Polymer Dynamics 2017 School on Soft Matters and Biophysics Instructor: Alexei P. Sokolov, e-mail: [email protected] Text: Instructor's notes with supplemental reading from current texts and journal articles. Please, have the notes printed out before each lecture. Books (optional): M.Doi, S.F.Edwards, “The Theory of Polymer Dynamics”. Y.Grosberg, A.Khokhlov, “Statistical Physics of Macromolecules”. J.Higgins, H.Benoit, “Polymers and Neutron Scattering”. 1 Course Content: I. Introduction II. Experimental methods for analysis of molecular motions III. Vibrations IV. Fast and Secondary Relaxations V. Segmental Dynamics and Glass Transition VI. Chain Dynamics and Viscoelastic Properties VII. Rubber Elasticity VIII. Concluding Remarks 2 INTRODUCTION Polymers are actively used in many technologies Traditional technologies: Even better perspectives for use in novel technologies: Example: Light-weight materials Boeing-787 “Dreamliner” 80% by volume is plastic Materials for energy generation Example: polymer solar cells Materials for energy storage Example: Polymer-based Li battery Polymers have huge potential for applications in Bio-medical field “Smart” materials, stimuli-responsive, self-healing … 3 Polymers Polymers are long molecules They are constructed by covalently bonded structural units – monomers Main difference between properties of small molecules and polymers is related to the chain connectivity Advantages of Polymeric Based Materials: Easy processing, relatively cheap manufacturing Light weight (contain mostly light elements like C, H, O) Unique viscoelastic properties (e.g. rubber elasticity) Extremely broad tunability of macroscopic properties 4 Definitions o Monomer – repeat unit, structural block of the polymer chain o Oligomer – short chains, ~ 3-10 monomers o Polymers – long chains, usually hundreds of monomers o Degree of polymerization – number of monomers in the polymer chain Molecular weight: Weight of the molecule M [g/mol]. Usually there is a distribution of M. Two main characteristics: Number average Mn and weight average Mw. Polydispersity is usually characterized by the ratio PDI=Mw/Mn. Properties of a polymer (Tg, viscoelastic properties) can be changed by variation of molecular weight, without change of monomer chemistry. Properties can be also changed by changing macromolecular architecture: • Linear chain • Branched chain • Cyclic molecule •Star • Dendrimer 5 Structure of a polymer chain Polymer chain has a backbone (the main atoms of the chain, e.g. C-C-, or Si-O-Si) and side groups (e.g. CH3, phenyl ring, etc.) Tacticity – the way monomers are connected also affect the polymer properties. Example poly(methyl methacrylate) (PMMA): OCH3 H3CH3C C CO CH3 C C C C C C C Random tacticity is called C C CO CO atactic polymer (e.g. aPMMA) CO OCH H3C OCH3 3 OCH3 syndiotactic (sPMMA) isotactic (iPMMA) Chain formed by different monomers - co-polymer.Canberandom (random position of monomers A, B in the chain) or block co-polymers (blocks of monomers A connected to blocks of monomers B). Mixtures of two polymers called polymer blend. 6 n Chain Statistics The end-to-end vector: R n r i ; its average value: R 0 i1 n n n n n n R2 r r rr l 2 cos n i j i j ij i1 j1 i11j i11j Here l is the bond length and is the bond angle. N 1 l Center of mass: RCM Rn ; N n1 N 2 1 2 Mean-squared radius of gyration: Rg (Rn RCM ) N n1 Freely joined chain assumes no correlations between directions of different bond vectors, <cosij>=0 for ij. R2 nl 2 Freely rotating chain assume a constant angle 1 cos R2 nl 2 n 1 cos o 2 2 For polyethylene (PE) =68 : <Rn > 2nl Distribution of the end-to-end distances : 3 / 2 3 3R2 P (R) exp N 2 2 2 R 2 R 7 Characteristic ratio, C, and Statistics for Real Chains 2 2 2 M 2 Traditional definition:R N RlR Cnl C l ; here n is the number, l is the m0 length and m0 is the mass of bonds (or monomers) in a backbone; M is the molecular weight of the chain. C is a characteristic ratio, known for most of the polymers. 2 Another important parameter is the Kuhn segment length defined as lK=<R >/Rmax. In many cases, another simplification is assumed: Rmax=NRlK=nl0,i.e.Rmax is equal to completely extended chain. It leads to lK=Cl0. This approximation is traditionally used for characterization of real chains and bead size (Kuhn segment). However, C characterizes just the coil size and not the bead (segment) length. It means that the bead size, lR,isdefinednotbyC only, there is at least one parameter more. BeadandSpringmodelis a traditional approximation in Polymer Physics. It presents a chain as a line of beads connected by springs. Equilibrium length of springs is assumed to be b~lK. Angles b between the springs are free to change. 8 Kuhn Segment 2 2 2 In order to bring real chain to the freely jointed chain: Rn Cnl Nb here N is the number of random steps (Kuhn segments) and b is the length of the random steps (Kuhn segment length). R2 The contour length of the chain L=Nb, so the Kuhn length b n L The assumption in many textbooks: L=R nl;e.g.forall max R2 trans PE ~0.83. Then n b Cl Rmax Parameters for known polymers 9 Entanglement According to chain statistics, R2M. It means that chain density, ~M/R3M-1/2, decreases with increase in molecular weight. We know, however, that polymers have normal density. It happens because different chains interpenetrate each other. As a result, they become entangled above particular molecular weight Me. Me is important polymer parameter. It depends on polymer chemistry and rigidity. For example, Me~15,000 in polystyrene, and Me~2,000 in polybutadiene. There is an argument that Me depends on packing efficiency of the chain: if chain can pack well Me is large. Entanglement results in very interesting polymer-specific properties, such as rubber-like elasticity even in a non-crosslinked state, extremely high viscosity and very long relaxation times. 10 Different Types of Molecular Motion I. Vibrations Oscillations around fixed positions (Center of mass does not move). II. Translation motion Change of the position (Center of mass shifts), Diffusion is one of the examples. III. Rotation IV. Conformational jumps. Example – transitions of cyclohexylene ring. The last three (II-IV) can be considered as a relaxation motion. Usually one can separate contributions from these motions because in most cases they occur on different time scales. However, these motions can be coupled and one should consider them all together or use some approximations. 11 Frequency map of polymer dynamics Chain dynamics: Rouse and terminal relaxation Structural relaxation, -process Vibrations Secondary relaxations Fast relaxation FREQUENCY, 10-2 1102 104 106 108 1010 1012 1014 Hz Mechanical relaxation G Dielectric Spectroscopy, *() Quasi-optics,TDS Traditional dielectric spectroscopy IR-spectr. Light Scattering, Iij(Q,) Interferometry Photon – Correlation Spectroscopy Raman spectroscopy Spin-Echo Neutron Scattering, S(Q,) Back-sc. Time-of-Flight Inelastic X-ray Scattering, S(Q,) High-Resolution IXS Scattering techniques have an advantage due to additional variable – wave-vector Q What is Measured in Scattering: Time-Space Correlation Function In an isotropic medium, time-space pair correlation function: G(r,t) n(0,0)n(r,t) N where n(r,t) (r rj (t)) presents coordinate of all atoms at time t. j G(r,t) has self and distinct part, G(r,t)=GS(r,t)+Gd(r,t). If at time t1=0 a particle was at position r1=0, GS(r,t) gives a probability to find the same particle around position r at timet,andGd(r,t) gives a probability to find another particle around position r at time t. 3 Intermediate scattering function is a space-Fourier transform of G(r,t): I(q,t) G(r,t)exp(iqr)d r V Definition from quantum mechanics: I(q,t) exp[iqri (0)]exp[iqrj (t)] exp[iq{ri (0) rj (t)}] ij ij 1 Time-Fourier transform of I(q,t) gives dynamic structure factor: S(q,) exp(it)I(q,t)dt 2 1 3 Sinc (q,) dt GS (r,t)expi(qr t) d r There are coherent and incoherent S(q,): 2 V 1 3 Scoh (q,) dt Gd (r,t)expi(qr t) d r 2 V These equations were first introduced by van Hove [van Hove, Phys.Rev. 95, 249 (1954)]. Separation of translation, rotation and vibration c (t) – center of mass position; ri (t) ci (t) bi (t) ui (t) i bi(t) – rotation about the center of mass; ui(t) – distance of the particle from its average position. S(q,t) exp[iq{ci (0) c j (t)}]exp[iq{bi (0) bj (t)}]exp[iq{ui (0) u j (t)}] ij If we assume that all 3 vectors are independent: S(q,t) sTR (q,t) s ROT (q,t) sVIB (q,t) ij Thus translation, rotation and vibration can be considered separately. Examples: TR 2 Diffusion: [x x(t)]2 Dt ; D is a diffusion coefficient; s (q,t) exp(q Dt) TR 2 TR N N Sinc (q,t) N exp(q Dt) N exp(t) Sinc (,t) exp(it)exp( t )dt 2 2 2 An exponential decay for S(q,t) A Lorentzian for S(q,) 1.0 q > q S(q,) The decay rate (or width) q2 0.8 1 2 q 2 0.6 q 2 0.4 S(q,t) q 1 0.2 q 1 0.0 0 t 15 Rotation: Let’s assume that there are two equal positions and molecule makes rotational jumps between r and r positions.