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198 7,2 0, 2289-2296 2289

neously performing Brownian (jittering) motion. Non-

Newtonian systems often show nonlinear hydrodynam ics.

If the mo tion of a body has two velocity components, th e

total drag on that body in a non-Newtonian fluid may not

be the s um of the drags experienced if the body performed

each of the compo nent motions separately. Th e sedi-

men tat ion of a Brownian part ic le need not be the same

as the sedimentation of an elsewise similar particle whose

Brownian motion has been suppressed . Einsteins argu-

ments refer to the drag coefficient of a particle which

simultaneously sediments and diffuses, not

to

a body which

sediments without undergoing Brownian motion.

In conclusion, we have observed probe diffusion in a

polyelectrolyte solution, noting the dependence of D on

the m ajor experimental variables: polymer concentration,

polymer neutralization, ionic strength, probe radius, and

solution viscosity. Relatively simple relations conn ect these

variables. T h e available theoretical models for

D

are not

entire ly sat isfactory, in th at th ey predict incorrect ly the

depend ence of

D I D ,

on

R

and d o not predict accurately

the depend ence of D on

I

or c. Identification of the re-

lationship between

D

and viscosity requires a more ex-

tensive study of the rheological properties of the polye-

lectrolyte solutions used for the probe diffusion mea-

surements .

Registry

No.

Polystyrene, 9003-53-6;poly(acry1ic acid) sodium

salt, 9003-04-7.

Supp lemen tary Mater ial Available: Tables of D values for

polystyrerie spheres a t various electrolyte concen trations as a

function of polymer concentr?tion an d a t various pHs and polymer

concentrations as a function of the concentration of adde d NaCl

and tables of results a t various pHs on polystyrene spheres in

different poly(acry1ic acid) and NaCl concentrations (13 pages).

Ordering information is given on any current masthead page.

References and Notes

(1) Ullmann, G. S.; Phillies, G.

D.

J. Macromolecules 1983, 16,

1947.

(2)

Ullmann, G. S.; Ullmann, K.; Lindner, R.; Phillies,

G . D.

J.

J.

Phys . Chem.

1985,89, 692.

(3) Phillies, G. D. J. Biopolymers 1985, 24, 379.

(4)

Lin, T. H.; Phillies, G.

D. J. J.

Phys . Chem.

1982, 86, 4073.

(5) Lin, T.-H.; Phillies, G. D. J. J.Colloid Interface Sci. 1984,100,

82.

(6) Lin, T.-H., Phillies, G. D. J. Macromolecules 1984, 17, 1686.

(7 ) Ullmann. K.: Ullmann. G. S.: Phillies. G.

D.

J.

J.

Colloid Zn-

,

terface Sci.

1985, 105,315.

8 ) Phillies, G.

D.

J.; Ullmann, G.

S.;

Ullmann,

K.;

Lin, T.-H.

J.

Chem. Phys.

1985 ,82, 5242.

(9) Phillies, G . D. J.

Macromolecules

1986, 19, 2367.

(10) Gorti, S.; Ware, B. R. J . Chem. Phys. 1985,83, 6449.

(11) Callec, G.; Anderson, A. W.; Tsao, G. T.; Rollings, J.

E.

J.

App l . Polym. Sc i . 1984, 22, 287.

(12) Kato,

Y.;

Komiya, Y.; Iwaeda,T.;Sasaki, H.; Hashimato, T.

J .

Chromatogr. 1981 ,208, 105; 206, 135;

208,

71; 211, 383.

(13) Yau, W. W.; Kirkland, J. J.; Bly, D.

D.

Modern Site-Exclusion

Chromatography; Wiley: New York, 1975,

p

61.

(14) McConnell, M. L. A m . L a b . 1978, 10 3 , 3.

(15)

Ouano,

A.

C.; Kaye, W. J . Polym. Sci.

1974, 12, 1151.

(16) Yu, L.-P., Rollings, J. E. J . Ap pl. Polym. Sci. 1987, 33, 1909.

(17)

Gorti, S.;Plank, L.; Ware, B. R. J.Chem. Phys.

1984,81, 909.

(18)

Koppell, D. E. J. Chem. Phys .

1972 ,57 ,4814 .

(19)

Phillies, G.

D. J.

J . Chem. Phys .

1974, 60, 983.

(20) Phillies, G. D. J. Macromolecules 1976, 9, 447.

(21) Jones, R. B. Physica 1979, 97A, 113.

(22)

Kops-Werkhoven, M.-M.; Pathmamanoharan, C.; Vrij,

A.;

Firjnaut, H. M. J.Chem. Phys. 1982, 77, 5913.

(23)

Lodge, J.

P.

Macromolecules

1983,

16,

1393.

(24) Noggle, J. H.

Physical Chem istry on

a

Microcomputer;

Little,

Brown, and Co.: New York, 1985.

(25)

Ogston,

A.

G.; Preston, P. N.; Wells, J.

D.

Proc. R. OC. o n -

don, A 1973,

333,

297.

(26) Langevin,

D.;

Rondelez,

F. Polymer

1978, 14, 875.

(27) Altenberger, A. R.; Tirrell,

M.

J.

Chem. Phys.

1984,80, 2208.

(28) Cukier, R. I. Macromolecules 1984, 17, 252.

(29) Malone, C. M.S. Thesis, Worcester Polytechnic Insti tute, 1986.

(30) Odijk, T. Macromolecules 1974, 12, 688.

Reptation

of

Living Polymers: Dynamics

of

Entangled Polymers

in the Presence

of

Reversible Chain-Scission Reactions

M. E. Cates

I n s t i tu t e f o r T h e o r e t i c a l P h y s ic s , U n iv e r s i t y o f Ca l i fo r n ia , S a n t a B a r b a r a , Ca l i fo r nia 93106.

Rece ived December

12, 1986

ABSTRACT: A theoretical stu dy is made of the dynamics of stress relaxation in a dense system of living

polymers. The se are linear chain polymers tha t can break and recombine on experimental time scales. A

simple model for the reaction kinetics is assumed, in which (i) a chain can break with equal probability per

unit time per unit length a t all points in th e chemical sequence and (ii) two chains can combine with a rate

proportional to t he product of their concentrations. The chain length distribution is then exponential with

mean

t;

s taken t o be large enough that

a =

L e / t

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m

0 (4

Figure 1. Rep tation of a chain in its tube. The tube is defined

at time

t

= 0 (a ). C urvilinear diffusion causes parts of the tube

to be lost (b,c); he new tube that is formed (not drawn) is in

equilibrium and does not support stress. At

t

Z

rlep,none of the

original tube remains (d).

Here Do is a mobil i ty co nstant , independ ent of L; t h e

curvilinear diffusion constant of a chain along its own

contour is

D,

= Do/L

Since each chain has to diffuse a curvilinear distance L to

disengage from its tub e, we have

L2 zz

D,r

which gives th e result 1). Since successive acts of disen-

gagement are uncorrelated and each results in a dis-

placem ent of th e center of mass of the ch ain by a distance

of order its gyration radius

R

-

L1)I2

wi th t he Kuhn

length ), the diffusion c onstant of the chain in real space

is

(2)

Th e result (2) is well confirmed experimen tally,2 or chains

longer than a cri tical length Le, he entanglement length.

For sh orter chains, the topological obstruction of on e chain

by ano ther is much less important; the dom inant relaxation

me c ha n ism is no t r e p t a ti on , bu t a R ouse- li ke m ~ t i o n . ~

It follows1 from eq

1

hat the zero-shear viscosity, p ,

should obey

(3)

for a homologous series of melts of differen t chain length s

L t e.

Here Go s th e instantaneous shear modulus, which

is proport ional to t he d ensi ty of degrees of freedom th at

couple to the reptat ive relaxation mechanism (Le. , the

num ber densi ty of entanglements) and is independ ent

of

L

for

L t e.

Notice th at by dem anding continuity (at

L r Le)

of eq

2

a nd 3 with th e corresponding results from

the Rouse model3

( D

- 1/L; - L;Go- / L ) , ne ob-

tains the fol lowing dependence o n Le of th e param eters

in the rep tat ion model:Ib

D

R 2 / r r e p z DolL-2

p r G,yrrep-

L 3

Do - Le;

Go - 1/L,

Th e re su l t (3)

is

somewhat less sat isfactory than (2),

since the expe rimental d ata is traditionally described2 by

th e law q - L 3 . However, various explanations for this

discrepancy have been proposed within th e framework of

the tub e model .4 (In particular,

it

is possible that the

-

(4

Figure 2. (a ,b) Histo ry of a given portion of tube (shaded) can

be traced by imagining the tube to diffuse relative to the chain

(for clarity this is drawn fully extend ed). As soon as the shaded

portion passes eith er end of the chain (of length L ) , he stress

associated with it is lost. The probab ility ~ ( t )hat this has not

happened by time

t

is therefore the survival probability of a

particle diffusing on the line segment [0,L]with absorbing

boundary conditions at each end

(c).

Averaging over all tube

segments present at time zero is equivalent to averaging over all

starting positions of the particle. In the presence of breakage and

recombination, the process is modified (eq 10) so that the ab-

sorbing walls can make discrete jumps toward or away from the

particle (corresponding to breakage (d ) and recombination (e) ,

respectively). The instantaneous diffusion constant of the particle

varies inversely with the

tota

length of the chain and so fluctuates

in time.

observed behavior is simply a misleading crossover from

the Rouse-l ike regime a t L Le.)

Doi an d Edw ards5 examined in detail th e consequences

of the reptation hypothesis, and constructed a constitutive

equation t ha t describes the l inear and nonlinear viscoe-

lastic response of polymer melts under a variety of de-

formation conditions.2 In the Doi-Edwards theory, a

central object is th e stress relaxation function

p t ) ,

which

describes the fraction of imposed stress remaining a t time

t after an infini tesimal stress is imposed a t t ime 0. Thi s

is simply th e average fraction of tu be (existing at time zero)

th at has no t been lost by disengagement by t ime t :

(4)

where Td= L2/D,x2 = T,,,,. Th is result is found by ob-

serving that t he parts of the original tub e remaining a t

time t are those through which neither end has passed. By

imagining the chain to be a t rest in a moving tube (Figure

2a-c), it is easy to see tha t p(t) is the survival probability

a particle of diffusion constant D,, launched a t t = 0 with

uniform probabil i ty on the l ine segment

[0,

L ]

with ab-

sorbing boundary con ditions a t either end. T he result (4)

is easily found by Fou rier decomposition. T he zero-shear

viscosity p is then given by

~ ( t )( S / d 2 P - ~ xp(-tp2/Td)

p=odd

In deriving

p t ) ,

an important a ssumption i s tha t the

tub e constraint imposed on a ny given chain by its neigh-

bors remains in tact on th e t ime scale ire f the disen-

gagement. Since these neighbors are themselves disen-

gaging from their own tub es on the same t ime scale, this

assumption is not obviously self-consistent. Nonetheless,

the Doi-Edwards theory is remarkably successful a t de -

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Reptation of Living Polymers 2291

any positive value.)

It

is fur ther assumed th at th e reverse

react ion proceeds a t a ra te proport ional t o the produ ct of

the concentrat ions of the two react ing subchains and,

moreover, that the rate con stant involved is independen t

of the m olecular weights of these two subchains. Th is

latter assumption is based on the idea that, a t high enough

concentrat ion, the rate-l imit ing step is no t the diffusion

of the reacting chains over a large distancels bu t som e local

kinetic process. We also assume successive breakage an d

re c ombi na t i on e ve n t s fo r a g i ve n c ha i n t o be

uncorrelated-there is no higher probability for a chain

end to l ink u p with th e chain from which i t was recently

detached th an w ith any other chain end in th e vicini ty.

Th e conditions und er which these assum ptions are self-

consistent will be discussed in section 3.2. Finally we note

that since only the linear viscoelastic response will be

considered, it is unnecessary to consider whether or not

these reaction kinetics are significantly altered by the

application of a strain field.

With these assumptions, the equation governing the

time development of the number density

N(L)

dL f chains

of len gth L 1/2 dL may be wri t ten as

(L)

= -c1LN(L)

- c2N L)J+dL L? +

0

scribing the experimental results for high polymers of

narrow molecular weight distribution.2 For polydisperse

systems, however, a simple superposition of the Doi-Ed-

wards p t ) for different chain lengths proves to be inade-

qua te ,6 and explic i t account must be take n of the tub e

renewal effects7 th at resu lt from rearr angem ent of

neighboring chains. Non etheless, unless th e system con-

tains (in significant numbers) chains of two or m ore com-

pletely dissimilar lengths, the characterist ic re laxation

t ime

7

tha t en ters the viscosity, 7

zi 07,

should remain

comparable to the reptation time

7,, , L)

of a chain of some

average length,

L.

2. Living Polymers

In the present paper, we discuss the modifications that

mu st be made to th e reptation picture in order to describe

th e dynam ics of long linear chains (L

>>

Le) ha t can break

and re -form on exper imenta l t ime sca les ( l iv ing

polymers). Our principal motivation comes from three

types of experiment:

(i) Recent studies of su rfact ant organization in micellar

systems and microemulsions8 have demonstrated that ,

under special conditions, the equilibrium structu re consists

of a concentrated solution of tubu lar m icelles th at are very

much longer th an an appropriately defined K uh n length;

they c an therefore be trea ted theoretically as polymers.1

Preliminary exp erimental dat a on shear behavior: which

indicates a stron g dependence of viscosity on the volume

fraction of surfacta nt , appear s to be quali ta tively com-

patible with the results from a reptat ion theory.l1 How-

ever, theor etical work on m icellization1J2 suggests th at,

under appropriate conditions, there may be no u pper limit

to the length of the micelles in equilibrium. Pre sum ably

therefore, ther e exists a regime of parameters in which the

reptatio n time of the micelles is so long tha t it exceeds the

time scale characterizing the dynamic equilibrium of their

breakage an d reformation; in anticipation of experiments

in this regime, we ask here wh at can be said theoretically

abo ut the result ing dynamics.

(ii) Sys tem s of end-functionalized linear polymers are

of increasing expe rime ntal interest,13 especially insofar as

they provide well-characterized ex amples of a much larger

class of associating polymers. Th ese are polymers tha t

can form tr ansien t reversible l inkages w ith one another;

they are of considerable technological significance as

flow-modifying additives.14 If both ends of each polymer

are capped wi th a fun c t iona l group tha t t ends to form

dimers (such as carboxylic acid groups in a nonpolar en-

vironment), then a t high densi ty the polymers will them-

selves be predominantly polymerized into longer linear

chains. Th e reptat ion t ime can then become very great ,

and one certainly expects the kinetics of chain breakage

and recombination t o be relevant in determining the time

scale of stress re laxation in the melt and concentrated

solutions.

(i i i) Th ird , one may be interes ted in th e dynamics of

chemically homogeneous equilibrium polymers, as seen in

systems such as liquid sulfur.15J6 In fact

it

is found th at

the present m odel (as extended in sect ion 4 o cover the

case of very small

7break)

gives an excellent account of the

experimentally observed viscosity of sulfur over a wide

temp erature range above the polymerization point. Details

of the relev ant calculation will be presen ted elsewhere.

2.1 Kinetics

of

Breaking and Recombination. W e

sta rt from th e simplest possible model for the kinetics of

chain breakage and recombination. In this model i t is

assumed th at a chain can break with a fixed probability

per u nit t ime per un it length anywhere along i ts length,

L .

(Thus

L

is treated as a continuous variable and can take

C ~ S ~ = S ~ - ~ ~ ~ N ( L ? N ( L ) G ( L + - L) 6)

Here the f i rs t t e rm represents the decrease in N (L ) by

breakage, the second is the decrease by reaction of chains

of length L with oth ers to form longer chains, the th ird is

the r ate of creation of chains of length

L

by breakage of

longer chains, and the final term is the rate of creation by

reaction of two shorter chains to produce one of length

L.

Th e parameters

c1

a nd c2 are (appropriately dim ensioned)

rate con stants for the breakage and recombination pro-

cesses, respectively.

This equation has the steady-state

(I\j(L)=

0) solution

N ( L )

= (2c1/c2)

exp(-L/L)

(7)

where L depend s as follows on the rate constants an d the

overall arc concentration of polymer,

p =

S t L N ( L ) dL:

2 P

=

p c z / c , (8)

7break =

( C l L ) - l

(9)

It

is convenient to define the characteristic time

which is th e expected survival time of a chain of the m ean

length ,C before i t breaks into two pieces. No te that the

survival time of a typical chain end, before it is lost by

recombination, is of the sam e order as 7break; the equiva-

lence of thes e two tim e scales is a dir ect consequence of

the principle of detailed balance.

2.2 Dynamics of Stress Relaxation. We imagine a

system of chains in a steady sta te of dynam ic equilibrium

as described b y eq

6-9.

In a sm all time interval 6t, a chain

of

length L (t) can break in any small interval

61

along its

length, with probability c16l6t, or eithe r of its ends ca n join

with anothe r chain of length L 6L, in each case with

probability

c1

exp(-L

/L)6L6t/ .

It is simple enough in principle to couple this stocha stic

process with that describing the curvilinear diffusion of

a chain in its tube, as described after eq

4.

We recall that,

in tha t case, i t was convenient to consider a section of the

original tube as a particle of diffusion constant D, aunched

with uniform probability on the line segment

[O,L]

with

absorbing walls at either end (Figure 2 ;

p(t)

i s then th e

survival probability at time

t

after launch. In the present

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case we m ust allow the abso rbing walls themselves

to

make

jumps according to the stochastic process described in th e

previous paragraph (F igure 2d,e). L et ua denote th e length

of the line segment to the left of the particle by Lle,(t), and

that to the right by Lright(t); he sum of these is

L ( t ) .

W itho ut loss of generality, we can se t in eq 7 L = 1 (by

choosing it as the un it of length), in which case (Ll eft)

=

(LQht) =

1.

(Note that each half of the chain has average

length L, not L/2 . Th is is because we are averaging over

all points on all chains, which is a weight average.)

This gives the following transition probabilities in a

small time interval 6 t :

p = 72: h e f t - eft

+ A; Lright

- right - A

( l oa )

p = y :

h e f t

-

e f t

- A;

Lright

-

right

+

A ( l o b )

p = 2c16L6t: h e f t

- left

* 6L

( L eft < e d

(10c)

(10d)

(10e)

p = 2c16L6t:

Lright - L

:ight

*

6L

(L

ight

< Lright)

=

c1

exp(-L?GLGt:

Lleft

-

left

+

L

f

6L

p =

c1

exp(-L?GLGt:

Lright - right +

L

f

6L

(10f)

In these equations,

f6L

epresents symbolically an infin-

itesimal length interval. Equations 10a and 10b represent

the D oi-Edwards diffusion process, in which a small

portion of the chain to th e left of the particle is transferred

to th e right, (lo a) , or vice versa, (l ob ); for simplicity we

have defined the transition probability

as

independent

of the time step b t , and accordingly taken a diffusive step

length

A

=

(2D,6t)i2 = [2 (DO/ L( t ) )6 t ]1 / 2

Not e t ha t

A

depends on the instantaneous curvil inear

mobility, D , = Do/L(t ) , and

so

changes stochastically with

time. Eq uation s 1Oc and 10d describe th e kinetics of the

breaking process, and eq 10e and 10f th at of recombina-

t ion; each of these occurs ind ependently for the left and

right half of the chain. T o complete the specification of

the system, we state th at a t t ime t

= 0,

an d Lright are

chosen independently from an exponential distribution

with mean

t 1;

moreover, if in any time interval

6 t

either

L1, or L ht becomes negative, the par ticle dies. W ith these

definitions, the stress relaxation functio n p t ) is simply the

survival probability of th e particle to tim e t , ust as in the

simpler case of o rdina ry reptation.

Since the ju mp s in Lleft and Lright orresponding to

breakage and recombination are not infinitesimal (as 6 t

- ,

the stochastic process described by eq

10

is no longer

purely diffusive, and we do not expect an analytic solution

to the problem. Nonetheless, various things can be said.

First,

if

Tbreak

- ,

we are left with the Doi-Edwards

stress-relaxation calculation for a system of chains with

exponential polydispersity, as given by eq 7. Th us eq 10a

and 10b will give

p t ) = ~ - 1 1 ~ ~xp(-L/L)p(L, t )

d~

11)

where p(L ,t) is given by eq 4. A steepest descents analysis

then suggests that to a good approximation 11)can be

replaced by

0

@(t)

-

e x ~ [ - ( t / ~ ~ ~ ~ ) I (12)

where T , , ~ s tands for T ~ ~ ~ ( L )D 0 / L 3 . This is a ra ther

disperse relaxation function, which in practice one would

expect to be significantly modified by tu be renewal effects,

c )

(4

Figure

3. Relaxation mechanism proposed in the text. The

relaxation of the shaded segment proceeds when

a

break occurs

within a distance X in the chemical sequence (a). The break must

be close enough that the

new

end

can

pass through the shaded

segment

(b,c)

before

it

recombines with tha t

of

a neighboring chain

(d).

as discussed in section

1.

Nonetheless, one does not expect

the power law dependence on L of the viscosity,

7 - J a p t ) d t - T , ~ ~ ( L )L3 (13)

to be affected by tube renewal, since this occurs on a

comparab le time scale to reptatio n itself. Generalizing to

the case when T , , ~ break m, we expect the relaxation

function in (12) to be reduced at long times

( t

2 Tbreak),

by which time it is, however, already small. Th us the

viscosity 7 is still given by (13).

Th e interesting new regime occurs when Tbre , ,~. In

this regime, chain breakage and reformation will both occur

often, for a typical chain, before i t has disengaged from

its tube by ordinary reptat ion. We propose tha t the

dominant relaxation mechanism for the relaxation of a

typical tube segment x initially not close to th e en d of a

chain, is the n as follows: (i) Th e chain breaks within a

certain distance X of x along the chemical sequence. (ii)

Curvilinear diffusion brings the new chain end past

x

before this chain en d is lost by recombination (Figure 3).

Since the recombination time is comparable to Tbre. (by

detailed balance; see section 2.11, we find tha t

A,

the largest

distance a t which a break can usefully occur for relaxation

by this mechanism, obeys

A 2 D c ( L ) T b r e a k

where

L

is the length of the newly broken chain.

Pre-

suming this to be of order the mean, L , we obtain

X z Do L-2/c1 = Lq- ({

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Th is may be rewri t ten as

(7rep7break)

l J 2 { S

1) (16b)

which emphasizes th at th e domina nt re laxation process

involves

a

cooperation between the breaking an d rep tation

mechanisms. Notice th at the result expressed in this form

has n o explic it dependence on th e overal l chain concen-

tration, the K uhn length, and other param eters; these only

ente r through

7break

a n d 7rep Th us we expec t the resul t

to apply w henever the rep tat ion model is appropriate for

the description of chain dynamics on scales of order

A

(Th is condition is quantified in sect ion 4 below.)

One might imagine th at a tub e segment on a very long

chain could relax faster th an th is by w aiting for two breaks

to occur nearby (o ne on either side), Th is would leave the

segm ent on a chain of length

L

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involving th at en d will make a n unbiased sampling of the

ensemble of oth er chains, in proport ion to their number

density and with no particular preference for the chain end

that was involved in the preceding dissociation event.

Since exploration by a re ptatin g chain end is compact,18

th e volume Ve is of order X1)3/z; in a melt, the condition

for consistency is then

( X l ) 3 1 2 / ( L l 2 ) >>

1

Th e factor 1/

(LP)

n this expression is just th e den sity of

chain ends. Since

X

=

L[l2

this may be rewrit ten as

[

>> ( L / 1 ) - 2 / 3

This, for long chains, is a rather weak restriction on [ =

Tb,, /Trep; nly for very sm all values will th e effects of chain

diffusion on reaction rates be noticeable.

Of course, for

part ia lly di luted systems, they may become m uch more

important .

In any case i t should be noted that the fundamental

scaling result for th e relaxation time, eq

16,

does not de-

pend in detail on our assum ptions for the recom bination

kinetics. Th e only requirements are that a chain can break

with roughly equ al probabil i ty per unit t ime anywhere

along its length an d

that

the recombination kernel depends

sufficiently smoothly on chain lengths that, by detailed

balance, the lifetime of a broken end b efore recomb ination

is comparable to th e lifetime of th e original chain before

breaking. Thus, for example, the argument leading to eq

16 would not necessarily be invalidated even if recomb i-

nation were to often involve the sam e end as took pa rt in

the preceding breakage event .

4. Crossover to Regimes Involving Breathing

and Rouse-Like Motion

It

is impo rtant to discuss und er w hat range of para m-

eters the combined reptat ion/ breakage mechanism pro-

posed above will dominate over those involving (i) the

breathing m odes of a chain in its tub e (leading to fluc-

tuations in the tube-length) an d (ii) the Rouse-like motion

of stretches of chain shor ter than t he e ntangle men t length,

0 0

2. 0 40

6. 0 8 0

Figure

5. log-linear plots of the stress-relaxatio n function p t )

against time

for

> Le

However, though necessary, this is not sufficient. In fact

we require

2

L e L ) / 2

=

La/*

(19)

LY = L e/ L 1

where we have introduced th e parameter

Th e condition (19) arises because an en tangled chain un-

dergoes

a

constrained Rouse-like motion within its tube,

which results in fluctuations in the tu be length of order

AI z La1/* ,on a time scalelbJ8J9

Tbreathe a7rep (20)

When X 5 AL he newly created chain end need not

reptate to th e posi t ion occupied by our chosen tube seg-

me nt b ut typically gets there more quickly by means of

a b reathing fluctuation in which the chain contracts

down i ts tube. It is known that the mean curvil inear

distance [ raveled by a chain end (or any other monom er)

as a function of time under such a fluctuation scales

aSlb,18,19

Do1/2t1/4

7/23/2019 Mac-1987-2289 Reptation of Living Polymers: Dynamics of Entangled Polymers in the Presence of Reversible Chai

7/8

Macromolecules,

Vol. 20, No. 9, 1987

Hence we define a characterist ic arc distance

A

by

A

= D01/Z7break1/4

5 /451/4

(21)

which is the m aximum distance away along the chemical

sequence tha t a break can typically occur for a breat hing

fluctuat ion to carry i t through our chosen tub e segment

before recombination.

Th e rate-l imit ing s tep for stress

relaxation is then w aiting for a break within th e distance

A, with a corresponding fu ndam ental re laxation t ime

(22)

Th e crossover between th e regime described by eq 16 and

that of eq 22 is when A = A, th a t i s, when { = CY.

Th is breathing picture is i tself only applicable a t arc-

length scales larger than the entanglement length,

Le.

Hence eq 22 breaks down when 7break is

so

small tha t

A

5 Le,

th at is , when

7 z

(c1A)-1 = 7repp4CY-1/4

{,