Nonequilibrium Steady States in Fluid Transport through Mesopores

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Nonequilibrium Steady States in Fluid Transport through Mesopores: Dynamic Mean Field Theory and Nonequilibrium Molecular Dynamics Ashutosh Rathi, Stathis Kikkinides, David M. Ford, and Peter A Monson Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00112 • Publication Date (Web): 28 Mar 2019 Downloaded from http://pubs.acs.org on March 28, 2019

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Nonequilibrium Steady States in Fluid Transport through Mesopores: Dynamic Mean Field Theory and Nonequilibrium Molecular Dynamics A. Rathi,† E. S. Kikkinides,‡ D. M. Ford,¶ and P. A. Monson∗,† †Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003-9303, U.S.A. ‡Department of Chemical Engineering, Aristotle University of Thessaloniki, University Campus, 54124 Thessaloniki, Greece ¶Ralph E. Martin Department of Chemical Engineering, University of Arkansas, Fayetteville, AR 72701-1201, U.S.A E-mail: [email protected] Abstract We present a dynamic mean field theory (DMFT) and nonequilibrium dual control volume grand canonical molecular dynamics (GCMD) simulation study of steady state fluid transport in slit-shaped mesopores under an applied chemical potential gradient. The main focus is on states where the bulk conditions on one side of the pore would lead to a capillary condensed state in the pore at equilibrium while those on the other side would lead to a vapor state in the pore. This choice of conditions is motivated by certain separation applications in which condensable vapors permeate through mesoporous membranes. Under these circumstances we have found partially filled states with a liquid-like state at the high chemical potential end of the pore and a vapor-like state at the low chemical potential end. This phenomenon is accompanied by hysteresis.

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The existence of partially filled states has been hypothesized in previous work but the present paper reveals them as an emergent feature of the systems. We find that predictions of DMFT are in good qualitative agreement with the GCMD results overall. However, the GCMD results demonstrate that the transport is faster through the partially filled pore than through the unfilled pore, a feature not captured by DMFT.

Introduction The transport of pure fluids in mesopores (pore diameter between 2 and 50 nanometers) under nonequilibrium conditions has been widely studied both experimentally and theoretically. 1–11 Much of the interest is centered around the application area of membrane separations. 12–17 Mesoporous membranes have the potential for gas separations with both high permeability and high selectivity, if they operate under conditions that take advantage of adsorption and capillary condensation inside the mesoporous structure of the membrane. In such cases there is preferential transport of the condensable component(s) over the noncondensable one(s), resulting in significant enhancement of the respective membrane selectivity. Burggraaf and coworkers 3 studied permeation of propylene and nitrogen in a gamma alumina membrane. Uchytil and coworkers 5,6,10 studied permeation of butane and isobutane through vycor porous glass and attempted to correlate it to the arithmetic mean of inlet and exit pressure. Sidhu and Cussler 9 studied transport of nitrogen through Anodisc and track etched polycarbonate membranes. The flux through the membranes was found to depend on the bulk conditions applied at either end of the membrane, presumably through the influence of these conditions on the state of the fluid in the membrane pores. The limitations of current experimental apparatus prohibit us from being able to confirm and investigate these states at microscopic level. There have been attempts to correlate experimentally observable quantities like flux, pressure and mesoporous structure 8 to these states through transport models. Lee and Hwang 2 attempted to describe the possible states shown in Fig. 1, through a combination of continuum models, assuming that the porous structure is represented by 2


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a bundle of cylindrical capillaries. They used Knudsen diffusion for gas flow. The flow of condensate was modeled using Darcy’s model for flow through porous media, which reduces to the Hagen-Poiseuille model if the porous medium is represented by a bundle of cylindrical capillaries, and the surface flow was modeled using Gilliland’s hydrodynamic flow model. 18 Ever since then going on 30 years now, people have made numerous attempts to reconcile the experimental observations with continuum models. 5,6,8,9,19,20 Recently, Petukhov and coworkers 11 used a combination of Hagen-Poiseuille model for liquid flow and Knudsen diffusion model for gas flow in an attempt to explain states similar to the one shown in Fig. 1(b) for anodized alumina membranes. Among the various states of fluid distribution inside mesopores under nonequilibrium conditions, the one where capillary condensed fluid does not completely fill the pore and is confined to the high pressure side as shown in Fig. 1(b), is of particular interest to us because it is found only under nonequilibrium conditions. Equilibrium adsorption studies show complete filling of mesopores once the vapor nucleates at high enough relative vapor pressures of pure fluid. The final equilibrium steady state achieved is similar to that shown in Fig. 1(c) with Pin = Pout . This state is in stark contrast with the nonequilibrium steady state with capillary condensation confined to high pressure side (Fig. 1(b)). Experimental differential permeability studies 3,11 have attributed an increase in permeance (ratio of flux to pressure difference across membrane) to the attainment of this state. (A differential permeability experiment 4 is the measurement of permeance across a membrane under steady state with small relative pressure difference between inlet and exit.) An increase in permeance as result of capillary condensation seems counter intuitive but, in the absence of any in-depth transport analysis at microscopic scale, has been the basis of all proposed hypotheses and models in literature. In the last two decades molecular simulation methods and classical density functional theory have dramatically increased the understanding of the thermodynamics of fluids confined in mesoporous materials, and especially capillary transitions and hysteresis. 21–23 Recently,

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this work has been extended to the study of transport in such materials through the development of dynamic mean field theory (DMFT). 24–30 This allows a treatment of the transport through porous materials which is fully consistent with the thermodynamic treatment from density functional theory. In the present study, we have employed DMFT and molecular dynamics to study the transport of condensable fluids in model porous membranes under nonequilibrium conditions. The model porous membrane is represented by a single slit pore to decouple the complexity of the pore structure from the complexity of the transport mechanism. A comparative study of dynamics of a system starting from an initial equilibrium state in approach to another equilibrium or nonequilibrium state is conducted. Furthermore, we have observed the presence of multiple steady states and hysteresis in nonequilibrium systems and investigated the factors and conditions affecting this phenomenon. We have investigated the transport in these states containing both vapor and liquid phases by comparing the results obtained from DMFT with more detailed off-lattice dual control volume grand canonical molecular dynamics (GCMD). 31–35 The results of this study can be extended and generalized to more complex porous media, in the form of pore networks and amorphous structures (e.g. Vycor glass).

Models and Methods Lattice Gas Model and Mean Field Theory Our theoretical approach makes use of a lattice gas model. The lattice sites are subject to constraints of single occupancy and nearest-neighbor interactions only. The external field {φ} is taken into account by means of interaction between the pore wall and fluid sites nearest to it. The attractive interaction between neighboring occupied fluid sites is defined by the parameter . The grand free energy of the system in the mean field approximation

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is 30 Ω = kT

X

[ρi ln ρi + (1 − ρi ) ln (1 − ρi )] −

i

X XX ρi ρi+a + ρi (φi − µ) 2 i a i

(1)

where i denotes a lattice vector, a denotes a nearest neighbor lattice vector defined relative to i and ρi is the density, or more precisely the dimensionless average occupancy, at site i. µ is the chemical potential, k is the Boltzmann constant and T is the absolute temperature. We can minimize the grand free energy of the system with respect to {ρi } in order to find the distribution of fluid density at thermodynamic equilibrium. The minimization leads to a set of nonlinear equations Eq.2 with variables as ρi and parameters {φ} and equilibrium chemical potential µ. X ρi − ρi+a + φi − µ = 0 ∀ i kT ln 1 − ρi a

(2)

These nonlinear equations are solved iteratively for a specific set of external field {φ} and chemical potential µ to find the equilibrium density distributions in the system. We can derive the pressure and chemical potential of the fluid under bulk conditions where density is uniform (ρb ) and there is no external field. 23

Ω/M = −φP = kT ln (1 − ρb ) + 3ρb 2 i h ρb µ = kT ln 1−ρb − 6ρb

(3) (4)

where φP = P νs is lattice equivalent of bulk pressure in lattice model, νs is physical volume equivalent of a lattice site, ρb is the bulk density and M is system size in lattice sites.

Dynamic Mean Field Theory Dynamic mean field theory can be derived by applying mean field approximation to the master equation of Kawasaki dynamics for the lattice model. 26,30,36 The Kawasaki dynamics is essentially nearest neighbor hopping and the hops are accepted or rejected in this case 5


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in accordance with Metropolis transition probabilities. In this study, the formulation of dynamic mean field theory that we have used is based on previous implementations. 26,30,36 The evolution of density distribution in the system with time (ρi ) can be written as: X ∂ρi =− Ji,i+a ∂t a

(5)

where Ji,i+a is the total flux from site i to a neighboring with i + a neighbor. Theses fluxes can be written in the form

Ji,i+a = ωi,i+a ρi (1 − ρi+a ) − ωi+a,i ρi+a (1 − ρi )

(6)

where ωi,i+a is the transition probability of hopping from site i to site i + a and can be expressed as: −Ei,i+a ωi,i+a = ω0 exp kT    0, Ei+a < Ei Ei,i+a =   Ei+a − Ei , Ei+a ≥ Ei P Ei = − ρi+a + φi

(7)

(8) (9)

a

where ω0 is the hopping frequency in the absence of interactions and sets the timescale for the calculations. The above expressions amount to a mean field approximation to Metropolis transition probabilities. Euler’s method is used to integrate Eq. 5 yielding the time evolution of the density distribution of fluid in the system. The dimensionless time scale for the system is defined as ω0 t. In previous work 30 DMFT was used to study the evolution of the density distribution for a fluid in a porous material following a change in the bulk chemical potential, thus modeling the dynamics of adsorption and desorption in the transition between two equilibrium (or metastable equilibrium) states. In both the initial and the final states the chemical potential 6


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was uniform throughout the system. In the present case we consider a pore in contact with two bulk phases with different chemical potentials at either end, thus simulating a nonequilibrium steady-state membrane permeation process. We also study the evolution of the density distribution in the approach to the nonequilibrium steady state. The system geometry for our DMFT calculations is shown in Fig. 2. There is bulk on both sides of the pore as well as control volumes to impose boundary conditions corresponding to inlet and exit. The control volumes consist of two layers of lattice sites where the density is fixed at the bulk fluid density for the given value of the chemical potential. The dimensionless chemical potential is defined as µ∗ =

µ

while the fluid-wall interaction parameter is taken as

3.0f f . The dimensionless temperature is defined as T ∗ =

kT

while dimensionless pressure

is P ∗ = Pvs where vs is volume of a lattice site. The relative activity is defined as λR = ∗ ∗ µ −µsat exp where µ∗sat is the chemical potential of the bulk saturated vapor, with µ∗sat = T∗ −3.0 for a simple cubic lattice gas model. We consider a slit pore with width of H = 6 lattice sites and pore length L = 40 lattice sites. Bulk regions on either side of the pore are 10 lattice sites in length with one lattice site long control volume. We have studied the system at T ∗ = 1.0 which is 66% of the critical ∗ temperature Tcritical = 1.5. All the boundary conditions at inlet or exit correspond to vapor

state of the fluid and thus λR,inlet , λR,exit < 1.

Dual control volume grand canonical molecular dynamics Molecular simulations were performed using GCMD 31–35 allowing the study of steady state transport between dual bulk control volumes with different chemical potentials. LennardJones(LJ) potentials were used for both the wall-fluid and fluid-fluid interactions and the LAMMPS code 37 was used for these calculations. A schematic of the slit pore system used in our studies is shown in Fig. 3(a). The slit pore is at the center of the system with bulk regions on each side. At each end of the system there is a wall of atoms which interact repulsively with the fluid molecules. Each bulk region is divided into two subregions. One 7


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of these, furthest from the pore, is the control volume in which grand canonical Monte Carlo (GCMC) moves are made, in addition to molecular dynamics, in order to control the chemical potential. In the other region, denoted as buffer in the figure, as well as in the pore, only molecular dynamics is performed. A dissipative particle dynamics thermostat 38–40 was used for temperature control in the system. This thermostat was used in order to minimize the impact of temperature control on velocity distribution of fluid atoms in the system. A cycle of GCMC was performed every 150 molecular dynamics steps. Each cycle of GCMC consisted of 100 attempted creations/destructions of the atoms in the control volumes. The system dimensions were 234σ, where σ is the LJ size parameter, in the x-direction, 11σ in the z-direction and 10.39σ in y-direction. The system was periodic in the y- and zdirections. The slit pore was of length 100σ, with a width of 8σ, and the control volume lengths were 40σ. The buffer zone between control volume and slit pore is 5σ. All the walls are made up of atoms packed in an FCC(111) configuration. The slit pore walls are placed at z=1.5σ and z=9.5σ in order to prevent interaction of atoms near one pore wall with atoms near the other pore wall due to periodicity of system in z-direction. Repulsive walls were placed along the y-direction in the gap between z-boundary and the edge of the closest slit pore wall. A pair of repulsive walls are placed at 2σ from the ends of simulation box in the x-direction. These repulsive walls sprevent the mixing of atoms from one side of pore to another in x-direction, thus allowing the flow of atoms between the two control only through the slit pore. A buffer zone of 20σ, not shown in the figure, is used between the repulsive walls at the ends of simulation box and the respective control volumes. A timestep of 0.004τ 2

1

(τ = ( σ m ) 2 ) was used for the molecular dynamics where m is the mass of the particles and is the LJ well depth. The LJ well depth for the fluid-fluid interaction was the same as LJ well depth for the fluid-pore wall interaction. The LJ potential was cut and shifted at 2.5σ for fluid-pore wall and fluid-fluid interactions. The LJ potential for interactions between 1

repulsive wall and fluid was cut and shifted at 2 6 σ. The molecular simulations were carried out at a temperature of kT / = 0.75.

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Results and Discussion DMFT We begin by considering the dynamics in the approach to a nonequilibrium steady state and contrast that with the approach to equilibrium (adsorption dynamics). As a reference to the processes studied, Fig. 4 shows the adsorption/desorption isotherm for a fluid in the slit pore. We focus on two states either side of the hysteresis loop associated with capillary condensation (state A in with λR,inlet = λR,exit = 0.719 and state B with λR,inlet = λR,exit = 0.998). In both calculations the system is initialized with the mean field density distribution for state A. For the approach to a nonequilibrium steady state study we then change the bulk density in control volume on the inlet side of the pore to the value associated with state B. For the approach to equilibrium study we change the bulk density in both control volumes to the value associated with state B. In both cases we then evolve the density distribution by integration of Eq. 5. Fig. 5 shows the evolution of the pore average density in the two cases. For the approach to equilibrium we see that the density remains initially steady and then a sharp increase occurs as the pore fills with liquid. In the approach to the nonequilibrium steady state we see a two stage filling process with a more gradual initial stage followed by a sharp increase as the density increases to its final value. We see that in this case the final average density is substantially less than for the approach to equilibrium, indicating that the liquid does not entirely fill the pore. We note that the dynamics in approach to the nonequilibrium steady state is much slower. This is because the flux into the pore only comes from one side (inlet) and there is a flux out of the pore on the exit side as well. For the approach to equilibrium case there is a flux of fluid into the pore from both sides. Fig. 6 shows visualizations of the density distributions from the two processes. Those on the left are associated with the approach to equilibrium. This process has been described before, 30 and involves local build up of density near the walls followed by nucleation of liquid

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bridges at each end of the pore. These bridges grow in size finally merging to fill the pore with liquid. The images on the right are associated with the approach to the nonequilibrium steady state. We see a local build up of density near the pore entrance followed much later by the nucleation of a liquid bridge which then grows in size but reaches a limiting size so that the steady state is a partially filled pore. Just as there is hysteresis in adsorption/desorption isotherms for mesoporous materials such as that shown in Fig. 4, we should anticipate that it can also be found in the context of these partially filled steady states. To investigate this we carried out two sets of calculations for a pore with dimensions L = 40 and H = 6. The inlet of the pore was kept at constant relative activity of 0.99 (near saturation). The exit relative activity in one case was varied from 0.99 to 0.0508 while in another case it is varied from 0.0497 to 0.97. The initial condition for each state after the first was taken as the steady state density distribution from the previous calculation in the sequence. The pore average density from the two sequences of calculations is shown in Fig. 7. When the exit state relative activity is increased starting from a low value we see that the pore averaged density increases slowly up to a point where there is a sudden large increase associated with condensation in the pore. On the other hand when the exit state relative activity is decreased starting from a high value we see that most of the density loss occurs steadily across a range of activity, until a point where the density drops discontinuously by a small amount. There is a quite wide range of exit activity where there are two steady states of the system. Visualizations of the density distributions for the states shown in Fig. 7 are shown in Fig. 8. The images are consistent with the formation of partially filled steady states. The sequence of states for decreasing exit relative activity show that the higher density branch of the hysteresis loop in Fig. 7 represents partially condensed states with progressively shorter liquid bridges. This is similar to the equilibrium case where the increasing activity branch requires a nucleation process to fill the pore, while the decreasing activity branch proceeds by evaporation of fluid from the pore exit and movement of the vapor-liquid menisci towards the center.

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We now move on to consider the nature of the transport in the system for these nonequilibrium steady states. For this purpose it is useful to focus on the two states shown in Fig. 9, which are high and low density steady states with the same entrance and exit conditions. Fig. 10 shows the average density along the x-direction for the two steady states. The sharp changes in density at the entrance and exit to the pore are evident as well as the vapor-liquid interface within the pore for system 1. We observed that the steady state flux in System 2 was greater than that in System 1 by approximately 19%. The density and velocity distributions in the z-direction are illustrated in Fig. 11 for the two regions denoted Region 1 and Region 2 in Fig. 9, for each system. Although there is no explicit velocity variable in the formulation of the DMFT, the velocities can be obtained by dividing the local flux by the local density. 41 The local density in the lattice model used in DMFT cannot display the fine structure associated with layering so we only see high density near the pore walls changing smoothly to a lower density in the center of the pore. In region 1, system 1 has a much higher density than system 2 in the middle of the pore, as seen in Fig. 11(a), because system 1 is filled with liquid in that region. The velocities are higher in region 2 than in region 1 reflecting the lower densities in that region. It is interesting that this phenomenon is seen even in system 2, where the small enhancement of density caused by multilayer adsorption in region 1 (Figs. 9(b) and 11(c)) is coupled with a significant lowering of the velocity (Fig. 11(d)).

GCMD To assess the predictions from DMFT we conducted a similar study of the transport using GCMD. We considered two nonequilibrium steady states with λR,inlet = 0.9345 and λR,exit = 0.14. Under equilibrium conditions a slit pore with λR,inlet = λR,exit = 0.9345 would correspond to a pore completely filled with capillary condensed fluid while λR,inlet = λR,exit = 0.14 would correspond to a pore with vapor phase and strongly adsorbed monolayer of atoms on the pore walls. The nonequilibrium steady states formed by the combination of these relative 11


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inlet and exit activities result in two systems: (1) a pore partially filled with capillary condensed fluid (GCMD1) and (2) a pore with adsorbed monolayer and vapor phase (GCMD2). These states are illustrated in Fig. 12. In order to achieve GCMD1 and GCMD2, a procedure similar to that in case of DMFT was followed. The starting point for GCMD1 was an equilibrium system with both control volumes at λR = 0.9345. The exit relative activity was set to λR,exit = 0.14. GCMD was performed on the system to achieve a nonequilibrium steady state. The starting point for GCMD2 was an equilibrium system at λR = 0.14. We increased the λR,inlet to 0.9345 and performed GCMD to achieve the steady state. The density and velocity profiles were collected every 10 MD time steps. The results shown are calculated by averaging over 3.11×106 steady state profiles for GCMD1 and 7.31×106 steady state profiles for GCMD2. A comparison of fluxes in the two nonequilibrium steady states, indicated that the flux of system with capillary condensed fluid (GCMD1) was greater than flux of system with only adsorbed and vapor phase (GCMD2) by about 19%. This is counter to the prediction from DMFT where System 2 had higher flux than System 1. The density profiles along x-direction in the two nonequilibrium steady states found using GCMD are shown in Fig. 13. These are qualitatively quite similar to those obtained from DMFT and shown in Fig. 10. The significant difference lies in region 2 where we see that in the GCMD case the density for the GCMD1 state is significantly higher than that of the GCMD2 state. They are almost identical for System 1 and System 2 in the DMFT results. The z-direction density distributions for regions 1 and 2 together with the associated velocity profiles are shown in Fig. 14. These density profiles exhibit additional structure associated with layering effects that are not seen in the DMFT as noted above. But the overall picture of high density near the pore walls and a lower density in the pore interior are common to both. The overall shape of the velocity profiles is quite similar to those from DMFT, with a lower velocity near the pore wall. However, for region 2 in the GCMD1 results and both regions for the GCMD2 results there is evidence of a slight plateau in the velocity profile that coincide with the second peak in the density profile. This is another example of an increase

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in local density causing a reduction, or in this case a reduction in the rate of increase, in local x-velocity.

Origin of the discrepancy between the flux difference obervations in DMFT and GCMD We believe that that the discrepancy in the predicted flux difference for systems 1 and 2 between DMFT and GCMD is caused by the lack of hydrodynamic interactions in DMFT. We have recently shown the importance of hydrodynamic interactions, which are absent in the DMFT model for the case of spinodal decomposition. 41 To further support this argument, we have employed Computational Fluid Dynamics (CFD) to solve numerically the continuity and the momentum equations for the case of a lattice fluid flowing in a slit pore with the same geometry as that of Figure 2, including a mean field description of the thermodynamics consistent with DMFT. Details on the CFD methodology and results are presented in the Supporting Information. It is seen that the CFD simulations produce similar density configurations with the ones produced by DMFT and GCMD for both systems 1 and 2 (see figure S2). Axial velocity profiles, on the other hand, are in better agreement with those resulting from the GCMD simulations than those of the DMFT model. In these cases we observe a distinct drop in the velocity gradient near the wall, due to the presence of viscous forces. A comparison of the fluxes from CFD simulations in the two systems shows that the flux in the system with capillary condensed fluid is greater than the flux in the system with only adsorbed and vapor phase by about 21%. This result is in accord with GCMD simulations and in contrast to the DMFT predictions for the same systems.

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Summary and Conclusions We have presented a study of nonequilibrium steady states for fluids in slit-shaped mesopores using lattice-based theory and off-lattice simulation. Our principal finding is existence of steady states where the pore is partially filled with liquid near the high chemical potential end of the pore. While such states have been hypothesized in previous modeling work the present study shows them to be emergent properties of the system. We have observed these states in both DMFT and GCMD. Accompanying this phenomenon is a hysteresis that allows for steady states of high and low density at the same set of entrance and exit conditions. The hysteresis has the same origin as that for adsorption/desorption and is associated with the metastability of the lower density state prior to nucleation of a capillary condensed state. Comparisons of the DMFT predictions with the GCMD results suggest that DMFT provides a qualitatively accurate description. One important feature of the systems that is not captured by DMFT is that the flux through systems with condensed states is faster than for uncondensed states with the same entrance and exit conditions. We have argued that this is related to how transport is described in the lattice model via Kawasaki dynamics and presented CFD results in support of this. A useful avenue for additional research on this will be the inclusion of hydrodynamics in DMFT. 41 Our results have implications for understanding the performance of mesoporous membranes for separating gases with wide difference in volatility. Such separations depend on the species fractionation produced by having a liquid phase in the pore. The fraction of the pore space occupied by the liquid phase will have significant effect upon the fluxes of the species through the pore and also the degree of separation achieved.

Supporting Information A description of CFD calculations and the results that were referred to in the results and discussion section. 14


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Acknowledgments This material is based upon work supported by the National Science Foundation under Grant No. CBET 1403542.

References (1) Rhim, H.; Hwang, S. T. Transport of capillary condensate. Journal of Colloid and Interface Science 1975, 52, 174 – 181. (2) Lee, K. H.; Hwang, S. T. The transport of condensible vapors through a microporous vycor glass membrane. Journal of Colloid And Interface Science 1986, 110, 544–555. (3) Uhlhorn, R. J. R.; Keizer, K.; Burggraaf, A. J. Gas-transport and separation with ceramic membranes .1. Multilayer diffusion and capillary condensation. Journal of Membrane Science 1992, 66, 259–269. (4) Tzevelekos, K. P.; Romanos, G. E.; Kikkinides, E. S.; Kanellopoulos, N. K.; Kaselouri, V. Experimental investigation on separations of condensable from noncondensable vapors using mesoporous membranes. Microporous and Mesoporous Materials 1999, 31, 151–162. (5) Uchytil, P.; Petrickovic, R.; Thomas, S.; Seidel-Morgenstern, A. Influence of capillary condensation effects on mass transport through porous membranes. Separation and Purification Technology 2003, 33, 273–281. (6) Uchytil, P.; Petrickovic, R.; Seidel-Morgenstern, A. Study of capillary condensation of butane in a Vycor glass membrane. Journal of Membrane Science 2005, 264, 27–36. (7) Lee, H. J.; Yamauchi, H.; Suda, H.; Haraya, K. Influence of adsorption on the gas permeation performances in the mesoporous alumina ceramic membrane. Separation and Purification Technology 2006, 49, 49–55. 15


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(8) Uchytil, P.; Loimer, T. Large mass flux differences for opposite flow directions of a condensable gas through an asymmetric porous membrane. Journal of Membrane Science 2014, 470, 451–457. (9) Sidhu, P. S.; Cussler, E. L. Diffusion and capillary flow in track-etched membranes. Journal of Membrane Science 2001, 182, 91–101. (10) Loimer, T.; Uchytil, P.; Petrickovic, R.; Setnickova, K. The flow of butane and isobutane vapors near saturation through porous Vycor glass membranes. Journal Of Membrane Science 2011, 383, 104–115. (11) Petukhov, D. I.; Berekchiian, M. V.; Pyatkov, E. S.; Solntsev, K. A.; Eliseev, A. A. Experimental and Theoretical Study of Enhanced Vapor Transport through Nanochannels of Anodic Alumina Membranes in a Capillary Condensation Regime. The Journal of Physical Chemistry C 2016, 120, 10982–10990. (12) Huang, P.; Xu, N.; Shi, J.; Lin, Y. Recovery of Volatile Organic Solvent Compounds from Air by Ceramic Membranes. Industrial & Engineering Chemistry Research 1997, 36 . (13) Baker, R. W. Future directions of membrane gas separation technology. Industrial and Engineering Chemistry Research 2002, 41, 1393–1411. (14) Baker, R. W.; Low, B. T. Gas Separation Membrane Materials: A Perspective. Macromolecules 2014, 47, 6999–7013. (15) Bernardo, P.; Drioli, E.; Golemme, G. Membrane Gas Separation: A Review/State of the Art. Industrial and Engineering Chemistry Research 2009, 48, 4638–4663. (16) Burggraaf, A.; Cot, L. Fundamentals of Inorganic Membrane Science and Technology; Membrane Science and Technology Series; Elsevier Science B.V., 1996; Vol. 4.

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(17) Pandey, P.; Chauhan, R. S. Membranes for gas separation. Progress in Polymer Science 2001, 26, 853–893. (18) Gilliland, E. R.; Baddour, R. F.; Russell, J. L. Rates of flow through microporous solids. AIChE Journal 1958, 4, 90–96. (19) Kainourgiakis, M. E.; Stubos, A. K.; Konstantinou, N. D.; Kanellopoulos, N. K.; Milisic, V. A network model for the permeability of condensable vapours through mesoporous media. Journal Of Membrane Science 1996, 114, 215–225. (20) Choi, J. G.; Do, D. D.; Do, H. D. Surface diffusion of adsorbed molecules in porous media: Monolayer, multilayer, and capillary condensation regimes. Industrial and Engineering Chemistry Research 2001, 40, 4005–4031. (21) Evans, R. Fluids adsorbed in narrow pores: phase equilibria and structure. Journal of Physics: Condensed Matter 1990, 2, 8989. (22) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Phase separation in confined systems. Reports on Progress in Physics 1999, 62, 1573–1659. (23) Monson, P. A. Understanding adsorption/desorption hysteresis for fluids in mesoporous materials using simple molecular models and classical density functional theory. Microporous and Mesoporous Materials 2012, 160, 47 – 66. (24) Edison, J. R.; Monson, P. A. Dynamic mean field theory of condensation and evaporation in model pore networks with variations in pore size. Microporous And Mesoporous Materials 2012, 154, 7–15. (25) Edison, J. R.; Monson, P. A. Dynamic Mean Field Theory for Lattice Gas Models of Fluid Mixtures Confined in Mesoporous Materials. Langmuir 2013, 29, 13808–13820. (26) Matuszak, D.; Aranovich, G. L.; Donohue, M. D. Lattice density functional theory of molecular diffusion. Journal of Chemical Physics 2004, 121, 426–435. 17


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(27) Matuszak, D.; Aranovich, G. L.; Donohue, M. D. Thermodynamic driving force for molecular diffusion-lattice density functional theory predictions. Journal of NonEquilibrium Thermodynamics 2006, 31, 355–384. (28) Matuszak, D.; Aranovich, G. L.; Donohue, M. D. Modeling fluid diffusion using the lattice density functional theory approach: counterdiffusion in an external field. Physical Chemistry Chemical Physics 2006, 8, 1663–1674. (29) Matuszak, D.; Aranovich, G. L.; Donohue, M. D. Single-component permeation maximum with respect to temperature: A lattice density functional theory study. Industrial and Engineering Chemistry Research 2006, 45, 5501–5511. (30) Monson, P. A. Mean field kinetic theory for a lattice gas model of fluids confined in porous materials. The Journal of chemical physics 2008, 128, 084701. (31) Heffelfinger, G.; Ford, D. Massively parallel dual control volume grand canonical molecular dynamics with LADERA I. Gradient driven diffusion in Lennard-Jones fluids. Molecular Physics 1998, 94, 659–671. (32) Cracknell, R. F.; Nicholson, D.; Quirke, N. Direct Molecular-Dynamics Simulation of Flow Down a Chemical-Potential Gradient in a Slit-Shaped Micropore. Physical Review Letters 1995, 74, 2463–2466. (33) Heffelfinger, G.; Vanswol, F. Diffusion in Lennard-Jones Fluids Using Dual ControlVolume Grand-Canonical Molecular-Dynamics Simulation (Dcv-Gcmd). The Journal of Chemical Physics 1994, 100, 7548–7552. (34) Macelroy, J. Nonequilibrium Molecular-Dynamics Simulation of Diffusion and Flow in Thin Microporous Membranes. The Journal of Chemical Physics 1994, 101, 5274–5280. (35) Papadopoulou, A.; Becker, E.; Lupkowski, M.; Vanswol, F. Molecular-Dynamics and

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Monte-Carlo Simulations in the Grand Canonical Ensemble - Local Versus Global Control. The Journal of Chemical Physics 1993, 98, 4897–4908. (36) Gouyet, J.-F.; Plapp, M.; Dieterich, W.; Maass, P. Description of far-from-equilibrium processes by mean-field lattice gas models. Advances in Physics 2003, 52, 523–638. (37) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. Journal of Computational Physics 1995, 117, 1 – 19. (38) Groot, R. D.; Warren, P. B. Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. The Journal of Chemical Physics 1997, 107, 4423. (39) Soddemann, T.; D¨ unweg, B.; Kremer, K. Dissipative particle dynamics: A useful thermostat for equilibrium and nonequilibrium molecular dynamics simulations. Physical Review E 2003, 68, 046702. (40) Yong, X.; Zhang, L. T. Thermostats and thermostat strategies for molecular dynamics simulations of nanofluidics. The Journal of Chemical Physics 2013, 138, 084503. (41) Kikkinides, E. S.; Monson, P. A. Dynamic density functional theory with hydrodynamic interactions: Theoretical development and application in the study of phase separation in gas-liquid systems. The Journal of Chemical Physics 2015, 142, 094706.

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Page 20 of 34

(a) Pout < Pin < Pcap

(d) Pout < Pcap < Psat < Pin

(b) Pout < Pcap < Pin < Psat

(e) Pcap < Pout < Psat < Pin

(c) Pcap < Pout , Pin < Psat

(f) Pcap , Psat < Pout , Pin

Figure 1: Indicative density visualizations of fluid distribution inside a mesopore for various possible equilibrium and nonequilibrium steady states. Here Pin is the inlet bulk pressure, Pout is the outlet bulk pressure, Psat is the bulk saturation pressure, and Pcap is the bulk pressure at which capilary condensation is observed.

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Figure 2: Geometry of the slit pore used to study fluid behavior under equilibrium and nonequilibrium conditions

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(a) Schematic

(b) Actual system

Figure 3: Schematic and actual system for molecular simulations

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1

Pore averaged density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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State B

0.8

0.6 State A

0.4

0.2

0 0

0.2

0.4 0.6 0.8 Relative activity(λR)

1

Figure 4: Isotherm calculated for H = 6, L = 40 using mean field theory at T ∗ = 1.0. States A and B referred to the text are shown on the isotherm.

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1 0.9 0.8 ρpore averaged

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.7 0.6 0.5 0.4 2

10

3

4

10 10 time(ω t)

5

10

6

10

0

Figure 5: Evolution of pore averaged density with time during the relaxation dynamics of fluid from one state to another. Solid line - Dynamics in the approach to equilibrium; Dashed line - Dynamics in the approach to nonequilibrium steady state.

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(a) ω0 t = 100

(f) ω0 t = 200

(b) ω0 t = 6300

(g) ω0 t = 190400

(c) ω0 t = 6500

(h) ω0 t = 1901200

(d) ω0 t = 14100

(i) ω0 t = 217600

(e) ω0 t = 40000

(j) ω0 t = 396800

Figure 6: (a)→(e) Visualizations of density in the pore during the dynamics of system starting from one equilibrium state in approach to another equilibrium state (f)→(j) Visualizations of density in the pore during the dynamics of system starting from an equilibrium state in approach to a nonequilibrium state

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1 0.9

Pore averaged density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 34

R,exit,initial

=0.99

R,exit,initial

=0.0498

0.8 0.7 0.6 0.5 0.4 0.3

0

0.2

0.4

0.6

R,exit

0.8

1

Figure 7: Pore averaged density for a set of nonequilibrium steady states corresponding to λR,inlet = 0.99, solid line - λR,exit − 0.99 → 0.508 , dashed line - λR,exit − 0.0497 → 0.97.

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(a) λR,exit = 0.99

(g) λR,exit = 0.0497

(b) λR,exit = 0.73

(h) λR,exit = 0.46

(c) λR,exit = 0.66

(i) λR,exit = 0.74

(d) λR,exit = 0.29

(j) λR,exit = 0.75

(e) λR,exit = 0.28

(k) λR,exit = 0.80

(f) λR,exit = 0.0508

(l) λR,exit = 0.97

(m) λR = 0.79

(n) λR = 0.97

Figure 8: (a)→(f) Visualizations of density distribution inside slit pore for nonequilibrium steady states corresponding to solid line in Fig. 7, λR,inlet = 0.99,λR,exit − 0.99 → 0.0508 (g)→(l) Visualizations of density distribution inside slit pore for nonequilibrium steady states corresponding to dashed line in Fig. 7, λR,inlet = 0.99, λR,exit − 0.0497 → 0.97. (m) Visualization of density distribution inside slit pore of width H = 6 at equilibrium at λR,inlet = λR,exit = 0.79 (Fig. 4). (n) Visualization of density distribution inside slit pore of width H = 6 at equilibrium at λR,inlet = λR,exit = 0.97 (Fig. 4).

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(a) System 1

(b) System 2

Figure 9: Nonequilibrium steady states as obtained from DMFT with λR,inlet = 0.99, λR,exit = 0.67, T ∗ = 1.0, = 1.0 (a) State with partial capillary condensation (b) State with only vapor and adsorbed fluid phase

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Page 29 of 34

1 System 1 System 2

0.8 Average density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.6

0.4

0.2

0

10

20

30 40 X coordinate

50

60

Figure 10: Density distribution along x-direction for two systems as shown in Fig. 9

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Langmuir

−3

1

7

x 10

Region 1 Region 2

6 5

Region 1 Region 2

0.6

x−velocity

Average density

0.8

0.4

4 3 2

0.2 1 0 2

3

4 5 z−coordinate

6

0 2

7

3

(a)

4 5 z−coordinate

6

7

(b) −3

1

7 Region 1 Region 2

0.8

x 10

Region 1 Region 2

6 5 x−velocity

Average density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 34

0.6

0.4

4 3 2

0.2 1 0 2

3

4 5 z−coordinate

6

0 2

7

3

(c)

4 5 z−coordinate

6

7

(d)

Figure 11: Density and x-velocity profiles along z-coordinates for System 1 (top) and System 2 (bottom) in Region 1 and Region 2 as defined in Fig. 9.

30


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(a) GCMD1

(b) GCMD2

Figure 12: Nonequilibrium steady states as obtained from GCMD with λR,inlet = 0.9345, λR,exit = 0.14 (a) State with partial capillary condensation (b) State with only vapor and adsorbed fluid phase

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0.7 GCMD1 GCMD2

0.6 Average density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 34

0.5 0.4 0.3 0.2 0.1 0 0

50

100 150 X−coordinate

200

Figure 13: Density distribution along x-direction for two systems as shown in Fig. 12

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Page 33 of 34

3

0.03

Region 1 Region 2

2 1.5 1 0.5 0 0

Region 2 Region 1

0.025 Average x−velocity

Average density

2.5

0.02 0.015 0.01 0.005

2

4 6 z−coordinate

8

0 0

10

(a) Density profile

4 6 z−coordinate

8

10

0.03 Region 1 Region 2

Region 1 Region 2

0.025 Average x−velocity

2.5 2 1.5 1

0.02 0.015 0.01 0.005

0.5 0 0

2

(b) X-velocity profile

3

Average density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2

4 6 z−coordinate

8

0 0

10

(c) Density profile

2

4 6 z−coordinate

8

10

(d) X-velocity profile

Figure 14: Density and x-velocity profiles along z-coordinates for GCMD1 (top) and GCMD2 (bottom) in Region 1 and Region 2 as defined in Fig. 12.

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Case 1 Pin

Mesopore ~ 2-50 nm

Pout Adsorption Desorption

Partially condensed state

Case 2 Pout

Pin

Average mesopore density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 34

Pin-Pout

Complete vapor state

Lattice-based theory (DMFT) - fluxcase 1 < fluxcase 2 Molecular simulations (DCV-GCMD) - fluxcase 1 > fluxcase 2

Figure 15: Table of contents graphic. On the left are visualizations from GCMD and DMFT calculations showing steady states with higher density (partial condensation) and lower density (vapor) steady states. The right hand side shows the pore geometry and a schematic graph of the average density vs. the exit pressure at fixed inlet pressure, illustrating the hysteresis.

34


Nonequilibrium Steady States in Fluid Transport through Mesopores

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