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Effect of oligomers extraction in polycondensation reactions on number and weight average degree of polymerization
European Polymer Journal 38 (2002) 1925–1927 www.elsevier.com/locate/europolj
Short communication
Effect of oligomers extraction in polycondensation reactions on number and weight average degree of polymerization M. Zatloukal *, F. Rybnik ar, P. S aha Faculty of Technology, Tomas Bata University in Zlın, T.G. Masaryk Square 275, 76272 Zlın, Czech Republic Received 13 September 2001; received in revised form 28 December 2001; accepted 11 January 2002
Abstract In many polycondensation reactions, the isolated polymer does not contain molecules with low degrees of polymerization. The absence of oligomers affects significantly the average degrees of polymerization, the number-average value, X n , in particular. Using the Flory theory of linear polycondensation, formulas have been derived for the calculation of the number-average and weight-average degree of polymerization, X n and X w , respectively, as a function of the extent of reaction, p, and the minimum degree of polymerization of molecules present in the polymer. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Extraction of oligomers; Flory theory of linear polycondensation; Average degree of polymerization
1. Introduction The kinetics of linear polycondensation reactions can be properly described according to the Flory theory [1]. Based on this theory, the number-average, X n , weightaverage, X w , degree of polymerization and distribution function of degrees of polymerization of the final polymer product as a function of the extent of reaction, p, are widely used for polymer characterization. The extent of reaction is simply measured by the amount of the byproduct [2–5] or by calculating X n assuming one type of end groups per chain [6–8]. At higher polymerization temperatures, however, degradation reactions can change the amount of end groups [9,10] and affect the X n value calculated only on the basis of p values derived from the number of polymer end groups. In such cases, it is desirable to use also a direct method for X n determination which is independent of the type and amount of end groups. Bulk, melt or solution polycondensation
of aromatic polyesters and polyamides yield mostly crystalline, insoluble materials where X n can be assessed only from p calculated from the amount of by-product [11–14] because direct methods for the determination of the degree of polymerization, using dilute polymer solutions, are not operative. If, in such case, X n and X w are calculated through the p value, the results can be distorted because such procedure does not take into account the fact that, during solution polycondensation, the formed low-molecular-weight oligomers are removed by extraction from the polymer. The aim of this paper is to derive generalized equations for computation of real X n and X w values, taking into account both the p value and the absence of oligomers in the polymer. Such equations are not available in the open literature.
2. Problem formulation, results and discussion X n and X w are defined as
*
Corresponding author. Fax: +420-67-7541444. E-mail address: [email protected] (M. Zatloukal).
Pb Xi Ni X n ¼ Pi¼a b i¼a Ni
0014-3057/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 2 ) 0 0 0 6 0 - 5
ð1Þ
1926
M. Zatloukal et al. / European Polymer Journal 38 (2002) 1925–1927
and b P
Xi mi i¼a Xw ¼ P b i¼a mi
ð2Þ
where Xi is the degree of polymerization of molecules of type i, Ni is the number of molecules of that kind, mi is their mass fraction, and a and b represent minimum and maximum degrees of polymerization of macromolecules present in the polymer, respectively. According to the Flory theory, the number of macromolecules of type i, Ni , and their mass fraction mi are Ni ¼ N½pXi 1 ð1 pÞ
ð3Þ
mi ¼ Xi pXi 1 ð1 pÞ2
ð4Þ
respectively; here N is the total number of macromolecules with all degrees of polymerization in the polymer and p 2 h0; 1i is the extent of reaction. Substitution of Eqs. (3) and (4) into Eqs. (1) and (2) yields: Fig. 1. Error function for number-average degree of polymerization.
Xn ¼
pa ½aðp 1Þ p þ pbþ1 ½1 bðp 1Þ ðpa pbþ1 Þðp 1Þ
Xw ¼
pa ½a2 ðp 1Þ2 þ 2apð1 pÞ þ pðp þ 1Þ pbþ1 ½b2 ðp 1Þ2 þ 2bð1 pÞ þ p þ 1 ½pa ðaðp 1Þ pÞ þ pbþ1 ð1 bðp 1ÞÞðp 1Þ
ð5Þ
For b ! 1, the equations simplify to X n ¼ lim X n ¼ a þ b!1
1 1 1p
ð7Þ
X w ¼ lim X w
ð6Þ
where subscript 1 means that the particular variable is calculated as function of p only (a ¼ 1) and subscript 2 specifies the variable that is calculated as a function of both p and a. It can be seen that, ignoring the absence of oligomers, the error in X w and, in particular, in X n is
b!1
¼
1 1 þ þa ½aðp 1Þ pðp 1Þ aðp 1Þ p 1 1 þ 1p
ð8Þ
Thus, X n and X w can easily be calculated as a function of the extent of reaction, p, and the minimum degree of polymerization of molecules present in the polymer, a. In order to illustrate how much higher the calculated values of X n and X w are in comparison with the values ignoring the absence of oligomers, we have solved Eqs. (7) and (8) for various values of p and a. The results are shown (Figs. 1 and 2) in terms of error functions: Err X n ¼
X n;2 X n;1 100 ¼ ð1 aÞðp 1Þ100 X n;1
Err X w ¼
X w;2 X w;1 aðp 1Þ2 ða 1Þ 100 100 ¼ ½p aðp 1Þðp þ 1Þ X w;1
ð9Þ
ð10Þ
Fig. 2. Error function for weight-average degree of polymerization.
M. Zatloukal et al. / European Polymer Journal 38 (2002) 1925–1927
considerable especially for decreasing p and increasing a. In this calculation, a values have been chosen for real liquid-crystal polymers described in the literature. For example, oligomers of p-oxybenzoate up to pentamers (a ¼ 6) are soluble [13]. Similarly, polymerization of pacetoxybenzoic acid in solution (Marlotherm S) results in extraction of oligomers with a degree of oligomerization of 5–7 (a ¼ 6–8), depending on concentration and temperature [14]. In both cases, low-molecular oligomers remain in solution.
3. Conclusion The Flory theory has been used to derive equations for computation of X n and X w values of the final polymer product of a polycondensation which take into account both the extent of reaction p and the extraction of oligomers from polymer. The absence of these components may significantly influence X n and X w . Thus, for precise determination of these molecular characteristics, equations derived in this paper should be employed rather than equations using only the p value for calculation.
1927
References [1] Flory PJ. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. [2] Park SS, Jun HW, Im SS. Polym Eng Sci 1998;38:905. [3] Vulic I, Schulpen T. J Polym Sci Polym Chem 1992;30: 2725. [4] Mathew J, Bahulekar RV, Ghadage RS, Rajan CR, Ponrathnam S. Macromolecules 1992;25:7338. [5] Williams PA, Han XH, Padias AB, Hall HK. Macromolecules 1996;29:1874. [6] Uphade BS, Patil PS, Pandit SB, Rajan CR, Nadkarni VM. J Polym Sci Polym Chem 1994;32:2003. [7] Economy J, Volksen W, Viney C, Geiss R, Siemens R, Karis T. Macromolecules 1988;21:2777. [8] Han XH, Williams PA, Padias AB, Hall HK, Linstid HC, Sung HN, et al. Macromolecules 1996;29:8313. [9] Peineschuessel HK, Dege GJ. J Polym Sci 1970;8:3265. [10] Van Bennekom ACM, Willemsen PAAT, Gaymans RJ. Polymer 1996;37:5447. [11] Iannelli P, Yoon DY. J Polym Sci Polym Phys 1995;33: 977. [12] Rybnikar F, Yuan BL, Geil PH. Polymer 1994;35:1863. [13] Ballauff M, Wu D, Flory PJ, Barrall EM. Ber Bunsen Phys Chem 1984;84:524. [14] Schwarz G, Kricheldorf HR. Macromolecules 1991;24: 2829.
Short communication
Effect of oligomers extraction in polycondensation reactions on number and weight average degree of polymerization M. Zatloukal *, F. Rybnik ar, P. S aha Faculty of Technology, Tomas Bata University in Zlın, T.G. Masaryk Square 275, 76272 Zlın, Czech Republic Received 13 September 2001; received in revised form 28 December 2001; accepted 11 January 2002
Abstract In many polycondensation reactions, the isolated polymer does not contain molecules with low degrees of polymerization. The absence of oligomers affects significantly the average degrees of polymerization, the number-average value, X n , in particular. Using the Flory theory of linear polycondensation, formulas have been derived for the calculation of the number-average and weight-average degree of polymerization, X n and X w , respectively, as a function of the extent of reaction, p, and the minimum degree of polymerization of molecules present in the polymer. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Extraction of oligomers; Flory theory of linear polycondensation; Average degree of polymerization
1. Introduction The kinetics of linear polycondensation reactions can be properly described according to the Flory theory [1]. Based on this theory, the number-average, X n , weightaverage, X w , degree of polymerization and distribution function of degrees of polymerization of the final polymer product as a function of the extent of reaction, p, are widely used for polymer characterization. The extent of reaction is simply measured by the amount of the byproduct [2–5] or by calculating X n assuming one type of end groups per chain [6–8]. At higher polymerization temperatures, however, degradation reactions can change the amount of end groups [9,10] and affect the X n value calculated only on the basis of p values derived from the number of polymer end groups. In such cases, it is desirable to use also a direct method for X n determination which is independent of the type and amount of end groups. Bulk, melt or solution polycondensation
of aromatic polyesters and polyamides yield mostly crystalline, insoluble materials where X n can be assessed only from p calculated from the amount of by-product [11–14] because direct methods for the determination of the degree of polymerization, using dilute polymer solutions, are not operative. If, in such case, X n and X w are calculated through the p value, the results can be distorted because such procedure does not take into account the fact that, during solution polycondensation, the formed low-molecular-weight oligomers are removed by extraction from the polymer. The aim of this paper is to derive generalized equations for computation of real X n and X w values, taking into account both the p value and the absence of oligomers in the polymer. Such equations are not available in the open literature.
2. Problem formulation, results and discussion X n and X w are defined as
*
Corresponding author. Fax: +420-67-7541444. E-mail address: [email protected] (M. Zatloukal).
Pb Xi Ni X n ¼ Pi¼a b i¼a Ni
0014-3057/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 2 ) 0 0 0 6 0 - 5
ð1Þ
1926
M. Zatloukal et al. / European Polymer Journal 38 (2002) 1925–1927
and b P
Xi mi i¼a Xw ¼ P b i¼a mi
ð2Þ
where Xi is the degree of polymerization of molecules of type i, Ni is the number of molecules of that kind, mi is their mass fraction, and a and b represent minimum and maximum degrees of polymerization of macromolecules present in the polymer, respectively. According to the Flory theory, the number of macromolecules of type i, Ni , and their mass fraction mi are Ni ¼ N½pXi 1 ð1 pÞ
ð3Þ
mi ¼ Xi pXi 1 ð1 pÞ2
ð4Þ
respectively; here N is the total number of macromolecules with all degrees of polymerization in the polymer and p 2 h0; 1i is the extent of reaction. Substitution of Eqs. (3) and (4) into Eqs. (1) and (2) yields: Fig. 1. Error function for number-average degree of polymerization.
Xn ¼
pa ½aðp 1Þ p þ pbþ1 ½1 bðp 1Þ ðpa pbþ1 Þðp 1Þ
Xw ¼
pa ½a2 ðp 1Þ2 þ 2apð1 pÞ þ pðp þ 1Þ pbþ1 ½b2 ðp 1Þ2 þ 2bð1 pÞ þ p þ 1 ½pa ðaðp 1Þ pÞ þ pbþ1 ð1 bðp 1ÞÞðp 1Þ
ð5Þ
For b ! 1, the equations simplify to X n ¼ lim X n ¼ a þ b!1
1 1 1p
ð7Þ
X w ¼ lim X w
ð6Þ
where subscript 1 means that the particular variable is calculated as function of p only (a ¼ 1) and subscript 2 specifies the variable that is calculated as a function of both p and a. It can be seen that, ignoring the absence of oligomers, the error in X w and, in particular, in X n is
b!1
¼
1 1 þ þa ½aðp 1Þ pðp 1Þ aðp 1Þ p 1 1 þ 1p
ð8Þ
Thus, X n and X w can easily be calculated as a function of the extent of reaction, p, and the minimum degree of polymerization of molecules present in the polymer, a. In order to illustrate how much higher the calculated values of X n and X w are in comparison with the values ignoring the absence of oligomers, we have solved Eqs. (7) and (8) for various values of p and a. The results are shown (Figs. 1 and 2) in terms of error functions: Err X n ¼
X n;2 X n;1 100 ¼ ð1 aÞðp 1Þ100 X n;1
Err X w ¼
X w;2 X w;1 aðp 1Þ2 ða 1Þ 100 100 ¼ ½p aðp 1Þðp þ 1Þ X w;1
ð9Þ
ð10Þ
Fig. 2. Error function for weight-average degree of polymerization.
M. Zatloukal et al. / European Polymer Journal 38 (2002) 1925–1927
considerable especially for decreasing p and increasing a. In this calculation, a values have been chosen for real liquid-crystal polymers described in the literature. For example, oligomers of p-oxybenzoate up to pentamers (a ¼ 6) are soluble [13]. Similarly, polymerization of pacetoxybenzoic acid in solution (Marlotherm S) results in extraction of oligomers with a degree of oligomerization of 5–7 (a ¼ 6–8), depending on concentration and temperature [14]. In both cases, low-molecular oligomers remain in solution.
3. Conclusion The Flory theory has been used to derive equations for computation of X n and X w values of the final polymer product of a polycondensation which take into account both the extent of reaction p and the extraction of oligomers from polymer. The absence of these components may significantly influence X n and X w . Thus, for precise determination of these molecular characteristics, equations derived in this paper should be employed rather than equations using only the p value for calculation.
1927
References [1] Flory PJ. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. [2] Park SS, Jun HW, Im SS. Polym Eng Sci 1998;38:905. [3] Vulic I, Schulpen T. J Polym Sci Polym Chem 1992;30: 2725. [4] Mathew J, Bahulekar RV, Ghadage RS, Rajan CR, Ponrathnam S. Macromolecules 1992;25:7338. [5] Williams PA, Han XH, Padias AB, Hall HK. Macromolecules 1996;29:1874. [6] Uphade BS, Patil PS, Pandit SB, Rajan CR, Nadkarni VM. J Polym Sci Polym Chem 1994;32:2003. [7] Economy J, Volksen W, Viney C, Geiss R, Siemens R, Karis T. Macromolecules 1988;21:2777. [8] Han XH, Williams PA, Padias AB, Hall HK, Linstid HC, Sung HN, et al. Macromolecules 1996;29:8313. [9] Peineschuessel HK, Dege GJ. J Polym Sci 1970;8:3265. [10] Van Bennekom ACM, Willemsen PAAT, Gaymans RJ. Polymer 1996;37:5447. [11] Iannelli P, Yoon DY. J Polym Sci Polym Phys 1995;33: 977. [12] Rybnikar F, Yuan BL, Geil PH. Polymer 1994;35:1863. [13] Ballauff M, Wu D, Flory PJ, Barrall EM. Ber Bunsen Phys Chem 1984;84:524. [14] Schwarz G, Kricheldorf HR. Macromolecules 1991;24: 2829.