On the N-scaling of the Ginzburg number and the critical amplitudes in various compatible polymer blends

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On the N-scaling of the Ginzburg number and the critical amplitudes in various compatible polymer blends

D. Schwahn, G. Meier, K. Mortensen, S. Janßen

To cite this version:

D. Schwahn, G. Meier, K. Mortensen, S. Janßen. On the N-scaling of the Ginzburg number and the

critical amplitudes in various compatible polymer blends. Journal de Physique II, EDP Sciences, 1994,

4 (5), pp.837-848. �10.1051/jp2:1994168�. �jpa-00248005�

Classification

Physic-s

^Abstracts

05.705 36.20 64.60F

On the N-scaling of the Ginzburg number and the critical

amplitudes in various compatible polymer blends

D. Schwahn

('),

G. Meier

(~),

K. Mortensen

(3)

and S. Jan6en

(')

(') Forschungszentrum Jdlich GmbH, Institut for Festksrperforschung, ^Postfach 1913, D-52425 JUlich,

Germany

(2) Max-Planck-Institut for

Polymerforschung,

Postfach 3148, D-55021 Mainz, Gerrnany (3) Ris# National

Laboratory-DK-4000

Roskilde, Denmark

(Received 3 November J993, received in final form J7 January J994, accepted 2 Februaiy J994)

Abstract. The

susceptibility

related _to

composition

fluctuations in polymer blends _can be described by a crossover function derived by Belyakov et al. [2, 3] _near and far from the critical temperature T~. ^{This has} quite recently been demonstrated by _us using neutron and light scattering data Ii. In this paper we discuss consequences of this

analysis

with respect to the accomplishment of the mean-field

approximation

far from T~ and the N

scaling

(N is the

degree

of

polymerization)

of the critical

amplitudes

of the bare correlation

length

and

susceptibility

respectively and the Ginzburg number, Gi. A

quite unexpected

N

dependence

of Gi _was found

polymer

blends with N < 30

give

the _same Gi _as low molecular

liquids

and _at about N

=

30 _an increase of Gi of _more than _one order of

magnitude

which is synonymous to an increase of the

3d-Ising

regime ^is

observed. For N _a 30 _a Gi _cc I/N~

scaling

is found in _a range up to N m400. The observed N scaling of Gi _can be

explained

by the large

entropic

contribution of the Flory-Huggins parameter

r~ related _to the

compressibility

_or free volume of the system ⁱⁿ comparison with the

combinatorial entropic term, which _was obtained from the analysis of the scattering ^{results with}

the _crossover function.

1. Introduction.

In _a

recenttpublication _[I]

_{we have used a crossover} function which is

appropriate

_to describe the static _structure factor

S(Q

_~

0),

with

Q being

the

scattering

vector, for various

binary polymer

blends at critical

composition

_~b~ over the whole measured temperature range far from and _{near to} the critical

point

of

phase separation.

The _crossover function used

[2, 3]

is _an

explicit

solution to first order in the

perturbation

_{parameter e}

=

4 d, with d

being

the

dimensionality

of the system, ^based on a renormalization group

analysis.

^This function links the _two

limiting

_cases,

namely

the mean-field

regime

valid _at temperatures ^far from the critical T~ and the

Ising regime,

which is found close _to T~ when fluctuations of the

composition

become dominant. The _crossover function reads

+

=

(1

+ 2.333

((0)~~Y)lY~'fl~[(~'(0)

+

(1

₊ 2.333

((0)~Y)~

_Y'~]

₍₁₎

with +

=

r/Gi where

r is the reduced

temperature

r :=

I/T~ I/Tj/(I/T~),

Gi the

Ginzburg

number and

((0) _=aoGis(0).

_{The critical exponents}

are those for the

3d-Ising

_case

y m 1.24 and A

m 0.51. The _constant ao is

given by

the coefficient of the

squared

term of the order parameter # r)

=

~P

r)

_{~b~ in} the

Landau-Ginzburg-Wilson

form of the Gibbs free

energy of

mixing

for second order

phase

transitions and normalised with RT

(R

_gas constant), hence

with r'

=

I

T~(MF)/T being

the reduced mean-field temperature. ^It can be shown that the

Ginzburg

number Gi controls the _crossover such that for

r « Gi the

susceptibility obeys Ising

and for

r »Gi mean-field behaviour is observed. Gi is

given

in _terms of the _constants

ao, uo, and

c-o = ao

fj,

with

fo being

the bare correlation

length, appearing

in

equation (2)

as

13j

~j

~~

~~ ^{4 4}

~6 /q2

^~~~

" ~0 0 A

Using

the results from the

Flory-Huggins theory [4]

for ao one gets ao =

2(r~

₊

r~)

=

2

r~/T~

with r the

Flory-Huggins

interaction parameter r

=

r~/T r~ (r~, _r~ being

the

enthalpic

and

entropic

part,

respectively). r~

indicates the value for r at the critical

point

with

r~

= + and uo = + with

V, being

the mole-

2

Vi

_~bc

v~(i

_~b~> ³

v, ~b/ v~(i

_~b~)3

cular volumes of the constituents. Then we can derive from

equation (3) [5, 16]

Gi

=

1.79 _x 10~ ^~ ₊

~

~/[(I

+

r~/r~) r~

^A~

N( _{]. (4)}

V,

_~b

Vj(I WI

A is the

flexibility

of the chain

experimentally

obtained

by

the

slope

of the Zimm

plots

_e-g-

S~'(Q)

=

S~'(0)+A.Q~.

A is identical to c-o in

equation (2).

With respect to the N-

dependence

^of Gi, with N the number of _monomers per chain, it follows from

equation (4)

^that

Gi _~

N~'

(5)

if

r~

«

r~,

which is identical _to the results from

Joanny [6]

and de Gennes

[7].

In _case of

r~

»

r~

Gi _~ N~ ^~ The N

scaling

of Gi is determined

by

the chain

length

t,ia

r~,

_uo ^{and the}

entropic

term

r~

of the

Flory-Huggins

parameter. This shows that the mean-field like behaviour of

polymer

^mixtures

originates

in the entropy ^{of the Gaussian chain}

[8]

and the entropy ^related to the

compressibility

^{of the} system

[19].

However, close to T~ ^where

r m Gi the mean-field

theory

is _no

longer

^valid

[1-3]

and the crossover to the 3-d

Ising regime

occurs. In this

regime

_{we use} the

expansion

of

equation (I)

for

((0)

_{» 1,} which is

essentially

the pure

Ising

case to find

[3]

s~

'(o

_j

= T ~/c ~

(6)

with the critical

amplitude

C+

=

Gi~Y

~'~(l

₊

2.333Y'~)Y/(ao. 2.333Y'~). (7)

This

equation

_can be

rearranged

to solve for Gi

thereby using

the fact that ao is

given by

the

inverse of the mean-field critical

amplitude C[~

in

analogy

with

equation (6)

of the

susceptibility

to

give

Gi

=

0.069(C+ /C[~)"~Y

^'~

₍₈₎

Hence the

Ginzburg

number _can be

expressed by

the ratio of the

Ising

to the mean-field critical

amplitudes

of the

susceptibility.

The N

dependence

of Gi in this form is

given

_as follows the N

dependence

of C+ has been derived

by

Sariban and Binder

[9]

to be

C+ _~

NI~~Y' (9)

whereas

C[~

_~ N, ^if

r~

«

r~

is valid. Then _we also obtain Gi _~

N~'

_as

already given by

equation (5). If, however, r~

$

r~

^{the situation} becomes _more

complicated

and Gi in

equation (4)

and

equation (8)

_seems to be

contradictory

in the _sense that Gi _~

l/r~

and Gi _~

rj'~Y

^'~ for Gi in

equation (4)

and

equation (8), respectively.

In our data

analysis

we will _use

equation (8)

rather than

equation (4)

because _we have shown

[I]

that the mean-field

description

for _our

polymer

blends under

study

^is rather

inadequate

^{and the} relation for uo is

only

correct in _terms of the

Flory-Huggins theory.

Since the _use of the _crossover function

has shown to model

accurately

the observed

scattering

data _we will _use them _to determine the N

scaling

of the Gi number

(Eq. (8))

and of the critical

amplitudes

of the bare correlation

length

and the

susceptibility.

The paper is

organized

in the way that first the main

properties

of

the _crossover function used _are

presented

and its

implications

with respect to the traditional

mean-field

picture

are

discussed,

and

second,

the N

dependence

of the Gi number is

presented

and discussed _on the basis of

equation (8).

2. Results.

2.I THE _{CROSSOVER FUNCTION.} In order to demonstrate the

validity

of the _crossover

function used

[2, 3],

we have chosen three

representative

data _sets from the studied systems whose characteristic parameters are collected in table I. These data are very different in their

behaviour. The system d-PB/PS with

(N)

=23

(the

mean N is defined _to be

(N)

=

(~b~/N~

+

~b~/N~

)~ ^'

according

to

r~),

PPMS/d-PS with

(N )

=

50,

and PVME/d-PS with

(N)

= 4 200. All data _{sets are}

plotted

in

figure

I in terms of the inverse reduced

susceptibility (~

'

(0)

and the reduced temperature ⁺ as defined in the _course of

equation (I).

The solid line represents ^the crossover function

equation (I ).

The

figure

is divided into three parts

by

three vertical lines. One is the line

corresponding

to +

= I

(r

=

Gi

)

which marks the

crossover from

3d-Ising regime (+

« I _to the _crossover range in the middle of the

figure.

The

other line marks the _crossover from the _crossover range to the mean-field

regime

and is

given by

+

= 300 _r

=

300 Gi

),

which arises from the

comparison

of _our

prefactor

in

equation (4)

with the

prefactor

in the

corresponding equation originally given by

^Bates et al.

[10] including

their constant _c

=

0.3. As _a further consequence it appears ^to be difficult _even

experimentally

to reach the _true mean-field

regime

Most of the data _are in the _crossover range or even below where

r < Gi. This

finding

is in

quantitative

agreement ^{with the} computer simulations

by

Deutsch and Binder

[11],

where it _was found that _even for N

m 512 _{no true} mean-field behaviour _was found. In

figure

I various other relevant

limiting

^lines are

plotted. Especially,

Table 1.

~ sy~t~m ~ N ~ ~ ~ B ~ ~' ~

~~ ~ ~' ~

~~

l d-PB/PS 1 731 29 1 912 19 0.47 23 1.12 _x 10-2 184 120 7

2 PPMS/d-PS 2 214 19 8 761 88 0.69 41 3.5 _x lo-' 346 512 8.6

3 PPMS/d-PS 2 214 19 17 500 176 0.69 50 3.6 _x 10-' 357 530 10.5

4 PPMS/d-PS 2 214 19 32 743 328 0.84 91 8.8 _x 10-2 65 483 0.2

5 PDMS/PEMS 19 856 260 30 624 40 0.47 297 .17 x 10-2 ₃ 333 2 1806.7

PB

poly(butadiene)

PS

poly(styrene)

PVME

poly(vinylmethylether)

_;

PDMS _:

poly(dimethylsiloxane)

PEMS

'poly(ethylmethylsiloxane)

PPMS

poly(phenylmethylsiloxane).

All mixture8 _are made from component8 ^which were

anionically polymerized,

hence the

polyd18per8ity

18

typically

_u

=

1.05 except the PVME which ha8

u =

2.2.

io~

'

io~ _3d-lsing

_{Crossover MF}

lo'

6

-

o ^,

lo ./

c~a ./

~

~~_~

io _~

~°_~

io~~ io~~ io° io' io~ io~

T

Fig.

I. Double logarithmic plot

(~'(0)

versus + for three polymer blends with

varying

molecular

weight

(Tab. I).

System

N° I d-PB/PS (O), System N° 6 PVME/d-PS (+), System N° 3 PPMS/d-PS (*).

The full line is the _crossover function equation (I). Dashed line 18 the pure 3-d Ising behaviour, dotted

line is the pure mean-field behaviour and dash-dotted line is the mean-field expansion (Eq. (8)) ^of the

crossover function. The vertical lines _are explained in the text.

as

already

mentioned

before,

the dotted line indicates the _true mean-field limit. Even the system ^with

(N)

=

4 200 does reach the mean-field _case

only

for _r

m 300Gi.

Thus,

in

figures

2a, b _we extracted _two

typical

data sets,

namely

N° I and 6 from table I

differing

in

,

~~

d-PB/PS /"

_,

_

'

m '

E '

g

^/

o i

E

~

j

~ ⁱ

£

lo

@

~ m

'

_~ . l

lo '

'

10~~ 10~~ 10~~

al

, ,

~~ ^{/ ,}

, _,

d-PS/PVME ,,'

^'

, , -

,'

m ,

~ ,

~

,,

fl 10° ,,'

E ,

,

u~ ,

~O

@

7

1°

/"

,'

_~

~~

0~

~

b)

Fig.

2. Plot of

S~'(0

₎

versus r for 8y8tem ^{N° I} d-PB/PS. Full line is the _crossover function (Eq. ii.

The dashed line 18 the pure Ising case and the dash-dotted line is the mean-field

expan8ion (Eq.

(8)) of the

crossover function, b) Like 2a) ^but System ^N° 6 PVME/d-PS instead.

(N ) by

two orders of

magnitude.

The data are

plotted

in terms of S~ ^'

(0)

_versus

r. For the d-

PB/PS system

(Fig. 2a)

one

recognizes

that _over the whole temperature range measured the data _are almost

properly

matched

by

the strict

Ising

case

(dashed line).

The _crossover function

(Eq. (I)),

however,

gives

a much better fit for the

higher

temperatures. ^{The situation is}

completely

different for the

PVME/d-PS

_system

(Fig. 2b)

where the

Ising

law fails

(dashed line)

almost

completely.

The

Ising

law has been evaluated from the _crossover fit via

equation (7).

The _crossover function fits the data _over the whole

temperature

range studied.

With respect to the _{true mean} field behaviour

(c.f. Fig. I)

it is

interesting

to

expand equation (I)

for

r » Gi _to get

[2, 3]

(~

'

(0)

=

(+ 1)(1 1.098(+ -1)~°~" _). (10)

Thus the

e-expansion

does _{not recover} the proper square-root ^{of the}

temperature

^correction to the classical limit

implied by

the Omstein-Zemike

theory [3]

and

given by [12] (~'(0)

=

(+ 1)(1 (+ 1)~°~).

_{However, this becomes}

unimportant

in

practice

_{as one can see} in

figure

2b, where

equation (10)

is

plotted

as a dotted line.

Obviously,

^the

expansion

is

equivalent

to the _crossover function within the

experimental

_error

already

for

r values smaller

than those

beyond

_r 5S300Gi in

figure

I where the deviation from

mean-field, namely

((0)

= +, is smaller than 10 iii with respect to the _crossover function

[1].

2.2 THE

(N )

SCALING OF Gi. After

having

establ18hed the _crossover function _we first focus

our attention _on the

problem

whether the

(N )

calculated and listed in table I is _a

properly

and

well defined

quantity.

One

might

_argue that all systems ^under

study

are

naturally

not

completely symmetric

with respect to

N,

_~b, and the segment

length

of the

components.

^In

principle,

_one could

speculate

that tedious corrections with respect to the ideal

symmetric

case

are necessary,

especially

if _{one wants to} compare our results with those obtained

by

simulations

[11].

To test that _we have used the

predictions

of the

(N) scaling

of

to given

in reference

ill

fo~N~'~"' ₍₁₁₎

with u m 0.63. The results _are

plotted

in

figure

3. From the obtained agreement ^between

experiment

and

theory

we conclude that _our

(N )

is well defined and _no further corrections _are

necessary. However, the data

point

^for system ^{N° 6}

(PVME/d-PS)

has been

dropped

because it is

completely

off. The very

asymmetric

character of this mixture and the very

polydisperse

PVME component seems to disturbe the mean-field type ^of

equation

for

determining

ioo

?

^~~~

=

N°'~~

o

~'

~

io ioo <N> iooo

Fig. 3. Double logarithmic plot of the critical

amplitude

fo of the correlation length f fo r~ ^~ _as a

function of the _mean

degree

of

polymerization.

For the theoretical

prediction

_Bee

equation

(I1).

(N )

which otherwise worked well. All data

points

of PVME/d-PS will therefore be

dropped

from the

following

considerations.

From the fit of

equation (I)

to the

scattering

data

(Fig. I)

_{two parameters are}

obtained, namely,

the mean-field critical

amplitude C[~

=

liar

and the

Ginzburg

number Gi. Both parameters are listed in table I. After

having

established the _mean

degree

of

polymerization (Fig. 3) (N)

_we will _now concentrate on the

plot

of Gi _versus

(N)

in _a double

logarithmic

scale _as shown in

figure

4. First we will discuss the data for

(N )

_a 30.

Here,

the data follow _a

straight

line which scales _as Gi _~ N~ ^~ This is _not

predicted by theory (Eq. (5)

and

(8))

for

r~

«

r~.

In order _to add further

experimental findings

to our data listed in table I _we have

incorporated

_a data

point by

Theobald

[13]

obtained

by light scattering

on a similar system to

our N° 5 in _terms of the _crossover function

equation (I ).

The data

point

for

(N )

_s 30 from _our

system N° I has _a much smaller Gi value than the systems ^2-3

beyond (N)

_a 30

by

about _a factor of 30. In order to understand that _we first calculate the value for Gi in the Omstein-

Zemike definition

[3].

It is estimated _to be Gi

= 0.01

using

values for ao and uo

appearing

in

equation (2)

and in the definition for Gi

(cf. Eq. (3))

from the _van der Waals ga8 model _to be ao =

9/4 and uo =

27/8. Anisimov et al.

[3]

have furthermore listed values for Gi for small molecules like

H~O, CO~

etc.

They

_range between 0.009 _< Gi

< 0.04.

0.1

~5

0.01

~

io ioo

~~~ iooo

Fig. 4. Double logarithmic plot of the Ginzburg number Gi _i,ersus the _mean degree of polymerization.

Data

point

(O) from Theobald [13]. The theoretical value for Gi _a8 calculated from _an Ornstein Zernike model [3] is indicated _on the left side. For

explanation

of the shaded vertical line _at

(N

30 _{Bee text.}

The other besides Gi-

experimentally

^accessible

quantity

is

(as already mentioned)

the

amplitude C[~

=

liar

as listed in table I and

plotted (and

further discussed) in

figure

7. Both values,

namely

Gi and

C[~ immediatly give

the

Ising

^critical

amplitude according

to

equation (7).

Its

N-dependence

is

predicted by equation (9)

_to be C+ _~

N°.~~

as

plotted

in

figure

5. Within _a factor of _two the data show the

predicted

N

scaling

of the crital

amplitude

C+ From that we furthermore conclude that the determination of ao and Gi is ^not

ambiguous,

although

the

product

_ao. Gi _enters in

fitting equation (I)

to the data.

3. Discussion.

One fundamental

question

which arises while

interpreting figure

5 lies in the

physical

nature of the shaded _area at

(N)

_m 30. This line

obviously

separates two

completely

different

regions

with respect to the widths of the

Ising regime.

In order _to

subsequently

discuss all relevant

points

of the

quite complex figure

4 _we start with what is

relatively

settled, ^that is the behaviour _at low

(N)

S 30.

There,

all available data behave

according

to the theoretical

prediction [12]

which further _means that _a

polymer

mixture with

(N )

= 23

(our

system I is in

this _{sense not a}

polymer

mixture.

According

to Deutsch and Binder I I

scaling predictions

for

N < 32

polymer

^mixtures are in better agreement ^{with those for semidilute}

binary

blends in _a

common solvent. Therefore the shaded _area separates ^the

region

where for N

< 32 excluded volume and end group effects may

play

an

important

role. This behaviour

might

be similar _to the _case of the determination of the

glass

transition T~ ^of

polymers

_{as a} function of the

degree

of

polymerization.

Since the number of molecular ends is

proportional

_to

I/N,

their influence diminishes with

increasing

N and

accordingly

_one finds T~ =

T~(N

_~ m

)

₊ Const./N

[14].

For N

m 30 _{we can} reach the

T~(N

~ m within lo iii

[15]

which _means that

already

for such

relatively

low molecular

weights

end effects _are

getting

less

important.

There _are furthermore data

by

Hobbie _et al.

[17]

on a system ^with

(N )

₌ 20 similar _{to our} system I which also has been

analyzed

with

equation (I) yielding

_a value for Gi

=

0.003

being

even smaller than the theoretical value. This too low value _{seems not to} be

reasonable, however,

the

given

trend is confirmed.

E m~

i coo

f

E

~2-y_

_~ ^o.76

ioo

io ioo <~> iooo

Fig. ^5. Double logarithmic plot of the critical amplitude ^C+ (Eq. (7)) of the Susceptibility as a

function of the _mean

degree

of polymerization. The theoretical slope according to

equation

(9) is shown.

A relevant

question

remains is the _« step » in Gi _at about

(N )

=

30

already

mirrored

by

the temperature

dependence

of S~

'(0)

_{or to say} do _we find in _our data this substantial difference in the

Ising

widths between

(N )

< 30 and

(N)

m 30 ? This is _{as more as an}

important question

because from computer simulations

ill]

such _a behaviour is not found.

According

to these _we would have

expected

a smooth _crossover between the

(N )

smaller and

(N ) larger

than 30

regions.

In order _to show that this step ^{is indeed} our

experimental

result _we have chosen the data sets of _our systems ^{N° I and N° 3} which _are

according

to

figure

I almost in the _same + range of _{our crossover}

plot

and have

plotted

S~ ^'

(0

) 1,ersus I IT in

figure

6. The

fitted

(linear)

^lines

according

to the mean-field

prediction, although

not correct, is sufficient to show the differences of the

Ising

widths. As _{one can see} from the

figure,

the width of the

Ising regime

indicated

by

AT* of the

(N)

=

50 system ^is

higher by

at least _a factor of five in

comparison

to the

(N )

=

23 system ⁱⁿ

qualitative

_agreement with the observed step ^{in Gi.}

Data

by

Bates et al.

[10]

with _a

(N )

=

57 and _a

chemically

different system ^than our systems

2-4 show a crossover width of about AT*

= 30 K which is similar to our result of the N m

(50)

_systems. From all these

findings

we have _to conclude that the

quantitatively

different behaviour of Gi above and below

(N)

_m 30 is

experimentally

well settled. This

finding

also

explains why

our

system

^{I with}

(N)

= 23 had _a constant c

[16], being

proportional

to the

prefactor

in

equation (4),

which is _at least _one order of

magnitude

smaller than the universal value of this

prefactor

_as

predicted by

Bates et al, ii

0]

for

poly.

.~r mixtures.

In order _to

explain

the nature of the behaviour of the

(N) dependence

Ji above

(N )

=

30 _as

depicted

in

figure

4 _we will

analyze

the

(N) dependence

of the

respective

critical

amplitudes appearing

in

equation (8).

In

figure

5 the N

dependence

of C+ has been

shown _to follow the

prediction

of

equation (9).

Then

accordingly

the

(N) scaling

of

C

[~

which is derived from the ao values as listed in table I is of immediate interest. The results

are shown in

figure

7. The data follow within _a factor of _two the

expected

N

powerlaw.

This is about the _same accuracy found for the N

scaling

of the critical

amplitude

C + in

figure

5. The N

dependence

of Gi therefore _comes from the deviations of both

amplitudes

within the shaded

areas in

figures

5 and 7 and the

high

_exponent of

I/(y -1)

=

4.17 in

equation (8).

To understand the N~ ^~

scaling

of Gi above

(N )

= 30 we have to

analyze

further the values of

ao =

2(r~

+

r~).

Because the

purely

combinatorial entropy part

r~

is determined

by

the

molecular volumes and the

composition

of the components

r~

_can be evaluated from

ao. ^In

figure

8 both

r~

and~

r~

^of our systems 1-5 _are

plotted

_versus

(N).

The

(N ) dependence

of

r~

is

r~

_~ N ^' _as

expected.

The

point

at

(N )

=

91 is _a little off which is correlated with its

relatively high asymmetric components (Tab. I).

The

really

remarkable effects _are the absolute values and the

(N ) dependence

of

r~.

First of all the absolute values of

r~

and

r~

_are

comparable

_over the whole

(N)

_range studied and for systems 2-4

r~

is

larger

than

r~ by

about _a factor of three. Hence the

inequality r~

«

r~

^{which leads} to

a Gi _~ N~ ^' is _never fulfilled. The

straightforward

calculation of

r~

in the _mean field

picture

leads _to values

roughly

one order of

magnitude

lower than those shown in

figure

8. However,

io

8

mj j

~

_~_AT*

AT*

E I

£

_~

64

'm

2

o

2.6 2.8 3.0

lfr

[10~~ lN]

Fig. 6. Plot of S~ ^'(0) _versus I/T for systems ^{N° I} (-O-) and N° 3 (-A-) from table I. The two different Ising widths AT* _are indicated. An about 5 times

larger Ising

_range is found for the

higher

molecular

weight

system ^{3 in}

qualitative

agreement to the observed

jump

of Gi in figure 4.

+

E loco

> £ _~l

~

+x LJ

ix

io ix

<N>

Fig.

7. Double logarithmic plot ^of the critical amplitude

C[~

of the

susceptibility

_{as a} function of the

mean

degree

of polymerization. ^The theoretical

slope according

to

equation

(9) is shown.

'«

",

Q/

¹⁰

i~,

E

"~

U '

, '

-

'

O '

~ _~ ^'

'

y I

# I

~ i

U~ ⁱ

d

I i

io ioo

<

fi>

Fig. 8. Double logarithmic plot of the combinatoric part r~(-O-)) and the

entropic

part r~ (-x-) of the

Flory-Huggins

parameter as a function of

(N),

r~ _seems to be responsible for the observed N dependence ^{of Gi because it is of the} same size _{or even}

larger

than

r~.

this

discrepancy

vanishes within 20 iii for _our systen> 6, when

(N )

is _so

high

that _we

really

can measure near the mean-field

regime

_as indicated in

figure

I for _r a 300 Gi. We

speculate

when

(N)

is

getting

much

larger

than of the order of the order of 100 that then

r~

is

getting

more and _more

unimportant

with respect to

r~

and Gi _~ N~ ^'

scaling

will then

probably

hold. Th18 _can be _seen from

figure

8.

Secondly,

_we notice that there 18 _an

intermediate

(N)

_range

(between

30<

(N) <200)

where

r~

_e.g. the free volume

requirements

of the

quite

short chains _are dominant. We _{are not aware} of any theoretical

description

for the N

dependence

of

r~

but could claim that this effect studied here is considered to be of

general

nature because it is related to the expense to which _a

given

volume is

occupied by

chain segments ^and

by

that characteristic for

polymers.

These

findings

_get

support ^from

quite

recent lattice cluster

theory computations by

Dudowicz et al.

[18]. They

find _{a two to} three orders of

magnitude higher

Gi for

compressible polymer

mixtures in

comparison

to results from

equation (4)

for

incompressible

_systems. These calculations have been

performed

for

PVME/d-PS

in the _same

(N)

_range of _our system 6. Even

though

their calculated Gi values _seems somewhat _too

high

ⁱⁿ

comparison

to our Gi in table I for PVME/d- pS these results confirm the influence of

r~

onto Gi and the width of the

Ising regime [20].

In

future work it _{seems to} be

interesting

to influence the

r~

term

by applying

_pressure and

furthermore,

_{to use systems to span} the

(N)

_range studied here

[19].

4. Conclusions.

In this

publication

we have discussed the consequences of the

analysis

of the structure factor at

Q

₌ 0 obtained from

scattering experiments

with _neutrons and

light

for

polymer

blends in

terms of the _crossover function derived

by Belyakov

et al.

[2, 3].

^This

analysis

has been

demonstrated in

publication [I]

and has been

performed

for various

polymer

blends with _an average

degree

of

polymerization

between

(N )

=

23 and

(N)

=

4 200

(Tab. I).

Our first main result

is,

_as demonstrated in

figure

and

figure

2, that the mean-field

regime

_even for _our system ^{6 with}

(N)

=

4200 is _never reached

Therefore,

this work

helps

to

clarify

the

meaning

of the

Flory-Huggins

_parameter r under the influence of fluctuations.

Usually,

this

quantity

has been determined

by

_an

extrapolation procedure (r~'~~)

_as described in refer-

ence

[5].

This

procedure implies

a linear behaviour of

S(0)~,'

_vs,

r which

usually

has been identified with the

validity

of the mean-field

theory

in the

region

of the

extrapolation.

The

observed

consistency

of this

description

_was

supported experimentally by

the reliable

reproduction

of the

phase diagrams [21].

However, ^it has been demonstrated in _our work

(Fig. I)

that the mean-field

theory

holds

only

at temperatures ^{rather far from} T~, ⁱⁿ a

region normally

inaccessible

by

the

experiment.

Instead, the

Flory-Huggins _parameter

_was evaluated

by

_a fit of

S(0)~

^' i,s, r

by

the _crossover function derived

by Belyakov

et al.

[2]

which

yields

the

Ginzburg _parameter

Gi and the critical

amplitude C[~

in

equation (I).

This leads _to the

« true meanfield value _»

r~[

_{Its value may differ}

considerably

from the

extrapolated

apparent meanfield value r~"~ _{described before} (see Tab. I in Ref. ii

])

and demonstrated in

figure

8 for

r~

and

r~, respectively. Only

the value of

r~~

_{could be identified with}

a calculation which, e.g., use the mutual interaction between the _monomers as in

[18].

Furthermore, we have discussed the

(N) scaling

of the

Ginzburg

^{number Gi and of the}

critical

amplitudes

of the bare correlation

length

and

susceptibility.

While the critical

amplitudes

lead _to the

expected scaling

as seen in

figures 3,

5, 7

(for

the

susceptibility

within a

factor of two) the

IN ) dependence

of Gi is

quite unexpected.

^For

(N )

< 30 _we find for _our

system ^{I the Gi for low molecular}

liquids

and in agreement ^{with the calculated value from the}

van der Waals model

[3].

At

(N )

_m 30 _we find _a sudden increase of Gi

by

more than _one order of

magnitude

and then for

larger (N)

_a Gi ~

N~~ scaling

law. The increase of Gi _near N

= 30 is

clearly

observed from the

scattering

data

by

the

larger

^{width of the}

Ising regime

in

figure

^{6 also in} agreement ^{with results from other} groups

[10,17].

^Also the too small

c parameter ^{in the}

Ginzburg

^{criterion of}

equation (4)

of _our system ^I as discussed in reference

[16]

in

comparison

with those of other

(N )

m 30

polymer

mixtures is indicated

by

the increase of Gi _at

(N )

=

30. A Gi _~

N~' scaling

is

only expected

if

r~

«

r~,

a relation which is

usually

^believed to be fulfilled from conventional

analysis

in terms of the mean-field

approximation.

However,

according

to our

analysis

^of

S(0)

with the _crossover function of

equation (I) r~

is found to be of the _same order of

magnitude

_as, or even

larger

than

r~

for N up to 300. There is _an indication that

r~

«

r~

is fulfilled for N considerable

larger

than 300. Because Gi is

proportional

to the ratio of the critical

amplitudes

of the

susceptibility

in

Ising

and mean,field

asymptotic regimes

to the power

I/(y 1)

₌ 4.17 Gi very

sensitively depends

on the detailed N

dependences

^{of C+} and

C[~

and therefore _on

r~

^{which is} the

entropic

term of the

Flory-Huggins

parameter ^{and related} to the

compressibility

_or the free volume of the system

[18].

Because the

large

value of

r~

in

comparison

to

r~

_seems to be

responsible

for the increase of Gi _or of the width of the

Ising regime

at about

(N )

=

30,

Gi _must

depend strongly

on pressure in this N range. Monte Carlo simulations I I seem to support our results, ^{because the}

scaling

behaviour similar _to

good

solvent systems ^is found below N

m 30.

Acknowledgements.

The authors _are indebted to Professors K. Binder, J. V.

Sengers

and T.

Springer

^for

helpful

discussions.

References

[ii Meier G., Schwahn D., Mortensen K., JanBen S.,

Europhys.

Lett. 22 (1993) 577.

[2] Belyakov M. Y., Kiselev S. B., Physic-a A 190 (1992) 75.

[3] Anisimov M. A., Kiselev S. B.,

Sengers

V. J.,

Tang

S., Physic-a A 188 (1992) 487.

[4] Flory P. J., Principles of

Polymer Chemistry

(Cornell Univ. Press, Ithaka, 1953).

[5] Schwahn D., JanBen S.,

Springer

T., J. Chem.

Phys.

⁹⁷ (1992) 8775.

[6] Joanny J. F., J. Phys. A II (1978) l17.

[7] de Gennes P. G., J.

Phys.

Franc-e Lett. 38 (1977) 441.

[8] Binder K., J. Chem. Phys. 79 (1983) 6387.

[9] Sariban A., Binder K., J. Chem.

Phys.

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[10] Bates F. S., Rosedale J. H.,

Stepanek

P.,

Lodge

T. P., Wiltzius P., Fredrickson G. H.,

Hjelm

R.,

Phys.

Rev. Lett. 65 (1990) 1893.

Ii ii Deutsch H. P., Binder K., J. Phys, if Franc-e 3 (1993) 1049.

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[13] Theobald W., PhD Thesis

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Flory

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lI6] JanBen S., Schwahn D.,

Springer

T., Phys. Ret'. Lent. 68 (1992) 3180.

[17] Hobbie E. K., Reed L.,

Huang

C. C., Han C. C., Phys. Rev. E 48 (1993) 1579.

[18] Dudowicz J., Lifschitz M., Freed K. F.,

Douglas

J. F., J. Chem.

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[19] JanBen, S., Schwahn D., Mortensen K.,

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T., Macromolecules 26 (1993) 5587.

[20] The Gi number in reference [18] is defined with five times

higher

prefactor than _our Gi. Furtherrnore their equation (4.2) has _to be divided

by

9 because of uo = [II

(Vj ~/)

_{+ I/(V~(1}

~~)~)].

[21] Schwahn D., Mortensen K.,

Springer

T., Yee-Madeira H., Thomas R., J. Chem. Phys. 87 (1987) 6078.