1506
Znd. Eng. Chem. Res. 1991,30, 1506-1515
Calculations of Phase Diagrams and Thermochemistry; Chang, Y . A,, Smith, J. F., Eds.; TMS-AIME: Warrendale, PA, 1979. Kaufman, S. M. Origins of Surface Tension Extrema in Metallic Solutions. Acta Metall. 1967,15, 1089. Kaufman, S. M.; Whalen, T. J. The Surface Tension of Liquid Gold, Liquid Tin, and Liquid Gold-Tin Binary Solutions. Acta Metall. 1965,13, 797. Kazakova, I. V.; Lyamkin, S. A.; Lepinshkikh, B. M. Density and Surface Tension of Pb-Bi Melts. Zh. Fir. Khim. 1984,58,1534. Kelly, A. The Strengthening of Metals by Dispersed Particles. Proc. R. SOC.,A, 1964,282,63. Kelly, A.; Davies, G.J. The Principles of the Fibre Reinforcement of Metals. Met. Reu. 1965a,I O , 1. Kelly, A.; Davies, G. J. Experimental Aspects of Fibre-Reinforced Metals. Met. Reu. 1965b,10,79. Kingery, W. D. Metal-Ceramic Interactions. I.: Factors Affecting Fabrication and Properties of Cermet Bodies. J . Am. Ceram. SOC. 1953,36, 362. Korol’kov, A. M.; Igumnova, A. A. Izu. Akad. Nauk SSSR,Met. Topl. 1961,No. 6,95. Lancaster, J. F. Metallurgy of Welding, 3rd ed.; George Allen & Unwin: London, 1980. Laty, P.; Joud, J. C.; Desre, P. Surface Tensions of Binary Liquid Alloys with Strong Chemical Interactions. Surf. Sci. 1976,60,109. Lazarev, V. B. Experimental Study of the Surface Tension of Indium-Antimony Alloys. Zh. Fiz.Khim. 1964,38,325. Leonida, G. Handbook of Printed Circuit Design, Manufacture, Components and Assembly; Electrochemical Publishers: Scotland, 1981. MacKenzie, J. D. “Ceramic-to-Metal Bonding for Pressure Transducers”; Final Report to National Aerospace Administration, NASA-Lewis Research Center, Contract No. NAG3-295, Report No. N84-22753, April 1984. Manning, C. R., Jr.; Stoops, R. F. High-Temperature Cermets. 11.: Wetting and Fabrication. J . Am. Ceram. SOC.1968,51,415. Matuyama, Y.On the Surface Tension of Molten Metals and Alloys. Sci. Rep. Tohoku Imp. Uniu. 1927,16,555. Mohn, W. R. Your MMC Product Will Stay Put. Res. Deu. 1987, July, 54. Morgan, C. S.; Moorhead, A. J.; Lauf, R. J. Thermal-Shock Resistant Alumina-Metal Cermet Insulators. Am. Ceram. SOC. Bull. 1982, 61,975. Okajima, K., Sakao, H. Equations for Surface Tension Related with Thermodynamic Enthalpy and Activities of the Binary Molten Alloys. Trans. Jpn. Inst. Met. 1982,23,121. Pelzel, E. Die Oberflachenspannung flussiger Metalle and Legierungen 11. Berg- Huettenmaenn. Monatsh. 1949,94, 10.
Pelzel, E.; Sauerwald, F. Dichtemeseungen bei hohen Temperaturen XII. 2 . Metallkde. 1941,33, 229. Predel, B. Association Equilibria in Liquid Alloys and Their Influence on the Formation of Metallic G h . Calculation of Phase Diagrams and Thermochemistry; Chang, Y . A., Smith, J. F., Eds.; TMS-AIME: Warrendale, PA, 1979. Predel, B.; Oehme, G. Kalorimetrische Untersuchung fliissiger Indium-Antimon-Legierungenunter dem Aspekt eines Assoziations-gleichgewichts. 2. Metallkde 1976,67,826. Prigogine, I.; Defay, R. Chemical Thermodynamics; Longmans, Green: London, 1954. Scatchard, G. Equilibria in Non-Electrolyte Solutions in Relation to the Vapor Pressures and Densities of the Components. Chem. Rev. 1931,8, 321. Shimoji, M. Liquid Metals; Academic Press: New York, 1978. Sommer, F. Association Model for the Description of the Thermodynamic Functions of Liquid Alloys. I: Basic Concepts. 2. Metallkde 1982a, 73, 72. Sommer, F. Association Model for the Description of the Thermodynamic Functions of Liquid Alloys. 11: Numerical Treatment and Results. Z. Metallkde 198219,73,77. Sommer, F. Homogenous Equilibria in Liquid Alloys and Glasses. Ber. Bunsen-Ges. Phys. Chem. 1983,87,749. Steeb, S.;Entress, H. Atomverteilung sowie spezifischer elekrischer Widerstand geschmolzener Magnesium-Zinn-Legierunger. 2 . Metallkde. 1966,57, 803. Steeb, S.; Hezel, R. Rontgenographische Strukuruntersuchungen an schmelzflussigen Silber-Magnesium-Legierungen. 2. Metallkde. 1966,57, 374. Stoicos, T. A Chemical-Physical Model for the Thermodynamics of Binary Metallic Solutions. M.S. Thesis, University of Illinois, 1980. Taylor, J. W. The Surface Tensions of Liquid-Metal Solutions. Acta Metall. 1956,4 , 460. Thomas, A. G.; Huffadine, J. B.; Moore, N. C. Preparation Properties and Applications of Metal/Ceramic Mixtures. Met. Rev. 1963,8, 461. Vinson, J. R.; Chou, T. Composite Materials and Their Uses in Structures; Wiley: New York, 1975. Williams, J. C.; Nielsen, J. W. Wetting of Original and Metallized High-Alumina Surface by Molten Brazing Solders. J . Am. Ceram. SOC.1959,42, 229. Wilson, J. R. The Structure of Liquid Metals and Alloys. Met. Rev. 1965,I O , 381.
Received for review October 30, 1990 Accepted November 9, 1990
Prediction of Solute Partition Coefficients between Polyolefins and Alcohols Using the Regular Solution Theory and Group Contribution Methods Albert L. Baner and Otto 0.Piringer* Fraunhofer Institute for Food Technology and Packaging, Munich, Federal Republic of Germany
The regular solution theory using group contribution solubility parameter estimation methods was applied to the estimation of partition coefficients of solutes between polyolefii polymers and alcohol solvents. Quantitative prediction was improved by using only the Hansen dispersive type solubility parameters and adding an empirical correction term to account for polar type interactions between the solute and the solvent and polymer phases. The method fails to fully account for multiple functional groups and stearic hindrances. The correction term is a function of the solute’s functional groups, the solute molecular weight, and the solvent and polymer phases. The group contribution methods of Hoy and of Van Krevelen and Hoftyzer gave equivalent results.
Introduction The prediction of solute partition coefficients between polymers and liquids is important in a number of applied fields such as protective clothing (Mansdorf et al., 1988), biomedical studies (Dunn et al., 1986), chromatography (Barton, 1983),chemical separations (Lee et al., 1989) and,
of major interest for this study, packaging (Hotchkiss, 1988). Solubility coefficients are important in package design and food shelf-life prediction because they are used in modeling the migration of substances from the packaging into the food and from the food into the package (Vom Bruck et al., 1986; Reid et al., 1980; Chatwin and
0888-588519112630-1506$02.50/0 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 7,1991 1507 Katan, 1989). These migrations can affect the safety and quality of the packaged product as well as the mechanical properties of the package. Although a great amount of research studying the partitioning of aromas between foods and their polymeric packaging exists (Becker et al., 1983; Kwapong and Hotchkiss, 1987; Ikegami, 1987; DeLassus et al., 1988; Koszinowski and Piringer, 19891, very little work has been done on estimating these partition coefficients. The estimation of partition coefficients of substances between foods and polymeric packaging is a complex problem. Foods can contain both solid and liquid phases containing a variety of macro- and microconstituents with varying polarities and chemical properties. The macroconstituent properties can range from very polar hydrogen-bonded systems, e.g., water and acids, to very nonpolar systems, e.g., oils and fats. The microconstituents of foods, such as flavor and aroma constituents, with concentration ranges of approximately 0.001-200 ppm (w/v) include all possible chemical compounds but mainly unsaturated and oxygenated compounds. Commonly used food packaging polymers can be semicrystalline (e.g., polyethylene), be oriented (e.g., polypropylene), have surface treatments (e.g., fluorination, sulfonization, metallization), contain various additives (e.g., plasticizers and antioxidants) and have a range of polarities (e.g., nonpolar, polyethylene, to polar, ethylenevinyl alcohol). There are several methods in the literature that can be used for estimating partition coefficients between liquids. Reid et al. (1987) presents methods for estimating activity coefficients from which partition coefficients can be calculated. There are numerous examples of correlations developed for estimating partition coefficients (Bao et al., 1988; Kamlet et al., 1988; and Kasai et al., 1988). Computer modeling (Jorgensen et al., 1990) is rapidly developing as a means of predicting partition coefficients directly from molecular structure. However, computer modeling is not yet ready for general application, and its use is dependent on the availability of mainframe computers and software. Of the methods presented in Reid et al. (1987) many are not applicable to this problem because they cannot be applied to polymer systems. Many also require experimental data that are not available for many aromas or they cannot be used with the wide variety of chemical substances found in aromas. Group contribution methods overcome the problem of estimating model parameters by assigning contributions to each of the functional groups making up the polymer, aroma, and solvent phases. The sum of these group contributions give an estimate of the parameter. Van Krevelen and Hoftyzer (1976) and Barton (1983) have reviewed the regular solution theory group contribution method applied to activity coefficient estimation in polymers. Goydan et al. (1989) have reviewed the use of three other group contribution methods that can be used for estimating solute activity coefficients in polymers. Recently Chen et al. (1990) have developed a group contribution equation-of-state method for estimating solute activity coefficients in mixtures containing polymers. The regular solution theory is by far the simplest of the estimation methods to apply. However, the regular solution theory is not necessarily the most accurate and is applicable in theory only to regular solutions. In fact, Barton (1983) in his extensive review says that one should not expect that partition coefficients can be predicted in detail by solubility parameters, particularly for polar molecules. However, considering the large amount of literature devoted to the use of the solubility parameter
and its sound theoretical basis, the accuracy of the regular solution theory should be tested on experimental data. Polyolefin polymers (mostly low-density polyethylene, LDPE) are the most widely used polymers for food packaging due to their low cost and useful physical and mechanical properties. With their importance and simple chemical structure, the development of any partition model should start with the polyolefins. Foods have complex compositions and are difficult to work with; because of this is it often necessary to use food-simulating solvents. Alcohols, particularly methanol and ethanol, are good simulants for the migration of substances between polyolefins and fatty type foods (Piringer, 1990,Schwartz, 1988). The alcohols do not swell the polyolefins, flavors and polymer additives are readily soluble in them, and they have clear analytical advantages over oil food simulants. The purpose of this paper is to test the effectiveness of the regular solution theory for estimating partition coefficients of solutes between polyolefin polymers and alcohol food simulants.
Regular Solution Theory Review The regular solution theory is by far one of the oldest methods that can use group contribution methods which can be applied to polymers. The theory and its usefulness have been reviewed by Van Krevelen and Hoftyzer (1976), Barton (1983), and Rider (1985). The method has been achieved its widest acceptance in the paint and coatings industry, where it is used as a means of predicting the tendency of polymers to dissolve in solvents (Rider, 1985). Very few direct applications of the regular solution theory for the prediction of partition coefficients have been found in the literature (Barton, 1983). Most applications require modification of the regular solution theory using empirical terms, and there are many more examples of correlations using only solubility parameters (Barton, 1983). The regular solution theory expression for predicting activity coefficients (y)in a binary mixture of solute 1in solvent 2 is given by eq 1 (Hildebrand et al., 19701, where y1 = ~ ~ P P / R T [ V ~ -@62)211 ~~(S~
(1)
Vl is the liquid molar volume of the pure liquid component at temperature T , R is the gas constant, and the volume fraction, iPz, is defined as @2
= xzVz/(x1V1
+ XZVZ)
(2)
where x1 and x 2 are the molar fractions of components. The solubility parameter for component i ( S i ) is defined as Si = C i i 1 / 2 = ( V i /Vi)’/2
(3)
Where cii is the cohesive energy density of the pure liquid i and Viis the molar internal energy, which is defined as the energy required isothermally to evaporate liquid i from a saturated liquid to the ideal gas phase (Reid et al., 1987). The internal energy at temperatures well below critical can be approximated by Vi == AH,,, - RT (4) where AHvi is the molar enthalpy of vaporization of pure liquid i at temperature T. The solubility parameter defined in this manner is often referred to as the “Hildebrand” solubility parameter after its originator. Equation 1is a result of the geometric mean assumption in eq 5, which implies that activity coefficients c12
=
(c11c22)1’2
(5)
1508 Ind. Eng. Chem. Res., Vol. 30, No. 7,1991
can be predicted by using only pure component data and properties. The importance of the geometric mean assumption is that it assumes that the interactions between different molecules in a mixture are similar to those the molecule experience between themselves in the pure substance. This assumption has been shown to be true for solutions of nonpolar molecules of similar sizes where only London or dispersive type interactions exist (Hildebrand et al., 1970). Further assumptions of the regular solution theory are as follows: (1) The volume change on mixing is quite small. (2) The excess entropy per mole of mixture is essentially zero (meaning only similar size molecules are present in the mixture). The regular solution theory best models mixtures of nonpolar molecules. As such it gives good prediction for aliphatic hydrocarbon mixtures but only a broad qualitative indication of behavior for mixtures containing polar molecules (Prausnitz et al., 1986). Numerous modifications have been proposed to eq 1 to allow the theory to be extended to a wider class of mixtures (Barton, 1983). Hansen proposed for polar and hydrogen-bonding compounds on a semiempirical basis that solubility parameters can be broken down into a linear combination of nonpolar or dispersive type interactions (bd), polar interactions (6 ), and hydrogen bonding interactions (ah) (Barton, 1983f: b2 = 6d2 + ;6 + 6h2 (6) Equation 1 using Hansen type parameters becomes 71 = exPll/RT[V1@2~((hj - 62d)2 4- (61, - 6 (61h
+ - &d2)13 (7)
2 ~ ) ~
Calculations Derivation of Partition Coefficient Equations. At equilibrium the fugacity (f) of the solute (i) above the polymer (P)and the liquid (L) phases is given by fiP = fiL (8) Where the fugacity is defined for the polymer and liquid phases respectively as = yipxipfi"
(9)
= YiLXiLfi0 (10) where fi" is the fugacity of the pure liquid solute. ComfiL
yiPriP
(11)
= 7iLxiL
For dilute concentrations of solute in the polymer and liquid phases the mole fraction can be approximated by the following equations: X? X?
ss
cTVp/M,j
(12)
ci'VL/M,,,
(13)
where Mm,i is the molar mass (g/mol), VL is the molar volume of the liquid, and Vp is the molar volume of one polymer repeat unit by convention (Van Krevelen and Hoftyzer, 1976). Combining eqs 12 and 13 with eq 11 gives c ~ / c ? = (YPVL)/ (7iPVp)
(14)
The equilibrium partition coefficient (Kp 3 can be defined as the ratio of the concentration (w/v) o#the solute in the polymer (c?) to the concentration (w/v) of the solute in the liquid (c?). Combining this definition with eq 1 gives the equation for calculating the partition coefficients: KpIL = C ? / C ~ = VL/Vp exp(Vi/RT[@L2(bi- bLI2 - 9p2(Si - 6p)2]) (15) Using Hansen type solubility parameters (eq 7), the equation for the partition coefficient becomes KpIL =
Barton (1983) has also reviewed several other systems for dividing solubility parameters into other linear combinations of dispersive, induction, and Lewis acids and bases to account for intermolecular interactions that the regular solution theory does not model. An empirical correction to the geometric mean assumption for c12in eq 5 adds a binary interaction parameter to better reflect the intermolecular forces between molecules (Hildebrand et al., 1970). Not much success has been made in correlating this parameter for polar and hydrogen-bonding mixtures, so that evaluation of this parameter must be done experimentally (Reid et al., 1987). For systems where there are large size difference effects between molecules, the Flory-Huggins equation has achieved success in predicting phase equilibrium (Reid et al., 1987). The Flory-Huggins theory was developed for polymer solutions where the polymer structure is substantially altered due to dissolution of the polymer in a solvent and does not apply to systems such as this case where the polymer is not solved in a solvent. More complete discussions of this theory can be found in Flory (1953), Flory (1970), and Prausnitz et al. (1986).
fiP
bining eqs 9 and 10 with 8 gives
V ~ / v pexp(Vi/RT[@L2((8di -
(bpi
- ~ P L +) ~(6E - )',8
(bpi
a d 2 -k
- @p2((bdi - bdp)' - 6pp)' - (hi - 6 ~ ) ~ ) 1(16) 1
Estimation of Solubility Parameters by Group Contribution Methods. The group contribution methods for estimating the values of solubility parameters at 25 "C from Hoy (1985), Van Krevelen and Hoftyzer (19761, and Fedors (1974a,b) were used. The estimation methods of Hoy and of Van Krevelen and Hoftyzer can be used to estimate Hansen type solubility parameters, whereas the method of Fedors cannot. The polymer solubility parameter is calculated for the repeat unit of the polymer in all three methods. The polymer solubility coefficient can be calculated in theory only for amorphous polymers. There is no method for estimating the solubility parameters for semicrystalline polymers (Van Krevelen and Hoftyzer, 1976). For calculations using Hoy's method all alcohols were treated as individual molecules and the primary OH group contribution parameter was used. Table I gives examples of the solubility parameters used in these calculations. All three group contribution methods give approximately the same solubility parameter values. When solubility parameters are calculated by using the group contribution methods, molar volumes are needed. Molar volumes at 25 "C can be calculated by dividing the molecular weight of the compound by ita density at 25 "C. For polymers, preliminary calculations showed the experimental density of the polymer bulk phase should be used rather than the amorphous density as called for by a strict interpretation of the regular solution theory. Where no experimental molar volume data or density data at 25 "C exist for a substance, the molar volumes can be estimated by using the molar volume group contribution method of Fedors (1974a,b), Although the accuracy of Fedors is sometimes rather poor, it does give a first approximation of a substance's molar volume (Van Krevelen and Hoftymr, 1976). In an effort to improve the prediction of Fedors' method the average molar contribution value of 16.45 cmg/mol for the CH2group was used based on the
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1509 Table 1. Solubility Parameters (J1la/cmala)at 25 "C" HOY compound polymer LDPE HDPE HPP COPP solvent methanol ethanol acetone hexane solutes n-alkanes c12 C14 C16 C18 c20 c22 polar d-limonene diphenylmethane linalyl acetate camphor diphenyl oxide isoamyl acetate eugenol menthol phenylethyl alcohol cis-hexenol
V, cmg/mol
d
30.60 29.39 46.78 46.67
18.3 19.1 16.7 16.7
40.73 58.69 74.05 131.6
12.3 12.8 12.9 14.9
228.6 261.7 294.1 326.0 359.6 392.4
15.6 15.7 15.8 15.9 15.9 16.0
162.4 168.1 219.3 153.3 159.0 150.4 154.0 173.6 120.4 119.0
15.3 16.5 14.8 15.5 16.4 14.5 14.9 14.7 15.8 14.5
P 0 0 0 0
13.2 11.4 9.89 0 0 0 0 0 0 0
5.59 10.0 7.64 8.65 11.7 7.36 11.4 7.81 13.2 7.81
t
h
Van Krevelen and Hoftyzer d P h t
0 0 0 0
18.3 19.1 16.7 16.7
17.6 18.4 16.5 16.5
24.9 19.4 10.2 0
30.8 25.9 19.2 14.9
15.5 15.3 15.3 14.6
0 0 0 0 0 0
15.6 15.7 15.8 15.9 15.9 16.0
15.5 15.6 15.7 15.8 15.9 15.9
0 0 0 0 0 0
16.5 19.3 16.8 19.1 20.2 17.1 21.9 19.1 25.9 17.7
16.4 18.6 15.6 17.5 18.6 15.1 16.9 16.1 15.5 15.5
0 1.30 2.23 5.02 3.90 3.26 6.56 2.88 5.07 4.22
2.08 0.10 2.08 6.93 1.39 5.07 11.4 9.51 15.8 6.51
0 0 0 0
12.3 8.52 10.4 0
0 0 0 0
22.2 18.5 5.22 0 0 0 0 0 0 0
0 0 5.65 3.61 4.34 6.82 3.87 10.7 12.9 13.0
Fedors t
17.6 18.4 16.5 16.5
18.0 18.3 16.7 16.7
29.7 25.4 19.2 14.6
29.1 25.9 19.0 14.6
15.5 15.6 15.7 15.8 15.9 15.9
16.0 16.2 16.3 16.5 16.5 16.6
16.4 18.7 16.7 18.6 19.5 16.9 18.5 19.5 20.8 20.7
16.3 20.2 17.3 19.1 20.6 17.4 22.1 19.4 24.4 22.1
a Hoy (1985), Van Krevelen and Hoftyzer (1976), and Fedors (1974a,b) solubility parameter group contribution methods. d, Hansen dispersive solubility parameter. p , Hansen polar solubility parameter. h, Hansen hydrogen bonding solubility parameter. t , Hildebrand solubility parameter. V , molar volume at 25 "C. LDPE, low-density polyethylene, density = 0.918. HDPE, high-density polyethylene, density = 0.956. HPP, homopolymer polpropylene, density = 0.902. COPP, copolymer polypropylene, density = 0.900.
recommendation of Van Krevelen and Hoftyzer (1976). Experimentally measured molar volumes for polymers can be found in Van Krevelen and Hoftyzer (1976). Densities and molar volumes for pure liquids can be found in Weast (19901, Windholz (19831, Synowietz (1983), and Arctander (1969). For substances that are normally solid at 25 OC the method of Fedors was used to estimate their liquid molar volume. Calculation of Equilibrium Partition Coefficients of Solutes between Polymer and Liquid Phases. The regular solution theory is intended to be used only for predicting the behavior of liquids. The theory can be extended to amorphous polymers because amorphous polymers in the first approximation behave like liquids, However, it is not clear how best to apply the theory to semicrystalline polymers (Van Krevelen and Hoftyzer, 1976). Comparisons between different group contribution methods for estimating solubility parameters from Fedors (1974a,b), Van Krevelen and Hoftyzer (19761, and Hoy (1985) were made. Consistent use of group contribution methods was made for estimating the solute, solvent, and polymer solubility parameters used for calculating the partition coefficient. The effectiveness of eqs 15 and 16 for predicting partition coefficients was compared. All calculations made here can be easily carried out on a hand-held calculator. Although solubility parameters depend on temperature, the regular solution theory itself assumes that the excess entropy is zero. At constant composition the activity coefficient remains constant. This allows the activity coefficient to be calculated at any convenient temperature, which is customarily 25 "C. All calculations were carried out for 25 "C and the molar volumes are the molar volumes of the substances as liquids at 25 "C. By convention (Van
Krevelen and Hoftyzer, 1976) the molar volume for polymers is taken to be that of one repeat unit of the polymer. Partition Coefficient Data. Experimental equilibrium partition coefficient values for solutes partition between polyolefins and alcohols were obtained from Becker et al. (1983), Koszinowski (1986a,b) and Koszinowski and Piringer (1989,1990). The partition coefficient measurements were carried out using a liquid measuring cell adapted from the method of Till et al. (1982). Briefly, a 35-cm3glass vial with Teflon-coated septa cap was filled with the solutesolvent solution with an average concentration of 1 X M. Several disks of polymer film (4.45 cm2)were cut and mounted on a glass rod separated by glass beads and placed in the vial. The vials were kept in a temperature-controlled roller shaker. The concentration of the solute in the solvent was monitored over time by using gas chromatography until no change in solvent concentration with time was observed (Le., equilibrium was reached). The polymer disks were then removed from the vial and quickly washed with pure solvent phase (e.g., ethanol) to remove any absorbed solute on the surface of the polymer. The concentration of the solute sorbed in the film was measured by extracting the film twice with a large volume of hexane relative to the volume of film to ensure that all solute is extracted. The concentration of solute in hexane was measured by gas chromatography. Experiments were also made to ensure that equilibrium was reached in the polymer phase during the experiment where the concentration of solute was measured as a function of time in the polymer phase. Interaction effects between the solvents and polymers were determined by calculating the solute diffusion coefficients in the polymer by using the relationship P, = DK, where the partition coefficient ( K )is determined by using the experiments described in the preceding paragraph and
1510 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991
the relative permeation coefficient (P,)is determined from the permeation pouch method described in Koszinowski (1986a). The pouch method uses a sealed pouch made from polymer film that is fded with solvent and solute and placed in a jar containing pure solvent. As the solute permeates through the film into the solvent in the jar, it is measured as function of time until it reaches steady-state permeation by using gas chromatography. Calculating diffusion coefficients using this method, Becker et al. (1983) observed that the solute diffusion coeficients in the polyolefins in contact with methanol and ethanol solvents have practically the same diffusion coefficients as with water as solvent. Water has essentially no interaction with polyolefins. Solutes in acetone showed a slight increase in the diffusion coefficient, and solutes in hexane, which swells the polymer, showed an increase of 1-2 orders of magnitude in the diffusion coefficient compared to water solvent.
Results and Discussion n -Alkanes. Hildebrand's regular solution theory (eq 15) and the Hansen multicomponent solubility parameters (eqs 6 and 16) were compared against experimentally measured partition coefficients for six n-alkanes((212422) between LDPE and ethanol from Koszinowski (1986a). Equation 15 using the Hildebrand solubility parameters was tested by using the group contribution solubility parameter estimation methods of Hoy (1985), Van Krevelen and Hoftyzer (1976), and Fedors (1974a,b). All of these three solubility parameter estimation methods seriously overpredicted these partition coefficients. On average the calculated partition coefficient values were 1 X 104-1 X lo5 times that of the experimental value. The failure of the regular solution theory to predict the partition of n-alkanes between ethanol and LDPE using Hildebrand solubility parameters appears to be mainly due to the difference between the alcohol and alkane solubility parameters (see Table I) being too large. I t is clear that for n-alkanes only dispersive attractive forces can exist between them and the LDPE and ethanol phases. Polyethylene is essentially a hydrocarbon, and it is observed that the difference between it and the n-alkane's solubility parameter is quite small relative to the difference between the n-alkane and the ethanol. This coupled with the grossly overpredicted partition coefficient suggests the error lies in the magnitude of the liquid solubility parameter term. Polar and hydrogen-bonding interactions are implicitly included in the ethanol Hildebrand solubility parameter even though these attractive force do not exist between the ethanol and the n-alkanes. The failure of the regular solution theory to model interactions between dissimilar molecules is a recognized limitation (Prausnitz et al., 1986). This failure can be largely traced to the geometric mean assumption (eq 5). The Hansen solubility parameters offer a way of separating dispersive interaction forces from polar and hydrogen-bonding type interactions containing in the alcohol Hildebrand solubility parameters. With the dispersive Hansen solubility parameters (see eq 6) from Hoy and Van Krevelen and Hoftyzer, the partition coefficient can be predicted to within an order of magnitude as shown in Figure 1. However, the regular solution theory coupled with the dispersive solubility parameters does not predict the increase in the partition coefficient with increasing carbon number of the experimental data well. In Table I the calculated solubility parameters are relatively constant as the n-alkanes become larger, which cannot be true since the experimental partition coefficients show an increase with chain length. An empirical correction factor
1
I
c c
k n
l -1
I
160
1
1
I
I
I
I
I
I
180
200
220
240
260
280
300
320
molecular weight Figure 1. Linear regression curves of experimental and calculated partition coefficienta of n-alkanes partitioned between LDPE and ethanol as a function of n-alkane molecular weight at 25 O C : ( 0 ) Koszinowski (1986a); (A) Hoy (1985) solubility parameters; (m) Van Krevelen and Hoftyzer (1976) solubility parameters.
.Q)
-2500 I-3000
I
160
,
I
I
I
1
I
I
180
200
220
240
260
280
300
320
molecular weight
Figure 2. Linear regression curve of empirical correction (a) versua the molecular weight of n-alkanes partitioned between LDPE and ethanol. a back-calculated by using eq 17, Hoy (1985) solubility parameters, and experimental data from Koszinowski (1986a).
(a) with the units J/mol can be back-calculated by using the experimental partition coefficient and solubility parameters. In Figure 2 the empirical correction factor varies linearly with molecular weight. Equation 16 for these calculations then becomes
KPIL =
where a = b + cMi, with Mi the solute molecular weight (relative molar mass),b the y intercept (J/mol), and c the slope (J/mol). The linear relationship of the empirical factor with molecular weight comes from the experimentally observed linear relationship between the log of the activity coefficient with carbon number for a homologous series of hydrocarbons which is directly proportional to molecular weight (Pierotti et al., 1959; Herington, 1967). The a is a result of the solute behavior in the polymer and solvent phases. It can be thought of as a sum of the empirical corrections for the solute-solvent (UL) and the solute-polymer (ap) interactions: a = UL + u p (18) It is clear from the preceding diecuseion that for n-alkanea
-; -
> w
7 W
g .+ v
Ind. Eng. Chem. Res., Vol. 30, No. 7,1991 1511
z-
I--
’Ooo 6000 +
5000 4000
--
3000 L 2000 1000
-
c *
.
*I
/*
b
0 -1000 r-2000 +
.-.L
-3000 -4000 r -5000 Ir
..-__...--.--A-
-6000
.!---c---~-
2
g @
-
-7000
160
0
e
0
A
E ‘1.
7000
6000
r--1 *
.....---A
__.----
~&~/.---.-
__.---___.--....--__-- A
-
7-
-4----J
I
I
I
,
I
1
I
180
200
220
240
260
280
300
.L
? 0,
320
..-A
I
280
300
__L________...
0 -1000 1 160
180
200
220
240
260
molecular weight Figure 3. Linear regression curves of empirical correction (a)versus molecular weight of n-alkanes partitioned between LDPE and various solvents. a back-calculated by using eq 17, Hoy (1985) solubility parameters and experimental data from Koszinowaki (1986a): ( 6 ) methanol solvent; ( 0 )ethanol solvent; (A)acetone solvent; (m)hexane solvent.
molecular weight Figure 4. Linear regression curves of empirical correction (a) versus molecular weight of n-alkanes partitioned between methanol solvent and various polyolefm. a back-calculated by using eq 17, Hoy (1985) solubility parameters, and experimental data from Koszinowaki (1986a): ( 6 )LDPE (0)HDPE (A)homopolymer polypropylene (HPP); (B) copolymer polypropylene.
partitioned between ethanol and LDPE up must be small in comparison to the aL term. The n-alkane-polymer activity coefficient is likely to be significantly less than 1. Alessi et al. (1982a) measured infinite dilution activity coefficients for n-alkanes in n-alkane solvents that were less than 1. They also found that the larger the difference in molecular weight of the alkanes, the smaller the activity coefficient. The activity coefficients calculated here where n-alkanes are solved in LDPE (which is made up of very long hydrocarbon chains) are likely to be much less than 1. Another of the recognized deficiencies of the regular solution theory is ita inability to predict activity Coefficients less than 1. Calculated activity coefficients for n-alkanes in LDPE range from 1.9 for C12 to 2.4 for C22 by using the Hoy solubility parameters. Therefore, an empirical correction to the regular solution theory for n-alkanes in polymers (ap)must be negative and increase with molecular weight. In comparison to the activity coefficients for n-alkanes in LDPE, the activity coefficients for n-alkanes in ethanol are greater than 1and also increase with molecular weight. Data from Cori and Delogu (1986) shows that the activity coefficients of n-alkanes in ethanol increase with molecular weight. The regular solution correctly predicts this behavior with activity coefficients of 2.1 for C12 and 4.8 for C22 using Hoy’s dispersive solubility parameters. However, the size of the regular solution theory activity coefficients are too small compared to the observations of Pierotti et al. (1959). As a result of these observations the empirical solute-solvent correction factor (aL)must be positive and increase with molecular weight. Figure 3 shows the empirical correction for n-alkanes to be a function of the solvent phase while keeping the polymer phase (LDPE) constant. The y intercept decreases with the polarity of the solvent, and the slopes of the curves remain relatively constant as function of molecular weight. The decreasing empirical correction comes from the decreasing polarity of the solvent. The less polar the solvent (e.g., methanol > ethanol > acetone > hexane) the better the regular solution theory predicts the liquid activity Coefficient. However, this a t the same time worsens the underprediction effect from the solutepolymer activity coefficient. The qualitative prediction by the regular solution theory of the decrease in solutesolvent activity coefficient with the decrease in solvent polarity is illustrated for C20 with 7.0,4.0, 3.7, and 1.2 by
using Hoy’s solubility parameters for methanol, ethanol, acetone, and hexane solvents, respectively. The deviation of the slope for hexane from the other solutes shown in Figure 3 is explained by the hexane solvent swelling the LDPE. Interactions between the polymer and solvent phase violate the assumption made here of no interaction between the two phases. The empirical correction factor in addition to varying with solute molecular weight and solvent also varies with the polymer phase. Figure 4 shows the variation of empirical correction terms for n-alkanes partitioned between methanol and polypropylene (PP) and high- and lowdensity polyethylene (HDPE and LDPE) polymers. Polar Solutes. Equation 15 using the Hildebrand solubility parameters and eq 16 using the Hansen solubility parameters were applied to the prediction of partition coefficients for polar solutes partitioned between methanol and polymers with different polarities. As could be predicted from the previous discussions, both equations regardless of the group contributions used grossly overpredict and show no qualitative prediction of the partition coefficient. This failure is largely due to the regular solution theory not being applicable to polar substances. Prediction with eq 16 was better without difference terms where one of the solubility coefficients was 0. Better qualitative and quantitative predictions, to within 2 orders of magnitude, for these polar solutes are obtained by using only the dispersive Hansen solubility parameter for the solute, solvent, and polymer. With the data of Koszinowski and Piringer (1990) empirical correction factors were back-calculated by using the experimental partition coefficient, the Hansen dispersive solubility parameters, and eq 17. As seen in Figure 5, which is similar to that for the n-alkanes, the n-aldehyde’s homologous series has a good linear correlation between the empirical correction and the number of saturated carbon atoms. The slope of the correlation is again positive and similar in magnitude to that of the n-alkanes. The y intercept is significantly smaller than that of the h e a . The positive slope means the functional group’s effect is diminished, and the molecule’s behavior becomes more like that of a hydrocarbon as the number of saturated carbon atoms increases. The negative y intercept indicates the regular solution overpredicts the partition coefficient for the n-aldehydes. This is mainly due to the activity coefficient of the aldehyde in the LDPE being too small. Experimental ob-
1512 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 -5000
0 -6000
-
-7000
-
-8000
-
-9000
-
E :
-
d
0
d
0
5 2L 0
-0
.-
0 L *-
P t Q)
E k
-
4000
-
2000
-
-2000
y
-4000
r
d
I ’ I I
0
0 1
c
,-.’
A
-lEo4 -1.lE04
j - 1 2E04
-: .-
A
e
-1.2E04
-1 4E04
,
-1 6E04
Q
S
-1.3E04I 120
*
Ob
d
0
t
0
1 j
I
140
I
160
I
180
1
200
I
220
’
servations by Alessi et al. (1982b)show polar solutes in long-chain hydrocarbons can have large activity coefficients, and the more polar the solute the larger the activity coefficient. The activity Coefficients for CS and C12 aldehydes in LDPE from the regular solution theory are 2.4 and 3.0, respectively, using Hoy’s solubility parameters, whereas the activity coefficients for C8 and C12 aldehydes in methanol are 1.4 and 1.8,respectively. The experimental aldehyde-methanol activity coefficients from Pierotti et al. (1959)are 10.4 and 45.6 for CS and C12, respectively. Remembering that the partition coefficient is essentially a ratio of the solute-liquid activity coefficient to the solute-polymer activity coefficient leads to the conclusion that the aldehyde activity coefficients in the LDPE must be grossly underpredicted. The LDPE solubility coefficient and molar volume must not be very well estimated by using only the repeat unit of the polymer. In Figure 5 the empirical correction fador for aldehydes with different degrees of unsaturation and those containing phenyl groups are shown in comparison to the n-aldehyde homologous series. Relative to the homologous series for solutes with the same molecular weight the presence of one double bond (A)shifts the empirical correction by roughly -1600 J/mol, and the presence of two double bonds (A) shifts the empirical correction by about -3600 J/mol. The presence of a phenyl group ( 0 )lowers the empirical correction by an average of -3500 J/mol with respect to the n-aldehydes. The scatter of the empirical correction for these aldehydes is due in part to structural effects, the group contribution parameters, and experimental error. From this example it can be seen that the empirical correction can be thought of as roughly the sum of contributions of individual functional groups. Figure 6 shows linear regression empirical correction curves for solute partition coefficients between LDPE and methanol versus the solute molecular weight for solutes with different functional groups and well-defined structures using the Hoy solubility parameters. The group contribution method of Van Krevelen and Hoftyzer gives similar results. As the polarity of the solute functional group increases, the y intercept decreases significantly, but the slopes of all the curves remain relatively the same for all solutes. The more polar and greater ability to form hydrogen bonds the functional group has, the larger the molecular weight needed to offset the molecule’s increased polarity. The effect of increased polarity of the solute’s
’
’
’
’
‘ I
@-l.8E04L1 I I 100 120 140 160 180 200 220 240 260 280 300
molecular weight
molecular w e i g h t
Figure 5. Linear regression curve of empirical correction (a) versus molecular weight of n-aldehydes partitioned between methanol solvent and LDPE. a back-calculated by using eq 17,Hoy (1985) solubility parameters, and experimental data from Koazinowski and Piringer (1990): ( 0 )n-aldehydes; (A)aldehydes with one double bond; (A)aldehyde with two double bonds; (0) phenyl aldehydes.
+
Figure 6. Linear regression curves of empirical correction (a) versus solute molecular weight of polar and nonpolar solutes partitioned between methanol solvent and LDPE. a back-calculated by using eq 17,Hoy (1985)solubilityparameters, and experimental data from Koszinowski (1986b)and Koszinowski and Piringer (1990): (0)nalkanes; (0) unsaturated hydrocarbons; (A)unsaturated aldehydes; (*) linear esters; (A)n-aldehydes; (+) phenyl esters; (e) primary alcohols; (0) phenyl alcohols; ( 0 ) phenols (ethanol solvent). Table 11. Empirical Correction ( a ) for Solum Partitioned between LDPE and Methanol (cySlope; kyy Intercept)? a = b + cMi b = Ebi i
solute functional group base value n-alkane functional group increments:
alkene (>=
