D. Nicholson

Capillary condensation of adsorbates in porous materials.

Hysteresis in capillary condensation is important for the fundamental study and application of porous materials, and yet experiments on porous materials are sometimes difficult to interpret because of the many interactions and complex solid structures involved in the condensation and evaporation processes. Here we make an overview of the significant progress in understanding capillary condensation and hysteresis phenomena in mesopores that have followed from experiment and simulation applied to highly ordered mesoporous materials such as MCM-41 and SBA-15 over the last few decades.

Molecular transport in nanopores: a theoretical perspective

Molecular transport in nanopores plays a central role in many emerging nanotechnologies for gas separation and storage, as well as in nanofluidics. Theories of the transport provide an understanding of the mechanisms that influence the transport and their interplay, and can lead to tractable models that can be used to advance these nanotechnologies through process analysis and optimisation. We review some of the most influential theories of fluid transport in small pores and confined spaces. Starting from the century old Knudsen formulation, the dusty gas model and several other related approaches that share a common point of departure in the Maxwell-Stefan diffusion equations are discussed. In particular, the conceptual basis of the models and the validity of the assumptions and simplifications necessary to obtain their final results are analysed. It is shown that the effect of adsorption is frequently either neglected, or treated on an ad hoc basis, such as through the division of the pore flux into gas-phase and surface diffusion contributions. Furthermore, while it is commonplace to assume that cross-sectional pressure is uniform, it is demonstrated that this violates the Gibbs-Duhem relation and that it is the chemical potential that essentially remains constant in the cross-section, as near-equilibrium density profiles are preserved even during transport. The Dusty Gas model and Maxwell-Stefan model for surface diffusion are analysed, and their strengths and weaknesses discussed, illustrating the use of conflicting choices of frames of reference in the former case, and the importance of assigning appropriate values for the binary diffusivity in the latter case. The oscillator model, developed in this laboratory, which is exact in the low density limit under diffuse reflection conditions, is shown to represent an advance on the classical Knudsen formula, although the latter frequently appears as a fundamental part of many transport models. The distributed friction model, also developed in this laboratory for the study of multi-component transport at any Knudsen number is discussed and compared with previous approaches. Finally, the outlook for theory and future research needs are discussed.

Tractable molecular theory of transport of Lennard-Jones fluids in nanopores

We present here a tractable theory of transport of simple fluids in cylindrical nanopores, which is applicable over a wide range of densities and pore sizes. In the Henry law low-density region the theory considers the trajectories of molecules oscillating between diffuse wall collisions, while at higher densities beyond this region the contribution from viscous flow becomes significant and is included through our recent approach utilizing a local average density model. The model is validated by means of equilibrium as well nonequilibrium molecular dynamics simulations of supercritical methane transport in cylindrical silica pores over a wide range of temperature, density, and pore size. The model for the Henry law region is exact and found to yield an excellent match with simulations at all conditions, including the single-file region of very small pore size where it is shown to provide the density-independent collective transport coefficient. It is also shown that in the absence of dispersive interactions the model reduces to the classical Knudsen result, but in the presence of such interactions the latter model drastically overpredicts the transport coefficient. For larger micropores beyond the single-file region the transport coefficient is reduced at high density because of intermolecular interactions and hindrance to particle crossings leading to a large decrease in surface slip that is not well represented by the model. However, for mesopores the transport coefficient increases monotonically with density, over the range studied, and is very well predicted by the theory, though at very high density the contribution from surface slip is slightly overpredicted. It is also seen that the concept of activated diffusion, commonly associated with diffusion in small pores, is fundamentally invalid for smooth pores, and the apparent activation energy is not simply related to the minimum pore potential or the adsorption energy as generally assumed.

Molecular transport in nanopores

Simulation of the transport of methane in cylindrical silica mesopores have been performed using equilibrium and nonequilibrium molecular dynamics (NEMD) as well as dual control volume grand canonical molecular dynamics methods. It is demonstrated that all three techniques yield the same transport coefficient even in the presence of viscous flow. A modified locally averaged density model for viscous flow, combined with consideration of wall slip through a frictional condition, gives a convincing interpretation of the variation of the transport coefficient over a wide range of densities, and for various pore sizes and temperatures. Wall friction coefficients extracted from NEMD simulations are found to be consistent with momentum transfer arguments, and the approach is shown to be more meaningful than the classical slip length concept.

On the Henry constant and isosteric heat at zero loading in gas phase adsorption.

The Henry constant and the isosteric heat of adsorption at zero loading are commonly used as indicators of the strength of the affinity of an adsorbate for a solid adsorbent. It is assumed that (i) they are observable in practice, (ii) the Van Hoffs plot of the logarithm of the Henry constant versus the inverse of temperature is always linear and the slope is equal to the heat of adsorption, and (iii) the isosteric heat of adsorption at zero loading is either constant or weakly dependent on temperature. We show in this paper that none of these three points is necessarily correct, first because these variables might not be observable since they are outside the range of measurability; second that the linearity of the Van Hoff plot breaks down at very high temperature, and third that the isosteric heat versus loading is a strong function of temperature. We demonstrate these points using Monte Carlo integration and Monte Carlo simulation of adsorption of various gases on a graphite surface. Another issue concerning the Henry constant is related to the way the adsorption excess is defined. The most commonly used equation is the one that assumes that the void volume is the volume extended all the way to a boundary passing through the centres of the outermost solid atoms. With this definition the Henry constant can become negative at high temperatures. Although adsorption at these temperatures may not be practical because of the very low value of the Henry constant, it is more useful to define the Henry constant in such a way that it is always positive at all temperatures. Here we propose the use of the accessible volume; the volume probed by the adsorbate when it is in nonpositive regions of the potential, to calculate the Henry constant.

On the Cavitation and Pore Blocking in Slit-Shaped Ink-Bottle Pores

We present GCMC simulations of argon adsorption in slit pores of different channel geometry. We show that the isotherm for an ink-bottle pore can be reconstructed as a linear combination of the local isotherms of appropriately chosen independent unit cells. Second, depending on the system parameters and operating conditions, the phenomena of cavitation and pore blocking can occur for a given configuration of the ink-bottle pore by varying the geometrical aspect ratio. Although it has been argued in the literature that the geometrical aspects of the system govern the evaporation mechanism (either cavitation or pore blocking), we here put forward an argument that the local compressibility in different parts of the ink-bottle pore is the deciding factor for evaporation. When the fluid in the small neck is strongly bound, cavitation is the governing process, and molecules in the cavity evaporate to the surrounding bulk gas via a mass transfer mechanism through the pore neck. When the pore neck is sufficiently large, the system of neck and cavity evaporates at the same pressure, which is a consequence of the comparable compressibility between the fluid in the neck and that in the cavity. This suggests that local compressibility is the measure of cohesiveness of the fluid prior to evaporation. One consequence that we derive from the analysis of isotherms of a number of connected pores is that by analyzing the adsorption branch or the desorption branch of an experimental isotherm may not lead to the correct pore sizes and the correct pore volume distribution.

Molecular Simulation of Excess Isotherm and Excess Enthalpy Change in Gas-Phase Adsorption

We present a new approach to calculating excess isotherm and differential enthalpy of adsorption on surfaces or in confined spaces by the Monte Carlo molecular simulation method. The approach is very general and, most importantly, is unambiguous in its application to any configuration of solid structure (crystalline, graphite layer or disordered porous glass), to any type of fluid (simple or complex molecule), and to any operating conditions (subcritical or supercritical). The behavior of the adsorbed phase is studied using the partial molar energy of the simulation box. However, to characterize adsorption for comparison with experimental data, the isotherm is best described by the excess amount, and the enthalpy of adsorption is defined as the change in the total enthalpy of the simulation box with the change in the excess amount, keeping the total number (gas + adsorbed phases) constant. The excess quantities (capacity and energy) require a choice of a reference gaseous phase, which is defined as the adsorptive gas phase occupying the accessible volume and having a density equal to the bulk gas density. The accessible volume is defined as the mean volume space accessible to the center of mass of the adsorbate under consideration. With this choice, the excess isotherm passes through a maximum but always remains positive. This is in stark contrast to the literature where helium void volume is used (which is always greater than the accessible volume) and the resulting excess can be negative. Our definition of enthalpy change is equivalent to the difference between the partial molar enthalpy of the gas phase and the partial molar enthalpy of the adsorbed phase. There is no need to assume ideal gas or negligible molar volume of the adsorbed phase as is traditionally done in the literature. We illustrate this new approach with adsorption of argon, nitrogen, and carbon dioxide under subcritical and supercritical conditions.

On The Cavitation and Pore Blocking in Cylindrical Pores with Simple Connectivity

We present a grand canonical Monte Carlo (GCMC) simulation of argon adsorption in connected cylindrical pores at 87.3 K. A number of pore models are constructed from various components: finite cylinder, finite cone, and flat surface. In the case of two cylinders of different sizes connected to each other with open ends, the adsorption isotherm can be described by a combination of two independent pores, the smaller of which is opened at both ends while the larger one is closed at one end. The adsorption isotherm depends on the relative size between the two sections of the connected pore. In the case of a cavity connected to the bulk surrounding gas via one or two narrower cylindrical necks, the phenomenon of either pore blocking or cavitation is observed, depending on the relative size between the neck and the cavity. If the neck size is smaller than a critical size, D(c), we observe cavitation, while pore blocking is observed when it is greater than D(c). This is due to the dominance of one of two mechanisms for removal of the adsorbates: either the receding of the menisci or the stretching of the fluid in the cavity. We also explore the effects of neck length and cavity length on the adsorption isotherm and conclude that while the neck length has a negligible effect on cavitation, it is of considerable importance when pore blocking occurs, because this process is controlled by the formation and movement of the meniscus in the pore neck. The effect of cavity length is found to be negligible in both cases.

On the existence of negative excess isotherms for argon adsorption on graphite surfaces and in graphitic pores under supercritical conditions at pressures up to 10,000 atm.

In this paper, we consider in detail the computer simulation of argon adsorption on a graphite surface and inside graphitic slit pores under supercritical conditions. Experimental results in the literature for graphitic adsorbents show that excess isotherms pass through a maximum and then become negative at high pressures (even for adsorption on open surfaces) when a helium void volume is used in the calculation of the excess amount. Here we show that, by using the appropriate accessible volume (which is smaller than the helium void volume), the excess isotherms still have a maximum but are always positive. The existence and the magnitude of this maximum is because the rate of change of the adsorbed density is equal to that of the bulk gas, which has a large change in bulk gas density for a small variation in pressure for temperatures not far above the critical point. However for temperatures far above T(c), this change in the bulk gas density is no longer significant and the maximum in the surface excess density becomes less pronounced and even disappears at high enough temperatures. The positivity of the adsorption excess persists for all pressures up to 10,000 atm for adsorption on surfaces and in slit pores of all sizes. For adsorption on a surface, the surface excess density eventually reaches a plateau at high pressures as expected, because the change in the adsorbed phase is comparable to that of the bulk gas. Positive excess lends support to our physical argument that the adsorbed phase is denser than the bulk gas, and this is logical as the forces exerted by the pore walls should aid to the compression of the adsorbed phase.

On the irreversibility of the adsorption isotherm in a closed-end pore

We present a simulation study of argon adsorption in a closed-end mesopore of uniform diameter in order to investigate the occurrence of hysteresis and propose two principal reasons for its existence: the variation in the shape and radius of curvature of the meniscus and the change in the packing of adsorbate during adsorption and desorption. This interpretation differs from classical theories that neglect both of these factors, and therefore find that adsorption-desorption in a closed-end pore is reversible. A detailed simulation study of the effects of temperature on the microscopic behavior of the adsorbate supports the interpretation proposed here.