Download Lecture 8 - Intro Polymers

Last Lecture:
• The Peclet number, Pe, describes the competition
between particle disordering because of Brownian
diffusion and particle ordering under a shear stress.
• At high Pe (high shear strain rate), the particles are more
ordered; shear thinning behaviour occurs and h
decreases.
• van der Waals energy between colloidal particles and
between parallel slabs can be calculated by summing up
the intermolecular energy between constituent molecules.
• Macroscopic interactions can be related to molecular.
• The Hamaker constant, A, contains information about
molecular density (r) and the strength of intermolecular
interactions (via the London constant, C): A = p2r2C
Polymer Structure and
Molecular Size
3SCMP
10 March, 2005
Lecture 8
See Jones’ Soft Condensed Matter, Chapt. 4, 5 and 9
Definition of Polymers
Polymers are giant molecules that consist of many
repeating units. The molecular weight of a molecule, M,
equals ______, where mo is the the molecular weight of
a repeat unit and N is the number of units.
Polymers can be _________ (such as poly(styrene) or
poly(ethylene)) or ________ (such as starch (repeat units
of amylose) or proteins (repeat unit of amino acids)).
Synthetic polymers are created through chemical
reactions between smaller molecules, called “________”.
Synthetic polymers ________ have the same value of
N for all of its constituent molecules, but there is a
____________ distribution of N.
Examples of Repeat Units
Molecular Weight of Polymers
The molecular weight can be defined by a ___________
average:
Mi
Total mass of all molecules
=
MN =
Ni
Total number of molecules
The molecular weight can also be defined by a _________
average:
MW=
 (Total mass of molecules with mass Mi)(Mi)
Total mass of all molecules
 Mi
The ___________ ________ describes the width of the
distribution:
MW/MN > ____
Polymer Architecture
Linear
Branched
Side-branched
Star-branched
Types of Copolymer Molecules
Within a single molecule, there can be “permanent
disorder” in copolymers consisting of two or more
different repeat units.
Diblock
Random or
__________
Alternating
Can also be multi (>2) block.
Polymer Structures
Crystalline: molecules show some degree of
ordering
Lamellar growth
direction
Lamella thickness
Glassy: molecules in a _________ ___________
conformation
Polymer Crystals
AFM image of a crystal of
high density poly(ethylene)
- viewed while “looking
down” at the lamella.
15 mm x 15 mm
Lamella grows
outwards
Polymer Crystals
Polymers are usually
polycrystalline. They are
usually never completely
crystalline but have
some amorphous
regions and “packing
defects”.
5 mm x 5 mm
Several crystals of poly(ethylene oxide)
Polymer Conformation
In a “________-________” chain, each repeat unit
can assume any orientation in space.
Shown to be valid for polymer
glasses and melts.
N repeat units
Size of each unit = ___
N
3
2
a
  1 

R =a1+a2 +a 3 +...aN = _____
An average R for ensemble of polymers
is 0.

R
2
But what is the mean-squared displacement R ?
Random Walk Statistics
N
N
 
R • R = (ai ) • a j
( )
i=1
j=1
1
3
2
2
 
R = ai • a j
 
By definition:ai • a j = ___________
 
Those terms in which i=j a • a = a 2 cos  = ____
i
j
i
ij
can be simplified:
NN
2
2
2
R = Na + 2a  cos  ij
ij
The angle  can assume any value between
cos  ij = ___
0 and 2p and is uncorrelated. Therefore:
Finally,
2
R = Na 2
Defining the Size of Polymer Molecules
2
 2 12
2
R = Na
We see that
and R
= ______
(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 described1 as
2 2
spheres with a radius of R
Typically, “a” has a value of 0.6 nm or so. Hence, a
molecule with 104 repeat units will have a radius of _____.
On the other hand, the __________ ________ of the same
molecule will be much greater: aN = 6x105 nm or 0.6 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 _____ times as big.
If the molecule is straightened out, then its length will
be proportional to ____.
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 ___________
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
2
function:

3R
2 2/3
P (R ) = [( 2p / 3)Na ] 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
2
radius, R:

3R
2
2 2/3
g (R ) = 4pR [( 2p / 3)Na ] exp(
2)
2Na
R 2 = Na
From Gedde, Polymer Physics
Entropic Effects
Recall the Boltzmann equation for calculating the entropy of
a system by considering the number of ways to arrange it:
S = _______
In the case of arranging a polymer’s repeat units in a
coil shape, we see that  = P(R).
S (R ) =
+ 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 _______ (be less disordered).
Concept of an “Entropic Spring”
Fewer configurations
Decreasing entropy
Helmholtz free energy:
F = U - TS
Internal energy, U, does not change significantly with stretching.
F (R ) = +
3kR 2
2Na
2 T + const .
dF
=
dR
Restoring
force
F
U = ( 1 2 )ks x 2
F
x
Spring
Polymer
Entropy change is ________,
but ____ is large, providing
the restoring force.
Entropy change is
________; it provides
the _________ force.
Molecules that are Not-Freely Jointed
In reality, most molecules are not “freely-jointed”, but their
conformation can still be described using random walk
statistics.
Why? (1) Covalent bonds have preferable _______
_________.
(2) Bond ________ is often __________.
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 ______
statistical step units.
The mean-squared end-to-end distance then
2
becomes:
N 2
R = s
g
An Extreme Example….
Carbon Nanotubes
Not as flexible as polymers: does not form random coils
Example of Copolymer Morphologies
Polymers that are immiscible can be “tied together” within
the same molecules. They therefore cannot phase
separate on large length scales.
2mm x 2mm
Poly(styrene) and poly(methyl
methacrylate) copolymer
Poly(ethylene) copolymers
Self-Assembly of Copolymers
__________ copolymers are very effective “building
blocks” of materials at the nanometer length scale.
They can form “__________” in thin films, in which the
spacings are a function of the sizes of the two blocks.
At equilibrium, the block with the lowest surface energy,
g, segregates at that surface!
The system will become “frustrated” for some film
thicknesses, if one block prefers the air and substrate
interfaces.
Thin Film Lamellae
Thermodynamic competition between polymer
chain stretching and coiling to determine lamellar
thickness, d.
d
The addition of each layer creates an interface
with an energy, g. Increasing the lamellar
thickness _________ the ______ _______ per
unit volume and is therefore favoured by g.
Increasing the lamellar thickness, on the other hand,
imposes a free energy_______, because it perturbs the
random coil conformation.
The value of d is determined by the ___________of
the free energy.
Interfacial Area/Volume
V =e
Area of each
interface: A = e2
e
d=e/3
3
Lamella thickness: d
e
Interfacial area/Volume:
V
A=
3
= =
e
Determination of Lamellar Spacing
• Free energy _______ caused by chain _________:
Fstr  kT
d2
Na 2
Ratio of (lamellar spacing)2 to
(_________ _______ ______)2
• The interfacial area per unit volume of polymer is 1/d, and
hence the interfacial energy per unit volume is ______.
The volume of a molecule is approximated as _____,
and so there are 1/(Na3) molecules per unit volume.
• Free energy increase (per polymer molecule) caused
by the presence of interfaces:
Fint 
Total free energy change: Fstr + Fint
Free Energy Minimisation
Ftot  kT
d2
Na 2
+
gNa3
d
Two different
dependencies on d!
dFtot
=0 
d (d )
2kT
d
Na 2
d =(
ga5
=
gNa3
d2
d3 =
fully
)1 / 3 N 2 / 3 Chains are NOT
2kT
stretched (N1) - but nor are
they randomly coiled (N1/2)!
Micellar Structure of Diblock Copolymers
When diblock copolymes are asymmetric, lamellar
structures are not favoured.
Instead the shorter block segregates into small
spherical phases known as “micelles”.
Interfacial “energy
cost”: ___________
Reduced stretching
energy for _______
block
Density within phases
is maintained close to
bulk value.
Diblock Copolymer Morphologies
Gyroid
TRI-block
Lamellar Cylindrical Spherical micelle
Pierced Lamellar
Gyroid
Diamond
“Bow-Tie”
Copolymer Micelles
5 mm x 5 mm
Diblock copolymer of poly(styrene) and
poly(viny pyrrolidone): poly(PS-PVP)
Applications of Self-Assembly
Thin layer of poly(methyl
methacrylate)/ poly(styrene)
diblock copolymer. Image from
IBM (taken from BBC website)
Nanolithography to make
electronic structures
Creation of “photonic
band gap” materials
Nanolithography
Used to make nanosized “flash memories”
From Scientific American,
March 2004, p. 44