Important Points from Last Lecture:
• The root-mean-squared end-to-end distance, <R2>1/2, of a freelyjointed polymer molecule is N1/2a, when there are N repeat units,
each of length a.
• The radius-of-gyration of a polymer, Rg, is 1/6 of its root-meansquare end-to-end distance <R2>1/2.
• Polymer coiling is favoured by entropy. The elastic free energy of a
polymer coil is given as
3kR 2
F (R ) = +
2 T + const.
2Na
• Thinner lamellar layers in a diblock copolymer will increase the
interfacial energy and are not favourable. Thicker layers require
chain stretch and likewise are not favourable! A compromise in the
lamellar thickness, d, is reached as:
5
d =(
a
2kT
)1 / 3 N 2 / 3
Last Lecture:
• Elastic (entropic) effects cause a polymer molecule to coil up.
• Excluded volume effects cause polymer molecules to swell (in a
self-avoiding walk).
• Polymer-solvent interactions, described by the c-parameter, also
have an effect, depending on whether polymer/solvent
interactions are more favourable than self interactions.
• Thus there is a competition between three effects!
• When c = 1/2, excluded volume effects are exactly balanced by
polymer/solvent interactions. Elastic effects (from an entropic
spring) lead to a random coil: <R2>1/2 ~ aN1/2
• When c < 1/2, excluded volume effects dominate over
polymer/solvent interactions. They dominate over elastic effects
and result in a swollen coil: <R2>1/2 ~ aN3/5
• When c > 1/2, unfavourable polymer/solvent interactions are
dominant over excluded volume effects. They lead to polymer
coiling: a globule results.
PH3-SM (PHY3032)
HE3 Soft Matter
Lecture 10
Polymer Elasticity, Reptation,
Viscosity and Diffusion
13 December, 2011
See Jones’ Soft Condensed Matter, Chapt. 5
Rubber Elasticity
A rubber (or elastic polymer = elastomer) can be created by
linking together linear polymer molecules into a 3-D network.

Covalent bonds between polymer molecules are called
“crosslinks”. Sulphur can crosslink natural rubber (which is
liquid-like) to create an elastomer.
To observe “stretchiness”, the temperature should be > Tg for
the polymer.
Affine Deformation
With an affine deformation, the macroscopic change in dimension
is mirrored at the molecular level.
We define an extension ratio, l, as the dimension after a
deformation divided by the initial dimension:

l=
o
Bulk:

o

l
l= =
 o lo
Strand:
lo
l
Transformation with Affine Deformation
z
Bulk:
lz z
z
y
x
z
Single
Strand:
zo
ly  y
y
x
lx  x
If non-compressible (volume conserved): lxlylz =1
Ro = xo+ yo+ zo
Ro
z R = lxxo + lyyo + lzzo
R
y
xo
x
y
yo
x
R2 = x2+y2+z2
Entropy Change in Deforming a Strand
The entropy change when a single strand is deformed, DS, can be
calculated from the difference between the entropy of the deformed
coil and the unperturbed coil:
DS = S(R) - S(Ro) = S(lxxo, lyyo, lzzo) - S(xo, yo, zo)
We recall our expression for the entropy of a polymer coil with endto-end distance, R:
S (R ) =
3kR 2
2Na 2
Initially:
+ const. ~
S(Ro ) ~
Finding DS:
S(R ) S(Ro ) ~
3k
2Na
3k
2Na
3k
2Na
2 2
2 2
2 2
(
l
x
+
l
y
+
l
x o
y
o
z zo )
2
2
2
2
(
x
+
y
+
z
o
o
o )
2
2
2
2
2
2
2
[(
l
1
)
x
+
(
l
1
)
y
+
(
l
1
)
z
x
o
y
o
z
o ]
2
Entropy Change in Polymer Deformation
DS ~
3k
2Na
2
2
2
2
2
2
[(
l
1
)
x
+
(
l
1
)
y
+
(
l
1
)
z
x
o
y
o
z
o ]
2
But, if the conformation of the coil is initially random, then
<xo2>=<yo2>=<zo2>, so:
DS ~
3kxo 2
2Na
2
2
2
[(
l
1
)
+
(
l
1
)
+
(
l
1)]
x
y
z
2
For a random coil, <R2>=Na2, and also R2 = x2+y2+z2 = 3x2, so we
see:
Na 2
< xo 2 >=
Substituting for xo2: DS ~
This simplifies to:
3
Na 2
2
2
2
(
)(
l
+
l
+
l
x
y
z
2
3
2Na
3k
k
DS ~
(lx 2 + ly 2 + lz 2
2
3)
3)
DF for Bulk Deformation
If there are n strands per unit volume, then DS per unit volume for
bulk deformation:
DSbulk
nk
~
(lx 2 + ly 2 + lz 2
2
3)
If the rubber is incompressible (volume is constant), then lxlylz =1.
For a one-dimensional stretch in the x-direction, we can say that lx
= l. Incompressibility then implies
ly = lz =
1
l
Thus, for a one-dimensional deformation of lx = l:
DSbulk
nk 2 2
~(l  - 3)
2
l
The corresponding change in free energy: (F = U - ST) will be
DFbulk
nkT 2 2
~+
(l +
2
l
3)
Force for Rubber Deformation
At the macro-scale, if the initial length is Lo, then l = L/Lo.
DFbulk
nkT L 2 2Lo
~+
(( ) +
2 Lo
L
3)
In Lecture 3, we saw that sT = Ye. The strain, e, for a 1-D tensile
deformation is
Substituting in
L/Lo = e + 1:
DL L Lo L
e=
=
=
1
Lo
Lo
Lo
nkT
2
2
DFbulk ~ +
((e + 1) +
3)
2
(e + 1)
Realising that DFbulk is an energy of deformation (per unit volume:
Nm/m3), then dF/deT is the force,  , per unit area, A (units: N/m2)
for the deformation, i.e. the tensile stress, sT.
dF 
nkT
2
  sT 
[2(e  1) ]
2
de AA
2
(e  1)
Young’s and Shear Modulus for Rubber
s T  nkT [(e  1) 
1
1

]

nkT
l


(e  1)2
l2 

This is an equation of state, relating together  , L and T.
In the limit of small strain, sT  3nkTe, and the Young’s modulus is
thus Y = 3nkT.
The Young’s modulus can be related to the shear modulus, G, by a
factor of 3 to find a very simple result: G = nkT
This result tells us something quite fundamental. The elasticity of a
rubber does not depend on the chemical make-up of the polymer nor
on how it is crosslinked.
G does depend on the crosslink density. To make a higher
modulus, more crosslinks should be added so that the lengths of
the segments become shorter.
Experiments on Rubber Elasticity
Rubbers are elastic over a large range of l!
s T  nkT [l -
1
l
2
Strain hardening region:
Chain segments are fully
stretched!
]
Treloar, Physics of Rubber
Elasticity (1975)
Alternative Equation for a Rubber’s G
We have shown that G = nkT, where n
is the number of strands per unit
volume.
For a rubber with a known density, , in
which the average molecular mass of a
strand is Mx (m.m. between crosslinks),
we can write:
Looking at the units makes this
equation easier to understand:
# strands
n=
(
=
N A
Mx
g
m
strand
# strands
)(
3
mole )
g
( mole )
N A
RT
G = nkT =
kT =
Mx
Mx
m3
Substituting for n:

This is a useful equation when constructing a network from “strands”.
Network formed by Hbonding of small molecules
Blue = ditopic (able to associate
with two others)
Red = tritopic (able to associate
with three others)
H-bonds can re-form when surfaces are brought into contact.
P. Cordier et al., Nature (2008) 451, 977
For a video, see:
http://news.bbc.co.uk/1/hi/sci/tech/7254939.stm
Viscoelasticity of Soft Matter
With a constant shear stress, ss, the shear modulus G can change
over time:
s
G(t ) =
s
 s (t )
G(t) is also called the “stress relaxation modulus”.
G(t) can also be determined by applying a constant strain, s, and
observing stress relaxation over time:
s
s s (t )
G(t ) =
s
t
Example of Viscoelasticity
High molecular weight polymer dissolved in water. Elastic recovery under
high strain rates, and viscous flow under lower strain rates.
Relaxation Modulus for Polymer Melts
Elastic
tT = terminal
relaxation time
Viscous
flow
tT
Gedde, Polymer Physics,
p. 103
Experimental Shear Relaxation Moduli
Poly(styrene)
GP
High N
Low N
G.Strobl, The Physics of Polymers,
p. 223
~ 1/t
Relaxation Modulus for Polymer Melts
• At very short times, G is high. The polymer has a glassy
response.
• The glassy response is determined by the intramolecular
bonding.
• G then decreases until it reaches a “plateau modulus”, GP. The
value of GP is independent of N for a given polymer:
GP ~ N0.
• After a time, known as the terminal relaxation time, tT, viscous
flow starts (G decreases with time).
• Experimentally, it is found that tT is longer for polymers with a
higher N. Specifically, tT ~ N3.4
• Previously in Lecture 3, we said that in the Maxwell model, the
relaxation time is related to ratio of h to G at the transition
between elastic and viscous behaviour. That is:
tT~h/GP
Viscosity of Polymer Melts
Extrapolation to low
shear rates gives us
a value of the “zeroshear-rate
viscosity”, ho.
ho
Shear thinning
behaviour
For comparison: h for water
is 10-3 Pa s at room
temperature.

Poly(butylene terephthalate) at 285 ºC
From Gedde, Polymer Physics
Scaling of Viscosity: ho ~ N3.4
h ~ tTGP
ho ~
N3.4
N0
Viscosity is shear-strain rate
dependent. Usually measure in
the limit of a low shear rate: ho
~
N3.4
3.4
Universal behaviour for
linear polymer melts!
Applies for higher N:
N>NC
Why?
G.Strobl, The Physics of Polymers, p. 221
Data shifted
for clarity!
An Analogy!
There are obvious similarities between a collection of snakes
and the entangled polymer chains in a melt.
The source of continual motion on the molecular level is
thermal energy, of course.
Concept of “Chain” Entanglements
If the molecules are sufficiently long (N > ~100 - corresponding to the
entanglement mol. wt., Me), they will “entangle” with each other.
Each molecule is confined within a dynamic “tube” created by its
neighbours so that it must diffuse along its axis.
Tube
G.Strobl, The Physics of
Polymers, p. 283
Network of Entanglements
There is a direct analogy between chemical crosslinks in rubbers
and “physical” crosslinks that are created by the entanglements.
The physical entanglements can support stress (for short periods
up to a time tTube), creating a “transient” network.
Plateau Modulus for Polymer Melts
• Recall that the elastic shear modulus of a network depends on
molecular weight between crosslinks, Mx. In a polymer melt, GP
therefore depends on the molecular weight between entanglements, Me.
• That is, GP ~ N0 (where N is the number of repeat units in the
molecule).
• Using an equation for the polymer melt that is analogous to a
crosslinked network:
GP =
RT
Me
• It makes sense that Me is independent of N - consistent with
experimental measurements of GP versus t for various values of M.
Entanglement Molecular Weights, Me,
for Various Polymers
Me (g/mole)
Poly(ethylene)
1,250
Poly(butadiene)
1,700
Poly(vinyl acetate)
6,900
Poly(dimethyl siloxane)
8,100
Poly(styrene)
19,000
Me corresponds to the Nc that is seen in the viscosity data.
Reptation Theory: Molecular Level
x
x
x
x
x
x
x
x
x x
x
x
x
x
x
x
x
x
x
x x x
x x
x
x
x
x
x
x
x
x
x
x
• Polymer molecules “dis-entangle” after a time, tTube.
• Chain entanglements create restraints to other chains, defining a
“tube” through which they must travel.
• The process by which a polymer chain moves through its tube
formed by entanglements is called “reptation”.
• Reptation (from the Latin reptare: “to crawl”) is a snake-like diffusive
motion that is driven by thermal motion.
• Models of reptation consider each repeat unit of the chain as
diffusing through a tube with a drag coefficient, xseg.
• The tube is considered to be a viscous medium surrounding each
segment.
• For a polymer consisting of N units: xpol = Nxseg.
Experimental Evidence for Reptation
Fluorescently-stained DNA molecule
Initial state
Stretched
Chain follows the path of the front
Chu et al., Science (1994) 264, p. 819.
Development of Reptation Scaling Theory
Pierre de Gennes (Paris) developed the concept of polymer
reptation and derived scaling relationships.
Sir Sam Edwards (Cambridge) devised tube models and
predictions of the shear relaxation modulus.
In 1991, de Gennes was awarded the Nobel Prize for Physics.
Polymer Diffusion along a Tube
In our discussion of colloids, we defined an Einstein diffusion
coefficient as:
D = kT
x
If we consider the drag on a polymer molecule, we can
express D for the diffusion of the molecule in a tube created
by an entangled network as:
Dtube =
kT
x pol
kT
=
Nxseg
Hence, the rate of 1-D tube diffusion is inversely related to
the length of the molecules.
Tube Relaxation Time, ttube
The polymer terminal relaxation time, tT, must be comparable to
the time required for a polymer to diffuse out of its confining
tube, ttube.
The length of the tube must be comparable to the entire length
of the polymer molecule (contour length): Na
By definition, a diffusion coefficient, D, is proportional to
the square of the distance travelled (x2) divided by the time
of travel, t.
x 2 (Na )2
Dtube ~
~
For a polymer escaping its tube:
t
t tube
Comparing to our previous Einstein definition:
We thus can derive a scaling relationship for ttube:
t tube ~ N 3
kT
N 2a 2
~
Nxseg t tube
Scaling Prediction for Viscosity
We can think of tT as the average time required for chains to
escape the confinement of their tube, ttube.
We see that t tube ~ N 3 which is comparable to
experiments in which tT ~ N3.4
We have also found that GP ~ N0
Recalling that h ~ GtT
Then: h ~ GPt tube ~ N 0N 3 ~ N 3
But, recall that experiments find h ~ N3.4. Agreement is not
too bad!
Polymer Self-Diffusion
Time = t
Time = 0
X
Reptation theory can also describe the self-diffusion of polymers,
which is the movement of the centre-of-mass of a molecule by a
distance x in a matrix of the same type of molecules.
In a time ttube, the molecule will diffuse the distance of its
entire length. But, its centre-of-mass will move a distance on
the order of its r.m.s. end-to-end distance, R.
In a polymer melt:
<R2>1/2
~
aN1/2
R
Polymer Self-Diffusion Coefficient
X
A self-diffusion coefficient, Dself, can then be defined as:
Dself
x 2 (aN1 / 2 )2 a 2N
~
~
=
t
t tube
t tube
t tube ~ N 3
But we have derived this scaling relationship:
Substituting, we find:
Dself ~
Na 2
N3
~N
2
Larger molecules are predicted to diffuse much more slowly
than smaller molecules.
Testing of Scaling Relation: D ~N -2
Experimentally, D ~ N-2.3
-2
Data for poly(butadiene)
Jones, Soft Condensed
Matter, p. 92
M=Nmo
“Failure” of Simple Reptation Theory
• Reptation theory predicts h ~ N3, but experimentally it varies as
N3.4.
• Theory predicts Dself ~ N-2, but it is found to vary as N-2.3.
• One reason for this slight disagreement between theory and
experiment is attributed to “constraint release”.
• The constraining tube around a molecule is made up of other
entangled molecules that are moving. The tube has a finite
lifetime.
• A second reason for disagreement is attributed to “contour length
fluctuations” that are caused by Brownian motion of the molecule
making its end-to-end distance change continuously over time.
• Improved theory is getting even better results!
Application of Theory: Electrophoresis
• DNA is a long chain molecule consisting of four different types
of repeat units.
• DNA can be reacted with certain enzymes to break specific
bonds along its “backbone”, creating segments of various sizes.
• Under an applied electric field, the segments will diffuse into a
gel (crosslinked molecules in a solvent) in a process known as
gel electrophoresis.
• Reptation theory predicts that shorter chains will diffuse faster
than longer chains.
• Measuring the diffusion distances in a known time enables the
determination of N for each segment and hence the position of
the bonds sensitive to the enzyme.
Application of Theory: Electrophoresis
One common technique: polyacrylamide gel
electrophoresis (PAGE) (or SDS-PAGE)
From Giant Molecules
Relevance of Polymer Self-Diffusion
When welding two polymer surfaces together, such as in a
manufacturing process, it is important to know the time and
temperature dependence of D.
R
Good adhesion is obtained when the molecules travel a
distance comparable to R, such that they entangle with
other molecules.
Stages of Interdiffusion at
Polymer/Polymer Interfaces
Interfacial wetting: weak
adhesion from van der
Waals attraction
Chain extension across
the interface: likely
failure by chain “pullout”
Chain entanglement
across the interface:
possible failure by chain
scission (i.e. breaking)
Example of Good Coalescence
Environmental SEM
Immediate film formation
upon drying!
Hydrated film
Tg of polymer  5 °C;
Bar = 0.5 mm
• Particles can be deformed without being coalesced. (Coalescence means
that the boundaries between particles no longer exist!)
J.L. Keddie et al., Macromolecules (1995) 28, 2673-82.
Strength Development with Increasing
Diffusion Distance
Full strength is achieved
when the diffusion distance,
d, is approximately the
radius of gyration of the
polymer, Rg.
s
Rg
d
K.D. Kim et al, Macromolecules (1994) 27, 6841
Relaxation Modulus for Polymer Melts
Viscous
flow
tT
Gedde, Polymer Physics, p.
103
Problem Set 6
1. A polymer with a molecular weight of 5 x 104 g mole-1 is rubbery at a temperature of 420 K. At this
temperature, it has a shear modulus of 200 kPa and a density of 1.06 x 103 kg m-3. What can you conclude about
the polymer architecture? How would you predict the modulus to change if the molecular weight is (i) doubled
or (ii) decreased by a factor 5?
2. Two batches of poly(styrene) with a narrow molecular weight distribution are prepared. If the viscosity in a
melt of batch A is twice that in a melt of batch B, what is the predicted ratio of the self-diffusion coefficient of
batch A over that of batch B? Assume that the reptation model is applicable.
3. The viscosity h for a melt of poly(styrene) with a molecular weight of 2 x 10 4 g mole-1 is given as X. (This
molecular weight is greater than the entanglement molecular weight for poly(styrene)). (i) According to the
reptation theory, what is h for poly(styrene) with a molecular weight of 2 x 10 5 g mole-1. (ii) Assuming that
poly(styrene) molecules exist as ideal random coils, what is the ratio of the root-mean-square end-to-end
distance for the two molecular weights?
4. The plateau shear modulus (GP) of an entangled polymer melt of poly(butadiene) is 1.15 MPa. The density
of a poly(butadiene) melt is 900 kg m-3, and the molecular mass of its repeat unit is 54 g mole -1.
(i) Calculate the molecular mass between physical entanglements.
(ii) The viscosity (in Pa s) of the melt can be expressed as a function of the degree of polymerisation, N, and
temperature, T (in degrees Kelvin), as:
h = 3.68x10 3 exp[
1404
]N 3.4
T 128
Explain why h has this functional form.
(iii) Estimate the self-diffusion coefficient of poly(butadiene) in the melt at a temperature of 298 K when it has
a molecular mass of 105 g mole-1.