Madras et al

Continuous distribution kinetics for ultrasonic degradation of polymers Giridhar Madras *, Sanjay Kumar, Sujay Chattopadhyay Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India Received 12 November 1999; received in revised form 11 January 2000; accepted 21 January 2000 Abstract The ultrasonic degradation of polystyrene and poly(vinyl acetate) in chlorobenzene was studied. The time evolution of the molecular weight distributions (MWDs) were determined by gel permeation chromatography of the degraded samples. The data were analyzed with a continuous distribution kinetics model that treats the molecular weight as a continuous variable. Since the chain cleavage in the ultrasonic degradation of polymers occurs preferentially near the midpoint of the polymer backbone, a stoichiometric kernel that describes scission at themiddle of the chain was used. The degradation rate coecients of the polymers were determined using ourmodel and was found to be 0.032 minÿ1. The model successfully simulated the number average molecular weight, the polydispersity of the degraded polymer and the time evolution of the molecular weight distributions.# 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ultrasound; Polymers; Degradation; Continuous distribution kinetics 1. Introduction The ultrasonic polymer degradation has several unique characteristics that make it interesting both from practical and theoretical viewpoints. Polymer can be degraded by thermal, photochemical or ultrasound methods. The scission of the polymer backbone can occur at the end of the chain or at the midpoint of the chain or randomly at any bond in the chain. While thermal degradation is primarily due to chain-end and/ or random-chain scission [1], the chain cleavage in ultrasonic degradation of polymers is preferentially near the middle of the chain [2]. Degradation of polymers by ultrasound was carried out as early as 1939 [3]. Since then, ultrasonic degradation has been studied for a wide variety of polymers and the literature has been reviewed by Price [4]. Recently, Price and Smith [5] studied the e€ect of temperature, ultrasound intensity and dissolved gases on the degradation of poly- styrene. However, fundamental studies and models on the time evolution of MWDs are lacking. Many early studies on polymer degradation were based on changes in molecular weight averages. However, these do not yield meaningful rate constants since the time evolution of the molecular weight distribution is unaccounted for. The value of polydispersity is a useful quantity that can shed information on the degradation mechanism [6]. However, details of the degradation process are still obscure and little information has been published on polydispersity [6]. Some models for ultrasonic polymer degradation have been proposed. Glynn et al. [7] defined a degradation index based on number average molecular weights and developed a basic model in terms of probabilities, P and Q. They represented the breakage of a molecule of a cer- tain length as P and denoted the probability that a mole- cule of a particular length will arise from that cleavage as Q. Hence, Q represented the probability of scission at various points along the polymer backbone. The average number of chain breaks was assumed and the predicted distributions were compared with the experimental MWD of polystyrene degraded in tetrahydrofuran. However, the predicted distributions assuming midpoint scission were in poor agreement with experimental data. The model was later modified [8] to account for the limiting molecular weight. However, others [9,10] have found that the degradation index defined above was not a satisfactory parameter when applied to polymers with a wide distribution. 0141-3910/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PI I : S0141-3910(00 )00042-2 Polymer Degradation and Stability 69 (2000) 73–78 * Corresponding author. Tel.: +91-80-309-2321; fax: +91-80-334- 1683. E-mail address: giridhar@chemeng.iisc.ernet.in (G. Madras). Administrator Callout 基于连续分布动力学的高聚物超声降解 Population balance equations can be applied to describe the time evolution of the molecular weight dis- tribution. Many mathematical models on polymer degradation account only for the average properties (like the first two moments) of the molecular weight distribu- tions (MWDs). The time- evolution of the molecular- weight distribution (MWD) is fundamental to the study of polymer degradation, since the MWD contains much more information than the lumped concentration. Con- tinuous-distribution population balances provide the governing integrodi€erential equations, which are con- verted to ordinary di€erential equations for MW moments. The stoichiometric kernel provides the dis- tribution of the chemical reaction products through the type of chain scission. Some papers [11–13] have dis- cussed mathematical solutions for polymer degradation. The objective of this paper is to formulate solutions for ultrasonic polymer degradation following a mathe- matical treatment similar to that of McCoy and Madras[13]. The mathematical model is compared to the experimental data for the ultrasonic degradation of polystyrene and poly(vinyl acetate). Our model success- fully predicts the time evolution of MWD and provides an e€ective way to obtain the degradation rate coe- cients. Our model also has the capability to simulate a bimodal distribution, which has been observed in some cases of ultrasonic polymer degradation [2]. 2. Experiments 2.1. Degradation experiments Polystyrene with an initial Mn of 157,000 and poly- dispersity of 1.2 was degraded in a horn-type sonicator (Vibronics). 20 ml of 2 g/l solution of polystyrene in chlorobenzene was irradiated at a frequency of 25 kHz for 10, 20, 30 60, 120, 180 min. Power input was fixed by controlling the supply voltage at 180 V. Poly (vinyl acetate) of Mn=270,000 and polydispersity of 1.1 was dissolved in chlorobenzene and degraded in a similar fashion. All experiments were conducted at 23�C. A 200 ml aliquot of the sample was injected into the HPLC–GPC system to obtain the chromatograph. The chromatograph was converted to MWD using the calibra- tion curve. 2.2. Determination of MWDs The MWDs of the sonicated samples were determined by GPC coupled with a HPLC (Waters). Tetra- hydrofuran (S.D. Fine Chem.) was pumped through the columns at 1 ml/min using a Waters 501 HPLC pump. Three waters columns (7.8�300 mm) packed with cross- linked poly(styrene-divinylbenzene) (HR 4, HR 3, HR 0.5) were used in series for ecient separation. The col- umns were maintained at 40�C with a column heater (Eldex). The refractive index of the sample was deter- mined continuously with the di€erential refractometer (Waters R401). The system was calibrated using narrow MW poly- styrene standards of MW of 500–0.3 million (Waters Corporation and TSK Corp). Fig. 1 shows the calibra- tion curve of the retention time versus molecular weight. 3. Theoretical model Continuous-distribution kinetics provides a straight- forward technique to determine the rate coecients of Fig. 1. Calibration curve for retention volume versus molecular weight for polystyrene standards. 74 G. Madras et al. / Polymer Degradation and Stability 69 (2000) 73–78 Administrator Line Administrator Line polymer degradation. TheMWD, p(x,t), is defined so that p(x,t) dx is the molar concentration of the polymer in (x, x+dx). For a binary scission occurring with a rate coe- cient, k, the products of the scission [11] are governed by 2 …1 x k…x0†p…x0; t† …x; x0†dx0 The stoichiometric kernel [12] for a polymer of MW x that fragments into two products of MW, x’ and xÿx’, can be written as …x; x0† ˆ xm…xÿ x0†mÿ…2m‡ 2†= ÿ…2m‡ 2†=‰ÿ…m‡ 1†2…x0†2m‡1Š …1† where m=0 and m!1 correspond to random and midpoint chain scission, respectively [13]. The moments, p…n†, are defined as p…j†…t† ˆ …1 0 xjp…x; t†dx …2† The number- and weight-average molecular weights are Mn ˆ p…1†=p…0† andMw ˆ p…2†=p…1† …3† and the polydispersity is D ˆMw=Mn …4† Degradation with r scissions in sequence can be repre- sented as [12], x1 ! …x1 ÿ x2† ‡ x2 x2 ! …x2 ÿ x3† ‡ x3 xrÿ1 ! …xrÿ1 ÿ xr† ‡ xr The governing balance equations for j=0, 1,..., rÿ1 (with k0=kr=0) is dpj‡1=dt ˆ ÿkj‡1…xj†Pj‡1 ‡ 2 …1 x kj…xj†pj …xj‡1; xj†dx0 …5† Since kr=0, the di€erential equation for j=rÿ1 is dp=dt ˆ 2 …1 x krÿ1…xrÿ1†prÿ1 …xr; xrÿ1†dx …6† Usually in polymer degradation, depending on the extent of the reaction, the rate coecient is considered either a constant or linearly dependent on the chain length. If the change in average MW is not large, an average rate constant independent of MW is satisfactory [1], as is this case. For a batch reactor, application of the moment operation to Eqs. (5) and (6) gives, dp…n†1 =dt ˆ ÿkp…n†1 …7† dp …n† i =dt ˆ ÿkp…n†i ‡ 2Znmkp…n†iÿ1 …8† dp…n†r =dt ˆ 2Znmkp…n†rÿ1 …9† where Znm ˆ …m‡ 1†n=…2m‡ 2†n, is the ratio of Poch- hammer symbols …m†n ˆ …m‡ n†=…m†. For n=0 and 1, Znm is 1 and 1 2, respectively, for all m. For the second moment, Zn0 ˆ 1=3 and Zn ˆ 1=4. Thus, the di€erence between the midpoint chain scission and the random chain scission is only reflected in higher moments (n52). This also indicates that a study of the time evo- lution of polydispersity would di€erentiate between the two chain mechanisms. The initial conditions for solving the above equations are p …n† 1 …t ˆ 0† ˆ p…n†0 …10† p …n† i …t ˆ 0†ÿ ˆ 0 for i > 1 …11† The solution of the di€erential Eq (7) is P …n† 1 ˆ p…n†0 eÿkt …12† Representing the product polymers as q, the solution of Eq. (8) is q …n† i ˆ p…n†0 eÿkt…2ktZnm†iÿ1=…iÿ 1†! …13† For the terminal scission [Eq. (9)], one has the following sequence q …n† rˆ2 ˆ p…0†0 2Znm…1ÿ eÿkt† …14† q …n† rˆ3 ˆ p…n†0 …2Znm†2…1ÿ …1‡ kt†eÿkt† …14a† The moments of all products of scission (j=2 to r) can be summed q…n†…t† ˆ X q …n† j …t† …15† The limiting value of the zeroth moment, q…0†…t ! 1† ˆ p…0†0 2rÿ1=…rÿ 2†! …16† 4. Results and discussion We determined the degradation rate coecients from experimental data by analyzing the time evolution of the G. Madras et al. / Polymer Degradation and Stability 69 (2000) 73–78 75 MWDs. The total polymer MWD for chain scission is ptot …x; t† ˆ p1…x; t† ‡ q…x; t† and the total moments can be written in terms of dimensionless number-average molecular weight, XMn …ˆMn=Mn0† and weight average molecular weight, XMw…ˆMw=Mw0† as XMn ˆ Pavgtot =pavg0 ˆ ‰p…1†1 ‡ q…1†Šp…0†0 f‰p…0†1 ‡ q…0†Šp…1†0 g …17† XMw ˆ ‰p…2†1 ‡ q…2†Šp…1†0 =fp…2†0 ‰p…1†1 ‡ q…1†Šg …18† XD, defined as ratio of observed polydispersity to the original polydispersity is then given by XMw=XMn . The mass of the final product is equal to the mass of the initial reactant and the number average molecular weight is the ratio of the first to the zero moment. Thus, Eq. (16) indicates XMn …t ! 1† ˆ …rÿ 2†!=2rÿ1 …19† From the experimental data, the limiting molecular weights were 40,000 for polystyrene and 68,000 for poly(vinyl acetate). The limiting molecular weight for polystyrene determined in this study agrees well the empirical relation, based on the intensity of the radia- tion, derived by Price and Smith [2]. Using Eq. (19), we calculate r (=3) for both poly- styrene and poly(vinyl acetate). There is evidence to suggest that the chain scission occurs preferentially at the centre of the chain for ultrasonic degradation. Thus, Eqs. (12), (13) and (14a) for r=3 and m!1, we obtain, XMn ˆ …4ÿ 3eÿkt ÿ 2kteÿkt†ÿ1 …20† XD ˆ …1‡ 3eÿkt ‡ kteÿkt†…4ÿ 3eÿkt ÿ 2kteÿkt†=4 …21† Fig. 2 shows XMn as a function of time for the degra- dation of polystyrene and poly(vinyl acetate). The exis- tence of a limiting molecular weight and an exponential decrease of molecular weight with time has been observed by many investigators and is a characteristic of ultrasonic degradation [4]. However, the continuous distribution kinetic analysis of the experimental data provides a straightforward technique of determining the decrease in the molecular weight with time by evaluating the number of sequential scissions. Fig. 3 shows the variation of polydispersity (XD) of polystyrene and poly(vinyl acetate) with time. These results are consistent with Wu et al. [14], who studied the degradation of poly(methyl methacrylate) in tetra- hydrofuran. They observed that initially narrow poly- dispersity fractions became broader on degradation before narrowing again at long sonication times. The experimental data, represented in Figs. 2 and 3, were used to obtain a regressed value for the degradation rate coecient, k. The degradation rate coecient for both polystyrene and poly(vinyl acetate) was found to be 0.032 minÿ1. As seen from the figures, the theoretical pre- diction of the experimental data is quite satisfactory. To study the time evolution of the MWD, the dis- tribution is represented by a gamma distribution in terms of y ˆ …xÿ xs†=�, p…x; t† ˆ p…0†y…�ÿ1†eÿy=‰�ÿ…�†Š …22† whose mean and variance are given by xs+ab and ab2, respectively. The reactant and product MWDs are represented as gamma distributions and added together. Fig. 2. Variation of XMn with time for polystyrene and poly(vinyl acetate).* Polystyrene,^poly(vinyl acetate), — model prediction. 76 G. Madras et al. / Polymer Degradation and Stability 69 (2000) 73–78 Moments are calculated as a function of kt based on Eqs. (12), (13) and (14a). The sum of p1(x,t) and q(x,t) yields the molar MWD, ptot (x,t). The mass MWD, ptot1 (=xptot (x,t)), is then computed as a function of kt and plotted with the mass MWD measured by gel permea- tion chromatography. Fig. 4 shows the model predic- tion and the experimental MWD for the ultrasonic degradation of polystyrene. Our model prediction, which assumes that the scission is at the midpoint, satisfactorily predicts the variation of the average molecular weight, polydispersity with time and the time evolution of the MWD. Thus, continuous distribution kinetics provides a simple technique to monitor the time dependence of MWDs and elucidates considerable information beyond the molecular weight averages. References [1] Madras G, Smith JM, McCoy BJ. E€ect of tetralin on the degradation of polymer in solution. I&EC Research 1995;34:4222–8. [2] Price GJ, Smith PF. Ultrasonic degradation of polymer solutions: 1. Polystyrene revisited. Polym Int 1991;24:159–63. [3] Jellinek HHG, editor. Degradation of vinyl polymers. New York: Academic Press, 1955. [4] Price GJ. The use of ultrasound for the controlled degradation of polymer solutions. In: Mason TJ, editor. Advances in sono- chemistry, vol. 1. Cambridge: Jai Press, 1990, p. 231–85. Fig. 3. Dependence of XD on time for polystyrene and poly(vinyl acetate).* Polystyrene, ^poly(vinyl acetate), — model prediction. Fig. 4. Evolution of the mass MWD, ptot1(x,t), for polystyrene before degradation and after ultrasonic degradation for 20 min. * Experimental data, — model prediction. G. Madras et al. / Polymer Degradation and Stability 69 (2000) 73–78 77 [5] Price GJ, Smith PF. Ultrasonic degradation of polymer solutions: 2. The e€ect of temperature, ultrasound intensity and dissolved gases on polystyrene in toluene. Polymer 1993;34:4111–17. [6] Koda S, Mori H, Matsumoto K, Nomura H. Ultrasonic degra- dation of water soluble polymers. Polymer 1993;35:30–3. [7] Glynn PAR, Van der Ho€ BME, Reilly PM. General model for prediction of molecular weight distributions of degraded poly- mers. 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Ultrasonic degradation of poly (methylmethacrylate). Polymer 1977;18:822–4. 78 G. Madras et al. / Polymer Degradation and Stability 69 (2000) 73–78