Linear response of self assembling systems: mean field solution

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Linear response of self assembling systems: mean field solution

F. Lequeux

To cite this version:

F. Lequeux. Linear response of self assembling systems: mean field solution. Journal de Physique II,

EDP Sciences, 1991, 1 (2), pp.195-207. �10.1051/jp2:1991155�. �jpa-00247507�

J

Phys.

II1

(1991)

195-207 ', F#VRIER 1991, PAGE 195

Classification

Physics

Abstracts

05 20 36.20 61 25

Linear response of self hssembling _systems

_{: mean}

_{field solution}

F

Lequeux

Laboratoire de

Spectiomktrie

_et

d'Iii1agerie

Ultrasonores

(*),

Universitk Louis

Pasteur,'Institut

Le Bel, 4 _rue B1alse Pascal, 67070

Strasbourg

Cedex,

Franci

(Received

3 August 1990,

accepted

_m

final form

29

Octobei

₁₉₉₀₎ ''

~ ^~.

Rksumk. Nous ktudions

lq

rkponse hndaire de

systdmes

vivants, c'est-I-dire s~associant _{et se} coupant ^I

tkqulhbre thermodynamlque

Une approximation de champ moyen donne _une solution

analytique

_pour l'kvolution

temporelle

de la reponse. Nous _en ddduisons _{une expression}

analytique

de la viscositk des

polymdres

vivants Nous trouvons de nouvelles lois d~kchelles _pour

cette vlscositk dans la hmlte des temps ^de vie courts devant les temps de relaxations propres Ces

lois

dkpendent

de

fagon

non kvidente de~

l'exposant

du temps de relaxation _par rapport I la

longueur

de la chaine Nous vknfions _{par un} calcul exact l'aspect critique de,la

rkponse

dans cette hmlte.

,,

Abstract. We

study

the linear response of

living

systems, which break and recombine

randomly

The _mean field approximation gives an

analytical

solution for the time evolution of the response As _an example, the viscosity ^of living polymers _is computed New

scaling

laws for viscosity are found _in the _case of very short lifetimes The laws do not

depend

_{in an} obvious way

on the exponent of the chain relaxation time with respect to its

length

The critical behaviour _is confirmed

by

_an exact calculation _in the limit of very short lifetimes

,

1. Inboduction.

New _«

living

_» systems, such _as

living polymers (sulfur [1],

selenium

[2],

grant ^micelles

[3]j

_or

physical gels (See [4],

for

instance)

_are now available for

physicists.

All these

Systems

contain

elementary

^bodies

(e

_g

monomers)

These bodies self-assemble _in various

geometnes (we

say in various

states)

The word

«living»

_means that _a given

body,

at

thermodynamical

equilibrium, changes randomly

and

discontinuously

of _state

during

time For instance, a

monomer in a

living polymer system belongs

to _a chain whose

length

_{varies in} time

by

breaking

and

recombining

with _an other chain.

Hence,

at

thermodynamical equilibnum,

every ^state has _a finite

lifetime,

and _{we can} define _{a jump} rate from _one state of another We

are interested _in the linear _response of iuch

systems

to any ^extensive

perturbation,

such _as strain stress

relaxation,

electric

birefnngence,

time correlation functions

There is a fundamental difference between the effects of finite lifetime and of the

polydispersity

A

polydisperse

system response is the

averaged

_{response over} all the states.

(*) CNRS UA 851

196 JOURNAL DE

PHYSIQUE

II lV° 2

Conversely,

for _a

living system,

the response is a

complex

combination of

jump

rates and of the

kesponse

of each state _{: a}

body

_{in a} state A may relax in its _own state, ^but also in any other

states after _some

jumps

We _assume the linear

approximation

Hence _we consider that both the

density

of states and

the transition

probabilities

are

always

the

equilibrium

_ones. It _is then crucial _to obtain _a conservation law for the response

during

_a recombination. This gtves the

equations

^{for the} response relaxation.

In order to solve this set of equations, we make _{a mean} field

approximation

^This consists in

taking

the

body

that

changes

of _{state as}

carrying

the _mean response of the

system.

^We

give

also the expression of the _mean field

approximation

_error. We solve the equations ^and get a

general

_expression of the Fourier transform of the macroscopic response. We then deduce the

viscosity

and the vlscoelastic modulus of

living _systems.

We obtain _{an expression} for the response function that is _a

generalisation

of that of _a

polydisperse _system.

We calculate the

viscosity

of

living polymers.

Our

approach,

is limited to

single exponential decays

for each aggregate, ^and gtves different results from those obtained

by

Cates

[5]

in the _case of the reptatlon process We discuss the limit _cases, where the life time _is small

compared

to the

mean relaxation time. We denve the

scaling

laws of the

viscosity

_{as a} function of the

exponent

of the relaxation time with

respect

to the

length.

We confirm _some of these results

by

_an exact

calculation of the response in the lbnit _case where the lifetime vanishes.

2. Definition of the system.

We consider _a

system

constituted

by

bodies

(monomers

_or

elses)

which self-assemble in different

patterns

at

thermodynamical equihbnum.

Let _us call A the _state of _a

body,

and S the ensemble of the states.

Typically,

A is a

geometncal

_parameter which characterizes _a

morphology

of the

aggregates

^It can be the

length

of

polymer,

_or the size of a cluster

(see

Fig, I).

For each state A, there _{is a}

density n(A )

of

aggregates.

^{We normalize}

n(A) by

:

j.h(A ^{) V(A )}

^{dA = I}

⁽¹⁾

s

where

V(A )

_is the number of bodies that constitute _an

aggregate

^of type A. The

integral

_over

S _is the

integral

_over all the _states.

In the absence of

change

^between states, I-e- for _an ideal

aggregate

^{which would} stay its

Fig

I

-Living

systems are systems ^{where bodies self} assemble _in aggregates that break and recombine at the thermodynarnical

equihbriurn

A describes the

morphology

of each type of aggregate B _is the elementary

body.

N 2 LINEAR RESPONSE OF SELF ASSEMBLING SYSTEMS 197

pnmary state, the _response to _a

perturbation

would

decay

in time We hmJt ourselves to a

single exponential decay

The response _«~ of _a

body

state A

decays according

to

bW~ _W~

~

T~~

~~~

and _{T~~ is} the relaxation _time of the state A Let _{us now} define the transition

probabilities.

We will first consider the _case where _one

body changes

state

alone,

without

comb1nlng

with others. We will show later that the _case of _{a many}

body

association amounts to that of _a

single body

state.

For each

pair

of states

(A, A'),

there _{is a} flux of bodies from A to A' Let

~

be the transition

probability.

The

equation

of conservation for the number of bodies _is then

bn(A V(A n(A V(A n(A') V(A')

dA'

~~ ~(A-s) ~~~

~

s T(A-A)

where T~~ _~~ ^is the lifetime of state A and _is gtven

by

~

A~-

s)

S'

~

~~

')

~~~

We suppose here that the _measure

(in

the

meaning

of the

integral)

of _a state _is

infinitely

small with respect to that of the whole system.

Thus, integrals

_over S _are

equivalent

to

integrals

over S-

(A).

We have to _{assume a} law for the response conservation

dunng

the state transition. This _{is a}

crucial

hypothesis

_as it _is necessary ^{to set} a

simple

rule _in order to continue the

study.

We limit ourselves to the _case where this rules _can be

expressed

_{as :}

trA<(t)

=

~r~(t) (5)

where _{« is} the response

by body,

A and A' _are the initial and final state of _a

jump taking place

at time t. This amounts to assume for instance that transitions take

place

_very fast

Let _us consider _now the _case where two _{or more}

aggregates

can combine. The transitions

occur either

by

combination of two

aggregates,

or

by splitting

of _an

aggregate.

First,

two

objects

with states A

i and A

~ can stick and _jump to the state A~

(see Fig. 2).

The combined

probability

of this event _is :

n(Ai) n(A2)

~(AI,A~-A~)

For the inverse transition, we have

s1mllarly

_:

~(~3)

~(A3~AI,~2)

One could also take into account other _non linear terms without

large changes

in what

198 JOURNAL DE

PHYSIQUE

II N 2

if' ^~~

^~'

~(i,,iz~~~)

^~

~3

_

<

_

"~

fi ^lJ3-1~,ij

~

,

Fig. 2 Two aggregates

(at

state A

j and A~) combine into _an aggregate at state A

(and reciprocally,

the aggregate at state A~ separates m two aggregates ^of type Al and A~) ^{~ ~}

follows. We _can then define the transition rate of _a

body

from _one state to another

by

_:

i

n(A~) dA~

,

=

(6)

~(Ai-A~)

s ~(Ai>A2-A~) In

addition,

the conservation rule for the response is :

~r~~(t).

^V

^jA~)

= ~r~~ V

(A i)

₊

_~r~~.

V

(A~)

Hence the rates of the response

traitsition _during

_the interaction _can be

expressed

for each state

simply

_using

(6)

_:

n(A1) _«~~.

V

(A i) n(A~) _«~~.

V

(A~)

for state I and for, _state 2

(7)

~(AI~~3) ~(A2-A3)

For many

body interactions,

and in the _case of _an extensive response, we

can'get

a

single

state like expression for

thiiesponse

fluxes

by

_averaging, for each

aggregate,

over all the others aggregates which _are able to associate with it We _can do this

only

because _{we are} at

equilibrium,

and all the quantities ^involved

(state

densities and transition

rate)

_are constant We _now

compute

^{the evolution of the} macroscopic response.

3.

Equations

for the time evolution of the system.

An intuitive way to

compute,

^{the evolution of such}

living

_{systems is} to consider the time

response of _one

body

that

changes

_states, and _to

averaged

_over all the

possible

time evolutions This leads of _course to the

following

functional

integral

for ~the macroscopic response

i~r(t)>

=

PA

_(i) ~r

(t

=

0)

_ar

_IA It))

D

(A (t)) (8)

where

P~

is the _response propagator ^for a

body belongtng

to a state A

varying

~ _in time, _ar the

dinsity

of

time-dependent

state

(t), D(A (t))

_is the

integrand

_over all the

possible

time evolutions. We _can

expand

^the

integral

^and _express it as a converging series of classical

integral.

We will _use this formalism _{in a}

simple limiting

_case in section 8

Another way for

computing

^the response evolution _is to

baljnce

the response fluxes for all

lV° 2 LINEAR RESPONSE OF SELF ASSEMBLING SYSTEMS 199

states. Bodies

leaving

the'state at~time f carry the _{response «~}

(t).

We _wnte the balance of

the trans1tlon rates and of the response relaxations

Using (2), (3)

and

(7)

_we obtain _:

b~r> n(A V(A

_~r>

n(A V(A )

_~r>

n(A V(A

_{~r> n}

(A') V(A')

dA'

bt ^~~ T«> T(A-s)

~

s>

T(A-A)

and

becadse n(A

and

V(A )

are constant, _we have _:

3«A

_{«~ «~, n}

(A ') V(A ')

dA '

~~ T

~

s' ^~(A' ^~

(~ )

" ~ ~~~

with the rule _:

= +

(10)

~A ^~ "A ~(A s)

We have obtained the

complete

set of

linear'equations

for the evolution of the

rdsponse.

A direct solution would require an inversion of _{a non}

symmetncal _operator,

and would be very

difficult except ^for some

peculiar

_cases

When writing equation

(9),

_we recall that _we consider each

body

at _a state as carrying the

mean response of bodies _{at state A.} We

neglect

both the

disinbuilon

_{of responses at}

a

given

state, and

[he spatial

correlation _into

thi

aggregates. ^{The second}

assumptiin'is justified by ihe

fact that internal decorrelitions _are

generally

faster than response

rilaxations.

For instance in the Rouse model

[6,

_7~

aid

_{even for reptation} of

living polymers

[7~ ends effects _are

generilly

small

compared

to the whole chain

relaxatiin

_We

_postulate

_that

in

general

^both

approii-

mations _are not constraining. ^In

fact,

_in

the,

case of

rejtation,

end effects have _a mare

important relaxation,

_as

exp1alned

in 7

4. Mean field

approximation.

Equation (9)

describes how bodies at state A

relax,

leave the state, and how _some others reach A carrying their _own response. The _mean field

approximation

consists in assuming that the

body

_{coming m} state

j

^{carries the} average response of the

system.

^This

approximation

^does

not take into account correlations between several states It can

onlj describe

a system

fir

which transitions between states _are

discontinuous,

like

living polymers

with

briakmg,

_or

living Cayley

_-trees, but not

living polymers

which _grow

longer

and shorter

incrementally'or

'living'networks.

We will discuss the

validity

of the mein field approximation ^later.

The _mean field approximation consists into

substituting

_in

(9)

^~

«A,

n(A') v(A')

dA

~ i

~,~ ~~,~

~

~~,~

~~, ~~~~

s< TjA<-A)

~

T(s-A)

s,"

^~

which _is

equal

to

(«)

_/T~~

~ where

(«),

the _mean response is

averaged

_over all the bodies :

j«>

=

«(A _{n(A ) v(A}

_dA

₍₁₂₎

In order _to compute ^{the value} of T~~_ ~j, ^we

uie

_{the same}

mein

_field approximation in the balance for bodies at _a gtven ^state Thls

corresponds

to

substituting

_in the

equation (3)

i~~~'~ ^{~'~~'~~~~ by} ln(A') ^V(A')dA',

s' T(A>-A) T(s-A)

s<

200 JOURNAL DE

PHYSIQUE

II lV° 2

Since-the state

density

_is normalized

(I),

relation

(3)

at

equihbnum gives,

in the _mean field

approximation

_:

,

~

nlA VIA

~

l

~i~~

~(A -s)

T(s-A)

Thls gtves a definition of T~~__~~ which _is coherent with

body

number balance

and

_response balance. We deduce from

(13), (11)

and

(9)

the _mean field

equation

b£rA _£rA

(£r)

_£r

A £rA

j ~)

= + = +

(14)

bt T~A T(A _~) f T(A _~)

We _see

immediately

the

specific

effect of the

living _systems

_{if «~} relaxes faster

(without

state

transitions)

than

(

_«

)

_in the

living system,

^then

(

_{« ~ «}

~ and the relaxation at state A will be slowed down

by

state jumps. A contrario, if the isolated state A tends to relax slower than all the system, relaxation will be accelerated

by

state transitions.

5. Discussion of the _mean field

approximation.

The _mean field

approximation

consists in

reality,

in

postulating

that

T~~ ~,~ does not

depend

on A

',

but

only

_on A It _is clear that _we

negleut

correlation

during

the transitions. For instance, for _a

living polymer

which grows

longer

and shorter

by

increments, there _are transitions

just

between the

polymer

of

neighbounng lengths.

In such _{a case,} the _mean field

approximation

is

completely

irrelevant.

Conversely,

the approximation is

quite gold

_for

_{living polymers}

which break and recombine at random

It _is easy ^to compute from

[8] (see

also Annex

I)

the transition rates from _a chain of

length ii

_{to a} chain of

length i~

2 ci if

ii

~

i~

T~i~_i~~ c~n(f~ ii)

if

f~

_z~

ii

where ci and c~ are

respectively

the break and the association rates Thus the _mean field

approximation

_is

good

for small

f~,

but _not for recombination

(see Fig. 3).

^We now

give

_an

analytical

_expression for the _{error on} the response due _{to mean} field

approximation.

Mean field self consistency error

We _can find _an

analytical

_expression from

(9)

for the

exalt-

mean response.

Integrating (9)

on A gtves

~(~

"

~~

~~~ ^~'~~

dA ₊

~~'~~~'( ^~'~~'~

^~~'~~

⁽¹⁵⁾

S A s s' (A'-A) f

Integratlng

_now the _mean field

equation (14)

_{on A, we} have

aj«)

=

I«A ^{n(A V(A )}

^«~

^{n(A V(A )}

^dA

dA ₊

(16)

bt

s

fA

s

~(A s)

So the

systematic

error E due _{to mean} field

approximation

_on the _time denvative of the macroscopic response is denved from

(13)

and

(14)

_:

E=

~~~~~~'~~~~~~~~~~'~~~~(«Aj- «A~)dAidA~dA~

sj s~ s~ ^~(A_i ~)

lV° 2 LINEAR RESPONSE OF SELF ASSEMBLING SYSTEMS 201

ii _I

Fig

3. The _mean field approximation consists in

substitutlnj

a constant to I/T~t~ t~j The continuous line represents the actual value of the _transition

probability

for living

polymers

and the broken line, the

value taken _in the _mean field approximation

which vanishes if

T~~, ~ does not

depends

on A'. We will show later how to

compute

« in the

mean field

approximation. Equation (7)

allow to

compute

^the error once the _mean field response is calculated. The formalism descnbed here _can then be

developed

ⁱⁿ order to

perform

_a

perturbation

calculation about the _mean field

6. Resolution of the _mean field

equation.

It is _now

simple

to compute self

consistently («)

from

(14),

because the integration ^of

(14)

over S gtves the time denvative of

(« )

in function of

(«)

But _we first

perform

the Fourier transform of

(14) multiplied by

the Heaviside function

Y(t)

_:

tW~A(W )

_"A

(t

"

°)

"

~

~~~

+

~~(~) ~~

)

^~~~~

A

where _WA

(w )

is the Founer transform of

Y(t). «A(t).

«~

(t

=

0)

_is the1nltial value of the _response In

general,

_one takes _«~

(0)

constant, but for

some

special systems,

^it can be _necessary to _use other initial conditions. Hence the Fourier transform of the response ^at a state A _{is :}

WA

(w )

=

~~~~~

+

~~~ ~°'~

(18)

lW +

~/~A

_~(A

-s)(lW

+

I/TA) "(~ )

and integrating over all the A,

weigthlng by n(A) V(A),

_we obtain the self consistent

expreision

for

(&)

_:

n(A V(A ) trA(o)

~~

,~ n(A ) V(A )

dA

~"~ ~°'~

~~~~

~ iw +

I/Y~

^{~ ~ ~°}

s T(A

-s)(iw

₊

I/fA)

This gtves the modulus G* in the _case where _« is the stress response, assuming

«~

(0) independent

of

IWT~A T(A

-s)n(A ) V(A )dA G*(iw)

=

Go

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

(20)

(iwT~~

+ I T~~ _~~

n(A ) V(A )

dA

lW~aA ~(A -s) ⁺ ~aA + ~(A-s)

202 JOURNAL DE PHYSIQUE II N 2

where

Go

is a constant

(Go

is the

plateau modulus),

and the _zero shear rate viscosity ^{has the}

simple

expression

n

(A V(A ) _f~

dA

n ₌

Go (21)

~

~)jl~ n(A ) V(A )

dA

(See fop

^instance in

[9],

the

K~amers Kjoenig

relations _in linear

vlscoelasticity, for

the

precise rilations

_{between response and}

_modulus)

,

Limiting

cases.

Long I)etime.

In the _case

qf a,system

^with very slow trans1tlons, in the

ljmit

T~~ _~~ ~ co,

equ(tion (19) becoiles

,

n(A ) V(A )

_trA

(0)

~ ~

(22)

l~)~~°'~"

iw+I/T~A

This _is of _course the response of _a

polydisperse _system,

and the Fourier _inverse transform

gtves the classical result

,.

" l _J

~ " ~

(~ )

"

(~ )

_"A

(°)

_eXp

_{(- t/T~A} )

dA

.~ ,

Cqnstant

relaxation time. In the _case where the relaxation time does not

depend

_on the state A, the modulus must not

depend

6n the lifetime Indeed _in

(20)

_{T~~ can} be factorized and

we obtain

, iwT~~

Go

^~

i G*

(iw )

= .,-.

,

iwT~~ + I

which _is of _course the modulus of _a

single

time relaxation

syiteni

,(nfimte

time relaxation In the _case of infinite _time

relajations

of all the states, we have

(&)

~p ₌ 0 infinite and _in fact the Founer transform _is

irrelevant'because _{integrals diverge}

This _is evident because the

system

^does not relax

.~ <,

7.

Viscosity

of

living polymers.

,~

lfe

_use the model of~random

breaking

and ends recombination~ _{as in}

[8]

The lifetime of _a chain of

length ^f

_is calculated _{in annex} I The rule of stress co@ervation _is

consistint

_with

(7), V(f) being simply ^f.

We suppose that the relaxation time of the chain scales with

length

_:

~'

(fjf)d

^'~

T~ " T

f ~

where the

exponent

_«

equals

3 for _a

reptation

model

[10,

7] in the _case of melt _or semi-dilute

living polymers,

2 for the Rouse

model,

and

3/2

or 3

v/2

^for the Zimm model

[6, 11,

_{7~ in} ^the

case of dilute solution

Imphcitely,

while

taking

these

models,

_{we assume} that end contribution _is

negligible

We will discuss _{m a}

forthcoming publication

the consequences of the

specific

end

dynamics

lV° 2 LINEAR RESPONSE OF SELF ASSEMBLING SYSTEMS 203

Thus the

viscosity

is deduced from

(21),

and _we get, ^with x =

iii, ₍

=

T~/T~

and for the lifetime

T =

T~/(x

₊

2) (see

_in Annex

I,.Eq.'(33))

_:

lm

~«+l~-x~~

q

~GOT~

^°

^~~~~'~~~~~ ₍₂₃₎

j*

^xe~~dx

o~t+x~. (x+2)

in the limit

t

_{~ co, q} reduces to

lw ^x~+~e~~dx

llm _q

=

Go

_T~ ^°

~

=

Go

_T~

r(«

₊

2) (24)

'~"

o

xe~~dx

Thus

I

_{scales like}

T~ iq ^the case T~ ^« T~ We _{are in} the limit of

a,polydispersj _system

On the other hand _in the limit

(

~

0,

_one obtains m~~-x~~

~'

~

x+2

GOT«C

q((~0)"G0~« (25)

m

~j-x~~

^T ~I '~

:

~

(+x~(2+x)

where C _is the

integral

at

the'numerator

_{and I that of the} denominator The

intejral

_I

diverges

for

(~0 when,«

m2. We have then three

scaling

laws for the viscosity in~the limit T~ « T

~

a)

For

« <

2,

_q

((

~

0)

~ T~ and has then _a finite value for

(

=

0.

In the _{case «,=} 0 _we found

(as previously explained)

_q

=

Go

_T~ whatever _T~,

b)

For _.«

=

2,

_{using a}

change

of

variable,

_we have

, , ~

j*

^{y e~ ~~}

^dy j"

^{e ~~}

^dY

ln

(a )

o +

(2

₊

aY)

_Y~

i Y

with _a

=

(~'~.

Hence the

scaling

law for _K

= 2 _is

~J q

(<

_- 0

) T~/In (T~/T~ (26)

c)

For

K ~

2,

_we

hive.

, j

1=j(2-«)/«, j"'

^Ye

^~~dY

.. o +

(2

+

ay) y~

with _a

=

( ~'~.'Thi

_{second factor}

is an

integral

which has _a finite limit for _a

= 0. We have

then the

scaling

law for _K & 2

'l

(~

- 0

)

~ T

i~ _T)~

^~~~~

₍₂₇₎

Figure

4 illustrates these different _cases.

204 JOURNAL DE PHYSIQUE II lV° 2

Fig.

⁴ The different behaviours, _{in a}

log log

representation, of the viscosity q as a function of ( the

ratio of the _mean lifetime and the _mean relaxation time, ^for different values of the exponent _«, ^the mean

relaxation time remainJng ^constant For _w

=

0, _{q is} constant For _{w =} 2 there _{is a}

loganthmlc

divergences,

for _w

~ 2 the slope of log _q as a function'of

log

( is (w

2)/w

Remark the

scaling

law for viscosity, in the _{case Km} 2 _is obtained

by estimating

the denominator

divergence

for

(

= 0 In tits

limit,

the effect of _very small chains _{is very}

important.

If the

scaling

law for the relaxation time of the _very small chains _is different from that for the

longer chains,

the above

predictions

_{are no}

longer

valid

Also,

if the _mlnlmum

chain

length

_{is non zero,} there is _a cut off _{near x}

= 0 _in the

integral,

which does not

diverge

for

(

=

0,

and the

scaling

law _is invalid This effect _can be important in semidilute

systems,

because for chains

shorter

_{than the mesh}

size

f,

the relaxation will be

completely,

different from that for chains

longer

than

f.

The

approach

above allows _us to compute viscosity ^{in such}

cases

Our results differ from those of Cates

[5],

who find

physically

in the _case of

reptation («

= 3

),

_q

~ T

('~

Tj'~, whereas _we find _{q T}

)'~ T('~.

^This is due to the fact that _we do _not take into account the whole

reptatlon

_process. In the tube model

[10],

the ends and the middle

part

of the chain relax with different rates.

Moreover,

the relaxation function _is not _a

single exponential,

but _{a more}

complicated

function. This _is considered

by

Cates

[5],

but not in _our

paper.

However,

in the _case of

single exponential decay

_{process, our} model should well

applied,

_as for instance _in the _case of rod-like

living polymers [12].

8. Exact results for

(

= 0.

We

present

_now another way ^to derive the critical behaviour of the response near

(

= 0 and

« =

2. We _use the direct functional

integral

calculation

(8) presented

in 3. In the limit

(

=

0, during

_any finite space of

time,

a

body

will _runs

successively

_{over every state} of the system, ^and can therefore be considered _{as a «mean »}

object

_{: we can} substitute the

integration paths (t) by

a mean

path

_{running over} the entire

system,

and

occupying

each state

during

at time

proportional

to the

body density. Actually

_{we use} the

Ergodic theorem,

the whole system is

fully

described

by

each

body, dunng

a time infinitesimal

compared

to that

of the _response relaxation. We also make _use of the

property

^of commutation of the

propagators P~~i~.

^The

propagator

is a

product

of

propagators

of

(2)

and _is

l'A(t)

_" eXp

(- _j~

^~~

l. (28)

0 ^{~ WA}(t)

N 2 LINEAR RESPONSE OF SELF ASSEMBLING SYSTEMS 205

Hence,

in the limit

(

=

0, equation (8)

becomes _:

j~r(t)j

= ~r

(0),

_exp

(- ^@

^dA

⁽²⁹⁾

s

~«A

where

IT (A )

dA

= t and _T

(A )

_is

proportional

to the

body de~lsity n(A ) V(A )

We _use the

s

normalization convention of

equation (I)

and obtain _:

j~r(t)>

=

~r(0)

_exp

(-

^t

~(~ ) "(~ )

dA

(30)

s ~«A

The response for

(

=

0

decays exponentially

if the

integral

converges In the _case of

living polymers,

the

integral diverges

for _{« m} 2

(see

Sect.

7)

We find also _a cntical behavlour _near

« = 2. This shows

clearly

that the results of the _mean field

theory

are consistent with the exact solution _in the limit

(

= 0.

Moreover,

_in this

limit,

the

viscosity

_can be deduced from

(30)

and is _:

~

~~

_~

~~

_~

~°

n(A V(A

~~

~~~~

s

A

For

comparison,

the

viscosity

in the limit

(

= co derived from equation

(22)

_is

q

(<

= co

)

=

Go n(A ) V(A

_T~~ dA

This leads to the

following physical

remark for

(

= co the response is a

superposition

of all individual _{responses m}

parallel

On the other

hand,

for

( =0,

the response is a

superposition ^{in series of the} individual _responses.

Let _us remark that the _exact

viscosity

in the limit

(

= 0

(31)

is

exactly

the _same than the

expression denved in the _mean field

approximation

^from

equation (21)

in the limit

T~A -s) ^" 0 This confirms that these _mean field results for

hung polymers

q((=0)=0

for «m2

and

q

((

= 0

)

_{# 0} and finite for

K < 2

are not artefacts.

The cntical behaviour of the _{response near}

(

= 0 and _K

=

2 is confirmed

by

this exact calculation. We have also shown that whatever the

complexity

of the system, the response

decays exponentially

_in the limit

(

=

0. This result _is in

agreement

^with

expenmental

data gtven for _{instance in}

[13] (for

some

concentrations)

_; for _more

details,

_see

[14].

Let _us remark that _we have used the

properties

of the P

propagators

which requires that

one

aggregate

^has a

single exponentiel decay

Generahsation to more

complex

_systems must be

carefully developped.

9. Conclusion.

In this _{paper, we} have studied the linear response of

living systems,

such _as

living polymers.

A

206 JOURNAL DE

PHYSIQUE

II lV° 2

mean field

approximation

can be

applied only

if the aggregates break and recombine

randomly,

and if the response is extensive

during

transitions.

According

to the _mean field

theory,

_we calculate _an

analytical

_expression ^{for the}

complex

modulus. We also

suggest

a

perturbative

calculation _near the _mean field

solution,

in order to take into account transition correlations. We

apply

_our model to

living polymers

and observe that the viscosity, in the _case

where the _mean

hfetimq

_{T~ is} small

compared

to the _mean stress relaxation time

T~, scales _as

T)~

^~)'~ _T

)'~,

where _{K is} the

scaling

law

exponent

^for _T~ ^with respect to the chain

length,

for _«

~ 2

Conversely,

for _{K <}

2,

the

viscosity

has _a finite hmite for T~.- 0. These results _are confirmed _in the limit _T~

= 0

by

_an exact calculations. The response has _an

exponential decay,

and the _mean field viscosity in this limit _is the exact one. All these results evidence the _existence of

complex

interferences _in

living

_processes, between _size,

lifetime,

and relaxation time.

In this

work,

_we have limJted the model to _a

single exponintial dicay

_{for isolated}

ajgregatei. ~our approich

_{will be extended}

in a

forthcoming pubhiation,

_in

order'to

_include

end

effects'in living jolymers, _{by taking}

_into, acc6unt

thi poiitions

_{of monomers} within the chains

Acknowledgments.

I _am

grateful

to R.

Hocquart

and Y

Thinet,

for many discussions and _a careful

reading

of the

manuscript, to S J. Candau for

stimulating encouragements

^{We benefit of}

helpful

discussions from M. E.

pates

within the framework of, E-E C Grant number SO*0288-C

(E.D B.)

Annex I

Recombinafion

dynandcs

of

living po1jnlers.

^{' '~}

~~

We take the model of Cates and Turner end associations This leads to the

following

_equation

(see _[5]) (where

_{ci is} the break

probability

and c~ the combination

probability)

_:

,

<

~~

~~~ ~

= cl

I

p

(I

_{2 c~}

p

(I ) j~

p

(x)

dx ₊ 2

cl j~

p

(I

₊

_x)

_{dx +}

at

~ ~

f

+ C2

o

P

(X)

_p

(~ X)

dX

(32)

The first term _is due to destruction

by

_scission, the second to destruction

by combination,

the third to creation

by

scission and the fourth to creation

by

combination.

At, equlhbnum,

_we have the

density

_:

P

(I)

= P exp

(- iii)li~

^'

,

with

I

A

~~)~

^where is the

density

in

polymer

ci

~~~

~ -,:

The lifetime of _a chain of

length ^I

_is calculated

by

suppressing the two last

tennis

of equation

(32).

^The term in cj ^is the

dispantion

rate due _{to scission,} the _{term in} c~ ^to recombination _:

im

^~

^(~

^{+ 2}

^I)

jj

~ ~~

jp

^~~~

-

^{2 c~ P}

(I)

~

P ~~~ ~~

".~

N 2 LINEAR RESPONSE OF SELF ASSEMBLING SYSTEMS 207

So the lifetime of _a chain is T~~ _~~ ⁼

l/(ci(I

₊ 2

I)).

It _can also be calculated from

(4)

and

(6).

We take T~ =

I/ci ^I

and have then

~~~ _~~ ⁼

~~/(x

+

2)

with _x

=

iii (33)

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[I]

PETSCHEK R G, PFEUTY P, WHEE~ER J. C,

Phys

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(1986)

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[3] ^JINDA~ V, KA~US J., PI~SI H, HOFFMANN H, LINDNER P, J

Phys

Chem 94

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references therein

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PORTE G, APPE~~ J,

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[4] ^PEzRON E, LEIB~ER L., PINCUS P A,

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[5] CATES M E., Macromol 20 (1987) 2289.

[6] ROUSE P. E, J Chem

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[7] DOI M., EDWARDS F, The Theory of polymer dynamics (Oxford University Press) 1986

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