1050
Ind. Eng. C h e m . R e s . 1992,31, 1050-1060
Eglinton, G.; Hamilton, R. J. Leaf Epicuticular Waxes. Science 1967, 156, 1322-1335. Fish, R. H. Speciation of Trace Organic Ligands and Inorganic and Organometallic Compounds in Oil Shale Process Waters. In Thirteenth Oil Shale Symposium Proceedings; Gary, J. H., Ed.; Colorado School of Mines: Golden, CO, 1980; pp 385-391. Fookes, C. J. R.; Walters, C. K. A Chemical Investigation of Shale Oil Ageing. Fuel 1990, 69, 1105-1108. Furimsky, E. Chemistry of Catalytic Hydrodeoxygenation. Catal. Rev.-Sci. Eng. 1983,25, 421-458. Guenther, F. R.; Parrish, R. M.; Cheder, S. N.; Hilpert, L. R. Determination of Phenolic Compounds in Alternative Fuel Matrices. J. Chromatogr. 1981,207, 256-261. Harvey, T. G.; Matheson, T. W.; Pratt, K. C.; Stanborough, M. S. Catalyst Performance in Continuous Hydrotreating of Rundle Shale Oil. Znd. Eng. Chem. Process Des. Dev. 1986,25, 521-527. Harvey, T. G.; Matheson, T. W.; Pratt, K. C.; Stanborough, M. S. HvdroDrocessinnof Shale Oil Using- Ruthenium-based Catalysts. Fuel f987, 66, 766-770. Hertz, H. S.; Brown, J. M.; Guenther, F. R.; Hilpert, L. R.; May, W. E.; Parris. R. M.; Wise, S. A. Determination of Individual Organic Compounds in Shale Oil. Anal. Chem. 1980,52,1650-1657. HP 1989 Analytical Supplies Catalog and Chromatography Reference Guide. Hewlett-Packard: Palo Alto, CA, 1989; p 80. Jaffe, R.; Albrecht, P.; Oudin, J. L. Carboxylic Acids as Indicators of Oil Migration I1 Case of Mahakan Delta, Indonesia. Geochim. Cosmochim. Acta 1988,52, 2599-2607. Kaluny, R. K. M. R.; Restivo, W. M.; Tidwell, T. T.; Boocock, D. G. B.; Crimi, A.; Douglas, J. Hydrodeoxygenation of Hydroxy, Methoxy and Methyl Phenols With Molybdenum Oxide/Nickel Oxide/Alumina Catalyst. J. Catal. 1985, 96, 535-543. Mackenzie, A. S.; Wolff, G. A,; Maxwell, J. R. Fatty Acids in Some Biodegraded Petroleums. Possible Origin and Significance. In Advances in Organic Geochemistry; Bjoroy, M., et al., Eds.; Wiley: Chichester, UK, 1983; pp 637-649. McClennen, W. H.; Meuzelaar, H. L. C.; Metcalf, G. S.; Hill, G. R. Characterization of Phenols and Indanols In Coal-derived Liquids. Fuel 1983,62,1422-1429. McCreham, W. A.; Brown-Thomas, J. M. Determination of Phenols In Petroleum Crude Oils Using Liquid Chromatography With Electrochemical Detection. Anal. Chem. 1987, 59, 477-479. Odebunmi, E. Q.; Ollis, D. F. Catalytic Hydrodeoxygenation I: Conversion of o-, m- and p-cresols. J. Catal. 1983, 80, 56-64. Regtop, R. A.; Crisp, P. T.; Ellis, J. Chemical Characterization of Shale Oil From Rundle, Queensland (Australia). Fuel 1982, 61, 185-194.
Riley, R. G.; Shiosaki, K.; Bean, R. M.; Schoiengold, D. M. Solvent Solubilization, Characterization and Quantitation of Aliphatic Carboxylic Acids in Oil Shale Retort Waters Following Chemical Derivatization with Boron Trifluoride in Methanol. Anal. Chem. 1979, 51, 1995-1998. Rollmann, L. D. Catalytic Hydroprocessing of Model Nitrogen, Sulphur and Oxygen Compounds. J. Catal. 1977, 46, 243-251. Rovere, C. E.; Crisp, P. T.; Ellis, J.; Bolton, P. Chemical Class Separation of Shale Oils By Low Pressure Liquid Chromatography on Thermally-Modified Adsorbants. Fuel 1990,69, 1099-1104. Satterfield, C.; Smith, C. S.; Ingalls, M. Catalytic Hydrodenitrogenation of Quinoline: Effect of Water and Hydrogen Sulphide. Ind. Eng. Chem. Process Des. Dev. 1985,24,1000-1004. Schmal, M. "Hidrogenaggo do Oleo de Xisto, loRelaMrio"; COPPETEC/COPPE/UFRJ; Technical Report ET-11150, 1988; pp 32. Silva, J. C. DSc. Thesis, Universit6 Paris VI, France, 1986. Silverstein, R. M.; Bassler, G. C.; Morril, T. C. Spectrometric Identification of Organic Compounds; Wiley: New York, 1981; Chapter 3. Souza, G . L. M.; Silva, M. I. P.; Schmal, M. Hidrotratamento Catalitico de Oleo de Xisto em Leito Fluidizado Trifasico. Third Brazilian Symposium on Catalysis, Aug 21-23,1985, SalvadorBA; Instituto Brasileiro de Petroleo: Rio de Janeiro, 1985; pp 209-223. Stenhagen, E.; Abrahamson, S.; McLafferty, E. W. Atlas of Mass Spectral Data; Interscience Wiley: London, 1969; Vols. I and 11. Ternan, M.; Brown, J. R. Hydrotreating a Distillate Liquid Derived from Sub-bituminous Coal Using a Sulfided Cobalt (11) Oxide Molybdenum Trioxide Alumina Catalyst. Fuel 1982, 61, 1110-1118. Tissot, B. P.; Welte, D. H. Petroleum Formation and Occurrence; Spring-Verlag: Berlin, Heidelberg, 1984; p 105. Van Meter, R. A.; Bailey, C. W.; Smith, J. R.; Moore, R. T.; Allbright, C. S.; Jacobson Jr., I. A.; Hylton, V. M.; Ball, J. S. Oxygen and Nitrogen Compounds in Shale-Oil Naphtha. Anal. Chem. 1952, 24, 1758-1763. White, C. M.; Jones, L.; Li, N. C. Ageing of SRC-I1 Middle Distillate from Illinois no. 6 Coal 11. Further Evidence of Phenolic Oxidative Coupling in Coal-Derived Liquids. Fuel 1983,62,1397-1403. Zweig, G.;Sherma, J. CRC Handbook of Chromatography; Chemical Rubber Co.: Boca Raton, FL, 1972; p 90. Receiued for reuiew September 3, 1991 Reuised manuscript received November 22, 1991 Accepted December 14, 1991
Diffusion of Hydrocarbons in Polyethylene Shain J. Doongt and W. S. Winston Ho* Corporate Research, Exxon Research and Engineering Company, Route 22 East, Annandale, N e w Jersey 08801
The diffusivities of a series of aromatic hydrocarbons in semicrystalline polyethylene were obtained by the use of a modified gravimetric sorption technique with a flow system capable of a wide range of activities (or vapor pressures) and temperatures. The effects of penetrant concentration and size and temperature on the diffusivities were investigated. The experimental data can be fitted quite well by our proposed hybrid model that combines the key features of the free-volume and molecular models. In the hybrid model, polymer parameters and penetrant molecular thickness are used to determine the term equivalent to the preexponential factor of the free-volume model, and the free-volume expression is used to relate penetrant diffusivity to penetrant concentration and size. The hybrid model fitted the data better than the free-volume model, and it avoided the complexity of the molecular model. Introduction Understanding of diffusion of molecules in polymers is important for separation, packaging, and polymer processing. It also provides valuable information on molecular
* T o whom correspondence should be addressed. 'Present address: The BOC Group, Technical Center, Murray Hill, N J 07974. 0888-5885/92/2631-1050$03.00/0
motions and structures of polymers. A comprehensive review on diffusion in polymer-penetrant systems can be seen in Frisch and Stern (1983). A recent review on the theoretical models of diffusion is given by Aminabhavi et al. (1988). A widely used method to interpret the diffusion process or mechanism of penetrants in polymers above the glass transition temperature is the "free-volume model". This model assumes that the penetrants diffuse through the free 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1051 volume, Le., empty spaces or holes, in the polymer. Thus, penetrant diffusivity depends on the free volume. Although this model has found success in explaining the concentration and temperature dependence of diffusivity, it is phenomenological and unable to relate penetrant diffusivity to the molecular parameters of polymer and penetrant (e.g., the chain spacing of polymer and the molecular thickness of penetrant). Following the theoretical basis of Cohen and Turnbull (1959), several versions of the free-volume model applied to polymer-penetrant systems have been suggested. These include Fujita et al. (1960), Frisch et al. (1971), Vrentas and Duda (1977a,b, 1985a,b), Kreituss and Frisch (1981), and Paul (1983). Another approach to describe the diffusion process is the “molecular model”. The model considers the detailed motions and interactions of penetrant molecules and polymer chains. Penetrant diffusivity depends on activation energy, penetrant jumping distance, and jumping frequency, and it can be related to the molecular parameters of polymer and penetrant. Classified into this category are the versions of the model by Meares (1954), Barrer (1957), Brandt (1959), Kumins and Roteman (19611, and DiBenedetto (1963). Although these versions are able to relate activation energies to polymer structures and penetrant diameters, none of them can be used to correlate diffusivity data in terms of concentration and temperature dependence as the free-volume model does. Furthermore, most of them are only limited to simple penetrants (inert gases, Nz, 02,etc.). Recently, Pace and Datyner (1979a-e) developed a statistical mechanical molecular model for diffusion of both simple and complex penetrants in polymers. They offered an explicit expression for diffusivity as a function of penetrant concentration and temperature. However, their model has not been widely tested against experimental data to justify the validity, and it is complicated in the correlation of diffusivity data for concentration and temperature dependence. For all the existing models, at least three adjustable parameters are usually needed to fit the experimental data The consensus of most of the models is that the diffusion jumps involve the cooperation of several polymer segments and penetrants pass through the space opened by the fluctuation of polymer chains. It is still not clear how the polymer chain mobility is affected by the penetrants, nor is it clear whether the penetrants have been “activated” before making a jump. The understanding of penetrant diffusion in polymers is still in its primitive stage. Diffusion of small molecules in polyethylene has been extensively studied, partly because of the simple structure of the polymer. Compilation of diffusion data of organic compounds in polyolefins can be found in Flynn (1982). McCall and Slichter (1958) measured the diffusion of paraffins in polyethylene. Their data show that the longer paraffins have lower diffusivities than shorter paraffins, and branched paraffm have lower diffusivities than linear paraffins. Michaels and Bixler (1961) studied the diffusion of a series of simple gases in polyethylene. They were able to correlate the activation energies with the gas diameters. Kulkarni and Stern (1983) used Fujita’s free-volume model, modified for semicrystalline polymer, to correlate their diffusivity data for COz, CHI, CzH4,and C,H8 in the polymer. Diffusion data of benzene and toluene in polyethylene have also been reported in the literature (Rogers et al., 1960; Kwei and Wang, 1972; Aboul-Nasr and Huang, 1979; Liu and Neogi, 1988). In this paper, the diffusivities of a series of aromatic hydrocarbons in semicrystalline polyethylene were obtained through the sorption of the aromatic vapors in the
polymer by the use of a modified gravimetric sorption technique with a flow system capable of a wide range of activities (or vapor pressures) and temperatures. The effect of penetrant size or shape for the aromatic hydrocarbons on diffusivity in the polyethylene was demonstrated through the different methyl and alkyl substituents on the benzene ring. The experimental data were fitted by a proposed hybrid model that combines the key features of the free-volume and molecular models. This hybrid model was found to be better than the free-volume model, Fujita’s model, or Vrentas-Duda’s model, and it avoided the complexity of the molecular model. As molecular parameters are included in this hybrid model, the insight about the mechanism of diffusion in polymers may be elucidated.
Theoretical Section Free-Volume Model. The basic assumption of the free-volume model is that the mobilities of both polymer segments and penetrant mfolecules in a polymer-penetrant system are primarily determined by the available free volume in the system. This concept of free volume has been used to describe the viscosity-temperature relationships for simple liquids (Doolittle, 1952) and polymers (Williams et al., 1955). Fujita and his co-workers (Fujita et al., 1960; Fujita, 1961) extended this concept to the diffusion of small molecules in polymers:
D = AT exp(-B/f)
(1)
where D is the thermodynamic intrinsic diffusivity of the penetrant, T is the absolute temperature, f is the fractional free volume in the given polymer-penetrant system, and A and B are constants characteristic of the system. In eq 1,f may be calculated by (Fujita et al., 1960; Fujita, 1961, 1968)
f = f g + 4 T - TJ + P9
(2)
where Tgis the glass transition temperature of the polymer, is the fractional free volume of the dry polymer at Tg, (Y is the thermal expansion coefficient of the polymer, is the parameter representing the contribution of the penetrant to the increase on the free volume of the system when it is sorbed into the polymer, and 9 is the volume fraction of the penetrant in the amorphous phase of the polymer which is responsible for the transport of the penetrant. There are three issues associated with this free-volume model. The first one is the definition of the free volume that is actually available for the diffusion process to take place. The second is how the free volume is distributed among the penetrants and polymer chains. Finally, the third is how much energy is required for the redistribution of the free volume. Vrentas and Duda (1977a) addressed these issues by suggesting that the total volume of a liquid or polymer consists of occupied volume, interstitial free volume, and hole free volume. The occupied volume is defined to be the volume of the liquid or polymer at 0 K. The interstitial free volume is the volume increase due to the increasing ampltiude of vibrating molecules as temperature increases. Its distribution, which is continuous among the molecules of a given species, requires a large amount of energy. The hole free volume is the holes or vacancies formed discontinuously throughout the material. Its redistribution requires no or little energy. This hole free volume is the one that is responsible for the diffusion process. They assumed that the free volume is equally distributed among all the polymer and penetrant jumping units in the polymer-
fg
1062 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 (a) Penetrant Embedded in Locally Parallel Chain Bundle Can Have Two Modes of Motions: (1) Longitudinal Movement and (2) Transverse Movement Transverse Movement
Crosslink
Crystallite
(b) Chain Separation Allows Penetrant to Pass Through in Transverse Movement
Chain Separation
A penetrant molecule can proceed along the axis until a barrier such as entanglement, crystallite, or cross-link is reached, where the penetration through the barrier requires more energy than chain separation. The transverse movement gives rise to the activation energy and jumping frequency, and it is the rate-determining step. The longitudinal movement accounts for the mean jumping distance of penetrant (the mean diffusion distance before the jumping, i.e., before the transverse movement takes place). According to this picture, Pace and Datyner (1979a) derived the activation energy E for a simple penetrant with a spherical diameter d. The activation energy is the minimum energy necessary to produce a symmetrical chain separation sufficient to allow the transverse passage of a penetrant molecule with the molecular dimension d: E = j23( 3/4 (,.077[ ($)'(p - 10d9 -
i,)'4( 7)
Figure 1. Diffusion mechanism according to Pace-Datyner's molecular model.
- 0.58[
p*( Lp*) O+] d
penetrant system. Following the theory of Cohen and Turnbull (1959), Vrentas and Duda (1985a) expressed the thermodynamic intrinsic diffusivity of the penetrant as
($)"(p
- 4d9 -
where where Do is the preexponential factor, E'is the activation energy, R is the gas constant, T is the absolute temperature, w1 and w2 are the weight fraction of the penetrant and that of the polymer in the system, ul* and u2* are the specific volume of the penetrant and that of the polymer at 0 K, ufl and un are the free volume per gram of the penetrant and that of the polymer, respectively, X is the overlap factor that is introduced because the same free volume is shared by more than one molecule, 5 is the ratio of the molar volume of the penetrant (Vl* = ul*Ml)to the molar volume of the polymer jumping unit (V2*= u2*M2), i.e., 5 = Vl*/Vz* = vl*M1/u2*Mz,and M1 and M2 are the molecular weight of the penetrant and that of the polymer jumping unit, respectively. In Fujita's model it is implicated that the molecular weight of the penetrant is equal to that of the polymer jumping unit, i.e., 5 = u1*/u2*. This major difference has been discussed in Vrentas and Duda's papers (1977a,b, 1985a,b). The free-volume model suggests an explanation about how the penetrant diffusivity changes with concentration and temperature. Increasing penetrant concentration or temperature increases the free volume in the system, and this results in the increase of the penetrant diffusivity. However, the free-volume model offers no information whatsoever about the absolute magnitude of the diffusivity. This is due to its incapability to characterize the preexponential term (see eqs 1 and 3). Fujita used the linear temperature dependence, whereas Vrentas and Duda used the exponential temperature dependence, both of which are essentially arbitrary. Molecular Model. The most useful molecular model was proposed by Pace and Datyner (1979a,d). They assumed that amorphous polymer regions possess an approximately paracrystalline order with chain bundles locally parallel over distances of several nanometers. A penetrant molecule may diffuse by two modes of motions illustrated in Figure 1: (1)sliding parallelly along the axis of a chain bundle (tube), which is called longitudinal movement, and (2) jumping perpendicularly to this axis when adjacent chains are sufficiently separated, which is called transverse movement. The longitudinal movement is generally much faster than the transverse movement.
d ' = d + p* - p
(5) E* and p* are the average effective Lennard-Jones energy and distance parameters per polymer backbone element, respectively, X is the average distance between two polymer backbone elements in a polymer chain, p is the equilibrium polymer chain spacing, and 7 is the effective chain-bending modulus per unit length. If thermal expansion is assumed to occur by a uniform increase in interchain spacing without the expansion along the chain, p can be calculated from
where a' is the thermal expansion coefficient. Generally, p*/p ranges from 1 to 1.02 for most polymers at room temperature, and p is approximately equal to p*. The diffusivity can be expressed as (Chandrasekhar, 1943)
D = f/gL2u
(7)
where L is the root-mean-square jumping distance of the penetrant and u is the average jumping frequency. Using the statistical mechanics, Pace and Datyner (1979a) obtained the following expression for u: u =
~
5.46 x
.)"I4( x2
P*
$ ) l 2 d ( G )dE
-l
exp(-E/RT)
where m* is the molecular weight per polymer backbone element. The model contains no adjustable parameters except L, which is not predictable within the limits of this model. Pace and Datyner (1979d) also extended their model to include complex molecules, such as organic solvents and dyes. There is more than one enfolding chain of the polymer draped across the penetrant length for complex molecules. The diffusivity is also determined by the displacement of the enfolding chains, which again is controlled by the chain mobility. It is evident that the chain
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1053 motions are influenced by the nearby penetrant molecules. Allowing for this effect, they derived a complicated expression for the diffusivity as a function of penetrant concentration. Two additional adjustable parameters are introduced in the model. These parameters all have precise physical meanings, and they all are lengths. Their model, however, has not been extensively tested against experimental diffusivity data. Hybrid Model: Combination of Free-Volume and Molecular Models. In this paper, a hybrid model that combines the key features of the free-volume and molecular models is proposed to correlate diffusivity data. I t includes the advantages from both models. For the diffusion of complex molecules in polymers, the diffusivity is expressed in the hybrid model as D = D,(d,T) D,(wi,Ui*,T) (9) Dd is the diffusivity calculated from eqs 4-8 according to the molecular model for a spherical penetrant with diameter d. Because Dd is determined by the activation energy needed to produce a chain separation d , for complex penetrants d is the smallest dimension or the thickness of the penetrant molecule so that the activation energy can be minimum. For complex penetrants, the actual value of Dd is influenced by the shape of the penetrant. D, is a factor to account for the concentration and size of the penetrant. For complex penetrants, on which there are more enfolding chains of the polymer lying, the diffusivity is reduced as compared with simple spherical penetrants. For high penetrant concentrations, which induce the swelling of the polymer matrix, the diffusivity is generally increased. Swellii of the polymer matrix is accompanied by increasing the free volume in the system. Thus, a phenomenological free-volume approach is used to express D,. The assumption of free-volume distribution is the same as that of Vrentas-Duda’s model (see eq 3). However, the total free volume in the system is represented by a relationship similar to that of Fujita’s model in eq 2. Thus, D, may be expressed as
where fl = dvf/dwl is the concentration coefficient (Stern et al., 1972) for the sorption of the penetrant to increase the free volume of the polymer-penetrant system, up This coefficient is assumed to be independent of concentration and temperature. Thus,in the hybrid model, the polymer parameters and the penetrant dimension (molecular thickness) are used to determine the Dd term in eq 9, with the penetrant jumping distance being the adjustable parameter. This term is equivalent to the preexponential factor of the free-volume model. As shown in eq 10, the free-volume expression is used to relate the diffusivity to the penetrant concentration (wl*) and the penetrant size (vl*). Both the preexponential term and the free-volume expression are a function of temperature. The molar volume of the polymer jumping unit at 0 K (V2* in [ = V,*/ V2*) and the concentration coefficient are the other two adjustable parameters. Although eq 9 is based on the molecular model of Pace and Datyner (1979d), the expression of D , in eq 10 is semiempirical. On the other hand, if eq 10 is considered as derived from the free-volume model, then the Dd term in eq 9 simply attempts to relate the preexponential term to the molecular parameters. The empirical nature of Dd still appears in the adjustable parameter L, the penetrant jumping distance.
Determination of Model Parameters. In order to correlate diffusivity data, the Fujita, Vrentas-Duda, and hybrid models require the information about the free volume of the pure polymer, vfl (or fg). Vrentas-Duda’s model also calls for the free volume of the pure penetrant, Ufl.
Both vfl/y and vn/y can be determined from the viscosity-temperature relationships for the penetrant and polymer, respectively. They are related to temperature by the following general equation with two constants, X i and Yi: vfi/y = Xi(Yi + T ) (11) For the penetrant (component l),the relationship between the penetrant viscosity pl and self-diffusivity suggested by Dullien (1972) is used to determine X1and Y,, The free-volume expression for the penetrant self-diffusivity is the same as eq 3 with w2 = 0. Substitution of this equation and eq 11into Dullien’s equation results in the following expression for the temperature dependence of the penetrant viscosity (Vrentas et al., 1985b): In p1 = In (0.124
X
lOlO;,2/sRT
1,
In [Doexp(-E’/RT)]
*
+ X l W“,l + T ) (12)
where V,, is the critical molar volume of the penetrant and V , is the penetrant molar volume at absolute temperature T. ul* can be estimated from Haward (1970) by a group contribution method. The parameters Do, E’, X1, and Yl can then be determined from viscosity-temperature and density-temperature data for the penetrant by using a nonlinear regression method based on eq 12. Nonetheless, as pointed out by Vrentas et al. (1985b), the presence of the exp(-E’/RT) term leads to an unacceptable parameter interaction effect. To alleviate this problem, we replace the Do exp(-E’lRT) term in eq 12 by DOT as in Fujita’s equation form for the preexponential factor (see eq 1). As a result, only three parameters Do, X,, and Y , need to be determined in the data fitting. Density-temperature data for penetrants are available in tabular form (Thermodynamics Research Center, 1987). Viscosity-temperature data for penetrants are obtained from Reid et al. (1977) based on the data correlation by van Velzen et al. (1972). The obtained values of X 1 and Yl are used in eq 11 to calculate the penetrant free volume, ufl. The value of Do determined in this way (without the use of E’as another fitting parameter), unfortunately, cannot be used as a substitute for the preexponential term in eq 3, because Do and E’are both considered as adjustable parameters in Vrentas-Duda’s model. For the polymer (component 2), X 2 and Y2 can be evaluated from the Williams-Landel-Ferry (WLF) constants of the polymer, c1 and c2 (Williams et al., 1955):
Y2 =
~2
- Tg
(1%)
The above equations have been derived from vn/y = [f + CY(T- T )]v2*/(1 - fJ and eq 11with f g = 1/(2.3O3clJ and CY = fgfc2. For polyethylene, c1 is 17.44 and c2 is 35.6 K, which correspond to f g = 0.025 and CY = 7 X lo4 K-’ in Fujita’s equation (Liu and Neogi, 1988). The glass transition temperature of polyethylene is 228 K, as measured from differential scanning calorimetry. In a way similar to ul*, u2* can be estimated from Haward (1970)
1054 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 Table I. Parameters Used in Diffuslvity Modeling component ui*. cm3/a X i , cm3/(pK) Yi,K benzene 0.8885 10.8 X -45.96 12.3 X -40.07 toluene 0.9018 -26.68 0.9193 9.86 X lo4 n-propylbenzene 13.0 X -36.84 mesitylene 0.9193 0.9251 9.14 X lo4 -25.64 n-butylbenzene 7.06 X -20.15 0.9251 prehnitene -192.4 polyethylene 0.9764 7.00X Table 11. Molecular Thicknesses for Aromatic Penetrants penetrant In, nm lM, nm a, nm 0.320 benzene 0.72 0.0 0.65 0.2 0.339 toluene 0.356 n-propylbenzene 0.56 0.45 mesitylene 0.45 0.4 0.358 n-butylbenzene 0.56 0.58 0.361 prehnitene 0.0 0.99 0.400
by a group contribution method. The estimated values of ui*, along with the obtained values of Xiand Yi, are listed in Table I for six aromatic penetrants and polyethylene. The geometric dimensions of the aromatic penetrants in this work are determined from the length of chemical bonds and the van der Waals volume of the atomic groups (Pauling, 1967; Pace and Datyner, 1979e). The thickness of the aromatic ring is 0.32 nm, the length of the C-C bond in the benzene ring is 0.14 nm, the length of the C-H bond is 0.1085 nm, and the length of the C-C bond between the benzene ring and the alkyl group is 0.153 nm. The axial distance in the alkyl group is 0.13 nm every two carbon atoms. The van der Waals diameter is 0.22 nm for the H atom, and it is 0.4 nm for the CHBgroup. To minimize the energy of chain separation for diffusion, the penetrant orients itself with its length perpendicular to the polymer chains and its width parallel to the polymer chains. As a result, the penetrant thickness is the required chain separation for diffusion. Two types of the thickness exist in the aromatic compounds studied in this work: (1)the thickness of the benzene ring and (2) the diameter of the methyl or alkyl substituents. The effective thickness of the whole penetrant ais then calculated from their average:
2 = (d&
+ d&)/l
(14)
where lB is the length of the penetrant section that has the benzene thickness dBand lM is the length of the penetrant section that has the diameter (thickness) of the methyl or alkyl group, dm 1 is the total length of the penetrant, Le., 1 = lB + lm dBis 0.32 nm -and dMis 0.4 nm. Table 11lists the values of LB, LM,and d for these aromatic penetrants. The d value is used in the calculation with eq 4. The polymer parameters required for eq 4 are taken from Pace and Datyner (1979b) for polyethylene. They are c* = 308 J/mol, p* = 0.463 nm, 71 = 12000 J.nm/mol, X = 0.1267 nm, p / p * = 1.0 at 298 K, a' = 8.2 X K-l, and m* = 14 g/mol.
Experimental Section Sorption Experiments. The diffusivities of a series of aromatic hydrocarbons in semicrystalline polyethylene were determined through the sorption of the aromatic vapors in the film of the polymer by the use of a gravimetric sorption apparatus. Figure 2 shows the schematic of this apparatus. A Cahn 2000 electrobalance was used to monitor the weight change. A flow system was employed in the apparatus so that it could handle the condensable vapors at much higher partial pressures and temperatures. With the flow system, the electronic components of the balance chamber were purged with an inert
D.tr kqubnlon Cahn ZOO0 Balance
Nz
kuurn
N1
N1
Oven
* 0.05%
Figure 2. Schematic of gravimetric sorption apparatus with flow system.
N2 gas stream, which exited through the side port of the hangdown tube. Penetrant vapor was carried into the system by N2 gas from the bottom of the hangdown tube, and the penetrant vapor/Nz gas stream exited also through the side port of the hangdown tube. For the penetrant stream, one Nz stream was bubbled through a liquid penetrant saturator and mixed with a second stream of pure N2 gas to give the desired partial pressure of the penetrant. Nz flows were measured and controlled by maw flow controllers. A constant-temperature oven (f0.05 O C ) was used to heat the liquid saturator. The partial pressure of the penetrant vapor carried was calculated from the Nz flow rates and the saturation pressure of the penetrant at the saturator temperature. The hangdown tube, in which the polymer sample was placed in a pan with a hangdown wire connected to the balance, was enclosed in a split tube furnace (fO.l "C). All the incoming lines were wrapped with heating tapes to prevent any condensation of vapor. The measured weight change, system preasure, and sample temperature were stored in a Hewlett Packard (HP) computer via a data acquisition unit. The experiments were conducted in a stepwise manner. At the end of each sorption run,i.e., when the equilibrium was reached, the inlet flow to the sample was stopped by switching it via a valve to a by-pass exhaust to determine the hydrodynamic effect, which resulted in the reduction of the weight measured by the microbalance. With the flow stopped, the equilibrium weight was then measured without the hydrodynamic effect in a short period of time (less than 0.5 min). The difference of the equilibrium weights measured with and without the flow was the hydrodynamic effect. During this short period of time, the nitrogen flow rates were readjusted to give a new partial pressure of the penetrant. Switching the inlet flow back to the sample via the valve, which could be achieved instantaneously, resulted in a step change of the penetrant partial pressure. The time zero was determined by taking into account the volume between the polymer sample and the valve position. The size of the step change ranged from a penetrant activity (partial pressure/saturation pressure at sample temperature) of 0.04 to 0.07. The transient data, which were stored in the HP computer, were fitted to Fick's diffusion equation (eq 4.18 in Crank (1975)) by a nonlinear regression method. The diffusivity (mutual D") was the fitting parameter. The polymer sample thickness was corrected for swelling. Thus, the thickness used in Fick's equation was z = Zd[l + (P2w/P1)11'3 (15) where Z is the polymer film thickness when the polymer
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1055 contains the penetrant, 2, is the thickness of the dry polymer film, p2 is the density of the dry polymer, p1 is the density of the penetrant, and w is the equilibrium weight gain (g of penetrant/g of dry polymer). Equation 15 assumes that there is no change of volume upon mixing and the penetrant has the same density as in the liquid phase. Both sorption and desorption runs were made for the same step change, and no significant differences were seen in the diffusivities obtained. The diffusivities reported were, therefore, the average values from the sorption and desorption runs. The runs were conduced at 30,40, 50, and 60 OC for each penetrant. To apply Fick's equation in this flow system, sorption equilibrium was assumed to be established instantaneously at the polymer surface. This could be indirectly verified by increasing the inlet nitrogen flow rates so that the external mass-transfer resistance in the gas phase was small compared with the diffusional resistance inside the polymer. For the nitrogen flow rates between 100 and 200 cm3/min, which corresponded to 0.88 and 1.66 cm/s for a hangdown tube with an inside diameter of 1.6 cm, the observed transient curves were essentially independent of the flow rates. Materials. A commercial polyethylene film obtained from Exxon Chemical Company was used in this study. It had a thickness of 43 pm, a density of 0.920 g/cm3, and an average degree of crystallinity of 45% as determined by differential scanning calorimetry (DSC) and X-ray diffraction. The melting point measured by DSC was about 101 OC. The glass transition temperature was about -45 OC. Prior to experiments, the sample was immersed into liquid toluene to wash the film surface. It was then outgassed at 60 "C overnight in the sorption apparatus. Since the experiments were conducted at 30,40,50, and 60 "C,the polyethylene film never encountered temperatures higher than 65 "C ta ensure there was no morphology change. The sorption results were found to be independent of the film history in this temperature range. Six aromatic hydrocarbons were used as penetrants, and they were benzene, toluene, n-propylbenzene, mesitylene (1,3,5-trimethylbenzene),n-butylbenzene, and prehnitene (1,2,3,44etramethylbenzene). These hydrocarbons were used as received from Aldrich Chemical Co., Milwaukee, WI,and their purities were at least 98%. Their saturation vapor pressures at various temperatures were calculated from Antoine equations, with their constants available from the thermodynamic tables of the Thermodynamics Research Center (1987).
Results and Discussion The experimentally measured mutual diffusivities Dv can be related to the thermodynamic intrinsic diffusivities D by the following expression (Fujita, 1961; Stern et al., 1983; Frisch and Stern, 1983):
where 4 is the volume fraction of penetrant in the amorphous phase of polymer and a is the activity of the penetrant. The 1- 4 corrects for the bulk flow due to the concentration gradient. The d In $/a In a term converts the concentration gradient for the mutual diffusivity to the chemical potential (or activity) gradient for the thermodynamic intrinsic diffusivity. This term can be evaluated from the data of sorption equilibrium. For example, if the Flory-Huggins equation In a = In 4
+ (1- 4) + x(1 - 4)2
(17)
loo0 500
9 40°C 0-0
50
10
0.00
I
I
0.05
0.10
Hybrld Model 0.20
0.15
Volume Fraction
Figure 3. Diffusivity of benzene in semicrystalline polyethylene modeled. 1000
I
I
500
I
I
I
50%
.,A*-
-Hybrid Model "b.00
0.h
O.\O
0.'15
0.hO
0.h5
040
Volume Fraction
Figure 4. Diffusivity of toluene in semicrystalline polyethylene modeled.
is used to describe the solubility data, Le., to evaluate a In 4/a In a, eq 16 becomes
D=
D' (1 - 4)2(1 - 2x4)
(18)
where x is the Flory-Huggins interaction parameter. 4 can be related to the measured equilibrium weight gain w by the following equation:
4=
1 PI%
(19)
1+-
Paw
where a, is the weight fraction of the amorphous phase in the polymer and pa is the density of the amorphous phase of the polymer which is 0.85 g/cm3 for polyethylene (van Krevelen and Hoftyzer, 1976). This equation has the same assumptions as eq 15. AU the diffusion models mentioned in this paper are based on thermodynamic intrinsic diffusivity. The diffusivities of six aromatic hydrocarbons in the semicrystalline polyethylene at different temperatures as a function of concentration (volume fraction) are shown in Figures 3-8. These are the thermodynamic intrinsic diffusivities converted from the measured mutual diffusivities by the use of eq 18. Used in this equation, the Flory-Huggins parameters x obtained from the gravimetric sorption experiments for the penetrant solubilities are given in Table I11 (Doong and Ho, 1991). In Figures 3-8, the concentrations are expressed in terms of the penetrant volume fraction in the amorphous phase of the polymer, because only the amorphous phase is responsible for the transport of the penetrants in the semicrystalline poly-
1056 Ind. Eng. Chem. Res., Vol. 31, No. 4,1992
*
loo0
100
1
I
I
Hybrid Model
I
I
0.10
0.15
0.20
Hybrid Model
boo
0.25
Oh5
o.;o
Volume Fraction
-0
I
I
I
I-
6
.y!
Er
100
'6
50
.-
=-
100
3
Hybrid Model
*soot -
+
*/*--.-*-*A
A;. -& *,-
A-
A
40%
-
A
-
g
- * 'A
*-*30DC
,*/*-
I
I
I
Figure 6. Diffusivity of mesitylene in semicrystalline polyethylene modeled.
-
t 0 .-E
3
60%
e -.*A
A
A
A-A-A~OOC
5ooc 0
-=
30%
/A-
.-
g
*-*+
1005 o : z
5 -
-Hybrid
-
--
-
Model 1
I
I
I
040
I
!I $
I
Table 111. Flory-Huggins Interaction Parameters x for Aromatic Penetrants in Semicrystalline Polyethylene x at penetrant 30 "C 40 "C 50 "C 60 "C 1.10 0.93 benzene 1.20 0.86 1.23 1.10 0.95 0.78 toluene 1.24 1.09 1.09 n-propylbenzene 0.96 mesitylene 1.09 1.01 0.87 0.77 n-butylbenzene 1.20 1.04 0.98 1.08 prehnitene 1.10 1.03 0.92 0.77
ethylene. As shown in these figures, the diffusivities all increased with increasing penetrant volume fraction (concentration) or temperature. Also, the concentration dependence appeared to become stronger with decreasing temperature or penetrant size in the order prehnitene, n-butylbenzene,mesitylene, n-propylbenzene,toluene, and
I
I
1
Benzene A
.'
5
t
3.0
d
I
1 2.9
I
Volume Fraction
5
.\
10
B
I
I
2
-/* 10
0:25
Figure 8. Diffusivity of prehnitene in semicrystalline polyethylene modeled.
I
I
L
500
o.ho
0.115 Volume Fraction
Figure 5. Diffusivity of n-propylbenzene in semicrystalline poly. ethylene modeled. 1000
I
-
I
I
3.1
3.2
n-Propylbenzene Mesitylene
Prehniiene
3.3
3.4
3.5
in xi03 Figure 9. Diffusivities at zero concentration showed Arrhenius dependency on temperature.
benzene (the molecular thicknesses are shown in Table 11). As expected, the diffusivities decreased with increasing penetrant size. This can be seen from Figure 9, which shows the diffusivities at zero concentration as a function of temperature. These diffusivities were obtained by extrapolating the experimental data to the zero volume fraction of the penetrants. It is noteworthy that prehnitene diffused slower than n-butylbenzene. These two penetrants have the same molecular weight but different molecular shapes. Presumably, the different shapes gave rise to different diffusivities. These results were consistent with the molecular thickness of the penetrants. As seen in Table 11,the effective molecular thickness of prehnitene is larger than that of n-butylbenzene. The same phenomenon was seen for mesitylene and n-propylbenzene, As shown in Figure 9, the linb for the penetrants investigated have about the same slope. The apparent activation energy calculated from this figure based on the Arrhenius equation was about 60300 J/mol. This was about the same as that reported by McCall and Slichter (1958) for several organic liquids in polyethylene. Data Correlation with Fujita and Vrentas-Duda Models. The experimental data were first correlated by the Fujita free-volume model, eqs 1and 2. The values of the three adjustable parameters A, B, and ,f? obtained by a nonlinear regression method are summarized in Table IV along with the absolute relative errors of the fitting results. These results were not satisfactory despite the three adjustable parameters. Comparison with the experimental data for mesitylene is shown in Figure 10. As shown in this figure, the Fujita free-volume model could not account for the large increase of the diffusivitiesat low concentrations. Similar behavior was seen for other penetrants, especially for toluene and benzene. Furthermore,
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1057
pI
50 cl
500
0
30%
0
5
E
1 0
100
to 0
-
Vmntas-Dude’s Model
-
Model 0
lo L.00
0.05
0;lO
0.115
0.iO
O./O
0:25
Volume Fraction
Figure 10. Fujita’s free-volume model could not follow diffusivities of mesitylene at low concentrations in semicrystalline polyethylene. Table IV. Modeling of Diffusivity Data Free-Volume Model penetrant A, cm*/(s.K) B 0.886 benzene 7.78 X lo4 toluene 6.64 X low8 0.475 0.596 n-propylbenzene 1.63 X lo-’ 0.569 mesitylene 8.65 X 0.702 n-butylbenzene 2.48 X lo-’ 0.603 prehnitene 4.41 X ~
for Fujita’s
~~~
p’ 0.113 0.166 0.0686 0.559 0.0348 0.0148
error,” % 31.9 33.0 16.2 15.9 17.8 11.0
a Error = (l/n)C,((calculated value - experimental value)/ experimental valuel.
Table V. Modeling of Diffusivity Data for Vrentas-Duda’s Free-Volume Model DO, E’, V2*, penetrant cm2/(s.K) J/mol cm3/mol error, % 25600 107 28.5 benzene 3.76 7780 173 30.4 toluene 4.44 X lo4 15 190 239 15.0 n-propylbenzene 3.56 X 21800 307 13.5 9.71 X mesitylene 25090 286 16.6 5.10 X n-butylbenzene 275 10.5 1.91 X 17470 prehnitene
the obtained parameters shown in Table IV could not be correlated well with the molecular sizes or other properties of these series of aromatic hydrocarbons, even though parameter B showed some increasing trend with the number of methyl substituents and the length of alkyl substituents on the benzene ring provided that the data of benzene were neglected. Since three adjustable parameters were used in the data correlation with the Fujita model for each penetrant, three adjustable parameters were also used with the VrentasDuda model. However, Vrentas-Duda’s model with the three adjustable parameters Do,E’, and f yielded only slightly better results. The correlation results with these three adjustable parameters are summarized in Table V. In this table, the molar volume of the polymer jumping unit V2*is listed instead of 5 (5 = Vl*/V2*; VI* is the molar volume of the penetrant at 0 K). As shown in this table, the preexponential factor Doand the activation energy E’ from the data correlation were essentially in random fashion. Only the molar volume of the polymer jumping unit V2*showed a corresponding trend with increasing sizes of the penetrants, except for mesitylene. The fact that Doand E’ have no physical meaning can be seen in Vrentaa et al. (1985b), where a negative value of E’ has been reported for the polystyrene-toluene system through data correlation. Figure 11shows the comparison of the experimental data with Vrentas-Duda’s model for toluene. Again, similar to Fujita’s model, Vrentas-Duda’s model gave a concentration dependence that was pretty much
Table VI. Modeling of Diffusivity Data for Hybrid Model: Combination of Free-Volume and Molecular Models L, v2*, penetrant nm cm3/mol V2*/13.67” cm3/g error, % 3.45 20.4 2.8 255 18.7 benzene toluene 2.7 376 27.6 3.62 18.0 48.8 1.80 12.5 n-propylbenzene 2.4 666 52.4 mesitylene 1.8 717 2.72 9.2 1.15 19.0 1.8 825 60.4 n-butylbenzene prehnitene 1.9 863 63.2 0.80 10.3 $1
a Number
of methylene segments.
flat, and it could not account for the large increase of the diffusivities at low concentrations. Data Correlation with the Hybrid Model. Our data were then correlated with the hybrid model that combines the key features of both the free-volume and molecular models. The correlation results with the three fitting parameters, the penetrant jumping distance L, the molar volume of the polymer jumping unit Vz*,and the penetrant concentration coefficient 0, are listed in Table VI. Figures 3-8 show the comparisons of the hybrid model with the experimental data for the penetrants. With this hybrid model, not only the overall errors were reduced in comparison with the free-volume model, but also the initial increase of the diffusivities at low volume fractions was followed reasonably well. The succesa of this hybrid model appears to lie in its capability in the use of the polymer parameters and the penetrant molecular thickness to determine the preexponential term (Ddin eq 9). On the other hand, the free-volume model fits the data in a global way; the preexponential terms in eqs 1 and 3 are totally empirical. As pointed out by Kreituss and Frisch (1981),the parameters of the free-volume model are very sensitive to the penetrant diffusivity at zero penetrant volume fraction. The three parameters extracted from the correlation of the hybrid model all have physical meanings. The first parameter L is the root-mean-square distance of the penetrant traveling along the chain axis (longitudinal movement) between two transverse jumps. The frequencies of the transverse jumps for simple penetrants are calculated according to eq 8, and the frequencies depend on the activation energy for the chain separation, E. This activation energy, in turn, increases with increasing penetrant diameter (or thickness). The larger penetrant would travel longer longitudinal distance before it makes a transverse jump, because of larger activation energy required to separate chains. However, the presence of the chain entanglements, crystallites, and cross-links would reduce this distance more for larger penetrants than for small ones (Pace and Datyner, 1979a,d). In this work, the penetrants
1058 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992
were complex alkylbenzenes and multimethylbenzenes. It is likely that the jumping distances L are controlled by the frequencies of these barriers that the penetrants encounter, not by the activation energy. Longer or bulkier penetrants would have more chances to be stopped by these obstacles, so their jumping distances L should be shorter. This is shown in Table VI, where the obtained values of L decreased when the length of the alkyl group in the benzene ring increased or the number of methyl substitutents increased (except for prehnitene, whose L value was very close to the value of mesitylene). The magnitudes of L were around 2 nm (20 A), which implied that the barriers were the chain entanglements, not the crystallites. Although little is known about the distribution of intercrystalline spacings in polyethylene, there is strong evidence that crystallites in polyethylene take the form of lamellae, approximately 100 A in thickness and a few microns in the other two dimensions (Bovey, 1979). For a polyethylene with 45% crystallinity as used in this study, the distance between two lamellae should be at least 100 A, if the lamellae are assumed to be perfect single crystals. Even allowing for about 20% crystalline defect in the lamellae, the intercrystalline distance should still be at least about 80 A, which is much greater than 20 A. Thus, the jumping distance results suggested that the major diffusion barriers were the chain entanglements in the amorphous region, not the crystallites. The second parameter V,* is the molar volume of the polymer jumping unit. The size of the polymer jumping unit is not a well-defined parameter. According to Vrentas-Duda’s model, each polymer jumping unit shares the same free volume as one penetrant molecule. Therefore, the polymer jumping unit may be considered as the total number of the polymer segments (repeating units) that are perturbed for the transverse movement of one penetrant molecule. Note that the polymer sections that are swept by the longitudinal movement of the penetrant molecule are not counted in the polymer jumping unit, as no activation energy is required for this movement. Naturally, the polymer jumping unit increases with increasing penetrant molecule size. The values of V,* listed in Table VI are also consistent with this reasoning. The volume of the jumping unit for either polymer or penetrant is based on the occupied volume of the molecule (the volume of the material at 0 K). For polyethylene, the molar volume of each methylene (CH,) segment at 0 K is 13.67 cm3/mol estimated from the group contribution method (Haward, 1970). Consequently, the number of the methylene segments that are perturbed by one penetrant molecule can be calculated and is listed in Table VI. The number ranged from 20 to 60, and it increased with increasing penetrant size. For simple gaseous diffusion in polymers, Barrer’s zone theory suggested that the number of the polymer segments involved in the diffusion process is about 16 for ethylene polymer (Barrer, 1957; Brandt, 1959). The theory of DiBenedetto and Paul (1964) suggested that the number of the segments involved in a diffusion jump for a variety of the polymers ranges from 10 to 20. It is not surprising that for more complex penetrants the number of the polymer segments involved in the cooperative motions for the penetrant diffusion is much higher. The third parameter is the concentration coefficient 0, which indicates the effectiveness of the penetrant to increase the free volume of the polymer-penetrant system. Table VI shows that B appeared to decrease with increasing penetrant size. For simple liquids, generally the larger penetrants have smaller free volumes or their free volumes are closer to the free volume of polymer. Consequently,
it is reasonable that the large penetrants are less effective to increase the free volume of the system. The 0 values listed in Table VI were significantly higher than the specific free volumes of their corresponding penetrant liquids, which may range from 0.2 to 0.3 cm3/g. Initially, this might seem unreasonable. However, the free volume associated with each penetrant molecule is arbitrarily defined as the average free volume equally distributed among the jumping units. When penetrant molecules are sorbed into the polymer, most of the free volumes could be surrounding the penetrant molecules. The addition of the penetrant molecules could cause a tremendous increase of the free volume in the neighborhood of the penetrant molecules, although the total free volume of the system may not be increased as much. Thus, it could be the local free volumes that account for the large values of 6. The assumption that free volume is equally distributed may not be applicable to the local regions of the penetrant molecules. The above molecular interpretation was solely based on the fact that the obtained parameters showed a trend with the penetrant size or shape. Because this hybrid model was not derived from the first principles theoretically, it is of empirical nature. Consequently, more evidence is needed to support the above picture of the diffusion mechanism. Conclusions The diffusivities of a series of aromatic hydrocarbons, benzene, toluene, n-propylbenzene, mesitylene, n-butylbenzene, and prehnitene, in semicrystalline polyethylene were obtained via the sorption of the aromatic vapors in the film of the polymer by the use of a modified gravimetric sorption technique with a flow system capable of a wide range of activities (or vapor pressures) and temperatures. The diffusivities all increased with increasing penetrant (volume fraction) or temperature. The concentration dependence appeared to become stronger with decreasing temperature or penetrant size (molecular thickness). As expected, the diffusivities decreased with increasing penetrant size. The experimental data can be fitted quite well by our proposed hybrid model that combines the key features of the free-volume and molecular models. The hybrid model fitted the data better than the free-volume model, and it avoided the complexity of the molecular model. With this hybrid model, the obtained values of the three adjustable parameters, penetrant jumping distance, molar volume of polymer jumping unit, and concentration coefficient, may be used to elucidate the mechanism of the diff,usion process in the polymer. However, more evidence is needed to support the molecular picture suggested by this hybrid model. Nomenclature A = preexponential factor in Fujita’s model, cm2/(s.K) a = penetrant activity cl, c2 = WLF constants B = constant in Fujita’s model Do= preexponential factor in Vrentas-Duda’s model, eq 3, cm2/s D = thermodynamic intrinsic diffusivity of penetrant in polymer, cm2/s D’ = mutual diffusivity of penetrant in polymer, cm2/s d = diameter of spherical penetrant or molecular thickness of penetrant, nm d’ = distance parameter defined in eq 5, nm E = activation energy, J/mol E’ = activation energy constant in Vrentas-Duda’s model, eq 3, J/mol
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1059
f = fractional free volume in the polymer-penetrant system f g = fractional free volume of polymer at glass transition
temperature
L = penetrant jumping distance, n m le = length of the penetrant section that has the benzene thickness dg, nm 1, = length of the penetrant section that has the methyl thickness dM,nm M1= molecular weight of penetrant, g/mol M z= molecular weight of polymer jumping unit, g/mol m* = molecular weight per polymer backbone element, g/mol R = gas constant, 8.314 J/(mol.K) T = absolute temperature, K T = glass transition temperature, K = molar volume of penetrant, cm3/mol Vcl = critical molar volume of penetrant, cm3/mol VI*= molar volume of penetrant at 0 K, cm3/mol Vz* = molar volume of polymer jumping unit at 0 K, cm3/mol ul* = specific volume of penetrant at 0 K, cm3/g uz* = specific volume of polymer at 0 K, cm3/g uf = total free volume per gram of the polymer-penetrant
fl
system, cm3/g un = free volume per gram of penetrant, cm3/g un = free volume per gram of polymer, cm3/g w = equilibrium weight gain, g of penetrant/g of dry polymer X i= constant defined in e q 11
Yi= constant defined in eq 11 Z = polymer f i i thickness when polymer contains penetrant, cm
Zd = thickness of d r y polymer film, cm Greek Symbols a = thermal expansion coefficient (see e q 21, K-' a' = thermal expansion coefficient defined in e q 6, K-' a, = weight fraction of the amorphous phase in polymer = parameter (in Fujita's model) representing the contribution of penetrant to the increase on the free volume of the polymer-penetrant system y = overlap factor for free volume c* = Lennard-Jones energy parameter per polymer backbone
element, J/mol = bending modulus of polymer chain, J.nm/mol 9 = concentration coefficient (auf/aol) which accounts for the sorption of penetrant to increase the free volume of the polymer-penetrant system, cm3/g X = average distance between two polymer backbone elements in a polymer chain, nm p = viscosity, g/(cm.s) Y = average jumping frequency, l / s .( = molar volume ratio of penetrant to polymer jumping unit p = equilibrium polymer chain spacing, n m p* = Lennard-Jones distance parameter per polymer backbone element, nm p1 = density of penetrant, g/cm3 pz = density of d r y polymer, g/cm3 pa = density of the amorphous phase of polymer, g/cm3 4 = penetrant volume fraction in the amorphous phase of polymer x = Flory-Huggins interaction parameter o1= weight fraction of penetrant in the polymer-penetrant 7
wz
system = weight fraction of polymer in the polymer-penetrant system
Subscripts 1 = penetrant 2 = polymer f = free volume g = glass transition i = component
Registry No. Polyethylene, 9002-88-4;benzene, 71-43-2; toluene, 108-88-3;n-propylbenzene, 103-65-1; mesitylene, 108-67-8;
n-butylbenzene, 104-51-8;prehnitene, 488-23-3.
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Pace, R. J.; Datyner, A. Statistical Mechanical Model for Diffusion of Simple Penetrants in Polymers: 11. Applications-Nonvinyl Polymers. J. Polym. Sci., Polym. Phys. Ed. 1979b, 17, 453. Pace, R. J.; Datyner, A. Statistical Mechanical Model for Diffusion of Simple Penetrants in Polymers: 111. Applications-Vinyl and Related Polymers. J. Polym. Sci., Polym. Phys. Ed. 1979c, 17, 465. Pace, R.J.; Datyner, A. Statistical Mechanical Model for Diffusion of Complex Penetrants in Polymers: I. Theory. J. Polym. Sci., Polym. Phys. Ed. 1979d, 17, 1675. Pace, R. J.; Datyner, A. Statistical Mechanical Model for Diffusion J. Polym. of Complex Penetrants in Polymers: 11. Applications. .. Sci., P d y m . Phys. Ed. 1979ei 17, 1693. Paul. C. W. A Model for Predictinn Solvent Self-Diffusion Coefficients in Nonglassy Polymer/Soivent Solutions. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 425. Pauling, L. The Chemical Bond; Cornel1 University: Ithaca, NY, 1967;Chapter 7, pp 135-155. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, NY, 1977; Chapter 9,pp 391-469. Rogers, C. E.;Stannett, V.; Szwarc, M. The Sorption, Diffusion and J. Polym. Sci. Permeation of Organic Vapors in Polyethylene. 1960, 45, 61. Stern, S. A,; Fang, S.-M.; Frisch, H. L. Effect of Pressure on Gas Permeabilitv Coefficiente: A New Application of Free-Volume Theory. J. Polym. Sci., Part A-2 1972, 10, 201. Stern, S. A.; Kulkarni, S. S.; Frisch, H. L. Tests of a Free-Volume Model of Gas Permeation through Polymer Membranes: I. Pure COP,CH,, CZH4, and C3H8in Polyethylene. J.Polym. Sci., Polym. Phys. Ed. 1983, 21, 467.
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Sorption Studies on Ion Exchange Resins. 1. Sorption of Strong Acids on Weak Base Resins Vinay M. Bhandari, Vinay A. Juvekar,* and Suresh R. Patwardhan Department of Chemical Engineering, Indian Institute of Technology, Bombay 400076, India
Sorption equilibria and batch dynamics of strong acids (HC1 and HNOJ on weak base ion exchange resins, in free base form, are studied. The sorption process shows significant reversibility even a t high concentrations of acids. It is also concluded from the dynamic studies that the extent of exclusion of hydrogen ions from the resin pores is far less than that predicted on the basis of complete dissociation of counterions from the protonated ionogens. An attempt is made to correlate the sorption equilibria and sorption dynamics by considering an electric double layer a t the pore walls. A reversible sorption model, which accounts for sorption equilibrium at ionogenic sites of resin, fits the experimental dynamics satisfactorily over the entire range of the resin conversion. Values of the effective pore diffusion coefficient of the acids, regressed from the observed dynamics, have been satisfactorily correlated on the basis of the developed theory.
Introduction Weak base ion exchange resins are commonly used for removal of acids from aqueous streams. Experimental work on the uptake of acids on weak base resins has been reported by Kunin (1958),Adams et al. (19691, HOll and Sontheimer (1977), Hubner and Kadlec (1978), Rao and Gupta (1982a,b),and Helfferich and Hwang (1985). The reaction of an acid with a weak base resin involves protonation of the ionogenic sites of the resin by the acid. The mechanism of ion exchange involving ionic reactions was first postulated by Helfferich (1965). He stated that whenever there is an irreversible sorption on a resin, the dynamics of sorption can be explained on the basis of the shrinking core model. Helfferich and Hwang (1985) suggested that acid sorption by most of the commercial weak base ion exchangers can be modeled considering sorption to be irreversible under most conditions. However, the work of Kunin (1958) and also the present work on sorp-
tion equilibria on weak base resins indicate that the sorption is significantly reversible, especially at low acid concentrations and for the resins with low basicity. In such situations the shrinking core model is not expected to be valid. So far, the most general treatment of the dynamics of the sorption of acids on weak base resins has been presented by Helfferich and Hwang (1985). They have also suggested a rate-controlling-ion model, which is a simplified version of their generalized model. According to these authors, the rate-controlling step in the sorption of strong acids (such as HC1) is the diffusion of H+ ions, which are excluded to a significant extent form the resin pores as a result of Donnan exclusion. Their model assumes complete dissociation of the acid anions from the protonated ionogenic species of the resin. Concentration of H+ in the resin is computed using the ideal Donnan exclusion principle. Analytical solution of the shrinking core model
0888-5885/92/2631-1060$03.00/00 1992 American Chemical Society
