Page 38 - Adsorption by Powders and Porous Solids

CHAPTER I. INTRODUCTION 23

decay, rather than the inverse 12th power inEquation (1.4). For this reason, low-cov-
erage isotherm and adsorption energy data are used to refine the evaluation of the
adsorbent-adsorbate interaction energy Another source of uncertainty is the magni-
tude of three-body effects, e.g. involving two adsorbate molecules and a substrate
&om, which with some systems are likely to be significant even at very low coverage
(Nicholson, 1996).

1.9.2. Molecular simulation
The two simulation methods in general use for solving the statistical mechanical
equations are Monte Carlo (MC) and moiecular dynamics (MD). The two techniques
have several common features, but each has certain advantages and limitations.

Monte Carlo (MC) simulation
~n this method a random number generator is used to move and rotate molecules in a
random fashion. If the system is held under specified conditions of temperature,
volume and number of molecules, the probability of a particular arrangement of mol-
ecules is proportional to exp(-U/kn, where U is the total intermolecular energy of
the assembly of molecules and k is the Boltzmann constant. Thus, within the MC
scheme the movement of individual molecules is accepted or rejected in accordance
with a probability determined by the Boltzmann distribution law. After the genera-
tion of a long sequence of moves, the results are averaged to give the equilibrium
properties of the model system.
An advantage of MC simulation is that it is not difficult to program. Aiso, the ther-
modynamic, canonical, variables may be readily changed. For gas adsorption studies
it is generally more useful to specify p, V, T (the grand canonical variables of chem-
ical potential, volume and temperature) rather than N, V, T (the number of molecules,
volume and temperature), so that p is an independent variable. For this reason grand
canonical Monte Carlo (GCMC) molecular simulation has been favoured by most
investigators.

Molecular dynamics (MD)
By the application of Newton's equations of motion. the trajectories and velocities of
the molecular motion can be obtained. Averaging over time then gives the properties
of the system. Since it is possible to simulate a few nanoseconds of real time, trans-
port properties can be evaluated as well as equilibrium states. MD is more difficult to
program than MC, but the molecular motions are more realistic and therefore
computer graphics can be used to give a more accurate impression of the actual
movement of molecules'moving through or into pores.

1.9.3. Density functional theory (DFT)
As a means of establishing the density profile, p(r), two free energy functionals are
introduced: Q(p(r)] and F[p(r)]. 52 is a form of thermodynamic potential and is gen-
erally known as the grand potential, or grand free energy. In general, for a system