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Discrete and continuous models for polymerization and depolymerization
Chemical Engineering Science 56 (2001) 2831}2836
Discrete and continuous models for polymerization and depolymerization Benjamin J. McCoy , Giridhar Madras * Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, USA Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India Received 16 June 2000; received in revised form 24 September 2000; accepted 17 October 2000
Abstract A fundamental continuous model for the distribution kinetics of polymerization and its reverse, depolymerization, is shown to be consistent with the well-known discrete model. Solutions for discrete and continuous models are compared and shown to have identical expressions for molecular weight (MW) moments. Representing the continuous molecular-weight distribution (MWD) as a distribution compares well with exact polymerization and depolymerization MWD solutions of the discrete model, except when the number of repeat units in the polymer is small. None of the solutions for chain end polymerization or depolymerization has the self-similar form, for which the MW and time would appear together in the same similarity variable. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Discrete kinetics; Continuous distribution kinetics; Chain end scission; Chain end addition; Similarity solutions; Gamma distributions
1. Introduction Describing how molecular weight distributions evolve during polymerization or depolymerization reactions is an important fundamental issue of polymer science and engineering. The importance of polymerization is well understood, but depolymerization is also signi"cant for plastics recycling and polymer characterization. Moreover, depolymerization is the reverse of polymerization, and a!ects the steady-state or equilibrium limitations on polymerization. Conversely, depolymerization processes can involve polymerization or repolymerization reactions that in#uence their reaction course, such as in pyrolysis (McCoy, 1999; Sezgi, Cha, Smith, & McCoy, 1999). Several publications have commented on the advantages and disadvantages of solving the equations for polymerization and depolymerization as discrete or continuous models (Dotson, Galvan, Laurence, & Tirrell, 1996; Ray, 1972; Kostoglou, 2000). In this work we focus on monomer addition or dissociation of monomers at the end of polymer chains. Because there is no doubt that monomers add or split o! individually, one should * Corresponding author. Tel.: #91-080-309-2381; fax: #91-080360-0683. E-mail address: [email protected] (G. Madras).
consider that the discrete approach is accurate, serving as a standard for judging other approaches. The continuous model, however, is straightforward to manipulate mathematically, whereas only special irreversible cases of the discrete reversible model can be solved. The continuous model would be preferred for this reason, if its limitations were completely understood. An obvious condition for the discrete distribution to be approximated by a continuous distribution is that the number of repeat units (monomers) in the polymers should be very large. The usual derivation of polymerization moments for a discrete distribution depends on assuming an in"nite number of polymers in the MWD, as we will show. In spite of numerous applications of continuous models, however, even when the results are identical to the special discrete cases that can be solved, some critics have challenged their application to polymer reactions models (Dotson et al., 1996; Kostoglou, 2000). Frequently the arguments are based on approximations to what we call below the fundamental continuous theory. Our primary objective is to examine in detail the issues involved in this controversy. A second objective is to examine if chain polymerization or chain end depolymerization have similarity (scaled, self-similar, or self-preserving) solutions, which combine with MW x and time t into one similarity variable. As discussed in numerous works (Madras &
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McCoy, 1998; McCoy & Madras, 1998; Zi! & McGrady, 1986; Zi!, 1991; Friedlander & Wang, 1966; Goren, 1968), the usual form of a similarity solution is when x appears in the expression for the MWD combined with time as the similarity group, xH/(t) (McCoy & Madras, 1998; Zi! & McGrady, 1986). Earlier, Madras and McCoy (1998) presented solutions for chain combination and degradation (random, parabolic, and midchain scission), demonstrating when and how these solutions evolved to similarity solutions. Even though chain-end addition of monomer (chain polymerization) was not discussed, we showed that chain-end scission, yielding monomers or oligomers, did not conform to the similarity solution form. This conclusion has also been questioned (Kostoglou, 2000). Chain polymerization is denoted by the addition of monomer, M(x ), of MW x to the polymer, P(x), of K K MW x by the second-order rate coe$cient k . The de? polymerization (reverse) step is chain-end scission with the "rst-order rate coe$cient k . Thus, the reversible B reaction (Kodera & McCoy, 1997) is IX M(x )#P(x) & P(x#x ). (1.1) K K IB The rate coe$cient for scission can be considered a composite coe$cient for a radical mechanism (Kodera & McCoy, 1997). The continuous molecular-weight distribution (MWD) is de"ned such that p(x, t) dx is the molar concentration of polymer at time t in the MW range (x, x#dx). Moments of the MW are de"ned as
p(x, t)xL dx, (1.2) and provide basic properties of the MWD, including total molar concentration, p, average MW, p " p/p, and variance, p "p/p!(p ). For many polymer reactions, only these moments are needed to interpret typical behavior. For a discrete model with j"0,1,2,2, N monomers in the polymers, the MW for an individual polymer is x"jx . The continuous MWD K is then made up of discrete polymers with MWD p (t), H , p(x, t)" p (t)(x!jx ). (1.3) H K H It follows from Eq. (1.2) that for the discrete distribution, the moments are given by pL(t)"
, pL(t)" p (t)(jx )L, (1.4) H K H which is the standard expression (Dotson et al., 1996). 2. Population balance equations The population balance for a batch reactor is written so that accumulation of a polymer of mass x is represented by the time derivative. Gain and loss terms in the
population balance are represented by the second-order forward reaction and the "rst-order reverse reaction. For the continuous distributions, integration over all reaction partners is required. Recent theoretical and experimental research on polymer degradation (Kodera & McCoy, 1997; Wang, Smith, & McCoy, 1995; Madras, Smith, & McCoy, 1996a,b) have analyzed and discussed the chainend scission kernels, (x!x ) and (x!(x!x )), for K K the monomer loss process. The population balance equation for polymer is
p(x)[x!(x!x )] dx K V V !k p m(x) dx#k p(x)m(x!x) dx, ? ? (2.1)
p(x, t)/t"!k p#k B B
and for monomer, m(x, t)/t"k B
p(x)(x!x ) dx!k m(x) p(x) dx. K ? V (2.2)
We will call these two equations the fundamental continuous model. The fundamental continuous model could be considered a semicontinuous model because of the way the monomer is treated. The delta functions provide the discontinuities needed for a realistic description of monomer addition or scission. The monomer is distinguishable from polymer by having exact molecular mass x . K Its molar distribution is, therefore, a Dirac delta distribution, m(x, t)"m(t)(x!x ), (2.3) K where m(x, t) is the monodisperse distribution and m(t) is the time-dependent molar concentration of the monomer. Eqs. (2.1) and (2.2) are an accurate representation of the continuous theory for polymerization}depolymerization. It is instructive to recognize that the population balance equation for the continuous MWD is simply related to the population balance equation for the discrete MWD. The integrals over the Dirac deltas exercise their selection property, f(x)(x!x ) dx"f(x ), because the integration intervals contain the delta position at x . From Eq. (1.1) we see that p(x)/t"!k p(x)#k p(x#x )!k p(x)m B B K ? #k p(x!x )m. (2.4) ? K De"ning x"jx , so that p "p(x!x ), p " K H\ K H> p(x#x ), gives the di!erence-di!erential equation, K dp /dt"!k (p !p )!k m(p !p ), (2.5) H B H H> ? H H\ which describes the discrete approach to chain polymerization and depolymerization. The special case of
B.J. McCoy, G. Madras / Chemical Engineering Science 56 (2001) 2831}2836
Eq. (2.5) for chain polymerization with its Taylor expansion has been discussed previously (Dotson et al., 1996; Ray, 1972; Kostoglou, 2000; Kodera & McCoy, 1997; Falkovitz & Segel, 1982). The discrete model thus can be considered a special case of the more general fundamental continuous model. As the derivation involves no approximations, we see that the fundamental continuous model is entirely consistent with and in fact contains information identical to the discrete model. Now, if p(x!x ) and p(x#x ) are expanded in their K K Taylor series, one "nds p(x)/t"k x [p/x#(x /2)p/x#2] B K K !k x m[p/x!(x /2)p/x#2]. ? K K (2.6) This is the (approximate) equation that has led to confusion about continuous models. The Taylor-expanded di!erential equation and its solutions have been used (Dotson et al., 1996) to argue that such an approximate continuous model is invalid for polymerization. Kostoglou (2000), on the other hand, has claimed that Eq. (2.6) is the only correct continuous model for depolymerization, and that the more fundamental model (Eqs. (2.1) and (2.2)) is inappropriate.
3. Solutions for discrete MWD With time-dependent monomer concentration, m(t), Eq. (2.5) is a general di!erence}di!erential equation for polymerization}depolymerization, which unfortunately has no known solution. Thus, to determine exact solutions for this discrete model, we consider polymerization and depolymerization separately. For polymerization (Dotson et al., 1996), we set k "0, and substitute B d"k m(t) dt for time. Di!erence}di!erential equations ? can be solved by Laplace transformation (to eliminate the time derivative, d) followed by substitution of p "Aj@. Allowing the underline to denote the Laplace H transformed MWD, one obtains p "p (0)/(s#1)H> for j*1, (3.1) H when the constant A has been determined from the boundary condition, dp /d"!p at j"0, and from H H the initial condition, p (0). Inverting the Laplace trans form gives p ()"p (0)He\F/j! for j*1. H
(3.2)
This is a Poisson distribution, which has MW moments p"p (0), p/x "p (0), p/x "p (0)(#). The K K dimensionless average and variance therefore are p ()/x "p ()/x ", K K
(3.3)
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which are usually determined by a z-transform or generating function method (Dotson et al., 1996), based on the assumption of an in"nite number of polymers of all MWs. The equality of the average and variance is characteristic of the Poisson distribution. It is signi"cant to our discussion to note that the Poisson distribution does not have the form of a similarity solution, which would combine the MW x"jx and time t into one K similarity variable. Madras, Smith, and McCoy (1996a,b), following Jellinek (1955), derived the discrete distribution results for depolymerization. Setting k "0 in Eq. (2.5) and ? applying Laplace transformation with respect to k t B yields a di!erence equation that can be solved as above to obtain p "p (0)(s#1)H/(s#1),>, (3.4) , H where the initial condition is a monodisperse distribution, p (t"0)"p (0). Solving the discrete case for H , a more general initial distribution is more di$cult and not usually done. Transforming back into the time domain gives the discrete MWD, p (t)"p (0)(k t),\He\IB R/(N!j)! for 2)j)N. (3.5) H , B This exact solution does not have the similarity form. Moment expressions can be obtained by a generating function method that apparently has not previously been published. De"ne , , g()" p (t)H"p (0)e\IB R (k t),\HH/(N!j)! B H , H H "p (0)e\IB R (k t)G,\G/i!"p (0),e\IB R\ N, , B , G (3.6) where in the last steps we have substituted i"N!j, considered NPR, and used the series expansion for eIB R N. The moments can be computed in the limit as P1, p"lim g()"p (0), , p/x "lim dg/d"p (0)[N!k t], K , B p/x "lim dg/d"p (0)[k t#(N!k t)]. K , B B It follows that p /x "N!k t p ()/x "k t. K B K B
(3.7) (3.8) (3.9)
(3.10)
4. Solutions for continuous MWD We next prove the moments found for polymerization and for depolymerization by the discrete model are the same as found from the fundamental continuous approach. When the moment operation is applied to
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Eq. (2.1) (see the appendix), one obtains in terms of the binomial coe$cient (L), H dpL/dt"!(k #k m)pL B ? L # (L)xL\HpH[(!1)L\Hk #k m]. (4.1) H K B ? H For n"0,1, and 2, we have
ation, the fundamental continuous model has no analytic solution for the MWD, p(x, t). Exact expressions for moments can be determined, however, and these can be used to construct the MWD. The distribution, commonly used to describe MWDs when MW moments are available, is written in terms of y"(x!x )/, K p(x, t)"[p/( )]y?\ exp(!y). (4.10)
dp/dt"0,
For the monomer addition and scission processes we are considering, the smallest MW in the distribution is x . K The moments (Abramowitz & Stegun, 1968) of the distribution are
(4.2)
dp/dt"!x (k !k m)p, (4.3) K B ? dp/dt"x (k #k m)p!2x p(k !k m). K B ? K B ? (4.4) Solving the moment equations with the polydisperse initial condition, p(x, t"0)"p (x), yields p(t)"p, (4.5) p (t)"p #x (m!m)[1!exp(!k pt)]/p, K ? (4.6) p (t)"p #x 2mk t#(m!m) K ? [1!exp(!k pt)]/p , (4.7) ? where m is the initial molar concentration and m"k /k is the equilibrium (tPR) value of the mo B ? lar concentration of the monomer. It is important to note that the fundamental continuous model can be solved for the reversible case, whereas the discrete model apparently can be solved only for the separate irreversible polymerization and depolymerization cases. Moreover, unlike the discrete model, the continuous model can be solved for a polydisperse initial MWD. It is not di$cult to show that for irreversible polymerization (m"0), and for irreversible depolymerization (m<1) the expressions for average and variance are identical to the discrete case, Eqs. (3.3) and (3.10). Thus, the discrete and fundamental continuous models are identical forms of the same population balance equation, and give the same results, suggesting that comments that criticize the fundamental continuous model for polymer reactions are misguided. The mass balance for monomer is the same for discrete and continuous models,
p (t)" #x
and p (t)" . (4.11) K We can compare the distribution to the Poisson distribution by matching moments. First, let x"jx , which is K already de"ned for Eq. (3.2). Next, we note that dx/ substitutes for j"1 in the equated de"nitions of the MWDs, p (t) j"p(x, t) dx. For polymerization the paraH meter is computed from the ratio, " / " p /(p !x )"x /(!1). The "nal step is to use the K K expression for p to solve for "x (!1)/. The disK tribution would be a similarity solution if were constant, but clearly it is not. It should also be emphasized that the general zero moment, p, appears in Eq. (1.3), whereas the monodisperse initial condition, p, appears in the Poisson distribution, thus placing a constraint on the discrete model. The two distributions are compared in Fig. 1, where p(x/x ) is plotted for several K values of degree of polymerization, . For all but the smallest , where the range of MW in the distribution is not large relative to x , the distributions are in excelK lent agreement. For depolymerization the distribution parameters are "x /(N! !1) and " K B B x (N! !1)/, where "k t. Because is not K B B B
dm/dt"(k !k m)p. (4.8) B ? Solving with the initial condition, m(t"0)"m, yields [m(t)!m]/[m!m]"exp(!k pt), (4.9) ? which can be shown to have the following special forms for depolymerization and for polymerization: (i) for k " ? 0, m(t)"m#k pt; and (ii) for k "0, m(t)" B B m exp(!k pt), which leads to "[m!m(t)]/ ? p, also known as the degree of polymerization. Whereas the discrete model has analytic solutions, p (t), for special cases of polymerization or depolymerizH
Fig. 1. Time evolution of molecular-weight distributions, p(x/x ) for K chain polymerization (monomer addition). The solid line is the exact solution of the discrete model, the dashed line is the -distribution approximation based on moments from the continuous model. Values of degree of polymerization (dimensionless time, ) starting with the MWD nearest the left axis are "2, 10, 20, 30, 40, 50, 60, 70, and 85.
B.J. McCoy, G. Madras / Chemical Engineering Science 56 (2001) 2831}2836
Fig. 2. Time evolution of molecular-weight distributions, p(x/x ) for K chain end depolymerization (monomer loss). The solid line is the exact solution of the discrete model, the dashed line is the -distribution approximation based on moments from the continuous model. The starting average MW is Nx . Values of dimensionless time starting with K the MWD nearest the right axis are k t"2, 10, 20, 30, 40, 50, 60, 70, B and 85.
constant, this -distribution approximation is not a similarity solution. Fig. 2 shows p(x/x ) for N"100 and K several values of . The agreement is excellent for small B time but deteriorates as the average MW decreases. Thus, as the average number of repeat units in the remaining polymer becomes small (less than about ten), the -distribution representation is less accurate. The results for polymerization and depolymerization underscore the importance of having a large number of repeat units in the polymer for the continuous model to agree with the exact discrete theory. The fundamental continuous model deviates from the exact discrete model only when we have to reconstruct the MWD. The moments are, however, identical.
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ing a Riemann integral as a summation. The fundamental continuous approach has been favored in analytical work because integrals are simpler to manipulate than summations, and because exact moment solutions can be obtained, usually under much more general conditions than for the solutions of the discrete model. We have demonstrated the di$culties and restrictions of the discrete solution, and the relative ease in mathematical manipulations for the fundamental continuous model. Only discrete models for simple irreversible polymerization or depolymerization can be solved for monodisperse initial distributions, whereas the reversible continuous model can readily be solved for moments when the initial condition is polydisperse. Adding higher order derivatives to the Taylor expansion of the discrete di!erence}di!erential equations gives a partial di!erential equation. Solving this approximate partial di!erential equation is not advisable because the fundamental continuous model has a more secure foundation and is easier to solve for moments. We have presented exact solutions for discrete polymerization and depolymerization models, some of which are well known, and have demonstrated that they are not similarity solutions. Any reasonable solution, of course, must be consistent with these exact solutions, and therefore will not be a similarity solution. One concludes that, unlike the chain end processes under investigation here, only models (Madras & McCoy, 1998; McCoy & Madras, 1998) involving random chain combination (Dotson et al., 1996) or random chain scission (Yoon, Chin, Kim, & Kim, 1996; Madras & McCoy, 1997) or mid-point chain scission (Shah & Ramakrishna, 1973) have similarity solutions. These conclusions are quite general in that they apply also to particulate systems (Peterson, 1986), where random fragmentation or aggregation, and single-unit (monomer) loss or gain are recognized as distinguishable processes. The former have similarity solutions, the latter do not.
5. Concluding remarks We have shown that the theory of reactions involving polymers can be described by either discrete or continuous distribution kinetic models. If the distribution is continuous in MW, an integro}di!erential equation is obtained. If the distribution of the molecular weight (MW) is discrete, a di!erence}di!erential equation is obtained. Converting from the discrete to the continuous model, however, requires an in"nite number of polymers of unlimited MW, an obvious condition for replacing the discrete distribution with a continuous distribution. The two approaches are consistent with each other, and in fact, we have shown that the discrete model is a special case of the fundamental continuous model. The moment expressions obtained by the two formulations are identical, assuming that the number of polymer molecules in a distribution is in"nite, the same requirement for de"n-
Notation g() j k B k ? m m m m(x, t) p p H
generating function, de"ned by Eq. (3.6) number of monomer units in the polymer chain-end scission rate coe$cient chain polymerization rate coe$cient time-dependent molar concentration of the monomer initial molar concentration of the monomer equilibrium molar concentration of the monomer monodisperse distribution of monomer polydisperse initial condition of the polymer, p(x, t"0) p(x)
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p G pL(t) p p p or p(x, t) x x K
Laplace transformed MWD nth moment of the polymer MWD molar concentration of the polymer initial molar concentration of the polymer MWD of the polymer molecular weight of the polymer, x"jx K molecular weight of monomer
Greek letters , Gamma distribution parameters time derivative, d"k m dt ? Acknowledgements This work was supported in part by NSF grant CTS9810194.
Appendix As several steps are required to derive the moments for the fundamental continuous model, we provide calculation details for one term. When the moment operation, de"ned by Eq. (1.2), is applied to Eq. (2.1), we have k B
dx xL
V
dxp(x!x)m(x) dx
dx m(x)
dx xLp(x!x), (A.1) VY where on the right-hand side we have interchanged the order of integration (Gradshteyn & Ryzhik, 1965). Let y"x!x and eliminate x, "k B
dx m(x) dy(y#x)Lp(y) L "k (L) dxm(x!x )xHpL\H, (A.2) K B H H where on the right-hand side we have used the binomial expansion for (y#x)L and the de"nition of the moment pL\H. Finally, the integration over x yields the result incorporated into Eq. (4.1).
k B
References Abramowitz, M., & Stegun, I. A. (1968). Handbook of mathematical functions, National Bureau of Standards, (Chapter 26).
Dotson, N. A., Galvan, R., Laurence, R. L., & Tirrell, M. (1996). Polymerization process modeling. New York: VCH Publishers Inc. Falkovitz, M. S., & Segel, L. A. (1982). Some analytic results concerning the accuracy of the continuous approximation in a polymerization problem. SIAM Journal of Applied Mathematics, 42, 542. Friedlander, S. K., & Wang, C. S. (1966). The self-preserving particle size distribution for coagulation by Brownian motion. Journal of Colloid and Interface Science, 22, 126. Goren, S. L. (1968). Distribution of lengths in the breakage of "bres of linear polymers. Canadian Journal of Chemical Engineering, 46, 105. Gradshteyn, I. S., & Ryzhik, I. M. (1965). Table of integrals series & products. New York: Academic Press (p. 615). Jellinek, H. H. G. (1955). Degradation of vinyl polymers. New York: Academic Press. Kodera, Y., & McCoy, B. J. (1997). Continuous-distribution kinetics of radical mechanisms for polymer decomposition & repolymerization. A.I.Ch.E. Journal, 43, 3205. Kostoglou, M. (2000). Mathematical analysis of polymer degradation with chain-end scission. Chemical Engineering Science, 55, 2507. Madras, G., & McCoy, B. J. (1997). Oxidative degradation kinetics of polystyrene in solution. Chemical Engineering Science, 52, 2707. Madras, G., & McCoy, B. J. (1998). Time evolution to similarity solutions for polymer degradation. A.I.Ch.E. Journal, 44, 647. Madras, G., Smith, J. M., & McCoy, B. J. (1996a). Thermal degradation of poly( -methylstyrene) in solution. Polymer Degradation and Stability, 52, 349. Madras, G., Smith, J. M., & McCoy, B. J. (1996b). Degradation of poly(methyl methacrylate) in solution. Industrial and Engineering Chemistry Research, 35, 1795. McCoy, B. J. (1999). Distribution kinetics for temperature-programmed pyrolysis. Industrial and Engineering Chemistry Research, 38, 4537. McCoy, B. J., & Madras, G. (1998). Evolution to similarity solutions for fragmentation and aggregation. Journal of Colloid and Interface Science, 201, 200. Peterson, T. W. (1986). Similarity solutions for the population balance equation describing particle fragmentation. Aerosol Science and Technology, 5, 93. Ray, W. H. (1972). On the mathematical modeling of polymerization reactors. Journal of Macromolecular Science * Reviews in Macromolecular Chemistry, C8, 1. Sezgi, N. A., Cha, W. S., Smith, J. M., & McCoy, B. J. (1999). Polyethylene pyrolysis: Theory & experiments for molecular-weight-distribution kinetics. Industrial and Engineering Chemistry Research, 37, 2582. Shah, B. H., & Ramakrishna, D. (1973). A population balance model for mass transfer in lean liquid}liquid dispersions. Chemical Engineering Science, 28, 389. Wang, M., Smith, J. M., & McCoy, B. J. (1995). Continuous kinetics for thermal degradation of polymer in solution. A.I.Ch.E. Journal, 41, 1521. Yoon, J. S., Chin, I. J., Kim, M. N., & Kim, C. (1996). Degradation of microbial polyesters * A theoretical prediction of molecular weight and polydispersity. Macromolecules, 29, 3303. Zi!, R. M. (1991). New solutions to the fragmentation equation. Journal of Physica A: Mathematical General, 24, 2821. Zi!, R. M., & McGrady, E. D. (1986). Kinetics of polymer degradation. Macromolecules, 19, 2513.
Discrete and continuous models for polymerization and depolymerization Benjamin J. McCoy , Giridhar Madras * Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, USA Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India Received 16 June 2000; received in revised form 24 September 2000; accepted 17 October 2000
Abstract A fundamental continuous model for the distribution kinetics of polymerization and its reverse, depolymerization, is shown to be consistent with the well-known discrete model. Solutions for discrete and continuous models are compared and shown to have identical expressions for molecular weight (MW) moments. Representing the continuous molecular-weight distribution (MWD) as a distribution compares well with exact polymerization and depolymerization MWD solutions of the discrete model, except when the number of repeat units in the polymer is small. None of the solutions for chain end polymerization or depolymerization has the self-similar form, for which the MW and time would appear together in the same similarity variable. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Discrete kinetics; Continuous distribution kinetics; Chain end scission; Chain end addition; Similarity solutions; Gamma distributions
1. Introduction Describing how molecular weight distributions evolve during polymerization or depolymerization reactions is an important fundamental issue of polymer science and engineering. The importance of polymerization is well understood, but depolymerization is also signi"cant for plastics recycling and polymer characterization. Moreover, depolymerization is the reverse of polymerization, and a!ects the steady-state or equilibrium limitations on polymerization. Conversely, depolymerization processes can involve polymerization or repolymerization reactions that in#uence their reaction course, such as in pyrolysis (McCoy, 1999; Sezgi, Cha, Smith, & McCoy, 1999). Several publications have commented on the advantages and disadvantages of solving the equations for polymerization and depolymerization as discrete or continuous models (Dotson, Galvan, Laurence, & Tirrell, 1996; Ray, 1972; Kostoglou, 2000). In this work we focus on monomer addition or dissociation of monomers at the end of polymer chains. Because there is no doubt that monomers add or split o! individually, one should * Corresponding author. Tel.: #91-080-309-2381; fax: #91-080360-0683. E-mail address: [email protected] (G. Madras).
consider that the discrete approach is accurate, serving as a standard for judging other approaches. The continuous model, however, is straightforward to manipulate mathematically, whereas only special irreversible cases of the discrete reversible model can be solved. The continuous model would be preferred for this reason, if its limitations were completely understood. An obvious condition for the discrete distribution to be approximated by a continuous distribution is that the number of repeat units (monomers) in the polymers should be very large. The usual derivation of polymerization moments for a discrete distribution depends on assuming an in"nite number of polymers in the MWD, as we will show. In spite of numerous applications of continuous models, however, even when the results are identical to the special discrete cases that can be solved, some critics have challenged their application to polymer reactions models (Dotson et al., 1996; Kostoglou, 2000). Frequently the arguments are based on approximations to what we call below the fundamental continuous theory. Our primary objective is to examine in detail the issues involved in this controversy. A second objective is to examine if chain polymerization or chain end depolymerization have similarity (scaled, self-similar, or self-preserving) solutions, which combine with MW x and time t into one similarity variable. As discussed in numerous works (Madras &
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 1 6 - 9
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McCoy, 1998; McCoy & Madras, 1998; Zi! & McGrady, 1986; Zi!, 1991; Friedlander & Wang, 1966; Goren, 1968), the usual form of a similarity solution is when x appears in the expression for the MWD combined with time as the similarity group, xH/(t) (McCoy & Madras, 1998; Zi! & McGrady, 1986). Earlier, Madras and McCoy (1998) presented solutions for chain combination and degradation (random, parabolic, and midchain scission), demonstrating when and how these solutions evolved to similarity solutions. Even though chain-end addition of monomer (chain polymerization) was not discussed, we showed that chain-end scission, yielding monomers or oligomers, did not conform to the similarity solution form. This conclusion has also been questioned (Kostoglou, 2000). Chain polymerization is denoted by the addition of monomer, M(x ), of MW x to the polymer, P(x), of K K MW x by the second-order rate coe$cient k . The de? polymerization (reverse) step is chain-end scission with the "rst-order rate coe$cient k . Thus, the reversible B reaction (Kodera & McCoy, 1997) is IX M(x )#P(x) & P(x#x ). (1.1) K K IB The rate coe$cient for scission can be considered a composite coe$cient for a radical mechanism (Kodera & McCoy, 1997). The continuous molecular-weight distribution (MWD) is de"ned such that p(x, t) dx is the molar concentration of polymer at time t in the MW range (x, x#dx). Moments of the MW are de"ned as
p(x, t)xL dx, (1.2) and provide basic properties of the MWD, including total molar concentration, p, average MW, p " p/p, and variance, p "p/p!(p ). For many polymer reactions, only these moments are needed to interpret typical behavior. For a discrete model with j"0,1,2,2, N monomers in the polymers, the MW for an individual polymer is x"jx . The continuous MWD K is then made up of discrete polymers with MWD p (t), H , p(x, t)" p (t)(x!jx ). (1.3) H K H It follows from Eq. (1.2) that for the discrete distribution, the moments are given by pL(t)"
, pL(t)" p (t)(jx )L, (1.4) H K H which is the standard expression (Dotson et al., 1996). 2. Population balance equations The population balance for a batch reactor is written so that accumulation of a polymer of mass x is represented by the time derivative. Gain and loss terms in the
population balance are represented by the second-order forward reaction and the "rst-order reverse reaction. For the continuous distributions, integration over all reaction partners is required. Recent theoretical and experimental research on polymer degradation (Kodera & McCoy, 1997; Wang, Smith, & McCoy, 1995; Madras, Smith, & McCoy, 1996a,b) have analyzed and discussed the chainend scission kernels, (x!x ) and (x!(x!x )), for K K the monomer loss process. The population balance equation for polymer is
p(x)[x!(x!x )] dx K V V !k p m(x) dx#k p(x)m(x!x) dx, ? ? (2.1)
p(x, t)/t"!k p#k B B
and for monomer, m(x, t)/t"k B
p(x)(x!x ) dx!k m(x) p(x) dx. K ? V (2.2)
We will call these two equations the fundamental continuous model. The fundamental continuous model could be considered a semicontinuous model because of the way the monomer is treated. The delta functions provide the discontinuities needed for a realistic description of monomer addition or scission. The monomer is distinguishable from polymer by having exact molecular mass x . K Its molar distribution is, therefore, a Dirac delta distribution, m(x, t)"m(t)(x!x ), (2.3) K where m(x, t) is the monodisperse distribution and m(t) is the time-dependent molar concentration of the monomer. Eqs. (2.1) and (2.2) are an accurate representation of the continuous theory for polymerization}depolymerization. It is instructive to recognize that the population balance equation for the continuous MWD is simply related to the population balance equation for the discrete MWD. The integrals over the Dirac deltas exercise their selection property, f(x)(x!x ) dx"f(x ), because the integration intervals contain the delta position at x . From Eq. (1.1) we see that p(x)/t"!k p(x)#k p(x#x )!k p(x)m B B K ? #k p(x!x )m. (2.4) ? K De"ning x"jx , so that p "p(x!x ), p " K H\ K H> p(x#x ), gives the di!erence-di!erential equation, K dp /dt"!k (p !p )!k m(p !p ), (2.5) H B H H> ? H H\ which describes the discrete approach to chain polymerization and depolymerization. The special case of
B.J. McCoy, G. Madras / Chemical Engineering Science 56 (2001) 2831}2836
Eq. (2.5) for chain polymerization with its Taylor expansion has been discussed previously (Dotson et al., 1996; Ray, 1972; Kostoglou, 2000; Kodera & McCoy, 1997; Falkovitz & Segel, 1982). The discrete model thus can be considered a special case of the more general fundamental continuous model. As the derivation involves no approximations, we see that the fundamental continuous model is entirely consistent with and in fact contains information identical to the discrete model. Now, if p(x!x ) and p(x#x ) are expanded in their K K Taylor series, one "nds p(x)/t"k x [p/x#(x /2)p/x#2] B K K !k x m[p/x!(x /2)p/x#2]. ? K K (2.6) This is the (approximate) equation that has led to confusion about continuous models. The Taylor-expanded di!erential equation and its solutions have been used (Dotson et al., 1996) to argue that such an approximate continuous model is invalid for polymerization. Kostoglou (2000), on the other hand, has claimed that Eq. (2.6) is the only correct continuous model for depolymerization, and that the more fundamental model (Eqs. (2.1) and (2.2)) is inappropriate.
3. Solutions for discrete MWD With time-dependent monomer concentration, m(t), Eq. (2.5) is a general di!erence}di!erential equation for polymerization}depolymerization, which unfortunately has no known solution. Thus, to determine exact solutions for this discrete model, we consider polymerization and depolymerization separately. For polymerization (Dotson et al., 1996), we set k "0, and substitute B d"k m(t) dt for time. Di!erence}di!erential equations ? can be solved by Laplace transformation (to eliminate the time derivative, d) followed by substitution of p "Aj@. Allowing the underline to denote the Laplace H transformed MWD, one obtains p "p (0)/(s#1)H> for j*1, (3.1) H when the constant A has been determined from the boundary condition, dp /d"!p at j"0, and from H H the initial condition, p (0). Inverting the Laplace trans form gives p ()"p (0)He\F/j! for j*1. H
(3.2)
This is a Poisson distribution, which has MW moments p"p (0), p/x "p (0), p/x "p (0)(#). The K K dimensionless average and variance therefore are p ()/x "p ()/x ", K K
(3.3)
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which are usually determined by a z-transform or generating function method (Dotson et al., 1996), based on the assumption of an in"nite number of polymers of all MWs. The equality of the average and variance is characteristic of the Poisson distribution. It is signi"cant to our discussion to note that the Poisson distribution does not have the form of a similarity solution, which would combine the MW x"jx and time t into one K similarity variable. Madras, Smith, and McCoy (1996a,b), following Jellinek (1955), derived the discrete distribution results for depolymerization. Setting k "0 in Eq. (2.5) and ? applying Laplace transformation with respect to k t B yields a di!erence equation that can be solved as above to obtain p "p (0)(s#1)H/(s#1),>, (3.4) , H where the initial condition is a monodisperse distribution, p (t"0)"p (0). Solving the discrete case for H , a more general initial distribution is more di$cult and not usually done. Transforming back into the time domain gives the discrete MWD, p (t)"p (0)(k t),\He\IB R/(N!j)! for 2)j)N. (3.5) H , B This exact solution does not have the similarity form. Moment expressions can be obtained by a generating function method that apparently has not previously been published. De"ne , , g()" p (t)H"p (0)e\IB R (k t),\HH/(N!j)! B H , H H "p (0)e\IB R (k t)G,\G/i!"p (0),e\IB R\ N, , B , G (3.6) where in the last steps we have substituted i"N!j, considered NPR, and used the series expansion for eIB R N. The moments can be computed in the limit as P1, p"lim g()"p (0), , p/x "lim dg/d"p (0)[N!k t], K , B p/x "lim dg/d"p (0)[k t#(N!k t)]. K , B B It follows that p /x "N!k t p ()/x "k t. K B K B
(3.7) (3.8) (3.9)
(3.10)
4. Solutions for continuous MWD We next prove the moments found for polymerization and for depolymerization by the discrete model are the same as found from the fundamental continuous approach. When the moment operation is applied to
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Eq. (2.1) (see the appendix), one obtains in terms of the binomial coe$cient (L), H dpL/dt"!(k #k m)pL B ? L # (L)xL\HpH[(!1)L\Hk #k m]. (4.1) H K B ? H For n"0,1, and 2, we have
ation, the fundamental continuous model has no analytic solution for the MWD, p(x, t). Exact expressions for moments can be determined, however, and these can be used to construct the MWD. The distribution, commonly used to describe MWDs when MW moments are available, is written in terms of y"(x!x )/, K p(x, t)"[p/( )]y?\ exp(!y). (4.10)
dp/dt"0,
For the monomer addition and scission processes we are considering, the smallest MW in the distribution is x . K The moments (Abramowitz & Stegun, 1968) of the distribution are
(4.2)
dp/dt"!x (k !k m)p, (4.3) K B ? dp/dt"x (k #k m)p!2x p(k !k m). K B ? K B ? (4.4) Solving the moment equations with the polydisperse initial condition, p(x, t"0)"p (x), yields p(t)"p, (4.5) p (t)"p #x (m!m)[1!exp(!k pt)]/p, K ? (4.6) p (t)"p #x 2mk t#(m!m) K ? [1!exp(!k pt)]/p , (4.7) ? where m is the initial molar concentration and m"k /k is the equilibrium (tPR) value of the mo B ? lar concentration of the monomer. It is important to note that the fundamental continuous model can be solved for the reversible case, whereas the discrete model apparently can be solved only for the separate irreversible polymerization and depolymerization cases. Moreover, unlike the discrete model, the continuous model can be solved for a polydisperse initial MWD. It is not di$cult to show that for irreversible polymerization (m"0), and for irreversible depolymerization (m<1) the expressions for average and variance are identical to the discrete case, Eqs. (3.3) and (3.10). Thus, the discrete and fundamental continuous models are identical forms of the same population balance equation, and give the same results, suggesting that comments that criticize the fundamental continuous model for polymer reactions are misguided. The mass balance for monomer is the same for discrete and continuous models,
p (t)" #x
and p (t)" . (4.11) K We can compare the distribution to the Poisson distribution by matching moments. First, let x"jx , which is K already de"ned for Eq. (3.2). Next, we note that dx/ substitutes for j"1 in the equated de"nitions of the MWDs, p (t) j"p(x, t) dx. For polymerization the paraH meter is computed from the ratio, " / " p /(p !x )"x /(!1). The "nal step is to use the K K expression for p to solve for "x (!1)/. The disK tribution would be a similarity solution if were constant, but clearly it is not. It should also be emphasized that the general zero moment, p, appears in Eq. (1.3), whereas the monodisperse initial condition, p, appears in the Poisson distribution, thus placing a constraint on the discrete model. The two distributions are compared in Fig. 1, where p(x/x ) is plotted for several K values of degree of polymerization, . For all but the smallest , where the range of MW in the distribution is not large relative to x , the distributions are in excelK lent agreement. For depolymerization the distribution parameters are "x /(N! !1) and " K B B x (N! !1)/, where "k t. Because is not K B B B
dm/dt"(k !k m)p. (4.8) B ? Solving with the initial condition, m(t"0)"m, yields [m(t)!m]/[m!m]"exp(!k pt), (4.9) ? which can be shown to have the following special forms for depolymerization and for polymerization: (i) for k " ? 0, m(t)"m#k pt; and (ii) for k "0, m(t)" B B m exp(!k pt), which leads to "[m!m(t)]/ ? p, also known as the degree of polymerization. Whereas the discrete model has analytic solutions, p (t), for special cases of polymerization or depolymerizH
Fig. 1. Time evolution of molecular-weight distributions, p(x/x ) for K chain polymerization (monomer addition). The solid line is the exact solution of the discrete model, the dashed line is the -distribution approximation based on moments from the continuous model. Values of degree of polymerization (dimensionless time, ) starting with the MWD nearest the left axis are "2, 10, 20, 30, 40, 50, 60, 70, and 85.
B.J. McCoy, G. Madras / Chemical Engineering Science 56 (2001) 2831}2836
Fig. 2. Time evolution of molecular-weight distributions, p(x/x ) for K chain end depolymerization (monomer loss). The solid line is the exact solution of the discrete model, the dashed line is the -distribution approximation based on moments from the continuous model. The starting average MW is Nx . Values of dimensionless time starting with K the MWD nearest the right axis are k t"2, 10, 20, 30, 40, 50, 60, 70, B and 85.
constant, this -distribution approximation is not a similarity solution. Fig. 2 shows p(x/x ) for N"100 and K several values of . The agreement is excellent for small B time but deteriorates as the average MW decreases. Thus, as the average number of repeat units in the remaining polymer becomes small (less than about ten), the -distribution representation is less accurate. The results for polymerization and depolymerization underscore the importance of having a large number of repeat units in the polymer for the continuous model to agree with the exact discrete theory. The fundamental continuous model deviates from the exact discrete model only when we have to reconstruct the MWD. The moments are, however, identical.
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ing a Riemann integral as a summation. The fundamental continuous approach has been favored in analytical work because integrals are simpler to manipulate than summations, and because exact moment solutions can be obtained, usually under much more general conditions than for the solutions of the discrete model. We have demonstrated the di$culties and restrictions of the discrete solution, and the relative ease in mathematical manipulations for the fundamental continuous model. Only discrete models for simple irreversible polymerization or depolymerization can be solved for monodisperse initial distributions, whereas the reversible continuous model can readily be solved for moments when the initial condition is polydisperse. Adding higher order derivatives to the Taylor expansion of the discrete di!erence}di!erential equations gives a partial di!erential equation. Solving this approximate partial di!erential equation is not advisable because the fundamental continuous model has a more secure foundation and is easier to solve for moments. We have presented exact solutions for discrete polymerization and depolymerization models, some of which are well known, and have demonstrated that they are not similarity solutions. Any reasonable solution, of course, must be consistent with these exact solutions, and therefore will not be a similarity solution. One concludes that, unlike the chain end processes under investigation here, only models (Madras & McCoy, 1998; McCoy & Madras, 1998) involving random chain combination (Dotson et al., 1996) or random chain scission (Yoon, Chin, Kim, & Kim, 1996; Madras & McCoy, 1997) or mid-point chain scission (Shah & Ramakrishna, 1973) have similarity solutions. These conclusions are quite general in that they apply also to particulate systems (Peterson, 1986), where random fragmentation or aggregation, and single-unit (monomer) loss or gain are recognized as distinguishable processes. The former have similarity solutions, the latter do not.
5. Concluding remarks We have shown that the theory of reactions involving polymers can be described by either discrete or continuous distribution kinetic models. If the distribution is continuous in MW, an integro}di!erential equation is obtained. If the distribution of the molecular weight (MW) is discrete, a di!erence}di!erential equation is obtained. Converting from the discrete to the continuous model, however, requires an in"nite number of polymers of unlimited MW, an obvious condition for replacing the discrete distribution with a continuous distribution. The two approaches are consistent with each other, and in fact, we have shown that the discrete model is a special case of the fundamental continuous model. The moment expressions obtained by the two formulations are identical, assuming that the number of polymer molecules in a distribution is in"nite, the same requirement for de"n-
Notation g() j k B k ? m m m m(x, t) p p H
generating function, de"ned by Eq. (3.6) number of monomer units in the polymer chain-end scission rate coe$cient chain polymerization rate coe$cient time-dependent molar concentration of the monomer initial molar concentration of the monomer equilibrium molar concentration of the monomer monodisperse distribution of monomer polydisperse initial condition of the polymer, p(x, t"0) p(x)
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p G pL(t) p p p or p(x, t) x x K
Laplace transformed MWD nth moment of the polymer MWD molar concentration of the polymer initial molar concentration of the polymer MWD of the polymer molecular weight of the polymer, x"jx K molecular weight of monomer
Greek letters , Gamma distribution parameters time derivative, d"k m dt ? Acknowledgements This work was supported in part by NSF grant CTS9810194.
Appendix As several steps are required to derive the moments for the fundamental continuous model, we provide calculation details for one term. When the moment operation, de"ned by Eq. (1.2), is applied to Eq. (2.1), we have k B
dx xL
V
dxp(x!x)m(x) dx
dx m(x)
dx xLp(x!x), (A.1) VY where on the right-hand side we have interchanged the order of integration (Gradshteyn & Ryzhik, 1965). Let y"x!x and eliminate x, "k B
dx m(x) dy(y#x)Lp(y) L "k (L) dxm(x!x )xHpL\H, (A.2) K B H H where on the right-hand side we have used the binomial expansion for (y#x)L and the de"nition of the moment pL\H. Finally, the integration over x yields the result incorporated into Eq. (4.1).
k B
References Abramowitz, M., & Stegun, I. A. (1968). Handbook of mathematical functions, National Bureau of Standards, (Chapter 26).
Dotson, N. A., Galvan, R., Laurence, R. L., & Tirrell, M. (1996). Polymerization process modeling. New York: VCH Publishers Inc. Falkovitz, M. S., & Segel, L. A. (1982). Some analytic results concerning the accuracy of the continuous approximation in a polymerization problem. SIAM Journal of Applied Mathematics, 42, 542. Friedlander, S. K., & Wang, C. S. (1966). The self-preserving particle size distribution for coagulation by Brownian motion. Journal of Colloid and Interface Science, 22, 126. Goren, S. L. (1968). Distribution of lengths in the breakage of "bres of linear polymers. Canadian Journal of Chemical Engineering, 46, 105. Gradshteyn, I. S., & Ryzhik, I. M. (1965). Table of integrals series & products. New York: Academic Press (p. 615). Jellinek, H. H. G. (1955). Degradation of vinyl polymers. New York: Academic Press. Kodera, Y., & McCoy, B. J. (1997). Continuous-distribution kinetics of radical mechanisms for polymer decomposition & repolymerization. A.I.Ch.E. Journal, 43, 3205. Kostoglou, M. (2000). Mathematical analysis of polymer degradation with chain-end scission. Chemical Engineering Science, 55, 2507. Madras, G., & McCoy, B. J. (1997). Oxidative degradation kinetics of polystyrene in solution. Chemical Engineering Science, 52, 2707. Madras, G., & McCoy, B. J. (1998). Time evolution to similarity solutions for polymer degradation. A.I.Ch.E. Journal, 44, 647. Madras, G., Smith, J. M., & McCoy, B. J. (1996a). Thermal degradation of poly( -methylstyrene) in solution. Polymer Degradation and Stability, 52, 349. Madras, G., Smith, J. M., & McCoy, B. J. (1996b). Degradation of poly(methyl methacrylate) in solution. Industrial and Engineering Chemistry Research, 35, 1795. McCoy, B. J. (1999). Distribution kinetics for temperature-programmed pyrolysis. Industrial and Engineering Chemistry Research, 38, 4537. McCoy, B. J., & Madras, G. (1998). Evolution to similarity solutions for fragmentation and aggregation. Journal of Colloid and Interface Science, 201, 200. Peterson, T. W. (1986). Similarity solutions for the population balance equation describing particle fragmentation. Aerosol Science and Technology, 5, 93. Ray, W. H. (1972). On the mathematical modeling of polymerization reactors. Journal of Macromolecular Science * Reviews in Macromolecular Chemistry, C8, 1. Sezgi, N. A., Cha, W. S., Smith, J. M., & McCoy, B. J. (1999). Polyethylene pyrolysis: Theory & experiments for molecular-weight-distribution kinetics. Industrial and Engineering Chemistry Research, 37, 2582. Shah, B. H., & Ramakrishna, D. (1973). A population balance model for mass transfer in lean liquid}liquid dispersions. Chemical Engineering Science, 28, 389. Wang, M., Smith, J. M., & McCoy, B. J. (1995). Continuous kinetics for thermal degradation of polymer in solution. A.I.Ch.E. Journal, 41, 1521. Yoon, J. S., Chin, I. J., Kim, M. N., & Kim, C. (1996). Degradation of microbial polyesters * A theoretical prediction of molecular weight and polydispersity. Macromolecules, 29, 3303. Zi!, R. M. (1991). New solutions to the fragmentation equation. Journal of Physica A: Mathematical General, 24, 2821. Zi!, R. M., & McGrady, E. D. (1986). Kinetics of polymer degradation. Macromolecules, 19, 2513.