# [lammps-users] Lennard Jones 9-3 potential

Thank you all for your advice. Assuming that the LJ9-3 model in LAMMPS
is "incomplete", how can we include the terms that are missing?
Depending what value sigma is, I'm afraid the LJ9-3 in LAMMPS and the
LJ9-3 we derived ourselves would give very different answers.

Jay

Dear all,

Instead of "incomplete", I would say that the potential implemented in
lammps is generic, with an effective epsilon that depends not only on
the inter-atomic 12-6 LJ epsilon, but also on the density rho_s of the
wall you want to represent:

epsilon(9-3) = (2pi/3)(rho_s sigma^3) epsilon(12-6).

So keeping a generic value epsilon seems important to me, since this
way the user can choose to model a wall with a given density and 12-6
LJ interactions. But maybe one could add something in the
documentation to explain these subtelties...

Best,
Laurent

2010/8/4 Jay Casey <[email protected]>:

Thank you for your link. The reason why I want to find out how the LJ9-3
model in LAMMPS manual is derived because in the manual it says:
LJ93=epsilon((2/15)(sigma/r)^9-(sigma/r)^6)
But if you derive LJ93 directly from LJ12-6 via integration method (also
proven by the link you sent), LJ93 should be:
LJ93=(2/3)*pi*epsilon*sigma^3*((2/15)(sigma/r)^9-(sigma/r)^6)

I think that last line is missing a rho = N/V, so what LAMMPS is "missing"
in its formula is 2/3 pi rho*, where rho* = N/V sigma^3 is the reduced
density in 3d.
I haven't worked out what it would be in 2d.

I can't remember now why we decided to
drop the prefactors in the LAMMPS formula, but Laurent below is correct
that we are sort of leaving it up to the user to choose the rho* for the wall
region since it is an independent choice, which can be effectively done
by rescaling the specified epsilon appropriately.

This is all somewhat complicated, so I'm open to suggestions. Maybe LAMMPS
should have the user input rho* as well as sigma and epsilon for the
9/3 wall case,
and use the exact formula with the 2/3 pi rho* prefactor.
At a minimum the doc page should clarify this prefactor issue.

Steve

PS: Paul, I CCd you in case you remember any of the thoughts that went
into these choices.

Dear all,

Thank you for all your help. I would hope that LAMMPS would allow user to input rho* as well as sigma and epsilon for the 9/3 wall case, and use the exact formula with the 2/3 pi rho* prefactor. This would make it more self-explanatory. However, having said that, then shouldn’t the 12/6 LJ include the density (rho) as well?

Casey Jay

Mark can comment, but the wall 12/6 potential is not derived from
integrating over a half-volume of particles, but is just a straight LJ 12/6
formula. So rho* for the wall isn't part of any derivation.

Steve

Dear all,

As Steve wrote, while fix wall/lj93 derives from a model of a
particule interacting with a wall of 12-6 LJ particules with a given
density, fix wall/lj126 don't originate in any particular model, so in
this case epsilon is just an arbitrary interaction energy, and there
is no reason to express it as a function of the density.

Concerning fix wall/lj93, a first source of confusion is maybe that
the same symbol -epsilon- has a different meaning in the
inter-particule 12-6 LJ pair interaction energy, and in the
wall-integrated interaction energy. Maybe one could use a distinctive
symbol, for instance epsilon_wall, and add the expression of this
integrated interaction energy as a function of the pair interaction
energy epsilon and the reduced density of the wall rho*:

\\varepsilon\_\\mathrm\{wall\} = \\frac\{2\\pi\}\{3\} \\rho^\* \\varepsilon.

In the present documentation, this sentence is particularly confusing:
"For the wall/lj93 and wall/lj126 styles, epsilon and sigma are the
usual Lennard-Jones parameters, which determine the strength and size
of the particle as it interacts with the wall."

Another possible source of confusion could be that the same symbol -r-
denotes the inter-particule distance for 12-6 LJ pair style, and the
distance between the particule and the wall in the 9-3 LJ wall fix. A
standard symbol for distance to a wall could be D, but alas this would
be conflicting with the D in wall/colloid style...

To conclude on the documentation, after the sentence "The wall/lj93
interaction is derived by integrating over a 3d half-lattice of
Lennard-Jones 12/6 particles", a convenient reference where the
integration of particule-wall interaction is detailed (at least for
the 1/r^6 part) can be "Intermolecular and Surface Forces" by J.
Israelachvili.

Best,
Laurent

PS: Paul, I CCd you in case you remember any of the thoughts that went
into these choices.

After reviewing this thread, I mostly agree with Laurent's and Steve's comments, specifically, here's what I think should be done:

1) Clarify the documentation. LAMMPS shouldn't have to justify the model, it should only have to clearly state the model and implement it correctly. So in the case of LJ93, whether or not the docs explain where it comes from is not as important as that they explain exactly what is implemented in LAMMPS and how to use it. So regarding the confusing statements about LJ93: if in doubt, leave it out. Might be nice to include references that explain the model's history.

2) It would be nice to allow LJ93 users to optionally input the rho* prefactor. And we'd need clear documentation with references that explain why a person would want to use the prefactor.

3) I disagree with the idea about changing the sigma, epsilon, and r to different symbols for the wall fixes. It might actually cause more confusion than the change intends to solve.

Paul