1. Glassy Polymers, Copolymer Self-Assembly, and Polymers in Solutions
PH3-SM (PHY3032)
Soft Matter
Lecture 9
Glassy Polymers, Copolymer Self-Assembly, and Polymers in Solutions
6 December, 2010
See Jones’ Soft Condensed Matter, Chapt. 5 & 9

2. Polymer Conformation in Glass
In a “freely-jointed” chain, each repeat unit can assume any orientation in space.
Shown to be valid for polymer glasses and melts.
i=1 N
Describe as a “random walk” with N repeat units (i.e. steps), each with a size of a: 1 2 3 N a
The average R for an ensemble of polymers is 0.
But what is the mean-squared end-to-end distance, ?

3. Random Walk Statistics
j=1 N
q12
q23
q34 a3 a2 a1
By definition:
Those terms in which i=j can be simplified as:
ij
N N
The angle q can assume any value between 0 and 2p and is uncorrelated. Therefore:
Finally,
Compare to random walk statistics for colloids!

4. Defining the Size of Polymer Molecules
We see that
and
(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 with a characteristic size of
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!

5. Scaling Relations of Polymer Size
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.

6. 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”.

7. 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):
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:

8. g(R)
P(R)
From U. Gedde, Polymer Physics

9. 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 R
A polymer coil is less dense than a hard, solid sphere. Thus, its Rg is significantly less than the rms-R:

10. Entropic Effects S = k ln
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:
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 (be less disordered).

11. 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.
Restoring force, f

12. Difference between a Spring and a Polymer Coil
In experiments, f for single molecules can be measured using an AFM tip! f x
Spring
Polymer
S change is large; it provides the restoring force, f.
Entropy (S) change is negligible, but DU is large, providing the restoring force, f.

13. 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:

14. 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.
Poly(styrene) and poly(methyl methacrylate) diblock copolymer
Poly(ethylene) diblock copolymers
2mm x 2mm

15. 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.

16. 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.
Poly(styrene) and poly(methyl methacrylate) copolymer
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.

17. Interfacial Area/Volume
Area of each interface: A = e2 e e
d=e/3
Lamella thickness: d e
Interfacial Area/Volume:
In general, d = e divided by an integer value.

18. Determination of Lamellar Spacing
• Free energy increase caused by chain stretching (per molecule):
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: 
Total free energy change: Ftotal = Fstr + Fint

19. Free Energy Minimisation
Fstr
Fint 
Two different dependencies on d!
Ftot
Finding the minimum, where slope is 0: d
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).

20. Experimental Study of Polymer Lamellae
Small-angle X-ray Scattering (SAXS)
Transmission Electron Microscopy
Poly(styrene)-b-poly(isoprene)
T. Hashimoto et al., Macromolecules (1980) 13, p
 (°)

21. Support of Scaling Argument
2/3
T. Hashimoto et al., Macromolecules (1980) 13, p

22. 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.

23. Copolymer Micelles AFM image 5 mm x 5 mm
Diblock copolymer of poly(styrene) and poly(vinyl pyrrolidone): poly(PS-b-PVP)

24. Diblock Copolymer Morphologies
Gyroid
TRI-block
“Bow-Tie”
Lamellar
Cylindrical
Spherical micelle
Pierced Lamellar
Gyroid
Diamond

25. Copolymer Phase Diagramf N
~10
From I.W. Hamley, Intro. to Soft Matter, p. 120.

26. 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.

27. Nanolithography
Nanolithography to make electronic structures, such as “flash memories”
Thin layer of poly(methyl methacrylate)/ poly(styrene) diblock copolymer. Image from IBM (taken from BBC website)
From Scientific American, March 2004, p. 44

28. 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 “self-avoiding 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?

29. 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 :
where c is a constant

30. Entropy with Excluded Volume
Hence, the entropy for each repeat unit in an ideal polymer coil is R
Unit vol. = b
Nth 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:
But if x is small, then ln(1-x)  -x, so: 

31. 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:
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
Larger R values reduce the free energy. Hence, expansion is favoured by excluded volume effects.

32. Elastic Contributions to F
Earlier in the lecture, however, we saw that the coiling of polymer molecules increased the entropy. This additional entropy contributes an elastic contribution to F:
Reducing the R by coiling will decrease the free energy.
Coiling up of the molecules is therefore favoured by elastic (entropic) contributions.

33. 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
Fexc
Ftot
Fel
At equilibrium, the polymer coil will adopt an R that minimises Ftot. At the minimum, dFtot/dR = 0:

34. Characterising the Self-Avoiding Walk
Re-arranging:
So,
The volume of a repeat unit, b, can be approximated as a3.  
This result agrees with a more exact value of n obtained via a computational method:
Measurements of polymer coil sizes in solvent also support the theoretical (scaling) result.
But when are excluded volume effects important?

35. Visualisation of the Self-Avoiding Walk
2-D Self-avoiding walks
2-D Random walks

36. 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.

37. 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 derive a c-parameter for polymer units in solvent:
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:
Observe that smaller coils reduce the number of P-S contacts because more P-P contacts are created. For a +ve c, DUint is more negative and F is reduced.

38. Significance of the -Parameter
We recall that excluded volume effects favour coil swelling:
Possibly opposing the swelling will be the 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:
The value of c then tells us whether the excluded volume effects are significant or whether they are counter-acted by polymer/solvent interactions.

39. Types of Solvent
• When c = 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.
The solvent is called a “theta-solvent”.
• When c < 1/2, the term is positive, and the excluded volume/energetic effects contribute to determining the coil size: Fexc + DUint > 0.
as shown previously (considering the balance with the elastic energy). The molecule is said to be swollen in a “good solvent”.

40. Types of Solvent
• When c > 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 coiling.
Both terms lower F (which is favourable) as R decreases. The molecule forms a globule in a “bad solvent”.

41. Determining Structure: Scattering Experimentsq l d
= characteristic spacing
Scattered intensity is measured as a function of the wave vector, q:

42. Determination of Polymer Conformation
Scattering Intensity, I  q -1/n or I -1  q1/n
Good solvent: I  q1/(3/5)
Theta solvent: I  q1/(1/2)

43. Applications of Polymer Coiling
Nano-valves
Bad solvent: “Valve open”
Good solvent: “Valve closed”
Switching of colloidal stability
Good solvent: Sterically stabilised
Bad solvent: Unstabilised

44. A Nano-Motor?
• The transition from an expanded coil to a globule can be initiated by changing c.
c < 1/2
A possible “nano-motor”!
c > 1/2
Changes in temperature or pH can be used to make the polymer coil expand and contract.

45. 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”.

46. Comparison of Particle Response in Solution and at an Interface
Ellipsometry of adsorbed particles
Bad solvent: particle is closed
Light scattering from solution
Good solvent: particle is open
V. Nerapusri, et al. , Langmuir (2006) 22, 5036.

48. Interfacial Width, w, between Immiscible Polymers
loop w
Consider the interface between two immiscible polymers (A and B), such as in a phase-separated blend or in a diblock copolymer.
The molecules at the interface want to maximise their entropy by maintaining their random coil shape.
Part of the chain - a “loop” – from A will extend into B over a distance comparable to the interfacial width, w. Our statistical analysis predicts the size of the loop is ~ a(Nloop)1/2

49. Simple Scaling Argument for Polymer Interfacial Width, w
But every unit of the “A” molecule that enters the “B” phase has an unfavourable interaction energy. The total interaction energy is:
At equilibrium, this unfavourable interaction energy will be comparable to the thermal energy:
In which case:
Substituting in for Nloop: