Last Lecture:

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 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

2. Polymer Structure and Molecular Size 3SCMP 10 March, 2005 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 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.

4. Examples of Repeat Units

5. Total mass of all molecules MN = Total number of molecules S(Total mass of molecules with mass Mi)(Mi) MW= S Mi Total mass of all molecules The ___________ ________ describes the width of the distribution: MW/MN > ____ Molecular Weight of Polymers The molecular weight can be defined by a ___________ average: The molecular weight can also be defined by a _________ average:

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

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

8. Polymer Structures Crystalline:molecules show some degree of ordering Lamellar growth direction Lamella thickness Glassy:molecules in a _________ ___________ conformation

9. 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

10. 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)

11. N repeat units Size of each unit = ___ N 3 a 2 1 But what is the mean-squared displacement ? Polymer Conformation In a “________-________” chain, each repeat unit can assume any orientation in space. Shown to be valid for polymer glasses and melts. An average R for ensemble of polymers is 0.

12. q3 q2 q1 N N i=1 j=1 N N ij Finally, Random Walk Statistics By definition: Those terms in which i=j can be simplified: The angle q can assume any value between 0 and 2p and is uncorrelated. Therefore:

13. In a simple approach, “freely-jointed molecules” can be described as spheres with a radius of 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. 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!

14. 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 _____ times as big. If the molecule is straightened out, then its length will be proportional to ____.

15. 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”.

16. In the limit of large N, there is a Gaussian distribution of end-to-end distances, described by a probability function: 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: Distribution of End-to-End Distances In an ensemble of polymers, the molecules each have a different end-to-end distance, R. Larger coils are less probable, and the most likely place for a chain end is at the starting point of the random coil.

17. From Gedde, Polymer Physics

18. 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). 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).

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

20. F F x Entropy change is ________, but ____ is large, providing the restoring force. Entropy change is ________; it provides the _________ force. Spring Polymer

21. The mean-squared end-to-end distance then becomes: 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.

22. An Extreme Example…. Carbon Nanotubes Not as flexible as polymers: does not form random coils

23. Poly(styrene) and poly(methyl methacrylate) copolymer Poly(ethylene) copolymers 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

24. 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.

25. Thin Film Lamellae Thermodynamic competition between polymer chain stretching and coiling to determine lamellar thickness, 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. d 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.

26. d=e/3 Lamella thickness: d Interfacial Area/Volume Area of each interface: A = e2 e e Interfacial area/Volume:

27. Determination of Lamellar Spacing • Free energy _______ caused by chain _________: 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: Total free energy change: Fstr + Fint

28. Free Energy Minimisation Two different dependencies on d! Chains are NOT fully stretched (N1) - but nor are they randomly coiled (N1/2)!

29. 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.

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

31. Copolymer Micelles 5 mm x 5 mm Diblock copolymer of poly(styrene) and poly(viny pyrrolidone): poly(PS-PVP)

32. 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 Applications of Self-Assembly

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