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Important Points from Last Lecture

Important Points from Last Lecture. The Peclet number, Pe , describes the competition between particle disordering because of Brownian diffusion and particle ordering under a shear stress.

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Important Points from Last Lecture

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  1. Important Points from 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 andhdecreases. • 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. Forces are found by differentiating the interaction energy with respect to the distance of separation. • The Hamaker constant, A, contains information about molecular density (r) and the strength of intermolecular interactions (via the van der Waals’ constant, C): A =p2r2C

  2. PH3-SM (PHY3032) Soft Matter Lecture 8 Introductions to Polymers and Semi-Crystalline Polymers 29 November, 2011 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. = Total mass divided by the total 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. Diblock Random or Statistical Alternating Types of Copolymer Molecules Within a single molecule, there can be “permanent order/disorder” in copolymers consisting of two or more different repeat units. 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 in the range from 20 to 60%. The chains surrounding polymer crystals can be in the glassy state, e.g. in poly(ethylene terephthalate) or PET The chains can be at a temperature above their glass transition temperature and be “rubbery”, e.g. in poly(ethylene) or PE 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 Lamella stacks L 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. 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 http://www.kth.se/fakulteter/TFY/kmf/lcd/lcd~1.htm

  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. Crystal Growth Mechanisms Crystals from small molecules grow one molecule at a time. Entire chains must join the polymer crystal at one time. L Xu et al., Nature Materials (2009) 8, 348.

  19. Melt to crystal (below Tm): Decrease in Gibbs’ free energy because of the enthalpy differences between the states (Enthalpy change per volume, H m) x (volume) x (fractional undercooling) 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 gf is an interfacial energy L At equilibrium: energy contributions are balanced and DG = 0.

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

  21. 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

  22. 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

  23. 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:

  24. 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

  25. 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 from before (slide 19) - 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.

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

  27. 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

  28. Chains Can Re-organise to Reduce the Number of Folds Xu et al., Nature Materials (2009) 8, 348.

  29. Temperature Dependence of Crystal Growth Velocity, n The rate at which a chain attempts to join a growing crystal, t -1, is expect to have the same temperature-dependence as the viscosity of the polymer melt (h ~ Gt): 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:

  30. Temperature Dependence of Crystal Growth Velocity, n 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. Recall that T0 is approximately 50 K less than the glass transition temperature.

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

  32. 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 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.

  33. T-Tm() (K) T-Tm() (K) T-Tm() (K) 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! n (cm s-1)

  34. 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

  35. 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

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

  37. Effects on Branching on Crystallinity Linear Poly(ethylene) Branched 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

  38. 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

  39. Crystal Nucleation from “Seeds” Original crystal Re-crystallised from “seed crystals” Xu et al., Nature Materials (2009) 8, 348.

  40. Summary • 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 below the melting temperature (Tm) but above the glass transition temperature, Tg. • 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

  41. 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 417.8 K. Temperature, T (K) Lamellar thickness, L (nm) 358.95 8.9 368.95 9.9 385.75 12.0 396.15 14.1 397.55 16.1 399.15 15.9 400.85 17.3 401.65 18.2 403.05 17.9 404.15 20.1 405.55 22.2 • Are the data broadly consistent with the predictions of theory? • Predict the melting temperature of crystals grown at a temperature of 400 K.

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