0.57191
132
  0.28595722359534864       0.16509748001949254        0.0
  0.85787167078604587        1.1556823601364479        0.0
   1.4297861179767433       0.16509748001949254        0.0
...
   12.296160614599994        10.401141241228032        0.57191
   11.724246167409296        11.391726121344988        0.57191

Note

The z-coordinate of the interaction positions of the cleaving walls in the file must be given with respect to the position of the cleaving plane. That is to say, in the case of a single cleaving wall for each side of the cleaving plane, the z-coordinate will be zero for all the interaction points. The fix will then determine the (z-)position of the interaction points as: position of the cleaving plane (specified in the input cleavplane) \(\pm\) \(z_w\).

1

J. Hansen and I. R. McDonald. Theory of simple liquids: with applications to soft matter. Academic press, 2013.

2

R. L. Davidchack and B. B. Laird. Direct calculation of the crystal–melt interfacial free energies for continuous potentials: application to the lennard-jones system. The Journal of chemical physics, 118(16):7651–7657, 2003.

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