Remarks on the dynamics of random copolymers

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Remarks on the dynamics of random copolymers

J. Bouchaud, M. Cates

To cite this version:

J. Bouchaud, M. Cates. Remarks on the dynamics of random copolymers. Journal de Physique II,

EDP Sciences, 1993, 3 (8), pp.1171-1177. �10.1051/jp2:1993189�. �jpa-00247894�

Classification Physics Abstracts

64.60C 05.40

Short Communication

Remarks

_on

the dynmdcs of random copolymers

J.P.

Bouchaud(*)

and M.E. Cates

TCM Group~ Cavendish Laboratory,

Madingley

Road, Cambrdige CB3 OHE, U-K-

(Received

7 May1993, accepted 15

June)

Abstract. We discuss the dynamics of random AB copolymers, and show that _a glassy,

quasi-frozen phase is reached _{as soon as XAB}

@

" I

(where

N is the number of _monomers,

N«

Ne the entanglement length and _XAB the Flory interaction

parameter).

This _{criterion is} satisfied for much smaller _XAB than the value for which segregation occurs: XAB " I. The relaxation

time of

weakly

disordered chains is furthermore found to take the empirical form

N~+",

_where

~1 ^can reach significant values for certain choices of _XAB and the fraction of B-monomers.

In recent years, several _papers have aimed _at

describing

the

thermodynamic properties

^of

random

copolymers

_or

heteropolymers [1-5].

^The motivation for these studies _comes from biol- ogy

(heteropolymers

_as models of

biological molecules)

and also from the

technological

interest of

synthetic copolymers.

A _very

interesting

work in this respect in the _one of Fredrickson and

MiIner [3], ^who

investigate

^the

phase diagram

of random A-B

copolymer

melts _as the

Flory

parameter XAB and the

polymerisation reactivity

ratios

(I,e,

^the correlations between A and B _sequences

along

the

chains)

_are varied. Such _an

equilibrium phase diagram

could be

helpful

for

designing copolymers by

control of temperature or the

polymerisation procedure

[3]. ^From

a theoretical

point

of

view,

reference [3] also report the _appearance of _an

interesting

multicrit- ical

l'Lifschitz')

_point in the

phase diagram.

All these

predictions, however,

_are

only

valid if

equilibrium

_can indeed be

reached~

I,e. if _some kind of

dynamical freezing

does not _occur. The aim of this note is to discuss the

corresponding dynamical elfects~

and to show

that,

because of the

randomness,

diffusion and relaxation _are

drastically

slowed down for

long

chains. New

reptation

laws _are derived. In the

interesting quasi-homogeneous

limit

(I,e,

when the fraction

#

of B _monomers is

small),

_we show that the

disentanglement

time grows with the molecular

weight

N _as

roughly N~+",

where ~1> ^{0 is} an effective exponent ^which

depends

both _on

#

_an

XAB.

Let _us start

by discussing

the motion of _one mobile random

copolymer

of

length

N in _a

melt of

similar,

but immobile

(for example,

much

longer)

chains. As

usual,

_{we assume} that the

(*)

^also at: Service de Physique ^{de l'Etat}

Condens4,

CBA, Orme des Merisiers, 91191 Gif-sur- Yvette cedex, France

1172 JOURNAL DE PHYSIQUE II N°8

chain _moves

by

curvilinear diffusion

along

its _own contour, ^and suppose for the moment that the

entanglement length

^is Ne ₌ I. If

#( I-S)

_XAB <

0.5,

the

equilibrium

state is

homogeneous

[3], ^{and it is reasonable to} assume no

particular

correlations in the relative positions ^A's and

B's,

The interaction _energy of the 'marked' chain in _a

given configuration

C is

given by V(C)

=

L$iui(Ri),

where

Ri

is the position ^{of the i~~} monomer and

ui(R)

is the

potential

created

by

all other chains at site

R,

which also

depends

_on the _nature of the i~~ _monomer

(A

_or

B).

One

can

always

write _vi

(R)

₊

u(R)

₊

~zw(R),

^where _~j = +I if the i~~ _monomer is of type

A,

^and

-I if it is of type B. Now let _us

bring

the

polymer

to _{a new}

configuration C'[s] by translating

the whole chain

along

its tube

by

_a curvilinear

length

_s.

Defining AVIS)

₌

V(C) V(C'[s])

and

using

< > to mean

averaging

_over the 'chemical'disorder

(not

the Boltzmann

average),

the averages ^<

AVIS)

> and _<

AV(s)~

_{> are}

given by:

<

AVIS)

_>= 0

(the problem

is

statistically

translational

invariant) and,

for 0 < _s <

N,

<

AV(s)~

_>=

^2si~

₊

^2Nlii~ ii)

We have assumed for

simplicity complete

chemical randomness

along

the

chain,

I.e. < ~, >=

l

2#

_{a m} and _< ~i~j ^> -m~

=

6ij.

In

equation (1),

² =< u~ _{> <}

u

>~

and

112 =< w~ _{> < w}

>~

_measure the fluctuations in

potential

_energy. In order of

magnitude,

I

~w I

~w

~/4(1 4)(z" 2)x~~kT (2)

where z" is the effective number of

neighbours

to a

given

_monomer

(2

of which

belong

to the

same chain and hence _cannot contribute to the fluctuation of

potential energy).

For _s >

N,

the tubelike environment of the chain is

completely replaced

^and

equation (I)

saturates at the value <

AV(s

>

N)~

>=

2N(ii~

₊

lii~).

~S _---' _-'

,'

I' I'

/

s

Fig. 1. Typical variation of the

potential

_energy

V(s)

_seen by an random copolymer _{as a} function of the 'distance' travelled along the chain, _{s :}

V(s)

_ci

P@~.

Equation (I)

has _a

simple meaning:

the mobile chain evolves in _a random

potential landscape schematically represented

in

figure

I. In

fact,

this

problem

is very close to the 'one dimensional

random force model' [6-9] ^{for which}

quite

_a number of results _are known. For _{our purpose,}

however,

it is sufficient _to understand from

(I)

that the characteristic _energy

barrier,

which the chain will have to overcome in order _to travel _a distance _s < N, _grows

roughly

^like

~AB

-Segregation

N

Fig. 2. Phase

diagram

in the plane

(N,

XAB). The dot-dashed line corresponds to the segregation

line for homopolymers, the thick dashed line is the dynamical _crossover line above which freezing

occurs, while the plain horizontal line is the equilibrium segregation line [3].

e(s)

+~ F

(s

+

N). Hence,

_a

simple

Arrhenius-Kramers argument

(see

_{e-g- [9])} leads _to the

following

law for time

t(s) required

for the

polymer

to move a distance _s

along

its tuber

where _{D is} the usual curvilinear diffusion _constant of the whole chain

ID

_oc

N~~),

and the expression between

curly

braces is the inverse 'trial

frequency'

_on scale _s. From equations

(2, 3),

_{one sees} that _a

dynamical

_crossover line _appears in the

plane N,

_XAB

(see Fig. 2)

for XAB Cf

N~~/~, i,e.,

much below the

equilibrium segregation

line _{XAB >} [3]. ^{Below this}

crossover

line,

the usual

reptation

laws _are

only slightly affected,

whereas above this

line,

the

chain is

trapped

in

deep

_energy

valleys,

which

drastically

slows down its

progression.

It is

possible

to understand this _crossover line _{as an} 'aborted

segregation

line' in the

following

sense: if _a chain is divided into segments ^of

length P,

each segment ^will present

(through

statistical

fluctuations)

_a relative _excess of A

(or B)

_monomers, of order

P~~/~

_{One could} think _{to arrange} the chains in such _{a way} that the 'A-rich' segments ^sit

together

in

regions

of size _ci

/P land _similarly

for the 'B-rich'

segments), leading

to _a micro

phase separation.

This

sorting

out of segments costs an entropy m P~~ _{per monomer,} while the

resulting

_energy

gain

is of order

XABkTP~~/~

_{per monomer.} The

optimal

P would thus be of order P" _ci

x(~,

and the condition P" < N would coincide with XAB ^>

N~~/~.

_The

reason

why

the system ^does

not

phase

separate is because _a local increase of

density

of A must

by exactly compensated by

_a decrease of

density elsewhere,

_so that the total _energy

gain

_{per monomer} is in fact of

order

XABkTP~~

and not

XABkTP~~/~. Hence,

the fluctuation _energy

XAB4

_cannot induce

strong

static correlations.

However,

it does lead to

important dynamical

effects.

1174 JOURNAL DE PHYSIQUE II N°8

In the _case where iii

= 0

(corresponding

to a

homopolymer

in _{a sea} of

heteropolymers),

_one

finds from

equation (3)

that the chain

'creeps' logarithmically ii,

_9] rather than reptates

along

its tube:

/$

_{for t < t"}

~ ~

s"

log~

for t" _< t < t*" ^~~~

t"

where s"

=

~~ )~ ^{is the}

length

such that the _energy barrier

e(s)

is of order kT and t" _e

G

_is

the time

need~d

_{to diffuse}

a distance s"

along

the tube.

Finally

the _escape time from the tube t""

obeys s(t"")

= N, which

gives:

N2 _~

t""

= -exp

(/fl-j ₍₅₎

D kT

Note that the condition s" _< N for the existence of the

logarithmic regime

in

equation (4)

is

equivalent

to XAB »

N~~/~,

_as described above. For time scales

longer

^than

t"",

the tube

configuration

is

completely

^{renewed and the effective} energy barrier _ceases to grow

(see Fig.

I). Assuming

Brownian statistics for the chain in real _space

(though

_see remark

(a) below),

the _center of _mass motion then reads:

(2Dt)~H

for t < t"

R(t)

_oc

/~log t"

_{for t" « t <} t""

(6)

fi

for t"" _< t

with the

asymptotic

real _space chain diffusion constant D~r~~p is

given by:

Dcreep

"

Drep

~XP

l/Twj

^~

₍₇₎

where

Drep

oc N~~ _{is the diffusion constant obtained from the usual}

reptation theory

_[10,

iii.

When iii

#

0, ^the

logarithmic regime

in equation

(6)

is

only

observable for N _{> s >}

Nll,

and the

asymptotic

diffusion constant is obtained

by replacing

ii~ with ii~ ₊ lii~ in

equation )7).

Several

important

remarks _{are now necessary:}

a)

The motion of the other chains. Of _{course, our}

assumption

that the other chains _are immobile is incorrect if _one is to discuss _a situation where all chains have

approximately

the _same size. The disorder is thus 'annealed' rather than

'quenched',

and the

potential

_seen

by

_a

given

chain becomes time

dependent: V(C, t).

The slow motion of all the

chains, however,

induces strong ^time correlations in

V(C, t).

A self consistent

argument,

in the

spirit

of those

developed

in [9, 12] ^{shows that in fact the Sinai diffusion in}

log~t

is

unchanged (although

the

prefactors

will

change).

The basic idea is that since the correlation time of the medium is the renewal time t""

itself,

the energy barriers

hindering

the motion of the chains will

persist

until t"" Hence _we believe that

equations

_{(4~7) are}

qualitatively

correct,

although possibly

with _a smaller coefficient of the chain

length

N. The fact that the disorder is annealed rather than

quenched

_was of _course

implicitly

^assumed in

obtaining equation (6). Indeed,

in the

case of _a

quenched disorder,

the

configuration

of _a chain is _no

longer

Gaussian

(R

_oc

@)

but is characterized

by

R _oc N"

(with

u

# 1/2,

the

precise

value of which is still the matter of _some debate

[13])

[14].

b)

Chain contour

length

fluctuations. Above _we have assumed that

reptation

_occurs

only by rigid

translation of the chain _{as a} whole

along

the tube. This is the conventional

reptation

description

and omits fluctuations in both the tube

length

and the local _monomer

density

within the

tube, arising

from the internal

dynamical

^{modes of the chain}

ill]. However,

_we believe that the above results _{are more}

general:

indeed _{suppose one moves} the chain from

configuration

C to

configuration

^C' such that _{si monomers} do not

change position

at

all,

_s2

monomers occupy

positions already

taken

by

^different _monomers of the _same chain in C. In

this _case

< AV

(si,

_s2)~ >= 2s2@~ + 2

(N

_si

^d~ (8)

which in effect does not differ

greatly

^from

equation (I).

c)

The

entanglement

blobs. If the

entanglement length

N~ is not

equal

_{to one,} then N should

presumably

be

replaced by £

in the above formulae.

However,

_one should also

replace

the

strength

of the

fluctuating pitentials _P,

iii

by

effective

potentials

P~,iii~

acting

_{on a} 'blob'of size

Ne.

If _no strong correlations exist _on the scale of the blob

(which

is

certainly likely

_as

long

_as

XABNj/~

_<

l),

these effective

potentials

_can be estimated

using

the

replica

method

and

perturbation theory~

and

noting

^{that within} each

blob,

the chain behaves _{as a} ideal

polymer.

One finds that the effective

i~,

_{lii~ are}

given by N)Ni,

_{iii. This differs from the}

result

N)/~i,iii

_found

by assuming

that the

potential

_energy is the _sum of N~

independent

random variables

115]. ^{In other}

words,

the thermal _motion preaverages the random

potential

and reduces the fluctuations.

[This preaveraging

cannot take

place

at scales >

N~

because of the tube

constraint.]

We thus find that

equation (7)

must be modified to:

D~r~ep =

Dr~p

_exp

1/fl _~~ (9)

N~ kT

d)

The semi-dilute _case, As _is well known [10], ^the role of the _monomers is in this _case

played by

the 'concentration

blobs',

each of which contains _{g ct}

c~~H

monomers

(where

_c is the

concentration). Contrarily

to the above _case,

however,

the number of contacts per blob is

a order _one, and thus the

fluctuating potentials

do _not get

multiplied by g~~~H.

One thus expect ^that

D~r~~p(N, c)

+

D~r~~p(NC~H, I)

_as is the _case for

ordinary

reptation

(see

_[16] for

a recent

experimental discussion).

e)

The

Quasi-homogeneous

^limit. Let _us

finally

consider in

slightly

_more detail the _case of _a melt of random

copolymers

such that the fraction of B is very small:

#

< I. Then the above arguments suggest that the renewal time grows as t"" _oc N~

exp(q@),

_where

q =

q

#(z* 2)XiBN7~/~,

with

XiB

_"

f@~

the difference between the reduced interaction energy of AA contact and that of _an AB contact, _{and q a} number of order _one

(the

calculation of which would

require

a more precise

model).

We have

plotted

the above

prediction

in

log- log coordinates,

for z"

=

6,

and different values of

#

and

qx[~N7~M

The range of N _was the

experimental

_one, I,e, 10~ 10~.

As _can be _seen in

figure

3, _one observes in

general

quite _a

good straight line,

with _an effective exponent

slightly

greater ^than 3.

Hence, empirically

_one has:

t""(N)

_oc

N~+",

where the value of

~1for

different choices of and

x[~

is

reported

in table I. These values _are

obviously only

indicative,

since ~ is

quite

sensitive to the range of N chosen for the linear fit.

Nevertheless,

the

interesting

conclusion of this

'toy-model'

is that _{a very} small

proportion

of _monomers of _a

different chemical nature is

enough

to increase the effective exponent ^{for the terminal relaxation} time. It is conceivable that residual

'impurities'

in the chemical sequence of

homopolymers

might play

_an

important

role in melts at

large

molecular

weight,

and interfere with attemps to

1176 JOURNAL DE PHYSIQUE II N°8

* *

t

(arb, units)

io ioo iooo ioooo

~ iooooo

Fig. 3. Log-log plot of the relaxation time t""

versus N for _a rather small proportion of B _monomers

(#

=

10~~),

and

~x(BNi~~

" 0.5. The best _power law fit for 100 < N < 10~ is found to be

t** « N~'~~

Table I. Values of the effective

exponent

_~ ^{for different choices of and}

qx[~N7~R

= 0.05.

The ^*

corresponds

to values of _~ less than

o,05,

^while the

corresponds

to cases where the power law fit it

unacceptable.

The _range of N chosen for the fit is 100 < N <

10~, although

formula

(7) only

mattes _sense for

#N

> 1.

iii ^{-10~~ 10~~ 10~~ 10~~}

0.5 0Al 1.29

0.1 0.08 0.26 0.82

0.05 0.04 0.13 0Al 1.29

0.01 _{* *} 0.08 0.26

0.005 _{* *} 0.04 0.13

precisely

determine this exponent

(see

_e-g-

iii] _).

The fact that the

viscosity

exponent ^{could be} increased

by trapping

effects _was

already

advocated in several _papers

[18-21], although

^with

underlying

mechanisms

quite

different from _{the one}

proposed

here.

In

conclusion,

let _us

emphasize

the two main

points

of this _paper:

I) ^The

dynamics

of random

copolymers

melts should be

extremely

^slow as soon as #AB

@

_>

N«

I, I-e- for values of #AB ^{much smaller}

(for long chains)

than the _one

corresponding

to the

equilibrium segregation

line #AB ~ l. The

dynamics

of _a labelled chain could reveal _an

interesting logarithmic

diffusion behaviour for intermediate times.

ii Even in the _case of _very

weakly

disordered

copolymers,

_{one may} expect

significant departures

from the 'ideal'

reptation theory

with effective exponents ^{for the renewal} time

significantly

greater ^{than 3.}

We

hope

that these remarks will

trigger

_some

experiments

_on this

subject.

In

particular,

it would be

extremely interesting

to monitor the fraction

#

of

'foreign'

atoms in order to check that the effective

viscosity

_or ^diffusion exponent ^shows a

systematic dependance

_on this

quantity,

_as

predicted

above.

Acknowledgements.

J.P.B, wishes _to thank M. Adam for _{a very}

enriching

discussion.

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exp

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