Macromolecules 1986,19,2513-2519
Applying the empirical Verlet criterion that simple fluids order when the peak in the structure factor reaches approximately 3-4, we would expect crystallinity with only a few hairs. Of course, a glassy phase might intervene and mask the order but nevertheless we would predict a definite shear modulus for small shear rates.
Acknowledgment. We are very pleased to acknowledge important discussions with Dr. M. Cates. References and Notes (1) For a general reference, see: Napper, D. H. Polymeric Stabi-
lization of Colloidal Dispersions: Academic: London, 1983. (2) Scheutjens,J. M.H. M.; Fleer, G. J. Adu. Colloid Interface Sci. 1982, 16, 341. de Gennes, P.-G. Macromolecules 1981, 14, 1637. de Gennes, P.-G.; Pincus, P. J. Phys., Lett. 1983,44, L2241. (3) Scheutjens, J. M.H. M.; Fleer, G. J. Colloid Interface Sci. 1982,16,361.de Gennes, P.-G.Macromolecules 1982,15,492. Klein, J.; Pincus, P. Macromolecules 1982, 15, 1129. Israelachvili, J. N.; Tirrell, M.; Klein, J.; Almog, Y. Macromolecules 1984,17,204. Almog, Y.; Klein, J. J. Colloid Interface Sci. 1985,106,33.
2513
(4) Alexander, S.J. Phys. (Les Ulis. Fr.) 1977,38,983.de Gennes, P.-G.Macromolecules 1980,13, 1069. (5) de Gennes, P.-G. C . R . Hebd. Seances Acad. Sci. 1985, 300, 839. (6) Hadziioannou, G.; Patel, S.; Granick, S.; Tirrell, M. J. Am. Chem. SOC., accepted for publication. (7) As with most scaling arguments, the emphasis is placed on functional dependences on physical parameters. Numerical prefactors are generally ignored. (8) Boltzmann’s constant is taken to be unity; i.e., T has units of
energy.
(9) Leibler, L.; Pincus, P. A. Macromolecules 1984,17,2922. (10) Watanabe, H.;Kotaka, T. Macromolecules 1984,17,342. (11) Daoud, M.;Cotton, J. P. J. Phys. (Les Ulis, Fr.) 1982,43,531. (12) Birshtein, T. M.;Zhulina, E. B. Polymer 1984,25, 1453. (13) Huber, K.; Burchard, W.; Fetters, L. J. Macromolecules 1984, 17,541. (14) des Cloizeaux, J. J. Physique (Les Ulis, Fr.) 1980,41,223. (15) See for example: de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University: Ithaca, NY, 1979. (16) Witten, T. A.; Prentis, J. J. J. Chem. Phys. 1982,77,4247. (17) Witten, T. A.; Pincus, P. A.; Cates, M. E., to be published. (18) Here we assume for simplicity that the colloidal particle is itself invisible to neutrons; i.e., only the monomers are scattering centers.
Kinetics of Polymer Degradation Robert M. ZifP and E. D. McGrady Department of Chemical Engineering, University of Michigan, A n n Arbor, Michigan 48109. Received April 1 , 1986
ABSTRACT Explicit size distributions for polymers undergoing degradation are found for systems where the rate of bond scission depends on total chain length as well as on the position of the bond within the chain. Previously, the only model solved explicitly was the case of purely random bond scission (by Montroll and Simha). A new model is solved where chain bonds break preferentially in the center with a parabolic probability distribution, F ( x , y ) = x y . Here F ( x , y ) is the rate of fragmentation of polymers length x + y into chains of length x and y . A ternary breakup model with equal bond reactivity is also solved. The general solution is given for the case where the rate of breakup is proportional to the total chain length raised to a power a, F ( x , y ) = ( x + y ) ” . When a < -1 it is f?und that mass is not conserved because of a cascading breakup rate as the fragments get smaller. The long-time scaling behavior of the models is studied and found to differ for the different models. Discrete models corresponding to a = -2 and -3 are also solved and help to show the “shattering”transition is a spontaneous breakup of a fraction of the system into monomers.
I. Introduction Polymer chains undergo degradation (depolymerization) through a variety of mechanisms, including shear action,l2 chemical a t t a ~ kand , ~ nuclear, ultraviolet, and ultrasonic irradiation.4~~ It is of great interest to predict theoretically the evolution of the size distribution during such processes. For that purpose, two approaches have been used. One has been through the use of statistical and combinatorial arguments, as fist used by Kuhn,G Mark and Simha,’ and Montroll and Simha6to solve the equireactivity model, in which polymer bonds break randomly and independently. The second approach has been through a kinetic equation for depolymerization. Such an equation was first introduced by Blatz and TobolskyB(in combination with polymerization) and solved for the case of size-independent polymerization and fragmentation rates. The kinetic equation of fragmentation was also studied by Jellinek and Whitelo and Saito.” Others who have considered the problem of depolymerization kinetics include Charlesby,12 Nanda and Pathria,13Simha and Wall,14and Simha, Wall, and Blatz.15 Demjanenko and DuEek16 have recently considered the problem of random degradation in conjunction with cross-linking. In all these works, explicit solutions were found for only the case where the breakup 0024-9297/86/2219-2513$01.50/0
rate is a constant, or all bonds break with equal probability, which is essentially the model first solved by Montroll and Simha 46 years ago. The equireactivity assumption is analogous to Flory’s model of chain polymerization where all bonds are equally probable.“ In many polymer systems, however, the breaking of bonds does not occur randomly but depends upon the position of the bond within the chain and/or the total chain length. Experimental studies on systems undergoing degradation through shear,18elongation,lg or irradiation4 have found that the bonds in the center of the chains break preferentially to those a t the ends. For many systems, it has been proposed that the breakage rate along the chains is a truncated Gaussian d i s t r i b ~ t i o n . ~One ~~~ would J ~ also like to be able to understand the depolymerization kinetics of these systems. Since the equations are linear, the discrete form of the fragmentation equation can be solved in principle for any breakup function.10p20Yet there have been no cases other than equireactivity where an explicit solution has been found, and other solutions have been obtained only numerically with computers. We note that a form of unequal scission has been considered in detail by AmemiyaZ2in connection with cross-linking kinetics. In his models, inhomogeneity was introduced by having 0 1986 American Chemical Society
Macromolecules, Vol. 19, No. 10, 1986
2514 Ziff and McGrady
the bonds of different breaking probability dispersed throughout the chain, which is different from the models considered here. In this paper we use the kinetic equation to find explicit solutions to both continuous and discrete models of fragmentation for two classes of models of unequal scission probability. The general kinetic equations that describes depolymerization is given by -an-(x,t ) at
in continuous form, where n(x,t)is the number of polymers of continuous molecular weight (or length) x, and by
in discrete form, where nk(t)is the time-dependent number of k-mers, k = 1, 2 , 3, ... These kinetic equations describe systems in which scission occurs without initiation, depropagation, and The matrix Fij gives the rate that (i + j)-mers break into i-mers and j-mers, and F(x,y) is the continuous version of this function. For random scission,where chains break independently of their length and bonds break independently of their position within those chains, the rate kernels Fij and F(x,y)are both constants. An overall rate factor in F can be removed by rescaling the time, and we will assume that this has been done. For the continuous case we consider two forms of the rate kernel F(x,y). First, we consider the case where the polymer chains break preferentially in the center with F ( x , y ) = xy. Here the chains break with an overall rate proportional to the cube of the length and with a parabolic distribution within a chain. Second, we consider F(x,y) = (x + y)*. Here the rate of breakup along the chain is uniform while the overall rate of chain breakup is proportional to the length of the chain to the power a. Such a reaction rate dependence on chain size has been observed in systems where elastomers are undergoing mechanical d e g r a d a t i ~ n .When ~ ~ a = 0, the equireactivity model of Montroll and Simha8 is recovered. In a recent paper,26to be referred to as I, we have given explicit solutions for rate kernels of the form F(x,y) = (x + y)* for the special cases a = 1,0, -1/3, -2/4, -3/5, ..., -1. In this paper we give the general solution of all a and find that when a < -1 the fragmentation rate cascades without limit and a finite fraction of the mass breaks into polymers of zero size. This phenomenon, which we call “shattering”, is analogous to (in an opposite sense) the gelation phenomenon that occurs during polymerization. We note that the shattering is also reminiscent of a phase transition, especially the BoseEinstein condensation of a quantum ideal gas in which a fmite fraction of the molecules condense into a state of zero moment~m.~’,~~ We also consider the discrete kinetic equation of depolymerization and discuss two models which correspond to shattering models in the continuous system. We show that the mass-nonconserving shattering seen in continuous systems corresponds to spontaneous formation of monomers in a discrete system with a minimum size cutoff. To characterize the asymptotic behavior for large t and small x of our solutions, we investigate the scaling behavior of our solutions. For large times, one expects
-
n(x,t) m w - 2 ~ ( x / m , ) (3) where m, is the time-dependent weight-average molecular
-
-
weight and where C$(F) 5‘ as [ 0. The exponent -2 in (3) ensures that mass is conserved. Furthermore, one usually expects that m, t-Z as t for some z. If the latter is true, then the scaling relation can also be written as
-
-
n(x,t)
-
tWxrcp(xtz)
(4)
for x 0 and t m with xtZ = constant, and by conservation of mass, w = z(2’+ 7). These scaling relations are written in analogy to the kinetic scaling forms of aggregation and polymerization (where the scaling limit concerns x m rather than x The scaling exponents z , 7,and w allow one to classify the asymptotic behavior of solutions into “universality classes” (models sharing common exponents). In I, we have shown that z = 1/(1 + a) and 7 = 0 for the model F(x,y) = ( x + y)*. In this paper we extend the discussion of scaling to the model F(x,y) = xy. Most of this work involves the continuous equation. Even though polymer systems are discrete rather than continuous, the continuous fragmentation equation is useful for finding explicit solutions. Its solutions will be valid when the average molecular weight of the sample is high and will break down when a substantial number of monomers have accumulated. Indeed, even if one is able to find the solution for a discrete system, one usually ends up taking the continuous limit to put the result in a more convenient form of analysis and use. The solutions presented here are derived for monodisperse initial conditions, n(x,O) = 6(x-1), where 1 is the initial chain length. These results are sufficient to find the solution for any initial distribution, since if n(’)(x,t)is the solution for n(x,O) = 6(x-1), then it follows from the linearity of (1)that the solution for an arbitrary initial condition n(x,O) is given by
-
+
O).2692*33
For example, for the model F(x,y) = x
+ y, we foundz6
+
n(”(x,t) = 8(x-l)e-tL2 2t1e-tx2
(6)
for x I1. According to (5), the solution for an arbitrary initial condition n(x,O) is given by
n(x,t) = e-‘xz[n(x,O)+ 2 t L m y n ( y , 0 )dy]
(7)
A relation similar to (5) holds for discrete solutions. 11. F ( x , y ) = xy
A simple model where bonds are more likely to break in the center than at the ends is the parabolic distribution F(x,y) = xy, where the rate of reaction is proportional to the product of the lengths of the fragments. As was shown by F ~ - e n k e lfor , ~ ~polymers in pure extensional flow, the tension is parabolically distributed along the chain length. If it is also assumed that the probability of the chain breaking at any given point along its length is proportional to the tension at that point, then the F(x,y) = x y model describes the degradation rate. If the breakup process is thermally activated, then a Gaussian rate kernel would be expected, and the xy model is an approximation to this when the variance is roughly the size of the chain. For the xy model, (1)becomes an(x,t) -at - -(x3/6)n(x,t)+ 2Jmx(y-x)n(y,t) dy ( 8 ) In Appendix A we find the solution to (8) by a series of transformations starting with the method Charlesby used
Kinetics of Polymer Degradation 2515
Macromolecules, Vol. 19,No.10,1986
solved by the techniques used in Appendix A. The kinetic equation for ternary breakup can be written as
8 F
t Lo
3 S l d y n(y,t)Sy-'dz 0 F(x,z,y-x-z) (13)
LI
I
where F(y,z,x-y-z) represents the rate a t which a chain of length x breaks up into chains of length y, z , and x y - z. The factor of 3 in the second term in (13) reflects that x can be any of the three products formed by the breakup process. For the simple case of F(y,z,x-y-z) = 1,or where the position of the breaks along the chain are equally probable, (13) reduces to
3
0.0
1.0
x/l
Figure 1. Plot of the mass distribution, xn(x,t),for the model
F = xy as a function of x / l for various values of the extent of
depolymerization,B ( t ) = Mo(t)/Mo(0)- 1. The corresponding values of tZ3/6 are 0.33 ( B = 0.3),1.434 ( B = l), 10.87 ( B = 3), and 226.1 ( B = 10). The marks on the right-hand side show the amount of l-mers still present at B = 0, 0.3, and 1 (same scale as on left).
Following the procedure of Appendix A, we find
for the equal reactivity model.12 While for the xy model the analysis is not as straightforward as in the F = 1model, we are able to find the explicit solution:
and
1
n(x,t) = e+la/%(x-l) + 2 t l x 2 S y-2e-tyS/6dy
(9)
X
It can be seen that the initial concentration of l-mers decays exponentially while the smaller chains develop a peaked distribution described by the second term, which can also be written in terms of the incomplete gamma function (see (A12)). The distribution is plotted in Figure 1as a function of the chain length x , for various values of time or, equivalently, the number of bonds broken per polymer molecule B ( t ) = Mo(t)/Mo(O)- 1 (10) (the degree of depolymerization). Here Mo(t)is the zeroth moment of the distribution, where the nth moment is defined as
M" = 1"
-3
2 [ -tp
m=O
(n - l ) ( n + 2m + 2) m!(n+ 2)(n + 2m - 1) (15) 1
n(x,t) = e-t'z/26(x-1)+ 3 t l x S y-2e-y2t/2dy
(16)
X
Evidently, the solution is formally more similar to the xy model than to the F ( x y ) = 1model, for which the solution is given by8Joi26
n(x,t) = e-%(x-Z)
+ e-tx[2t + t2(Z-x)]
(17)
Taking the scaling limit of (16), holding t x 2 = constant, we find
n(x,t)
-
(~(5= ) 2-3/21[S2m2u-3/2e-u du E/
tcp(xt'/2)
-
61
as 5
-
(1W 0
(18b)
which implies z = '/2 and 7 = 0. This scaling function holds for an arbitrary initial condition, with 1 replaced by the mass, All. These scaling exponents are the same as for the F(x,y) = x + y (or a = 1)model.
+
To find the scaling behavior of (9) we consider the behavior for small x and large t. A scaling form exists if we keep x t 3 = constant while taking the limit x 0 and t m. (Note that only the second term in (9) contributes to the asymptotic solutions.) We find
- -
n(x,t)
-
-
(12a)
xt&t'/3)
-
Which satisfies 1 = constant as [ 0. For a general initial condition, the scaling form will be given by (12) with 1 replaced by the weight-average mass, Ml. Thus we find that the scaling behavior is independent of the initial distribution, which is necessary for the scaling concept to be significant. For any initial distribution, a plot of n(x,t)/(xt)against x t 1 / 3 should fall on a single curve given by (12b) (for small x and large t ) . Equation 12 implies z = 1 / 3 and T = 1, so that xy model is of a different universality class than the ( x + y)" model. 111. Ternary Breakup Model We note that a model in which the chains break up into exactly three pieces for each breakup event can also be
IV. F ( X J ) = (x y)" In this class of models, the rate of scission depends upon the length of the chain to which the bond belongs but is independent of the position of the bond within the chain. While in I we gave solutions for some special values of a, here we give the solution for all a. Equation 1 becomes d n ( x , t ) / d t = -x"+'n(x,t)
+ 2Smy"n(y,t)dy
(19)
The general solution can be found by writing n(x,t)as the following series: m
n(x,t) = e-A(L)t[6(x-l) + Ctk/k!Bk(x)]
(20)
k=l
where A(1) is given in (B2). Substituting (20) into (19) and equating like powers of t , we find (see Appendix B)
n(x,t) = exp(-tl"+l) L
J
exp(-tla+')(6(x-l)
+ 2tZ"M[ (a+3)/ ( a + l ) , 2 , ( t l " + 1 - t x a + 1 ) ] ~ (21)
Macromolecules, Vol. 19, No. 10, 1986
2516 Ziff and McGrady
+
3 r
2
---
v
E X
1
0 0.
D
0.5
x/l
1.0
Figure 2. Mass distribution as a function of x / l for B = 3 for F = (x + y)* with a = -1, 0, 1,2, for times tP+' = 1.386,3,5.09, and 8.62, respectively. The curve for F = xy at B = 3 is ah0 shown.
where M(a,b,z) is Kummer's hypergeometric function.35 When a = 2/m - 1, m = -1, -2, -3, ..., the series in eq 21 terminates and the solution can be written in terms of the associated Laguerre polynomial: n(x,t) = exp(-tP/m)(6(x-L) - 2tm-'12/m-'L~~-1(t12/m-tX2/m)J (22) These are the solutions found in I by a much different method. Using Kummer's transformation,35we can rewrite (21) as
n(x,t) = exp(-tla+l)6(z-l)
+
2lUtexp(-tx*+l) m r(k-2/(a+l)) (tX"+l-tp+lk 1 = ) I?(1-2/(a+l)) k = l k!(k - l)! exp(-tIa+l)6(x-l) + 2tl" exp(-tx"+')M[ (a-1)/(a+1),2,(txa+'-tl"+')I (23)
c
In this form, we find terminating solutions for a = 2/m
- 1 with m = 1, 2, 3, ...:
V. F ( x , y ) = (x y ) "Where a < -1 For polymerization kinetics, it has been found that gelation, or the formation of an infinite size network in a finite amount of time, occurs when the homogeneity of the growth kernel K(x,y)is greater than 1,where K(x,y) is the rate of bond formation between x-mers and y-mers, and the homogeneity X is defined by K(ai,aj) = a A K ( i j ) .Gelation is indicated by one of the moments of the solution (normally the second moment M,, which represents the weight-average polymer size) becoming infinite in a finite amount of time and is a consequence of a cascading of the reaction rate as the polymers get bigger. Analogously, a cascading in the breakup behavior might be expected for the depolymerization model if the chains fragment sufficiently fast as they get smaller. In this case, an infinite number of zero-size particles will form in a finite amount of time and the mass, as represented by Ml, will not be conserved. For the class of models ( x + y)-*, we find that this "shattering" process occurs when a < -1. We first describe the solution for the case of a = -1, which is on the borderling of this behavior. By taking the limit of (22) as m m (using x 2 l m 1 (2/m) In x ) , we find in I
-
-
M , = 1" exp[(l - n ) t / ( l
+
+ n)]
(27b)
where I , is the first-order modified Bessel function. The exponential increase in the tatal number, Mo,is indicative that this case is at the boundary of singular behavior. Note that (27) does not show scaling behavior. Now we consider the solution for a = -2, which is given by (24) for m = -2:
n(x,t) = e-t/L[6(x-l)+ 2t/Z2
+ t 2 / 1 3 ( l / x - l)] (28)
The moments are given by
n(x,t) = exp(-tl2/")6(x-2) + e ~ p ( - t x ' / ~(2tm-112/m-1L2!, ) ( t ~ ~ / ~ - t (24) l~/~)) for n > 0, with Mo = m for all t > 0. For n = 1 (29) gives The general scaling behavior can be found from either (21) or (23). Letting x 0 and t m with txa+l = constant, M, = le-t/l[l + t / l + t2/212] = I(1 - t3/613...) (30) we find and the mass is a decreasing function of time. This system n(x,t) = t2/(*+')&tl/(a+')) (254 goes through a cascading fragmentation process in which a finite fraction of the mass shatters into particles of zero with size. Note that (28) does not have a typical scaling becp(t)= 21/I'((a+3)/(a+l)) exp(-("+l) (25b) havior and cannot satisfy (3) because the mass is not conserved. which goes to a constant as $. 0. Equation 25 gives z = For F(x,y) = l / ( x + Y ) ~ the , solution is given by (24) with l / ( a + 1) and 7 = 0, which was derived in I by using a m = -1: special solution. Equation 25 gives the scaling behavior n ( x , t ) = e-t/l2[6(x-I) + 2t/P] for an arbitrary initial condition, with 1 replaced by mean (31) mass of the chains, M I . with The moments of the distribution are given by
- -
-
M , = In e~p(-tl"+~)M(2/(a+l),(~+l)/(a+l),tl*) = PM( (n-1) / (a+l),(n+l)/(a+l),-t I") (26)
-
-
and satisfy M , t-("-I)/("+l)as t m. In Figure 2 we show the behavior of these models for various values of a. We plot the mass distribution, xn(x,t), as a function of x f 1 for B = 3. We also plot the xy curve for comparison. It can be seen that as a increases, the distribution tends to get more peaked. The curve for the F = xy model has a more peaked character and is especially different as x 0, where it is cubic in contrast to linear for the ( x + y)" models.
-
M , = l"e-t/'2(1+ 2t/[12(n
+ l)])
(32)
for n > -1. Again, we find that MI decreases in time: Ml = l(1- t 2 / Z4...). This decrease is faster than in (30),since in this model the smaller particles break even faster than in the a = -2 case. In this case, Mo remains finite. Likewise, for all models with a < -1, shattering takes place and the mass is a decreasing function of time. It is interesting to note that when shattering occurs, a fractal Cantor dust is formed. The fractal dimension, D, of this dust can be found from the relation36
Nr(x>X)
- XD
as X
-
0
(33)
Macromolecules, Vol. 19, No. 10, 1986
-
Kinetics of Polymer Degradation 2517
where Nr(x>X) is the number of fragments with length greater than X. In the limit x 0 for a < 0, (23) gives35
n(x,t) x-(“+3) (34) which implies Nr(x>X) X-(a+2),so D = a 2 for -2 < (Y < -1. Thus the depolymerization model can produce fractal dusts of all possible values of D by varying the fragmentation rate. For a = -1 we find D = 1, meaning that the distribution is not yet a fractal. For a = -2 the power law behavior does not hold, since Nr is logarithmic. When a < -2 we find D < 0 and the distribution seems not to be describable in terms of fractal sets. While no real system can fragment without bound, these shattering models can represent the continuum limit of a system in which large polymers rapidly fragment into monomers. To make this point clear, we study some related discrete models.
-
+
VI. Discrete Models For polymer systems, where the size or molecular weight of a particle is a discrete function, the discrete form of the fragmentation equation is perhaps more appropriate, especially for long times. For a discrete system the fragmentation process will eventually stop as the system becomes composed entirely of monomers. Here we look at discrete solutions for two models which are discrete analogues of the a = -2 and -3 continuous models. Since ( 2 ) is just a set of coupled linear differential equations which can be solved by matrix diagonalization,1° the discrete equation (2) is soluble in principle for any Fi,. However, no explicit solutions have been found, except for Fi, = l8J0 and Fij = i + j in I. In Appendix C we show that, starting with a monodisperse system of 1-mers, one can solve explicitly for the concentration of I-mers, then ( 1 1)-mers, ( I - 2)-mers, etc. We give the general recursion formulas and describe a heuristic procedure to search for solutions using a computer. We find solutions for the following interesting cases: (1)For Fjj = 1/(k - l)(k + 1)where k = i + j , we find nl = e-t/(1+1) nk = ( l / k ) ( l - k
+ l)e+/(l+l)+ (21/k)(l - k)e-t/l + ( I / k ) ( l - k - l)e-t/(l-l)
(1 < k
< 1)
nl = 1 - ( P / 2 ) ( 1 - l)e-t/(’+’)+ 1(1 - 1)(1 - 2)e-til (Z/2)(1 - 2)(1 - 3)e-t/(c1) (35) In the continuum limit, (35) becomes the solution (28) for the model F ( x , y ) = (x Y ) - ~which , shows a loss of mass, while in (35) mass is conserved. In (351, it can be seen that monomers show a different behavior than the other k-mers, and as t 0 3 , n, 1. In the continuum limit, those monomers are lost in the distribution (“a point of measure zero”), resulting in an apparent loss of mass. However, this loss of mass is redly an artifact of the process of taking the limit. (This is reminiscent of the Bose-Einstein gas.) In the finite system, there is no loss of mass. ( 2 ) For Fjj = l / k ( k - l)(k 1)where k = i + j , we find nl = e-t/41+1)
+
- -
+
nk = e-t/l(l+l)- e-t/l(l-l) (1
< k < 1)
nl = I - y2(1+ 2)(I - l)e-t/L(l+l)+ y2(L + 1)(1- 2)e-t/L(l-1) (36) This model is analogous to the continuum model (F(x,y) = (x + Y ) - ~and , indeed (36) gives (31) in the continuum limit. In both the continuum and discrete models, the distribution is completely flat (independent of x or of k)
between 1 and monomers. Again, there is an accumulation of monomers in the discrete system. For both these system, we see that shattering in the continuum model corresponds to rapid production and accumulation of monomers in the discrete system. In I, we give the solution for Fi,= i + j . The continuum limit can be taken and the F(x,y) = x + y solution is found. In this case there is no point of measure zero that is lost, as the mass is conserved in this continuous system. We also give the solution for Fij = ij in Appendix C.
VII. Conclusions Because of the many practical applications of the equations of depolymerization, they have been the object of much study. We have shown that analytical solutions can be found for a model where the rate of breakup depends on the position of the bonds within the chain, F(x,y) = xy. This model approximates systems where the chain tends to break near the center with greater probability, such as occurs in shear degradation and some forms of chemical attack. Thus for these problems it is possible to gain insight into the physical mechanism of the degradation process with an explicit solution. Coupled with the solution of the ( x + y)” model and the discrete models, we have a wide variety of possible models to describe depolymerizing systems. The scaling behavior of the solutions can be summarized as follows: F(x,y) = (x
+ y)”,
a
> -1:
z = l/(a
+ 1)) 7 = 0
F(x,y) = xy:
z=y3,
7 = l
F ( x , y , z ) = 1:
z=y2,
7 = 0
(37)
The methods of solution presented here should also be applicable to finding solutions for other models for both discrete and continuous depolymerization. While we have used the language of linear chain breaking as our model, that model should be interpreted as a metaphor for a wider variety of possible systems. For example, we can have branched or cross-linked polymers undergoing depolymerization, and in the average the rate should be describable in terms of a net F(x,y), which will most likely not be uniform. Beyond the polymer system, the same mathematical formalism and solutions presented here can apply to particle fragmentation, grinding, comminution, etc. The shattering phenomena observed for the a < -1 models represent a cascading fragmentation to monomers. Generalizing what we found for the two discrete models, we can conclude that when a loss of mass in a continuous model is seen, it corresponds to a “delta function” peak of monomers. For the corresponding discrete model this appears as a rapid increase in monomers. When the continuum limit is taken of the discrete system, this accumulation of monomers gets lost. In these cases, one should identify the missing mass (Ml(0) - Ml(t)) as the mass of the system belonging to the monomers. This is much like Flory’sI7identification of the missing mass in branched polymers as molecules belonging to the gel. It appears that the shattering phenomenon occurs when the homogeneity of F(x,y) is less than -1. This is an interesting counterpoint to the condition for gelation, which is that the homogeneity of the growth kernel is greater than + l . Added Notes: A. Kerstein has kindly sent us a preprint37in which he discusses the general scaling behavior + (for for a class of kernels including F(x,y) = x@y@(x a > -1). His result agrees with the scaling of the special cases studied here. We also would like to add a reference to Bak and Bak,38in which the fragmentation model of
Macromolecules, Vol. 19, No. 10, 1986
2518 Ziff and McGrady
chains breaking exactly at the center is solved explicitly.
where I'(a,x) is the incomplete gamma function.
Appendix A. Solution for F ( x j ) = xy Multiplying (8) by x n and integrating over x , we find
Appendix B. Series Solution for the Continuous System For a system with monodisperse initial conditions, we write n(x,t) as a series development of the form
-dM, - dt
( n- l ) ( n+ 6 ) Mn+3 6(n + 2 ) ( n + 3)
(AI)
m
This relation can be used to find M,(t) by Charlesby's method.12 Write
+ M,'(O)t/l! + M,"(0)t2/2! ...
M,(t) = M,(O)
m
1°j=o C-
(-t13/6)j (n - l ) ( n+ 3 + 3j) j! ( n + 3 ) ( n - 1 + 3j)
(A3)
Now we want to deduce the n(x,t) from the M,. In contrast to the relative ease for the F ( x y ) = 1 case,12we must do many manipulations here. For a monodisperse initial condition we expect a solution of the form
n(x,t) = e-t%(x-i)
+ f(x,t)
(A4)
where the first term satisfies the initial condition and satisfies (8) a t x = 1. Equation A4 implies
~ , ( t =)
ine-tl3/6
+ J 1 x n f ( x , t ) dx
We rearrange (A3) to this form by writing ( n - l ) ( n+ 3 + 3j) 12j = I ( n 3)(n - 1 + 3j) ( n 3)(n - 1 + 3j)
+
+
and thus find
k=l
(Bl)
where
(A21
The j t h derivative of M , at t = 0 can be found by iterating ( A l )j times and using MJO) = 1" (assuming monodisperse initial condition). Substituting the result into (A2),we find
M,
n(x,t) = e-A(l)t[d(x-l)+ C B k ( x ) t k / k ! ]
(A5)
Note that the &function term in ( B l )satisfies (1) a t x = 1 and satisfies the monodisperse initial condition. Substituting ( B l ) into (1) and equating like powers of t , we find Bl(x) = 2F(x,l-x) 033) 1
&+1(x) = [A(1)- A(x)]Bk(x)+ 2 1 Bk(y)F(x,y-x) dy 034) assuming A(x) # A(1). For F(xyy) = ( x + y)", we have A(1) = la+l, B1 = 21", and 1
Bk+i(X)= (I"+' - X"+')Bk(X)+ 2$ y"Bk(y) dy X
which can be solved recursively as follows: a+3 B&) = 21"(1"+' - X"+1
)Ly+1
(A61
2
12ln(-tl3/6) (-t P /6)'-' M , = lne-t13 (A7) n + 3 j=lj!(n+ 3j + 2 )
B3(x) = 21"(1"+1- x a + y
( a + 2)(a + 3) (a
+ 1)2
We hypothesize that Bk has the form Bk(X) = Dk(l"+l-X"+l) -
Using the identity
Substituting (B7) in (B5) we find k + 2/(a Dk+l = Dk k
which can be verified directly, we find
and, by virtue that D1 = 21", find k-lj 2 / ( a 1) - 2 ~ r ( k + 2 / ( ~ + 1 ) )(B9) Dk = 21"n j=1 I'(2/(a+l)+l)(k- l ) !
+
+ 1)
+
Putting this result into ( B l )and (B7),we thus find (21). Substituting (A9) into (A7),we find
+
M,, = Zne-t13/6 2tl/(n
+ 3)$
1 0
x-n-1e-tx3/6dx
(-410)
To get M , in the form of (A5) we must still eliminate the l / ( n + 3) factor above. This can be accomplished by an integration by parts, which yields the following result:
Appendix C. General Solution for Discrete Systems We write (2) in the form
where k-1
Ak (All) where we have also written the first term of M , as an integral over dy. Comparing ( A l l ) with (A5),we deduce that n(x,t) must be given by (9) in the text. We can also write the solution as
e-t'3/66(x-l)+ 4 1 ~ ~ ( t / 6 ) ~ / ~ [ I ' ( - ' / ~- ,I'(-Y3,tZ3/6) tx~/6) J (A12)
,EFi,k-i
1=1
For nk(0)= bkl, nl satisfies the closed equation dnl/dt = -Alnl or nl = exp(-Alt)
t13/6
n(x,t) = e-t'3/66(x-l)+ 41x2(t/6)4/3Ax3/6 u-4/3e-udu =
E
Then for k = 1 - 1, ( C l ) gives dnl-l/dt = -Al-lni-i or
+ 2Fi-1,lnl
Kinetics of Polymer Degradation 2519
Macromolecules, Vol. 19, No. 10, 1986 nl-I
= Cl-1,I-l exp(-Al-,t)
+ Cl-l,l exp(-A$)
(C5)
Where c h i (i 1 k) are numerical coefficients. It can be seen that the general formual for nk will be of the form J
assuming Ai # Aj for all ij. Inserting (C6) into ( C l ) ,we find n
and from the initial condition nk(0) = 0 for all k follows that
< 1 it
1
Ckk = -
(k
cki
i=k+l
< 1)
(CB)
Note Cli = 1. Thus, for any Ai,the c k i can be found recursively, and by (C6) the solution is determined.3J0J8 While in principle this procedure will work for any nondegenerate Fij, it usually leads to complicated expressions for the c k i which are not obviously expressible in closed form. We have however been able to find many cases where an explicit solution can be found by the following empirical procedure. The above recursion equations were programmed on a small computer and solved successively, first for 1 = 2, then for 1 = 3, etc. The resulting Cki (which are a function of I ) were then examined to determine if a closed-form solution is possibly by inspection. For the cases F = l / ( k + l ) ( k - 1)and l / k ( k + 1)(k - 1) the cki were in a particularly simple form and thus we were led to the solutions (35)and (36) given in the text. We note that for similar rate matrices F = l / k 2 and l / k 3 , the coefficients appeared to be very complicated and no explicit solution was obvious. We have also found the solution for Fjj = ij: nr(t)= exp(-Art)
nk(t) = 41k2C i=k
exp(-Ait) - exp(-Ai+lt)
i2(i
+ 1)2
-
(C9)
where Ai = i(i - l)(i + 1 ) / 6 . When t a,only monomers remain. In the continuum limit this solution goes over to the solution for F(x,y) = xy. Likewise, many other interesting discrete models can be solved by this method.
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