Light scattering by dilute solution of polystyrene in a good solvent

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Light scattering by dilute solution of polystyrene in a good solvent

M. Adam, M. Delsanti

To cite this version:

M. Adam, M. Delsanti. Light scattering by dilute solution of polystyrene in a good solvent. Journal

de Physique, 1976, 37 (9), pp.1045-1049. �10.1051/jphys:019760037090104500�. �jpa-00208500�

LIGHT SCATTERING BY DILUTE SOLUTION

OF POLYSTYRENE IN A GOOD SOLVENT

M. ADAM and M. DELSANTI

Service de

Physique

^{du Solide et} de Résonance

Magnétique,

Centre d’Etudes Nucléaires de

Saclay,

^{BP n°}

2, 91190 Gif-sur-Yvette,

^France

(Reçu

^{le 11}

février

1976,

accepte

^{le 22 avril}

1976)

Résumé. ²⁰¹⁴ Par une technique de battement de photons ^{nous mesurons} le coefficient de diffusion translationnel du

polystyrène

^en solution diluée dans un bon solvant (benzène). Le rayon

hydrodyna-

mique n’a pas

la

même dépendance ^{en masse}

(RH ~ M 0,55 ± 0,02)

que le rayon de

gyration

^{de la}

macromolécule

(Rg ~ ^M0,6).

^{Nous montrons qu’une} solution diluée de macromolécules peut ^être décrite _par un modèle de

sphères

dures où l’on tient compte des interactions

hydrodynamiques

^et

thermodynamiques.

Abstract. ²⁰¹⁴ We study ^the

light

^scattered by ^a dilute solution of

polystyrene

^{in a}

good

^solvent (benzene) using ^a

light

scattering spectrometer. ^The

experimental

value of the translational diffusion coefficient shows that the molecular weight

dependence

of the effective

hydrodynamical

^radius

(RH ~ M0.55±0.02)

is different from that of the mean radius of

gyration (Rg ~ M0.6).

^{Otherwise we}

show that a model of hard spheres

taking

^{into account the}

hydrodynamic

^and thermodynamic interactions accounts satisfactory ^{for a dilute} solution of polymer ⁱⁿ

good

^solvent.

Classification

Physics ^Abstracts

5.660 ^- 7.146 ^- 7.610

1. Introduction. ^- We

present

^{results on the trans-} lational diffusion coefficient for

polystyrene

^{in a}

good solvent,

^{in the dilute}

regime.

The molecular

weights

^{of the}

polymers

range from 2 x 104 to 4 ^x

106.

In the first section we

briefly analyse

^the

theory

^of

light scattering by

^{a dilute}

polymer

solution. The

experimental

conditions are described in section two.

Finally,

^we discuss the results and _compare them with the

theory

and with other

experiments.

2.

Light scattering by

^{a dilute}

polymer

^{solution. -}

The spectrum ^{of the}

light

^scattered

by

^{a mixture}

contains a

component

^{due to} concentration fluc- tuations

[1].

The usual

theory

shows that the

measured-photo-

count autocorrelation function is related to the scattered field autocorrelation function which is :

The

notation

represents the ensemble _average, k is the

scattering

^vector,

bC(k, t)

^{is the}

spatial

^Fourier

transform of the local fluctuation at time t and A is a

parameter independent

of the concentration. For

solutions,

^a

good starting point

^{is to assume that the}

concentration fluctuations are

proportional

^{to the}

fluctuations of the number

density

^of

polymer

seg- ments. In the dilute

regime,

the distances between

chains are

large

^{and the}

polymers

behave like coils.

The

density [2]

^is

where N is the number of

macromolecules, rim(t)

^the

position

of the ith segment of the mth macromolecule at

time t, Z(m)

the number of segments ^{of the mth}

chain,

po the average number

density.

With this

approximation,

^{for a}

given

^{non zero}

scattering angle,

^{we obtain :}

If the radius of

gyration

^{of the}

macromolecule Pg

is small

compared

^{to the}

wavelength

²

nlk

the internal motion does not affect the

spectrum

of the scattered

light [2]

^and

equation (3)

^{becomes :}

where rm

^{is the}

projection along

^{k of the}

position

^of

the centre of mass.

In very dilute solutions the cross correlation terms

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019760037090104500

1046

of

equation (4)

^{vanish in a first}

approximation [3],

and :

The

validity

of this factorization has been demons- trated for N

particle

^diffusion

equations [4].

We assume that the macromolecule motion is diffusive and

obeys

^the

equation :

where

D.

^{is the} diffusion coefficient of the mth macromolecule.

Starting

^{from the}

expression (6)

^{it can be}

easily

shown that the autocorrelation function is _propor- tional to the well known

Z-average [5]

When the

weight

distribution is

relatively

^narrow

[6],

we have.

In

short,

^{at a}

scattering angle

^{chosen to} respect

kRg ^1,

^the

experimental photocount

autocorrela- tion obtained from

light

^scattered

by

^a dilute solution of

polymer

^{with a narrow}

weight

distribution is an

exponential

^{with a} characteristic time

inversely

pro-

portional

^{to the}

mean

^D

>z.

3.

Experimental procedure.

^- 3.1 LIGHT BEATING SPECTROMETER. ^- The

technique

^and

theory

^{of homo-}

dyne

spectrometer have been described elsewhere

[7].

Here,

^we

give only

^the

principal

characteristics of the

apparatus.

^The monochromatic

light

^{source is an}

argon ion laser used ^at 4 880

A.

A system

comprising

a lens and two

pinholes

selects the

scattering angle

and excludes stray

light

^scattered

by

^{the cell.}

The scattered radiation is detected

by

^{an I.T.T.}

photomultiplier

tube. The

homodyne photocount signal

^is

analysed

^{on a} 24 channel real time

digital

autocorrelation

[8] (P.D.S.

^Ltd

Malvern)

^interfaced

to a Hewlett Packard 9820 A calculator.

The autocorrelation function is

immediately

^fitted

to a

single exponential by

^{means of least} squares program. This

permits

^{one to} determine the corre-

lation

time,

’tc, of the

photocount

^{with an} accuracy of a few

percents.

3.2 EXPERIMENTAL CONDITIONS. ^- The macromo-

lecules studied ^are of atactic

polystyrene,

^whose

weight

distribution characteristics are listed in table I

[9].

The solutions are

prepared by weight

^{in the}

sample

cell. Dust

particles

^can introduce deformation on the spectrum

profile,

but the low

viscosity

of the solution allows them to fall to the bottom of the cell if the mixtures are

prepared

^several

days

^{in advance.}

TABLE I

The weight-average ^molecular weights, M )w, and the number- average molecular weights, ( ^M )n, ^were respectively ^obtained by light scattering and osmotic measurements. The Z-average ^mole-

cular weights, ^M )z, ^were calculated assuming ^{that the number}

distribution is symmetrical : ( M)z = ^M )n(3 - 2 ^M )n/ ^M )J.

The

good quality

^{of the}

exponential

fit and the

precision (5

^x

10-3)

with which the

decay

^{time can}

be measured confirms the condition that dust

particles

are

effectively

eliminated from the

scattering

^volume.

The

signal

^is

purely homodyne.

The solvent used is

benzene;

this mixture has a very low 0 temperature

(100 K) [5]

^{and small}

change

^near

room temperature ^{does not introduce}

experimental

errors; measurement ^on very dilute solution can be done because of the

significant specific

^refrative

index

ðn/ðc.

The radius of

gyration

^of

polystyrene

^{in benzene}

is known from

light scattering intensity

^measure-

ment

[11].

^{We can} evaluate k* ⁼

Rg 1

^{and the}

number N* where the chains

begin

^to

overlap [10] :

where V is the volume of the solution. The criterion

adopted

for the dilute

regime

^is

N % N*/2. Experi- mentally,

^we find that for k k* the coherence time of the

clipped photocount

autocorrelation function is

inversely proportional

^to

^k2 (Fig. 1)

^within

experi-

mental accuracy. We ^can conclude that :

1)

the internal motions do not affect the spectrum of the scattered

light [3] ;

FIG. 1. ^- Inverse characteristic time ^{as a} function of k’.

2)

^{the diffusion} coefficient is ^not

frequency depen-

dent

[4] ;

3)

^the

polydispersity

^{of the}

sample

is small and the autocorrelation

photocount

^{can be described}

by

^a

single exponential [6].

^An

example

of the fit is

given

in

figure

^2.

FIG. 2. ^- An example ^of photocount autocorrelation: 8 experi-

mental point. --- ^curve fitting.

We obtain the

experimental D >z

average diffusion coefficient from _’l’c with the

following

^{relation :}

Ts

is the standard

temperature

^chosen

equal

^{to 293}

K ; T,

^the

experimental temperature,

^and

q(7J

^the

viscosity

of the solvent. This method

gives

^{a value}

of

^D

)z

with an

accuracy ~

³

^%.

4. Results and discussion. ^- In the- dilute

regime,

the translational diffusion

coefficient,

^D

>z,

^{for a}

,given

^{mass increases}

^slightly

^and

^linearly

^{with concen-}

tration.

Figure

^{3 shows an}

example

^{of this concen-}

FIG. 3. ^- Concentration dependence ^{of the} diffusion coefficient for the sample ³ which has a concentration N*/VA equal ^to

9 x 10-8 cm-3. The results obey ^to

tration

dependence.

^The

experimental

^{results can be}

represented by

^the

following expression :

A is the

Avogadro

^number.

The second term is _very small

compared

^{to the}

^first,

it describes the weak interactions between the coils.

The diffusion coefficient of free

macromolecules, Do(M) )z, corresponds

^to infinite dilution.

4.1 DIFFUSION COEFFICIENT OF MACROMOLECULES AT INFINITE DILUTION. ^- The diffusion coefficient of

non

interacting macromolecules, Do >z

is obtained from

extrapolation

^{to zero} concentration. If we assume

that the Stokes-Einstein relation is

valid,

^{we can}

deduce from the diffusion data the values of the effective

hydrodynamic radius, RH >z :

where kB

^is the Boltzmann’s constant. The

experi-

mental

results, Do >z,

^{as a function}

of M >z obey

the relation

(Fig. 4) :

FIG. 4. ^- Diffusion coefficient of non interacting macromolecules

versus mass ^M >z : + this point corresponds ^to sample ^{5. The}

dotted line corresponds to D N M w.6.

This result must be

compared

^{with the}

theory [12]

which

predicts

^that the radius of the chains scales like

MV

with :

In the case of the 0

solvent,

measurements of static correlation

[11, 13]

^and

spectral study

^{of the}

Rayleigh

line

[16]

confirm this law. For a

good

^solvent

only

static

experiments yield

^{v ~ 0.6}

[11, 13],

^inelastic

light scattering

in benzene and in butanone 2

[14, 15]

1048

gives

values of v between 0.53 and 0.56. In a

good

solvent i.e. when the excluded volume effect is present, the observed

exponent

^{of the}

hydrodynamic

^radius

differs

only slightly

^{from the}

expected

value. At this

point,

^{we can ask if it is correct to} transpose ^{the static}

to the

dynamic

^results.

In the

previous

part, ^we have assumed that the macromolecule is

impermeable

^to the solvent. The

permeability

^{model of}

Debye

and Bueche

[17]

^can

only explain

deviations in the direction of

higher

values of the

exponent.

^{This is}

obviously

^{not the fact}

in

good

^{solvent as well as in theta} conditions.

4.2 CONCENTRATION DEPENDENCE. ^- In order to

explain

^the effect of concentration

Altenberger

^and

Deutch

[4] (A. D.)

^have

recently proposed

^an

adequate

model for dilute

polymer

solution. The macromo-

lecules

are treated as hard

spheres

of radius R where

repulsive

interactions

(excluded

^volume

effects)

^as

well as

long

range

hydrodynamic

interactions are

taken into account. The effective diffusion

coefficient

results from the resolution of a full N

body

^diffusion

equation.

^{A. D. find :}

where the

non-interacting

diffusion coefficient is

This

theory predicts

that the interaction term

depends

on the volume

occupied by

the macromolecule.

In

light scattering experiments (formula (8)),

^we

have access to the

Z-average

^diffusion

coefficient,

^the

experimental

^{results can be}

represented by

the follow-

ing expression :

in the case of a narrow

weight

distribution.

The

theory [10] predicts

that in dilute

regime

^for

good

solvent the chains behave like a gas of hard

spheres

^{with a}

geometrical

^{radius R =-}

Rg ~

^{Mv with}

v ⁼ 0.6.

Figure

⁵

presents

^a

plot

^{of the}

exponential

values

[

^D

>z/ Do >z - 1] ^V·A/N

^versus

^Mz

^which

are in

good agreement

with the theoretical variation in

M1.8.

FIG. 5. ^- Plot of the relative increment of diffusion coefficient

versus mass. The dotted line scales like the theoretical law Mlg.

i

In their calculations A. D. suppose that the

hydro- dynamical

radius and the

geometrical

radius interfer-

ing

^{in the}

repulsive

interaction are the same. This is not self consistent with the first

part

^{of our work.}

We have modified the A. D. calculation

taking

^into

account that R

(geometrical radius)

is different from

RH (hydrodynamical radius).

^The

experimental

results must then

obey

^the

following

^{relation :}

If we suppose that R is identical to

Rg ^{( ~ MO.6)}

^and

RH )z

^{to the}

experimental

^value

RH >z (~ M >0-55), z

the term

( RH >z/ R >z

⁼

RH >z/ Rg >z

^decreases

slowly

^{with mass}

like >

^M

zo.os.

The relative varia- tion for the _range of mass considered is not

significant

and the

plot

^of

experimental quantity gives essentially

the

dependence

upon the ^mass of

C Rg ^)z.

^{In short}

the

light beating scattering experiment

^on

polystyrene

in

good

solvent far from 0 conditions is able to detect the

difference

between the

hydrodynamical (RH ~ M0.55)

and the

geometrical

^{radius of}

gyration (Rg ~ ^Mo.6).

Acknowledgments.

^- The authors

gratefully

^thank

F. I. B.

Williams,

^G. Jannink and J. des Cloizeaux for

stimulating

discussions.

References and notes

[1] DUBIN, ^S. B., Ph. D., Thesis Massachusetts Institute of Techno- logy (1970).

[2] PECORA, R., ^{J. Chem.} Phys. ⁴⁰ (1964) ^1604.

[3] PUSEY, P. N. (in ^ref. [7]).

[4] ALTENBERGER, ^{A. R. and} DEUTCH, J. M., ^J. Chem. Phys. ⁵⁹ (1973) 894.

[5] FLORY, ^P. J., Principles of Polymer chemistry (Cornell ^Univ.

Press. Ithaca) ^1953.

[6] BROWN, J. C., PUSEY, ^P. N., DIETZ, R., ^{J. Chem.} Phys. ⁶² (1975) 1136.

[7] Photon correlation and light beating spectroscopy ^edited by

H. Z. Cummins and E. R. Pike (Plenum press. New York and London) ^1974.

[8] PIKE, ^E. R., ^Rev. Phys. Technol. 1 (1970) ^180.

[9] ^These polymers and their characteristics are obtained from Centre de Recherches sur les Macromolécules (C.N.R.S.) Strasbourg (France).

[10] DAOUD, ^{M. et} al., Macromolecules 8 (1975) ^804.

[11] DECKER, D., ^Thesis Strasbourg (1968).

[12] ^DE GENNES, ^P. G., Phys. ^{Lett. 38A} (1972) ^339.

[13] COTTON, ^{J. P. et} al., Phys. ^{Rev. Lett. 32} (1974) ^1170.

[14] FORD, ^N. C., KARASZ, ^F. E. and OWEN, ^{J. E.} M., ^Discuss.

Faraday ^{Soc. 49} (1970) ^228.

[15] KING, ^T. A., KNOX, ^{A. and Mc} ADAM, ^{J. D.} G., Polymer ¹⁴ (1973) 293.

[16] KING, T. A., KNOX, A., LEE, ^{W. and Mc} ADAM, ^{J. D.} G., Polymer ¹⁴ (1973) ^151.

[17] DEBYE, ^{P. and} BUECHE, A. M., ^{J. Chem.} Phys. ¹⁶ (1948) ^573.