On the Validity of the Maxwell Model and Related Constitutive Equations; a Study Based on the First Normal Stress Coefficient

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On the Validity of the Maxwell Model and Related Constitutive Equations; a Study Based on the First

Normal Stress Coefficient

H.-P. Wittmann

To cite this version:

H.-P. Wittmann. On the Validity of the Maxwell Model and Related Constitutive Equations; a Study

Based on the First Normal Stress Coefficient. Journal de Physique I, EDP Sciences, 1997, 7 (12),

pp.1523-1533. �10.1051/jp1:1997153�. �jpa-00247469�

On the Validity of the Maxwell Model and Related Constitutive

Equations;

_a

Study Based

_on

the First Normal Stress Coefficient

H.-P. Wittmann

(*)

Max-Planck-Institut ffir

Polymerforschung, Ackermannweg

10, 55128

Mainzi Germany

(Received

4

April

1997, revised 21

August

1997,

accepted

2

September 1997)

PACS.47.15.-x Laminar flows

PACS.47.27.-I Turbulent flows, convection, and heat transfer PACS.47.50.+d Non-Nevionian fluid flows

Abstract. In _a previous

publication (J.

Phys. ^f £hance 4

(1994) 1791)

the semi-microscopical Rouse model _was

projected

onto macroscopic

Langevin equations

for concentration and stress

tensor fluctuations.

Adding

afterwards appropriate convective terms due to _a simple shear

flow the

resulting

equations _were shown to be _a

generalization

of the upper convected Maxwell model. Based _on the projected equations the first normal stress coefficient flfi,o is _now calcu- lated in this publication: flfi,o ₌

2~]/kBTco

where ~o and _co denote the zero-shear

viscosity

and

the _monomer concentration

respectively.

The

resulting steady

state

compliance

Js is found to

be Js

=

1/kBTco.

This _{turns out to} be in

disagreement

with

experimental findings.

Based _on the derivation of the

Lodge

equation _{one can} conclude the

following:

In order _to get the right first normal stress coefficient it is necessary ^to

incorporate

the shear flow at first into the Rouse

model and to project afterwards the modified Rouse model.

Introduction

A lot of constitutive

equations

for

polymer

melts and solutions _were accumulated _over the

course of time.

They

_are reviewed

by

Larson

ill.

Most of these constitutive

equations

refer to

a stress tensor

averaged

_over the

sample.

This

corresponds

from _an theoretical

point

of view

to the restriction of the

hydrodynamic

limit.

Moreover,

most of these constitutive

equations

known

today rely

_upon

phenomenological

elements like stress relaxation rates.

On the other hand shear enhanced concentration fluctuations in

polymer

solutions have attracted _a lot of attention in recent years

[2-llj.

From _an

experimental point

of view _one

was interested for

example

in

measuring

the

non-equilibrium steady

state structure factor.

As demonstrated

by

Wu et al. [4j ^this

quantity

turned out to become

anisotropic

due _to the presence of _an external shear flow.

In order to be able to calculate the structure factor under shear at first _a reliable

equation

of motion for the concentration fluctuations is needed. As shown

by

Helfand and Fredrickson [2j elastic forces characteristic for _a

polymeric _system

lead to a

type

^{of elastic} stress-induced diffusion _term that has to be added to the

equation

of motion for the concentration fluctuations.

In this _way the concentration fluctuations and the fluctuations of the stress tensor field become

* e-mail:

wittmann@th21.mpip-mainz.mpg.

de

@

Les

#ditions

_de

Physique

1997

coupled.

In order to close the

equations

of motion _a constitutive

equation

^is needed. This fact has revitalized the interest in constitutive

equations.

To be _more

specific,

Helfand and Fredrickson [2j ^{formulated for sheared}

polymer

solutions _a

phenomenological theory

where

coupled equations

of motion for _monomer

concentration,

the

velocity

field of the

solvent,

and the deviatoric

polymeric

stress tensor field _were

presented.

For the

equation

of niotion for the stress tensor variable these authors utilized the so-called second order fluid

[lj.

In

modeling

the viscometric coefficient _as functions linear in the concentration fluctuations

they

_were able to

explain

the

anisotropic growth

of concentration fluctuations in the presence of shear flow.

Simultaneously

Onuki [3j

published

_a

theory

that could be used _to describe

polymer dy-

namics in

solution, assuming

either Rouse _or

reptation dynamics depending

_on the

polymer

concentration. In this

context,

he used Marrucci's constitutive model

[12j

^{in which} a

long

lived strain variable is used _{as an} alternative to stress in the second order fluid model. Later _on MiIner

[6,9j presented

_a

dynamical theory

for

entangled polymer

solutions under shear based

on the two-fluid model introduced

by

Brochard and de Gennes

[13j

^and discussed

by

Onuki

[14j

and Doi and Onuki

[lsj. Finally

Ji and Helfand

ill]

used for their

theory

also the two-fluid

model to obtain

Langevin equations

for

concentration, velocity

and _a defined

polymer

strain.

Since the fluctuations of the concentration and the strain tensor _are slow

compared

to the

velocity variables,

these authors eliminated the

velocity

variables from their set of slow vari- ables. Both MiIner [9j as well _as Ji and Helfand

ill]

_were able to calculate

mimerically

the

non-equilibrium steady

state structure factor in excellent

agreement

with results obtained

by

Wu et al. [4j ^~ia

light scattering experiments.

As

already

mentioned above the constitutive

equations

_as reviewed

by

Larson

[lj

are usu-

ally

based _on

phenomenological arguments.

^{In order} to _overcome this

phenomenology

it _was

speculated early

_{on [2,}

6j

^{that it should be}

possible

to

project

^the

semi-microscopical

Rouse niodel

[16,17j

onto

macroscopic equations

of motion for

polymers

under shear. The set of _re-

sulting coupled Langevin equations

for collective variables like concentration and stress tensor fluctuations _can then be considered _{as a}

generalized

constitutive relation. For _a

polymeric

_sys-

tem under shear there _are in

principle

two

possibilities

to

proceed:

The first _one is to

project

at first the Rouse model where the shear flow is turned off _onto

macroscopic equations

and to add in _a second step

appropriate

convective terms. This

approach

_was

adapted by

Wittmann and Fredrickson

[18j.

In order to deduce

macroscopic equations

of motion from the Rouse model

one has to know how to calculate the matrix of

decay

rates _or memory kernels and the matrix of

Onsager

coefficients in _a

systematic

fashion. Based _on the

projection operator technique

of

Zwanzig

and Mori

[19-22j

Wittmann and Fredrickson

[18j

^found the

following:

The matrix of

decay

rates is

given by

the matrix of the

dynamical

_response functions

multiplied by

the

inverse of the matrix of

dynamical

correlation functions.

Furthermore,

since the

projection

_op-

erator

technique

_assumes

implicitely

_a free energy that is bilinear in the collective variables the matrix of

Onsager

coefficients is

given by

the matrix of

decay

rates times the matrix of static

correlation functions. Due _to the Gaussian nature of the Rouse model

[16,17j

the

required

correlation and response functions _can be determined

[18j

^{at least in the}

hydrodynamic

limit.

The

Onsager

coefficients calculated in this way can be used to formulate

Langevin equations

for concentration and stress tensor fluctuations. These

equations

contain _as the

hydrodynaniic

limit the _upper convected Maxwell model

[lj.

Therefore these

equations

shall be called the

generalized

Maxwell model.

On the other hand there is _a second _{way to}

project appropriate equations:

At first convective

terms _are added in the

equation

of motion for individual Rouse modes. These _so modified Rouse

triodes [17] are then used for _a

projection

onto

macroscopic equations

^of motion.

Up

to now

such _a

procedure

_was not

persued. However,

there is _a constitutive

equation

known _as the

Lodge equation

or the rubberlike

liquid

model

[I,17, 23]

^{which -relies} on the Rouse modes under

shear.

Starting point

for this constitutive

equation

is the thermal average of _a

microscopic expression

^for the stress tensor in the

hydrodynamic

limit where this

quantity

is

essentially

_a

sum over second _moments of individual Rouse modes. If these moments _are calculated for the

case that _a shear field is

present

one finds _a stress tensor which _can be

expressed

in terms of the shear modulus and the

finger

strain tensor

[17].

^The

deficiency

of the

Lodge equation

however is that these constitutive

equations

start from the very

beginning

with the

hydrodynamic

limit.

As described

above,

constitutive

equations

_are needed _{as an}

input

for the calculation of the

non-equilibrium steady

state structure factor under shear.

However,

constitutive

equations

_can also be used to

predict

normal stress differences

[1,17,23]. By comparing

these

predictions

with

experimental

results the normal stress differences _can be used to decide whether _a constitutive

equation

^is reasonable or not. This

approach

will be

adapted

in the

present

paper. In Section the

generalized

Maxwell model introduced in reference

[18]

^{shall be recalled. In Section 2 the}

viscometric coefficients _are calculated in the framework of the

generalized

Maxwell model. As _a

byproduct

the

Lodge

Meissner

relationship [24-27]

is rediscovered. Section 3 is used to review the

Lodge equation including

the associated viscometric functions

following

reference

[17].

^As it will be discussed in Section 4 the

generalized

Maxwell model and the

Lodge equation yield

the

same result for the

viscosity

^but differ in the

prediction

with respect to the first nornial stress coefficient. Based _on the

viscosity

and the first normal stress coefficient the so-called

steady

state

compliance

_can be defined. From

experiments

done

by Onogi

et al.

[28]

on melts of _narrow- distribution

polystyrenes

it is known that the

compliance

is for

sufficiently

short

polymers (Rouse polymers) proportional

to the

degree

of

polymerization,

while the

compliance

beconies

independent

of the

degree

of

polymerization

for

sufficiently long polymers (reptation regime).

As it is well known the

compliance

_as

predicted by

the

siniple Lodge equation

is in

agreement

with the

experimental findings.

The _more elaborate

generalized

Maxwell model however

yields

for Rouse

polymers

_a

conipliance

^which is

independent

of the

degree

of

polymerization.

This is in clear

disagreement

with the

experimental

result. Based _on these

findings

_one can conclude the

following:

If _one is interested in _a

projection

of the Rouse model onto

macroscopic equations

of motion _one has at first to

incorporate

^the effect of _a shear field into the

semi-microscopical

Rouse model. The Rouse modes under shear _can then be used for _a

projection

onto

macroscopic Langevin equations

for collective variables like concentration and stress tensor fluctuations.

1. Generalized Maxwell Model

In reference

[18]

^for a melt of Rouse

polymers coupled Langevin equations

were formulated for collective variables

c(q, w)

and

Q(q,

++ _w defined _as follows:

c(q, w)

₌

/ ^~ dte~~"~

^~°

/

_ds

_{e~~~'~~~'° coda} ^l

N

~

In this context

r(s, t)

denotes the

position

of _a Kuhnian

segment

at contour

position

_s and time t whereas b, ^{N and} co are the

segment size,

the number of Kuhnian

segments

_per ^chain and the

averaged

_mononier concentration

respectively.

Therefore

c(q, _w)

is the fluctuation of

++

the _nionomer concentration about its average value _co and

Q(q,w)

is the fluctuation of the

deviatoric stress tensor field measured in units of

kBT

times the

polymer

concentration

co/N.

As

pointed

out in reference

[18j

^the

equation

^of niotion for the concentration fluctuation _con- tains _a term

proportional

to the

divergence

of the elastic force

density

field which is

essentially

++

qq

kBT Q(q, w).

In reference

[18] advantage

_was taken from this

specific

sort of

coupling by decomposing

++

Q(q, w)

into

longitudinal

and _transverse

components Q"(q, w)

₌

#~#po~p(q, w)

and

Q~(q,w)

=

(d~p (~(p)Q~p(q, t),

with

(~

= q~

/q.

This leads to

(c, Q", Q~)

_{as a set} of collective variables. Based _on the

projection operator technique

^of

Zwanzig

^and Mori

[19-22j

the

Langevin equation

for _a collective variable A _e

(c, Q", Q~)

in the _presence of _an external shear flow reads:

iw iqx )lAiq w)

_-

iQ14 ^rt~ i~ ~d)iBi~

_~°~ +

H~i~

_{~°~. i~~}

For

siniple

shear flow characterized

by

_a flow field

u(x)

=

§vex

with shear rate

§

the differential

operator §qxb/bqy

describes the convection of Kuhnian

segments by

the flow field whereas the

frequency

matrix

in((

with the

following

_non

vanishing

niatrix elements

jajjc

1

=

jai(Q"

=

-iQ)$~

~

~~~~~

^~~

~'~~~~~

appears as a consequence of the reorientation of Kuhnian

segments by

the flow field. The

quantity El~(q, w)

is _a randoni force with _zero mean,

(B~(q, _w))

=

0,

and covariance

ie~lq, w)e~ _{I-q, -w))}

=

2Rel©~~'"~l

where the

right

hand side denotes _up to _a factor of 2 the real

part

^of the

Onsager

coefficient

A(~(q,w)

with

A,

B _e

(c,Q",Q~).

Both the

Onsager

coefficient

A(~(q,w)

_as well _as the

decay

rate

r)~ (q, w)

which appears in the

equation

of motion _were calculated in reference

[18j

in the framework of the

semi-microscopical

Rouse model.

However,

in order to have _access to

++

the normal stress coefficients it proves convenient to undo the

decomposition

of

Q

into Q" and

Q~

This leads to the

following equations

of motion

iw iqx)lclq,

uJ) =

-rl~lq, uJ)clqiuJ) rl~lq, _uJ)=iilq,

_{uJ) +}

_{e~lq,uJ) 12)}

»

~

l~

^H ~ H H ~T _~

iW

iqx~) ^Qlq,

_W) ~

Qlq, uJ)- Qlq,

_{uJ) ~} 2 _e

clq,

_uJ)

=

y

=

-ri~(q, _{w): ~(q,}

w) ri~(q, _{w)c(q, w)}

₊

El§q, _{w). (3)}

The left hand side of the last

equation

denotes the _upper convected time derivative. The term

2$c(q,w)

_appears in order _to compensate ^{the fact that not the stress itself but the deviatoric} stress is used

[18j.

^The

quantity I

=

(i7u)~

with ~~p =

§d~xdpy

^{is in} this context the

velocity gradient

tensor whereas

$= (1/2)(I

₊

i~)

with e~p =

(§/2)(d~~dpy

+

d~ydpx)

is known _as the rate of strain _tensor. Due to

equation (20)

of reference

[18]

^the

decay

rates are in

leading

order

given by

r[~(q~w)

=

+Dq~ r[~(q~ _w)

=

-(kBT/()qq

,

rf~(q, _w)

=

-2Dqq (4)

where

(

is the friction constant and D the center of _mass diffusion constant. In the

hy-

drodynamic

--

_ _

no(q,w):Q (q,w) might

an

appropriate

scalar. Furthermore it proves convenient to _use instead of

no(q, _w)

_{the Green's}

function

~~~~~'"~

iuJ +

~~(q,

w) ~~~

~

~~~~~

_~~~

In the last

relationship

which follows also from

equation (20)

of reference

[18] ~*(w)

denotes the

complex viscosity

which is related to the shear modulus

Gjt)

=

~~~~° f~-2t/rp _j~)

N p=1

where Tp denotes the relaxation time of Rouse mode _p ~ia _a

Laplace

transform:

~*(w) =

pling

of

the _equation

_of

_niotion

_for

concentration and

stress

_tensor

luctuations. Since _both

quantities are of

order

O(qq) the quations of _motion decouple in the strict ydrodynamic

limit.

Furthermore,

if equation

(5)

is

not

_used but

if

nstead the quantity

no(q,w)

placed

by

ome

phenomenological tress _ecay rate I/TQ,

equation (3) yields in

real space

_in

the

^rodynamic limit

the

following

relationship:

~ ++

_ H ++ _T i H _{_}

bikBT Q +q i7kBT Q

_~

kBT Q -kBT Q

_~

=

-kBT Q

+2 e

G(0) (7)

TQ where the noise _was

suppressed

and where

G(0)

=

kBTc

is

interpreted

_as the shear modulus for t

= 0. The second term _on the left hand side of the last

equation

describes the convection of Kuhnian segnients due to the flow field whereas the third and fourth ternis niimic the

reorientation of Kuhnian

segments

due to the flow field.

However,

since

equation (7)

is

nothing

else than the _upper convected Maxwell model

ill,

the set of the

coupled equations (2, 3)

can

be considered _{as a}

generalized

Maxwell model.

2. Viscometric Functions

In the

following

the considerations of reference [18] ^{shall be extended}

by calculating

the vis-

-- _ _

cometric functions. _{For this purpose} the

replacement no(q, _{w):Q(q, w)}

~

no(q, _{w) Q(q, w)}

shall be used

again.

In terms of

ili~(q, _w)

=

I/GiQ(q, _w)

_the

_equation

_{of niotion for the stress}

variable I.e.

equation (3)

_niay be rewritten _as

§q

^~

Q jq ~)

=

_~yQQjq ~) Q jq ~)

₊

gojq _{~) j8)}

~bqy

~

' ° '

~

'

~

'

where the

quantity

°~lq, _w)

=

t. llq, _{w)+ 41q,}

w) t~

+

12? -ri~lq,w)iclq,w)

₊

6ilq, _w)

has the

nieaning

of _an

inhomogeneity.

^In order to see this _more

clearly

it is instructive to write

_ ++ ++ _T ₊₊

down the elements of the tensors ~.

Q

+

Q

_.~ and _e

explicitly:

_ _ ~

ilo~»+Q»x) ^{iQ»» iQ»z}

^o

^I

^o

~. _Q

₊

_Q .~

=

§Qyy

0 0 and 2

$= _§

_{0 0}

iozy

0 0 0 0 0

Equation (8)

^{formulated for the} tensor element

Qxx(q,w)

for

example

contains the tensor element

0~~~(q,w).

This

quantity

is _a functional of

Qxy(q,w)

and

Qyx(Q, w)

^{but it is inde-}

pendent

of

Q~~(q, w).

Therefore

0~~~(q, _w)

_can be considered _{as an}

inhomogeneity.

A similar

++

behavior is also true for all the other tensor elenients of

Q(q,w).

As _a result the methods of characteristics

[2,29]

^allow one to rewrite

equation (8)

_as

llq, w)

₌

/~ ^dt'exPl- _/)dt" 4~i~lqlt"), w)1°~lqlt'), _{w) 19)}

where the time

dependent

_wave vector

q(t')

= q

t'§q~ey

_was introduced. In the

following

the noise shall be

neglected again.

Then _one finds for

off

e

(yy,

_yz,

zy,zz)

QaPlq, w)

₌

/~ ^dt' exPl- /)dt" ^i~i~ lqlt"), w)in°~~ lqlt'), w)Clqlt'), _w).

This

expression

can be

expanded

in _a power series with

respect

to the shear rate

§.

The

zero order contribution is obtained

by

the

replacements q(t')

_{~ q} and

q(t")

_{~ q.} Then the

integrations

^with

respect

to t' and t" _can be done:

/~~dt' ^~XPI

)~~"~i~~~'~°~ _Yi~)q,

°~)

~~~~~~ ~~ ~~~~

~

~~~~

where

~*(w)

was for convenience

replaced by

its low

frequency

limit ~o =

limw~o ~*(w). Taking equation (4)

into account it _can be _seen that for

off

e

(yy,yz,zy,zz)

the deviatoric stress

tensor in the

hydrodynamic

limit is

given by:

QaPlq,1°)

"

2fiDqnqpClq,1°)

⁺

^{Oli) 11°)}

At next the shear stress

Q~y(q, w)

^{shall be} considered. At first

equation (9) implies:

Qxyiq,w)

=

f~dt'expj- f)dt"ilyojqjt"),w)j

X

i)Qyyiqit'), ))

+

ii ^Fi~~~ iqit'), ~°)iciqit'),

_L°)1

However,

^since

Qyy(q,w)

₌

O(qyqy)

due to

equation (10)

^and

ri~~~(q,w)

=

O(q~qy)

due to

equation (4)

both

quantities

shall be

neglected

in the

following

for the

hydrodynamic

limit.

Analogous

to the

reasoning given

above _a power series

expansion

with

respect

to the shear rate

I yields immediately

kBTQxy(~,td)

_"

kBTQyx(~,td) ~'f~0~~~~~~

+

°l'f~) ₍₁₁₎

Co

since the tensor

~j

is

symmetric. Finally,

if ternis

containing ri~~~(q,w)

and

ri"~(q,w)

are

again neglected

because

they

are of order

O(q~)

_one

obtains,

due to

equation (9),

kBTjoxxlq, _w) Qyylq,

_{w)1 =}

2i ~j)~~ ^{kBToxylq, w)}

⁺

°li~) ₍₁₂₎

Such _an

equation

^is known _as the

Lodge

Meissner

relationship [24-27j.

Note that T~ =

~o/kBTco

is _a characteristic time and that the

product _§T~

_can be considered _as

the Deborah number

[lj. Furthermore,

in

combining equations (11, 12),

one finds for the first normal stress difference

[lj

kBT(Qxx(~> td)

QyY(~>

td)j ~'f~ /~~~ ~~j~~

~~~~

Finally

it shall be nientioned that in the framework of the

approxiniations

^made above the second normal stress difference

[lj

vanishes:

kBTioyylq, _w) Qz=lq,

_w)1

= °.

l14)

The viscometric functions which are the

viscosity dqo

_(cj ^and the first and second normal stress coefficients

dill,o[cj

and

d1l2,o[cj respectively [lj

are in the context with the deviatoric stress

++

tensor field

kBT Q

defined _as

d~o(c]

₌

lint(kBToxyli) (15)

i~o

di~l.0(cj

_#

11r$(kBT(Qxx QyYjl'l~) (16)

~~

~~y~~~jcj =

j~IkBTIQYY ^Q~ll'~

~ ^~~~~

From

equation (12)

one can

identify: d~o[c]

=

~oc/co

where _~o

=

ffdtG(t)

denotes the

viscosity.

In _a similar fashion _{one can} write:

dili,o(c]

=

iii,oc/co

and

d1l2,o(c]

=

1l2,oc/co,

where

iii,o

and

1l2,o

are known _as the _zero shear limit normal stress coefficients. In addition

a

quantity Js

known _as the

steady

state

compliance

is defined via

Js

=

iii,o/2~(, _Equations (12-14) yield:

~° ~~ ~~~~

'

~~'°

~

k~ico

'

~~'°

^~

'

~ B~co

~~~~

3.

Lodge Equation

The

perspective

of this section is to review the calculation of the viscometric coefficients _on the basis of the

Lodge equation

also known _as the rubberlike

liquid

model

following

reference

[17].

For _a melt of Rouse

polymers

the

fluctuating

stress tensor field

Six, _t)

_at

_position

x and time

t reads:

~ _~~'~~ ~~~

~~~

~~~'~~~

~~~ ~~~

^~~

~~~

^~~ ~~~~

Note that the stress tensor

$ (x, t)

is related to the deviatoric stress variable

lx, t) through

_ ++ ++

a

(x, t)

=

kBT[Q (x, t)

₊

c(x, t)

d]. ^{For the} constitutive

equation

known _as the

Lodge equation

one considers

usually

not

$ _{(x, t) itself,}

_{but the stress tensor}

IS it))

=

/ _{d~xi?ix t)i}

=

j _~(i~ j~dsi~~l'~~ ~~l'~~1 ¹²⁰⁾

From _an

experimental point

of view this _means to average ^over the

sample

which

corresponds

from _a theoretical

point

of view to consider the

hydrodynanlic

limit.

Representing

the

position r(s, t)

of a Kuhnian

segment

^{in terms of the} set

(xp(t))p

of Rouse modes [17] as

cc

r(s, t)

= xo

It)

+ 2

~j

_xp

_(t)cos(~~~

N P=1

the stress tensor _as

given by equation (22)

can be rewritten [17] ^{in the}

following

_manner

IS _it))

=

) f

kpixplt)xpit)) 121)

where

kp

=

(67r2kBT/Nb~)p~.

In order to derive from here _an

expression

for the stress tensor in the presence of

simple

shear

flow,

the

Langevin equation

for _an individual Rouse mode

xp(t)

is modified

by adding

_an

appropriate

term due to the shear flow

[17].

^This

implies

a first order differential

equation

for

lip _it)

=

(co/N)kp(xp(t)xp(t))

which is

given explicitly by _[17]

bi imp It)) t _imp lt)) _imp it)) t~

=

-( (i?p ^{It)) ~~(~°} ij 122)

~ ~T

While Tp denotes the relaxation time of Rouse mode _{p, ~} and _{~ are} the

velocity gradient

tensor and his

transposed counterpart.

As

pointed

out in reference

[17j

^the differential

equation (24)

can be solved without

difficulty.

In ternis of the shear modulus

G(t)

_as defined

by equation (6)

^{and the}

finger

strain tensor ++

Bit) _[23]

the solution of

equation (22) yields

in combination

with

equation (21) immediately:

ii _it)

~

f~ dt'°Gij~j ^t') tit _t')

123)

For

simple

shear flow which will -be considered here

only

the tensor elements of the

finger

strain tensor are

given explicitly by [17j

~

B~p(t)

=

d~p

+

§t(d~~dpy

+

d~ydp~)

+

i~t~da~dp~

Equation (23)

is known _as the

Lodge equation

_or the rubberlike

liquid

model

[lj

and _re- sults also from the

Green-Tobolsky

model

[1,17, _23, 30-33j.

The viscometric coefficients which

are the

viscosity

_~o

"

limi~o((a~y) Ii)

and the first and second normal stress coefficient

iii,o

₌

limi~o (a~~ ayy) li~)

and

1l2,o

"

linii~o (ayy a~~) li~) respectively

_are

given

in the framework of the

Lodge equation (25)

as

~o "

/

_dt

_G(t)

,

iii,o

_" 2

/

_dt

_tG(t)

,

1l2,o

_" 0

,

Js

=

/

_dt

_{tG(t) (24)}

~ ~

~o

~

where the

steady

state

compliance Js

=

iii,o/2~(

_was added

again.

4. Discussion

At first the

conceptual

differences of the

generalized

Maxwell model and the

Lodge equation

shall be discussed. In order to obtain

equations (2,3)

which

represent

^the

generalized

Maxwell model at first the

decay

rates

r)~(q,w)

_as well _as the

Onsager

coefficients

A(~(q,w)

_with

A,

B _e

(c, Q", Q~)

_were calculated for the _case that _no external shear field is

applied.

As shown in reference

[18j

^{both the}

decay

rates as well _as the

Onsager

coefficients _can be related to correlation and response functions. These

quantities

_can be calculated

explicitly

in the

framework of the

semi-microscopical

Rouse model

[16,17j.

In _a second

step

^convective terms that mimic convection and reorientation of Kuhnian

segments

^due to the shear flow _were added

yielding equation (8). By

the method of characteristics this

equation

could be transformed into

equation (9).

From _a mathematical

point

^of view the convection of Kuhnian

segments

^is now

accounted for

by

the tinie

dependent

wavevector

q(t)

= q

t§q~ey.

In

setting q(t)

to q in the

subsequent equations

the effect of convection is turned off. The _occurrence of the shear rate

§

in

equation (11)

to

equation (13)

is thus

entirely

due to the effect that _a flow field is able to

reorient Kuhnian segments.

The

strategy persued

in context with the

Lodge equation

differs. There _one starts out front the very

beginning

with the stress variable

Ii _(t))

_{which is the}

hydrodynaniic

liniit of

the stress variable

Ii _{lx, t)).}

_While

I _lx, t) depends

both _on the

position

_as well _{as on} the

orientation of Kuhnian

segments, Ii _(t)) depends only

on the orientation of Kuhnian

segments.

Due to

equation (21)

the term

lip _It))

=

(co/N)kp(xp(t)xp(t)) represents

the contribution of Rouse mode _p to the stress tensor

Ii _(t)).

_The

corresponding quantity

for the stress variable

++ ++ ++ ++

(kBT Q (t))

=

(kBT Q (q

=

o, t))

is

given by (kBT _Q~ (t))

=

(co IN) [kp(xp(t)xp(t)) kBT dj.

In terms of this stress variable

equation(22)

_can be rewritten _as follows:

bi ikBT (~ _it) t _jkBT Iv _{it)j jkBT} (~ jt)j t~

=

=

-?jkBT _{4p itjj}

+ 2

t Gpj0) 125)

where

Gp(0)

=

kBTco/N

is considered to be the contribution of Rouse mode _p to the shear modulus _at time t

= 0. The last

equation

_{can now} be

compared

with the upper convected Maxwell niodel _as

given by equation (7).

The

right

hand sides of both

equations correspond

to one another,

However,

the second term _on the left hand side of

equation (7) mimiking

the convection of Kuhnian

segments by

the flow is

missing

_on the left hand side of

equation (25).

This behaviour is in agreement ^{with the} fact that the considerations

leading

to

equation (23)

started out from the

hydrodynamic

limit

Ii _(t))

_of

Ii _{lx, t)).}

In reference

[18]

^it was

explained

how the

transport

coefficients of the

niacroscopic Langevin equations (2, 3)

_can be calculated in the franiework of the

semi-niicroscopical

Rouse niodel. In the

hydrodynamic

limit the

macroscopic transport

coefficients could be

represented

in _a

siniple

way in ternis of the center of _mass diffusion constant and the shear modulus of the Rouse

model. In

particular

it _was shown that the Green's function

G)Q

_can be related either to the

complex viscosity ~*(w)

or to the shear modulus

G(t) respectively:

Gi~lq,1°)

~

)(j~~

⁺

^°lq~)

^Or

^Gi~lq,t)

^~

/jj~

₊

°lq~)

This result

originally

obtained in the framework of the Rouse model _can be transfered _to other models _on the level of _{a one}

polymer approximation

like the Zimm model

[34]

or the

reptation

model [35]

by exchanging

the shear modulus. As it is well known the shear modulus of the Zimm model has the _same structure _as that of the Rouse model

given by equation (6).

The

relaxation time Tp,

however,

scales _now like _Tp

mJ

N~p~~

where N denotes the number of Kuhnian

segments

and where

/L ⁼

3/2

for _a 0 solvent and _/L

= 3v for _a

good

solvent. In the

following

the _mean field value _v

=

3/5

of the self

avoiding

walk

exponent

_v valid in 3 dimensions shall be used. On the other hand for the

reptation

model the shear modulus reads

[17]

G(t)

=

kBTco

^~

~

~ ( exp(-p~t/Td)

~~

p,odd l~ ~

where _a and b denote the

step length

of the

priniitive

chain and the size of _a Kuhnian seg- ment

respectively.

The

disentanglement

time _Td scales _{as Td}

~ N~ where the theoretical value

of the

exponent

is _z

= 3 whereas the

experiments yield

_{x m} 3.4. After

specifying

the _ex-

pressions

for the shear modulus it is easy ^to derive how the viscometric coefficients scale with

N,

the

degree

of

polymerization.

In order to

distinguish

the first normal stress coefficient _as

given by equation (18, 24)

the

following

notation shall be introduced:

ilf~

= 2Y/(

/kBTco

and

ill

~ ⁼ 2

ffdt tG(t)

where the

superscripts

M and L indicate whether the

generalized

Maxwell

midel

or the

Lodge equation

was used to calculate the first normal stress coefficient. Analo-

gously

_one has

Jfl

=

I/kBTco

and

J)

=

(1/~() flidt tG(t).

For the Rouse model _one finds:

w~N, il(~mJN~, il[,~mJN~, JfImJN°, J)mJN.

For the Zimm model _one has to

distinguish

between the _case of the 0 solvent with

~~ ^rW

~/~

~

, il)~Q rW

~/

,

il)Q

rW

~/~

,

~~

rW

~/~ ~~

rW ~/

and the _case of _a

good

solvent which results for _v

= 0.6 in

~QrW~/~~, ilforw~/~~, il)QrW~/~~, ~~rW~/~, ~~rW~/.

Finally

the results of the

reptation

model shall be

given

in terms of the exponent x introduced above:

w ~ N~

,

ilf~

mJ

N~~

,

ill

~ ^mJ

N~~

,

Jf

mJ

N°, J)

mJ

N°.

Note

that, by accident,

the

scaling

behavior of

ilf~

and

ill

~

coincide for the

reptation model,

whereas

they

differ for the other niodels.

Froni'the expiriments

_done

_{by Onogi}

_{et al.}

_[28]

on melts of narrow-distribution

polystyrenes

it is known that the

compliance

for

sufficiently

short

polymers (Rouse polymers)

is

proportional

to the

degree

of

polymerization,

while the

compliance

becomes

independent

of the

degree

of

polymerization

for

sufficiently long polymers (reptation regime).

This well known

experimental

result rules out the

generalized

Maxwell

model. In other

words,

in order to

get

^the

right

first normal stress coefficient _or the

steady

state

compliance respectively

it is necessary ^to

incorporate

the shear flow at first into the Rouse model and to

project

afterwards the modified Rouse niodel.

Acknowledgments

thank Glenn Fredrickson for _a critical

reading

of the

manuscript.

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