J . Phys. Chem. 1989, 93, 5931-5937
5931
Nonequilibrium Processes in Polymers Undergoing Interchange Reactions. 1. Relaxation Rates Dilip
K. Kondepudi,*
Department of Chemistry, Wake Forest University, Box 7486, Winston-Salem, North Carolina 271 09 John A. Pojman,+ Department of Chemistry, University of Texas at Austin, Austin, Texas 78712 and Mohamad Malek Mansour FacultP des Science, UniversitP Libre de Bruxelles, 1050 Bruxelles, Belgium (Received: September 7 , 1988; In Final Form: December 19, 1988) Relaxation processes due to interchange reactions in polymer systems are analyzed by use of the analogy with the Boltzmann equation for a gas of hard spheres. The linear rate of relaxation of a perturbation in the n-mer concentration is obtained. By use of Monte Carlo simulation, the range of validity of the linear approximation is shown to be fairly large. Introduction Nonequilibrium chemical systems are known to exhibit a great variety of interesting behavior.I4 The current understanding of polymer systems under nonequilibrium conditions, in particular the nature of molecular weight and sequence distribution, is rather rudimentary. On the one hand, in model chemical systems one can easily see how a set of nonequilibrium conditions under which polymerization is made to take place can leave an imprint in the polymer sequence;5 on the other, it is not yet known if some of the commonly encountered polymer kinetics can give rise to such nonequilibrium “imprinting” or, like other known nonequilibrium chemical systems, to a transition to an organized state with interesting molecular weight and sequence distributions. In general it is interesting to see how the decrease of entropy in the nonequilibrium states will be manifested in the molecular and sequence distributions. Our study is aimed at developing a better understanding of the relation between molecular weight and sequence distribution and the nonequilibrium conditions under which the polymerization reactions take place. For this, however, we need to first understand some basic nonequilibrium aspects such as linear relaxation rates. In this article we develop a general mathematical formalism with which we obtain the equilibrium distribution and the relaxation rates for a perturbation from the equilibrium state. Once this is established, sequence distributions in flow reactors can be studied. In the context of flow reactors some interesting features of catalytic polymerization in a flow reactor, under pulsed flow conditions, have been recently discussed by Peacock-Lopez and Lindenberg6 We begin our study with the consideration of interchange reaction that occurs in several polymer systems. These reactions involving condensation polymers alter the molecular weight distribution without affecting the average length of the polymers. Such reactions have useful applications in industrial polymer synthesis,’ and their effect on the molecular weight distribution of polyesters was discussed by Flory.* There are three main mechanisms for interchange reactions among polyester^:^
Reaction conditions may be manipulated such that the interchange reactions will be the dominant reaction and polymerization may be neglected. In this study, we consider the first two mechanisms because not enough is known regarding transesterification in solution under acidic conditions. Also, there is an interesting analogy between this system and a gas of hard spheres (described by the well-known Boltzmann equation) that is helpful in considering relaxation processes. We formulate the basic equations of the system with this analogy in mind.
-
(1) Nicolis, G.; Prigogine, I. Selforganization in Nonequilibrium Systems; Wiley: New York, 1977. (2) Haken, H. Synergerics-An Introduction; Springer: Heidelberg, 1977. (3) Vidal, C.; Pacault, A., Eds. Non-Linear Phenomena in Chemical Dynamics; Springer: Berlin, 1981. (4) Field, R. J.; Burger, M., Eds. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. (5) Nicolis, G.; Rao, S.; Rao, S.; Nicolis, C. In Proceedings of the International Workshop of Structure, Coherence and Chaos; Nicolis, G.,Sanglier, M., Eds.; Manchester University Press: Manchester, 1987. (6) Peacock-Lopez, E.; Lindenberg, K. J . Phys. Chem. 1984,88,2270; J . P h y s . Chem. 1986, 90, 1725. (7) Kotliar, A. M. J. Polym. Sci.: Macromol. Rev. 1981, 16, 367. ( 8 ) Flory, P. J. Chem. Rev. 1946, 39, 166. (9) Koskikallio, J. Chemistry of Carboxylic Acids and Esters; Patai, S., Ed.; Interscience: New York, 1969; p 104.
(i) alcoholysis RlCOOR2
+ R30H
RlCOOR3
+ RzOH
R4COOR2
+ RlCOOH
R,COOR,
+ R4COOR2
(ii) acidolysis RlCOOR2
+ R4COOH
(iii) transesterification RICOORZ + R4COOR3
+
-+
‘Current address: Department of Chemistry, Brandeis University, Waltham, MA 02254-91 10.
0022-3654/89/2093-5931$01 .50/0 , I
,
Reaction-Diffusion Equation for Polymers Undergoing Interchange Reactions Unlike most nonequilibrium chemical systems that have been studied extensively in the past two decades, polymer systems have a large number of species. In general, dealing with a large number of species is a formidable task, both analytically and numerically. For the case of interchange reactions, however, the situation is simpler. One of the simplifying features is that the interchange reactions are analogous to energy-conserving collisions in a gas of hard spheres (see Figure 1). The conserved quantity in the case of polymers is, of course, the total number of monomer units. Also, the total number of polymer molecules remains unchanged by the reactive collisions,just as the total number of gas particles remains unchanged. Hence we can derive a Boltzmann-like equation for the polymers. For the hard-sphere gas in thermodynamic equilibrium, the number of particles with kinetic energy E can be shown to be proportional to exp(-@’E), where p’ = l / k T . Similarly, for polymers the number of polymer molecules with n bonds turns out to be proportional to exp(-(?n) in which (? is a parameter similar to l / k T . This is an alternative way of expressing the Flory distribution; the extent of reaction p is equal to exp(-(?). The Boltzmann-like equation for a polymer system undergoing interchange reaction can be obtained by defining the following quantities. For mathematical convenience we shall identify the polymers of different lengths by the number of bonds, rather than the number of monomer units. The summations are then from
0 1989 American Chemical Society
5932
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
Figure 1. Analogy between collisions in a gas of hard spheres and interchange reactions in polymers. Conservation of energy in the gas, V12 + V22= V’12 + V1,2,leads to the Maxwell-Boltzmann distribution P(V, a exp(-@’p). In an interchange reaction between an ,”,-mer and an N2-mer that results in an N;-mer and N>-mer,we have the conservation of total number of units: N , + N 2 = N ; + N l 2 . This leads to a distribution P(N) a exp(-BN). By setting the extent of polymerization p = exp(-@).one can obtain the Flory distribution.
0 to rather than from 1 to m. We define the following: f(n,r,t) concentration of polymer molecules with n bonds (i.e., (n + 1) monomer units) at the point r, at time t (1) u(n’l,n’2lnl,n2)
=
rate constant for the reaction in which a polymer with nl bonds reacts with a polymer with n2 bonds to produce polymers with n’, and n’2 bonds. In such a reaction we have the conservation law n’l + n‘2 = nl n2 (2)
Kondepudi et al. change, we could as well have written the above equation in terms of a probability P(n) =f(n)/N of finding a polymer with n bonds in the total sample of N polymer molecules. Equation 6 and its solutions under nonequilibrium conditions will be the focus of our present and future study. In the following section, however, we will first obtain the equilibrium distribution using the principle of detailed balance. Equilibrium Distribution When a = y = 0 the system can reach thermodynamic equilibrium. When in equilibrium, the system will be homogeneous and, in accordance with the principle of detailed balance, each reaction will be balanced by its reverse. Iffo(n) is the equilibrium distribution, then the principle of detailed balance implies that If we assume that u(n’l,n’21nl,nz)= ~ ( n , , n ~ l n ’ , , n and ’ ~ ) use the conservation equation (nl + n2) = (dl+ n$) in the above equation, then it follows that at equilibrium fo(nl) f o ( n 2 ) = fo(n’,) X fo(nl+n2-n’,). Hence the equilibrium distribution fo(n) should satisfy the condition
+
At any time t , the rate at which polymers with n bonds are converting to polymers of different length can be written as a “loss term“: (loss term), =
2 5 a(n’l,n+n2-n’l~n,nz) Jn,r,t) f(n2,r,t) (3)
n‘l=O n2=0 n+n2-”‘l 2 0
+
+
In the above equation we have used the fact n n2 = n’l Here the summations are restricted by the conservation law and the requirement that all n’s should be greater than zero. Similarly we can write a “gain term“ due to polymers of length other than n bonds converting to polymers with n bonds: (gain term), = m
m
n‘l=O n‘l+n‘2-n>0
n;=O
~(n,n’l+n’2-nln’~,n’2) f(n’,,r,t)f ( r ~ ’ ~ , r ,(4) t)
From time reversal symmetry it follows that a(n’l,n’21nl,n2)=
~ ( n ~ , n ~ l n ’ , , n (Further, ’~). to a good approximation it may be assumed that u is independent of the lengths of the polymer^.^^^) Thus, for a homogeneous system in whichfis independent of the position r. we have d f( n, 1 - (gain term), -f
- (loss term),
dt
=
c c u(n’1,n+n,-n’,In,n2)
n‘,=O n2=0 n+nz-fl’120
x
U(n’1,t) f(n+n2-n’,,t) - f ( n , t ) f(n2,t)l ( 5 )
in which we have suitably redefined the dummy summation indices. Equation 5 has strong resemblance to the Boltzmann equation for the hard-sphere gas. For nonequilibrium situations we should, in general, consider inhomogeneous systems with inflow and outflow of polymers. If diffusion were the only transport process, and a(n,r,r)and y(n,r,t) are respectively the rates of inflow and outflow for the n-bond polymer, then the complete reaction-diffusion equation for the system is
f o ( n ) = (constant) exp(-@n) (9) Using the condition that the total number of molecules N = C,”=af(n), the constant can be determined; thus we obtain f o ( n ) = N[1 - exp(-P)I exp(-Pn) (10) This equation is an alternative way of writing the Flory distribution:
= N( 1 - p)pm-’ (11) in which p ( m ) is the number of polymer molecules with m units (Le., ( m - 1) bonds) a n d p is the extent of reaction. By making the identification p = exp(-P) we can see that (10) and (1 1) are the same. For the hard-sphere gas, the Boltzmann equation for the number of molecules with energy E must satisfy the law of conservation of energy. This condition and the principle of detailed balance lead to an equation similar to (8),whose solution gives the familiar Maxwell-Boltzmann equation P ( E ) = (constant) exp(-pE) in which @’ is identified as l / k T . This naturally leads us to question if there is any advantage in introducing a parameter T, similar to temperature, for polymers undergoing interchange reactions especially when considering nonequilibrium transport processes. This point will be discussed in our study of transport processes. p(m)
Relaxation to the Equilibrium State The first nonequilibrium process we analyze is the rate at which from the equilibrium distributionfo(n) a small perturbation, An,?), (given by (10)) relaxes to the equilibrium state. Note that C,g(n,t)= 0 since the total number of molecules, N = C,f(n), does not change; Le., the positive perturbation in some polymer species is compensated by a negative perturbation in others. Also, we shall assume that u is independent of the size of the polymers interacting as a first simplifying approximation, an assumption that was found to be good in the previous We consider a well-stirred reactor in which spatial inhomogeneities can be ignored. By defining and substituting it into (6) (in which we drop the diffusion term and set a = y = 0), we can obtain the following linear equation for g(n,t) by neglecting the quadratic terms because they are small:
n+n2-n‘,20
f(n+n2-n’1,t) - f ( n , t ) f(n2,t)l
It is clear (and it can be shown rigorously) that the solution of this equation can be written as
+ D,Vtf(n,r,t) + a(n,r,t) r(n7J) ( 6 )
in which D, is the diffusion coefficient for a polymer with n bonds. does not Since the total number of molecules N = E,“Eaf(n,t)
The Journal of Physical Chemistry. Vol. 93, No. 15, 1989 5933
Nonequilibrium Processes in Polymers Now using (lo), this equation may be written as
11
n+n~-d,?O
exp(-/3(n
+ n2 - n;)) + g(t~+n~-n’~,t)exp(-/3n’I) g ( w ) exp(-bn,) - g(n23t) exp(-bn)l (14)
Note that setting a = y is not sufficient to obtain this equation; since we also need the assumption thatfo(n) is a Flory distribution (as defined in (lo)), we must have both a = y = 0. Setting a = y corresponds to a general nonequilibrium steady state. Also it is important to note that these summations are restricted by the condition n n2 - n’, L 0 (15)
+
as indicated under the summation signs because the number of bonds on a polymer cannot be a negative number. This equation can be simplified by performing one of the summations and writing it as a system of linear equations, Le., a matrix equation. This simplification does not involve any special features that need to be pointed out. The details of this simplification are given in the Appendix. The result is the equation
2
-dg(n’t) -dt - aN k=O [2G(n,k) - F(n,k) in which G(n,k) =
+ C(n,k)]g(k,t)
-1
(16)
k n
\ (a) (b) Figure 2. The two dominant processes contributing to the relaxation of a perturbation, g(n), in the concentration of ( n + 1)-mer. The average number of bonds of the system is denoted by (n);the total number of molecules in the system is N . Process a is a schematic representation of the interchange reaction in which a terminal unit of a molecule of the perturbed species (with n bonds) reacts with a bond of one of the polymers molecules in the rest of the system. For this process the reaction rate is proportional to g ( n ) N ( n ) . Process b is a schematic representation of the opposite of process a. Here the terminal unit of one of the polymer molecules in the system reacts with one of the bonds of a molecule belonging to the perturbed species. For process b, the reaction rate is proportional to g(n)Nn.
value. It is a result that one expects intuitively; the rate is proportional to the sum of the number of bonds in the polymer, n, and the average number of bonds, (n),of the polymer system that is being perturbed. These two terms correspond to the two processes shown in Figure 2 which result in the decay of g(n,t); the process shown in Figure 2a is proportional to n, while the process shown in Figure 2b is proportional to (n). For process a there is a small additional contribution, 4nfo(n)/N, due to the interaction of the perturbed (n 1)-mer with polymer species of the same length. A closer look at the off-diagonal elements of the matrix (2G - F C) reveals the processes that are being ignored in approximating the eigenvalue by the expression (21). The offdiagonal elements of G correspond to the gain in the number of (n + 1)-mer molecules due to perturbation in the concentration of polymers other than the (n + 1)-mer. Further, since the condition C,g(n)= 0 must be satisfied, a perturbation g(n) in the concentration of an (n 1)-mer is accompanied by a perturbation in other species; this contributes to the off-diagonal terms in F. The elements of C can be similarly interpreted. In a typical situation in which we have an initial Flory distributionfo(n) to which we add a perturbation of (n + 1)-mer and measure the rate of decay, these corrections will be negligible. To get a better idea of the range of validity of this approximation, we have performed a numerical simulation of the interchange reactions, as described in the following section. The results show that the relaxation rates obtained (using (21) for the eigenvalue) have a broad range of applicability.
+
+
where Bnk is the Kronecker delta and ( n ) is the average length expressed as number of bonds per molecule: total no. of bonds = Cnf(n) ( n ) = total no. of molecules , Note that ( n )does not change during the reaction because the total number of bonds and the total number of molecules do not change. In the Appendix it is shown that
The elements of this matrix are small compared to the elements of G and F. The eigenvalues of the matrix (2G - F + C), the elements of which are given by (17)-( 19), determine the rate of relaxation of g(n,t). Though the exact eigenvalues are difficult to obtain analytically, a good approximation can easily be obtained by noting that in the matrix (2G - F + C) the diagonal elements are dominant. The off-diagonal elements are derived from the elements of G and C (F being a diagonal matrix), and from (17) and (19) it can easily be seen that these elements are small compared to n and ( n ) in general. The diagonal elements are
(20) Since the last three terms are small compared to the first two and may be neglected, to the leading approximation the eigenvalues A, are A,, = Nu[n (n)] (21)
+
+
Because in the above approximation (2G - F C) is a diagonal matrix, the corresponding eigenvectors are the unit vectors e, = (0,O ,...,1,0,0 ,...), in which the nth element equals 1 and the rest are zero. The expression (21) along with the eigenvectors e,, give us the first approximation to the rate at which a small perturbation in the concentration of a (n + 1)-mer will decay to the equilibrium
+
Numerical Simulation of Relaxation to Equilibrium
To develop an understanding of the range of validity of the approximations made in obtaining the above relaxation rates, a Monte Carlo numerical code was developed. When a large number of species are involved, such a code is more convenient to use than a differential equation solver. For the system we have studied, the distribution includes polymers up to those containing 100 units though only concentrations of polymers containing about 40 units were significant. Longer polymers can easily be handled. Also, the Monte Carlo code was chosen over a differential equation solver with future studies involving fluctuations in mind. Monte Carlo codes have also been used in simulating similar processes in coagulation.” The flowchart for the numerical code is shown in Figure 3. The following algorithm was employed for the Monte Carlo simulation of the interchange reaction: Two molecules, with N, and N2 bonds, are randomly chosen with the probability of selection proportional to the number of molecules of each of the species,f(N,)/N andf(N2)/N, respec(10) Sridharan, R.; Mathai, I. M. J . Sci. Ind. Res. 1974, 33, 170. (1 1 ) Garcia, A,; Van Den Broeck, C.; Aertsens, M.; Serneels, R. Physica A 1987, 143A, 535.
5934
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
INITIALIZE PSEUDORANDOM NUMBER GENERATOR
I INITIALIZE POLYMER DISTRIBUTION I
Kondepudi et al. number generator of Vax Fortran whose seed was initialized from the machine clock before each run. This process is repeated ( N , + N2)times, which is the total number of bonds of the two reacting species. In a sample that contains a total of N molecules, this means the rate of reaction between N , and N 2 (for all possible outcomes) is (simulation rate) =
r
DETERMINE WHICH
SUM(K) C 1R C SUM(K+l).
I
1
MOLEC(1) = K+l
*
+I CHOOSE RANDOM INTEGER. IT:
II
1
IT AND MOLECO)
+ MOLECW - IT
"SCATfZRING" PROCESS BEEN PERFORMED MOLECOB + MOLECW
-
2
.
CALCULATE % CHANGE IN PERTURBATION OF SELECTED SPECIES.
(N,
+ N 2 )per iteration
This is because the probabilities of selecting molecules with N , and N 2 bonds aref(N,)/N andf(N2)/N, respectively, and each such pair can be chosen in two ways. The factor ( N , + N 2 ) appears because the rate of reaction is assumed to be proportional to the total number of bonds. Also, we have assumed that the probability of an interchange reaction occurring at a particular bond is independent of the position of the bond. This approximation has proven to be a good one in previous studies.'** The relation between the parameters in the Monte Carlo simulation and the theoretical parameters can now be easily established. According to the theory presented in the previous sections, the total rate (Le., for all possible outcomes) for the reaction between molecules with N , and N 2 bonds is u f ( N , ) f ( N 2 ) ( N , N 2 ) ;comparing this with (22), we obtain the relation
+
u =
s IT C MOLECW + MOLECO)
PRODUCT MOLECULES ARE
2 f ( N , )A N , )
2/Nz
(23)
From (21), it follows that the expected relaxation rate for a perturbation in the concentration of ( n + 1)-mer is
In the simulation we have a total of lo6 molecules undergoing interchange reactions. To test the applicability of the linearized theory, the rate of relaxation of a perturbation from equilibrium for a number of species for various average lengths was obtained through simulation. The determination of the relaxation rate of the perturbed species was done as follows: The system was initialized to have the equilibrium Flory distribution for a set average length (Le., for set extent of reaction p = e-8). For a selected species, a perturbation, AN, of 25% of the equilibrium value was added, and the reaction simulation begun. The relaxation of this perturbation to the new equilibrium value was monitored at regular intervals of time. The perturbation, g(n,O) with respect to the new equilibrium was obtained by the following equation: g(n,O) =f(n,O) - ( N + AN)(l
- e-@')e-p"
(25)
in which e-8'
Figure 3. Flow chart for the code used to determine the relaxation rate for a perturbation to the equilibrium value of a selected polymer species. The simulations were performed on lo6 polymers with lengths from 1 to 100.
tively. The lengths of the resulting product molecules are determined by choosing a random integer, R, between 0 and the total number of bonds, ( N , N2),of the reacting species. The number of bonds in one of the product molecules is this random number, R, while the length of the other is ( N , N 2 - R ) . Random numbers employed were generated by using the pseudorandom
+
+
= 1-
N + AN N ( n + 1) ANn
+
In the simulation the unit of time is the number of iterations. Results were averaged over 10 runs. For the case of 10-mer, a plot of the exponentially decaying perturbation ratio g(9,t)/g(9,0) in the concentration is shown in Figure 4. In Figure 5 , the ratio g(9,t)/g(9,0) versus number of iterations is shown. The relaxation rate of 0.000035 7 per iteration obtained from the linearized theory (using (24)) agrees well with the value 0.000036 5 per iteration obtained from the simulation. The relaxation rates for polymers of various lengths were similarly obtained through numerical simulation. The comparison between theory and simulation is provided in Figure 6 . The calculated values are represented with the line. Within the statistical limits of the simulation, there is excellent agreement between the values for a 25% perturbation. To obtain a simple estimate of the error due to the statistical fluctuations of the Monte Carlo simulation, the time evolution of the standard deviation for an ensemble of 20 runs was obtained. This is shown in Figure 7. Initially, the deviation is zero because all the runs are initialized with the same equilibrium distribution, but it increased to about 10%of g(n,t) after 20000 iterations. The
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989 5935
Nonequilibrium Processes in Polymers
5
0
4
8 L 0)
c
e
3
C
-zE c
2
0.0
=
200
100
300
400
I
10
0
600
500
n
Iteratlons'lOO
Figure 4. Relaxation of a perturbation of a 10-mer. A 25% perturbation was added to the equilibrium value for a 10-mer. The simulation with lo6 molecules was allowed to proceed for 50000 iterations, with the fraction of the perturbation remaining, g(9,r)/g(9,0), calculated every 100 iterations. Results were averaged over 10 runs. Error bar indicates a standard deviation of *4% for data averaged over 20 runs.
y
D
20
Figure 6. Relaxation rate of a 25% perturbation plotted against the number of bonds, n, in the perturbed species. The straight line indicates
the theoretical value for the rate from the expression for the eigenvalues in (21). The error bar corresponds to the error in Figure 7, for 10-mer. 0.2
-
2.27t2 3 . W X RA2 I 0.997
-. C
U
0.1
0
U
0 0
100
200
300
400
500
100
ltoratlons*lW
iteration.
ratio of the standard deviation to g(n,O) is about 4% at 2 0 0 0 0 iterations as indicated in the error bar in Figure 4. This gives us an idea of how much the value of the perturbation g(n,t) fluctuates from run to run. The error bar shown in Figure 6 is based on this result for the 10-mer. The size of the error for other species can be expected to be about the same. The validity of the theory remains for perturbations as large as 75%. A plot of the log [g(9,t)/g(9,0)] versus number of iterations is shown in Figure 8 for a 75% perturbation. The relaxation rate is virtually identical with the simulation of a 25% perturbation. Thus, though the kinetic equation (5) is quadratic in the concentration, the linear approximation seems to have a good range of validity.
300
400
500
600
bnllOM*1W
f
Figure 5. Logarithm of the ratio g(9,t)/g(9,0) plotted versus number of iterations. The slope of this line provides the relaxation rate of 3.65 X per iteration compared to the theoretical value of 3.57 X per
200
Figure 7. Time evolution of the standard deviation during the relaxation of a 10-mer perturbation. The initial perturbation g(n.0) was 25% of the
equilibrium value; the plot shows the ratio of standard deviation and the perturbation g(n,t) as a function of time, t , measured by the number of iterations. The standard deviation shown is obtained from 20 runs. The initial deviation is zero because each of the 20 runs was begun with the same initial values. 1
y = 1.0862 - 3 . 5 2 d X W2;: 1.OOO
Concluding Remarks
In this study we have obtained the general diffusion-reaction equation for polymers undergoing interchange reactions and pointed out the analogy between interchange reactions and collisions in a gas of hard spheres. From this we can see clearly the relation between the conserved quantity and the equilibrium distribution. Our broader goal is to look at various nonequilibrium processes in this system. The simplest among these is the relaxation process, and we have shown that the linear approximation gives us a value for the rate that has good range of applicability.
0
100
200
300
400
500
600
Iteratlons'lOO
Figure 8. Relaxation rate for a 75% perturbation of 10-mer. The value per iteration is very close to 3.65 X per iteration from of 3.52 X per iteration. the 25% case. The theoretical value is 3.57 X
The rate, to a very good approximation, is proportional to the sum of n, the number of bonds of the perturbed polymer, and ( n ) , the
5936
The Journal of Physical Chemistry, Vol. 93, No. 15, 1989
(a) (b) (c) Figure 9. Three types of processes that could contribute to the increase of the number of polymers with n bonds: (a) the general process in which the restriction nl + n2 1 n applies; (b) process in which two molecules with n bonds are produced; (c) the process that amounts to a nonreactive collision.
Kondepudi et al. Note that the second term takes care of the additional contribution due to process b. The first term includes (c) but it will cancel with an identical contribution in the expression for the decrease infln). The restriction on the summations is indicated under the summation signs. The three types of processes for the decrease offln), which are the inverse of the processes shown in Figure 9, are shown in Figure 10. The loss term can now be written in a way similar to the gain term.
n+n2-n’]>0 n+n2
=
C C n2=0 n’l=O
.f(n)f(n2) + (2n)crf(n)f(n)
(A.3)
Combining (A.2) and (A.3) we can now write:
-df(n,t) - - (gain term),, - (loss term),, dt
2n
Figure 10. Three types of processes that could contribute to the decrease of the number of polymers with n bonds: (a) the general process in which the restriction dl 5 n + n2 applies; (b) the process in which two molecules with n bonds are consumed; (c) the process that amounts to a nonreactive collision.
average number of bonds of the polymer system. The first-order correction to this relaxation rate is given by (20). Monte Carlo code developed for simulation of interchange reactions showed that this rate is valid for a perturbation of as much as 75% of the equilibrium value. Similar results were obtained for the dependence of the relaxation rate on (n),the average number of bonds for the polymer system. The Monte Carlo simulation code that we have used can easily be enlarged to include more reaction mechanisms such as polymerization and transport mechanisms such as diffusion. These aspects will be studied in the future. Acknowledgment. We thank Prof. I. Prigogine for his unfailing encouragement for this work. We also thank the U.S. Department of Energy, Office of Basic Energy Sciences, for supporting this work through Grant DE-AS05-81ER10947. J.A.P. thanks the Welch Foundation for supporting him through a predoctoral fellowship. We thank Wake Forest University for providing funding for publication and reprint charges. D.K.K. thanks the North Carolina Board of Science and Technology for support for computer software.
Appendix: The Linearized Kinetic Equation for the Exchange Reactions As described in the main text, we assume that u is the independent of the size of the reacting molecules. The gain and loss in f ( n , t ) f(n,t) = number of molecules with n bonds per unit volume at time t (A.1) due t o the interchange reaction can be written in terms of three types of processes shown in Figures 9 and 10, respectively; (a) contributes one molecule, (b) contributes two, and (c) contributes none. Since process c appears in both the gain and the loss terms, the corresponding contributions cancel, so we need not explicitly exclude them. The increase off(n,t) due to processes a and b is: (gain term), = 2n
5 5 uf(n’~)f(n’d+ c
n‘l=O
n’2+n’l-n>0
n;-O
nJl=O
~f(n’~)f(2n-n’1)(A.2)
C n;=o
C n‘l=O
=
uf(n’Jf(n’2) + d C l = O uf(n’1)f(2n-n’l) -
To obtain the decay of a small perturbation g(n,t) from the equilibrium distribution, fo(n) = N( 1 - exp(-@)) exp(-np), we first write f(n,t) =fo(n)
+g(w)
(A.5)
Then we substitute this form off(n,t) in (A.4) and retain terms up to the first order in g(n,t). This leads to dg(n,t) -dt
-
n‘l+n‘z-nLO 2n
C
n‘]=0
uB(g(n’l) exp(-p(2n - n’,)) + g ( 2 n - t ~ ’ ~exp(-pn’,)l ) n+nl
C 2 UBMn) exp(-pn2) + g(n2) exp(-8n)lnl=O d l = O uB(2n)2g(n) exp(-pn) (A.6) Here B = N( 1 - exp(-P)). To obtain (A.6) from (A.4) and (A.5) we have used the fact that fo(n) is the equilibrium solution. The first term in the first summation can be simplified as follows: m
c
n’,=O n’,+n\Ln
m
X
uBg(n’d e ~ p ( - p n ’ ~ )=
n;=O
”
-
n-l
C
C uBg(n’l) exp(-pn’d
n;=n-n;
=fn’]=n
oBg(n’1)
nJl=O
+
(1 - exp(-P)) m
n-1
c
a=O n‘]=O
uBg(n’1) exp(-p(n - n’l
+ a ) ) (A.7)
+
in which a is defined by n’2 = n - n’l a. The summation over a can now be performed to obtain the factor 1/(1 - exp(-@)). Now since x g ( n ) = 0, we may write n-1
5- g(n’1) = - x g(n’1)
n;=n
n‘l=O
(A.8)
The Journal of Physical Chemistry, Vol. 93, No. 15. 1989 5937
Nonequilibrium Processes in Polymers
2 exp(-Pn)kg(k)]
- aB(4n) exp(-Pn)g(n) (A.13)
k=O
The second term in the first summation in (A.6) is the same as first. Thus the total gain in g(n,t) (i.e., the first two summations in (A.6)) can be written as
2”
2 a B z exp(-P(2n
Now since B = N( 1 - exp(-P)) andf,(n) = B exp(-pn) we can write the above expression in a matrix form as (loss), =
- k ) ) g(k) (A.lO)
(A.14)
k-0
in which we have replaced n’] in the second summation by k = 2n - n’]. Here
Combining (A.lO) and (A.14) we can write dg(n,t) - - (gain), dt
The third summation in (A.6) can easily be simplified because the summation over n’l can be performed. The loss term (i.e., the third summation plus the last term in (A.6)) can be simplified as follows. The summation over n’] gives -aB
-
m
u N c exp(-@n)kg(k) (A.15) k=O
c (n + n,)[g(n)
exp(-Pn,)
n2=0
+
exp(-Pn)l -
By defining
uB(4n)g(n) exp(-pn) = c
n
+ (n) + -
(A.16)
and
m
c An,) - c nzg(n2) exp(-Pn)
n2=0
- (loss),
n2=0
- aB(4n)g(n) exp(-Pn)
(A.12) Noting that for the Flory distribution the average values of n, (n), is given by
2fo(n) exp(-P(n - k ) ) - k exp(-pn) N we can write (A.15) as:
C(n,k) =
(A.17)
m
( n ) = Cn(1 - exp(-P)) exp(-Pn) n=0
and using the fact Cg(n) = 0, we can write the loss term (A.12) as
which is (16) in the main text. In the above expression it can be seen that C(n,k) is very small compared to G(n,k) and F(n,k); the first term in C is reduced by the factor 1/N while the second term decreases exponentially with n.
