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  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 molecular. The Hamaker constant, A, contains information about molecular density (  ) and the strength of intermolecular interactions (via the London constant, C): A =  2  2 C

2. 3SM Polymer Structure and Molecular Size 12 March, 2009 Lecture 8 See Jones’ Soft Condensed Matter, Chapt. 4, 5 and 9

3. Definition of Polymers Polymers are giant molecules that consist of many repeating units. The molar mass (molecular weight) of a molecule, M, equals m o N, where m o is the the molar mass of a repeat unit and N is the number of units. Synthetic polymers never have the same value of N for all of its constituent molecules, but there is a Gaussian distribution of N. 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”. The average N (or M) has a huge influence on mechanical properties of polymers.

4. Examples of Repeat Units

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

6. Molecular Weight of Polymers The molecular weight can be defined by a number average that depends on the number of molecules, n i, having a mass of M i : The polydispersity index describes the width of the distribution. In all cases: M W /M N > 1 The molecular weight can also be defined by a weight average that depends on the weight fraction, w i, of each type of molecule with a mass of M i : MWMW MNMN = Total mass divided by number of molecules

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

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

9. Polymer Structures Glassy Polymers: molecules in a “random coil” conformation Crystalline Polymers: molecules show some degree of ordering Lamellar growth direction Lamella thickness

10. Polymer Spherulites 15  m x 15  m High density poly(ethylene) From I.W. Hamley, Introduction to Soft Matter, p. 103.

11. Polymer Crystals Several crystals of poly(ethylene oxide) 5  m x 5  m Polymers are usually polycrystalline - not monocrystalline. They are usually never completely crystalline but have some glassy regions and “packing defects”.

12. Thermodynamics of Glass Transitions V T Crystalline solid TmTm Liquid Glass TgTg Crystals can grow from the liquid phase (below the melting temperature, T m ) but not in the glassy phase (below T g ).

13. Temperature Dependence of Crystal Growth Rate, u From Ross and Frolen, Methods of Exptl. Phys., Vol. 16B (1985) p. 363. T-T m (K) T m = crystal melting temperature

14. Why is crystal growth rate maximum between T g and T m ? As T decreases towards T g, molecular motion slows down. Viscosity varies according to the V-F equation: Temperature Dependence of Crystal Growth Rate, u Growth rate, u, is inversely related to viscosity, so u ~ 1/  ~ exp (- B/(T-T o )) Hence, u decreases as T decreases toward T o, because of a slowing down of configurational re-arrangements.

15. Above T m, the crystal will melt. The liquid is the most favourable state according to thermodynamics. Crystallisation becomes more favourable with greater “undercooling” (i.e. as T decreases below T m ) because the free energy difference between the crystal and glass increases. There is a greater “driving force”. Hence u increases exponentially as the amount of undercooling (defined as T m - T) increase, such that: Temperature Dependence of Crystal Growth Rate, u Considering the previous argument, there is an intermediate T where u is maximum.

16. Data in Support of Crystallisation Rate Equation J.D. Hoffman et al., Journ. Res. Nat. Bur. Stand., vol. 79A, (1975), p. 671. V-F contribution: describes molecular slowing down with decreasing T Undercooling contribution: considers greater driving force for crystal growth with decreasing T

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

18. i=1 j=1 NN Those terms in which i=j can be simplified as: ijij N The angle  can assume any value between 0 and 2  and is uncorrelated. Therefore: By definition:    Random Walk Statistics Finally, Compare to random walk statistics for colloids!

19. Defining the Size of Polymer Molecules We see that and 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 10 4 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 = 6x10 3 nm or 6  m! (Root-mean squared end-to-end distance)

20. 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.

21. 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”.

22. 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:

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

24. 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).

25. Concept of an “Entropic Spring” Decreasing entropy Fewer configurations Helmholtz free energy: F = U - TS Internal energy, U, does not change significantly with stretching. Restoring force, f R R

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

27. 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:

28. 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. Poly(styrene) and poly(methyl methacrylate) diblock copolymer Poly(ethylene) diblock copolymers 2  m x 2  m

29. 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 spacings are a function of the sizes of the two blocks. At equilibrium, the block with the lowest surface energy, , segregates at the surface! The system will become “frustrated” when one block prefers the air interface because of its lower , but the alternation of the blocks requires the other block to be at that interface. Ordering can then be disrupted.

30. Thin Film Lamellae: Competing Effects There is thermodynamic competition between polymer chain stretching and coiling to determine the lamellar thickness, d. d The addition of each layer creates an interface with an energy, . Increasing the lamellar thickness reduces the free energy per unit volume and is therefore favoured by . Increasing the lamellar thickness, on the other hand, imposes a free energy cost, because it perturbs the random coil conformation. The value of d is determined by the minimisation of the free energy. Poly(styrene) and poly(methyl methacrylate) copolymer

31. Interfacial Area/Volume e e Area of each interface: A = e 2 Interfacial area/Volume: d=e/3 Lamella thickness: d In general, d = e divided by an integer value.

32. 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  /d. The volume of a molecule is approximated as Na 3, and so there are 1/(Na 3 ) molecules per unit volume. Total free energy change: F str + F int  Free energy increase (per polymer molecule) caused by the presence of interfaces: 

33. Free Energy Minimisation Chains are NOT fully stretched - but nor are they randomly coiled! Two different dependencies on d! The thickness, d, of lamellae created by diblock copolymers is proportional to N 2/3. Thus, the molecules are not fully-stretched (d ~ N 1 ) but nor are they randomly coiled (d ~ N 1/2 ).   Finding the minimum, where slope is 0: d F tot F str F int F

34. Experimental Study of Polymer Lamellae Small-angle X-ray Scattering (SAXS) Transmission Electron Microscopy  (°) T. Hashimoto et al., Macromolecules (1980) 13, p. 1237. Poly(styrene)-b- poly(isoprene)

35. Support of Scaling Argument 2/3 T. Hashimoto et al., Macromolecules (1980) 13, p. 1237.

36. 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”. Density within phases is maintained close to the bulk value. Interfacial “ energy cost ”:  (4  r 2 ) Reduced stretching energy for shorter block

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

38. Diblock Copolymer Morphologies LamellarCylindricalSpherical micelle GyroidDiamondPierced Lamellar TRI-block “Bow-Tie” Gyroid

39. Copolymer Phase Diagram  NN ~ 10 From I.W. Hamley, Intro. to Soft Matter, p. 120.

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

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

44. Interfacial Width, w, between Immiscible Polymers AB w loop 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(N loop ) 1/2

45. Simple Scaling Argument for Polymer Interfacial Width, w In which case: Substituting in for N loop : 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: