Using Thin Zeolite Membranes and External Electric Fields To

Molecular dynamics has been used to study the separation of supercritical solutions using zeolite membranes. In the simulations, two thin membranes cu...
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Using Thin Zeolite Membranes and External Electric Fields To Separate Supercritical Aqueous Electrolyte Solutions S. Murad* and J. Lin Department of Chemical Engineering, University of Illinois at Chicago, Chicago, Illinois 60607

Molecular dynamics has been used to study the separation of supercritical solutions using zeolite membranes. In the simulations, two thin membranes cut from a cubic cell of ZK-4 zeolite were used as the semipermeable membranes to separate water from aqueous NaCl solutions. The results indicate that molecular dynamics is a viable technique for studying such separation processes at the fundamental molecular level. The study also showed that ZK-4 zeolite membranes show promise for use in membrane-based separation of aqueous electrolyte solutions, as well as other similar systems. Our simulations have also shown the important role external electric fields can play in enhancing the separation rate in such systems. Finally, through our simulations, we found the serious risks in applying principles of macroscopic hydrodynamics to nanoscale systems. Introduction Zeolites and related microporous materials have found widespread applications in many scientific and technological disciplines.1-3 These include catalysis in the petroleum and chemical industries, molecular sieve sorbents, ion exchange in separation processes, and so on. The remarkable chemical properties of zeolites arise from their complex porous crystalline aluminosilicate structure,2 in which the pore sizes are comparable to the molecular dimensions of many chemical substances. The framework thus plays a major role in the diffusion mechanisms and molecular selectivity inside the cavities and pores of a zeolite.3 In view of their thermal and chemical stability, zeolites are attractive candidates for use as inorganic membranes for a wide range of separations processes.1-3 This paper reports a molecular simulation study on the possibility of using thin zeolite membranes for separating supercritical aqueous electrolyte solutions based on the phenomenon of reverse osmosis/gas permeation. Osmosis and reverse osmosis are membrane-based separation phenomena of considerable interest because of their higher energy efficiency compared to traditional separation processes such as distillation.4-6 The applications of reverse osmosis include a wide range of processes such as desalination of seawater, separation of gas mixtures, removal of chemical and radioactive pollutants in wastes from many chemical processes, and so on. In our previous studies on liquid solutions, we have shown the viability of using molecular simulations to study membrane-based separation processes.7-8 We have in this study investigated the separation of supercritical aqueous electrolyte solutions (gas mixtures). We view this study as a molecular-level screening study to investigate the possibility of separating such mixtures using thin zeolite membranes. There has been considerable recent progress in the manufacture of zeolite membranes; they can therefore be realistically viewed as potential candidates for many separation processes.9-10 * Telephone: 312-996-5593. Fax: 312-996-0808. E-mail: [email protected].

The simulation scheme used is an adaptation of the one used for simple membranes in our previous studies.11-12 It was modified to model the separation of supercritical aqueous NaCl mixtures using ZK-4 zeolite membranes, in which reverse osmosis separations were then studied. The pore network of ZK-4 consists of a cubic lattice of cavities connected by narrow windows, so that the nearly spherical shape of the pore walls that interact with the sorbed molecules enhances the confinement effect.13-14 We have performed a series of molecular dynamics (MD)15 simulations to study the separations of water from aqueous NaCl solutions using ZK-4 as the separating membranes. Our research reported here has shown that thin zeolite membranes show considerable promise for use in separating supercritical (dense gas) aqueous electrolyte mixtures. In view of the unique behavior of supercritical gases,16 there is a lot of interest in using them for removing pollutants from a wide range of sources, especially solid. Method and Model Used The simulation method for studying membrane-based separations of liquid solutions or gas mixtures is based on the method developed by us, which has been described in detail elsewhere.7,11-12 It is therefore only summarized here. To set up the simulation system, the two zeolite membranes were placed centered at x ) Lx/4 and 3Lx/4 (Lx is the system dimension perpendicular to the membranes), using the known coordinates of the atoms that constitute the ZK-4 membrane17 (see Figure 1). The region between the two membranes was then filled with water molecules at the desired density, in an FCC configuration (because we are studying rather high temperatures, the FCC structure is randomized quite quickly; in other cases, configurations from a previous simulation could be used). The regions to the right of the right membrane and left of the left membrane were then filled with NaCl and water molecules at the appropriate desired density. Periodic boundary conditions were then used to allow these two sections to behave as one continuous solution compartment. All the membrane molecules were tethered to their equilibrium positions by a simple harmonic potential; the

10.1021/ie010425+ CCC: $22.00 © 2002 American Chemical Society Published on Web 10/31/2001

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Figure 1. Schematic of the simulation system.

magnitude of the harmonic potential constant, K, was 900 in reduced unit, based on the parameters of Na+. The initial topology of the framework bonds of the membranes is thus retained during the simulation. For the systems studied here, the results were not found to be qualitatively sensitive to the value of K, unless it was reduced significantly; quantitatively, if the value of K was decreased (which decreases the vibrational frequency of the tethered molecules), there was a slight but measurable increase in the membrane permeability. We have also tested models in which the membrane molecules were tethered to each other and found the results to be qualitatively similar.18 The simulation system in most studies consisted of either 2176 or 1214 particles in the basic cyclically replicated parallelepiped, 1152 or 704 of which constitute the two semipermeable membranes, as shown in Figure 1. In the larger system (2176 particles), the membrane is one unit cell wide, whereas in the smaller system (1214 particles), the unit cell was cut in half along the x direction to form the membrane. The molecules were given a Gaussian velocity distribution corresponding to the system temperature being investigated. In the solution compartment, NB molecules are

designated as solute molecules, while the remaining (512 - NB) or (256 - NB) are designated as solvent molecules; the number NB is determined by the concentration of the NaCl in the supercritical mixture being studied. The density of the solution being studied determines the width of the solution compartment. The solvent compartment has 512 or 256 molecules in the initial setup and a volume equal to that of the solution compartment (see Figure 1), but we can remove an arbitrary number between 0 and 512/256 to give the solvent compartment any desired initial pressure or density. The initial densities of the solution and solvent compartments were fixed to correspond to the desired pressure difference between the two compartments to drive the osmotic flow between the solution and solvent compartments. The initial pressures were fixed (approximately) by using the known/estimated densities of water and NaCl mixtures under pressure. During the simulation (till steady state is reached), these pressures change as a result of movement of the solvent molecules into and out of the zeolite membranes. All simulations were carried out using the molecular dynamics method. The method consisted of a 5th-order Gear’s predictor-corrector algorithm for translational

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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 Table 2. Range of System Variables Investigateda

Table 1. Interaction Parameters for Potential Models Used water ions ZK-4

interacting sites

σ (1010 m)

 (kJ/mol)

Q (e)

O H Cl Na Si O

3.17 0.0 4.42 1.9 4.009 2.890

0.65 0.0 0.49 6.69 0.5336 0.6487

-0.82 0.41 -1.0 1.0 0 0

motion and a 4th-order predictor-corrector algorithm for rotational motion using the quaternion method.15 Temperature was kept constant using a Gaussian thermostat.15 Because our objective was only to obtain a qualitative understanding of membrane separations in such system, we chose previously developed models for water, ions, and the zeolite membranes that are known to provide a realistic representation of such systems. These models were not further modified in these studies. Water was modeled using the simple point charge (SPC) model,19 the so-called primitive model20 was used for the ions, and the zeolite was modeled using the parameters suggested by Lee et al.21 The site-site interaction potential used is of the form

uij ) 4ij[(rij/σij)-12 - (rij/σij)-6] + qiqj/rij

(1)

where rij is the scalar distance between sites i and j, and ij and σij are the Lennard-Jones (LJ) interaction parameters. The values qi and qj represent the charges on sites i and j, although not all sites have charges. These interactions are then summed over all the active sites on the molecules to obtain the intermolecular interactions. Lorentz-Berthelot mixing rules were used for cross interactions between the different sites. The reaction field method22,23 was used to model long-range interactions. The parameters for the models used are given in Table 1. Computer simulations consisted of 560 000 time steps, after 60 000 steps of equilibration. Each time step was 3.52 × 10-16 s (0.001 in reduced units based on Na+ parameters); thus a typical simulation consisted of about 0.25 ns. The smaller time step was necessary to account for the vibrating motion of the zeolite membrane. The method was adapted to study steady state processes based on a technique suggested recently,24 and similar in spirit to two recent methods developed for steady-state simulations,25,26 and involves recycling solvent molecules (that permeate the membrane) periodically. If done often enough, this would result in a system approaching steady state. The molecules to be recycled were chosen randomly, one molecule at a time from a designated section of the solvent or solution compartment (as dictated by the direction in which solvent molecules permeate the membrane). This section was initially 2 units thick (in reduced units based on Na+ parameters), and in the middle of the compartment (as far away from the membranes as possible as shown in Figure 1). They were then replaced (again one molecule at a time) at a random location in a similar section of the other compartment (solution or solvent as required), but only if they passed the usual Metropolis particle displacement requirement widely used in Monte Carlo.15 This ensured that the recycling process, even though it was being carried out rather far away from the membrane (usually 5-10 solvent molecule layers away) did not produce any unusual disturbance in the system to affect the solvent permeation rate

system sizes studied temperature solution density solvent density number of ions in solution a

1214-2176 particles 1.0-2.25 0.08-0.175 0.02-0.08 6-52

Reduced using Na+ parameters.

across the membrane. This procedure is a simplification of the more rigorous schemes suggested in MacElroy25 and Heffelfinger and van Swol.26 We were able to successfully use such a simplified scheme because we explicitly recycle the correct number of solvent molecules. This is done at a sufficient distance from the membranes so that the slight inconsistency introduced in our hybrid scheme does not affect the solvent permeation rate being calculated in our studies, while allowing us to use a much simpler and faster algorithm. We carried out tests that showed that this did result in a system in steady state, or very close to it, since normal statistical fluctuations make it impossible to make an exact determination. In our MD simulations, a cation-free analogue of zeolite A (called ZK-4) was used as the membrane to separate water from supercritical aqueous NaCl solutions. In ZK-4 based membranes, the ratio of silicon to aluminum is equal to infinity. This is a rather attractive membrane for molecular simulation studies, in which we need to have two identical membranes and simple rectangular shape. The additional complications due to diffusion anisotropy and cation interaction can thus be avoided in this first set of simulations, although we plan to study membranes that do include such effects in the future. The side of the unit cubic cell for ZK-4 is a ) 24.555 Å (the topological net has a cell with a′ ) a/2).14,17 The framework structure of zeolite ZK-4 results by deleting all the Na+ and changing all the Al3+ into Si atoms from the structure of NaA zeolite. Since the cations are eliminated in such a zeolite, the effective size of the aperture for adsorption is increased. The diameter of each cavity is about 11.4 Å, and each one is connected to six neighboring cavities by windows about 4.2 Å in diameter. Its crystal structure is represented as a fm3c space group, which is the same as that of the NaA zeolite. One unit cell contains eight R-cages and consists of 192 Si and 384 O atoms. In the simulations, each membrane consisted of a unit cell or a unit cell that was cut into two parts at the center, along the y-z plane. Each such subcell would then contain four R-cages and form the two identical membranes in the computer simulations for the smaller simulation system (see Figure 1). Each subcell membrane would consist of 352 atoms (128 Si and 224 O atoms). A summary of the range of state conditions studied in our investigations is summarized in Table 2. Results and Discussion The main objective of this investigation was to use molecular simulations to examine if zeolite membranes can be used effectively to separate supercritical aqueous NaCl solutions. The secondary objective was then to determine the main osmotic driving forces in such separations. This information could help in finding the optimal set of system parameters/state conditions for any separation process. We hope that our results will convince the reader that the state of art of molecular simulations is at a stage where it can, if properly used

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Figure 3. Typical ionic cluster formed during the simulation. Figure 2. Density profiles of water and the ions in the simulation system at steady state.

and its limitations are clearly understood, be a valuable tool for carrying out qualitative screening studies to determine the suitability of not only a membrane for a separation process but also the variables that should be adjusted to improve the efficiency of the separation process. Since simulations are significantly less expensive than experiments in general, they could prove to be useful tools in preliminary process design. Since the membranes being investigated are rather thin (half to one unit cell thick), direct comparisons of the permeation rates with experiments are not possible. However, we have validated many aspects of our simulations and models indirectly here and in previous studies. These include agreement with experimental values for diffusion coefficients inside the zeolite8 and in bulk solutions;7,8 osmotic pressure for ideal solutions;11 and experimental trends for aqueous solutions in permeation rates for changes in temperature, pressure, concentration, polarity of solvent, and external electric fields.7-8 In addition, the models used for the solvents and ions in this study were shown in previous studies to be satisfactory for the range of properties being investigated here.19-21 Finally, we tested the sensitivity of the results to the harmonic coefficient used in tethering of the membranes. We found that an order of magnitude change in the coefficient resulted in a change of permeation rate of 10-20%, but it did not change any qualitative behavior reported here. We also carried out tests to check the sensitivity of the results to the initial system configuration and system size, and we found no significant dependence (within 2-5%). Results for a typical separation process involving supercritical aqueous NaCl solutions are shown in Figures 2-4. All results are shown in reduced units based on Na+ intermolecular parameters (see Table 1), unless otherwise stated. Figure 2 (density profiles in the simulation system) shows that during the simulation no ions were able to permeate the zeolite membrane (there were in fact never any ions in the center compartment during the entire simulation), while water was able to permeate the membrane quite readily. This is quite remarkable considering the fact that the molecular diameter used for Na+ in our simulations is 1.9 Å, whereas that for water it is 3.2 Å. Our investigation of the configurations of the molecules in the simulation showed that this was caused by ions being surrounded by water molecules which then formed rather stable ionic clusters (even at these supercritical conditions). This effectively increased the size of the ions so that

Figure 4. Number of solvent molecules permeating the membrane during the simulation as a function of time. -dNc/dt corresponds to the solvent flux across the membrane.

they were unable to permeate the 4.2 Å pores in the zeolite membranes being studied here. In the past, it was generally thought that the reason ions are unable to permeate membranes is related to surface interactions (not well understood) between the ions and the membrane surface.4-6 In our simulations, as mentioned earlier there are no Coulombic interactions between the ions and the membrane (since the membrane is uncharged), and no such interactions can exist. It is therefore likely that these clusters play a significant role in such separations, which in the past was not fully appreciated.4-6 These clusters were found to have a high energy of desolvation, which effectively prevented the ions from breaking away from such clusters. A typical ion cluster observed in our simulations is shown in Figure 3. Ion pairs of Na+ and Cl- are also formed in supercritical solutions that also effectively increase the size of the ions27 and that also prevents the ions from permeating the membrane. Figure 4 shows the number of water molecules permeating the zeolite membrane as a function of time. In our sign convention, a positive number shows osmosis while a negative number shows reverse osmosis. In the results shown in Figures 2 and 4, the initial density of the solution and solvent compartments were Fsoln ) 0.10 and Fsolv ) 0.03, respectively, and the temperature was T ) 1.0. This resulted in the pressure difference between the solution and solvent compartment being considerably larger than the osmotic pressure, and thus led to solvent flow from the solution compartment to the solvent compartment (reverse osmosis). The results in Figure 4 also clearly show that our simulation scheme was successful in establishing steady state. The flow rate of the solvent molecules across the membrane can be obtained from the gradient dNc/dt in Figure 4, which we find to be constant within the normal fluctuations in a simulation. Our results

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thus quite clearly show that zeolite membranes show promise for separating a wide range of supercritical polar electrolyte solutions. Supercritical fluids are currently being widely considered for a wide range of separation processes including the removal of pollutants and other chemicals especially from solids,28 such as contaminated soils and so on. Once we established that ZK-4 membranes could possibly be used to separate aqueous supercritical solutions, we investigated a range of osmotic driving forces that affect the flow rate of the solvent molecules across the membrane. The energy costs are usually a significant fraction of the total cost of operating membrane-based separation processes, so it is quite important to determine the optimal conditions for operating such processes. Molecular simulations as we will show can also be useful for studying such problems qualitatively. The main driving forces in a membrane-based separation process are pressure, temperature, concentration, and external force fields.4-5 The resistance to flow is primarily from the membrane (its thickness/pore structure). In our simulations, we investigated the effect of all these variables in some detail. We found our results to be most interesting and, at first glance, somewhat unexpected for the effect of electric fields and the thickness of the zeolite membrane. We will therefore discuss these effects in some detail. The remaining results, while important in validating the simulation method developed by us, will only be summarized, since they were found to be neither unusual or surprising. To investigate the effect of temperature, we examined systems with Fsoln ) 0.09, Fsolv ) 0.03, NaCl mole fraction ) 3%, and T ) 1.0, 1.2, and 1.4. For the three temperatures, the solvent flux across the membrane was found to be 0.08, 0.11, and 0.19 molecules/unit time, respectively. The increase results from two important factors. At the higher temperatures, the pressure of both compartments increases, and as a result, the pressure difference across the membrane is increased. This increases the driving force across the membrane. Another factor is the increased diffusion coefficient of the solvent as a function of temperature. It will be noted that the increase in the flux is not linear. This should not be unexpected since temperature has a rather complex effect on the stability of clusters and ion-pairs that play a key role in these separations. We also observed that at higher temperatures the solute molecules (ions) also begin to permeate the membrane as a result of destabilization of some of the ionic clusters/ pairs. We then examined the effect of solution and solvent densities on the mass transfer rate across the membrane. We examined this in two ways. In the first set, we changed the absolute densities of the solution/solvent while keeping the density ratios constant. In the second set, we kept the solution density constant but only changed the solvent density. At T ) 1.4, for solution densities of 0.11, 0.09 and 0.08 (solvent density are 30% of these values), and 3 mol % NaCl, we observed solvent flux of 0.246, 0.187, and 0.115. Once again, the results are not linear. One reason for this nonlinear dependence is that the flux across the membrane is determined by the relation

J ) -dNc/dt ) A(∆P - ∆Π)

(2)

where J is the flux across the membrane in molecules/

Figure 5. Effect of an external electric field on the number of solvent molecules permeating the membrane as a function of time.

time, ∆P is the pressure difference between the solution and solvent compartments at steady state, ∆Π is the osmotic pressure of the solution, and A is the transport coefficient of the membrane. When the density is reduced, the pressure difference between the solvent and solution compartment is reduced (linearly only in the case of an ideal gas); however, the osmotic pressure remains almost unchanged. In addition, the diffusion coefficient has a nonlinear dependence on density. This leads to a nonlinear decrease in the driving force for mass flow. In the case when the solvent density was only decreased while the solution density was kept constant, we found that for a solution density of 0.15 (T )1.5) when the solvent density was lowered from 30% of the solution density to 20%, the flux increased from 0.4 to 0.46. This increase is also attributable to the increase in ∆P in eq 2. Finally, we also looked at the effect of concentration on the solvent mass transfer rate. For T ) 1.5, Fsoln ) 0.11, and Fsolv ) 0.03, we studied NaCl mole fractions of 3 and 1.5%. With the higher concentration the flux was found to be 0.25, while with the lower concentration a flux of 0.26 was obtained. This difference is almost all due to the change in osmotic pressure in eq 2; the pressure difference is nearly unchanged. As mentioned earlier, even though none of the results summarized above can be viewed as unusual or unexpected and all have been observed experimentally or could be predicted using existing theories,4-6 they do and have served an important purpose of validating molecular simulations as a viable tool for studying such membrane-based separation systems. One of the most dramatic results observed in our simulations was the effect on the separation rate due to an external electric field, sometimes referred to as electro-osmosis.29-31 Electro-osmosis is not a well understood phenomenon, but its applications are quite diverse and include soil purification, chemical separations, water desalination, and drug delivery.29-34 It has also been suggested that electro-osmosis may be responsible for transport processes in the membranes of the human eye.35 Figure 5 shows the increase in the number of molecules permeating the molecules as a function of time with a range of electric fields (a reduced field of E ) 0.1 corresponds to a voltage difference across the membrane of about 465 mV) for Fsoln ) 0.11, Fsolv ) 0.03, and a NaCl mole fraction of 3%. The fields used were alternating fields with a frequency corresponding to 10 000 time steps. This was necessary to prevent the

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Figure 6. Density profile of the solvent (water) molecules in the simulation system at steady state for with an electric field. Figure 8. Effect of external electric fields on the mean-squared displacement of solvent molecules perpendicular to the membrane plane.

Figure 7. Effect of external electric field strength on the density profile of Cl- in the simulation system. Note that when E ) 0.25 the Cl- ions can permeate the membrane.

accumulation of ions near the membranes and thus “fouling” the membrane.32 The results do not appear to be sensitive to frequency in the vicinity of this value. Our results show that the flux across the membrane increased almost fifteen fold between E ) 0 and 0.25. We realize that the fields used here are somewhat higher than those generally used, but this was necessary to observe the effects in the very small time scale of the simulations, about 0.25 ns. In the usual time scale of experiments, these effects would probably be observable with much smaller electric fields as well. This has been found to be generally true in most simulations using external fields.36 At the higher electric fields, we also observed that the ions were sometimes able to permeate the membranes. This is caused by the electric field making clusters of ions and water molecules referred to earlier less energetically favorable; the electric field aligns the dipole moments in the direction of the electric field, which in our studies was along the direction of flow (perpendicular to the membranes). The electric field also weakens or separates ion-pairs (the interaction of the field leads to forces in the opposite direction on oppositely charged ions in the pair). We also observed that in the time scale of the simulations, the electric field had to be larger than 0.025 to observe statistically meaningful increases in the solvent flux (this is somewhat analogous to a yield stress in non-Newtonian flow). Keeping this in mind and the usual fluctuations likely in such simulations, the effect of the electric field is linear up to about 0.125. This is then followed by a region where the effect is not linear: a well-known effect referred to as dielectric saturation.37 To understand the reasons for the large increase due to an external electric field we examined the density profiles, the diffusion coefficients, and the structure of

Figure 9. Radial distribution function gI-water with and without an external electric field.

the ion-water clusters. The results are shown in Figures 6-9. The density profiles in Figure 6 show that as a result of the electric field the loading in the zeolite cavity increases almost 3-fold. This can in part explain the rather large increase in the solvent flux due to the field. Since there are more solvent molecules in the zeolite, it is obvious that many more will be able to leave the zeolite membrane as a result of this increase. As pointed earlier, at higher electric fields (generally in the nonlinear region), in addition to the increase in the solvent flux, there is an undesirable effect of solute permeation that also results. In these cases such as the one shown in Figure 7 for an electric field strength of 0.25, some of the Cl- ions are able to permeate the membrane and into the “solvent” compartment. It is also interesting to note that even though Na+ ions are smaller than Cl- ions, we have observed that Cl- ions are more likely in general to permeate the membrane first. This has to do with the greater stability of Na+ water clusters compared to Cl-. For this case, for example, the Na+ ions did not permeate the membrane. Another reason for the increase in the solvent flux due to the electric field can be seen from the results of the mean-squared displacement of the solvent molecules perpendicular to the membrane plane (x coordinate) shown in Figure 8. The mean-squared displacement (MSD) of a particle is the second moment of the particle distribution spatially in a Gaussian about the origin at any time t > 0, from which the transport properties can be related to time or ensemble averages performed

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Figure 10. Number of solvent molecules permeating the membrane when the thickness is varied. The thick membrane is twice as wide as the thin membrane.

on systems at equilibrium. The mean-squared displacement of the particles perpendicular to the membranes is related to the permeability of the membranes, and the self-diffusion coefficients perpendicular to the membranes (D⊥).15

) Σ[x(t) - x(0)]2/N ) 2D⊥t

(3)

This equation is valid when time t is large compared to the average time between the collisions of particles. Similarly, the overall and parallel diffusion coefficient can be related to the MSD by the following equations:15

) Σ[r(t) - r(0)]2/N ) 6Dt ) Σ[(y(t) - y(0))2 + (z(t) - z(0))2]/N ) 4D|t (4) We would like to point out that eqs 3 and 4 are strictly valid for homogeneous systems only. However, they can be used in inhomogeneous systems such as ours, to obtain “effective” diffusion coefficients. The results shown in Figure 8 clearly show that the diffusion rate of solvent molecules increases significantly, as a result of the external electric field. For the diffusion coefficient, the effect appears to be almost linear up to the highest field strengths we studied. The cause of this increase is related to the water clusters, the ion-pairs and the ionwater clusters weakening as a result of the external field, and thus allowing the water molecules to move more freely. In addition, since the ions move around as a result of the electric field (the interaction of an electric field leads to not only an additional torque on the solvent molecules but also a force on the ions), the water molecules would also get dragged along with the associated ions. Finally, we show the effect of electric field on the radial distribution functions of Na+ and Cl- ions and water in Figure 9. The results show that as a result of the electric field the first peak of the distribution functions increases in both cases. This has to do in part with fewer ion pairs, which allow more water molecules to surround each ion on the average. The more randomized orientation of the water molecules surrounding the ions as a result of the field may also lead to more molecules being able to fit in the first neighbor shell, even though the clusters are now weaker (lower energy of desolvation). In Figure 10, the effect of membrane thickness on the mass transfer rate across the membrane is shown. The “thick” membrane is one unit cell (24.555 Å) wide, while

the “thin” membrane is half that thickness. The density and temperature of the solvent and solution compartments are the same in both cases. In macroscopic membranes, it is always observed that the transport coefficient of the membrane (A in eq 2), is inversely proportional to the thickness of the membrane. Thus, doubling the thickness of the membrane would reduce the flow rate by half. It will be seen quite clearly that this is certainly not the case in our simulations. The results point to the important differences in macroscopic and mesocopic (nanoscale) hydrodynamics. In the macroscopic systems that most engineers are used to, the entrance and exit effects are generally negligible compared to the resistance to flow inside the membrane. In nanoscale systems such as the one being examined here, the entrance and exit effects dominate the resistance to flow, and the resistance inside the membrane is a much smaller fraction. Thus even though the resistance inside the membrane is almost doubled, it does not seem to increase significantly the overall total resistance to flow. The results shown are for T ) 1.5, Fsoln ) 0.11, Fsolv ) 0.03, and NaCl mole fraction of 1.5%, but we found the results to be similar for several other state conditions and electrolyte concentrations we studied as well. The results obtained for the effect of membrane thickness can be compared to those for the external electric field. The electric field, as stated previously, makes the ionic clusters ion pairs and water clusters less stable (and consequently fewer and smaller). This makes the entrance (and to a smaller extent the exit there are no ions in general inside the membrane) resistance smaller to solvent flow. Thus while the electric field significantly affects the solvent flow rate, a change in the thickness of the membrane does not, since the entrance and exit effects are essentially unchanged when the membrane thickness is changed. Conclusion We have reported a simulation study on the separation of supercritical solutions using thin zeolite membranes. By using the method of molecular dynamics, we have demonstrated the feasibility of using ZK-4 zeolites as membranes to separate water from aqueous NaCl solutions. On the basis of our results we also believe that zeolites may be suitable for separating a wide range of electrolytes in polar supercritical solvents. Recent developments in zeolites are making the manufacture of zeolite membranes technologically feasible.9-10 Our results also show that molecular simulations can be a valuable tool for conducting screening studies for establishing the suitability of membranes for separation processes. Finally, our results point to the dangers and risks of extrapolating macroscopic hydrodynamic theories to the meso- or nanoscale. We have found that at the nanoscale the transport coefficients for mass transfer are almost insensitive to the thickness of the membrane, whereas in macroscopic systems, they are well-known to be inversely proportional. Acknowledgment This research was supported by a grant from the Division of Chemical Sciences, U.S. Department of Energy (No. DE-FGO2-96ER14680). Literature Cited (1) Breck, D. W. Zeolite Molecular Sieves-Structure, Chemistry, and Use; Wiley: New York, 1974.

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Received for review May 9, 2001 Revised manuscript received August 6, 2001 Accepted August 8, 2001 IE010425+