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>.
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LE JOURNAL DE
TOME
PHYSIQUE
37,
SEPTEMBRE
1976, :
1045
Classification
Physics Abstracts
5.660
-
7.146
-
7.610
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.
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 hydrodynamique 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
2014
thermodynamiques.
Abstract.
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.
2014
1. Introduction.
We present results on the translational 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.
-
Light scattering by a dilute polymer solution.
The spectrum of the light scattered by a mixture
2.
-
contains a component due to concentration fluctuations [1].
The usual theory shows that the measured-photocount autocorrelation function is related to the
scattered field autocorrelation function which is :
The notation represents the ensemble average,
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 segments. In the dilute regime, the distances between
chains are large and the
The density [2] is
polymers
behave like coils.
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 :
k is the
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
and :
(4) vanish
in
a
first
The validity of this factorization has been demonstrated 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 proportional to the well known Z-average [5]
When the weight distribution is relatively narrow [6],
have.
we
short, at a scattering angle chosen to respect
kRg 1, the experimental photocount autocorrelaIn
tion obtained from light scattered by a dilute solution
of polymer with a narrow weight distribution is an
exponential with a characteristic time inversely proportional to the mean D &#x3E;z.
3.1 LIGHT BEATING
Experimental procedure.
The technique and theory of homodyne spectrometer have been described elsewhere [7].
Here, we give only the principal characteristics of the
3.
SPECTROMETER.
TABLE I
approximation [3],
-
-
apparatus. The monochromatic light source is an
argon ion laser used at 4 880 A. A system comprising
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 correlation time, ’tc, of the photocount with an accuracy
of a few percents.
a
3.2 EXPERIMENTAL CONDITIONS.
The macromolecules 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.
The weight-average molecular weights, M )w, and the numberaverage molecular weights, ( M )n, were respectively obtained by
light scattering and osmotic measurements. The Z-average molecular weights, M )z, were calculated assuming that the number
distribution is symmetrical : ( M)z = M )n(3 - 2 M )n/ M )J.
of the exponential fit and the
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 measurement [11]. We can evaluate k*
Rg 1 and the
number N* where the chains begin to overlap [10] :
The good quality
precision (5 x 10-3)
=
where V is the volume of the solution. The criterion
adopted for the dilute regime is N % N*/2. Experimentally, we find that for k k* the coherence time
of the clipped photocount autocorrelation function
is inversely proportional to k2 (Fig. 1) within experimental 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’.
1047
2) the diffusion coefficient is not frequency depen[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
dent
in
figure 2.
dependence. The experimental results
represented by the following expression :
tration
can
be
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.
A is the
4.1 DIFFUSION
COEFFICIENT OF MACROMOLECULES
The diffusion coefficient of
interacting macromolecules, Do &#x3E;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 &#x3E;z :
AT INFINITE DILUTION.
-
non
is the Boltzmann’s constant. The experimental results, Do &#x3E;z, as a function of M &#x3E;z obey
the relation (Fig. 4) :
where kB
FIG. 2.
-
An
of photocount autocorrelation: 8
mental point. --- curve fitting.
example
experi-
We obtain the experimental D &#x3E;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 %.
In the- dilute regime,
4. Results and discussion.
the translational diffusion coefficient, D &#x3E;z, for a
,given mass increases slightly and linearly with concentration. Figure 3 shows an example of this concen-
FIG. 4.
versus
-
Diffusion coefficient of non interacting macromolecules
M &#x3E;z : + this point corresponds to sample 5. The
dotted line corresponds to D N M w.6.
mass
This result must be compared with the theory [12]
which predicts that the radius of the chains scales
like MV with :
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
-
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.
In order to
4.2 CONCENTRATION DEPENDENCE.
the
concentration
effect
of
explain
Altenberger and
Deutch [4] (A. D.) have recently proposed an adequate
model for dilute polymer solution. The macromolecules 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 i
equation. A. D. find :
-
FIG. 5.
-
Plot of the relative increment of diffusion coefficient
The dotted line scales like the theoretical law Mlg.
versus mass.
In their calculations A. D. suppose that the hydrodynamical radius and the geometrical radius interfering in the repulsive interaction are the same. This is
not self consistent with the first
where the
non-interacting diffusion coefficient is
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 :
This theory predicts that the interaction term depends
on the volume occupied by the macromolecule.
In
have
light scattering experiments (formula (8)), we
to the Z-average diffusion coefficient, the
experimental results can be represented by the following expression :
access
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 &#x3E;z/ Do &#x3E;z - 1] V·A/N versus Mz which
are in good agreement with the theoretical variation
If we suppose that R is identical to Rg ( ~ MO.6) and
RH )z to the experimental value RH &#x3E;z (~ M &#x3E;0-55), z
the term ( RH &#x3E;z/ R &#x3E;z
RH &#x3E;z/ Rg &#x3E;z decreases
slowly with mass like &#x3E; M zo.os. The relative variation 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
and the
the hydrodynamical (RH ~ M0.55)
geometrical radius of gyration (Rg ~ Mo.6).
=
in M1.8.
Acknowledgments.
The authors gratefully thank
Williams, G. Jannink and J. des Cloizeaux for
stimulating discussions.
-
F. I. B.
References and notes
[1] DUBIN, S. B., Ph. D., Thesis Massachusetts Institute of Technology (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.
1049
[6] BROWN, J. C., PUSEY, P. N., DIETZ, R., J. Chem. Phys. 62 (1975)
[7]
[8]
[9]
1136.
Photon correlation and light beating spectroscopy edited by
H. Z. Cummins and E. R. Pike (Plenum press. New York
and London) 1974.
PIKE, E. R., Rev. Phys. Technol. 1 (1970) 180.
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.