Swelling and Collapse
of Single Polymer Molecules and Gels
Single polymer molecules
Coil-globule transition
If polymer chains are not ideal, interactions
of non-neighboring monomer units ( the
so-called volume interactions ) should be
taken into account. If these interactions are
repulsive, the coil swells with respect to its
ideal dimensions. If monomer units attract
each other, contraction leads to the
“condensation” of polymer chain upon itself
with the formation of a ”dense droplet”
conformation, which is called a polymer
globule.
Polymer gels
The gel as a whole is actually one
giant three-dimensional molecule.
Swelling
of a gel
Repulsive interactions
of monomer units
Collapse
of a gel
Attractive interactions
of monomer units
Advantage of studying gels vs. dilute
polymer solutions: possibility of direct visual
observation of conformational transitions.
Main disadvantage: very slow
equilibration in macroscopic gel. For the
sample of 1 cm it takes several days.
Equilibration time  can be diminished by
using smaller size L of the sample, one can
show that
 ~ L2
DNA macromolecules
DNA coil and globule
as revealed by fluorescent microscopy
The size of DNA coil and even globule can
be of order of the wavelength of visible light
(400-700 nm), therefore conformational
changes with DNA can be seen by optical
microscope. Normally in such experiments for
better contrast DNA chain is doped with
fluorescent dyes.
Single Polymer Chains with Volume Interactions
Main polymer chain models for the consideration of volume interactions:
1. Model of beads on a Gaussian filament

a
Beads of volume  on immaterial filament.
The connectivity of beads is given by a
condition that the the probability distribution

for a vector r between neighboring beads is
3
  3 2
 3r 2 
g (r )  
exp   2 
2 
 2a 
 2a 
  
 Comment: chain connectivity

as in the model of beads can
 
be obtained for any chain by
selecting the dividing points separated by
several persistent lengths and by assigning all
the mass of each chain segment to the
dividing point.
The value of a is the average distance
between neighboring beads. It is possible
to have   a 3
or   a 3
The beads interact with the interaction

potential U (r ).Typical interaction potential
U
repulsion due to
self-volume of the
beads (excluded volume)
r

Van-der-Waals
attraction
Note: in principle this potential is
renormalized by the presence of the
solvent, so its form can be much more
complicated.
2. Lattice model
Polymer chain is modeled as a random walk
on a lattice. The random walk cannot visit
each site more than once (the excluded
volume condition); the lattice attractive
energy   is assigned to each pair of
neighbor sites visited by two nonconsecutive monomer units of the chain.
Concept of  -temperature
It should be noted that for both models:
1. At high values of T, the ratio  / kT << 1
and only repulsion matters. The coil should
swell with respect to the ideal dimensions;
this phenomenon is called the excluded
volume effect. In this case the so-called
swelling coefficient of the coil, , is larger
than unity:
2

R

2
  2 1
R 0
2. At low values of T, the ratio  / kT >> 1,
and attraction dominates. The coil should
shrink and form a condensed globule
( coil-globule transition ).
3. At intermediate values of T, the effect of
repulsion and attraction should
compensate each other and the coil should
adopt ideal-chain ( unperturbed ) size. This
happens at the so-called -temperature.
The free energy of a coil is the sum of
energetic and entropic contributions:
F  E  TS
As to the energy E, contrary to the case of the
ideal coil, for the present case it is not equal to
zero.
To write down the expression for E we
should recall that the concentration of
monomer units inside the polymer coil is very
small for long chains. Therefore, we can write
the expression for E in terms of the expansion
in the powers of n:
E  NkT ( Bn  Cn 2  )
Here B, C, … are second, third, … virial
coefficients; these coefficients are responsible
for binary, ternary, etc. interactions of
monomeric units. These coefficients depend
only on the interaction potential between the
units, e.g.
3
1

B(T )   1  exp( U ( r ) kT )  d r
2
From the expression for B(T ): at high T
(  kT  1 ) B  , with the decrease of T
the value of B diminishes, until it reaches
zero at some temperature .
T-Θ
At T   : B   , where  
T
The characteristic
dependence of the
second virial
coefficient
on temperature
At the -temperature B  0  F  TS 
chain adopts the conformation of ideal coil.
At T   repulsion dominates, coil swells
due to the excluded volume effect - region of
good solvent.
At T   attraction dominates, coil shrinks
into globule - region of poor solvent.
Remarks:
1. The fact of complete compensation
of interactions at the -point is a
specific polymer property (not valid
e.g. for gases) connected with low
polymer concentration in the coil.
2. Repulsion for T   and attraction
for T   is valid for the usual form of

the potential U (r ). For more
complicated forms the situation may
be reverse, or it may be many -points
(see below).
The Excluded Volume Problem
Let us consider the polymer coil
far above the -point, in the good
solvent region, and let us calculate the
swelling of the coil due to the excluded
volume.
F  E  TS  NkTBn  TS 
3N
3kTR 2
 NkTB

 const
3
2
4 R
2 Na
For the second term we used the
expression for the entropy of a polymer
coil expanded up to the size R (derived
earlier).
Excluded volume repulsion (first
term) induces the coil swelling, while
entropic elasticity (second term)
opposes the swelling. The balance
(minimization of F with respect to R)
gives the equilibrium coil size.
The free energy of a coil is
N
3kTR 2
F  NkTB 4

3
2

R
2
Na
3
Minimization of F with respect to R :
F
0
R
kTBN 2 kTR


0
4
2
R
Na
Therefore,
R  Ba
2
)
1
5
N
3
5
  a
2
)
1
5
N
3
5
Also swelling coefficient is
1
  5 110

  1   3  N  1
N 2a  a 
R
The polymeric coil swells essentially due
to the excluded volume. In spite of the low
polymer concentration in the coil, the
swelling is large because of high
susceptibility of long polymer chains.
Swelling of polymer gels
We will consider the gels synthesized in the
presence of a large amount of solvent. For
such gels entanglements between the gel
chains are not important.
A swollen
polymer gel
Since each subchain of a gel swells
independently in its own subvolume:
1. The swelling degree of each subchain is
equal to the swelling degree of the gel as
a whole.
2. The swelling degree of each subchain is
the same as the swelling degree of an
isolated chain in the good solvent.
 a)
 
1
5
3
N
1
10
 1
Superabsorbing properties of
polyelectrolyte gels
In real case the value of  varies between
1.2 and 2.5 which corresponds to a 15-fold
increase of the volume of the gel upon
swelling in the maximum, except for the case
of polyelectrolyte gels.
counter ions
charges on
polymer chains
Schematic picture of a polyelectrolyte gel
Polyelectrolyte gels normally increase their
volume in water several hundred times. This
property is used for the design of
superabsorbing polymer systems.
The
superabsorbing
properties
of
polyelectrolyte gels in water are quite
universal: they are observed for any gel,
independently of its chemical structure,
provided that the gel contains charged
monomer units.
Since there are charges on polymer chains,
there should also be counter ions, because the
overall system should be electrically neutral.
For these counter ions it is thermodynamically
advantageous to abandon the gel and travel in
the whole external volume, because in this way
they gain significant entropy of translational
motion. However, this is impossible, because
this would violate the condition of macroscopic
electroneutrality of the gel. The counter ions
are therefore forced to be confined within the
gel, and they produce a significant osmotic
pressure in the gel. This osmotic pressure
induces a very essential gel swelling: in this
way there is more space for the translation
motion of each counter ion.
Coil-Globule Transition
Now let us consider the whole range of
temperatures. When the temperature is
lowered below the Θ-point, the coil-globule
transition(or polymer chain collapse) should
take place.
T >  ;  >1
good solvent
T <  ;  <1
poor solvent
The interest to globular form of
macromolecules is induced by molecular
biophysics: proteins-enzymes are polymeric
globules. Denaturation of proteins is
sometimes considered to be analogous to the
transition from globule to coil.
To determine the characteristics of the
coil-globule transition, let us write F  E  TS,
as before, but now the expression for both E
and S should be different.
Energy: For dense globules higher
virial coefficients may be of importance:
E  NkT Bn  Cn 2  )
The term Bn gives the attraction at T   ,
while the term Cn2 gives repulsion, since
normally C  2 > 0 in the region of interest.
The higher-order virial coefficients can be
neglected in the coil-globule transition region.
Thus, omitting, as usual, all numerical
coefficients, we have
 BN CN 2 
E  NkT  3  6 
R 
R
Entropy: We must take into
account the possibility of chain compression
(not only extension).
1
In case of coil R  N 2 a
3kR 2
R2
S
 2
2
2 Na
Na
For the globular state
1
R  N 2 a
N
Na 2
S  k  k 2
g
R
The subchain of g links with ga2  R2
is practically free ( ~ one encounter with the
“wall”  loss of entropy  1 per this
2
2
subchain); g  R a .
Interpolation formula valid (in the orders
1
of magnitude) for R  N 12 a , R  N 2 a
and in the intermediate region:
 R 2 Na 2 
S  k  2  2 
R 
 Na
Finally we have for the free energy:
F  E  TS 
 BN CN 2 
 R 2 Na 2 
 NkT  3  6   kT  2  2 
R 
R 
R
 Na
Minimization of F with respect to R
( F R  0 ) gives
3
R5
CN
2
2

RNa


BN
Na 2
R3
2
C
υ
2
2
2
Or in terms of   R Na , y  6  6 ,
a
a
1
1
2
BN
υτ N 2
x

3
a
a3
we have
 5   y  3  x

1
1
1
2
3
2
3
y=1
y = 1/ 60
y = 1/ 300
x
-2
poor solvent
-1
0
 - point
1
2
good solvent
Conclusions:
1 Coil-globule transition takes place at x  1,
1
1
3
3
2
i.e. N   a  1    a N 2  1,
this is only slightly lower the -point. It is
enough to have a very weak attraction to
induce the transition into globule (this is not
the case for condensation of gases).
Reason: due to the chain connectivity the
independent motion of monomer units is
impossible (polymer coil is poor in
entropy).
2 For y<<1 the chain collapse is discrete,
while for y ~ 1 it is continuous. The value
of y depends on /a3 ( y  2/a6 ). For <<
a3  y << 1, while for   a3  y ~ 1.
For realistic chain models y << 1
corresponds to stiff chains, and y  1 to
flexible chains. Indeed, it can be shown that
C  d 3 l 3; a  l , therefore
y = C / a6  d 3/ l 3. Thus, for stiff polymer
chains the collapse is discrete, while for
flexible chains it is continuous.
 )
3 In the limit of small , B < 0 ( globular
region ) the size of the globule is defined
by the following two terms:
CN 3
 3  BN 2
R
Thus the size of the globule is
CN
R 
B
3
or
R  C B ) N
1
3
1
3
The volume fraction of monomer units
within the globule is
nυ
B υυ τ
N
υ

 2  τ
3
R
C
υ
1
(a) For the globule R ~ N 3 (cf. with
3
1
R ~ N 2 for ideal coil and R ~ N 5
for the coil with excluded volume).
3 (b) In the globule far from -point ( ||  1)
the volume fraction of monomer units
is generally not small. This is a dense
liquid droplet.
3 (c) The globule swells as the -point (and
the coil-globule transition point) is
approached, so the description in
terms of B and C only in the vicinity of
transition point is valid.
Φ
Θ
τ
The volume fraction of
4 Experimentally the coil-globule transition
was observed for many polymer-solvent
systems. A very convenient system is
polystyrene in cyclohexane, since the
-temperature for this case corresponds to
 = 35 oC.
Main difficulty for the experimental
observation of the coil-globule transition is
connected with the possibility of
intermolecular aggregation and formation
of precipitate, rather than coil-globule
transition. To avoid this the concentration
of polymer in the solution should be very
small ( e.g. for polystyrene-cyclohexane
system it should be less than 10-4 g/l ).
Formation of precipitate in the poor solvent region
5 One of the most interesting possibilities to
study the coil-globule transition for the
particular case of DNA molecules is to use
direct visualization via fluorescence optical
microscopy. In this case one may work in the
range of much lower polymer concentrations
(up to 10-5 g/l) and the problem of chain
aggregation is not very serious. One may
observe the bimodality of the histogram of
observed sizes which is the indication of the
first ordered coil-globule transition. This is
in full agreement with the theory outlined
above, since DNA is a stiff-chain polymer.
Histogram of DNA sizes as a function of concentration of added
poly(ethylene oxide) - agent making solvent quality poorer.
Collapse of polyelectrolyte gels
At some critical solvent composition the
several-hundred-fold jumpwise decrease of
the gel volume is observed.
V/V0
1,0
The dependence of a
volume of a polyelectrolyte
gel V on the volume
fraction  of the poor
solvent added to water.
0,8
0,6
0,4
0,2
0,0
0
20
40
60
80
100
, %
The amplitude of the jump is directly
connected with the polyelectrolyte nature of
the gel: it increases with an increase of the
degree of charging of the gel chains.
The gel shows high responsiveness to the
external actions in the region of jumpwise
collapse. In this region it is enough to change
the external conditions ( solvent quality, salt
concentration, pH, temperature ) only slightly
to obtain a very strong reaction of the system.