Kinetics of polyesterification of adipic acid with 1,3-butanediol.
Polymeric plasticizers consist of polyesters based on a divalent acid such as adipic, sebacic or azelaic condensed with diols, with a molecular weight between 850 and 3500. Commercially, polycondensation is carried out with a stoichiometric excess of acid and the carboxylic end groups are esterified with monoalcohols or with excess diol, in which case the terminal hydroxy groups are reacted with long chain monocarboxylic acids. Although they have a relatively low share of the plasticizer market ([approximately]2%), polymerics are very important for special uses when increased extraction resistance to hydrocarbons, oils, and fats is required or when products are processed or used at relatively high temperatures. All polyesters are more or less compatible with PVC and most are also compatible with cellulose acetate, butyrate, or nitrate.
A high purity polymeric plasticizer, with no unreacted acid and/or alcohol, is required for applications such as film, sheet, floor covering, cable insulation, and sheathing resins that require good volume resistivity and low exudation. This means that the esterification process should be as complete as possible using a minimum of very active catalyst. This study of the kinetics and thermodynamics of model esterification reactions for polymeric plasticizers is aimed to establish the best reaction conditions for obtaining the desired product quality in the shortest possible time.
Adipic acid (DuPont Adipure [greater than] 99.9%), 1, 3-butanediol (Hoechst Celanese [greater than] 99.5%), nitrogen extra dry (AGL Welding Supply), monobutyltin oxide (MBTO, Aldrich [greater than] 97%), dibutyltin oxide (DBTO, Elf Atochem [greater than] 97%), monobutyltin trichloride (MBT, WITCO [greater than] 98%), monomethyltin trichloride (MMT Aldrich [greater than] 97%) and a mixture of 75% monomethyltin trichloride and 25% dimethyltin dichloride (MDC 75 WITCO) were used as received.
A total of 873 g (5.97 moles) of adipic acid and 592.2 g (6.57 moles) of 1, 3-butanediol were charged into a 2-liter resin kettle with bottom drain valve, mechanical stirrer, a heating mantle, and thermometer with thermowatch to maintain the temperature to an accuracy of 2 [degrees] C. The reactor was connected to a 5 plate Oldershaw column of 35 cm, a 50 ml Barrett trap and a Friedrich condenser for water distillation and avoiding the loss of diol. A nitrogen sparge with a flow rate of 500 ml/min was used to facilitate water elimination and prevent oxidation of the product. The materials were melted together at about 130-140 [degrees] C and the first evolution of water started at about 160 [degrees] C. Catalyst was added at the reaction temperature (180 [degrees] C or 215 [degrees] C) and at intervals of 0.5-2 hours a sample was withdrawn and analyzed. For the vacuum experiments, vacuum was applied when the acid number reached [approximately]20 mg KOH/g using a connection with a vacuum pump and a manostat to maintain the pressure constant. Samples were taken from the bottom valve using a vacuum takeoff adapter.
The unreacted carboxylic groups were potentiometrically titrated in tetrahydrofuran solution with 0.1 M sodium hydroxide, using a Brinkmann titrimeter (Titrino 716 DMS, software: Titrino Workcell Ver. 3) and a glass electrode. Hydroxyl groups were determined by reaction with acetic anhydride in pyridine and potentiometric titration with 0.5 M sodium hydroxide of acetic acid generated after water addition against a blank sample. A correction for free carboxylic acid was made. Water was determined by the Karl Fischer method using a Brinkmann titrimeter (682 Titroprocessor and 665 Dosimat) and Aquastar Comp 5 solution (EM Science).
RESULTS AND DISCUSSION
Selection of the reaction model. The esterification of acids with alcohols is a reversible reaction with a well-known mechanism (1, 2) Two reaction mechanisms through a tetrahedral intermediate and acyl-oxygen cleavage BAc-2 and AAc-2 are most common and require a nucleophilic attack of a hydroxylic nucleophile on the carbonyl double bond of the carboxylic or protonated carboxylic group. In relatively dilute solutions and at moderate temperatures the reaction is the sum of the uncatalyzed (BAc-2) and acid catalyzed (AAc-2) processes (Scheme 1):
[Chemical Expression Omitted]
The carboxylic acid may play the role of the acid catalyst (Scheme 1) if no other stronger acid is added and the reaction rate becomes the sum of a bimolecular uncatalyzed and trimolecular autocatalyzed process.
V = [k.sub.2][RCOOH][ROH] + [k.sub.3] [[RCOOH].sup.2][ROH] (1)
Economically advantageous manufacture of esters and polyesters requires reactions in carboxylic acid-alcohol melts at high temperatures, in the absence of a strong acid that may promote secondary reactions and solvents. Flory (3, 4), considering data for relatively high conversion to avoid large changes in polarity during the reaction, found that ethylene glycol-succinic acid polyesterification is predominately a trimolecular process with a partial order of two with respect to carboxylic acid and one with respect to alcohol. Subsequent research in similar conditions confirmed the overall order of three (5-11) for the autocatalytic process or a combination of orders two and three (12-15) representing a sum of noncatalytic and autocatalytic processes. However, in some low polarity systems, a more adequate order of reaction is 1.5 with respect to the acid and I with respect to the alcohol (16-24). This fractional order of reaction with respect to the acid was considered to be due to a very strong association of carboxylic acids in a dimer (22-24), although it may also result as a sum of an order one and an order two process. As the synthesis of polymeric plasticizers is based on the reaction of glycols and diacids with relatively low molecular weight and the polarity of the reaction medium is relatively high, a sum of a noncatalytic bimolecular and an autocatalytic trimolecular reversible process represents a reasonable kinetic model.
The use of polymeric plasticizers requires a very low residual acidity and a small or no excess of glycol. The reaction rates of the noncatalytic and autocatalytic processes are very low at the end of the process, and consequently the reaction time to reach low acidity is quite long. It is best to add a very effective catalyst that in very small amounts does not affect the quality of the product, even if it is not eliminated after the reaction. The best such catalysts are the organotin compounds. It is well known that mono and dialkyltin derivatives, especially oxides or chlorides, are very good catalysts for esterification and transesterification reactions. The reaction mechanism for these catalyzed processes is complexation of the carboxylic acid with the tin alkoxide followed by insertion (Scheme 2).
[Chemical Expression Omitted]
This mechanism is supported by the observed first order kinetics with respect to acid and alcohol (9, 25, 26) although a 0.5 order was also observed in some nonpolar systems where the carboxylic acids are highly associated (25, 27). Tin alkoxides were formed by the reaction of dibutyltin oxide with alcohols in conditions similar to those used in esterifications and trans-esterifications and were characterized by infrared spectroscopy and HPLC (26). They were also identified as intermediates in lactone polymerization by infrared spectroscopy (28). It has been proved by infrared spectroscopy (29) and crystal structure analysis (30) that carboxylic acids and esters form coordination compounds with organotin derivatives. A clear indication of a non-ionic insertion mechanism (Scheme 2) is the absence of racemization in the polymerization of L, L-lactide catalyzed by tin compounds (31). [Omega]-Hydroxycarboxylic acids form cyclic lactones of 7 to 15 atoms with excellent yields in the presence of di-n-butyltin oxide, even at high solution concentration that can be explained only by insertion of the carboxyl group into alkoxide O-Sn bonds in an intermediate having hydroxyl and carboxyl groups coordinated to the same tin atom (32, 33). We have found no comprehensive studies of correlation between catalytic activity of organotin catalysts and structure. However, the existing data support the insertion mechanism. As expected, alkoxytin derivatives are the best catalysts in polymerization of lactones (34). In esterification processes, all the tin derivatives that can generate alkoxides such as oxides or chloro derivatives are also very active (29, 35-37). However, carboxylates that are not easily transformed into alkoxides, and sulfides, which cannot be transformed into alkoxides at all, have little or no catalytic activity (26, 29, 37). Steric hindrance of tin substitutents and/or alkoxy or carboxyl substituents reduces the catalytic activity (29, 34, 38). In the polymerization of [Epsilon]-caprolactam with different dialkyktin alkoxides, the alkoxide group has been proven by NMR to constitute the polymer end group (34). Finally, as expected, the catalytic esterification process has a very negative activation entropy, corresponding to a highly ordered transition state (27, 39).
The simplified mechanism presented in Scheme 2 does not account for the fact that mono and dialkyl tin oxide/alkoxides and tin oxide/carboxylates are usually in oligomeric or polymeric form as a result of formation of oxygen bridges between two or more tin atoms (40). It has been shown that some oligomers, especially distannoxanes, exhibit a remarkably high catalytic activity in esterification and transesterification reactions (41-44). The postulated mechanism of this catalytic action is very similar to the mechanism from Scheme 2. The difference is that the alkoxy group is bonded to one tin atom and the coordination of the carboxylate group takes place at a second tin atom, bonded with the first through an oxygen bridge. The transition state of the nucleophilic attack of the alkoxide on the carboxyl group has a cyclic system of 6 atoms (Scheme 3), sterically favorized compared with the transition state of four centers from Scheme 2. This mechanistic aspect does not change the kinetic order of the reaction, which is first order in acid and first order in alcohol. The kinetic order in tin, owing to the association of tin oxide molecules, is more complex and depends on the compound type, concentration, and temperature.
It is consequently necessary to include a bimolecular term, monomolecular in acid and in alcohol, to represent the esterification catalyzed by organotin derivatives. As the concentration of catalyst is constant, it is not possible to distinguish kinetically between the non-catalyzed ([k.sub.n]) and organotin catalyzed ([k.sub.c]) reactions. However, the non-catalyzed bimolecular rate constant ([k.sub.n]) can be determined separately in a system without organotin catalyst. The bimolecular rate constant ([k.sub.t]) is expressed as a sum of non-catalyzed and organotin catalyzed constants:
[k.sub.t] = [k.sub.n] + [k.sub.c] (2)
We conclude that the model for the esterification of glycols with diacids should include two terms: a bimolecular term for the noncatalytic and organotin derivative catalyzed processes and a trimolecular, autocatalytic term. All three processes are reversible and the reverse reactions have to be considered. Finally, the eliminations of water and glycol at higher temperatures have to be included in the model (Scheme 4).
This scheme is simplified by assuming that no change in reactivity of the carboxylic and hydroxylic groups occurs as the macromolecule progressively increases in size. Detailed studies for the autocatalytic (3) and organometallic catalyzed esterification (9) showed that this assumption is perfectly valid. The change in viscosity of the reaction system due to polymerization is also neglected as it was shown that this change has a minor effect on the rate (3, 9). The change in polarity of the reaction medium by the progressive transformation of the carboxylic acids and glycols to esters is not negligible.
[Chemical Expression Omitted]
To avoid large changes in medium polarity, the kinetics was followed only for conversions higher than 70%. As a matter of fact, the reaction starts when the acid and glycol are heated to 120-140 [degrees] C, and a conversion of 65-70% is reached by the time a temperature of 180 [degrees] C or 215 [degrees] C is reached. The catalyst was added at this temperature, to avoid excessive conversion before the study started. We used 1, 3-butanediol in our kinetic studies for which the reactivities of the primary and secondary hydroxyl groups are different. The rate constants we measured are consequently a mean value of the rates of these two groups.
[Chemical Expression Omitted]
The most complete published investigation of polycondensation of polyols with adipic acid, by Chang and Karalis (45), uses a model similar to ours. They assume that the reverse reactions are negligible as the water is stripped off completely during the process. This assumption reduces the model from Scheme 3 to a system of a second ([k.sub.n] + [k.sub.c]) and a third order ([k.sub.a]) parallel reaction that can be solved analytically. The system is further simplified by Chang and Karalis, who assume that the uncatalyzed reaction is negligible, compared with the autocatalytic process ([k.sub.n] = 0). As we were interested especially in studying the reaction to very high conversions appropriate for synthesis of polymeric plasticizers, we chose not to eliminate a priori the reverse reactions and the non-catalytic process from the model, since these processes would be more important at high conversions. based on our results (see below), inclusion of the additional terms is necessary in the systems of interest.
Investigation of the equilibrium and calculation of equilibrium constants. We define A as
A = [RCOOR[prime]][[H.sub.2]O]/[RCOOH][R[prime]OH] (3)
where the concentrations of carboxyl groups, hydroxyl groups, and water were determined experimentally and the concentration of ester was calculated by difference between the initial acid concentration, corrected for water loss, and the measured acid concentration.
Figures 1 and 2 represent plots of A values versus time at two temperatures (180 [degrees] C and 215 [degrees] C). For the organotin catalyzed processes, when the system reaches the working temperatures of 180 [degrees] C or 215 [degrees] C, the values of A are small because the system is not at equilibrium and the ester and water concentrations are low. The A values increase with time as the concentration of ester increases, although most of the water is lost by distillation. At around 15,000 to 20,000 s (4.5 to 5.5 h, conversion 92-97%, acid value 20-25 mg KOH/g) the rate of water elimination starts to be relatively slow compared with the reaction rates of catalyzed processes. This is because the very small amount of water is strongly associated with ester molecules present in large excess. The system is no longer controlled kinetically but becomes thermodynamically controlled and the progress of the reaction is then determined by the rate of water elimination. The change [TABULAR DATA FOR TABLE 1 OMITTED] from kinetic to thermodynamic control does not seem to be dependent on temperature between 180 [degrees] C and 215 [degrees] C. The increase of reverse reaction rate with temperature is offset by the increase of water elimination. In the absence of tin catalysts, the reaction reaches equilibrium at 215 [degrees] C after approximately 30,000 s (8.3 h) [ILLUSTRATION FOR FIGURE 1 OMITTED] but is far from equilibrium even after 50,000 s (14 h) at 180 [degrees] C [ILLUSTRATION FOR FIGURE 2 OMITTED]. The calculated equilibrium constants are presented in Table 1.
From the above experimental data, we see that we may neglect reversal of the esterification reactions (Scheme 4) at temperatures below 180 [degrees] C and/or conversions lower than 95%. However, if we are interested in a commercial process where very high conversions are required, the reverse reactions cannot be neglected. The last part, thermodynamically controlled, usually represents the longest part of an industrial process of polyesterification. It may be drastically shortened by applying vacuum to the system to accelerate water elimination. For example, a conversion of 99%, acid value of 5 mg KOH/g by the reaction of adipic acid (5.97 moles) with 1,3-butane diol (6.52 moles) in the presence of 0.2% dibutyltin oxide (8 x [10.sup.-3] molar) at 215 [degrees] C, is reached in 14.8 h. This time represents 2 h for heating to 215 [degrees] C, 1.9 h kinetic controlled esterification to 94.5% conversion (acid value 25 mg KOH/g) and 10.9 h thermodynamic controlled process to 99% conversion (acid value 5 mg KOH/g). If this second part is done under only 60 Torr, it takes just 2.3 h to complete and the total reaction time becomes 6.2 h. That is less than half the time for the same process at normal pressure. There is no use in applying vacuum from the beginning of the reaction as it will not increase the rate of the kinetic controlled process. As most of the water generated at the beginning is distilled, vacuum is also very difficult to apply and maintain.
Rate constants effect of organotin derivatives. The rate constants of such a complex process as described by Scheme 4 cannot be calculated from experimental data using analytical procedures, and a numerical procedure is required. Scheme 5 gives a short algorithm of GIT, a Gear iterator program (QCMP-022) (46,47), used for these calculations. The program integrates the system of differential equations representing the kinetics of the process and computes the concentration of components for known rate constants and initial conditions. The initial set of rate constants is optimized by the Fletcher-Davidson-Powell iterative method, based on the sum of deviations between experimental and calculated concentrations of components.
To obtain significant results there should be no more than two rate constants to calculate per component concentration-time dependence. The equilibrium constants determined experimentally represent the ratio between the reaction constants:
K = [k.sub.n]/[k.sub.-n] = [k.sub.a]/[k.sub.-a] = [k.sub.c]/[k.sub.-c] (4)
and can be used to optimize only the forward rate constants. The reverse constants are calculated from the above equation. As the concentration of the catalyst is constant, the non-catalytic and the organotin catalyzed processes may be considered together. That reduces the rate constants to be optimized to four where two, [k.sub.w] and [k.sub.ol], should be easily guessed as they are common to all processes. As we measured the values for the time dependencies of carboxyl and alcohol groups as well as water concentrations as a function of time, there was more than enough experimental data for the simulation.
Figures 3 and 4 represent two examples of data fit, the points being experimental and the lines the calculated values for concentrations of reaction components using the optimized rate constants. The fits allow the calculation of the rate constants from Table 1. As can be seen from this Table, the rate constant of the noncatalytic process is quite important in our system and cannot be neglected. To illustrate this fact we have calculated the rate of reaction as a function of time for one example (Table 2) using the rate constants from Table 1. In the first two hours the autocatalytic process is the most important. The fast decrease of the acid concentration makes all three processes, the non-catalytic, the autocatalytic, and the catalytic, more or less equivalent after 3.5 h of reaction and a conversion of 89%. After 18 h and a conversion of 99.4% the rate of the autocatalytic process represents only 5.6% of the non-catalytic reaction. At the very low concentration of 8 x [10.sup.-4] mol/kg of dibutyltin oxide, the catalytic rate is approximate equal to the non-catalytic one. At the beginning when the autocatalytic reaction predominates, the catalyst effect is not significant, but when the acid concentration decreases it becomes an important part of the process and will reduce the reaction time considerably. The esterification reaction is much faster than the hydrolysis at the beginning (ratio 3/1, Table 2), when the process is kinetically controlled. At conversions of 95.7% the process starts to be controlled thermodynamically, and the ratio of these two processes is close to 1. These results support the use of the catalyst from the beginning of the reaction and vacuum at the end of the process.
The rate of reaction increases with catalyst concentration but much less than proportionally (Table 1). This means that the catalysts in higher concentrations form a less active product, which presumably is a polymeric system. Spectroscopic data indicate that organotin oxides are crosslinked three-dimensional networks in a polymeric system (40). Halogeno organotin derivatives form alkoxides and hydroxides with alcohols and water at high temperatures that generate also oxides. Thus the activity of oxides and halogen derivatives of organotin compounds should be similar. In fact they are not very different (see monomethytin oxide and monomethyltin trichloride, Table 1) although the halogeno derivatives have a slightly higher catalytic activity. This can be explained by the fact that the in situ formed alkoxides-oxides are probably less polymeric than the commercial oxides. Halogeno tin derivatives are also expected to react faster than organotin oxides with alcohols to generate the alkoxides. The increase of steric hindrance at tin decreases the catalytic activity, as expected from the insertion reaction mechanism (see monobutytin oxide versus dibutyltin oxide, monomethyl tin trichloride versus monobutyltin trichloride and mixture of monomethyl and dimethyltin chlorides, Table 1). The difference in [TABULAR DATA FOR TABLE 2 OMITTED] reactivity between chlorides and oxides is not important enough to justify their industrial use, considering the problems that hydrogen chloride evolution may cause.
The synthesis of polymeric plasticizers by the polycondensation reaction of adipic acid and 1,3-butanediol in the presence of organotin catalysts is a reversible process that consists of a non-catalytic, an autocatalytic, and an organotin-catalyzed reaction. It is best to use the organotin catalyst from the start and to eliminate water by efficiently applying vacuum after approximately 95% conversion to reduce the reaction time to a minimum. Although monoalkyltin trichloride and especially monomethyltin trichloride are more active than organotin oxides, consideration of price and avoiding evolution of hydrogen chloride makes dibutyltin oxide the catalyst of choice from those considered.
Support from the Witco corporation for this work is gratefully acknowledged. We are especially grateful to Mr. Nirmal Jain and Dr. Lawrence R. Brecker for supporting this research and Dr. Wally L. Chang for valuable discussions.
1. C. K. Ingold, Structure and Mechanism in Organic Chemistry, p. 1129, Cornell University Press, Ithaca, N. Y. (1969).
2. J. March, Advanced Organic Chemistry, Reaction Mechanism and Structure, pp 334, 348, John Wiley, New York, (1985).
3. P. J. Flory, J. Amer. Chem. Soc., 59, 466 (1937); 61, 3334 (1939); 62, 2261 (1940).
4. P. J. Flory, Chem. Rev., 39, 154 (1946).
5. S. R. Rafikov and V. V. Korshak, Dokl. Akad. Nauk. SSSR, 64, 211 (1949).
6. V. V. Korshak and S. V. Vinogradova, Zh. Obshch. Khim., 22, 1176 (1952).
7. V. V. Korshak, T. M. Frunze, and I, Li, Vysokomol. Soedin., 3, 665 (1961).
8. S. D. Hamann, D, H. Solomon, and J. D. Swift, J. Macromol. Sci., Chem., 2, 153 (1968).
9. J. L. McCarty and B. J. R. Scholtens, J. Appl. Polym. Sci., 42, 2223 (1991).
10. J. Otton and S. Ratton, J. Polym. Sci. Part A Polym. Cherm., 26, 2183 (1988).
11. M. A. El-Safty, A. E. Shaabah, and H. Y. Mustafa, Acta Polym., 41, 504 (1990).
12. M. Davies, Trans. Faraday Soc., 34, 410 (1938).
13. M. Davies, Research (London), 2, 544 (1949).
14. Y. R. Fang, C. G. Lai, J. L. Lu, and M. K. Chen, Sci. Sin, 18/1, 72 (1975).
15. J. Otton and S. Ratton, J. Polym. Sci. Part A Polym. Chem., 29, 377 (1991).
16. A. C. Tang and K. S. Yao, J. Polym. Sci., 35, 219 (1959).
17. I. Vancso-Smercsanyi and E. Makay-Bodi, J. Polym. Sci. Port C., 16, 3709 (1968).
18. I. Vancso-Smercsanyi, K. Marcos-Greger, and E. Makay-Bodi, Eur. Polym. J., 5, 155 (1969).
19. E. Makay-Bodi and I. Vancso-Smercsanyi, Eur. Polym. J., 5, 145 (1969).
20. I. Vancso-Smercsanyi, E. Makay-Bodi, and E. Szabo-Rethy, J. Polym. Sci. Part A-1, 8, 2861 (1970).
21. A. Fradet and E. Marechal, J. Macromol. Sci. Chem., 17, 859 (1982).
22. M. Maties, R. Bacaloglu, I. Gross, R. Pape, and K. Laszlo, Rev. Chim. (Bucharest), 36, 213 (1985).
23. R. Bacaloglu, M. Maties, C. Csunderlik, L. Cotarca, A. Moraru, I. Gross, and N. Marcu. Angew. Makromol. Chem., 164, l (1988).
24. R. Bacaloglu, M. Maties, L. Cotarca, I. Gross, A. Moraru, and A. Tirnovenu, Angew. Makromol. Chem. 165, 9 (1988).
25. L. Nondek and J. Malek, Makromol. Chem., 178, 2211 (1977).
26. G. Hu, Y. Sun, and M. Lambla, Makromol. Chem., 194, 665 (1993).
27. O. M. O. Habib and J. Malek, Coll. Czech. Chem. Commun., 41, 2724 (1978).
28. J. Dahlmann and G. Rafler, Acta Polymer., 44, 103 (1993).
29. L. A. Hobbs and P. J. Smith, Appl. Orgaomet. Chem., 6, 95 (1992).
30. F. D. Lewis, J. D. Oxman, and J. C. Huffman, J. Am. Chem. Soc., 106, 466 (1984).
31. H. R. Kricheldorf and M. Sumbel, Eur. Polym. J., 25, 585 (1989).
32. K. Steliou, A. Szczyglelska-Nowosielska, A. Favre, M. A. Poupart, and S. Hanessian, J. Am. Chem. Soc., 102, 7378 (1980).
33. K. Steliou and M. A. Poupart, J. Am. Chem. Soc., 105, 7130 (1983).
34. H. K. Kricheldorf, M. V. Sumbel, and I Kreiser-Saunders, Macromolecules, 24, 1944 (1991).
35. F. Pilati, A. Munari, and P. Manaresi, Polymer Comm., 25, 187 (1984).
36. K. C. Kumara Swamy and S. Nagabrachmanandachart, Phosphor, Sulfur Silicon, 65, 9 (1992).
37. W. Hetterich, D. Joel, H. Becker, and C. Wehrstedt, Angew. Makromol. Chem., 179, 135 (1990).
38. A. K. Kumar and T. K. Chattopadhyay, Tetrahedron Lett., 28, 3713 (1987).
39. L. Nondek and J. Malek, Coll. Czech. Chem. Comm., 43, 1907 (1978).
40. P. G. Harrison, Chemistry of Tin, pp. 44, 91, Chapman and Hall, New York (1989).
41. J. Otera, Chem Rev., 93, 1449 (1993).
42. J. Otera, N.Dan-oh, and H Nozaki, J. Org. Chem., 56, 5307 (1991).
43. J. Bonetti, C. Gondard, R. Petiaud, M. F. Lauro, and A. Michel, J. Organomet. Chem., 461, 7, (1994).
44. I. Espinasse, R. Petiaud, M. F. Llauro, and A. Michel, Int. J. Polymer Analysis & Characterization, 1, 137 (1995).
45. W. L. Chang and T. Karalis, J. Polym. Sci. Part A. Polym. Chem., 31, 493 (1993).
46. R. J. McKinney and F. J. Weigert, QCPE Bull., 6, 141 (1986).
47. F. J. Weigert, Comput. Chem., 11, 273 (1987).
|Printer friendly Cite/link Email Feedback|
|Author:||Bacaloglu, Radu; Fisch, Michael; Biesiada, Keith|
|Publication:||Polymer Engineering and Science|
|Date:||Jun 1, 1998|
|Previous Article:||Heat transfer coefficient in a high pressure tubular reactor for ethylene polymerization.|
|Next Article:||Theoretical analysis of the thermally induced delamination of polymeric coatings in double-coated optical fibers.|