Molecular Simulation Methods; Monte Carlo Method - Micromeritics Gemini VII 2390a Operator's Manual

Appendix F
If the equilibrium distribution of the gas atoms near the surface could be described as a function of
pressure and the molecular properties of the components of the system, then a model could be con-
structed for the adsorption isotherm for the system. Modern physical chemistry provides several ways
to calculate this distribution. All these methods are based on the fundamental thermodynamic law that
such a system adopts a configuration of minimum free energy at equilibrium. Also needed is a descrip-
tion of the pairwise interaction energy between atoms, U(s), commonly given by a Lennard-Jones
potential:
where
ε =
=
s =

Molecular Simulation Methods

Two simulation techniques are commonly used to determine the distribution of gas molecules in a sys-
tem in equilibrium: the molecular dynamics method and the Monte Carlo method. Both of these are
used as reference methods because their results are considered exact.
Molecular Dynamics Method
In the molecular dynamics method, the position and velocity of individual gas particles are calculated
over time at very short intervals. This method takes into account both the forces acting between the gas
particles themselves and those acting between the gas particles and the atoms of the simulated surface.
As the simulated particles collide with each other and with the surface, the average concentration of
particles in the space near the surface is calculated; this calculation yields the amount of gas adsorbed.
This method can be thought of as a way to determine the chronological record of the movement of
each particle in the system using time steps of 10-14 seconds. Although the mathematics are simple,
the number of calculations required for a system of even a few hundred particles is astronomical and
challenges even the fastest computers.

Monte Carlo Method

In the Monte Carlo method, determination of the system equilibrium distribution begins with an
assumption (which may be only approximate) about the initial configuration of particles in the system.
The system is "equilibrated" through a process of randomly selecting one particle and conditionally
moving it a random distance in a random direction.
If the move results in a configuration of lower total energy, then the move is completed and another
particle is randomly selected to be moved.
F-2

 
U s  
-- -
= 4
 
s
a characteristic energy of the adsorptive,
the diameter of the adsorptive molecule, and
the separation distance.
12 6

 
-- -
–
 
s
Gemini VII
239-42828-01 - Dec 2012