Diffusion-controlled reaction of polymers

Chemiial Physics 9 (1975) 455-466 8 North-Holland Publishing Company

DIFFUSIONCONTROLLED

REACTION OF FOLYMERS

MASAO DO1 Departmentof Physics, Faculty of Science, Tokyo Metropolitan

University. Setagwa-ku.

Tokyo. Japatt

Received 7 April 1975

An analysis is made of diffusinn-controlled ring closure reaction of a polymer chain based on the general theory of WiIemski and Fixman. It is shown that the reaction rate is strongly affected by the short time behavior.of the segmental motion. and consequently the harmonic spring model and the Rouse model yield quite different results. This explains the characteristic feature of the numerical results of Wikmski and Fixman. A simple intuitive interpretation is made for this difference.

1, MIoduction Recently Wilemski and Fixman [1,2] have developed a general theory of diffusion-controlled intrachain reaction of a flexible polymer. As an application of their theory, they have discussed in detail the ring ciosure reaction [3], in which a pair of reactive groups are attached to both chain ends and react with certain intrinsic rate constant k if the reactive groups are within a distance R. This reaction is a fit order reaction and the fraction G(t) of the unreacted chain obeys the equation (1)

d@/dt = -kl@ ,

where kl is the tint order reaction rate. With certain closure approximation, they derived an equation fork, and solved this through an extensive numerical calculation for several models of polymers. The results include some qualitatively interesting features in comparison with the conventional results of the diffusion-controlled maction of simple molecules. In case that the reaction is purely diffusioncontrolled, i.e., the case that reaction takes place at every encounter of the reactive groups, one may expect that kl will decrease, with the decrease of R. the radius of the reaction region. However their results indicate that for the free draining Rouse chain, k, is almost independent of R and is about the same order of magnitude of inverse of the longest relaxation time ~~ of the end-to-end vector; kl = l/r,

.

(2)

On the other hand, for the harmonic spring model in which two reactive groups are COMeCted by a harmonic spring, their results indicate that kl decreases with decrease of R and is much smaller than 7;‘. For this mtxlel, the problem can be solved exactly [4], and k, is found to be approximately written as k, =2R/fiLr,

= (R/L&‘,

(3)

where L is the root mean square separation between the two reactive groups. Eq. (3) is smaller than eq. (2) by a factor R/L. which is very small for the usual polymers because R is an order of the bond lengthb. and L = &b (N being the degree of polymerization). Then a question arises what makes such qualitative and quantitative difference in the reaction rate between the Rouse model and harmonic spring model. Since Wilemski and Fixman carried out most of their actual calculation numerically for special values of R/L, they have not paid full attention on this difference and have not given a satis-

456

M. Doi/Di$xion-controlled

reaction of polymen

factory explanation for it. In this paper we shah clarity this question. We start from the formalism of Wilemski and Fixman and attempt to solve their equation for kr analyticahy. Since this equation is very complicated, we restrict ourselves to the investigation of the asymptotic form of k, in the limit of large degree of polymerization. We shall show that the specific result (2) is attributed to the characteristic behavior of the segmental motion of the Rouse chain in the short time region, which was pointed out by de Gennes and his coworkers [S,6].

.< I

2. Survey of the theory of Wilemski and Fixman

First we survey the theory of Wilemski and Fixman in a form appropriate for our later calculation. Their starting point is the general diffusion equation for the configurational distribution function $({r); r) of the whole system: atilar + 8r~ = -B({r))

+ .

(4)

In eq. (4), Q is the diffusion operator in the absence of the chemical reaction. The right hand side represents the sink term due to the chemical reaction: k is the intrinsic reaction rate constant. The function S({r)) restricts the configuration for the reaction to take place, i.e., S({r}) takes unity for the configuration where a pair of reactive groups are within a distance R, and zero otherwise. Here it should be mentioned that the above definitions of k and S are slightly different from those of Wilemski and Fixman. Following their definitions, k has a dimension of the second order reaction rate and S has that of inverse of volume. However this difference is not essential because in any case the final results include k and S in the combination of kS. Owing to the sink term -kS, the distribution function Jl({r); t) is not normalized and its integral over the whole confiiational space yieIds the unreacted fraction O(r), $0) = sd +I N(r); 0 .

(5)

According to the closure approximation

6(s) = ; -

of Wilemski and Fixman, the Laplace transform of@(f) is given by

kve,

sq1+ kb(s)/v,)

(6)

.

In eq. (6), kvq is the reaction rate at the initial equilibrium k% = k(S(IrI)) , and 3s) is the Laplace transform of the probability tirner=Oandt=r d(s) = $ df exp(-sr)(Sexp(-er)S) 0

state (7)

D(t) of finding the reactive groups within a distance R at both

_

In eqs. (7) and (8), the bracket (...) denotes the average over the equilibrium

(8) distribution

functionP_({r})

Co Throughout

this paper we denote the Laplace transform of any function g(f) by i(s):

i(s) = r dr exp (-sr)g(r) 0

.

M. Doi/Diffutin-conrrolled

reaction of polymers

451

The above formulation is quite general and can apply to any reaction system. Now we restrict to the pertinent ring closure reaction of an isolated polymer chain. We assume that the chain consists of N segments whose positions are labelled by rl ) r2, .... rN. Since the reactive groups are attached only to the chain ends, S is a function of the end-to-end vector u =rN - rl. Then eq.(8) is rewritten as

i(s)=

i

(10)

dr exp(-st)_rd3ulSd3u*S(ul)S(u2)G(ul,u2;t),

0

where G is the reduced Green function

;C(rr,.U2;t)=(6(ul-(rN--rl))exp(-St)6(u2-(rN-ft))),

(10

and S(u) is the step function

S(u)=H(R-

(

1

lul)=

a

lulS;R ,u,>R

(W

.

Wilemski and Fixman employed the boson representation method [7], to evaluate G(ul, ~2; f). Another elementary derivation of G is given in appendix 1. In any case G is given by

C(u,, u2; f) =

(1-

_ l+22p(t)U~*u2+4

P2(o)3’2 exp [

ww(0)3

where p(r) is the normalized time correlation p(t) = t-2
2g 0)

1,

function of the end-toend

(13) vector

L2 = (IA ,

and g(r) = (L2/3)(1 - ~~0)) . If we neglect the excluded volume effect, L2 is equal to M2 and p(t) is given by ~(1) =

$ pEap-2 exp (-X,t) ;

.

For the free draining Rouse chain, hP is expressed as ‘b = P%,

-

Here T,,, is the maximum relaxation time of p(r)

T, = cN2b2/3n2kBT

,

a being the friction constant of a segment). For the nondraining Rouse chain, AP can be obtained only approximately. Wiiemski and Fixman adopted a complicated expression of )b obtained from the diagonal approximation [7]. Here we employ simplified form of $, proposed by Dubois and de Cennes (61 hp = p3/21rm .

(19)

The longest relaxation time T,,, of the nondraining chain is different from eq. (IS), but the explicit form of rn, is not discussed here because it is not important in the following discussion. The p dependence of +_, is also &mged by the excluded volume effect [8]. Hence we proceed with our discussion assuming the general form

458

M. Doi~Diffision-contmlledreaction of polymers

On the other hand if we start from the harmonic spring model, we have the same equations (lO)-(lS). difference is that p(r) is given by

The only

p(f) = ex$ (- t/rm) .

(21)

implies that we approximate the sum in eq. (16) by the first term. This approximation is not so bad as far as the long tLne (t 2 T,) behavior is concerned. However as we shall see later, the R dependence of kt is mainly determined by the short time behavior of p(r) and consequently the Rouse model and the harmonic spring model yield quite different results. The setpf equations (6)-(21) determine e(f) completely. As was shown by Wilemski and Fixman [2], the singularities of e(s) are simple poles of real negative values. Hence Q(t) is expressed as the sum of exponentially decaying functions. Especially, the asymtotic decaying rate is given by the largest (or smallest in the absolute value) root of the equation hhtheIIIatiC~y.

this

1 +rti(S)/u,s

=0 .

(22)

However this equation is still very difficult to solve. Wilemski and Fixman have had recourse to the numerical calculations. Here we attempt to solve eq. (22) analytically. For this purpose we assume that N is sufficiently large. In that case it is possible to obtain the asymptotic form of kl. Note that B(s) includes only three parameters, L, R and 7,. If we choose L and T,,.,as the units of length and time, we have only one dimensionless parameter 7 = R/L, which approches zero in the limit of N+ Q). By use of this fact we can investigate the R or N dependence of kt .

3. Simplification of the problem In this section we shall further simplify our problem. In the previous section,d(u) is defined as the step function (see eq. (12)). However the choice of S is not unique. Function S merely indicates the restriction that reaction should occur in the reaction region whose dimension is about R. Therefore here we take the following Gaussian form for the sake of computational convenienos, S(u) = exp (- 3rr2/2R2) . Then substitution

of eq. (23) into eq. (7) yields

u~=Jd3y(&j~2w

Furthermore,

(23)

(-j$)ew(-$$)

(inthelimitofr_cO~24)

=(1+72)-3’273=73,

from eqs. (lo), (13) and (23), we have

&s)=rdrexp(-sr) 0

(l+g+$

g(r))-3’2=+5

[

dt exp (--st)

(1 -

p2 + 272

+ #A)-312

.

(25)

In the denominator of eq. (25). we can omit fl because it is always smaller than 2r2, but we cannot omit the term 2r2 because 1 -p2 becomes smaller than 2r2 for t = 0. Hence &s)=~~/drexp(-sf)(l-p~i-27~)-~~~.

(26)

Since our interest lies in the behavior of as) with small value of s, we may expand b(s) into a power series of s. Here care must be taken to the fact that the first term in the expansion is not a constant term but a term of s-l,

M. Doi/Diffision-contml;rd

459

reaction of polymers

because b(t) approaches a constant value y6 as t * -. If y6 is subtracted from D(t), the Laplace transfbrm of the remaining part can be expanded. Thus we may write (W

~(S)=y6[s-t~DO(y)-D1(y)S+Dt(y)S2-...], where D,(Y)=;

Jdt [(1-p2+272)-3~2-1]P.

(29)

0

The integral for D”(y) converges because the integrand behaves as 1” exp (-t/r,) into eq. (6) we have ((ky3)-‘+LIo) - D,s + ... d(s) = 1 +((ky3)-‘+D,)s--Dts~ + _.. -

for t 9 r,. Substituting

eq. (28)

- (30)

Note that J(s) has no pole at s =O. This must be so because in the intrachain reaction system, the umeacted fraction e(t) must go to zero as t + =. Thus the reaction rate kt is given by the root of the equation l-(llky3tDo)k,-D&

(31)

...=o.

Therefore our problem is reduced to solve eq. (31) in the limit of y + 0. In the subsequent section this is done for several models of polymers.

4. Harmonic spring model Let us first consider the harmonic spring model. In this case p(t) is given by eq. (21). To see the asymptotic form of D,(y) for y -+ 0, we consider the integral DJO)=$Jdt

[(1-p2)-3/2-1]t”. 0

Since the integrand behaves as integrand a

I

P exp (- t/r,) p-312

for for

t % T, f*r,’

(33)

the integral (32) converges for n 2 1, but Do(O) diverges owing to the t -3/* dependence of the integrand fort Z= 0. To determine the asymptotic behavior of&(y) in the limit of y -b 0, we separate the integral into two parts TC

Do(r) = j- d t I(2 t/r, 0

+2~*)-~/~

- l] + s dt [(l - p*t 27+3/2-

l] ,

(34)

TC

where T= is some appropriate cut off time which is chosen so that for t 5 TV, p(t) may be approximated by 1 -$T,. (Actually TC may be about 0.5~~). The second integral in eq. (34) may be suppressed because it is not resport.siiIe for the divergence of Do(y). Ln terms of the new variable x = t/y*r, , the first integral is evaluated as &(X

+

1)-3/z

(35)

M. DoifDifji&on-conmdIe#

460

warsion of polymers

Then L+,(y) becomes infinitely large as 7 +O. Thus eq. (31) asymptotically

approaches the following form

As ‘y approaches zero. the coefficient of kl becomes very large, so that kl must be very small. Then we may safely omit the higher order terms of k,. Hence eq. (3 1) is solved as k, = [(f+)-‘+

J+t;

kl- = &%r,,,

(37)

.

. Thus the reaction rate is If the re⁣r is purely diffusion controlled, i.e., ifk+m, k, approaches k,, = JT+, may be well compared with the exact one (3) proportional to R in this case. The asymptotic value k,, =&r/r,,, obtained from the boundary value problem of the general diffusion equation (4) with the boundary condition Jl(u,f)=Oatlul=R. ‘Ihe above technique to evaluate the asymptotic form of kl yields rather accurate values. In fact if we adopt the same sink function as that of Wilemski and Fixman, we can completely reproduce their results of detailed numerical calculation for rR = O.Ol_ This is discussed in appendix 2. The important point of the above example is that the property kl, Q:R is closely related to the divergence of DO(~) in the limit of 730, or more fundamentally to the short time behavior of D(r), not the long time behavior. As can be seen from eqs. (16) and (21), the short time behavior of the segmental motion of the Rouse chain is quite different from that of the harmonic spring model. Hence one may understand that this difference is responsible for the qualitative difference in k, between the two models. In the next section we shall discuss this point more quantitatively.

5. Rouse model

?;I~ .,

First we investigate the short time behavior of p(f) with $ given by eq. (20). In case oft 4 TV, those terms of in the sum of eq. (16) have a significant contribution to p(f). Hence we may replace the sum by the integral : l-&)=8

.2 Pz

s

(I-

exp(-Pt/r,))

=G {dpp-2(l n 0

-exp(-P&))

=&tlrra)“a

,

(38)

with A = (4/n2) r(l - l/a) .

(3%

Then the integrand for D&y) behaves as t- s/h in the short time region. Therefore ifcu> 3/2,&,(O) value and the asymptotic form of eq. (31) becomes l-(llk~3+Go(0)kl-D1(0)~-...=0. In the hit of k + 0, the solution of eq. (40) is independent is found to be written as kl- =x/r,

.

has a finite (W

of 7. From the dimension analysis, this solution k,, (41)

which agrees with eq. (2). The numerical factor K cannot be determined analytically. The numerical calculation of Wilemski and Fii indicates that K is about 0.46 for Q = 2 (free draining Rouse chain), but this value will change depending on the choice ofS(u). In any case it can be concluded that if a> 3/2, kl becomes independent of R if

the reactionis purelydiffusion-controlled. IFL 9 ftnite, the solution of eq. (40) cannot be written ina simple form. But a plausiile approximate expression

M. Doi/Diffusin-contmiIed macrion of polymers

461

for kt will be .

k*= [l/kr3+k;iJ-’

(42)

This equation is correct for the two limiting cases of k + 0 and k-+ 0. In case of a Q 3/2, we must examine the divergence of Do(-y) in detail. (One can easily verify that DJO) CORverges for n > 1.) By use of the similar technique described in the foregoing section, we obtain dx x*-1(x

t 1)-3/z ,

(43)

where c = A(T&,#~

.

WI

Therefore if a< 3/2, the integral converges even if we put the upper limit of the integral to infinity. Then the integral is easily evaluated, yielding Q-J(7) = B7&--3Tyn

,

(45)

with B =

(211)-lj2 r( I+ a) r(3/2

- a) [(4/a2) r‘( I-

l/a)] *

.

W)

If (Y= 3/2, the integral (43) can be estimated as 37, DO(T) =

cg

4fiA3f2

x1/2

oJ

dx(xtl)

cl? dx - = B ‘T, ln (~7-~) x

3rnl 3/2~4&A312

J1

,

(47)

(48) Hence in both cases of a = 312 and a < 3/2, the coefficient form of the solution of eq. (3 1) is written as ’9

kt = [(Ecr3)” t ki~1-l

k, = [(k73)-’

+

0.67,

k,, = l/Do(r)

.

chain (a = 3/2), by tentatively

Especially for the nondraining

of k, becomes very large as 7 + 0 and the asymptotic (4% setting c = 1 in eq- (47). we have

ln 7-2 ] -1 .

cm

This equation is found to agree with the result of Wilemski and Fixrnan within a error of 20%.

6. Intupretation

of the results

The results in the preceding two sections are summarized as follows: For smalt k, the y dependence most the same for all cases kl =k73.

of k, is al-

Gl) However for large k, the 7 dependence of k, is quite different depending on the short time behavior of p(r). if 1 -p(r) 0: tfi, the limiting value limk__kl = k,, is written as

462

M. Doi&tjiiiansontrolled

iI,+,

a

70

B < 213

(l117-~)-~

B = 213 .

73-218

B>2/3

reactionof &mm

(52)

For the Rouse model, 6 is related to u as fl= l/a. Notice that a is the parameter determining the relaxation spectrum [a]. Usually a is between 3/2 and 2. Hence k,, k almost determined by T, and the molecular weight dependence m . There is no additional molecular weight dependence arisiig from the parameter ofk,, is the same as that of 7-l TSr:AP’/2. Since all of the above analyses are based on the closure approximation (6). one may be dubious of these conclusions. Therefore we shall give an intuitive explanation for the reason why kl_ is independent of-y for the free draining Rouse chain. AS w.as shown in the foregoing section, the behavior of&) of the Rouse chain is quite different from that of the harmonic spring model in the short time region. To understand this more intuitively, we consider the Brownian motion of the end-to-end vector u(t). Let us express u(t) in terms of the normal modes (53) where QJr) is defined by

a,(t) =@

cos

((2n2-$)pn) f,(f).

As is well known the Brownian motion of Q,(t) is independent given by [8,91 N2b2

=-

exp (-p*fh,)

p2t?2

of each other and its time correlation function is

.

From eqs. (53) and (55) we can understand the characteristic features of the Brownian motion of u(r). Eq. (53) indicates that the stochastic motion of u(r) is expressed as the sum of the independent stochastic motion of Q,(r) (see fig. la). First we note that u(t) is primarily determined by the fust normal mode Q,(r), because the mean square amplitude (Q$ of the higher normal mode (p > 3) is proportional to pB2 and is much smaller than CQ:>. This corresponds to the fact that p(f) is well approximated by that of the harmonic spring model. However the ir@portant point is that there is fluctuation of u(t) around mm Qt(t) owing to the higher terms Qs, Qs, ... . The fluctuation amplitude may be estimated as R’2 = ((u(r) - a)Ql(t))2>

Furthermore

=

(1- n2/8)L2 = 0.2L2 .

from eq. (55). we find that this fluctuation

(5’3

is very rapid compared to the Brownian motion of Ql(r).

(b) Fig. 1.

X. DoifDif~ncontroUed

463

reactionof pdymers

Thus we may roughly understand the Brownian motion of u(r) as follows: u(f) is almost bound to&&%&(& but it fluctuates rapidly in the sphere of radius R’ centered on &@@Ql(r). (see fii. lb). If we do not take into account this fluctuation, we recover the harmonic spring model. For the global property such as the dynamic viscosi@, the harmonic spring model yields a good first approximation. However this is not the case of the present problem of the chemicaI reaction. In that case, the crucial point is that the reaction takes place if lu(t)l becomes smaller than R’, not R because the motion of u(t) is very fast in the sphere of radius R’. If we use R’ instead of R in the result of the harmonic spring model (eq. (3)). we have

(57) which is independent

of R. Of course the factor a

is merely a tentative one.

7. Discussion We have shown that the Rouse model and the harmonic spring model yield quite different reaction rates for the case of purely diffusioncontrolled reaction and that this is attributed to the characteristic behavior of the segmental motion of the Rouse chain in the short time region. However it must be remembered that such difference appears in the limit R/L < 1. In fact, XR = L, the two models wilI yield almost the same results because the ihrctuation amplitude R’ may be neglected compared to R. In connection with this, we should like to make a further remark on our conclusion. We have shown that the reaction rate of the free draining Rouse chain becomes independent of R in the limit + =, not R + 0, because in the whole of R/L + 0. We must, however, remember that this limit means L = fib our discussion, we have implicitly assumed that R S b. Such assumption is necessary as far as the real polymer chain is replaced by the Rouse chain. Moreover the assumption R S b i 1crucial for the conclusion (52) even for the hypothetical Rouse chain, i.e., N beads system successively connected by springs with mean square SepaatiOn b*. This may be understood as follows. For this system, the characteristic short time behavior of p(r) predicted by eq. (38) does not hold for actually very short time region. If N is finite, eq. (16) is expanded as 1 -p(r)

=

(

A’ 8 z-1 -&-P-2$

)

t= f-

(58)

If R
Acknowledgement The author appreciates Dr Seiichi Sunagawa and Dr Hzuhiko

Appendix 1: Derkatlon

Nakajima for valuable and stimulating

discussions.

of the reduced Green function

I_& us introduce the complete Green function Gu({r),{r’);

f) for the whole confwrational

space. it is defmed

hf. Doi/Diffusion-conrroUed rmxion of polymers

464

as the solution of the equation (

&+ Q C,(Jr).Ir’k 1

0 = S(r) 6(Ir) -9’))

-

(A-1)

In terms of Co, the reduced Green function is written as (A.21

GC,.uz;f)=~dir)dC~‘}6(ul-(‘N-‘1))6(ug-(f~-r;))GO(O.Cr’~;f)Psq(Il’~)-

For the Iinear system such as the harmonic spring model or the free draining Rouse model, the Green function Gu can be obtained exactly [lo]. The explicit form of G,-, is rather complicated, but an important point is that Go is given by a multivariate Gaussian function of {r) and {r’}_ Since the equilibrium distribution function Peg ({r’)) is also a Gaussian function of ir’), the integral (A.2) yields a multivariate Gaussian function of ut and ~2. Thus G(ut,142; t) is written as G(ut,u2;r)aexp[-3iit’Hl.ul-tiiz.Hz-u2

,

-3Gs-H3-u31

(A.3)

where HI, H2 and Hs are symmetric time dependent 3 X 3 matrices, and tilde denotes the transpose of the vector. If there appear nonlinear forces such as the hydrodynamic interaction or the excluded volume effect, eq. (A.3) does not hold strictly. However even in that case, if we take into account these nonlinear forces by changing the parameters L or tp and still asume a linear system, we recover eq. (A.3). The matrices HI, Hz and H3 are determined as follows: First we note that His must be proportional to unit matrix because the system is isotropic. Hence we rewrite eq. (A.3) as G(u,,u2;t)aexp[-~ht(r)u:

-*h2(r)u$

.

-~h3(t)u,~uzl

(A.4)

Next we make use of the fact that the integral of G(u,, u2; C) over u1 (or u2) yields an equilibrium function of u2 (or ul), i.e.,

s s

d3u1 G@,

u2; t) =P,,&)

d3u2 G(q,

u2;

r) =Pq(uI)

distribution

,

(AS)

.

(A-6)

Finally, we utilize the relation p(r) =L-2(u(r)=u(0))

= L-2 sd3u1 d3u2u,.u2G(ut,

Substituting eq. (A.4) into eqs. (A.S)-(A.7). lutions give the desired result of eq. (13).

u2; r) .

we have a simultaneous

(A-7) equation for h t (t), h2(r) and h3(r). These so-

Appeodbr 2: Analytic expmsion of k, of the harmonic spring model For the sink function S(u), Wilemski and Fixman employed the step function (12). For this function Us is calculated as

%I’

,JRd3U (2&)3’2eXP (-$) =(&)3’2,7LJ37u exp(--$7~) I

(A.8)

u

where 7, = u/L. Since 17, I is small, we may replace exp (- 37:/2)

by 1. Then the integral is easily evaluated as

M. Doi/Diffision-controlkd

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465

of polymers

uq = (6/n)‘,* 73 . Furthermore

w.9)

eq. (10) becomes (A.lO)

Wilemski and Fixman made a further simplification for 6(s). They approximated eq. (A.lO) as as)=

Jdiexp(-sr) 0

l

d3u1

lu#CR

4rrR3 OD d%+ G(u,, 0;r) = 3 s dtexp(-st) s 0 IuaKR

s lUll
d3ulG(ul,0;c). (A.11)

We first follow this expression. Substituting eq. (13) into eq. (A.1 1) and expanding i(s) into a power s&es of s, we have as)

= (6/n)+

(A.12)

[s-l - D,,(y) + Dl(-r)s - ...] .

Here D,,(Y) is now defmed by D,,(Y)

=

&,dCd3YU[df

2 [(I - exp(-2f/~,))-3/2

exp [2(L_e~~~_2~,T,j~

- ;] .

(A.131

As in the case of eq. (32). we find that only Do(r) diverges in the limit of Y+O. To obtain the asymptotic form of DO(Y), we follow the same procedure as in section 4. Since the integral for f+,(Y) is determined by the contribuas ltion of small t, we evaluate Do(?) by the integral from t = 0 to t = re, for which we expand exp (-2f/r,) 2t/7,.,, . Hence ~0(7)==

[dt(

&,Y‘(;rd)l. u

$)3’2exp

In terms of the new variable x = T,,, r$f, &t(r)

=3

4ny3

j

d3TU J

I7uKr

(-2yz)

.

(A-13)

this is rewritten as dx

Qlr~lsc

rm 2filY,l

exp (- 4x) .

(A:l4)

The lower limit of the integral may be replaced by zero, because the integral from x = 0 to x = T,Y E/r, is.negligiile in the limit of IY,l+O. Then the integral (A.14) is easily evaluated to yield

Hence the solution of eq. (3 1) becomes (A-16) This is rewritten in terms of D, B and rR defined in the paper of Wilemski and Fixman as

izrkl= A36

B73R [ 1 +

2&Z,] -I .

(A-17)

This simple equation is found to reproduce completely the result of more detailed numerical calculation of WiIemski

466

M. Doi/Diffrrrionj.ontdlerdkd reaction of polymcn

(111

forthecaseof7R=0.01. Finally, if we start from the ori@nal expression of &s)(eq.

(A.10)): we obtain

A more detailed numerical calculation [4] indicates that eqs. (A-16) and (A-18) yield satisfactory accurate values of k, for 7 < 0.1, inwhich case the deviation is within 2%.

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