Linear macroscopic properties of polymeric liquids and melts. A new approach

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Linear macroscopic properties of polymeric liquids and

melts. A new approach

H.R. Brand, H. Pleiner, W. Renz

To cite this version:

Linear

_{macroscopic properties}

of

_polymeric

liquids

and melts. A

new

approach

H. R. Brand

_{(1, 2),}

H. Pleiner

_{(1, 3)}

and W. Renz

₍₄₎

(1)

FB _7,

_Physik,

Universität Essen, D 4300 Essen 1, F.R.G.

(2)

Center for Nonlinear Studies, MS-B 258, Los Alamos National

_{Laboratory, University}

of

California, Los _Alamos, NM 87545, U.S.A.

(3)

Materials

Department, University

of

California,

Santa Barbara CA _93106, U.S.A.

(4)

Theorie II, IFF, KFA Jülich, Postfach 1913, D 5170 Jülich, F.R.G.

(Reçu

le 8 décembre 1989,

accepté

le 22

février 1990)

Abstract. 2014 We

present a new

_approach

to the linear

_macroscopic

behaviour of

_polymeric

_liquids

and melts in terms of the

_macroscopic

variables :

velocity

field,

density,

entropy

density

and strain field and we discuss how the classical Maxwell model emerges as a

_{limiting special}

case. The novel

approach

used here, which considers the strain field as an additional

macroscopic

variable that relaxes on the timescale of the Maxwell time, allows to

_incorporate

additional variables such as

the

_{macroscopic polarization}

or the concentration in mixtures

_easily.

We

_point

out a number of new

_{cross-couplings}

not

_given

before in connection with the Maxwell model and we outline four

experiments

to test some of the

suggestions

made. We

_predict

for

_example

that transverse acoustic

_phonons

can be excited in a transient network with a

_{frequency larger}

than the Maxwell

frequency

of the

polymer

melt or solution studied.

Classification

Physics

Abstracts

61.40K - 05.70L - 65.70

1. Introduction and motivation.

The

_dynamic

behaviour of

_polymer

melts and solutions is known to be

_complex

and

complicated,

both in the

_{macroscopic (Ref. [1]}

and Refs. cited

therein)

and in the

_microscopic

and

_mesoscopic

domain

_(Refs.

_[2,

_3,

_4]

and Refs. cited

_therein).

To describe the low

frequency, long wavelength

behaviour of

_polymer

melts and solutions the most

_commonly

used model is the Maxwell

_model,

which was considered first

_by

Maxwell

_[5]

to’characterize

the non-ideal behaviour of gases. To describe the

phenomenon

that

_polymer

melts and solutions react like a

liquid

below a certain

frequency

_(usually

called the Maxwell

_frequency)

and solid-like above this characteristic

_frequency,

in the classical Maxwell model a

_dynamic

equation

for the stress tensor is introduced. This model has been used in the

_past

to

_study

a

large

number of

_{physical phenomena (Ref. [1]}

and Refs. cited

_therein)

and more

_recently

for e.g. thermal convection

[6-8]

and surface waves

_[9].

To

_give

an accurate

_description

of the

experimental

results a number of modifications of the classical Maxwell model

_(e.g.

thé

Jeffreys model)

was introduced

_[1]

_and

the

_analogy

to electric circuits was

_emphasized

and

exploited.

For none of these models a discussion of the

_{cross-coupling}

to other

macroscopic

quantities

such as

_temperature

or

_polarization

seems to exist. No fundamental reason

_why

there should be a

_{dynamic equation}

for the stress tensor was ever

_given

to the best of our

knowledge.

Here we

_suggest

to use a different

_gateway

of attack to characterize the low

_{frequency, long}

wavelength

behaviour of

_polymer

melts and

_polymer

solutions in the small

_amplitude

limit. Based on the observation that in their

liquid

state

_polymers

behave like a

_simple

liquid

for

sufficiently

small

_frequencies

and like a solid for

_frequencies

above the Maxwell

_frequency,

we use the

_hydrodynamic

_{approach (compare}

for

_example

Refs.

_{[ 10]}

and

_{[ 11 ])}

to describe the

macroscopic

behaviour of

_polymer

melts and solutions. In

_writing

down

_hydrodynamic

équations

one focuses on the

_{long wavelength,}

low

frequency

response of a condensed

system,

which does not have

_long

_{range forces e.g.} due to Coulomb interactions. Then one

derives balance

_equations

for the

_truly

conserved

_quantities

such as

_density

and

_density

of linear momentum and for the

_quantities

associated with

_{spontaneously}

broken continuous

symmetries

such as the

_displacement

field in

_{crystals [10]}

or the deviations from the

equilibrium

direction of the director for nematic

_{liquid crystals [11].}

In some cases this

rigorous hydrodynamic approach,

which

_{keeps only}

the

_quantities

which do not relax in the limit of infinite

_wavelength,

is modified to

_keep

_variables,

which relax in a

_long

but finite

time,

such as for

_example

the modulus of the order

_parameter

close to a

_phase

transition. This

approach

is called

_{macroscopic dynamics}

and it has been used first to describe the

_dynamic

behaviour near the À -transition in

4He

[12]

and to _many other situations in the

_following

(compare

the discussion in Ref.

_{[13]). Recently}

this

concept

has been used to

_study

the static behaviour close to the

_{isotropic-nematic}

transition in

_{liquid crystalline}

elastomers

_[14].

In this paper we _propose to use the

_{macroscopic dynamic approach}

to

_polymers.

Apparently

this has not been done before. From the observation that

_polymer

melts and solutions behave like a solid or

_{liquid-like depending}

on the

_frequency

of the

perturbation,

we

conclude

_immediately

that the

_displacement

_field,

which is a

_{truly hydrodynamic}

variable in

crystals,

is the best candidate to serve as a

_{macroscopic dynamic}

variable in

polymer

dynamics.

Since

_polymers

behave like a

_liquid

for

_frequencies

smaller than the Maxwell

frequency,

the

_{dynamic equation}

for the

_displacement

field will

_acquire

a source term, which describes the fact that the

_displacement

acts like a

hydrodynamic

variable

only

for

frequencies

large compared

to the Maxwell

_frequency.

As the

_temperature

for the

_glass

transition is

approached,

the Maxwell time goes to

_infinity

and renders the

_displacement

field a

_truly

hydrodynamic

variable as the

_{glassy phase}

is entered.

In the

_following

it will become clear that the

_{incorporation}

of this

_macroscopic

_variable,

the

displacement

field,

will lead to a number of novel

_{cross-coupling}

effects not considered before in the

_dynamics

of

_{polymeric liquids.}

In

_addition,

the existence of the

_displacement

field as a

macroscopic

variable,

leads to additional

_support

for the

_concept

of a transient

network,

which has been

_put

forward in various versions as a

_microscopic

and as

_mesoscopic

_concept

in many

publications (compare

Refs.

[2]

and

[4]

for a

review).

In this paper we

point

out for the first time

th4t

one could also use it as a

_macroscopic

_concept

when viewed as outlined above.

The paper is

organized

as follows. In section 2 we

_give

the

macroscopic dynamic equations

and in section 3 we

_analyze

in detail how our

_approach

is connected to the classical Maxwell model. In section 4 we

_suggest

four

_experiments

to test our

predictions

and in section 5 we

briefly

summarize the main results and conclude with a

_perspective.

In an

_Appendix

we list

the correlation functions of the

_fluctuating

_parts

of the

_dissipative

currents and discuss the

structure

_they

assume in the classical Maxwell model.

2. The linearized

_macroscopic

_equations.

equations throughout

the

_present

paper. In the

truly hydrodynamic regime,

that is for all

frequencies

cv, which are small

_compared

to all

_{microscopic frequencies}

_{1/Tc :}

_{wr, «..c 1}

and for all wavevectors k small

_compared

to all

wavevectors

associated with

_{microscopic length}

scales

_{fc: ki,, «}

1,

one has as

_hydrodynamic

variables the densities of all conserved

_quantities.

These are for a

_simple

fluid : the

_density

p associated with the law of mass

_{conservation,}

the

density

of linear momentum _g

_{(conservation}

of linear

momentum)

and the energy

density

e

(energy conservation).

In addition one has in mixtures without chemical reactions the concentration c as a

_hydrodynamic

_variable,

since the concentration of one constituent in a

mixture cannot be

_destroyed,

but

_only

be redistributed in space.

In the introduction we have

_already

outlined

why

we want to

_keep

one additional

quantity

in the list

_{of macroscopic}

_variables,

_namely

the

_displacement

field

_{u (r, t ).}

This is not a

_truly

hydrodynamic

variable in a

_polymer

melt as the other variables introduced

_above,

since it relaxes in a

_large,

but finite time. Nevertheless we

_keep

the

_displacement

field in our list of

macroscopic

variables to account for the

_empirical

_fact,

that

_polymer

melts and solutions behave for small

_frequencies

as a

_{simple liquid,}

but for

_{sufficiently large frequencies}

like a

solid. On the other hand in a solid the

_displacement

field is a

_{truly hydrodynamic}

_variable,

since it does not relax in the

_long

time limit. We also

_keep

in our list of

_macroscopic

variables

the

_density

of the

_{macroscopic polarization}

P,

which will

_mainly

be

_{of practical importance}

for

highly polarizable polymer

melts. Our

_goal

is now to derive for all these

_macroscopic

variables balance

_equations

valid for

_{long wavelengths}

and for

_frequencies

smaller than all

_microscopic

frequencies.

The Maxwell

_frequency

is considered to be a

_{macroscopic frequency}

in this

context.

The

_approach

used to derive such balance

_equations

is linear irreversible

_{thermodynamics}

[15].

In the

_hydrodynamic

_regime

a

_given

_system

is in

thermodynamic

equilibrium locally

and

the

_validity

of the Gibbs relation

is

_guaranteed.

This relation can be viewed as a local formulation of the first law of

thermodynamics.

In contrast to the case of

_{global (that}

is

_everywhere

in

_space)

thermodyna-mic

_equilibrium,

the Gibbs relation

_{(Eq. (2.1 ))}

is formulated

_locally,

that is for the

_vicinity

of

a

fixed,

but otherwise

_{arbitrary point}

_{in space. This} means that - in contrast to

global

equilibrium

- all the

quantities

in

_{equation (2.1)}

are

fields,

which vary

_spatially

and

temporally

in the

_{hydrodynamic regime.}

In

_{deriving equation (2.1 )}

one has

_{averaged already}

over many atoms, or - to

phrase

it

_differently

- each fluid

particle

contains

_enough

molecules so that a continuum

_{approximation}

is

_{justified. Equation (2.1) gives}

a relation

between the increments in the

_{thermodynamic}

variables. It relates the variation of the

entropy

density a (r, t )

to the

_changes

of all the other

_{thermodynamic quantities}

_{p, g,} E, P and

eij,

where eij

is the strain tensor, that is the

_{symmetrized gradient}

of the

_displacement

field :

eij = 1/2

(VUj

+

_{O .u . ) . It is}

the

_gradient

of the

_displacement

field and not the

_displacement

field

itself,

which enters the Gibbs

_relation,

since

_{homogeneous displacements}

do not

_change

the energy of the

system.

Invariance under rotations is

_{guaranteed by}

the

_symmetric

form of

EU

(curl

u =

0).

Nevertheless one must make sure when

_writing

down

_{macroscopic equations}

in the

_following

that at all

_steps

one

_only

deals with three

_{independent macroscopic quantities,}

since the

_displacement

field

u (r, t )

has

_only

three

_components

_{(compatibility}

_condition,

cf.

equation (2.12) below).

_

other ones are

_kept

constant. For

_example

the chemical

_{potential »}

is thus obtained via

g =

5 £ /5 pl...,

, where the

ellipses

indicate that all other

thermodynamic

variables are

kept

fixed while the energy is varied with

respect

to the

_{density. Similarly}

one obtains the local

velocity v (r, t ) by varying

the _energy with

_respect

to the

_density

of linear momentum

g(r,

t )

or the electric field

_{E(r, t )}

_{by varying -}

with

_respect

to the

_{polarization density}

P(r,

t ) .

The

_{thermodynamic}

forces in

_{equation (2.1)}

we have not

_yet

alluded to are the

temperature

T(r, t ),

the relative chemical

_potential

of the mixture

_{Z(r, t ),}

and the elastic

stress l/Jij(r, t),

the

_quantity

which is the

_{thermodynamic conjugate}

to the strain field

£ij(r, t).

Since

_{eij (r, t )}

is

_symmetric,

_Oij

_{(r, t )}

can be chosen to be

symmetric

without loss of

generality.

The static behaviour of the

system

is now obtained

_{by connecting}

the

_{thermodynamic}

forces

to the

_{macroscopic thermodynamic}

variables. As we have seen above these forces can be obtained _{from the energy,}

_{by taking}

variational derivatives with

respect

to the variables. For linearized

_{thermodynamics}

one has two

_{equivalent possibilities}

to formulate the thermostatic

theory : i)

either one can

_expand

the

_{thermodynamic}

forces

_linearly

into the variables or

_ii)

one

_expands

_{the energy up} to second order in the variables. The derivation of the

thermodynamic

forces from the energy

guarantees

then

_{automatically}

that

they

are linear in

the variables. In

_pursuing

either

_approach,

_however,

the

_expansion

must _{preserve all}

fundamental invariance

_properties

of the

_system

under

_{investigation including}

for

_example

translational and rotational

_invariance,

time reversal

_symmetry

and

_symmetry

under

_spatial

parity.

Je

In the

present

case of a

_polymer

melt we obtain for the energy

where

_Eo

contains all the terms

_already

_present

in a

_{simple liquid}

and where the summation convention over

_repeated

indices is

_implied

in

_{equation (2.2)}

and all the

_{following equations.}

The tensors

_Cijkl

and

_(ijkt

take for an

_isotropic

medium

_generally

the form

_aijkt

=

al

_6ij&kl

+ a2

81k8 v

+ ’b it’bjk -2sii&kl),

where

cl, CI

and

c2,

C2

are connected with the trace and the

i

deviator of the

_{corresponding}

second rank tensors

_(Eij

and

_Zip/),

_{respectively.}

Cross-couplings

between the trace of the strain tensor and the

_density,

_entropy

_density

and concentration variations are

_{expressed by}

the static

_{susceptibilities}

_{X p, X u,}

and

X ;

respectively.

Note that in

_{equation (2.2)}

we have to use

_explicitly

the deviations of the variables from their

_equilibrium

values

_(e.g.

_5p

_{(r, t )}

=

p

(r, t ) -

p

eq),

while under

gradients

or time derivatives this distinction _{is unnecessary}

_(since

_{(a/at )}

_Pcq

= 0

= ViPeq

etc.).

The

tensor

_Xij

_{( = y P & j}

for an

_{isotropic medium)}

is the electric

_{susceptibility}

tensor.

_By

inspection

of

_{equation (2.2)}

we see that there is no static

cross-coupling

between strains and

spatially homogeneous

electric

fields,

but

_only

the

_higher

order

_{cross-coupling}

to field

gradients.

As outlined above the forces are then evaluated

by taking

the variational derivative of the

For the conservation laws for

_{density, density}

of linear momentum and concentration we have

where

_{U ij}

is the stress tensor

_{and j f}

the concentration current. As we can read off from

equations (2.4)

and

_{(2.1 )}

the

_density

of linear momentum serves as

both,

as a variable and as

the current of the

_density.

The balance

_equations

for the non-conserved fields read

In

_{equations (2.5) j t}

is the

_{entropy current, j r}

the

_polarization

current,

_Xij

the

quasi-current

associated with the strain tensor and the source term

_{R/ T}

in the

_{dynamic equation}

for the

entropy

density

is the

_entropy

_production.

The second law of

_{thermodynamics}

reads R > 0 for

_dissipative

_{and reversible processes,}

_{respectively.}

To

guarantee

rotational invariance of the

_{dynamic equation}

for the strain

field,

its

_{quasi-current}

_XiJ

must be

_symmetric

Xij = Xji.

From

_{equations (2.4)}

and

_(2.5)

_{the conservation law for the energy}

_density

follows

by making

use of the Gibbs relation

(2.1).

Having

written down

_already

the

_dynamic

equations

for all the

other

dynamic

variables,

the form of the energy current is thus fixed

completely.

To close the set of

_{macroscopic equations}

it is now necessary to connect the

currents and

_{quasi-currents}

_{g 1,}

_oii,

_{j t, j f, j rand}

_Xij

to the

_{thermodynamic conjugate}

quantities

T,

g,

Z,

v, E and

_tPij

taking

into account all

_symmetry

_properties

of the

_system.

This

_step

contains as a

_{simple special}

case for

_example

Fourier’s

law,

which connects the heat

current jf’

to the

_{appropriate thermodynamic}

_force,

the

temperature

T via

for an

_isotropic

_system

with thermal

_diffusivity

K. In the

_present

_paper we concentrate on the

linear connection between forces and

_fluxes,

since we are

_focusing

on the linearized

macroscopic dynamics.

All the currents written down in

_{equations (2.4)}

and

_(2.5)

can be

_split

into

_dissipative

and

into réversible

_parts,

_depending

on whether

_{they give}

rise to a finite amount of

_dissipation

_(or

positive

entropy

production)

- such as for

_example

the heat current

_(2.6)

- or to a

vanishing

entropy

production

(R

=

0).

Using

general

symmetry

and invariance _arguments we obtain for the

_linear,

reversible

_parts

where p

is the

_hydrostatic

pressure and where

_A¡j

is the

symmetric

velocity

gradient

Ai . _ 1

(VVj

+

_Vjv).

The reversible current for the strain field reflects the fact that for 2

homogeneous

translations

_{(solid body translations)}

ù = v. The

_{term 4Jj}

in the stress tensor is the elastic contribution from the transient network. The

_quantities

_{p o, co} and _ao are the

equilibrium

values of

density,

concentration and

_entropy

_density

around which the linearized

equations

are derived. All the reversible currents

_given

in

_{equation (2.7)}

lead to

_vanishing

entropy

production :

R = 0.

For the derivation of the

_dissipative

_parts

of the currents in

_{equation (2.6)}

one has two

equivalent possibilities : i)

either one

_expands

the currents _up to linear order in the

_forces,

or

ii)

one

_expands

the

_dissipation

function to second order in the

_{thermodynamic conjugates}

and then obtains the

_dissipative

currents

_{by taking}

the variational derivatives of the

_dissipation

function with

_respect

to the forces.

We find for the

_dissipation

function

where

_Ro

is the

_part

of the

_dissipation

function

_already

_present

in

_simple

fluids. All the terms

in

_{equation (2.8)}

have been chosen in such a _way that

they

_guarantee

the

positivity

of the

entropy

production

in accordance with the second law of

_{thermodynamics,}

if the

phenomeno-logical

transport parameters

satisfy

a number of

positivity

conditions

including

T:>

0,

y >

0, uE :>

0,

’YuE:>

(C E)2.

In addition all contributions to R transform like a

_scalar,

that is

they

must be invariant under

translations, rotations,

under time reversal and under

_operations

including spatial parity.

The

_quantity

T is the relaxation and y the selfdiffusion constant of the strain field and

0» E

is the electric

_{conductivity. Inspecting equation (2.8)}

we see that there are several

dissipative cross-couplings

between the strains in the transient network on one hand and

between the

_gradients

of

_temperature

and concentration and of an electric field on the other.

It seems worthwhile to notice that none of these terms has been considered in the classical version of the Maxwell model. Some

_{experimentally}

testable consequences of these

cross-coupling

terms will be

_highlighted

in section 4.

As

_already

mentioned above we

_get

the

_dissipative

_parts

of the currents

_{by taking}

the variational derivative of the

_dissipation

function with

_respect

to the

_appropriate

thermodyna-mic force. We find for

_example

for the

_dissipative

current associated with the

_dynamic

equation

for the strain tensor

From

_{equation (2.9)}

we read off

immediately

that it can be

_split

into a source term and into a

part

which can be

_expressed

as the

gradient

of a vector :

where we have written

Xij

in an

_{explicitly symmetric}

form and where we have made use of the

relation curl E = 0 of electrostatics. The

requirement

that the

dynamic equation

for the strain tensor transforms the same _way as the strain tensor itself

_implies

a restriction. From classical

elasticity theory

one knows that the strain tensor must

_satisfy

a

_{compatibility}

condition to

guarantee

that a continuous

_displacement

field u

(r, t )

exists. This condition reads

and

_applies,

because of

_{equation (2.5),}

also to

_Xij

_(Eijk

_eimn

_VjVm

_{X kn}

=

0).

Obviously,

the

reversible

_part

_xC

and

_Xij

fulfill this condition

_{automatically,}

but not the source term

tP ij / (2 T)

in

_{equation (2.10).}

The latter

_{obeys e ijk e imn Vj V m}

_ek,,

= 0

_only

if the elastic moduli

_{(cf. Eq. (2.2)) satisfy}

the condition

that is bulk and shear modulus of a

_polymer

melt above the Maxwell

_frequency

are not

independent,

but

_satisfy

within a linearized

_{macroscopic dynamic .theory equation}

_(2.13).

Condition

_(2.13)

_guarantees

that the relaxation times for

_longitudinal

strains and for

transverse

_strains,

3

_{cl/ (2}

_{T )}

and

_c2/T,

_{respectively,}

are

_equal.

For a true elastic

_solid,

there is no

_counterpart

to condition

_{(2.13) (there}

is no source term in

_X1D,

if there is no

_relaxation)

and the two elastic moduli are

_independent.

On the other hand we have

_kept

the

compressibility (i.e.

the

_{density dependence}

of _{the pressure in}

_Eq.

_(2.7)),

so that in our

description

of

_polymer

_{melts the response} to bulk and shear stresses is still

_{given by}

two

independent quantities.

3.

_Comparison

of the

_présent

_approach

with the Maxwell model in its classical formulation. In its classical formulation the Maxwell model takes the

_following

form

_using

a

_dynamic

equation

for the

_off-diagonal

_part

of the stress _tensor,

_Ufj

with Maxwell time

_to

and shear

_viscosity

v. From the

_point

of view of the derivation of

macroscopic equations

it is not clear

_why

the

_equation

for the stress tensor should be

_dynamic.

From

_{equation (3.1 )}

and the momentum conservation

_equation

it is also not clear how to

_get

couplings

to other

_macroscopic

variables

_including

_entropy

_density,

concentration in mixtures and

_{macroscopic polarization.}

To elucidate the similarities and differences between the two

_approaches

in detail we take a

Fourier transform of

_{equation (3.1 )}

which

_yields

Next we combine this

_equation

with the linearized momentum conservation

_equation

It is this

_{equation (3.4),}

which can be

compared

most

_easily

with the

_approach

_put

forward in the

_present

paper. To see this we start from the

_suitably

reduced

_{equation (2.5) (using}

Eqs.

(2.7)

and

_(2.9))

where we have discarded as discussed above all

_couplings

to the other

_macroscopic

variables and assumed

_{incompressibility}

in order to be able to _{compare with the classical Maxwell}

model. From the

_{(generalized)}

Navier-Stokes

_{equation (cf. (2.4)}

and

_(2.7))

with the

_help

of

_{equation (2.2)}

and after

_taking

Fourier

_transforms,

_{equations (3.5)}

and

_(3.6)

combine to

Equations (3.4)

and

(3.7)

are _very convenient for the

_comparison

of the

_predictions

of the

Maxwell model and of the

_{macroscopic dynamics approach. Looking}

at the coefficient of

vi k2 we

see that

JI (1 + i lù t 0) -

1 is

_{replaced by}

_{71 2 + T (1 + i lù T / C2 + y k 2 T ) -}

_using

the

present

macroscopic dynamics approach.

To carry out the

_{comparison explicitly}

it turns out to be very efficient to

_investigate

various

limiting

cases. For the low

_{frequency regime (w - 0)}

we obtain v for the classical Maxwell

model

_{and ’02}

+ T

_{( 1}

+

yk 2 T )-1

in

the

_present

framework. We read off

_immediately

that in

the

_{long wavelength}

limit the

_{appropriate expression is 712}

+ T and v in the classical model.

Neglecting

the

_dispersion

_{( yk2 T ) is}

_justified

as

_long

as it is small

_compared

to 1. Maximal

absorption

is obtained for

_{both, co}

_to

and Wr

_IC2

of order one for the two

_{approaches compared}

here,

which is in _agreement with one’s intuition. Both

_approaches

also agree in the limit of

vanishing

Maxwell

_time,

in which

_{they give}

rise to the behaviour of a

_{simple liquid.}

For the classical Maxwell model the

_{dynamic equation}

for the stress tensor reduces to the constitutive law of a

_{simple incompressible liquid}

and in the case of the

_macroscopic

dynamic approach

the

strain tensor no

_longer

acts as a

_{macroscopic (slow)}

variable in the Tu 0 limit.

By

construction the

_{macroscopic dynamics approach gives}

the correct limit for the case of

an infinite Maxwell time. In this case the

_{hydrodynamic equations}

for a solid result as it must

be the case. For the classical Maxwell model it is

_{quite complicated}

to obtain this

limit,

since the classical model

_{depends only}

_{on lù to}

thus

_making

it

_impossible

to

_investigate

the limit of a

solid,

since the

_macroscopic

behaviour can

only

be

expected

to be obtained

_correctly

for

sufficiently

small

_frequencies.

Otherwise

_microscopic

modes are

being

excited. In the

macroscopic dynamics approach.

and Tare

_{independent quantities}

and it is therefore

staightforward

to take the T - oo limit.

From the

_point

of view of

_{macroscopic dynamics}

there is no

_{justification}

for models which

incorporate

a whole sequence of

microscopic

times. In the

_{present gateway}

of attack

only

the

Maxwell time is

_{incorporated.}

All shorter time scales are considered to be

_{truly microscopic.}

However,

it is conceivable that there exist additional

_long

time scales

_originating

from

microscopic

or

_mesoscopic

_{processes. In that} case, additional

macroscopic

variables must be

kept.

We note that the

_present

_approach

loses its

_{appealing conceptual simplicity}

if there are

4. Possible

experimental

tests of the

_{macroscopic dynamics approach.}

Naturally

it seems _very

_important

to test the

concept put

forward above

_{experimentally.}

In

this section we outline four

_simple,

but nontrivial

_{possibilities.}

As we have seen in the last

_section,

the additional

_degrees

of freedom can be excited at

sufficiently high frequencies by applying

for

example

extensional flow or

_simple

_{shear. This} occurs

_{predominantly through}

the direct reversible

_coupling

between

temporal

strain variations and

_{velocity gradients. Having generated}

these strains in the transient network

_they

are

_dissipated

on the time scale of the Maxwell time. This

_dissipation

leads via the

corresponding

term in the

_entropy

_production

to a

_heating

effect on the

_sample

to

_quadratic

order in the

_amplitude

of the

_applied

field

Thus one can detect

_by

a dc measurement

_(observation

of the mean

_sample

_temperature

as a

function of

_time)

_{the consequences of the presence of the transient network. For}

_sufficiently

small

_shearing

rates with

_frequencies

smaller than the Maxwell

_{frequency only}

the

_ordinary

viscous

_heating

_present

in all

_simple

fluids arises whereas for

_{higher frequencies}

additional

dissipation

channels are

_{being opened}

due to _{the presence of the transient network. To make}

sure that the additional

_heating

observed is due to the mechanism

_presented

here and not due

to

_higher

order nonlinearities one can

_plot

for fixed

frequency

the

_temperature

increase as a

function of

_{shearing amplitude.}

It is then the

_quadratic

_part

of the small

_amplitude

response which is of interest here. If one

plots

the

_temperature

increase as a function of

_frequency,

one

should observe a

change

in the value of the

slope

at a

_{frequency comparable}

to the Maxwell

frequency.

There are indications from

_experiments

done in

_{engineering [16]}

that the effect described here

_might

be at least

_{partially responsible}

for the drastic

_heating

effects observed in

_polymer

melts under

_high

shear. Since the

_{reported heating}

effects are rather dramatic

_(up

to 1 000

_K/s)

it can be

_expected

that chemical reactions and even

_{decomposition}

take

_place

at

those elevated

_temperatures

thus

_{changing significantly}

the

_{physical properties}

of the melt. This would

_{explain why}

a second

_experiment

carried out on the same

_sample

leads to

_strong

hysteresis

effects and to a

_completely

different

_physical

_response.

Looking

at the

_approach

taken here which

_{closely parallels}

that used to derive

hydrodyna-mic

_equations

for a

_solid,

another

_possibility

to test the

_concept

of a transient network comes

into mind

_immediately.

To do this one

_investigates

the behaviour of a

_{polymeric sample}

in a

stationary

sound resonator.

_Depending

on whether the

_frequency

used for excitation is above or below the Maxwell

frequency,

a transient network and thus a finite shear modulus is

probed

or not

_{[1, 17].}

To test whether this

_picture

makes sense one can now excite transverse

acoustic

_phonons.

We

_predict

that this is

_{possible only}

for

_frequencies

above the Maxwell

frequency.

The detection of such

_propagating

transverse acoustic

_phonons,

which do not exist in a

_simple

_fluid,

would

_yield

the

_perhaps

most

_convincing

evidence for the

_validity

of the

picture

of a transient network. For

_frequencies

below the Maxwell

_frequency

such a

propagative

transverse mode could not be observed and one should rather see the

purely

dissipative vorticity diffusion,

which is well known from

_{simple liquids.}

To the best of our

knowledge

an

_experiment

to excite transverse sound waves has not been

done,

but it should be

_{fairly straightforward}

to fill this _gap. We should also like to

_point

out that

_only

the

_picture

given

here

_prompts

one to

_perform

such an

_experiment,

since it is not at all obvious how to

interpret

the outcome of such a measurement

_using

the classical Maxwell model.

spatial

and

_temporal

Fourier transform of the

_{resulting equation}

for the

_strain,

we obtain the

following equation

where the

_ellipses

indicate the terms irrelevant for the

_argument

_given

here. Thus we read off

from

_{equation (4.2)}

that a

spatially inhomogeneous

ac electric field leads to a strain in the

transient network which in turn leads to

_heating

effects. We stress that a

spatially

homogeneous

ac electric field is not

_sufficient,

since it does not

_couple

to strains to

_quadratic

order in the

_macroscopic

variables. Nevertheless a

_set-up

with a

_{spatially homogeneous}

electric field will be useful to evaluate the effects due to Ohmic

_heating

which are

_always

present.

A

_{highly polarizable polymer}

is

_mainly

needed to _carry out the

_{experiment suggested}

here because otherwise the effect

_{predicted might}

be masked

_{completely by}

the trivial effect of Ohmic

_heating.

In a sense this

_{experiment using}

ac

_{spatially inhomogeneous}

electric fields

is somewhat

_{complementary}

to the case of mechanical shear in that one now has a different

extemal

driving

field,

namely

an electric field

_gradient

instead of a

_{velocity gradient.}

Below

the Maxwell

_frequency

one should

_again

observe

_only

Ohmic

_heating

even in a

spatially

inhomogeneous

electric field.

For

_polymer

melts with a _very

_large

Maxwell

_time,

that is for

_systems

which behave _very

much like a solid

_already,

an even more direct test for the

_concept

of a transient network seems feasible

_{experimentally.}

In this case one could

investigate

the linear

coupling

between

velocity gradients,

strains and

_temperature

variations

_directly

for

_sufficiently

small

_frequencies

above the Maxwell

_frequency.

One would have to measure the

_temperature

variations in a

time-resolved

_experiment

and one should observe ac

_temperature

variations which are

proportional

to the

_amplitude

of the

_applied

shear for small shear rates, an effect which

cannot

_possibly

exist in a

_{simple liquid}

or in a

_polymer

below the Maxwell

_frequency.

Also for this

_experiment

we are not aware of _any

_{published experimental}

results nor of _any motivation

to _carry out such an

_experiment

based on the formulation of the classical Maxwell model. This is even more the case for the

_coupling

to the electric field

_gradients.

We close this section

_by

pointing

out that a

_magnetic

field cannot induce an effect

_analogous

to the electric field due to

the difference of the behaviour under time reversal of E and B.

5. Conclusions and

_perspective.

In _{this paper} we have

_analyzed

within the framework of

_{macroscopic dynamics}

the

_long

wavelength,

low

_frequency

behaviour of

_polymer

melts and

_polymer

solutions in the

limit

of small

_amplitude

motion. We have discussed in detail the relation of the classical Maxwell model to our

_approach

and we have

suggested

four

_experiments,

which could test the

_validity

and

_clarify

the limitations of our

_approach

and

_yield

clear-cut evidence for the

_picture

of a

transient network. It turns out that there is no

_physical

motivation to write down a

dynamic

equation

for the stress tensor as it is even

_being

done in the

_physics

textbook literature

_[18].

It also emerges that the use of the

_displacement

field as the

_macroscopic

variable reveals _very

easily

cross-couplings

to the other

_macroscopic

variables such as

_temperature

and

_macroscopic

polarisation,

which have not been

_previously

considered. From the

_{dynamic equation}

for the strain we can read off in our

approach

immediately

that the melt behaves like a solid for

_large

frequencies

and like a

_liquid

for

frequencies

small

_compared

to the Maxwell

_frequency.

A

resort to various modifications of

electric

circuit models is therefore _unnecessary

_using

the

present gateway

of attack

_leaving

alone that those circuit

type

models cannot

_incorporate

thermomechanical and electromechanical effects which have been shown to _{be very}

_important

The

_{biggest challenge}

at this time seems to be to

_get

reliable

_quantitative

estimates for the novel

_{cross-couplings.}

We note,

however,

that even for classical elastomers no

_quantitative

calculations for these coefficients _appear to be available

_[19].

We

_feel,

however,

that this task

might

be easier in some

_respect

for the transient

_network,

since it is

_{spatially homogeneous}

and thus not

_suffering

from

_problems

such as

_{inhomogeneous cross-linking}

densities,

etc.

Having

laid the foundation of a

_{macroscopic dynamic description}

of

_polymer

melts and

solutions in this paper, we will

_present

its

_{generalizations}

in the near future.

Acknowledgements.

H.R.B. and H.P. thank the Deutsche

_{Forschungsgemeinschaft}

for

_support.

The work done at

the Center for Nonlinear

Studies,

Los Alamos National

_Laboratory

has been

_performed

under the

_auspices

of the United States

_Department

of

_Energy.

Appendix.

In this

_appendix

we derive the formulas for the correlation functions within

fluctuating

hydrodynamics [20].

From statistical mechanics it is well known that

_dissipation

is

_necessarily

accompanied by

fluctuations.

_Having

_vanishing

mean values these fluctuations can be

neglected

in a deterministic _treatment, but for various

applications

of linear

hydrodynamic

equations

it is fruitful to allow for these

_fluctuating

_parts

in the

_dissipative

currents. In

equilibrium (or

near

_it)

the correlation functions of the

fluctuating

_parts

of the currents are

related to the

_dissipative

_{transport parameters}

_by

the

_{fluctuation-dissipation-theorem.}

Denoting by

Ji’,

_Xij,

./f,

_eij,

and Jp

the

_fluctuating

parts

of the

_{currents it,}

_0-jj,

if,

Xij,

and jf

respectively,

and

_assuming

that

_they

are described

_by

Gaussian,

white noise

stochastic processes, standard textbook

_{procédures [20]}

lead to the correlation functions

and

where

_kB

is Boltzmann’s constant,

_Dijkl

=

& ik’6jf

+

_{’6 il Sik}

and where we have

_neglected

the

diffusion-like

y-contribution

against

the relaxation

_{( ~}

_{1 / T )}

of the transient network.

Equation (A.2) applies

for the

_{pairs (a, /3 ) =}

_(0-,

_{T), (c, c),}

and

_{(P, E )}

_only.

All other correlation functions are as in

_{simple liquids,}

e.g.

where

_7Jijkt

is the

_{(isotropic) viscosity}

tensor

_having

the same tensorial structure as

aijkl,

defined after

equation (2.2).

From

equations (A.1)-(A.3)

it is

straightforward

to derive the correlation functions of all

_hydrodynamic

variables desired in a

_given

situation.

In order to _compare with the classical Maxwell

_model,

we will eliminate the elastic

part

in the stress tensor with the

_help

of

_{equation (2.7).}

After

_spatial

and

_temporal

Fourier transformation we are left with an effective stress tensor, whose

_fluctuating

_part,

where

c-1

is

the inverse of the

_elasticity

tensor

_{(cf. (2.2)).}

For the correlation function of this effective

_fluctuating

stress tensor we find

with W =

WT /C2.

Equation (A. 5)

is the

appropriate

formula to be

compared

with the classical

Maxwell model.

References

[1]

FERRY J. D., Viscoelastic

Properties

of

Polymers (Wiley,

N. Y., 2nd

Edition)

1970.

[2]

GRAESSLEY W. _W., Adv.

_Polym.

Sci. 16

(1974)

1.

[3]

DE GENNES P. G.,

Scaling Concepts

in

_{Polymers (Cornell University}

Press : _Ithaca, N.

_Y.)

1979.

[4]

DOI M., EDWARDS S. F., The

_Theory

of

_Polymer

_Dynamics

_(Clarendon

Press :

_Oxford)

1986.

[5]

MAXWELL J. C., Phil. Trans. R. Soc. A 157

₍₁₈₆₇₎

49.

[6]

VEST C. M., ARPACCI V. _S., J. Fluid Mech. 36

₍₁₉₆₉₎

613.

[7]

BRAND H. R., ZIELINSKA B. J. A.,

Phys.

Rev. Lett. 57

₍₁₉₈₆₎

3167.

[8]

ZIELINSKA B. J. A., MUKAMEL D., STEINBERG V.,

Phys.

Rev. A 33

₍₁₉₈₆₎

1454.

[9]

PLEINER _H., HARDEN J. L., PINCUS P. A.,

Europhys.

Lett. 7

₍₁₉₈₈₎

383 and HARDEN J. L.,

PLEINER _H., PINCUS P. _{A., Langmuir} 5

₍₁₉₈₉₎

1436.

[10]

MARTIN P. C., PARODI O., PERSHAN P. S.,

Phys.

Rev. A 6

₍₁₉₇₂₎

2401.

[11]

FORSTER _D.,

_{Hydrodynamics,}

Broken

_Symmetry

and Correlation Functions

_{(Benjamin, Reading,}

Mass)

1975.

[12]

KHALATNIKOV I. M., Introduction to the

_Theory

of

_{Superfluidity}

_{(Benjamin, Reading, Mass)}

1965.

[13]

PLEINER H., Incommensurate

_{Crystals, Liquid Crystals}

and

_{Quasicrystals,}

J. F. Scott and N. A. Clark Eds.

(Plenum,

New

_York)

1987.

[14]

BRAND H. _R., Makromol. Chem.

_Rap.

Comm. 10

₍₁₉₈₉₎

67 and 441.

[15]

DE GROOT S. R., MAZUR P.,

Nonequilibrium Thermodynamics (North-Holland: Amsterdam)

1962.

[16]

GLEISSLE W., Ph. D. Thesis, Universität _Karlsruhe, West

_{Germany (1978).}

[17]

DE GENNES P. G., PINCUS P. A., J. Chimie

Phys.

74

₍₁₉₇₇₎

616.

[18]

LANDAU L. _D., LIFSHITZ E. M.,

Elasticity Theory (Pergamon

Press, New

_York)

1959.

[19]

TRELOAR C. R. G.,

Rep. Progr. Phys.

36

₍₁₉₇₃₎

755.