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Main Properties of Entangled
Polymer Fluids
Entangled polymer fluids are polymer
melts and concentrated or semidilute ( above
the concentration c* ) solutions. In these
systems polymer coils strongly overlap with
each other, and polymer chains are entangled.
Specific properties of entangled polymer
(i) Entangled polymer fluids normally
have a very high viscosity ;
(ii) Such fluids keep for a long time the
memory about the history of flow ;
(iii) the property of viscoelasticity:
at fast (high frequency) external action
the response is elastic, while for slow
(low frequency) external action the
response is viscous (i.e. flow starts).
Viscosity of Fluids
Simplest setup
for viscosity
Newton-Stokes law:
The proportionality coefficient  is called
the viscosity of the fluid.
In the differential form the Newton-Stokes
law can be written as follows
f 
  
where  is imposed stress and z is
the coordinate perpendicular to the plates.
Viscosity is measured in poise;
1 poise  1 g cm  sec  . The viscosity of water
at 200C is  10-2 poise, while viscosity of
polymer melts can be of order 10101012
poise, or even higher.
Flow in capillary viscosimeter
Viscosity is measured by viscosimeters.
Most common are capillary viscosimeters.
The method of measurements is then based
on the Poiseille equation:
r 4 Pt
where Q is the mass of the fluid flown through
the capillary during the time t, P is the
pressure difference at the ends of the capillary
of length l and radius r.
The Property of Viscoelasticity
ln 
ln 
ln t
Step-wise stress
starting at t =0
Reaction of the
ordinary liquid
ln * ln t
Reaction of the
polymer liquid
• The reaction of the ordinary fluid to such a
stress would be a normal flow (after some
initial equilibration period), i.e. the shear
angle  will vary with time as     t 
where  is the viscosity of the fluid.
• The typical reaction of polymer fluids is
qualitatively different. At t << * the value of 
is practically constant    E and
  t 
only at t >> * the flow starts
In other words for entangled polymer fluid
at t << * we have the elastic response    E
where E is the effective Young modulus,
while at t >> * we have     t  i.e. the
response is viscous. This is just the property
of viscoelasticity.
By comparing these two relations, we have
1 
E 
The described property of viscoelasticity is
a general property of all entangled polymer
fluids, as long as they are not crystalline,
glassy or crosslinked. Therefore, as for the
property of high elasticity, the general
molecular explanation should be possible
which is based on the fact of the chain
structure of polymer molecules, without the
explicit reference to the specific chemical
nature of monomer units.
Such explanation was developed by de
Gennes, Doi and Edwards (1971-1979); it is
called the theory of reptations.
Theory of Reptations
Let us consider one chain entangled with
many others and let us “freeze” for a
moment the conformations of the other
chains. This gives rise to a certain “tube”:
the given chain cannot escape in the directions perpendicular to the tube axis 
the only allowed type of motion is the
snake-like diffusion along the tube axisreptations.
If other chains are “defrozen” the competing
mechanism appears:“tube renewal”, but it
can be shown that reptations always give a
dominant contribution.
The neighboring chains forming the walls of
the tube create restrictions for the motion of a
given chain and are in this sense analogous to
But these are “quasi-cross-links” with finite
lifetime: they “relax” after some time  * ,
because the chain leaves the initial tube and
has new neighbor after this time  *.
Thus, the property of visoelasticity has now
molecular interpretation: at
t *
polymer liquid behaves as network with
“quasi-cross-links”, and the response is
elastic; while at t   * “quasi-cross-links”
relax, and the response is viscous.
What is the elastic modulus E for the network
of “quasi-cross-links”? According to the
classical theory of high elasticity E  kT ,
where  is the number of elastic chains per
unit volume,   1 Na 3, N being the number
of monomer units between two cross-links.
It is normally assumed that N  N e , where
N e is some constant for a given polymer,
most often in the interval 50 to 500; this is a
phenomenological parameter reflecting that
not each contact acts like cross-links.
Two neighboring
Two neighboring
chains not forming a chains forming a
“quasi-cross-link” “quasi-cross-link”
E  kT N e a 3
N.B. The model of reptations is valid only
for N  N e .
The picture of a tube
The chain is a sequence of subcoils, each
containing N e links and having the size
d  N e1 2.aThe width of the tube is
d  N e1 2 a . The length of the tube is
d  1 2  Na  L
The diffusion coefficient corresponding to
reptation along the tube is Dt  kT  ,
where  is the corresponding friction
coefficient;   N0 ( 0 - friction coefficient
for one monomer unit) 
Dt  kT  0 N
On the other hand, 2  Dt *, thus
2 N 2 a 2 0 N N 3 0 a 2 N 3
  
N e kT
N e kT
where  0   0 a 2 kT  10 12 sec is the characteristic microscopic time.
  0
Note the very strong N -dependence. For N  105
N e  102, this gives  *  10 sec - macroscopic relaxation time. This is the reason
for slow relaxation and high viscosity in
polymer liquids. Thus
  E * 
 0  2 0
Nea Ne
where 0  kT 0 a 3 is a characteristic viscosity for low molecular liquid 0  1 poise
Therefore, the reptation theory gives for
the viscosity  ~ N 3 . This relationship is
close to the best experimental fit  ~ N 3..4
The Method of Gel-Electrophoresis
Macromolecules containing charged monomer
units are called polyelectrolytes. Charged
monomer units appear as a result of dissociation
monomer unit
monomer unit
Most important polyelectrolytes are biological
macromolecules: DNA and proteins.
As an application of the reptation theory, let us
consider the gel-electrophoresis of polyelectrolytes, having in mind mainly DNA molecules.
This method is used in practice for the separa-
tion of DNA fragments of different length and
composition (for DNA sequencing).
Gel is swollen polymer network. The negatively
charged DNA molecules (total charge Q) move
through the gel in external electric field E . The
drift velocity  depends on the length of the
chains  the separation of the DNA molecules
of different lengths is achieved.
In the absence of the gel: F  QE and F   ,
where F is the electric force acting on the DNA
molecule and  is the friction coefficient. Thus,
  QE  . But Q ~ L and  ~ L (different
parts of DNA molecule exhibit independent friction), therefore  is independent of L and DNA
molecules cannot be separated in the solution.
In the gel DNA molecules are in the effective
“tubes” and move via reptations.
Let us divide the moE
fragments and count
only the forces acting
S i
along the tube.
QS i  Q 
Ft  
E    S i E 
L i
The drift velocity along the tube  t  Ft ,
where    0 N,  0 being the friction coefficient for one monomer unit. Thus,
t 
L 0 N
We have Q ~ L, R ~ N 1 / 2 (in weak field),
1/ 2
N ~ L , therefore  t is proportional to 1 L
(or 1 N 1 2), shorter chains move faster, and
the separation of DNA fragments of different
length is possible.
However, in stronger fields DNA molecules
stretch, R becomes ~ N, and the resolution
of gel-electrophoresis vanishes. To deal with
this problem the direction of the field is
turned at 900 after each interval  *(  * being
the time of renewal of initial tube via reptations).
Then the chain does not have time to stretch
during one cycle. In this way it is possible to
keep a good resolution of the method of
DNA gel-electrophoresis even in strong
enough field.
Gel Permeation Chromatography
This method is also based on the idea of
the separation of chains of different length
in the process of their motion through
chromatographic column. The differences
from gel-electrophoresis:
• The driving force for the motion is pressure gradient (due to the pumping of
polymer solution through chromatographic
column), not electric field. Therefore, the
method can be applied to all polymers, not
only to charged ones.
•The gel medium is normally solid microporous material, rather than swollen soft gel.
Contrary to gel-electrophoresis, the size of
the largest pores is usually much higher
than the coil size (although the pore sizes
exhibit very wide distribution).
• Contrary to gel electrophoresis, in the
normal exclusion regime of gel permeation chromatography (no specific interactions of polymers with the column) larger
chains move faster.
Explanation: shorter chains can penetrate
even in small pores of microporous system,
while long chains move only through
largest pores. Therefore, the “effective
way” for the long chains is shorter.
There is another regime, called adsorption
chromatography, when polymer chains are
attracted to the walls of microporous
system of the column and “stick” to them.
In this case, the “sticking energy” for
longer chains is higher, and they move