The size of a polymer molecule in semi-dilute solution

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The size of a polymer molecule in semi-dilute solution

M. A. Moore, G.F. Al-Noaimi

To cite this version:

M. A. Moore, G.F. Al-Noaimi. The size of a polymer molecule in semi-dilute solution. Journal de

Physique, 1978, 39 (9), pp.1015-1018. �10.1051/jphys:019780039090101500�. �jpa-00208830�

THE SIZE OF A POLYMER MOLECULE IN SEMI-DILUTE SOLUTION

M. A. MOORE and G. F. AL-NOAIMI

Department

of Theoretical

Physics,

^Schuster

Laboratory,

^The

University, Manchester,

^M13

9PL,

^U.K.

(Reçu

le 28 avril

1978, accepté

^{le 23 mai}

1978)

Résumé. ²⁰¹⁴ En utilisant le formalisme de la théorie des champs, ^on calcule la distance quadratique

moyenne des extrémités d’un

polymère

^en solution semi-diluée. Cette distance croit avec la tempé-

rature quand ^on

s’éloigne

^du

point 03B8

^{vers la} limite du bon solvant et la fonction de crossover décrivant cette variation est calculée, correctement à l’ordre 03B5 = 4 - d, suivant la méthode du groupe de renormalisation. L’accord avec les résultats

expérimentaux

foumis par la diffraction des ^neutrons

par

des chaînes deutérées en solution n’est pas très bon.

Abstract. ²⁰¹⁴ The mean-square end-to-end distance for ^a

given polymer

^{in a} solution whose concen-

tration lies in the semi-dilute

regime

^has been studied using the field-theoretic formulism. The cross- over function which describes the increase in the end-to-end distance as the temperature ^is raised from the

03B8-point

towards the good solvent limit is calculated (correct ^{to order 03B5 = 4 -} d)

using

^{the renor-}

malization group. The agreement ^{with the} present

experimental

^{data obtained} by

scattering

^neutrons

off a few deuterated chains in the solution is poor.

Classification Physics ^Abstracts

36.20

1. Introduction. ^-

Scaling arguments [1, 2] imply

that the mean-square end-to-end

distance R 2 >

^{of a}

given

labelled flexible

polymer

^{chain in} semi-dilute solution has the form

where N is the number of effective segments, ^{1 their}

length, v

^the

segment-segment interaction,

^{c the} concentration of segments in the solution and

e ⁼

4 - d,

where d is the

dimensionality of

^the system.

In the

vicinity

^{of the}

0-temperature

^--

where T is the

temperature.

^The

large

^x

(good solvent)

limit of the crossover function

f(x)

^is

where v is the

polymer

^size

exponent

^{in the dilute}

good

solvent

regime where R2 >~ ^{l2 N2v} [1, 2].

^The

value of ^v in three dimensions is close to

3/5 [1].

Large

values of x

correspond

^to either small concen-

trations c or

large

^{values of v and occur in}

region

^II

of the classification of Daoud and Jannink

[2].

^Small

values of x

correspond

^to either poor ^solvents

(i.e.

small

v)

^{or to}

large

concentrations and occur in

region

^III of Daoud and Jannink

[3].

^{At the}

0-point

itself v ⁼

0, f(0)

^{= 1} and the familiar

result,

for a random chain is recovered. The small x limit

(which

^we shall call the _poor solvent

limit)

^{has been}

treated

by

^Edwards

[3] using

^a

perturbation expansion

in the interaction between the segments

(which

ⁱⁿ

semi-dilute solution is screened

by

^the presence of the other

polymer chains).

^{In this note we shall}

compute

the crossover function

f(x)

for all values of x correct to first order in a,

using

^{the same} renormalization group

(RG) procedure

^{as was}

previously

^used

by

^one

of us in

computing

^{the crossover} function for the osmotic _pressure

[4].

Since the calculation is a

simple extension

of the work in reference

[4] (to

be referred to as

I), only

the essential

points

of it will be

given. !

In section 2 we compu.te

f (x)

for small values of x

(the

poor solvent

limit) using

the des Cloizeaux field theoretic formulism

[5]

and then compare the results with the

physically

^more direct method of Edwards

[4].

In section 3 we use the RG to determine

f(x)

^{in both}

poor and

good

solvent limits and _compare it with

some

experimental

data of Cotton et al.

[6]

^{on deute-}

rated

polysterene

chains in a

cyclohexane

^solution

containing

^a finite concentration of undeuterated

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019780039090101500

1016

polysterene

chains. Some reasons

why

^{the fit to the}

data is so very poor ^are

finally

^discussed.

2. Poor solvent limit. ^- The

starting point

^{of the}

’ field-theoretic formulism of des Cloizeaux

[5]

^{is the}

Lagrangian density

gp is the

n-component

^{field variable}

(qJ(l),..., qJ(n»),

h is a

magnetic field

^{needed to}

generate

^a

non-vanishing

concentration of

polymers and fl

is related to the

ternary

^cluster

integral [4].

Given the effective

poten-

tial

r(M, t)

associated with

(2 .1 )

in the limit n ->

0,

one can calculate the concentration c of

polymer

segments

by

the relation

and the concentration of

polymer

^molecules

c/N by

where M

= Q(1) >.

^The transverse correlation func- tion

GT(q) == p(2)(q) p(2)( - q)

is related to the correlations between the two ends of a

given

^{chain in}

the solution and its

small q

behaviour is

simply

^related

to R2).

^{Thus at}

small q

because of the

0(n) symmetry

^{of the}

Lagrangian density

^of

equation (2 .1 ) [7].

^From-

(2.4)

^{it then}

follows that

By definition,

the effective

potential

is such that

so,

using (2.3),

In mean-field

approximation

the effective

potential is [4]

From

(2.2)

^and

(2. 3)

^one finds that in this

approxi-

mation

and

so

using (2. 7), R 2 >

⁼

N12,

which is the random- chain result.

The first corrections to mean-field

theory

^{are the}

one-loop

terms, ^which

give

^{for T’}

[4],

where,

^{in the} notation of

[4],

and

Hence by (2.2)

The

one-loop

corrections to the mean-field result

c

= j ^MI

^{can be}

approximated by replacing t

^{and MI}

in them

by

mean-field

values,

^when

(2.14)

^{reduces to}

In the semi-dilute _poor solvent

region (region III),

one supposes that 2 uc + 6

pC2 » 1 /N. Defining

one

gets

^from

(2.7)

For d ⁼

3, (2.17)

^becomes

which is similar to the result found

by

^Edwards

[3]

apart

from the numerical coefficient which he

gives

as

(2 J3/Tt).

^{We have}

^repeated

^his

calculation,

^which

makes use of a screened interaction between the

poly-

mer

segments

but obtain

(2.18)

rather than his result.

From

(2.16)

^{one can define a}

(concentration depen- dent)

effective 0

temperature, Oeff,

^{as that}

temperature

at which _veff ⁼ 0. The

temperature

^variation

of R 2 >

in

(2.18)

^{is then as}

(T - (Jeff)1/2

^{for T =}

(oeff.

^{This has}

been

experimentally

^verified

by

^{Richards et al.}

[8].

3. Good solvent limit. ^-

Ordinary perturbation theory

fails in the

good

solvent limit

(region II)

as here the effective

coupling

^{constant x is not small.}

Resort must be had to the renormalization _group.

It was shown in 1 that the RG

improved

^effective

potential

^was

given

correct to first order in e

by

where

and v* ⁼

e/8 Kd.

^{r* is}

given by

^the

matching

^condi-

tion

[10, 11] ]

where L1 ==

(n

⁺

2)/(n

⁺

8)

⁼

1/4

^{for n = 0. Note}

that to first order in e the usual

trajectory integral

contribution to the

improved

^value

of r(M, t)

^vanishes

for n ⁼ 0

[9]. fi

^is of the order

e’

and hence can be

ignored

^{in a} calculation to first order in e.

Using (2.2)

^and

working

^to first order in e

so from

(2.7)

From

(2. 3)

^{we have}

or

For semi-dilute solutions it is

again possible

^to

neglect

the

N-dependent

^{term in}

(3. 6)

^{when t = -}

vcQ 2Â - 1.

The

matching equation (3.3)

then reduces to

It is

invariably

^{the case that eet*} > 1 ^so that

(3.2)

reduces to

The crossover function

f(X)

^of

( 1.1 )

^is

Q d

^where

x =

(v/v*) (2 vc)-e/2

^is

proportional

^{to x. It may be}

obtained

by eliminating

T* between

(3.7)

^and

(3.8),

when one obtains

In the _poor solvent

limit, x

^{1 and}

f - 1

^and

equation (3.9)

^{can be solved as a} power series in x.’

The first correction to the random-walk result

R2 >

⁼

^N12

^is

given by

and as e -> 0

(3.10)

agrees with

(2.17),

^{the result}

obtained

by ordinary perturbation theory.

^{In the} very

good

solvent limit where x >

1, f >

^{1 and the}

solution of

(3.9)

^is

where we have eliminated d in favour of v

using 1 /v

^{= 2 - Ed .}

Equation (3 .11 )

^{is in} agreement ^with the

scaling prediction given

ⁱⁿ

equation (1.2).

To compare

theory

^with

experiment, equation (3.9)

was solved

numerically

^{for all x. e was set to one and}

d ⁼

1/4. Writing

^{the crossover function}

f

^as R 2

>/Ro2,

^with

Rg2

^---

N12,

^{the relation}

can be recovered from

(3.9),

^{where b is a}

parameter

which involves c, 1

and vo

^{and is chosen to}

give

^{a best}

fit to the data. It lies in the _range 0.02-0.04. In

figure

¹

two solutions of

(3.12)

^{are shown} for different values

FIG. 1. - Comparison ^of equation (3.12) ^with experiment. ^The top curve is for 0=303.5 K,. Ro = ¹⁰⁹ Â, b = 0.04 while the bottom ^curve is for 0 ⁼ 312.9 K, Ro ^{= 107.1} Â, b ^{= 0.02. The} experimental points ^{are taken} from reference [6] and refer to

polysterene ⁱⁿ cyclohexane.

1018

of 0 and

R,,.

^{The values 0 = 303.5 K and}

Ro

^{= 109}

^Á

are taken from reference

[6].

^The

best fit

^{values were}

0 ⁼ 312.9 K and

RB

^{= 107.1}

^A.

^{Neither set}

of para-

meters can be said to

give

^a

good

^{fit to the}

experimental

data.

4. Discussion. ^- There are several

possible

^reasons

for the _poor

agreement

^with

experiment.

^{The calcu-}

lation is

only

correct to first order in _{e ;} addition of further terms will

change

^the theoretical curves,

especially

^at

temperatures

well above the 0

point

ⁱⁿ

the

good

solvent limit.

However,

^the

experimental

curve looks to have the _wrong

shape

^{in the}

vicinity

of the

0-temperature.

^{It is concave}

upwards

^but

should be concave downwards as the

température

dependence

^of

[ R 2 > - Re2]

^{is as}

^{(T - Oeff)1/2,}

as T -

0 eff [8].

^{It is also difficult to}

pin

^{down a value}

for 0. In reference

[6],

^the

O-temperature, quoted

^as

303.5

K,

^{was obtained}

by using

^{the form}

for v, vo(1 - ^OIT)

^at

temperatures

well above the 0

point,

where it

surely

^{will need to be modified}

by

^additional

terms like

v 1 (1 - O/T)2.

In the calculation of the

crossover function in section 3 the

temary

^cluster

intégral

^was

neglected

since it is

of higher

^{order in E.}

Nevertheless,

below the dilute solution value for the

0-temperature,

^it

plays

^an

important

^role

by preventing collapse

of the chain. Another

possible

^{source of error}

(although probably small)

may arise from the diffe-

rence between the end-to-end

distance,

^calculated

here,

and the radius of

gyration,

which is what is

experimentally

determined.

What we have been

calculating

^{in this} paper ^are the dimensions of a

particular

^labelled

polymer

^molecule

in semi-dilute solution. This

polymer

^is

supposed

identical in _{every way} to the other

polymers

present.

Experimentally

^one gets ^{close to} this situation

by scattering

^{neutrons off a} small concentration of deuterated

polymers

^{added to} the semi-dilute solution.

However the work of Strazielle

and

^Benoit

^[12]

^shows

that the

0-point

may be shifted

by

^{as much as 5 OC}

by deuteration, suggesting yet

^another

possible

^source

of

discrepancy

^between

theory

^and

experiment.

Acknowledgments.

^{- We} would like to

acknowledge

the

help

^{of A.} Karabarbownis with the numerical work.

References

[1] DAOUD, M., COTTON, J. P., FARNOUX, B., JANNINK, G., SARMA, G., BENOIT, H., DUPLESSIX, R., PICOT, C.,

DE GENNES, P. G., Macromolecules 8 (1975) ^804.

[2] DAOUD, M., JANNINK, G., ^J. Physique ³⁷ (1976) ^973.

[3] EDWARDS, S. F., J. Phys. ^A 8 (1975) ^1670.

[4] MOORE, ^M. A., ^J. Physique ³⁸ (1977) ^265.

[5] ^DES CLOIZEAUX, J., ^J. Physique ³⁶ (1975) ^281.

[6] COTTON, J. P., NIERLICH, M., BOUE, F., DAOUD, M., ^FAR-

NOUX, B., JANNINK, G., DUPLESSIX, R., PICOT, C., J. Chem.

Phys. 65 (1976) ^1101.

[7] BREZIN, E., WALLACE, D. J., Phys. ^{Rev. B 7} (1973) ^1967.

[8] RICHARDS, ^R. W., MACONNACHIE, A., ALLEN, G., to be

published.

[9] RUDNICK, J., NELSON, D. R., Phys. ^{Rev. B 13} (1976) ^2208.

[10] RIEDEL, ^E. K., WEGNER, ^F. J., Phys. ^{Rev. B 9} (1974) ^294.

[11] BURCH, D., MOORE, M. A., J. Phys. A 9 (1976) 435.

[12] STRAZIELLE, C., BENOIT, H., Macromolecules 8 (1975) ^203.