Polymer Dynamics and Rheology

Transcription

1 Polymer Dynamics and Rheology 1

2 Polymer Dynamics and Rheology Brownian motion Harmonic Oscillator Damped harmonic oscillator Elastic dumbbell model Boltzmann superposition principle Rubber elasticity and viscous drag Temporary network model (Green & Tobolsky 1946) Rouse model (1953) Cox-Merz rule and dynamic viscoelasticity Reptation The gel point 2

3 The Gaussian Chain Boltzman Probability For a Thermally Equilibrated System Gaussian Probability For a Chain of End to End Distance R By Comparison The Energy to stretch a Thermally Equilibrated Chain Can be Written Force Force Assumptions: -Gaussian Chain -Thermally Equilibrated -Small Perturbation of Structure (so it is still Gaussian after the deformation) 3

4 Stoke s Law F = vς ς = 6πη s R F 4

5 Creep Experiment Cox-Merz Rule 5

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7 Boltzmann Superposition 7

8 Stress Relaxation (liquids) Creep (solids) ( ) = ε ( t) J t σ Dynamic Measurement Harmonic Oscillator: δ = 90 for all ω except ω= 1/ τ where δ= 0 Hookean Elastic Newtonian Fluid 8

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10 Brownian Motion For short times For long times 0 10

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17 The response to any force field 17

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23 Both loss and storage are based on the primary response function, so it should be possible to express a relationship between the two. The response function is not defined at t = or at ω = 0 This leads to a singularity where you can t do the integrals Cauchy Integral 23

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25 25 W energy = Force * distance

26 D = ε 0 E + P = ε 0 ε E Dielectric Displacement Parallel Analytic Technique to Dynamic Mechanical (Most of the math was originally worked out for dielectric relaxation) Simple types of relaxation can be considered, water molecules for instance. ε 0 Free Space Creep: ε Material ε u Dynamic material Dynamic: Instantaneous Response Time-lag Response 26

27 Rotational Motion at Equilibrium A single relaxation mode, τ relaxation 27

28 Creep Measurement Response K = 1 τ 28

29 Apply to a dynamic mechanical measurement dγ 12 ( t) dt = iωσ 0 12 J * ( ω )exp( iωt ) Single mode Debye Relaxation Multiply by ( iωτ 1) iωτ 1 ( ) 29

30 Single mode Debye Relaxation Symmetric on a log-log plot 30

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32 Single mode Debye Relaxation More complex processes have a broader peak Shows a broader peak but much narrower than a Debye relaxation The width of the loss peak indicates the difference between a vibration and a relaxation process Oscillating system displays a moment of inertia Relaxing system only dissipates energy 32

33 Equation for a circle in J -J space 33

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35 Time Dependent Value Equilibrium Value ( ) dγ dt = γ t τ + ΔJσ 0 Lodge Liquid Boltzmann Superposition 35

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38 Boltzmann Superposition 38

39 Flow Rouse Dynamics 39

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43 Newtonian Flow Entanglement Reptation Rouse Behavior ω 2 ω 1 1 ω 2 ω

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50 Newtonian Flow Entanglement Reptation Rouse Behavior ω 2 ω 1 1 ω 2 ω

51 Lodge Liquid and Transient Network Model Simple Shear Finger Tensor Simple Shear Stress First Normal Stress Second Normal Stress 51

52 For a Hookean Elastic 52

53 For Newtonian Fluid 53

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56 Dumbbell Model t ( ) = dt 'exp k( t t ') x t ξ g t ( ) 56

57 Dilute Solution Chain Dynamics of the chain Rouse Motion Beads 0 and N are special For Beads 1 to N-1 For Bead 0 use R-1 = R0 and for bead N RN+1 = RN 57 This is called a closure relationship

58 Dilute Solution Chain Dynamics of the chain Rouse Motion The Rouse unit size is arbitrary so we can make it very small and: With dr/dt = 0 at i = 0 and N Reflects the curvature of R in i, it describes modes of vibration like on a guitar string 58

59 x, y, z decouple (are equivalent) so you can just deal with z ς R dz l dt = b R(z l+1 z l ) + b R (z l 1 z l ) For a chain of infinite molecular weight there are wave solutions to this series of differential equations z l ~ exp t τ exp ilδ ( ) Phase shift between adjacent beads Use the proposed solution in the differential equation results in: τ 1 = b R ζ R ( 2 2cosδ ) = 4b R ζ R sin 2 δ 2 59

60 For N R = 10 τ 1 = b R ζ R ( 2 2cosδ ) = 4b R ζ R sin 2 δ 2 Cyclic Boundary Conditions: z l = z l+nr N R δ = m2π NR values of phase shift δ m = 2π m; m = N R N R 2 1,..., N R 2 60

61 For N R = 10 τ 1 = b R ζ R ( 2 2cosδ ) = 4b R ζ R sin 2 δ 2 Free End Boundary Conditions: z l z 0 = z NR 1 z N R 2 = 0 dz dl ( l = 0) = dz dl ( N R 1)δ = mπ ( l = N R 1) = 0 NR values of phase shift NR Rouse Modes of order m δ m = π ( N R 1) m; m = 0,1, 2,..., ( N R 1) 61

62 Lowest order relaxation time dominates the response τ R = 1 3π 2 ζ R 2 a R kt R 4 0 This assumes that ζ R a R 2 is constant, friction coefficient is proportional to number of monomer units in a Rouse segment This is the basic assumption of the Rouse model, ζ R ~ a R 2 ~ N N R = n R 62

63 Lowest order relaxation time dominates the response τ R = 1 3π 2 ζ R 2 a R kt R 4 0 Since R 0 2 = a 0 2 N τ R ~ N 2 kt 63

64 The amplitude of the Rouse modes is given by: Z m 2 = 2 2 R 0 3π 2 m 2 The amplitude is independent of temperature because the free energy of a mode is proportional to kt and the modes are distributed by Boltzmann statistics ( ) = exp F p Z m 90% of the total mean-square end to end distance of the chain originates from the lowest order Rouse-modes so the chain can be often represented as an elastic dumbbell kt 64

65 Rouse dynamics (like a dumbell response) dx dt = x t du dx ζ + g(t) = k spr x ζ ( ) = dt 'exp t t ' τ = ζ k spr Dumbbell t τ g( t) + g(t) Rouse ζ τ R = R δ 4b R sin 2 2 π δ = N R 1 m, m=0,1,2,...,n R-1 65

66 Rouse dynamics (like a dumbell response) g( t 1 )g( t 2 ) = 2Dδ ( t) where t = t 1 t 2 and δ ( ) is the delta function whose integral is 1 Also, D = kt ζ x( t) x( 0) = kt exp t τ k spr τ = ζ k spr For t => 0, x 2 = kt k spr 66

67 Predictions of Rouse Model ( ) ~ t 1 2 G t G' ( ω ) ~ ( ωη 0 ) 1 2 η 0 = ktρ p τ R π 2 12 ~ N 67

68 Newtonian Flow Entanglement Reptation Rouse Behavior ω 2 ω 1 1 ω 2 ω

69 Dilute Solution Chain Dynamics of the chain Rouse Motion Predicts that the viscosity will follow N which is true for low molecular weights in the melt and for fully draining polymers in solution Rouse model predicts Relaxation time follows N 2 (actually follows N 3 /df) Diffusion constant follows 1/N (zeroth order mode is translation of the molecule) (actually follows N -1/df ) Both failings are due to hydrodynamic interactions (incomplete draining of coil) 69

70 Dilute Solution Chain Dynamics of the chain Rouse Motion Predicts that the viscosity will follow N which is true for low molecular weights in the melt and for fully draining polymers in solution Rouse model predicts Relaxation time follows N 2 (actually follows N 3 /df) 70

71 Hierarchy of Entangled Melts Chain dynamics in the melt can be described by a small set of physically motivated, material-specific paramters Tube Diameter dt Kuhn Length lk Packing Length p 71

72 Quasi-elastic neutron scattering data demonstrating the existence of the tube Unconstrained motion => S(q) goes to 0 at very long times Each curve is for a different q = 1/size At small size there are less constraints (within the tube) At large sizes there is substantial constraint (the tube) By extrapolation to high times a size for the tube can be obtained dt 72

73 There are two regimes of hierarchy in time dependence Small-scale unconstrained Rouse behavior Large-scale tube behavior We say that the tube follows a primitive path This path can relax in time = Tube relaxation or Tube Renewal Without tube renewal the Reptation model predicts that viscosity follows N 3 (observed is N 3.4 ) 73

74 Without tube renewal the Reptation model predicts that viscosity follows N 3 (observed is N 3.4 ) 74

75 Reptation predicts that the diffusion coefficient will follow N 2 (Experimentally it follows N 2 ) Reptation has some experimental verification Where it is not verified we understand that tube renewal is the main issue. (Rouse Model predicts D ~ 1/N) 75

76 Reptation of DNA in a concentrated solution 76

77 Simulation of the tube 77

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79 Plateau Modulus Not Dependent on N, Depends on T and concentration 79

80 Kuhn Length- conformations of chains <R 2 > = lkl Packing Length- length were polymers interpenetrate p = 1/(ρchain <R 2 >) where ρchain is the number density of monomers 80

81 this implies that dt ~ p 81

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84 McLeish/Milner/Read/Larsen Hierarchical Relaxation Model 84

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