Dynamical properties of dilute flexible polymer solutions

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Dynamical properties of dilute flexible polymer solutions

M. Daoud, G. Jannink

To cite this version:

M. Daoud, G. Jannink. Dynamical properties of dilute flexible polymer solutions. Journal de Physique,

1978, 39 (3), pp.331-340. �10.1051/jphys:01978003903033100�. �jpa-00208766�

331

DYNAMICAL PROPERTIES OF DILUTE FLEXIBLE POLYMER SOLUTIONS

M. DAOUD and G. JANNINK

DPh-G/PSRM,

^CEN

Saclay,

^{BP N°}

2,

⁹¹¹⁹⁰

Gif-sur-Yvette,

^France

(Reçu

^{le 29}

septembre 1977,

^{révisé le 21 novembre} 1977

accepté

^{le 28 novembre}

1977)

Résumé. ²⁰¹⁴ Nous examinons les propriétés

dynamiques

^des polymères ^en solution diluée à l’aide des lois d’ échelle. Nous

généralisons

l’analyse faite dans le cas statique ^en introduisant un exposant

dynamique,

^{z. Cet} exposant prend deux valeurs différentes, ^bien que proches ^{l’une de} l’autre, respec- tivement pour le bon solvant ^et le solvant theta. Nous supposons que ^{z est} égal à la valeur de Zimm

(z ⁼ 3) ^{en solvant} theta, ^{et a une valeur z =} 2,9 ^en bon solvant. Cette dernière valeur est tirée de résultats expérimentaux récents. Nous calculons les dépendances ^en température, ^masse moléculaire et transfert de moment des différentes observables telles que le coefficient de diffusion, ^la viscosité,

les temps

caractéristiques.

^{Comme dans} l’ expérience, ^{nos résultats} indiquent ^un changement ^de comportement

significatif

^{de certaines} observables en passant du solvant theta au bon solvant.

En particulier, ^{on trouve} que le coefficient de diffusion n’ obéit pas à la loi d’Einstein ^en bon solvant.

L’analyse ^est étendue à certaines

propriétés dépendantes

^du temps, telle que la viscosité

dynamique

et la partie réelle du module complexe d’élasticité. On montre que ^ces

propriétés

^{ont un} changement

de comportement

caractéristique.

Des coordonnées universelles sont

proposées

^{pour une for-}

mulation plus générale ^{de leur} comportement. Enfin, ^nous faisons la critique de la méthode des variables réduites telle qu’elle ^est appliquée ^{aux solutions} diluées. Nous définissons un nouvel ensemble de variables, valables à haute et basse fréquence, ^alors que la méthode usuelle ne peut exprimer ^que

les phénomènes ^basse

fréquence.

Abstract. ²⁰¹⁴ We analyse ^the dynamical

properties

^{of dilute} Polymer ^solutions by

using

scaling arguments. ^{To this} end, ^we generalize ^the scaling analysis proposed for the static

properties

^{of these}

solutions by introducing ^a dynamical exponent ^{z. This} exponent ^{has two}

slightly

^{different values}

for theta and good solvents. In order to compare ^our results with experiments, ^we suppose that ^z has its Zimm value (z ⁼ 3) ^{in theta} solutions, and another

value (z

⁼ 2.9) ^which we

pick

^{from recent}

experimental

results, for solutions in good solvents. We calculate the temperature, ^molecular

weight,

and

scattering

^vector dependences ^{of different} observable

quantities

^{such as the} diffusion constant, the characteristic times and

frequencies,

^the viscosity, ^{etc. Our results are in} good agreement ^with

experiments.

They show how these dependences deviate from the classical patterns of behaviour

as we

change

^{from z to z. In}

particular,

it is found that the diffusion coefficient in a

good

^solvent

does not obey Einstein’s law.

The

analysis

is then extended to some time dependent properties, ^{such as the} dynamic viscosity

and the real part ^{of the}

complex

modulus. It is shown that these

properties

^exhibit cross-overs.

Universal coordinates are proposed ^{for their} study. Finally, ^a criticism of the so called reduced variables coordinates is

given.

^{Another set} of variables is

given,

which is valid for both low and

high frequencies,

whereas the usual method failed for

high

frequencies.

LE JOURNAL DE PHYSIQUE ^TOME 39, ^MARS 1978,

Classification Physics ^Abstracts

05.20 ^- 61.40

1. Introduction. ^- Considerable _progress has been made

recently

^{in the}

understanding

^of

polymer

solutions

[1, 2, 3, 4]

^Small

Angle

^Neutron

Scattering

has

proved

^{to be an} essential tool for the

experimental study

^{of static}

properties [4],

^while

light scattering

has been more suitable for

dynamical properties [5].

In this _paper, we wish to

give

^a

scaling approach

to dilute

polymer solutions, mainly

^{in the}

good

solvent

regime.

^Some

properties

may

(hopefully)

be checked

by

^Neutron

Scattering,

^{but our} purpose is to

give

^{some more}

general properties

^of

frequency

dependent

functions of dilute

polymer

^{solutions in}

good

^solvent.

Different

theoretical

models have been

given [6, 7, 8, 9, 10]

^{for the}

’dynamics

^of

polymer solutions, taking [I1]

^{or not}

taking [12]

^{into account the}

hydro- dynamic

interactions. From an

experimental point

of

view,

^{it seems} that the Zimm model is a

good description

^of

reality

in ’the theta

régime [ 13],

^which

means that

hydrodynamic

interactions ^are dominant in this

regime.

^For solutions in

good solvents,

^there

is no theoretical model which describes

reality

^with

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003903033100

any accuracy. The Zimm model still seems to be

appropriate

^{but as we will see, it}

completely

^fails

to

give

^{the molecular}

weight dependence

^{of the}

diffusion coefficient or

viscosity,

for instance. For this reason

(and

other related

ones)

^{we are led to}

discard it. Our _purpose here is not to

give

^{a new}

approximate theory,

but rather to

pick

^from

experi-

mental results one exponent, ^{which we}

need,

^{and to} derive a

scaling approach

^for

dynamical properties using

^that exponent ^{and what we} know from static

properties.

^{We will see that}

although

^this exponent is _very close to the Zimm

prediction

^the

slight

^diffe-

rence between these two

exponents

is sufficient to lead to

significant

differences in other

dynamical properties,

^{such as} the diffusion

coefficient,

^{or the}

intrinsic

viscosity

^for

example.

So in the first part of the paper, ^we will

briefly

recall the results for static

properties.

^{In section}

3,

^we will extend these results to

dynamical properties by introducing

^a

new exponent z, which we call the

dynamical

expo- nent.

Scaling properties

^can then be discussed both in the theta

regime

and in the

good

^solvent

regime by assuming

^{that the}

dynamical

exponent ^{has res-}

pectively

the Zimm value of 3

(we

restrict ourselves to the usual three-dimensional

solutions)

^{in the}

theta

regime

^{and a value of}

2.9,

^{which we} get ^from

recent

experimental results,

ⁱⁿ

good

^solvents.

Finally,

some other consequences will be discussed in

part

⁴

including frequency dependent properties.

^{We will}

see that for these

properties,

^{whereas a uniform}

behaviour is

predicted

in the theta

regime,

^{one has to}

introduce a cross-over

frequency

^w*

separating

^two

different behaviours for solutions in

good

^{solvent :}

- for

frequencies higher

^{than w*}

(co » a)*) thé

behaviour is the same as in the theta

regime ;

- for lower

frequencies,

^one gets the characte- ristic

good

solvent behaviour.

2. Static

properties.

^{- Dilute flexible}

polymer

solutions have been

extensively

^studied

[14,

15,

6].

The

only

interaction taken into account here is the excluded volume interaction. This interaction is described

by

^a parameter v, called the excluded volume

parameter [15] :

where

V(r)

is the interaction

potential

^{between two}

monomers. This parameter ^is temperature

dependent.

There is a temperature ^{0 at} which it vanishes. So for

temperatures

^{not too far from}

0,

^{we shall assume}

that v is

proportional

^to

(T - 0).

^{Thus we will des-}

cribe the solvent effects

by

^a dimensionless tempe-

rature

Starting

^from

high

temperatures, différent behaviours

are observed for

long

macromolecular solutions when the

teniperature

is decreased.

a)

^For

high enough temperatures,

the excluded volume effect is dominant. The solvent is called

good

solvent. The effect of volume exclusion is ^to swell the

coil,

and there is an

exponent v

for the molecular

weight dependence

of the radius of

gyration R :

where N is the number of statistical units of the chain,

of length 1,

^{and is}

proportional

^to the molecular

weight.

^v is the excluded volume

exponent.

^{It is} very close to the

Flory [14]-Edwards [15]

^value

3/5.

^It

has been shown

by

de Gennes

[1]

that it is in fact the correlation

length

exponent ^{of an n-vector} model

[16]

in the limit when n _goes to zero.

b)

^For temperatures ^around

0,

^the excluded volume

parameter

^is very small and can be treated as a per- turbation

[17].

The behaviour of ^a chain is of mean-

field type :

It has been shown that this is related ^to tricritical behaviour

[18, 19].

^The witdh of this

region depends

on the

length

of the chain and it has been shown that it is of order

N-1/2.

Let us recall that no

sharp

^transi-

tion is

expected

^{at the} passage from

regime (a)

^to

regime (b),

but that there is a cross-over from one

behaviour to the

other,

^{all the}

physical properties being perfectly

^smooth

during

^this cross-over.

c)

^{For lower} temperatures, the coils contract to

a more dense form

[20]

and

finally

^{if we} go ^on

lowering

temperature, ^a

phase separation

^occurs.

In the

following,

^{we will} restrict ourselves to

regimes (a)

^and

(b)

^and

especially

^to

regime (a)

where the situation is _very rich

[21].

Before we go into

dynamical properties,

^{let us}

briefly

recall the static

properties

of dilute solutions.

To this end let us consider for

example

^the

scattering

function

by

^{a chain}

where

Rij

^{is the vector}

connecting

monomers i and

j, and q

^the

scattering

^vector

(q

^{= 4}

n/À

^sin

0/2,

Â

wavelength,

⁰

scattering angle).

For dilute solutions it can be shown that

S(q,

^r,

N)

can be put into the scaled form

[22]

which indicates that the characteristic

length

^{in the}

theta

regime (j T N 1/2 «1)

^is

proportional

^to

^{N 1/2}

as mentioned before.

333

In order to

study

^the

good

^solvent

regime,

^we

have to consider an

equivalent

^{form for}

S(q,

^r,

N).

In the

good

^solvent

regime,

^{we have}

Ni2 >

^1.

Relation

(8) clearly

^{shows thât there are two} types of behaviour

depending

^{on the}

length

scale which is considered.

For small

distances,

^i.e.

large

^{values for} q, the behaviour is

expected

^{to be the same as} in the theta

regime :

The mean square distance between two monomers

separated by n

^monomers, for instance is

For

larger distances,

i.e. for smaller values

of q,

both variables of the R.H.S. of _eq.

(8)

^{are less than}

unity.

One is then led to make a

change

^{in the}

variables

[23] :

The function

S(q’, N’)

^{has to} exhibit the excluded volume exponent v

[1]

which shows the variation of the radius of

gyration

of a chain in a

good

^solvent

quoted

^above.

The main features exhibited

by

^this

approach

^are

the

following :

1)

^{In the theta}

^solvent,

the behaviour of a chain is

uniformly q-2,

^as

long

^as

possible entanglement

effects are

neglected [24].

2)

^When temperature ^is

raised,

^we get ^{into the}

good

^solvent

regime.

^{In this}

régime,

the excluded volume

exponent

^v appears for

large

^scale

properties,

whereas for

properties conceming

^small

distances,

the behaviour is identical to that of the theta

regime.

For

example,

^the

scattering

function has a

q-1/v

behaviour for small values of the

scattering

^vector

and a

q-2

behaviour for

large

^{values of} q. The cross- over between these two characteristic behaviours

occurs for a value of the scattering vector q* - - ’ - r

as can be seen from relations

(8)

^and

(9).

^This

typical two-regime

behaviour has been

recently

^checked

by

^small

angle

^neutron

scattering [25].

^{As a}

simplified visualization,

^{one can}

imagine

^{the chain as a succes-}

sion of blobs. Inside each

blob,

the behaviour is that of the theta

régime.

The chain is a succession of blobs with excluded volume

[26].

It is this

special

property ^of

polymer

^{chains in a}

good

solvent which

we

wish to discuss for

dynamical properties.

^{We are}

going

^{to see}

that,

^{in the same} way, the

dynamical properties

^of

polymer

^{chains are not}

the same when one is interested in local

properties necessitating

parts ^{of a} chain inside a blob or

global properties including

several blobs. This

naturally

appears when

frequency-dependerit properties

^are

studied.

3.

Dynamical properties.

^{- In order to discuss}

the

dynamical properties

of dilute

solutions,

^we wish first to

generalize

^{our static}

scaling equation (7)

to

dynamics.

^{To do so, we are}

going

^{to assume in}

all that follows that in the ^case of _very

long polymer

chains

(i.e.

in the limit N -+

oo)

the characteristic times

(relaxation times),

^{which are much} greater than all the other times involved

(hopping times, ...),

have the same behaviour in N. In other words we assume that if one looks at

scattering

^{vectors much}

greater than the inverse radius of

gyration,

^{one can}

define a characteristic

frequency

WC. We

postulate

where z is a new exponent, called the

dynamical

exponent

[27, 28] (this frequency

^{can be}

imagined

as the half width at half maximum in a

quasi-elastic scattering experiment

^for

instance).

Now,

^{in the same} way that the exponent ^{for the} radius of

gyration (cf.

^relation

(3))

has different values when

temperature

^{is varied}

(namely

^{v =}

1/2

when

rN’/’ «

^{1 and v z}

3/5

^if

TN1/2 » ₁₎

^we

naturally

expect ^{that z will have two} different values in theta and

good

^solvents

respectively [29].

Relation

(10)

^{is a} well known result in theta solvents where it has been shown a

long

^time ago that the Zimm model is valid in such solvents :

1

(theta solvents) .

Recent

light scattering experiments by

^{Adam and}

Delsanti

[30]

indicate that relation

(10)

^is still valid for solutions in a

good

solvent. Their

experimental

value for the

dynamical

exponent ^{z is}

slightly

^different

from the Zimm value

(good solvent) .

As this value is in

agreement

with other

experi-

mental results

[13, 34],

^we shall take z ⁼ 2.9 for the

dynamical exponent

ⁱⁿ

good

^solvents.

Having

^{now all the}

required ingredients

^{for the}

puzzle,

^we may discuss the

frequency dependence

of the

scattering

^{law and}

generalize

^our

scaling

relation

(7) :

^{in the}

vicinity

of the theta

point,

^we

assume that the

scattering

function has the scaled form :

One can check

easily

^that

equation (7)

is recovered

by integrating

both sides of _eq.

(11)

^over

frequency.

Let us

briefly

^{comment on} this last relation. We have seen above that ^we have to introduce two different values for the

dynamical

exponent ^{z in} the case of a theta or a

good

^{solvent. It has been}

shown

[25]

that the actual

polymer

^{chain in a}

good

solvent is a delicate

mixing

of both of

them, including

a theta like behaviour at short distance. Relation

(11)

indicates the same structure for the

dynamical properties,

^{as we will see} below. The

important point

^{to be} mentioned here is that the

scaling

^{of the}

characteristic times

depends strongly

^{on the scale}

under consideration : if n is a contour

length along

the

chain,

the characteristic times do not have the

same structure for _every value of n.

For

large

values of n, ^we have the

good

^solvent

behaviour

(wc’" q2.9).

^.

For values of n which are small

(although

^much

greater

^{than the} step

length 1),

^{we recover the theta-}

like behaviour

(co,,, - q3).

The cross-over value for n

depends

^on temperature.

When

TN 1/2 _ 1, the

theta-like behaviour extends to the whole chain. These seem to be rather subtle considerations about a very weak and

probably non-directly-measurable

difference. But we are

going

to see that

they

^{lead to}

significant

differences in other related

properties.

Let us extract from relation

(11)

the characteristic features of

dynamical properties

^of

polymer

^solutions

in theta solvents.

i)

The characteristic

frequency

^{has the}

usual q3

behaviour as mentioned above.

Equation (11)

^{can be} put ^{into an}

equivalent form,

which we can discuss more

directly :

In a theta

solvent,

^{we can}

neglect

the second variable of the R.H.S.

ii)

^{If one looks at}

scattering

^{vectors of order}

N-1/2

(i.e. qR N 1)

^{we find a} characteristic

frequency

coi -

N- 3/2,

^which

corresponds

^{to the terminal}

time

Ti

in viscoelastic

experiments [13].

iii) Finally,

for small values of the

scattering vector q,

^one is sensitive to the diffusion of the chain.

There results a

broadening

This

relation, together

^with eq.

(12’)

^{shows that}

Do

has to be

So we recover Einstein’s law for the diffusion coeffi- cient in a theta solvent.

From these fundamental

results,

^{one can also} get the intrinsic

viscosity

where il,

^{is the}

viscosity

^{of the} solvent and ^c the

monomer concentration. It has been shown

[31] ]

that

M

^{is related to}

T,

which leads to

One can check that all the

previous

results of the , Zimm

theory

^can be recovered

by

^this

approach.

Let us tum now to the more

interesting

^{case of}

good

solvents. As we have

already

^{seen in} part

1,

tempera-

ture effects are

expected

^{to be}

important

^{for the}

overall

properties.

^{So let us} consider the more appro-

priate

^{form to} eqs.

(11)

^and

(12)

where the first variable of the R.H.S. is less than

unity

^{since we consider a}

good

^solvent

(,rN 1/2 » 1).

We have seen in section 2 that S as a function of the second variable

depends

^on whether it is greater than or less than

unity. Similarly,

^{one is led to intro-}

duce a cross-over

frequency.

and to consider two kinds of behaviour

depending

on the relative value of ^w

compared

^{to w*.}

If one of the last two variables of the R.H.S. of eq.

(17)

^is greater ^than

unity (i.e.

^{co » cu* or}

q » - ’ - i),

^{a behaviour} characterized

by

^the

exponent ^{z = 3}

(Le. Zimm-like,

^or

theta-like)

^is

observed.

If _{co «} w*

and q «

^{r, a} behaviour characterized

by an,exponent z

^{= 2.9 must be observed.}

We are thus led to

generalize

the blob notion to

dynamics :

- Inside a

blob,

^the

hydrodynamic

interaction is dominant and a Zimm behaviour is

expected (see Fig. 1).

- The interaction between different blobs is more

complicated including

^both

hydrodynamic

and exclud- ed volume interactions.

Another value for the

dynamical

exponent ^results.

Although

the value for this exponent ^{is close to} the Zimm value

(2.9 compared

^to

3),

^{we shall see}

that it leads to

significant

deviations from the Zimm- type ^behaviour.

335

FIG. 1. ^- Domains of characteristic dynamical behaviours in the (q, co) plane ^for good solvents. The critical exponent ^{z = 2.9} is observable in the shaded area. The classical Zimm exponent

z ⁼ 3 is valid elsewhere.

Relation

(17)

shows the local

dynamical

^behaviour

of the chain :

i)

^{If one looks at}

q/r -

¹

(i.e. q - 1, where ç

is the radius of the

blob),

^{one is led to a characte-}

ristic time

ii)

The characteristic

frequency

^for

large

^values

of q (q

^»

r)

^is

This last relation shows that the behaviour inside the blob is the same as in the theta

regime. Incidentally

this shows that T* has the ^same

meaning

^as

Ti,

the difference between them

being

^{that for}

Tl

^the

whole chain is

considered,

whereas for

T*, only

a part ^{of it}

(the blob)

is taken into account.

iii)

For small values of the

scattering

^vector

(R -1 q r)

the diffusion of the blob can be observed :

and,

^from eq.

(17)

So Einstein’s law for the diffusion coefficient is still satisfied for the blob diffusion.

All these features show that within the

blob,

^the

behaviour is

Zimm-like,

^{i.e. the}

hydrodynamic

^inter-

action ^is

dominant. This is

easily

understandable because ive know that the excluded volume inter- action is weak inside ^a blob.

For the overall

properties

^{of the}

chain,

the excluded volume interaction is _very strong ^{and has to be taken} into account. As we have mentioned

above,

^this

results in a

dynamical

exponent which will be called z below.

Retuming

^to eq.

(17),

^{we are now}

looking

^{at the}

case where all the variables of the R.H.S. are less than

unity.

’We are then led to make a

change

in the variables

[23]

So _eq.

(17)

^reads

which in tum can be put ^{into the} scaled form

with v ⁼ 0.6 and z ⁼ 2.9 as discussed above.

From relation

(21),

^{we can extract the} main features of the overall

dynamical properties just

^{as we did}

for the theta solutions.

i)

^The characteristic time is measured for

qR ’"

¹

(i.e. q’ N IV ’" 1)

This time

corresponds

^to the terminal time in viscoelastic

experiments,

^{as was the case for}

Ti

ⁱⁿ

theta solutions.

ii)

^For

large

^values

of q,

the characteristic fre- quency is exhibited

This relation has been checked

recently by light scattering experiments [30].

The value for z which is used

throughout

^this paper ^comes from these

experiments.

iii) Finally,

for small values of the momentum

transfer q (qR « 1),

^{there is a}

broadening

^characte-

ristic of the diffusion of the coil

and,

^from eq.

(21)

^.

Three remarks can be made about relation

(23) : 1)

The diffusion coefficient of a

polymer

^chain

in a

good

solvent does not

obey

Einstein’s law.

Rough- ly,

^{we have D -}

R -(z- 2) and,

^{as z is not}

equal

^to

3,

D is not

inversely proportional

^to the radius of

gyration.

2)

^Relation

(23)

^{allows us to}

give

^{bounds to the}

numerical values of z : The exponent ^{in the N}

depen-

dence of the diffusion coefficient has to be more

than

1/2.

This last value

corresponds

^to the radius of

gyration

^{of a chain in a} theta solvent :

In the same way, this exponent ^{has to} be less than

or

equal

^{to v}

0.6,

^which

corresponds

^{to Einstein’s}

law,

so

3)

The intrinsic

viscosity

^can be calculated

simply by combining

eq.

(15)

^and

(22) :

One can

easily

check that

by lowering

^the

tempe-

rature, ^we get back relation

(16)

^{at the cross-over}

temperature

Tc ’"

N - 1/2 separating

^{the theta}

regime

from the

good

^solvent

regime [19].

So before we come to viscoelastic

properties,

^{let us}

summarize the

dynamic properties

that have been exhibited.

i)

^In the theta

regime

^the

dynamical

^behaviour

is uniform and ^can be defined

by

^a

dynamical

expo- nent ^{z =} 3.

(We

^have

completely neglected possible entanglement

effects in this

regime [24].)

ii)

^{In the}

good

^solvent

regime,

^{one has to consider}

the scale which is involved :

a)

For small scales

(small

^distances

compared

to the radius of the blob or small times

compared

to the characteristics time

T*)

the behaviour is the

same as in the theta

regime.

b)

^For

large scales, including

the overall

properties

of the

chain,

^the

dynamical

behaviour is characterized

by

^an

exponent z

^{= 2.9.} This last

exponent

^is introduced

empirically, by

^a

simple analysis

^{of a lot of}

experiments.

^{It leads to some} well-known results :

The diffusion coefficient of a chain is

That it does not

obey

Einstein’s law is related to the concentration fluctuation

along

^{the chain}

[26].

The characteristic

frequency

^is

Wc ’" q2,9.

The characteristic time

(or

^terminal

time)

^is

4. Some remarks about viscoelastic

properties.

^-

The

scaling properties

of dilute solutions have

béen

discussed in the

preceding

^{sections with}

special

reference to the

scattering

^function

S(q,

^{co, r, N -}

1).

They apply,

^of course, to other functions. In this

section,

^we would like to extend the

preceding

^dis-

cussion to the viscoelastic

functions, and,

^more

generally,

^{to time}

dependent properties

of the chain.

Two effects will be

approached, namely

^the

frequency dependence

^{and the}

temperature

^effects.

When the system ^{is known to exhibit a}

given

^number

of characteristic

lengths,

^{one can}

expect

^{from the} above considerations that it will present ^an

equal

number of characteristic

frequencies,

^{to which the}

frequency

range under

study

^{has to be}

compared.

In the theta

regime,

^one

has only

^one

length, namely

the radius of

gyration R

of the chain. As a

consequence, the characteristic features of the

poly-

meric nature will be observed in the _range

In the

good

^solvent

regime,

^relation

(26)

^{has to be}

replaced by

Moreover,

^this range has to be divided in two parts, due to the appearance of another

length,

^the

radius j

of the blob. This introduces another

frequency

m* -

i3

as we have said above.

For

frequencies

^{lower than}

co*(Oî ’ « co « ro*),

one

expects

the characteristic features of the chain in a

good

^{solvent to be}

exhibited,

whereas for

higher frequencies (m

^»

ro*),

the characteristic

properties

of the blob must appear. As we have seen in the

preceding ^sections,

^these

properties

^are

independent

of the

length

of the chain and

depend only

^on tempe-

rature

(and naturally

^on

frequency).

As a first

example,

^{let us} consider the

dynamical viscosity il(co).

^It is related to the

imaginary

part ^of the

complex

^modulus

[13] by

with the

requirement

Let us consider the

dynamical

^intrinsic

viscosity

This can be put ^{into a} scaled form

or,

equivalently

337

From _eq.

(30),

^{one can}

easily

deduce the intrinsic

viscosity

in the theta

regime rN 1/2 «

¹

In order to

study

^the

good

^solvent

regimern

^{1/ 2}

»1 ,

eq.

(30’)

^{is more}

appropriate.

^{Thus one can} separate the

high frequency

from the low

frequency

ranges.

a)

The usual intrinsic

viscosity

is measured at very low

frequencies (co « Oïl).

^{In this}

frequency

range, ^{one can} make a

change

in the variables of eq.

(30’)

Then

which leads to an intrinsic

viscosity :

when Co _«

N-vi -r3-2vi ’" 0-1 (with

^{v x 0.6 and}

z m

2.9).

b)

^For

higher frequencies,

^{one used to}

predict

that the

viscosity

^tends

simply

^{towards the}

viscosity

fis of the

solvent,

^{as m} increases. The new

scaling

^shows

however

that there is a

specific

contribution to

viscosity

^at

high frequencies, coming

from the blobs.

According

^to eq.

(30’),

^one has the residual

viscosity :

which is the

corresponding

theta-like behaviour.

So for intermediate

frequencies

one expects ^a contribution of the

polymer

^chain

proportional

^to

^{T - l .}

Let us notice that this contribution is still theta like. This means that it does not scale like the overall

part

which needs the exponents v and z. The ^corres-

ponding

behaviour is sketched on

figure

^{2. Such a}

cross-over in the

viscosity

has indeed been observ-

ed [13, 22].

In the same way, let us consider the real part

G’ ( Q) )

of the

complex

^modulus

[13].

The scaled form of this

quantity

^{will be}

given

below for the discussion of temperature ^{effects. But from}

simple scaling

FIG. 2. ^- Reduced dynamical viscosity ^versus frequency. ^{The low} frequency regime gives ^the contribution of the overall chain whereas the high frequency regime gives the contribution of the blobs.

One recovers the viscosity of the solvent at very high frequencies.

arguments, ^{one can} show that in the theta

regime,

and in the

interesting

^domain

(26)

^{we have}

[4]

where C is the monomer

concentration,

^{and T the} temperature

(we

take Boltzman’s constant

equal to unity).

In the

good

^solvent

régime,

and for low

frequen-

cies

(1) «(Ji l

^{« m «}

co*)

eq.

(28)

^{has to be}

replaced by

where v m 0.6 and J m 2.9 as before

whereas for

higher frequencies

^{one has to recover}

an ideal behaviour

So when

temperature

^is

varied,

^different

regimes

occur. In the theta

regime,

the behaviour of G’

is

expected

^{to be}

^uniform,

whereas in the

good

^solvent

regime (TN’I’ » 1)

^a cross-over occurs which _sepa- rates a

high frequency regime

^{where a} theta behaviour is

expected,

^{from a low}

frequency regime

^{where a}

good

solvent behaviour is

expected,

^{as shown in}

figure

^3a.

4.1 TEMPERATURE EFFECTS. ^- As we have seen

above,

^{these effects are}

important mainly

^{in the}

good

solvent

regime. They

^are

usually

^described

by

^intro-

ducing

^a parameter

[33]

aT :

(1) Eq. (29) is obtained by ^{the same method as before} by using

the scaled form for G’(co) (refer ^{to eqs. (34) and} (34’)).

where T and

To

^are

respectively

the actual temperature of the solution and a reference temperature, ^and

[ ]T

means that the

quantity

^between brackets is measured at temperature ^T.

This

method,

called the reduced variable method has

proved

^{to be very} fruitful for

relatively

^low

frequencies

^{in the}

good

^solvent

regime. However,

for

higher frequencies

^{it failed to}

give

^universal

curves. We wish here to discuss such a method within the framework of what has been said above. We are

going

^to discuss the real part

G ’(m)

^{of the}

modulus,

FIG. 3. ^- a) Behaviours of the real part G ’(w) of the modulus in the (q, w) space ^as derived from the rules in figure ^1. b) Logarithmic plot ^{of the real} part G’(w) ^{of the} complex ^{modulus versus} frequency.

The coordinates are normalized in such a way that the curve is universal : 1) pure theta behaviour (N1/2 ^{r «} 1); 2) hypothetical

pure good ^solvent behaviour ; 3) actual behaviour : there is a cross- over for co/,r’ - 1. The characteristic times

T1/i3

^and 01/r3 depend

on N. For tNII 12 ’" 1, both of them are of order unity, ^{and we}

recover curve (1) (see text).

but the results _may be extended

quite straightforward- ly

^to any function of the

frequency.

In the

vicinity

of the theta

point

^G’ may be put into the scaled form

We have seen that in the

good

^solvent

regime,

one has to separate ^a

high frequency

^{from a low}

frequency regime.

^We

argued

^{that the}

scaling

^is

not the same in these

regimes.

^{So two remarks can}

be made at this level about the usual reduced variable method :

1)

^{One must}

require

^{both T and}

To

^{to be in the}

good

^solvent

regime (io ^N1/2

^and

rN1/2 » 1).

2)

The method cannot describe the behaviour of the blobs in a universal _way. This means that it has to fail at

high frequencies.

In order to

study

^the temperature

effects,

^{we are}

going

^to use a more

practical

^{form than} eq.

(34)

This

equation

shows that the curves

G’/TCr;2

versus

(co/ r3)

^{should be} universal for the

asymptotic behaviours,

^{as shown on}

figure

^{3b. Let us comment}

this

figure.

In the theta

regime,

^we have found

(see

^eq.

(28))

which can be written

This behaviour is also

expected

^{in the}

high

^fre-

quency

regime

^{in the case of a} solution in a

good

solvent.

In the low

frequency regime

^{for a} solution in a

good ^solvent,

^we have found

(see

^eq.

(29)).

which can also be put in the form

Thus the

asymptotic

behaviours for G’ have the stated

scaling

behaviour. There is still another _para- meter

7:NI/2

which is

important

^{for the cross-over}

from one of the

previous

behaviours to the other : For a pure theta behaviour

(7:N1/2 1)

^one expects eq.

(28’)

^to hold in the entire range. For ^a

hypothetical

pure excluded volume

behaviour,

^one

expects

eq.

(29’)

to hold. For the actual chain in a

good solvent,

^we expect both of these behaviours to hold when fre- quency is varied

(see Fig. 3b) :

^{For low}

frequencies (W/7:3 1),

^we expect ^relation

(29’)

^to hold whereas for

higher frequencies

^a theta-like behaviour is

expected

^and eq.

(28’)

should be valid.

The

importance

^{of the}

parameter (7:N1/2)

^{lies in}

the fact that the lower bounds

01 ’/,r’

^and

T1-1/7:3

depend

^{on it.} As mentioned

above,

the lower bound which has to be used

depends

^on the value of