Vapor and Liquid Sorption in Matrimid Polyimide: Experimental

Article pubs.acs.org/IECR

Vapor and Liquid Sorption in Matrimid Polyimide: Experimental Characterization and Modeling M. Minelli, G. Cocchi, L. Ansaloni, M. Giacinti Baschetti, M.G. De Angelis, and F. Doghieri* Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali (DICMA), Alma Mater Studiorum-Università di Bologna, via Terracini 28, 40131 Bologna, Italy ABSTRACT: The sorption of organic vapors of different chemical nature, molecular weight and polarity in glassy Matrimid 5218 polyimide films is measured and modeled in the entire activity range up to saturated vapor conditions. Experimental isotherms show several different peculiarities according to the nature of the solute component, consistent with the glassy character of the polymeric matrix. These features range from high vapor solubility coefficient at infinite dilution to swelling and plasticization of the polymer matrix depending on temperature and solute affinity. The various behaviors are then discussed by means of a thermodynamic model for properties of glassy polymeric phases. Indeed, the Non-Equilibrium Lattice Fluid (NELF) model is used to interpret the Vapor−Liquid-Equilibrium below Tg, while the corresponding equilibrium Lattice Fluid model was considered to the same purpose for the case of plasticization conditions in the system. This representation is able to describe consistently all different observed features and allows for a predictive procedure to reproduce solubility isotherms over a wide activity range.

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INTRODUCTION Polyimides are synthetic polymers adopted in several industrial applications due to their high chemical, mechanical, and thermal stability.1,2 The mass transport properties of several polyimide polymers have been widely investigated for application in gas separation membranes as they can provide good selectivity and low swelling.3−7 Polyimides are also characterized by high glass transition temperatures and in many cases glassy is the only stable state for the polymer. Recently, nanoporous polyimide membranes have been commercially developed and sold for the purpose of liquid filtration as solvent-resistant membranes in processes of organic solvent nanofiltration (OSN).8,9 In interpreting gas separation through polymeric membranes, the solution-diffusion model can be confidently used to represent the mass transport across the film, according to which the gas molecules are absorbed onto the membrane at the upstream side, diffuse inside it, and then desorb from the film at the downstream side. Furthermore, in other membrane separation processes, such as pervaporation, the separation mechanism is dictated by the solubility and diffusivity parameters. Furthermore, some studies indicate that, more precisely than the pore flow model, the solution-diffusion model can describe the permeation of solvent mixtures through polyimide commercial membranes in organic solvent nanofiltration processes.10 The solubility and diffusivity parameters in polyimide membranes are thus essential, in all the above-mentioned processes, to properly design the membrane separation devices. The determination of fluid sorption in polyimides can be of interest also for other applications, such as the development of chemical sensors and protective coatings. We focus here on the experimental characterization and thermodynamic modeling for sorption of several fluids in a polyimide of interest in several membrane separation processes as well as in other applications. The polyimide here considered © 2013 American Chemical Society

is commercially sold as Matrimid 5218 and is based on 5(6)amino-1-(4′ aminophenyl)-1,3,-trimethylindane, fully imidized. The structure formula is reported in Figure 1.

Figure 1. Repeating unit of Matrimid.

Results from the fundamental characterization of this polyimide available in the literature are rather poor, just as for the case of most commercial polymers in the same class. Indeed, no volumetric measurements have been reported; only the values of density at room temperature are included in few experimental works.6,11,12 Despite this very narrow physical and morphological characterization, a large amount of sorption and permeation data is available in the literature.6,7,13−17 In gas separation, Matrimid shows moderate gas permeability and high gas selectivity, its fractional free volume being equal to about 17% according to the estimate of the occupied volume calculated with Bondi’s method18 or about 12% according to molecular simulation studies.19 At all the temperatures of interest, in the dry state, Matrimid is a glassy polymer very far from the thermodynamic equilibrium achieved above the glass transition, and as it refers to sorption characteristics, it shows all the phenomena typically Special Issue: Giulio Sarti Festschrift Received: Revised: Accepted: Published: 8936

October 12, 2012 January 23, 2013 January 24, 2013 January 24, 2013 dx.doi.org/10.1021/ie3027873 | Ind. Eng. Chem. Res. 2013, 52, 8936−8945

Industrial & Engineering Chemistry Research

Article

achieved at high concentration of penetrants due to swellinginduced glass transition of the matrix, the equilibrium Lattice Fluid (LF) equation of state is used. In the NELF model, the glassy polymer is considered as a homogeneous structure with a macroscopic density, which univocally defines its out of equilibrium degree.32−34 In this framework, indeed, the value of polymer density accounts for the different thermal, solvating, and processing history experienced by different glassy polymer samples, and it allows a representation of the corresponding different solubility values.32−34

encountered in glassy polymers: relaxation, aging, historydependent behavior.20,21 In particular, the physical and sorption properties depend on the thermal history, that is, the cooling rate experienced by the sample as well as the subsequent thermal treatments performed to remove the solvent, or specific protocols used to obtain the membrane. This is indeed a relevant issue, as Matrimid can be used in the form of dense film, nanoporous film, or hollow fiber, and consequently, several properties such as the fluid solubility, which depend on the processing conditions, may differ from case to case. Although a wide experimental investigation has been carried out on Matrimid for mass permeation properties, the process of fluid sorption has been rarely analyzed with rigorous thermodynamic models. Different gas solubility isotherms in Matrimid have been modeled with the well-known Dual Mode (DM) model, which, however, is a semiempirical tool that cannot be used for predictive purposes and does not provide a full insight into the sorption process. The DM model assumes that the gas sorption in a glassy polymer is the sum of a Henry’s law contribution, typical of the solubility in liquids or equilibrium solids, and of a Langmuir contribution, due to adsorption of penetrant molecules on the excess free volume of the glassy matrix.22 This equation contains three adjustable parameters that have to be determined from the gas solubility isotherms for every gas-polymer couple of interest. Furthermore, they depend on the experimental conditions such as temperature and history of the glassy polymer (i.e., conditioning, annealing, etc.) but also on the pressure range of investigation. It must also be mentioned that, as far as organic vapor sorption is considered, the dual mode model cannot easily account for complex shapes of the sorption isotherms, such as the sigmoidal shape frequently encountered when alcohols are absorbed into glassy polymers.22−24 Also, the dual mode model cannot represent the sorption behavior of rubbery polymers and, consequently, the penetrant-induced glass transition, which is often observed in glassy polymers at room temperature, when a large amount of penetrant is absorbed.25,26 Molecular simulations techniques were used also for the prediction of gas solubility in glassy polyimides,27,28 but their agreement with the experimental results is not always satisfactory. The characteristic relaxation times of glasses are much longer than those current computers can simulate, and the situation is complicated by the presence of penetrantinduced swelling phenomena. In a recent work by Hesse and Sadowski,29 the solubility of organic liquids in two polyimides including Matrimid is modeled using the Perturbed Chain−Statistical Associating Fluid Theory (PC−SAFT) equation of state. Although it is a rigorous approach, it cannot be used to describe the sorption behavior of the corresponding vapor phase of the penetrants, in the low and intermediate activity regions, when the matrix is still in a glassy non-equilibrium state. In this work, solubility isotherms in Matrimid were determined, in the entire activity region, from the low pressure vapor up to the liquid phase, for several organic fluids, at different temperatures. The sorption process is described by means of a generalized approach, which uses the Lattice Fluid representation of pure substances and mixtures.30,31 In particular, in the glassy region the Non-Equilibrium Lattice Fluid (NELF) model32−34 is adopted to describe the solubility isotherm, while in the equilibrium region, which can be

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MATERIALS AND SAMPLE PREPARATION Matrimid 5218, used in all sorption experiments presented in this work, was kindly provided by Huntsman Advanced Materials. The polymer has an average molecular weight of 80 000 g/mol35 and a polydispersity index of 4.5,36 whereas Tg was about 320 °C.35 The low molecular weight organic species used for the sorption experiments in this work were purchased from Aldrich. They were reagent grade purity and were used without further purification. Double distilled deionized water (conductivity lower than 0.01 μS/cm) was used for both water vapor and liquid experiments. The Matrimid films used in vapor sorption experiments were prepared by cast deposition, dissolving the polymer in a 1 wt % solution of dichloromethane, and casting it on a clean Petri dish that was then covered and placed under a hood until complete solvent evaporation has occurred. The samples used for liquid sorption experiments were prepared through a similar procedure, starting from 5 wt % solution of dichloromethane, in order to obtain thicker films able to sustain the manual manipulation required by the procedure. After the casting process, membranes prepared were annealed under vacuum at about 200 °C for 24 h to obtain homogeneous samples with a similar history and properties. It is indeed well-known that glassy materials can show different behaviors when pretreated through different protocols.11,21 The mass density of the samples was measured by using a measurement kit on a Sartorius microbalance (Sartorius Density Kit YDK01). To increase the accuracy of the measurements, specimens were weighed in air and in ndodecane,37 since previous sorption experiments of this liquid in Matrimid showed negligible absorption even after several days. As a result, an average density of 1.238 ± 0.002 g/cm3 was obtained for Matrimid films at 27 °C with a standard deviation lower than 0.16%.

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EXPERIMENTAL SECTION Sorption experiments with vapors of dichloromethane (DCM), water, methanol (MeOH), and methyl acetate (MeAc) were performed with a quartz spring sub-atmospheric microbalance (QSM), whose components are shown in the scheme in Figure 2. The quartz spring is held in a glass column whose temperature is controlled through a water jacket, while its elongation is continuously monitored in time by means of a CCD camera. The weight change of the sample is thus recorded during all phases of vapor sorption experiments, which can be run in this apparatus in a wide range of activities, at temperature ranging from 5 to 50 °C. Acetone (Ac) vapor sorption experiments were performed using a different apparatus to monitor weight uptake in time, making use of the pressure decay (PD) technique, in which the 8937

dx.doi.org/10.1021/ie3027873 | Ind. Eng. Chem. Res. 2013, 52, 8936−8945


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were immersed in a vial filled with the liquid of interest and placed in a thermostatic bath. Each sample was then weighted at regular intervals of time by removing it from the vial, quickly drying it with a towel to remove the excess of liquid, and placing it on the balance to record the weight. The measurements were repeated until the steady state mass uptake was reached.

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MODEL In the present work, the description of thermodynamic properties of Matrimid−solute systems was obtained through the use of the Lattice Fluid (LF) model introduced in 1976 by Sanchez and Lacombe30,31 and its extension to the glassy states of polymeric systems, represented by the non-equilibrium lattice fluid model (NELF) presented by Doghieri and Sarti in 1996.32−34 The theoretical bases and the mathematical development of both LF and NELF models have been reported in cited papers, and here, only the key results of the two models will be presented for the sake of clarity. The LF model describes the polymeric phase as a lattice partially filled by the different molecules, which occupy one or more lattice sites depending on their structure and molecular weight. Statistical analysis of the corresponding picture for polymer−solid binary mixtures allows to derive an expression for the Helmholtz free energy density in equilibrium states aLFEQ as a function of temperature T, polymer mass density ρpol, and solute mass fraction ωsol of the following form:46

Figure 2. Scheme of the quartz spring microbalance (QSM) apparatus for vapor sorption experiments.

pressure change of the vapor phase upon sorption is measured in a calibrated volume compartment where the sample is placed. The scheme of the PD apparatus is shown in Figure 3. The pressure decay apparatus allows for a better control of pressure over long run times, with respect to the quartz spring apparatus.

(mix)EQ * , p* , ρ* , Msol , Tsol *, aLF = a(T , ρpol , ωsol ; M pol , Tpol pol pol

* , ρ* , ks/p) psol sol

(1)

in which pure component model parameters molecular weight Mi, characteristic density ρi*, pressure pi*, temperature Ti* are evidenced, as well as binary interaction parameter ks/p. Characteristic density, ρi*, represents mass density for pure component lattice in closed packed states, while Ti* and pi* are related to energy density and energy per cell, respectively, estimated for the same conditions. By using well-known results from classical thermodynamics, an equation of state (EoS) for equilibrium pressure describing the behavior of the polymer−solute system is derived:

Figure 3. Scheme of pressure decay (PD) apparatus for vapor sorption experiments.

⎛ (mix)EQ ∂a p EQ = −a(mix)EQ (T , ωsol , ρpol ) + ρpol ⎜⎜ ⎝ ∂ρpol

Reliability of solubility data from vapor sorption experiments was tested and the maximum error was estimated to be well below 10% for both experimental techniques. The uncertainty in absorbed mass is basically related to uncertainty in the temperature control and thus in the actual vapor activity. Gravimetric and manometric techniques used in measuring the mass uptake in Matrimid for the case of vapor sorption experiments are examples of the apparatuses designed and set up in the laboratory for diffusion in polymers that was built and developed by Giulio Sarti at the University of Bologna and details on both apparatuses can be found in several papers presented by his research group.38−45 For all the penetrants considered, measurements were also performed of mass uptake from the liquid phase, with the only exception of liquid DCM, in which Matrimid is completely soluble at room temperature. Mass uptake was measured within a Sartorius analytical balance model CPA225D-OCE with a precision of 10−5 g. The latter experiments were performed with the classical blot and weight method: the different samples

⎞ ⎟ ⎟ ⎠T , ω

sol

(2)

The NELF model moves from the results of LF theory indicated above, extending the mapping of free energy density to the glassy states of the polymer−solute systems. The extension rule is obtained by using the thermodynamic tools of systems endowed with internal state variables, after assuming that the density evolution toward equilibrium is kinetically hindered in glassy states. The simple conclusion is finally drawn for Helmholtz free energy density in non-equilibrium states of the system indicated in the following relation:32−34 a(mix)NE(T , p , ωsol , ρpol ) = a(mix)EQ (T , ωsol , ρpol )

(3)

A specific expression for the chemical potential of the solute component is finally derived in NELF model to be used in phase equilibrium problems for the evaluation of vapor solubility in glassy polymer matrixes:32−34 8938



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All sorption isotherms reporting the measured solute mass ratio Ωsol as function of the solute activity a are shown in Figures 4. In Figure 4(a), data are compared for the case of

(mix)NE * , p* , ρ* , Msol , Tsol *, μsol,LF = μ(T , ρpol , ωsol ; M pol , Tpol pol pol

* , ρ* , ks/p) psol sol

(4)

It must be stressed that while in equilibrium conditions the mass density of polymeric species ρpol is related to temperature, pressure, and composition by means of eq 2, in non-equilibrium glassy states, the mass density of polymeric species is not unequivocally related to the same set of variables, and it needs to be independently estimated in corresponding NELF formulations of phase equilibrium problems. Since its introduction by the group of Prof. Sarti at the University of Bologna, more than fifteen years ago, the NELF model proved to offer a consistent physical picture of the process of gas or vapor sorption in glassy polymers. Consequently, reliable procedures for the prediction of corresponding solubility coefficient were set up allowing for a fruitful discussion of several features of thermodynamic properties of polymer−solute mixtures below the glass transition temperature.32−34,41,44,47−57 For the case of sorption from pure vapor phase, the pseudoequilibrium solute content can be calculated after the following set of equations: ⎧ μ (mix,NE)(T , ω , ρ ; M , T * , p* , ρ* , sol pol pol pol pol pol ⎪ sol ⎪ (gas) * * * ⎨ Msol,Tsol , psol , ρsol , ks/p) = μsol (T , p) ⎪ ⎪ ρpol = ρpol 0 (T )ψ ⎩

(5)

where ρpol0 is the mass density of pure glassy polymer at temperature T and ψ is the volume swelling factor for the polymer matrix in sorption conditions. The solubility coefficient in glassy polymers can be reliably predicted from eqs 5 for the case of sorption of nonswelling gases or in the limit of infinite dilution mixtures, assuming negligible volume dilation in any of these conditions (ψ = 1).47 It was also shown that a linear expression for the dilation factor as function of solute pressure allowed for a detailed description of solubility of swelling gases in traditional glassy polymers through eqs 5, in a wide pressure interval.47 ψ (p) = 1 − kswp

Figure 4. (a) Experimental data for vapor sorption isotherms in Matrimid at 35 °C for different penetrants. (b) Experimental data for sorption isotherms of dicholoromethane and water in Matrimid at different temperatures.

sorption of different penetrants in Matrimid at 35 °C, while in Figure 4(b) dichloromethane and water isotherms are reported for the case of different temperatures. It must be first noted that solutes considered in this work are absorbed into Matrimid in the following solubility order: dichloromethane > methyl acetate > acetone > methanol > water. In particular, it must be observed that, while the first three components exhibit solubility in the same order, water is absorbed in Matrimid in an amount lower by 2 orders of magnitude and finally the measured solubility of methanol lies at an intermediate level. Although similar trends appear for the solubility isotherms pertinent to different penetrants plotted in Figure 4(a), relevant differences can be put in evidence. The slope in logarithmic plots for solubility vs activity at low solute content in Figure 4(a) is lower than 1 for the three most soluble components here considered, thus resulting in a decreasing value for the solubility coefficient (Ω/a) when activity increases, in the same region. The latter is a typical feature shown by gas and vapor solubility isotherms in glassy polymers, which can be easily associated to relatively low values of partial molar volume for the penetrant component in the polymer−solute mixture at low solute activity, which ultimately results in a decrease of the free

(6)

In the simple relation reported above, the swelling coefficient ksw is used to represent the effect of solute fugacity on volume dilation of polymer matrix, and it is assumed to depend on solute nature and sorption temperature, as well as preparation protocol for the glassy polymer sample. From a different point of view, the swelling factor ψ for assigned temperature and solute pressure can be calculated after eqs 5 once dry polymer density and pseudo-equilibrium solute mass fraction ωsol are known from direct measurements.

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EXPERIMENTAL RESULTS As for the study of Matrimid−organic vapors mixtures, sorption isotherms for dichloromethane at different temperatures (10, 20, and 35 °C), as well as those for methanol, acetone, and methyl acetate at 35 °C were characterized experimentally in this work. The case of solubility isotherms for water vapor in Matrimid at different temperature (25, 35, and 45 °C) was also examined through specific sorption experiments. All sorption isotherms were analyzed in the widest activity range that the experimental set up described above allowed to consider. 8939



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equilibrium volumetric properties is not available at all. For these reasons, solubility data for gases in glassy Matrimid presented in the literature were here considered to retrieve the characteristic lattice fluid parameters (T*, p*, and ρ*) of the pure polymeric species. Predictions from NELF model for the infinite dilution solubility coefficient of several gases at different temperatures in Matrimid were first considered and compared to available experimental data,7,15,20,59 following the procedure already employed in previous works.50,52,56 From the solution of the corresponding phase equilibrium problem in eqs 5, the solubility coefficient (Ω/p) is derived in the limit of vanishing penetrant pressure, accounting for the polymer density values ρpol0 (T) estimated from data in original papers and after the assumptions of negligible dilation (ψ = 1) and zero value of binary interaction parameter ks/p, in all cases. From the best fitting procedure, results for lattice fluid parameters of Matrimid were retrieved and the corresponding value are reported in Table 1, while the correlation results are shown in parity plot presented in Figure 5.

volume of the polymeric systems induced by the vapor sorption process. The expected negative value of mixing volume for polymer− solute mixtures, in this case, is also confirmed by the exothermic nature of the corresponding mixing process, which is put in evidence by the clear, although slight, increase in solubility coefficient observed for a decrease in temperature, in the low activity range, for DCM sorption data reported in Figure 4(b). Conversely, the sorption of water in Matrimid shows a linear relation between solute mass ratio and activity, in the region of lower solute content. Data in Figure 4(b) also indicate that a negligible thermal effect is associated to the Matrimid−water mixing process, as the solubility isotherms measured at different temperature do not show appreciable differences in the Ω vs a plots. An intermediate behavior, also in terms of slope of the solubility isotherm at low activity, is finally shown for the Matrimid−MeOH system in data illustrated in Figure 4. The slope of solubility isotherms in logarithmic plots increases at higher activity for all components considered, reaching values higher than 1 for all them at the highest activities considered. For the cases of most soluble species, this is the effect of the significant volume swelling of the polymer matrix induced by the sorption process at mid activity values. This is confirmed by the reduced value of absolute enthalpy of mixing that can be estimated by the solubility data for DCM in Figure 4(b) in the intermediate activity range. An abrupt increase in slope at the highest activity is finally shown by the solubility plot for DCM, for which only data at the lowest temperature considered are available. It is easy to recognize that this is the effect of plasticization of polymer matrix, which is apparent at solute content above the threshold value of approximately 40% by weight at the temperature of 10 °C. The increase in the slope of the solubility plot evidenced in Figure 4 at higher solute content, for the case of water and methanol, on the other hand, is rather associated to the more hydrophilic character of the polymer−penetrant mixture at higher solute activity, while induced volume swelling plays a minor role in this case.23,41 While observations about the relevance of swelling and plasticization at different activity values are offered above based on reasonable assumptions for the state of polymer−solute systems across the glass transition temperature, more robust foundations to the same conclusions, as well as additional details to the picture, can be obtained through a quantitative thermodynamic analysis of the same results, which makes use of a suitable model. The following section is devoted to this analysis, for which the NELF model briefly described above has been considered.

Table 1. Lattice Fluid Characteristic Parameters for Polymer and Penetrants Matrimid CO2 CH4 O2 N2 Ar Kr Xe DCM H2O MeOH Ac MeAc

T* (K)

p* (MPa)

ρ* (g/cm3)

ref.

880 300 215 170 145 190 230 304 487 670 510 584 480

450 630 250 280 160 180 290 351 560 2400 1080 533 520

1.350 1.515 0.500 1.290 0.943 1.400 2.700 3.360 1.540 1.050 0.900 0.917 1.120

this work 32 30 52 33 52 52 52 62 52 52 30 62

To further validate the values of LF characteristic parameters retrieved, they were then used to describe, through the NELF approach, the sorption isotherms of selected gases in a relatively wide pressure range and compared again with available

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MODELING ANALYSIS Retrieving Model Parameters. The thermodynamic study of the solubility data in glassy Matrimid through NELF model requires that the pure component lattice fluid parameters for the polymeric species are first retrieved. This procedure is typically performed through the analysis of pure component equilibrium pVT data which, unfortunately, are not available for the polymer of interest in this work. Indeed, the known description of volumetric properties of pure Matrimid is limited to few temperatures, and it is also rather scattered.11,13,58,59 Moreover, the temperature range above Tg was never characterized by any author, and a description of the

Figure 5. Infinite dilution gas solubility coefficient in Matrimid: comparison between experimental data7,15,20,59 and NELF model predictions. 8940



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Chung et al., respectively, well compare with experimental dilation measured by Punsalan.60 Modeling Results. Results from NELF thermodynamic analysis of the complete set of solubility data measured for DCM sorption in Matrimid are shown in Figure 7. With

experimental data. The detailed representation of sorption isotherms in this cases were obtained using both the binary interaction parameter ks/p and the swelling coefficient ksw as adjustable parameters. The resulting values for ks/p and ksw are indicated in Table 2, while the results from the model description of experimental data are shown in Figure 6. Table 2. Polymer−Penetrant Binary Parameters for NELF Model ks/p

ksw (MPa−1)

CO2

−0.03

0.0165 (23 °C) 0.0174 (35 °C)

CH4 O2 N2 DCM

0.068 0.065 0.11 0.016

H2O MeOH Ac MeAc

−0.093 −0.035 0.0035 0.003

penetrant

8.9 (10 °C) 6.5 (20 °C) 3.2 (35 °C) cubic 9.0 9.0

Figure 7. Dichloromethane sorption in Matrimid at 10, 20, and 35 °C: experimental data together with LF-EoS and NELF model.

reference to the values for pure component lattice fluid parameters independently retrieved and indicated in Table 1, the binary interaction parameter ks/p for Matrimid−DCM pair is first determined through a best fit procedure for LF EoS predictions toward the solubility data measured at the highest activity examined, for temperature equal to 10 °C. The value retrieved this way for the binary parameter is indicated in Table 2. The good correlation obtained by means of the equilibrium model for this part of the solubility isotherm (see dashed line in Figure 7) confirms the plasticized character of Matrimid−DCM mixtures at 10 °C for solute content higher than 40 wt %. The experimental solubility data measured for lower activity values, at the same temperature, exceed corresponding predictions from the LF EoS, while they are correctly represented by the NE version of the model (NELF, see solid line in Figure 7) when the same pure component and binary model parameters are used for the calculations. For the NELF formulation of the phase equilibrium problem represented in eqs 4, reference was made to dry polymer density value ρpol0 = 1.239 g/cm3 as calculated from measured dry polymer density at 27 °C and estimated cubic dilation coefficient 63 × 10−3 K−1 from Tin et al.58 A linear form for swelling factor ψ was also assumed, for which swelling coefficient ksw was retrieved through the analysis of experimental data. A rather satisfactory representation of solubility isotherm at 10 °C below the solute content of 40 wt % was indeed obtained by adjusting ksw to 8.9 MPa−1, as indicated in Table 2. The comparison between equilibrium and non-equilibrium isotherm described by LF EoS and NELF model, respectively, shows the threshold point on the isotherm, from glassy to rubbery conditions, for which the pressure of 250 mbar at 10 °C for the system can be recognized. Similar analyses were performed for the two solubility isotherms of DCM in Matrimid at higher temperature. In these cases, counting on the same value of binary interaction parameter ks/p estimated for the lower temperature, the solubility predictions for the equilibrium phase were obtained straightforwardly, and the corresponding results are reported in dashed lines in Figure 7. Accurate representations of non-

Figure 6. Gas solubility in Matrimid: experimental data from Moore and Koros15 and from Chung et al.59 together with NELF model predictions.

It can be concluded that the model is able to describe accurately the sorption isotherms of different gases (CO2, CH4, N2). The two sets of data, reporting gas solubility at 35 °C from Moore and Koros15 and at 23 °C from Chung et al.,59 respectively, were obtained from PI samples of different densities, as reported in the original papers,15,59 suggesting that Matrimid specimens were subjected to different preparation protocols, ultimately resulting in higher or lower free volume of the glassy polymer. This effect is nicely predicted by the model as a result of the different density value ρpol0 input to the phase equilibrium problem, while the same binary interaction parameter ks/p has been considered for the same polymer−penetrant couple, as reported in Table 2. Furthermore, in the pressure range inspected, up to 30 bar, CO2 is able to produce a significant dilation of the polymer matrix and a nonzero swelling coefficient ksw has been taken in account. The values of 0.017 and 0.018 MPa−1 for ksw, which allowed for the representation of the isotherms from Moore and Koros and 8941



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equilibrium solubility data at both 20 and 35 °C were finally obtained through the same procedure described above, after accounting for the corresponding estimates of dry polymer density and adjusting the swelling coefficients to values indicated in Table 2. It must be observed that all solubility data collected at temperature 20 and 35 °C lie above values predicted by the equilibrium equation and that all explored conditions actually refer to glassy state for the polymer solute mixtures. In Figures 8 and 9, measured solubility data in Matrimid at 35 °C, for MeAc and Ac respectively, are compared with values

Figure 8. Methyl acetate sorption in Matrimid at 35 °C: experimental data together with LF-EoS and NELF model.

established formulation of phase equilibrium problem in melt phases. The predictions from the same isotherms in the entire activity range are shown in dashed lines for MeAc and Ac in Figures 8 and 9, respectively. The corresponding prediction for non-equilbrium solubility isotherms from NELF model was then obtained in both cases, accounting for the already estimated mass density of dry glassy Matrimid at 35 °C and for a linear form of swelling factor for which the coefficient was conveniently adjusted. Data from the resulting model correlations are shown in dark solid lines in Figures 8 and 9, respectively, while corresponding values retrieved for the swelling coefficient are indicated in Table 2. To appreciate the role of the swelling factor ψ in phase equilibrium problem for glassy state conditions in eq 4, data were also reported in Figures 8 and 9, as they are predicted by the model when volume swelling is neglected, that is for the case in which ψ is set to 1. The resulting values are reported in light solid lines in the same figures, and their comparison with predictions from complete formulation of the model allow understanding that the effect of swelling factor is negligible as long as the estimation of solute fugacity in polymeric mixture is of interest up to about 2% by weight. This is a peculiar feature of sorption in glassy polymers, in which the excess free volume in dry matrixes dominates the thermodynamic of mixing process at very low solute concentrations. The contribution by the swelling factor to the overall mass uptake in glassy polymers is relevant at intermediate vapor pressures for the case of methyl acetate and acetone in Matrimid. Figure 10 illustrates the comparison between experimental data and model predictions for the case of sorption of methanol

Figure 9. Acetone sorption in Matrimid at 35 °C: experimental data together with LF-EoS and NELF model.

Figure 10. Methanol sorption in Matrimid at 35 °C: experimental data together with LF-EoS and NELF model.

predicted from LF EoS and NELF models. Together with data from vapor sorption experiments, values are reported for final solute content measured in corresponding sorption runs from pure liquid phases (see filled symbols in Figures 8 and 9). As for both MeAc and Ac, the mass uptake in Matrimid from liquid phase at 35 °C was higher than the threshold value measured for glass transition in DCM at 10 °C, the thermodynamic analysis moved from the assumption that the polymer matrix was plasticized at unit solute activity. The binary interaction parameter for LF EoS was thus determined for both Matrimid−MeAc and Matrimid−Ac pairs to the value that allows for the correct prediction of the corresponding liquid solubility measured at 35 °C, through the well

in glassy Matrimid. The mass uptake measured for sorption from liquid phase is compared with that obtained from vapor sorption experiments at various pressures. As long as it refers to model predictions, the relatively low amount of sorbed mass, even at saturation conditions, suggests the non-equilibrium approach should be used over the entire pressure range. In view of the negligible effect of the swelling factor on the solubility coefficient at low solute content, adjustment of binary interaction parameter ks/p for the polymer−penetrant pair is here obtained aiming at the correct model representation of experimental solubility data in the low pressure range. The prediction of vapor solubility coefficient at higher pressures can only be represented through the proper account of swelling 8942



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behavior only for the case of water concentration in Matrimid lower than 1 wt %. Deviations of experimental observations from model prediction at higher pressure at 25 °C do not seem to be related to unpredicted volume dilation, in view of the still very low solute content but rather to the limit of the equilibrium thermodynamic model used, which does not address all possible interactions between the water molecules and functional groups in Matrimid. Indeed, the same limit does not allow the Lattice Fluid EoS model to correctly predict the volumetric properties of pure water in a wide range of temperatures. The above conclusion is also supported by the observation that, unlike the case of methanol sorption examined above, the nonlinear behavior of solubility coefficient is not evidenced in prediction from equilibrium model (see dashed lines in Figure 11).

factor, and in this case, a correct representation of solute content in this range requires the use of a more complex function of ψ: ′ p3 ψ (p) = 1 − kswp − ksw −1

(7) −3

where ksw = 3.4 MPa and ksw ′ = 5300 MPa . The eq 7means to interpret the actual volume swelling behavior of Matrimid− MeOH systems, and its reliability is supported by the good representation the LF EoS model is able to give of pVT properties of pure methanol and by the similarity between the variation induced by solute pressure on solubility as predicted by the equilibrium model (see dashed lines in Figure 10) and that measured in sorption experiments. The just-mentioned similarity is actually interpreted in more fundamental terms by the model development recently presented by Minelli and Doghieri.61 The last case considered in this thermodynamic analysis is that of water vapor sorption in glassy Matrimid. Experimental data from vapor sorption at 25, 35, and 45 °C are shown in Figure 11, together with the predictions from LF EoS and

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CONCLUSIONS The experimental characterization of Matrimid−vapor systems was performed, for the case of different low molecular weight species, at solute concentrations ranging from infinite dilution up to more than 50% by weight, testing different temperatures around room conditions. The solute fugacity in the polymeric solution was measured as a function of temperature and concentration, collecting the results from vapor and liquid sorption experiments. The cases of several organic vapors as well as that of water vapor in Matrimid were considered, emphasizing the difference in absolute value of fugacity and in its sensitivity to temperature and concentration. Corresponding values for solubility, solubility coefficient, and mixing enthalpy were first qualitatively identified for the different cases, as they result from the experimental raw data. A brief discussion of these data was also attempted, with reference to key polymeric properties such as excess free volume in dry conditions, volume dilation, and plasticization induced by vapor sorption, as well as affinity to vapor species. A quantitative thermodynamic analysis was then performed through the interpretation of fugacity results in terms of the original lattice fluid model for polymer−solute melt mixtures proposed by Sanchez and Lacombe, and of its extension to glassy states, as developed by our research group. The results of this analysis allowed a recognition of the different roles of volume dilation and plasticization in distinct fugacity ranges, for penetrants of different affinity to the polymer matrix. The work reported in this paper directly addressed the problem of retrieving reliable parameters of equilibrium thermodynamic models for pure polyimide species, for which equilibrium volumetric properties are not available. The resulting thermodynamic picture for polymer−penetrant mixtures offers an interesting interpretation of their properties, useful in predicting the behavior of Matrimid−solute systems different from those directly considered in this case, or at least to identify proper tools for their analysis.

Figure 11. Water sorption in Matrimid at 25, 35, and 45 °C: experimental data together with LF-EoS and NELF model.

NELF models. The experimental datum from liquid water sorption measurement in Matrimid at 25 °C is also included in the same figure. Prediction from NELF model were here obtained first accounting for variation of polymer dry density ρpol0 with temperature as measured by glassy cubic dilation coefficient.58 Accounting for the actual polymer mass density, also in this case, the NELF approach is able to correctly estimate the higher solubility value for water vapor in Matrimid measured at low pressures, with respect to pure equilibrium EoS predictions. Moreover, comparison in Figure 11 reveals that the use of a common value for the binary interaction parameter ks/p at all temperatures considered allows for a correct representation of the effect of temperature on infinite dilution solubility coefficient, thus correctly predicting the negligible heat of mixing in the system already underlined from the results of experimental data. It must be stressed also that the correlation results have been obtained without accounting for the volume dilation induced by water sorption (ψ = 1), consistent with what observed in previous cases for the low concentration range. Finally, it should be noticed that the linear increase of water content in Matrimid at intermediate vapor pressure, as described by the model, is representative of experimental

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +39 (0) 51 2090270. Fax.: +39 (0) 51 2090247. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 8943



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ACKNOWLEDGMENTS This paper is meant to be a tribute to the work by Giulio Sarti from the people of his research group, in recognition of all his efforts and results in building up the experimental and modeling expertise of our laboratory at the University of Bologna.

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Vapor and Liquid Sorption in Matrimid Polyimide: Experimental

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