VOC sorption in glassy polyimides—Measurements and modeling

Journal of Membrane Science 415-416 (2012) 596–607

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VOC sorption in glassy polyimides—Measurements and modeling Lisa Hesse, Shahbaz Naeem, Gabriele Sadowski n Laboratory of Thermodynamics, Department of Biochemical- and Chemical Engineering, TU Dortmund University, Emil-Figge-Strasse 70, 44227 Dortmund, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 January 2012 Received in revised form 16 May 2012 Accepted 18 May 2012 Available online 29 May 2012

The sorption behavior of volatile organic compounds (VOCs) in polymeric membrane materials such as polyimide, has a strong effect on the separation efficiency of dense polymeric nanofiltration membranes. To investigate the sorption behavior, this work presents gravimetrically measured and modeled sorption profiles for a series of VOCs (like n-hexane, ethanol, toluene, 2-propanol, and ethyl acetate) in two polyimides (P84 and Matrimid) at 25 1C. The experimental results show significant differences in the sorption speed of the VOCs in the two polyimides. The sorption of toluene, ethyl acetate, and ethanol in Matrimid is very fast compared to the sorption of 2-propanol, but the sorption in P84 is very slow for all the VOCs studied. The sorption behavior of the polyimide/VOC systems has been modeled using an improved one-dimensional Maxwell–Stefan diffusion model. The measured and modeled results provide detailed insight into the specific sorption behavior of the VOCs in the glassy polyimides P84 and Matrimid. & 2012 Elsevier B.V. All rights reserved.

Keywords: Polyimide Glass transition temperature Diffusion coefficient Organic solvent nanofiltration

1. Introduction High consumption of energy, resulting in a faster depletion of energy resources and increasing costs, forces the chemical industry to strive for new economical purification processes. The ‘organic solvent nanofiltration’ (OSN) [1–3], also called ‘solvent resistant nanofiltration’ (SRNF) [1,4–6] or ‘organic nanofiltration’ (oNF) [7], process performs fluid purification and offers an economic benefit because of the mild process conditions that are used. In addition to the fluid purification and subsequent permeate recycling, OSN also facilitates the enrichment of highvalue products like pharmaceuticals dissolved in an organic solvent stream [2,8]. Asymmetric integrally dense polymeric membranes with a dense active separation layer made of polyimide [2,4] such as P84 or Matrimid [2,9–16] are an interesting group of OSN membranes. In recent literature, these polyimides are also used as cross-linked material [17,18] as well as material for porous supporting layers [19]. A systematic characterization of this active layer membrane material (P84 or Matrimid) is an essential step to predict the membrane performance to achieve an improved membrane development [20]. This characterization requires the identification and investigation of the complex interactions between the material of the active separation layer and the feed stream consisting of volatile organic compounds (VOCs) and dissolved

n

Corresponding author. Tel.: þ49 231 755 2635; fax: þ 49 231 755 2572. E-mail address: [email protected] (G. Sadowski).

0376-7388/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2012.05.054

compounds [20,21], all of which influence the separation efficiency. Two important factors influencing the separation efficiency are the solubility and diffusivity of VOC into the dense polymeric active separation layer. The solubility of the VOCs ethanol, ethyl acetate, n-hexane, 2-propanol, and toluene in the two polyimides (P84 and Matrimid) [2,9,10] commonly used as OSN membrane materials has already been measured and modeled in previous work [22]. In this work, the same VOCs have been selected to investigate their diffusion behavior by measuring the sorption isotherms of the polyimide/VOC systems. Subsequent modeling of the sorption isotherms by an improved one-dimensional (1D) Maxwell–Stefan (MS) diffusion model allows the estimation of mutual MS diffusion coefficients. The polyimides that have been examined have very high glass transition temperatures (see Section 2). Therefore, the polyimide/ VOC mixtures investigated within this work are assumed to be glassy during the whole sorption process. As early as 1949 [23–26], the anomalous diffusion behavior of a VOC in a glassy polymer was described mathematically. The VOC uptake into glassy polymers is influenced by the diffusion of the VOCs as well as by the swelling of the polymer chains during VOC sorption. In contrast to diffusion in rubbery polymers above their glass transition temperature (Fickian diffusion), the swelling of glassy polymer chains is very slow, and this swelling becomes the ratedetermining step in the VOC uptake. The relative mass uptake, which is calculated within this work by the mass uptake of the VOC in the polymer, is not linearly related to the square root of the time (t1/2) but shows deviations from this Fickian linearity. At small VOC uptakes in the glassy polymer, the shape of the sorption isotherm is sigmoidal or the sorption isotherm forms

L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607

two stages with the square root of the time [27]. At high amounts of VOC, the relative mass uptake of the VOC is linear with time [26]. The case of linear mass uptake with time is commonly referred to as Case II diffusion behavior. Two main effects influence the behavior of diffusion in glassy polymers and result in different shapes of the sorption isotherms [27,28]. The first effect is the slow changes in the polymer structure during the VOC uptake [28]. The rearrangement or swelling of the polymer chains is assumed to be viscoelastic, where the time dependence is called relaxation (viscous volume relaxation, VVR) [27]. The slow swelling of the polymer chains leads to a tension between the chains and to a pressure buildup in the polymer. Because this increased pressure also affects the driving force (chemical potential) of the VOC sorption, only a small amount of the VOC is absorbed at the beginning. More and more VOC can be absorbed in the polymer when the pressure is reduced by swelling. The second effect that influences the diffusion behavior is the different degree of swelling in different parts of the polymer because of the variable VOC loadings in those parts [28]. Because of the non-uniform swelling, different swelling potentials or differential swelling stresses (differential swelling stresses, DSS) [27] occur. Usually both effects appear during the VOC sorption below the glass transition temperature of the mixture [27]. In some cases, one effect dominates the sorption behavior while the other effects can be neglected. Long and Richman [29] determined that for the two-stage behavior, only the VVR effect must be considered. Moreover, Crank and Park [25] as well as Long and Richman [29] were able to describe the different types of sorption behavior in glassy polymers while neglecting the DSS effect. In this work, the sorption behavior of different VOCs into the industrially important polyimides is modeled by taking only the VVR effect into account. In this work, the sorption curves of five different VOCs (nhexane, toluene, 2-propanol, ethyl acetate, and ethanol) in P84 and Matrimid have been measured gravimetrically at 25 1C. The experimental measured sorption data are modeled using a 1D-MS diffusion model by fitting the model parameters.

2. Experimental section 2.1. Materials Polyimide P84 (molar mass Mn ¼26,000 g/mol, polydispersity index 2.3) was kindly supplied by HP Polymer GmbH (Lenzing, Austria) (see Fig. 1a). Matrimid 9725, which is a micro-pulverized version of Matrimid 5218, (molar mass Mn ¼48,000 g/mol, polydispersity index 2.2) was kindly supplied by Huntsmann (Basel, Switzerland) (see Fig. 1b).

597

The Tg values for P84 and Matrimid were measured in Hesse and Sadowski [22] to be 306 1C and 311 1C, respectively. Both Tg are in the same range of the temperatures already stated in literature for P84 and Matrimid 315 1C [30,31] and 320 1C [32]. The polymers were dried and stored for 24 h at 100 1C under vacuum before use. VOCs N,N-dimethylformamide (DMF), nhexane (analytical grade), toluene (LiChrosolv), 2-propanol (LiChrosolv), ethanol (LiChrosolv, not denatured), and ethyl acetate (LiChrosolv) were purchased from Merck. All chemicals were used without further purification. 2.2. Polymer film preparation Dense dry polymer films with a thickness of 80–95 mm were produced with a 25 wt% polymer/DMF solution. To obtain a homogeneous polymer solution in DMF, the mixture was heated to 55 1C and stirred overnight. After the polymer was fully dissolved, the solution was cast onto a glass surface by means of a coating knife. The slit between the glass and the coating knife surface was 750–850 mm. The coated glass plates were placed into a vacuum oven for the evaporation of the DMF. A detailed description of the polymer film preparation and drying process is reported elsewhere [22]. 2.3. Sorption measurements The sorption isotherms of five VOCs (n-hexane, ethanol, toluene, 2-propanol, and ethyl acetate) in two membrane active layer materials (polyimide P84 and Matrimid) were measured gravimetrically. An 80–95 mm thick dry polyimide film of known initial mass was immersed in liquid VOC in a closed glass tube at a constant temperature of 25 1C. The glass tube was placed in an incubator to control the temperature within a temperature deviation range of 70.1 K. To prevent moisture impurities in the VOCs inside the glass tubes, the inner atmosphere of the incubator was kept at a low humidity by means of silica gel. To measure the mass uptake of VOC in the film, the immersed films were removed from the liquid at regular time intervals. The polymer film surface was dried between two sheets of the filter paper, weighed and placed back into the glass tube containing the VOC. The polymer film was weighed with an accuracy of 70.1 mg. This procedure was repeated until the mass of the polymer film remained constant, indicating that equilibrium had been reached. The sorption isotherms were calculated using the mass uptake of VOCs. The longest time needed for equilibrium was four months in the case of P84/ethyl acetate. The same procedure was applied for all the polymer/VOC systems presented in the Results section. The measurements had a maximum standard deviation of 70.0116 g/g corresponding to a maximum deviation of 73% in polymer weight fraction (measured for Matrimid/ethyl acetate). The average deviation was 71.9% in polymer weight fraction. 2.4. Young’s modulus measurement The Young’s modulus of Matrimid was measured by creep measurements using a DMA 2980 at 25 1C. A detailed description of the measurement procedure and the evaluation of the measurements is given by Mueller et al.[33].

3. Diffusion modeling

Fig. 1. Molecular structure of (a) P84 and (b) Matrimid monomers.

The objective of the sorption modeling is the determination of binary MS diffusion coefficients of the polymer/VOC inside the polymer film. The model used here follows the principles of

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Fig. 2. Division of the half polymer film into N ¼1,2yn small discrete volume elements each of length (dz). I and II stand for the polymer-free VOC phase I (outside the film) and polymer/VOC mixture phase II (inside the film), respectively. The first sub-index for all the variables inside the film (phase II) is the number of the volume element while the second sub-index stands for the component (1¼ polymer, 2¼ VOC).

¨ Durning and Russel [34] and Kruger [35]. Based on these models an improved one-dimensional (1D) Maxwell–Stefan (MS) diffusion model was developed which consists of three different equations. First, the flux equation considers VOC diffusion; second, a mechanical equation of state describes the swelling effects below the Tg; and third, the overall mass balance of all the species in the system is considered in the third equation. In contrast to the models mentioned above, Durning and Russel [34] and Krueger [35], the PC-SAFT equation of state was used to calculate the mixture properties of the binary polyimide/ VOC systems. By applying this model, the diffusive driving force, which is the gradient of the VOC chemical potential as well as mixture volumes can be consistently calculated as function of concentration and pressure [36,37]. To model the polymeric swelling below the glass-transition temperature, the viscoelastic swelling is considered but the differential swelling is neglected. For the modeling of the VOC diffusion into the polyimide film, the film is divided into n small discrete volume elements each having the length dz. Every volume element is characterized by its pressure (p), the weight fraction (w) and the chemical potential (m) of the VOC (see Fig. 2). Here component 1 is the polymer and component 2 is the VOC. The temperature is constant throughout the whole film. Assuming a symmetrical VOC sorption into the polymer film from both sides [38,39], a film with thickness (L(t¼0)) can be divided in the middle into two identical parts. Only one sorption side, e.g., (0ozoL(t¼0)/2), must be considered. 3.1. Flux equation The diffusive VOC flux within the polymer is modeled using the MS diffusion equation [35,40]. In the general MS equation, the driving force (Fi) acting on a component i in a mixture is equal to the sum of the frictional forces between the component i and all other species j [40,41]. In this work, the difference of the chemical potential (mi) in the VOC-phase I and the polymer-film II is assumed to be the only driving force which causes the sorption of the VOC. It is assumed that the driving force or the diffusive flux takes place only in z direction of a Cartesian coordinate system.   dmi Fi ¼ ð1Þ dz T This equation allows the calculation of VOC diffusion in glassy polymers under continuous consideration of the swelling pressure by the chemical potential where Fick’s equation is not applicable. The MS equation for an component i can be written as [42]: !   C C X X wj wi ðui uj Þ wk wi dmi ¼ with Œ ij ¼ Œ ji M RT dz Œ ij Mj k T j¼1 k¼1 ð2Þ

w is the weight fraction, M is the molar mass of component j, Œ ij is the MS diffusion coefficient, T is the temperature, R the real gas constant, and z the z direction in the Cartesian coordinate system. By inserting the definition of the diffusive flux (j) [42] for a binary polymer/VOC system ji ¼ ri ðui nÞ ¼ rt wi ðui nÞ

ð3Þ

where ri is the density of component i, rt is the total density and v is the reference velocity, Eq. (2) is simplified to !   C X wk w2 dm2 ðw1 j2 w2 j1 Þ ¼ ð4Þ M k RT dz T Œ 21 M1 rt k¼1 The chemical potential for VOC (m2) can be calculated by summing the chemical potential of the pure VOC (m id 02 ) at the reference pressure (p þ ), the ratio of the pressure inside the polymer film (p) and the reference pressure (p þ ), the mole faction of the VOC in the membrane (x2), and the fugacity coefficient of the VOC in the membrane (j2), according to Eq. (5).   p þ mII2 ðT,pÞ ¼ mid ð5Þ þRTlnx2 þRTlnj2 02 ðT,p Þ þ RTln pþ Inserting this expression into the MS expression (Eq. (4)) results in the following expression for the flux: !   C X wk dðlnðp  x2  j2 ÞÞ ðw1 j2 w2 j1 Þ ¼ ð6Þ w2 dz Mk Œ 21 M1 rt T k¼1 For each volume element (N ¼1,2yn), the following flux equation must be fulfilled for the polymer-rich phase II. ! ! C wII X dðlnðpIIN  xIIN,2  jIIN,2 ÞÞ wII jN,2 wIIN,2 jN,1 N,k II wN,2 ¼  N,1 dz Mk Œ 21 M1 rt k¼1 T

ð7Þ 3.2. Mechanical equation of state The mechanical equation of state for simple elongation (e) of a linear viscoelastic material is shown below:[43] Z t  eðtÞ ¼ F  sðtÞ þ F ðtxÞUsðxÞdx ð8Þ 1

In the equation above, F(t) represents the creep compliance of the polymer matrix which causes the elongation (e(t)) at time (t) and s is the stress between the polymer chains at the time t [43]. Below Tg of the neat polymer and Tg,mixture of the polymer/VOC system, the creep compliance (F) can be described by a linear viscoelastic model, e.g., a spring-dashpot model. The simplest of these models is the so-called Maxwell element [44] (one spring, one dashpot): FðtÞ ¼

1 t þ E0 Z

ð9Þ

L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607

where E0 is the Young’s modulus, and Z represents the viscosity of the mixture. By means of Eqs. (8) and (9), the following relationship between stress (s) and elongation (e) results [35]: de 1 ds 1 ¼ þ UsðtÞ E0 dt dt Z

¼ pIIN pI

DL dzN ðtÞdzN ðt ¼ 0Þ ¼ Lðt ¼ 0Þ dzN ðt ¼ 0Þ

ð11Þ

ð12Þ

The difference in the length can be expressed as a function of volume fractions.

FN ðtÞ ¼

dz2 ðt ¼ 0Þ  dzN ðtÞ V N ðtÞ ¼ N 3 ¼ eN ðtÞ þ 1 V N ðt ¼ 0Þ dzN ðt ¼ 0Þ

dFN 1 dðpIIN pI Þ 1 II ¼ þ ðpN pI Þ E0 dt Z dt

ð14Þ

3.3. Mass balances

arises in the polymer film during VOC sorption below the glass transition temperature. This swelling pressure corresponds to the normal stress (szz). In the further simplification, the stress is therefore replaced by the swelling pressure (pswell ). N The elongation of the polymer during a one-dimensional swelling in the z-direction (DL) corresponds to the relative difference between the initial length of the polymer (dzN(t ¼0)) at the time t ¼0 and the length of the polymer (dzN(t)) at time t during VOC sorption into the polymer membrane.

eN ðtÞ ¼

equation of state can be obtained.

ð10Þ

A swelling pressure (pswell ) N pswell N

599

ð13Þ

where VN(t ¼0) is the volume of one unswollen volume element N and VN(t) is the volume of a swollen volume element at time t. Using the definition of the swelling pressure Eq. (11) and the expression for the elongation (Eq. 12), the following mechanical

During VOC sorption into the polymer, the thickness of the polymer film changes with time. Because both the flux equation and the mechanical equation of state depend on the z-coordinate, this problem is called a moving boundary problem. To solve this problem, Bausa and Marquardt [42] proposed a polymer mass fixed coordinate system in which each position of the swollen (VOC-loaded) polymer film is related to a corresponding position of the dry film (before VOC sorption), as shown in Fig. 3. In the case of the polymer mass fixed coordinate system, the mass of the polymer is fixed for each volume element. The transformed length dz of each volume element stays constant during VOC sorption. In addition to the transformed coordinate (z), a transformed density (ri ) is also used [42]:

ri  dz  ri  dz

ð15Þ

dz r V N ðt ¼ 0Þ ¼ 1 ¼ ¼ fN,1 dz V N ðtÞ r1

ð16Þ

fN,1 is the volume fraction of the polymer in each volume element. For the 1D flux, the mass balance for each volume element in the transformed coordinate system of each component can be

Fig. 3. One dimensional swelling of a polymer film in (a) and (c) Cartesian and (b) and (d) transformed (polymer mass fixed) coordinate systems [42]. The unswollen dry polymer film at time t¼ 0 in the (a) Cartesian and (b) transformed (polymer mass fixed) coordinate systems [42]. A VOC-loaded polymer film (swollen polymer film) at time t is shown in an (c) Cartesian and (d) transformed (polymer mass fixed) coordinate systems.

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3.4. Boundary conditions

written as[35]: drN,i dðrN,i  uN,i Þ þ ¼0 dt dz

ð17Þ

Eq. (17) can be set up for both the polymer and the VOC. If the polymer is considered (i¼1), the right half of the left side of Eq. (17) is zero because the polymer has no velocity in a polymer mass fixed coordinate system (uN,1 ¼ 0). The transformed density of the polymer over time stays constant in each volume element: II N,1

dr dt

¼0

ð18Þ

For the VOC, the balance equation for the volume elements N (N ¼2, 3yn) is

For the interface between the outer liquid phase and the first volume element (N ¼1) of the polymer-rich phase, an instantaneous chemical equilibrium is assumed. The first boundary condition is shown below:

mI2 ðT,pI ,xI2 Þ ¼ mII1,2 ðT,pII1 ,xII1,2 Þ

ð29Þ

For the last volume element (N ¼n) in the middle of the polymer film of thickness (L(t ¼0)/2), no flux is leaving the element because of symmetric VOC sorption from both sides of the film. Thus: II j~n,2 ¼ 0

ð30Þ

II

drIIN,2 d jN,2 þ ¼0 dt dz

ð19Þ

whereas the diffusive (transformed) VOC flux polymer velocity: II

jN,2 ¼ rIIN,2 ðuN,2 uN,1 Þ ¼ rIIN,2  uN,2

II jN,2

is based on the

ð20Þ

The following conversions were used to replace the transformed VOC density (rIIN,2 ) in Eq. (19) by the weight fraction (w):

rIIN,i ¼ wIIN,i U

rIIN,1

ð21Þ

wIIN,1

wIIN,i ¼ wIIN,i

ð22Þ

A detailed description of the conversions can be found in Bausa [42]. Finally, the mass balance equation for the VOC can be formulated in a polymer mass fixed coordinate system for each volume element N (N ¼1, 2yn).  II  r ! II d wN,1 II rIIN,1 dwIIN,2 d jN,2 N,1 II wN,2 þ þ ¼0 ð23Þ dt dt dz wIIN,1 This procedure is repeated to convert the MS as well as the mechanical equation of state into the transformed coordinate system for each volume element N (N ¼2, 3yn) leading to Eqs. (24) and (25): 0 1 ! II C wII X dðlnðpIIN  xIIN,2  jIIN,2 ÞÞ jN,2 ðwIIN,1 Þ2 N,j 2 @ AwII ð f Þ ¼  N,1 N,2 dz Mk Œ 21 M1 rIIN,1 j¼1 T

3.5. Properties of pure components and mixtures PC-SAFT equation of state. The fugacity coefficient (j) or the chemical potential (m) of each component is outside the film as well as in each volume element required in Eqs. (24) and (29). The fugacity coefficient is calculated for each pressure (p) and mass fraction (wi) by the PC-SAFT equation of state proposed by Gross and Sadowski [36,37].The PC-SAFT equation of state is also used to calculate the volume of each dry (VN (t¼0)) and VOC-loaded II (VN (t)) volume element and the volume fraction (fN ), which is needed in Eqs. (24) and (25), for each pressure (p) and mass fraction (wi). The general expression of the PC-SAFT equation [36,37] is shown in Eq. (31), where the residual Helmholtz energy (Ares) is calculated as a sum of the hard-chain term (Ahc), a term representing the dispersion contribution (Adisp) and an association contribution (Aassoc). Ares ¼ Ahc þ Adisp þAassoc

For non-associating components, only the hard-chain term as well as the dispersion term is needed to model their thermodynamic properties. A non-associating component i is characterized by three pure-component PC-SAFT parameters: the segment diameter (si), the segment number (mi), and the dispersionenergy parameter (ei). For associating VOCs like ethanol, two additional parameters (the association energy parameter (eAiBi) and the association volume (kAiBi)) are used within Aassoc. To calculate thermodynamic properties for more than one component, mixture parameters like the segment diameter (sij) and dispersion energy (eij) can be calculated by the conventional Berthelot–Lorentz combining rules using pure-component PCSAFT parameters:

ð24Þ 1 E0

dðpIIN ðtÞpI Þ dt

1

þ

Z

 ðpIIN ðtÞpI Þ

II dð1=fN,1 Þ

dt

¼0

ð25Þ

To simplify the numerical solution of the equations, dimensionless variables were introduced [28,38,39].

t¼ 

D0  t 2

II

II j~N,2

z¼

dimensionless time

ð26Þ

Lðt ¼ 0Þ 2

¼

jN,2





Lðt ¼ 0Þ 2

D0

z2 Lðt ¼ 0Þ

standardized f lux

dimensionless length

ð27Þ

ð28Þ

D0 is a diffusion constant, L (t ¼0) is the length of the total dry polymer film and VN (t ¼0) the volume of the each dry polymer volume element.

ð31Þ

1 2 pffiffiffiffiffiffiffi

sij ¼ ðsi þ sj Þ

ð32Þ

eij ¼

ð33Þ

ei ej ð1kij Þ

In Eq. (33) above, kij is the binary interaction parameter required to correct the cross-dispersion energy parameter of each binary system. The use of PC-SAFT to calculate glassy polymer/VOC mixtures was introduced by Hesse and Sadowski [22]. The advantages of Table 1 Pure-component PC-SAFT parameters for VOCs used within this work. Component

m/M [mol g  1] r [A] ˚

e/k [K] eAiBj/k [K] jAiBj/k [-] Ref.

Ethyl acetate n-hexane 2-propanol Toluene Ethanol

0.040149 0.035479 0.060000 0.030545 0.051719

230.80 236.77 240.00 285.69 198.24

3.3079 3.7983 3.7000 3.7169 3.1771

2653.4

0.032384

[36] [36] [45] [36] [46]

L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607

the PC-SAFT model, like the thermo-physical bases and the large number of published PC-SAFT pure-component parameters, can be exploited. The pure-component and binary interaction PCSAFT parameters of all five VOCs and the two polymers as well as their mixtures are listed in Tables 1–3. Viscosity. The viscosity of the polymer-VOC mixture for different VOC concentrations below Tg,mixture is calculated using a VOCconcentration dependence according to Eq. (34).

Z ¼ Z0  expðaZ  wIIN,2 Þ

ð34Þ

The exponential decrease of the viscosity of glassy polymers with increasing VOC concentration was introduced by Thomas and Windle [47]. The term Z0 is the viscosity parameter, and aZ is the concentration dependence factor. Thomas and Windle [47] used the ratio of the time-dependent volume fraction of the VOC to the equilibrium volume fraction while mass fraction has been used within this work as introduced by Krueger [35]. 3.6. Parameters and iteration variables The sorption isotherms were calculated by fitting the mixture viscosity of the mechanical equation (the concentration dependence factor of the viscosity (aZ) and the viscosity parameter (Z0) in Eq. (34)), as well as the binary diffusion coefficient (Œ ij ) of the flux equation (Eq. (24)) to the measured sorption isotherms. The iteration variables in the volume elements, N ¼1 to n, of the model are the time-dependent pressure (pIIN ) inside the polymer film and the weight fraction of the VOC (wIIN,2 ). The weight fraction of the polymer wIIN,1 in each volume element can be calculated by Eq. (35). wIIN,1 ¼ 1wIIN,2

ð35Þ

4. Results and discussion 4.1. Measurement and modeling of sorption isotherms for the P84/VOC systems To examine the sorption and the diffusion kinetics of ethyl acetate, toluene, 2-propanol, ethanol, and n-hexane in polyimide P84, sorption measurements were performed as described above and are illustrated in Fig. 4. The non-Fickian shapes of the sorption isotherms, suggest that the observed polyimide/VOC mixtures were below Tg,mixture during all of the sorption measurements for all systems performed within this work. Therefore, the investigated P84/VOC mixtures were assumed to be glassy during all measurements. The sorption isotherms of the systems P84/toluene, P84/2-propanol, as well as P84/n-hexane show typical nonFickian two-stage behavior. In contrast, the sorption isotherms of the P84/ethyl acetate and P84/ethanol systems are s-shaped. All five sorption isotherms were modeled by means of the 1D-MS diffusion model described above and are illustrated in Fig. 4. Table 2 Pure-component PC-SAFT parameters for P84 and Matrimid [22]. Component

m/M [mol g-1]

˚ r [A]

e/k [K]

Association sites per monomer

P84 Matrimid

0.038 0.038

3.042 3.100

290 320

10 10

601

The s-shaped and the two-stage sorption isotherms could be described well by the developed model except for the P84/toluene system. For this system, the measured sorption isotherms indicate only a slight two-stage behavior whereas the model calculates a more distinctive behavior. For each system, one set of significant parameters (diffusion coefficient (Œ 21 ) and concentration-dependency factor (aZ)) was found (listed in Table 4). The Young’s modulus (E0) of P84 was set to 8.5  109 Pa which is the same as the measured Young’s modulus of Matrimid. The viscosity parameter (Z0) was fitted to the value 1  1015 Pa s and is constant for all five binary systems. As is typical for MS diffusion coefficients[40], the diffusion coefficients determined within this work are VOC-concentration independent (see Table 4). The PC-SAFT pure-component and binary interaction parameters applied to calculate the chemical potentials and volumes for each volume element are presented in Tables 1–3. Each property influences a distinct part of the sorption isotherm. The initial slope (first stage) of the two-stage sorption profiles of the toluene, n-hexane, and 2-propanol systems is highly influenced by the value of the MS diffusion coefficients. The first stage of the sorption profile is assumed to be affected mainly by the diffusion in the glassy polymer and not by the slow swelling of the glassy polymer chains (already noted by Watt [48]). Until the first stage is reached, the polymer/VOC system is in a quasi-equilibrium state where no polymer swelling has yet taken place. The swelling pressure (pswell) in the polymer film reaches its maximum value due to the initial VOC uptake. As an example, Fig. 5a) and b) shows the modeled swelling pressure in the different volume elements for the system P84/n-hexane. The maximum swelling pressure of 135 bars is reached after 4.7 h (130 s1/2). At this time, the quasi-equilibrium state and, therefore, the first stage of the P84/n-hexane sorption curve are reached (Fig. 4).The pressures are identical in each volume element inside the whole P84 film. After this first stage, the glassy polymer chains start to swell, and the swelling pressure decreases simultaneously in all volume elements down to zero. The comparison of the fitted diffusion coefficients in Table 4 shows that the toluene diffusion in P84 is the fastest whereas n-hexane and 2-propanol diffuse more slowly. This behavior can be attributed to the different VOC molecular structures. Toluene might fit very well into the polymer structure even without remarkable swelling. After passing the first diffusion stage, the polymer starts to swell, and the swelling pressure (pswell) in the polymer decreases. The faster the polymer swells, the faster the equilibrium concentration is reached [22]. This swelling velocity depends directly on the mixture viscosity and on the concentration-dependence factor (aZ) (Z0 is the same for all P84/VOC systems). The smaller this concentration-dependence factor, the faster the polymer swells with increasing VOC concentration in the polymer film. The concentration-dependence factor varies from  310 for toluene to  22 for ethyl acetate. In particular, the s-shaped sorption isotherms of P84/ethyl acetate and P84/ethanol have only slightly negative concentration-dependence factors (aZ). In both cases, the VOC sorption takes a rather long time until reaching the equilibrium concentration. For these systems, the diffusion coefficients are very small. The diffusion of these VOCs is so small that even the swelling of the polymer chains starts before a quasiequilibrium concentration is reached in the polymer film. These sorption curves are therefore s-shaped. The profiles of the calculated swelling pressure for the system P84/ethyl acetate inside the volume elements of the polymer film are shown in Fig. 6a) and b). The swelling pressure of the first volume element is the darkest line. As soon as the sorption begins, the swelling pressure within this first volume element reaches its maximum because of the boundary condition that the chemical potential of the surrounding liquid (phase I) is always equal to the chemical potential of the VOC within the first volume element. At this point, the driving force, respectively the difference of the chemical potential of the pure VOC in phase I and in the first volume element, is at maximum. This high driving force decreases towards the middle of the polymer film. In the polymer film, the VOC flux is caused only by the difference in chemical potentials of the VOCs in the two adjacent volume elements. The maximum swelling pressure in a volume element depends on its position within the polymer film. The closer the volume element to the middle of the film, the lower its maximum pressure. In comparison to the twostage behavior of the P84/n-hexane system (Fig. 4), the swelling pressure in each volume element of the P84/ethyl acetate system reaches its maximum value at different times. Fig. 5 shows that the swelling pressure of each single volume element decreases immediately after reaching the respective maximum value. The swelling pressure of the P84/ethyl acetate system does not decrease homogeneously in the whole polymer film as observed for the two-stage sorption

Table 3 PC-SAFT binary interaction parameters (kij)a [22]. Component

P84 Matrimid a

kij Ethanol

Ethyl acetate

n-hexane

2-propanol

Toluene

 0.0854 þ 1.308  10  4  T  0.1115 þ 1.230  10  4  T

 0.0503 þ8.3  10  5  T  0.0765 þ9.5  10  5  T

0 -

0.01  0.07

0.035  0.0677 þ9  10  5  T

The temperature T is given in [K].

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L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607

Fig. 4. Sorption isotherms of different VOCs in P84 at 25 1C. Symbols represent the measured points, and lines are the modeled sorption isotherms.

Table 4 Model parameters used within this work.b parameter

Ethanol

2

Œ 21 [m /s] 8.5  10 ag [–]

 35

 16

Ethyl acetate 4  10  22

 16

n-hexane 6  10  100

 14

2-propanol 9  10  60

 15

Toluene 3  10  13  310

b Fitted diffusion coefficients and viscosity parameters for the VOC sorption in polyimide P84.

behavior of the P84/n-hexane system but decreases separately in each single volume element. For the binary systems where the sorption is very slow, the equilibrium concentration of the VOC in the polymer film is very high (e.g., wequilibrium,ethylacetate ¼ 18 wt%). The order of the equilibrium concentrations for the VOCs at 25 1C being considered is shown below [22]: ethyl acetate4ethanol4toluene4n-hexane42propanol. In contrast, the order of the diffusion coefficients in the P84 film is: toluene4n-hexane42-propanol4ethanol4ethyl acetate. The order of the solubility (equilibrium concentrations) is not equal to the order of the diffusion coefficients. Therefore it is not possible to deduce from one property to the other, but it is important to measure diffusion as well as solubility. The particular order of solubilities and diffusion coefficients results from both, the different structure of the VOC molecules as well as their different intermolecular (e.g., polar) interactions with P84.

The ethanol molecule is more polar and smaller than the 2-propanol molecule which might cause the higher solubility of ethanol compared to 2-propanol. The smaller polarity of 2-propanol and the therefore smaller interaction with P84 is also an explanation for the fast diffusion of 2-propanol in P84. Thus, the diffusion coefficient of 2-propanol is higher than that of ethanol. The solubility of toluene in P84 is smaller than that of ethanol. First reason might be the more complex molecular structure of toluene. In addition to that, toluene is less polar than ethanol which could explain the smaller solubility as well as its bigger diffusion coefficient caused by weaker interactions with P84. However, the solubility of toluene is higher compared to the one of n-hexane. Hence, the phenyl ring of toluene fits better through the P84 chains than the linear n-hexane molecule. The order of the toluene and n-hexane solubility is equal to the order of their diffusion coefficients. Obviously, the better fitting of the toluene phenyl ring also causes a faster diffusion compared to that of the linear n-hexane molecules.

4.2. Measurement and modeling of sorption isotherms in Matrimid/VOC systems Measured sorption isotherms of the binary Matrimid/toluene, Matrimid/ethyl acetate, Matrimid/2-propanol, and Matrimid/ethanol systems are illustrated in Fig. 7. Like the sorption curves of the P84/VOC systems, all measured Matrimid/ VOC sorption isotherms show typical non-Fickian sorption behavior. Since this behavior appears in cases when VOC diffuses in glassy polymers, the assumption that the Matrimid/VOC mixtures remain glassy during the sorption measurements

200

200

150

150

pswell [bar]

pswell [bar]

L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607

100 50 0

0

500

1000 t1/2

1500

100 50 0

2000

603

0

100

[s1/2]

200

t1/2

300

[s1/2]

600

600

500

500

400

400

pswell [bar]

pswell [bar]

Fig. 5. Swelling pressure profiles within the polymer film as calculated by the 1D-MS diffusion model for the P84/n-hexane system. The different shades of gray indicate different volume elements of the film. The darkest curves indicate swelling pressure profiles of the first volume element. The brighter the curve, the closer the element is to the middle of the polymer film. (a) Whole swelling pressure curves of the respective volume elements. (b) Zoom view of the first part of the swelling pressure curves.

300 200 100 0

300 200 100

0

1000

2000 t1/2

[s1/2]

3000

4000

0

0

200

400 t1/2

600

800

1000

[s1/2]

Fig. 6. Swelling-pressure profiles within the polymer film as calculated by the 1D-MS diffusion model for the P84/ethyl acetate system. The different shades of gray indicate different volume elements of the film. The darkest curves indicate swelling pressure profiles of the first volume element. The brighter the curve, the closer the element to the middle of the polymer film. (a) Whole swelling-pressure curves of the respective volume elements. (b) Zoom view of the first part of the swelling-pressure curves.

Fig. 7. Sorption isotherms of different VOCs in Matrimid at 25 1C. Symbols represent the measured points, and lines are the modeled sorption isotherms.

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is again confirmed. The shapes of the isotherms of the Matrimid/ethyl acetate systems are s-shaped, but the isotherms of Matrimid/toluene, Matrimid/2-propanol as well as of the Matrimid/ethanol systems seem to be typically two-staged. The Matrimid/VOC sorption isotherms were again modeled using the 1D-MS diffusion model. Fig. 7 shows the measured and the modeled sorption isotherms. All isotherms appear to be in good agreement with the experimental data. The fitted diffusion coefficients (Œ 21 ), viscosity parameters (Z0) and concentration dependency factors (aZ) are listed in Table 5. In contrast to the constant viscosity parameter (Z0) for P84, the viscosity parameters had to be fitted for each binary Matrimid/VOC system. The Young’s modulus for Matrimid was determined experimentally to be 8.5  109 Pa. All fitted diffusion coefficients of the Matrimid/VOC systems are again VOC concentration independent, typical for MS diffusion coefficients [40]. The needed PC-SAFT pure component and binary interaction parameters to calculate the chemical potentials and the volume factions are presented in Tables 1–3. The sorption isotherm of Matrimid/toluene is slightly two-staged. The sorption curves of Matrimid/ethanol and Matrimid/2-propanol have two distinct stages. Fig. 8a) and b) shows the modeled pressure profiles of the swelling pressure (pswell) in the volume elements of the system Matrimid/ethanol. The

Table 5 Model parameters determined within this work.c Parameter 2

Œ 21 [m /s] ag [–] g0 [Pa s]

Ethyl acetate  13

Toluene -13

4.6  10  14.2 2.3  1011

1.0  10  9.5 3.2  1012

Ethanol

2-propanol

-13

5.9  10  24.0 1.8  1013

1.0  10  13  35.0 1.0  1015

500

500

400

400 pswell [bar]

pswell [bar]

c Fitted diffusion coefficients and viscosity parameters for the VOC sorption in Matrimid.

swelling pressure inside the whole polymer film increases very rapidly to the maximum value of 390 bars at the time 0.11 h (20 s1/2). At this point, the quasiequilibrium, the first stage of the sorption curve, is reached, seen by comparing the sorption curve in Fig. 7 and the swelling pressure profile in Fig. 8a) and b). Prior to the second stage, a homogeneous decrease of the swelling pressure to zero is observed in all volume elements. At the same time, the glassy polymer chains start to swell. The fitted diffusion coefficients of the ethanol diffusion in Matrimid indicate that ethanol diffuses as fastest and toluene and 2-propanol as slowest. All fitted diffusion coefficients of the Matrimid/VOC systems are in roughly the same range of magnitude. The differences in the molecular structure of the VOC molecules influence the diffusion in Matrimid, although not as much as for the P84/VOC systems. The time to reach the equilibrium concentration corresponds to the swelling time of the polymer. This swelling velocity depends directly on the mixture viscosity calculated by Eq. (37). The smaller the mixture viscosity (Z) of a specific Matrimid/VOC system compared to the others, the faster the polymer swells with increasing VOC concentration in the polymer film. The lightly two-stage sorption curve of the system Matrimid/toluene and the s-shaped sorption isotherms of Matrimid/ethyl acetate have only slightly negative viscosity concentration dependence factors (aZ) of  9.5 and  14.2 compared to the Matrimid/ethanol and Matrimid/2-propanol systems with  24 and  35. Therefore, the change in the mixture viscosity with VOC concentration is slow. By observation of the mixture viscosities of the Matrimid/toluene and Matrimid/ethyl acetate systems calculated by Eq. (37), these viscosities are, in general, very small: the polymer matrix can swell faster compared to the Matrimid/ethanol and Matrimid/2-propanol systems. For the Matrimid/toluene and Matrimid/ethyl acetate systems, the diffusion is slower than the polymer chain swelling, so the sorption curves are lightly two-stage and s-shaped. The profiles of the swelling pressure within the volume elements for the system Matrimid/ethyl acetate are shown in Fig. 9a) and b). At the beginning of the sorption process, the maximum swelling pressure within this first volume

300 200 100 0

300 200 100

0

100

200 300 t1/2 [s1/2]

400

0

500

0

10

20 t1/2 [s1/2]

30

40

600

600

500

500

400

400

pswell [bar]

pswell [bar]

Fig. 8. Swelling-pressure profiles within the polymer film as calculated by the 1D-MS diffusion model for the Matrimid/ethanol system. The different shades of gray indicate different volume elements of the film. The darkest curves indicate swelling pressure profiles of the first volume element. The brighter the curve, the closer the element is to the middle of the polymer film. (a) Whole swelling-pressure curves of the respective volume elements. (b) Zoom view of the first part of the swellingpressure curves.

300 200 100 0

300 200 100

0

20

40

60

t1/2 [s1/2]

80

100

0

0

5

10

15

20

t1/2 [s1/2]

Fig. 9. Swelling-pressure profiles within the polymer film as calculated by the 1D-MS diffusion model for the Matrimid/ethyl acetate system. The different shades of gray indicate different volume elements of the film. The darkest curves indicate swelling pressure profiles of the first volume element. The brighter the curve, the closer the element to the middle of the polymer film. (a) Whole swelling-pressure curves of the respective volume elements. (b) Zoom view of the first part of the swelling-pressure curves.

L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607 element is reached for the same reason as already explained for the P84/ethyl acetate system. It can be observed that the maximum pressure of the volume elements is different for each specific volume element. Hence, the same statement, as already mentioned for the P84/VOC systems, can be note here: the closer the distance of an element to the middle of the polymer film, the lower is its maximum pressure. In addition the maximum value of the swelling pressure in each volume element of the Matrimid/ethyl acetate system is reached at different times, which is shown in Fig. 8a) and b). Immediately after reaching the respective maximum value, the swelling pressure of each single volume element decreases. The decreasing of the swelling pressure of the Matrimid/ethyl acetate system is not homogeneously in the whole polymer film. Here it decreases separately in each single volume element. The order of the VOC solubility in Matrimid at 25 1C is as follows [22]: toluene4ethyl acetate4ethanol4 2-propanol. In contrast, the order of the diffusion coefficients is: ethanol4ethyl acetate4 toluene/2-propanol. Again, the order of the solubility (equilibrium concentrations) is not equal to the order of the diffusion coefficients. Therefore it is not possible to derive one property from the other, but it is important to measure both, the diffusion as well as the solubility. The high solubility of toluene can be explained by the molecular structure of the monomers of the polyimide Matrimid. One monomer consists of four phenyl rings and three CH3 groups. It can be assumed that the toluene molecule has a high affinity to one Matrimid monomer which leads to a high equilibrium concentration. The solubility of ethanol is a little bit higher than the solubility of 2-propanol. The order of these two solubilities can be explained by the size and polarity of both molecules. Ethanol is smaller and more polar than 2-propanol. Therefore, the smaller molecule fits better through and interacts more with the Matrimid monomer. This small size of the ethanol molecule might also cause the high diffusion coefficient compared to the other compounds. Thus, the diffusion coefficient of the slightly larger 2-propanol is smaller than the diffusion coefficient of ethanol. The value of the diffusion coefficient of toluene might be so small because of its phenyl ring. Hence, the diffusion velocity is small because of the slow swelling and rearranging of the polymer chains.

4.3. Comparison of the P84/VOC and Matrimid/VOC sorption behavior A comparison of the diffusion coefficients of Matrimid/VOC vs. P84/VOC systems reveals significant differences. An ascending order of diffusion coefficients in Matrimid/VOC systems for: ethanol4 ethyl acetate4 toluene/2-propanol and in P84/VOC systems for: toluene4n-hexane 42-propanol4ethanol 4ethyl acetate is found. By comparing the order of the diffusion coefficients of the P84/toluene and Matrimid/toluene systems the molecular effect of the molecular structure of, for example, toluene on the diffusion in the polyimide P84 cannot be observed for the toluene diffusion in Matrimid. Toluene diffuses as fastest in the polyimide P84 and as slowest in Matrimid. Matrimid has much more CH3 side groups compared to P84 (see Fig. 1). This side groups probably hinder the toluene molecules in the polymer which would cause a slower diffusion. Compared to the slow diffusion velocity of toluene in Matrimid, the solubility of toluene in Matrimid is very high compared to the solubility in P84. Therefore, the affinity of toluene might be higher to Matrimid than to P84. The solubility of all VOCs is higher in Matrimid than in P84. Matrimid has more side groups and, hence, a less regular structure than P84. Maybe, this less regular structure allows only for a loose packing of the polymer chains. The free space between the Matrimid polymer chains (higher free volume) might cause the higher solubility of the VOCs. In addition, the diffusion coefficients of all VOCs in Matrimid are very high compared to this in P84, which could also be caused by the different free volume of the both polyimides. Hence, a fast diffusion of the VOC molecules in P84 is prevented because of the small free volume of this polymer.

5. Conclusion In this work, the sorption isotherms of five different VOCs (nhexane, ethyl acetate, 2-propanol, ethanol and toluene) in two industrially important polyimides were calculated. Matrimid and P84 have been measured at 25 1C by means of gravimetric sorption. The shapes of the measured isotherms show typical non-Fickian sorption behavior. The isotherms of the systems P84/ ethyl acetate, P84/ethanol, and Matrimid/ethyl acetate are s-shaped, but the other isotherms show two-stage behavior. The non-Fickian shape of the Matrimid/VOC and P84/VOC sorption curves suggests that all of the measurements were conducted below the glass transition temperature of the investigated polyimide/VOC mixtures.

605

The comparison of the VOC sorption in both polyimides showed that the VOC sorption in Matrimid is, in general, faster than the sorption in the polyimide P84. The only exception is the sorption of the 2-propanol, for which the sorption velocity in Matrimid is slower than in P84. The order of the sorption velocities to reach equilibrium concentration in the Matrimid films is: ethyl acetate4 toluene 4ethanol42-propanol. In contrast, the order of the sorption velocities to reach equilibrium concentration in the P84 films is: toluene4n-hexane4 2-propanol4ethanol 4ethyl acetate. All isotherms were modeled by means of a 1D-MS diffusion model. Within this model, the diffusive fluxes in the polymer were calculated using the Maxwell–Stefan diffusion equation. A mechanical equation was used to model the viscoelastic swelling of the glassy polymer chains below their Tg. An overall mass balance of all species in the system is accounted for. The PC-SAFT equation of state is implemented in the 1D-MS diffusion model to calculate the chemical potential of each component and the volumes at each system pressure, concentration and temperature. Model parameters required to describe the diffusion and the mechanical properties are the diffusion coefficient, the viscosity concentration dependence factor, and the viscosity. All fitted MS diffusion coefficients are VOC-concentration independent, which is typical for these coefficients [40]. All measured sorption isotherms could be calculated to be in good agreement with the experimental data. By means of this modeling, diffusion coefficients for very important industrial binary systems could be determined. Acknowledgment ¨ The authors would like to thank Kai-Martin Kruger for fruitful ¨ ¨ ¨ discussions and ‘EUROPAISCHE UNION Europaischer Fonds fur regionale Entwicklung’ for supporting this work within the Chek.NRW program (Project No. w0711ck038c).

Nomenclature A dz dz

Πij D0 E0 F(t) Fi j j j~ k L Mn m n N p R Tg T t u u V

Helmholtz energy (J) transformed coordinate system (m) length (m) Maxwell–Stefan diffusion coefficient (m2 s  1) diffusion constant (m2 s  1) Young’s modulus (Pa) creep compliance (Pa  1) driving force (m s  1) diffusive flux (kg m  2 s  1) standardized flux (kg m  2 s  1) standardized flux (kg m  3) interaction parameter length (m) molar mass (g mol  1) segment number number of volume elements volume element pressure (Pa) gas constant (J mol  1 K  1) glass transition temperature (1C K) temperature (K) time (s) velocities (m s  1) transformed velocity (m s  1) volume (m3)

606

w x z

L. Hesse et al. / Journal of Membrane Science 415-416 (2012) 596–607

weight fraction (  ) weight percent (%) mole fraction ( ) direction of a Cartesian coordinate system

Greek letters

m B ri f

t z

j e s Z Z0 eAiBi aZ I,II

kAiBi

chemical potential (J/mol) friction coefficient transformed density (kg m  3) volume fraction dimensionless time dimensionless length fugacity coefficient elongation dispersion-energy parameter (J) ˚ stress (Pa) segment diameter (A) viscosity (Pa s  1) viscosity parameter (Pa s  1) association energy parameter (J) concentration dependence factor phase association volume

Subscribes i,j,k þ

0i 1 2 assoc disp hc res swell t

species reference pure component polymer VOC association dispersion hard-chain residual swelling total

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