A New Molecular Simulation Technique for Determining Vapor-Liquid
Equilibrium in Nanopores
Minoru Miyahara* and Hideki Tanaka
Department of Chemical Engineering, Kyoto University, Nishikyo, Kyoto 615-8510, Japan
*Presenting and corresponding author (Email: [email protected])
Scope and Summary
Because of the incorrectness of the Kelvin model in nanometer scale, the nanopore-size characterization
needs local isotherms, which are to be determined, desirably, by molecular simulations, or by DFT if
computational costs are to be reduced. The grand canonical Monte Carlo (GCMC) simulation inevitably
suffers from difficulty in determining thermodynamic equilibrium pressure because of pronounced
adsorption-desorption hysteresis, which may in some case be an artifact. In an earlier work, we
developed a unique molecular dynamics (MD) simulation cell that is opened to a bulk vapor phase [1]:
Metastable states are easily suppressed because the open-ended nature gives ease in holding stable
menisci within the cell, thus avoiding the hysteresis. Another benefit of this open-cell MD was that, in
contrast to usual MD method, we can easily determine the equilibrium vapor pressure just by counting
the frequency of the molecules that are able to escape from attractive pore space and to arrive at the
vapor phase. In some cases with lower temperatures, however, the computational cost of waiting for
sufficient number of such molecules has arisen up as a problem.
As a typical and rigorous way to determine the equilibrium points, Peterson and Gubbins had
proposed thermodynamic integration method based on GCMC simulations [2], while Neimark and
Vishnyakov proposed gauge-cell method utilizing canonical MC principle [3]. Both of the methods,
however, need intensive simulation runs around hysteresis region and/or isotherms at elevated
temperatures.
In this study we have developed a new simulation method in which the open cell and Monte Carlo
technique are combined to suffice the two requirements; to avoid metastable states or hysteresis loop,
and to determine equilibrium pressure easily without significant computational load. We demonstrate
that the equilibrium point can be determined by just one simulation, without suffering from the
hysteresis or complicated procedure.
Simulation
For LJ-nitrogen in slit pore made of graphite wall of Steele's 10-4-3 potential with width H, at T=77 K,
the following three methods are compared.
Potential buffering field (PBF)
Full potential field
H
PBF
Pore cell
Particle exchange
Vapor cell
Fig. 1. Adsorption isotherms obtained by
GCMC and gauge cell methods, and
vapor-liquid equilibrium point obtained by
open cell method.
Fig. 2. Schematic sketch of open cell method.
The full potential energy in FPF of the pore cell
attenuates in PBF to vanish at the border.
The pore and vapor cells exchange particles
under a fixed total number of particles.
<Thermodynamic Integration Method> Using GCMC isotherm, the grand potential for adsorption
branch Ωads is given by an integration along the path A→C (Fig. 1). The intersection of Ωads with the
Ωdes for desorption branch (D→F) gives the equilibrium point μeq. The latter, however, requires a
continuous isotherm at a sufficiently elevated temperature (Th = 140 K, μA→μD) and variation of
adsorbed amount against temperature (Th→Tl) at fixed chemical potential μD, which must be obtained by
additional GCMC runs.
<Gauge Cell Method> A vapor cell and a pore cell, as a whole, form a canonical ensemble. The mass
exchange between the two cells attains the equality in μ. The trick in the method is to limit the number
of molecules in the vapor cell to be small, so that the density fluctuation in pore fluid is severely
suppressed: the system follows not only the metastable states (B→SV) but also unstable ones to yield a
continuous S-shaped isotherm, with which one can determine the μeq by Maxwell's rule. Reliable and
sufficient number of data points in the hysteresis region, however, must be produced for accurate
determination.
<Open Cell Method> Contrary to the case of the gauge cell method, we intend to let the system be
settled in a equilibrium state by employing an open-ended cell: In the middle of the cell is the pore space
with a given potential energy, (Fig. 2, full potential field: FPF). At each end of the cell, we set a border
plane, beyond which an imaginary vapor phase is assumed to exist. Since the potential energy in the
vapor phase must be zero, also must be equipped is a connecting space with slope of potential energy
between the border and FPF, which is called potential buffering field (PBF). The important feature is
that almost no metastable state emerges because vapor-liquid interface easily stands in the cell, which
enables us to find out the equilibrium μeq by only once of MC simulation run: The equilibrium vapor
pressure can be determined by introducing a vapor cell that exchanges mass with the pore cell. One of
the results is shown by the hatched square key in Fig.1, which is exactly aligned with the equilibrium μeq
obtained by the above two methods.
Typical Results
For slit pores with H = 7, 10, 13σff, typical results are
summarized in Table 1. The difference of μeq given by the
open-cell method from those by thermodynamic
integration method are in all cases less than 0.06%, which
corresponds to an error in P/Ps of only 0.8%. The results
thus demonstrate an excellent degree of accuracy in
determination of V-L equilibrium points in nanopores.
Based on this technique, a set of local isotherms are
calculated, and used for pore-size characterization, which
will be discussed in the Conference.
Table 1. Vapor-liquid equilibrium points
obtained by open cell method
H [σff]
μeq [-]
difference [%]
7
10
-10.852
-10.114
0.058
0.055
13
-9.835
0.011
Reference
[1] M. Miyahara, T. Yoshioka and M. Okazaki, J. Chem. Phys., 106, 8124 (1997).
[2] B. K. Peterson and K. E. Gubbins, Mol. Phys., 62, 215 (1987).
[3] A. V. Neimark and A. Vishnyakov, Phys. Rev. E, 62, 4611 (2000).