The capture and sequestration of carbon dioxide in deep coal beds including enhanced methane recovery is an attractive option for reducing atmospheric emissions. Microscopic simulation approaches such as Monte Carlo and Molecular Dynamics are powerful techniques that can be used to probe the physical and chemical interactions between the carbon dioxide, methane and coal-pore surfaces. Information, including adsorption isotherms, diffusivity, and permeability can be extracted using these computational methods and then can be used to calibrate larger-scale macroscopic models. These modeling techniques have been widely used on idealized pore structures, generally represented by smooth graphite surfaces or carbon nanotubes. A key issue preventing the implementation of these simulation techniques to study the heterogeneous structures which are more representative of the actual coal surfaces is the lack of efficient long-range force-field schemes.

The real surface of the coal pores includes both polar and nonpolar functional groups making the interactions between the walls and the molecules highly anisotropic and limiting the accuracy of the commonly used 10-3-4 Steele potential[1]. In this work an efficient force field scheme will be introduced to account for the anisotropic distribution of the electrostatic forces in real coal pores. Although novel to this field, the scheme is based on the particle-particle particle-mesh (P3M) technique and was originally developed and validated for use in simulating ionic transport in biological ion channels[2]. Within the P3M approach, the forces are divided into a short-range force that accounts for neighboring particle interactions and a smoothly-varying long-range force.

Short-Range Interactions: The short-range, particle-particle force is defined within a small sphere surrounding each particle and includes a Coulomb term, a van der Waals term and a reference term that corrects for any double counting due to the overlap of the short- and long-range domains. The van der Waals interactions are typically included using Lennard-Jones potentials. In this work, to account for the possibility of stronger interactions, i.e., chemisorption, data generated from first-principle calculations using Gaussian03 or VASP will be incorporated. These data contain the interaction information, such as potential energy and partial charges, as a function of distance between the surface groups on the coal and CO2 and CH4 molecules. To reduce computational time interactions can be stored in look-up table and accessed during run-time.

Long-Range Interactions: The second part of the P3M method, the particle-mesh term, contains the long-range interaction which is from the Coulomb force between far charges and is computed on a finite difference mesh in real space using Poisson's Equation and the iterative multigrid method. The mulitgrid method is one of fastest and efficient iterative solvers available for solving large sparse matrix equations, scaling as O(N)[3]. The approach can be shown in the following steps:

1. Assign particle charges to mesh points in the computational domain

2. Using the approximated charge density, calculate the electrostatic potential at each mesh point by solving Poisson's equation using the multigrid method

3. Calculate the electric field at each mesh point by differentiating the potential numerically

4. Interpolate the force from the mesh point to particle

It is important to note that this approach does not assume periodic boundary conditions, making it more flexible for simulating anisotropic systems.

The force-field scheme described will be validated against the Steele potential for ideal smooth graphite layers and pores structures and the error associated with excluding the long-range portion of the electrostatic interactions will be examined within more realistic pore structures that include polar functional groups representative in the coal matrix. Extensive timing comparisons will be made to illustrate the effectiveness of the approach for use in Monte Carlo and Molecular Dynamics simulations.

[1] W. A. Steele, Sur. Science, 36 (1973) 317

[2] D. Marriero, M. Saraniti, S. Aboud, “Brownian Dynamics simulation of charge transport in ion channels”,, J. Phys.: Condens Matter 19 (2007) 215203.

[3] W. Hackbush, Multi-Grid Methods and Applications, Springer-Verlag, Berlin (1985).