PH3-SM (PHY3032)
Soft Matter
Lecture 9
Glassy Polymers, Copolymer SelfAssembly, and Polymers in Solutions
6 December, 2011
See Jones’ Soft Condensed Matter, Chapt. 5 & 9
Polymer Conformation in Glass
In a “freely-jointed” chain, each repeat unit can assume any
orientation in space.
Shown to hold true for polymer
glasses and melts.
Describe as a “random walk” with N
repeat units (i.e. steps), each with a
size of a:
N
3
2
a
  1 
N

R =a1+a2 +a 3 +...aN = ai

R
i=1
The average R for an ensemble of polymers is
0.
2
But what is the mean-squared end-to-end distance, R ?
Random Walk Statistics
a4
 
N
N
R • R = (ai ) • a j
34
( )
i=1
12
j=1
2
 
R = ai • a j
By definition:
23 a3
a2
a1
 
ai • a j = ai a j cos  ij
Those terms in which i=j can
be simplified as:
 
2
ai  a j  ai cos ii  a 2
2
2
2 NN
R = Na + 2a  cos  ij
ij
The angle  can assume any value between 0
cos  ij = 0
and 2p and is uncorrelated. Therefore:
Finally,
2
R = Na 2
Compare to random walk
statistics for colloids!
Defining the Size of Polymer Molecules
We see that
2
R = Na 2
and
2
R
1
2
=N
1
2a
(Root-mean squared end-to-end distance)
Often, we want to consider the size of isolated polymer
molecules.
In a simple approach, “freely-jointed
molecules” can be described as spheres
 2 12
with a characteristic size of R
Typically, “a” has a value of 0.6 nm or so. Hence, a very large
molecule with 104 repeat units will have a r.m.s. end-to-end
distance of 60 nm.
On the other hand, the contour length of the same molecule will be
much greater: aN = 6x103 nm or 6 mm!
Scaling Relations of Polymer Size
2
R
1
2
1
~N 2
Observe that the rms end-to-end distance is proportional to
the square root of N (for a polymer glass).
Hence, if N becomes 9 times as big, the “size” of the
molecule is only three times as big.
However, if the molecule was straightened out, then its
length would instead be proportional to N.
Concept of Space Filling
Molecules are in a random coil in a
polymer glass, but that does not mean
that it contains a lot of “open space”.
Instead, there is extensive overlap
between molecules.
Thus, instead of open space within a
molecule, there are other molecules, which
ensure “space filling”.
Distribution of End-to-End Distances
In an ensemble of polymers, the molecules each have a different
end-to-end distance, R.
In the limit of large N, there is a Gaussian distribution of end-to-end
distances, described by a probability function (number/volume):
2
3
R
P (R )  [3 /( 2pNa2 )]3 / 2 exp(
)
2
2Na
Larger coils are less probable, and the most likely place for a chain
end is at the starting point of the random coil.
Just as when we described the structure of glasses, we can construct
a radial distribution function, g(r), by multiplying P(R) by the surface
area of a sphere with radius, R:
3R 2
2
2 3/2
g (R )  4pR [3 /( 2p / Na )] exp(
)
2
2Na
R 2 = Na
P(R)
g(R)
From U. Gedde, Polymer Physics
Radius of Gyration of a Polymer Coil
The radius of gyration is the root-mean square distance of an
objects' parts from its centre of gravity.
For a hard, solid sphere of radius, R, the
radius of gyration, Rg, is:
R
Rg =
R
2
10
R=
R = 0.632R
5
5
A polymer coil is less dense than a hard, solid
sphere. Thus, its Rg is significantly less than the
rms-R:
1 2
Rg = R
6
12
a 12
= N
6
Entropic Effects
Recall the Boltzmann equation for calculating the entropy, S, of a
system by considering the number of microstates, , for a given
macro-state:
S = k ln
In the case of arranging a polymer’s repeat units in a coil shape,
we see that  = P(R), so that:
S (R ) =
3kR 2
2Na
2
+ const .
If a molecule is stretched, and its R increases, S(R) will decrease
(become more negative).
Intuitively, this makes sense, as an uncoiled molecule will have more
order (i.e. be less disordered).
Concept of an “Entropic Spring”
Fewer configurations
R
R
Decreasing entropy
Helmholtz free energy: F = U - TS
Internal energy, U, does not change significantly with stretching.
F (R ) = +
3kR 2
2Na
2T
+ const .
Restoring
force, f
dF 3kTR
f 

dR
Na2
Difference between a Spring and a Polymer Coil
f
U = ( 1 2 )ks x 2
x
Spring
In experiments, f
for single
molecules can be
f measured using
an AFM tip!
Polymer
Entropy (S) change is negligible,
but DU is large, providing the
restoring force, f.
S change is large; it
provides the restoring
force, f.
Molecules that are Not Freely-Jointed
In reality, most molecules are not “freely-jointed” (not really
like a pearl necklace), but their conformation can still be
described using random walk statistics.
Why? (1) Covalent bonds have preferred bond angles.
(2) Bond rotation is often hindered.
In such cases, g monomer repeat units can be
treated as a “statistical step length”, s (in place of
the length, a).
A polymer with N monomer repeat units, will
have N/g statistical step units.
The mean-squared end-to-end distance then
becomes:

N
R2 = s2
g
Example of Copolymer Morphologies
Immiscible polymers can be “tied together” within the same
diblock copolymer molecules. Phase separation cannot then
occur on large length scales.
2mm x 2mm
Poly(styrene) and poly(methyl
methacrylate) diblock copolymer
Poly(ethylene) diblock copolymers
Self-Assembly of Di-Block Copolymers
Diblock copolymers are very effective “building blocks” of materials
at the nanometer length scale.
They can form “lamellae” in thin films, in which the spacing is a
function of the sizes of the two blocks.
At equilibrium, the block with the lowest surface energy, g,
segregates at the surface!
The system will become “frustrated” when one block prefers the air
interface because of its lower g, but the alternation of the blocks
requires the other block to be at that interface. Ordering can then be
disrupted.
Thin Film Lamellae: Competing Effects
The addition of each layer creates an interface with an
energy, g. Increasing the lamellar thickness reduces
the free energy per unit volume and is therefore
favoured by g.
d
Increasing the lamellar thickness, on the other hand,
imposes a free energy cost, because it perturbs the
random coil conformation.
There is thermodynamic competition
between polymer chain stretching and
coiling to determine the lamellar
thickness, d.
The value of d is determined by the
minimisation of the free energy.
Poly(styrene) and
poly(methyl methacrylate)
copolymer
Interfacial Area/Volume
Area of each
interface: A = e2
ee
d=e/3
V =e
3
Lamella thickness: d
e
In general, d = e divided by
an integer value.
Interfacial Area/Volume:
3e 2
3 1
A =
V e3 = e = d
Determination of Lamellar Spacing
• Free energy increase caused by chain stretching (per
molecule):
Fstr

 kT
d
2
Na 2
Ratio of (lamellar spacing)2 to
(random coil size)2
• The interfacial area per unit volume of polymer is 1/d, and
hence the interfacial energy per unit volume is g/d.
• The volume of a molecule is approximated as Na3, and so there
are 1/(Na3) molecules per unit volume.
• Free energy increase (per polymer molecule) caused by the
presence of interfaces:
Fint 

gNa 3
d
Total free energy change: Ftotal = Fstr + Fint
Free Energy Minimisation
F
Fstr
Fint
Ftot kT
Ftot
Finding the
minimum, where
slope is 0:
d2
gNa3
Na
d
2+
Two different
dependencies on d!
Ftot
d
gNa3
 0  2kT
 2
2
d
Na
d
d
d
2kT
d =(
ga5
2kT
Na 2
1/ 3
)
N
=
gNa3
2/3
d2
d3 =
gN 2a 5
2kT
Chains are NOT fully stretched - but
nor are they randomly coiled!
The thickness, d, of lamellae created by diblock copolymers is
proportional to N2/3. Thus, the molecules are not fully-stretched (d ~
N1) but nor are they randomly coiled (d ~ N1/2).
Experimental Study of Polymer Lamellae
Small-angle X-ray Scattering (SAXS)
Transmission Electron Microscopy
Poly(styrene)bpoly(isoprene)
 (°)
T. Hashimoto et al.,
Macromolecules (1980) 13, p.
Support of Scaling Argument
2/3
T. Hashimoto et al., Macromolecules (1980) 13, p.
1237.
Micellar Structure of Diblock Copolymers
When diblock copolymers are asymmetric, lamellar structures are not
favoured – as too much interface would form!
Instead the shorter block segregates into small spherical
phases known as “micelles”.
Interfacial “energy
cost”: g(4pr2)
Density within phases is
maintained close to the
bulk value.
Reduced stretching
energy when the shorter
block is in the micelle.
Copolymer Micelles
AFM image
5 mm x 5 mm
Diblock copolymer of poly(styrene) and poly(vinyl pyrrolidone):
poly(PS-b-PVP)
Diblock Copolymer Morphologies
Gyroid
TRI-block
Lamellar
Pierced Lamellar
Cylindrical
Spherical micelle
Gyroid
Diamond
“Bow-Tie”
Copolymer Phase Diagram
N
~10
f
From I.W. Hamley, Intro. to Soft
Matter, p. 120.
Applications of Self-Assembly
Creation of “photonic band gap” materials
Images from website of Prof. Ned Thomas, MIT
In photovoltaics for solar
cells, excitons decay into
electrons and holes.
Controlled phase separation
of p-type/n-type diblock
copolymers could allow a
large contact area between
the two phase.
http://crg.postech.ac.kr/korean/viewforum.php?f=90
Nanolithography
Thin layer of poly(methyl
methacrylate)/ poly(styrene)
diblock copolymer. Image from
IBM (taken from BBC website)
Nanolithography to make electronic
structures, such as “flash
memories”
From Scientific American,
March 2004, p. 44
The Self-Avoiding Walk
In describing the polymer coil as a random walk, it was tacitly
assumed that the chain could “cross itself”.
The conformation of polymer molecules in a polymer glass and
in a melted polymer can be adequately described by random
walk statistics.
But, when polymers are dissolved in solvents (e.g. water or
acetone), they are often expanded to sizes greater than a
random coil.
Such expanded conformations are described by a “selfavoiding walk” in which <R2>1/2 is given by aNn (instead of
aN1/2 as for a coil described by a random walk).
What is the value of n?
Excluded Volume
Paul Flory developed an argument in which a polymer in a
solvent is described as N repeat units confined to a volume of
R3.
Each repeat unit prevents other units from occupying the same
volume. The entropy associated with the chain conformation
(“coil disorder”) is decreased by the presence of the other units.
There is an excluded volume!
From the Boltzmann equation, we know that entropy, S, can
be calculated from the number of microstates,  , for a
macrostate: S = k ln .
In an ideal polymer coil with no excluded volume, , is
inversely related to the number density of units, r :
c
c
cR 3
where c is a
~ ~
~
r N 3
N
constant
R
Entropy with Excluded Volume
Hence, the entropy for each repeat unit in an ideal polymer
coil is
cR 3
Sideal = k ln  = k ln(
)
N
Unit vol. =
bNth unit
In the non-ideal case, however, each unit is excluded from the
volume occupied by the other N units, each with a volume, b:
c(R 3 Nb )
cR 3
Sni = k ln(
) = k ln(
cb )
N
N
cR 3
bN
cR 3
bN
Sni = k ln[
(1
)] = k [ln(
) + ln(1
)]
3
3
N
N
R
R
bN
Sni = Sideal + k ln[(1
3 )]
R
kbN

But if x is small, then ln(1-x)  -x, so:
Sni Sideal
R3
R
Excluded Volume Contribution to F
For each unit, the entropy decrease from the excluded
volume will lead to an increase in the free energy, as
F = U - TS:
bN
Fni = Fideal + kT 3
R
Of course, a polymer molecule consists of N repeat units, and
so the increase in F for a molecule, as a result of the excluded
volume, is
Fexc (R ) = kT
bN 2
R3
Larger R values reduce the free energy. Hence, expansion is
favoured by excluded volume effects.
Elastic Contributions to F
Earlier in the lecture (slide 18), however, we saw that the
coiling of polymer molecules increased the entropy. This
additional entropy contributes an elastic contribution to F:
Fel (R ) = +kT
3R 2
2Na 2
Reducing the R by coiling will decrease the free energy.
Coiling up of the molecules is therefore favoured by elastic
(entropic) contributions.
Total Free Energy of an Expanded Coil
The total free energy change is obtained from the sum of the
two contributions: Fexc + Fel
Ftot (R ) =
Ftot
kbN 2
R
3
T+
Ftot
3kR 2
2Na
2T
+ const .
Fel
Fexc
R
At equilibrium, the polymer coil will adopt an R that minimises Ftot.
At the minimum, dFtot/dR = 0:
dFtot
=0=
dR
3kbTN 2
R4
+
3kRT
Na 2
Characterising the Self-Avoiding Walk
Re-arranging:
3kbTN 2
R
So,
4
=
3kRT
Na 2
R 5 = a 2bN 3
The volume of a repeat unit, b, can be approximated as a3.
R 5  a5N 3
R = aN  aN 3/ 5
This result agrees with a more exact value of n obtained via a
computational method: 0.588
Measurements of polymer coil sizes in solvent also support
the theoretical (scaling) result.
But when are excluded volume effects important?
Visualisation of the Self-Avoiding Walk
2-D Random walks
R
2 1/ 2
= aN
2-D Self-avoiding walks
1/ 2
R
2 1/ 2
= aN 3 / 5
Polymer/Solvent Interaction Energy
So far, we have neglected the interaction energies between the
components of a polymer solution (polymer + solvent).
Units in a polymer molecule have an interaction energy with
other nearby (non-bonded) units: wpp
wss
wps
There is similarly an interaction energy between the solvent
molecules (wss). Finally, when the polymer is dissolved in the
solvent, a new interaction energy between the polymer units
and solvent (wps) is introduced.
Polymer/Solvent -Parameter
When a polymer is dissolved in solvent, new polymer-solvent (ps) contacts are
made, while contacts between like molecules (pp + ss) are lost.
Following arguments similar to our approach for liquid miscibility, we can
write out a -parameter for polymer units in solvent:
z
(2w PS w PP
=
2kT
w SS )
where z is the number of neighbour contacts per unit or solvent molecule.
We note that N/R3 represents the concentration of the repeat units in the
“occupied volume”, and the volume of the polymer molecule is Nb.
When a polymer is added to a solvent, the change in potential energy (from
the change in w), will cause a change in internal energy, DU:
N
bN 2
DUint  (2w PS - w PP - w SS )(no.units )  - 2kT (Nb) 3  2 kT 3
R
R
Observe that smaller coils reduce the number of P-S contacts because more P-P
contacts are created. For a +ve , DUint is more negative and F is reduced.
Significance of the -Parameter
We recall (slide 31) that excluded volume effects favour coil swelling:
Fexc (R ) = kT
bN 2
R3
Also, depending on the value of , the swelling will be opposed by
polymer/solvent interactions, as described by DUint. (But also elastic effects, in which Fel ~ R2, are also still active!)
As the form of the expressions for Fexc and DUint are the same, they
can be combined into a single equation:
Fexc + DUint = kTb(1 2  )
N2
R3
The value of  then tells us whether the excluded volume effects
are significant or whether they are counter-acted by
polymer/solvent interactions.
Types of Solvent
Fexc + DUint = kTb(1 2  )
N2
R3
• When  = 1/2, the two effects cancel: Fexc + DUint = 0.
The coil size is determined by elastic (entropic) effects only, so it
adopts a random-coil conformation.

R2
12
= aN
1
2
The solvent is called a “theta-solvent”.
• When  < 1/2, the term is positive, and the excluded
volume/energetic effects contribute to determining the coil size:
Fexc + DUint > 0.
3
 12
R2
= aN
5
as shown previously (considering the balance with the elastic energy).
The molecule is said to be swollen in a “good solvent”.
Types of Solvent
Fexc + DUint = kTb(1 2  )
• When
N2
R3
 > 1/2, the term goes negative, and the
polymer/solvent interactions dominate in determining the coil
size.
Fexc + DUint < 0.
Energy is reduced by coiling up the molecule (i.e. by
reducing its R).
• Elastic (entropic) contributions likewise favour
Ftot (R ) = Fint + Fel ~
N2
R
3
+
coiling.
3kR 2
2Na
2T
Both terms lower F (which is favourable) as R decreases. The
molecule forms a globule in a “bad solvent”.
Determining Structure:
Scattering Experiments


d
= characteristic spacing
Scattered intensity is measured as a function of
the wave vector, q:
2p 4p
q=
=
sin 
d

Determination of Polymer Conformation
Scattering Intensity, I  q -1/n or I
 q1/n
Good solvent:
1/(3/5)
I 
-1
Theta solvent: I 
1/(1/2)
Applications of Polymer Coiling
Nano-valves
Bad solvent:
“Valve open”
Good solvent:
“Valve closed”
Switching of colloidal stability
Bad solvent: Unstabilised
Good solvent:
Sterically stabilised
A Nano-Motor?
• The transition from an expanded coil to a globule can be
initiated by changing .
A possible “nano-motor”!
 > 1/2
 < 1/2
Changes in temperature or pH can be used to make the
polymer coil expand and contract.
Polymer Particles Adsorbed on a
Positively-Charged Surface
1 mm
100 nm
Particles can contain small molecules such as a drug or a
flavouring agent. Thus, they are a “nano-capsule”.
Comparison of Particle Response in
Solution and at an Interface
1500
1250
Ellipsometry of
adsorbed particles
size (nm )
1000
thic knes s
Bad solvent:
particle is closed
750
hy drody namic dia.
Light scattering
from solution
500
Good solvent:
particle is open
250
0
15
25
35
45
te m pe r atur e ( oC)
55
65
Fig.1 Com parison between Hydrodynam ic diam eter of
P oly(NIP A M )/1% B A m icrogel particles by P CS and the
V. Nerapusri, et al. , Langmuir (2006) 22, 5036.
thickness of m icrogel film by E llipsom etry as a function of