Density Functional Formulation - Micromeritics Gemini VII 2390a Operator's Manual

Gemini VII
If the move results in a configuration of higher energy, a probability for that event is calculated, and a
random number between zero and one is generated. If the generated number is smaller than the proba-
bility of the event, then the move is accepted; otherwise, another particle 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 accumulated to yield the mean density distribution of parti-
cles in the system.
Monte Carlo simulations require considerably less computation time than molecular dynamic simula-
tions and can yield the same results; however, neither method provides a really practical 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 pro-
vides 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 separated by a
distance H. The pore is open and immersed in a single component fluid (adsorptive) at a fixed temper-
ature and pressure. Under such conditions, the fluid responds to the walls and reaches an equilibrium
distribution. In this condition (by the definition of equilibrium), the chemical potential at every point
equals the chemical potential of the bulk fluid. The bulk fluid is a homogenous system of constant den-
sity; its chemical potential
The fluid near the walls is not of constant density; its chemical potential is composed of several posi-
tion-dependent contributions 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 ther-
modynamic grand potential as a functional of the single-particle density distribution; therefore,
calculating the density profile that minimizes the GPE yields the equilibrium density profile. The cal-
culation method requires the solution of a system of complex 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 supercom-
puting speed, the calculation of many isotherm pressure points for a wide range of pore sizes is a
feasible task. The complete 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.
Figure D-1 shows the density profile for argon at a carbon surface as calculated by density functional
theory for a temperature of 87.3 K and a relative pressure of about 0.5.
1.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).
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is determined by the pressure of the system using well-known equations.
Appendix F
F-3