Density Functional Formulation - Micromeritics ASAP 2020 Confirm Operator's Manual

ASAP 2020 Confirm
If the move results in a configuration of higher energy, a probability for that event is calcu-
lated, and a random number between zero and one is generated. If the generated number is
smaller than the probability of the event, then the move is accepted; otherwise, another par-
ticle is selected and the process is repeated. This process continues until the average total
energy of the system no longer decreases; at this point, average configuration data are accu-
mulated to yield the mean density distribution of particles in the system.
Monte Carlo simulations require considerably less computation time than molecular dynamic
simulations and can yield the same results; however, neither method provides a really prac-
tical way to calculate complete isotherms.

Density Functional Formulation

Density functional theory offers a practical alternative to both molecular dynamic and Monte
Carlo simulations. When compared to reference methods based on molecular simulation, this
theory provides an accurate method of describing inhomogeneous systems yet requires fewer
calculations. Because the density functional theory provides accuracy and a reduced number
of calculations, it is the basis embodied in the DFT models.
The system being modeled consists of a single pore represented by two parallel walls sepa-
rated by a distance H. The pore is open and immersed in a single component fluid
(adsorptive) at a fixed temperature and pressure. Under such conditions, the fluid responds to
the walls and reaches an equilibrium distribution. In this condition (by the definition of equi-
librium), the chemical potential at every point equals the chemical potential of the bulk fluid.
The bulk fluid is a homogenous system of constant density; its chemical potential
mined by the pressure of the system using well-known equations. The fluid near the walls is
not of constant density; its chemical potential is composed of several position-dependent con-
tributions that must total at every point to the same value as the chemical potential of the bulk
fluid.
As noted previously, at equilibrium, the whole system has a minimum (Helmholtz) free
energy, known thermodynamically as the grand potential energy (GPE). Density functional
theory describes the thermodynamic grand potential as a functional of the single-particle den-
sity distribution; therefore, calculating the density profile that minimizes the GPE yields the
equilibrium density profile. The calculation method requires the solution of a system of com-
plex integral equations that are implicit functions of the density vector. Since analytic
solutions are not possible, the problem must be solved using iterative numerical methods.
Although calculation using these methods still requires supercomputing speed, the calculation
of many isotherm pressure points for a wide range of pore sizes is a feasible task. The com-
plete details of the theory and the mathematics can be found in the papers listed under
References at the end of this appendix.
The following graphs and accompanying text illustrate the results of using density functional
theory to predict the behavior of a model system.
*Chemical potential may be thought of as the energy change felt by a probe particle when it is inserted
into the system from a reference point outside the system. It can also be defined as the partial derivative
of the grand potential energy with respect to density (or concentration).
02-42811-01 - Mar 2011
Appendix F
*
is deter-
F-3