Polymer Dynamics and Rheology - PDF Free Download

Polymer Dynamics and Rheology 1

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

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

Stoke s Law F = vς ς = 6πη s R F 4

Creep Experiment Cox-Merz Rule 5

Boltzmann Superposition 7

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

Brownian Motion For short times For long times 0 10

13 http://mathworld.wolfram.com/first-orderordinarydifferentialequation.html

The response to any force field 17

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

25 W energy = Force * distance

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

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

Creep Measurement Response K = 1 τ http://mathworld.wolfram.com/first-orderordinarydifferentialequation.html 28

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

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

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

Equation for a circle in J -J space 33

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

Boltzmann Superposition 38

Flow Rouse Dynamics 39

Newtonian Flow Entanglement Reptation Rouse Behavior ω 2 ω 1 1 ω 2 ω 5 7 43

Newtonian Flow Entanglement Reptation Rouse Behavior ω 2 ω 1 1 ω 2 ω 5 7 50

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

For a Hookean Elastic 52

For Newtonian Fluid 53

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

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

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

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

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

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

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

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

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

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

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

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

Newtonian Flow Entanglement Reptation Rouse Behavior ω 2 ω 1 1 ω 2 ω 5 7 68

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

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 http://www.eng.uc.edu/~gbeaucag/classes/morphologyofcomplexmaterials/sukumaranscience.pdf 71

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

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

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

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

Reptation of DNA in a concentrated solution 76

Simulation of the tube 77

78 Simulation of the tube

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

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

this implies that dt ~ p 81

McLeish/Milner/Read/Larsen Hierarchical Relaxation Model http://www.engin.umich.edu/dept/che/research/larson/downloads/hierarchical-3.0-manual.pdf 84