Entropy-Driven Molecular Separations in 2D-Nanoporous Materials

Article pubs.acs.org/JPCC

Entropy-Driven Molecular Separations in 2D-Nanoporous Materials, with Application to High-Performance Paraffin/Olefin Membrane Separations Kylen Solvik, Jessica A. Weaver,† Anna M. Brockway,‡ and Joshua Schrier* Department of Chemistry, Haverford College, Haverford, Pennsylvania 19041, United States S Supporting Information *

ABSTRACT: Nanometer-scale pores in carbon-based materials such as graphene, carbon nanotubes, and two-dimensional polymers have emerged as a promising approach to high permeance, high selectivity gas separation membranes. In previous studies, quantummechanical mass-dependent tunneling, classical size-exclusion and differences in surface adsorption have been used to obtain high selectivity. Here, we illustrate a new classical approach in which an entropic barrier causes the selective separation of gas molecules. Using atomistic molecular dynamics simulations, we study the separation of ethane, ethene, propane, propene, n-butane, isobutane, 1-butene, cis-2-butene, trans-2-butene, isobutene, and 1,3butadiene through a novel nanoporous two-dimensional hydrocarbon polymer (denoted PGTP1), as a function of temperature and pressure. Despite the absence of a potential energy barrier for both types of species and the greater surface adsorption of the paraffins, selective passage of olefins results from the greater number of possible conformational and orientational configurations possible within the small pores. This entropic barrier allows for differentiation of similar gases, such as ethane and ethene or propane and propene. Larger branched alkanes and alkenes are completely rejected by size exclusion. The PG-TP1 selectivity exceeds practical requirements for economical separations of propene and 1,3-butadiene; moreover, the high permeances (10 × 106 and 17 × 106 GPU (gas permeation units), respectively) greatly exceed all existing membrane materials by 5 orders of magnitude.

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INTRODUCTION Ethene, propene, and butadiene are essential industrial feedstocks used in plastics and chemical production. These olefins are produced from paraffins (ethane, propane, and butane) by steam cracking; this process results in a paraffin/ olefin mixture, which must then be separated through cryodistillation of the liquified gas mixture. The multiple phase changes make this separation highly energy-intensive; paraffin/olefin distillations consume more than 1011 MJ annually in the United States.1 Membrane-based gas separation methods have a much lower thermodynamic cost than distillation and adsorption processes2 and thus can substantially reduce the energy needed.3 In principle, membrane-based or hybrid distillation-membrane processes have the potential to reduce the energy usage and cost of paraffin/olefin separations by 20−30% compared to existing distillation methods.4 Several recent reviews discuss the use of membranes in the petrochemical industry (including paraffin/olefin separation),5,6 and a history of the membranes used for paraffin/olefin separations is presented in a technical report by Merkel et al.7 A growing number of theoretical studies have investigated the use of two-dimensional nanoporous membranes to perform chemical,8−21 ionic,22,23 and isotopic10,21,24−28 separations. Recent experiments have demonstrated the use of oxidative etching to produce small size-selective pores for gas separation,29 and intrinsic defects in CVD graphene for separating solutes in solution.30 These studies have predom© 2013 American Chemical Society

inantly relied on size exclusion, or more broadly, the creation of a large potential energy barrier for the passage of one gas versus a smaller potential energy barrier for passage of another gas, by changing the size and chemical functionalization of the nanopore, to perform the separation.8−11,13,15,17,18,20,21 A second approach uses the mass-dependence of quantum tunneling to separate isotopes of helium and hydrogen.10,21,24−28 A third approach uses differences in surface adsorption to selectively enrich the local concentration of one of the gases near the pore to enhance its transport rate.14−16,19 We initially hypothesized that the surface adsorption differences between the unsaturated olefins and the saturated paraffins would allow for gas separation. However, as we show below, the selective separation actually results from gases having a sizedependent entropic barrier for passage, resulting from the relative differences in the loss of conformational and orientational degrees of freedom as they move through a narrow pore. Entropic barrier separations have been studied theoretically,31,32 used to explain gas separation using zeolites, carbon molecular sieves, and polymer membranes,33 and used to separate DNA and small proteins in microfluidic devices.34,35 These previous implementations involved passage through Received: May 18, 2013 Revised: July 24, 2013 Published: July 26, 2013 17050

dx.doi.org/10.1021/jp404903g | J. Phys. Chem. C 2013, 117, 17050−17057

The Journal of Physical Chemistry C

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many barriers, but here we find that even a single entropic barrier can result in highly selective gas separations. As a concrete example, we consider a generalization of the two-dimensional polyphenylene synthesized by Bieri et al.36 (referred to by the authors as porous graphene (PG)) in which the biphenyl-like subunits in one direction are replaced with para-terphenyl (TP)-like group to expand the size of the pore. We denote the resulting material, shown in Figure 1, as PG-

The membrane permeance and selectivity were calculated using the Langmuir adsorption based model introduced in our previous work16 and described briefly below. A series of equilibrium molecular dynamics simulations were performed under constant particle number, volume, and temperature (NVT) conditions using a Nose−Hoover thermostat, with periodic boundary conditions in all directions, and integrating Newton’s equations of motion with a velocity Verlet algorithm and 0.5 fs time step. Following a 1 ns equilibration period, data collected during a 10 ns simulation was used to compute the number of barrier crossings and pressure. The total number of gas molecules, rtotΔt, that cross the barrier in both directions during the simulation of duration Δt, is given by ⎛ αP ⎞ ⎟ rtotΔt = 2k 0[S0]⎜ ⎝ 1 + αP ⎠

(1)

where k0 is an effective barrier passage rate (accounting for both diffusion along the surface and passage through the pore), [S0] is the concentration of possible surface sites for adsorption, and (αP)/(1 + αP) is the fractional surface coverage, where α is a species-, surface-, and temperature-dependent adsorption equilibrium constant and P is the pressure in the system. The pressure is determined using the ideal gas law in the region ±20 Å away from the nanoporous sheet. The k0[S0] and the α parameters are determined by counting the number of crossings that occur during the simulation, as a function of the gas pressure. As in refs 16 and 21, crossing events were counted using the criterion of Du et al.,14 whereby a molecule is said to have crossed the barrier if it starts on one side of the barrier and then crosses to the other side and remains there for 1 ps. A particle is defined as being outside the pore when it is ≥2 Å left or right of the polymer’s plane. After the 1 ps time limit, a recrossing back to the original side is counted as another crossing event; however, we note that in all of the trajectories we visualized, we only saw desorption from the surface, without recrossing. The fitted parameters can be found in Table S1 of the Supporting Information. When there is a pressure difference across the membrane, the net crossing rate is the difference between the left → right and right → left rates:

Figure 1. (a) PG-TP1; (b) example geometry from molecular dynamics simulation.

TP1. This results in a nearly flat sheet, with the two-ring sides of the hexagons forming an extended, roughly planar surface for adsorption of gas atoms. Along the direction of the three-ring sides of the hexagon, the center benzene ring rotates to minimize steric interactions between hydrogen atoms. This creates a small pore for gas passage: the distance between the two rotated benzene rings ranges from 6.8 to 7.2 Å in the MD simulation.

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COMPUTATIONAL METHODS Atomistic molecular dynamics simulations were performed with LAMMPS 2012.02.37,38 Interactions between all of the atoms were treated using the AIREBO force-field.39 The noncovalent parameters in AIREBO were originally parametrized to reproduce the liquid pair-correlation function and the phase behavior of small paraffins and olefins (including ethane, ethene, propane, and butane); thus, it is an appropriate forcefield for describing the gas−gas and gas−surface interactions in our simulations. Additionally, the covalent and intramolecular noncovalent interactions in AIREBO were originally parametrized to describe the torsional barriers of small hydrocarbon molecules and elastic properties of graphite and diamond; thus, it is also an appropriate force-field for describing the conformational changes of the PG-ES1 surface and the gas molecules in our simulation. While other force-fields such as OPLS/AA40 and TraPPE-EH41,42 are also designed to describe covalent and noncovalent interactions in hydrocarbons, AIREBO has the additional advantage of being a reactive force-field capable of describing bond breaking and formation. This allowed us to simultaneously investigate the possible degradation of PG-ES1 on the time scale of the total 400 ns of simulation at each temperature used to study all of the different gases and pressures. We observed neither defect formation in PG-ES1 nor covalent adducts between PG-ES1 and olefins in any of our simulations. All atoms are allowed to move freely in our simulations; this is needed to allow the longer hydrocarbons to adopt different conformations needed for barrier passage, and previous studies have found it important to allow for relaxations of the nanoporous material that reduce the potential energy barrier for gas passage.18,43

⎛ αPL αPR ⎞ − rnet = k 0[S0]⎜ ⎟ 1 + αPR ⎠ ⎝ 1 + αPL

(2)

This nonequilibrium net crossing rate is computed using the k0[S0] and α parameters obtained by fitting eq 1 to the (equilibrium) molecular dynamics simulations described above. The permeance of a species A, pA, is the ratio between the number of species A crossing the membrane per unit time and area, a, and the applied pressure difference across the membrane: rnet,A pA = a(PL − PR ) (3) The units of our simulation are number in molecules, area in Å2, time in ns, and pressure in bar. These can be converted to macroscopic units by dimensional analysis (1 molecule Å−2 ns−1 bar−1 s−1 = 16.6054 mols cm−2 bar−1 s−1). In chemical engineering, permeance is often reported in gas permeation units (GPU), where 1 GPU = 10−6 (cm3 STP) cm−2 (cm Hg)−1 s−1, and where cm3 STP denotes the volume of the gas at standard temperature and pressure (273.15 K, 1 bar). Assuming the ideal gas equation of state for performing the conversion 17051



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α ∝ exp[−ΔHads/kBT], where ΔHads is the adsorption enthalpy. Since adsorption is an exothermic process (ΔHads < 0), increasing temperature will decrease the adsorption equilibrium constant, α. For simplicity, consider the Henry’s Law limit of eq 1 where αP ≪ 1, leading to a total number of crossings, ntot, of

between moles of gas and volume of gas at STP, the conversion factor is 1 mol cm−2 bar−1 s−1 = 2.988 × 108 GPU. Details of dimensional analysis calculations are in the Supporting Information. In the Henry’s Law limit, αP ≪ 1, the permeance is pressure-independent,21 but this is not the case here. Nudged elastic band (NEB) calculations were performed using LAMMPS 2012.02 and the AIREBO force-field as described above, using the fire minimization algorithm of Bitzek et al.44 Twenty-four replica points were used, with an interreplica spring constant of 0.1 eV Å−2, an energy stopping tolerance of 0.001 eV, and a force stopping tolerance of 10−7 eV/Å.

ntot = 2k 0[S0]αP ∝ e−(Ea +ΔHads)/ kBT P

(4)

Thus, the temperature-dependence seen in Figure 2 is consistent with surface adsorption being the rate-limiting step. Drahushuk and Strano have recently derived kinetic mechanisms for gas transport through nanopores, which consider each of the possible rate limiting steps, including this one,19 but this is the first time the temperature-dependence consequences have been discussed. We point this out to suggest a simple experimental way to discern whether surface adsorption or barrier crossing dominate the transport process. While surface adsorption effects explain the temperaturedependence, they do not explain the preferential transport of ethene. Figure 3 shows the probability density of gas molecules

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RESULTS AND DISCUSSION To illustrate the basic physical mechanism, we consider the C2 hydrocarbons, ethane and ethene, as a model case. The barrier crossing results as a function of pressure and temperature are shown in Figure 2; note the different ranges of the vertical axes.

Figure 3. Probability density of C2 hydrocarbons as a function of distance perpendicular to the plane of the PG-TP1 center of mass, as a function of temperature (273, 300, and 325 K): (a) ethane (at 7.5 ± 0.6, 8.9 ± 0.7, and 10.1 ± 0.7 bar) and (b) ethene (at 8.1 ± 0.6, 9.4 ± 0.7, and 10.5 ± 0.7 bar).

as a function of distance perpendicular to the PG-TP1 sheet, sampled every 0.02 ps. (The small left−right asymmetry in Figure 3 is due to limited sampling.) For both ethane and ethene, increasing temperature decreases the number of molecules adsorbed on the surface, as described above. Thus, the decrease in the number of crossings with increasing temperature is consistent with the crossing rate being proportional to the number of surface-adsorbed gas molecules. However, at a given temperature, more ethane is adsorbed on the surface than ethene, consistent with previous simulation results for adsorption of ethane and ethene on carbon nanotubes,45 due to the greater dispersion interaction. (Because we ran isochoric simulations, the ethane pressures shown in Figure 3a are slightly less than the ethene pressures shown in Figure 3b, yet more ethane is adsorbed on the surface despite the lower pressure.) The observation that ethene molecules cross more frequently despite the adsorption of fewer molecules on the surface is surprising because it is the opposite of what was observed in recent work studying the role of surface adsorption on nanopore gas separation.14−16,19 In all of those previous cases, stronger surface adsorption leads to an increase in the local concentration of gas molecules near the pore, which in turn increases the crossing rate through the pore. However, this cannot be the effect that is leading to enhanced ethene crossing because the local concentration is lower than the corresponding case of ethane. To cross the PG-TP1 sheet, the gas molecules must pass through the center of the sheet. However, the probability

Figure 2. Barrier crossings of C2 hydrocarbons as a function of temperature and pressure during molecular dynamics simulations; curves show the fit to eq 1 with parameters in Table S1 in the Supporting Information: (a) ethane and (b) ethene.

Both species show the same qualitative behavior, but at any given temperature and pressure, about three times as many ethene molecules cross the PG-TP1 barrier. Both ethane and ethene show a decreasing slope of the number of crossings with increasing pressure, which is characteristic of the Langmuir isotherm model. Both species also show a decrease in the number of crossings with increasing temperature, which is the opposite of what one would expect for an Arrhenius-type activated barrier crossing model. This is surprising because Arrhenius theory arguments are widely used to estimate the rates of gas transport through nanopores.8−11,18,20 However, as we will show below, the anti-Arrhenius behavior is consistent with the effects of surface adsorption. The barrier crossing rate, k0, in an Arrhenius-type model behaves as k0 ∝ exp[−Ea/kBT], where Ea is the activation barrier height energy, kB is the Boltzmann constant, and T is the absolute temperature. By the Clausius−Clapeyron equation, the gas-surface adsorption equilibrium constant behaves as 17052



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density at the center (shown in Figure 3) is zero for ethane, but not for ethene. This indicates that the ethene molecules have a longer residence time within the pore, consistent with a potential energy well, rather than a potential energy barrier. To study this, we computed the potential energy between a single transiting gas molecule and the PG-TP1 sheet taken from the molecular dynamics simulations above. The results from some representative crossings are plotted in Figure 4. Notice how the

Suppose that we have two different gases (e.g., ethane and ethene) labeled A and B, with corresponding crossing rates, ntot,A and ntot,B, respectively. The ratio of the two crossing rates (at the same T and P conditions) is ntot,A ntot,B

‡

‡

eΔSA / kB e−ΔHA / kBT e−ΔHads,A / kBT

=

‡

‡

eΔSB / kB e−ΔHB / kBT e−ΔHads,B/ kBT ‡

‡

‡

‡

= e(ΔSA −ΔSB)/ kB e−(ΔHA −ΔHB)/ kBT e−(ΔHads,A −ΔHads,B)/ kBT (7)

labeling the entropy and enthalpies with the corresponding A and B subscripts. For the case of ethane and ethene, we showed above that the activation enthalpy for the two species is approximately the same, ΔH‡A ≈ ΔH‡B, and the adsorption enthalpy terms favor ethane passage. Thus, it must be the entropy term, exp[(ΔSA‡ − ΔSB‡ )/kB], that leads to the preferential passage of ethene observed in our calculations. The reason can be understood by considering a microcanonical expression of the entropy, S = kB lnΩ, where Ω is the number of accessible microstates. The entropy of activation, ΔS‡A = S‡A − SA, is the difference between the entropy of the gas at the transition state (inside the PG-ES1 pore, S‡A) and in the gas phase (SA). Equivalently, this can be expressed in terms of the ratio between the number of accessible microstates in the pore (Ω‡A) and in the free gas (ΩA) as

Figure 4. Potential energy as a function of molecule distance from the center of the PG-TP1 sheet: (a) ethane and (b) ethene.

potential is attractive, not repulsive, throughout the entire barrier crossing process. The potential energy well-depths obtained from these simulations, as well as via nudged elastic band (NEB) calculations, are tabulated in Table 1. The well-

ΔSA‡ = kB ln(Ω‡A /ΩA )

Table 1. Potential Energies (in eV) between the Gas Molecule and the PG-TP1 Sheet When the Molecule Is at the Center of the Pore; MD and NEB Denote Values Observed from Molecular Dynamics Trajectories and Nudged-Elastic Band Calculations, Respectively ethane ethene propane propene n-butane 1-butene cis-2-butene trans-2-butene 1,3-butadiene

MD

NEB

−0.273 −0.244 −0.335 −0.338 −0.409 −0.393 −0.373 −0.404 −0.374

−0.278 −0.253 −0.336 −0.343 −0.403 −0.424 −0.389 −0.402 −0.377

Thus, the entropy term is ‡

kBT ΔS‡/ kB −ΔH ‡/ kBT e e h

(5)

where h is Planck’s constant, ΔS is the entropy of activation and ΔH‡ is the enthalpy of activation. Following the same logic leading to eq 4, but instead using eq 5 for k0 leads to kBT ΔS‡/ kB −ΔH ‡/ kBT −ΔHads/ kBT e e e P h

(Ω‡A /ΩA ) (Ω‡B/Ω B)

(9)

Because of the small size of the pore, only a limited number of orientations of the gas molecule are allowed, many fewer than the possible orientations in the gas phase (Ω‡ ≪ Ω). Since ethene is smaller, it can fit inside the pore in more ways than ethane (Ω‡ethane < Ω‡ethene). Ethene therefore has a lower entropic barrier, which makes it more likely to enter the pore from the gas phase and gives it a faster transport rate than the larger ethane. This separation mechanism is distinct from the size-selection process described in earlier papers on nanoporous gas separation, where the larger potential energy barrier of larger molecules was used to control the selectivity.8−11,13,17,18,21 The trends in our atomistic simulations are consistent with simplified model calculations of spherical particles in linear channels.46−48 Moreover, the presence of a potential energy well in our calculations is consistent with the intermediate-state model studied by Mondal,49 which leads to enhanced sizeselective transport. Entropic barriers also contribute to nonArrhenius temperature dependence,48,50 and the relative separation by size increases with a decreasing chemical potential gradient (which in our case is the pressure difference across the membrane).46 Our method for calculating the permeances is based on infinitesmal displacements from equilibrium, and thus, we see the maximum size selectivity. Future calculations using the nonequilibrium approach of Liu et al.51 would be necessary to observe how this changes. Barrier crossing results for the C3 and C4 hydrocarbons shown in Figures 5 and 6, respectively, follow the same qualitative patterns discussed for the C2 results. In all cases, the paraffins transit with lower probability than the olefins, and the same anti-Arrhenius behavior with respect to temperature is

‡

ntot ∝

‡

e(ΔSA −ΔSB)/ kB =

depths for ethane and ethene are comparable, and in fact, ethane has a deeper potential energy well. However, the absence of a measurable ethane probability density at the center of the system (Figure 3a) indicates that ethane is not trapped in the potential energy well. This leads to the conclusion that the ethane/PG-TP1 system does not have a potential energy barrier, but rather an entropy barrier of the type described by Zwanzig.31,32 Instead of using the Arrhenius rate expression, consider the Eyring−Polanyi transition-state theory rate k0 =

(8)

(6) 17053



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Figure 5. Barrier crossings of C3 hydrocarbons as a function of temperature and pressure during molecular dynamics simulations; curves show the fit to eq 1, with parameters in Table S1 in the Supporting Information: (a) propane and (b) propene.

observed. Note that the number of crossings for propane (Figure 5a), n-butane (Figure 6a), 1-butene (Figure 6b), and cis-2-butene (Figure 6c) results are 2 orders of magnitude smaller than that of the other species we have considered. As a result, the agreement between the data points and the fit of eq 1 for these cases does not appear as good as the other cases because the limited sampling time of 10 ns makes small fluctuations of ±5 crossings appear larger at the scale these are plotted. No crossings were observed for the branched species, isobutane and isobutene, at any temperature or pressure, even during extended simulation time of 20 ns, which can be attributed to a molecular-sieving effect due to the small size of the pores. The maximum pressures of 4.5, 7.9, and 10.5 bar for the isobutane simulations and 6.1, 8.9, and 11.5 bar for the isobutene simulations (at 273, 300, and 325 K, respectively) allow us to calculate an upper bound on the permeance. The nbutane and 1-butene crossing rates are comparable, which is consistent with the similarity of the molecules; for most of the barrier crossing process the size of the two molecules is the same, so the conformational reduction constraints are nearly the same for both molecules. trans-2-Butene has a much higher crossing rate than cis-2-butene because there are more conformations that can pass through the pore. The fastest crossing rate among the C4 species occurs for 1,3-butadiene; this is consistent with the entropy barrier considerations discussed above, as it is the C4 species with the smallest volume and hence has the most possible configurations available for crossing through the pore. Like the ethane/ethene example discussed above, the potential well depths for each of the paraffin/olefin pairs is qualitatively the same; see Table 1.52 We note that 1,3-butadiene can undergo torsional transitions between several different conformers; the lowest energy conformer is the s-trans conformer, but the two gauche conformations are also possible and correspond to local minima 0.123 eV higher in energy than the s-trans ground state according to Raman experiments53 and 0.125−0.139 eV in quantum chemistry calculations.54 However, the AIREBO

Figure 6. Barrier crossings of C4 hydrocarbons as a function of temperature and pressure during molecular dynamics simulations; curves show the fit to eq 1, with parameters in Table S1 in the Supporting Information: (a) n-butane; (b) 1-butene; (c) cis-2-butene; (d) trans-2-butene; and (e) 1,3-butadiene. Isobutane and Isobutene results are not shown, as no crossings were observed in extended 20 ns simulations at any pressure or temperature.

force-field predicts the difference between the local minima to be only 0.067 eV, and thus, from Boltzmann’s law, one expects the percentage of trans conformers to be artificially low 17054



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in our calculations. On the basis of the energy minima, AIREBO predicts 87% s-trans, compared to the >99% observed in experiment and more detailed calculations. In our MD simulations, we started with all molecules in the s-trans conformation, and at equilibrium at 300 K, we observed that 78% were in the s-trans conformation. This was observed regardless of pressure or the presence or absence of the PGTP1 sheet. Nevertheless, since we observed crossings occurring with both 1,3-butadiene conformations, we do not expect this to have a major impact on the permeance results. The C3 and C4 1-alkenes can enter the pore from either the saturated or unsaturated (CC) end; the smaller size of the latter should bias entry by the unsaturated end. Examining a random sample of 40 successful crossing events for propene at 300 K, we found that 26 (65%) of them enter via the unsaturated (CC) end. Similarly, out of the total of 24 successful 1-butene crossings at 300 K, 15 (62.5%) occurred via the unsaturated entry. This demonstrates a slight bias in the entry direction of the 1-alkenes. Using the fitted values of k0[S0] and the α (listed in Table S1, Supporting Information), the permeances can be computed at any feed (the upstream or high chemical potential side of the membrane) and permeate (the downstream or low chemical potential side of the membrane) pressure combination by substituting these pressures for PL and PR into eq 3. Because of surface adsorption effects, the permeances depend on the feed and permeate pressures used. For concision, here we will only discuss a feed compression separation carried out with feed pressure of 5 bar and permeate pressure of 1 bar; the temperature-dependent permeances are listed in Table 2. Note

Figure 7. (a) Propane/propene and (b) n-Butane/1,3-butadiene mixtures. The curves show the predicted number of olefin crossings in the ideal (noncompetitive and noninteracting) case.

curves showing the predictions based on the single-gas permeance data. A complete agreement between the two indicates that the ideal selectivity is valid. At lower temperatures and pressures, the observed number of olefin crossings is systematically less than the prediction, which is consistent with surface adsorption site competition. However, the pure-gas model is within the error bars of the numerical results, which justifies our use of the ideal selectivity in the discussion below. Finally, we use the results of Motelica et al.’s recent technical and economic feasibility study of membranes and hybrid distillation−membrane processes for paraffin/olefin separation to evaluate the potential application of PG-TP1.4 In their model, the operational cost is a monotonically decreasing function of olefin permeance, so the very large (10−25 MGPU) permeances predicted for PG-TP1 are exceptionally favorable when compared to the 10−6−10−4 MGPU permeance of the best existing membranes.5−7,55 Thus, the limiting property of PG-TP1 will be the selectivity of the separation, which we estimate using the ideal selectivity obtained by taking the ratio of the permeances of the two species. Motelica et al. found that membrane separations save energy relative to distillation only when the ethene/ethane selectivity exceeds 60 , and that hybrid separations become favorable to distillation when ethene/ ethane selectivity exceeds 10. From the data in Table 2, we see that the PG-TP1 ethene/ethane selectivity is predicted to be only 2.3−2.5 in the temperature range considered here; therefore, despite the high ethene permeance, this is not a suitable application for PG-TP1. Motelica et al. did not report similar calculations for propene/propane separations, but we will assume that a similar criterion is needed for energy- and cost-efficient separations of these species. The propene/ propane selectivity of 59−93 for PG-TP1 in the temperature range of 273−325 K is suitable for either membrane-only or hybrid separations. For C4 hydrocarbons, Motelica et al. found that the butadiene/mono-olefin selectivities ≥ 7.5 or butadiene/saturated hydrocarbon selectivities ≥ 15 are required for membrane separation to be advantageous. With the

Table 2. Single-Gas Permeances of PG-TP1, in 106 Gas Permeation Units (MGPU), Computed under 5 Bar Feed Pressure and 1 Bar Permeate Pressure ethane ethene propane propene n-butane 1-butene cis-2-butene trans-2-butene isobutane isobutene 1,3-butadiene

273 K

300 K

325 K

39 100 0.27 25 0.66 0.45 1.0 36 3400

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CONCLUSIONS Our simulations reveal two generally applicable insights pertaining to gas separations using nanoporous materials. First, the temperature dependence of the membrane permeance can be used to discern the relative importance of surface adsorption effects; when surface adsorption dominates, increased temperatures result in decreased permeances. Second, entropic barriers allow for selective passage of a smaller gas, even when its surface adsorption is weaker and its potential energy minima are similar to that of another, larger gas. From a technological perspective, our results indicate that PG-TP1 is a promising material for membrane separation of propene and butadiene, and greatly outperforms all existing membranes in both selectivity and permeance. As an aside, we note that the PG-TP1 pore size of 7 Å was recently predicted to be optimal for water desalination.23 It is our hope that the properties described above will stimulate synthetic efforts, which may be feasible following the same surface-catalyzed Ullmann coupling reaction used to synthesize PG.36

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

REFERENCES

* Supporting Information S

Dimensional analysis calculations, table of fitted isotherm parameters, and animations showing example crossings for the gases discussed above. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Author

*(J.S.) E-mail: [email protected]. Present Addresses

† Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States. ‡ SunShot Initiative, U.S. Department of Energy, Washington, DC 20024, United States.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS K.S., J.A.W., and A.M.B. were supported by grants to Haverford College from the Howard Hughes Medical Institute, the Haverford College Provost’s Student Research Fund, and the ACS Petroleum Research Fund, respectively. This work was also supported by the Research Corporation for Science 17056



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Entropy-Driven Molecular Separations in 2D-Nanoporous Materials

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