Interval Sorption of Alkyl Acetates and Benzenes in Poly(methyl

Article pubs.acs.org/IECR

Interval Sorption of Alkyl Acetates and Benzenes in Poly(methyl acrylate) Dustin W. Janes,† Ji Seung Kim, and Christopher J. Durning* Columbia University, Department of Chemical Engineering, 500 W. 120th Street, Room 801 Mudd Hall, New York, New York 10027, United States ABSTRACT: We describe a simple, low-cost, gas flow vapor sorption/desorption apparatus that enables “interval” or “differential” experiments on supported polymer films, with control of small step changes in diffusant weight fraction, ω1, to intervals as small as 0.1%. The apparatus permits accurate determination of the binary mutual diffusion coefficient D12(ω1) for ω1 up to at least 3% for “condensable” organic species that typically exhibit very sharp increases in D12 with the diffusant’s weight fraction. We report results for two homologous series of diffusants, n-alkyl acetates and alkyl benzenes, in monodisperse poly(methyl acrylate) (PMA). The Vrentas−Duda free volume theory adequately accounts for the strong concentration dependence of D12 seen in all cases. Nearly constant values of the jump-unit size ratio, ξ, for the homologous set of n-alkyl acetates in the range ξ = 0.61−0.67 closely match those reported previously. For the alkyl benzenes, ξ for benzene (ξ = 0.78 ± 0.04) was higher than those for the larger homologues toluene, ethylbenzene, and cumene (ξ = 0.58−0.60). The ξ values for both homologous series are well-predicted by a free volume based estimate for eccentric diffusants, which jump as a single unit in a mixture whose components contribute dif ferent amounts of free volume per jump unit, according to their geometries.

1. INTRODUCTION Low cost and/or high selectivity drive the development of new gas and liquid separations membranes, especially in connection with the energy related technologies of natural gas treatment,1 biofuel refining,2,3 and fuel cells.4,5 These applications impose demanding criteria for species flux and selectivity. Systematic studies of sorption and diffusion of low molar mass organic species in polymers provide the basis for such developments. Such data also underlie estimation of the removal rate of residual monomers and casting solvents from polymers6−8 and enable the assessment of materials for advanced sensing methods.9 These development efforts are well served by improved experimental methods and streamlined data analysis procedures that allow convenient, accurate determination of sorption and diffusion coefficients for a variety of polymer/ penetrant systems. Toward these ends, we describe here a simple, low-cost carrier-gas flow vapor sorption/desorption apparatus that enables “interval” or “differential” experiments10 in thin cast films, on the order of micrometers in thickness. The device allows accurate control of penetrant partial pressure in the carrier gas and real-time monitoring of experiments with very small changes in penetrant mass fraction in the cast film, ω1, as small as 0.1 wt %. We report results from the technique for two homologous series of diffusants, n-alkyl acetates, and alkyl benzenes, in monodisperse poly(methyl acrylate) (PMA). Analysis of the interval sorption allows easy extraction of the binary mutual diffusion coefficient, D12(ω1). Application of Vrentas−Duda free volume theory explains the rather strong increase of D12 with ω1. Values of the jump-unit size ratio, ξ, for the homologous set of n-alkyl acetates match the values reported previously. These, plus new values of ξ for alkyl benzenes in PMA, are in good agreement with the picture of penetrant structure effects on D12 suggested by Vrentas et al.,11 which asserts that the jump units of the two components © 2012 American Chemical Society

contribute different amounts of hole free volume to a mixture. It does not appear necessary to invoke segment-wise diffusion of penetrants in either homologous series in order to explain the dependence of ξ on penetrant size. From sorption isotherm data, we also determine partition coefficients, K(ω1), and the Flory−Huggins interaction parameters, χ(ω1), in the concentrated polymer regime (0.05 ≤ ω1 ≤ 0.1). All penetrants studied partition favorably into PMA (K ≫ 1). Additionally, all penetrants were good solvents for PMA (χ < 0.5), except for n-butyl acetate (χ > 0.5).

2. TRANSPORT THEORY For concentrated polymer/penetrant mixtures well above the glass transition, Fick’s laws govern one-dimensional interdiffusion:10,12,13 Jx = −D

∂C ∂x

∂C ∂ ⎛⎜ ∂C ⎞⎟ = D ∂t ∂x ⎝ ∂x ⎠

(1a)

(1b)

Here, J is the mass flux of penetrant (solvent) relative to polymer, C is the penetrant mass concentration per unit volume polymer, x is the polymer-fixed frame distance, t is time, and D is the polymer fixed frame diffusion coefficient, simply related to the binary mutual diffusion coefficient D12 of the penetrant against the polymer: D = ϕ22D12 with ϕ2 being the polymer volume fraction.14 The partition coefficient, K, defined as the Special Issue: Giulio Sarti Festschrift Received: Revised: Accepted: Published: 8765

February 28, 2012 August 14, 2012 August 15, 2012 September 19, 2012 dx.doi.org/10.1021/ie300536c | Ind. Eng. Chem. Res. 2013, 52, 8765−8773

Industrial & Engineering Chemistry Research

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When the driving force is large enough that D is not constant (i.e. D = D(C) must be accounted for), the experiment is termed an “integral” sorption. In polymer−solvent systems, D typically increases very strongly with penetrant content, making true differential experiments a challenge. Free volume theory accurately captures the effects of composition, temperature, T, and molecular architecture on D12 for binary polymer−penetrant mixtures. The free volume theory result of Vrentas and Duda15 provides a working relation for analysis of the D12(ω1) reported here

ratio at equilibrium of C in the mixture to the equivalent in the phase contacting the mixture, also shows up in models for diffusion processes involving phase boundaries. Note that the product KD = P is the permeability, P, which defines the steadystate rate of diffusion of the low molar mass component through a membrane of the polymer and is the key metric of membrane performance in separations.10,13 In a sorption experiment, a thin film sample of the polymer is exposed to a step increase in the penetrant chemical potential at its outer surface and the mass increase of the film is recorded as the penetrant diffuses into the film (see Figure 1). In a

⎡ (ω V̂ * + ω ξV̂ * ) ⎤ 2 2 ⎥ D12 = D01ϕ2 2(1 − 2χϕ1) exp⎢ − 1 1 ̂ /γ ⎢⎣ ⎥⎦ VFH (5a)

where V̂ FH is the mixture’s specific “hole” free volume, enabling diffusive motions, which depends on temperature and composition: ̂ VFH K K = 11 ω1(K 21 + T − Tg1) + 12 ω2(K 22 + T − Tg2) γ γ γ

Figure 1. Schematic of the sorption experiment described by eqs 1−4. The film swells in the direction normal to the substrate.

(5b)

eqs 5a and 5b include the component volume fractions, ϕi (i = 1 indicates the penetrant and i = 2 the polymer), weight fractions ωi, and the specific free volume necessary for a diffusive jump of a segment or “jump unit” of component i, V̂ i*, estimated as the specific volume at 0 K. The free volume parameters K11/γ, K21 − Tg1, K12/γ, and K22 − Tg2, essentially the WLF constants for the components, can be extracted from pure component viscosity−temperature data. The Flory− Huggins interaction parameters χ can be found from sorption isotherm data. Therefore, only the pre-exponential factor D01 and “jump unit” size ratio ξ are used in eq 5 to fit the effect of ω1 on D12 data at a fixed temperature. The size ratio ξ is the main parameter in the Vrentas−Duda theory characterizing the effect of penetrant architecture on D12(ω1) in a given polymer.16 Typically ξ < 1. The original development by Vrentas and Duda15,17 presumes each mobile segment or “jumping unit”, whether polymer or penetrant, contributes the same amount of free volume to the mixture. As a result, ξ represents the ratio of the (molar) volume necessary for a diffusive “jump” of a penetrant’s freely mobile segment, Ṽ 1*, to that of a polymer’s, Ṽ 2*:

“differential” or “interval” sorption the initial penetrant chemical potential jump is small enough that the relevant properties D and K stay nearly constant during the sorption.10 Equation 1 governs with constant D, and subject to the auxiliary conditions, ∂ C(0, t ) = 0 ∂x C(S , t ) = KC1 C(x , 0) = KC0

(2)

Note that x ranges from 0 at the substrate surface to the (dry) film thickness S at the outer edge. At t = 0, the potential step change in the environment corresponds to an equivalent concentration change C1 − C0. The measured mass uptake per unit area during sorption M(t) is M(t ) − M(0) =

∫0

S

[C(x , t ) − KC0] dx

(3a)

ξ = V1̃ */V2̃ *

Defining M(t ) − M(0) M̃ (t ) = M(∞) − M(0)

Accordingly, for relatively low molar mass species expected to jump as a single unit, ξ values for a set of penetrants in a particular polymer should correlate with any valid measure of the diffusant’s molecular or hard-core volume, such as the molar volume at 0 K. This expectation fails for diffusants beyond even moderate molecular weights11 (≈ 100 Da), leading to the assertion of segmental diffusive motions for relatively small, compact species. A revision11 accounts for the components contributing different amounts of free volume per jump unit because of the jump unit geometries. In this case,

(3b)

one can show12 ∞ ⎛ Dt ⎞1/2 ⎡ n S ⎤⎥ M̃ (t ) = 2⎜ 2 ⎟ ⎢π −1/2 + 2 ∑ ( −1)n i erfc ⎝ S ⎠ ⎢⎣ Dt ⎥⎦ n=1

(4a)

or, equivalently ∞

M̃ (t ) = 1 −

∑ n=0

(6)

⎛ −D(2n + 1)2 π 2t ⎞ 8 exp⎜ ⎟ 2 2 (2n + 1) π 4S2 ⎠ ⎝

ξ=

(4b)

V1̃ *VFH2 ̅ * ̃ V V̅

2 FH1

These two solutions are complementary in that the sum on the right of eq 4a converges quickly at short times, during the initial stages of sorption, while that on the right of eq 4b converges quickly at long times, as the sorption approaches equilibrium.

(7)

where V̅ FHi is the average hole free volume contributed to the mixture by a jumping unit of component i. Vrentas et al.11 prescribed a method for predicting ξ for eccentric, axisym8766

dx.doi.org/10.1021/ie300536c | Ind. Eng. Chem. Res. 2013, 52, 8765−8773


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Figure 2. Experimental apparatus used in vapor sorption.

elevates the transducer’s resistance, ΔR. In the limit that the added film is rigid and Δm is relatively small, the linear response is given by Sauerbrey,19

metric penetrants which should jump as a single unit. Their development gives ξ=

MW1V1̂ * V2̃ * + MW1V1̂ * 1 −

(

A B

)

(8)

Δm =

Here, MW1 means the molecular weight of the penetrant, while A/B is the aspect ratio (A/B ≤ 1). While Vrentas et al.11 suggested a particular computational chemistry software and methodology to determine values of A/B, presumably, any realistic method suffices. Note that in the case of spherically symmetric penetrants that jump as a single unit, A/B = 1, Ṽ *1 = MW1V̂ *1, and eq 8 reduces to eq 6, predicting that ξ increases linearly with the penetrant molar volume at 0 K. We use penetrant diffusion data reported here and in previous works to test the capabilities of eq 6 vs eq 8.

− ρq μq 2fq2

Δf ;

ΔR = 0 (9)

where ρq and μq are the density and shear modulus of quartz, respectively. Δf and ΔR are recorded continuously, giving in situ measurement of mass loading on a crystal (from Δf) and its damping (from ΔR) . If one passes penetrant vapor over a thin polymer film cast on a crystal the transient mass uptake of penetrant into the film can be measured in situ. Our equipment includes quartz crystal transducers fitted into a set of flow cells having Kalrez O-rings (CHK-100, Inficon Inc.), each driven at resonance by dedicated oscillators (Maxtek PLO-10i and RQCM, Inficon Inc.), which in the case of the PLO-10i are connected to a frequency counter (Model H384A, Agilent Technologies, Santa Clara, CA) and an analog/digital converter (Model NI USB-6008, National Instruments, Austin, TX). Standard, 1-in. diameter, AT-cut, 5 MHz quartz crystals bearing gold electrodes were purchased from Stanford Research Systems (Sunnyvale, CA) or Inficon Inc. (East Syracuse, NY). The quantities Δf and ΔR are continuously recorded for each transducer by data acquisition software. For the PLO-10i the data acquisition software was Labview v7.1 (National Instruments), while the manufacturer’s provided software was used for the RQCM system. Vapor sorption experiments on polymer films using a QCM have been done by others to determine D,20−24 the activity coefficient,25 water solubility,26 and the Flory−Huggins interaction parameter.27 Our apparatus (Figure 2) allows improved control of the penetrant partial pressure in a carrier gas stream passed over the quartz crystals to enable differential sorptions. Best results were obtained when the QCM flow cells were arranged in series, meaning that the vapor flowing out of one cell is directed into the following cell. Up to four cells could be used simultaneously in a single experiment. The flow rates of carrier gas (>20 mL/min) and vapor-phase pentrant diffusivities (∼10−2 cm2 s−1) were always high enough relative to rates of sorption into the polymer film that the external (gas phase) mass transfer resistances did not affect the results significantly and true step changes in penetrant vapor activity could be effected for all polymer films.

3. EXPERIMENTAL SECTION 3.1. Materials. Poly(methyl acrylate) (PMA, Mw = 55,960, Mn/Mw = 1.04) was synthesized via atom transfer radical polymerization (ATRP) according to a procedure detailed elsewhere.18 The solvents 2-butanone (MEK), 99+%, n-propyl acetate, 99+%, n-butyl acetate, 99+%, ethyl benzene, 99+%, and cumene, 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%, and toluene, 99+%, were purchased from Sigma-Aldrich. Benzene, 99+%, was purchased from Amend Chemical (Irvington, NJ). Zero-grade nitrogen gas was purchased from Tech Air (White Plains, NY). The polymer casting solutions were 50−100 mg/mL of PMA in MEK. The antioxidant Irganox 1010 was supplied by BASF (Ludwigshafen, Germany), and was then dissolved in MEK to make a concentrated stock solution of 20 mg Irganox/mL MEK. 50 μL of the Irganox stock solution was then added to the PMA solution resulting in a concentration of 0.1 wt % Irganox relative to polymer, to minimize oxidation during subsequent thermal annealing of polymer film samples. 3.2. Quartz Crystal Microbalance (QCM) Based GasFlow Sorption Apparatus. Our vapor sorption apparatus employs quartz crystal “microbalance” (QCM) transducers driven at the fundamental resonant frequency, fq, by a feedback controlled oscillator. Mass added (per unit area) onto a crystal’s surface, Δm, depresses the resonant frequency, Δf, while external damping and/or dissipation within the added layer 8767



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removing a portion of the film with a razor blade and averaging at least five thickness measurements made on the remaining film with a KLA-Tencor profilometer (Milpitas, CA). One should note that the uncertainty in f due to removing and reloading the crystal was negligible (fq − f F is typically on the order of 104 Hz). Also, RF − Rq was typically less than 0.5 Ω, indicating that the cast films are very nearly rigid, thin, and lossfree, justifying the use of Sauerbrey’s relation (eq 9). Thus, with a measured frequency change due to solvent uptake Δf S, the component weight fractions are

Control of the penetrant partial pressure in the carrier gas was accomplished as follows. Incoming nitrogen, dried by an adsorbent cartridge (Matheson Model 451), is split into two streams, one saturated with penetrant, S1, and a second that remains penetrant free, S2. The flow rates of these streams are adjusted by the rotameters R1 and R2. Penetrant vapor saturates S1 by bubbling the carrier gas through two successive porous glass frits submerged in the pure liquid penetrant. A previous work confirmed that the carrier gas exiting this bubbler was saturated with penetrant vapor for flow rates 20− 170 mL/min.28 Temperature control of the gas bubblers by a cooling bath (Neslab model RTE-210D) fixes the penetrant partial pressure at its vapor pressure, p1, in S1, known from Antoine’s coefficients.29 One can pass pure nitrogen carrier gas over the crystals in series by directing S1 into a chemical hood. Conversely, one can pass solvent vapor at a known partial pressure (equal to the vapor pressure) over the crystals by shutting R2 completely. Adjustment of R1 and R2 to different relative rates permits intermediate penetrant partial pressures (activities) to pass over the crystals. Rapid switching enables a near step change in partial pressure (i.e., in the gas phase chemical potential of penetrant), allowing sorption or desorption experiments. For all experiments reported here, the QCM probes and their feed gas stream were held at 25 °C by a temperature controlled water bath. 3.3. Crystal Cleaning and Film Casting. As received quartz crystals with gold electrodes were first immersed in piranha solution (3:1 ratio solution of concentrated sulfuric acid to 30% hydrogen peroxide solution) for less than one minute. Each was then washed twice in deionized water and ethanol successively and then cleaned for 20 min in a UV/O3 oven (Jelight Company, Model No. 342, Irvine, CA). After a final rinse in deionized water, each crystal was blown dry with filtered nitrogen gas. Crystals were handled with Teflon tweezers outside of the electrode surface and stored in sealed wafer holders to prevent contamination. After cleaning, crystals were loaded in the flow cells to record f and R for the bare crystal, effectively “taring” it for use as a microbalance. These quantities, referred to hereafter as fq and Rq, are typically near 5 MHz and 8 Ω, respectively. Values of Rq larger than 10 Ω indicated damage, contamination, or inappropriate crystal loading. When this condition was encountered, the crystal was reloaded into the flow cell, or the cleaning procedure was repeated, or the crystal was discarded. After recording fq and Rq, the crystal was removed from the flow cell and a PMA film was cast on it by spin coating with a PMA/MEK solution. Spinning at a rate of 1000−1300 rpm for ≈1.5 min, yielded a film thicknesses ∼1 μm. The coated crystal was then annealed at 150 °C under a N2 atmosphere for two days and reloaded into the flow cell. f and R were recorded again (denoted hereafter to as f F and RF). The film was then “solvent annealed” in the sorption apparatus (see Figure 2) under n-propyl acetate vapor for at least two days by directing S1 over the crystal, turning R2 completely closed, and setting the temperature controlled bath around the gas bubblers containing the penetrant to a constant temperature within the range 15−20 °C. The film was then dried by redirecting S1 to the hood, and turning R2 back to ∼20 mL/min so that only pure nitrogen gas flowed over the crystal. Sorption experimentation began after reaching a constant frequency, characteristic of the dry film. The dry film thickness, S , was determined after all experimentation was completed by

ω1 =

ΔfS ΔfS + fq − fF

and ω2 = 1 − ω1 (10a)

Using bulk values for the mass density of component i, ρi, and assuming no volume change upon mixing, the component volume fractions are ϕ1 =

ω1ρ2 ω1ρ2 + (1 − ω1)ρ1

and ϕ2 = 1 − ϕ1

(10b)

3.4. Analysis of Sorption Isotherm Data. The sorption isotherm data was found by measuring the difference in the equilibrium frequencies, Δf S, between when only S2 flows over the coated crystal and when only S1 flows over (see Figure 2). By adjusting the temperature of the cooling bath for the gas bubblers in S1, typically between −5 and 5 °C, the effect of p1 on ω1 was determined. K was determined from K=

RTω1ρ1ρ2 MW1p1 [ω1ρ2 + (1 − ω1)ρ1]

(11a)

where R is the gas constant, T is absolute temperature, and MW1 is the molecular weight of the penetrant (solvent). The Flory− Huggins interaction parameter, χ, was also calculated from the isotherm data according to p1 p°1

= ϕ1exp(ϕ2 + χϕ22)

(11b)

where p1° is the vapor pressure of pure penetrant at the experimental temperature.14 3.5. Analysis of Interval Sorption Data. Interval sorption or desorption data was obtained by incrementally raising or lowering the penetrant content in the vapor flowed over the crystal by sudden adjustment of R1 or R2 and subsequently tracking the frequency shift until equilibrium. Gas flow rates were typically in the range 20−40 cm3 min−1; no significant change in frequency resulting solely from raising or lowering the gas flow rate was observed. The frequency drop for each interval was typically 10 to 30 Hz, corresponding to changes in 100ω1 of about 0.2, but this could easily be made larger. The penetrant mass uptake was normalized to the total change in uptake from prior to sorption until equilibrium, yielding plots of M̃ vs t for different (equilibrium) values of ω1. Values of D extracted from each interval determine D(ω1). These were found by fitting the M̃ (t) curve with the transport model summarized earlier. Complementary approximations to eq 4 accurate at short and long-times are lim M̃ (t − t0) ≅

t→0

8768

2 S

DSTt π

(12a)


Industrial & Engineering Chemistry Research ⎛ −D π 2t ⎞ 8 LT ⎟ lim M̃ (t − t0) ≅ 1 − 2 exp⎜ 2 t →∞ π ⎝ 4S ⎠

Article

penetrant heat of vaporization. Figure 3 shows good agreement with this expectation, except for ethylbenzene, based on literature values31 for ΔHvap. The values of χ indicate that methyl acetate, ethyl acetate, and n-propyl acetate are “good solvents” for PMA as are all of the alkyl benzenes, but that nbutyl acetate is a nonsolvent. This is at odds with Fujita et al.32 who reported values of χ > 0.5 for all four of the n-alkyl acetates in PMA. The increasing trend of χ with ω1 in the concentratedpolymer regime agrees with previous reports27 and is seen in many polymer−solvent systems. 4.2. Diffusion Coefficients (D12(ω1)). Figure 4 shows representative raw data for nine successive sorption intervals of

(12b)

The D were found by nonlinear regression fits (Microcal Origin software) of these formulas to the data between appropriate cutoffs. We found a “short-time” result, DST, by applying eq 12a in the range 0.1 ≤ M̃ ≤ 0.4, and a “long-time result”, DLT, by applying eq 12b in the range 1.7 ≤ M̃ ≤ 0.95 (see Figure 5b, for example). For a linear sorption process where D is truly constant, one should find DST = DLT. The resulting D12(ω1) data were analyzed by Vrentas−Duda free volume theory to obtain the parameters ξ and D01 by a nonlinear regression of eq 5 to the data. Best results were obtained when the fitting was done with weighting inverse to D12(ω1). The sources used for all other free volume parameters used in the fit are given in Tables 1−3.

4. RESULTS 4.1. Partition Coefficients K(ω1) and Interaction Parameters χ(ω1). K(ω1) and χ(ω1) for the n-alkyl acetates and alkyl benzenes in a PMA melt are shown in Figure 3. The

Figure 4. Representative QCM data showing f and R vs time for the 8th through 16th sorption intervals of ethyl acetate in a PMA film of dry thickness S = 1.55 μm.

ethyl acetate in PMA (0.005 ≤ ω1 ≤ 0.018). In Figure 4, showing crystal frequency f and resistance R versus time, each sorption interval begins with a sharp drop in f, corresponding to an increase of film mass due to the step change in penetrant vapor concentration over the crystal. With increasing time, f relaxes toward a constant value consistent with the vapor− liquid equilibrium; eventually, the mass of the film is no longer increasing, and the sorption interval has ended. Note that, throughout these intervals, R is nearly invariant with time and penetrant uptake. Figure 5 shows representative results from an exhaustive set of sorption intervals for ethyl acetate in PMA. The first 24 intervals of the data set appear in Figure 5a. A typical interval from that group, plotted in “reduced” format, is shown in Figure 5b. The data clearly indicate ordinary diffusion: The relative weight gain linearly increases with √t initially from the origin, always remains concave down, and approaches a welldefined equilibrium shortly after the initial rise. The values of −2 D12,ST ≡ ϕ−2 2 DST and D12,LT ≡ ϕ2 DLT lie very close to one another, although they are not identical. We typically found D12,LT > D12,ST, consistent with a slight increase in D12 with ω1 during a sorption interval. The step changes in ω1 shown in Figure 5a (⟨100Δω1 ⟩ ≈ 0.12) were small enough that δD = | D12,LT − D12,ST|/D12,ST was, on average, 11%. That is, the diffusion behaviors observed under these conditions are very nearly the linear response. Figure 6 shows the effect of the sorption interval step size, Δω1, on the values of D12,ST and D12,LT for ethyl acetate in more detail. The two values determined from eq 12 are plotted against the equilibrium value of ω1 in Figure 6a for sets of intervals with several different average step changes in penetrant concentration, ⟨100Δω1⟩. Included also are results from several “integral” sorptions, using initially dry films. The interval data show that the results agree within 9% on average for ⟨100Δω1⟩ ≤ 0.5, but

Figure 3. (a)K(ω1) in pure PMA, S = 1.55 μm, for methyl acetate (◇), ethyl acetate (□), n-propyl acetate (○), and n-butyl acetate (△). (b) K(ω1) in pure PMA, S = 0.69 μm, for benzene (◆), toluene (■), ethylbenzene (●), and cumene (▲). The solid lines in (a) and (b) show anticipated trends in K from literature values31 of ΔHvap. (c) χ(ω1) calculated from eq 8b using the same data; symbols are as in (a) and (b).

penetrants partition favorably into the PMA, that is, K ≫ 1 in all cases. Further, K is remains nearly constant with ω1 over the range probed, that is, Henry’s law is observed. The thermodynamic relation30 for the temperature dependence of K, ((∂ ln K)/∂T) = ΔHvap/RT2, anticipates the trend in K among the penetrants: log K ∼ ΔHvap, where ΔHvap is the 8769



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they become distinct with D12,LT > D12,ST and shift down as ⟨100Δω1⟩ increases above 0.5. In these cases, D12 evidently varies enough during the interval to make analysis using a linear model inappropriate. Figure 6b shows δD = |D12,LT − D12,ST|/ D12,ST vs ⟨100Δω1⟩ for the interval and integral sorptions in the ethyl acetate/PMA system. A monotonic increase is evident. Provided 100Δω1 ≤ 0.6 where δD is within the experimental reproducibility (see next paragraph), one can safely assume a linear response for the ethyl acetate−PMA system. We used this criterion to set the interval size for all the penetrants studied. The concentration dependence of D12 is shown for ethyl acetate in Figure 5c; it increases strongly with ω1, which is typical of polymer melt−condensable penetrant systems.15 Note that the data include D12,ST found from three samples with different S . These superpose on the plot to within the experimental uncertainty of ±10%, which primarily arises from uncertainties in S (4%) and D (≤3%) . This fact, combined with the previously described characteristics in Figure 5b, indicate beyond doubt that the diffusion observed obeys Fick’s second law (eq 2). The D12(ω1) of Fujita et al.32 for ethyl acetate in PMA lie above the data reported here. Note, however, that Fujita et al. reported a glass transition for their PMA samples of Tg = 0−5 °C, significantly lower than the value for our material of Tg ≈ 17 °C. This difference explains the discrepancy qualitatively. Figure 7 shows the diffusivity data from sufficiently small intervals for all the penetrants studied in PMA. D12(ω1)

Figure 5. Diffusion coefficients of ethyl acetate in pure PMA . (a) The first 24 sorption intervals for a film of dry thickness S = 1.55 μm, in which the fourth interval is plotted in reduced format in (b). For visual clarity, only every 20th point is shown. (b) Example of extraction of D values for the interval ω1 = 0.0122 → 0.0203, where data points (○) are fit using eq 12a (―) and eq 12b (---). The horizontal lines represent the range where eq 12a (―) and eq 12b (---) were fit to the data. The values of DST and DLT and the uncertainties arising from the fit are shown for this interval. (c) Values of D12 extracted from the data set partially shown in (a) via short time (△) or long time (□) analysis are shown in comparison to values extracted via short-time analysis for films of thickness S = 1.88 μm (×) and S = 0.69 μm (+). For comparison, the results of Fujita et al.32 for the same system are also displayed (○).

Figure 7. Diffusion coefficient data for PMA at 293 K, where the lines are fits of eq 5 to the data. (a) contains data for the n-alkyl acetates, with methyl acetate (S = 1.55 μm, ◇), ethyl acetate (S = 1.55 μm, □), n-propyl acetate (S = 1.88 μm, ○), and n-butyl acetate (S = 1.55 μm, △). (b) contains data for the alkyl benzenes (all have S = 0.69 μm), with benzene (◆), toluene (■), ethylbenzene (●), and cumene (▲).

Figure 6. Effect of interval step change size on the value of D12 for ethyl acetate in PMA at 25 °C. (a) D12 plotted at the interval’s final (equilibrium) value of ω1 extracted from short-time (filled symbols) and long-time (unfilled symbols) asymptotic analysis for a film of thickness S = 1.88 μm. The average step change size for the sorption intervals, ⟨100Δω1⟩, was varied ≈ 0.26 ± 0.11 (●), 0.48 ± 0.05 (■), 1.0 ± 0.2 (▲), and 2.0 ± 0.2 (▶). Also shown are D12 from “integral” sorptions beginning with a dry film, ⟨100Δω1⟩ ≈ 4.5 ± 0.5 (▼). Note the values from short time analysis (filled symbols) generally lie below those from the corresponding long time analysis (unfilled symbols). (b) 100δD is plotted against 100Δω1. Note the convergence between D12,ST and D12,LT (i.e., δD → 0) at small interval step change sizes. Data from Figure 5c with ω1 < 0.03 is also displayed (○) in addition to the data from (a), which retains the same symbols.

increases strongly with ω1 in all cases. For both homologous series, D12(ω1) generally decreases with increasing penetrant size. In all cases, eq 5 with D01 and ξ values determined by regression fitting and all other constants known from reference data gave excellent representations of the experimental D12(ω1) trends (see solid lines in Figure 7). Tables 1−3 collect Table 1. Free Volume Parameters for PMA15 V2̂ * 3

−1

0.749 cm g 8770

K12 γ

× 104

3.99 cm3 g−1 K−1

T + K22 − Tg2 67.15 K



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Table 2. Free Volume Parameters for n-Alkyl Acetates in PMA parameter

V1̂ * K11 γ

× 104

T + K21 − Tg1 χ̅ D01 (T = 298.15 K) × 105 ξ

units

methyl acetate

ethyl acetate

n-propyl acetate

n-butyl acetate

source

cm3 g−1

0.830

0.863

0.886

0.904

36

cm3 g−1 K−1

5.08

7.43

7.35

8.98

33, 34

K

396.85 0.25 89 ± 42 0.63 ± 0.02

319.09 0.40 11 ± 3 0.61 ± 0.01

309.10 0.35 11 ± 4 0.64 ± 0.02

270.14 0.61 9±4 0.67 ± 0.02

33, 34 this study this study this study

cm2 s−1

Table 3. Free Volume Parameters for Alkyl Benzenes in PMA parameter

V̂ * 1 K11 γ

× 104

T + K21 − Tg1 χ̅ D01(T = 298.15 K) × 105 ξ

units

benzene

toluene

ethyl benzene

cumene

source

cm3 g−1

0.914

0.932

0.946

0.956

36

cm3 g−1 K−1

7.02

8.61

10.85

13.34

33, 35

K

208.84 0.16 ± 0.03 251 ± 184 0.78 ± 0.04

265.74 0.43 ± 0.05 1.61 ± 1.14 0.60 ± 0.03

241.37 −0.22 ± 0.16 0.24 ± 0.21 0.58 ± 0.04

205.04 0.21 ± 0.12 0.29 ± 0.27 0.60 ± 0.04

33, 35 this study this study this study

cm2 s−1

invites an alternate view. This motivates a comparison between the experimentally measured values of ξ and those predicted by eq 8. To determine a value of Ṽ *2 for PMA needed in eq 8, we applied eq 6 to the three noble gas penetrants in Figure 8 and benzene. Due to their monatomic structure, the noble gas penetrants are symmetric and diffuse as a single jump unit, meaning they should obey ξ = MW1V̂ 1*/Ṽ 2* according to eq 6. Additionally, the relatively compact and rigid penetrant benzene is nearly symmetric, with A/B = 0.92 used in ref 11, and may also be expected to diffuse as a single jump unit and closely obey eq 6. Indeed, the noble gases and benzene clearly show a linear increase of ξ with MW1V̂ 1* in Figure 8. A linear regression constrained to pass through the origin of the data for these in Figure 8 yields Ṽ 2* = 95 ± 23 cm3 mol−1. In the subsequent application of eq 8, we set A/B = 1 for the noble gases. Ref 11 provides a self-consistent set of A/B for all the remaining compounds in Figure 8, except water and cumene, which were excluded from the comparison with eq 8. Values of ξ predicted by both eq 6 and eq 8 are compared to the experimental values in Figure 9, where agreement between theory and experiment corresponds to a line of unit slope

parameters from the free volume analysis, which involved the regression fitting of our D12(ω1) results and of pure solvent viscosity data. The latter, used for the determination of K11/γ and T + K21 − Tg1, were from literature33−35 for temperatures greater than 250 K. In Figure 8, the values of ξ determined in this work are added to a plot of ξ against the penetrant molar volume at 0 K,

Figure 8. Variation of jump unit size ratio ξ with the estimated36 penetrant molar volume at 0 K for penetrant/PMA systems. MW1 × V̂ 1* is the product of the penetrant’s molecular weight and specific volume at 0 K. Values shown are from this work (▲) and Vrentas and Duda16 (■). The solid line is a linear regression of neon, argon, krypton, and benzene to eq 6 and results in Ṽ 2* = 95 ± 23 cm3 mol−1 (see text for explanation). The dashed line serves to guide the eye.

MW1V̂ 1*, published previously by Vrentas and Duda.16 The values of ξ for the n-alkyl acetates and benzene in PMA extracted by fitting our D12(ω1) results lie within 5% of those reported by Vrentas and Duda.16 For the n-alkyl acetate series, ξ is invariant with MW1V̂ 1* within the uncertainty. On the other hand for the alkyl benzene series, ξ for benzene lies about 30% higher than for the larger homologues toluene, ethylbenzene, and cumene. The trends in ξ for both homologous series do not follow the linear increase with MW1V̂ 1* predicted by eq 6 and the assertion that the diffusants jump as single units. While segmental diffusion16 provides a possible explanation for the nalkyl acetate results, owing to their chain-like structure, the behavior seen for the more rigid, eccentric alkyl benzenes

Figure 9. Comparison of jump unit size ratio ξ presuming the diffusant jumps as a single unit predicted by eq 6 (unfilled points) and eq 8 (filled points) with experimental values in penetrant−PMA systems. The dashed line has a unit slope and represents agreement between predicted and experimental values. Values shown are from this work (△,▲) and Vrentas and Duda16 (□,■). 8771



Article

through the origin. For penetrants with A/B near unity (neon, argon, krypton, and benzene) eqs 6 and 8 are nearly identical, and the results lie close to the line of unit slope, as expected. For eccentric penetrants, with A/B < 1, values of ξ predicted by eq 8 agree with experimental results to within the uncertainty, except for ethylbenzene, which lies only 1% outside the uncertainty. Note that the relatively large error bars on the abscissa for these higher molar volume penetrants in Figure 9 come primarily from the uncertainty in Ṽ *2 . Clearly, eq 8 provides much better agreement between theory and experiment than does eq 6, if one presumes that all of the penetrants studied jump as single units.

disrupts local molecular packing sufficiently that a greater free volume per penetrant jump unit results. Equation 8 predicts values of ξ that agree within uncertainty with the experimental values in each case a comparison was made, except for ethylbenzene. Even for this compound, the value of ξ predicted by eq 8 lies just outside the experimental uncertainty. On the other hand, eq 6 fails badly for the larger eccentric diffusants, if one presumes they jump as single units. We conclude that eq 8 provides a satisfying explanation of the effects of penetrant structural changes on D12(ω1) for both homologous series in PMA, without the need of asserting segmental diffusion.

■

5. CONCLUSIONS AND DISCUSSION The QCM vapor-sorption apparatus described here proves to be a reliable, low-cost method for extracting the key parameters characterizing sorption (K) and diffusion (D12) of moderate molecular weight organic species in polymer film samples. The apparatus was able to access very small sorption interval sizes, and thus linear diffusion behavior, by diluting a vapor stream high in penetrant concentration prior to passing it over the QCM sensor probes. A limitation of our current apparatus is the range of ω1 values that can be accessed for measurement of K, χ, and D. First, K(ω1) and χ(ω1) values are needed at lower ω1 than we could access. This demands the ability to prepare flow-streams with a low vapor phase penetrant activity that is known precisely. We could dilute the stream saturated with solvent (S1) with one of pure nitrogen (S2) to achieve such low penetrant activities, but we could not know the activity precisely since the current setup did not allow accurate measurement of the gas flow rates of S1 and S2. A simple improvement to the apparatus would be the installation of appropriate gas flow rate controllers, which are insensitive to backpressure. This would enable measurements of K(ω1) and χ(ω1) at much lower ω1. Measurement of D at higher ω1 was impeded by the QCM becoming excessively noisy (unstable), due to damping associated with viscous losses in the film coating and a corresponding increase in crystal resistance R. This could be remedied by utilizing pulsed QCM probes rather than steady resonators, which are less susceptible to damping effects. Using this method, we reported values of K(ω1), χ(ω1), and D12(ω1) for the homologous series of n-alkyl acetates and alkyl benzenes in monodisperse poly(methyl acrylate) at 25 °C. These penetrants were observed to partition favorably into PMA according to Henry’s law in the range 0.02 ≤ ω1 ≤ 0.15. Values of χ indicate all penetrants tested except n-butyl acetate are good solvents for PMA, phenomenologically consistent with their ability to readily dissolve the polymer. All penetrants showed strong increases of D12 with ω1, consistent with the Vrentas−Duda free volume theory.15 The values extracted of the jump unit size ratio ξ for the n-alkyl acetates and benzene are identical, within uncertainty, to those reported in literature. While the ξ for n-alkyl acetates in PMA are nearly constant, the three largest alkyl benzenes had ξ values significantly lower than that of the lighter homologue benzene. To understand the trends in ξ for both homologous series within a simple framework, we compared all but one (cumene) of our experimental values to those predicted by eq 8 from Vrentas et al.11 They developed this equation to explain nonlinear trends of ξ with penetrant volume observed for large, eccentric penetrants without the presumption of segmental motion. Rather, they asserted that a penetrant’s eccentric geometry

AUTHOR INFORMATION

Corresponding Author

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

The University of Texas at Austin, Austin, TX 78703.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully thank the National Science Foundation for funding through the IGERT Program Grant for the Study of Multiscale Phenomena in Soft Materials at Columbia University and City University of New York. The authors thank Shane Harton for providing the polymer samples used in this study, Jad Cooper for technical assistance with data acquisition, and the Pall Corporation for a grant that enabled purchase of the RQCM system.

■

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Interval Sorption of Alkyl Acetates and Benzenes in Poly(methyl

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