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

R 
R
mN 3
M 3
where M is molecular mass of the polymer
chain, M=mN, and NA is Avogadro number .

4 R3 N A
   2.5  2.5 

3
M
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
valid. Then
1
4 R3N A
3
   2.5 
  N 2a3
3 mN
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.
Conclusions
1. Indeed, if the measurements are performed
at the -point we have
   6 0
S
2
3
2
0
M
(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
segment.
2
l
6S
0
L
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.
1
3





     


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*
Dilute polymer
solutions
Overlap
concentration
Semidilute
polymer solutions
The average polymer concentration in the
solution is equal to that inside one polymer
coil at the overlap concentration c*. Thus,
N
1

c  3  3 12 3
R
 N a
Since c*  N/R3 , and []  R3/N , we have
[]c*  1. For the practical estimations it is
normally assumed
c  1  
4. At the -point
  ~ R3 M ~ M 3 2 M . Thus, we can
write    KM 1 2, where K is some
proportionality coefficient.
In the good solvent
  ~  3M 1 2 ~ M 3 10 M 1 2 ~ M 4 5 , i.e.
   KM 4 5, where K is another
proportionality coefficient.
In the general case
   KM a
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 .
L
The characteristic distance L connected
with the draining is
1
  2
L 
 c 
where  is the viscosity of the liquid
and  is the friction coefficient of each
obstacle.
In the application of these results to
polymer coil, we identify obstacles with
monomer units. Then
1
c 3 12 3
 N a
Thus, we have
1
3 12 3 12
2
  N a 
 

L     

 c 


For -solvent
3 12
 a 
 N 1 4  R  aN 1 2
L  
  
For good solvent
3 12
 a 
 N 2 5  R  aN 3 5
L  
  
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
J(  ).
1
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
scatterer.
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
0 r
2
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
moment
 acquired by the coil in the external
field E ).
Experimental results are normally
expressed in terms of reduced scattering
intensity
J r 2 16 4 2 1  cos2 
I
 4  c0
J0 V
2
0
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
M
where M is the molecular mass of a polymer, and
NA is Avogadro number.
Therefore
J r 2 16 4  2  1  cos2 
I
 4
NA
J0 V
2
0 M
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.
n M n
,
 0
2N A 
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,
2
4 n  n 
1  cos 2 
I 4
  M
0 N A   
2
2
2
0
2
1

cos
 ,
We have I  HM
2
2
2
4 2 n0  n 
where H 
  is so-called optical
4 N A   
0
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 
M
H
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
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 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
1
I    I 0 
N2
1
 I 0 
N2
N

 exp  i k rj 
j 1
2
N N
 
  exp i k rj  rl 
j 1 l 1
 2 u
 4
k
sin 
0

u
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,
this gives
I 
1
 1 S 2  k 2
I 0
3
where
is the mean square
S 2 
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,  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
follows
I 0  iwt 3 ikr

I  w0  w 
dt e  d r e δc0,0 δcr , t 
2 

 kwhere is the scattering wavevector

( k  4  sin ) , and  δс 0,0  δc r , t 
2
0
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 .
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.
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

equation
cr , t 

 Dcr , t 
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
2
Dk
I  w0  w  I 0
2 2
Dk
 w2
 
The dependence I(w) is called a Lorenz curve.
I
The spectrum
of the light
scattered at
some angle 
w
w0
w0+ w
The characteristic width of the Lorenz
curve is w  Dk 2  (16 2 D 20 ) sin 2  .
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
kT
D
6R
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
1
1

RH 2 N 2
1
 
i 1,i  j , j 1 rij
N
N
2 12
The values of
 SR2 H1 ,2
 Rand

are of the same order of magnitude; the
difference is in only numerical coefficients.