Hydrodynamics of polymer solutions via two-parameter scaling

HAL Id: jpa-00248044

https://hal.archives-ouvertes.fr/jpa-00248044

Submitted on 1 Jan 1994

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Hydrodynamics of polymer solutions via two-parameter scaling

Ralph Colby, Michael Rubinstein, Mohamed Daoud

To cite this version:

Ralph Colby, Michael Rubinstein, Mohamed Daoud. Hydrodynamics of polymer solutions via two-parameter scaling. Journal de Physique II, EDP Sciences, 1994, 4 (8), pp.1299-1310.

�10.1051/jp2:1994201�. �jpa-00248044�

Classification Ph_vsic.s A h.iiiat I-I

83. ION 83.20F 61.25H

Hydrodynamics of polymer solutions via two-parameter scaling

Ralph

H.

Colby ('),

Michael Rubinstein

(')

and Mohamed Daoud

(2)

(1) Imaging ^Research and Advanced

Development,

Eastman Kodak

Company,

Rochester, New York 14650-2109, U.S.A.

(2) Laboratoire Leon Brillouin (*), C.E.N. Saclay, 91991Gif-sur-Yvette Cedex, France

(Receii'ed10 Januar~ 1994, accepted 5 May 1994)

Abstract. We discuss the temperature and concentration regimes for the

hydrodynamic

properties

of linear

polymer

solutions. New regimes are reported ^{in addition} to the known

regimes

for the static conformations of the coils. The _new

regimes

_appear because of the ex18tence of _a second length scale, the tube diameter, which is _not proportional to the screening

length.

In the

theta regime, the polymer volume fraction where entanglement becomes important is

~c _m la

j/hl'?

N- "~~ where

a_j, h, and N _are the tube diameter in the melt, the Kuhn length, and the number of Kuhn segments ^{in the} chain, respectively. Except for very large N, ~~ is larger than the

overlap concentration ~ *. In the good solvent region. ~ ^* and ~~ have the _same

scaling

for their

dependence

_on N, but have different temperature

dependences.

The results _are ~ummarized _{on a}

diagram

in the

temperature-concentration plane.

Introduction.

It has been

previously

shown that static

properties

^{of linear}

polymer

^solutions can be described

rather

directly using 8caling

for effects of both concentration and temperature

[1, 2].

Presumably,

the latter is

directly

related _to the so-called solvent

quality.

A

simple phase diagram

_was

_suggested

Ii that summarized the static

scaling properties

of solutions of flexible

linear

polymers. Dynamic properties

of these

polymers

_were also studied in the

good

solvent

case

[3]

with

scaling

arguments, ^{and the results}

compared favorably

with

experiments.

When

generalized

to theta

solvent, however,

these

scaling

ideas based _{on a}

single length

scale do _not agree with

experiments [4~6].

This led to the two-parameter

scaling [4]

for theta solvent

dynamics,

which showed that there _are two

important length

scales the

screening length [2,

7, 8] and the tube diameter

[9, 10],

which scale

differently

with concentration in theta solvent.

The

resulting

two-parameter

scaling predictions

_were shown to compare

favorably

with

experimental

results for

viscosity

and modulus of semidilute theta solutions

[4-6].

(*1 Laboratoire _commun C-N-R-S--C-E-A-

In this paper we

explore

the consequences of two-parameter

scaling

for the full range of temperature ^and concentration. We find _new results for the temperature

dependence

of the tube diameter in the

good

solvent

regime.

Thus in

good

solvents the

screening length

and the tube diameter have identical concentration

scalings

but different temperature

scalings.

Another

consequence of two-parameter

scaling

is that there _are

regimes

for both

good

^{and theta} solvent, which _are

semidilute,

but

unentangled (because

the tube diameter is

larger

than the

screening length).

The full temperature ^and concentration

dependences

of criteria for chain

entanglement

will be elucidated.

After _a brief review of

scaling

for static

configurations

of chains, we write

general

expressions

for relaxation

times, modulus,

and

viscosity

in _terms of the

appropriate length

scales and known parameters ^such as solvent

viscosity

_7~~, Boltzmann's _constant

k,

and

absolute temperature T. We _next define the various

regimes

on the

temperature-concentration plane

and obtain

scaling

laws for the concentration and temperature

dependences

of the various

length

scales.

Finally,

_we tabulate _our results for concentration and temperature

dependence

of relaxation

times, modulus,

and

viscosity

in the different

regimes

and compare with

light scattering

and linear

viscoelasticity experiments.

Static

scaling.

In dilute

solution,

the root-mean-square ^end-to-end distance,

Ro,

of the

polymer obeys

_a

scaling

law

[2,

II in _terms of the number, N, and

length,

b, of Kuhn segments

(hereafter

called

monomers)

in the chain.

RombN~' (1)

The

Flory

_exponent

v is 1/2 in solvents (we

neglect logarithmic

corrections _to

scaling)

and

roughly

3/5 in the

good

solvent limit. As concentration is increased, the dilute

regime

ends

when the coils start to

overlap.

The volume fraction of

polymer

at which this _occurs is

determined when it reaches the volume fraction inside _a

single

coil

[I Ii.

#i*iNb~/R(wN'~~' ₍₂₎

Above

overlap,

the semidilute solution is characterized

by

a

screening length [2, 7, 8]

(or blob

size), f, beyond

which

hydrodynamic

interactions and any excluded-volume interactions _are screened. The concentration

dependence

of f was determined from a

scaling

argument

[2, 8],

which

requires

f to be

independent

of N.

it Ro(~b/~b *)-

^{~"~~ ~'- ' '} (3)

Note that _at #i = #i

*,

the

screening length equals

the dilute solution coil size

Ro.

Inside _a blob any excluded-volume is felt and thus the

screening length

can be related to the number of

monomers in the blob~ _g,

by

a relation

analogous

to

equation

(1).

f _I

bg

^~

(4j

On

length

scales

larger

than

f

the chain

obeys

random walk statistics (a random walk of

N/g blobs).

(R/f

)~ _m

N/g (5)

At _some concentration #i~,

larger

than _~b

*,

the coils become

entangled.

The

length

^scale

describing

this

entanglement

is the tube diameter from

reptation theory [9, 10],

_a. The tube

diameter has been

suggested

to scale with the

following

concentration

dependence [4]

(aj

is the tube diameter in the

melt).

away ~b~'

(6)

The exponent,r ^{has been determined from the assertion that} a fixed number of

binary

contacts in _a volume

a~

_controls

entanglement. Binary

contacts are determined

by

the

screening length

in the

good

solvent

limit,

where _we expect a~f ^and thus _x

=

v(3

_v

Ii

=

3/4. In solvent the

screening length

^reflects ternary contacts

[2, 12]

and _a

simple

_mean field argument

[4] gives

_x

=

2/3. In

good

solvent _a and

f

scale with the same concentration

dependence,

but in theta solvent

they

do not, and _a and f

actually

intersect

[4]

at

#i ₌

(aj/b

)~^~ However, except ^for

polymers

_{of very}

high

molecular

weight,

^this concentration is below the

overlap

concentration, and _we

ignore

such

ultra-high

molecular

weights

in this

paper.

Since _a

~ f, an

entanglement

^{strand of}

N~

_monomers is _a random walk of

N~/g

blobs.

(a/f

_)~

i N

~/g (7)

Combining equations

(5) and

(7) provides

a final static relation that will be used in what follows _to

replace

all static parameters

by

the three

length

scales f, _a, and R.

(Rla

_j~

w

N/N~ (8)

Dynamic scaling.

Below the

overlap

concentration (~fi _~ #i *

) polymers

^exist as separate ^{coils and}

dynamics

are

known to be described

by

the Zimm model

[13]

with full

hydrodynamic

interaction

[10].

The

longest

relaxation time in dilute solution follows Zimm

scaling.

Tz,mm _{I 1~}

~

R(/kT (9)

The modulus for the Zimm model is determined _as kT of stored energy per chain.

Gw

#ikT/Nb~ (10)

The

product

of

equations (9)

and

(10) yields

the

viscosity

of _a dilute

polymer

solution.

7~ w

GTz~~~

w 7~

~

#iR(/Nb~

I1)

This is the basis of the earlier

Fox-Flory equation

[I

I].

According

to two-parameter

scaling [4]

there should be _a

regime

^of concentration above

#i ^* that is semidilute but

unentangled.

This

regime

occurs for f ^~R _~ a, and the

physical picture

of the chain is that of _an

unentangled

random walk of

N/g

blobs of size f.

Hydrodynamic

interactions dominate up to size f, _so the relaxation time of _a blob is

determined

by

the Zimm model.

T~ I 7~~

f~/LT. (12)

The

longest

relaxation time is the Rouse time

[10,

14] of the chain of

N/g

blobs.

T~~~,~ _-

(N/g

_{)~ T~}

-

(R/f

)~ T~ w 7~

~

R~/fkT. (13)

The

unentangled

modulus is still determined

by

kT of stored energy per chain

(Eq. (lo)).

The

viscosity

of the

unentangled

semidilute solution is thus obtained from the

product

of

equations (ID)

and

(13).

7~ m 7~

~

#iR~/fNb~ ₍₁₄₎

At still

higher concentrations,

when R _{> a,} the

polymers

become

entangled.

The

physical picture

^{of the chain is} now an

entangled

random walk of blobs. The

dynamics

inside the blob

are as

before,

and the Zimm relaxation time of the blob is

given by equation (12j.

Entanglement

strands of

N~/g

blobs relax

by

Rouse

dynamics (with

relaxation time

T~).

T~ m

(N~/g

j~T~ m

(a/f

_{)~ T~}

m 7~,

a~/fkT. ₍₁₅₎

The

longest

relaxation time is controlled

by reptation

of the chain of

N/N~ entanglement

strands [10].

T~~~ ^~ IN IN~)~ T~ m

(Rla

_)~T~

w 7~

~

R~la~ fkT

II

6)

The

entangled (plateau)

modulus is determined

by

kT of stored energy per

entanglement

strand

[4].

G

w

kTla~ _{f (17)}

The

viscosity

of the

entangled

semidilute solution is determined from the

product

of

equations (16)

and

(17).

7l m 7l

~

R~la~

_f^~ _(l

8)

We next examine the

scaling

of

lengths

f, _a, and R as functions of temperature and

concentration _to make

specific predictions

for the various

regimes.

Regimes

on the

temperature-concentration plane.

For

temperature scaling Ii

_we make _use of the standard reduced variable

[2]

_r, defined below

in _terms of the excluded volume

[2,

1II, v, such that

r =

0 in _a theta solvent

(T

_v

=

0) and _r

=

I in the

good

^{solvent limit}

(T

~ oJ ; ^v =

I)

r =

v/b~

I

(T )IT (19)

Previous works

[1, 2]

have demonstrated that in semidilute solutions, _r

~ #i means that the

two-body

interactions

(v)

dominate and thus

r ~ ~fi

corresponds

_to

good

^solvent

scaling.

Conversely,

_{r ~ ~fi}

implies

that

three-body

interactions dominate and therefore theta solvent results govern

scaling.

In other words, the criterion

[1, 2]

~~ ~ ~b

(20)

defines the _crossover from theta _to

good

solvent

regimes

in the

r-~fi plane.

In the theta solvent

regime equation (I gives

the coil size

(with

_v

=

1/2).

RibN"~

_r~~fi.

(21)

Equation (3) gives

the

screening length (with

_v

= 1/2).

I m b~b ^' r _~ ~b

(221

Equation (6) gives

the tube diameter

(with,t

=

2/3).

a w a ~fi

~'~

r ~ #i

(23)

The

good

solvent

regime

is _more

complicated

because of the presence of the thermal

blobs

[2]

that

bring

in an

explicit

temperature

dependence

to these

length

scales. In dilute solution the

physical picture

of the chain is

a random walk up ^to the thermal blob size

f~

and _an excluded~volume walk

beyond f~. Therefore,

there _{are two}

length

scales in the

dilute

good

solvent

regime.

The thermal blob size

[2]

is

only

a function of temperature.

f~

_w ^{br '} _{r >}

#i1 (24)

The coil size

[1, 2]

is determined from _a

self-avoiding

walk of thermal blobs.

The

overlap

concentration

[1, 2]

is determined from

equation (2).

~ _415

~- ³¹⁵ _{~ >} Qi

$

~~~~

The coil size

(Eq. (25))

_{crosses over} the standard theta solvent result

(Eq. (21))

when

r = #i

i,

and also _{crosses over to} the standard result

(Eq. II )

^with _{v =}

3/5)

in the

good

^solvent

limit

(T

~

).

In semidilute solutions in the

good

solvent

regime,

the chain is

again

a random walk up to the thermal blob size

f~,

and thus

equation (24)

is valid for all concentrations in the

good

solvent

regime.

Excluded volume is effective _on

length

scales between

f~

and the

screening length,

f. ^The

screening length

is obtained

using equations (3), (25),

and

(26) (with

v =

3/5) [1, 2].

f _w b~fi ^~'~ _r ^"~ _{r > ~fi}

(27)

There _{are g monomers} in _a blob of size f.

g ~

iiliTi~'~ (iT/b)~

i ~b ^{~'~ r~ ~'~} r ~ ~b

(281

Beyond

f the chain is _a random walk of swollen blobs.

R

w f

(N/g )"~

w hN ^"~

~fi

"~

r

"~

r ~ ~fi ~ #i ^*

(29)

We determine the

scaling

for the tube diameter in the

good

solvent

regime

with _a

simple scaling

argument at a fixed

r. Since the tube diameter is

always

controlled

by two-body

interactions

[4],

and the

screening length

crosses over from

two-body

to

three-body

interaction dominance _at #i

= r, we

anticipate

_{a crossover} for the tube diameter at #i

= r. For

~fi > r,

equation (23j

holds (it

effectively

_{uses a mean} field _count of

binary contactsj.

^For

~fi ~ r we know that the

density

^of

binary

contacts scales with the

screening length.

a WA

~fi

~'~

~fi ~ r

(30j

We evaluate A

by matching equations (23)

and

(30)

at _~fi = r.

-213 ~ ^-3'4

(~~)

aj T I T

Thus A

w aj r

"'~, giving

_{us a}

prediction

for the tube diameter in the

good

solvent

regime.

a w aj ~fi ~'~

r

"'~

r ~ ~fi

(32)

JOURNAL DE PHYSIQUE ii T 4 N' 8 AUGUST 1994

49

We _now

identify

six

regimes

in the

temperature-concentration plane,

shown in

figure

I.

Regimes

I and I' _are the dilute

good

and dilute theta

regimes, respectively. Regimes

II and

II' _are the semidilute

unentangled regimes

in

good

and theta solvent.

Regimes

III and

III' _are the

corresponding

semidilute

entangled regimes.

Dilute and semidilute solutions _are divided

by

_~fi

*, given by equations (2) (for

_{r ~ ~b,} with _v

=

1/2)

and

(26),

^{which is}

equivalent

to the concentration where the

screening length

and the coil size _are the _same.

fit ^l12 _~§ *

~§ ^* ~

~ ~

(33)

fit ⁴¹⁵ _T ³¹⁵ _{~ > ~§} ^*

Good and theta

regimes

_are

separated by

a critical reduced temperature

II-

~bi

_{~b ~}

~b1

~~~~

~~~

~b

~b>~bi

Unentangled

and

entangled regimes

are divided

by

_#i~ that is determined _as the concentration where the tube diameter and the coil size _are the _same.

Matching equations (21)

^and

(23),

and

equations (29)

and

(32)

thus

yields

the

entanglement

^criterion.

~b~ w

~'~~)~~~

_N~ ~'~

~ ~ ~

(a

_{16 )~'~ N} ^~'5 _~ ⁱⁱ¹⁵ _~

~

> ~b

e

(35j

Equations (33-35j

_are the solid _curves in

figure

I.

We _use the results for f

(Eqs. (22j

and

(27jj,

_a

(Eqs. (23j

^and

(32))

and R

(Eqs. (21), (25)

and

(29))

_to calculate the

viscosity, modulus,

and various relaxation times

predicted

^{in the}

t~ te

III

I

_II

III'

,

II'

Fig,

I.- Phase

diagram

in the space of

polymer

volume fraction ~ and reduced temperature

r w (T )/T, for the

example

of

aj/b

₌ 5 and N loo. Primes denote theta solvent regimes and _no

prime

denotes

good

solvent

regimes. Regimes

I and I' _are dilute regimes II and II' _are semidilute- unentangled

regimes

III and III' _are semidilute-entangled.

dynamic scaling

section in the six

regimes.

The results _are summarized in table I. In dilute solution

(regimes

I and

I')

_{we recover} the standard results of the Zimm model

[10, 13]

for

relaxation

time,

modulus and

viscosity.

In

semidilute-unentangled

solution

(regimes

II and

II')

Shiwa

[15]

has

predicted

the theta solvent results and the

good

solvent results in the

good

solvent limit

(with

_r

=

I)

but _no temperature

dependence.

We have

previously

derived the

entangled

theta solvent results

[4]

and the

good

solvent limit results in the

entangled

_{case are}

known _as well

[2, 3]

without the temperature

dependence.

Table I.

Summary of predictions for dynamic scaling.

Reg,me Tz,~~ iTl~, ^h~ I') TRou~ kTl~, h~ (~) _T,~~ kTl~, ^h~ Gh'liT ~l~,

' N~~ N~' _~ N"~ _~

l' N~'~ N-' _~ N~'~~

II ~ ^~'~ _r ^"~ N ~ ^''~ _r^~'~ N ^' ~ N~ ^~'~ r

"~

ii' ~-' N2~ N-'~ N~2

-9>4 -~'4 -9'4 7n2 ~'

)~

^{~i ~i12 ~'6 h ~ ~q,4 1<i2 h 2 ~, j~i4 11<12 h}

)~

~~~ ~ _~ ~ ~

h ^~ aj ^~ Uj

~ ~

aj

(') Zimm time of the entire chain for regimes I and I'. Zimm time of _a blob (Tz,mm T~) ^for regimes II, II', III and III'.

(2) Rouse time of the entire chain for regimes II and II'. Rouse time of _an entanglement strand (TRou« _" T~) ^for regimes III and III'.

Note that

scaling predicts

_a concentration

regime

that is semidilute and

unentangled,

as first

suggested by Graessley [16].

The width of this

semidilute-unentangled (Rouse) regime

^is

determined from the ratio of

equations (33)

and

(35).

(aj16

)~'~ N~ ^"~ r ~ ~b ^*

~(

m

(aj/b)~'~ ^{N"~° r~'~}

#i ^* _{~ r ~} #i

~

(36)

~ (a,/b)~'~ ^r~"~

_{r >}

#i~

Comparison

with

experiments.

The concentration

dependence

of

viscosity

in

entangled good

solvents has been reviewed

recently [17],

with

7~

#i~°~°~ _(see

_{Tab. III} of Ref.

[17]

and Refs,

therein),

in excellent agreement ^{with the}

scaling prediction

of

7~ #i ^'~'~

Experimentally,

the

plateau

modulus in

good

solvent

[17]

scales as G #i ^{~.~ ~ °}_~,

again

in excellent agreement ^{with the}

scaling theory.

The theta solvent results for

plateau

modulus have also been reviewed

recently [18],

with G

#i~~~°

^' in excellent agreement ^{with the}

predicted

_exponent of 7/3.

Viscosity

data in

entangled

theta solution _{are scarce,} but appear ^to be consistent with the

scaling prediction

(7~

#i')

with exponent ^values x =

5.4

[5, 4],

_x

=

4.8

[6, 4],

and _x

=

4.7

[19]

in

good

agreement ^{with the}

expected

exponent 14/3.

There is

experimental

^{evidence for} the

semidilute-unentangled regime

of concentration _as well.

Typically a~/b

_m

5, implying

that the

good

solvent limit

(r

=

I of

equation (36) predicts

the width of the

semidilute-unentangled regime

to be

roughly

_a factor of _ten in concentration,

in reasonable agreement ^with

experiment [19]

(see

Fig.

8 of Ref.

[19]).

Another group has

reported

that the

entangled scaling

for the

viscosity

in

good

solvent holds down _{to a} concentration of four times the

overlap

concentration

[20]. However,

their estimation of the

overlap

concentration if _a factor of three

larger

than the usual _one

[21] (the reciprocal

of intrinsic

viscosity).

Thus _we believe the

viscosity

data of reference

[20]

deviate from the power law for

entangled

solutions below _a concentration of

roughly

twelve times the _true

overlap

concentration,

in excellent agreement ^with

equation (36).

The rather limited _extent of the

semidilute-unentangled regime

and the small amounts of data make

quantitative comparison

with exponents

pointless

at present.

Clearly

a detailed

study

^of this concentration

regime

is needed.

Notice that _a

typical

_«

good

_» solvent should not

correspond

to the _true

good

solvent limit (r

= I means T~

oJ)

and therefore should have four

regimes

of concentration for the

viscosity (I,

II, III and III' in

Fig. I)

_as observed

experimentally (see Fig.

8 of Ref.

[19]j.

The temperature

dependence

of osmotic

compressibility [22], plateau

modulus

[5],

and

viscosity [5]

have been measured

by

Adam and coworkers for

polystyrene

^solutions in

cyclohexane

from the theta

point

to T

= + 25 K. We _now make _use of these data to test our

predictions regarding

temperature

dependence.

Scaling predictions

^for the osmotic

compressibility [2]

(c dar/dc, where _c is concentration and _ar is osmotic

pressure)

conclude it is kT per correlation blob

(I,e.,

_c d ar/dc _m kT f ~). Thus

osmotic

compressibility

measurements can determine the blob size

(within

_a

prefactor).

The

prefactor

can be eliminated

by taking

the ratio of osmotic

compressibilities

at theta and _{at some}

(higherj

temperature.

Equations (22j

and

(27)

_suggest the

following scaling

function.

(d&T/dC

_)@ ^"3 I _T

~ #i

m

#if/b

_m

(37j

d&T/dC (T/#i )~~~~ _{T ~} #i

We therefore

plot #if/b

_i>s. r/#i for the data of reference

[22]

in

figure

2. Above

r/#i

of 0,I the data

obey

_{a power} law

[23].

#i

fib

=

0.57 I ^{° ~~ ~°°~}

(38)

~b

These data _are in excellent agreement ^{with the}

scaling prediction

[I] of

equation (37),

as also noted in reference

[22].

Below

r/#i

of 0, the data _{crossover to} the

expected

theta behavior (#i

fib

m

I)

_as the theta temperature ^is

approached.

We next compare the

plateau

modulus data of reference

[5] (in

the

semidilute-entangled regime)

with _our

predictions.

While the osmotic

compressibility changed by

an order of

magnitude

between and ₊ 25

K,

the

plateau

^{modulus is} much less temperature sensitive,

changing by only

20 %. This is consistent with

comparisons

of

plateau

modulus in

good

solvents and theta solvents

[5, 19, 20],

which indicate that the

plateau

modulus does _not

depend

on solvent

quality.

The

plateau

^{modulus data of reference}

[5] actually

decrease

weakly

as temperature ^{is increased}

(the

data above

r/#i

_m 0.I in

Fig.

8 of Ref.

[5] obey

a power law G

r~° '),

whereas _we

predict

_a weak increase with temperature

(see

Tab. I, G

r"'~

_is

predicted).

However, these data _are

actually

in

qualitative

agreement ^with our

theory,

as our

scaling

_{exponents may} not be the

simple

rational powers we have

suggested.

For

example,

the exponent v in

equation

I is known to be 0.588 from renormalization group calculations

[24],

as

opposed

to the

3/5

_we have used here.

Using

_v

=

0.588 leads _{to an}

expected

value of the

exponent

ⁱⁿ

equation (38)

of

0.23,

in _even better agreement ^with

experiment

than Il.

Similarly,

_we cannot claim _to know other exponents

(such

as the value of

x =

2/3 in

Eq. (6))

with _too much

precision.

e . o

o.7

It

b

ig.

from ^smotic data [221 _using the ~caling form of equation

(37).

Filled

symbols

for M = 3.84 x 10~ and ~ = 0.0 loo. Open symbols are _data for

line is linear fit to ata r/~ ~

0.5

(Eq. _(38)).

We _can extract the apparent temperature

dependence

of the tube diameter

by combining

the

osmotic

compressibility

determination of the

screening length [22] (Fig. 2)

with the

plateau

modulus data

[5]

_on the _same system,

through equation (17). Equations (23)

and

(32)

_suggest the

following scaling

form.

Ii _l~

^'~

m ~b ^~~

alai

_m

~~~

ji~ i1 (391

We therefore

plot

#i ^~'~

alai

_i>s. r/#i for the data of reference

[5]

in

figure

3,

using interpolation

and

extrapolation

of the data in

figure

2 for the

screening length.

Above r/#i of 0, I the data

obey

a power law

[23].

~b~~ala~

1.4

I

^{° ~~ ~°°~}

~b

(40)

Exponents

of

v = 3/5 in

good

solvent and _x

=

2/3 in theta solvent led _{us to} expect a weaker temperature

dependence

of the tube diameter

(a

_r

"'~, Eq. (39jj. Using

the renormalization group value

[24]

of _v

= 0.588, we conclude from

equation (40j

that the

experimental scaling

for the tube diameter in theta solvent is _a

#i°~~~°",

_{in excellent agreement} with the exponent 2/3

(the

exponent ^{for Tin}

equation (40)

is

v/(3

_v

Ii

-J. for

general

_v and x). More data are needed at

higher

r/#i on the temperature

dependence

of

plateau

modulus in

near-theta

solutions,

_as the data of reference

[5] only

include

r/#i

_~ 2.

Given the _success of

scaling

in

predicting

the temperature

dependence

of osmotic

compressibility, plateau

modulus,

screening length,

and tube

diameter,

we would

naturally

expect

equally good prediction

of the temperature

dependence

of

viscosity.

However, this is

far from the _case. The temperature

dependence

of

viscosity [5]

in

semidilute-entangled

solutions demonstrate _some fundamental flaw in _our

reasoning.

Our

scaling theory predicts

2

o

Q ~

2/3

Qi

o

i

iO-2 10"1 io

'C/#

Fig. ^3. Temperature dependence of the tube diameter for polystyrene ⁱⁿ

cyclohexane

calculated from plateau modulus data [5] (M 6.77 x10~ _and ~ ₌ 0.0535) and the

screening length

(obtained from

Fig.

2

by interpolation

and

extrapolation

for the lowest r/~

point) using

the

scaling

^{form of}

equation

(39). Solid line is _a linear

regression

fit to data with r/~ _~ 0.07

(Eq.

(40)).

that both the relative

viscosity

and the terminal relaxation time in the

good

solvent

regime

should increase with

increasing

temperature

(7~/7~~~r""~, T~~~~r~'~). Experimentally,

however, both relative

viscosity

and terminal time decrease with temperature

by nearly

a factor of two between and ₊ 25 K

(see Fig.

5 of Ref.

[5]j.

This result is itself _a

paradox,

because in

semidilute-entangled

solutions the

good

solvent limit

clearly

has _a relative

viscosity

that is

slightly higher

than in theta solvent

[5, 20],

as

predicted by

our

simple scaling theory.

Apparently,

instead,of the monotonic increase in relative

viscosity

that _we

expected,

the relative

viscosity

_goes

through

_a minimum _{as a} function of concentration.

Where is _our

reasoning

flawed ? Static

quantities (osmotic compressibility

and

plateau modulus)

allowed _{us to} calculate the _two static

length

scales in the

problem (screening length

and tube

diameter).

The temperature ^and concentration

dependence

of these

length

scales agree very well with _our

simple theory

^and

they

both _seem to crossover from the theta

regime

to the

good

^solvent

regime

at the same

point (r/#i

_m 0,I, _see

Figs.

2 and

3).

Our calculation of

dynamic quantities (terminal

relaxation time and

viscosity) required

these two

length

scales, but also

required assumptions regarding dynamic scaling.

The

hierarchy

of relaxation times

(Eqs. (15)

and

(16)) might

be suspect, ^{but such}

reasoning

is at the heart of molecular theories of

polymer dynamics [10],

and these theories _seem to have been well-tested.

We also assumed the

hydrodynamic screening length

behaves in the _{same manner as} the

static

screening length.

While it _{seems as}

though

these

lengths

have the _same concentration

dependence

in both

good

solvent and theta solvent, it is _not

necessarily

true that

they

_are identical. In

particular, they

_may well _crossover with temperature

differently.

In

fact,

other

crossovers between

dynamic

and static

quantities (such

_as the coil size and

hydrodynamic

size

in dilute solution _as functions of chain

length [25])

_are known _to be different. However, to

explain

the

pronounced

^{minimum in the} temperature

dependence

of

viscosity,

the

hydrodyn-

amic

screening length

would need to have a nonmonotonic temperature

dependence (I,e.,

a

strong

maximum),

^{There is} no reason to believe such _a maximum should exist.

Certainly

data

in dilute solutions show _no

pronounced

differences between the temperature

dependences

of radius of

gyration

and

hydrodynamic

size

[26].

Clearly

_more data _are needed for the temperature

dependence

of

viscosity

in near-theta solutions. In

particular,

it would be nice to have data in _a

high-boiling

theta system, ^such as

polystyrene

in

dioctylphthalate [27],

which would allow for _a much wider temperature _range to be covered,

possibly getting deep

into the

good

solvent

regime (well beyond

the minimum in

viscosity

_vs.

temperature).

Conclusions.

We have

presented

_a two-parameter

scaling theory

for the concentration and temperature

dependences

of

rheological quantities

such _as

plateau

modulus, ^relaxation

time,

and

viscosity.

The presence of _two

independent length

scales

(screening length

f and tube diameter

a)

implies

six

regimes

of behavior

(see Fig, I).

Good solvent

scaling applies

at

high

temperature and theta solvent

scaling applies

at temperatures ^close

enough

to H. At low concentration there

are dilute

regimes (I

and

I')

where R

~ f. At intermediate concentration there _are semidilute-

unentangled regimes (II

and

II')

where f _~ R _{~ a.} At

high

concentration there _are semidilute-

entangled regimes (III

and

III')

where R

> a.

The concentration

dependences

of all

predicted rheological quantities

in the semidilute-

entangled regime

_appear to be consistent with

experiment. Using

data for osmotic compress-

ibility

and

plateau modulus,

_{we were} able _to

verify

that the

expected

temperature

dependences

of _{our two}

length

scales

(f

and

a)

_are observed.

However,

the

viscosity

and terminal relaxation time exhibit nonmonotonic temperature

dependences

that _are

qualitatively

different from _our

simple scaling predictions.

The

origin

of this

discrepancy

is unknown. It is worth

noting

that similar difficulties _are observed in other

polymer problems.

An

example

^is

gelation,

where static

properties

such as molecular

weight

distribution

[28], gel

^fraction

[28], swelling [28, 29]

and modulus

[29, 30]

are

well-understood,

but

dynamic properties

such _as

viscosity [30] (and

relaxation

time)

are not.

References

[ii Daoud M., Jannink G., J. Phys. Franc-e 37 (1976) 973.

[2] de Gennes P. G., Scaling Concepts ⁱⁿ Polymer

Physics

(Cornell University Press, Ithaca, 1979).

[3] de Gennes P. G., Macromolecules 9 (1976) ^{587 and} 594.

[4] Colby R. H., Rubinstein M., Macromolecules 23 (1990) 2753.

[5] Adam M., Delsanti M., J. Phys. ^{Fiance 45} (1984) 1513.

[6]

Roy-Chowdhury

P., Deuskar V. D., J. Appl. Polym. ^Sci. 31(1986) 145.

[7] Edwards S. F., Proc.

Phys.

Soc.. 88 (1966) 265.

[8] Daoud M., Cotton J. P., Famoux B., Jannink G., Sarma G., Benoit H.,

Duplessix

R., Picot C., de Gennes P. G., Macromolecules 8 (1975) 804.

[9] de Gennes P. G., J. Chem. Phys. 55 (1971) 572.

[10] Doi M., Edwards S. F., The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986).

[I ii Flory P. J., Principles of Polymer Chemistry (Comell University Press, Ithaca, 1953).

j12] Famoux B., Baud F., Cotton J. P., ^Daoud M., Jannink G., Nierlich M., de Gennes P. G., J. Phys.

Franc-e 39 (1978) 77.

[13] Zimm B. H., J. Chem. Phys. 24 (1956) 269.

[14] Rouse P. E., J. Chem. Phys. 21 (1953) 1272.

[15] Shiwa Y., J. Phys. II France 3 (1993) 477.

[16] Graessley ^W. W., Polymer 21 (1980) 258.

[17] Pearson D. S., ^{Rubb. Chem. Tech. 60} (1987) 439.

[18] Colby R. H., Rubinstein M., Viovy ^J. L., Macromolecules 25 (1992) 996.

[19] Colby R. H., Fetters L. J., Funk W. G., Graessley W. W., Macromolecules 24 (1991) 3873.

[20] Adam M., Delsanti M., J. Phw. Fiance 44 (1983) l185.

[21] Graessley W. W., Adi,. Polym. Sci. 16 (1974) 1.

[22]

Stepanek

P.,

Perzynski

R., Delsanti M., Adam M., Macromoleiules 17 (1984) 2340.

[23] While _our theory naively _expects the _crossover to occur at r/~

m I, scaling

actually

cannot be

expected

to

predict

the _crossover

point

exactly. Note that, _as expected, ^all quantities cross at

roughly the _same point (r/~ _m 0.1).

[24] Le Guillou J. C., Zinn-Justin J., Phy.I. Rev. B 21 (1980) 3976.

[25] Weill G., des Cloizeaux J., J. Phy.I. Fiance 40 (1979) 99.

[26] Miyaki Y.. Fujita H., Macromolecules 14 (1981) 742.

[27] Park J. O., Berry ^G. C., Macromolecules 22 (1989) 3022.

[28] Colby R. H., Rubinstein M.. Gillmor J. R., Mourey T. H., Macromolecule-I 25 (1992) 7180.

[29] Rubinstein M., Colby ^R. H., Macromolecules 27 (1994) 3184.

[30]

Colby

R. H., Gillmor J. R., Rubinstein M., Phys. ^Rev. E 48 (1993) 3712.