Ind. Eng. Chem. Res. 1991,30,1351-1361
1351
Sorption of Organic Vapors in Polyethylene Shah J. Doongt and W.5. Winston Ho* Corporate Research, Exxon Research and Engineering Company, Route 22 East, Annandale, New Jersey 08801
The solubilities of a series of aromatic vapors in semicrystalline polyethylene were obtained by a modified gravimetric sorption technique with a flow system capable of a wide range of temperatures and activities. The experimental data were fitted well by a theoretical model. This model takes into account the free energy contributions in the penetrant-polymer system from combinatorialentropy, free-volume, interactional-enthalpy, and elastic factors. Both UNIFAC group contribution and solubility parameter methods were used to calculate the interactional effects. In semicrystalline polyethylene, the elastic contributions due to the inhibition of chain deformations in amorphous regions from crystalline domains were very significant in determining the temperature dependence of the solubilities. Unlike molten polyethylene, the effects of the number of methyl substituents and the chain length of an alkyl substituent in the benzene ring on solubilities were found to be insignificant for semicrystalline polyethylene. The elastic factor, which is absent in molten polyethylene, decreased the solubilities more significantly for the bulkier penetrants, resulting in the insignificant effects observed for semicrystalline polyethylene. Introduction An understanding of equilibrium sorption of gases and vapors in polymers is important in various areas such as separation, packaging, and polymer processing. Prediction of solubilities of small molecules in polymers has attracted special attention in recent years (Misovich and Grulke, 1988; Tseng et al., 1985; Iwai et al., 1981; Oishi and Prausnitz, 1978;Biros et al., 1971;Flory, 1970). Traditionally, the penetrant-polymer system has been described by the Flory-Huggins theory (Flory, 1953),which uses the lattice model to account for the entropy effect and the interaction parameter x for the enthalpy effect. Although the Flory-Huggins equation has found some success in correlating the experimental sorption data (Prager et al., 1953),the strong temperature and concentration dependence of the interaction parameter x has greatly reduced its predictive capability (Orwoll, 1977). Flory (1970)introduced an equation of state for liquids and polymers. Extension of this equation of state to a polymer-penetrant mixture results in an additional "free-volume effect" or "equation of state effect" in addition to the entropy and enthalpy effects mentioned above. Sophisticated as it is, this theory requires the information of pure components that is usually difficult to obtain. These include the thermal expansion coefficient, and thermal pressure coefficients of polymer and penetrant. To fit experimental data, one or two adjustable parameters sometimes have to be introduced (Sharma and Lakhanpal, 1983;Flory and Shih,1972;Chahal et al., 1973). Group contribution concepts have been successfully used for the prediction of activity coefficients in solution. These were demonstrated by the analytical solution of groups (ASOG) model (Derr and Deal, 1969)and the UNIQUAC (universal quasi-chemical activity coefficient) functional group activity coefficient (UNIFAC) model (Fredenslund et al., 1975). Oishi and Pransnitz (1978)extended the UNIFAC method to polymer solutions. With the incorporation of the free-volume effect in the UNIFAC-FV model, they showed that the solvent activities in various polymer solutions can be estimated with an uncertainty of no more than 10%. Also, as was shown recently by High and Danner (1988),this model predicted well the activities of various solvents in polystyrene. The advantage of the ~~
UNIFAC method is that no experimental data for the solvent-polymer mixture are required. Hildebrand's regular solution theory, in conjunction with the Flory-Huggins lattice theory, also provides an estimation of the equilibrium sorption of penetrants in polymers. The Flory-Huggins interaction parameter x is closely related to the solubility parameters of the penetrants and polymers (Barton, 1983). The concept of the solubility parameters has been widely applied to the solvent selection in the polymer industry. Quantitatively, it offers a simple method to calculate penetrant solubilities in polymen without any adjustable parameter and mixture data. While the Flory theory, the UNIFAC method, and the solubility parameter method are only good for amorphous polymers, the equilibrium sorption in semicrystalline polymers requires some different considerations. It is generally believed that the crystalline region of a polymer is not accessible to penetrants. Nonetheless, the sorption characteristics in the amorphous domain of a semicrystalline polymer are not the same as those in a totally amorphous polymer. The presence of the crystalline region imposes a constraint on the polymer chains in the amorphous domain, as suggested by Michaels and Hausslein (1965). Rogers et al. (1959,1960) considered the crystalline region as cross-links restricting the swelling or sorption of penetrant in a network. Apparently, the prediction of sorption in semicrystalline polymers is more complicated than that in amorphous polymers. In this paper, the solubilities of a series of aromatic vapors in semicrystalline polyethylene were obtained by a modified gravimetric sorption technique with a flow system capable of a wide range of temperatures and activities (or vapor pressures). The experimental data were fitted by a theoretical model. This model takes into account the free energy contributions in the penetrantpolymer system from combinatorial-entropy,free-volume, interactional-enthalpy, and elastic factors. This model contains only one adjustable parameter, which is characteristic of each individual polymer and independent of temperature and penetrant concentration. As long as this single parameter is determined, this model can be used to predict the solubilities of other penetrants over a wide range of concentrations and temperatures. Both the UNIFAC erouD contribution and solubilitv parameter methods were used to calculate the interaciihal effects and their results were compared. The dependence of the solubilities on temperature is elucidated via the elastic I
* To whom correspondence should be addressed.
'Present address: The BOC GrouD. Technical Center. Murrav Hill, NJ 07974.
.
0888-5885/91/2630-1351$02.50/00 1991 American Chemical Society
1352 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
factor. The effects of the methyl and higher alkyl substituents in the benzene ring on solubilities are discussed in terms of the factors of this model. Different sorption characteristics between semicrystalline and molten polyethylenes are also elucidated.
Theoretical Details For a semicrystalline polymer, given the penetrant volume fraction #,, the activity of the penetrant a, can be calculated from the following expression: lna, = lna, + Inat, combinatorial freevolume
+
1nainr interactional
+
1nael (1) elastic
These are the free energy contributions in the penetrant-polymer system from combinatorial-entropy, freevolume, interactional-enthalpy, and elastic factors. The first three factors are similar to the UNIFAC-FV model (Oishi and Prausnitz, 1978). The fourth factor takes into account the elastic contribution due to the crystalline domain of the polymer (Rogers et al., 1959,1960). These factors are discussed as follows. 1. Combinatorial-EntropyFactor. This factor takes into account the differences in size and shape of two dissimilar molecules. The Flory-Huggins lattice theory is usually assumed to be adequate (Flory, 1953): In a, = In + (1- 4,) (2) In the UNIFAC method, Staverman's combinatorial formula, eq 3, is used instead of the Flory-Huggins expression (Fredenslund et al., 1975):
where &' is the segment weight fraction of penetrant, MI the molecular weight of the penetrant, 2 the lattice coordination (set to be equal to lo), q, the surface area parameter of the penetrant, and 8, the surface fraction of the penetrant. If the segment fraction is used in eq 2, the difference between the calculated results from eq 2 and those from eq 3 is insignificant. 2. Free-Volume Factor. Flory (1970) developed an equation of state for pure liquid or amorphous polymer. Adaptation of this equation of state to a penetrant-polymer mixture results in the following expression for penetrant activity:
where 3C1is the number of external degrees of freedom per penetrant molecule. For hydrocarbon penetrants with carbon numbers 4-10, C1 is about 1.1(Oishiand Prausnitz, 1978). In eq 4 D* = u,/u,*,
Dm = u,/u,*
(5)
where u1 and u, are the specific volumes for the penetrant and the penetrant-polymer mixture, respectively. ul* and u*, are the specific hard-core volumes for the penetrant and the penetrant-polymer mixture, respectively. Assum-
ing the volumes are additive, u, can be obtained from and u,*
can be calculated from (Flory, 1970):
-1- u,*
a1 u1*
+ -u2* 02
(7)
where w, and w2 are the weight fractions for the penetrant and the polymer, respectively, and u2 and u2* the specific volume and the specific hard-core volume for the polymer, respectively. The second and third terms on the right-hand side of eq 4 are called the "free volume" or "equation of state" contributions, Le. In a*. = I.
3C, In
[ -1
0,113
-1 -
(% - 1)( 1 -
Cl[
um
L)] 41/3
(8)
Because polymer molecules are much more tightly packed than penetrant molecules, the reduced volume of the polymer O~ usually is closer to 1 and smaller than the reduced volume for the penetrant Ol. The origin of the free-volume effect comes from the different liquid structures between penetrants and polymers. The last term on the right-hand side of eq 4 includes a parameter X12,representing the exchange enthalpy contribution from neighbor interactions. The interactionalenthalpy factor will be discussed next. 3. Interactional-Enthalpy Factor. According to Flory's theory, the parameter X12has to be specified in order to evaluate the interactional effect. Furthermore, a surface area parameter is also required to calculate the surface fraction of the polymer, O2 (see eq 4). Since X12 is characteristic of each penetrant-polymer pair, the reliable value of it depends on experimental data. In most cases (Chahal et al., 1973; Eichinger and Flory, 1968),X12 was considered as an adjustable parameter to best fit the sorption data. In this paper, both UNIFAC group contribution and solubility parameter methods were adopted to evaluate interactional effects. The UNIFAC group contribution method (Fredenslund et al., 1975; Oishi and Prausnitz, 1978) considers a liquid mixture as a solution of functional groups rather than a solution of molecules. According to this method, the interactional part of the activity is In aint= C d [ l n k
rk- In r j ~
(9)
where v'; is the number of group k in molecule i, rk the interaction activity of group k in the penetrant-polymer mixture, and rl, the interaction activity of group k in pure component i. rk can be calculated from the UNIQUAC equation
where qk' is the surface area parameter for group k, 8,' the surface area fraction of group m (related to the surface area parameter of the group, i.e., similar to q1 and 8, for penetrant l), and I),,,,, is given by the group interaction parameter An,,,: I)nm = exp(-Anm/T) (11) The UNIQUAC equation (eq 10) contains two adjustable parameters, A,,,,, and A,,,,,, per pair of functional groups,
Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 1383 which were evaluated from phase equilibrium data for mixtures containing various functional groups. From a data bank for the surface area parameter qh' and the interactional parameters A, and A,,,,,, the UNIFAC method can be applied to calculate the interactional activity In aht for the penetrant-polymer system. In the second approach to evaluate the interactional effect, the solubility parameter method suggests that the energy of mixing in binary solutions can be calculated by the square of the difference between penetrant and polymer solubility parameters, 6. The solubility parameter is defined as the square root of the energy of vaporization per molar volume, i.e., 6 = (AE/V)'/~. This enthalpy effect is precisely what the last term of eq 4 in Flory's theory attempts to evaluate. Consequently, we can express the activity due to interaction as @22
In aint= (Ml~,*Xl2/o,)- = V1&?(6,- b2)2/RT (12) RT where VI is the molar volume of the penetrant. In the presence of polar or hydrogen bonding interactions, the solubility parameter can be expanded into the three-dimensional form, as suggested by Hansen (1967): In aht = The solubility parameters 6,+ a, and ah are listed in tabular form for most of the liquids. Solubility parameters for common polymers have also been estimated in the literature (Barton, 1983). In case those values are not available, they can also be calculated from various group contribution methods (van Kreveln and Hoftyzer, 1976). All these features make the solubility parameter method quite attractive in calculating the interactional effect without any adjustable parameter. One of the problems with this method is the geometric mean assumption implicit in eq 12 or 13. The geometric mean assumption may be suitable for nonpolar molecules; however, their applicability to polar or specific interactions is doubtful. In the system studied here, alkylbenzenesand polyethylene, only nonpolar or slightly polar interactions were involved. The solubility parameter method was, therefore, used to calculate the interactional effect. 4. Elastic Factor. In semicrystalline polymers such as polyethylene, although the crystalline domain is not accessible to the small molecules, it affects the tension or tightness of amorphous chains. This restraining action imposed by the crystalline domain results in an additional elastic free energy contribution. Rogers et al. (1959,1960) were first to model this effect. They considered that the crystalline lattices behave like giant cross-links and so decrease the sorption. This assumption enabled them to apply the Flory-Rehner theory of swelling of cross-linked rubbers to semicrystalline polyethylene. According to the Flory-Rehner theory, the elastic contribution to the activity of the penetrant is (Flory and Rehner, 1943): In a,l = ( ~ ~ V 1 / M ~ ) ( d d ~ ' ~ (14) where pa is the density of the amorphous polymer and M, the molecular weight between cross-links or crystalline domains. This theory in combination with the traditional Flory-Huggins equation was used to explain the solubility data in semicrystalline polyethylene, and the Flory-Huggins interaction parameter x and M , were estimated by data correlation (Liu and Neogi, 1988; Brown, 1978). However, in some cases the correlations were not good and the obtained values of x and M , were unreasonable. Moreover, the strong temperature dependence of solubility
at constant activity cannot be modeled by eq 14. The unusually large enthalpies of sorption in Semicrystalline polyethylene relative to those in completely amorphous polyethylene are also difficult to explain by the FloryRehner theory (Kwei and Kwei, 1962). To overcome the problems mentioned above, Michaels and Hausslein (1965)proposed a theory suggesting that the tension on the intercrystalline tie chains is in equilibrium with the free energy driving force of crystallization. Thermodynamic equilibrium requires the quality of the polymer chemical potential in the amorphous phase pa and that in the crystalline phase pi.
d = Pus
(15)
may be considered as the sum of two terms, p?, the chemical potential of the amorphous chain in the presence of penetrant, plus &, the chemical potential due to the elastic constraints imposed by the crystallites on the chain segments. If cl? is expressed by -RT(Vz/V1)(91- ~ 9 1 ~ ) from the Flory-Huggins equation (assuming x is independent of concentration) and cli is expressed by the theory of the melting point depression for a crsytalline polymer, -AHz[l- T/T,], eq 15 can be rewritten as pi
where AHz is the molar heat of fusion and T, the melting point of the crystalline polymer. By the same reasoning, the penetrant chemical potential in the amorphous phase p; should be the same as that in the vapor phase p:, and clt may be considered as the sum of p y and F!. P? + P? = CCY
(17)
the sum of RT In a,, RT In afi,and RT In aht, as described above, and p: is RT In al. Evaluation of p! and pt is based on the assumption that the force-elongation relations follow the Hookean behavior for polymer chain segments. With this assumption, Michaels and Hausslein (1965)have obtained the final expression for & or RT In ael: pf" is just
p:
= RT In ael=
(18) where f is the fraction of elastically effective chains in amorphous regions. The amorphous regions of the polymer are composed of two types of segments-elastically effective and elastically ineffective. The former are responsible for chain deformation by sorption. The latter consist of loops, isolated chains, and chain ends terminating in the amorphous regions; they are not elastically deformed when the penetrant is sorbed by the polymer. f is characteristic of each semicrystalline polymer and independent of temperature and penetrant concentration. The Flory-Huggins equation is In al = In
c.q j
+ (1 - dl) + x(1 - d1)2
(19)
where x is the interaction parameter and 1 - 91= $2 Since x takes into account the interactional and free-volume contributions, it can be evaluated from In aht + In at, X = (20) 92
1354 Ind. Eng. Chem. Res., Vol. 30, No. 6,1991 Substituting eq 20 into eq 18,we obtain
Data Acquisition T, P, W --------- --------
Cahn 2000 Balance
Computer
where m2is the heat of fusion per gram of crystalline polymer. AH2/V2has been equated to Unfortunately, independent evaluation of parameter f is not possible at present. The theory of Michaels and Hausslein is still considered as a correlation method with a single parameter. However, this theory is able to account for the temperature dependence of solubility of the large heat of sorption in semicrystalline polyethylene. When the temperature of the polymer is raised or brought close to its melting point, the constraint of the amorphous chain imposed by crystallite is relieved, presumably due to the thermal motion. Equation 21 predicts a smaller penetrant activity at higher temperature. In their original paper, Michaels and Hausslein (1965) used eq 18 to interpret the enhancement of penetrant activity due to the elastic factor for the xylene and polyethylene system. The required Flory-Huggins parameter x in eq 18 was estimated from the sorption data above the melting point of polyethylene, for which there was no elastic contribution. Equation 19 was then applied to calculate x for molten polyethylene. The value of x for semicrystalline polyethylene was obtained by extrapolation to the low-temperature range. Thus, they were able to use a single value off to fit the sorption data for semicrystalline polyethylene. In this paper, eq 21 was employed for solubility modeling, with the free-volume and interactional contributions calculated in a predictive way. The only adjustable parameter in this modeling is f , which is characteristic of the polymer and independent of temperature and penetrant concentration. In this study, the solubility data of a series of aromatic vapors in semicrystalline polyethylene were fitted by the following four solubility modeling methods, and their results were compared. These methods were (1)the UNIFAC and Michaels-Hausslein method, (2)the UNIFAC and Flory-Rehner method, (3) the solubility parameter and Michaels-Hausslein method, and (4)the solubility parameter and Flory-Rehner method. In the first method the UNIFAC method was applied to calculate the interactional effect, In aw. The required group parameters were obtained from the tables given by Oishi and Prausnitz (1978)and Skjoid-Jorgensen et al. (1979).The combinatorial contribution was calculated from eq 3. The freevolume effect was calculated from eq 8. According to Oishi and Prausnitz (1978),the specific hard-volume ui* was calculated from 1 Mi
vi* = 19.14-rj
(22)
where Mi is the molecular weight of component i (penetrant or polymer) and ri the hard-core volume parameter. The elastic contributions were computed from eq 21 based on the Michaels-Hausslein theory. The second method was the same as the first method except that the FloryRehner theory, eq 14,was used to compute the elastic contributions. In the third method, the solubility parameter method was used to estimate the interactional contribution instead of the UNIFAC method, and eq 2 was employed to calculate the combinatorial effect instead of
U Liquid Feed
$q--Na M a u Flow Controllers
Liquid Saturator
Oven f 0.05OC
Figure 1. Schematic of gravimetric sorption apparatus with a flow system.
eq 3. The calculations for free-volume and elastic effects were the same as in the first method. The fourth method was the same as in the third method except that the Flory-Rehner theory, eq 14,was used to compute the elastic contributions.
Experimental Section Apparatus. Figure 1 shows the schematic of the gravimetric sorption 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 temperatures and partial pressures. With the flow system, the electronic components of the balance chamber were purged with an inert N2 gas stream, which exited through the side port of the hang-down tube. Penetrant vapor was carried into the system by N2 gas from the bottom of the hang-down tube, and the penetrant vapor/N2 gas stream exited also through the side port of the hang-down tube. For this penetrant stream, one N2 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. N2flows were measured and controlled by mass flow controllers. A constant-temperature oven (f0.05"C) was used to heat the liquid saturator. The partial pressure of the penetrant vapor carried was calculated from the N2 flow rates and the saturation pressure of the penetrant at the oven temperature. The hang-down tube, in which the polymer sample was placed in a pan with a hang-down wire connected to the balance, was enclosed in a split tube furnace (iO.l "C). All the incoming lines were wrapped with heating tapes to prevent any condensation of vapor. The measured weight change, sample temperature, and system pressure were stored in a Hewlett Packard (HP) computer via a data acquisition unit. Unlike the traditional gravimetricapparatus, the current setup was not operated under vacuum, although it was also equipped with a vacuum system to be operated in the traditional way. With the heating tapes wrapped around the feed lines, the flow system was capable of operating at penetrant activities (partial pressure/saturation pressure of penetrant at sample temperature) close to 1 and saturator temperatures close to the boiling point of the penetrant liquid. In a traditional closed system, the penetrant vapor was introduced into the balance, which was previously evacuated. The maximum vapor pressure was limited to the saturation pressure of the penetrant at ambient temperature, as the vapor would condense in the balance chamber. An alternative is to enclose the whole balance
Ind. Eng. Chem. Rea.,Vol. 30,No.6,1991 1355 I
e
1
1
I
I
I
I
'qooc
-UNlFAC and Michaels-Haudeln Method
0
Michaeir.Hauriein Method
-i
m 0.10
-
3
I
0.05
-
-
u)
-
a
o'ooOfil
Of2
Ob
0!4 015 0:s 0:7 Penetrant Activity, a
d.8
d.9 1.0
Figure 2. Solubility of benzene in semicrystalline polyethylene.
in a constant-temperature box (Liu and Neogi, 1988). Nevertheless, the box temperature cannot exceed the maximum allowable temperature of the electrobalance (usually 60 "C). The major complexity of the flow system was the hydrodynamic effect, which reduced the weight measured by the microbalance. This could be overcome by switching the inlet flow to a bypass exhaust outlet at the end of the sorption equilibrium measurement. The equilibrium weight was then measured without the hydrodynamic effect in a short period of time (less than 0.5 min.). Materials. A commercial polyethylene film obtained from Exxori Chemical Co. was used in this study. It had a thickness of 43 pm, a density 0.920 g/cm8, 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. Prior to experiments, the sample was immersed i n t ~ liquid toluene to wash the film surface. It was then outgassed at 60 OC overnight in the sorption apparatus. The experiments were conducted at 30,40,50,and 60 "C for each penetrant. The polyethylene film never encountered temperatures higher than 65 "C to ensure there was no morphology change. The sorption results were found to be independent of the film history in this temperature range. Six aromatic compounds were used as penetrants, and they were benzene, toluene, mesitylene (1,3,5-trimethylbenzene), prehnitene (1,2,3,4-tetramethylbenzene),npropylbenzene, and n-butylbenzene. These compounds 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 constanta available from the thermodynamic tables of Thermodynamics Research Center (1987).
Results and Discussion The solubilities of six aromatic compounds in the polyethylene sample at four different temperatures and a wide range of activities are shown in Figures 2-7. These solubilities showed a similar upward bend against activities, which is a typical sorption behavior in rubbery polymer. Furthermore, the solubilities increased with increasing temperature at the same activities. Thii indicated that the sorption was of endothermic nature.
o'wO.O
0.2
0.1
0.3
0.4 0.5 0.6 0.7 Penetrant Activity, a
0.8
0.9
1.0
Figure 3. Solubility of toluene in semicrystalline polyethylene.
l .
-
g!* 0.20 --
I
I r
e '0
1I
I
I
I
I
1
I
I
Raw v
6OoC 0 5OoC A 40°c o 30°C -UNiFAC and Michaels-Haursiein Method -J
0.15
L
-i
0.10
-
3 s . c
-
-
0.05 v)
0.00 0.0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Penetrant Activity, a
0.8
0.9
1.0
Figure 4. Solubility of mesitylene in semicrystalline polyethylene.
A
40.C
0
30.C UNiFAC and
Penetrant Activity, a
Figure 5. Solubility of prehnitene in semicrystalline polyethylene.
The data were first correlated with the Flory-Huggins equation, eq 19. In the correlation the experimentally obtained solubilities,which were e x p r d in terms of the weight gain w (g of penetrant/g of dry polymer), were
1356 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
v 60%
5OoC
-30% UNIFAC and Michaels-HaussleinMethod 0
a
E
f : - P
0*15
L
e
a
0
0.10
1
-I"s
I
2 0.05 -
8
0.00
A 0.1
0.0
Penetrant Activity, a
Figure 6. Solubility of n-propylbenzene in semicrystalline polyethylene. Table I. Flory-Huggins Interaction Parameters x Aromatic Vapors in Semicrystalline Polyethylene x at various temp penetrant 30°C 40°C 50°C 1.20 1.10 0.93 benzene 1.10 0.95 toluene 1.23 0.87 1.09 1.01 mesitylene 1.03 0.92 prehnitene 1.10 n-propylbenzene 1.24 1.09 1.09 1.08 1.04 n-butylbenzene 1.20
for
60°C 0.86 0.78 0.77 0.77 0.96 0.98
related to the volume fraction of the penetrant in the amorphous phase of the polymer, &, by $1=
1,
[+ J 1
where p1 is the liquid density of the penetrant, a, the weight fraction of the amorphous phase in the polymer (0.55 for the polyethylene sample), and pa the density of the amorphous phase of the polymer. Equation 23 assumes that there is no change of volume upon mixing and the penetrant in the polymer has the same density as in the liquid phase. For eq 23, we used the densities of the penetrants at various temperatures available from the thermodynamic tables of the Thermodynamic Research Center (1987) and a value of 0.85 g/cm3 at 25 "C for the density of the amorphous phase of polyethylene (pa) with its thermal expansion coefficient of 7 X 10"' K-' (van Krevelen and Hoftyzer, 1976) to account for the temperature dependence of pa. It is generally believed that only
0.2
0.3 0.4 0.5 0.8 0.7 Penetrant Actlvlty, a
0.8
0.9
Figure 7. Solubility of n-butylbenzene in semicrystalline polyethylene.
the amorphous phase is accessible to penetrant. So only the penetrant volume fraction in the amorphous phase of the polyethylene sample was considered and used in the modeling in this paper. The obtained values of FloryHuggins parameters x are listed in Table I. It was found that, for each individual penetrant at a temperature, a single parameter x was able to fit the data within 5% error. The Flory-Huggins parameters were independent of concentration but dependent on temperature in this case. Furthermore, x varied from penetrant to penetrant. To account for the variations of x among the penetrants and temperatures, we need a more elaborate theory. Solubility Modeling Methods with UNIFAC: Comparison of Michaels-Hausslein and Flory-Rehner Theories. The two solubility modeling methods, (1) the UNIFAC and Michaels-Hausslein method and (2) the UNIFAC and Flory-Rehner method, were used to fit the solubility data shown in Figures 2-7. The MichaelsHausslein theory contains the unknown parameter f , the fraction of elastically effective chains in the amorphous phase, whereas the Flory-Rehner theory contains the unknown parameter M,,the chain molecular weight between crystalline domains. With the combinatorial, free-volume, and interactional contributions calculated as stated (UNIFAC-FV) in Theoretical Details for these two solubility modeling methods, the experimental data were fitted to eq 21 based on the Michaels-Hausslein theory with the parameter f and to eq 14 of the Flory-Rehner theory with the parameter M,. For eq 21, the heat of fusion for the crystalline domain of polyethylene, A& used was 65 cal/g, which was calculated from the group contribution method
Table 11. Comparison of Solibility Modeling Methods error: %
solubilitv data all benzene toluene mesitylene prehnitene n-propylbenzene n-butylbenzene a
UNIFAC and Michaels-Hausslein method (f = 0.373) 7.96 6.74 5.54 9.25 7.23 8.38 8.64
1.0
UNIFAC and Flory-Rehner method (Me= 216) 12.40 14.08 15.57 11.87 11.02 11.53 9.28
solubility param and Michaels-Hausslein method (f = 0.380) 8.69 6.60 9.25 8.75 9.40 10.19 7.58
solubility param and Flory-Rehner method (M,= 211) 14.62 14.52 17.04 13.39 11.56 16.81 13.52
Error = ((l/n)~,,[(calculatedactivity value experimental activity value)/ (experimental activity val~e)]~1'/~.
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1357
I.' n
I
1
I
I
I
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I
1
1
-f, 0.20 -g
6OoC 5OoC A 4OoC 0 3OoC UNIFAC and Flory-Rehner Method P
0.20 1
8P
n
&
0
-
E
U
P
U
30°$$1
01
3
LL
3e 0*15
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0.15
-
v 6OoC 0 50°C A 40°C o 30% -UNiFAC and Flory-Rehner Method
60°C v\
c
c
-
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CI
P 0
I
P
a
0.10 -
e 0.10 D
D
-9s
r"
'0 0.05 -
0.05
u)
u)
I 0.00 0.0
0.1
I
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0.2
0.3
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0.4 0.5 0.6 0.7 Penetrant Activity, a
I
0.8
-
I
0.9
1.0
o.oo/
'
a'
n
I
I
I
I
1
I
I
I
0
Figure 8. Solubility of mesitylene in semicrystallinepolyethylene as fitted by the UNIFAC and Flory-Rehner method.
Figure 9. Solubility of n-butylbenzene in semicrystalline polyethylene as fitted by the UNIFAC and Flory-Rehner method.
(van Krevelen and Hoftyzer, 1976). This value is consistent with that used by Michaels and Hausslein (1965), which was AH2 = 910 cal/mol of CH2 unit. A total of 240 data points from 6 penetrants at 4 temperatures were all used in the single-parameter fitting. The obtained value of parameter f or M,was then used to compute the relative root mean square error of the corresponding solubility modeling method for each penetrant, as shown in Table 11. As shown in this table, the Michaels-Hausslein theory fitted the data within about 10% error. This demonstrated that the parameter f is a property of the semicrystalline polyethylene sample and independent of temperatures and penetrants. The results are also shown in Figures 2-7 for benzene, toluene, mesitylene, prehnitene, n-propylbenzene, and n-butylbenznee, respectively. The obtained f value, 0.373, is consistent with the results of Michaels and Hausslein (1965), ranging from 0.265 to 0.365 for various polyethylenes. It is possible that the parameter f depends on temperature because the crystallinity changes with temperature. Therefore, the assumption of constant value off is only applicable to the temperature range far below the melting point of the polymer, where the crystallinity is a weak function of temperature. For the temperatures from 30 to 60 "C used in this study, the crystallinity of polyethylene does not change more than 10% (Charlesby and Callaghan, 19581, so a constant value off is justified. On the other hand, the Flory-Rehner theory did not fit the data too well. Its inability to account for the temperature dependence of solubilities is also shown in Figures 8 and 9 for mesitylene and n-butylbenzene, respectively. Obviously, both the interactional effect calculated by UNIFAC and the free-volume effect were also unable to explain this temperature dependence. Thus, the elastic contribution, due to the presence of the crystalline domain, was responsible for the temperature dependence of solubilities. When penetrants are sorbed into the amorphous phase, they expand or stretch the polymer chains. Energy is stored in the form of elasticity. This is consistent with the endothermic sorption found in the experimental data. If the Flory-Rehner equation was employed to correlate the data for each penetrant at an individual temperature, the correlation results were much better than the singleparameter fitting, as shown in Table 111. However, the magnitudes of M,showed an unreasonable variation with temperature. It was unlikely that the chain molecular weight between two crystallites could increase more than
Table 111. Modeling of Solubility Data by the UNIFAC and Flory-Rehner Method for Each Penetrant at an Individual Temwrature penetrant temp, "C Me error, 5% 30 138 5.08 167 6.02 benzene 40 50 252 9.76 299 2.79 60 30 138 4.38 40 173 2.83 237 5.80 toluene 50 60 363 2.43 600 270 5.49 30 178 3.05 205 2.74 mesitylene 40 1.76 261 50 327 3.37 60 171 1.94 30 189 3.42 prehnitene 40 223 2.38 50 290 4.88 60 30 141 4.88 174 5.74 n-propylbenzene 40 179 4.82 50 226 5.13 60 30 160 3.58 193 7.64 n-butylbenzene 40 50 211 7.97 60 239 7.95 ~
a 35%
~~
~
crystallinity.
twice in the temperature range 30-60 "C. In this temperature range, the crystallinity of polyethylene, determined by the DSC thermal analysis based on the technique of the heat of fusion (Runt, 19861, did not change more than 10%. An extensive study on polyethylene crystallinity by Charlesby and Callaghan (1958) also demonstrated that the crystallinity for low-density polyethylene only decreased from 50% to 45% when the temperature changed from 30 to 60 "C. Such a small change of crystallinity should not be sufficient to reflect the temperature dependence of solubilities and the large variation of M,. Table I11 also includes a result calculated from 35% crystallinity at 60 "C (instead of 45% ) for toluene. The obtained M,was still almost twice the value of M,at 30 "C. The difference between the M,values should be about 10% corresponding to their crystallinity difference, if the
1358 Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 Table IV. Comparison of Solubility Modeling Methods for Paraffin Data' error, % UNIFAC and Flory-Rehner method (M,= 109) 21.75 24.65 17.33 19.40 25.27
UNIFAC and Michaels-Hausslein method (f = 0.411) 8.07 8.82 3.36 10.14 4.31
solubility data all butane propane hexane heptane
solubility param and Michaels-Hausslein method (f = 0.325) 9.90 14.07 9.72 7.87 6.40
solubility param and Flory-Rehner method (M,= 147) 20.71 25.69 19.08 19.43 17.36
'Paraffin data taken from Castro et al. (1987): butane at 0, 10.20, and 40 O C ; propane at 0, 10,20, and 35 OC; n-hexane at 0,20,30, and
E op 2
-0
0.07
-
v 4OOC
0
ooc
I
I
-
Castro et ai. data (1987)
0*06 - -Solubility Parameter and Mlchaeis-Haus8ieln Method 0.05 -
s 0.04 e
n
-
-
-5 0.02 -
-
E
0.03
1
I
v)
o ' o o O ~ 0:l
012
Oi3
014
015
0!6
d.7
0:s
Oh
1.0
Penetrant Activity, a
Figure 10. Solubility of n-hexane in semicrystalline polyethylene.
Flory-Rehner equation was applicable. To further compare the Michaels-Hausslein and Flory-Rehner theories, the solubility data of n-butane, npentane, n-hexane, and n-heptane in a polyethylene film at different temperatures reported by Castro et al. (1987) were modeled by the use of the UNIFAC and MichaelsHausslein method and the UNIFAC and Flory-Rehner method. Our modeling results of their data are presented in Table IV. Again, a single value of parameter f fitted all the data quite well with the Michaels-Hausslein theory. However, the fitting with the Flory-Rehner equation was very poor. It should be noted that their data also showed the increase of the solubilities with increasing temperature at the same activity (see Figure 10). Solubility Modeling Methods with Solubility Parameters. The solubilitiesshown in Figures 2-7 were also modeled by the use of (1) the solubility parameter and Michaels-Hausslein method and (2) the solubility parameter and Flory-Rehner method. In the modeling, the Hansen three-dimensional solubility parameters used are listed in Table V for each penetrant and amorphous polyethylene. The modeling results are shown in Table I1 for comparison with those for the solubility modeling methods with UNIFAC. Examining this table shows that the solubility parameter and Michaels-Hausslein method gave only slightly worse results than the UNIFAC and Michaels-Hausslein method. Also, in the modeling methods with solubility parameters, the Michaels-Hausslein theory for the elastic effect was superior to the Flory-Rehner theory, the same conclusion found in the modeling methods with UNIFAC. The success of both UNIFAC and solubility parameter methods can be attributed to the lack of strong polar interaction and hydrogen-bonding effects between these aromatic compounds and polyethylene.
Table V. Hansen's Three-Dimensional Solubility Parameters compound hd, MPa1J2 6,, MPa1J2 6hr MPa1/2 benzene 18.4 0 2.0 1.4 18.0 2.0 toluene 0.0 18.0 0.6 mesitylene 0.74b 18.00 0.6' prehnitene 0.7gb 17.0b 0.6' n-propylbenzene 16.95b 0.70b 0.6' n-butylbenzene 0 0 14.1 butane 0 14.5 0 pentane 0 14.9 0 hexane 0 0 15.3 heptane 0 0 16.5b polyethylene aAssumed the same aa mesitylene. bCalculated from group contribution method (van Krevelen and Hoftyzer, 1976).
The solubility modeling methods with solubility parameters were applied to the data of Castro et al. (1987). The modeling results are included in Table IV for comparison with those from the solubility modeling methods with UNIFAC. As shown in this table, again, the solubility parameter and Michaels-Hausslein method and the UNIFAC and Michaels-Hausslein method were much better than the solubility modeling methods with the Flory-Rehner theory. Both modeling methods with the Michaels-Hausslein theory fitted all the data satisfactorily to about 10% error. However, the obtained values of the parameter f were 0.411 for the UNIFAC and MichaelsHausslein method and 0.325 for the solubility parameter and Michaels-Hausslein method. This difference is due to the fact that for the sorption of paraffin in polyethylene, the UNIFAC method predicts no interactional contribution but the solubility parameter method predicts a finite value for interactional contribution. The UNIFAC method assumes no difference between methyl and methylene groups in terms of interactional effect. On the other hand, the solubility parameter increases with the increasing chain length of paraffins (see Table V). This different interpretation of the interactional contribution is absorbed into the values of the adjustable parameter f . Nevertheless, both methods can fit the data quite well. Figure 10 shows the modeling of n-hexane solubility data for polyethylene by the use of the solubility parameter and MichaelsHausslein method. Effects of Methyl and Higher Alkyl Substituents on Solubility. From the analysis of the theoretical model, significant insights into the sorption of small molecules in polymers can be revealed. One of the examples is the effects of methyl and higher alkyl substituents in the benzene ring on solubility. Inspection of Figures 2-5 and Table I shows that the solubilities in this semicrystalline polyethylene only increased slightly with the number of methyl substituents in the aromatic rings for benzene, toluene, mesitylene, and prehnitene. In this table, the
Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991 1359
0.6 -
I
+,= 0.1
I
I
40-C
I V
V V
V
0.4 -
0.4
V
-
A
0.2 -
0
A
A
-
0
0.2
A 0
-
V
-
A
-
A
-
A
0
A
A
0
0
0
0
0.0 In a
-o.2
-0.4
-
Tot01 o Combinatorial 0 Free-Volume A lnteractlonal 0 Elastic From UNIFAC and Michrele-Hauaslein Method
O.O
0
in a
-
-1.6 -1'4/
Ben
0
Total Combinatorial Free-Volume -0.2 - A Interactlonrl v Elastic - From UNIFAC and -o.4 - Michaels-HaussleinMethod
-0.6 *-
-
-
0
0
.
0
-
-
0
e
-0.6
-
0
0
-
-
To1
1 M? pre 4 Number of Methyl Subatltuents In Benzene Ring 0
Figure 11. Various contributions to penetrant activities that counteract for the effect of the number of methyl Substituents in the benzene ring on solubilities in semicrystalline polyethylene.
smaller the Flory-Huggins parameter x is, the higher the solubility is. As shown in this table and Figures 2,3,6, and 7 for different chain lengths of alkyl substituents in the benzene ring, the solubilities for benzene, toluene, n-propylbenzene, and n-butylbenzene did not show significant variations. One would expect that the solubilities in polyethylene increased with the increasing number of methyl substituents and the chain length of an alkyl substituent in the benzene ring, i.e., decreasing aromaticity, as the methyl or higher alkyl groups should be more compatible with polyethylene. The concept of aromaticity was not applicable here. Other factors had to be taken into consideration. Figures 11 and 12 depict the four free energy contributions to sorption in the polyethylene sample in terms of In a for these two aromatic series with respect to the number of methyl Substituents and the chain length of an alkyl substituent in the benzene ring, respectively. These were calculated from the UNIFAC and Michaels-Hausslein method (f = 0.373)for the penetrant volume fraction in the amorphous phase t$l = 0.1 at 40 OC. p-Xylene and ethylbenzene are also included in Figures 11 and 12,respectively, for comparison. As shown in these figures, the combinatorial contribution accounted for about 60% of the total penetrant chemical potential or activity, the free-volume contribution about 590, the interactional contribution about 5-15%, and the elastic effect about 20-30%. The contributions that increase the activity at a given penetrant volume fraction dl (solubility)decrease the solubility at a given activity. For the same penetrant volume fraction, the combinatorial contribution should be the same for each compound according to eq 2. However, if the segment fraction $1' was used as in eq 3 in the UNIFAC method, the combinatorial contribution would increase slightly with the larger molecular size, as shown in these figures. The free-volume contribution reduced with increasing molecular size, as the free volume of the larger penetrant should be smaller and closer to that of polyethylene. The interactional contribution decreased with the increasing number of methyl substituents and the
Table VI. Flory-Huggins Interaction Parametere x for Aromatic and Paraffinic Penetrants in Molten Pol.vethvleneo x at given temp penetrant 120 O C 145 "C toluene 0.34 0.34 ethylbenzene 0.33 0.33 m-xylene 0.29 0.29 mesitylene 0.24 0.25 n-octane 0.31 0.30 n-nonane 0.28 0.28 n-decane 0.25 0.26 n-dodecane 0.23 0.24 (I
Measured by Schreiber et al. (1973).
chain length of an alkyl substituent owing to the decreasing aromaticity. Finally, the elastic contribution increased with increasing molecular size, because the bulkier penetrant should introduce more stress to the polymer chains and cause more elastic deformations. The above contributions to the penetrant activities counteracted for the effects of the methyl and higher alkyl substituents in the benzene ring on the solubilities in semicrystalline polyethylene. The penetrant activities calculated from all these countributions did not show any appreciable variations, and they were in line with the experimental data. However, in molten polyethylene without the presence of crystallinity, the solubilities of aromatic compounds show remarkable differences. Table VI lists the FloryHuggins parameters x for aromatic and paraffinic penetrants in molten low-density polyethylene measured from the gas-liquid chromatographic technique by Schrieber et al. 1973). As shown in Figure 11, both free-volume and interactional contributions are favorable for the sorption of mesitylene in molten polyethylene, as compared to the sorption of toluene. These two contributions counteract the combinatorial contribution. The combined contributions result in a much smaller value of x for mesitylene, as shown in Table VI. Similar results exist for the sorption of normal paraffins
1360 Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991
lS0l +,
I
I
20%= 0.1
Y
V
i
Total Combinatorial Free-Volume interactional v Elastic From UNIFAC and Michaels-Haussiein
V
0.8
0 0 A
V
0.4
0.0
-0.4
-1.6
I
? But
"
0
0
T
I
I Hex
I
Hep
1
Pen
n-Paraffin Chain Length
Figure 13. Various contributions to penetrant activitiea that counteract for the effect of the n-paraff'ii chain length on solubilities in aemicryatalline polyethylene.
in polyethylene. Figure 13 shows various contributions to the penetrant activities for the penetrant volume fraction dl = 0.1 at 20 OC. Without interactional contributions in the paraffin-polyethylene system, the competition of combinatorial, free-volume, and elastic contributions results in a slightly higher solubility for butane than heptane. The Flory-Huggins parameters are 1.28 for butane and 1.40 for heptane a t 20 "C from the data of Castro et al. (1987) in semicrystalline polyethylene. However, in molten polyethylene, a longer chain paraffii compound has higher solubility, as indicated in Table VI, where n-dodecane has much lower x values at 120 and 145 OC than n-octane. The elastic contribution from the crystalline domain reduces solubility more significantly for a longer paraffin. This results in this different sorption behavior between semicrystalline and molten polyethylenes.
Conclusions The solubilities of a series of aromatic compounds, benzene, toluene, mesitylene, prehnitene, n-propylbenzene, and n-butylbenzene, in semicrystalline polyethylene were obtained by the use of a modified gravimetric technique with a flow system over a wide range of temperatures and activities. The experimental data can be fitted quite well by the theoretical model with the UNIFAC group contribution method in conjunction with the MichaelsHausslein theory. Replacement of the UNIFAC method by the solubility parameter method can also fit the data reasonably well. The solubilities increased with increasing temperature at a given activity. The elastic factor was very critical for the determination of the temperature dependence of the solubilities. The use of the Michaels-Hausslein theory to calculate the elastic factor explained the temperature dependence quite well. However, the FloryRehner theory for the elastic factor was unable to interpret this temperature dependence. Unlike molten polyethylene, the effects of the number of methyl substituents and the chain length of an alkyl substituent in the benzene ring on solubilities were found to be insignificant for semicrystalline polyethylene. The
solubility difference between semicrystalline and molten polyethylenes was due to the elastic factor, which is absent in molten polyethylene. The elastic factor decreased the solubilities more significantly for the bulkier penetrants, resulting in the insignificant effects observed for semicrystalline polyethylene. Nomenclature al: penetrant activity a,: penetrant activity from combinatorial contribution aG penetrant activity from free-volume contribution aht: penetrant activity from interactional contribution a,fi penetrant activity from elastic contribution An,: group interaction parameter in the UNIFAC method C1: total number of external degrees of freedom for penetrant hE: energy of vaporization, cal/mol f: fraction of elastically effective chains in the amorphous region AH : molar heat of fusion for crystalline polymer, cal/mol specific heat of fusion per gram of crystalline polymer, cd/g M1: molecular weight of penetrant, g/mol M,: chain molecular weight between two crystallites in polymer, g/mol qi: _. surface area parameter for molecule i in the UNIFAC method, l / g qk': surface area parameter for group k in the UNIFAC method, l / g R: gas constant ri: core volume parameter for molecule i in the UNIFAC method, l/mol T: absolute temperature, K T,: melting point of crystalline polymer, K Vl: molar volume of penetrant, cm3/mol V,: molar volume of polymer repeating unit, cm3/mol ul: specific volume of penetrant, cm3/g u2: specific volume of polymer, cm3/g u:, specific volume of penetrant-polymer mixture, cm3/g (see eq 6) ul*: specific hard-core volume of penetrant, cm3/g uz*: specific hard-core volume of polymer, cm3/g u,*: specific hard-core volume of penetrant-polymer mixture, cm3/g (see eq 7) D: reduced specific volume, u/u* (see eq 5) w: weight gain, g of penetrant/g of dry polymer Xlz: interaction parameter in Flory's equation of state theory, cal/cm3 2 coordination number
d,:
Greek Symbols
amorphous weight fraction in polymer interactional activity of functional group k in penetrant-polymer mixture :'I interactional activity of functional group k in pure component i "1: number of functional group k in molecule i 0,: surface fraction of component i in mixture 0,': surface fraction of group m in mixture 6d: dispersion component of solubility parameter, MPal/, 6,: polar component of solubility parameter, MPa1I2 66 hydrogen-bonding component of solubility parameter, MPa112 p: density, g/cm3 4; volume fraction of component i in mixture &': segment weight fraction of component i in mixture +:, group interaction parameter in the UNIFAC method (w cy,:
rk:
eq 11)
chemical potential of component i x: Flory-Huggins interaction parameter w: weight fraction in penetrant-polymer mixture
p;
Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991 1361 Subscripts
1: penetrant 2: polymer a: amorphous i: component or molecule k: functional group
Registry No. H 2 W H 2(homopolymer), 9002-88-4; benzene, 71-43-2; toluene, 108-88-3; mesitylene, 108-67-8; prehnitsne, 488-23-3; propylbenzene, 103-65-1; butylbenzene, 104-51-8.
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Received for review August 16, 1990 Revised manuscript received December 26, 1990 Accepted January 21, 1991
