I am working on a very simple model of a granular solid made of Lennard-Jones particles, i.e. the non-bonded interaction of these particles is given by the simple, standard shifted, purely repulsive LJ potential, which reads: 4\epsilon ( (\sigma/r)^12 - (\sigma/r)^6) + epsilon. The potential, as usual is cut off at its minimum distance sqrt[6]{2}. Of course, as usual, sigma determines the length scale, i.e the diameter of particles and epsilon determines the energy scale.
Now, a non-trivial problem occurs, when simulating, say a cube of LJ particles (which in our case are subject to shock waves via impact) and you want to map this chunk of matter to a real material, say some kind of rock material, or to be more specific, aluminum for example.
The thing is, that the length scale can easily be mapped by just claiming that the simulation cube corresponds to, say a cube of aluminum (which has a certain density in absolute units) of 1 m^3. That defines (via the number of particles) the length sigma.
But... How does one obtain a mapping of the energy scale, i.e. what is the energy of epsilon, when I want to map it to a real material, e.g. aluminum with known density and volume ? We need this epsilon to determine our time scale in the simulation, i.e. to map our simulation time unit t* to a unit of time t in absolute (real units). The formula for that is t* = t (epsilon / m / sigma^2)^1/2. So, if we knew epsilon, we could map our time scale to a real system.
In literature, I discovered, that the authors NEVER seem to disclose the method of mapping their LJ-unit to a real system. They just state their simulation time step in absolute units as a matter of fact (which means they must have mapped epsilon as well). That is why one does not know how, e.g. the authors can claim that their simulation timestep is supposed to be, e.g. 10^{-12} seconds when they simulated a generic fcc crystal and then claim that this is supposed to be a chunk of Fe of such and such dimensions.
I'd appreciate any specific ideas how to perform this mapping of a LJ simulation time scale to a real time scale (which involves first mapping epsilon to a real system, which seems to be the actual problem).
I am not sure, How about this.
For example, Al.
1 ) Calculate average mass of one atom of Al:
26.9815 g/mol ÷ 6.022*10^23 atoms/mol = 4.48049*10^-23 g/atom
2) Atoms in 1cm3
2.7g/cm3 ÷ 4.48049*10^-23 g/atom = 6.02613*10^22 atoms/cm3
For epsilon.(In case of Aluminum)
Energy unit is [KJ/mol]
A [KJ/mol] ÷ 6.022*10^23 [atom/mol] =( A÷ 6.022*10^23) ≡B [KJ/atom]
B [KJ/atom] ÷ 6.02613*10^22 [atom/cm3] = B ÷6.02613*10^22 [KJ/cm3]
then The unit of Epsilon can be calculated. Like this. B ÷6.02613*10^22 [KJ/cm3]
Of course, to watch specific again. maybe.
Dear Kyunghwan,
many thanks for your suggestion. I'm not sure if I understand 100% of your proposal, though. Maybe you could expand a little bit. Let me see, how I understand what you've suggested:
So, in essence you are saying, that one gets the energy in LJ-units when I divide the number of LJ-particles (say, this number is X) that I have in my simulation model in 1cm^3 by the number 6.02613*10^22, right ? And this number X we can get from claiming, e.g. that our 10^6 LJ-particles, arranged in a cube are supposed to represent, e.g. 1m^3 of Aluminum. From that we get the size of ONE LJ-particle and from that the number X, the number of LJ particles in 1cm^3
So, then we divide X by 6.02613*10^22 and this is the energy epsilon of ONE LJ-particle in a cube of volume 1 cm^3.
So, epsilon is actually this number: X ÷ 6.02613*10^22, i.e. X * 6.02613*10^(-22), right ?
I mean that the unit of epsilon same energy Unit. [KJ/mol]
You wanted to get the practical absoulte unit.
So, I tried to change KJ/mol to KJ/cm3.
The first step, I explained average mass of one atom of Al
We know atom weight of aluminum and Avogdro's number.
I used two given constants and can get mass per one atom, (it is called constant T1),
and furthere calculate atoms per unit volume(it is called contant T2) by using experiment density data and mass per atom (constant T1).
The second step, I applied to translate KJ/mol to KJ/cm3.
First, I translate KJ/mol to KJ/atom (Epsilon divided by Avogadro's number; it is called constant B) and then, we calculate like this the constant B divided by constant T1. Finally, I got KJ/cm3 unit.
It is just my suggestion, I hope that it is helpful to you and if you don't mind, could tell me later the working result ?
I am wondering, too. :)
Thanks
OH, KYUNGHWAN
Dear Martin,
Can you please tell me how one should decide epsilon parameter in LJ potential?
I mean I have plotted potential Energy curve as a function of inter-atomic distance varied between two atoms and I have minimum energy at -16.8 eV (basis cell of NaCl-type structure; comprising 8 atoms), in this case will my epsilon parameter be 16.8 or 16.8/2=8.4 (eV/atom)? I appreciate your help. Thank you.
Sincerely,
Krutarth Patel
- Badges
- Science topic