End-Group Interchange Reaction in a Homopolymer Melt

Macromol. Theory Simul. 2001, 10, 355–362 355

End-Group Interchange Reaction in a Homopolymer

Melt

Yaroslav V. Kudryavtsev

Topchiev Institute of Petrochemical Synthesis of the Russian Academy of Sciences,Leninskii pr. 29, Moscow B-71, 117912, RussiaFax: 7(095)2302224; E-mail: [email protected]

Introduction

In preceding work,[1] we developed the theoretical

description for a direct interchange reaction in a homopo-

lymer melt. This reaction brings an arbitrarily taken

initial molecular weight distribution (MWD) to the sta-

tionary geometrical distribution. For linear polymers, it is

known as the most probable (Flory) distribution.[2]

Describing the reaction kinetics, two problems were con-

sequently solved. First, we derived the equation for the

MWD function in the most simple form and then found

its analytical solution.

In this work, we study the kinetics of an end-group inter-

change reaction, such as alcoholysis, aminolysis or acidoly-

sis (see examples in the literature[3, 4]), in the same manner.

Some previous theoretical studies on this topic should

be mentioned. An equation for the MWD evolution in the

course of end-group interchange was derived long ago by

Hermans[5] and Abraham[6] who showed that this reaction

results in the Flory MWD. Hermans obtained a general

solution for the generating function of the transient

MWD. However, considering special cases, he encoun-

tered mathematical difficulties that he could not over-

come.

Kotliar[7, 8] considered different interchange reactions in

a blend of two condensation polymers that may be chemi-

cally different. His theory has been based on the state-

ment that an interchange reaction may be treated as a

two-step process, namely, several random chain cleav-

ages followed by the same number of random couplings

of chain ends. Being relatively simple, Kotliar’s approach

makes it possible to calculate different averages over the

transient MWD and to follow the evolution of sequence

distribution in the course of a reaction. However, we

demonstrated[1] that the order of cleavages and couplings

affects the MWD and can not be chosen arbitrarily. In

this connection, we believe that more consistent results

may be obtained with the assumption that each chain

cleavage is followed by an immediate coupling. Note that

this is exactly the case for interchange in a system with-

out catalyst.

Full Paper: Kinetics of end-group interchange reaction ina homopolymer melt is studied in theory. The relaxationof the molecular weight distribution to its most probable(Flory) stationary form is considered. To this end, thetime-dependent generating function of the transient distri-bution is calculated analytically. Peculiarities of therelaxation process are investigated for two types of theinitial distribution, namely, the sum of two Flory distribu-tions with different number averages N1 and N2, and thedelta-function. In each case, the dependencies of the dif-ferential molecular weight distribution and the weight-and z-average polymerization degrees on the number ofinterchanges per end group are obtained in the explicitform. The reaction kinetics is compared with that of directinterchange studied in preceding work.

Macromol. Theory Simul. 2001, 10, No. 4 i WILEY-VCH Verlag GmbH, D-69451 Weinheim 2001 1022-1344/2001/0404–0355$17.50+.50/0

Time evolution of the reduced weight-average polymeriza-tion degree Nw (s)/N. Initially we take an 1 :1 wt/wt mixtureof two homopolymer fractions each having the Flory MWDbut different N2 /N1 = 5 (curve 1), 10 (2), 20 (3).

356 Y. V. Kudryavtsev

Lertola[9] numerically solved the kinetic equation for

end-group interchange with the initial conditions corre-

sponding to a monodisperse melt. In our work, we consider

the same special case and confirm his result analytically.

An original derivation procedure for the kinetic equa-

tion was proposed by Kondepudi et al.[10] It is based on the

analogy between interchange reactions and collisions in a

gas of hard spheres. From the kinetic equation, the linear

relaxation rate of a perturbation in the n-mer concentration

was calculated. The validity of this analytical result was

checked by Monte Carlo simulation of the reaction. Later,

the diffusive intermixing of chains of different lengths was

incorporated into the kinetic equation.[11]

In this work, we explicitly solve the kinetic equation

for end-group interchange. First, we reproduce the deriva-

tion of the kinetic equation using the notation introduced

in our previous work,[1] where direct interchange was con-

sidered. Unlike Hermans,[5] we use continuous MWD

and, correspondingly, a Laplace transform as its generat-

ing function. Then, we consider the kinetics for two spe-

cial cases of initial conditions, i.e., a sum of two Flory

distributions describing a blend of two condensation

polymers, and the Dirac delta-function corresponding to a

monodisperse melt. Since the same special cases were

studied in previous work,[1] we compare the MWD evolu-

tion in the course of end-group interchange with that of

direct interchange.

In all earlier considerations, a uniform one-phase melt

was considered and intra-chain interchange reactions

leading to the formation of cyclic macromolecules were

neglected. We also adopt these basic assumptions.

Theory

Consider a melt of m homopolymer chains with the total

number of repeat units being equal to n. Let each end-

unit of any chain contain a reactive end group. All reac-

tive groups are assumed to be identical. A contact

between an end-unit and another unit may result in the

breakage of the latter. Correspondingly, the chain to

which it belongs cleaves into two segments. Right away,

the end group detaches from its chain and attaches to one

of the segments. The chain that lost the end group couples

with the other segment. In such a case, we say that an

end-group interchange reaction takes place. Just as in our

previous work,[1] we identify an interchange with a

change in chain length. Therefore, each reaction event

includes two interchanges.

Assume that each end group reacts with probability c

per unit time. Obviously, the reaction keeps m and n con-

stant, thus, the number-average polymerization degree

N = n/m does not change as well.

We introduce mi = m (i, t) for the total number of chains

consisting of i units (i-chains, i F 1) at the time t. A new

i-chain appears if

a) An end group of any chain attacks an l-chain with

l F i so that the l-chain cleaves into (i–1)-segment and

(l–i)-segment, then, the (i–1)-segment joins to the end

group of the attacking chain becoming an i-chain. The

probability that one of two randomly chosen contacting

units contains an end group and another one belongs to

an l-chain is 4mlml /n2. The total number of neighbor unit

pairs in the melt is nz/2, where z is the coordination num-

ber. Then, we must take into account that only 2 of l pos-

sible positions of the reacting unit in an l-chain are appro-

priate to obtain an i-chain. Besides, the end group

attaches to the (i–1)-segment with the probability 1/2.

(For an actual condensation polymer, only one of two

segments is able to attach an end group but this segment

would be of proper length with the same probability 1/2.)

Thus, the described mechanism changes mi byPv

l¼i c (nz/

2)(4mlml /n2)(2/l)(1/2) = 2cz

Pv

l¼iml /N per unit time.

b) The attack of an l-chain by an end group of a j-chain

with j f i results in the cleavage of the l-chain into (i–j)-

and (l–i + j–1)-segments, then, the j-chain replaces its

end group by the (i–j)-segment thus becoming an i-chain.

In this case, the number of i-chains increases with the ratePi

j¼1

Pv

l¼iÿjþ1 c (nz/2)(4mj lml /n2)(2/l)(1/2) = 2czPi

j¼1Pv

l¼iÿjþ1 mj ml /n.

c) An i-chain disappears if it is attacked by another

chain. In this case, it is indifferent, which one of the two

segments generated after i-chain cleavage joins to an

attacking end group. It doubles the rate of the i-chains

population decrease that has the form c (nz/2)(4mimi /n2)

= 2czimi /N.

d) An end group of any i-chain reacts. By analogy to

the previous item that the i-chain disappears with the uni-

tary probability, we obtain the rate of i-chains decay:

c (nz/2)(4min /n2) = 2czmi .

Gathering two gain terms and two loss ones, we may

write down the kinetic equation for mi

qmi=qt¼ kXv

l¼i

ml=NþXi

j¼1

Xvl¼iÿjþ1

mlmj=nÿmiÿ imi=N

!ð1Þ

where k = 2cz is the effective rate constant. Exactly this

equation had been derived previously.[5, 6] We should note

that gain and loss terms in Equation (1) were calculated

without exclusion of situations where i-chains react pro-

ducing new i-chains. It is clear, however, that these reac-

tions do not change the value of mi . Thus, Equation (1) is

exact and surplus terms in gain and loss terms cancel

each other.

It is easy to check whether Equation (1) satisfies the

conservation laws for the total number of chains and

units. Performing trivial summations, we obtain

qPv

i¼1 mi

ÿ �/qt = qm/qt = 0 and q

Pv

i¼1 miiÿ �

/qt = qn/qt = 0.

The stationary solution of Equation (1) mð0Þi is the geo-

metrical, or normal, distribution. Indeed, if we put qmi /

End-Group Interchange Reaction in a Homopolymer Melt 357

qt = 0 in Equation (1), substitute mð0Þi = ch i and perform

trivial summations, then we have:

1

Nð1ÿ hÞþ cih

nð1ÿ hÞ ÿ 1ÿ i

N¼ 0 ð2Þ

As this relation is valid for any i, it enables to find both

constants c and h at once:

h¼ 1ÿ1=N; c¼ nð1=hÿ1Þ=N ¼m=ðNÿ1Þ ð3Þ

Hence:

mð0Þi ¼

m

ðN ÿ 1Þ1ÿ 1

N

� �i

ð4Þ

For the polymeric case N S 1, it turns into the most

probable (Flory) distribution mð0Þi = i exp(–i/N)/N

2.

To find an explicit time-dependent solution of Equa-

tion (1) for N S 1, we introduce the continuous distribu-

tion b(x,s)dx = m(i, t) /m, where x = i/N and s = kt are

new reduced variables. Replacing sums by integrals, we

arrive at the equation:

qbðx; sÞ=qs ¼Z v

x

dybðy; sÞ þZ x

0

dybðy; sÞZ v

xÿy

dhbðh; sÞ ÿ ðxþ 1Þbðx; sÞ ð5Þ

Here b(x,s) is the differential number fraction of

chains, i.e., the density of probability that a randomly

chosen chain has a polymerization degree falling into the

interval [Nx, N (x + dx)]. The constancy of the total num-

ber of units and chains provides normalization conditionsRv

0bðx; sÞdx = 1 and

Rv

0bðx; sÞxdx = 1, respectively.

It is appropriate to apply the Laplace transformation

bðp; sÞ ¼Z v

0

eÿpxbðx; sÞdx ð6Þ

to both sides of Equation (5). Thus, we obtain:

qbðp; sÞ=qs ¼ ð1ÿ b2ðp; sÞÞ=pÿ bðp; sÞ þ qbðp; sÞ=qp ð7Þ

Equation (7) is the first order quasi-linear partial differ-

ential equation. Its solution may be found from the equa-

tion F(C1, C2) = 0, where C1, C2 are independent first inte-

grals of the system of ordinary first-order differential

equations:

db

bÿ ð1ÿ b2Þ=p¼ ÿds ¼ dp ð8Þ

Using standard methods,[12] we find:

C1 ¼ sþ p; C2 ¼ eÿp N1ÿ bðpþ 1Þbþ pÿ 1

ð9Þ

The function F is to be determined from the initial con-

dition b (p,0) =R v

0eÿpxb (x,0)dx = b0 (p). After some alge-

bra we obtain:

bðpÿ s; sÞ

¼ b0ðpÞþpÿ1þeÿsð1ÿpþsÞð1ÿb0ðpÞ N ð1þpÞÞðb0ðpÞþpÿ1Þð1þpÿsÞþeÿsð1ÿb0ðpÞ N ð1þpÞÞ ð10Þ

Going over to the object function, we have:

bðx; sÞ ¼Z aþiv

aÿiv

bðpÿ s; sÞeðpÿsÞxdp ð11Þ

Using Equation (10), (11) we can describe the evolu-

tion of the MWD for any given initial conditions.

Now we should make two important remarks. First,

note that the reduced time s = kt = 2czt is just the number

of interchanges per chain end during the time t. The value

of s determines the shape of the MWD function b (x, s).

Since s is independent of N, we may conclude that the

rate of the MWD evolution towards its stationary form

does not depend on the average molecular weight of a

melt. At first sight, it may seem obscure because the num-

ber of chain ends and, hence, the total number of inter-

changes per unit time is proportional to N. However, the

number of chains that should undergo the reaction

changes with N in the same proportion. As a result, melts

with different N would be characterized by identical

MWD curves b (x, t) at any given time t.

Second, one can notice that b (p,s) is a generating func-

tion for the MWD. Hence, any average over the distribu-

tion may be easily calculated. For example, the weight

average polymerization degree is

NwðsÞ ¼ N

Z v

0

bðx; sÞx2dxZ v

0

bðx; sÞxdx

¼ÿNd2bðp; sÞ

dp2

�dbðp; sÞ

dp

� �p¼0

¼ 2N

1ÿ eÿsð1þ ðsÿ1 þ ðb0ðsÞ ÿ 1Þÿ1Þÿ1Þð12Þ

and the z-average polymerization degree is

NzðsÞ ¼ N

Z v

0

bðx; sÞx3dxZ v

0

bðx; sÞx2dx

¼ÿNd3bðp; sÞ

dp3

�d2bðp; sÞ

dp2

� �p¼0

¼3Nð1ÿsÿb0ðsÞÞ2ÿeÿsððð1ÿb0ðsÞÞ2þs2ðdb0ðsÞ=dsÞÞð1ÿsÿb0ðsÞÞ2ÿeÿsð1ÿsÿb0ðsÞÞ N ð1ÿð1þsÞb0ðsÞÞ

ð13Þ

where b0ðsÞ =R v

0eÿsxb (x,0)dx. Now let us consider two

special cases.


A The Blend of Two Chemically Identical Polymers

Having the Most Probable MWD but Different

Number Averages N1 and N2

Initially we may write

mði; 0Þ¼ m1

ðN1ÿ1Þ1ÿ 1

N1

� �i

þ m2

ðN2ÿ1Þ1ÿ 1

N2

� �i

ð14Þ

where m1 and m2 are the total number of chains of each

component. Introducing g as the fraction of units of the

component with number average N1 so that m1 = ng/N1

and m2 = n(1 – g)/N2, we easily calculate the number

average over all chains:

N ¼ n=ðm1 þ m2Þ ¼ N1N2=ðN1ð1ÿ gÞ þ N2 gÞ ð15Þ

The corresponding continuous distribution has the

form b (x,0) = ((1 – b)a2 exp(–ax) – (1 – a)b2 exp(–bx))/

(a – b), where x = i/N, a = N/N1, b = N/N2. Its Laplace

transform is

b0 (p) = ((1 – b)a2/(p + a) – (1 – a)b 2/(p + b))/(a – b) (16)

Substituting (16) into (10) we obtain

bðpÿs; sÞ¼ pð1þeÿsðkÿ sÞÞþkÿðsþ1Þeÿsðkÿ sÞp2þpðkþ1ÿsÞþkð1ÿsÿeÿsÞþ seÿs

ð17Þ

where s = ab, k = a + b – 1.

We see that the denominator of Equation (17) is a

quadratic polynomial in p. Its discriminant is D = (k –

1 + s)2 + 4(k – s)exp(–s). Using the definitions of s, k, a,

b, we easily obtain that k – s = g (1 – g)(N1 – N2)2/

(N1(1 – g) + N2 g)2 F 0. Thus, D F 0, so the polynomial

in the denominator of Equation (17) has two real roots

p1,2 = (s – k – 1 l D1/2)/2.

Applying the inverse Laplace transformation we

obtain:

bðx; sÞ ¼X2

i¼1

ðÿ1ÞieðpiÿsÞx N ðpið1þeÿsðkÿ sÞÞþkÿðsþ1Þeÿsðkÿ sÞÞp2 ÿ p1

ð18Þ

The transient MWD may be characterized either by

b (x,s) or by q (x,s) = xb (x,s), where q (x,s) is the differ-

ential weight fraction, i.e., the density of probability for a

unit to belong to a chain with the polymerization degree

falling into the interval [Nx, N(x + dx)]. The dependence

q (x,s) on x is plotted in Figure 1 for different values of s,

the reduced time s being the number of interchanges per

end group. One can see that at s L 1, the maximum of the

distribution shifts to the position characteristic for the

most probable MWD, whereas two interchanges per end

group almost bring the whole curve to its stationary form.

Using Equation (15), (16) we can easily calculate the

weight-average polymerization degree

NwðsÞ ¼2N

1ÿ eÿsðkÿ sÞ=ðsþ kÞ ð19Þ

and the z-average polymerization degree:

NzðsÞ ¼ 3Nðsþ kÞ2 þ eÿsðkÿ sÞ

ðsþ kÞ2 ÿ ðsþ kÞ eÿsðkÿ sÞð20Þ

The dependencies Nw (s)/N and Nz (s)/N are plotted in

Figure 2 and 3 for three different blends with constant

g = 0.5 but different N2 /N1 ratio. The highest evolution

rate is observed at early stages (up to s L 0.5). In spite of

large differences in the initial values, only one inter-

change per end group is sufficient to bring all averages

close to their equilibrium values (Nw (v) = 2N,

Nz (v) = 3N for N S 1). At late stages (s A 1), the evolu-

tion of averages proceeds very slowly and similarly for

all blends.

Figure 1. Differential weight fraction of chains q plotted ver-sus reduced polymerization degree x = N/N ((a): 0 a x a 5; (b):5 a x a 10). Curves correspond to the different values of thereduced time s = k t. Dotted line: s = 0 (1 :1 wt/wt mixture oftwo homopolymer fractions each having Flory distribution, N2 /N1 = 10). Hairline: s = v (the final Flory distribution,q (x) = xexp(–x)). Solid lines: s = 0.2 (1), 0.5 (2), 1 (3), 2 (4).


B Interchange Reaction in the Initially Monodisperse

Blend

In this case, all of m chains initially consist of N inter-

changeable units. It yields b (x,0) = d(x – 1). Going over

to the Laplace transform b (p,0) = exp(–p), we obtain

from Equation (10):

bðpÿ s; sÞ

¼ eÿpð1ÿeÿsð1ÿpþsÞð1þpÞÞþseÿsÿð1ÿpÞð1ÿeÿsÞeÿpðð1þpÞð1ÿeÿsÞÿsÞþeÿsÿð1ÿpÞð1þpÿsÞ ð21Þ

The object function b (x,s) can not be found directly

from Equation (21) using the residue theorem, since

b (p – s,s) does not uniformly converge to zero as

|p| e v (Re p A a A 0, a = const). However, eliminating

the term Vp2 from the numerator, we may present Equa-

tion (21) in the form:

bðpÿ s; sÞ¼ eÿpÿs

þ eÿ2pÿsðsÿð1þpÞð1ÿeÿsÞÞþeÿpð1ÿ2seÿsÿeÿ2sÞþseÿsÿð1ÿpÞÞð1ÿeÿsÞeÿpðð1þpÞð1ÿeÿsÞÿsÞþeÿsÿð1ÿpÞð1þpÿsÞ ð22Þ

The first term on the right-hand side of Equation (22)

is the image of the generalized function exp(–s)d (x – 1),

whereas the second term behaves like 1/p as |p| ev and,

consequently, its object function may be found using

standard methods.[13] This formal trick finds its clear

physical interpretation. Indeed, for the initial condition

b (x,0) = d (x – 1), the singularity of the transient MWD

at x = 1 should correspond to unreacted chains at any s. It

follows from Equation (5) that the differential number

fraction of unreacted chains bu (x,s) obeys the equation

qbu (x,s)/qs = –(x + 1)bu (x,s). Taking into account the

initial condition bu (x,0) = b (x,0) = d (x – 1), we find that

bu (x,s) = d (x – 1)exp(–(x + 1)s). Applying Laplace

transform, we obtain bu (p,s) = exp(–p – 2s), so that

bu (p – s,s) = exp(–p – s) coincides with the first term on

the right-hand side of Equation (22).

For s A 0 the denominator of the fraction in Equa-

tion (22) has two real roots: a simple root p1, which is

dependent on s, and a double root p2 = 0. The function

p1 = p1(s) is drawn in Figure 4. At the point s = s*, where

s* L 1.3 is the root of the equation exp(–s) – s + 1 = 0,

roots p1 and p2 coincide and one triple root p = 0 exists.

Applying the residue theorem, we finally obtain

bðx; sÞ ¼ eÿsxðdðxÿ 1Þeÿs þ b1ðx; sÞ þ b2ðx; sÞÞ

b1ðx; sÞ ¼ep1x

eÿp1ðsÿ p1ð1ÿ eÿsÞÞ þ 2p1 ÿ s

ðseÿs þ ðp1 ÿ 1Þð1ÿ eÿsÞþ eÿp1ð1ÿ 2seÿs ÿ eÿ2sÞgðxÿ 1Þþ eÿ2p1ÿsðsÿð1þp1Þð1ÿeÿsÞÞgðxÿ2ÞÞð23Þ

where g (x) is the unit step function g (x) = 1, x A 0 and

g (x) = 0, x f 0,

Whereas the first term on the right-hand side of Equa-

tion (23) is the fraction of unreacted chains, the other

terms correspond to the fraction of chains having under-

gone at least one interchange. The differential weight

fraction of reacted chains qr(x,s) = x exp(–sx)(b1 (x,s)

+ b2 (x,s)) was plotted versus x in Figure 5 for different

values of reduced time s. One can see (curve 1) that, at

early stages, qr (x,s) grows almost linearly with x for 0 a

Figure 2. Time evolution of the reduced weight-average poly-merization degree Nw (s)/N. Initially we take an 1 :1 wt/wt mix-ture of two homopolymer fractions each having the Flory MWDbut different N2 /N1 = 5 (curve 1), 10 (2), 20 (3).

Figure 3. Time evolution of the reduced z-average polymeriza-tion degree Nz (s)/N. Curves correspond to the same initial con-ditions as in Figure 2.

b2ðx; sÞ ¼

2ð1ÿ eÿsÞð1ÿ xÞ þ xseÿs

1þ eÿs ÿ sÿ 2

3

ðseÿs ÿ 1þ eÿsÞðsþ 2ÿ 2eÿsÞð1þ eÿs ÿ sÞ2

; 0 a x a 1

2eÿsð1ÿ eÿsÞðxÿ 1Þ þ sð2ÿ xÞ

1þ eÿs ÿ sÿ 2

3eÿsð1ÿ sÿ eÿsÞðsþ 2ÿ 2eÿsÞ

ð1þ eÿs ÿ sÞ2; 1 a x a 2

0; x A 2

8>>>><>>>>:


x a 2 since end-group interchange between two N-chains

yields (N – a)- and (N + a)-chains, where all values of a

between the limits 0 and N are equally probable. Before

s = 1, the shape of the MWD curve is far from the most

probable one, with the maximum being close to x = 2.

Within the interval 1 a s a 2, the maximum moves to its

stationary position at x = 1. Afterwards, a slow relaxation

of the whole MWD curve takes place.

The singular initial conditions cause a discontinuity of

q (x,s) that is easily visible in Figure 5 at x = 2. Indeed,

no chain longer than 2N can be formed in the reaction

between two N-chains. The jump at x = 2 disappears at

later stages as the weight fraction of unreacted chains

xbu (x,s) exponentially decreases with time. Another

minor discontinuity located at x = 1 is nearly impercepti-

ble. The transient MWDs numerically calculated by Ler-

tola[9] are in good agreement with the results of the pres-

ent consideration.

Finally, we obtain the average polymerization degrees

from Equation (12), (13):

NwðsÞ ¼ 2N1ÿ sÿ eÿs

ð1ÿ eÿsÞð1ÿ sÿ eÿs ÿ seÿsÞ ð24Þ

NzðsÞ ¼ 3N

1ÿ eÿs

1ÿ sÿ eÿs

ÿ sð1þ eÿ2sÞð1ÿ eÿsÞð1ÿ sÿ eÿs ÿ seÿsÞ

!ð25Þ

The dependencies Nw (s)/N and Nz (s)/N are plotted in

Figure 6. Contrary to case A, the averages increase in the

course of end-group interchange. Their values become

close to the stationary ones after 3 interchanges per end

group.

Discussion

In this section, we compare the kinetics of end-group

interchange with that of direct interchange. Both reac-

tions lead to the Flory MWD, however, the relaxation

processes may have distinctive features. To find the dif-

ference, we should consider the transient distributions

originated under the same initial conditions. Therefore,

we study the same special cases A and B in this work as

in the previous one.[1]

First, note that for both reactions the MWD functions b

and q depend on reduced molecular weight x = N/N and

reduced time s only, however, s is introduced differently.

For direct interchange, sd = Ncd zt, where sd stands for

the number of interchanges per average chain during the

time t. For end-group interchange, se =2cezt, se being the

number of interchanges per chain end. Since each chain

has two end groups, an average chain undergoes 2se inter-

changes during the time t. Thus, it makes sense to com-

pare distributions qd (x,sd) and qe(x,se /2) corresponding to

the same number of interchanges per average chain

sint = sd = se /2.

To this end, we evaluate average x (sint) and mean

square deviation r(sint) of variable x over the distribution

q(x,sint). By the definition of averages Nw (sint) and Nz (sint),

we may write that x(sint) = Nw /N and r(sint) = (Nw (Nz –

Nw))1/2/N. These dependences are plotted in Figure 7 for

Figure 4. Numerical solution of the equation exp(–p)((1 +p)(1 – exp(–s)) – s) – (1 – p)(1 + p – s) + exp(–s) = 0: p1 ver-sus s; another root p2 = 0 at any s.

Figure 5. Differential weight fraction of chains having under-gone at least one interchange qr versus reduced polymerizationdegree x = N/N for initially monodisperse melt: s = 0.1 (curve1), 0.5 (2), 1 (3), 2 (4), infinite time (hairline).

Figure 6. Change of the reduced weight- and z-average poly-merization degrees Nw (s)/N (thick line) and Nz (s)/N (thin line)with reduced time s = kt for initially monodisperse melt.


case A and in Figure 8 for case B yielding the values of x

and r at a given number of interchanges per average

chain. One may see that end-group interchange is more

effective in both cases. The difference is much more pro-

nounced if the reaction broadens the MWD (case B),

whereas the rate of the MWD narrowing (case A) is

nearly insensitive to the type of interchange. Note that

these conclusions do not directly concern the real-time

relaxation of the MWD that depends on N and rate con-

stants ce , cd .

The dependence on N exists for direct interchange

only, as the relaxation by means of end-group interchange

appeared to be insensitive to the average molecular

weight of a melt. Formally, the difference consists in the

definition of the reduced time: se = ce zt for end-group

interchange, and sd = Ncd zt for direct interchange. In the

latter case, a melt with greater value of N would approach

the Flory MWD more rapidly as longer chains go through

more interchanges at the same time than shorter ones.

Consequently, if end-group interchange and direct inter-

change concurrently took place in the system, the relative

contribution of the latter process to the real-time relaxa-

tion of the MWD would increase with N.

Conclusions

In two articles, this one and ref.,[1] we described explicitly

the kinetics of end-group interchange and direct inter-

change reactions in a homopolymer melt. We obtained a

general solution for the generating function of the transi-

ent MWD.

We found that MWD relaxation in the case of end-

group interchange is insensitive to the average molecular

weight of a melt. On the contrary, direct interchange pro-

vides faster relaxation of the MWD in a melt with greater

N.

The comparative analysis of two special cases demon-

strated that the evolution of the MWD may proceed dif-

ferently depending on the type of interchange and the

initial conditions as well. However, the Flory MWD was

confirmed to be the stationary one in every case.

Having replaced the time by the number of inter-

changes per average chain, we demonstrated that end-

group interchange provides “faster” relaxation of the

MWD than direct interchange does, especially, if the

MWD broadens in the course of the reaction.

In this work, we neglected intra-chain reactions leading

to the formation of cyclic species. The problem of incor-

porating them into the theory will be considered else-

where.

Acknowledgement: Dr. E. N. Govorun and Prof. A. D. Litma-novich are thanked for the valuable comments on the manu-script. Financial support from the Russian Fund for BasicResearch (project no. 00-03-33193a) is gratefully acknowl-edged.

Received: September 13, 2000Revised: December 6, 2000

[1] Ya.V. Kudryavtsev, Macromol. Theory Simul. 2000, 9, 675.[2] P. J. Flory, J. Am. Chem. Soc., 1942, 64, 2205.[3] A. M. Kotliar, J. Polym. Sci.: Macromol. Rev. 1981, 16,

367.[4] C. Koning, M. van Duin, C. Pagnoulle, R. Jerome, Prog.

Polym. Sci. 1998, 23, 707.[5] J. J. Hermans, J. Polym. Sci. C 1966, 12, 345.[6] W. A. Abraham, Chem. Eng. Sci. 1970, 25, 331.

Figure 7. Differential weight distribution parameters: averagex (thick lines) and mean square deviation r (thin lines) vs thenumber of interchanges per average chain sint. The initial state isan 1 :1 wt/wt mixture of two homopolymer fractions each hav-ing Flory distribution with N2 /N1 = 10. The reaction is directinterchange (solid lines) or end-group interchange (dashedlines).

Figure 8. Differential weight distribution parameters: averagex (thick lines) and mean square deviation r (thin lines) vs thenumber of interchanges per average chain sint . Initially, we havea monodisperse melt. The reaction is direct interchange (solidlines) or end-group interchange (dashed lines).


[7] A. M. Kotliar, J. Polym. Sci.: Polym. Chem. Ed. 1973, 11,1157.

[8] A. M. Kotliar, J. Polym. Sci.: Polym. Chem. Ed. 1975, 13,973.

[9] J. G. Lertola, J. Polym. Sci., Part A: Polym. Chem. 1990,28, 2793.

[10] D. K. Kondepudi, J. A. Pojman, M. Malek-Mansour, J.Phys. Chem. 1989, 93, 5931.

[11] J. A. Pojman, A. L. Garcia, D. K. Kondepudi, C. Van denBroeck, J. Phys. Chem. 1991, 95, 5655.

[12] E. Kamke, “Differentialgleichungen. Losungsmethodenund Losungen”, Akad. Verlagsgesellschaft, Leipzig 1943.

[13] G. Doetsch, “Anleitung zum praktischen Gebrauch derLaplace Transformation und der Z-Transformation”, 3rd

edition, Oldenbourg Verlag, Munchen 1967.

Documents

End-Group Interchange Reaction in a Homopolymer Melt