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Last Lecture:

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.

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Last Lecture:

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  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 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. 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 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. = Total mass divided by number of molecules MN 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 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:

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

  8. Diblock Random or Statistical 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.

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

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

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

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

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

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

  15. Temperature Dependence of Crystal Growth Rate, u • Above Tm, 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 belowTm) 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 Tm - T) increase, such that: •Considering the previous argument, there is an intermediate T where u is maximum.

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

  17. N i=1 N 3 a 2 1 But what is the mean-squared end-to-end distance, ? 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. Describe as a “random walk” with N repeat units (i.e. steps), each with a size of a: The average R for an ensemble of polymers is 0.

  18. q3 q2 q1 N N By definition: i=1 j=1 Those terms in which i=j can be simplified as: N N ij The angle q can assume any value between 0 and 2p and is uncorrelated. Therefore: Finally, Random Walk Statistics Compare to random walk statistics for colloids!

  19. In a simple approach, “freely-jointed molecules” can be described as spheres with a characteristic size 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 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!

  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. In the limit of large N, there is a Gaussian distribution of end-to-end distances, described by a probability function (number/volume): 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.

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

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

  26. f f x Entropy (S) change is negligible, but DU is large, providing the restoring force, f. S change is large; it provides 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! Spring Polymer

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

  28. Poly(styrene) and poly(methyl methacrylate) diblock copolymer Poly(ethylene) diblock 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

  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, 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.

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

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

  32. •Free energy increase caused by chain stretching (per molecule): •Free energy increase (per polymer molecule) caused by the presence of interfaces:  Determination of Lamellar Spacing 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. Total free energy change:Fstr + Fint

  33. Finding the minimum, where slope is 0:  Chains are NOT fully stretched - but nor are they randomly coiled! Free Energy Minimisation F Fstr Fint Two different dependencies on d! Ftot d 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).

  34. 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. 1237.  (°)

  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”. Interfacial “energy cost”: g(4pr2) Reduced stretching energy for shorter block Density within phases is maintained close to the bulk value.

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

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

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

  40. 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 Images from website of Prof. Ned Thomas, MIT

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

  42. Interfacial Width, w, between Immiscible Polymers A B 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

  43. 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: Simple Scaling Argument for Polymer Interfacial Width, w

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