A Predictive Model for Vapor Solubility and Volume ... - ACS Publications

Article pubs.acs.org/IECR

A Predictive Model for Vapor Solubility and Volume Dilation in Glassy Polymers Matteo Minelli and Ferruccio Doghieri* Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali Alma Mater Studiorum, Università di Bologna, via Terracini, 28 I-40131 Bologna, Italy ABSTRACT: The nonequilibrium lattice fluid (NELF) model is here implemented with a simplified relation for bulk rheological behavior of glassy polymers, to obtain a predictive model for pseudoequilibrium solute content and volume dilation induced by swelling agents. The rheological model discussed in this work is ultimately used to derive a relation for the pseudoequilibrium volume of the polymeric system as a function of temperature and solute fugacity. The model makes use of two nonequilibrium parameters which, in turn, can be determined through the analysis of nonequilibrium pVT properties of the polymeric species. The predictive ability of the new model for solubility and volume swelling was tested by means of the comparison of model predictions and correlations with experimental data, available in the literature, for different polymer/penetrant pairs. The results obtained from the comparison allow one to conclude that the model is not just useful to describe the swelling effect of penetrants but also to recognize conditions at which plasticization by solute species induces a glass to rubber transition in the polymeric system.

■

INTRODUCTION The swelling induced by exposure to gas, vapors, or low molecular weight liquids is a fundamental characteristic of polymeric materials, which plays a crucial role for their performances in a number of applications, from gas/oil piping1 to drug delivery2 or paint formulation.3 In more specific terms, the ability to predict solubility in glassy or rubbery polymers is a key factor, for example, for the design of membrane for fluid mixture separation4,5 for the development of barrier materials for packaging6,7 or for sensor applications.8,9 Much effort has been applied in last decades to build up models for the representation of thermodynamic properties in polymeric mixtures above the glass transition temperature, and many reliable equations of state (EoS) are now available for the solution of phase equilibrium problems in this specific field.10−19 On the other hand, for the representation of solubility isotherms of gases and vapors in glassy polymers, we still rely on empirical tools for which predictive abilities have severe limitations. Among the latter, the dual mode model20−23 is the most popular one, because of its simplicity and flexibility, which is paid for with a lack of physical meaning of its parameters. Indeed, notwithstanding efforts performed, reliable procedures are not available to retrieve values of dual mode parameters from pure component properties or from data independent from sorption experiments in the glassy state. Many of the difficulties in description of glassy polymeric systems lie in the out-of-equilibrium nature of their state, for which classical thermodynamic tools are not directly applicable. Several attempts have been made in recent years to address the representation of out-of-equilibrium conditions through empirical approaches, such as the so-called continuous site,24 gas− polymer matrix,25 or concentration−temperature superposition models.26 Interesting results in providing an accurate representation of sorption isotherms in glassy polymers are those due to Lipscomb27 and Kirchheim,28 who made use of © 2012 American Chemical Society

solid thermodynamic tools to describe the deformation induced by the sorption of solute molecules. In this work, a model is in focus that was first proposed fifteen years ago by Doghieri and Sarti,29 which also accounts on an empirical approach for the description of out-ofequilibrium condition in glassy polymer/solute mixture. Indeed, the model offers a general procedure, known as nonequilibrium theory for glassy polymer (NET-GP), to extend the results attainable from arbitrary equilibrium EoS for the polymer/ solute pair to the case of glassy pseudoequilibrium conditions. In the present work a simple rheological assumption has been used to implement the NET-GP approach with the necessary relation for system volume as a function of solute fugacity, which makes the model suitable for the description of solubility isotherm of swelling agents in glassy polymer through a pure predictive procedure. The description of pseudoequilibrium conditions reached for a polymer mixture below the glass transition temperature when exposed to assigned solute fugacity was specifically in focus in the modeling efforts performed in this work. The rheological model proposed allows one to recognize the pseudoequilibrium conditions as those in which only short-term relaxation modes reach equilibrium with respect to prevailing conditions for temperature and solute chemical potential. As the new rheological assumption is here combined with the original NET-GP model to obtain a new tool for the description of solubility isotherm in glassy polymers, the introduction of the specific rheological considerations is preceded, in next section, by a brief discussion of essential statements and relations for the thermodynamic model which the resulting tool is based on. Received: Revised: Accepted: Published: 16505

August 6, 2012 October 19, 2012 October 23, 2012 October 23, 2012 dx.doi.org/10.1021/ie3021076 | Ind. Eng. Chem. Res. 2012, 51, 16505−16516

Industrial & Engineering Chemistry Research

■

Article

MODEL DESCRIPTION NET-GP Approach. The NET-GP approach was originally proposed29 to represent thermodynamic properties of glassy polymers for the case of homogeneous, amorphous, and isotropic materials. The details of the model have already been described in previous publications29−31 and only main results are recalled hereafter. On the basis of the use of polymer mass density (ρpol) as an order parameter to measure the departure from equilibrium conditions in a glassy polymeric system and assuming it behaves as an internal state variable, simple extensions for free energy density and solute chemical potential in nonequilibrium states are finally obtained from proper maps of the corresponding property in thermodynamic equilibrium conditions, as indicated in eq 1:

expression of Helmholtz free energy density is derived in terms of several parameters, which together with pure component molar mass (Mi) and characteristic temperature (Ti*), pressure (pi*), and density (ρi*) for both species, include one binary energy interaction parameter (ks/p) as indicated in the equation below: EQ aLF = a(T , ω , ρpol ; M p , Tp*, pp* , ρp* , Ms , Ts*, ps* , ρs* , ks/p)

(3)

in which subscripts s and p label solute and polymer pure parameters, respectively. It should be finally considered that, while the details of molecular weight distribution of polymer species are often unknown, the sensitivity of the Helmholtz free energy of the binary system to the molecular weight of polymeric species can be neglected in most cases. Indeed, in all thermodynamic calculations performed in this work, the effect of polymer molecular weight on thermodynamic properties of polymer/solute mixtures is not addressed. As it refers to necessary information about the volume per polymer mass in nonequilibrium conditions, and specifically to its dependence on solute fugacity, several approaches were used in past analysis that rely on independent experimental measurement for volume swelling or account for adjustable parameters pertinent to a volume swelling coefficient. The latter procedure will be briefly addressed in the Results and Discussion, while in the following subsection a new approach is proposed that relates the volume behavior of glassy polymers in sorption experiments to their rheological properties, and it can be used to predict volume dilation in glassy polymer samples as induced by sorption of swelling agents, with no need of adjustable parameters. Rheological Assumption. With respect to the complexity of the swelling process induced by solute sorption into glassy polymers, the approach here proposed to address the description of bulk rheology in glassy polymers is highly simplified and empirical in character. The present effort is aimed to develop a simple model that could be used to interpret the volume swelling behavior of a glassy polymer/ solute mixture as induced by an increase in solute fugacity or by a decrease in mean normal mechanical stress. To this specific purpose, a simple viscoelastic model could be considered, which proved to be effective in representing the overall kinetics of the swelling process observed for long time experiments of pentane sorption in glassy PS microspheres.47 However, constant fugacity experiments lasting for years at conditions just below the glass transition temperature Tg, such as those by Enscore et al.,48 which were addressed in the mentioned paper, are rare and refer to rather peculiar conditions. In fact, solubility isotherms used to characterize the thermodynamic properties of polymer/solute mixtures below the Tg are built after a series of subsequent ordinary sorption experiments and refer to overall durations much shorter than the higher relaxation time for the polymeric system. We will here refer to the condition reached at the end of constant fugacity ordinary experiments in glassy polymers as “pseudoequilibrium” condition, to distinguish this from that obtained in experiments whose duration exceeds the higher relaxation time in the system, for which true thermodynamic conditions are reached. In this work, we are interested in the simplest possible rheological tool addressing the polymer behavior in a sorption experiment below the glass transition temperature, up to the attainment of pseudoequilibrium conditions. A corresponding

a(mix)NE(T , p , ω , ρpol ) = a(mix)EQ (T , ω , ρpol ) μsNE (T , p , ω , ρpol ) = μsEQ (T , ω , ρpol )

(1)

It is useful to underline here that, on the other hand, the volumetric equation of state, which relates pressure, temperature, and density in equilibrium conditions ⎛ (mix)EQ ∂a p = −a(mix)EQ + ρpol ⎜⎜ ⎝ ∂ρpol

⎞ ⎟ ⎟ ⎠T , ω

(2)

does not apply to nonequilibrium states. Indeed, in order to apply the model to the description of nonequilibrium states, specific additional information about polymer density is needed. As it refers to representation of equilibrium properties, different EoS have been already successfully applied to the calculation of gas and vapor solubility in nonequilibrium glassy polymers. In its first application, the model was used to extend to the nonequilibrium conditions, the expression for free energy obtained by the Sanchez−Lacombe lattice fluid (LF) model,13,32−34 leading to the so-called NELF relations for solute solubility in glassy polymers.29−31,35 The same approach was later applied to different versions of both SAFT and PHSCT EoS,14−19 obtaining the corresponding nonequilibrium models NE-SAFT36,37 and NE-PHSCT36,38 for the description of the polymeric mixture in glassy states. Results from the application of the NET-GP model to the calculation of gas or vapor solubility in several glassy polymers have been reported in a number of papers, which proved the reliability of this approach.39−46 For those cases in which assumptions for the isotherm of polymer mass per system volume as function of solute fugacity were necessary, these were assumed as linear functions and the corresponding coefficients were treated as adjustable parameters.35 The Sanchez−Lacombe EoS and corresponding NELF model have been considered in this work to evaluate the equilibrium and nonequilibrium properties of all systems taken in account. Sanchez−Lacombe is a lattice fluid theory that allows for the description of a pure substance by three characteristic parameters: the close packed characteristic density, ρi*; the characteristic temperature, Ti*; and the characteristic pressure pi*. The specific details, the mixing rules, and the complete derivation to evaluate the chemical potential of a chemical species for this theory can be found in the original papers.13,32−34 It is useful here just as a reminder that, for the case of binary solute polymer mixtures, the LF 16506

dx.doi.org/10.1021/ie3021076 | Ind. Eng. Chem. Res. 2012, 51, 16505−16516


Article

where κEQ(T, p) represents the compressibility of pure polymer in true thermodynamic equilibrium conditions and the superscript 0 labels pure polymer specific volume. When eq 6 is applied to glass transition pressure pg(T), for which equilibrium and pseudoequilibrium specific volumes have the same value, a useful relation can be derived for model parameter χ, which is reported below:

simplification of the spectrum of relaxation times is then here considered, for which the distribution is assumed to be bimodal with a constant ratio between weighting factors of “soft” and “hard” elements, respectively characterized by short-term and long-term relaxation modes. It is also assumed that only contributions from short-term relaxation modes are relevant in description of ordinary sorption experiments below the glass transition temperature. Just as evidenced in a previous paper,49 the model represents the total volume per unit polymer mass (V̂ pol) as the sum of two contributions respectively labeled “S”, for short-term modes, and “L”, for long-term modes, as indicated below S

̂ = χV̂ + (1 − χ )V̂ Vpol

̂S

χ (T ) =

L

(4)

⎡ ⎛ dV ̂ 0 ⎞ ⎤ ⎢ 10 ⎜ pol ⎟ ⎥ ̂ ⎝ dp ⎠ ⎥ ⎢⎣ Vpol T ⎦ p+ g

⎡ ⎛ ̂ EQ0 ⎞ ⎤ ⎢ 1 ⎜ dVpol ⎟⎟ ⎥ 0⎜ ⎢ Vpol ̂ EQ ⎝ dp ⎠ ⎥ ⎣ T ⎦ p−

EQ0

0

Vĝ (T ) =

(5)

1−χ

(8)

It is worthwhile to specify that, while dependence on temperature or on preparation protocols for model parameters is not directly addressed in what follows, their relevance for the evaluation of parameter V̂ g, to be used for the specific sample and sorption run of interest, cannot be neglected. In pertinent calculation from eq 8 it is thus necessary to refer to the nonequilibrium value for V̂ 0pol(T, p → 0) at the specific conditions of interest, when available.

Equation 5 is the key equation of the model here proposed, which allows for a direct representation of volume swelling isotherms in polymer/solute glassy mixtures from dry conditions up to the glass transition point, after results for V̂ EQ pol (T, p, f) from a proper equilibrium EoS for the same system. Equation 5 is also the basis for the calculation of corresponding sorption isotherms, which, starting from the knowledge of pseudoequilibrium volume V̂ pol, can be pursued through the NET-GP model. Weighting Factor χ. Nonequilibrium model parameters χ and V̂ g need to be first retrieved in order to apply the procedure for swelling and sorption isotherms sketched above. In what follows, simple relations are proposed to estimate both model parameters from pure polymer nonequilibrium data. Assuming eq 5 correctly represents polymer swelling behavior for pseudoequilibrium conditions at constant temperature, the following relation holds for the apparent compressibility of dry glassy polymer κg:

■

RESULTS AND DISCUSSION Evaluation of Model Parameter χ. With the aim toward the application of the model for the calculations of interest in this work, the temperature dependence of model parameter χ is neglected, and pure polymer experimental pVT data obtained from isothermal measurements have been considered for its evaluation, as available in the technical literature.50 Indeed, when the temperature effect is ignored, all terms in eq 7 can be evaluated in the case of atmospheric pressure conditions, ultimately resulting in the following relation:

⎛ ∂V̂ 0 ⎞ ⎜ pol ⎟

1 ̂ 0 ⎜⎝ ∂p ⎟⎠ Vpol T

χ=

⎛ ̂ EQ0 ⎞ χ ⎜ ∂Vpol ⎟ =− 0 ⎜ ̂ ⎜ ∂p ⎟⎟ Vpol ⎝ ⎠T =−

(7)

̂ (T , p → 0) − χVpol ̂ (T , p → 0) Vpol

EQ

0 ̂ EQ Vpol χ 0 κ EQ (T , ̂ Vpol

κ EQ (T , pg )

When its dependence on solute content is neglected, model parameter χ can be evaluated through eq 7 above, from the analysis of pure polymer pVT data around the glass transition point. Once a value for parameter χ has been retrieved for a given polymer, model parameter V̂ g for the specific sample at the temperature of the sorption run can be finally estimated again from the use of eq 5. Indeed, V̂ g is finally related to the experimental value of low pressure specific volume for dry glassy polymer V̂ 0pol at the temperature T, according to the following expression:

where V and V are specific volume for soft and hard elements, respectively, while χ and 1 − χ are the corresponding weighting factors. It is then assumed that, at pseudoequilibrium conditions as defined above, the specific volume for short-term relaxation time element (V̂ S) corresponds to the equilibrium value of specific polymer volume at the given temperature T, pressure p, and solute fugacity f. On the other hand, the specific volume (V̂ L) for long-term relaxation element is here assumed to be constant during ordinary sorption experiments below Tg. In what follows, the latter parameter is meant to be dependent on temperature and the specific history for the polymer/solute system, and it is indicated as V̂ g

κ g ( T , p) = −

κg(T , pg )

g

̂L

̂ = χVpol ̂ (T , p , f ) + (1 − χ )Vĝ Vpol

=

⎡ ⎛ dV ̂ 0 ⎞ ⎤ ⎢ 10 ⎜ pol ⎟ ⎥ ̂ ⎝ dp ⎠ ⎥ ⎢⎣ Vpol T ⎦T − g

⎡ ⎛ ̂ EQ0 ⎞ ⎤ ⎢ 1 ⎜ dVpol ⎟⎟ ⎥ 0 ⎜ ⎢ Vpol ̂ EQ ⎝ dp ⎠ ⎥ ⎣ T ⎦T + g

=

κg(Tg − , p → 0) κ EQ (Tg + , p → 0) (9)

As a first step, a convenient empirical relationship able to describe the volumetric behavior of the polymer has to be selected to calculate the polymer compressibility, either above or below the glass transition temperature. The Tait equation has been here used for this purpose, and when not directly

p) (6) 16507



Article

available, pertinent parameters were retrieved from the best fit of experimental pVT data. This procedure is described in detail in the Appendix. The model parameter χ has been here retrieved for several different polymeric species for which reliable data for sorption isotherms of swelling agents were identified in the literature and results are reported in the Appendix. Among these species, there are poly(carbonate) (PC),51 poly(ethyl methacrylate) (PEMA),50 poly(sulfone) (PSU),50 and poly(styrene) (PS).50 The weighting factors χ that have been obtained for this set of polymers change in a rather wide interval from 0.56 of PSU to 0.83 of the PEMA, as discussed in the Appendix. Thermodynamic Properties. Equilibrium properties of polymer/solute mixtures are here calculated with reference to the Sanchez−Lacombe (SL) EoS,13,32−34 for which pure component characteristic parameters of low molecular weight species are evaluated from vapor pressure and saturated liquid densities data, whereas melt state pressure−volume−temperature data are employed to retrieve those of polymeric species. The specific values of lattice fluid parameter T*, p*, and ρ* used in this work, which are listed in Table 1, were mainly taken from the literature, although in few cases a specific analysis of pure component pVT data were performed in this work to retrieve them. Table 1. Characteristic Parameters of Pure Polymers and Penetrants Species for the SL EoS and for NELF Calculations

PC PEMA PSU PS CO2 m-xylene p-xylene

T* (°C)

p* (MPa)

ρ* (g/cm3)

source

755 602 820 750 300 561 560

534 567.5 560 360 630 381 384

1.275 1.221 1.318 1.099 1.515 0.949 0.952

27 this work, data from ref 50 this work, data from ref 50 31 27 35 35

Figure 1. Solubility and volume swelling isotherm of CO2 in PC at 35 °C: experimental data from refs 52, 53 and predictions given by SL EoS and NELF model.

In this work, the equilibrium SL EoS was then first used to describe the sorptive behavior at higher concentrations. To this aim, at any temperature T and solute pressure p of interest, through a well-established procedure, the phase equilibrium and volume equation of state are coupled to obtain a set of two equations that allow one to simultaneously derive polymer mass per system volume ρpol and the solute mass fraction ω:

Description of Sorption and Volume Swelling Isotherms. The procedure through which the model provides the description of solubility and volume swelling isotherm is best presented by discussing in detail the first example of correlation of experimental data considered in this work. This is the case of CO2 sorption in PC at 35 °C and, specifically, the pertinent experimental data reported in two different works, by Wissinger and Paulaitis52 and Fleming and Koros.53 The sorption isotherms reported in the mentioned papers are illustrated in Figure 1, together with corresponding data for volume per polymer mass. It must be specified that just data for relative volume swelling of polymer samples as a function of CO2 pressure were reported in either work by Wissinger and Paulaitis52 or by Fleming and Koros53 and that values for volume per polymer mass indicated in Figure 1b were here estimated in all cases on the basis of a mass density value for dry polymer PC equal to 1.194 g/cm2, corresponding to that measured by Zoller51 for the same polycarbonate. In both experimental works,52,53 a wide pressure range had been investigated, up to the supercritical region for gaseous CO2, at which rather high values of penetrant content were reached in the polymeric phase, which are responsible for the plasticization of polymer matrix to rubbery conditions.

⎧ μ(mix)EQ (T , ω , ρ ) = μ(gas)EQ (T , p) pol ⎪ ⎪ ⎛ (mix)EQ ⎞ ⎨ ∂a ⎟ ⎪ p = − a(mix)EQ + ρpol ⎜ ⎜ ∂ρ ⎟ ⎪ ⎝ ⎠T , ω pol ⎩

(10)

Preliminarily to the solution of the above set of equations, the solute chemical potential μ(gas)EQ in pure gaseous gas phase at system temperature T and pressure p need to be estimated. While several different procedures could be used toward this specific purpose, the same equilibrium SL EoS was used in all calculations performed in this work for the evaluation of this term of the problem. For the case in which pure component lattice fluid parameters of both polymeric and solute species are known, different solution of the set of eqs 10 may be obtained according to the value chosen of polymer/solute energy interaction parameter ks/p. 16508



Article

fraction ωEQ, which is the solubility that the penetrant would have in the polymer if the latter was at equilibrium. This value may significantly differ to the actual nonequilibrium solubility in the glassy polymer, and it is an output of the EoS, which describes the volumetric behavior of the “soft” elements. The above calculation scheme results in a pure predictive procedure once pure component and binary lattice fluid parameters are known, together with parameters of rheological model χ and V̂ g. In turn, as mentioned in previous section, the weight coefficient χ can be estimated after values of compressibility factors κEQ and κg above and below the glass transition (see eq 9), respectively, while value of specific volume of long-term relaxation element V̂ g can be retrieved from that of dry polymer sample V̂ 0pol (see eq 8). It is not trivial to observe that model parameters χ and V̂ g should be regarded as nonequilibrium properties, and they are expected to be characterized by different values for different glassy polymer samples, according to their specific preparation, just as is the case of properties κg and V̂ 0pol that can be used for their estimation. In the case here considered, as specific values of κg and V̂ 0pol for sample considered by Fleming or Wissinger were not available, the latter were evaluated from pVT data measured by Zoller51 for bisphenol-A polycarbonate. Corresponding values calculated for dry polymer density at 35 °C and weight parameter χ for PC are listed in Table A1 of the Appendix. Results of solute content and volume swelling isotherm for PC/CO2 mixtures at 35 °C from NELF calculations are finally reported in solid red lines in Figure 1, up to the penetrant gas pressure value already identified as the limit for the glass− rubber transition in this case. Agreement between experimental results and NELF calculation, performed in this case through the straight predictive procedure described above, is remarkable indeed, with reference to both solubility and volume swelling. A comparison between the approach introduced in this work to the description of volume swelling behavior and that used in previous applications of NELF model is in order and it will be here discussed in general terms, first. Significant results for prediction of low pressure gas solubility in glassy polymers were first obtained through the assumption of negligible volume swelling. Indeed, it was shown that the infinite dilution solubility coefficient in glassy polymer can be predicted in a rather accurate way by assuming the solute partial molar volume is vanishing in the same limit.44,54 Within the assumption of constant polymer mass per unit volume, appreciable predictions of pseudoequilibrium solute content in glassy polymers were obtained also at high pressures, for the case of sorption of light gases.30,31 In past applications of NELF model, when dealing with sorption of swelling agents in glassy polymers, it was assumed that polymer mass per unit volume in pseudoequilibrium conditions could be interpreted as a linear function of solute pressure psol according to the following expression35

In Figure 1, the results of calculation are shown as solid black lines as obtained from the use of ks/p as best fit parameter for the representation of solute content in the high solute pressure range, while in Table 2 the value of binary parameter retrieved Table 2. Characteristic Binary Interaction Parameters of Polymers/Penetrants Pairs for the SL EoS and for NELF Calculations polymer

penetrant

T (°C)

ks/p

source

PC PEMA PS PS PSU

CO2 CO2 m-xylene p-xylene CO2

35 15 25 25 35

0.022 0.020 0.049 0.0515 0.013

52 55 58 58 63

from the procedure is reported. As evident from data in Figure 1, the use of SL EoS allowed for a correct description of the CO2 fugacity in PC/CO2 mixture for penetrant concentration values higher than 8% by weight (CO2 pressure higher than 5 MPa). It is worthwhile to observe that, for the same model parameters, also the volume swelling behavior is correctly represented by the SL EoS, above the same critical value for solute fugacity. On the other hand, use of an equilibrium equation of state would not be suitable for the representation of thermodynamic properties of PC/CO2 mixture at low CO2 content, where true thermodynamic equilibrium conditions are not reached within the duration of the sorption experiment. With a somehow simplified notation, in what follows, we will refer to the high solute content range as the “rubbery” region of the experimental isotherm and to the lower solute content range as the “glassy” region. Within the same notation, the threshold value of solute pressure (content) dividing the two regions can be indicated as the “glass−rubber transition” value. It is necessary to specify that the latter term is here used with strict reference to the conditions described above for the results of sorption experiments. After the above analysis of binary equilibrium data, the NELF model was used in this work to describe thermodynamic properties under pseudoequilibrium conditions. The corresponding procedure is, indeed, straightforward when the rheological model described in previous section is embodied. Results for pseudoequilibrium solute mass fraction ω and polymer mass per unit volume ρpol, at the given value of temperature T and solute pressure p, are obtained from the solution of the set of equation reported below ⎧ μ(mix)NE (T , p , ω , ρ ) = μ(gas)(T , p) (a) pol ⎪ ⎪ ⎨ 1 ̂ EQ (T , p , f |(gas) ) + (1 − χ )Vĝ (b) = χVpol ⎪ T ,p ⎪ ρpol ⎩

(11)

where V̂ EQ pol indicates the value of volume per polymer mass predicted by EoS at temperature T, pressure p, and solute chemical potential equal to the prevailing value in the gas phase. It should be underlined that, at any temperature and pressure conditions, the term V̂ EQ pol is first calculated after the solution of a phase equilibrium problem equal to that represented in eq 10, the latter value is then introduced in the eq 11b to calculate the pseudoequilibrium value ρpol, and finally eq 11a is solved to derive the pseudoequilibrium solute content ω. Note that eq 11b implicitly encloses the solute mass

ρpol =

1 (1 − kswpsol ) ̂0 Vpol

(12)

where ksw is a coefficient interpreting the volume swelling in the glassy state. Through the use of ksw as an adjustable parameter, satisfactory representation of solubility isotherms was obtained for the case of several polymer−gas or −vapor mixtures up to significant gas pressure or solute content. 16509



Article

equal to the value estimated for dry conditions. The corresponding phase equilibrium problem is described below

When examining the above approach in terms proposed in previous section, eq 12 can be here recognized as a first-order approximation of the dependence of polymer volume V̂ pol on solute pressure psol described in eq 5. In this view, the following expression can be retrieved from the swelling coefficient ksw, when attention is focused on the sorption behavior in the low pressure range: ksw

EQ 1 ∂V̂ =χ 0 ∂f ̂ Vpol

f →0

⎧ μ(mix)NE (T , p , ω , ρ ) = μ(gas)EQ (T , p) pol ⎪ ⎨ 0 ⎪ ρ = 1/V̂ pol ⎩ pol

(14)

Results from first-order approximation in nonequilibrium volume−pressure relation (data in green line in Figure 2) are obtained by assuming a linear variation of polymer mass per unit volume as indicated in phase equilibrium formulation which follows

(13)

According to the relation above, the simplified rheological model proposed in this work allows for the estimation of swelling coefficient ksw in eq 13 based on nonequilibrium 0 properties χ and V̂ pol of the polymeric species and on equilibrium volumetric properties of polymer/solute mixture. For the sake of discussing the relevance of volume swelling behavior in gas sorption of glassy polymers, in Figure 2

⎧ μ(mix)NE (T , p , ω , ρ ) = μ(gas)EQ (T , p) pol ⎪ ⎪ ⎪ ⎡ ⎛ ⎞ ⎤ ⎨ ⎢ ⎜ 1 ∂V̂ EQ ⎟ ⎥ 1 ⎪ρ = ⎢1 − ⎜χ 0 ⎟ psol ⎥ pol 0 ⎪ ∂f ̂ ⎢ ̂ ⎜ Vpol ⎟ ⎥ Vpol ⎪ ⎝ f →0 ⎠ ⎣ ⎦ ⎩

(15)

Results from phase equilibrium problems in eqs 14 and 15 are compared to those from complete model formulation (eq 11) in Figure 2, revealing that the infinite dilution solubility coefficient is indeed correctly predicted already from the simplified analysis of negligible solute partial molar volume and that the difference between the prediction from the NELF model after first-order approximation and complete formulation for volume swelling behavior is appreciable only at higher pressure, close to the glass−rubber transition. It should be finally stressed here that while linear density dependence on solute pressure was already considered in a previous application of the NELF model, the approach proposed in this work allows for a prediction of swelling coefficient ksw based on nonequilibrium properties of pure polymeric species and on equilibrium EoS for polymer/solute mixtures. The second case here considered for the application of the model refers to the analysis of solubility and volume swelling data for CO2 in PEMA at 15 °C as measured by Kamiya et al.55 This belongs to a rather interesting set of data from the same research lab and includes results for solubility and volume swelling for a number of different probes in the same polymer matrix.55−57 Experimental results for CO2 mass content and volume per polymer mass are indicated by symbols in Figure 3. Just as in the first case considered, as far as experimental data for system volume are concerned, it must be specified that only data for relative volume increase as function of solute pressure were reported in ref 55 and that results for experimental volume per polymer mass reported in Figure 3 were evaluated from them, based on the value for the specific volume of dry polymer samples at the same temperature equal to 1.124 g/cm3. The latter, in turn, was estimated from the value for dry polymer density at 25 °C indicated in the paper by Kamiya et al.55 and that for thermal expansion coefficient for the same glassy polymer reported by Zoller.50 The solid black line in Figure 3 shows the data for solute content and volume per polymer mass as they result from a best fit procedure for solubility data in the higher range of solute pressure through the use of the SL EoS, by means of the phase equilibrium problem in eq 10. Pure component parameters considered for SL EoS for both PEMA and CO2 are listed in Table 1, while the binary interaction energy parameter ks/p was used as an adjustable parameter in the best

Figure 2. Solubility and volume swelling isotherm of CO2 in glassy PC at 35 °C: experimental data from refs 52, 53 and predictions given by the NELF model through different assumptions for volume behavior.

isotherms are compared as predicted by the NELF model for PC/CO2 mixture at 35 °C under pseudoequilibrium conditions, under different approximation for the representation of volume swelling. Results from zero-order approximation (data in blue line in Figure 2) are obtained by simply assuming that the polymer mass per unit volume at all pressures of interest is 16510



Article

relation for density and solute pressure, offers results pretty close to those obtained from the complete model formulation for volume swelling. In the literature, there are just few examples of characterization of sorption isotherms for swelling components in glassy polymers that are as complete as those considered in previous examples. Indeed, the combined measurement of both solute content and volume swelling in a wide vapor pressure range, including both glassy and rubbery regions, is a rather peculiar characteristic of the examples discussed so far, which allowed for a complete comparison with model calculations and for a exhaustive discussion of features of the model here introduced. On the other hand, solubility or volume swelling predictions from the model proposed here rely only on pure component pVT data for both components and on a minimum amount of binary data that are necessary to identify binary parameter(s) of the specific EoS equilibrium model considered in the procedure. It is thus possible to apply the above analysis also for the case of a less complete set of experimental data, and in what follows, discussion is offered for the characterization of sorption properties for xylenes in polystyrene at 25 °C, as measured by Tsutsui et al.,58 for which experimental data for corresponding volume swelling are not available. In Figure 4 experimental data for m-xylene content in PS reported in ref 58 are shown as a function of solute activity,

Figure 3. Solubility and volume swelling isotherm of CO2 in PEMA at 15 °C: experimental data from ref 55 and predictions given by the SL EoS and NELF model.

fit procedure, and the corresponding retrieved value is indicate in Table 2. It should be first observed that the volume per polymer mass as predicted by the SL EoS in the same procedure well compares with the experimental value, although the method underestimated values for system volume results from model predictions at the higher pressures considered. In the lower pressure range, measured solute content and system volume clearly show values exceeding the predictions from the solution of the phase-equilibrium problem in eq 10, but they are well-described, in both cases, by results shown as solid red curves in Figure 3, corresponding to results from the pseudoequilibrium formulation in eq 11. In either solubility or volume plot, curves calculated for equilibrium and corresponding pseudoequilibrium problems overlap at a gas pressure of approximately 2.5 MPa (corresponding to 12 wt % for CO2 solubility in PEMA), in reasonable agreement with the threshold value for solute-induced glass transition as it appears from experimental data shown in the same figure. Predictions obtained from zero-order or first-order approximation of volume swelling behavior, corresponding to solutions for problems in eqs 13 or 14, respectively, are also shown in Figure 3 for the same polymer/solute system. Also in this case, the zero-order approximation, corresponding to the assumption of negligible volume swelling, returns a valuable prediction only at very low gas pressure, while the first-order approximation, corresponding to the assumption of a linear

Figure 4. Solubility and volume swelling isotherm of m-xylene in PS at 25 °C: experimental data from ref 58 and predictions given by the SL EoS and NELF model. 16511



Article

those resulting from different approaches is left to a future work.60−62 It is only of interest here to consider that a complete solubility isotherm for a plasticizing component in glassy polymers could be finally computed after the above assumption in eq 16 for pg, through the solution of the pseudoequilibrium problem in eqs 11 for solute pressure p lower than pg and through the solution of the equilibrium problem in eq 10 for higher values of solute pressure. Once proper pure component pVT properties are retrieved for both low molecular weight and polymeric species, also accounting for necessary data at the glassy state in the latter case, a complete solubility isotherm at any temperature can be computed just after suitable assumptions for binary parameters required by the specific equilibrium model chosen. An application of the procedure is discussed hereafter with reference to the case of data measured again by Tsutsui et al.58 for sorption of p-xylene in PS at 25 °C. Pure component model parameters preliminarily retrieved are all reported in Table 1, as well as key pVT data used for their estimation. In Figure 5,

calculated as the ratio between solute pressure in the gaseous phase and solute vapor pressure at the system temperature. As in the previous cases, modeling analysis of the same data was obtained by considering the SL EoS as the proper tool for the description of equilibrium thermodynamic properties, for which the pure component parameters used are indicated in Table 1. Volumetric data for pure glassy PS which were used to derive nonequilibrium parameters to be considered in the procedure were taken again from the collection of Zoller,50 and key original data, together with retrieved model parameters, are reported in Table A1. In Figure 4, model results for equilibrium solute content and volume swelling (black line) are reported as obtained from the best fit of high pressure solubility data, in turn obtained for the binary interaction parameter ks/p value indicated in Table 2. Similarly to the cases discussed above, in the same figure model results are reported for pseudoequilibrium solute content and volume swelling as function of solute pressure, up to the conditions for which predictions from pseudoequilibrium problems match corresponding values calculated for equilibrium formulation [glass−rubber transition solute activity (ag) = 0.6]. It should be noted that also for this case, the latter threshold value well compares with solute pressure, corresponding to the apparent glass transition for experimental solubility data. For the sake of comparison with different approximation of volume behavior, solute predictions obtained neglecting volume swelling and assuming a linear dependence on pressure/activity are also reported in the same figure as green and blue lines, respectively. It can be appreciated that in this case, the zero-order approximation returns values comparable with those resulting from complete procedure for solute activity up to 10% of ag, while for the first-order approximation of volume swelling, good results in the same terms are obtained for solute activity up to one-half of ag. Predicting Solute Pressure Glass Transition and Complete Solubility Isotherm. Results from previous analysis encourages one to consider a proposal for a procedure devoted to the prediction of the solute pressure glass transition. Indeed, in cases examined above, the solubility curves from equilibrium and pseudoequilibrium problem formulations match at a solute pressure that is consistent with the value of the apparent glass transition as it results from solubility experimental data. When the latter is assumed as a condition of general validity, the following relations are derived for the calculation of solute pressure glass transition pg:

Figure 5. Solubility isotherm of p-xylene in PS at 25 °C: experimental data from ref 58 and model predictions for different values of the binary interaction parameter.

pertinent experimental data are compared with predictions from the above-mentioned procedure for a complete isotherm embracing both rubbery and glassy regions. To better describe the best fit procedure used, few additional isotherms are plotted in the same figure as elaborated for the LF binary interaction parameter ks/p being either slightly higher or lower than the optimal value. Although experimental data for vapor solubility at very low pressure are not available and comparison with corresponding prediction is thus missing, the correct representation of pg and ωg for the apparent glass transition is clear and the value of solubility coefficients above and below the threshold values are represented in a pretty satisfactory way. The fact that the only adjustable parameter(s) in the procedure sketched above corresponds to the binary parameter(s) envisaged by the specific equilibrium EoS considered should also be accounted for. In view of their autonomous origin, the value of the involved binary parameter(s) may be known by analysis of binary experimental data independent from those under analysis. That is the case of the example discussed below, corresponding to the analysis of data for high pressure CO2 sorption in polysulfone (PSU) at 35 °C as measured by Holk et al.63 As in previous analyses, calculations were preformed on the basis of LF and specific model pure

EQ

̂ (T , p , f |(gas) ) = Vĝ Vpol T ,p g g

(16)

Corresponding solute glass transition ωg then identifies the equilibrium penetrant mass fraction for solute pressure pg. For a correct evaluation of the meaning of the above equation, one should account for the complexity of the glass behavior and glass−rubber transition. In fact, the transition occurs at different pressure/temperature conditions, according to the thermodynamic potential (thermal or chemical) driving the process, to the direction of the transition (from glass to rubber or from rubber to glass), and to specific protocols of glass preparation.59 The indication in eq 16 above should be then considered as proposed for the glass to rubber transition induced by an increase in solute chemical potential in isothermal processes for conventionally prepared glassy polymers. An extensive comparison of results for glass transition solute content as calculated after the procedure proposed above with 16512



Article

can be equally used, starting from LF versions embodied with more complex mixing rules up to EoS resulting from hard spheres chain theories, which can host specific contribution for different kinds of binary interactions involved. Together with the representation of equilibrium thermodynamic properties that result from the EoS selected, the model relies on pure polymer pVT properties in glassy states to retrieve two nonequilibrium model parameters χ and V̂ g. It should be considered that the latter parameters are representative of the specific out-of-equilibrium conditions of the polymer sample and they need to be evaluated from pVT properties for glassy state representative of the specific preparation protocols. In this respect the model parameter V̂ g, which ultimately is evaluated from the polymer sample mass density under dry conditions ρ0pol, must be regarded as the real measurement of the out-of-equilibrium degree of the polymeric material. The model can be applied in an entirely predictive mode when, together with pure component pVT properties, binary equilibrium data are available that are suitable for the evaluation of all binary parameters of equilibrium EoS. The model proved to be effective not only in representing the volume dilation induced by sorption of the swelling agent in the glassy matrix but also in predicting the threshold value for solute pressure and content, which correspond to plasticization conditions. The ability of the model to correlate glass transition temperature to solute content deserves to be specifically analyzed in due developments of this work. Through the implementation of the volume swelling model introduced in this work, NET-GP theory (NELF model for the specific case here considered) reaches the remarkable ability of describing, through a purely predictive procedure, solubility isotherms for swelling and even plasticizing agent in glassy polymers. It represents, indeed, a fundamental upgrade of the original model, as the ability of predicting solubility was so far confined to the case of infinite dilution conditions, where volume swelling phenomenon can indeed be neglected in solubility calculations. In the discussion offered in this work, a comparison has also been given between the volume dilation description proposed in this model and the approach previously considered in the same NET-GP context for volume swelling. For the different cases considered, correctly predicted volume swelling proved to be approximately proportional to solute pressure, as assumed in previous papers. It should be observed, however, that the swelling coefficient considered in previous papers to discuss the volume dilation induced by swelling agents in glassy polymers was always used as an adjustable parameter for the procedure of correlating the solubility isotherm. Through the model here proposed, the same coefficient could be estimated a priori on the basis of pure polymer glassy volumetric properties and on binary equilibrium data. In addition, in its complete formulation, the model proved to be able to represent also deviations from linear dependence of volume swelling on solute pressure which was observed in the cases examined here. The NET-GP theory has been already proved to be able to describe the sorption behavior of fluid mixtures in glassy polymers, and the present approach can be applied in a straightforward manner, conferring to the model a predictive fashion. Furthermore, this latest development of the NET-GP approach is applied to evaluate low molecular weight species solubility, and when coupled with a convenient model to

component parameters indicated in Table 1, as retrieved a priori from corresponding pVT data. In this case, however, also LF binary interaction paramenter ks/p was estimated preliminarily with respect to the model calculation. In fact, for the polymer/gas pair of interest, equilibrium high pressure vapor− liquid data are available at 40, 50, and 60 °C from the work by Tang et al.,64 from which values of the LF binary interaction parameter at the corresponding temperature were estimated, on the basis of the same pure component LF parameters in Table 1. The PSU/CO2 binary interaction parameter at 35 °C indicated in Table 2 and used for the predictive calculation for solubility that are of interest in this work was actually estimated by linear extrapolation of the corresponding data at higher temperatures. Results from pure predictive calculation for CO2 content in PSU at 35 °C are compared in Figure 6 with experimental data reported in ref 63.

Figure 6. Solubility isotherm of CO2 in PSU at 35 °C: experimental data from ref 63 and model predictions.

The glass transition solute mass fraction and solute pressure are estimated, by the predictive procedure, at 6.8% and 42 bar, respectively, and this seems to be consistent with experimental results, although not enough high pressure solubility data are available to be conclusive. More clearly, remarkably good results were obtained for the estimation of excess solute content below the glass transition pressure both at infinite dilution, where binary interaction parameter and dry polymer density play major roles, and at higher pressure, where the correct description of volume swelling behavior is crucial. The proper description of the latter feature is, in fact, a specific result of the model proposed in this paper.

■

CONCLUSIONS A model was introduced for the description of volume swelling behavior in polymer/solute mixture below the glass transition temperature that, coupled with nonequilibrium theory for glassy polymer, allows for the description of the solubility and volume swelling isotherm under pseudoequilibrium conditions. The volume swelling model can be applied to the nonequilibrium version of any EoS suitable for the representation of equilibrium properties of the polymer/solute pair in focus. In all calculations performed in this work, the simplest version of the Sanchez−Lacombe lattice fluid theory was selected as equilibrium EoS, and the NELF model consequently results for the representation of properties of polymer/solute mixture in glassy states. Different formulation for equilibrium properties 16513



Article

Table A1. Tait Equation Parameters for the Considered Polymers, Glass Transition Temperatures, and Weighting Factor χ PC 3

a0 (cm /g) a1 × 104 (cm3/g °C) a2 × 107 (cm3/g °C2) B0 (MPa) B1 × 103 (°C−1) Tg (°C) χ ρ (g/cm3) ref

PSU

PEMA

PS

glass

rubber

glass

rubber

glass

rubber

glass

rubber

0.830 2.20 − 388 2.60

0.808 3.06 5.5 351 4.42

0.8091 0.78 2.3 358 1.11

0.7376 5.21 − 262 2.62

0.8820 3.13 − 337 7.50

0.8923 0.007 30 260 5.05

0.952 2.41 − 366 3.65

0.918 5.96 − 268 4.30

150 0.69 1.194 (35 °C) 51

177 0.56 1.238 (35 °C) 50

calculate the corresponding diffusivity, a description can be obtained of the transport properties of the glassy polymers, suitable for the prediction of gas permeability and selectivity. The latter would be of great relevance to the analysis and design of materials for gas separation applications. In this respect, it should be noted that while analysis of diffusion coefficient for swelling gases in glassy polymers includes questions that have not been addressed in this papers, the contribution the current model can give to this specific analysis is remarkable. Indeed, it is well-established that a major factor in determining the value of diffusion coefficient in polymer− solute systems is the free volume available in the mixture,65 which in turn depends on volume dilation. As the latter quantity is a specific output of the predictive procedure here proposed, the analysis of diffusive properties in the same systems appears as a natural development of the approach, which will be indeed considered in future works.

61 0.83 1.124 (15 °C) 50

κ(0, T ) =

92 0.68 1.044 (25 °C) 50

C C = B (T ) B0 exp( −B1T )

(a4)

For the sake of exemplifying the results of the analysis of pVT data behind the procedure, in Figure A1 values of isothermal

■

APPENDIX The evaluation of the weighting factor χ is a key point for the present approach. In this approximation, χ is a material parameter and depends only on the behavior of the different polymers. In order to determine the value of χ, according to eq 12, isothermal pVT data at different pressure for the pure polymer above and below Tg are needed. This data can be thus used to calculate the isothermal compressibility at different temperatures, and the κg and κEQ are then extrapolated at the glass transition; their ratio will return the weighting factor. The Tait equation was employed to have fit the pVT properties of the polymer and to have a precise description of the temperature behavior of the compressibility: ⎛ ⎛ p ⎞⎞ V (p , T ) = V (0, T )⎜1 − C ln⎜1 + ⎟⎟ B (T ) ⎠⎠ ⎝ ⎝

Figure A1. Isothermal compressibility for Lexan polycarbonate calculated as the numerical derivative of pVT data (points)51 and calculated with the Tait equation (lines).

compressibility as function of temperature are reported for Lexan poly(carbonate) both in the glassy and melt state. Points are calculated as a numerical derivative of the isothermal pressure−volume−temperature data after the method of Zoller,51 whereas lines represent values calculated with the Tait equation, whose parameters are retrieved from the same data set. The κg and κEq compressibility values could then be evaluated at the glass transition by extrapolation procedures from the glassy and from the melt region, respectively.

(a1)

■

where V (0, T ) = a0 + a1T + a 2T 2

(a2)

B(T ) = B0 exp( −B1T )

(a3)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

and C is a universal constant for all the polymers and it is equal to 0.0894. The characteristic parameters of the Tait equations are reported in Table A1, as taken from the literature or from the best fit of the experimental pVT curves. The isothermal compressibility values can be easily derived from eq a1:

REFERENCES

(1) Flaconnèche, B.; Klopffer, M. H. Transport properties of gases in polymers: Experimental methods. Oil Gas Sci. Technol. 2001, 56, 245. (2) Kanjickal, D. G; Lopina, S. T. Modeling of drug release from polymeric delivery systemsA review. Crit. Rev. Ther. Drug Carrier Syst. 2004, 21, 345.

16514



Article

(3) Brown, W. R.; Park, G. S. Diffusion of solvents and swellers in polymers. J. Paint Technol. 1970, 42, 16. (4) Wijmans, J. G.; Baker, R. W. The solution-diffusion model: A review. J. Membr. Sci. 1995, 107, 1. (5) Matteucci, S.; Yampolskii, Y. P.; Freeman, B. D.; Pinnau, I. In Materials Science of Membranes for Gas and Vapor Separation; Yampolskii, Y. P., Pinnau, I., Freeman, B. D., Eds.; John Wiley & Sons: New York, 2006; Chapter 1, pp 1−47. (6) Masi, P.; Paul, D. R. Modeling gas transport in packaging applications. J. Membr. Sci. 1982, 12, 137. (7) Del Nobile, M. A.; Mensitieri, G.; Manfredi, C.; Arpaia, A.; Nicolais, L. Low molecular weight molecules diffusion in advanced polymers for food packaging applications. Polym. Adv. Technol. 1996, 7, 409. (8) Grate, J. W.; Abraham, M. H. Solubility interactions and the design of chemically selective sorbent coatings for chemical sensors and arrays. Sensor Actuat. B-Chem. 1991, 3, 85. (9) Adhikari, B.; Majumdar, S. Polymers in sensor applications. Prog. Polym. Sci. 2004, 29, 699. (10) Vimalchand, P.; Donohue, M. D. Thermodynamics of quadrupolar molecules: The perturbed-anisotropic-chain theory. Ind. Eng. Chem. Fundam. 1985, 24, 246. (11) Chen, F.; Fredenslund, A.; Rasmussen, P. Group-contribution Flory equation of state for vapor−liquid equilibria in mixtures with polymers. Ind. Eng. Chem. Res. 1990, 29, 875. (12) Panayiotou, C.; Vera, J. H. Statistical thermodynamics of r-mer fluids and their mixtures. Polym. J. 1982, 14, 681. (13) Sanchez, I. C.; Lacombe, R. H. An elementary molecular theory of classical fluids. Pure fluids. J. Phys. Chem. 1976, 80, 2352. (14) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation-of-state solution model for associating fluids. Fluid Phase Equilib. 1989, 52, 31. (15) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New reference equation of state for associating liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (16) Huang, S. H.; Radosz, M. Equation of state for small, large, polydisperse, and associating molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (17) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (18) Peters, F. T.; Laube, F. S.; Sadowski, G. Development of a group contribution method for polymers within the PC-SAFT model. Fluid Phase Equilib. 2012, 324, 70. (19) Song, Y.; Lambert, S. M.; Prausnitz, J. M. Equation of state for mixtures of hard-sphere chains including copolymers. Macromolecules 1994, 27, 441. (20) Barrer, R. M.; Barrie, J. A.; Slater, J. Sorption and diffusion in ethyl cellulose. Part III. Comparison between ethyl cellulose and rubber. J. Polym. Sci. 1958, 27, 177. (21) Michaels, A. S.; Vieth, W. R.; Barrie, J. A. Solution of gases in polyethylene terephthalate. J. Appl. Phys. 1963, 34, 1. (22) Vieth, W. R.; Howell, J. M.; Hsieh, J. H. Dual sorption theory. J. Membr. Sci. 1976, 1, 177. (23) Koros, W. J.; Paul, D. R.; Rocha, A. A. Carbon dioxide sorption and transport in polycarbonate. J. Polym. Sci. Polym. Phys. Ed. 1976, 4, 687. (24) Weiss, G. H.; Bendler, J. T.; Shlesinger, M. F. Continuous-site model for Langmuir gas sorption in glassy polymers. Macromolecules 1992, 25, 990. (25) Raucher, D.; Sefcik, M. D. In Industrial Gas Separations, Whyte, T. E., Yon, C. M., Wagener, E. H., Eds.; ACS Symposium Series 223; American Chemical Society: Washington, DC, 1983. (26) Mi, Y.; Zhou, S.; Stern, S. A. Representation of gas solubility in glassy polymers by a concentration−temperature superposition principle. Macromolecules 1991, 24, 2361. (27) Lipscomb, G. G. Unified thermodynamic analysis of sorption in rubbery and glassy materials. AIChE J. 1990, 36, 1505.

(28) Kirchheim, R. Sorption and partial molar volume of small molecules in glassy polymers. Macromolecules 1992, 25, 6952. (29) Doghieri, F.; Sarti, G. C. Nonequilibrium lattice fluids: A predictive model for the solubility in glassy polymers. Macromolecules 1996, 29, 7885. (30) Doghieri, F.; Sarti, G. C. Predicting the low pressure solubility of gases and vapors in glassy polymers by the NELF model. J. Membr. Sci. 1998, 147, 73. (31) Sarti, G. C.; Doghieri, F. Predictions of the solubility of gases in glassy polymers based on the NELF model. Chem. Eng. Sci. 1998, 53, 3435. (32) Lacombe, R. H.; Sanchez, I. C. Statistical thermodynamics of fluid mixtures. J. Phys. Chem. 1976, 80, 2568. (33) Sanchez, I. C.; Lacombe, R. H. Statistical thermodynamics of polymer solutions. Macromolecules 1978, 11, 1145. (34) Sanchez, I. C.; Rodgers, P. A. Solubility of gases in polymers. Pure Appl. Chem. 1990, 62, 2107. (35) Giacinti Baschetti, M.; Doghieri, F.; Sarti, G. C. Solubility in glassy polymers: Correlations through the nonequilibrium lattice fluid model. Ind. Eng. Chem. Res. 2001, 40, 3027. (36) Doghieri, F.; Ghedini, M; Quinzi, M.; Rethwisch, D.; Sarti, G. C. Gas solubility in glassy polymers: Predictions from non-equilibrium EoS. Desalination 2002, 144, 73. (37) Doghieri, F.; Quinzi, M.; Rethwisch, D. G.; Sarti, G. C. In Advanced Materials for Membrane Separations; ACS Symposium Series 876; Pinnau, I., Freeman, B. D., Ed., American Chemical Society: Washington, DC, 2004; Chapter 5, pp 74−90. (38) Doghieri, F.; De Angelis, M. G.; Giacinti Baschetti, M.; Sarti, G. C. Solubility of gases and vapors in glassy polymers modelled through nonequilibrium PHSC theory. Fluid Phase Equilib. 2006, 241, 300. (39) Banerjee, T.; Lipscomb, G. G. A comparison of analytic thermodynamic models for gas solubility, volume dilation and heat of sorption in glassy polymeric materials. Comput. Theor. Polym. Sci. 2000, 10, 437. (40) Giacinti Baschetti, M.; De Angelis, M. G.; Doghieri, F.; Sarti, G. C. In Chemical Engineering: Trends and Developments; Galan, M. A., Martin del Valle, E., Ed.; J. Wiley: Chichester, U.K., 2005; Chapter 2, pp 41−61. (41) Grassia, F.; Giacinti Baschetti, M.; Doghieri, F.; Sarti, G. C. In Advanced Materials for Membrane Separations; ACS Symposium Series 876; Pinnau, I., Freeman, B. D., Ed.; Washington, DC, 2004; Chapter 4, pp 55−73. (42) Giacinti Baschetti, M.; Ghisellini, M.; Quinzi, M.; F. Doghieri, M.; P. Stagnaro, M.; Costa, G.; Sarti, G. C. Effects on sorption and diffusion in PTMSP and TMSP/TMSE copolymers of free volume changes due to polymer ageing. J. Mol. Struct. 2005, 739, 75. (43) De Angelis, M. G.; Sarti, G. C.; Doghieri, F. NELF model prediction of the infinite dilution gas solubility in glassy polymers. J. Membr. Sci. 2007, 289, 106. (44) De Angelis, M. G.; Sarti, G. C. Solubility and diffusivity of gases in mixed matrix membranes containing hydrophobic fumed silica: Correlations and predictions based on the NELF model. Ing. Eng. Chem. Res. 2008, 47, 5214. (45) Minelli, M.; Campagnoli, S.; De Angelis, M. G.; Doghieri, F.; Sarti, G. C. Predictive model for the solubility of fluid mixtures in glassy polymers. Macromolecules 2011, 44, 4852. (46) De Angelis, M. G.; Sarti, G. C. Calculation of the solubility of liquid solutes in glassy polymers. AIChE J. 2012, 58, 292. (47) Piccinini, E.; Gardini, D.; Doghieri, F. Stress effects on mass transport in polymers: A model for volume relaxation. Composites: Part A 2006, 37, 546. (48) Enscore, D. J.; Hopfenberg, H. B.; Stannett, V. T. Diffusion, swelling, and consolidation in glassy polystyrene microspheres. Polym. Eng. Sci. 1980, 20, 102. (49) Carlà, V.; Hussain, Y.; Grant, C.; Sarti, G. C.; Carbonell, R. G.; Doghieri, F. Modeling sorption kinetics of carbon dioxide in glassy polymeric films using the nonequilibrium thermodynamics approach. Ind. Eng. Chem. Res. 2009, 48, 3844. 16515



Article

(50) Zoller, P.; Walsh, D. Standard Pressure−Volume−Temperature Data for Polymers; Technomic: Lancaster, 1995. (51) Zoller, P. A study of the pressure−volume−temperature relationships of four related amorphous polymers: polycarbonate, polyarylate, phenoxy, and polysulfone. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 1453. (52) Wissinger, R. G.; Paulaitis, M. E. Swelling and sorption in polymer−CO2 mixtures at elevated pressures. J. Polym. Sci. B: Polym. Phys. 1987, 25, 2497. (53) Fleming, G. K.; Koros, W. J. Dilation of polymers by sorption of carbon dioxide at elevated pressures. 1. Silicone rubber and unconditioned polycarbonate. Macromolecules 1986, 19, 2285. (54) De Angelis, M. G.; Sarti, G. C.; Doghieri, F. Correlations between penetrant properties and infinite dilution gas solubility in glassy polymers: NELF model derivation. Ind. Eng. Chem. Res. 2007, 46, 7645. (55) Kamiya, Y.; Mizoguchi, K.; Hirose, T.; Naito, Y. Sorption and dilation in poly(ethyl methacrylate)−carbon dioxide system. J. Polym. Sci. B: Polym. Phys. 1989, 27, 879. (56) Kamiya, Y.; Mizoguchi, K.; Hirose, T.; Y. Naito, T.; Bourbon, D. Argon sorption and partial molar volume in poly(ethyl methacrylate) above and below the glass transition temperature. J. Polym. Sci. B: Polym. Phys. 1991, 29, 225. (57) Kamiya, Y.; Bourbon, D.; Mizoguchi, K.; Naito, Y. Sorption, dilation, and isothermal glass transition of poly(ethyl methacrylate)− organic gas systems. Polymer J. 1992, 24, 443. (58) Tsutsui, K.; Katsumata, T.; Fukatsu, H.; Yoshimizu, H.; Kinoshita, T.; Tsujita, Y. The isomer effect on complex formation in syndiotactic polystyrene−xylene system. Polymer J. 1999, 31, 268. (59) Alcoutlabi, M.; Banda, L.; Kollegodu-Subramanian, S.; Zhao, J.; McKenna, G. B. Environmental effects on the structural recovery responses of an epoxy resin after carbon dioxide pressure jumps: Intrinsic isopiestic, asymmetry of approach, and memory effect. Macromolecules 2011, 44, 3828. (60) Gibbs, J. H.; Di Marzio, E. A. Nature of the glass transition and the glassy state. J. Chem. Phys. 1958, 28, 373. (61) Chow, T. Molecular interpretation of the glass transition temperature of polymer−diluent systems. Macromolecules 1980, 13, 362. (62) Condo, P. D.; Sanchez, I. C.; Panayiotou, C. G.; Johnston, K. P. Glass transition behavior including retrograde vitrification of polymers with compressed fluid diluents. Macromolecules 1992, 25, 6119. (63) Hölck, O.; Siegert, M. R.; Heuchel, M; Böhning, M. CO2 Sorption induced dilation in Polysulfone: Comparative analysis of experimental and molecular modeling results. Macromolecules 2006, 39, 9590. (64) Tang, M.; Huang, Y. C.; Chen, Y. P. Sorption and diffusion of supercritical carbon dioxide into polysulfone. J. Appl. Polym. Sci. 2004, 94, 474. (65) Vrentas, J. S.; Duda, J. L. Diffusion in polymer−solvent systems. I. Reexamination of the free-volume theory. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 403.

16516


A Predictive Model for Vapor Solubility and Volume ... - ACS Publications

Recommend Documents