Part III : M6 Polymeric Materials

Transcription

1 16/1/004 Part III : M6 Polymeric Materials Course overview, motivations and objectives Isolated polymer chains Dr James Elliott 1.1 Course overview Two-part course building on Part IB and II First 6 lectures by JAE, final 6 lectures by AHW First half of course deals with fundamental issues including: Isolated polymer chain conformations Interactions between polymers and solvents Blending of polymers, and block copolymers Polymer dynamics Diffusion and permeation of polymers Latter half deals with high performance polymers, including ultra-high strength/stiffness fibres, liquid crystalline polymers, conducting and barrier polymers 1

2 1. ecommended reading Introduction to Polymers,.J. Young and P.A. Lovell, (AN6a.40) Chapter 3 (Mixing and Permeation). Fundamentals of Polymer Science, P. Painter and M. Coleman, (AN6a.58), Chapter 9 (Mixing). The Physics of Polymers, nd Edition, G. Strobl (AN6c.14) Chapters, 3, 6 and Appendix. (Single chains, Mixing and Dynamics). The Physics of Glassy Polymers,.N. Haward (ed.), (AN6c.17 or 131), Chapters 9 (Permeation) and 10 (Block copolymers). 1.3 Supplementary reading The Coming of Materials Science,.W. Cahn, (S139), Chapter 8 (Historical discussion). Statistical Mechanics of Chain Molecules, P.J. Flory, (AN6c.47) Chapters 1,, 4 (Single chains). Scaling concepts in Polymer Physics, P.G. de Gennes (AN6c.70) Chapters 1,, 3 (Single chains, mixtures)

3 1.4 Motivations Why bother to study polymers? Versatile and inexpensive materials for manufacturing Increasing demand for designer materials Biopolymers (biomaterials) the future of Materials Science Why bother to study polymers in molecular detail? Single chain conformations and dynamics often dominate materials behaviour, e.g. rubber elasticity, liquid crystallinity Understanding of scaling laws allows us to understand common features of many chemically different systems 1.5 What do we already know about polymers? T g viscoelasticity WLF glassy T-T superposition toughening Considère mechanics crazing plastic flow POLYMES terminal time dynamics diffusion reptation spherulites conformations crystalline lamellae tacticity 3

4 1.6.1 Single polymer chains Polymers consist of long sequences of monomeric units (poly-mer) Statistical description of chain, with several associated length scales Coarse-grained representation Bond length ~ 1Å Persistence length ~ 10 Å Coil diameter ~ 100 Å 1.6. Single polymer chains Chemical detail vanishes at low resolution piece of string - many conformations can understand common features: scaling laws, exponents can explain properties in a generic way: rubber elasticity, scaling of average end-to-end distance Chemical detail important for actual conformations understanding detailed structure: prefactors specific properties: isotropic-nematic transition, actual end-to-end distance 4

5 1.6.3 Single polymer chains How far does orientation of chain at one point persist? What is the typical end-to-end distance? andom flight or freely jointed chain Model for an ideal polymer chain (also Brownian motion) l N 1 l N l l 1 i = 0 p( r ) ij 3 π rij 3/ 3r ij exp rij = Gaussian distribution of segments for infinitely long chain. Exact distribution for finite length chains is skewed. 5

6 1.7. Properties of a freely jointed chain Freely jointed chains are characterised by a number of length scales, usually in decreasing order of magnitude: Contour length L = Nl End-to-end distance N = l l i, j i j = Nl adius of gyration g N 1 = ( N + 1) s i= 0 i = / 6 g can be measured from particle scattering experiments Properties of a real polymer chain eal polymers can be mapped onto a random chain of segments with uncorrelated orientation l N 1 l N l l 1 i = 0 These are called Kuhn segments a / L K = 6

7 1.8. Properties of a real polymer chain A measure of the stiffness of a polymer chain is the characteristic ratio C C = Nl = a / l / K Polymer Ideal random chain Poly(ethylene oxide) [PEO] Poly(ethylene) [PE] Poly(methyl methacrylate) [PMMA] Poly(styrene) [PS] DNA C Other models for polymer chains Problems with freely jointed chain model Neglects interactions between polymer segments, and the effect of solvents gets < > scaling wrong (see lecture ) Polymer conformations unrealistic gets C wrong Need a better model: Freely rotating chain Worm-like (or Kratky-Porod) chain 7

8 Freely rotating chain Constrain the valency angle γ of adjacent segments, but allow free rotation about chain axis γ a ϕ Call ϕ the dihedral or torsional angle. = Nl 1+ cosγ 1 cosγ C 1+ cosγ = 1 cosγ Freely rotating chain Consider how the orientational correlations between segments decay down the chain: cosθ ij exp ( a ) l ij K Exponential decrease in orientational correlation, with a characteristic decay constant The decay constant is known as the persistence length, and is the distance measured down the chain for orientational correlation function to drop to 1/e of its value a p = a K / 8

9 1.11 Kratky-Porod chain Like freely rotating chain, but with continuous chain curvature, therefore referred to as worm-like chain A good model for stiff chains, but not necessarily restricted to these Also has exponential decay of orientations down the chain true for any model 1.1 Problems with freely rotating chain For PE, γ = 70.53, which gives C =.0!! This is still lower than the expected value so what is going wrong? Answer is that there are also dihedral forces acting to stiffen the chain still further These are due to steric repulsion between the atoms substituted on next-nearest neighbour carbons along the chain Need to take account of the torsional states of the polymer chains in order to generate realistic conformations 9

10 Torsional conformations V k φ φ (φijkl ) = + ( 1 cos3φ ) ijkl V ϕ G T G + ϕ Using the standard cos3φ potential, there are three equilibrium positions: ϕ=180º (trans state) and ±60 (gauche states). In practice, the energies of the gauche states are slightly different than that of the trans state, depending on the atoms involved in the torsion Torsional conformations Large degree of freedom in the torsional states 10

11 otational isomeric state (IS) model Description (due to Flory) of polymer conformations by representing them as sequences of torsional configurations Each distinguishable sequence (symmetry) has an associated probability: 1 p( φ,,φ N ) = exp 1 Z { V (φ,,φ ) k T} 1 N / IS where Z IS is the IS partition function: { V (φ,,φ ) k T} ZIS = exp 1 N / B ϕ B otational isomeric state (IS) model In its simplest form (one dimensional approximation), the IS model can be invoked by considering each torsional state to be independent of its neighbours In this case, the probability of each conformation is just the probability of the statistical weights of each torsional state 1 p( φ1,,φ N ) = exp{ V (φi) / kbt} ZIS i For PE, t(trans) = 1, t(gauche + ) = t(gauche ) 0.5 C 1+ cosγ 1+ = 1 cosγ 1 cosφ cosφ = 3.4 Still wrong!! 11

12 1.15 The pentane effect Conformation resulting from the sequence t g + g t leads to close interatomic distances and hence a raised energy Two dimensional IS model Need a transfer matrix of coefficients to represent couplings between adjacent torsional states tt tg + tg 1 w 0 w 0 T= g + t g + g + g + g = 1 w 0 w 0 w 1 g t g g + g g 1 w 0 w 1 w 0 For PE: T = C =

13 Lecture 1 summary This lecture, we introduced the course and spent a little time recapping on what we know already about polymers, and how this course will build on that knowledge We then introduced the random flight or freely jointed chain model and derived scaling laws for the end-to-end distance and radius of gyration We saw that the random chain is not stiff enough to be realistic, and tried to rectify the problem by first considering the fixed valence angle chain, and then by taking into account torsional energies A rotational isomeric state model with couplings between adjacent torsions is required to give correct characteristic ratio for polyethylene 13