Development of Thermodynamic Methods for the study of Nanopores R. Paul * and S. J. Paddison ** * Department of Chemistry, The University of Calgary, Calgary, T2N 1N4 Canada, rpaul@ucalgary.ca **Computational Nanoscience Group, Motorola Inc., Los Alamos Research Park, 87544 USA, s.paddison@motorola.com 1 ABSTRACT A statistical mechanical formalism is presented for the calculation of the thermodynamic properties of the water in nanopores of polymer electrolytes, with particular emphasis on the polarization. We describe a method that exploits the variational properties of the Ornstein-Zernicke integral equation. 2 INTRODUCTION The main thrust of the work to be presented in this paper will be the development of statistical thermodynamic methods for application to polymer electrolyte membranes commonly used in a variety of fuel cells [1]. In these devices the anode and cathode are separated by a polymer membrane, which conducts the ionic current (as protons) while the usable electronic current passes through the external circuit. The efficiency of the cell operation, clearly, must be a function of the ionic conductivity of the membrane and thus of the morphology of the latter. A considerable body of structural information about these membranes is now available and it is known that the transport of protons occurs through a network of hydrated channels or pores supported by a hydrophobic matrix or medium that provides the requisite mechanical strength and chemical and thermal stability of the material. The system of pores may be modeled as a network of channels each of which due to their size are referred to as nanopores. The structures of these hydrated nanopores are quite complex and vary from one polymeric material to another, however, for the present study the most relevant feature is the presence of negatively charged anionic groups attached to the pore walls. These groups are either distributed along the interior of the pore walls or may be present at the termini of short chains protruding into the pore interior. Since in the majority of cases the ionic species being conducted are hydronium ions it is evident that these negatively charged groups, which are most often 3 SO - groups, will play a critical role in determining the proton diffusion [2]. The state of water within the nanopores is, in general, different from that of bulk water. Neutron scattering experiments carried out by Lee et al [3] on water confined in nanopores of perfluorinated ionomer membranes shows that the radial distribution functions agree with that of bulk water for only the fully hydrated pores. The precise form that the water adopts under the influence of both pore confinement and the electrical field due to the anionic groups is still not understood [4]. From the above discussion, it is evident that the thermodynamic properties of water in these nanopores will differ from that of bulk water. In this paper we present the framework for calculating the Helmholtz free energy for water in the pore and apply this method for the computation of the polarization of the water. The latter is a very important electrical property that plays significant role in the transport of the protons. 3 THE HELMHOLTZ FREE ENERGY In order to compute the Helmholtz free energy, for which we assume that a classical mechanical formalism to be adequate, a suitable Hamiltonian must be first adopted. In general such a Hamiltonian, , will be composed of both particle kinetic and potential energies, the former, however, cancel out and will therefore not be displayed. The remaining terms must appropriately account for all the interactions and energy sources in the system. Each nanopore for simplicity is assumed to be cylindrical in shape with radius and length in which the fixed sites are modeled as a sequence of negatively charged rings, attached to the pore wall and constituting a lattice. An analytic expression for the field due to such a charge arrangement is available in literature and will not be reproduced here (see [5]). Unlike most traditional fluid systems, in which only an external field is present, in this case the energy of the field due to the anionic sites is a part of the system and must be included: (1)