On the θ-behaviour of a polymer chain

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On the θ-behaviour of a polymer chain

A.R. Khokhlov

To cite this version:

A.R. Khokhlov. On the

θ-behaviour of a polymer chain. Journal de Physique, 1977, 38 (7), pp.845-849.

�10.1051/jphys:01977003807084500�. �jpa-00208647�

ON THE 03B8-BEHAVIOUR OF A POLYMER CHAIN

A. R. KHOKHLOV

Physics Department,

Moscow State

University ;

^Moscow

117234,

^U.S.S.R.

(Reçu

^{le 21 decembre}

1976,

révisé le 8 mars

1977, accepté

^{le 15 mars}

1977)

Résumé. ²⁰¹⁴

L’application

^{d’une méthode de}

développement

^{en amas à l’étude d’une}

pelote polymérique

^montre que la

présence

^des collisions d’ordre élevé entraîne : a) ^une renormalisation du coefficient du second viriel de l’interaction des monomères;

b)

l’existence de corrections

spéciales

liées au fait que la chaîne ^est finie. Ces corrections sont près ^du

point

0, elles entraînent l’existence d’une

région

⁰ finie pour des chaines finies. Le comportement ^{0 de}

polymères hétérogènes

(polymères

avec défauts,

polymères ramifiés)

^est aussi considéré.

Abstract. ²⁰¹⁴ By ^{means of a cluster}

expansion

method it is shown that the presence of

higher-order

collisions in a

polymeric

^{coil leads to :}

a)

^the renormalization of the second virial coefficient of

monomer interaction;

b)

^{the existence of}

specific

corrections due to chain finiteness which are the main corrections near the

03B8-point.

These corrections lead to the existence of a finite

03B8-region

^{for finite}

chains. The 03B8-behaviour of

inhomogeneous polymers (polymers

^with defects, branched

polymers)

is also discussed.

Classification

Physics ^Abstracts

5.620

1. Introduction. ^- Considerable attention has been

paid recently

^to the effects which take

place

^{in the}

vicinity

^{of the}

0-temperature

in dilute

polymer

solutions

[1-4].

Thus it is

important

^{to consider more}

carefully

^the

concept

^{of the}

0-temperature

^itself

which was introduced

initially by Flory by

^means

of the

semi-qualitative

arguments

(see [5]).

^{The main}

questions

^{discussed in this} connection in this paper

are follows :

a)

^{How is the}

0-temperature

connected with the characteristics of ^monomer interaction ?

(section 2).

b)

^{To what extent the}

0-point

concept ^is

approxi-

mate, i.e. how ^narrow is the temperature ^interval within which _any characteristic of ^a

polymer

^chain

takes its

unperturbed

^{values ?}

(section 3).

c)

^{What is} the 0-behaviour of

inhomogeneous polymers (polymers containing

inclusions of other sort of _monomers; branched

polymers; etc.) ? (section 4).

Concerning

the first

question

^{it is}

usually

^assu-

med

[6]

^{that the}

0-temperature

coincides with the temperature of inversion of the second virial coefficient of monomer interaction B

(1).

This statement is based on the

following.

^{If we}

consider ^a

polymer

^{chain as the} gas of monomers

uniformly

distributed in the volume -

R 2 > 1/2,

where

( R 2 >

^{is the mean} square end-to-end distance for the

chain,

then it is _easy to obtain that at the 0-

point,

^where

( R’ >

⁼

^Na2 (N

^is the number of

monomers in the

chain,

a, the mean distance between two

subsequent

^monomers

along

^the

chain),

^the

number of

simultaneously occurring binary

^collisions

of monomers is ^-

N 1/2 .

The number of

three-body

and

higher-order

collisions _appears to be - 1.

Thus the

binary

collisions

play a prevailing

^{role in a}

polymeric coil;

^{if B =} 0 then the contribution in all the characteristics of the chain from these collisions vanishes and the

higher-order

collisions remain the

only perturbing

factor. The number of these collisions is

relatively

^{small and}

they weakly perturb

^{the cha-}

racteristics of the chain ^- in

[7],

^{it was shown that}

at the

0-point they

^lead

only

^to small corrections

1/ln

^{N -} thus when B ⁼ 0 the chain takes

(appro- ximately)

^its

unperturbed

dimensions.

But these

arguments

^{do not take into account that} the monomers are connected in the chain and ^are not distributed

uniformly

^{in the}

region - R’ >’/".

In fact both the number of

binary

collisions and the number of

higher-order

collisions is ^- N due to the collisions between the monomers which are close to each other

along

^{the chain}

(neighbour collisions).

^So

the

higher-order

collisions _may

play

^an essential role.

(1) ^To be definite here and further we shall note that we consider the model Gaussian chain (with ^{Gaussian bond} probabilities;

see [6], p. 16) ; ^this assumption ^{is not} obligatory, ^because analogous

results can be obtained for other models of the chain, but it sim-

plifies the consideration.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003807084500

846

The 0-behaviour of ^a

polymer

chain will be consi- dered

by

^means of the usual cluster

expansion

method

[6],

^but

taking

^{into account the}

higher-order

collisions.

Usually

this method

gives

^{for the} expan- sion factor

a2

of a

polymer

^{chain :}

Here and ; B

^mean averages ^over the confi-

gurations

^of

self-interacting

and free chains _respec-

tively ; Ki

^are the numerical coefficients and

z ⁼

BN1/2ja3.

^{The form}

(1) ^{of a’}

^{is due to the} assump- tion :

where

Vij(ri

^-

r)

^{is the}

pair potential

^{of the} interaction between monomers i and

j.

^The

assumption (2)

means the

neglect

of all interactions in a

polymer

chain besides the second virial

coefficient,

^{i.e. the}

neglect

^of

higher-order

collisions. In the next section

we shall consider the cluster

expansion

^method

without this omission.

2. The renormalization of the second virial coef- ficient. ^- Let us consider a

polymer chain,

^{in which}

monomers interact with the virial coefficients

W1

⁼

B, W2...,

^etc. The radius of this interaction is assumed to be small

(ro a),

but finite ^- in this

case we may ^use the standard cluster

expansion techniques

and avoid

divergences [8].

^{We shall}

assume that

WP - ^vP,

^where

v - rg (if

^{there is no}

additional

requirements

^for

WP

^{to be close to}

0).

In the

expression

^for

^a2

besides the terms propor- tional to

B, B 2,

^{etc. there} appear also terms propor- tional to other virial coefficients and their various combinations. For

example,

^{the term -}

W2 (contri-

bution from one

three-body collision)

is of order

W2 N 1/2 la , 3

i.e. of the same order with respect ^{to N}

as the term

proportional

^to

Wl.

This is due to the fact that the main contribution to

a2

from one three-

body

^collision appears from the collisions between two

neighbour

^{monomers and one monomer situated}

far from them

along

the chain. The

corresponding diagram

is shown in

figure

^{1 a. One} may ^see

already

FIG. 1. ^- a) ^and b) Typical diagrams contributing ^{to the term}

- N 1/2. Shaded intervals correspond ^to the distances along ^the chain of order a, nonshaded - of order Na, c) ^{one of the} diagrams

breaking ^the independence of summation.

from here that when

W1

^{= 0 there are terms in}

^a2 proportional

^to

^N’I’

^{and so}

W, 0

^{0 when T = 0.}

The term -

wf (contribution

^{from two}

binary collisions)

^{is to} first order in N -

wf Nja6;

^the

first correction is

- Wf N1/2ja3.

^{This correction}

describes the mistakes

owing

^to

approximations

^made

during

the summation of

corresponding diagrams

to first order in N

(for example,

when the summation is

replaced by integration).

^A

typical diagram

^contri-

buting

^to these mistakes is shown in

figure

lb. It is

important

^{to note that} now, when

Wl

^{is not close to}

zero in

0-conditions,

^the correction -

w,2 N1/2ja6

is of the same order of

magnitude

^{as the term}

- W, N 112 la . 3 Obviously,

^each of the other

diagrams including

^one collision between the monomers situated far from each other

along

the chain dressed in various

neighbour

collisions

(as

ⁱⁿ

Fig. la)

^or in collisions between the monomers

neighbour

^{to the}

initially colliding

^ones

(as

ⁱⁿ

Fig. I b) gives

^a contribution to

a2 proportional

^to

^{N .}

^Such

diagrams correspond

^to

the

p-th

^order correction to the term -

N(p + 1»2, where p

^{+ 1} is the number of collisions in the

diagram

under consideration.

Let us now collect in the

expression

^for

^a2

^{all the}

terms -

N l2 .

where

Cit...ip

^are numerical coefficients. We call the

quantity

in brackets in

(3)

the effective second virial coefficient of the interaction between the

pieces

^{of the}

chain B * and assume that the

expansion

^for

^rx2

^{has the}

form

(1)

but with renormalized B : ^{z =} B*

N 1/2 la’.

The

simple

combinatorial

analysis

shows that this

assumption

^is

equivalent

^{to the}

following

^statement.

Let us consider a

diagram including n

arcs corres-

ponding

^{to n}

independent

collisions between the

monomers situated far from each other

along

^the

chain

(distant collisions).

^{If we dress one of these}

collisions so that the dressed collision will include m1

binary collisions,

m2

three-body collisions,

etc., then the contribution from this

diagram

^{will be}

equal

^to

first order in N to the contribution of the initial bare

diagram, multiplied by C(ml, m2 ...),

^{where the}

factor C is

independent

^on the initial

diagram

^and

on the choice of the collision to be dressed. The last statement may be seen from the fact that one can sum

first _over the

positions

^{of those monomers which}

take

part

^{in the} collisions

dressing

the initial collision and then over the

positions

^{of monomers} of the bare

diagram.

^{The main} contribution to the first sum

appears

from the summation over short distances

along

the chain - a and to the second sum from the summation over

long

distances -

Na,

^{so the} influence of the first summation on the second one is

expressed only

^{in small}

edge effects,

^{which are not of}

first order in N. Of _course, the

independence

^of

summation we have described breaks down for

topological

^{reasons for the}

diagrams

^{similar to one}

shown in

figure 1 c,

^{but these cases}

correspond

^{to the}

three-body

^and

higher-order

collisions of the distant parts of chain and lead

only

^to the corrections of relative order -

1/ln

^N

[7].

^{If we are not} interested in corrections of this

order,

the summation over the bare

diagram

^is

independent

^{of the}

previous

^summa-

tion.

Hence,

^to within the distant

three-body

^collisions

the virial

expansion

has the form

(1),

^{but with renor-}

malized coefficient B*. This renormalization reflects the nature of collisions in a

polymer :

^collisions

between

pieces

of the chain and not between isolated

monomers. In the limit of an infinite chain the 0-tem- perature

corresponds

^{to B * = 0.}

The renormalization of the second virial coefficient due to the

higher-order

^cluster

integrals

^{is well-}

known in

magnetic problems (see,

^for

example, [9]).

The

expression (3)

^{shows the concrete} form and the

physical

^reason for this renormalization for a

polymer

chain ^- a system

analogous

^{to the}

magnetic

system in one of the

limiting

^cases

[10].

It is seen from

(3)

^that

(B - B *)IB v/a3,

^thus

the difference between the

0-point (J?* = 0)

^{and the}

point

of inversion of the second virial coefficient

(B

⁼

0)

decreases with the decrease of the ratio

vla’

(in particular,

this decrease _may

correspond

^{to the}

increase of the chain stiffness : the

correspondence

between the

gaussian

chain and the

persistent

^model

can be established

by

^means of substitution

[4]

v/a3

^-

d/a,

where d is the width of the

persistent

chain and a is its

persistent length). Therefore,

^{in the}

Lifshitz

theory

^of

polymeric globules [II],

^{which is}

the zeroth-order

theory

^with respect ^to

vla3,

^the

temperature ^{of the}

coil-globule

transition in the limit of infinite N coincides with the

temperature

when B ⁼ 0

[4].

3. Corrections due to N finiteness. ^- The

repla-

cement of monomer interaction

by

^{the effective}

interaction of the

pieces

of the chain does not in itself lead to new effects. New effects in the

9-region

are due to the deviations from

(1).

One of these deviations is connected with the distant

three-body

collisions

[7],

^{which are not} taken into account in

(1).

Another

deviation,

^{which is}

significant

^{near the}

0-point

^{is due to N} finiteness. This deviation _appears to be of the

opposite sign

^{to the}

previous

^one.

The corrections due to N finiteness are connected with the fact that ^even when B * ⁼ 0 the collisions of the chain end

pieces give

^{a non-zero} contribution to

a2 owing

^{to the}

change

ⁱⁿ

composition

^and

quantity

of

possible dressing diagrams

^{near the ends. It is} easy to estimate the order of these corrections near the

0-point.

^{We note} that the relative difference between the contributions to

a2

from the collision

occurring

at a distance of i monomers from the chain end and from the collision in the case of infinite N is of order

In order to estimate the correction to the term -

N 1/2

in

(1)

^{it is} necessary ^to attribute to each of the collisions the

weight - ^{i- 1/2} during

the calculation of the

corresponding

^{sum. This sum} per ^monomer is

ro.I N -

ir2,

^so the correction due to the chain finiteness

N

is

of order L (Ni) - 1/2 ro.I

const. - -

Co (2).

i= 1

It is essential that

Co ’#

^{0 when B * = 0.}

The calculations in the cluster

expansion

^method

confirm these

simple appreciations.

^{The terms}

giving

-

Co

^are the corrections to terms

proportional

^to

N

1/2 ;

^{we must} choose from all these corrections

only

those which are connected with N finiteness

(others

^{= 0}

when B * ⁼

0).

^For

example,

^{for the}

diagram

^shown

in

figure ^la,

the corrections mentioned are connected with the restriction of the

region

of summation over

the distance MN. The bare

diagram - ^{BN 1/2}

^{does not}

contribute to these

corrections,

^so

Co :0

^{0 when}

B * ⁼ 0. There are also corrections due to N finiteness from the

higher-order

^{terms of}

expansion (1),

^but

they

^{are small}

compared

^to

Co

^{in the}

region

^of

validity of (1).

Thus,

in 0-conditions

(B* N 12/a3 1)

^{we can}

write the

following expression

^for

^a2 taking

^into

account all corrections :

One _{may see,} that the last

item,

obtained in

[7],

^is

Co

^for

large

^{N. Further we shall assume that}

Co

>

C’/In N

for all

N,

^{so that at the true}

0-point (the point

^{where B * =}

0) ^a2

^{1. If in some}

region

of variation of N this is not the _case, our conclusions may

easily

^be reformulated.

Let us consider first the deviation of

temperature

when (X2 = 1 from the true

0-point.

^We notice that

Co - (v/a3)2

and B * - vr, where i ⁼

(T - 0)/0,

(2) ^{We note} that for the 0-temperature Wi ^{0 as it} is, appa-

rently, ^{for most real} polymers ^{due to} (3) ^{and to} the fact that for usual interactions Wi ^{is the first among} the virial coefficients to

change ^its sign ^{when the} temperature is lowered. Then it is natural that at the 0-point the end effects give the effective attraction due to the increase of the role of one bare binary collision in comparison

with the collisions dressing the initial collision (some of these last collisions cannot be realized near the ends). So the correction in a2is 0, ^and Co > 0.

848

because

Co

^{does not} include the term - B - v and 2?* does include it. Hence

a2

⁼ 1 when

or i N

vla N1/2. So,

^{if in an}

experiment (real

^or

computer)

^the

0-temperature

is determined from the condition

OC2 =

1 with N

finite,

then its value is overestimated

by

^{the amount AO -}

Ðvja3 ^N1/2.

^In

usual conditions ^{N -}

104

and v -

a3 ;

^this

gives

A0 - 1 °.

Now let us assume that the

0-point

is determined from the condition that the ratio of _any other moments

RP >/ RP >0

^is

unity

^or from the condition that the osmotic second virial coefficient is zero. Each time we shall obtain near the

0-point

^an

expression

similar to

(4),

but with other numerical coefficients.

Thus each time we shall obtain other A0 of order

Ðvja3 ^{N 1/2 .}

This enables us to make the

following

conclusions :

a)

when N is finite there exists a

0-region

of width A0 -

Ovla 3 ^{N 1/2 .}

^For any characteristic of a

real

polymer

chain there exists a

points

within this

region

where this characteristic takes its free chain value. The true

0-point

is situated at the lower boun-

dary

^{of this}

region; ^b)

^the difference between the

0-temperature

determined from the

corresponding

characteristic and the true

9-point decreases,

^{when N}

increases,

^as

IjN1/2.

This is in agreement ^{with com-} puter

experiments (Fig.

^{2 of}

[12], Fig.

^{11 of}

[13]).

It may be noted also that at the true

0-point (B *

⁼

0) a2 1,

^{i.e. the}

polymeric

coil is contracted. When

v N

a3,

i.e. for flexible

chains,

^{1 -}

(X2 ’" 1,

^{so this} contraction _may be

significant.

^The

0-temperature

may be characterized

by

the fact that at this

tempe-

rature the end-to-end distance of a

polymeric

^coil

depends linearly

^{on N}

(although

^{it is not}

equal

^{to the}

unperturbed

end-to-end

distance).

4. The C-behaviour of

inhomogeneous polymers.

^-

Let us now

apply

the results obtained to a

qualitative

consideration of the influence of defects in a

polymer (for example,

inclusions of other sorts of

monomers)

on the

properties

^{of the}

0-region.

^{Let C be the defect}

concentration in a chain of N monomers. In this case

there is an additional

broadening

^{of the}

0-region

due to the defects. Let us consider the

magnitude

of this

broadening.

The relative difference between the contributions to

a2

from the collision

occurring

^{at a distance of i}

monomers from the defect and from the collision in the chain without defects is of order

i- 3/2 .

Ana-

logously

^{to section}

3,

^{we can} then estimate

thp

^correc-

tion in

a2

due to the

defects;

^it appears ^to be

-

N1/2 Cvla3.

^Let

Ccr

^{be the} defect concentration

when the corrections in

a2

due to the chain finiteness and to the defects are of the same

order;

^then

C,, - vla 3 ^{N 1/2.}

^{When C}

Cer

^the

0 -region

^width

is defined

by

^N finiteness :

AO/O - vla 3 ^N’I’

^and

when C >

Ccr

^-

by

^the presence of defects

A0/0 -

^C.

Similar statements may be formulated for the devia- tion of the

experimentally

determined

0-point

^{for a}

finite chain with.defects from the true

0-point

^{for an}

infinite chain without defects.

It is also _easy to show the behaviour of the _expan- sion factor

a2

of a

polymer

chain with defects at the true

6-point.

^{When C}

Cer

the defects

play a

^non-

essential role and the behaviour of the chain is ana-

logous

^{to the one} described in the

previous

^section.

When C >

Cer

^the

expansion

of the chain is deter- mined

by

the defects. If we repeat ^{for this case the} well-known

Flory

evaluation of the

equation

^for

cx 2(Z) ([5],

^ch.

14), taking

^{into account that now}

the most

important

collisions are the collisions of defects with _any other parts ^{of the}

chain,

^{then we shall}

obtain that

a2

is determined from

Flory’s equation

but with z’ -

N 1/2 Cvla3

^{in the}

place

^{of z. In}

parti- cular,

when z’ >> 1

Any

^other

inhomogeneities

^{in a}

polymer

^{chain can}

be treated

analogously (near

^the

0-point,

^where

they

are most

significant).

^We have considered here the unavoidable

inhomogeneity

^{of a} chain due to its finiteness and the

inhomogeneity

^{due to} the inclusions of other sorts of monomers. One more

example

^of

inhomogeneous

chains is the branched

polymers.

The existence of additional branches in the chain leads to the

changes

ⁱⁿ

composition

^and

quantity

^of

the

dressing diagrams

^{near the}

branching points.

^The

absence of the

pieces

of the chain

(the

effects of the chain

finiteness)

^{led to} the effective

attraction ;

^the

presence of extra

pieces

of the chain

(branching)

would lead to the effective

repulsion,

^{i.e. to the}

increase of

a2

^at the

0-point.

^Another

independent

reason for this increase is the existence of

three-body

distant collisions which _may

play a significant

^role

in the branched

polymers.

^{The correct}

theory

^{of the}

expansion

factor of branched chains in 0-solvents must take account of both these factors. In

[14] only three-body

collisions were considered and the

quanti-

tative agreement ^{with the}

experiment

^{was not obtained}

(for

^comb-like

polymers).

^The

corresponding theory

which considers the two effects

simultaneously

^{is the}

field of the current work.

The author wishes to thank Acad. I. M.

Lifshitz,

Dr. A. Yu.

Grosberg

^{and Dr. J.} Ya. Yerukhimovich for valuable discussions.

References

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[3] COTTON, J. P., NIERLICH, M., BOUE, F., DAOUD, M., FAR- NOUX, B., JANNINK, G., DUPLESSIX, R., PICOT, C., ^J. Chem.

Phys. ⁶⁵ (1976) ^1101.

[4] LIFSHITZ, I. M., GROSBERG, ^A. Yu., KHOKHLOV, ^A. R., ^Zh.

Eksp. Teor. Fiz. 71 (1976) ^1634.

[5] FLORY, P. J., Principles of polymer chemistry (Cornell ^Uni- versity Press, N. Y.) ^1953.

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