Polymer melts: a theoretical justification of double reptation

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Polymer melts: a theoretical justification of double reptation

J. Des Cloizeaux

To cite this version:

J. Des Cloizeaux. Polymer melts: a theoretical justification of double reptation. Journal de Physique

I, EDP Sciences, 1993, 3 (1), pp.61-68. �10.1051/jp1:1993112�. �jpa-00246712�

Classification

Physics

Abstracts

05.40 46.608 61.25H 81.60J

Polymer melts:

_a

theoretical justification of double reptation

J. des Cloizeaux

Service de

Physique Th40rique(*),

Ceiitre d'Etudes de

Saclay,

F-91191 Gif-sun-Yvette Cedex, France.

(Received

29 AToveniber 1991, revised 10

September

1992,

accepted

22

September 1992)

Abstract The concept of double

reptation

iii

polymer

melts is

supported by experiment

and

seems to be

logical.

llowever the

dissymetry

of the

disentanglement

_process cannot be

denied,

and therefore double

reptation

has _to be _proven. This is done here in the framework of two models in which either the

polymer

is fixed at stress points _or it _may slip

through

these points.

The results _can be summarized in the _same way: the stress relaxation function is

proportional

to the

density

of

polymer

segments limited

by

stress

points.

These

points

_can be relaxed either

by reptation

_or

by

"tube release".

1 Introduction.

A

long

time _ago,

trying

to describe the motion of

entangled polymers

in

melts,

de Gennes

[I]

recognized that,

ⁱⁿ the _presence of fixed

obstacles,

the

polymers

_move

by reptation,

and he extended this kind of

propagation

to

long polymers

in _a melt.

However,

in this _{case, a}

polyiuer

cannot be considered _as

moving

in _a fixed medium and the motion of the other

polyiuers

has to be tal;en into account.

Consequently,

the

polymer

has been assumed to be confined in _a tube which is not fixed but is

continuously

renewed.

Thus,

in

a strained

material, simple reptation

is corrected

by calculating,

"tube constraint release"

[2].

Another

appproach

has been

proposed

_on

experimental grounds (by

_{us [3]} and

previously by

other

people [4]).

^{~iTe noted that}

siniple reptation

and tube release _can be

replaced by

double

reptation.

In other terins, we assumed that stress can be associated with stress

points involving

two

polyiuers

A and B and that these stress

points

_can be released

by reptation

of

either A _or B.

Thus,

froiu the start, the situation is considered _as

symmetrical

and the stress relaxation function of _a melt

consisting

of

polymers

of different _masses is

given immediately

as a

quadratic

function of the volume ii-act ions _i>A

(or i>B)

^{of the} constituents

~(l)/~0

"

~ i'Ai'BPA(I)PB(I)

_"

(~j _i'APA(1))~

A B A

(* Laboratoire de la Direction de la science de la Matilre du Commissariat I

l'Energie Atomique.

IOURNAL DE PHYS,QUE i T 3. N. ,, JANUARY >993 3

62 JOURNAL DE

PHYSIQUE

I N°1

where

Go

is _{a constant} and

pA(t)

is the _mean fraction of unrenewed tube of

type

A at

time t.

The

dispute

could be viewed _as

semantic, by considering

double

reptation

as a

simple (and imperfect)

_means of

accounting

for tube release.

But,

in

fact,

_we claim that double

reptation

is the basic correct

approxiniation

and _a theoretical

proof

will be

given. Thus,

the

advantages

of the method _are threefold.

I)

It is _more correct

conceptually. 2)

It is _more exact

experimentally.

3)

It is

simpler

since it suppresses the difficulties associated with the calculations of "constraint release".

Certainly,

in _a strained

material,

the _occurrence of stress is _a

twc-body

_process but this does _not

imply

the

validity

of double

reptation.

As _was

pointed

out

by

Edwards

is],

at a stress

point,

the forces exerted between _a

polymer

A and

a

polymer

B _are

equal

and

opposite,

but

again

this fact does not

give

_any

compelling

indication _on the _nature of the stress

dependence

with

respect

to the

polymers.

In

fact,

the result of _a calculation of the stress relaxation function is model

dependent.

So

we must first discuss the model w,hich _we

regard

_as

giving

_a reasonable

description

of _a

polymer melt,

but which differs to _some ext.ent from the conventional _one

[2].

Most

physicists

_rea~son in terms of

entanglements. Entanglements

_are static

(even

when

they

_are involved in

dynamical processes). They

have _never been

properly defined,

there is _no

unique

_way to

recognize them, and,

at

present,

^{there is} no way ^to observe them. It is believed that their number is

approxiiuately

constant and

that,

_{as soon as one}

entanglement disappears,

another _appears in its

place. Moreover,

this number is considered _as

constant,

not

only

when the

polymer

melt is at

rest,

but ,vhen it is strained and

during

the whole relaxation _process

(in

linear

deformation).

Our model does not _use

entanglements

but stress

points.

The notion of stress

point

is _a

dynamical

_one. In _a relaxed

melt,

there _{are no} stress

points (but

there _are

entanglements

whatever

they

_may

be).

In _a

polymer

iuelt with _{a mass} which is lower than the

entanglement

mass, there _{are no} st.ress

points either,

and the rela~~ation is of the Rouse

type.

In _an

entangled

polymer melt,

_an initial strain

produces

stress

points

which later _{on are}

destroyed by reptation:

therefore,

the nuiuber of stress

points

diiuiiiishes with time like the _stress itself. We believe that these stress

points

are due to the existence of

entanglements,

but these

concepts

^do not coincide. More

precisely

these stress

points

could be defined _as "effective

entanglements".

Therefore,

in the

melt, they

_are

present

but in _a virtual state, and

they

become "activated"

by applying

_a stress to the iuelt. This

explains why

the initial nuniber of stress

points

must be considered _as constant for a

given polyiuer

_even in the limit of infinitesimal stress.

Moreover, during

the relaxation _process and for siuall

motions,

the destruction of _a stress

point

in the middle of _a chain

"by

tubc release" does not

produce

the appearance of another stress

point

at the _same

point

because this destruction iuust be considered

simultaneously

_{as a} kind of disactivation _process.

Actually,

_we rule out the appearance of _{a new} stress

point

_as

being unlikely

and for the

folio,ving

_reasons:

I)

the

efficiency

of

ent~nglements

is overestiiuated because in tw,c-diiuensional

drawings,

_a

chain

always

_appears

strictly

confined

by perpendicular chains;

2)

^{the chains} are sometimes

parallel

and _may have

a

tendancy

to be _so;

3)

^the deformation of _a melt _{on a} small scale is

uniform;

_so it is

quite possible

that _many

entanglements

do not

produce

stress

points (they

_are not

active);

4) finally,

^{it should be realized that the stress tensor is calculated in the} zero stress

limit,

and therefore it would be hard to belie,~e that the removal of _one stress

point

leads to the appearance of another stress

point since, theoretically,

_~ve deal here with infinitesimal

displacements.

This _means that. ,,>hen _{a stress}

point

is

destroyed,

the

portion

of chain _to which it

belongs

relaxes

freely (I.e.

_{as a} chain below the

entanglement point),

with the

only

constraints

produced

by

the

remaining undestroyed

stress

points.

In

brief,

_stress

points

_are considered _as accidents which

prevent

viscous relaxation fi.om

taking place.

In _a way, in

spite

of the fact that

they

have

specific properties, they

are somewhat similar to the

"slip rings"

which _are

currently

used to define the constraints

resulting

from

entanglements.

In the

following, starting

front these

assumptions,

_{we prove} that the stress tensor is propor- tional to the

density

of

segiuents comprised

between _stress

points;

this result

gives

_a

justification

of double

reptation

since the result

depends

_on the number of stress

points

but does not

depend

on the _way

they

_are

destroyed.

Note that _stress is not located at stress

points

but around

it,

in the

subchains,

between two

stress

points.

At this

point,

the reader _may wonder

why

the former

theory

_{[2] was} not

good enough

and

why

these _new

assumptions

_were needed. In

fact,

_our motivation

originates

from

practical

purposes. Two facts

require

_a

conceptual change

I)

as indicated

above,

double

reptation

is _a milch better

approximation

_[3]

2)

from _a theoretical

point

of

view,

it _seems natural _{to argue} in favour of _a

two-body

_process,

and this leads

immediately

to double

reptation.

In fact the

iiupossibility

of

crossing corresponds

to _a correlation bet,veen _two

polymers

in

iuotion,

and the calculation of

G(t)

does not

give

any

compelling

_reason for

specifying

one

polymer

with respect to the other _one.

Thus,

for

calculating G(t), assuiuing that,

in first

approximation, only

_one

polymer

is

moving

_among

fixed obstacles

representing

the other

polymers

does not _seem

right;

in first

approximation,

one has to take into account the motion of the

polymers,vith respect

to _one another. This is

what is done here. The situation would be different if,ve _were interested in _an autc-correlation function.

The introduction of stress

points proceeds

from the _same motivation. These stress

points

_are

more

directly

observed than the

coninionly

used

entanglements;

in

fact,

^for

high

_masses

(and

slow

times),

their nuiuiJer is

proportional

to

G(t);

therefore the introduction of these

objects

is natural.

In section

2,

_w,e review the main diaracteristics of _a melt made of Brownian chains. In section

3,

_we define model I and model 2. In section

4,

_we calculate the contribution of _one

segment

^{of chain}

(bet,,,een

stress

points)

to the constraint. In section

5,

the

special

features of

model 2 _are

analysed

in

detail;

thus this section _can be

skipped

at first

reading. Finally

the

stress tensor and the stress relaxation function _are calculated in section 6

and,

from the

result,

we deduce the

validity

of "double

reptation".

2. General characteristics.

The melts consist of

polymers

which _are

represented by

continuous Brownian chains and the

probability density

that _a

segment

of

polymer

chain of "Brownian area" _s,

starting

from _ro, arrives at _ri, is:

Piri

_co

isi

₌

~,j~~/~e;P I-in _ro)2/2sl

with

((ri _ro)~)

" 3s and _{s oc} "number of links" between _ro and _ri This is

a conditional

probability

distribution for _a

given

s. ii'e have

J

d~i~

P(r(s)

₌ I.

The

corresponding entropy

is

S = -In

P(ri

_ro

_Is)

₌ ^~~~

2s'

^~°~~ +

~ln(2xs)

2

64 JOURNAL DE

PHYSIQUE

I N°1

The forces _are of

entropic

nature; ^let

Fi

and

Fp

be the _z

components

of the forces

applied

at ri and _ro

pj~+

^~~

^{~~l ~0~}

z fi

~s~~ _{j~~ [~~} Ii)

p

j~-

~

~~0

₈

These

entropic

forces _are

compensated by

random forces.

The

points

_ri and _{ro are} stress

points;

in the

following,

it will be assumed that _a stress

point joining

_a

polyiuer

A and _a

polymer

B remains

unchanged,

_as

long

_as it is not

destroyed by

the _passage of _one end

point

of either A _or B

(this

is Rubinstein's

assumption [6]).

The initial

probability

distribution of the stress

points

is at

random,

on the

chain,

and _c will be the average number of _stress

points

_per miit "Brownian area" of the chain. The

probability

distribution of _s

obeys

Poisson la~v; therefore

P(s)

_{= c} ^e~~~

(2)

with

/

~ _ds

_P(s)

= I

o

3. Models I and 2.

At tiiue _{zero, a} shear St-rain is

applied

for which the

displaceiuents

_are AZ ₌ ~cy,

Ay

₌

0,

Az ₌ 0. Then at tiiue t, the shear stress is

?xy(t)

_{= K}

G(t)

where

G(t)

is the stress relaxation function which _{we are}

going

to calculate.

Let _us consider _a chain with _{ii + I} stress

points

of coordinates _{ro, ...,rn.} We _assume that

they

_move with the strained material. In the

unperturbed

state, these

points

_are

separated by

Brownian _{areas si>}

,

s,i

(proportional

to the numbers of

monoiuers). Now,

we consider two models

I)

Afodel I. The _stress

points

_are fixed _on the

polyiuers

and the Brownian _{areas si,}

...,

sn are

independent

of stress and strain. Thus the

probability P(s)

remains

unchanged

but

P(r(s)

becoiues

P,(r(s).

2)

Model 2. At tinie _zero, ,vhen t-he strain is

applied

the

polymer slips through

the stress

points;

the stress

points

_are

separated by

_new Bro,vnian _areas which _are still called _si,

..., sn

but

P(s)

is

replaced by P~(s).

On the other

hand, P«(r(s)

is

replaced by

the _same function

as in model I. It is assumed that this effect takes

place only

when the strain is

applied,

and that the _new values of _si,

,

s,i reiuain fixed afterwards. In this

approximation, equations (I)

remain valid.

4. Model I. The fun<:tion

P<(r(s)

and the value of the _stress tensor.

For model

I,

_we define the

probability P~(r(s) by set.ting

P«(its)

₌

P(z KYIS)P(vls)P(zls) 13)

We have

d~l' P~(I'(S)

_#

Now,

_{on a}

polymer,

_we consider _ii + I stress

points

_{ro, ri,}

, rn

separated by

_{si, s2,}

, sn. We

set

n

P<(ro,ri,

..,

rn(si, ..,sn)

=

fl Px(rj

_rj-i

(sj)

j=1 and therefore

/ _d3ri /

<'~r,~

P~(r~,

_vi,

,

r,~

is

_i,

, s~ = i

We want now to caculate a~y in a

plane

of ordinate _ya

belonging

to the interval

(y; y;-i)

I-e- _we want to calculate t-he force exerted

lJy

the _upper half

plane (y

>

ya)

_on the lower _one

(y

<

ya).

Let

azy(P, j)

be the contribution of the

j'~ _segment (subchain)

of

polymer

P. The forces exerted

by

the _upper half

plane

will be either

Ffj

_or

_Fzj.

~ zj zj-i

i yj ^> vu > yj-i

a~y(P,J)

_«

_F~,j

₌

p

_~

J

zj zj-i

and it yj-i ^> Vu > Vi

'£Y(P>j)

"

F~,j

"

p

_s.

The randoiu forces do not

play

_any r61e. Thus the contribution of _a

segment j

of _a

polymer

P

is,

_on the average,

equal

to the

product

of

(xj

_z~-i

)£(yi yj-i) by

the

weight _(yj yj-i(

and therefore

,

~p

_.~

(xi

_xi-i

_)(Yj Yj-i)

~y ,J "

p

_{~_}

J

Finally,

_we have

+m +m +m

(a~ylP,j))

₌

/« dzj /« dyj /« dzj p(zj

_z~-i

«(vi vi-i) lsj) p(yj yj-iisj~p(~j zj-iis~~(IJ ~J-j)[jJ

vi-i)

Let _us

integrate

_{on zj,}

replace (xi

_zj-i and

(yj

_yj-i

respectively by

_z and _y and make the transforniat.ion _{x z} + _iv, y _- y. ~iTe

get

(Y,y(P, j ))

_"

/~~ dY /~~

d~

P(~'~J _)PIY'~J

^~~

i _/~~

_~ -«

=

i )

dJ i~,hit?

ex'J

iY~/~~Ji

and

find"Y

i,~~ip, ji)

= ~

p-1

_14~

We obtain _a result

independent

of sj.

66 JOURNAL DE

PHYSIQUE

I N°1

5. Model 2: value of the sti~ess tensoi~ and function

P~(s).

In the _case of model

2,

the derivation of

P~(r(s)

remains

valid,

if the Brownian _{area s} which appears in this formula is considered _{as stress}

dependent.

Therefore,

for model

2,

the

equation

la~y(P,j))

_{= ~C}

fl~~

is still valid. This result _appears

surprising

at first

sight,

because model 2 is "softer" than model

I,

since the _stress

points

are not fixed but _may

slip,

when the strain is

applied. However,

it

must be realized that the

probability

law of the actual value _s~ of _s does not coincide with the initial

probability

of _s.

According

to

equation (3),

_we have for model

~~~~~~~

(2xs)3'2~~~

2s

For model

2,

we must consider that

~

_~~~~~

2xs~)3'2

~~~

~~

'~~~i

^{~~ ~ ~~}

where _s~ is different fi.om _s.

Now,

_we must calculate the

probability P,(s~)

of _s<, front the

probability

of _s, I.e.

P(s)

=

c e~~~ Thus

we must relate _s and _s<.

It will be assui»ed that

Ss

(Z Ky)~

+

y~

+ Z~

7

_{Z~ + g~}

+ Z~

In

fact, by

this

foriuula,

_{we express} that the _{new square} distance

ix

_~cy)~ +

y~

+

z~

is

proportional

to the extension _s,

just

_as the old _square distance

z~

₊

y~

+

z~

was

proportional

to s, and _we shall calculate

P~(s,) by averaging

over z, y and _z. In this _way,

we take into

account the fact

that,

_on the _{average, no} stress appears ^on the chain. We _may set

I = sin fl _cos p

y = cos fl

z = sin fl sin p

and _We fi"d

S~

= 1- 2~ sin « _cos o _cos v +

«2cos~fl _l~~

~

From this

equality,

_we deduce the value of

P~(s)

~~~~ lx _%'~~

""

~~

~

~'l

« sin 2fl

c~

i> + ^~c2 cos2 fl

~~~

l _« sin 2fl

~s

+ ~c2 cos2 fl~

Expanding

with

respect

to K, we

get

p~jsj

₌

dfl sin 0

di> (I

+ « sin 20 _cos i> +

«~(sin~

^2fl

^cos~

_i>

^cos~ _fl)]

4x

%' ^~~

xP(s (I

+ _« sin 20 _cos i> +

«~(sin~

2fl

cos~

i>

cos~ _fl)]

~2

K

P~(S)

#

P(S)

+ dfl _Sin flX

4T

2w 2

x

di>[sin~

^2fl

cos~i> (P(s)

+

2sP'(s)

₊ ^~

P"(s)) -cos~fl(P(s)

₊

sP'(s)))

o 2

From

which,

_we

get

P~(s)

=

P(s)

+ K~

1- ^P(s)

⁺

~

sP'(s)

₊ ^~

~P"(s)j

16 16 16

and _more

explicitly

P,(s)

= c

1

+

K~(-

^~

cs +

~s~)j

e~~'

(6)

16 16

From

(5)

_or

(6),

_~ve deduce

m

~2

(s)~

₌ ds

sP«(s)

=

c-~

_{l +}

3

Thus,

_we obtain the result which _we

expected;

the distribution of

probabilities

of _s is broader in the strained state

(model 2)

than at rest

(model

I

). ^Therefore,

^{the nuiuber of} stress

points

is smaller in model 2 than in model

I,

in

spite

of the fact that

formally

_we find the _same

answer

(Eq.(4))

in both _cases.

Nevertheless,

_we deal here with _a correction of order K~ it is

negligible

for small strains.

6. Shear stress and stress relaxation function.

Now let _{us come} back to the contribution of _a

segiuent (I.e.

a

subchain) j given by equation (2) (Y(P,j))

#

Nfl~~

To obtain

a~y(t),

p,e have to _suiu the contributions of the

polymer segments;

^then

n~(t) being

the number of

segiuents

^between stress

points

_per unit

voluine,

_at tiiue

t,

we have

?~y(t)

= K

fl~lll~(t)

and theref°~~

G(ij

=

fl-~n~(i)

_~~~

This is _our main result.

Now,

from this result and

by applying

Rubinstein's

assumption,

_we deduce the

validity

of

double

reptation,

^since the relaxation function

depends

_on the number of

polymer

segments

^between stress

points

and not on the _way

they

_are

destroyed.

More

explicitly,

consider two chains A and B ,vhich have _{a common} stress

point P;

the contributions of the chains _to G _are

respecti;ely proportional

to

(NA I)

and

(NB I),

where

NA

and

NB

are the number of _stress

points

_on each chain. No,v let _{us assuiue} that the stress

point

P is

disentangled by reptation

of A. Then

(NA I)

becomes

(NA 2),

this is the

simple reptation

effect,

and

siiuultaneoiisly (NB I)

becomes

(NB 2),

this is the double

reptation

effect. This

68 JOURNAL DE

PIIYSIQUE

I N°1

second _process could be called "tube

release",

but in

fact,

_we observe that the effect is identical

in both _cases.

Thus,

the

validity

of

equation (7)

_proves double

reptation.

Denoting by fif~(t)

the nuiuber of stress

points

_per unit volume and

remarking

that _a stress

point belongs

to two

polymers,

_we

get

G(t

_m

2fl-~fif~(t)

For model

I, fif~ (t

is

independent

of _« and therefore

fit,(t)

=

fit(t)

for model

2, fit,(t) depends

on stress but for low stress, we can write

fit,(t)

_ci

fit(t).

Thus in both _{cases, we} have

G(i)

_m

2p-iw(1)

References

ii]

de Gennes P-G-, J. C'llem.

Pliys.

55 (1971) 572.

[2] Doi M. and Edwards S-F-, The

theory

of

Polymer Dynamics (Clarendon Press, Oxford, 1986)

Ch.7.

[3]des

Cloizeaux

J., Eiirophys.

Lett. 5

(1988)

437; Erratum

Europllys.

Lett. 6

(1988)

_475;

Macromolecules 23

(1990)

4678; Alakromol. Chein. Macroniol.

Symp.

45

(1991)

153.

[4] ^Marucci G., J.

Polym.

Sci.

Polyni.

Pb_>'s. 23

(1985)

159.

Tsenoglou C.,

J. Pot_ym. Sci. B, Pol~i,m.

Phys.

2G

(1988

2329.

Viovy

J-L-, J. Pb_ys.

(France)

46

(1985)

847.

[5] ^Edwards S-F-, Private communication. The author is indebted _to Professor Edwards for many

stimulating

discussions, in

particular

for

formulating objections

with great

clarity;

this article is _an attempt to answer them.

[6] Rubiiisteiii hi. and Colby R., J. Chew. Pbys. 89

(1988)

5291. See the

beginning

of section V.