Dynamic form function of a long polymer constrained by entanglements in a polymer melt

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Submitted on 1 Jan 1993

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entanglements in a polymer melt

J. Des Cloizeaux

To cite this version:

J. Des Cloizeaux. Dynamic form function of a long polymer constrained by entanglements in a polymer

melt. Journal de Physique I, EDP Sciences, 1993, 3 (7), pp.1523-1539. �10.1051/jp1:1993198�. �jpa-

00246814�

Classification Physics Abstracts

05.40 61.128 61.40

Dynamic form function of

_a

long polymer constrained by entanglements in

_a

polymer melt

J, des Cloizeaux

Service de

Physique Thdorique (*), CE-Saday,

91191 Gif-sur-Yvette

Cedex,

France

(Received

2

December1992,

revised 3

February1993, accepted

26 March

1993)

Abstract. The

dynamic

form function of _a

long polymer

in

a

polymer

melt is calculated

by assuming that,

in the intermediate time

region,

the

entanglement points

remain fixed. The

calculation agrees with the reptation concept of de Gennes. The results recall those of Ronca but _{are more} exact, as we show

by comparing

them with the

experimental

spin echo neutron

scattering

data

by

D. Richter et al. The form

function,

is

expressed

in terms of _pure numbers

depending only

_on two parameters, namely the _{mean square} distance between

entanglements

and _a coefficient

characterizing

the continuous Rouse

equation

of the

polymer.

1 Introduction.

The

dynamic

^{form function} of _a

polymer,

the motion of which is

given by

_a Rouse

equation,

was calculated in 1967

by

de Gennes

iii.

In another

connection,

in

1971,

de Gennes described the motion of _a

polymer

_among fixed obstacles [2]

by introducing

the

concept

^of

reptation.

Since

then,

the idea _was

developped

to describe the

dynamics

of

polymer melts,

in

particular by

himself and

by

Doi and Edwards

[3].

^More

recently,

it _was shown

by

the author [4] that this idea could lead to very

precise

results in the framework of double

reptation. Nevertheless,

the effect of

entanglements

_on the time

dependent

structure factor has not received that much attention.

In

1983, starting

from _a different

point

of

view,

Ronca [5] introduced _a semi

phenomenolog-

ical

equation

obtained

by adding

_{a memory} term to the Rouse

equation.

In this _way, he _was able _to

predict

the behaviour of the

dynamic

form function

ti(q, t)

of _a

polymer

of intermediate times I-e- for times which _are somewhat shorter than the

reptation

time.

Thus,

in this

region,

the ratio

ti(q,t)/ti(q, 0) acquires

_a ^{finite value} as t goes to

infinity,

instead of

going

to _zero.

However,

the

experimental

result for the

time~dependent

form function obtained

by

the

spin

echo neutron

scattering

_covers

precisely

this intermediate

region. Very recently

Richter et al.

(* Laboratoire de la Direction des Sciences de la Matikre du Commissariat k

l'Energie Atomique.

[6] ^measured

ti(q, t)

for various

polymers

and

they

found that the data _were

incompatible

with Ronca's

predictions.

Thus,

in order to obtain _a

good fit,

we must

replace

Ronca's

theory by

a more

precise

and direct _one,

and,

in the

present article,

_we obtain this result

by introducing

the

concept

of

entanglement (or

stress

points)

in the

original

calculation of the form function

by

de Gennes

iii.

For this _{purpose, we} have

only

to _assume

that,

at _a

given time,

the

polymer

in _a melt is

a Brownian chain and

that,

in this intermediate time

regime,

the

entanglements

remain fixed

on the chain. The

principle

of the calculation is therefore _very

simple

but its realization is somewhat involved. It will be

presented

in the next sections.

Thus,

in the

following,

_we shall determine

ti(q,t)/ti(q,0) (q

₌ momentum

transfer,

t ₌

time)

for _an infinite chain in terms of _a dimensionless parameter

proportional

to

q~. (Note that,

in this

approximation 5(q, 0)

is

proportional

to

q~~).

The formalism is made

explicit

in section 2. The ratio

ti(q, t) /ti(q, 0)

is calculated in section 3. The limit

ti(q, cc)/5(q, 0)

is studied in section 4. A transformation of the

general

result and

a value of

5(q,t)/5(q, 0)

are obtained for _a

given

value S of the "Brownian area"

(or length)

of _a

polymer,

in section 5. The

dispersion

of the _stress

points

is accounted for in section 6 and

5(q, t)/5(q, 0)

is obtained for _a

given IS).

In section

7,

the formula of de Gennes is recovered in the limit

q~S

-

0;

the

limiting

behaviour is shown to be the _same, if

q~(7t)~/~

_-

0,

in the

general

_case ^and for de Gennes formula. In section

8,

_we comment _on the results obtained

by

Ronca and _we note that

they

cannot account for the most recent

experimental

results

[6].

In section

9,

_{we compare} the _curves

representing ti(q,t)/ti(q, ₀₎

with the _new

experimental

data of Richter et al.

[6].

^This

comparison supports

our

approach.

2. Notation and

parameters.

The

theory depends

on two

parameters

S and _7. The _{mean square} distance between entan-

glements

is

R~

= 35

(in

accordance with _a classical notation

[7]).

^The

parameter

₇ ^is the coefficient of the continuous Rouse

equation

which describes the motion of the chain between

entanglements.

For each

component z(s,t), y(s,t)

and

z(s,t)

of the

position r(s,t)

of

a

point

of the

chain,

_we have

if«iS, t)f«

_iS~,

_t~))

₌

276 _IS

_{S~) 6}

it

_t~)

Therefore, according

to this

equation

(lzls> 0) z10, 0)l~)

= S

12)

in

agreement

with the

preceding

definition of S.

Actually,

_we shall _express

ti(q,t)/ti(q, 0)

in terms of two dimensionless parameters which _are functions of 7 and S

Y ₌

q~17t)~/~

Z

j~~

=

q~S/2

3 Form function

5(q, t).

The form function is

given by

the _average

i ^{ST ST}

slqi t)

"

m /

_d~

/

_d~'

l~~P liq'

_l~(~<

t)

_~(~~<

_o)11)

T 0 0

where

ST

is

proportional

to the total number of links _on the chain. Since _{we are}

considering only

random Gaussian

variables,

_{we can} also write

~(~< l) ~

/~~

^~~

/~~

^{~~~ ~~P}

(i~(S<1)

^~

^(s~i °)j~) ^(~)

T

In

particular, according

to

equations (2)

^and

(4)

~~~'~~ /( ^)~

^~~

^~~

~~~ _~~~

~~ ~~

=

~

ds

(ST s)

_exp ^~~_s

S~

2

In this

work,

_we consider

only

the limit

q~ST

> 1 where

ti(q,0)

₌

£ ₍₅₎

Tq

Combining (4)

and

(5),

we obtain for

q~ST

_»

Ii(q, t)/Ii(q, 0)

₌

(

/~~

^dS

/~~ ^d/

exP

(

(lz(S> t)

_{z lS~,}

°)l~) ⁽⁶⁾

The

polymer

is

represented by

_a Gaussian chain

pinned

at fixed

points

which _are called _stress

points

_or

entanglements (see Fig. I).

So f_1

r~

~

',

~~~

S-i

^'

t~t' _j

p~

t-I"

~l

Fig. Fig

2

Fig,

I. Motion of _a

polymer

chain in _a melt. The stress

points (or entanglements)

remain fixed.

Fig.

2.

"Length"

of segments _{on a} chain

(Sp)

and _vectors

(rp) joining

the intervals between stress

points.

(Actually Sp

₌

(r() /3

has the dimension of the _square of _a

length).

JOURNAL DE PH~SIOUE i -T I N' 7 JULY 1991 V

Now,

consider the chain at time t

(see Fig.2).

It will be assumed that there _are

n stress

points

and that their

positions along

the chain _are

Si,

,

Sn.

The number _n is

large and,

for

simplicity,

we shall _assume that the

origin (So)

and the

extremity (Sn+2

_"

ST)

_are also fixed.

Thus,

_{we may} write

5(q<1)/slq> o)

_"

~~ f f ^~~~'

d~

~~~'

d~~ ~~P

(~ (l~l~<1)

_~

l~'i )I))

n-Q fl~0 & fl

Now,

_we have two

types

^of contributions: contribution 1 where _{a =}

fl

and contribution 2 where _o

# fl.

We

separate

^the two contributions

s(q,i)/s(q, o)

₌

ii

+

i~ (7)

and for

simplicity (see

discussion in Sect.

8),

_{we assume} that the

entanglement

"distance"

along

the

polymer

is _a constant

S~+i S~

₌ S

(8)

Contribution

gives

Ii

"

~~₄₅

/~

ds

/~

^ds' _exp

(-

^~~

_([z(s, t)

z

(s', 0)]~) ⁽⁹⁾

o o 2

where the coordinates

z(0,t)

and

z(S, t)

remain fixed in time

(see Fig.3).

t«L t-L

~~~ t=0

a~

b)

Fig.

3.

a) Image

of contribution I: the

points

_z

is,

to and _z

(s', ₀₎ represented by

_crosses

belong

to the _same segment which is shown at t

= 0 and at t

= tot the stress

points

_on each side of the segment remain fixed.

b) Image

of contribution 2: the

points

_z

is, to)

and _z

Is', 0)

represented

by

_{crosses are}

on two segments

separated by (at least)

_one stress point and the segments t ₌ 0 and t

= to

correspond

to

independent

Brownian chains.

To evaluate contribution

2,

_{we use} the fact that two branches of

polymer

which _are

separated

by

stress

points

_are

completely independent

from each other and that each branch remains

Gaussian at all times

(see Fig.3). Thus,

contribution 2

gives (without forgetting

_a factor

2)

12 =

~~

/~

^ds exp

(-

^~~

_([z(s, t) z(0, )]~)) /~ ds'

_exp

(-

^~~

_{([z(0, 0)}

_z

(-s', _0)]~)

25 o 2

o 2

where

again

^the coordinates

z(0,t)

and

z(S,t)

_are fixed.

Nevertheless,

the chain remains Gaussian _at all

times,

and the

preceding expression

is the

product

of two

independent quantities

which _can be calculated

easily.

We have

~

dS _exP

I-I _{ii~(St) ~(0} ^)i~il

-

£~

^{dS exP}

l~l~l

₌

I

exP

I-~l~ll

On the other

hand,

_we

get

%"

^{dS' exP}

I

l~~(°> °)

_~

(-S' °)i~l

=

I°~

^{dS' exP}

(- _~l~'l

-

Therefore,

_we obtain

12 =

A

exP

I-~l~ll ^(lo)

Finally, equations (7), (9)

and

(10) give

5(q,t)/5(q,0)

₌

( _(1-

exp

~- ^~~)j

~

(~~)

+

~( /~

dS

/~ dS'

_exP

(- (lzls> t)

^z IS'>

0)l~))

o o

where

z(0, t)

and

z(S, t)

remain fixed.

I-t,

Fig.

4.

Representation

of Ii the

points X(s,to)

and X

(o,s')

_are

represented by

_crosses; this

figure

is similar to

figure

3a but _now the stress

points

coincide.

Let _{us now}

displace

the

extremity,

in order to make it coincide with the

origin,

at all times

(see Fig.(4)).

Since

Z(S,t) Z(0,t)

%

Z(S, 0) Z(0, 0)

We Set

~l~>t)

_"

)l~ls> ^o) ~lo< o)I

+

xl~<~) Using

the relations

lizis, _{0) z10, 0)i~l}

= S and

iizis, 0) z10, 0)1ixis, t)

x

is', ₀₎₁₎

₌ 0

we see that

izis, i)

_z

(St, o)12)

⁼

l~ jl'~~ ^{iizis, o) zio,} o)121

⁺

(ixis, i)

x

(St, o)12)

=

~~

l'~~

+

(lXlS, ^t)

^X IS'>

0)l~)

where _{we may} set

X(0, t)

=

X(s,t)

₌ 0.

In this _way,

equation (11) gives

~i~ t)/~i~ ₀₎

₌

A

exP

(- _~l~ll i12)

+

( _j~

ds

j~ ^ds'

_exp

[-( _is ^')~]

exp

(- ( _{(ixis, i)}

x

is', _o)i~)

Since,

we have

X(0, t)

₌

X(S, t)

₌

0,

_{we can} resolve

X(s, t)

into modes and from

Appendix A,

_we find that

11~°~<

^{~) X}

IS', 0)l~) ~( f

~ _~

j2n~rsj

~~~~,

lr

~ ~

n2 ^" ~ ^cos ~

+2 _cos ^{~'~ ~~ +}

~')

_{nit s} ^{~ ~}

(13)

s ^cos

~

)j

^{n ir}

~ ^~~~ s2

1j

We kn°'~~ ~~~~

jjxj~

_~~ x

~/ _o~j~i

-

is

^s"

isi ^{'~ /'~}

(see Appendix B).

Consequently, equation (13)

becomes

(lXlS, ^t)

^X

^IS', °)l~)

=

~~ ~'~

~~)

^{~~ ~'~~}

~

25

f 2nir (s s') 2nir (s

+

s')j

~ n~ir~

~)j

~2 ~2 ^~~~ s ^~~~ s ^~~P

~7

n=1

Thus

equation (12)

_can be

expressed

in the

following

_way

~i~ t)/~ii~> 0)

_{= exP}

I- _~l~ll

+

£~

^dS

i~ ^dS'

_exP

_1~~ ^{~~ "l}

q~S

^°' 2nir

(s s') 2nir (s

+

s') ^n~ir~

~ ~~P

~2

~

~2 ^~~~ s ^~~~ s ^~~~

~71 (~~)

~=i

4. Calculation of the limit

ti(q, cc) / ti(q, 0).

Before

transforming ti(q,t)/ti(q,0),

let _us calculate the limit

ti(q, cc)/ti(q, 0).

~ii~ C~)/~iiS 0)

_{= exP}

I- _~l~l

+

i~

^dS

i~

^ds' _exP

_1~~ ^~~

^~"

q~sf

^I

[nit[s-s'[j jnir(s+s')jj

~ ~~P

(~W

~~~

7

^~" _s ^~" _s

o S(q.«)/S(q,0)

a6ymptot,c behaviour o 8

~ ~ _~

~, (,

o 4 _,,

o z

"$

,,~

~', '.

~'~raged

, Rancn

~~~ ^°~~~~ged

Q Q

',

~.~

',,

° 5 lo q<s>'/2

or qs'/2

Fig.

5. Plot of

§(q, cc)/§(q, 0)

_versus

q(S)~/~ (averaged: Eq.(22));

the _curve appears as a solid line.

It is

compared

with the _curve

representing §(q,cc)/§(q, ₀₎ (dot-dashed line)

in terms of _q

S~/~ (non averaged: Eq.(19)),

and also with the

prediction

of Ronca

(Eq.(24):

dashed

line).

Using

the classical formula [8]

~ £ ^~ _cos(2irnz)

=

z~ (z(

+ < z < 1

(15)

ir

~

n 6

we have that

°'

2nir is s'[ 2nir is

+

s')

^ss' _s + s'

is s'[

7 (

n~ ^~" 25 ^~" 25 S~ ^~ 25 25 ~~~~

and therefore

~~~~~~~~ P

~~~ _~~ ~~~~2~~~~~~~~~~~~~ ~

~~~

^~~

q _° ~

~~l(I'll/~[iq,o)

=

j

exP

I- _{~l~ll ~l~} ^~

^d~

'~'~~

_{~' ~~P}

1~/~l

ii~~

"

j[~ ~~P~

~~~

~

~~ ~~~~~ _~~

~~~~ ~~~

(see Eq.(3)).

This result _appears in

figure

5

(non averaged:

dot-dashed

line).

5. Transformation of

ti(q,t) / ti(q, 0).

We shall _{now use} the

following

Fourier formula which _can be obtained

by elementary

_means

f cos(2irnz) ^II

^exp

(-2ir~n~z))

~~

n=1

n2

~ ~ ~~~~ ^~~~

~ ~~2

~f~

^~~P

^{[~(~ p)~/2w)}

^~~~~

l +W

~ ~

~~~~~~~~~

^~~

~

~~P

(~ (~ p)~ /2zu~)

Pm-m

(by putting

w =

zu~).

We transform

equation (14) by using

^this

^identity; putting

_{z =}

7t/25~,

we obtain

~i(q

C~)/~i(q 0)

_-

A

exP

(- _{~l~ II}

₊

( _i~

dS

£~ ^dS'

^exP

1-

^~~

^~~

^~~'

x exP

~~Sl~~ i~ j ~i~ ^exP (-

^~

l

^~~

PSI

~

7tlL~j

~~~

(-

^{~ ~~}

^{'~) ?~'~~j}

or

by introducing

the dimensionless constants Y and Z

ti(q, t)/ti(q, ₀₎

= [1

exp(-Z)]

+

~

/~

^da

/~

^db

^exp(-Z[a

hi x exp

(- ^j~

Z 2

o o ^ir

/~

^du

f

(exp (-

^~~

~

#~~~~

_exp

(-

^{~~ ~}

~

#~~~~

o

~=_~

Y _'L Y _'L

Taking symmetries

into

account,

_{we can} now reduce the domain of

integration

and

setting A=a-b=(s-s')/S

B=a+b= (s+s')/S

we can

integrate

over the domain 0 < A _< B _< 1.

The ratio

ti(q,t)/ti(q, ₀₎

is thus

given by

ti(q, t)/ti(q, 0)

₌

[l exp(-Z)]

₊ Z

/~

^dA

/~

^dB

z

~ ~

exP

lzA & _i~

d1L

[( exP

1-

~~ l'~~~~

^exP

1-

~~ l'~~~~ II

~~~~

The _curves

representing

formula

(19)

is drawn for various values of Z in

figure

6

(non

averaged:

solid

line)

and in

figure

7

(non averaged:

dotted

line).

In the _same

figures,

the

asymptotic

^formula

(Z

=

cc)

of de Gennes is also indicated.

o s(q,t)/s(q,o)

o-a

z.5

°.6

3 5

0. 4

4 non averaged

',,

", "' 45

0.~ ",,

Roncn, dnehea C© "'~,- 5

Fig.

6. Plot of

5(q~t)/ti(q,0)

_versus

q~~~/~t~/~

for various values _{of q}

S~/~ (solid line, Eq.(19)).

For the

same values of _q

S~/~,

the dashed lines

give

the

corresponding

results of Ronca.

6. Limits

q~S

_{~ cc}

(de Gennes)

and lin~it

q~(7t)~/~

_~ 0.

By taking

the limit Z ₌

q~S/2

_{~ cc,} we recover the result of de Gennes

iii. Equation (19)

can be written in the form

siq1)/siq, o)

=

(ii _expi-z)i

₊

j _j~

_da

_j~

_db

_expi-a)

~ ~~~

~~2 /~

^~~

~ ~~~

_~~

~)~~~~

~~~ ~~

~~~~~j ~

o p=-«

In this

expression,

let _us take the limit Z _{-- cc; we} obtain

m y ¹ ~2

~(~i1)/~(~> °) /

_~~

~~P

~~

@ /

_~~

~~~

~y2q~2j~

0 0

which is the de Gennes formula.

On the other

hand,

the limits at t ₌ 0

(Y

--

0)

of

b ([5(q, t) /5(q, _0)] -1)

_are the _same for _our formula

(19)

^and for the

simplified

de Gennes

formula,

since for very low

t,

the

entanglements

do not

play

_any ^{r61e. For Y} < 1, ^de Gennes formula

gives

5(q>i)/s(q, o)

~ i

$

/°~

^da

^exp(-a) j~

^du exp

~-))

= l

~

o

o s(q,t)/s(q,o)

q<S>'I'=Z o-a

0.6

,_

4

0.4

",,,j"'

'~/jjj)j)"~"---,---,

averaged

""'~~~~["~~'~~---

0 2 "~~'~'--~---

cc

non averaged dashed

-o.o

0 5 0 q?7'/~ti/?

Fig.

^{7. Plot of}

5(q,t)/5(q, 0)

_versus

q~~~/~t~/~

for various values of

q(S)~/~ (solid line, Eq.(23)).

The _non

averaged

results

(Eq.(19))

for the _same values of _q

S~/~

appear in dashed lines.

Thus,

the

asymptotic

limit

y2 ~4~~

5(q,t)/5(q, 0)

ci 1-

= 1-

(20)

is valid in both _cases.

7.

Dispersion

of S: value of

ti(q, t) / ti(q, 0).

Until _{now, we} calculated

ti(q,t)/ti(q, 0)

in terms of S I-e- the

"length"

between

entanglements (measured along

the

polymer),

but _we can also account for the

dispersion

of S.

Neglecting

correlations between

entanglements,

_{we may} assume that these

points

_are chosen at random

on the

polymer,

and therefore that the

probability

law of S or Z is

given by

a Poisson law

Pois)

₌

) exPi-S/iS))

or

j21)

PIZ)

=

)exPi-Z/iZ))

with

~ ~

/

_dZ

_P(Z)

= and

/

_dZ

_ZP(Z)

=

(Z)

o o

where

(Z)

₌

q~(S)/2

is the _mean value of Z.

Then

by retracing

all the

steps

^{in the derivation}

ofequation (14),

_{we see} that

ti(q,t)/ti(q,0)

_may

be obtained

by averaging ti(q,t)/ti(q,0) directly

_over

S,

_{or over} Z.

(This

is not a

priori obvious).

Thus,

it is _easy to calculate the average of

ti(q,cc)/ti(q,0) (Eq.(17)) by using equation (21) 5(q, co)/§iq, o)

₌

%" )

exP

I- II _IA

exP

I- _{~l~ll ~l~ i~}

d~

'n12~ ii

_exP

I- _~~l~l1

and _we obtain

(with (Z)

₌

q~(S)/2)

§jq, cc) /§jq, 0)

=

jInjl

₊

jZ)) () f~

^dz

~)'~jj ^{j~ (22)}

This formula is

plotted

in

figure

5

(averaged:

solid

line).

In the

general

_{case, we} set

(see Eq.(19))

ti(q,t)/ti(q,0)

=

~ _(1-

exp

(-@)j

+ F

(q~(7t)~/~, q~S/2)

q

We obtain after

averaging

5(q,t)/5(q, 0)

₌

)In(I

+

iZ))

₊

/~ _dp e~PFIY,p(z)) 123)

where

F(Y, Z)

is

given by equation (19)

F(Y, Z)

_" ~

/~

^~~

^~

^~~

y p=+« A 2

~Z~j

exp

(- ^{~~ ~~~~ ~}

l

f

_d

_fl

^~

x exp

-ZA

@

o

~p=-«

^{~~~ ~~ ~}

Unfortunately,

_no

simplification

_occurs in

equation (23). Nevertheless,

^the

integral

_can be calculated _on

a computer ^{and the}

resulting

_{curves are}

plotted

in

figure

7

(averaged:

solid

line).

The limit Z

= cc found

by

de Gennes appears in this

figure.

The results _are

compared

with the

simple

_case where

we do not average

putting

S

=

IS)

(non averaged:

dashed

line).

This

averaging

has two consequences:

a)

the distance between the _curves is

reduced;

b)

_a softer

dependence

of the _curves with

respect

to Y

=

q~(7t)~/~

is obtained.

8. Comments _on Ronca's

theory.

In

1983,

Ronca [5] ^devised a

theory,

valid in the intermediate

region

and therefore amenable to _a

comparison

with

experimental

data and _our results. For this _{reason, we} summarize here the main

predictions.

First,

Ronca _expresses the

asymptotic

value

ti(q,cc)/ti(q,0)

in _terms of

(r()

which is the

mean square distance between

entanglements. Therefore,

_we

put

jrjj

= 35

and,

in this _way, the

asymptotic

value of Ronca's result is

[ti(q, cc) /ti(q, 0)]R

_" ^~

/~

_du

exp

-

~

(u

+

~~)j

~

«° ~z

_~~

¹²⁴⁾

~~ ~~~ ~

~ ^~~~ ~

~

This result _appears in

figure

5

(dashed lines).

We _see that the result is not very very far from

our non

averaged result,

but

steeper,

and _very far from _our final

averaged

result.

Ronca's

general

result reads

lsiq, t)/siq, 0)lR

=

( _/~

du _exP

(- @g (vi ot)]

where

(setting erfc(z)

=

) Jf

dt

e~~')

_we have

g(u, 9)

₌ 2u +

£exp(-u)

erfc

16

Vi) £ _exp(u)

erfc

(fi

+

Vi)

lr 2 9 1r 2 9

~~ ~

r~/~

~~~~

~~

%~

^{~~ ~~~~}

^~~~~~~~

^~~~

^~

^~

²ⁱ¹² ~

In this

formula,

_wo

according

to Ronca has _an

expression _(wo

"

1/16 _6~T)

which makes _a

comparison

difficult.

Therefore,

_we

ignore

it and

we

put (see Eq.(AA))

~~ ~~7

~°°

~2~° @

where _{p is a} coefficient which will be determined

by requiring

the

validity

of Ronca's formula for small t

(to

the order

Y~

=

q~7t).

Now,

_we set

q~S

Zu

~ 8 ^~ 4

~~~ _"~~ ~~~~

oi/2

=

jiwi)1/2

⁼

^17j~/~

⁼

iii- _~)

@

⁼

~l= ^°)

Consequently,

if _we know _{p, we can}

plot [ti(q, t)/ti(q, 0)]R

in terms of Y ₌

q~(7t)~/~

for various values of

Z~/~

= q

S~/~

_{We find}

~~~~'~~~~~~'~~~~ /~

~~ _{~~~ ~}

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

f(a p)

_{= exp}

(-fl~) /~

_dt

_sinh(2tfl)

exp

[-(t

₊

_a)~)

'

~ri/2

~

Now,

let _us determine _p

by calculating [§(q, t)/ti(q, _0)]R

for small values oft. For Y

-

0,

we

can write

lslq, t)/slqi _o)lR

~

"

~(~' /

d£Y

fl~, _fl)

0

~~~~~

j~~ _p~

_-~

^~fl f~

_{dt exp}

_[-(t

₊

_a)~]

'

~l/2

~

and

fl

=

pY/22

Thus

~ ~

@lq

i

t)/~(q, 0)lR

Ci

(~(( /

_da

/

_dt

t exp

j-(t

₊

a)

_~j

~y2

^{° °}

⁽²⁵⁾

m 1- ~

By comparing,

with the exact result

given

in formula

(20), equation (25) gives

P = 4

126)

Therefore,

_our

interpretation

of Ronca's

theory

leads to

@lq,t)/~lq, _0)lR

"

/~

_da

eXP

(-a ~f (), _{~)) (27)}

with

f(o, fl)

₌

£

~

exp

(-fl~) /

_dt

_sinh(2tfl)

exp

(-(t

₊

_a)~)

ir o

A

plot

of this formula is

given

in

figure 6,

where it is

compared

with _our result

(Eq.(24)).

We _see that Ronca's

theory predicts

curves which _are much _more

widespread

than _ours

(see Fig. 5). Actually

these _{curves are} unable to fit

properly

the

experimental

data of Richter et al. [6]

(see Fig.

8 in this article _or the

Fig.

13

top,

in their

article) though

this

discrepancy

did not _appear

previously

_so

clearly.

This

difficulty

_occurs to _a smaller extent for the _{curves cor-}

responding

to

equation (24)

but the

averaging

_process described in section 9

again

reduces the distance between the _curves

(for

_a

given S);

thus

they

lead to an

agreement

with

experiments

as will be _seen in the next section.

9.

Comparison

with

experiments

and conclusion.

Curves

representing

the form factor of

polyethylene (hydrogenated polybutadiene)

measured at T ₌ 509 K

by

D. Richter et al, _are

plotted

in

figure

9 _versus

q~7~/~t~/~

and

compared

with

our

theory.

The values of ₇ and

IS)

_are chosen _{so as} to obtain the best fit. Tbe theoretical

curves are functions of

q~(S) /2

and obtained

by averaging

_over

S; they correspond

to

equation

(28).

The _mean distance between

entanglements

is

IS)

₌ 12.25

nm~.

The coefficient ₇ of the Rouse

equation

is for

polyethylene

_{7 =} 0.8

nm~(ns)~~. (In

Richter's notation this

gives W£~

=

97

I-e-

W£~ (10~~~i~s~~j

^{= 7.2 whereas he finds}

^W£~

^{= 7.0 +}

0.7).

The

agreement

^is

quite good considering

the uncertainties. The

discrepancy

observed with Ronca's

theory

does not appear. This proves the

validity

of _our

approach.

One _may still doubt

s(q,t)/s(q,o)

i o

7"0 8 nm~ (ns)~~

<S>~/~=3 ₅

S (Q,f) IS (Q,0)

nm

~ ~

iooio ^~

o 6

B

D

~ °~aO~

~

0.lsskl

^{° ~}

o o

°

~

~

~

_fi

~o ~~ ~8

^A

^q=078nm~~ q<S>'/?=273

Q2 j2 @fi

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

~<~>'~~"~

°~

C q=1 55 nm~'

q<S>~/?=5

₄₃

~~~.

~ ^°'°

o 5 io q2yi/2t/2

Fig.

9

Fig.

8.

Comparison

of

experimental

data

by

Richter with Ronca results: the

discrepancy

is obvious

(according

to Richter et al.

[6]).

Fig.

9.

Comparison

of

ti(q, t)/ti(q, ₀₎

with

experimental

data of Richter

(the

_{same as} in

8).

^The solid line

correspond

to _our theoretical results

(Eq.(23)).

The _{curves were} fitted with the values

~ = 0.8

nm~(ns)~~

and

IS)

= 12.25

nm~.

that

polymers

remain

completely

fixed

(during

_a certain

time)

at stress

points;

however there is _no evidence for

assuming

_any motion.

Thus,

this

calculation,

which is based _{on very}

transparent assumptions,

is in _very

good agreement

^with

experiments.

It shows

that,

in this

domain,

the notion of

entanglements

is valid and it is _an additional

justification

of the

primary importance

of this

concept

^{for the}

interpretation

of the mechanical

properties

of

polymers. Thus,

it _may be

expected that,

in the

future, entanglements (or

stress

points)

will be used

jointly

with Rouse

equation

to describe all

types

^{of motions that do not}

depend

on the

chemistry

of

polymers.

At

present,

there remain

difficulties with this _program but their solution is

beyond

the _scope of this article.

Acknowledgements.

The author would like to thank D. Richter for communication of his results before

publication.

Appendix

A. Mode

expansion.

Since, X(0,t)

=

X(S,t)

=

0,

we can resolve

X(s,t)

into modes _as follows

XIS, t)

=

AL _in (t) Sinln~rs/S) IA-1)

and therefore

XIS, t)

X

IS', °)

=

4~ [(nit) Sinln~rs/S) in lo)

Sin

In~rs'/S)I lA.2)

Using

definition

(A.I)

in Rouse

equation (I),

_we find

((nit)

₌

^-n~W (nit)

+

infit) iA.3)

where

w = 7

~r~/S~ (A.4)

and

~

fn,~(t)

=

v5

_ds

sin(nits/S) fx(s, t)

Hence,

_we deduce

ifn,fit))

= °

ifn,f it)fn<,f it'))

"

)6n,n's it t') ^i~'~~

The solution of the

preceding equation (A.3)

is

t

fn(t)

₌

in e~"'~~

₊

/

_dt

e~"'~(~-~')

f~,~ (t') (A.6)

o

Putting

this

expression

in

equation (A.2),

_we find

X(S,t)

X (S~,

0)

"

If Sin

l'~l~l ^e-"~~~

^Sin

(~]~'II

^{+ sin}

(~i~) f~ ^di' e-n~wit-t')Jn~

₍₁₁₎

from which _we obtain

(~~~~'~~

^~

~~~'°~~~l

^{~ ~}

~

i~~i ^~~~ l~l~l ^~~~~~

^~~~

(~l~'II

^{~ ~}

~]

_~~~~

_l~l~l

Now, by modifying equation IA-G),

one write

o

in

₌

/

_dt

e+"'~~ fn,~(t)

-«

and

therefore,

from the

preceding equation

and from

equations IA-S),

_we find the classical result

j2

~

~/~2 ~2

n) Finally,

_we

_get

(~~~~~~~

^{~ ~~'~}

~~~~ ~

l

~ ~

~~~

~~~~~

^~~~

^~~~~'~

~_?)

+ 2

C°S '~'~

~i+ /~l

C°S

'~'~ ~i ^/~ _II

exP

1-'~li~7tl1

Appendix

B.

From

equation (13),

_we know that

([X(S, ⁰⁾

^X

_{is', 0)]~)}

"

$ jj _/2 ^~

~~~

~~~~)

^~~~

^~~~'~~

^{~ ~~~}

^{~~ ~s}

^{~'~ ~ ~ ~~~}

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

~=

By applying

the classical formula [8]

(Eq.15)

we find

~~~~~'~~ ~~~'~~~~~

^~

~~

^~

^{~ ~~}

^~

^~

(s s')~ _is s'[ (s

+

s')~ (s

+

s')

~~ 45~ 25 ^~ 6 ~ ~

45~ 25 ^~ 6

~ ~~

~

)2 ~2

^~

~~

~ ^~~

s~

and th~~~~°~~

(jx~s,o~

^x

^~/, _o~j~j ^is S"~Sj

^IS

^{S"i (B.1)}

References

[II

de Gennes

P-G-, Physica

3

(1967)

37.

[2] ^{de Gennes}

P-G-,

J. Chem.

Phys.

55

(1971)

572.

[3] ^Doi

M.,

Edwards

S-F-,

J. Diem. Soc.

Faraday

Trans. 74

(1978)

1789, 1802, 1818; The

Theory

of

Polymer Dynamics (Clarendon Press, Oxford, 1986) Chapter

6.

[4]des

Cloizeaux J.,

Europhys.

Lett. 5

(1988)

437; Erratum

Europhys.

Lett. 6

(1988)

475;

Macromolecules 23

(1990)

4678, 25

(1992)

835; ^J.

Pllys.

^I France 3

(1993)

61.

[5] Ronca G., J. Chem.

Phys.

79

(1983)

1031.

[6] Richter D., Butera

R.,

Fetters

L.J., Huang J-S-, Farago

B., ^Ewen

B.,

Macromolecules ₂₅

(1992)

6156

(the

lack of agreement ^{with Ronca results} _was less obvious in

previous publications;

_see

Phys.

Rev. Lett 64

(1990)1389.

[7] ^des Cloizeaux

J.,

Jannink

G., Polymers

in solution

(Clarendon

Press,

Oxford, 1990).

[8]Gradshtein I-S-, Ryzhik J.M.,

Table of

Integral

Series and

Products, (Academic

Press,

1965)

formula IA 44-3.