Polymer Dynamics. Tom McLeish. (see Adv. Phys., 51, , (2002)) Durham University, UK

Transcription

1 Polymer Dynamics Tom McLeish Durham University, UK (see Adv. Phys., 51, , (2002)) Boulder Summer School 2012: Polymers in Soft and Biological Matter

2 Schedule Coarse-grained polymer physics Experimental probes of polymer dynamics Local friction - the Rouse chain Hydrodynamics - the Zimm chain Entangled Dynamics

3 Coarse-grained Polymers n=0 n R(n,t) n=n

4 Polymers as Random Walks R Force, f, from free energy F(R)=-k B TlnP G 0 =k B TC mon /N x

5 Experiment - techniques that probe polymer dynamics: Bulk information orthogonal to rheology Direct Molecular information Dielectric Spectroscopy Simulation T1/2-NMR PFGNMR SANS NSE PCS Self-Diffusion Linear and non-linear rheology

6 Rheology Shear log G(t) N G 0 N τ max log t Extension tension viscosity / Pa.s time /s 3

7 Frequency-dependent Shear rheology G( t) = G0e G' G'' ( ω) = G 0 t / τ ωτ ω τ 2 2 ω τ 1+ ω τ ( ω) = G0 2 2

8 Stress Tensor ~ ( n, t) R( n t) R, n n

9 Linear Polymers Branched Polymers

10 ~ exp(ν M a /M e ) η (Poise) ~ M w Span M w Fetters and Pearson (1983)

11 Dielectric Spectroscopy for Z=16 stars (Watanabe 2002) R( n', t) R( n, 0) n' n normalised response functions rheology frequency /s

12 Neutron Scattering I ( q) = FourierTransform( density _ correlations) q = 4π sin θ λ I i θ I Sees inside the chain ( higher q )

13 SANS on Deformed Chains 1 dn dn' exp t 2 N Linears - Muller et al. (1993) N 0 N 0 { iq. [ R( n, t) R( n', )]} Hs - Heinrich et al. (2002)

14 Neutron Spin Echo 1 N N N dn dn' exp n { iq. [ R( n, t) R( ',0)]} Wischnewski et al. (2002) ns

15 Transverse (T2) NMR Klein et al. (1998) ( ) ( ) n t n n t n dt b t b ',.. ' ', ' 2 3 cos 0 2 R L R

16 Diffusion Measurements (NMR, NR, SIMS..) Komlosh and Callaghan (1999)

17 Lodge (1999)

18 Simulation ( R( n, t) R( n,0) ) 2 (in this case ) 100 slope 1 g 2 (t), g 3 (t) [ σ 2 ] 10 1 slope 0.26 slope 1/2 slope 1/2 0.1 Putz et al. (2000) t [ τ ]

19 Summary of Probes of Polymer Dynamics

20 The Rouse Model R(n,t) R(n,t+ t) ( R) 2 Dt n=0 n D eff 1 n( t) 1 ( R) ( R) ( R) t n=n ( R) t 1/ 4

21 Diffusion in the Rouse Model Rouse Time τ R R D 2 cm 2 N b kt 2 Nb 2 kt Nζ 0

22 Stress Relaxation in the Rouse Model G( t) c c N N N kt n( t) τ R kt t 1 2

23 Polymer solutions and Hydrodynamics: The Zimm Chain Polymers in solution: scaling, correlations Dynamics: Zimm model, dilute and semi-dilute regimes References: M Rubinstein and R Colby, Polymer Physics (2003) R Larson, The Structure and Rheology of Complex Fluids

24 Polymers in Solution: excluded volume Real polymer chains: excluded volume parameter Flory: balance contact energy with chain entropy ) Swollen chains Phantom coil: Melt, near Θ-point

25 Semi-dilute regime Onset of multi-chain behaviour, interactions, enhanced viscoelasticity. e.g. Raise T ) swell ) induce overlap increase viscosity!

26 Chain correlations Sections of chain only see themselves at short distances: monomers in a blob Flory/single chain scaling inside blob blobs fill space Correlation length (e.g. light scattering)

27 Small distances: excluded volume negligible Large distances: Random walk of blobs 1/ /2 Crossover to melt when: (Edwards screening)

28 Semi-dilute solutions: osmotic pressure Slope = 1.32 Nocho et al., Macromolecules 1981

29 Diffusion: Zimm model Local drag coefficient in Rouse model: In solution, include long range hydrodynamic drag: (Stokes) Einstein Relation:

30 Zimm or Rouse who cares?! Relax by Zimm modes (faster) Monomer diffusion up to a Zimm time : Zimm regime ln<r 2 > Fickian diffusion 2 ln ( Nb ) 2 ln ( b ) Molecular diffusion diffusion sub-fickian diffusion 2/3 ln ( τ 0 ) ln ( τ ) Z 1 ln(t) (recall Rouse sub-fickian t 1/4 )

31 Stress relaxation G(t) Count number of unrelaxed chain segments N/n(t) [Decalin in Θ-solvent, Hair & Amis, 1989] 2/3

32 Stress relaxation at later times.. After Zimm time..not much stress left! ln(g(t)) -2/3 ln ( τ ) Z ln(t)

33 Intrinsic viscosity (dilute solution) Einstein calculation for colloids (spheres), only due to surrounding hydrodynamics: Polymeric contribution in dilute solution:

34 Dilute solution ) Melt in Good Solvent? dilute semi-dilute melt Zimm Rouse

35 Rouse/Zimm together ln(g(t)) -2/3 Zimm relaxation up to screening length ξ Rouse relaxation of blobs -1/2 ln(t)

36 Entangled Dynamics The Problem The Solution

37 Chain motions Linear polymers: reptation Branched polymers: arm retraction

38 Diffusion in the reptation Model lnφ(t) 1 R 2 g ar g a 2 1/4 1/4 1/2 1/2 τ e τ R τ d

39 Stress Relaxation in the reptation Model log G(t) N G 0 N τ max log t

40 Star Polymers End-retraction is an activated process over a thermal barrier ~M M τ ~ exp( νm )

41 The Star-arm fluctuation potential

42 1E7 1E6 G/ Pa 1E5 1E4 'G star G'' star G' linear G'' linear 1E3 1.0E-4 1.0E-3 1.0E-2 1.0E-1 1.0E0 1.0E1 1.0E2 1.0E3 1.0E4 1.0E5 1.0E6 1.0E7 Frequency /rad/s

43 Further Topics Non-linear Rheology Quantitative Linear Entangled Dynamics Rheology, Topology and the Pom-Pom model Workshop problems.

44 A non-linear view of entanglement: Bifurcation of Stretch and Orientation relaxation times lnτ τ orient τ stretch lnm e lnm

45 Startup of extensional flow shows two nonlinearities in rate:

46 Quantative Theory: New Physical Processes Contour length fluctuation Constraint Release R(n,t)

47 Linear regime - (Likhtman and McLeish, Macromolecules 2002, 35, ) numerical solution of CLF CR: Rubinstein and Colby (1988) longitudinal stress relaxation + + = = = N Z p t p Z p t p e R R e p Z e p Z c t R t M RT c t G τ τ ν ν µ ρ ), ( ) ( 5 4 ), (??? µ(t)=l(t)/l(0) - fraction of tube segments survived after time t Linear Rheology ( ) ( ) n t n R n t n R,,

48 Polystyrene, Shausberger et al, 1985, Mw=290K, 750K and 2540K, Me=13K G', G'' (Pa) E-5 1E-4 1E ω (s -1 )

49 Polybutadiene, Baumgaertel et al, 1992, Mw=20.7K, 44.1K, 97K and 201K 10 6 G', G'' (Pa) ω (s -1 )

50 G' Lo g G' Lo g Rheology and Topology óó óó óó óó ó ó óó ó óó ó ó ó ó ó ó ç óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó ç ç ç ç ç ç çç çç ç ç çç ç ó óó óó óó ç ç ç óó çççç ç ç çç ç çç çç ç ç çççç ç ç ç ç çççç ó ó ó ç óó ó ó ç ó -3 ó Logw ó (a) Lo G' g G' Lo g ç ç óó óó óó óó ó ó óó ó óó ó ó ó ó ó ó ç óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó ç ç ç ç ç ç çç çç ç ç çç ç ó ó óó óó ç ç ç óó çççç ç ç çç ç çç çç ç ç çççç ç ç ç ç çççç ó ó ó óó ó ó ó -3 ó Logw ó Lo G ' g Lo G' g óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó ó ó ó ó ó ó ó ó ó ó ç ç ç ç ç ç ç ç ç ç ç ç ççç ç ççç ççç ç ççç ç ç ó ó ó ó ó ó ó ç ç ç ó ç ó 3.5 ó ç çó ó Logw (b) Figure 8 Experimental and theoretical complex moduli for the two H-polymers of the table. Fitting parameters of G 0 and τ e were consistent with literature values, the number of entanglements along arm and crossbar, s a and s b together with their polydispersities ε a and ε b determined by GPC and SALS. Theoretical curves accounting for polydispersity are dashed; those without are solid.

51 LCB Polymers in non-linear flow Pom-pom constitutive equation highly entangled branch points long flexible backbone sections dangling ends

52 Stretch Orientation Stress

53 Represent a polydisperse (branched) polymer as a spectrum of pom-poms Linear relaxation spectrum => τ bi, g i decorate these modes using nonlinear extensional data => q i, τ si

54 Viscosity /Pa s Multi-mode pompom - an example Steady State Viscosity Extension Shear Viscosity /Pa s Transient Viscosity Extension Shear Rates s s s s s s s s s s Strain Rate /s Time /s First Normal Stress Difference in Shear Stress /10 5 Pa Rates 10 s -1 5 s -1 2 s -1 1 s -1 Data from Meissner (1972, 1975) and Münstedt and Laun (1979) Time /s

55 Classes of LCB and q-spectra LDPE IUPAC A and shifted IUPAC X pom-pom parameters Metallocene g i 10 2 X r i X q i X g i A r i A q i A g i 10 r i, q i Comb τ b