Light Scattering from Polymer Solutios
Light Scattering from Polymer Solutios
Viscosity of Dilute Polymer Solutions
For dilute polymer solutions, one is
normally interested not in the value of itself
but in specific viscosity s= (-0)/0 (0 is the
viscosity of pure solvent) and characteristic
viscosity  = (-0)/0 , where is the
density of monomer units in the solution.
For the solution of impenetrable spheres of
radius R Einstein derived
0 1 2.5
where is the volume fraction
occupied by the spheres in the solution. If each
sphere consists of N particles (monomer units)
of mass m, and their density is , we have
N A 4 3 N A 4 3
where M is molecular mass of the polymer
chain, M=mN, and NA is Avogadro number .
4 R3N A
For dense spheres N~R3 and  is
independent of the size of the particles
viscosity measurements are not informative
in this limit. E.g. for globular proteins we
always obtain  4 cm3/g independently of
the size of the globule.
However, polymer coils are very loose
objects with N R3 N 3 R03 1 3 N 1 2 a 3. If
they still move as a whole together with the
solvent inside the coil (non-draining
assumption), the Einstein formula remains
4 R3N A
therefore, for polymer coils there is an
N dependence. So by measuring  it is
possible to get the information on the size of
1. Indeed, if the measurements are performed
at the -point we have
(Flory-Fox law), where universal
constant 0 2.84 1021 if [ ] is expressed
in dl/g. From this relation, if we know M
(elastic light scattering, chromatography), it
is possible to determine <S2>0, and
therefore to obtain the length of the Kuhn
On the other hand, if we know l we can
2. By determining [ ] and [ ] in the good
solvent, we can calculate the expansion
coefficient of the coil.
3. Another important characteristics of
polymer solutions which can be
determined from the value of [ ] is the
overlap concentration of polymer coils c*
c < c*
c = c*
c > c*
The average polymer concentration in the
solution is equal to that inside one polymer
coil at the overlap concentration c*. Thus,
c 3 3 12 3
Since c* N/R3 , and  R3/N , we have
c* 1. For the practical estimations it is
4. At the -point
~ R3 M ~ M 3 2 M . Thus, we can
write KM 1 2, where K is some
In the good solvent
~ 3 M 1 2 ~ M 3 10 M 1 2 ~ M 4 5 , i.e.
KM 4 5, where K is another
In the general case
It is called a Mark-Kuhn-Houwink
equation. Its experimental significance
is connected with the fact that by
performing measurements for some
unknown polymer for different values of
M and by determining the value of a it
is possible to judge on the quality of
solvent for this polymer.
5. Assumption of non-draining coils.
Analysis shows that this assumption is
always valid for long chains.
Let us consider the system of obstacles of
concentration c moving through a liquid with
the velocity . Inside the upper half-space the
liquid will move together with the obstacles,
inside the lower half-space it will mainly
remain at rest.
System of obstacles
in the upper half-space
move through a liquid
with the velocity .
The characteristic distance L connected
with the draining is
where is the viscosity of the liquid
and is the friction coefficient of each
In the application of these results to
polymer coil, we identify obstacles with
monomer units. Then
c 3 12 3
Thus, we have
3 12 3 12
N 1 4 R aN 1 2
For good solvent
N 2 5 R aN 3 5
In both case the value of L is much smaller
than the coil size R (for large value of N),
thus the non-draining assumption is valid.
Analysis shows that the opposite limit (free
draining) can be realized only for short and
stiff enough chains.
Light Scattering from Polymer Solutions
It is well-known that all media (e.g. pure
solvent) scatter light. This is the case even for
macroscopically homogeneous media due to
the density fluctuations. If polymer coils are
dissolved in the solvent, another type of
scattering appears - scattering on the polymer
concentration fluctuations. This is called
excess scattering; it is this component which
is normally investigated for the analysis of the
properties of the coils.
In this section we will consider elastic (or
Rayleigh) light scattering (without the change
of the frequency of the scattered light) and the
scattering from dilute solutions of coils.
Let us assume that the incident beam of
light (wavelength 0 , intensity J0 ) passes
through a dilute polymer solution.
The detector is located at a distance r
from the scattering cell in the direction of
the scattering angle . The quantity which is
measured is the intensity of excess scattering
Normally the size of the coil R aN 2 is
less than 100nm and is, therefore, much
smaller than the wavelength of light 0 . In
this case the coil can be regarded as point
Scattering of normal nonpolarized light
by point scatterers has been considered by
Rayleigh. The result is :
16 4 2
1 cos 2
J 4 2 c0VJ 0
where c0 is the concentration of coils
(scatterers), V is the scattering volume, while
is the polarizability
of the coil ( defined
according to P E ; P being a dipole
acquired by the coil in the external
field E ).
Experimental results are normally
expressed in terms of reduced scattering
J r 2 16 4 2 1 cos2
The value of I does not depend on the
geometry of experimental setup.
Traditionally for polymer scatterers the value
of polymer mass per unit volume, , is used
instead of c0 :
c0 N A
where M is the molecular mass of a polymer, and
NA is Avogadro number.
J r 2 16 4 2 1 cos2
The polarizability can be directly expressed
in terms of the change of the refractive index of
the solution n upon addition of polymer coils to
n M n
where n0 is the refractive index of the pure
solvent. The value of n is called refractive
index increment; it can be directly
experimentally measured for a given polymersolvent system. Thus,
4 n n
1 cos 2
0 N A
We have I HM
4 2 n0 n
is so-called optical
4 N A
constant of the solution. It depends only on
the type of the polymer-solvent system, but
not on the molecular weight or concentration
of the dissolved polymer.
So, by measuring I( ) we can
determine the molecular mass of the
dissolved polymer. For example, if I( 900 ) is
the scattering intensity at the angle 900,
2 I 900
The physical reason for the possibility
of the determination of M from the light
scattering experiments can be explained as
follows. The value of I is proportional
to the concentration of scatterers c0 ( ~ 1/M)
and to the square of polarizability ( ~ M2 ) ,
thus I ~ M.
Whether it is possible to obtain from the
same experiment the size of the coil, R , in
addition to M ? The answer is yes, and this
can be explained as follows. It should be
emphasized that the coil actually can not be
regarded as point scatterer, as long as
R > /20. In this case it is necessary to take
into account the
light scattered by
scattered from the
monomer units A and B in the direction of the
unit vector u are shifted in phase with respect
to each other, because of the excess distance l .
This phase shift is small, as soon as l << , but
still it is responsible for the partially destructive
interference which leads to the decrease in I.
This effect should be larger for higher values of
From the scattering theory we know that
I I 0
exp i k rj
exp i k rj rl
j 1 l 1
For the light scattering always k r 1
(where r is the distance between two
monomer units), since k 1 0 . Thus, we can
expand the expression for I in the powers of k.
Since linear terms vanishes after averaging,
1 S 2 k 2
is the mean square
radius of gyration of polymeric coil.
1. By measuring the intensity at any
specified angle it is possible to obtain the
molecular mass of a polymer chain, M.
2. By measuring the angular dependence
of scattered light it is possible to obtain the
mean square radius of gyration of a polymer
coil, S 2 .
Inelastic light scattering
from dilute polymer solutions
In the method of inelastic light scattering
we measure not only the intensity, but also the
frequency spectrum of scattered light. For this
method the incident beam should be obligatory
a monochromatic light from laser ( reduced
intensity I0 , frequency w0 , wavelength 0 ).
The light scattered at the angle is actually
not monochromatic, since the scattering
objects are moving.
If I(w0 + w) is the intensity of the light of
frequency w0 + w scattered at the angle ,
then from the general scattering theory it
I 0 iwt 3 ikr
I w0 w
dt e d r e δc0,0 δcr , t
kwhere is the scattering wavevector
( k 4 sin ) , and δс 0,0 δc r , t
is so-called dynamic structure factor of
polymer solution; δc r , t c r , t c is the
deviation from the average polymer
concentration at the point r and the time
moment t .
(i) scattering is connected with the
dynamics of concentration fluctuations;
(ii) intensity of scattering is given by the
Fourier-transformation ( vs. time and
spatial coordinates ) of the dynamic
By studying the scattering at a given angle
(or k ), we investigate the dynamics of
polymer chain motions with the
wavelength 1 k .
For dilute solutions at k R 1 the
wavelength 1 k R and for this case we
can study the internal motions with the coil.
These condition can be realized for the
scattering of X-rays or neutrons. But for the
light scattering normally even at 1 ;
( k 1 0 ), k R R 0 1 .
In this limit the method of dynamic light
scattering probes the motion of the coils as a
whole. Coils move as point scatterers with the
diffusion coefficient D. The concentration of
coils (scatterers) с r , t obeys the diffusion
cr , t
Dcr , t
where 2 x 2 2 y 2 2 z 2 .
For the Fourier transform ( vs. time and
spatial coordinates ) of the dynamic structure
factor the diffusion equation is known to give
I w0 w I 0
The dependence I(w) is called a Lorenz curve.
of the light
The characteristic width of the Lorenz
curve is w Dk 2 (16 2 D 20 ) sin 2 .
Thus, by measuring the spectrum of the
scattered light it is possible to determine the
diffusion coefficient of the coils, D.
What should be expected for the value of D?
If coils are considered as impenetrable spheres
of radius R ( we will see below that this is the
case in many situations ), then according to
Stokes and Einstein
where is the viscosity of the solvent.
Thus, by measuring D it is possible to
determine R. This is a more precise method for
the determination of the size of a polymer coil
than elastic light scattering. Detailed analysis
shows that the value thus determined is the socalled hydrodynamic radius of the coil
RH 2 N 2
i 1,i j , j 1 rij
The values of
SR2 H1 ,2
are of the same order of magnitude; the
difference is in only numerical coefficients.