The mean field to Ising crossover in the critical behavior of polymer mixtures : a finite size scaling analysis of Monte Carlo simulations

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The mean field to Ising crossover in the critical behavior of polymer mixtures : a finite size scaling analysis of

Monte Carlo simulations

H. Deutsch, K. Binder

To cite this version:

H. Deutsch, K. Binder. The mean field to Ising crossover in the critical behavior of polymer mixtures :

a finite size scaling analysis of Monte Carlo simulations. Journal de Physique II, EDP Sciences, 1993,

3 (7), pp.1049-1073. �10.1051/jp2:1993182�. �jpa-00247882�

Classification Physics Abstracts

64.70J 64.60F 61.25H

The

_mean

field to Ising

_crossover

in the critical behavior of

polymer mixtures

_{: a}

finite size scaling analysis of Monte Carlo simulations

H. P. Deutsch

(~)

and K. Binder

Institut fur Physik,

Johannes-Gutenberg

Universitit Mainz,

Staudinger

Weg 7, 6500 Mainz,

Germany

(Received 3 Februaiy J993, accepted 6

April

J993)

Abstract. Monte Carlo simulations of the bond fluctuation model of

symmetrical polymer

mixtures (chain lengths N~ ₌

N~

= N) _are analyzed near the critical temperature

T~(N

) of their unmixing transition. Two choices of interaction range are studied, using _a square-well

potential

with effective coordination number

z~~~=14

_or

z~aws, respectively,

at a volume fraction

#

= 0.5 of

occupied

lattice sites, and chain

lengths

in the range 8 _w N _w 512. A linear relation between N and

T~(N

^is established, _T~(N

= AN + B, where the correction terra B is

positive

for z~rr = 14 but

negative

_{for z~ir}

= 5. The critical behavior of the models is

analyzed

via finite size

scaling techniques, paying

attention to the _crossover from three-dimensional

Ising-like

behavior _to mean-field behavior, using a forrnulation based _on the

Ginzburg

criterion. It is shown that the location of the _crossover does _not

depend

on z~r~, consistent with the

expected

entropic

origin

of the mean-field behavior for long chains. However,

despite large

numerical efforts only a crude estimation of the _crossover

scaling

function of the order parameter

(describing

the coexistence

curve of the blends) and of the

singular

chain

length dependence

of the associated critical

amplitude

is

possible.

These results _are discussed in the _context of

pertinent

theories and related experiments.

1. Introduction.

The critical behavior of

polymer

mixtures has become the

subject

of intensive research in

recent years

[1-29].

There are many reasons for this interest.

E,g.,

while

Ginzburg-type

criteria

[30-35] imply

that for

binary (AB)

mixtures of flexible

polymers

in the

symmetrical

_case

(chain lengths N~

=

N~

=

N)

for N

- ~xJ the critical behavior is described

by

the Landau

theory [36] (mean-field

critical

exponents),

the

universality principle [37, 38] implies

that very

near to the critical temperature T~ one should _see the _same critical exponents as for the three- dimensional

Ising

model

[38-40].

This

prediction

has been indeed verified both

by

computer

(*) Piesent address Andersen

Consulting,

Arthur Andersen & Co., _s-c-, Otto volger ^Str. 15, 6231 Sulzbach/raunus,

Germany.

simulation

[2-4, 17-21]

and

experiments [5-15].

The _crossover from fluctuations-dominated

lsing-like

critical behavior very near T~ ^to classical mean-field like behavior farther away from T~ has found

longstanding

attention

theoretically,

in

particular

for the

gas-liquid

critical

points

in

simple

fluids

[41-57J.

The _crossover behavior is found _to be controlled

by

two variables

[57

a

microscopic

cutoff wavenumber

Ao

^{of the} critical fluctuations at short

wavelengths,

and _a

system-dependent

parameter, ^the «

Ginzburg

number G~ », which may be

expressed

in _terms of the

(mean-field)

critical

amplitude /f~

_{of the} correlation

length

f ^of order parameter

fluctuations,

f

=

/f~(-

_t

_)~

^'~~

,

t ₌

T/T~ (1)

of the

scattering

function

S(q

-

0)

at _zero wavenumber _q

-

0,

at critical concentration (or

density respectively)

s

(q

_- o) ₌

?F(-

_t)-

(2)

and of the order parameter

(

_m

),

1lm II

=

b~~

_t'~~

₍₃₎

This

Ginzburg

number then can be written _as

(co

is the volume per molecule in the _case of

small molecules _or the volume per monomer in the _case of

polymer mixtures) [35, 57J

~

~" l~ ^BIF)4 ~°~~~l(I)F)3

~~~

For t » G~ the classical _mean field

theory

holds while for

it

« G the

Ising

^critical behavior

applies [37-39],

f ₌

I+ _(- t)-~ s(q

-

oi

=

©+ (- t)-Y, iimii

=

h(t)P ₍₅₎

with exponents

II 0]

v - 0.63

,

y i 1.24, p _- 0.325

(6)

If _one could

disregard

the cutoff parameter

Ao,

^the crossover between these two types ^of

critical behavior would

simply

be controlled

by

one universal function of the variable

t/G,

for each

quantity

of interest it appears that such a universal _crossover

scaling description

is _at least _a reasonable

approximate description

for

simple

fluids

[57].

However, there the

situation is obscured

by

the fact that the

Ginzburg

number

G,

is _never very small

(e.g.

G,

=

0.014 for

CO~ [53]

_or

G,

=

0.025 for

H~O [53]),

and since the _crossover spans several decades of the reduced variable

t/G,,

one leaves the critical

region

where

power-laws

such _as

equations (1-3), (5)

would be _a reasonable

description

for

t/G,

w

altogether.

Now

polymer

mixtures _are, in

principle, ideally

suited _to

study

such

problems,

since

G,

can be made

arbitrarily

small for

long enough

chains. This is _seen

inserting

the results of the

Flory-Huggins [58, 59] theory

for the critical behavior in

equation (4), namely [30-33]

~

(71

_~~

j'3

,

j, _©?~

= N"

'

~~~

' ~

'

which

yields

G,

=

3~/[2~ gr~N ]12.308/N (8)

Note that apart ^{from the chain}

length

there _{are no}

material-dependent

parameters ⁱⁿ

equations (7, 8),

which reflect the fact that the mean-field behavior of

polymer

mixtures

derives from the entropy ^{of Gaussian coils}

[30-34]. Consequently,

it is

implied

that in _terms of the variable

t/G,

=

Nt/2.308 all

symmetric polymers

should exhibit the _same behavior,

irrespective

of the chain

lengths

and their chemical nature, etc., if the universal _crossover

scaling description suggested by

the

theory [47, 56, 57]

holds. Since this

reasoning

_can be

extended _to somewhat

asymmetric polymer

mixtures

[7]

all what

happens

is _a

change

of the factor

relating G,

and the chain

lengths N~, NB

-, it is somewhat

disturbing

that the _recent

experiments

indicate

problems

with this universal _crossover

scaling description [10].

^In the present paper we address this

problem extending

our simulation

approach [17-21]

to the bond fluctuation model

[60-62]

of

symmetrical polymer

mixtures.

By studying

the critical

region

of this model for _a wide range of chain

lengths,

from N

=

8 to N

=

512,

we

provide

further evidence _to the fact

[18, 19]

that

Flory-Huggins theory [58, 59] yields

the correct exponent ⁱⁿ the variation of

T~(N

with N, _T~

(N

_cc

N, although

the

prefactor

in this linear relation is not

reliably predicted.

These results

[18,19]

have been corroborated

by

_a recent

elegant experimental study [22]

and _a reformulation

[26]

of the

polymer

reference interaction site

model

(PRISM) theory [23-25],

and thus it is clear that the earlier result

[23-25]

T~

(N

cc

,I

was due to an artefact of the _mean

spherical approximation decoupling

scheme.

It is

hoped

that the additional data

given

in the present paper will be useful _{as a} further test of various

approximate

theories

[26-28]

for the

thermodynamics

of

polymer mixtures,

and thus contribute _{to our} theoretical

understanding

of _an

important

class of materials

[29].

In section 2 _{we now} summarize the main characteristics of the present ^{model and the finite} size

scaling analysis

used _to extract information _on critical

properties

from the simulation data.

Section 3 surveys these critical

properties

for the various chain

lengths studied, discussing

also

our conclusions for

T~(N ).

Section 4 presents ^details on crossover

phenomena

ⁱⁿ finite size

scaling.

Section 5 discusses _our attempts to

numerically

estimate the _crossover

scaling

function of the order parameter,

using

the variable tN

(which

is

simply proportional

to

t/G,,

as discussed

above),

and discusses the related

problem

of the chain

length dependence

of critical

amplitudes. Comparing

data for _two different versions of the model

(in

the present model the interaction _range is _very short, effective coordination number _z~~~- 5, while in

previous

work _we used

[17-2 Ii

_=~~~

i

14),

_{we are} able _to

investigate

to what _{extent crossover}

phenomena

_are universal. Section 6 then contains _our conclusions.

2. A brief review of the model and the

techniques

of simulation and

analysis.

The bond fluctuation model

[60-62]

is _a

coarse-grained

model of flexible

polymers,

where

« effective _{monomers »} block all 8 sites of _an

elementary

cube of the

simple

cubic lattice for further

occupation,

and _are

jointed by

_« effective bonds _»,

represented by

vectors

I

_{from the}

set

P(200), P(210), P(211), P(300)

and

P(310).

Here and in the

following

all

lengths

are

measured in units of the lattice

spacing,

and the

symbol

P stands for all

permutation

and

sign

combinations of the Cartesian coordinates

(I, I, I=),

One

imagines

that each effective bond

corresponds

to the end-to-end distance of _a group of _n

-

3-5 successive chemical _monomers

along

the backbone of the chain

[63, 64].

Thus it is natural that the

length

^I of the effective

bond is not fixed but can fluctuate in _some range. Previous work

[61, 62]

has demonstrated that for _a volume fraction

4

=

0.5 of

occupied

lattice sites the static and

dynamic properties

of

long

chains

correspond

to chains in dense

polymer

melts, excluded volume interactions

being

screened out on a

length

f 16, ^I-e- about _two average bond

lengths.

At this volume fraction the acceptance rate of the

attempted

_moves, random

displacements

of the effective _monomers

by

one lattice

spacing,

^{is still}

reasonably large (A

1 0.15

[62] ),

^and hence the coils _renew their

configurations reasonably

fast. The athermal version of the model _can be vectorized rather

efficiently,

about 2 million

attempted

_{monomer moves per} second _are reached _at CRAY-YMP processors

[65].

This model of

polymer

^melts can be

straightforwardly

extended to simulate

symmetrical

blends

by associating

two types ^of

identities,

A and B, with the

chains,

and

introducing pairwise

interactions e~~

(r

), _e~~

(r )

and e~~

(r

between

AA,

AB and BB

pairs

of _{monomers at}

separation

r.

Only

two

special

choices of these interaction parameters, ^and two choices for the range of this interaction _are considered.

(I)

The interaction is _nonzero

only

for _monomers of different kind

(e~~

_{= e~~ =} ^{0 and for} the smallest distance that two monomers can have,

namely P(200)

_; note that smaller distances

between the center of _mass of two effective _{monomers cannot occur} due _to their extended size

and the excluded volume interaction. We then denote _e

= e~~.

(iii

^The interaction

e(r)

=

p~~(r)

₌

p~~(r)

₌

p~~(r)

^is _nonzero and _constant for all three bond _vectors

P(200), P(210), P(211)

that contribute _to the first

peak

of the radial

distribution function _g(r between _monomers

[6 Ii-

This medium range interaction is

probably physically

_more realistic, but the program ^executes about _a factor of two slower than the choice

(ii,

since in

computing

the energy

change

AE caused

by

_an

attempted hop

of _{a monomer one} has _to search for _«

neighbors

_» in _a

larger surrounding region.

Thus the

performance

of the code in the presence of these interactions is

only

about 500 000

attempted

_{monomer moves per}

second,

Since in the

Flory-Huggins theory [58, 59]

the critical temperature ^of a

polymer

mixture is written, for _a

selfavoiding

walk model of chains of N steps ^{in the dense limit}

(4

= II, _as

k~

_{T~ =} ziN/2, with P

= e~~

(e~~

+

p~~)/2,

where _z is the coordination number of the lattice, it is of interest to estimate the

analog

of _z for the present model

containing

these

extended _monomers. We define _an effective coordination number in _terms of the radial

distribution function

g(r)

around the center of

gravity

of _{a monomer as}

Zeff "

£ E(r) g(r)/F

,

(9j

p without

argument being

the

strength

of the

(constant)

interaction

potential,

as defined above.

Using

the results of reference

[61]

_we find

z~~~15

^for model

(ii

and _z~~~1 14 for model

(iii.

Note that these numbers include contributions from

neighbors along

the

chain,

which do not contribute to the

unmixing

of chains

[2-4].

Due to a small temperature

dependence

of the coil

configurations

there is also _a very weak

temperature dependence

of

z~~~ to be

expected,

but this

was not

investigated,

In _a

semi-grand

canonical ensemble

description

of _a

symmetric polymer mixture,

where

chains may switch their

identity

A -+B, but the total number of chains is constant, the

temperature ^{T and the chemical}

potential

difference AR between A and B-monomers _are the

(given) independent

thermal variables, Phase

separation

and also the critical

point

_occur at

AR

₌

0,

because the system ^is

symmetric against

_a

change

of chain _« labels _» A, B and

simultaneous

change

of volume fractions

4~

_-+

4~, Denoting

the numbers of A-chains in the cubic box of linear dimension L _as n~, the number of B-chains _as n~, the basic observable of

interest is the _« order parameter » m,

mmAn/n, nmn~+n~,

Anmn~-n~. (10)

A realisation of the

semi-grand

canonical ensemble is

easily

obtained if _one considers in the simulation

apart

^{from the local}

hops

^of monomers, which

only

lead _to relaxation of the chain

configurations

and diffusion of the

chains,

a second type ^of move, where _one attempts to

« relabel

» a chain A

- B. This _{means, a} chain of _one type is

replaced by

a chain of the other

type ^{but in} an identical

configuration,

if the _move is

accepted.

Of _course, in both kinds of motions the

Metropolis

transition

probabilities involving

the energy

change

AE have to be

considered,

_as is standard in Monte Carlo

sampling [2-4,

18,

20, 66].

Using histogram techniques [20, 67]

the

probability

distribution

P~(m)

of the order parameter as well _as its low order _{moments are} obtained from the simulation in _an effective way, as described in detail in

[20],

Due to a

large

investment of computer ^time on the CRAY- YMP

8/32

at the

Hochstleistungsrechenzentrum (HRLZ)

in Jiilich and due _to this economical data

analysis,

it has been

possible

to

study

_{systems up} to chain

lengths

of N

=

512 effective

monomers on lattices up to linear dimensions of L

=

160,

I,e. systems

containing

256 000

monomers.

Altogether

about 3 000 hours CPU time _were needed for the data discussed in the

present paper, As discussed in

[18, 20, 62, 65],

a

significant

fraction of computer time is needed _to prepare a

well-equilibrated

initial

configuration

of the

polymer

melt with the

largest

chain

length

at each lattice size.

Configurations

at the _same values of L but values of

N smaller

by

_{some power} of 2 _are

easily produced by cutting

the chains in

pieces

^of

equal

length

and further

equilibration.

These

configurations

_are stored and reused _as

starting configuration

for the blend simulations

(where arbitrary assignment

of labels

A,

B realize

fully

ordered _starts with

only

A-chains _or

fully

disordered starts, with random mixtures of A

and

B,

for

instance).

These data for

(

_m

), (m~), (m~)

and suitable ratios of these _{moments are} then

analyzed by

standard finite size

scaling techniques [2-4, 17-21, 66, 68-70], Figures1-3

_{present some}

examples

of _our data for the model _case

(I)

_more data for _case

(iii

can be found in references

[17, 18, 20].

It is _seen that for the choices of L which _were

analyzed

there is indeed _a very

strong ^{finite size}

rounding

of both

( (m )

and the response functions S~~jj, S[~,j ^{in the critical}

region,

rather than

displaying

the power laws written in

equation (5).

The response functions S~~jj, S[~jj are defined here in terms of fluctuation relations

S~~jj = n

(m~) (m) ~)

,

(=

_n

(m~)

for AR

=

0)

,

(11)

Siou

= n

( lm~) I

_m

) ~i (121

Note that

equations

(^II

), (12)

_are defined in such _a way

[66]

that S~~j, is

proportional

to the

scattering

function

S(q

- 0

)

above T~ for a critical volume fraction

4

_{(~" =}

4

_{(~'~ =}

(l 4 )/2

in the

thermodynamic

limit

(n

-

ml,

while S(~jj ^is

proportional

to the

scattering

function

S(q

-

0)

below T~ ^for a

path along

one branch of the coexistence _curve

(for

_{n -}

ml.

Of

course, due _to the

symmetry

^of our model both branches _are

equivalent,

and thus

S(q

_-

0)

_at the A-rich branch coincides with

S(q

_-

0)

at the B-rich branch at the _same

temperature.

Although

_a finite system ^does not show any

spontaneously

broken symmetry ^and thus for

AR

=

0 _we have _a

vanishing

order

parameter (m)

for all _nonzero

temperatures

^and finite _n, the _use of

( (m )

_{as measure} of the effective order

parameters

^{and in}

equation (12)

ensures a smooth

approach

to the

thermodynamic limit,

lim

(

_m

)

~~ ~ ⁼

lim lim

(m )

~~

n ~ co Ap ~o _co

The

histogram reweighting technique [20, 67]

allows _to obtain data at AR

#0

from

reweighting histograms

taken at AR ₌ 0 with

appropriate

factors exp

(AR

^mNn/2

k~ T).

^In this way, data for S~~jj ^{at constant}

(m)

# 0 have been

generated,

_as discussed in _more detail in references

II

8,

20].

Since

Flory Huggins theory [32,

58,

59] predicts

that

Sj~)j(4~ )

vanishes _at the

spinodal

Sco'l(~bA)CcXsp(~bA)~X, (13)

phase diagram

N=64, binodal=1 38[1.T/r~)°~

0.9

08

o.7

06

j ^0.5 E

v 04

,~ ^,,

0.3

~~ ~'~

_~

~'/"~-

0.2 ~°~ ^{"' '}

binodal

o i *---« spinodal

~ ~

7 0 7 S 8.0 8 S 9.0 9.S 10 5 11 0

k~T/E

a)

susceptibility

N=64 60

48~

50 64~ _",

80~ [

therm. limit

40 ',

«

A , I

E _, ,,

(

₃₀

",

_',

"E

',

",,

V ',

C 20 ^'

io

~8_{0 8.4 8.8} 9 2 9.6 lo-o 10.4 10.8 11 2

k~T/E

b)

Fig. I. Order parameter

ii

_m

)

(a) and order parameter susceptibilities _S~~j, n(

(ni~) (ni)~),

S/~,j _{= n}

(ni~) (

_ni

)

~) (b)

plotted

_vs. temperature ^for N 64 and three choices of L _a~ indicated _in the

figure. Open symbols (triangles,

_squares, and circles) denote the data directly observed _at tho,e

temperatures where the actual simulations _were made,

including

_error bars

indicating

the ~ize of statistical _errors, while broken _curves indicate the

multihistogram extrapolation

based _on the _use of all data _on P~(ni at all considered temperatures. ^Full curves are based _{on a} finite size

scaling

extrapolation

towards the

thermodynamic

limit. The broken _curve with stars (marked _«

spinodal

») in (a) is obtained from _an extrapolation _{of Sj~j',} to the

point

_Sj~)j 0 _{at constant}

(Jii )

_as linear function of ~/k~ T. In (b) the

upper set of _curves for k~ T/t _m lo refers to S~~jj while the lower ~et of _curves refers to S/,,jj.

cumulants U~~~~

N=64

1.5

,, ,

,' ,'

; ^,

1.4 ,' "

,"',,,

~

,',(:.'

A

,((."

~ ^l.3 ;."

V ) ,.

;~

~ ,"j/

E ₂

."",~'

v ,,~ ,,

:. ,.

, /

,' " 48~

1-1 ' / ~~3

j"

8

i.o

8 0 8.4 8.8 9.2 9.6 lo-o 10.4 lo 8 11 2

k~T/E a)

cumulants _U~~~~

N=256

1'[.-' ^~

l 5 ,'

"'

,-'

' _,"

/ ,'

/ /

lA ^/

«~

E 1.3

~

"~

~ ³

V 1.2 :.' 4~

,:" -II 80

,,"" ,."/ ^96~

,,,,~' ;~/

128~

,,,""

_.j,~ ^l60~

1-o

..----.--""""~'°'~'

32 34 36 38 40 42 44 46 48 50

k~T/E b)

Fig. ^{~. Ratio of} moments

(ni?)

/

(

_ni

)?

_vs. temperature ^{for N 64} (a) and N 256 (b). Different

curves result from histogram extrapolations for different choices of L _as indicated in the

figure.

from _a

plot

of the inverse _structure factor _versus the

Flory-Huggins-parameter

_X at constant

volume fraction

#~

of A-monomers _{one can} locate the

spinodal X,p(~bA)

^from a linear

extrapolation.

In _our model,

4~

₌ 4(1+

(ni) )/2

and

approximately

one can assume that X ^cc

p/k~

T_: thus X~p (4

corresponds

to a

spinodal

temperature T,~⁽

(ni )

) and it is the

quantity

p/k~

_T~~(

(ni)

obtained from linear

extrapolation

of

Sj~)j( (m)

_vs.

p/k~

T that is included in

scaled order

parameter

N=64, k~TJe=99264, u=1 526,v=0.5482

5.6

A 3.5

E

V

>

2.2

48(

84 80~

1.38X°~°~

1.4

0.3 3.0 30.0

L~t

al

scaled

susceptibil~, _T<T~

Nd4, k~TJe=9.9264, u=1.526, v=0.5482 3.3

2.6

j i

E

v

~

1.6 84~

---;

40X'?'"

2.0

L~

t

b)

Fig.

3. Finite size

scaling

plots of the scaled order parameter, L'

(

_m

),

(al, the scaled

susceptibility

S(~j, ^below T~,

L~~((m~)-([m[)~),

(b), and the scaled

susceptibility

_S~~,j above T~,

L~~

(m~) (m)

^~), versus the scaled temperature distance L" _t (a, b) _or L" t' (c), where t T/T~,

t'

= I T~/T, k~ T~/~ ₌ 9.9264, and the scaling _{exponents u}

= 1.526, v = 0.5482. Full

straight

lines _on these log-log-plots indicate

asymptotic

_power laws (where _x is the abscissa variable) _as quoted in the

figure.

scaled

susceptibility, _T>T~

Nd4, k~TJ£=9.9264, u=1.526,v=0.5482 15.8

io.o

"A

E v

I

"E

v

~

4.0 84~

-~--.

50x'?"~

2.0

L~

(I-TJfI

C)

Fig.

figure

I. Of _course, the _«

spinodal

_{curve »}

intrinsically

is _a mean-field concept and thus should be viewed with

precautions.

For

#~

₌

#

_(~'~,

equation (13)

^reduces to

equation (2),

of _course,

so the mean-field value for exponent _{y =} ^I is

automatically implied,

rather than the correct

value

(Eq, (5)).

^Therefore

T~~((m) ⁼⁰⁾

must lie above the actual critical temperature

T~, as also observed

experimentally [7-10],

cf.

figure

la. But _one expects ^{that the difference} T~~(

(m)

= 0

)

_T~ vanishes as N

- co,

We _now summarize the finite-size

scaling equations [68-71]

used _to

analyze

the _{« raw} data _» for

( (m ),

_S~~,j and S[~,j ^{obtained for different linear dimensions L from} our simulation _:

(

_m

_~)

=

L~ ^~~

/~ _(L"

_t

₎

k

=

1, 2,

(14)

where

/~

is _a suitable

scaling

function and _{u, u are} characteristic exponents, ^{In the}

Ising

critical

region [40]

u =

p/v

= 0.515

,

u =

I/v

=

1,587

(15)

while in the mean-field critical

region

for d

=

3

hyperscaling

does not

hold,

and then L does _no

longer

scale with the correlation

length f

cc

it

^" but rather _{a «}

thermodynamic

_»

length [7 Ii

f

_cc

((lml)-~scon)~/~cc (ltl~°~~~~~~~lN)~'~ =N~'~ltl~~'~

One therefore

obtains, using

^d ₌ ^{3 and}

pMF

"

, YMF "

2

u ₌

p~~ d/(2 p~~

+

y~~)

=

~,

u =

d/(2 p~~

+

y~~)

₌ ^~

(16)

4 2

In the mean-field critical

region,

bulk

(homogeneous)

fluctuations of the order parameter

dominate,

and control the finite size

rounding [7 Ii.

Therefore the mean-field correlation

length f~~cc it (~"~~

with v~~

=1/2

does not enter the finite size

scaling description,

^In the

asymptotic

non-mean-field critical

region,

however, the

important

order parameter fluctuations

in _a finite size system are

spatially inhomogeneous

_ones, and therefore then L scales with

f.

It _turns out that in the _crossover

region

from mean-field behavior there is _a smooth

matching

between all these

lengths,

_see

figure

4. It _now

depends

_on whether the linear dimension L exceeds _{f~~~~~ cc} N _or not if

equations (15)

_or

equation (16)

should be used in the finite size

scaling law, equation (14).

In view of the smoothness of the _crossover it is

clear,

that for L

- f~,~~~ neither of these _sets of exponents will lead _{to a} reasonable

scaling,

and it may then be

practically

_more useful _{to treat u, u as}

adjustable

_« effective exponents _» ^which may

smoothly interpolate

between their theoretical limits

equations (15), (16).

While for N _~ 32 the fit of the

data to master curves

by plotting ((m(~)L~~

_vs. L"t works very

well,

if _one fixes

u, v at the _«

Ising

values _»

quoted

in

equation (15),

this is _no

longer

true for N _m 64, _see

figure

3. But _a reasonable fit

(over

_a restricted range of L which

always

is

comparable

to

N, cf. Tab.

I)

is

possible

with these effective exponents, as

expected.

4A z~

o mean field

f ~t'~

N~~~

fnf~mt'~'~

~

non-mean-tied

i

t"~'~ ^' -log 6i

log

^t

Fig. 4. Crossover scaling of characteristic lengths for _a polymer mixture. For t»G, ^(I,e.

log t to the left of log G,) one has _mean field behavior, with _two characteristic lengths fMF cc

,'N

_t~ ^'~~ and I

cc N ^"~ t~ ^~'~ These

lengths smoothly

_merge in the _crossover regime (t _I G, and in

the non-mean-field critical

region

_a

single

characteristic length f _cc N " t~ ^" takes _over.

Table I. Critical temperatures,

ejfiective

critical exponents ^and

effective

critical

amplitudes

as a

function ofchain length.

Also shown is the interval

(L~,~,

L~~~

fi.om

which these _estiniates

al-e derii>ed.

~

~eff ~eff ~~ ~~~

Y~~~ kB

Tc/F /~'~

_U fi

max

8 1. 18 0.325 225 52 1.24 0.9693

(24,

40) 1.59 0.51

16 1.27 0.325 260 72 1.24

(24,48)

1.59 0.51

32 1.25 0.325 500 130 1.24 4.845

(40,64)

1.59 0.51

64 1.38 0.359 850 240 1.2473 9.9261

(48,80)

1.526 0.548

128 1.32 0.389 700 700 1.134 20.69

(48,80)

1.570 0.61

256 1.46 0.42 4 000 700 1.0787 40.9

(64,

1.563 0.567

512 1.35 0.402 8 200 3 500 1.042 82.27

(96,

1.626 0.653

m

,fi

_{0.5 1.0 3/2 3/4}

A delicate

point

^is the

precise

estimation of the critical temperature.

Choosing

ratios such _as

U[

=

llml~)~/llml~)~

With

kf

=

iJ (17)

one concludes from

equation (14)

that

they depend

_{on a}

single scaling

combination

L"t

only,

U))

₌

f~~(L" ^{t), (18)}

with

f)) being

another suitable

scaling

function.

Plotting

_now ratios

U))

vs. t for different choices of L, _one expects ^{that all such} curves should intersect in _{a common} intersection

point

f))(0)

for t

=

0.

Figure

2 shows that this works

reasonably

well for N

= 64, while for N

=

256 considerable _scatter is apparent. ^This must be

expected

since for N

=

256 _a

larger

range of L is covered and part ^{of the data fall in the}

region

where L is

distinctly

smaller than f~,~~~. Since part ^of the scatter seen in

figure

2b is due to statistical and

systematic

_errors

involved in the

histogram reweighting technique,

a more

sophisticated

data

analysis

(considering

corrections _to finite size

scaling)

does not _seem to be warranted.

Figure

3 then shows _an

example

for the

« data

collapsing

_» obtained for N

= 64 for the three

quantities (

_m

), (m~)

and

(ni~) (

_m

)

~. We estimate here that the

systematic

deviation

seen between the curves for the three different choices ol'L should be

mostly

due _to corrections to finite size

scaling

rather than statistical and

systematic

_errors of the _{« raw} data

». The _same

is _true for N _<

32,

where _u, u were not allowed to vary but rather fixed _at their

Ising

values. In contrast, for N _m 128 statistical and

systematic

_errors in the data become _more of _a

problem.

simply

because for the

larger

lattice sizes _at T~ ^critical

slowing

down of order parameter fluctuations is _{a severe}

problem [66]

and also the chain

configurations

remain

highly

correlated

over

large

times. Thus the non-monotonic variation of effective exponents ^{and effective}

amplitudes

with N in table I needs to be taken with

corresponding precautions.

An altemative way of

checking

for the effective exponent u to be used in

equation (14)

is _to

study directly,

the variation of the order parameter

((m()

at T~, ^which

according

to

equation (14)

should vary as

( _(m )

~~

cc L~ ~

Alternatively,

one can

study

^{the variation of} the

response function

fi[(ni~)~ ((m()(]

with L at T~, or

equivalently

the variation of

~ c

(S(~,j )~~~. ^Since n = L~ #/

(8

N ), _we expect that (S(~jj

)~~,

cc L~ ^~ "IN.

Figure

5 checks for such power laws. The data _are indeed

compatible

with such _an

interpretation

in _terms of _an effective exponent v. The estimates

resulting

for v from either

((ni( )~

_or

(S(~,j)~~,

are

roughly

in agreement ^{with each other} and in _{most cases} also

compatible wit~

the numbers which resulted from the data

collapsing

of the temperature

dependence

(Tab.

I).

A similar behavior for the

model

(it

) with

longer

range of interaction, mentioned

above,

has

already

been documented

[18].

A _more refined

description

of this _crossover will be

attempted

in section 4.

3. Chain

length dependence

of critical

properties.

The critical

temperature

_T~

(N )

is found _to scale

linearly

with N for both models

(I)

and

(it)

for

large

N ;

however, unexpectedly

the corrections _to this

leading

behavior _are different

(Fig. 6)

while for _case

(I)

one

approaches

^the

asymptotic

value

k~T~/Np-0.16124

for

large

N from

below,

for _case

(it)

_one

approaches

this value from above. In both _cases these data _can be fitted to

straight

lines with _a

« free-end

»-correction, k~ T~/p

₌ 0.16124 N 0.227

,

model

Ii) (19)

k~ T~/p

₌ ^{2.15 N} +

1.35,

model (11)

(20)

order

parameter

at the critical

point

swpe~ems expomm ,v(=,p/v) 0.6

oN=8

,~ ~ s~oe:,o.528 aN=16

6N=32

~_~

° ~

~~=~8

~°

o

~

*N=256

j

~

(

_+N=512

V1.2 o

f

a p +

1,4

store.-0.658

.6 ^*

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 S-o 5.2 5.4

ln(L)

a)

maximal

susceptibilties

s~oe yWMs exmmm d-2v (~d-2p~) 3.0

s~oe:1.73

2.6 ^°

~ o

woe;1 95

+

~/

_2.2

I o o

v ^{6 V} +

I _{O N=8}

"E

~.~

a N=16

~i

o 6

V 6 N=32

~$ ^°

+ ^o N=64

1.4 V N=128

* N=256

~

+ N=512

~'~3.0 _{3.4 3.8 4.2 4.6 5.0 5.4}

In(L)

b)

Fig. 5. Log-log plot of the order parameter at T~,

( ml

_{)~~ (a)} and of the maximum value of the response function, _(S(~jj)~~~

=

n((m~) ([m[~)

_{)~~~ (b) versus L.} All choices of N _are included.

Straight4ine

fits _are indicated for N

= J6 and N

= 256.

but the

sign

^of this correction differs. While _one could

physically interpret

_a

positive

correction

as a contribution due _{to extra} interactions that free ends _can have with nonbonded

neighbors II

9,

20],

we now doubt whether such _a

simple interpretation

is _correct in view of the different

sign

found for model

(I).

We have _no

interpretation

_to offer for this difference in

sign.

It is

chain

length dependence

of critical

temperatures

o.2o

0,16 @ & S ^0.16124

e ~

e

0,12 @

~

~l

~ 0.08

0.04

16 8

0.00

o-o 0.I

o-j

^{0.3 0.4}

N

a)

~-~--~

Ul z

~ j

~

256 128 64 32 16

~

0 0.05 0.1 0.15 0.2 0.3

N'~~~

b)

Fig.

6. Variation of the observed normalized critical temperature k~

T~/(

EN with N'~~, for model (I) (

only

_p~~ # 0, _z~r~

= 5 ) and for the _« most

symmetric

» model (11)

(pA8

_FAA

~ ~88 ~ F,

Zcff ~

14

).

interesting

_{to note,}

however,

that in the revised versions of PRISM

decouplings [26] slightly

different schemes

yield

different

sign

of the correction for the _same

model,

which indicates also that it is _not

straightforward

_to understand

precisely

the

origin

of this correction. It is also

interesting

that for model

(I)

_one has to go ^to

fairly large

N until _one reaches the

asymptotic behavior,

unlike model

(it).

This fact

might

be due to the smallness of

z~~~, which leads _to rather low

ordering

temperatures ^{for N} w 32 in _case

(I),

and hence rather strong fluctuation effects

leading

to a

suppression

of T~ may be

expected.

However, in the model of Sariban and Binder

[2-4]

which also has

z~~~ w 3 _a

positive

correction similar _to

equation (20)

_was nevertheless found.

Discussing

the

magnitude

of the coefficient of the

leading

term, we would expect a reduction

by

_a factor of 2 times the ratio of the effective coordination

numbers,

I-e-

roughly

_a factor of 6

(the

factor of 2 _comes from the fact that in _case

(it)

all

pairs

of _monomers between different

chains contribute but

only AB-pairs

contribute for _case

(I)).

In

reality

the ratio between both factors in

roughly 13, indicating

for _case

(I)

a stronger

discrepancy

between the model and

Flory-Huggins-prediction (k~

T~ (#

)/2

= z~~~N

#/2

1 1.25 N than in _case

(it).

A similar trend _was observed for the

simple self-avoiding

walk model of references

[2-4]

as well. If _we would take the

prefactor

in

equation (19)

_{as an} estimate of _an effective coordination number

between nonbonded

neighbors,

_we would obtain the very small value _z~~~

- 0.64. This shows that

Flory-Huggins theory

can

hardly

estimate the scale for the critical temperature

reasonably, although

it

yields

the exponent ^{of the chain}

length dependence correctly.

As discussed

already

in reference

[3],

the order parameter ^{should exhibit} a crossover

scaling

from mean-field behavior _to non-mean-field like behavior _as follows

( _(m )

=

h~'~ t"~ fi(t/G,)

=

,,I _{t~° M(2.3}

_Nt

₍₂₁₎

where in the last step we have used

h~~

=

,,5 _{(Eq. (7))}

_{and inserted}

_{equation (8).}

_{In order that}

equation (21)

reduces to

equation (5)

_{we must} have

[3, 18] M(()

cc

(P

^"~ and hence

h(N)CCN~P~"~~-N~°'~~ ₍₂₂₎

In order to check for

equation (22)

we followed the

procedure

described in reference

[20]

and fitted data for

( (m )

which _{are not} affected

by

finite size effects

(for

the

largest

choice of L in each case)

directly

to the

equation ((m( )

=

htP imposing

the

Ising

model value for

p

(p

=

0.325

). Figure

7

gives

our results. It is _seen that _two

regimes

_must be

distinguished.

For short chains

(N

_w 32

equation (22)

is not yet ^valid and rather _one finds that the data _are

compatible

with _a law

proposed

for semidilute

binary

blends in _{a common} solvent

[72-74],

where an exponent,<

describing

the reduction of the X-Parameter ^due to excluded volume enters

h(N

cc N ^{~~ "~~ "~'} +'Ii _N ~° °~'~

(>. =

0.22

(23)

Equation (23)

results if _one considers chains in the _crossover

regime

from dilute _to semidilute solutions, where chains have started

overlapping

and

interacting,

and _{one can} describe this

interaction

by rescaling

them in _terms of

« blobs _» which have the size of the

screening length

of excluded volume interactions. Inside _one blob, there _{are no monomers} of other chains, and thus the number of

pairwise

contacts that lead _to

unmixing

is

greatly

reduced. A

scaling theory

for these effects

yields

also

equation (23).

Previous work where also the volume fraction

#

of the _{monomers was} varied _{over a} wide range

[4]

was shown to be in

good

agreement with

equation (23) [75].

The fact that _our data for N

< 32 _are described

by equation (23)

_can be

understood that the

screening length

of excluded volume forc

/~6

^for

^#

^{= 0.5}

^[62]

^and

this is

comparable

to the

gyration

radii for small N

( (R(~~)

^{=12.26 for N}

^=16,

fi

= 17.5 for N

32). Unfortunately, only

in this

regime

^where the effective exponent p~~~ agrees with the

Ising

exponent p ₌ ^0.325

(Tab. I)

_can the associated critical

amplitude h accurately

estimated; for

larger

N _{we can} fit rather

precisely

_an effective

amplitude h~~ (corresponding

to a critical behavior described

by

p~~~, see Tab.

I),

but _an

amplitude h corresponding

to

p

=

0.325 _can be estimated

only

within

large

_{errors, as}

figure

7 shows.

This

problem

could

only

be resolved if data closer _to T~

(I.e.,

for

larger L)

_were available. It is

order

parameter

^critical

amplitudes

in

lsing

limes Nwoememeamumove<sorting

1.51

1.37

,,,,~~~~~~

( ^{.~~ ~} """,,,,

il

_E 1.04

"",,,,

jf

0.95 ',

",,,_

m

#

0.87 ',,,

',,,

~

~ ~~

l.39 N~ ^""

0.72

2.50N~'~~

0.66

~

~i.00

10.05 25 24 63.40 159.24 400 00

N al

w- E

~J

-2 53N"°~"

l.53 N'°°~~'

20 200

N b)

Fig. 7. Log-log

plot

of the critical

amplitude h

(N for the model with =~j w 5 (a) and the model with z~,, = 14 (b).

Straight

lines indicate the

predictions equations

(22) and (23).

obvious from table I that the effective

amplitude ^B~~

has _an

N-dependence

rather different from

equation (22)

_:

probably

it

smoothly interpolates

between the

Ising

value for short chains and the mean-field result

,,5

_that

one obtains for N

- co with p ₌ ^1/2.

The effective critical exponents p ^~~' y~~~ etc. and the associated effective critical

amplitudes quoted

in table I for model

(I)

with _z~~j

=

5 and in table I of reference

II 8]

for model

(it)

with

z~~~i14

are estimated from _a

relatively

narrow «window» of lattice sizes around

L~

=

(L~,~ +L~~~)/2

and

temperatures

^{where the} correlation

length f

is

comparable

to

L~.

Therefore the

N-dependence

of these effective critical exponents

simply

^reflects the fact that for different chain

lengths

N we work in different

portions

of the crossover

scaling diagram

for the characteristic

lengths, figure

4, _some of the data

correspond

to

ilf~~~,,

~ l and

other data _to

ilf~~~,,~

l. Note that

f~,~,,

_cc N and

I

cc

f

for

f

_>

f~,~,,

^{and the} rounded temperature

region

of _our data

corresponds

to f _i

L~,

and hence it makes _sense to

analyze

our