I n d . Eng. Chem. Res. 1995,34,4193-4201
4193
Structural Characterization of Polycarbonates for Membrane Applications by Atomic Level Simulation Frank T. Gentile: Shone Arizzi? and Ulrich W. Suter Znstitut fur Polymere, ETH-Zentrum, Zurich, Switzerland
Peter J. Ludovice* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
Two commercially important membrane polymers, the tetramethyl (TMPC) and tetrabromo (TBPC) derivatives of Bisphenol A polycarbonate, were studied with computer simulation. The volume available to various gas diffisants in these polymers was characterized by calculating the volume of clusters of Delauney tetrahedra between the atoms of a n ensemble of bulk molecular mechanics models of the polymer. The inverse of this available volume correlated with the diffisivity of various gases in these polymers. This correlation was able to qualitatively reproduce the gas diffusion consistent with the superior diffusivity and superior selectivity of TMPC and TBPC, respectively. Analysis of the structure of the two polymers suggests a more ordered packing of the TMPC chain which is consistent with the experimentally observed trend in which inhibited packing leads to increased selectivity for gas diffusion in polymers. Despite the model’s neglect of the thermal motion of the polymer, it has potential for use as a tool to suggest other perturbations in polycarbonate structure that may produce superior properties.
Introduction Polycarbonate polymers are desirable for use in membrane applications for separating gas mixtures due to their unique properties. The mechanical strengths of these polymers allow the surface area to be maximized in the filtration network, while the diffisivities of gases through them allow a large throughput. This is believed to be due to the larger distribution of pockets of available volume in polycarbonate relative to other polymers which has been observed experimentally as well is in simulations [Royal and Torkelson (1992); Arizzi et al. (1992)l. Of particular interest are the tetramethyl and tetrabromo derivatives of the polycarbonate of Bisphenol A (Figure 1). Tetramethylpolycarbonate (TMPC) has a larger throughput than polycarbonate [Muruganandam and Paul (198713, and tetrabromopolycarbonate (TBPC) [Muruganandam and Paul (1987); Hellums et al. (199111 has a higher diffusive selectivity for different gases. In particular, the selectivity for nitrogen and oxygen is superior for tetrabromopolycarbonate [Koros and Hellums (198911. We have analyzed the structure of these membrane polymers in an attempt to elucidate a connection between their structure and their diffusivity and selectivity. Given the low values of the diffusivity of gases in glassy polymers and the current time limits for molecular dynamics simulations, it is impractical t o simulate gas diffusion in polymers using brute force molecular dynamics simulations. Therefore, some type of meanfield or perturbation approach must be taken to circumvent this limitation. We intend to study the connection between the atomic level structure of these two polycarbonates and their diffusive properties using detailed molecular mechanics models of these polymers. Our
* Author to whom correspondence
should be addressed. Present address: Cytotherapeutics Inc., 4 Richmond Square, Providence, RI 02906. t Present address: DuPont de Nemours International S.A., Fibers Technology Laboratory, CH1218 Grand Saconnex, Switzerland. +
Br
Br
Figure 1. Structure of substituted polycarbonates ( n = 17).
goal is to determine whether the differences in the simulated static structure of these amorphous polycarbonate glasses correlate with experimentally observed differences in diffisivity and to elucidate the connection between structure and diffusivity. Previously, molecular mechanics has been used to find local energy minima of ensembles of structures of polymer chains in three-dimensional periodicity. Properties were then averaged from these ensembles of static structures. Initially this approach was applied to atactic polypropylene [Theodorou and Suter (1985a)l and subsequently t o atactic poly(viny1chloride) (PVC)[Ludovice and Suter (1992)1, Bisphenol A polycarbonate (PC) [Hutnik et al. (1991a)l, and aromatic polysulfone [Fan and Hsu (1992)l. These simulations have been used t o predict solubility parameters, structural parameters, and mechanical properties. More recently, these molecular mechanics models have been used to study the diffusion of gases through these glassy polymers [Arizzi et al. (1992);Gusev et al. (1993);Gusev and Suter (1991, 1993)I. We intend t o analyze the diffisivity of various diffusants, He, Nz, 0 2 , and CHI, in TMPC and TBPC by characterizing the unoccupied volume available to each diffusant. This unoccupied volume consists of the clusters of available tetrahedra defined by the atoms in the polymer glass and was previously used to analyze
0888-5885/95/2634-4193$Q9.QOIO 0 1995 American Chemical Society
4194 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 Table 1. Physical Properties of Polymer Structures degree of cube edge density polymer polym MW atoms (A) (g/cm3) 485 20.42 1.08 TMPC 35 5540 485 20.47 1.95 TBPC 35 10094
structural differences between Bisphenol A polycarbonate (PC) and atactic polypropylene [Arizzi et al. (1992)l. Although this volume is related to free volume, it is inherently different. Free volume includes dynamic fluctuations of the polymer glass. This model, based on the unoccupied volume, neglects the fluctuations of the polymer glass and the coupling of these motions to the diffusant as well as the transition state between these domains of unoccupied volume. We will discuss the degree t o which this available volume may be used to estimate and characterize diffusivity of gases through these polymers.
Simulation Procedure The substituted polycarbonates simulated here are the result of the condensation of 17 substituted Bispheno1 A units with 18 phosgene units t o produce the phenyl-terminated polymers seen in Figure 1. Relevant parameters for the periodic cubes of these polymers are contained in Table 1. The basic simulation procedure has been described previously [Theodorou and Suter (1985a); Ludovice and Suter (1992); Hutnik et al. (1991b)l. It involves the generation of amorphous conformations of the polymer chain in a periodic cube, followed by subsequent minimization of the potential energy of the system. An ensemble of these periodic cubes is considered representative of the structure of the glassy polymer. This ensemble consisted of 10 periodic cubes of TMPC and TBPC. The procedure for the simulation of tetramethyl- and tetrabromopolycarbonates is essentially the same as that used for the previous polycarbonate study [Hutnik et al. (1991b)l) except for minor changes in the initial guess procedure and the potential energy force field. After generating the initial conformations for the substituted polycarbonates only those with a radius of gyration (s2) within 15% of the experimentally determined (s2) value were subjected to energy minimization. This screening procedure was used previously [Ludovice and Qian (1993)lto counteract the fact that the average radius of gyration (s2) produced by this procedure is sometimes significantly less than the experimentally determined value [Ludovice and Suter (199213. Although the simulated radii of gyration for these polymers did not differ significantly from their experimental values, this screening procedure was employed to eliminate the adverse effects of the self-avoiding nature of the initial guess procedure. This particular procedure consists of a self-avoiding walk in periodic boundary conditions. We have observed that, by the time 80% or more of the chain is generated, the remaining chain is assembled in a very compact conformation. This occurs due to the self-avoiding nature of this initial guess procedure, which can produce an unrealistically low radius of gyration. Despite the fact that it may bias the simulation results somewhat, it represents a correction to the unrealistic conformations that are sometimes generated by this initial guess procedure. This screening procedure is important because (s2) varies only a small amount upon minimization [Theodorou and Suter (1985a); Ludovice and Suter (1992)l. Even molecular dynamics used in conjunction with energy
Table 2. Statistical Weights and Rotational Barriers Used in the RIS Model for Initial Guess Generation angle from RIS statesa phenyl group rotational barrier Figure 1 (deg) (kcaVmo1) TMPC 1 45, 135 TMPC 2 45,135 4.0 TMPC 3 80,100 20.0 TMPC 4 0,180 TMPC 5 0,180 80,100 TMPC 6 45,135 TBPC 1 TMPC 2 45,135 2.0 TMPC 3 90 32.0 TMPC 4 0,180 TMPC 5 0,180 90 TMPC 6 a In both TMPC and TBPC, the statistical weight parameters from Hutnik et al. (1991a) were used.
minimization fails to produce a noticeable change in (2) [Ludovice and Qian (1993); Lee and Mattice (199213. In an effort to reduce this self-avoiding effect, other researchers have employed a phantom chain model for the initial guess procedure [McKechnie et al. (1992)l. These phantom chain models are not afflicted by this self-avoiding problem, but they do not introduce the effect of the surrounding polymer until the initial guess is already generated. By not including the effect of the surrounding polymer at the earliest stage, these phantom chain models can bias the polymer to be more amorphous than it actually is. Recently, the effect of the surrounding polymers has been associated with structures that are not totally amorphous in polymers such as poly(ethy1ene oxide) [Smith et al. (1995)l. This effect appears to produce a type of order that is intermediate between the amorphous and crystalline states in polymers such as PVC. Most amorphous polymers exhibit the typical “amorphous halo” diffraction pattern with a large broad diffraction at a wavevector magnitude of approximately 1 A-1 and with another smaller broad diffraction a t 3 A-l. This intermediate order is usually manifested in polymers as a splitting in this first diffraction peak of the amorphous halo pattern [Mitchell (1987)l. Polycarbonate’sneutron diffraction pattern contains a subtle split of this first diffraction peak, suggesting that the effect of the surrounding polymer may be important to the formation of some type of intermediate order in it and its derivatives [Gentile and Suter (199311. Both the phantom chain and the approach used here may bias the initial guess, and more work is needed to optimize this initial guess procedure. However,because we believe the effect of the surrounding polymer plays a critical role in the structure of polycarbonates, we have opted for the technique that includes this in the initial guess. The torsional states used in the RIS model for the construction of the initial guess conformations were modified for TMPC and TBPC [Hutnik et al. (1991b)l. The values of the angles of these states were determined from AM1 semiempirical quantum calculations on a trimer of each substituted polycarbonate [AMPAC (1988)l. A full geometry optimization on the trimers of TMPC and TBPC was carried out. Since the AM1 calculations indicated a negligible difference in the energy, all of these states were assigned equal statistical weights in the initial guess procedure. The values of these angles and rotational barriers used are listed in Table 2. The force field was modified to reflect the geometry and energetics of the substituted polycarbonates. All
Ind. Eng. Chem. Res., Vol. 34, No. 12,1995 4195 Table 3. Additional Potential Parameters for TMPC and TBPC parameter average polarizability (a,) for bromide group van der Waals radius (r,)for bromide group effective number of electrons (Ne,,)for bromide group phenyl carbon-bromine bond length phenyl carbon-methyl bond length
value 3.50 A3 1.90 A 21.2 1.88 A 1.49 A
Table 4. Static Point Charges for TMPC and TBPC atom number from Figure 1 atom type charge (e)asb TMPC 1 sp2 C (bound to 0 ) +0.08 2 sp2C (bound to H) -0.14 3 H +0.14 4 sp3 c 0.00 5 CH3 (bound to sp3 C) 0.00 6 main chain 0 -0.18 carbonate C +0.50 7 8 carbonate 0 -0.30 9 sp2 C (bound to sp3C) 0.00 10 sp2 C (bound to CH3) -0.10 11 CH3 +Oslo TBPC 1 sp2 C (bound to 0) +0.09 2 sp2C (bound to H) -0.09 3 H +0.15 4 sp3 c 0.00 5 CH3 (bound to sp3 C) 0.00 6 main chain 0 -0.15 7 carbonate C f0.45 carbonate 0 -0.33 8 9 sp2 C (bound to sp3C) 0.00 10 sp2 C (bound to Br) -0.15 11 Br f0.09 a Force-field parameters were generated using AM1 calculations [AMPAC (1988)l. Electrostatic charges were consistent with ab initio calculations using both 431G* and 631G* basis sets [Luthi (199O)J.
Lennard-Jones parameters were the same as in the PC study [Hutnik et al. (1991b)l with the exception of the methyl of bromide substituents. Unified atoms were used for the isopropylidene methyl groups in both polymers and for the methyl groups in TMPC. The scalar polarizability, van der Waals radius, and effective number of electrons may be used to calculate the Lennard-Jones parameters using the Slater-Kirkwood equation [Pitzer (1959)l. These parameters, as well as the relevant bond lengths, are listed in Table 3. An electrostatic potential function, which was truncated and employed a distance-dependent dielectric, was also included in the potential energy of the system. The same electrostatic function was used here as was used in the PC study except that the static partial charges were different [Hutnik et al. (1991b)l. These partial charges, listed in Table 4,were calculated using the AM1 calculations above and were in qualitative agreement with direct SCF calculations [Luthi (1989)l. The structures were minimized using a constantvolume procedure similar to the minimization for PC [Hutnik et al. (1991a,b)l. However, the total number of steps in the minimization was double that of the PC minimization. The additional steps were implemented to overcome the increased difficulty in finding a stable minimum caused by the increased torsional potential in the force field for these substituted polycarbonates. The structures were minimized to gradients less than (kcaYmol deg).
Conformational and Energetic Properties Given that the initial guesses were screened for reasonable values of the radius of gyration, we would
expect that the minimized structures would have reasonable values of this parameter. This fact is illustrated on the comparison to the experimental values [Bayer (1988)lof the radius of gyration divided by the molecular weight ((s2)/M)seen in Table 5. The simulated solubility parameters (6) also reflect the correct experimental trend. The solubility parameter was calculated by adding the long-range tail correction (AUtail,) to the difference in energy between the periodic structures (v) and the parent chains (Up,,) [Theodorou and Suter (1985a); Hutnik et al. (1991a); Ludovice and Suter (1992)l. The uncertainty in the parent and periodic energies in Table 6 is equal to the standard deviation of the average values for the ensemble of structures. The tail correction (AUt,i1*) is calculated as an average over all the ensemble structures. The value reported in Table 6 is the average of two values obtained from assuming two extremes in the distribution of the dipoles in the bulk polymer. This method has been discussed previously [Hutnik et al. (1991a); Ludovice and Suter (1992)l. The solubility parameter calculated from the simulated structures compares reasonably well to the values predicted from group additivity methods [Van Krevelan and Hofiyzer (1976)l. Although no exact measurement of the solubility parameter was available, the solubility behavior of these polymers is consistent with a higher solubility parameter for the brominated derivative [Bayer (1988)l.
Volumetric Properties The unoccupied volume in the glassy polycarbonates was determined by calculating the volume of the Delauney tetrahedra between the atoms in the glassy matrix. The portion of the van der Waals volume of the atoms in each tetrahedra was subtracted from this volume, taking csire t o correct for the overlap of van der Waals spheres. This procedure and its underlying algorithms have been described previously [Arizzi et al. (1992); Tanemura et al. (1983); Mott (1990)l. The volume-weighted distributions of the volume of these tetrahedra may be calculated. Figures 2 and 3 exhibit the distribution of empty volume in TMPC and TBPC, respectively. The volume percentage represents the percentage of the volume in a tetrahedra of a particular size that is not occupied by the van der Waals volumes of the atoms. Figures 2 and 3 indicate there are no striking differences in the these volume distributions, yet there are noticeable differences in the diffisivity and permeability characteristics of these two polycarbonates. Diffusivity of TMF'C varies from 2 to 4 times the diffisivity of TBPC for a variety of gaseous diffusants [Muruganandam and Paul (198711. Obviously there is little hope of describing these diffusivity differences using the distributions in Figures 2 and 3. A more useful distribution to analyze is the distribution of volume that is accessible to a particular size diffusant. This approach was previously used in a Monte Carlo model [Shah et al. (1989)l. This Monte Carlo model inserted spheres at random points and grew them to a given size. The fraction of positions a t which a sphere of a given size could be inserted was then calculated for a number of polymer models. A similar quantity may be calculated by using the volume of clusters of tetrahedra that are accessible to a diffusant of a given size. As defined previously [Arizzi et al. (199211, these clusters are sets of tetrahedra with sufficient volume to hold a given diffusant, that are connected by faces of the tetrahedra sufficiently
4196 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 Table 5. Radius of Gyration for Initial Guess and Minimized Structures of TMPC and TBPC (X= 35) initial guess structure (s2)/M minimized structure (s2)/M experiment (s2)/M polymer no. of structures (A2 moVg) no. of structures ( A 2 mol/g) (A2 moVg) 12 0.12f 0.02 0.14 TMPC 100 0.15f 0.06 0.05f 0.03 0.07 10 0.07f 0.01 TBPC 100 ~~
Table 6. Calculated Energies (Upar, AUta*, U,and Uooh) and Hildebrand Solubility Parameters (6)for TMPC and TBPC TMPC TBPC parameter Up,,, kcal/mola 508.4f 22.5 629.1f 42.6 -46.4 f 5.1 -110.0 f 1.2 AUtails,kcal/molb 246.1 f 16.4 259.0f 49.9 U,kcal/mola 445.0f 68 308.8f 44 ucoh, kcaVmo1 15.9f 1.2 23.0 f 3.2 6, (J/rnL)lI2 18.5 21.0 group contribution calculation of 6, (J/mL)'I2 a Error is 1 standard deviation about mean for simulated structures. Mean of two different methods [Ludovice and Suter (1992)l. Van Krevelan and Hoftyzer (1976).
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Volume (A3) Figure 3. Volume distribution of Delauney tetrahedra in TBPC. Table 7. Diffusant Sizes diffisant He
diameter (AY 2.576 3.350 0 2 3.480 N2 3.790 CH4 a He diameter calculated from diffusivity data and other values from viscosity data from Hirschfelder et al. (1964).
large to permit a diffisant t o cross from one tetrahedra to another within a given cluster. As with the previous approach [Shah et al. (198911, the distribution of these cluster volumes is a function of the size of the diffisant. For simplicity we use spherical diffisant molecules with mean diameters given in Table 7. These diffisant diameters are appropriately taken from the kinetic cross section of the molecules from transport coefficient data [Hirschfelder et al. (1964)l. Although the structure of the polymer is determined from local energy minima based on the force field described above, our mean
spherical approximation does not include any detailed energetic interactions between the diffusant and the polymer. The clusters of Delauney tetrahedra are comprised of those tetrahedra into which a diffusant of a given diameter may fit. This is tantamount to modeling the interaction of the diffusants with the polymer as a square well potential that is zero if the polymer atoms remain outside the effective radius of the diffisant and infinite if the diffisant effective radius and the van der Waals radii of the polymer atoms overlap. A more sophisticated calculation of the energetic interaction is possible, however, this approach is more computationally expedient. Clearly, when strong polymer interactions with very polar gases occur, a more detailed potential may be necessary. The distributions for three diffusants are shown in Figures 4-6 for TMPC and TBPC, respectively. The observed constant increase in the height of the histogram bars is due to the fact that the distributions are based on a volume percentage and most of the larger clusters occur only once in the ensemble of 10 structures. A larger cluster will contribute a proportionally larger amount to the total volume. The histogram bars that are approximately 2 or 3 times the height of their neighboring bars indicate the presence of 2 or 3 clusters of that size. Unlike the distribution of individual tetrahedra volumes, the cluster volumes in Figures 4-6 show a more distinct difference between TMPC and TBPC. An identical difference is also observed for the clusters containing methane as a diffusant. The TMPC contains larger clusters of available volume than TBPC. This would imply a larger amount of available volume for diffusion in TMPC relative t o TBPC. This distribution of cluster volumes may be integrated to calculate the total available cluster volume. The arithmetic average of the available cluster volume for each of the ensembles of polycarbonate structures is listed in Table 8. The total fraction of unoccupied volume is also listed in Table 8. This quantity is the same regardless of the diffisant and represents the limit of the available cluster volume as the size of the diffusant becomes vanishingly small. This volume represents the volume in the simulated periodic cube that is not occupied by the polymer van der Waals volume. The maximum volume of the clusters found in the ensemble, the total number of clusters, and the number of tetrahedra comprising these clusters are also listed in Table 8. The error listed for the fraction of total and available volume is the 90% confidence limit about the mean of these values for the ensemble of 10 structures. All values listed in Table 8 are fractional volumes, but actual volumes would reflect the same trends because the total volumes of the simulated periodic cubes differ by less than 1%for the two polycarbonate derivatives. Table 8 indicates that TMPC has larger clusters of available volume than TBPC, and these clusters contain more tetrahedra. Therefore, these clusters are larger in TMPC as indicated by Figures 4-6. The available cluster volume in TMPC is statistically different (90% confidence) from the values for TBPC. This is more obvious in the plot of available cluster volume vs the
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diffisant diameter seen in Figure 7. The cluster volume may be calculated for any diffusant volume. Figure 7 contains additional points, with the cases for the diffusants from Table 5 labeled. The fact that the available volume is consistently greater for W C is qualitatively consistent with the fact that the diffisivity of a variety of gaseous diffisants in TMPC is greater than that for TBPC. The continuous lines in Figure 7 simply connect the calculated points which appear as error bars. The error bars in Figure 7 represent the 90% confidence interval calculated from the variance of the 10 simulated structures. The behavior of this cluster volume as a function of diffusant diameter is different than the function obtained by the aforementioned Monte Carlo model [Shah et al. (1989)l. In the aforementioned model, available volume decayed exponentially with increasing diffisant diameter while the cluster volume in Figure 7 levels off a t small diffisant diameters. This leveling off occurs because, as the diffusant diameter approaches zero, the volume available to that diffusant approaches the volume available in the ensemble of structures outside
the van der Waals radii of the atoms contained therein. The fact that the Monte Carlo model did not exhibit such behavior is probably due to the poor convergence of this model for small diffusants. The tessellation model presented here provides an unambiguous characterization of the available volume. Assuming that this simulated available volume is at least qualitatively related to the measured free volume in polymers provides us with a qualitative connection between this volume and the diffisivity. It can be shown that the logarithm of the diffusivity varies inversely with the free volume of many polymers meith (199111. Koros and co-workers have shown the logarithm of the diffisivity varies inversely with the fractional free volume for halogenated polycarbonates [Hellums et d. (1989b)l. Given the aforementioned assumption, a similar relationship between diffusivity and the available volume may exist. This relationship is plotted in Figure 8. The relationship in Figure 8 does appear to be fairly linear. The fitted line is the result of a linear least-
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4198 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 15
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squares fit of the logarithm of the diffusivity vs the inverse of the available volume (R2= 0.93). The fitted line from Figure 8 was used t o estimate various diffusivities and selectivities in TMPC and TBPC. These estimated values are compared to their experimental counterparts in Table 9 [Muruganandam and Paul (198711. Qualitative trends in the experimental values such as the higher diffisivity of gases in TMPC and the higher selectivity in TBPC are reproduced by the linear relationship between the diffisivity and the simulated available volume from Figure 8. However, given the fact that numerous values differ by a factor of 4 or more, this relationship cannot be said to quantitatively reproduce the experimental values. The comparison in Table 9 illustrates that the correlation in Figure 8 is clearly only qualitative. Since the effect of thermal motion is critical to the process of diffusion, one would suspect that the failure of this correlation to quantitatively reproduce experimental data is due to the failure of the model t o take into account the dynamic motion of the polymer. The data in Figure 8 flatten out as the inverse of the
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available volume increases. This~ deviation from linear~ ' ~ ~ " ~ ~ ity may occur because the simulated available volume neglects the effect of the polymer motion. This motion makes a larger contribution t o diffusants of larger volumes and would increase the available volume for larger diffisants. This may improve the linearity of the fit in Figure 8. Previous transition state models of gas diffusion through the clusters modeled from molecular mechanics such as the ones discussed here [Gusev et al. (1993)l were only slightly improved by the inclusion of the dynamic motion of the polymer [Gusev and Suter (199313. Dynamic simulations confirmed that the rootmean-squared fluctuations of the polymer matrix due t o the thermal motion assisted the larger diffusants more than small ones due to the longer residence time of these diffisants in a particular Delauney tetrahedron. However, molecular dynamic simulations of the polymer are required to characterize these fluctuations and these simulations are very computationally intensive. The dynamic simulations used in these transition state models [Gusev and Suter (1993); Gusev et al. (199311 would eliminate the need to fit experimental data, but the resulting correlation between the predicted and experimental diffusivities is no better than that obtained here. Given the computational requirements of these dynamic simulations, the correlation procedure here appears to be a more practical approach. For this reason the authors of these transition state models suggest that fitting of experimental data may be more expedient than dynamic simulations. The lack of a good quantitative correlation with experimental data for these models is most likely due to the approximate nature of the force field. Experimentally, PC should possess diffusivities intermediate between those of TMPC and TBPC. However, an analysis of 10 PC structures simulated previously [Hutnik et al. (1991a)l produced cluster volumes significantlylower than those for TBPC. Also noticeable was the significantly larger error associated with these cluster volumes [Hutnik et al. (1991a)l. The cluster analysis was carried out in the same manner so the discrepancy appears to be due t o the manner in which the polymer structures were simulated. The two differences in the simulation procedure are the fact that the original PC initial guesses were not screened by their radii of gyration and the substituted polycarbonates were minimized in twice as many steps as the original PC simulations. The effect of each of these differences on the resulting structure should be isolated and is currently under study. The additional steps in the minimization procedure were implemented because we had difficulty obtaining a minimum with the desired gradient tolerance for the substituted polycarbonates. These additional steps may have driven these structures to a set of lower energy minima, which may account for their difference in structure when compared to the original PC structures. If this is the case, then the PC structures did not sufficiently relax to open up a sufficient amount of free volume to produce a simulated diffusivity greater than that of TBPC. This study clearly shows the significant effect that the details of the simulation procedure have on the resulting structure. Only structures simulated by identical procedures may be compared, and the effect of the simulation procedure on the structure for this class of polymers requires significant further study.
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Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4199 Table 8. Quantitative Information about Clusters quantity
He
0 2
max cluster vol. (As) no. of available clusters total tetrahedra in clusters total unoccupied vol. fraction fraction of total available vol.
2229.1 109 4299 0.418f 0.003 0.267?c 0.004
max. cluster vol. (A31 no. of available clusters total tetrahedra in clusters total unoccupied vol. fraction fraction of total available vol.
1979.9 129 3897 0.406f 0.003 0.227f 0.008
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TMPC 1625.2 79 2214 same 0.185f 0.012 TBPC 1190.2 86 1762 same 0.139f 0.013
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Figure 8. Experimental diffusivity (ref 3)vs the inverse of the calculated available volume. Table 9. Comparison of Predicted and Experimental Diffusivities and Selectivities TMPC TBPC exptl predicted exptl predicted diffusivity (loscm2/s) He 1266 429 556 394 0 2 8.11 12.76 1.69 1.61 Nz 2.15 7.93 0.34 0.53 CHI 0.94 1.79 0.13 0.03 diffusive selectivity 1347 239.9 4277 13200 He/CH4 OD2 3.77 1.61 4.97 3.038
Structural Properties A model's prediction of experimentally consistent results is of little significance unless some insight into the mechanism behind those results is provided. The accuracy to effort ratio of most simulation studies is much lower than that of experimental studies for simple
Nz
CHI
1613.8 80 1954 same 0.172f 0.013
1521.4 55 1440 same 0.141 f 0.017
1168.2 70 1471 same 0.123 f 0.014
668.0 47 1019 same 0.095 f 0.015
properties such as diffusion. Therefore, the real use of a simulation study should be to determine what atomic level characteristics produce desirable properties such as the high diffisivity of gases through TMPC and the high selectivity of TBPC. Since the diffision model is based on static structure, some difference in the static structure between TMPC and TBPC might provide a clue to the desired structureproperty relationship. Since the polymers are basically amorphous, one would not expect a large degree of structure in the chain. After examination of all the binary radial distribution functions [Theodorou and Suter (1985bl1, very little difference was observed between the TMPC and TBPC structures. There was very little difference between most of these pair distribution functions and those of the former polycarbonate study, except, of course, for those that involve the new methyl and bromide substituents. The only pair distribution function g(r)that is noticeably different than the polycarbonate study is the hydrogen-hydrogen pair distribution function. This difference occurs because half of the hydrogens on the phenyl rings have been replaced by bromo or methyl groups. These phenyl hydrogens represent half of the hydrogens because unified atoms that contain no explicit hydrogen atoms are used for all the methyl groups. The only difference between TMPC and TBPC appears t o be a subtle difference in the intermolecular correlation of the chains. Figure 9 contains the intermolecular correlation function of the vectors connecting the main-chain oxygen atoms that flank the carbonyl carbon of the carbonate group. This intermolecular correlation function is described by the equation
(1) where O(r)is the angle between two carbonate vectors on different chains, where the centers of the these vectors are separated by a distance r [Theodorou and Suter (1985b)l. Because the models consist of one chain packed into a periodic cell, intermolecular interactions are derived from the single chain interacting with images of itself. Positive values of this correlation function that approach 1indicate parallel alignment of this carbonate vector, whereas negative values that approach -l/2 indicate perpendicular alignment. More alignment of the chains in TMPC relative to TBPC appears to be observed in Figure 9. The difference between this alignment may be on the order of the noise of calculation, and more convergent simulations are required to make conclusive statements about the origin of the diffusivity difference between TMPC and TBPC. The direction of the difference in alignment observed
4200 Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995
1
TMPC
Y
-0.5
"
0
'
2
'
"
"
"
4
6
8
10
12
14
Radial Distance (A)
TBPC
n
1
1
l
I
logarithm of the diffisivity of various gases in these polymers, and the nature of the deviation from linearity is consistent with the neglect of the polymer motion. This linear correlation is able to qualitatively reproduce the gas diffisivities. The principle purpose of this work is not to obtain a quantitative correlation but a model that can assist in the design of new membrane materials. Although preliminary, this work represents the first case in which qualitatively accurate differences in structure were predicted by a model for two commercial polymers this similar in structure. Given the amorphous nature of these polymer glasses, it is difficult to pinpoint differences in their structure that would cause the differences in properties. The pair distribution functions for the two polymers are essentially identical, but the intermolecular chain direction correlation function suggests a more ordered chain packing in TMPC. This result is consistent with the previous experimental finding that inhibition of chain packing leads to the increased selectivity. Additional work is required to characterize the sensitivity of this method to various aspects of the modeling procedure such as the force field, energy minimization protocol, and the generation of the initial conformation. Once these issues are addressed, however, this method shows promise for developing structure-property relationships that can assist in the design of membrane materials.
Acknowledgment
-0.5
0
2
4
6
8
10
12
14
Radial Distance (A) Figure 9. Intermolecular correlation function of carbonate groups for TMPC (a) and TBPC (b).
in Figure 9 suggests a possible source of the diffisivity differences in these two polycarbonate derivatives. There is an increased tendency for these carbonate vectors t o align parallel to each other when separated by approximately 3.8 A and perpendicular when separated by 5.3 A. This alignment is more pronounced in TMPC relative to both TBPC and polycarbonate [Hutnik et al. (1991a)l. This increased alignment could produce a locally denser packing of the TMPC chains in some regions, which would result in increased available volume in other regions. Previous experimental work with polycarbonate membranes supports the hypothesis that both inhibition of chain packing and restriction of chain mobility improve selectivity in these membranes [Hellums et al. (1989a,b, 199111. Because this model is statjc and has neglected the mobility of the polymer, we cannot comment on the effect of chain mobility on selectivity. However, increased order in the packing of TMPC relative t o TBPC is consistent with the experimental finding that inhibition of chain packing can enhance selectivity.
Conclusions Despite the fact that the motion of the polymer was neglected in this model, a correlation between the simulated available volume distribution and the diffusion properties of TMPC and TBPC was observed that was consistent with experimental results. The inverse of the total available volume varies linearly with the
We gratefully acknowledge the assistance of Dr. Hans-Peter Luthi from the ETH-Ziirich for his ab initio calculations and Dr. Peter H. Mott and Dr. Michelle Hutnik for the initial computational codes modified for these simulations. Financial support is also gratefully acknowledged from Dow Chemical of Midland, MI, the Petroleum Research Fund of the American Chemical Society (Grant No. 28597-G7), and the Bundesminesterium fur Forschung und Technologie of the German Government.
Special Abbreviations and Symbols TBPC = tetrabromopolycarbonate TMPC = tetramethylpolycarbonate Si,,&) = intermolecular bond direction correlation function Literature Cited AMPAC Computer Program (Department of Chemistry, Indiana University, QCPE Program #506,1988). Arizzi, S.; Mott, P. H.; Suter, U. W. Space Available to Small Diffusants in Polymeric Glasses: Analysis of Unoccupied Space and Its Connectivity. J . Polym. Sci., Part B Polym. Phys. 1992, 30,415-426. Bayer AG, Leverkusen, Germany, unpublished results, 1988. Fan, C. F.; Hsu, S. L. Application of the Molecular Simulation Technique to Characterize the Structure and Properties of an AMPAC Computer Program (Department of Chemistry, Indiana University, QCPE Program #506, 1988).Aromatic Polysulfone System. Macromolecules 1992,25,266-270. Gentile, F.T.; Suter, U. W. In Amorphous Polymer Microstructure from Materials Science and Technology: Structure and Properties in Polymers; Haaseen, P., Kramer, E., Eds.; VCH Publishers: Weinheim, Germany, 1993;pp 33-77. Gusev, A. A.; Suter, U. W. Theory for Solubility of Static Systems. Phys. Rev. A 1991,43,6488-6494. Gusev, A. A.; Suter, U. W. Dynamics of small molecules in dense polymers subject to thermal motion. J . Chem. Phys. 1993,99, 2228-2234. Gusev, A. A.; Arizzi, S.; Suter, U. W. Dynamics of light gases in rigid matrices of dense polymers. J . Chem. Phys. 1993, 99, 2221-2227.
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IE9502291
* Abstract published in Advance ACS Abstracts, November 15, 1995.
