Molecular Simulation Study of Gas Solubility and Diffusion in a

Article pubs.acs.org/JPCB

Molecular Simulation Study of Gas Solubility and Diffusion in a Polymer-Boron Nitride Nanotube Composite Congyue Wang,† Preeti Jagirdar,† Saber Naserifar,†,‡ and Muhammad Sahimi*,† †

Mork Family Department of Chemical Engineering & Materials Science, University of Southern California, Los Angeles, California 90089-1211, United States ‡ Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, United States ABSTRACT: We study the possibility of using polymer composites made of a polymer and boron nitride nanotubes (BNNTs) as a new type of membranes for gas separation. The polymer used is amorphous poly(ether imide) (PEI), and zigzag BNNTs are used to generate the composites with the PEI. The solubilities and selfdiffusivities of CO2 and CH4 in the PEI and its composites with the BNNTs are calculated by molecular dynamics (MD) simulations. The molecular models of the PEI and its composites with the BNNTs are generated using energy minimization and MD simulation, and the Universal Force Field is used to represent the interactions between all the atoms. The morhology of the composites are characterized and are compared with that of PEI. The accuracy of the computations is tested by calculating the gases’ solubilities and self-diffsivities in the pure PEI and comparing them with the experimental data. Good agreement is obtained with the data. The computed diffusivities and solubilities in the polymer-BNNTs composites are much larger than those in the pure polymer, which are attributed to the changes that the BNNTs induce in the polymer composite’s free-volume distribution. As the mechanical properties of the polymer-BNNTs composites are superior over those of the pure PEI, their use as a membrane for gas separation offers distinct advantages over the pure polymer. We also demonstrate that, calculating the diffusion coefficients with MD simulations in the NPT ensemble, as opposed to the common practice of utilizing the NVT ensemble, leads to much more accurate results.

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INTRODUCTION Two important classes of materials are polymers and nanotubes. Natural polymers and polymeric materials, ranging from amber, wool, silk and cellulose, to polysaccharides, polypeptides, and polynucleotides are critical to many aspects of living systems. Synthetic polymers exist in very large numbers and variety, without which technology would not have been in its curent advanced stage. Both groups of polymers are studied in biophysics, macromolecular science, polymer physics, and polymer chemistry, as well as for engineering applications. Fabrication of nanotubes, which constitute a class of quasione-dimensional materials, has been studied extensively over the past two decades. Among them, carbon nanotubes, which are essentially rolled graphene sheets, have been studied by many groups and are being tested for a wide variety of applications.1−3 Anther important class of nanotubes that has been fabricated4−7 includes those that are made of silicon carbide, a material with outstanding mechanical properties. Hydrogen sorption4,5,8 and flow of water at room temperature,9 as well as diffusion of supercooled water10 in such nanotubes have been studied recently, both experimentally and by atomistic simulation. These studies indicate the rich properties of such nanotubes, which are also being considered for several applications. Boron nitride (BN) is a traditional material with a variety of applications. Its layered hexagonal structure makes it © XXXX American Chemical Society

possible to fabricate another class of materials, including the BN nanotubes11−16 (BNNTs) and other types of nanostructured materials. Meng et al.17 provide a good review of the properties of a variety of BN-related nanostructured materials. Fabrication of nanotubes has motivated studies of yet another class of new materials with novel properties. They are composites that consist of a polymer and nanotubes. In principle, such composites should possess better mechanical properties than pure polymers and withstand higher temperatures. Several classes of such composites using carbon nanotubes,18−25 silicon-carbide nanotubes,26−29 and the BNNTs11,30,31 have been studied, either experimentally, or theoretically, or by atomistic simulations, in order to understand the mechanical and electrical properties of the composites. The BNNTs represent important candidates for fabrication of polymeric composites. In fact, several important properties of the BNNTs are superior over their carbon counterparts. For example, they are thermally stable up to 900−1000 °C in air.32 It is also worth noting that the thermal conductivity of the Received: October 26, 2015 Revised: January 11, 2016

A

DOI: 10.1021/acs.jpcb.5b10493 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B BNNTs has been predicted33,34 to be higher than that of carbon nanotubes. According to several theoretical and experimental studies, the BNNTs also possess such superb properties as structural stability, antioxidation resistance, chemical inertness, and surface polarization,32 which have the potential to be exploited for a variety of applications. In this paper we describe another application of polymer− nanotube composites, namely, their use as a new type of membrane for separation of gaseous mixtures, by computing diffusion coefficient and solubility of small gases in such materials. We test one class of such composites, namely, a polymer−BNNT composite. Past experiments indicated35 that membranes that are composed of polyimide and carbon molecular-sieve membranes possess much better permeance and selectivity for separation of CO2/CH4 mixtures than the pure membrane, and atomistic simulations provided evidence that diffusion of various gases in carbon nanotubes36−38 occurs high rates and speeds. Thus, we present the results of extensive molecular dynamics (MD) simulation of diffusion and solubility of the two gases in the polymer-BNNT composites. The polymer utilized in the study is poly(ether imide) (PEI),39 which has been utilized extensively in the fabrication of nanoporous inorganic membranes, as well as polymeric ones. Atomistic simulation of diffusion of small gases in polymers40−42 is highly intensive. We are aware of only a few past molecular simulation studies of diffusion of gases in polymers with flexible chains, repeating units and relatively simple structures,40−46 or those with stiff chains.47−50 The PEI that we utilize in our study is a glassy polymer, and to our knowledge there has been only one set of atomistic simulation of diffusion of small molecules through such polymers.51 This paper is organized as follows. In the next section we present the details of the molecular modeling, including the atomistic models of the BNNTs, the PEI, and its composites with the nanotubes. We then present the details of the MD simulation of the various properties of the polymer composites, after which we present the results and describe their implications, followed by in-depth discussion of the results. The last section summarizes the paper.

chain, the polymer’s atom were added to it one by one, following the growth algorithm.54−56 A bond b takes on an orientation relative to a neighboring bond b′ with a probability that depends on the orientational state of b′. The growth of the polymer must be in such a way that only the allowed orientational states are utilized. Such states are selected from the appropriate probability distribution functions in such a way that they produce the minimum increase in the polymer’s potential energy, corresponding to the maximum probability of its growth. To eliminate the end effects, 40 repeating units of the PEI together with one polymer chain were utilized in the simulation cell. Carbon dioxide was represented as a three-site molecule. The interactions were described by the Lennard-Jones (LJ) potential. Partial point charges were at the center of each site. Methane was represented by its full atomistic model with one carbon and four hydrogen atoms. All the parameters for the two gases, as well as those for the atoms in the PEI and the BNNTs were those given by the Universal Force Field (UFF), which we describe next. The nanotubes’ ends were saturated by hydrogen. B. The Force Field. While several accurate force fields (FFs) are available for polymers, to our knowledge no specific FF exists for the PEI-BNNT composites. Previously, the performance of the force field MM357 was evaluated58,59 for polymer-carbon nanotube composites for their interfacial properties. Taghavi Nasrabadi and Foroutan60 used the same FF to study the interfacial properties of the BNNTs with small portions (oligomers) of three types of polymers. None of the polymers had a structure similar to the PEI. In addition, in the absence of experimental data, our study is an exploratory one, as we wish to obtain insights into the properties of the polymer composite for our intended application. Thus, we utilized the UFF,61 the parameters of which have been estimated using general rules based only on the elements, their hybridization, and connectivity, and are available for the entire periodic table.61 Hence, the UFF provides a unified FF for the polymers, the BNNTs, and the gases that are considered in the present paper. But, in order to have an objective measure of the accuracy of the FF, we first computed the properties of the two gases in the pure PEI and compared the results with the experimental data. In the UFF, the interactions between the atoms are divided into a nonbond partbetween any pair of atoms that are within a given cutoff radiusand the bonded parts between the atoms connected chemically. In the case of nonbond (Coulombic and van der Waals) interactions, partial charges and parameters for the repulsive and attractive parts of the interactions are assigned to each atom. The bonded interactions consist of the stretching (two-body interactions), covalent bond angle (three-body interactions), and the dihedral (four-body interactions) terms. The total potential energy U is written as,

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MOLECULAR MODELS AND DETAILS OF THE SIMULATION In this section we describe the molecular models of the BNNTs and the PEI, and the force field that we utilized in the MD simulations. A. The Nanotubes and Polymer. We utilized the BNNTs of the zigzag type, namely, (3,0), (7,0) and (12,0) nanotubes. Their length is about 12.3 Å, while their diameters are, respectively, 9.5 Å, 5.6 Å, and 2.5 Å. The structure of the BNNTs is similar to that of carbon nanotubes, with the difference that the length of the B−N bond is 1.44 Å, slightly larger than the C−C length, 1.42 Å in carbon nanotube. The length of the N−H and B−H bonds is 1.9 Å. Ab initio calculations52,53 have indicated that the BNNTs are slightly buckled, with the nitrogen atoms moving slightly outward and the boron atoms slightly inward. We refer to the three polymer composites as PC3, PC7, and PC12. We utilized a version54,55 of the algorithm developed by Theodorou and Süter56 in order to develop the atomistic structure of the PEI, which is based on self-avoiding walks. Three polymer atoms in PEI’s backbone, connected together by bonds, were initially put in a unit cubic cell. The orientations of the bonds were selected at random. To grow the backbone

U = Us + Uθ + Uϕ + Uω + Unb

(1)

Here, Us is the bond-stretching potential, given by Us =

∑ bonds ij

1 kij(rij − r0)2 2

(2)

where rij is the distance between the centers of atoms i and j, with r0 being the equilibrium length of the bond, and kij a force constant. The potential Uθ represents the energy associated B


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The Journal of Physical Chemistry B with the change in the angle θ between pairs of bonds, and is given by Uθ = kijk ∑ Cn cos(nθ) n

well as the free-volume (FV) distribution around the tubes. As discussed shortly, our molecular simulations indicated that the gas molecules do not pass through the nanotubes. Indeed, the MD simulation indicated that the most important effect of the nanotubes on the composites’ morphology is generating large variations in the accessible FV of the composite. Such fluctuations are responsible for the better solubility and higher diffusivities of the gases in the polymer composites, as reported below. C. Molecular Dynamics Simulation. Generation of the atomistic structure of the PEI was initiated in a large simulation cell. The packing density at time t = 0 was 0.1 gr/cm3. We used the conjugate-gradient (CG) iteration to minimize the system’s energy, after which it was compressed by MD simulation in the NPT ensemble for 100 ps. The pressure during the NPT simulation was 0.6 GPs. The compression was done in order for the polymer’s density to approach its experimental value, 1.27 gr/cm3. To further relax the morphology of the polymer, its energy was minimized again, followed by MD simulation in the NVT ensemble, during which the temperature was 1000 K. Finally, MD simulations in the NPT ensemble were carried out for several ns at 1 atm and 300 K, which yielded the equilibrium morphology of the polymer. The self-diffusivities of two gases in the PEI were computed by the same procedure. The main difference was that, the gas molecules were inserted in the atomistic model of the PEI that had already been generated at 1 atm and 300 K and, then, the total energy of the system was minimized by the CG iteration and MD simulation. Needless to say, no overlap between the various atoms and molecules was allowed. We then estimated the diffusivities by simulation in both the NPT and NVT ensembles. As discussed shortly, we discovered important differences between the results obtained with the two ensembles. A similar procedure was followed for generating the atomistic structure of the polymer composites. First, the atomistic structure of the PEI at low density was generated (see above). Then, for each composite, PC3, PC7, and PC12, we inserted three BNNTs in the simulation cell, with their spatial locations and orientations selected at random. The equilibrium structures of the composites were then determined by the same procedure as before. In order to eliminate boundary effects, we used the minimum-image periodic boundary conditions. To compute the self-diffusivities of the gases in the polymer composites, after inserting the BNNTs into the polymer matrix, ten molecules of each gas were placed at random locations in the simulation cell. We then followed the same protocol, namely, minimized the total energy of the system by the CG iterations, followed by MD simulations and subsequent compression of the cell and, finally, relaxation of the entire structure. The time step for all the MD simulations was 1 fs. The Nosé−Hoover and Andersen algorithms were utilized to, respectively, keep the temperature and pressure constant. In the case of the pure polymer, the linear size of the cubic simulation cell was 31.3 Å, whereas in the case of the polymer-BNNT composites we used cells of lengths 31.8 Å, 32.4 Å, and 33.4 Å for simulating, respectively, the PC3, PC7, and PC12. The simulation succeeded in generating a molecular structure of the PEI with a density equal to its experimental value. Other structural parameters of the polymers, such as the angles between the bonds, were also checked against the data in order to ensure that the correct structure had been produced. The

(3)

with kijk being a force constant. The parameters Cn have been estimated so as to satisfy the appropriate boundary conditions, including the requirement that the function have a minimum at the natural bond angle θ0. The torsional rotational potential that we used is given by Uϕ = kijkl ∑ Dn cos(nϕijkl) n

(4)

where kijkl and the coefficients Dn are determined by the rotational barrier vϕ, the periodicity of the potential, and the equilibrium angle. For a given central bond jk, all torsions about the bond are considered, with each torsional barrier being divided by the number of torsions present about the bond jk. The inversion terms is expressed by Uω = kijkl[E0 + E1 cos ωijkl + E2 cos(2ωijkl)]

(5)

where kijkl is a force constant, and ωijkl is the angle between bond il and the plane ijk. For a given central atom i there are three unique axes, namely, ij, ik, and il. All three are considered, with each inversion barrier being divided by the number of inversions present (three) about the center i. Writing Unb = UvdW + Ue, the van der Waals part is given by UvdW

⎡⎛ ⎞12 ⎛ xij ⎞6 ⎤ xij ⎢ = dij⎢⎜⎜ ⎟⎟ − 2⎜⎜ ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

(6)

Here, dij is the depth of the potential well, and xij is the van der Waals radius. In the UFF, one uses, xij = xixj , where xi is the atomic van der Waals distance, although in certain cases one 1 may also use the standard arithmetic average, xij = 2 (xi + xj). Moreover, one uses the standard formula, dij = didj , where di is the atomic van der Waals energy. Finally, the electrostatic part of the nonbond potential energy is given by qiqj Ue = 332.0637 ϵrij (7) Here, qi is the partial charge of atom i, and ϵ is the dielectric constant (whose default value in the UFF is 1). The nonbond potential energy was cut off for rij > 12.5 Å. All the parameters of the UFF are given in the original reference61 and, therefore, need not be repeated here. Experience62−64 with single-wall carbon and silicon-carbide nanotubes8 indicated that, as their diameter decreases, their curvature structure changes the valence orbital hybridization and the effective bond order. In turn, this influence the dynamic fluctuations8,18,38,62−64 of the tubes out of their smooth cylindrical structure at zero temperature. This affects the strength of the interaction between the gases and the carbon nanotubes, which in turn influences their diffusion in the nanotubes. Thus, one may expect the same type of phenomena to happen in the polymer−BNNT composites that we study here, particularly since the diameters of the BNNTs that we use in the MD simulations are relatively small. Passage of the gas molecules through the nanotubes in the polymer matrix depends on the sizes of both the gases and the nanotubes, as C


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The Journal of Physical Chemistry B equilibrium densities of the polymer composites were lower than that of the pure polymer. There are currently no experimental data for the polymer composites that we study. But, as pointed out earlier, the nanotubes induce timedependent variations in the FV of the composites. Thus, one expects lower densities for the three composites.

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μe = −kBT ln S0 = −kBT ln⟨FvV exp( −U /kBT )/⟨V ⟩⟩ (8)

where T is the temperature, Fv is the accessible fraction of the FV, and kB is the Boltzmann’s constant. Here, ⟨V⟩ is the ensemble average of the volume V of the cell, with the averaging being over all the 20 configurations each of the pure polymer or its composites. The computed solubility S0 is related to the actual solubility S by

COMPUTED PROPERTIES

We calculated several morphological properties of the composite polymers, as well as the self-diffusivities and solubilities of the gases in the pure polymer and its composites with the BNNTs. A. The Radial Distribution Function. To understand the structural differences between the pure PEI and its three composites with the BNNTs, we computed the radial distribution function g(r) of the four classes of materials. g(r) is defined as the probability of finding atoms at a distance r from another atom, when compared with the ideal gas distribution. The computations are standard and need no explanation. B. The Accessible Free Volumes. An important issue is the effect of the nanotubes on the distribution of the FV of the polymer-BNNT composites. Thus, we utilized MD simulations in the NPT ensemble in order to calculate the FV distributions of the composites, as well as that of the pure polymer. For each case 20 snapshots of the materials were obtained from the MD simulation, each taken 100 fs after the last one. The accessible FVs were then calculated by the following procedure. (a) The simulation cell was divided into 1003 subcells, with 100 cells in each direction. (b) To determine whether each subcell contributed to the accessible FV, a spherical probe was used. Clearly, the accessibility is a function of the diameter of the probe. Thus, we inserted one particle of a fixed size at the center of each subcell, and determined the atom that was closest to each probe. If the distance between the probe’s center and the nearest atom was larger than the sum of the van der Waals radii of the probe and the atom, we counted the subcell as one that contributes to the accessible FV. We then used the Voronoi tessellation method65 to compute the size distributions of the cavities in the pure polymer and its composites. The details of the method are given by Tanemura et al.65 and need not be repeated here. The procedure was repeated for all the 80 snapshots, which allowed us to study the dynamic evolution of the distributions of the FVs. We also computed the same for several probe sizes. C. The Solubility. A gas solubility S0 is defined as its concentration in a volume element of a material that is in equilibrium with the same gas at the same pressure under bulk conditions. To compute S0 of the two gases in the pure polymer and its composite, the standard particle insertion method66 was utilized. The same 20 configurations each of the pure polymer and its composites, which were used for calculating the FV distributions, were also utilized with a test particle, with the potential parameters for the test particle also taken from the UFF. After placing the test particle in the system, the interaction energy U of the entire systemthe test particle plus the pure polymer or its compositeswas computed. The energy U was then used to compute the excess chemical potential μe that, in the NPT ensemble, is given by

⎛T ⎞ S = ⎜ 0 ⎟S0 ⎝ P0T ⎠

(9) 3

3

where S is in cm (STP)/(cm atm), T0 = 273.15 K, and P0 = 1.0 atm. D. The Self-Diffusivity. We already described the procedure for computing the self-diffusivity D of the two gases in the pure polymer, as well as its composites with the BNNTs. Thus, if R(t) is the position of a gas molecule at time t, estimates of its diffusivity D were obtained through the Einstein relation that relates D to the mean-square displacements of the gas, 1 D = lim ⟨|R(t ) − R(0)|2 ⟩ (10) t →∞ 6t where ⟨·⟩ represents an average over all the gas molecules and all the possible time origins of their diffusion, with the time intervals between them being 100 fs.

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RESULTS Figure 1 shows a sample configuration of the PEI with three BNNTs in its matrix. Two of the nanotubes are close to each

Figure 1. Three-dimensional view of the PEI and three boron nitride nanotubes in its matrix. The molecular structure of the nanotubes is shown on the right.

other, while the third one is at a position relatively far from the other two. The nanotubes’ locations are, of course, determined by the energy minimization and MD simulation procedure that was described earlier. Figure 2 presents the computed radial distribution functions g(r) for the four types of polymeric materials. The peak at 1.3 Å represents the C−H bonding, whereas the one around 1.4 Å is due to the C−C bonding in the phenyl rings of the pure polymer. It also represents the B−N bonding in the BNNTs. The magnitude of the second peaks in the polymer-BNNTs is larger than that in the pure polymer, which is clearly due to the contribution of the BNNTs. The computed g(r) demonstrate clearly that all the four types of materials are amorphous, as indicated by short-range order dominated by the fixed bond lengths that are associated with the backbone atoms. D


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The Journal of Physical Chemistry B

Figure 4. Distributions of the free-volume fractions of the PEI and its composites with the nanotubes, and their dependence on the size of the probe. The results represent an average over 20 configurations.

Figure 2. Computed radial distribution functions g(r) of the PEI and its composites with the nanotubes.

The fluctuations with the time of the accessible FV fractions Fv of the four materials are presented in Figure 3. The

lowest for the pure PEI, while there is not much difference between those of the PC7 and PC12. The distributions of the free or cavity volumes at three different times are shown in Figure 5. The results for PC12 are similar to those for PC7 and, thus, are not shown. It appears that the main difference between the PEI and its three composites is that the latter contain a few high-volume cavities that the pure polymer does not have. This is particularly true about PC7, which has a small but significant fraction of cavities with volumes larger than 40 Å3. Due to the interactions between the nanotubes and the polymer matrix, the latter is squeezed, causing the distribution to be more sharply peaked. The generation of a small number of large cavities in the polymer composites, which Figure 5 indicates are stable over time, opens the way for improved performance of the composites for transport and sorption of gases in them and, therefore, better separation of mixtures. The fluctuations in the excess chemical potentials are clearly correlated with those of the FVs. Because the polymer composites have high fractions of the FVs, the excess chemical potentials μe of the gases in them are lower than those in the pure PEI, which explain their higher gas solubilities presented in Table 1. In the case of the pure polymer, the solubilities are also compared with the experimental data.67 They both are larger than their experimental counterparts, and for both computed and measured solubilities, SCO2 > SCH4, indicating the consistency and relative accuracy of the molecular simulation and the UFF. The computed solubility of carbon dioxide in the pure PEI is about 50% higher than its measured value, whereas the computed SCH4 is about 38% larger than the reported value. The computed ratio, SCO2/SCH4 ≃ 4, is very close to the measured value of 3.7. Note that, the difference between the computed and measured solubilities that we are reporting here is about the same as those reported previously.41,42 Thus, it is clear that the MD simulation produces reliable estimates of the solubilities. Although we are aware of no experimental data for the solubilities of CO2 and CH4 in the polymer−BNNT composites, Table 1 does provide evidence that they have much higher gas solubilities than the pure polymer. Assuming that the difference between the computed solubilities in the three composites and the true experimental data, had they been available, are roughly the same as those for the pure PEI, Table 1 supports the view that gas solubilities in the polymer

Figure 3. Time-dependence of the fluctuations of the accessible freevolume fractions of the PEI and its composites with the nanotubes.

composites’ Fv are all greater that of the pure polymer, hence hinting at faster gas diffusion through the composites. We must point out that the ensemble fluctuations of Fv during the MD simulation in the NPT ensemble are very small. This is because the characteristic times in glassy polymers are so long that their accessible FVs fluctuate very little during the course of the MD simulation. The FVs of the PC7 and PC12 are almost the same, whereas the FV of the PC3 is higher than the other three types of materials, namely, the pure polymer, PC7 and PC12. On the other hand, the FVs of the PC3 also fluctuate more strongly than those in the other materials. This is interesting in view of the fact that the diffusivities of the two gases through the PC12 are higher than those in the PC7 and PC3 (see below). Inspecting the structures of the polymeric composites indicated, however, that the pressure used during equilibration, 0.6 GPa, had caused a squeezing effect on the nanotubes, as the size of the (12,0) nanotube is larger than that of the (7,0) BNNT. The repulsive interactions between the BNNTs and the polymers’ chain give rise to the fluctuations that are stronger in the case of the PC3. Figure 4 presents the dependence of the FVs on the size of the probe molecule. Note that one must account for the different densities of the PEI and its composites. As expected, Fv decreases sharply with increasing probe diameters. Consistent with Figure 3, the FVs are largest for PC3 and E


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Figure 5. Time-dependence of the cavity volume distributions in PEI and its composites with the nanotube.

Table 1. Computed and Measured Solubilities (in cm3/atm) of the Gases PEI S (computed) S (measured)

PC3

PC7

PC12

CO2

CH4

CO2

CH4

CO2

CH4

CO2

CH4

1.17 0.76

0.29 0.21

3.82

0.95

21.36

3.66

105.55

22.31

simulations were carried out in the NPT ensemble. In actual experiments in which CO2 or CH4 diffuses in a polymer or its composites, the volume of the polymer is also expected to increase due to swelling, unless it is somehow prevented from doing so. Thus, we believe that MD simulations of diffusion in the NPT ensemble correspond closely with experiments. But, because practically all the previous MD simulations of gas diffusion in various polymers were carried out in the NVT ensemble, we also carried out such simulations with this ensemble. To demonstrate that calculation of the diffusivities in the NPT ensemble does not generate any unphysical artifacts, we provide details of the computations. We carried out MD simulation in the NPT ensemble to compute the MSDs of the gases, but the aforementioned 0.6 GPa pressure was not maintained, which was used to generate the structure of the

composites are significantly larger than those in the pure PEI. As discussed below, the permeability K of a gas through a polymeric membrane is related to its self-diffusivity and solubility. Thus, higher solubilities of the gases in the polymer composites may imply that they also have higher permeabilities as well. As already described, we carried out MD simulation of diffusion of the two gases in the pure PEI, as well as in its composites with the BNNTs, in both the NVT and NPT ensembles. Simulation of diffusion in the latter ensemble implies that the polymer or its composite can expand and equilibrate at a lower density. Indeed, although energy minimization and MD simulation yielded a structure for pure PEI with density that was the same as its experimental value of 1.27 gr/cm3, during simulation of the gases in the same polymer the density decreased to 1.19 gr/cm2, when the F


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Figure 6. Root mean-square deviations of the size of the simulation cell during the computation in the NPT ensemble.

spatial distribution determined by energy minimization, the aforementioned nonuniformity decreases. In this case, the diffusivities are an accurate representative of the effect of the presence of the nanotubes on the morphology of the polymer composites. Studying the trajectories of the gases within the polymer composite provides clues to the effect of the nanotubes on the morophology of the polymer. We find, for example, that when compared with the pure PEI, the diffusion trajectories of carbon dioxide in the polymer composites occupy a more extended and elongated part of the polymer matrix, but they go around the BNNTs, rather than passing through them. In addition, the set of such trajectories in the polymer composites is much larger than the corresponding ones in the pure PEI. Thus, we may draw two conclusions: (a) The diffusivities of the two gases in the polymer composites must be larger than those in the pure polymer, and (b) the larger diffusivities are not because the gases pass through the nanotubes, but rather due to the effect of the nanotubes on the FV distribution of the composites, as already discussed. The evolution with time of the net displacements R(t) of the two gases in the PEI and the PC7 are presented, respectively, in Figures 7 and 8. These results were computed in the NVT ensemble. The notable aspect of these results is the difference between the displacement of CH4 in the pure polymer and in the PC7. In the former case, CH4 simply vibrates around 10 Å, indicating that it is trapped around its location (at least temporarily). In the polymer composite, on the other hand, the methane’s displacement is between 2 Å and 14 Å. It is the dense morphology of the pure polymer and, as discussed below, the MD simulation in the NVT ensemble that keeps the polymer’s volume fixed that make it difficult for CH4 to diffuse.

polymer and its composites, but the diffusivities were calculated at atmospheric pressure. We began the simulation at 0.6 GPa, and then relaxed the system to reach atmospheric pressure. Prior to the simulation in the NPT ensemble, we equilibrated the system in the NVT ensemble to remove all the strains from the polymer or its composites. Then, long simulations in the NPT ensemble were carried out in order to compute the MSD, but there were no significant changes in the size of the simulation cell, as we had already equilibrated the system in both the NPT and NVT ensembles. To demonstrate this, we computed the root mean-square deviations (RMSD) of the size of the simulation cells for the various cases over the course of the simulations. The results are presented in Figure 6. The average change in the size of the simulation cell is about 0.1 Å, which is completely negligible when compared with the simulation cell’s size of 35 Å. We note that there is a qualitative difference between a case in which a single nanotube is inserted in a polymer matrix, and one in which the polymer composite contains several of such tubes. In the former case, one would expect the accessible FV for diffusion of small gas molecules to be distributed nonuniformly about the nanotube’s axis. Thus, the diffusivities may differ if they are calculated in various segments of the polymer composite. In this case, they are correlated well with the local FV fraction through the Doolitles’ law,68 ⎞ ⎛ A ⎟⎟ D = C exp⎜⎜ ⎝ η − ηmax ⎠

(11)

with η and ηmax being, respectively, the packing fraction of the material and its maximum value. Here, A and C are two constants. However, if the polymer composite contains several nanotubes, which are distributed within the matrix with their G


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to the adjacent cavities, if they are not already occupied. The density of the polymer and its morphology control the amplitude of the vibrations and the hops’ frequency. In addition, the nanotubes are not rigid, but are allowed to deform as the MD simulation proceeds. The deformation further increases fluctuations in the FV of the polymer, allowing more hops. The increase with time of the mean-square displacements (MSDs) of the two gases in the pure polymer is shown in Figure 9. These results were obtained in the NVT ensemble. As

Figure 9. Time-dependence of the mean-square displacements (MSDs) of CO2 and CH4 in the pure PEI, computed in the NVT ensemble.

Figure 7. Time-dependence of the displacements of CO2 and CH4 from their initial positions in the pure PEI.

they indicate, and consistent with Figures 7 and 8, the MSDs of CO2 is larger than those of CH4. Moreover, methane cannot diffuse too far from its initial position in the polymer and, thus, its effective diffusivity calculated in the NVT ensemble is not representative of what one measures in experiments, since the polymer is not allowed to swell in the NVT ensemble. A similar phenomenon happened during diffusion of the two gases in the three polymer composites and, thus, the results are not shown. Figure 10 presents the resulting MSDs for the two gases in the pure PEI, obtained in the NPT ensemble. Because the

Figure 10. Time-dependence of the mean-square displacements (MSDs) of CO2 and CH4 in the pure PEI, computed in the NPT ensemble. Figure 8. Same as in Figure 7, but in the polymer composite of the PEI and the (7,0) boron itride nanotube.

polymer is allowed to expand, the self-diffusivities are larger than those computed with the NVT ensemble, which holds the volume and, therefore, the density of the polymer, fixed during the simulations. Figure 11 presents the same as in Figure 10, but for diffusion of the two gases in the three polymer composites. Once again, because simulation in the NPT ensemble allows the polymer composites to expand, the

These observations are in accord with the hopping mechanism that is believed to be responsible for transport of gases in amorphous polymers. In this mechanism, the molecules vibrate temporarily in a cavity until they can tunnel H


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determined for a large class of materials, they were not fitted so as they could be used for computing the solubility and diffusion coefficients of gases in polymers. Given such limitations, Table 2 indicates that there is reasonable agreement between the computed and measured solubilities and diffusivities of the two gases in the pure polymer. We also note that, consistent with the experimental data, DCO2 > DCH4. The computed self-diffusivities of CO2 and CH4 in the polymer composites, obtained with the NPT ensemble, are also listed in Table 2. No experimental data for diffusion of the two gases in the polymer composites have been reported. The selfdiffusivities of both gases in the PEI-BNNT composites are much larger than those in the pure polymer, and, consistent with the results for the pure polymer, for the PEI-BNNT composites, DCO2/DCH4 > 1. The permeability of gas through a polymer is usually written as, K = SD. Because the solubilities S and diffusivities of the two gases in the polymer composites are higher than those in the pure polymer, the permeabilties of the two gases in the polymer composites are also much higher. Thus, the PEI-BNNT composites have high potential for use as membranes with mechanical and separation properties that are superior to those of the pure polymer, consistent with a theoretical study by Hill.71

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DISCUSSION Since previous atomistic simulations36−38 provided evidence that diffusion of various gases in carbon nanotubes is very fast, one might expect that if nanotubes are used to fabricate polymer composites, the resulting material will also have much higher rates of diffusion of gases, because the gas molecules diffuse through the nanotubes. This is, however, naive because it had been assumed in the previous MD simulations36−38 that the nanotubes are rigid. If this assumption is relaxed, then it has been shown8 that diffusion in carbon and silicon-carbide nanotubes is significantly slower than what had been claimed.36−38 In addition, even if diffusion through the nanotubes were to occur, they must be large enough that the gas molecules can pass through them. Most importantly, even if the nanotubes have large enough diameters and significant diffusion does occur through them, percolation theory72 predicts that transport through the nanotubes will not be effective, unless the tubes form a sample-spanning cluster across the polymer matrix. In the present study, the volume fraction of the nanotubes in the composites is much smaller than the percolation threshold, or critical volume fraction for the formation of a sample-spanning cluster of the nanotubes, which is about 0.17.72 Therefore, the BNNTs do not form such a cluster, and Figure 1 confirms this. Thus, the improvement in the solubility and diffusion coefficient of the gases and, hence, permeation and separation properties of the polymer composite is due to the nanotubes generating large fluctuations in accessible FVs of the polymer, which give rise to cavity sizes that do not exist in the pure

Figure 11. Time-dependence of the mean-square displacements (MSDs) of CO2 and CH4 in the three polymer composites, computed in the NPT ensemble.

MSDs are much larger than those obtained in the NVT ensemble. If gas diffusion is Fickian, the slope of log (MSD) versus log (t) must be 1, and they are indeed ≈1. Note also that the time scale for diffusion to become Fickian is relatively short. In Table 2 we compare the self-diffusivities that we have computed in the pure polymer with the experimental data. For CH4 the two self-diffusivities differ by a factor of about 2.5. For CO2, on the other hand, there is wide scatter in the experimental data,69,70 varying by nearly 1 order of magnitude. The computed self-diffusivity of CO2 is consistent with the reported range of the experimental data. We should keep in mind that, although the parameters of the UFF were

Table 2. Computed and Measured Self-Diffusivities of the Gases (in cm2/s), Obtained with the NPT Ensemble PEI D × 107 (computed) D × 107 (measured)

PC3

PC7

PC12

CO2

CH4

CO2

CH4

CO2

CH4

CO2

CH4

6.7 1.3−11.4

3.2 1.13

14.4

4.2

23.1

7.2

47.1

9.73

I


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polymer. Our results presented in this paper indicate that this is indeed the case. As Figures 7 and 8 indicate, whereas the gas molecules may become temporarily trapped in the pure polymer, which is due to the entanglement of the polymer chains, the same does not happen in the polymer composites. The polarization effect of the BNNTs induces a much stronger adhesion of polymers to such nanotubes than to carbon nanotubes.60 The mechanical strength of the polymer− BNNT polymers is also much better than those with carbon nanotubes, as is their interfacial strength. Given such desirable properties and our promising MD simulation results, it would be of great interest to fabricate polymer composites with the BNNTs and measure the self-diffusivities and solubilities of gases in them. Due to their superior mechanical properties, such composite polymeric membranes will also tolerate much higher temperatures without their structure collapsing.

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SUMMARY Extensive molecular dynamics simulations were carried to generate molecular models of amorphous poly(ether imide) and its composites with boron nitride nanotubes. The solubilities and self-diffusivities of CO2 and CH4 in the pure polymer and its composites with the BNNTs at 300 K were computed using MD simulation. They are all larger than those in the pure polymer. This was explained by showing that the nanotubes induce larger fluctuations in free-volume distributions of the composites, which are manifested in the distributions of the cavity sizes of the composites. The distributions are similar to that of the pure polymer, except that they contain a longer tail toward the larger cavity sizes, indicating the presence of a larger number of cavities with large sizes that facilitate diffusion of gases through the composite. Thus, a polymer composite that consists of a polymer and boron nitride nanotubes possesses much improved gas permeabilities and separation properties for gaseous mixtures than the pure polymer.

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AUTHOR INFORMATION

Corresponding Author

*E-mail address: [email protected]. Phone: (213)740-2064. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to the National Science Foundation and the Department of Energy for partial support of the work reported in this paper.

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Molecular Simulation Study of Gas Solubility and Diffusion in a

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