Short time behavior and universal relations in polymer cyclization

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Short time behavior and universal relations in polymer cyclization

Barry Friedman, Ben O’Shaughnessy

To cite this version:

Barry Friedman, Ben O’Shaughnessy. Short time behavior and universal relations in polymer cy- clization. Journal de Physique II, EDP Sciences, 1991, 1 (4), pp.471-486. �10.1051/jp2:1991181�.

�jpa-00247531�

Classification

Physics

Abstracts

82 35 05 70L 61 40K

Short time behavior and universal relations in polymer cyclization

Barry

Friedman

(', _*)

and Ben

O'shaughnessy

₍₂₎

1') Institute for Solid State

Physics~

University of

Tokyo, 7-22-1Roppongi,

Minato-ku,

Tokyo

106, Japan

(2) Department of Chemlcal Engineenng and

Applied

Chemistry, Columbia University, New York, NY 10027, U S A

(Received 24 August 1990, revised 4 December 1990, accepted16 January 1991)

Abstract. This paper is a renormalization group study of the short time irreversible reaction

rate for _a

single polymer

with reactive groups attached to its ends

(cychzation)

In melts _we predict that for ttmes less than the entanglement time the _{reaction rate} k(t) scales _as

t~ ^" where _{a is a} nonuniversal power which

depends

_on the molecular weight This _power tends to

I/4 _as the chains become very long, the asymptotic ^behavior is «diffusion controlled»

independently

of the reactivity of the end groups In dilute theta solvents _we find

marginal

behavior _m which

k(t)

depends logarithmically _on t For

cychzation

_in good solvents the correlation hole

plays

_a crucial role, _{causing asymptotic} mean-field _{or «} law of _mass action »

behavior, _again

independently

of the chemistry of the _reactive groups

k(t)

scales _as the equilibrium

probability

of chain end contact with weak _time

dependent

corrections Eliminating the nonuniversal elements between these results for k(t) and

previously

calculated

long

time

reaction rates

k~

allows the derivation of universal relations between the

experimentally

observable quantities

k(t), k~

and the unperturbed

longest

polymer ^relaxation time r

1. Introduction.

Polymer cydization,

m which

polymer

chains

bearing

reactive end _groups react to form

rings (see Fig I),

is _an

important

_process m many

polymerizing _systems [1-4]

^Much

expenmental

effort has been invested _m the

photophysical

measurement of irreversible

cyclization

rates m

carefully prepared

dilute

polymer

solutions

[5-7],

such expenments ^not

only

tell _us about

cyclization

kinetics themselves but also test the current fundamental theories of

polymer dynamics.

^The

quantity usually

^measured is the

long-time

irreversible

cyclization

rate

k~

which _is defined _as the asymptotic

decay

rate of A

(t),

the fraction of the

polymers

which

are

uncychzed

after time t.

A

(t)

_exp

(- k~ t),

t » _r

(1)

(*) Permanent Address Department of Physics~ Sam Houston State University, Huntsville, TX

77341, US A

b) a)

Fig

I Each polymer has reactive groups ^at the chain ends Reactions _are «switched _on» at t ₌ 0~ when the chains _{are in}

equilibrium,

and the groups can react with _one another (with a certain

probability _per unit time) if

they

_come into contact After time t, each group has diffused _a distance of order

r(t)

and

explored

_a volume _m d-dimensional space of order

r(t)~ _(shaded

_regions) The

figure

depicts 2 different polymers at the _same time t (a)

Configuration

such _as these cannot

possibly

have

cyclized

since the

exploration

volumes do not

overlap

(b)Smce the exploration volumes of such

configurations

overlap

they

_may have reacted,

depending

_on the value ofthe exponent 0

(see Eq.

(5) _m text) If 0 <1, reaction occurs with

probability

of order unity (« diffusion-controlled

behavior),

if

0 ~ l reaction

probability

_is much less than unity («law of _mass action »)

This

long

time regime is

anticipated

for times

large compared

to _r, the

longest

relaxation time of the _«

unperturbed polymer (i

e without

reactive

_end

groups)

This paper addresses the short-time

cyclization

rate

k(t)

and the

relationships,

which _{m our} renormalization group framework _are

predicted

to be

universal,

between the

experimental

observables

k(t)

and

k~

We define

k(t)

as the _rate of decrease of

A(t)

^for short times

k(t)

m

~~

~~~,

_{t «}

r

(2)

dt

For these times the system has yet to reach

steady

state and _one expects

time-dependent

behavior

Though

_we consider both dilute and concentrated

polymer

solutions

(or melts),

the

polymers bearing

reactive groups are

always

assumed dilute

(e

_g a few reactive

polymers

in _a

background

of dense unreactive

chains)

Thus intermolecular reactions are

negligibly infrequent

and _{ours is}

essentially

a

single

chain

cyclization problem

While short time

cyclization

kinetics have been studied

theoretically

_m the _case of

entangled

melts

[8-10],

studies of dilute solutions have

emphasized long

time rates

k~ [I1-14]. Recently

the present ^authors calculated

k~

for dilute solutions and

nonentangled

melts using renormalization group

techniques [13-14]

_we refer to reference

[14]

as FO

throughout

^this _paper In contrast to previous work FO _{was a «} first

principles

_» calculation _m

the _sense that excluded volume and

hydrodynamical

interactions were

systematically

incorporated (without preaveragtng)

and _{no use was} made of Wilemski and Fixman's

«closure

approximation» ill] (an approximate

method for

including

the effect of the

reaction sink term m the Fokker-Planck

equation)

The _main conclusion from FO _is that the

asymptotic (high

molecular

weight)

form of

k~

_is

independent

of the

strength

and other details of the reacting ^end groups,

being

determtned

only by

the

universality

class to which the

polymer dynamics belongs (see below).

For

example,

_m the _case of dilute

polymers

_{m a}

good solvent,

the asymptotic behavtor _is of law of _mass action _» form with

k~ scaling

_as the

equibbnum probability

that the chain ends _are

together k~ ^~N~~~~+~~

where N _is the number of segments m a

polymer

Here _v and d _are

respectively

the

Flory

exponent and dimension of space, and g is the correlation hole exponent

[15-17]

The existence of the correlation hole _», the reduced

probability

of chain end contact, is the

underlying

_reason that

k~

does not scale _as the _inverse

polymer

relaxation time _as has been

widely

assumed _m the past ^If we refer to the asymptotic

(N

_very

large)

behavior _m

good

solvents

then,

_{we may}

say that there _{are no} such

thtngs

_as diffusion-controlled _» reaction kinetics _{: no} matter how

strongly

_reactive the end groups the system gets « dnven _» to law of _mass action behavior for

large enough

N

(corresponding

to the

vanishing

fixed point value of the sink

coupling constant) Similarly, cyclization

of

unentangled polymers

m a melt

(which

_are believed to

obey

_« Rouse»

dynamics [16, 18, 19])

is

intrinsically

diffusion-controlled _no matter how

weakly

reactive the end groups,

provided

the chains _are

long enough (Although entangle-

ments may

modify

the

dynamics

before realization of the asymptotic form for

k~,

^the

asymptotic diffusion-controlled form for

k(t)

_is

expected

to be

robust,

_even when

entangle-

ments _are present-see

below.)

In the present

study

_we will consider

cyclization

_m melts _(« Rouse _»

dynamics),

_m theta solvents _(« Zimm

dynamics [16-18])

and in

good

solvents _(« Zimm _»

plus

excluded volume

interactions) Fortunately

_our results for melts _are

expected

to

apply

_even when the chains _are

long enough

to be

entangled, provided

_one considers time scales less than the

entanglement

time _»

[16-18],

for such times Rouse

dynamics apply [16, 18, 19].

One

further, unphysical

case we will consider _is that of Rouse

dynamics plus

excluded volume

(no hydrodynamics)

This _is of academic interest

and,

_{moreover, is} testable

by

numerical simulation.

In the

following

section

simple scaling

_{arguments are} invoked to motivate the small time behavior

k(t)

for each of the

dynamical

classes to be treated here. As for

long

times, the correlation hole

plays

_{an important} role when excluded volume _is present In section 3 the renormalization group calculations of

k(t)

_are

presented,

and _m section 4 _we summarise _our

predicted

universal relations between

k(t)

and

k~

^{which should be}

expenmentally

_measur-

able Section 4 concludes with _a bnef discussion of the

experimental

outlook.

2.

Scaling

arguments.

In many ^situations one can deduce the

scaling

form of reaction rates

simply by

_testing whether

the reactive groups

explore

_space

compactly (r(t)~~t~'~, _d/z<I)

or

noncompactly (d/z

~

l)

where

r(t)

is the root mean square

(rms) displacement

of _a group after _time t This classification _was introduced

by

de Gennes

[20] (m

the _context of

mterpolymenc

reactions m melts for systems m which there _{are no} correlations between _reactive

groups).

He showed that _reaction rates

obey

laws of _mass action

(the

reaction rate

proportional

to the

equibbnum probability

the reactive groups are m

contact)

when the volume

explored by

a

group grows faster than t

(non-compact,

dilute

exploration

of

space),

reactive groups collide

infrequently

and the

system

^has

plenty

of time to relax and _{remain near}

equilibrium

In compact cases, on the other

hand,

_space is

densely explored

and all _reactive groups within each other's

exploration

volume will have reacted after time t

,

thus

k(t)

scales _as the rate of

increase of the volume

explored

Thts reasoning breaks

down, however,

when there _are strong

spatial

correlations between

the _reactive group as, for

example,

_m the _case of

cyclization

_m

good

solvents To _see

this,

consider first the

simpler

case of short time behavior _m

melts,

_ie Rouse

dynamics

We consider _an ensemble of chains which _{are m}

equilibrium

^when reactions _are « switched _{on »} at t

=

0 Let _{us assume «} diffusion-controlled

behavior»,

_ie all chain end _pairs which _are

initially separated by,

_{say, r} will have reacted after _{time t}

provided r(t)

> r

(see Fig I).

In

section 3 _we will show that very

long

Rouse chains

always

exhibit diffusion controlled

behavior Then for small times the fraction of unreacted chains should scale like

A

(t)

= I

d~r p(r)

= I

(r(t)/R)~ _(Rouse) (3)

1rl <r(1)

We have used the small _r behavior of the

equilibrium

end-to-end distribution

function, p(r)

=

R~~f(r/R)

R _is the _rms end-to-end distance An essential component ^{of this}

argument

_is the

good

behavior of

p(r)

at the origin, for Rouse chains the function

f

_is

proportional

to a

Gaussian,

which asymptotes a constant

(unity)

at _r

=

0 Then _since

[16]

r(t) r~'§

R _r

~'(

with _z

=

4 for Rouse

dynamics,

_we have

k(t)

=

~

()~~

=

i/r(t/r)d"-

^'

=

I/r(t/r)~~'~ (Rouse) (4)

where

e =

4 d Our

prediction

for real expenments

(d

=

3,

_{e. e}

=

I _on very

long

chains

m a melt _is thus

k(t)

t~ ^"~ In section 3 _we will derive this result _more

systematically,

as well

as its

generalization

to finite chains

Now consider the effects of excluded volume

(as

is relevant to dilute

solutions)

We will _see that the

scaling

form of equation

(4)

_is incorrect and that the criterion

determining

the

reaction behavior _is not

simply

_{a comparison} of

d/z

wtth unity ^{As is}

well-known,

p(r)

_now vanishes

algebraically

at the origin

[15-17], f(r/R) (r/R)~

This leads to

k(t)

=

d/dt d~r p(r)

= I

IT (t/r)~ ',

₀ <1

, r <r(t)

where

o

=

(d+ g)iz (5)

Unlike the Rouse _case the distribution

p(r)

_is not

evenly spread

_over the coil volume

R~,

there

being

_a

special penalty

for the

meeting

^{of chain ends. Hence the relevant} exponent

involves the correlation hole exponent g When R

~

l, k(t) t~~' _represents

« diffusion- controlled behavior _since the number

(~ t~)

of accessible reactive groups

(m

the

ensemble),

i e withtn the

exploration

volume

(see Fig I),

_is less than the number

(~ t)

of accessible

«steps» (one

could imagine

discretiztng

time into small units of _size At, so m a time t each group makes

(t/At)

steps ^of size Ar where Ar _is the _rms

displacement

_m the _time At

[20])

If it

happens

that o

>

I,

there _{are more} accessible reactive groups than could

possibly

have been vtsited _m the time t, a small fraction wtll have

reacted,

the system ^wtll

always

_remain close _to

equthbrium

and

k(t)

should scale _as the

equthbnum probability

that the chain ends _are wtthm the _« sink radius _a of _one another

(Q

_{is a} local reaction

rate)

_:

k(t)

=

Q (a~/R~) (a/R)~

N ^~~~+

~l,

_o

>

(6)

>

Let _us

apply

equations

(5)

and

(6)

to some

specific

_cases An interesting

(albeit unphysical)

case is that of Rouse

dynamics plus

excluded volume interactions for which

[21]

z =

2 ₊

II

_v whence o

= v

(d

+ g

)/(2

_{v +} I

)

= I

e/8

₊

O(e~)

_using the first order express-

ions

[15-17]

_v

=

(I

₊

e/8)/2,

_g

=

EM.

^Thus o

~ l and _we

identify

_{a «} diffusion-controlled _»

case

k(t)

t~ ^~'~

(Rouse

+ excl vol

(7)

When

hydrodynamics

_are included the situation _is more subtle In theta solvents the correlation hole effect vanishes

(g

=

0)

and _since

z =

d _we have o

= I. Tints _{is a}

marginal

case and

simple scaling

cannot tell _{us more} A natural

expectation, by analogy

with other

marginal phenomena

_m statistical mechanics

[22]

is

logarithmic

_time

dependence

A similar situation arises in «

ordinary

brownian

particle

diffusion which _is

marginal

m 2 dimensions

(o =d/z= I),

for small times

[23]

the reaction rate _can be

expressed

_as

(C

and C' _are

constants)

k(t)

=

(C/r (Brownian particles,

d

=

2

) (8)

In

(t/t~)

₊ C'

In tints expression r is the

longest

relaxation time of the

system

and t~

proportional

to

L~

is the diffusion _time for the sink of _{size L} The result _is for the limit of _an

infinitely

reactive sink In section 3 _we will arrive at _a result for

theta,

solvents of

precisely

this form.

Finally,

_m

good

solvents both excluded volume and

hydrodynamical

interactions

feature,

with _z

= d and o

=

I ₊

g/d

from equation

(5)

Since g > 0, thus o

~ l and _one identifies _a

law of _mass action case From equation

(6)

_we

anticipate

k(t)

N ~~~+ ~l

(good

solvents

) (9)

We expect at most a weak time

dependence

_m

k(t), certainly

_no

algebraic

behavior

k(t)

has the _same

form,

_m fact, _as

k~

as denved _m FO and does _not scale like

r

'

(The significance

of the exponent o of

equation (5)

_m

determining k~

is discussed _m FO

m terms of the

parameter

Z N ^{"l~ ~l} which _measures the number of chain end collisions

in one

polymer

relaxation _time

r

)

For

good solvents, then,

the correlation hole _« tips the balance _in favor of mean-field law of _mass action behavior away from the

apparently marginal

situation

suggested by

_z

= d.

3. Renormalization group calculations of

k(t).

The

scaling _arguments

of the previous section have

already suggested

_a

special

role for d

=

4,

for

example

equations

(4)

and

(7)

descnbe behavior which

changes

from diffusion- controlled to mean-field _as d passes

through

4 In this section _we

exploit

tints fact

by deriving

k(t)

for the _{various cases in an}

epsilon

_expansion about four dimensions

3 BARE SERIES FOR

k(t).

We describe the

cyclization

_process

by

the

following

Fokker-

Planck equation for

P( (c), t),

the

probability

of _a

polymer configuration (c(r))

$

= FP ₊ Uo ³

(r(0)

c

(No) )

P

(io)

F,

whose

explicit

form _is given m

appendix

A, _is the diffusion operator

[17] including

the effect of

hydrodynamics through

the Oseen tensor and excluded volume

through

the Edwards

Hamiltonian The _vanous

dynamical

classes considered here _are defined

by switching

_on and

off _various of these interactions _m F _{i e}

setting

^the

corresponding coupling

constants to zero

uo is the bare reaction rate and

No

the bare

polymer

contour

length

The reader _is referred to FO for _a fuller discussion of the model

The solution of equation

(10)

_can be

expressed

_m terms of the Green's function

(without sink)

G

( (c'), (c),

t

t')

_as

P( jcj, t)

=

P~~( jcj

+ +

~

dt' d

ic'l

G

ic'l, ICI,

_t

t')

_{uo 3}

(c'(0) c'(No))

P

jc> j,

t~

(i1)

where the chains _are assumed to be _m

equihbnum, P~~(c),

at t

=

0

By integrating equation (10)

_one obtains the

dynamics

of the normalization

A(t)

~

= uo

Id (c)

3

(c(0) c(No ))

P

( (c), t) (12)

t

Solving iteratively

for P m powers of uo from equation

(I I)

and

substituting

_m

(12)

_we find to second order _m uo

~

=

Uo13 (R))

₊

Ul ldt'(3 (Ro)

3

(R,,)) (13)

where

(.)

denotes _an average over the

equilibrium

distribution

P~~(nosink),

R

=

c(0)

_c

(No)

and

R,

=

c(0, t)

_c

(No, t)

_is the end-to-end _vector at time t The second order term _is the time

integral

of the return

probability

that _an

initially looped

chain _is again

looped

at time t m the absence of reactions

(see FO)

In

appendix

B _we outline the evaluation of the terms _m equation

(13)

for small times

(t

« rR where r~ =

to N(/w~

_is the

longest

relaxation _{time m} the Rouse

model).

In terms of the dimensionless

coupling

constants

womuoL~'~fo

_and

eowfoL~'~

~fo

and

to being respectively

the bare excluded volume and

hydrodynamic coupling

constants and L _a

phenomenological length

scale

[17])

the final result for the small time bare _series reads

k(t)Nl

" A

(Wolio) (No/L)~'~

₊

Wl/fo(No/L)~ Ble(t/rR)~'~

+ Wo

eo/tolNo/Ll~ lDle

₊

El

,

where

A

=

(2

_g~

)

^{(e/2) 2}

~

B

4(5

EM) 4

~

(e/4j 3

~~~~

D

=

6(2

_{w )~} ^~

,

~

and E _is dimensionless and

depends nonsmgularly

_on

spatial

dimension Since the

singularities

_m the bare _series for

k~ (see FO)

derive from the

integral

^of the small time behavior of the _return

probability

and excluded volume effects _m

(3 (R)),

_{one can see} from equation

(13)

that

k(t)

has the _same

singularities

to second order. Thus the beta functions and

the fixed point values of the renormahsed

coupling

constants _{w, e,}

t

are as for

k~

and _{we can use} the results from FO Of _course this must be true if the

theory

_is

renormahzable

3'2 _{MELTS We now}

speciahse

_to the _various types ^of

dynamics, begtnning

with the Rouse model for which eo =

0 _m

equation (14).

We

emphasize

that Rouse behavior _is believed to

apply

_{even to}

entangled

melts for the short _times which _we will consider in the

following, enabling

_{qs to}

employ

renormalization _group methods

Renormahsmg

the bare _series

(see Appendix C)

leads to the

following

expressions for the fixed point

coupling

constant w*

(to

first order _m

e)

and for the renormahsed _series for

k(t) _(to

order

w~

w*=-64we+O

(e~)

k(t) (N~

= Aw

(N/L)4~

₊

jBlei(t/rR)~~

l

i(N/L)~

₊

Hj ^w~+

O

(w~)

where H

=

O

(e°) _{(Rouse) (15)}

and A and B have been defined in

equation (14).

H

depends

_on N and L but the

important property

^here is

simply

that H is

independent

of time. Note that

to

=

t

and

No

=

N _are unrenormalised _in the Rouse _case and that the

lie singularity

has been renormalised _away.

From

(15)

_one has

k(t)

=

Aw*/ ((N~)

_to first order in

e at the fixed

point

_; we seem to have

a time

independent

result at odds with _our

scaling expectations, equation (4)

The

following observation, _however,

resolves the

apparent

contradiction. Whtle _we do not know

k(t)

to second order _{in e}

(for

which purpose we would require w* _to O

(e~)),

_{the time-}

dependent _part

_can be inferred to its

leading

order of e~, as may be _seen

by expanding

the expression in

equation (15)

k(t) tN~

=

Aw

(N/L)~'~

₊

(Em

In

(t/rR)

₊ G

j w~, (16)

where all time

dependent

terms _in G _are

O(e).

Since B

=

Ofie°)

and the Aw term cannot

generate

time

dependence, equation (16) yields

the

leading O(e~)

_time

dependent

term when w* is known to

O(e)

At the fixed

point equation (16)

_can be written

k(t)

= 16

e/(wN~ _t ii em

In

(t/rR)1+ (17)

Motivated

by

the

scaling arguments

we

exponentiate

this form _to obtain the

large

N behavtor to first order _{m e} for chatns in _a melt

k(t)

=

~~

( _(t/r~)~

~'~

(melts,

N very

large ) (18)

wN

t

The result agrees with the

scaling prediction

of equation

(4).

The

subtlety

involved _in this

type

of

exponentiation

_is discussed _m reference

[26].

For finite chains _we must consider _crossover effects. The renormalization group

equatton

dictates

(see Appendix D)

that

k(t)

has the

following

form

~f

_{is an}

arbitrary function)

k(t)

=

I/TRf(t/TR, X)

where

x=

(N/L)e/2w/(w* -w). (19)

Therefore _we rewrite the renormahzed _series

equation (16)

_in terms of the

good

^vanable

X,

the time

independent part

to

O(e)

and the time

dependent part

to

O(e~)

The result _is

k(t)

=

~~ _~ ~

[l

^~

^{e/4 log} (t/r~)j ⁽²⁰⁾

wN~(

I ₊ X I + X

Since the fixed

point

result must be recovered when X becomes

large, exponentiation

is

again

suggested

k(t)

=

k«(t/TR)~

^{~'~~~~' ~ "~}

k~

=

16

e/w~ ~

~~ (21)

~R "

~N§ir~ (nlelts)

The result is

expressed

in terms of

k~

from FO.

Very long

chains

correspond

to

X

becoming large

when the fixed

point

^result

equation (18)

_is recovered. The

expenmental prediction

_may be stated thus for

melts, entangled

_or

unentangled,

_we

predict k(t) decays

_as

a nonuniversal _power law for

sufficiently

short times

(less

than the

entanglement time),

wtth

JOURNAL I)F PHhSIQUF II T M 4 AVRIL (WI 23

that power law

tending

to

I/4

_as the chains become

longer.

Of _course if the chains _are

entangled

then the observed

k~

_is not

given by

the expression in

(21).

3.3 ROUSE PLUS EXCLUDED voLuME. Before

proceeding

to dilute

solution,

let us

investigate

the effects of the correlatton hole

by treating

^the

unphysical

but

interesting

_case of Rouse

dynamics plus

excluded volume

interacttons,

the bare series is then

given by

the full

expression, equation (14). Having

the _same

singularities

_as for the _senes for

k~

which _was

analysed

in detail in

FO,

it follows that

to, No,

wo and _{eo are} renormahsed in the _same way and

e and _w have the _same fixed

point values,

with _w*

= 32 _we

being

the stable sink ftxed

point

value to order _e

(see FO).

The crucial

point

is that the first order term and the

leadtng

time

dependent

term of the renormahsed series have the _same form _as for the Rouse _case, equation

(15).

This _is true because these two terms derive

respectively

from the wo and

w(

terms in

equation (14)

whose

leading

contnbutions to the renormahsed _senes denve from

the

leading dependence

_{of wo} on w and _e,

namely

_{wo =} w The wo eo ^term does _{not enter} the

picture, being

second order and time

independent. Furthermore,

_{one can} set

to N(

=

(N~

_to

this order _since

lfigher

order renormalization of

to N( produces only

second order time

independent

and third order time

dependent

terms.

Finally,

the choice of renormalization constants _{is so as} to subtract off the

Ifs divergence

_m the

w(

_term,

just

_as for the Rouse _case The renormahsed _series must therefore be of the _same form _{as m} equation

(16),

other than the presence of

higher

order terms

involvtng

_e

k(t) (N~

= Aw

[1

⁺

~~ ln

(t/r~)j

^{+ O}

^(w~)

^{+ O}

^{(ew) (22)}

4 A

The time

dependence

_m the

higher

order terms ii of

O(e~)

when _w and _{e are} of order

e. If _{we were} to set _w

equal

to the Rouse fixed point value

(w*

= 64 _we

)

in the above expression, the Rouse result would be recovered

(Eq. (18)).

The

only

difference lies _m the

new fixed

point value,

_w *

=

32 _we

being

half of the Rouse

value,

the final exponent is

halved from

EM

to

-e/8. Explicitly,

with

B/A =1/(64 w)

when

e=0,

we have

Bw*/(4

A

)

=

e/8. Nottng

that A

=

II (4

_{w ~)} when _e

=

0,

and

exponentiatmg

as

before,

we

have for _very

long

chains

k(t)

=

~ _~

~

(t/rR)~

^"~

(Rouse

_{+ ex.} vol.

) (23) w(N

Now m FO the renormalization group

equation

for

k~

_was

solved,

at the fixed point ^the

solution _was of the form

k~ proportional

to

I/r~x

where r~x

~N~~~+~~

is the

longest

relaxation time _m this model A similar calculation shows that

k(t)

=

f(t/r~x)/r~x

for _some function

f. Since,

to zeroth order _{m e, r~}

= r ~x

[24, 25]

the form of

k(t)

_m

equations (23)

_is

consistent wtth this result and _{one can}

simply replace

_r~ wtth r~x since the

prefactor

and the

exponent

are both valid

only

to order _e

Using

the expression for

k~

from FO _our final result is of

precisely

the form _in

equations (7) suggested by scaling

_arguments,

exhibiting

the

importance

of the correlation hole

"~ ~~'~~

~~ (24)

=

/~w _~x~~

(Rouse

_{+ ex.} vol

,

large

N

)

3 4 DILUTE SOLUTION We will treat the _cases of_« strong

hydrodynamic

interaction

(i.e

all expressions will be evaluated at the fixed

point

of the fnction constant

f).

Once

again,

the renormahsed _{senes is}

unchanged

from the Rouse _case

(to

the orders of interest to

us) by

vtrtue of simtlar arguments ^{to those}

expounded

in the Rouse

plus

excluded volume _case. Thus

equation (22)

is still valid. An

important

new

feature, however,

_is that _now

( plays

the role of

a

coupling

constant. More

precisely,

there _{is an} additional

coupling

constant

to

w

((of1~ )

L^~'~

[17]

and _our strong

hydrodynamtc

^results _are at the ftxed point value of

f, namely f*.

Therefore the renormahsed _senes reads

k(t) l~f

* L

~'~N~

=

Aw

1

+

~'~ ln

(t/r~)j

^+...

(hydrodynamics). (25)

4 A

Consider

firstly

theta solvents for which excluded volume interactions

vanish,

I-e-

e ₌ 0. Renormalization _{is as} for

k~

_in

FO,

where it _was shown that at the fixed

point, f*

=

8

w~ e/3,

the beta function

fl~

=

w~/(128 w)

_to order _e, I-e- to this order it has _no nontnvial fixed

point.

Thus to understand the

large

N behavior of

k(t)

_one must

study

crossover behavtor

describing

the

approach

of _w to _zero.

Now

k(t) obeys

the _same renormalization group equation as

k~.

In

appendtx

B of FO this

was solved at the

nondraimng

^fixed point to order _{e :} k

=

H(Lexp[128 w/w])

where His _a well behaved function. Thus to this order

k(t)

= G

(L

_exp

[128 w/w],

t

) (26)

for _some G

When

hydrodynamics

_{is present} the relevant dimensions _are

[ii

=

[k(t)]~

=

[L]~'~ (this

may be _seen from the form of the diffusion

operator F, (Eq (Al)),

and the Fokker-Planck

equation, (Eq. (10))),

Thus

k(t)/C~/~

= G

(CL

_exp

[128 w/w],

C ^~/~t

) (27)

Choosing

C

= I

IN dictates,

for _some function

I,

the form

k(t)

=

(I/N~/~) I((L/N) exp[128 w/w], t/N~'~) Recogntsmg

that

N~/~ equals

the

longest

relaxation time _r~

(the

_« Zimm _»

time)

to within _a dtmension

independent

constant

[26]

_we have for

some function

f

k(t)

=

I/r~ f(X, t/r~)

~28)

X

= 128

w/w

₊ In

(LjN) (theta

solvents

)

We _now express equation

(25)

in a form

compatible

wtth equation

(28) [17] Substituting

w =

128

w/[X

In

(L/N)]

into the

leading

order time

independent

and

dependent

terms

(in

powers of

e)

of

equation (25)

_we have

~ ~ 32 In

(t/r~)

~~~~'~~

~

" ~~~

w(X

In

(L/N))

~

~

2(X

In

(L/N))

^{' ~~~~}

where

f*

=

8 w

~

e/3, B/A

=

1/64

_{w +} O

(e),

A

=

I/4 ^w~

₊ O

(e)

and the fact that _w

= O

(e)

have been used. Now the

correspondence

of the above expression ^to that in

equation (25)

to

leading

order is _not affected

by adding

and

subtracting higher

order terms in

(29)

Thus _we

replace N~

with

N~/~,

and X In

(L/N )

with

X, thereby rendering

the _expression

compatible

wtth the

requirement

of equatton

(28)

_:

k(t)

=

~~

^~~~~

(l

₊

1/(2 X)

In

(t/r~)) (theta

solvents

). (30)

SW

'IN

X

Note that _w and therefore X _are

negattve,

ensunng the correct

sign

of

k(t).

Note also that the

argument

^{of the}

logarithm

is undetermined to within _a

multiplicative constant by

_our

derivation

(such

a factor induces

higher

order

time-independent terms).

In

equation (30)

_we

have made the

simplest choice,

with the

ambiguity

understood.

At this

point

_we are

guided by

the

arguments

of section 2 to exponentiate ^to a

loganthmtc

form

(cf. Eq. (8))

rather than _{a power} law.

Using

the result for

k~ (Eq. (40)

of

FO)

the final result _is

~ -12

~~ ~

ew~1~N~/~X(I 1/(2 X)

In

(t/r~))

~ l

" l

1/(2 X)

In

(t/r~)

where

k~

=

Ci/(Xr~)

,

C i ⁼ 1.08 and X

= 128

w/w

₊ In

(L/N) (theta

solvents

). (31)

The result

[26]

_r~

=

(w~

^~~/~

Ci/12)

1~ EN ^~'~ has been used. It _is remarkable that the above expression is consistent

(to

the orders to which _we

work)

with _a result which _is of

precisely

the

same form _as the Browntan

particle result,

equation

(8). Specifically, by explicitly substituting

for X and interpreting ²¹ⁿ

(NIL)

_as the zeroth order truncation of

(d/2)In (NIL)

_one

obtains

2

Ci k(t)

=

(theta

solvents

)

,

(32)

r

[In (t/t~ )

²⁵⁶

w/w]

where t~

proportional

to

L~/~

may

loosely

be

thougltt

of _as the relaxatton ttme of the

renormahsed

«sink», namely

the end «blobs» of the _coarse

gramed polymer

each

containtng

L segments.

The last _case to consider _is that of

good

solvents

(full

bare series,

Eq (14)). Again,

renormalization

proceeds

_as for

k~

for which

(see Eq.(39A)

of

FO)

to

leadtng

order

fl~

=

w(w*

_w

)/128

_w at the self

avoiding

non-free

draining

fixed

point (e*

= w

~

e/2, f*

=

2

w~e)

where w*

= 16 _{we is} the

unphysical positive

ftxed

point.

^The

physical

ftxed

point again vanishes, necessitating

crossover

analysis.

Since

k(t) obeys

the _same renormali-

zation group

equation

as

k~ (t

is not

renormalised),

the solution has _a similar form to that for

k~ (cf. Eq. (C3)

of

FO) k(i)

= G

(LN-

i/Y, L

j(w

w*

)/wj-81e, 1) (33)

where _y

=

y~r(e*), y~r(e)

=

L(3

In

Z~r/3L)/~

^and

^No

₌

Z~r

N

[17].

The relevant dtmensions

are

[ii

=

[k]~~

=

[L]~'~.

Thus under the

rescaling L,

N go ^to

CL,

CN _we have

G

(c

^{i i/Y} LN i/Y, cL

(w

w*

)/wj-

_81e, c

d/21)

₌ c ^d/2

k(1) (34) Choosing

C such that C ^~'Y LN ^~/Y

=

I,

i e C

=

(N IL

_)~/~Y ^l L

=

(L/N

_)~~ L

(since [17]

² _v

=

I/(I

_y

))

and _using

[17]

the fact that

r~

^is

proporttonal

to

(N/L)~~L~/~

where

r~

is the

longest

relaxation time for

good solvents,

_{we can} wnte

k(i)

=

i/r~ f(x, i/r~)

where

X

=

(N/L)~~'~ (w

w*

)/w (good

solvents

(35)

Substituting

w ₌

(NIL

)~~'~ w

*/ (NIL

)~~'~

Xi,

which to order _e becomes

w =

w*/(X-

I

),

into the renormahsed _series equation

(25)

and

insisting

on the form _in

equation (35)

_we obtain

k(t)

=

[ _(L/N)~~

~~~

[l

^{~ In}

(t/r~)j ^{(good solvents) (36)}

'r 'l L

(X- 1) 16(X 1)

Our

scaling

_arguments have led _us to expect no

algebraic

ttme

dependence

in this _{case, so we} leave

equation (36) unexponentiated

and

interpret

the

loganthm

_{as a} weak time

dependent

modification of _a result which

essentially

scales _as the

equihbnum loop probability.

This

follows if _{we assume} weak

N-dependence

of _w such that

X~N~~/~

_for

large N,

then

k(t) N~~~~+~/~~

_{to within}

logarithmic

time

dependence.

Since _g

=

EM

to first order _we

recognise

this

exponent

as

v(d+g)

to this

order,

_i-e this _is the

scaling

result of

equation (9).

Finally,

_we summarise _our

good

solvent result in terms of

k~ (Eq. (41)

of

FO)

and

[17]

r~

= el~

(N/L)~~ L~/~2(2

w

)~~+~

_exp

(eI/4

_e 3

J/8),

where I and J _are constants

given

in

equattons (340), (341)

of

[17]

~~~~ ~°'(~ 16(I-

1)

^~~

~~~~~j

C~

~'~~~~~

~°'

~

(X

I

r~'

^{~ ~ ~ ~~'~~}

and

X

=

(N/L)~~'~ (w

16 _we

)/w (good

solvents

(37)

The

simple

relation between

k(t)

and

k~

in

equation (37) suggests

there should be _a

simple

formula which

interpolates

between the short time reaction rate

k(t)

and the

long

time

reaction rate

k~.

We denve such _an

approximate interpolation

formula _in

appendix

E.

4. Universal relations and

experiment.

In this

study

_we have derived untversal

relationships

between 3

quantities

which _are

(in pnnciple) expenmentally

measurable the short

cyclization

reaction rate

k(t),

the

long

ttme

cyclization

rate

k~

and the

longest

relaxation time _r of the

unperturbed polymer (i

_e. no

reactive

groups)

These relations

apply

to the _crossover regtme i-e-

they

_are not limited to

«

infinitely long polymer chains,

and _are

independent

of the

strength

and

chemistry

of the

reacting

end groups.

For theta solvents this

relationship

_may be stated _as

(see Eq. (31)) k(t)

= ~

°~

(theta solvents) (38)

where

Ci

is _a constant defined _in

equation (31).

The

relationship

has _no free

parameters

other than _a

possible

constant

multiplying

the

t/r~

_argument of the

logarithm (see

discussion

following Eq. (30)).

For

good solvents,

_equation

(37) implies

the

no-free-parameter

relation

k~

_r~

k(t)

=

k~ 1

~~~

ln

(t/r~) (good solvents), (39)

"

which in practice may be

indistinguishable

from

k(t)

=

k~.

Turntng

_now to

polymer

melts below the

entanglement threshold,

equation

(21) predicts

that

k _t

)

=

k~ ( if

_r~

)~

^{~~ '~ "}

^~" (unentangled melts) (40)

The _presence of

entanglements

invalidates this

relationship

of _course in that _{case our} results

are confined to the short time regime

(less

than the

entanglement time).

We restate the result for 3 dimensions for

completeness (Eq. (21))

k(t)

=

~ X/(I

₊

X) (t/TR)

^{~~~~~~ ~~~~}

ir TR

which tends to

~ (t/rR)~

^~'~ _as N becomes

large (41)

w rR

Winmk

[5]

^{has reviewed} many of the

extsttng expenmental iieasurements

_of

_long

_time

cyclization

rates m dilute

polymer

solutions Fluorescence measurements of

k~

_{on pyrene}

end-capped polystyrene

_{m a} theta solvent

[5]

_are consistent with the theoretical

k~ (as

calculated _m

FO)

with _a value of the _crossover parameter ^X=-12. The

predicted

loganthmtc

time

dependence

of

k(t) (Eq. (31))

_may then be observable for this

system, though

the

prefactor

_is rather small. Simtlar measurements m

good

solvents

[5]

suggest a

value X

= 5

,

from

equation (36)

this

logarithm

_m

k(t

_may be unmeasurable. At least for this system,

then,

it _is

perhaps

_more realistic to

predict

^that

k(t)

and

k~

are very close to _one

another.

Evidently, then,

short time

cyclization

_m melts and solutions exhibits much nch behavior.

We _{are unaware} of any ^extstent

systematic

measurements

probing

this regtme and _we

hope,

for such

expenments

^{in the future}

Acknowledgements.

This work _was

supported by

the National Science Foundatton under grant ^No. CTS-89-08995

(B.O.'S.)

and the

Japan Society

for -the Promotion -of Science

through

a

postdoctoral fellowship (B.F.).

Appendix

A _: the di«udon operator ^F

The operator ^F

appearing

_in

equation(10), describing hydrodynamics through

the Oseen

tensor

T~~

^{and excluded volume} interactions

through thq

Edwards Hamiltontan

H,is given by

N0 .N~

F

= dT dT~

£ 3/3Ca(~) [3afl/~0

3

(~ ~')

₊

l~afl(C(~) ^C(~')))

X

~ ~ ~~

X

[3/3Cj(T')

₊

3H/3Cfl(T'))

,

T~~ (x)

=

(2

_w

)~~ ^d~

k

1/(l~k~) [3~~ k~ k~/k~] e~~'~,

_H

=

Ho

+

H~

Ho

=

1/2 j)~ dr[dc(r)/dr]~

H~

=

fo/2 ~~

dr

~~

dr' 3

(c(r ) c(r')). (Al)

o o