Silica Nanocomposites

Jan 14, 2013 - coefficients (D12) of a homologous series of n-alkyl acetates as a function of ... concentration dependence of D12 for the largest alky...
0 downloads 0 Views 1MB Size
Download PDF
Article pubs.acs.org/Macromolecules

Sorption and Diffusion of n‑Alkyl Acetates in Poly(methyl acrylate)/ Silica Nanocomposites Dustin W. Janes† and Christopher J. Durning* Department of Chemical Engineering, Columbia University, Room 801, S. W. Mudd Building, 500 West 120th Street, New York, New York 10027, United States S Supporting Information *

ABSTRACT: We report partition coefficients (K) and mutual diffusion coefficients (D12) of a homologous series of n-alkyl acetates as a function of particle loading in poly(methyl acrylate)/silica nanoparticle composites. The nanocomposites, prepared using nearly monodisperse poly(methyl acrylate) (PMA, Mw ≃ 56 kDa) and spherical silica (SiO2) nanoparticles of two different diameters (d ≃ 14 nm and ≃50 nm), are all well above their glass transition temperatures. The K values indicate that the addition of silica causes sorption in excess of that expected from a simple blocking effect. Also, for the three smallest penetrants in the series, the addition of nanofiller suppressed the D12, but hardly affected the diffusant concentration dependence of the diffusivity. Conventional composite theory can account quantitatively for the suppression of D12 with particle volume fraction, ϕP, for nanocomposites with the larger nanoparticles (d ≃ 50 nm) but not for those made with the smaller nanoparticles (d ≃ 14 nm), which showed a much stronger effect. Further, the addition of the smaller nanoparticles to PMA also notably altered the concentration dependence of D12 for the largest alkyl acetate studied, n-butyl acetate. This implies changes in selectivity with filler content for the composites made with the smaller nanoparticles, an effect not seen with the larger particles. Overall the results suggest an interfacial layer proximal to each nanoparticle with modified transport characteristics, whose influence on the average properties is much more obvious in composites made with smaller particles. theoretical analyses16−20 support this view, although the details of filler-induced matrix modifications are still debatable. At least some of the recent theoretical work asserts that the influence of nanofiller on glassy state characteristics cause the main effects seen,19 that is, such effects are peculiar to the glassy state. Despite progress a predictive understanding is still lacking. We report here the partition coefficients (K) and mutual diffusion coefficients (D12), of a homologous series of n-alkyl acetates in poly(methyl acrylate) (PMA)/silica (SiO2) nanoparticle composites as a function of particle loading. We employ the same vapor sorption technique used to determine these properties in PMA melts, described in detail elsewhere.21 The PMA/SiO2 composites used here were studied previously22 by differential scanning calorimetry, transmission electron microscopy, and small-angle X-ray scattering to determine the glass transition and dispersion morphology characteristics. All of the transport data reported were measured well above the glass transition temperature of the system. Also, the thermo-chemical history of the PNC samples employed here is essentially identical to one used in the previous morphological studies (the “thermal annealing with subsequent solvent annealing” procedure described in Table 1 of ref 22). We therefore know the dispersion morphology for these composites to be demixed into particle-rich and particle-lean phases as described

I. INTRODUCTION Permselective membranes can enable more efficient separations than current methods in several energy-related processing operations.1−5 A simple design strategy to develop membranes with optimal permeability and selectivity toward specific components exploits composite materials with components contributing desired permselective characteristics anticipated from the pure-component properties. This underlies a number of “mixed-matrix” composite membranes composed of a polymer matrix and a highly selective inorganic filler, such as zeolite particles6,7 or carbon nanotubes.8,9 In connection with mixed matrix membrane development for gas separations, recent studies of gas-transport in polymer/nanoparticle composites (PNCs) revealed surprising effects of added filler on membrane permselectivity that cannot be anticipated from pure component properties. Most remarkably, Merkel et al.10,11 found that the addition of fumed silica, presumably an inert, impermeable filler, to poly(4-methyl-2-pentyne) (PMP) tripled the methane permeability and increased the selectivity of nbutane over methane by a factor of 8. Other recent works12−14 show beyond doubt that the transport characteristics of PNCs do not obey simple additivity relationships adequate for conventional composites employing much larger-sized particles.15 Generally, the extremely high surface to volume ratio of nanoparticles can result in alteration of a significant fraction of matrix polymer proximal to the nanoparticle surfaces, possibly accounting for such effects. Experimental evidence by positron annihilation lifetime spectroscopy (PALS)10,11 and recent © 2013 American Chemical Society

Received: September 21, 2012 Revised: December 10, 2012 Published: January 14, 2013 856

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

previously.22 The experimental data reported here therefore provide a systematic and complete data set defining effects of nanoparticle volume fraction ϕP for a known dispersion morphology on K and D12, for a homologous set of penetrants. Note that the study of transport of the homologous series of nalkyl acetates (methyl, ethyl, n-propyl, and n-butyl acetate) allows detection of possible selectivity changes with nanoparticle addition, as observed dramatically by Merkel et al.10,11 in the PMP/fumed silica system. For several reasons, the diffusant/PNC systems studied here are particularly simple and appropriate for a systematic, fundamental study. Except at the very highest particle loadings studied, the mass transport is ordinary, Fickian diffusion, which is well understood23 in contrast with the non-Fickian transport found for sorption experiments in many glassy and/or unannealed composite systems. This allows for unambiguous determination of the key transport properties. Also, we anticipate that the strong favorable polymer segment-nanoparticle surface site interactions in this composite eliminates matrix-particle debonding and therefore the formation of void space at nanoparticle surfaces, frequently associated with a reduction of selectivity.24,25 Related to this, because the particles used here are relatively monodisperse and spherical they do not exhibit intraparticle void space as do fractal or structured nanoparticles, such as fumed silica, a feature which may also reduce selectivity.24,25 The overall aim of this work is to augment the current understanding of the effects of added nanoparticles on the transport properties of a polymer to possibly enable engineering of better gas separations membranes. The dramatic enhancements of permeability and selectivity observed by Merkel et al.10 encourage this. Their results were for a glassy polymer with transport properties atypical of most polymers (reverse selective on small hydrocarbons). While many studies have attempted to recreate those membrane performance enhancements in other polymer/nanoparticle combinations, clear conclusions regarding nanoparticle effects on the composite’s selectivity have not been forthcoming. In this work we study a system very different to that of Merkel et al.: A rubbery polymer with conventional size selectivity. Our results clearly show size-selective alterations to the transport properties, and indicate such effects are generally present in PNCs. The data presented subsequently makes clear that even in rubbery systems at relatively low particle loadings (ϕP ≈ 0.1), the addition of nanoparticles can modify the matrix polymer enough to produce changes in selectivity. It points to a design strategy for development of permselective membranes from PNCs, which relies on controlled modification of matrix polymer by addition of nanoparticles.

Pi and Sij are the key parameters for judging the suitability of a given membrane for separation of species i from j. As they increase, the membrane becomes more permeable and selective toward a target component i. Robeson26 first articulated that Pi and Sij often exhibited a “trade-off” relationship for penetrant pairs and, based on available data, there appeared to be an upper bound in plots of log Sij vs log Pi for many pairs of permeants ij in key applications. A new membrane material that goes beyond the (empirically) established “Robeson” permselectivity limit for existing membranes represents a significant advance for a given separation. As mentioned previously for the glassy polymer PMP, there have been recent reports that the addition of inert nanoparticles results in simultaneous and unexpected increases in both Pi and Sij relative to the pure polymer for low molar mass hydrocarbons.10,11 A few similar reports exist in the literature for other polymers.12−14 However, while simultaneous increases in Pi and Sij by inert nanoparticle addition appear in some systems, this has not been observed consistently.27−32 For example, adding fumed silica to poly(1-trimethylsilyl-1propyne) (PTMSP) increased PH2 and SH2CH4 but degraded the pure-polymer’s intrinsic reverse selectivity of n-butane over methane.29 Similarly a poly(1,2-butadiene)/MgO PNC exhibited increased CO2, CH4, and N2 permeabilities with increasing ϕP but reduced selectivities for diffusant pairs having Sij > 1 in the pure polymer.28 The effects seem to depend on the particular polymer/nanoparticle/penetrant combination, although a precise understanding is lacking in most cases. One prior experimental study clearly exposes a microstructural effect on permselectivity in PNCs. Takahashi and Paul24,25 systematically tested permeabilities of He, O2, N2, CH4, and CO2, in a set of poly(ether imide)/fumed silica PNC membranes prepared by several methods, including chemically modifying the silica surface,24 or chemically coupling the filler to the matrix polymer.25 Only melt-mixed, chemically coupled PNCs showed a statistically significant increase in SHeCO2 accompanied by very small changes in PHe.25 All other membrane preparations exhibited simultaneously increased Pi but lower Sij. This was convincingly attributed to the formation of internal void space, either at the polymer/nanoparticle interface, or within nanoparticle aggregates having internal void structure, resulting in relatively higher permeability but altered selectivity. Literature suggests that void space, whether at a debonded polymer/nanoparticle interface or within nanoparticle aggregates can be reverse-size selective,10−12 “Knudsen”-selective,14,29 or nonselective.24,25,29 The current theoretical understanding of mass transport in PNCs cannot fully rationalize these experimental findings. The effects of impermeable filler on K and D12 in a composite with no void formation and no alteration of the polymer matrix’s properties by the addition of filler are predictable. Since the diffusant cannot dissolve into the filled fraction, K falls with filler content. Also, impermeable particles force penetrant molecules to travel a longer, more tortuous path through the membrane than for the pure polymer, causing D12 to drop as well. Indeed in systems where one expects the matrix properties are unaltered by the filler (e.g., composites with relatively large, geometrically regular particles), theoretical predictions of P = KD12 can account quantitatively for this “blocking” effect.33,34 Since filler volume is inaccessible to penetrant molecules, the effective partition coefficient of the composite, Kϕ, is related to that of the pure polymer matrix, KB, by

II. SORPTION AND DIFFUSION IN PNCs WITH INERT NANOPARTICLES The essential properties controlling permselectivity, determined from transport experiments on individual diffusants in a membrane, are D12 and K. When they are constants, their product the permeability, P, defines the steady-state flux of that diffusant through the membrane. The ratio of permeabilities for two different penetrants in the same membrane is the selectivity, Sij: Sij =

Pi KD = i i2 Pj KjDj2

(1) 857

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules Kϕ = KB(1 − ϕP )

Article

An attempt to reckon the transport properties of a matrix layer proximal to nanoparticle surfaces from first principles was given recently by Hill.16 He attempted to explain the origin of the reverse selectivity and permeability enhancement in the poly(4-methyl-2-pentyne)/fumed silica system10,11 from a molecular perspective. He employed self-consistent mean field theory to predict the polymer segment density distribution around a particle, and assumed that penetrant diffusivities obey free volume theory.42 If the polymer−particle interactions are repulsive, a polymer segment density depletion layer forms around the particle which introduces a reverse size-selective free volume effect. The depletion layer thickness, found by adjusting model parameters to match the experimental transport data,10 matches with the size scale of free volume elements determined in the same system by PALS. The model agrees with experimental trends in selectivity and permeability as ϕP increases. Reverse selective interfacial material is predicted for any polymer−particle pair with repulsive interactions, so this theory may explain the apparent deterioration24,25 or even switching12 of selectivity observed with particle addition in polymers exhibiting normal selectivity. In particular the work suggests that PNCs made from polymers with normal size selectivity and particles having attractive interactions polymer segments should show a polymer segment density excess around the particles, which should yield a PNC with Dϕ suppressed by ϕP beyond passive “blocking” theories and a simultaneous enhancement of the normal selectivity. Hill’s ideas have been applied in multiscale calculations on composites by Ganesan and co-workers.18 Effective medium theories34,43−45 (EMTs) offer another approach relevant to analysis of transport data in PNCs. These provide the volume averaged properties of a two phased system with a defined microstructure. These are clearly relevant considering the results reported in ref22 on the PNC studied here. Sax and Ottino34 developed an EMT theory which can be adapted to describe transport in a demixed PNC with particlerich (volume fraction = ϕP + ϕδ) and particle-lean (volume fraction =1 − ϕP − ϕδ) phases. The resulting relation for Dϕ, is

(2)

Maxwell’s result35 for the effect of ϕP on D12 in a very dilute random dispersion of impermeable spheres is, Dϕ =

DB 1+

ϕP 2

(3)

where Dϕ is the mutual diffusion coefficient for the diffusant and composite and DB is the corresponding diffusion coefficient for the bulk polymer matrix (i.e., the unfilled material). Maxwell’s result is, in principle, only valid for ϕP ≪ 1 (no particle “interactions” with respect to the blocking effect). However, comparisons with data show eq 3 can work well up to much higher ϕP(∼O(10−1)) in many cases. For example, Pi and Dϕ of propane in rubber composites filled with impermeable, micrometer-sized ZnO particles can be described quantitatively by eq 2 and eq 3 up to ϕP ≅ 0.4. Equations 2 and 3 also account for changes with ϕP in Pi of ethyl p-aminobenzoate in silicone rubber/fumed silica composites,36 and for changes in Pi of CO2 and H2 in cross-linked poly(ethylene glycol) diacrylate/ fumed silica composites,37 for ϕP up to ≅ 0.15. Many treatments of Dϕ for ϕP beyond the very dilute limit have appeared in the literature, including perturbative developments for dilute systems up to O(ϕP2) by Jeffrey38 and Batchelor,39 and the cell model analysis by Sangani and Yao for much higher ϕP.40 All of the approaches mentioned presume that the filler does not cause void formation and has no effect on the intrinsic properties of the matrix. They can account quantitatively for the experimentally observed effects of filler in systems with relatively large filler particles (characteristic particle size ≥102 nm). An implication of all these theories is that the temperature and composition dependencies of Kϕ and Dϕ in composites are identical to those of the pure polymer, KB and DB, as is the selectivity of such composites. Obviously, these treatments are inadequate for the PNC systems discussed in the first part of this section. A reasonable model accounting for modifications of matrix material near the matrix/particle interface presumes a three“phase” structure comprised of particle, bulk matrix polymer, and an interfacial layer of matrix material with distinct properties, (see for example eq 1 of ref 22) with respective volume fractions ϕP, 1 − ϕP − ϕδ, and ϕδ. While the approach cannot predict results in PNCs from pure component properties, it can account for the unexpected experimental results found with a few adjustable parameters having clear physical significance. Mahajan and Koros7 considered the particle and attached interfacial layer to comprise a pseudo“particle” with an effective Kϕ and Dϕ given by relations akin to eqs 2 and 3. The dispersion of such pseudoparticles in the bulk polymer matrix modifies its solubility and diffusivity, also according to relations akin to eqs 2 and 3 but for which the dispersed phase is not impermeable. Along similar lines Xue et al.41 simulated systems of monodisperse spherical particles each with a bound layer having a permeability 100 times greater than the bulk polymer. The relative volume of the bound layer and particle was varied. The simulations could account for the increases in Pi with ϕP observed by Merkel et al.11 Also, greater permeability increases with increasing ϕP were observed if the interfacial regions were interconnected due to particle aggregation than if the particles were widely separated, reminiscent of some of the results found by Paul and Takahashi.24,25

⎡ = S −1⎢Y + ⎣ DB Dϕ

Y2 +

⎤ 2 SX ⎥ z−2 ⎦

⎛ ⎞−1 1−S 1+ (1 − ϕP − ϕδ)⎟ ⎝ ⎠ S

⎜

(4a)

where Y= (z /2)(1 − ϕP − ϕδ) − 1 + SX[(z /2)(ϕP + ϕδ) − 1] z−2 (4b)

S=

Kϕ(ϕP + ϕδ = 1) KB

(4c)

and X=

Dϕ(ϕP + ϕδ = 1) DB

(4d)

Here z is a “coordination” number describing the two-phase morphology. Thus, if S and X are set by knowing the properties of each phase, z may be adjusted to best fit the experimental data interpolating between the extreme cases of the sample 858

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

butyl acetate. However, we do not expect the order that the penetrants were subsequently tested to affect the results because the solvent annealing step usually represented the highest degree of swelling for each film. Additionally, the films were dried to a constant frequency between penetrant runs, which did not vary greatly from earlier runs. For example, for the film made with smaller nanoparticles and ϕP = 0.29, the starting frequencies for the four penetrants differed by less than 6 Hz from their average value, which represents only 0.07% of the frequency shift caused by film deposition. Data Treatment. We report Kϕ and Dϕ, normalized to the values found in the pure polymer to assess the effect of nanoparticles. Knowing WP, the mass fraction of particles in the dry PNC, allows correction of the overall penetrant weight fraction, ω1 (see eq 10 of ref 21), to the weight fraction of penetrant in the polymer matrix only, ω1°:

being composed of one or the other phase. When treating gas permeation in polymer blends, z = 4 was found to fit data on “compatible” blends, while “incompatible” blends were better fit by z = 6. In this way z may be considered to characterize the microstructure, as larger z values appear to be characteristic of larger-size scale demixed domains. In the case of PNCs studied by us, these are a particle-lean (nearly pure polymer) and a particle rich phase (nearly close packed particles, i.e., a phase composed entirely of particles and interfacial material (ϕP + ϕδ = 1)).

III. EXPERIMENTAL SECTION Materials. Poly(methyl acrylate) (PMA, Mw = 55,960, Mw/Mn = 1.04) was synthesized via atom transfer radical polymerization (ATRP) according to a procedure described previously.22 Silica (SiO2) nanoparticles suspended in 2-butanone (organosilicasols MEKST and MEK-ST-L) were supplied by Nissan Chemical America (Houston, TX). The silica nanoparticles were characterized by the manufacturer using electron microscopy as nearly monodisperse spheres with diameters in the range 10−15 nm (MEK-ST) and 40−50 nm (MEK-ST-L). Both suspensions were specified as 30−31 wt % silica and were stored after shipment at 5 °C. Toluene (reagent grade), 2-butanone (MEK, 99+%), n-propyl acetate (99+%), and n-butyl acetate (99+%), were purchased from Acros Organics (Geel, Belgium). Certified ACS grade ethyl acetate was purchased from Fisher Chemical (Fairlawn, NJ). Methyl acetate (anhydrous 99.5%) was purchased from Sigma-Aldrich. Zero-grade nitrogen gas was purchased from Tech Air (White Plains, NY). The antioxidant Irganox 1010 was supplied by Ciba (Basel, Switzerland), and was added to solutions of PMA in MEK at 0.1 wt % Irganox relative to polymer to minimize oxidation during preparation and subsequent thermal annealing of PNC membranes. Standard, 1” diameter, AT-cut, 5 MHz quartz crystals bearing gold electrodes were purchased from Stanford Research Systems (Sunnyvale, CA) or Inficon Inc. (East Syracuse, NY). Methods. PNC films were cocast from semidilute solutions of PMA in MEK with added silica nanoparticles according to a previously described procedure.22 The PNC solution was spin-coated onto clean, “tared” quartz crystal transducers (uncoated crystal resonance frequency = fq), thermally annealed at 150 °C for two days under an inert atmosphere, reloaded into the crystal holder (coated frequency = f F), and then “solvent annealed” in situ with n-propyl acetate. The solvent annealing step was necessary to obtain reproducible transport data. The details of these steps are described in ref 21. The volume and weight fractions of nanoparticles in the dry PNC, denoted ϕ P and W P respectively, were measured by thermogravimetric analysis as detailed in ref 22. We carried out transport measurements with a flow-through vapor sorption apparatus (see Figure 2 of ref 21) utilizing a piezoelectric quartz crystal microbalance system (Maxtek PLO-10i and RQCM, Inficon Inc., Syracuse NY) as the mass sensors. A frequency drop associated with penetrant sorption by the PNC membrane coated onto the quartz crystal, Δf S, is proportional to the diffusant’s mass uptake during sorption. By adjusting the relative flow rates in two gas streams that are mixed before contacting the membrane, one containing pure N2 and the other containing penetrant at a known partial pressure, the apparatus allows good control of the penetrant vapor activity flowing over the crystal. Sorption isotherms were determined from the steady frequency of a coated crystal in contact with different streams having a known penetrant vapor activity (see section 3.4 of ref 21), thus establishing the effect of penetrant weight fraction ω1 on Kϕ. Interval sorptions (see section 3.5 of ref 21) were performed by making the smallest possible step changes in the penetrant activity flowed over the crystal and tracking the frequency response, providing a direct measure of the kinetics of mass uptake during sorption, thus measuring the effect of ω1 on Dϕ. Each penetrant was tested sequentially on the same film, generally in the order of n-propyl acetate, ethyl acetate, methyl acetate, and n-

ω1◦ =

ΔfS ΔfS + (fq − fF )(1 − WP)

(5a)

The polymer weight fraction in the volume not occupied by particles is therefore ω°2 = 1 − ω°1 . The volume fraction of penetrant in the matrix polymer of the PNC, ϕ1°, is then easily calculated assuming no volume change upon mixing, and using bulk values for the mass density of penetrant and polymer, ρ1 and ρ2 respectively:

ϕ1◦ =

ω1◦ρ2 ω1◦ρ2

+ (1 − ω1◦)ρ1

(5b)

The polymer volume fraction in the volume not occupied by particles is then ϕ°2 = 1 − ϕ°1 . A third, necessary correction, requires an estimation of the dry PNC film mass density ρP ρ2 ρF = ρP (1 − WP) + ρ2 WP (6) where we assume no volume change upon mixing for the pure PMA (ρ2 = 1.22 g/cm3) and silica nanoparticles (ρP = 2.2 g/cm3). With these relations Kϕ follows from eq 11a of ref 21 but with ρF substituted for ρ2, resulting in eq 7:

Kϕ =

RTω1ρ1ρF MW1p1 [ω1ρF + (1 − ω1)ρ1]

(7)

where R is the gas constant, p1 is the partial pressure of penetrant vapor, T is absolute temperature, and MW1 is the molecular weight of the penetrant (solvent). To normalize the Kϕ, i.e. to determine Kϕ(ω1°)/KB(ω1°) we employ the least-squares linear fit of the KB(ω1) data found in Figure 3 of ref 21. The relation Dϕ = DST(1 − ϕ1)−2 converts the polymer fixed frame diffusion coefficient found by (short-time) analysis of the interval sorption data, DST (see eq 12a of ref 21), but with ρF substituted for ρ2 in eq 10b of ref 21 in the determination of ϕ1. Thus, the volume fraction of penetrant in the PNC film is ω1ρF ϕ1 = ω1ρF + (1 − ω1)ρ1 (8) To normalize the Dϕ, i.e. to determine Dϕ(ω1°)/DB(ω1°), we use the values from a free volume theory fit of the pure PMA D12(ω1) data discussed in ref 21 (see eq 5 and section 4.2 of ref 21). In this manner Kϕ and Dϕ are always normalized to pure polymer results at identical penetrant content in the polymer fraction to account for the effects of penetrant content on K and D12.

IV. RESULTS Sorption Isotherms. ln(Kϕ/KB) vs ω1° for methyl acetate, ethyl acetate, n-propyl acetate, and n-butyl acetate appear in Figures 1−4, respectively. For all cases, the partition coefficients are nearly independent of composition over the range studied. Also, for almost all penetrants, Kϕ > KB(1 − ϕP), that is more penetrant partitions into the polymer with nanoparticles 859

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

Figure 1. Partition coefficients for methyl acetate normalized by the regression fit values for the bulk polymer for PMA/silica composites, with pure polymer data also shown. The lines correspond to the ideal prediction given by eq 2 and the abscissas are the penetrant weight fraction in the polymer defined by eq 5a.

Figure 3. Partition coefficients for n-propyl acetate normalized by the regression fit values for the bulk polymer for PMA/silica composites, with pure polymer data also shown. The lines correspond to the ideal prediction given by eq 2.

Figure 4. Partition coefficients for n-butyl acetate normalized by the regression fit values for the bulk polymer for PMA/silica composites, with pure polymer data also shown. The lines correspond to the ideal prediction given by eq 2.

Figure 2. Partition coefficients for ethyl acetate normalized by the regression fit values for the bulk polymer for PMA/silica composites, with pure polymer data also shown. The lines correspond to the ideal prediction given by eq 2.

PMA from silica; i.e., they adsorb preferentially over polymer segments at nanoparticle surface “sites”.46 This suggests that the “excess” penetrant sorption may be physisorbed at the nanoparticle surface. To test this, we plot the average excess sorption, ⟨KϕKB−1(1 − ϕP)−1 −1⟩, against ϕP in Figure 5 where the average is over

present than anticipated from the melt data. The composite’s interfacial chemistry suggests a mechanism for this, since the carbonyl group present in n-alkyl acetates interacts strongly with the hydroxyl groups on a silica surface via hydrogen bonding. In fact, methyl and ethyl acetates displace adsorbed 860

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

loadings (e.g., ϕP ≅ 0.1). When this vapor preconditioning was not done, repeated interval sorptions from dry conditions were not reproducible. However, after vapor preconditioning the transport data for any diffusant (including nondisplacers) in moderate ranges of ϕP (0 ≤ ϕP ≤ 0.5) the transport was perfectly or very-nearly normal, Fickian, and well-reproduced in repeat runs. The reason this preconditioning has such an effect is clear from the microstructural studies summarized in ref 22: Exposure of thermally annealed PMA/silica nanocomposites to a good-solvent/displacer triggers demixing to a biphasic morphology of coexistent particle-rich and particle-lean phases. Hence vapor preconditiong with a good solvent/displacer establishes a stable composite morphology for subsequent diffusant transport. This behavior is akin to the Mullins effect,47 where the first strain on a carbon-black filled rubber alters its microstructure and performance in future strain cycles. Interestingly, preconditioning with the nondisplacing good solvent toluene did not enable reproducible sorptions subsequently, presumably since toluene cannot trigger significant demixing in the composite. Another distinct, reproducible feature emerged at the highest particle loadings (ϕP ≥ 0.5) in composites made from the smaller nanoparticles. Despite thermal annealing and solvent preconditioning, which results in a stable (demixed) composite morphology during subsequent sorptions, non-Fickian transport characteristics became evident for ϕP ≥ 0.5. In particular, “two-stage” sorption kinetics48−50 appeared for composites with ϕP ≥ 0.5. This feature generally results from microstructural relaxations in response to the osmotic perturbation applied in an interval sorption having a time scale matching that for the transport. In other words a diffusion Deborah number,59 De, evidently becomes a relevant system parameter under these conditions where the characteristic relaxation time is associated with the composite’s microstructure response to a sudden internal osmotic pressure.51 While the nature of this microstructural relaxation process is unclear,60 its influence is unequivocal. Details of this “two-stage” response are discussed further below. Figure 6 shows representative data for interval sorption of ethyl acetate in PMA/silica nanocomposites made from the smaller nanoparticles, illustrating the effects mentioned above. The figure compares results found for the pure PMA to those for moderate (ϕP = 0.29) and very high (ϕP = 0.56) loadings of smaller nanoparticles. Recall that the calorimetery results from ref 22, showing the glass transition’s breadth reached a plateau at ϕP = 0.56 in this system, suggests that matrix material is entirely interfacial or “bound” at this loading. For these highly loaded samples, for which a representative transmission electron micrograph is shown in ref 22, almost all polymer lies within a few nanometers of a nanoparticle surface. The sorption intervals found for the pure polymer, examined in detail in another work21 (plotted as normalized mass uptake M̃ vs √(t − t0)l−1 where l is the dry film thickness), clearly show all the characteristics53 of ordinary Fickian diffusion: the plots are initially linear from the origin and become concave down, then quickly relax to equilibrium. The data for composites at moderate loading begin to show weak deviations by not satisfying the final criterion. Such sorption results, shown in the middle graph of Figure 6, only reach true equilibrium very slowly after the initial transient, on a time scale much longer than the initial portion of the solvent uptake. Here, the overall amount of sorption corresponding to the slow, “second stage” of uptake was practically insignificant

Figure 5. Average “sorption excess” for n-alkyl acetates in PMA/silica nanocomposites. The ordinate represents the difference between the mean normalized partition coefficient from the previous four figures and the corresponding ideal prediction given by eq 2 and is plotted here against ϕP. Ideal behavior (no excess) corresponds to zero on the ordinate. The error bars represent the standard deviation of the data. Filled data points represent samples made with smaller nanoparticles (d = 10−15 nm, MEK-ST) and unfilled data points represent samples made with larger nanoparticles (d = 40−50 nm, MEK-ST-L). Colors enable visual comparisons with the previous four figures.

the (at most, weak) composition dependence. This quantity would be zero if the particles have no effect beyond excluding volume from the diffusant. Further, since the total nanoparticle surface area scales with 3ϕP/R for spherical particles, if the excess sorption results from surface immobilization, then ⟨KϕKB−1(1 − ϕP)−1 −1⟩, should increase with ϕP, and about 4-fold more quickly for composites made with the smaller nanoparticles (d = 10−15 nm, MEK-ST) than for those made from larger ones(d = 40−50 nm, MEK-ST-L). The data in Figure 5 clearly show an increasing trend, consistent with a surface adsorption mechanism, but the details are not completely consistent with expectations. In particular, the methyl acetate data clearly shows a sorption excess for all PNCs with smaller nanoparticles and not for PNCs with larger nanoparticles for ϕP = 0.11 and 0.19, however an excess of methyl acetate was observed for the PNC made with larger particles at the highest loading ϕP = 0.48. For ethyl acetate, a sorption excess was always observed, independent of particle diameter. The other two penetrants, n-propyl acetate and nbutyl acetate, exhibit weak or no sorption excess except at the very largest nanoparticle loadings. Overall, the excess sorption data give the impression that surface immobilization may indeed play a role in the sorption isotherm, evidently being a stronger influence for the two lower mass acetates. The obvious excess for all four penetrants at the very highest loadings may also be influenced by matrix defects in the tightly packed systems; the nominal particle loadings are near well-established packing limits for spherical particles. This issue is discussed more explicitly later in the context of the diffusivity data. Sorption Intervals. Despite thermal annealing of spin-cast PNC films, it was found that an additional “solvent annealing” step, essentially that described in section 3.3 of ref 21, was necessary to attain repeatable interval sorption kinetics. In particular, an extended in situ exposure af ter thermal annealing for at least 48 h of the spin-cast membrane to a high activity of a good-solvent/displacer (e.g., n-propyl acetate), followed by desorption to the dry state was necessary in order to subsequently achieve reproducible interval sorption data of any penetrant in the composites, even at relatively low particle 861

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

Figure 6. Interval sorption data for ethyl acetate in PMA/silica nanocomposites made with smaller (d = 10−15 nm, MEK-ST) nanoparticles. The left-hand plot is for the pure polymer. The middle plot has a moderate particle loading (ϕP = 0.29) while that on the right has a very high loading (ϕP = 0.56). Different colors for the highly loaded samples represent repeat runs done successively (1, 2, 3) on the same film.

Figure 7. Binary mutual diffusion coefficients normalized by a free volume theory fit to bulk polymer data for methyl acetate in PMA/ silica composites, with pure polymer data also shown. The abscissas are the penetrant weight fraction in the polymer defined by eq 5a. The half-filled points used for ϕP = 0.56 in part b indicate where interval sorption data had obvious two-stage, non-Fickian characteristics. In part b, the solid lines corresponds to the average value for the ϕP = 0.56 where the ordinate does not appear to change with increasing penetrant concentration.

compared to the fast, “initial stage” of uptake. In other words, these data are very nearly Fickian. In contrast, the two-stage characteristic becomes much more obvious for samples with highest loading of silica, as shown in the graph on the right of Figure 6. Here, the non-Fickian second-stage characteristic of a long, protracted approach to equilibrium was consistently observed for every penetrant over a broad range of composition, and shown to be very repeatable by successive runs for ethyl acetate. Further, the second stage of uptake was comparable in magnitude to the first stage. It is notentirely surprising to encounter relaxation-mediated diffusion in samples where the volume fraction of nanoparticles is very high. For monodisperse hard spheres, the maximum packing of a disordered, “molten“ state54 is ϕP = 0.494, and the maximum random packing is ϕP = 0.638. Our calculated (nominal) values of ϕP lie between these values. It is clearly possible that the osmotically driven adjustments of densely packed nanospheres52 are slower than or commensurate with the rate of diffusion of penetrant into the sample, leading to the experimentally observed protracted relaxation to equilibrium. Interestingly, non-Fickian behavior was not observed for any of the PNCs made with larger nanoparticles. A model for non-Fickian diffusion, including the coupling of structural relaxation to mass transport, may be applied to fully analyze the two-stage data.48,49 This effort lies beyond the scope of our study and we content ourselves here with the results from (short-time) analysis of the initial Fickian portion of the data using the conventional short time analysis (see ref 21). Non-Fickian data treated in this way is clearly demarcated subsequently (e.g., see Figures 7−10). The Dϕ determined from purely Fickian sorption data were normalized by DB(ω°1 ), a Vrentas−Duda free volume theory55 fit of the pure polymer D12(ω1) data in ref 21. ln(Dϕ/DB) vs ω1° for methyl, ethyl, n-propyl, and n-butyl acetate are shown respectively in Figures 7−10. Two important features are apparent. First, diffusivities in PMA are generally suppressed by the addition of nanoparticles. The effect is much stronger in composites made with the smaller nanoparticles. We analyze

Figure 8. Binary mutual diffusion coefficients normalized by a free volume theory fit to bulk polymer data for ethyl acetate in PMA/silica composites, with pure polymer data also shown. The half-filled points used for ϕP = 0.56 in part c indicate where interval sorption data had obvious two-stage, non-Fickian characteristics. In parts a and c, the solid lines corresponds to the average values for the ϕP = 0.60 and ϕP = 0.56 data where the ordinate does not appear to change with increasing penetrant concentration.

862

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

Figure 9. Binary mutual diffusion coefficients normalized by a free volume theory fit to bulk polymer data for n-propyl acetate in PMA/ silica composites, with pure polymer data also shown. The half-filled points used for ϕP = 0.56 in part c) indicate where interval sorption data had obvious two-stage, non-Fickian characteristics.

Figure 10. Binary mutual diffusion coefficients normalized by a free volume theory fit to bulk polymer data for n-butyl acetate in PMA/ silica composites, with pure polymer data also shown. The half-filled points used for ϕP = 0.56 in part c indicate where interval sorption data had obvious two-stage, non-Fickian characteristics. The lines are free volume theory fits to the PNC data described in the text.

this suppression quantitatively later. Second we point out that composites with smaller nanoparticles (d = 10−15 nm, MEKST) for all silica loadings below ϕP ≤ 0.45 show no systematic change in ln(Dϕ/DB) with ω1° for methyl, ethyl, or n-propyl acetate, while these same composites with a very high silica loading (ϕP ≥ 0.45) showed a decreasing trend in ln(Dϕ/DB) with ω1° for these penetrants. Furthermore, a steep decrease in ln(Dϕ/DB) with ω1° is observed over the whole range of ϕP for nbutyl acetate (see Figure 10). In sharp contrast, PNCs made with larger nanoparticles (d = 40−50 nm, MEK-ST-L), show no consistent change in ln(Dϕ/DB) with ω1° for all penetrants under any circumstances, including for n-butyl acetate. This second feature demonstrates unequivocally that the addition of fine nanoscaled particles alters the membrane’s transport properties in a penetrant-specific manner. In this system the most obvious effects are for n-butyl acetate and enhance the normal (conventional) selectivity of the pure polymer, which favors the lower mass diffusants. This finding is akin to, but not identical with, those of Merkel et al.10,11 of enhanced reverse selectivity in PMP with addition of nanosized silica. Put somewhat differently, recall that the concentration dependence of the diffusion coefficient results in part from the free volume characteristics of the matrix hosting the diffusant. Evidently, the addition of the smaller nanoparticles has substantially altered these features for PMA, so as to affect nbutyl acetate most strongly. We discuss free-volume interpretations of this effect subsequently, but first we return to analysis of “blocking” effects. For the cases with no significant trend in ln(Dϕ/DB) with ω°1 (see Figures 7−9 for ϕP ≤ 0.49), we determined the mean value with respect to ω1° and plotted the result against ϕP in Figure 11 to better quantify the “blocking effect” of

Figure 11. Diffusivity data showing the “blocking effect” of nanoparticles on the transport of n-alkyl acetates in a PMA/silica nanocomposite. The plot shows the average over penetrant concentration of the logarithm of normalized binary mutual diffusion coefficients for penetrants with no apparent concentration dependence of this quantity against ϕP. The error bars represent the standard deviation of the data. Filled data points represent samples made with smaller nanoparticles (d = 10−15 nm, MEK-ST) and unfilled data points represent samples made with larger nanoparticles (d = 40−50 nm, MEK-ST-L). Color is used to enable visual comparison to Figures 7−10. The dashed line is the result of Maxwell (eq 3) for a very dilute suspension of impermeable spheres. The results of a numerical analysis by Sangani and Yao40 for impermeable nanoparticles that do not alter matrix material properties is also shown (×).The solid line is the result of an effective medium theory fit (eq 4), considering the PNC as biphasic with particle-lean (ϕP = 0) and particle-rich (ϕP = 0.56) phases. A lattice coordination parameter of z = 10 is used, eq 4d is set by the experimental data, and eq 4c is set by eq 2. 863

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

of 2° for the PNC relative to the pure polymer was observed experimentally (see Supporting Information of ref 22).

nanoparticles. For the highly loaded samples (methyl and ethyl acetate with ϕP > 0.5) where ln(Dϕ/DB) initially decreases with ω1° before appearing to become nearly constant, the average of the region where ln(Dϕ/DB) is constant is included; the regions where ln(Dϕ/DB) appears constant are clearly marked by the solid lines in Figures 7 and 8. Smaller nanoparticles suppress ln(Dϕ/DB) to a greater extent at identical ϕP than do the larger nanoparticles. In fact, for the larger nanoparticles, the suppression effect is described quantitatively by Maxwell’s result35 (eq 3) with no adjustable parameters for ϕP ≤ 0.48 (see dashed line in Figure 11). On the other hand, eq 3 fails badly for ϕP ≥ 0.19 for PNCs made with smaller nanoparticles. The numerical analysis of Sangani and Yao40 designed specifically for “ideal“ composites at high loadings does a somewhat better job, but also fails to describe the blocking effect of these samples for ϕP > 0.29. To our knowledge, no theory that assumes the properties of the matrix polymer correspond to that of the unfilled material can adequately describe the suppression of diffusivity observed here for the PNCs made with smaller nanoparticles over the whole range of ϕP tested. To model these data we applied the effective medium theory (EMT) developed by Ottino et al.34 for diffusivities in immiscible polymer blends which exhibit biphasic morphologies reminiscent of those seen in our composite, as reported in ref 22. In particular, we employ eq 4 to adequately describe the data for PNCs made from the smaller nanoparticles. We set the parameters in eq 4d using the ϕP = 0 data to describe the diffusion in the particle-lean phase DB, and using the ϕP = 0.56 data to describe the diffusion in the particle-rich phase Dϕ(ϕP + ϕδ). By using eq 2 to set parameters in eq 4c, the data for the PNC made with smaller nanoparticles is adequately described with a lattice coordination parameter of z = 10, corresponding to values typically needed to fit diffusivity data in incompatible blends.34 We note that if different (larger) values for eq 4c are used, the data in Figure 11 can be described equally well but with different values for z; somewhat higher z are needed when values for eq 4c correspond to a “sorption excess”. While lacking predictive capability, the EMT does supply a rational framework for interpolation between the extreme values in Figure 11 based on a realistic microstructural picture. To explore further the nature of selectivity changes observed in Figure 10, we applied Vrentas−Duda free volume theory55 (see eq S1 in the Supporting Information) to the PNC diffusivity data for n-butyl acetate. Based on PALS evidence that the free volume of the matrix material is altered by addition of nanoscale filler,10,11 it is reasonable to adjust one of the Vrentas−Duda free volume parameters pertaining to the polymer (as opposed to parameters describing n-butyl acetate) to best fit D12(ω1°) data in PNCs. Indeed, adjusting any of ξ, K12/γ, or T + K22 − Tg2 in addition to D01 can account for the change in ln(Dϕ/DB) with ω1° observed for the n-butyl acetate/ PMA system when the smaller nanoparticles are added, with each pair providing equally good fits. In particular, the data may be explained by a drop from ξ = 0.67 in the pure PMA to ξ ≈ 0.55 in the PNC. Physically, this corresponds to a larger segment of the polymer chain being involved in diffusive jumps in the PNC due to confinement of chains near nanoparticles surfaces. Alternatively the PNC data may be explained by raising K12/γ from 3.99 × 10−4 cm3g−1K−1 in the pure polymer to ≈4.7 × 10−4cm3g−1K−1, or increasing T + K22 − Tg2 from 67.15 K in the pure polymer to ≈80 K. The latter parameter change corresponds to an effective increase in the polymer glass transition temperature, Tg2, of ∼13°, although only an increase

V. DISCUSSION AND CONCLUSIONS This work reports on the effect of nanoparticle loading and diameter on the partition coefficients and binary mutual diffusion coefficients of n-alkyl acetates in model PMA/silica nanocomposites. Generally, the measured partition coefficients in our PNCs revealed a “sorption excess” of methyl acetate and ethyl acetate relative to the pure polymer which increases with nanoparticle loading. The two lowest mass alkyl acetates may be expected to exhibit relatively stronger interactions with the polar surface hydroxyl groups of the nanoparticles, because this homologous series becomes more hydrophobic with increasing mass. If the surface chemistries of the different diameter particles are identical, it is reasonable to expect that a sorption excess for the same ϕP arising from surface interactions should increase with inverse particle radius, R−1. However, for methyl acetate and ethyl acetate the sorption excess appeared to be independent of nanoparticle diameter, in contrast with that expectation. Differences in the surface chemistry between the two nanoparticle samples, such as different surface silanol densities, could arise from structural effects56 and/or differences in the ligand composition.57 Also, due to limitations of our sorption apparatus described in ref 21, the partition coefficients could not be measured at very low penetrant contents, where the effects of a surface excess might be more apparent. Additional measurements of the partition coefficients at lower overall penetrant content are needed along with detailed surface chemical analysis of the nanoparticles employed. For the three smaller penetrants, the concentration dependence of the diffusivity was nearly unaffected by the addition of nanoparticles; the diffusivities were simply uniformly suppressed. This suppression was weak with ϕP for PNCs made with larger nanoparticles and easily accounted for by conventional composite theory. It was a much greater effect with ϕP than could be described by any existing conventional theory for PNCs made with smaller nanoparticles. The addition of smaller nanoparticles to PMA was seen to change the concentration dependence of the diffusivity of n-butyl acetate much more strongly than observed for methyl, ethyl, or n-propyl acetate. This selectivity effect points toward penetrant-sensitive changes in the polymer’s free volume characteristics, and was not found at all in PNCs made with larger nanoparticles. These two important effects on the diffusion coefficients of n-alkyl acetates in PMA/silica nanocomposites with ϕP < 0.6, of profound suppression and selectivity changes are unequivocally absent in the complementary data on composites with larger nanoparticle diameter. In the following we discuss the microscopic implications of these experimental observations. For the smaller three penetrants, methyl, ethyl, and n-propyl acetate, there was no concentration dependence of the normalized diffusion coefficients (Dϕ/DB) for ϕP < 0.56. The mean values of the normalized diffusion coefficients illustrate the blocking effect of nanoparticles (see Figure.11). For PNCs made with smaller nanoparticles the diffusivities are suppressed much more than any “passive” theory, which does not account for alteration of the matrix properties. Suppression of diffusivity beyond these passive theories may be the result an effective “densification” of the PMA due to confinement and strong favorable interactions of polymer segments with the silica surface. This effect is opposite to the local rarefactions implied 864

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

by Merkel et al.10,11 in PNCs with repulsive polymer/ nanoparticle interactions, which led to increased diffusion coefficients. However, another reasonable explanation for the abnormally slow diffusion is that penetrant transport is delayed by strong interactions of the diffusant with the silica surface sites, which effectively act as “traps”. Figure 10 reflects that the diffusivity of n-butyl acetate in PMA/silica nanocomposites made with smaller nanoparticles increased much less quickly with increasing penetrant content than it did in the pure PMA melt, a striking result for this diffusant. The free volume fit for DB from the melt, used to normalize the PNC data in that figure has the free volume parameter ξ = 0.67, which is also obtained from free volume fits of data in PNCs made with larger particles and low ϕP, and is identical to the original value calculated by Vrentas and Duda23 from analysis of the D(T,ω1 = 0) data of Fujita.58 However, the smaller particle composite data, and the larger-particle PNC data for ϕP = 0.48 suggest a modification to the polymer’s free volume characteristics. This is likely the result of a modified interfacial layer. Indeed one can estimate that the fraction of interfacial material is significant even at low ϕP for PNCs made with smaller nanoparticles, and is only becomes significant at the highest ϕP employed for PNCs made here with the larger nanoparticles. We note that the data were analyzed from a free-volume perspective here because the work of Merkel et al.10 suggested that addition of nanoparticles to their polymer altered its freevolume characteristics in a way sensitive to the size of the diffusant. Overall our data and analysis agree with this conclusion, in spite of the system being quite different from that studied by Merkel et al., i.e., a system with normal or conventional-size selectivity and above its glass transition temperature. These results make clear the possibility of engineering materials for gas separations applications by appropriate selection of nanoparticle filler. The addition of very large amounts (ϕP = 0.56) of the smaller silica nanoparticles to PMA resulted in dramatic alteration of the diffusion behavior. On the basis of calorimetric evidence22 we concluded that samples at this loading have matrix material almost entirely composed of interfacial (or “bound“) polymer. The sorption intervals for these samples were obviously non-Fickian, even after extensive vapor preconditioning with a high concentration of good solvent/ displacer. The non-Fickian, two-stage shape of the sorption intervals included a protracted approach to equilibrium that was shown to be highly repeatable. This behavior is likely the result of slow adjustments among densely packed nanoparticles, perhaps mediated or controlled by “bridging” among particles by matrix chains.

■

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS D.J. gratefully thanks the National Science Foundation for funding through the IGERT Program for the Study of Multiscale Phenomena in Soft Materials. The authors thank Shane Harton for providing the polymer sample used in this study, Jad Cooper for writing the LabView program used in data acquisition, and a grant from the Pall Corporation that enabled purchase of the RQCM system.

■

(1) Baker, R. W. Ind. Eng. Chem. Res. 2002, 41, 1393−1411. (2) Vane, L. M. J. Chem. Technol. Biotechnol. 2005, 80, 603−629. (3) Feng, X.; Huang, R. Y. M. Ind. Eng. Chem. Res. 1997, 36, 1048− 1066. (4) Jiang, L. Y.; Wang, Y.; Chung, T.-S.; Qiao, X. Y.; Lai, J. Y. Prog. Polym. Sci. 2009, 34, 1135−1160. (5) Lin, H.; Van Wagner, E.; Raharjo, R.; Freeman, B. D.; Roman, I. Adv. Mater. 2006, 18, 39−44. (6) Jeong, B.-H.; Hoek, E. M. V.; Yan, Y.; Subramani, A.; Huang, X.; Hurwitz, G.; Ghosh, A. K.; Jawor, A. J. Membr. Sci. 2007, 294, 1−7. (7) Mahajan, R.; Koros, W. J. Polym. Eng. Sci. 2002, 42, 1420−1431. (8) Rao, P. S.; Wey, M.-Y.; Tseng, H.-H.; Kumar, I. A.; Weng, T.-H. Microporous Mesoporous Mater. 2008, 113, 499−510. (9) Weng, T.-H.; Tseng, H.-H.; Wey, M.-Y. Int. J. Hydrogen Energy 2009, 34, 8707−8715. (10) Merkel, T. C.; Freeman, B. D.; Spontak, R. J.; He, Z.; Pinnau, I.; Meakin, P.; Hill, A. J. Science 2002, 296, 519−522. (11) Merkel, T. C.; Freeman, B. D.; Spontak, R. J.; He, Z.; Pinnau, I.; Meakin, P.; Hill, A. J. Chem. Mater. 2003, 15, 109−123. (12) Merkel, T. C.; He, Z.; Pinnau, I.; Freeman, B. D.; Meakin, P.; Hill, A. J. Macromolecules 2003, 36, 8406−8414. (13) Kim, H.; Kim, H.-G.; Kim, S.; Kim, S. S. J. Membr. Sci. 2009, 344, 211−218. (14) Matteucci, S.; Kusuma, V. A.; Kelman, S. D.; Freeman, B. D. Polymer 2008, 49, 1659−1675. (15) Barrer, R. M.; Barrie, J. A.; Rogers, M. G. J. Polym. Sci., Part A 1963, 1, 2565−2586. (16) Hill, R. J. Ind. Eng. Chem. Res. 2006, 45, 6890−6898. (17) Hill, R. J. Phys. Rev. Lett. 2006, 96, 216001. (18) Khounlavong, L.; Ganesan, V. J. Chem. Phys. 2009, 130, 104901. (19) Grazia De Angelis, M.; Sarti, G. C. Ind. Eng. Chem. Res. 2008, 47, 5214−5226. (20) Pryamitsyn, V.; Hanson, B.; Ganesan, V. Macromolecules 2011, 44, 9839−9851. (21) Janes, D. W.; Kim, J. S.; Durning, C. J. Ind. Eng. Chem. Res. 2013, DOI: 10.1021/ie300536c. (22) Janes, D. W.; Moll, J. F.; Harton, S. E.; Durning, C. J. Macromolecules 2011, 44, 4920−4927. (23) Vrentas, J. S.; Duda, J. L. J. Polym. Sci., Polym. Phys. Ed. 1979, 17, 1085−1096. (24) Takahashi, S.; Paul, D. R. Polymer 2006, 47, 7519−7534. (25) Takahashi, S.; Paul, D. R. Polymer 2006, 47, 7535−7547. (26) Robeson, L. M. J. Membr. Sci. 1991, 62, 165−185. (27) Matteucci, S.; Kusuma, V. A.; Sanders, D.; Swinnea, S.; Freeman, B. D. J. Membr. Sci. 2008, 307, 196−217. (28) Matteucci, S.; Raharjo, R. D.; Kusuma, V. A.; Swinnea, S.; Freeman, B. D. Macromolecules 2008, 41, 2144−2156. (29) Merkel, T. C.; He, Z.; Pinnau, I.; Freeman, B. D.; Meakin, P.; Hill, A. J. Macromolecules 2003, 36, 6844−6855. (30) Ahn, J.; Chung, W.-J.; Pinnau, I.; Guiver, M. D. J. Membr. Sci. 2008, 314, 123−133. (31) Shao, L.; Samseth, J.; Hägg, M.-B. J. Membr. Sci. 2009, 326, 285−292. (32) Patel, N. P.; Miller, A. C.; Spontak, R. J. Adv. Funct. Mater. 2004, 14, 699−707.

ASSOCIATED CONTENT

S Supporting Information *

List of symbols and summary of Vrentas−Duda free-volume theory. This material is available free of charge via the Internet at http://pubs.acs.org.

■

REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: (212)854-8161. Present Address †

The University of Texas at Austin 865

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866

Macromolecules

Article

(33) Barrer, R. M. Diffusion and Permeation in Heterogeneous Media. In Diffusion in Polymers, Crank, J., Park, G. S., Eds.; Academic Press: London, 1968; pp 165−215. (34) Sax, J.; Ottino, J. M. Polym. Eng. Sci. 1983, 23, 165−176. (35) Maxwell, J. C. Treatise on Electricity and Magnetism; Oxford University Press: London, 1873; Vol. 1. (36) Most, C. F. J. Appl. Polym. Sci. 1970, 14, 1019−1024. (37) Patel, N. P.; Miller, A. C.; Spontak, R. J. Adv. Mater. 2003, 15, 729−733. (38) Jeffrey, D. J. Proc. R. Soc. London A 1973, 335, 355−367. (39) Batchelor, G. K. Annu. Rev. Fluid Mech. 1974, 6, 227−255. (40) Sangani, A. S.; Yao, C. J. Appl. Phys. 1988, 63, 1334−1341. (41) Xue, L.; Borodin, O.; Smith, G. D. J. Membr. Sci. 2006, 286, 293−300. (42) Cohen, M. H.; Turnbull, D. J. Chem. Phys. 1959, 31, 1164− 1169. (43) Landauer, R. J. Appl. Phys. 1952, 23, 779−784. (44) Kirkpatrick, S. Phys. Rev. Lett. 1971, 27, 1722−1725. (45) Davis, H. T. J. Am. Ceram. Soc. 1977, 60, 499−501. (46) Kamiyama, F.; Inagaki, H. Bull. Inst. Chem. Res, Kyoto Univ. 1974, 52, 393−402. (47) Diani, J.; Fayolle, B.; Gilormini, P. Eur. Polym. J. 2009, 45, 601− 612. (48) Billovits, G. F. A Fundamental Investigation of Viscoelastic Diffusion in the Polystyrene-Ethylbenzne System; Columbia University: New York, 1989. (49) Billovits, G. F.; Durning, C. J. Macromolecules 1994, 27, 7630− 7644. (50) Kishimoto, A.; Fujita, H.; Odani, H.; Kurata, M.; Tamura, M. J. Phys. Chem. 1960, 64, 594−598. (51) Vrentas, J. S.; Duda, J. L. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 441−453. (52) Weeks, E. R.; Crocker, J. C.; Levitt, A. C.; Schofield, A.; Weitz, D. A. Science 2000, 287, 627−631. (53) Kishimoto, A. Prog. Org. Coatings 1972, 1, 91−112. (54) de Luzuriaga, A. R.; Etxeberria, A.; Rodriguez, J.; Pomposo, J. A. Polym. Adv. Technol. 2008, 19, 756−761. (55) Duda, J. L.; Vrentas, J. S.; Ju, S. T.; Liu, H. T. AIChe J. 1982, 28, 279−285. (56) Chuang, I. S.; Maciel, G. E. J. Phys. Chem. B 1997, 101, 3052− 3064. (57) Chen, C.; Justice, R. S.; Schaefer, D. W.; Baur, J. W. Polymer 2008, 49, 3805−3815. (58) Fujita, H.; Kishimoto, A.; Matsumoto, K. Trans. Faraday Soc. 1960, 56, 424−437. (59) De is defined

De =

characteristic material relaxation time characteristic diffusion time

(60) The microstructural relaxations evident in two-stage sorptions observed at relatively high particle loadings may manifest the response of a “bridging network” from chains adsorbed onto several particles each, or from a protracted, glass-like reponse of the composite at high loadings, akin to that observed in colloidal glasses.

866

dx.doi.org/10.1021/ma301990u | Macromolecules 2013, 46, 856−866