1. 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 acting between a colloidal particle and a semi- slab (or another particle) can be calculated by summing up the intermolecular energy between the constituent molecules.
Macroscopic interactions can be related to the molecular level.
The Hamaker constant, A, contains information about molecular density (r) and the strength of intermolecular interactions (via the London constant, C): A = p2r2C

2. 3SM Introductions to Polymers and Semi-Crystalline Polymers
29 April, 2010
Lecture 8
See Jones’ Soft Condensed Matter, Chapt. 5 & 8

3. Definition of Polymers
Polymers are giant molecules that consist of many repeating units. The molar mass (molecular weight) of a molecule, M, equals moN, where mo is the the molar mass of a repeat unit and N is the number of units.
Polymers can be synthetic (such as poly(styrene) or poly(ethylene)) or natural (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 “monomers”.
Synthetic polymers never have the same value of N for all of its constituent molecules, but there is a Gaussian distribution of N.
The average N (or M) has a huge influence on mechanical properties of polymers.

4. Examples of Repeat Units

5. Molecular Weight Distributions
Fraction of molecules M M
In both cases: the number average molecular weight, Mn = 10,000

6. Molecular Weight of Polymers
The molecular weight can be defined by a number average that depends on the number of molecules, ni, having a mass of Mi: MN
= Total mass divided by the total number of molecules
The molecular weight can also be defined by a weight average that depends on the weight fraction, wi, of each type of molecule with a mass of Mi: MW
The polydispersity index describes the width of the distribution. In all cases:
MW/MN > 1

7. Types of Copolymer Molecules
Within a single molecule, there can be “permanent order/disorder” in copolymers consisting of two or more different repeat units.
Diblock
Random or
Statistical
Alternating
Can also be multi (>2) block.

8. Semi-Crystalline Polymers
It is nearly impossible for a polymer to be 100% crystalline.
Typically, the level of crystallinity is 20 to 60%.
The chains surrounding polymer crystals can be in the glassy state, e.g. poly(ethylene terephthalate)
The chains can be at a temperature above their glass transition temperature and be “rubbery”, e.g. poly(ethylene)
The density of a polymer crystal is greater than the density of a polymer glass.

9. Examples of Polymer Crystals
Poly(ethylene) crystal
Crystals of poly(ethylene oxide)
15 mm x 15 mm
5 mm x 5 mm
Polymer crystals can grow up to millimeters in size.

10. Crystal Lattice Structure
The unit cell is repeated in three directions in space.
Polyethylene’s unit cell contains two ethylene repeat groups (C2H4).
Chains are aligned along the c-axis of the unit cell.
Polyethylene
From G. Strobl, The Physics of Polymers (1997) Springer, p. 155

11. Structure at Different Length Scales
Chains weave back and forth to create crystalline sheets, called lamella.
A chain is not usually entirely contained within a lamella: portions of it can be in the amorphous phase or bridging two (or more) lamella.
The lamella thickness, L, is typically about 10 nm.
Lamella stacks L
From R.A.L. Jones, Soft Condensed Matter, O.U.P. (2004) p. 130

12. Structure at Different Length Scales
Lamella usually form at a nucleation site and grow outwards.
To fill all available space, the lamella branch or increase in number at greater distances from the centre.
The resulting structures are called spherulites.
Can be up to hundreds of micrometers in size.
From G. Strobl, The Physics of Polymers (1997) Springer, p. 148

13. Hierarchical Structures of Chains in a Polymer Crystal
Chains are aligned in the lamella in a direction that is perpendicular to the direction of the spherulite arm growth.
Optical properties are anisotropic.
From I.W. Hamley, Introduction to Soft Matter, p. 103

14. Crossed Polarisers Block Light Transmission
An anisotropic polymer layer between crossed polarisers will “twist” the polarisation and allow some light to pass.
The pattern is called a “Maltese cross”.
Crossed polarisers:
No light can pass!
Parallel polarisers:
All light can pass

15. Observing Polymer Crystals Under Crossed Polarisers
Light is only transmitted when anisotropic optical properties “twist” the polarisation of the light.

16. Free Energy of Phase Transitions
The state with the lowest free energy is the stable one.
Below the equilibrium melting temperature, Tm(), the crystalline state is stable.
The thermodynamic driving force for crystallisation, DG, increases when cooling below the equilibrium Tm (). DG
Free energy, G
Crystalline state
Liquid (melt) state
Tm()
Undercooling, DT, is defined as Tm – T.
Temperature, T

17. Thermodynamics of the Phase Transition
Enthalpy of melting, DHm: heat is absorbed when going from the crystal to the melt.
Enthalpy of crystallisation: heat is given off when a molten polymer forms a crystal.
The melting temperature, Tm, is always greater than the crystallisation temperature.
The phase transitions are broad: they happen over a relatively wide range of temperatures.
DHm
Heat flows in
Heat flows out
From G. Strobl, The Physics of Polymers (1997) Springer

18. Thermodynamics of the Crystallisation/Melting Phase Transition
Melt to crystal: Increase in Gibbs’ free energy from the creation of an interface between the crystal and amorphous region. When a single chain joins a crystal: a2
g f is an interfacial energy L
Melt to crystal (below Tm): Decrease in Gibbs’ free energy because of the enthalpy differences between the states
(Enthalpy per volume, H m) x (volume) x (fractional undercooling)
At equilibrium: energy contributions are balanced and DG = 0.

19. Thermodynamics of the Crystallisation/Melting Phase Transition
From DG = 0:
Re-arranging and writing undercooling in terms of Tm(L):
Solving for Tm(L):
We see that a chain-folded crystal (short L) will melt at a lower temperature than an extended chain crystal (large L).

20. Lamellar Crystal Growth
Lamella thickness, L
Lamellar growth direction a
From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 145 L
Crystal growth is from the edge of the lamella.
The lamella grows a distance a when each chain is added.
From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 161

21. The Entropy Barrier for a Polymer Chain to Join a Crystal
Free energy
Crystalline state
TDS
Melted state DG
Re-drawn from R.A.L. Jones, Soft Condensed Matter, O.U.P. (2004) p. 132

22. The Rate of Crystal Growth, u
Melt to crystal: the rate of crystal growth is equal to the product of the frequency (t-1) of “attempts” and the probability of going over the energy barrier (TDS):
Crystal to melt: the rate of crystal melting is equal to the product of the frequency (t-1) of “attempts” and the probability of going over the energy barrier (TDS + DG):
Net growth rate, u: the net rate of crystal growth, u, is equal to the difference between the two rates:

23. Lamellar Crystal Growth
Lamella thickness, L
Lamellar growth direction a
From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 145 L
Crystal growth is from the edge of the lamella.
The lamella grows a distance a when each chain is added.
From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 161

24. The Velocity of Crystal Growth, n
The velocity of crystal growth can be calculated from the product of the rate of growth (u, a frequency) and the distance added by each chain, a. Also, as DG/kT << 1, exp(-DG/kT)  1 - DG/kT:
But DG is a function of L:
The entropy loss in straightening out a chain is proportional to the number of units of size a in a chain of length L:
Finally, we find:
We see that the crystal growth velocity is a function of lamellar thickness, L.

25. The Fastest Growing Lamellar Thickness, L*
L dependence
To find the maximum n, set the differential = 0, and solve for L = L*. L n L*
Solve for L = L*:

26. Lamellar Thickness is Inversely Related to Undercooling
Experimental data for polyethylene.
L* (nm)
L* (nm)
Jones, Soft Condensed Matter, p. 134
Tm() - T
Original data from Barham et al. J. Mater. Sci. (1985) 20, p.1625
Tm()-T

27. Temperature Dependence of Crystal Growth Velocity
The rate at which a chain attempts to join a growing crystal, t, is expect to have the same temperature-dependence as the viscosity of the polymer melt:
This temperature-dependence will contribute to the crystal growth velocity:
Recall that DG depends on T and on L as:
Finally, recall that the fastest-growing lamellar size, L = L*, also depends on temperature as:
We see that DG(L*) becomes:

28. Temperature Dependence of Crystal Growth Velocity
We can evaluate n when L = L* and when DG = DG(L*):
We finally find that:
Describes molecular slowing-down as T decreases towards T0
Describes how the driving force for crystal growth is smaller with a lower amount of undercooling, DT.
T0 is approximately 50 K less than the glass transition temperature.

29. Tm() = crystal melting temperature
Experimental Data on the Temperature Dependence of Crystal Growth Velocity
T-Tm() (K)
n (cm s-1)
Tm() = crystal melting temperature
From Ross and Frolen, Methods of Exptl. Phys., Vol. 16B (1985) p. 363.

30. Data in Support of Crystallisation Rate Equation
V-F contribution: describes molecular slowing down with decreasing T
Undercooling contribution: considers greater driving force for crystal growth with decreasing T
n exp (B/(T-T0)) [cm s-1]
1/(T(Tm()-T)) [10-4 K-2]
J.D. Hoffman et al., Journ. Res. Nat. Bur. Stand., vol. 79A, (1975), p. 671.

31. Why Are Polymer Single Crystals (Extended Chains) Nearly Impossible to Achieve?
Crystal with extended chains are favourable at very low levels of undercooling, as L* ~ 1/DT
But as temperatures approach Tm(), the crystal growth velocity is exceedingly slow!
T-Tm() (K)
n (cm s-1)

32. Factors that Inhibit Polymer Crystallisation
Slow chain motion (associated with high viscosity) creates a kinetic barrier
“Built-in” chain disorder, e.g. tacticity
Chain branching

33. Tacticity Builds in Disorder
Isotactic: identical repeat units
Easiest to crystallise
Syndiotactic: alternating repeat units
Atactic: No pattern in repeat units
Usually do not crystallise
R.A.L. Jones, Soft Condensed Matter (2004) O.U.P., p. 75

34. Polymer Architecture
Linear
Branched
Side-branched
Star-branched

35. Effects on Branching on Crystallinity
Branched Poly(ethylene)
Linear Poly(ethylene)
Lamella are packed less tightly together when the chains are branched. There is a greater amorphous fraction and a lower overall density.
From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 148

36. Determining Whether a Polymer Is (Semi)-Crystalline
Raman Spectra
“Fully” crystalline
Amorphous
Partially crystalline
From G. Strobl, The Physics of Polymers (1997) Springer, p. 154

37. Summary Further Reading
Polymer crystals have a hierarchical structure: aligned chains, lamella, spherulites.
Melting point is inversely related to the crystal’s lamellar thickness.
Lamellar thickness is inversely related to the amount of undercooling.
The maximum crystal growth rate usually occurs at temperatures between the melting temperature and the glass transition temperature.
Tacticity and chain branching prevents or interrupts polymer crystal growth.
Further Reading
Gert Strobl (1997) The Physics of Polymers, Springer
Richard A.L. Jones (2004) Soft Condensed Matter, Oxford University Press
Ulf W. Gedde (1995) Polymer Physics, Chapman & Hall

38. Problem Set 5
This table lists experimental values of the initial lamellar thickness for polyethylene crystallised at various temperatures. The equilibrium melting temperature was independently found to be K.
Temperature, T (K) Lamellar thickness, L (nm)
Are the data broadly consistent with the predictions of theory?
Predict the melting temperature of crystals grown at a temperature of 400 K.

40. 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: a1 a2 a3 N a
The average R for an ensemble of polymers is 0.
But what is the mean-squared end-to-end distance, ?

41. Random Walk Statistics
j=1 N
By definition:
q12
q23
q34 a2 a1
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!

42. 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!

43. 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.
If the molecule is straightened out, then its length will be proportional to N.

44. 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”.

45. 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:

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

47. S = k ln Entropic Effects
Recall the Boltzmann equation for calculating the entropy, S, of a system by considering the number of microstates, , for a given macrostate:
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).

48. 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

49. 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.

50. 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:

51. Example of Copolymer Morphologies
Immiscible polymers can be “tied together” within the same diblock copolymer molecules. Phase separation cannot occur on large length scales.
Poly(styrene) and poly(methyl methacrylate) diblock copolymer
Poly(ethylene) diblock copolymers
2mm x 2mm

52. 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.

53. 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.

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

55. 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: Ftot = Fstr + Fint

56. 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).

57. 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
 (°)

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

59. 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.

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

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

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

63. Applications of Self-Assembly
Creation of “photonic band gap” materials
Nanolithography to make electronic structures
Thin layer of poly(methyl methacrylate)/ poly(styrene) diblock copolymer. Image from IBM (taken from BBC website)
Images from website of Prof. Ned Thomas, MIT

64. Nanolithography Used to make nano-sized “flash memories”
From Scientific American, March 2004, p. 44

66. 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

67. 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: