Main Properties of Entangled Polymer Fluids

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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 fluids (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).
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Viscosity of Fluids
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
where ? is imposed stress and z is the
coordinate perpendicular to the plates.
Simplest setup for viscosity determination
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Viscosity is measured in poise
. 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. Viscosity is measured by
viscosimeters. Most common are capillary
viscosimeters. The method of measurements is then
based on the Poiseille equation
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.
2r
Flow in capillary viscosimeter
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• 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 where ?
  is the viscosity of the fluid.
• The typical reaction of polymer fluids is
  qualitatively different. At t ltlt ? the value of
  ? is practically constant and
  only at t gtgt ? the flow starts
  .

The Property of Viscoelasticity
?
t
Step-wise stress starting at t 0
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In other words for entangled polymer fluid at t
ltlt ? we have the elastic response where E is
the effective Young modulus, while at t gtgt ? we
have i.e. the response is
viscous. This is just the property of
viscoelasticity. By comparing these two
relations, we have
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.
or
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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 direc-tions perpendicular to the
tube axis the only allowed type of motion is the
snake-like diffusion along the tube
axis-reptations.
If other chains are defrozen the competing
mechanism appearstube renewal, but it can be
shown that reptations always give a dominant
contribution.
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It is normally assumed that , where
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 chains not forming
a quasi-cross-link
Two neighboring chains forming
a quasi-cross-link
Thus,
N.B. The model of reptations is valid only for
.
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The picture of a tube
The chain is a sequence of subcoils, each
containing links and having the size .
The width of the tube is . The length of
the tube is
The diffusion coefficient corresponding to
reptation along the tube is ,
where is the corresponding friction
coefficient ( - friction
coefficient for one monomer unit)
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Note the very strong N --dependence. For
, this gives -
macro-scopic relaxation time. This is the reason
for slow relaxation and high viscosity in polymer
liquids. Thus
where is a characteristic
vis- cosity for low molecular liquid Therefore,
the reptation theory gives for the viscosity
. This relationship is close to the best
experimental fit .
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The Method of Gel-Electrophoresis
Macromolecules containing charged monomer units
are called polyelectrolytes. Charged monomer
units appear as a result of dissociation
reaction
Most important polyelectrolytes are biological
macromolecules DNA and proteins.
As an application of the reptation theory, let
us consider the gel-electrophoresis of
polyelectro- lytes, 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).
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-

Gel is swollen polymer network. The
negatively charged DNA molecules (total charge Q)
move through the gel in external electric field
. 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 and
, where is the electric force acting
on the DNA molecule and is the friction
coefficient. Thus, . But
and (different parts of DNA
molecule exhibit independent fric-tion),
therefore is independent of L and DNA
molecules cannot be separated in the solution.
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In the gel DNA molecules are in the effective
tubes and move via reptations.
Let us divide the mo- lecule in small fragments
and count only the forces acting along the tube.
The drift velocity along the tube
, where , being the friction
coef-ficient for one monomer unit. Thus,
We have (in weak
field), , therefore is proportional to (or
), shorter chains move faster, and the
separation of DNA fragments of different length
is possible.
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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
after each interval ( being the time of
renewal of initial tube via repta- tions).
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.
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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 microporous (gel)
medium - chromatographic column. The differences
from gel-electrophoresis

• The driving force for the motion is pres-sure
  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 micropo-rous
  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).

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• Contrary to gel electrophoresis, in the normal
  exclusion regime of gel permea-tion
  chromatography (no specific interac-tions 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 slower.