Light Scattering from Polymer Solutios

1
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 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 where M is molecular mass of the
polymer chain, MmN, and NA is Avogadro number .

2
For dense spheres NR3 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 . If they still move as
a whole together with the solvent inside the coil
(non-draining assumption), the Einstein formula
remains valid. Then therefore,
for polymer coils there is an N dependence. So by
measuring ? it is possible to get the
information on the size of polymer coils.
3
Conclusions 1. Indeed, if the measurements are
performed at the ?-point we have
(Flory-Fox law), where
universal constant if ? is expressed in
dl/g. From this relation, if we know M (elastic
light scattering, chromatography), it is
possible to determine ltS2gt0, and therefore to
obtain the length of the Kuhn segment.
On the other hand, if we know l ? we
can determine M. 2. By determining ? ? and ?
in the good solvent, we can calculate the
expansion coefficient of the coil.
4
3. Another important characteristics of
polymer solutions which can be determined
from the value of ? is the overlap
concentration of polymer coils c Th
e average polymer concentration in the solution
is equal to that inside one polymer coil at the
overlap concentration c. Thus, Since c ?
N/R3 , and ? ? R3/N , we have ?c ? 1. For
the practical estimations it is normally assumed
c lt c Dilute polymer solutions
c c Overlap concentration
c gt c Semidilute polymer solutions
5
4. At the ?-point .
Thus, we can write , where K
is some proportionality coefficient. In the
good solvent , i.e. ,
where K is another proportionality
coefficient. 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.
6
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. The characteristic
distance L connected with the draining
is where ? is the viscosity of
the liquid and ? is the friction coefficient of
each obstacle.
?
System of obstacles in the upper half-space move
through a liquid with the velocity ?.
L
7
In the application of these results to polymer
coil, we identify obstacles with monomer units.
Then Thus, we have For
?-solvent For good solvent 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.
8
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.
9

• 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 J(
  ? ).
• Normally the size of the coil 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 scatterer.

10

• Scattering of normal nonpolarized light by point
  scatterers has been considered by Rayleigh.
  The result is where c0
  is the concentration of coils (scatterers), V is
  the scattering volume, while ? is the
  polarizability of the coil ( defined according to
  being a dipole moment acquired
  by the coil in the external field ).
• Experimental results are normally expressed in
  terms of reduced scattering intensity
• The value of I does not depend on the geometry
  of experimental setup.

11
Traditionally for polymer scatterers the value of
polymer mass per unit volume, ?, is used instead
of c0 where M is the
molecular mass of a polymer, and NA is Avogadro
number. Therefore 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 the solvent. ,
where n0 is the refractive index of the
pure solvent. The value of is called
refractive index increment it can be directly
experimentally measured for a given
polymer-solvent system. Thus,

12

• We have ,
• where is
  so-called optical
• 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,
• 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.

13

• 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 gt ? /20. In
  this case it is necessary to take into account
  the destructive interference of light
  scattered by different monomer units.
• The waves scattered from the monomer units
  A and B in the direction of the unit vector
  are shifted in phase with respect to each other,
  because of the excess distance l . This phase
  shift is small, as soon as l ltlt ? , 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
  ?.

14

• From the scattering theory we know that
• 
  
  
• 
  
  
• where I(0) is the light scattered at ? 0 (equal
  to the value of I discussed above), and
  
  
• is the scattering wave vector ( being unit
  vector pointing in the direction of scattered
  light).

15

• For the light scattering always (where is
  the distance between two monomer units), since
  . Thus, we can expand the expression for I in
  the powers of k. Since linear terms vanishes
  after averaging, this gives
  where is the mean square radius of gyration of
  polymeric coil.
• Thus
• 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,
  .

16
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.
17
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
follows where is the
scattering wavevector (
) , and
is so-called dynamic structure factor
of polymer solution
is the deviation from the average polymer
concentration at the point and the time moment
t . Thus (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 structure factor.
18
By studying the scattering at a given angle ? (or
), we investigate the dynamics of polymer
chain motions with the wavelength
. For dilute solutions at the
wavelength 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 (
), .

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)
obeys the diffusion equation whe
re .
19
For the Fourier transform ( vs. time and
spatial coordinates ) of the dynamic structure
factor the diffusion equation is known to
give The dependence I?(w) is called a Lorenz
curve. The characteristic
width of the Lorenz curve is
. Thus, by
measuring the spectrum of the scattered light it
is possible to determine the diffusion
coefficient of the coils, D.
The spectrum of the light scattered at some
angle ?
20
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 so-called
hydrodynamic radius of the coil
The values of RH ,
and are of the same order of
magnitude the difference is in only numerical
coefficients.