Ind. Eng. Chem. Res. 1998, 37, 177-184
177
Membrane Design for Pervaporation or Vapor Permeation Separation Using a Filling-Type Membrane Concept Takeo Yamaguchi,* Yosuke Miyazaki, Shin-ichi Nakao, Toshinori Tsuru,† and Shoji Kimura‡ Department of Chemical System Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
A new membrane design concept is proposed. Using the filling-type membrane, the membrane solubility and the suppressing effect of swelling can be controlled independently. The fillingtype membrane is composed of two materials: the porous substrate and the filling polymer which fills the pores of the substrate. The filling polymer exhibits permselectivity due to the solubility difference, and the porous substrate matrix suppresses the swelling of the filling polymer due to its mechanical strength. When membrane solubility, the suppression effect of membrane swelling, and solvent diffusivity in the membrane can be qualitatively predicted by model simulation, the filling-type membrane concept and the prediction method enable us to choose a suitable membrane material for a separation mixture. Pervaporation and vapor permeation flux through the membrane can be estimated by the calculated solubility, diffusivity, and suppression effect without any adjustable parameters. The calculated fluxes roughly agreed with experimental fluxes in a wide range between 0.002 and 48 kg/m2 h, although a fillinggrafted polymer was treated as a pure polymer. Effects of substrate strength and operating temperature were examined by the model. Introduction Organic liquid or vapor separation is one of the most important fields for membrane applications, and there are many kinds of organic mixtures to separate. A suitable membrane should be prepared for each organic mixture separation. At present, many researchers are trying to test many kinds of membrane materials for the separations because there is not yet a design concept to prepare a suitable membrane. In this paper, we propose a new design concept for membrane preparation. Pervaporation transport can be described by the solution-diffusion model. Membrane solubility and solvent diffusivity in a membrane determine the membrane transport properties. In general, solubility parameters (Hansen and Beerbower, 1971) were used to choose a membrane material for pervaporation separations based on solubility prediction (Cabasso, 1983; Yamaguchi et al., 1993), the diffusivity properties were not taken into account for the purpose. Moreover, although the solubility parameters supply a tendency toward polymer solubility, the prediction is not qualitative (Yamaguchi et al., 1992, 1993). Using the solubility parameters, solvents can be divided into two groups, good solvents or poor solvents for the polymer, and energy of mixing for the polymer and solvent cannot be obtained. For design, we need a qualitative prediction for both solubility and diffusivity. We proposed a new type of membrane for organicliquid separation, called the filling-polymerized membrane, which can control membrane swelling (Yamaguchi et al., 1991). To make a pervaporation membrane, † Present address: Department of Chemical Engineering, University of Hiroshima. ‡ Present address: Department of Chemical Engineering, University of Osaka.
the membrane swelling must be controlled because an excessive membrane swelling always leads to reduction of selectivity due to its plasticization effect. The filling membrane is composed of two materials: the porous substrate and the filling polymer, which fills the pores of the substrate. The porous substrate is completely inert to organic liquids, and the filling polymer is soluble with a specific component in the feed. The filling polymer exhibits permselectivity due to the solubility difference, and the porous substrate matrix restrains the swelling of the filling polymer due to its mechanical strength. The filling-type membranes showed high permeability and selectivity for organic-organic separation (Yamaguchi et al., 1991, 1992, 1993) and dissolved organics’ removal from water (Yamaguchi et al., 1996a). This type of membrane can be prepared by the plasmagraft filling-polymerization technique. Using this technique, the filling polymer can be linear polymer because of graft polymerization, and such a linear polymer can show a fast solvent diffusivity compared with a crosslinked polymer. Thus, this preparation technique has advantages for pervaporation membranes compared with the usual cross-linked membranes (Yamaguchi et al., 1996a). On the basis of the solution-diffusion model, membrane solubility, diffusivity, and suppression effect of membrane swelling will determine the separation and transport properties for membranes. For the filling-type membrane, membrane solubility and suppressing effect of membrane swelling can be independently controlled by choosing the filling polymer and porous substrate, respectively. When we can qualitatively predict these three factorssfilling polymer solubility for a solvent, solvent diffusivity in the filling polymer, and the suppressing effect of the filling polymer due to the substrate matrixesswe can predict pervaporation flux through the filling-type membrane. The filling-membrane concept
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178 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
and the prediction method enable us to choose a suitable filling-polymer material and porous substrate for the separation. This is the design concept we proposed in the present study. First, a separation mixture should be decided. Then, many kinds of filling-polymer and -substrates strength will be tested by calculation to design a suitable membrane for the separation. Thus, the prediction method should not include any adjustable parameters from the sorption, diffusion, or the transport experiment. Model Simulation Several pervaporation models were proposed (Greenlaw et al., 1977; Mulder and Smolders, 1984; Okada and Matsuura, 1992, etc.), and most of the models include adjustable parameters from pervaporation, sorption, or diffusion experiments. Fells and Huang (1970) described pervaporation fluxes using the free-volume theory proposed by Fujita (Fujita and Kishimoto, 1958; Fujita et al., 1959). This pervaporation model needs adjustable parameters determined by sorption and pervaporation or diffusion experiments, although the model can be adapted to multicomponent separations (Rhim and Huang, 1989; Yeom and Huang, 1992). These parameters fitting methods cannot be used for the design concept because membrane materials must only be chosen by calculation. Heintz and Stephan (1994) proposed a pervaporation model describing multicomponent fluxes. The model can predict solvent composition in a membrane using the UNIQUAC model for a multicomponent system. However, the model requires experimental data such as pure component vapor sorption isotherms, the pure component diffusion coefficient, and vapor/liquid equilibrium data of the binary mixture. Recently, Doong et al. (1995) successfully predicted multicomponent pervaporation fluxes through a polyethylene film. They employed the Unifac-FV model (Oishi and Prausnitz, 1978) for solubility, and a hybrid model of free-volume theory (Vrentas and Duda, 1977; Duda and Vrentas, 1982) and molecular theory (Pace and Datyner, 1979) for diffusivity. The hybrid model they proposed (Doong and Ho, 1992) could make better predictions than the free-volume theory proposed by Vrentas and Duda (1977; Duda and Vrentas, 1982) for these polyethylene film cases. However, the prediction method included some parameters for the polymer or solvent, and the parameters must be determined by a diffusivity experiment. In this design concept, the free-volume theory proposed by Vrentas and Duda was employed for diffusion prediction because parameters of the free-volume theory for many kinds of polymer and solvents were already determined (Zielinski and Duda, 1992; Hong, 1995), and we can predict the solvent diffusivity in polymers without any adjustable parameters from the experiment. The Unifac-FV model was used for a solubility prediction, and the suppressing effect of a porous substrate was estimated by a model which was made for the filling-type membrane (Yamaguchi et al., 1997). Assumption for Filling-Type Membranes. The filling polymers are linear polymer because the polymers are graft polymerized from the substrate pore wall (Yamaguchi et al., 1996b). The grafted polymer (the filling polymer) in the pores was assumed to be a normal linear polymer, and solubility and diffusivity of the filling polymer will be treated as pure polymer properties. The density of the grafted polymer is almost the
same as the pure polymer density (Yamaguchi et al., 1991). It is assumed that the solvent can only penetrate into the filling-polymer region, and the substrate supplies mechanical strength, suppressing the membrane swelling. Thus, properties of the pure polymer will be predicted for the filling polymer, and the suppressing effect of the substrate will be combined with the pure polymer prediction for the filling-membrane properties. Solubility Prediction. The Unifac-FV model (Oishi and Prausnitz, 1978) was used to predict the solubility in this study because the model does not include adjustable parameters, unlike the UNIQUAC model (Abrams and Prausnitz, 1975). The activity of the component i in a polymer solution is given below:
ln as ) ln aCs + ln aRs + ln aFV s
(1)
ln as ) ln vs + vp + vp2χs
(2)
Thus, χ can be estimated from the Unifac-FV model (eqs 1 and 2) using group contributions of the chemical formula. ln aCi and ln aRi are the combinatorial and residual parts of mixing, and the effects can be calculated using the ordinal Unifac model for vapor-liquid equilibrium (Fredenslund et al., 1977). For the polymer-solvent case, the free-volume effect, ln aFV i , must be considered.
v˜ s -1 v˜ m v˜ s - 1 ln aFV ) 3c ln c s 1 1 1 v˜ m1/3 - 1 1 - 1/3 v˜ s 1/3
(3)
where 3c1 is the number of external degrees of freedom per solvent, and in most cases, c1 is set to 1.1 (Oishi and Prausnitz, 1978). c1 was fixed to 1.1 in this study.
v˜ s )
vs v* s
(4)
The reduced volume for the solvent and the mixture are
v˜ s )
v˜ m )
vs 15.17br′s
∑j vjwj 15.17b
∑j r′jwj
(5)
(6)
where b can be fixed to 1.28 (Oishi and Prausnitz, 1978). Suppressing Effect of the Filling-Polymer Swelling. Swelling behavior of the filling-type membrane can be described using the following equations (Yamaguchi et al., 1997). This model was derived from a swelling model for a semicrystalline polymer such as polyethylene (Michaels and Hausslein, 1965).
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 179
ln as ) ln vs + vp + vp2χs Vs ∆Hm T 1- (vs - vs2 χp) Vm RT Tm
{
(
)
}
(7)
3(vp-1 + A - 1) A2f
A)
(nmVm - Vp) nmVm
(8)
where Vm and Vp are the molar volume of the methylene unit of the polymer and the pore volume of the porous substrate, respectively. The last term in eq 7 expresses the suppressing effect of membrane swelling by the substrate. The equation includes three parameters: χs, χp, and f. Both χs and χp express interaction between the filling polymer and solvent, and f expresses the tie segment density of the porous substrate which can suppress the filling polymer’s swelling. χs and χp are χ estimated from solvent and polymer activity, respectively. When χ is independent of solvent concentration, χs equals χp. Moreover, when χp changes between -1 and 1, the χp effect is small in eq 7, and it can be assumed to be χs. χs can be estimated from eqs 1 and 2 using the Unifac-FV model. f can be estimated by a vapor sorption experiment, and each substrate has a specific value. The porous high-density polyethylene film we used has a value of 0.15, and each filling-type membrane with the substrate should have the same value. Thus, once we set the f value for a substrate, any other adjustable parameters are not needed to estimate the filling-membrane swelling. Mutual Diffusion Coefficient. The mutual diffusion coefficient of a solute in polymer solution, D, can be predicted by the free-volume theory proposed by Vrentas and Duda (Vrentas et al., 1977; Duda et al., 1982).
D1 ) D0 exp
exp
{(
× (-E RT )
}
-(wsv*s + wpξv*p) K11 K12 ws (K21 - Tgs + T) + wp (K22 - Tgp + T) γ γ
)
( )
(9)
D′0 ) D0 exp
(-E RT )
(when E effect is negligible)
D ) D1(1 - vs)2(1 - 2 χvs)
(10)
where D1 is a self-diffusion coefficient of the solute.
ξ)
Vs(0) Vpj
Vpj ) 0.0925Tgp + 69.47
(11) (at T > Tgp)
(12)
The following parameter estimation methods were employed to avoid the use of an adjustable parameter (Zielinski and Duda, 1992; Hong, 1995). Solvent free-volume parameters, such as K11, K21, Tgs, Vs(0), and v* s, were already determined for more than 50 kinds of solvents, and we can find parameters for
the usual solvents in literature. The activated energy effect is negligible for the diffusion, and D′0 can be used instead of D0 as a pre-exponential factor. V* p can be estimated from the group contribution method, which requires a chemical formula of the monomer unit. Tgp is the glass transition temperature, and we can find many examples of data in literature. K12/γ and K22 can be estimated from WLF parameters derived from the temperature dependence of visco elastic properties of pure polymers. Some WLF parameters are available in literature. Parameters used in this study were referred from literature (Hong, 1995). Finally, χ can be predicted by the Unifac-FV model mentioned above. Pervaporation Flux. Pervaporation flux, Js, was expressed as follows (Paul, 1973; Paul and Ebra-Lima, 1971; Paul and Paciotti, 1975):
Js ) -
FsD
∂vs (1 - vs) ∂z
(13)
Equation 13 can be integrated across the membrane thickness, and permeability, Ps, can be estimated.
Ps ) Jsl ) Fs
∫vv
sf
sp
D dv (1 - vs) s
(14)
It is assumed that solubility equilibrium is reached at an interface between the feed liquid and the swollen membrane and solvent concentration in the filling polymer at the feed interface, vsf, can be treated by thermodynamic equilibrium equations. The solvent concentration in the grafted polymer at the permeate side, vsp, can be treated as zero for this case. Using eq 14, pervaporation flux, Js, can be calculated without any fitting parameters because vsf can be estimated from eqs 1, 2, 7, and 8, and the concentration dependence of the solvent mutual diffusion coefficient D can be calculated by eqs 9 and 10. Accuracy of the prediction will be discussed, and substrate effect and operation conditions will be examined using the prediction method. Experimental Section Materials. Porous high-density polyethylene (HDPE) film was used as a porous substrate. The HDPE substrate of 5 µm in thickness, 0.02 µm in pore size, was supplied by Tonen Chemical Co. Ltd. Methyl acrylate (MA) was used as the grafted monomer. The monomer was diluted in a methanol/water ) 50/50 wt % mixture to make a homogeneous membrane (Yamaguchi et al., 1996b). The grafting process has been described in detail elsewhere (Yamaguchi et al., 1991). Sorption Measurements. N-hexane solubility for poly(methyl acrylate) was measured by the quartz spring method described elsewhere (Yamaguchi et al., 1997). Diffusion Measurements. A filling membrane was used to measure a mutual diffusion coefficient of benzene in poly(methyl acrylate). In general, solvent diffusion coefficients in a polymer strongly depend on the polymer swollen state, in other words, solvent concentration in the polymer. Thus, a step-up method was employed for the diffusivity measurements. The apparatus is shown in Figure 1. Organic vapors, having the same vapor pressure, were introduced to both the feed and permeate side compartments of the membrane
180 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
Figure 1. Apparatus of mutual diffusion coefficient measurements. A: liquid feed. B: constant temperature bath. C: constant temperature oven. D: gas tank. E: membrane cell. F: baratron pressure sensor. G: computer. H: vacuum pump. I: stop cock.
cell, and the system was kept until the equilibrium was reached for the membrane swelling. Then, a small amount of additional vapor pressure was added to only the feed side compartment, and a time lag of pressure increase in the permeate side was measured using baratron pressure sensor. Concentration dependence of diffusivity is negligible in this method because the membrane swollen state at the feed and permeate sides are not different. Thus, the solvent diffusion coefficient in the membrane can be considered constant. Organic vapor pressure was controlled by changing the liquid feed temperature, and the membrane temperature was kept at 25 °C. The benzene concentration in the grafted polymer at fixed vapor activity can be calculated by eq 7 with measured benzene solubility for poly(methyl acrylate) (Yamaguchi et al., 1997). An equation to calculate the mutual diffusion coefficient from time lag is as follows (Barrer, 1937):
D)
ld2 6θ
Figure 2. Comparison between the experimental polymer solubility and calculated solubility by the Unifac-FV model. Relationship between the solvent activity and solvent volume fraction in poly(methyl acrylate) at 25 °C.
(15)
Membrane thickness change due to swelling was not considered in this calculation because this effect is not so serious. Transport Experiments. Pervaporation and vapor permeation experiments of single organic liquids or vapor through the membrane were carried out. The feed liquid or vapor was maintained at 25 or 50 °C. For the vapor permeation case, the temperature of the feed liquid reservoir was maintained below the membrane cell temperature to control the feed vapor activity contacted with the membrane. The flux of the permeate was measured by the weight of the condensed permeate liquid or gas chromatography. Benzene, toluene, ethylbenzene, o-xylene, chloroform, carbon tetrachloride, and acetone were used as the feed. Results and Discussion Prediction of Solubility. The dependence of solvent activity on solvent concentration in poly(methyl acrylate) is shown in Figure 2. Figure 3 shows the concentration dependence of the χs parameter between poly(methyl acrylate) and some solvents, which the results were converted from the data shown in Figure 2 using eq 2. Experimental results for benzene, chloroform, toluene, and carbon tetrachloride were referred
Figure 3. Comparison between the experimental polymer solubility and calculated solubility by the Unifac-FV model for poly(methyl acrylate). Relationship between the solvent activity and interaction parameters at 25 °C.
from literature (Saeki et al., 1983; Yamaguchi et al., 1997). The lines were calculated by the Unifac-FV model. In this prediction, the number of external degrees of freedom per solvent, 3c1, was fixed. Thus, there is no adjustable parameter in this prediction. For carbon tetrachloride and n-hexane, the calculated lines are in good agreement with the experimental results. For other organic cases, the calculated lines somewhat differ from the experimental results. Especially, calculated χs parameters for chloroform and toluene in the high solvent fraction region differ from the experimental results. Some parameters for the chlorinated carbon groups and toluene group or free-volume effect in the model need to be modified for a better prediction. For pervaporation or vapor permeation transport prediction, polymer solubility in the lower solvent
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 181
Figure 4. Relationship between the solvent volume fraction in poly(methyl acrylate) and mutual diffusion coefficient of the solvent at 25 °C.
fraction region is important because the down side of the membrane is kept dry to make a driving force for solvent permeation and solvent diffusivity at a lower concentration region has serious concentration dependence. The prediction is successful to some degree, especially in the lower solvent fraction region from Figure 2. For membrane design, the Unifac-FV model can supply much more important information compared with solubility parameters, although we need more accurate prediction in the future. Prediction of Diffusivity. Prediction of the solvent mutual diffusion coefficient in poly(methyl acrylate) by the free-volume theory at 25 °C is shown in Figure 4 with the experimental results of benzene. The diffusivities have a serious solvent concentration effect in the lower solvent concentration region. For benzene, the prediction agreed well with experimental results, although a MA-grafted membrane was used as a polymer sample. The assumptions we set for a filling-type membrane can be allowed in the diffusion prediction. The validity of the assumption will be rechecked by pervaporation or vapor permeation results through the membrane. On the basis of the calculation, the chloroform diffusivity is larger than the other organics diffusivity. Prediction of Pervaporation Flux through the Filling-Type Membrane. Predicted results of organic fluxes through the MA-grafted membrane were compared with experimental fluxes as shown in Figure 5. Benzene, toluene, ethylbenzene, o-xylene, and acetone were used as the feed solvents, and the temperature was fixed at 25 or 50 °C. Predicted pervaporation fluxes roughly agreed with the experimental values in the wide range between 0.002 and 48 kg/m2 h. Vapor permeation of benzene through the membrane could also be predicted by this method. Although the filling polymer was treated as a normal linear polymer for the estimation of solubility and diffusivity, the predicted results do not seriously differ from the experimental results. The grafted polymer in pores of the porous substrate can be treated as a pure polymer with the swelling suppression effect as shown in eq 7. This is a great advance of a filling-type membrane for the membrane design because usual prediction methods for linear polymers can be utilized for the practical membrane. At present, we could not predict accurate flux values because each prediction step includes small inaccuracies, and those small inaccuracies lead to difficulty in
Figure 5. Comparison between the experimental solvent flux through poly(MA) filling membrane and calculated flux by the model described in text at 25-50 °C.
Figure 6. Solvent concentration profile in the filling polymer between the feed and permeate interfaces during pervaporation operation at 25 °C.
obtaining accurate flux calculations. However, the prediction can help us to choose a suitable membrane material for a separation or removal of organic vapor, and we can examine the substrate strength effect or operation condition effect by the simulation. We are trying to predict binary mixture separation for organic mixtures and also trying to predict more accurate results for the present design concept. Simulation of Membrane Permeability through a MA-Grafted Membrane. Figure 6 shows the calculated concentration profile of the solvent in the membrane during pervaporation. The membrane at the feed side is in a swollen state, and the concentration gradient at the permeate side is steep. The reason why the membrane at the permeate side is dry is that the solvent diffusivity must be extremely low compared with that of the swollen feed side, as shown in Figure 4. At a steady state, the solvent flux must be the same at each point of the membrane. Thus, a larger driving force is needed at the permeate side because diffusivity is small in such a dry membrane. The suppressing effect of the membrane swelling affects the membrane permeability, and the effect is determined by the mechanical strength of the porous substrate. In the model of swelling suppression, the f value represents the mechanical strength of the crystalline polyethylene substrate. Each high-density poly-
182 Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998
Figure 7. Mechanical strength of porous substrate effect on pervaporation permeability through MA-grafted filling membrane. Relationship between the unit ratio of the tie segment to the filling polymer f and permeability at 25 °C.
Figure 9. Temperature dependence of chloroform permeability through the MA-grafted filling membrane.
swelling suppressing effect is more serious for a low swollen state like benzene permeability with a high f value. To suppress membrane swelling at high temperatures, we should choose a substrate material having a high melting point of the crystal. Conclusions
Figure 8. Temperature dependence of benzene permeability through the MA-grafted filling membrane.
ethylene substrate has a specific f value, and the substrate used in this study has an f value of 0.15. Figure 7 shows a prediction of solvent permeability through filling-type membranes with different porous polyethylene substrates. Although chloroform showed higher permeability with a large f value, other solvents displayed a serious suppressing effect on the permeability. The calculations are for single-solvent permeability; however, we can imagine the mixture separation from the results. Poly(methyl acrylate) filling membrane may show high selectivity and relatively high permeability for chloroform over other organics from the mixture. Experimental results through a MA-grafted membrane showed high-chloroform permselectivity from chloroform/n-hexane or chloroform/carbon tetrachloride mixtures (Yamaguchi et al., 1992). Temperature dependence of benzene and chloroform permeabilities are shown in Figures 8 and 9, respectively. Solvent permeabilities increase with an increase of temperature, especially for benzene with 0.3f case. The suppression effect of membrane swelling is due to tie segments between polyethylene crystals, and the elastic energy of the crystals decreased with an increase of temperature, and the effect will disappear above the melting point of the polyethylene crystal. As a result, the membrane will show high permeability at high temperature because of the high swollen state. Of course, solubility and diffusivity also have temperature dependence; however, temperature dependence of the
A new design concept for a pervaporation or vapor permeation membrane was proposed. The concept consists of a filling-membrane methodology and a prediction method. The prediction method needs parameters of tie segment density in a substrate, f, glass transition temperature, Tg, and WLF parameters for a filling polymer. The parameter, f, is an intrinsic parameter for a porous substrate, and Tg and WLF parameters for many pure polymers are available in reference tables. We tried to predict pervaporation fluxes through filling-type membranes without any fitting parameters and obtained the following results: (1) Solubility of the filling polymer to some solvents was calculated by the Unifac-FV theory, and quantitative prediction can be done in the low-solvent concentration region. This method is much more useful for the membrane design compared with solubility parameter methods. (2) Benzene diffusivity in a grafted polymer can be predicted by the free-volume theory, assuming that the grafted polymer is a pure polymer with elastic properties due to the substrate matrix. The grafted polymer in substrate pores can be predicted by the theories for linear polymers. (3) Pervaporation fluxes of several solvents through a poly(methyl acrylate) filling-type membrane can be predicted using the model simulation. Although the filling polymer was assumed to be a pure polymer, the prediction roughly estimates solvent fluxes in the wide range between 0.002 and 48 kg/m2 h. This prediction method can be utilized for the membrane design concept we proposed in this study. (4) The solvent concentration profile in the membrane during pervaporation was estimated. Concentration gradients at the permeate side were very steep because of poor diffusivity. (5) A porous substrate effect and operation temperature effect for pervaporation fluxes could be simulated by the simulation.
Ind. Eng. Chem. Res., Vol. 37, No. 1, 1998 183
1 A ) Ad v 2/3
Acknowledgment The authors thank Tonen Chemical Co. Ltd. for supplying the porous high-density polyethylene films and Mr. Takashi Sugawara for his helpful guidance of the diffusion experiments. Nomenclature A: membrane area during pervaporation Ad: dry membrane area as: solvent activity 3c1: number of external degrees of freedom per solvent D: mutual diffusion coefficient D0: constant pre-exponential factor D′0: constant pre-exponential factor when E is set equal to 0 D1: self-diffusion coefficient E: activation energy for diffusion f: unit ratio of tie segment to filling polymer J: pervaporation flux K: free-volume parameters l: membrane thickness during pervaporation ld: dry membrane thickness P: permeability Q: flux velocity R: gas constant T: temperature Tg: glass transition temperature Tm: melting temperature of polyethylene crystal v: volume fraction of solvent and polymer in a fillingpolymer region v*: core segment volume v˜ : reduced volume V: molar volume Vm: pore volume of porous substrate Vs(0): molar volume of pure solvent at 0 K Vp: molar volume of methylene unit w: weight fraction Greek Letters χ: Flory interaction parameter χs: Flory interaction parameter estimated from solvent activity χp: Flory interaction parameter estimated from polymer activity ∆Fel: elastic energy of polyethylene ∆Fm: mixing energy of amorphous polymer and solvent ∆Hm: molar heat of fusion of polyethylene crystal θ: time lag F: density γ: overlap factor which accounts for shared free volume ξ: ratio of critical molar volume of solvent jumping unit to that of polymer jumping unit
Appendix Flux calculations need membrane thicknesses at the permeation state. Paul and Paciotti (1975) considered the membrane thickness change at the permeation state. In this study, membrane area change during permeation was also considered. In this calculation, the volume change of mixing was not considered, and the local swelling was assumed to be isotropic. Equation 14 can be changed to the following equations considering the membrane area change:
Qs )
dv ∫vv (1 AD -v) s
Fs l
sf
sp
s
(16)
(17)
p
J′s )
Qs Ad
(18)
where Qs is the solvent flux velocity and Ad and A are membrane areas at a dry state and a swollen state during permeation, respectively. J′s is a solvent flux which corresponds to experimental flux. Thickness change can be described as follows. The total polymer volume can be treated as a constant at a dry state and a permeation state because a volume change by mixing is not considered in this case.
ldAd )
∫0lAvp dz
(19)
z l
(20)
∫01vp1/3 dt
(21)
t) Then,
ld ) l
From eq 14, the following relation can be obtained:
F AD
∫vv (1 s- v ) dvs st
sp
t)1-
s
Qs l
(22)
Combined with eqs 16, 17, 21, and 22, thickness l can be obtained. J′s was used in Figure 5 to compare with the experimental results. These calculations are for normal polymeric membrane cases, and we must consider the substrate volume effect for the filling membrane. The following effective filling-polymer area and membrane thickness should be used in the above equations, instead of the usual dry area and thickness.
A′d ) Ad{1 - vsub2/3} vsub )
Vsub (Vsub + Vgraft)
l′d ) l(1 - vsub)1/3 + ldvsub1/3
(23) (24) (25)
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Received for review July 3, 1997 Revised manuscript received October 14, 1997 Accepted October 15, 1997X IE970464E
X Abstract published in Advance ACS Abstracts, December 15, 1997.
