Molecular Simulations of Physical Aging in Polymer Membrane

J. Phys. Chem. B 2006, 110, 16685-16693

16685

Molecular Simulations of Physical Aging in Polymer Membrane Materials Xiao-Yan Wang,† Frank T. Willmore,† Roy D. Raharjo,† Xiaochu Wang,† Benny D. Freeman,† Anita J. Hill,‡ and Isaac C. Sanchez*,† Department of Chemical Engineering, UniVersity of Texas at Austin, Austin, Texas 78712, and CSIRO Manufacturing Science and Technology, PriVate Bag 33, Clayton South, MDC, Victoria 3169, Australia ReceiVed: April 10, 2006; In Final Form: June 8, 2006

Poly(1-trimethylsilyl-1-propyne) (PTMSP), the most permeable polymer known, undergoes rapid physical aging. The permeability of PTMSP to gases and vapors decreases dramatically with physical aging. Cavity size (free volume) distributions were calculated in as-cast and aged PTMSP, using an energetic based cavitysizing algorithm. The large cavities found in as-cast PTMSP disappear in aged PTMSP, which is consistent with the positron annihilation lifetime spectroscopy (PALS) measurements. We also characterized the connectivity of cavities in both as-cast and aged PTMSP membranes. Cavities are more connected in as-cast PTMSP than in aged PTMSP. The average cavity sizes calculated from computer simulation are in good agreement with PALS measurements. The transport and sorption properties of gases in as-cast and aged PTMSP are also measured by molecular simulation. Computer simulations showed the decrease of permeability and the increase of permeability selectivity in PTMSP membranes with physical aging, which agrees with experimental observations. The reduction in gas permeability with physical aging results mainly from the decrease of diffusion coefficients. Solubility coefficients show no significant changes with physical aging.

1. Introduction Poly(1-trimethylsilyl-1-propyne) (PTMSP), a high fractional free volume polymer, has the highest gas permeability of all known polymers.1-5 Its exceptionally high permeability has been attributed to a rigid, low cohesive energy density backbone composed of alternating double bonds and bulky methyl and trimethylsilyl substituents. PTMSP also exhibits very unusual gas and vapor transport properties for a glassy polymer. PTMSP is more permeable to larger organic vapors than to smaller permanent gases. For example, the mixed-gas selectivity of organic vapors over permanent gases for PTMSP is 27 for n-C4H10/CH4.5 Such high vapor/gas selectivity may be useful for the selective removal of higher hydrocarbons from natural gas or hydrogen streams or for the removal of volatile organic compounds from air.6 However, rapid physical aging has precluded practical application of PTMSP as a membrane material.7-14 Physical aging (structural relaxation accompanied by an increase in density and reduction in free volume) is a fundamental phenomenon of glassy polymers related to their nonequilibrium state.15 The permeability of PTMSP to gases and vapors decreases dramatically with physical aging, whether the films are stored in ambient conditions or in a vacuum. For example, the permeability of isobutane in this polymer decreases 2 orders of magnitude during 100 days in a vacuum.7 Oxygen permeability decreased by a factor of 20 after storage in the ambient atmosphere for several years.11 The aging process is accelerated by annealing PTMSP films in a vacuum, air, and inert atmospheres.8,14,16 The rate of aging also depends on the type of catalyst used in the polymerization reaction17-19 and the film thickness.20,21 The rate of aging is faster for thin films (1 and 3 µm) than for thick * Address correspondence to this author. Phone: (512) 471-1020. E-mail: [email protected]. † University of Texas at Austin. ‡ CSIRO Manufacturing Science and Technology.

films (85 µm). The effects of aging on gas permeability and selectivity in PTMSP membranes are retarded when the polymer is continuously exposed to a gas mixture containing a highly sorbing organic vapor such as n-butane.10 The aging process of PTMSP is accompanied by changes in many physical properties of the polymer. The density shows dramatic increases from 0.70-0.75 g/cm3, which is representative of freshly cast films, to 0.95-1.05 g/cm3 for aged materials.9,12,14 A reduction in free volume is also observed by positron annihilation lifetime spectroscopy (PALS).9,12,17 In addition, aging leads to a decrease in interchain spacing as measured by WAXD9 and to more restricted chain movements as measured by spin lattice relaxation times of NMR.18,19 Physical aging of PTMSP is also complicated by chemical aging (partial oxidation of the polymer) and absorption aging (induced by the capture of some nonvolatile impurities that can block free volume and thus reduce permeation rate).8-9,12,17-18,22 Several studies have been devoted to understand and inhibit the aging phenomena in PTMSP membranes. Different methods have been considered to improve physical aging and chemical resistance based on cross-linking,23 nanocomposites,24,25 copolymers,26-28 or blends.4,29-30 Molecular modeling techniques have been widely used to get a deeper understanding of the relationship between structure and properties of polymeric materials, and they are even capable of predicting sorption and transport properties of gases and solvents in rubbery and glassy polymers.2 In this work, we will investigate how the aging process affects properties such as free volume, cavity size distribution, and connectivity in PTMSP membranes. Sorption and transport properties of small molecules in PTMSP are also calculated by molecular simulation techniques. The free volume of a polymer could be determined experimentally by using photochromic labeling31 and spin probe32 methods as well as inverse gas chromatography33 and positron

10.1021/jp0622334 CCC: $33.50 © 2006 American Chemical Society Published on Web 07/29/2006

16686 J. Phys. Chem. B, Vol. 110, No. 33, 2006 annihilation lifetime spectroscopy (PALS).34 PALS provides the most direct and detailed information on the size and concentration of free volume elements in the materials. In PALS, the free volume cavities are measured by the lifetime of orthopositronium (o-Ps) before annihilation in the free volume regions of the materials. For most polymers, three exponential components were used to analyze PALS spectra. The longest lifetime component (τ3) of o-Ps is directly correlated to the free volume cavity size in a material. The intensity (I3) of the o-Ps annihilation is indicative of the concentration of free volume cavities. However, for high free volume polymers such as PTMSP and TFE/BDD87 (a random copolymer composed of 13 mol % tetrafluoroethylene and 87 mol % 2,2-bis(trifluoromethyl)-4,5-difluoro-1,3-dioxole, also known as AF2400 (Du Pont, Wilmington, DE)), a four-component spectrum (with two long-lived components having lifetimes τ3 and τ4 and intensities I3 and I4) gave a better description of the experimental data.35-38 Consolati et al.12 reported the time dependence of PALS results and gas permeability for PTMSP samples stored under vacuum and in air for up to 60 days. The concentration of large free volume elements, as characterized by I4, decreased with aging, but with only a slight decrease in their size (τ4). In contrast, the concentration of small free volume elements (I3) slightly decreased, and the size element (τ3) also decreased. Yampol’skii et al.9 performed gas permeability and PALS measurements on as-cast and aged (stored at ambient conditions for 4 years) PTMSP membranes. There is a significant decrease in both the small and large free volume elements, characterized by τ3 and τ4. Their study also showed the reduction in the concentration of large free volume elements as characterized by I4. Molecular modeling provides an alternative way to determine the free volume and free volume distribution of the materials. Free volume as well as free volume distribution can be obtained by either geometric or energetic methods. The most often used geometric methods include the Voronoi tessellation39-41 method, the Voorintholt42 method, which generates a van der Waals surface of the polymer chain, and the accessible free volume37,43,44 methods. The cavity energetic sizing algorithm (CESA)45 is a recently developed energetic method in our group. In CESA, the cavity is defined as a spherical volume with a well-defined center, and that center as a local minimum in a repulsive particle energy field. Interestingly, in most PALS studies, spherical shapes are also assumed for the free volume elements. Therefore, free volume and cavity size distribution obtained from CESA can be considered as a good comparison with the PALS measurements. The average spherical cavity size in PTMSP by CESA is 11.2 Å,46,47 whereas it is 13.6 Å from PALS.36,48 Hofmann et al.49 investigated the aging behavior of PTMSP by molecular simulation. The model system of PTMSP built with a density of 0.75 g/cm3 (representative for as-cast PTMSP films) showed density increases toward about 0.95 g/cm3, which is a typical value for PTMSP membranes of a few months age.3,49 In this paper, cavity size distributions of as-cast and aged PTMSP membranes are calculated by using the CESA algorithm to examine the physical aging behavior of PTMSP. Connectivity of the cavities found by CESA is also characterized. Furthermore, transport properties of N2 and CO2 in as-cast and aged PTMSP membranes are calculated by molecular simulation. Our simulation results are compared with experimental data.9,11,12

Wang et al. 2. Methodology 2.1. Permeability. The steady-state gas permeability coefficient of a polymer film to a penetrant is the pressure and thickness normalized flux, and it is given by the relationship:2

P)

Nl p2 - p 1

(1)

where P is the gas permeability coefficient (cm3(STP)‚cm/(cm2‚ s‚cmHg)), N is the steady-state penetrant flux (cm3(STP)/(cm2‚ s)), l is the film thickness (cm), and p1 and p2 are the downstream (permeate) and upstream (feed) pressure (cmHg), respectively. The permeability is often expressed in Barrers, and 1 Barrer ) 10-10 cm3(STP)‚cm/(cm2‚s‚cmHg), where STP represents standard temperature (273.15 K) and pressure (1 atm). When Fick’s law is obeyed, and when the downstream pressure is much lower than the upstream pressure, the permeability coefficient defined in eq 1 can also be expressed as a product of diffusivity and solubility, that is:

P ) SD

(2)

where S is the solubility coefficient, and D is an average diffusion coefficient. Permeability is often measured experimentally by using the constant pressure/variable volume method.50 In this study, we will calculate the diffusivity and solubility of N2 and CO2 in as-cast and aged PTMSP through molecular modeling. We then compute the permeability of CO2 in PTMSP based on eq 2. The ideal selectivity of a polymer for penetrant A relative to penetrant B is the ratio of their pure gas permeabilities:2

RA/B )

PA PB

(3)

Combining eqs 2 and 3 then gives:2

RA/B )

( )( ) SA DA SB DB

(4)

where SA/SB is the ratio of solubilities of penetrants A and B, which is called the solubility selectivity. DA/DB is the diffusivity selectivity, which is the ratio of the diffusion coefficients of penetrants A and B. Solubility selectivity is controlled by the relative condensability of the penetrants and the relative affinity between the penetrants and the polymer matrix, whereas diffusivity selectivity is strongly influenced by the size difference between the penetrant molecules and the size-sieving ability of the polymer matrix. 2.2. Solubility. Experimentally, the sorption2 of penetrants into a glassy polymer is usually described by the dual mode model:

C ) k Dp +

C ′Hbp 1 + bp

(5)

where C is the equilibrium penetrant concentration in the polymer at pressure p(atm), with units of (volume of penetrant (cm3)/volume of polymer (cm3)). kD is the Henry’s law parameter that describes penetrant dissolution into the equilibrium densified polymer matrix. C H′ is the Langmuir capacity of the glass. b is the Langmuir affinity parameter describing the affinity of a penetrant for a Langmuir site. The Langmuir capacity is equivalent to the maximum concentration of solute molecules in the unrelaxed (Langmuir) matrix of a glassy

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polymer. It can be viewed as a measure of excess free volume of a polymer. The solubility of a penetrant in a polymer is defined as the ratio of equilibrium penetrant concentration to penetrant pressure:51

S)

C b ) kD + C ′H p 1 + bp

(6) Figure 1. Trans and cis structure of PTMSP.

The infinite dilute solubility coefficient, S0, is calculated as follows:

S0 ≡ lim pf0

C ) kD + C ′Hb p

(7)

S0, also called the Henry’s law solubility coefficient, can be calculated by using the Widom insertion method52 through molecular simulation. In this method, we calculate the energy ∆u of randomly inserting a particle into a molecular configuration. ∆u is the interaction energy between the inserted particle and the rest of the particles in the system. The dimensionless solubility, also called the Widom insertion factor B, is defined as:52

B ) 〈exp(-∆u/kT)〉

(8)

where 〈‚‚‚〉 represents an ensemble average. The dimensionless insertion factor B is related to the experimentally determined infinite dilution solubility coefficient, S0, by:47,53-55

B)

T T S0 ) (k + C ′Hb)(1 atm) 273.15 273.15 D

(9)

2.3. Diffusion. Experimentally, diffusion coefficients are usually calculated from permeability and solubility data with eq 2. Permeability and solubility can be measured independently. In molecular simulation, diffusion coefficients can be estimated from the Einstein relationship:52

1 Di ) lim 〈[ri(t) - ri(0)]2〉 tf∞ 6t

(10)

where ri is the position vector of atom i, 〈[ri(t) - ri(0)]2〉 represents the ensemble average of the mean-square displacement of the inserted gas molecule trajectories; and ri(t) and ri(0) are the final and initial positions of the center of mass of the gas molecules over the time interval t. The diffusion coefficient can also be calculated from the velocity autocorrelation function.52 3. Model and Simulation Details 3.1. Building Amorphous Cells. The Materials Studio56 software of Accelrys Inc. was utilized to construct the amorphous packing structure. The COMPASS force field57 was used in all simulations. The nonbonded interactions of the COMPASS force field include a Lennard-Jones 9-6 function for the van der Waals interaction and a Coulombic function for the electrostatic interaction. For PTMSP, the initial polymer chain was constructed including 50 repeat units with a 50:50 probability for the occurrence of cis and trans monomers, mimicking what is believed to be the structure of PTMSP material polymerized in the presence of TaCl5 catalyst.3,19,37 Figure 1 presents the chemical structure of PTMSP. Two PTMSP chains (each with 50 repeat units) were folded in the Amorphous Cell at a density of 0.75 g/cm3, which corresponds to the well-accepted as-cast PTMSP experimental density, and at densities of 0.85 and 0.95

TABLE 1: Experimental Densities in As-Cast and Aged PTMSP reference Shantarovich et al.36 Consolati et al.12 Yampol’skii et al.9

film preparation history

storage time (days)

density (g/cm3)

as-cast as-cast aged as-cast aged

0 0 60 0 ∼1500

0.75 0.8467 0.8569 0.71 ( 0.07 0.91 ( 0.09

TABLE 2: Simulation Data polymer

no. of repeat units

density (g/cm3)

length of the periodic cell (Å)

PTMSP(as-cast) PTMSP-085(aged) PTMSP-095(aged)

100 100 100

0.75 0.85 0.95

29.2 28.0 27.0

g/cm3, which represent two aged PTMSP densities. Table 1 lists the as-cast and aged experimental PTMSP densities from several references for comparison.9,12,36 The resulting cell lengths for three simulation densities are presented in Table 2. Sixty initial states for each density were constructed and followed by 5000 steps of energy minimization to eliminate “hot spots”. Afterward, a 10 ps NVT MD run at 298 K was performed for each of the 60 states to equilibrate structures. The resulting equilibrated structures are presumed to be representative of the glassy polymers. 3.2. Simulation of Cavity Size Distribution. The CESA was then applied to each of the above equilibrated structures. Below is a quick review of CESA:45-47 (i) A polymer structure is created by MD (or MC) simulation. (ii) The force field used to generate the structure is replaced with a purely repulsive force field. All atoms remain in fixed positions. (iii) A trial repulsive particle is then randomly inserted into the repulsive structure, and a local energy minimum is located in the repulsive force field. (iv) After the minimum is determined, attractive interactions are turned on, and the size of the test particle is adjusted until its potential interaction with all other atoms becomes zero. This size is taken as the diameter of a spherical cavity. (v) A check is then made to determine whether the initial random inserting point is inside the cavity or not. The cavity is only accepted if the initial point is inside the cavity. This procedure leads to volume distribution rather than a number distribution of cavities. (vi) Steps iii to v are repeated enough times to obtain a representative distribution of cavity sizes for a given structure. Therefore, CESA is a Monte-Carlo technique to find cavities in the structures. The radius of the initial trial repulsive particle does not affect the cavity size distribution, which is an advantage of CESA. 3.3. Simulation of Connectivity and Shape of the Nanopores.58 By considering the overlap of spherical cavities obtained using the CESA, we also characterize the connectivity of the void spaces. Two cavities are considered to be overlapping

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Figure 2. A set of one or more contiguous overlapping cavities form a nanopore (cavity cluster).

Figure 3. Nanopore volume is determined by Monte-Carlo integration. Volume is the ratio of points inside one or more cavities to the total number of points sampled, times box size.

if the distance between their centers is less than the sum of their radii. Overlapping cavities are grouped into a new entity called a nanopore (or a cluster) (Figure 2). Each cavity belongs to only one nanopore, and each nanopore contains at least one cavity. The connectivity of these nanopores is then characterized in terms of span and radius of gyration, and the shapes of these nanopores are characterized by volume and surface area. 3.3.1. Span. The span is defined as the farthest distance between any two points that lie within the nanopore. Analytically, this is calculated by finding the two cavity centers in the nanopore with the greatest distance between them. The span is this distance between centers, plus the radius of each of the farthest cavity centers. 3.3.2. Radius of Gyration. In general, the radius of gyration of an object is defined as the mass averaged root-mean-square distance from the center of mass.58,59 The radius of gyration of the nanopores in this work was calculated by selecting points at random within the nanopore and assigning to each of them an equal “weight”. The center of mass of this set of points was then determined, and the radius of gyration was determined as the root-mean-square distance of the set points in the set from the center of mass of the set. This calculation was performed on a “per nanopore” basis. 3.3.3. Volume. The volume of the nanopore is calculated by performing a straightforward Monte-Carlo integration (Figure 3).52,58 Points are selected at random from within the simulation box, and are determined to lie within at least one of the component cavities of the nanopre or not. This ensures that the volume of overlapping regions is not overcounted. The nanopore volume obtained is the ratio of points inside the nanopore to the total number of points selected, times the box volume. 3.3.4. Surface Area. Nanopore surface area is determined by selecting points that lie on the surface of each component cavity within the nanopore.58 Sets of zenith and azimuth (longitude and latitude) are selected randomly to generate points uniformly on the surface of each sphere in the nanopore. Zenith angles are sampled from a sinusoidal distribution, to prevent divergence in the sampling density at the poles of the sphere. The exposed surface area fraction, φi, of each component sphere is calculated

Figure 4. A typical packing model for PTMSP. Large cavities are observed.

by counting the number of points which lie on the surface of each sphere (not inside the radius of any other sphere) and dividing by the total number of points sampled on the sphere. The surface area of the nanopore is then determined by multiplying this fraction by the surface area of that sphere σi and summing over all spheres. Thus, the nanopore surface area is

σcluster )

∑i φiσi

(11)

3.4. Simulation of Diffusion, Solubility, and Permeability Coefficients. Diffusion coefficients were determined by adding five N2 (or CO2) molecules to each of 10 independent states. After each state was equilibrated for 60 ps, the diffusion constants were calculated by molecular center of mass displacement over a 200 ps interval. Finally, an average value of D was calculated from 10 independent states. Charati and Stern71 have calculated diffusion coefficients of He, O2, N2, CH4, and CO2 in poly(dimethylsiloxane) (PDMS) by means of the Einstein equation with a simulation time of 200 ps. Their plots of mean square displacements (MSD) versus time are linear for all penetrant gases over a period time of about 100 ps, and all estimated diffusion coefficients, D, were determined from the linear parts of the plots. PDMS has a FFV of 0.18 based on the group contribution method, which is comparable to the aged PTMSP. The Widom insertion method52 was used to simulate the solubility of N2 and CO2 in as-cast and aged PTMSP. For each density’s 60 configurations, we performed 20 000 N2 (or CO2) configurations. The permeability is then obtained using eq 2. 4. Results and Discussion 4.1. Fractional Free Volume (FFV), Fractional Cavity Volume (FCV), and PALS. Figure 4 shows a typical structure of as-cast PTMSP from molecular dynamics simulation. Table 3 presents the fractional free volume (FFV), the fractional cavity volume (FCV), and the average cavity size in as-cast and aged

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TABLE 3: Comparison of Fractional Free Volume (FFV), Fractional Cavity Volume (FCV), Average Cavity Size, PALS Data in As-Cast and Aged PTMSP, and PSF FFV%

av cavity size (Å)

polymer

Bondi

CESA

CESA

PALS

PTMSP(as-cast) PTMSP-085(aged) PTMSP-095(aged) PSF

34 25.2 16.4 15.6

15.6 12.9 10.3 7.6

11.2 8.6 6.9 3.7

13.236 9.8312 7.469 5.6261

PTMSP as well as PSF.9,12,36,46,47,60,61 PTMSP has the highest permeability among all known glassy polymers.3 It has a rigid polyacetylene backbone and two bulky substituents attached to the backbone, which may contribute to inefficient chain packing and, in turn, high free volume and high permeability. The FFV is commonly used to characterize the efficiency of chain packing and the amount of free space in a polymer matrix. A common definition of FFV is:

FFV )

V - V* V

(12)

where V is the specific volume. Usually, V*, the so-called occupied volume, is determined from tabulated van der Waals’ volumes (Vw) and the Bondi recipe V* ) 1.3νw.59 The FFV values presented in Table 3 are determined with this method.59 As-cast PTMSP has greater free volume than conventional glassy polymers such as polysulfone (PSF), which has a FFV value of 0.156 by the Bondi method and 0.133 from zero pressure PVT data.46,62 The fractional cavity volume is the fraction of void space occupied by spherical cavities defined by CESA.45-47 In other words, not all of the free volume is in the form of well-defined spherical cavities. The average cavity size 〈x〉 is calculated as follows:

∫0∞x3P(V) dx 〈x〉 ) ∞ ∫0 x2P(V) dx

(13)

where x is the cavity size, and P(V) is the volume distribution obtained from CESA. Average cavity sizes in as-cast and aged PTMSP from PALS measurements are also included in Table 3. There is a very good agreement between simulations and PALS measurements. The FFV, FCV, and average cavity size of PTMSP decrease with increasing densities. The density of PTMSP could increase from 0.75 to 0.95 g/cm3 as a result of physical aging.3 It is interesting to note that the FFV and FCV of PTMSP-095 (aged PTMSP with a density of 0.95 g/cm3) is higher than the FFV and FCV of PSF, which is a conventional, low free volume glassy polymer. The FCV value of 10.3% in PTMSP-095 from our simulation is larger than the FCV value of 7.6% in PSF.60 The average cavity size of PTMSP-095 is also larger than that of PSF, which is in agreement with the experimental observation that the permeability of gases in PTMSP-095 is greater than that in PSF. Table 4 presents positronium components of the PAL spectra (lifetimes τ3 and τ4 in ns and intensities I3 and I4) of as-cast and aged PTMSP collected from several sources.9,12,36 The lifetimes and intensities of the two short-lived components have common values for polymers: τ1 and τ2 are about 0.2 and 0.4 ns, respectively. Assuming a spherical shape of free volume elements, the lifetimes of o-PS can be related to the radius of free volume elements as follows:36

τi ) 0.5[1 - R/R0 + (1/2π) sin(2πRi/R0)]-1

(14)

where τi ) τ3 or τ4 in ns, and Ri ) R3 and R4 in Å; R0 ) Ri + ∆R, where ∆R ) 1.656 Å is the fitted empirical electron layer thickness. To compare with molecular simulation data, the values of Ri are calculated and presented in Table 4. A decrease in the free volume sizes with physical aging is seen. Note that the cavities defined in CESA also have a spherical shape. 4.2. Cavity Size Distribution. The cavity size distributions of as-cast and aged PTMSP determined by using the CESA are presented in Figures 5-7. As-cast PTMSP shows a broader free volume distribution than aged PTMSP. To quantitatively compare the cavity size distributions of as-cast and aged PTMSP, the cumulative distributions are presented in Figure 8. The cumulative distribution for aged PTMSP is shifted toward smaller cavity sizes. Larger cavities disappear as PTMSP ages. Interestingly, PALS results also show that the free volume redistribution and reduction are due to a reduction in the number of large free volume elements in PTMSP.63 4.3. Connectivity: Span and Radius of Gyration. Distributions of spans and radius of gyration values for the nanopores in as-cast and aged PTMSP are presented in Figures 9 and 10. Both the span and radius of gyration distributions shift to smaller sizes upon physical aging. The average nanopore span size and radius of gyration, which are tabulated in Table 5, also decrease with physical aging. These results suggest that cavities in aged PTMSP are less connected, which may contribute to the decreased permeability and diffusivity in aged PTMSP. One speculation is that the size of the simulation box may not be big enough because of the nanoporosity of PTMSP. Nevertheless, the simulated nanopore span sizes do agree with the transport property observations. 4.4. Shape: Nanopore Volume and Surface Area. Distributions of nanpore volumes and surface areas in as-cast and aged PTMSP are presented in Figures 11 and 12. Both nanopore volume and surface area distributions show an exponential-like decay with increasing nanopore volume and surface area. This is expected since there are always more smaller nanopores than larger ones. There are more smaller nanopores in aged PTMSP than those in as-cast PTMSP. However, there are more larger nanopores in as-cast PTMSP than those in aged PTMSP. PALS results also show a reduction in the number of large free volume elements in aged PTMSP.63 The average nanopore volume and surface area, which are tabulated in Table 5, decrease with physical aging. 4.5. Transport Properties of CO2 and N2 in As-Cast and Aged PTMSP. Tables 6 and 7 present the simulation results for diffusivity, solubility, and permeability of CO2 and N2 in as-cast and aged PTMSP and the corresponding experimental data. A wide range of gas permeability values has been reported for PTMSP.3 The oxygen permeability values obtained in ascast PTMSP synthesized by using the same process varied from 1110 to 13200 Barrer.3 Such a large variation in gas permeability is not observed for conventional low free volume glassy polymers. Here, we compare our simulation results with experimental data from several literatures based on as-cast or aged membranes’ density, which is closely related to the free volume as well as the gas permeability in PTMSP. Our simulated diffusivity, solubility, and permeability of CO2 and N2 in as-cast PTMSP are in good agreement with the values reported by Srinivasan et al.64 However, our simulated data are smaller than the data reported by Yampol’skii et al.9 as shown in parentheses in Tables 6 and 7. For aged PTMSP-085 (with a density of 0.85 g/cm3), the only available experimental data

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TABLE 4: Comparison of the PALS Parameters in As-Cast and Aged PTMSP reference Shantarovich et al.36 Consolati et al.12 Yampol’skii et al.9

film preparation history

storage time (days)

τ3 (ns)

I3 (%)

2R3 (Å)

τ4 (ns)

I4 (%)

2R4 (Å)

as-cast as-cast aged as-cast aged

0 0 60 0 ∼1500

2.68 2.8 2.2 2.5 1.7

4.37 11.2 9.5 5.0 7.8

6.82 7.0 6.0 6.5 5.0

10.9 6.6 6.4 6.7 3.8

33.8 31 26 30 15

13.62 10.8 10.7 10.9 8.2

Figure 5. Cavity size distribution in as-cast PTMSP at T ) 298 K and F ) 0.75 g/cm3 from molecular simulation. The average cavity size is 11.2 Å, and the fractional cavity volume is 15.6%. Figure 8. Comparison of cumulative cavity size distributions in ascast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

Figure 6. Cavity size distribution in aged PTMSP at T ) 298 K and F ) 0.85 g/cm3 from molecular simulation. The average cavity size is 8.6 Å, and the fractional cavity volume is 12.9%.

Figure 9. Comparison of nanopore span distributions in as-cast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

Figure 7. Cavity size distribution in aged PTMSP at T ) 298 K and F ) 0.95 g/cm3 from molecular simulation. The average cavity size is 6.9 Å, and the fractional cavity volume is 10.3%.

is the N2 permeability, which is 2450 Barrers, and this value is in agreement with the simulation value 5500 Barrers. The simulated permeability of N2 and CO2 in aged PTMSP-095 (with a density of 0.95 g/cm3) compares fairly well to the experimentally measured permeability, though the simulated value is larger than experimental data. Note that permeability values in as-cast PTMSP measured by Yampolskii et al.9 are also smaller than those of Srinivasan et al.64 and other research groups.2,65,66 This may account for the disagreement between simulation and

Figure 10. Comparison of nanopore radius of gyration distributions in as-cast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

experimental values in aged PTMSP-095. Experimental diffusion coefficients listed in parentheses in Tables 6 and 7 in ascast and aged PTMSP-095 were measured using the time-lag method by Yampolskii et al.9 We then calculated solubility based

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TABLE 5: Comparison of Average Span Size, Average Radius of Gyration Size, Average Nanopore Volume, and Nanopore Surface Area in As-Cast and Aged PTMSP polymer

av nanopore span size (Å)

av nanopore radius of gyration (Å)

av nanopore vol (Å3)

av nanopore surface area (Å2)

PTMSP(as-cast) PTMSP-085(aged) PTMSP-095(aged)

15.06 11.97 7.25

4.03 2.91 2.12

557 391 181

475 325 150

Figure 11. Comparison of nanopore volume distributions in as-cast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

Figure 12. Comparison of nanopore surface area distributions in ascast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

on eq 2. In contrast, the solubility values reported by Srinivasan et al.64 were measured directly by using sorption experiments, and the diffusivity coefficients were calculated afterward based on eq 2. The diffusivity coefficients determined by the timelag method are smaller than those estimated by the ratio of steady-state permeability to solubility in as-cast PTMSP. We suspect that the uncertainty in the measured diffusion coefficient in PTMSP-095 is large since the solubility values calculated based on it are “suspiciously” large. We suggest that the simulation provides more reasonable solubility values in aged PTMSP. Our simulation results showed that the solubility did not change much with physical aging. However, there is a dramatic decrease of diffusion coefficients with physical aging: around seven times decrease for CO2 and 8 times decrease for N2. The solubility of N2 showed a slight decrease with physical aging. The solubility of CO2 showed almost no change with physical aging considering the error bars associated with the simulation. Experimental studies of propane sorption in aged PTMSP by Nagai et al.11,27 also showed that there is little change in the propane solubility with physical aging. The permeability of CO2 and N2 decreased around 6 and 11 times with physical aging, respectively. The permeability decrease in CO2 and N2 with physical aging is mainly due to the decrease in diffusion

coefficients. The decrease in gas permeability with physical aging predicted from simulation is consistent with the experimental observation. In Table 7, the diffusivity, solubility, and permeability selectivity values for CO2/N2 in as-cast and aged PTMSP are presented. Simulated diffusivity, solubility, and permeability selectivity values in as-cast PTMSP are in excellent agreement with experimental data from Srinivasan et al.64 Simulated solubility selectivity increases slightly with physical aging, and experimental data9 show the same trend. CO2/N2 diffusivity selectivity does not change much with physical aging. The overall simulated permeability selectivity also increases slightly with physical aging. The experimental data from ref 9 also showed an increase in permeability selectivity upon physical aging. Diffusivity selectivities in as-cast and aged PTMSP are close to one, which suggests that both as-cast and aged PTMSP membranes have very low size-sieving ability. The experimental study by Yampol’skii et al.9 also observed the weak size-sieving ability in both as-cast and aged PTMSP. Although there is a substantial decrease in the diffusivity of aged PTMSP, it remains weak size-sieving. High CO2/N2 solubility selectivity accounts for the high permeability selectivity. Both as-cast and aged PTMSP can be considered solubility-selective materials, or socalled reverse selective materials.67,68 In contrast, overall gas selectivity is dominated by diffusivity selectivity in conventional polymeric membrane materials such as polysulfone and Matrimid. Both our computer simulation results and Yampol’skii’s9 experimental study indicate that aged PTMSP remains among the most permeable polymeric materials known. 4.6. Diffusivity and Free Volume. In Figures 13 and 14, we plot the simulated diffusion coefficients as a function of 1/FCV and 1/FFV vs for CO2 and N2 in as-cast and aged PTMSP. On the basis of these figures, the diffusion coefficients obey a relationship consistent with free volume theory:69-70

D ) A exp(-B/FFV)

(15)

where A and B are constants. A slight change in FFV (or FCV) leads to a relatively strong change in diffusion coefficients. However, this calculation is tentative since the fits to eq 15 only involve 3 data points. Takeuchi and Okazaki72 have conducted molecular dynamics simulation of diffusion of simple gas molecules in a short chain polymer. Their simulated diffusion coefficients also obeyed free volume theory very well. 5. Conclusions Molecular modeling techniques have been applied to study cavity size distributions, sorption, and transport properties of physical aging of PTMSP. By using atomistic models, cavity size (free volume) distributions and the connectivity of those cavities in the as-cast and aged PTMSP were characterized by a combination of molecular dynamics and Monte-Carlo methods. Cavity size distributions showed a reduction in the number and size of large cavities with physical aging, which is consistent with PALS measurements.9,12,63 Cavities in the aged PTMSP

16692 J. Phys. Chem. B, Vol. 110, No. 33, 2006

Wang et al.

TABLE 6: Comparison of Diffusion, Solubility, and Permeability of CO2 in As-Cast and Aged PTMSP at T ) 298 K (Experimental Data from Refs 9 and 64) diffusivity (10-5 cm2/s)

solubility (cm3(STP)/(cm3 atm) polymer PTMSP(as-cast) PTMSP-085(aged) PTMSP-095(aged) a

simulation

exptl

simulation

exptl

9.8 ( 1.5

8.8a

3.5 ( 0.05

3.0a

(7.4)b

11.3 ( 2.2 10.8 ( 2.6

exptl

45000

35200a (19000)b

(2.5)b

1.5 ( 0.06 0.53 ( 0.09

(53.5)b

permeability (Barrer) simulation

22000 7530

(0.022)b

(1200)b

Reference 64: Srinivasan et al. J. Membr. Sci. 1994, 86, 67-86. b Reference 9: Yampolskii et al. J. Appl. Polym. Sci. 1993, 48, 1935-1944.

TABLE 7: Comparison of Diffusion, Solubility, and Permeability of N2 in As-Cast and Aged PTMSP at T ) 298 K (Experimental Data from Refs 9 and 64) diffusivity (10-5 cm2/s)

solubility (cm3(STP)/(cm3 atm) polymer PTMSP(as-cast) PTMSP-085(aged) PTMSP-095(aged) a

permeability (Barrer)

simulation

exptl

simulation

exptl

simulation

exptl

2.7 ( 0.24

1.35a

3.3 ( 0.05

a

11700

6400a (1600)b 2450 (58)b

(1.04)b

2.6 ( 0.31 2.1 ( 0.35

3.6 (1.5)b

1.6 ( 0.07 0.4 ( 0.1

(4.03)b

Reference 64: Srinivasan et al. J. Membr. Sci. 1994, 86, 67-86;

b

5500 1100

(0.014)b

Reference 9: Yampolskii et al. J. Appl. Polym. Sci. 1993, 48, 1935-1944.

TABLE 8: Comparison of Diffusivity, Solubility, and Permeability Selectivity of CO2/N2 in As-Cast and Aged PTMSP at T ) 298 K (Experimental Data from Refs 9 and 64) solubility selectivity (SCO2/SN2) polymer

diffusivity selectivity (DCO2/DN2)

permeability selectivity (PCO2/PN2)

simulation

exptl

simulation

exptl

simulation

exptl

PTMSP(as-cast)

3.63

6.53a (7.37)b

1.07

0.83 (1.6)b

3.8a

5.5a (11.8)b

PTMSP-085(aged) PTMSP-095(aged)

4.35 5.15

a

(13.31)b

0.92 1.3

(1.57)b

4.0 6.8

(20.9)b

Reference 64: Srinivasan et al. J. Membr. Sci. 1994, 86, 67-86. b Reference 9: Yampolskii et al. J. Appl. Polym. Sci. 1993, 48, 1935-1944.

Figure 13. Plot of DSimul vs 1/FFV and 1/FCV for the diffusion of CO2 in as-cast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

are also less connected then those in as-cast PTMSP. The average spherical cavity size in as-cast PTMSP is 11.2 Å; it is 6.9 Å in aged PTMSP with a density of 0.95 g/cm3. PALS measurements also showed a decrease in the average spherical cavity size from 13.2 to 7.5 Å. The transport and sorption properties of CO2 and N2 in ascast and aged PTMSP were also obtained through molecular simulation techniques. The simulated permeabilities decrease with physical aging, and they are in good agreement with experimental results. The decrease of permeability with physical aging results mainly from the decrease of diffusion coefficients. Solubility did not show significant changes with physical aging. There is a slight increase in the permeability selectivity with physical aging. As-cast PTMSP is a solubility selective material, and so is aged PTMSP.

Figure 14. Plot of DSimul vs 1/FFV and 1/FCV for the diffusion of N2 in as-cast (F ) 0.75 g/cm3) and aged (F ) 0.85 and 0.95 g/cm3) PTMSP.

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