Optimum Interparticle Porosity for Charge Storage in a Packed Bed of Nanoporous Particles

Abstract

Analytical and numerical methods are used to investigate ion transport and storage within the pore network of a porous bed of nanoporous particles. Under simplifying assumptions that electromigration dominates ion transport, that the pore capacitance is constant, and that the electrolyte exhibits ideal behavior, transport within this hierarchical pore network is modeled using a coupled pair of one-dimensional equations describing net charge motion into the bed by way of the interparticle void and subsequently into smaller pores within particles. These diffusion-like equations are solved analytically via Laplace transforms, and the results are used to determine the optimum interparticle porosity yielding the maximum energy deliverable in a specified time. Both step and periodic boundary conditions are considered, and closed-form expressions for the optimum porosity are derived for the periodic case. We find that the optimum porosity is remarkably similar for the periodic and step boundary conditions when the normalized bed thickness is very large or very small. We also find that the optimum porosity increases at least linearly with bed thickness when the thickness is small but is independent of both the thickness and prescribed time for delivery when the bed thickness is large. Sample results are presented and discussed. High performance double-layer capacitors typically employ porous electrodes having very large specific surface areas, 1,2 and these large surface areas are necessarily synonymous with very small pores. Although large surface areas yield high energy densities, and very small pores may enhance this, 3 small pores also impede transport into and out of electrodes when the pore size is comparable to or smaller than the Debye layer thickness. In such cases, a large fraction of ions reside in relatively immobile layers adjacent to the pore walls, leading to hindered transport and reduced ion conductivity. 4,5 As a result, the high energy densities offered by large surface areas may come at the expense of impaired ability to deliver stored energy quickly, forcing a trade-off between energy and power densities. One means to circumvent this trade-off is to employ very thin electrodes such that ion transport need only occur over a very small distance. This however leads to increased complexity, increased fabrication costs, and a significant fraction of dead volume associated with the separator and collector such that there remains a strong trade-off between power and energy densities. An alternative to this approach that is perhaps simpler, more effective, and less costly is to employ multiscale materials having a hierarchical pore structure. If properly designed, such materials can effectively combine the desirable high storage densities of very small pores with a network of larger pores providing high ion transport rates on the device scale. Previous studies of hierarchical structures have focused largely on steady transport in branching networks such as those found in vascular networks, 6 leaf veins, 7 and river drainage basins. 8 Hierarchical structures have also been investigated for optimized branching networks that supply water, gas, and electricity 9 and for engineered materials that are intended to provide optimal heat extraction, 10 fluid distribution, 11 or catalytic conversion. 12,13 The optimization of hierarchical structures for ion transport and charge storage in double-layer capacitors has received much less attention despite the growing importance of energy storage for both electric vehicles and electric grid management in support of intermittent renewable sources. Numerous experimental investigations have demonstrated the impact of electrode morphology on capacitor performance, 14-22 but these studies have mainly addressed the relative performance of various materials, not optimization per se. And, while several good models of capacitor performance have been developed, 23-28 only a few theoretical studies have directly considered optimization. Building on an early model of porous electrodes, 29 Dunn and Newman 30 developed a rather complete analytical model of charging for a packed-bed electrode, including the effects of the separator thickness and finite solid-phase conductance, and applied this model to compute the optimum porosity and electrode thickness as a function of the charging time and other systems parameters. Although particle porosity was included in their analysis, the effect of finite transport rates within the intraparticle pores was neglected. Neglecting such intraparticle transport is consistent with the conclusion of Lanzi and Landau that intraparticle transport has little effect on the overall behavior, 31 but this generally remains valid only so long as ion diffusivities in the intraparticle pores are comparable to those in the interparticle void. This, however, is not usually the case for the materials of greatest interest at present. Koresh and Soffer, for example , showed that conductivities or diffusivities in nanoporous carbons were up to 9 orders of magnitude smaller than bulk values. 32 More recently, Eikerling et al. 33 employed a Cantor-block material model, along with appropriate complex impedances for each block element, to maximize the overall capacitance of a two-level structure. Their results yield the optimum size of the smallest pores and the associated optimum Cantor hierarchy for a specified thickness , along with a closed-form expression for the maximum capaci-tance. These results thus provide good guidance for specialized engineered structures but give little insight into the optimization of the packed-bed electrodes more commonly employed. In the present study, we consider ion transport and storage in a hierarchical porous material consisting of nanoporous particles and the interparticle porosity between these. Under several significant assumptions outlined below, we employ a combination of analytical and numerical techniques to determine the optimum interparticle porosity yielding the maximum energy deliverable in a specified time, accounting for both interparticle transport and hindered transport within the particles. This is equivalent to maximizing the mean power density. Periodic and step boundary conditions are included in the analysis; for the periodic case, both apparent and reactive powers are examined. Governing Transport Equations Consider a packed bed of uniform porous particles that are roughly spherical. As shown in Fig. 1, this bed is notionally half of a double-layer capacitor having a separator at the left boundary and a current collector at the right boundary. Ions reside within these porous particles and within the interparticle void volume, and these ions are transported within particles, between the particles and void, and to or from the open bed surface adjacent to the separator. We assume that the ion diffusivity in the interparticle void is much larger than that within the particles owing to hindered diffusion within the small particle pores, that the bed thickness is much z