Fundamentals of Polymer Rheology Sarah Cotts TA Instruments Rubber Testing Seminar CUICAR, Greenville SC Rheology: An Introduction Rheology: The study of stress-deformation relationships =Viscosity =Modulus 1
Shear Flow in Parallel Plates Stress (σ) Strain (γ) Strain rate ( ) = M = θ = Ω r = plate radius h = distance between plates M = torque (µn.m) θ = Angular motor deflection (radians) Ω= Motor angular velocity (rad/s) Open Boundary Rotational Rheometer ARES G2 Measured Torque (Stress) Transducer Sample Controlled Strain Applied Strain or Rotation Direct Drive Motor 2
Closed Die Cavity Rheometer Measured Torque (Stress) Transducer RPA Elite Applied Strain or Rotation Direct Drive Motor Controlled Strain Viscoelasticity Viscoelasticity: Having both Viscous and Elastic properties. Elastic Behavior: Stress is dependent on Strain Hooke s Law of Elasticity: Modulus = Stress/Strain Viscous Behavior: Stress is dependent on Strain Rate Newton s Law of Viscosity: Viscosity = Stress/ Strain Rate Viscoelastic materials cannot be fully understood using only one of these relationships. Oscillation measurements are able to measure stress as a function of both strain and strain rate. 3
Hooke s Law of Elasticity For an Elastic Solid, Stress and Strain have a constant proportionality σ = E*ε If the material follows Hooke s Law, the deformation will be reversible when the stress is removed The modulus of a Hookean solid will not show any time dependence- the stress depends on the strain, but not the strain rate Newton s Law of Viscosity For a Viscous Liquid, Stress is proportional to Strain Rate dε/dt by a coefficient of Viscosity η σ = η* dε/dt The deformation of a liquid is non-reversible Stress η Strain Rate 4
Viscoelastic Behavior σ= E*ε+ η*dε/dt Kelvin-Voigt Model (Creep) Maxwell Model (Stress Relaxation) L 1 L 2 Viscoelastic Materials: Force depends on both Deformation and Rate of Deformation and vice versa. Time-Dependent Viscoelastic Behavior T is short [< 1s] T is long [24 hours] Deborah Number [De] = τ/t 5
9/26/2016 Pitch Drop Experiment Long deformation time: pitch behaves like a highly viscous liquid 9th drop fell July 2013 Short deformation time: pitch behaves like a solid Started in 1927 by Thomas Parnell in Queensland, Australia http://www.theatlantic.com/technology/archive/2013/07/the-3-most-exciting-words-in-science-right-now-the-pitch-dropped/277919/ Time-Dependent Viscoelastic Behavior Silly Putties have different characteristic relaxation times Dynamic (oscillatory) testing can measure time-dependent viscoelastic properties more accurately and efficiently by varying frequency (deformation time) 6
Frequency Sweep- Time Dependent Viscoelastic Properties Frequency of modulus crossover correlates with Relaxation Time Storage and Loss of a Viscoelastic Material RUBBER BALL TENNIS BALL X X LOSS (G ) STORAGE (G ) 7
Oscillation Testing The motor applies an oscillating deformation to the sample at a set Amplitude and Frequency. Amplitude: Degree of Arc, or % Strain Frequency: Hz (cycles per second) or CPM (cycles per minute) Deformation Torque, S* The torque transducer measures the response of the sample. Amplitude: Torque (S*), or Stress The phase lag between the deformation and the torque is used to determine the Elastic and Viscous response. Phase Angle δ Oscillation Testing In a purely Elastic material, Stress is in phase with the Strain Phase angle= 0 Phase angle= 90 In a purely Viscous material, Stress is out of phase with the Strain Viscoelastic material 0 < Phase angle< 90 8
Oscillation Testing- Lissajous-Bowditch Plots Elastic Material Viscoelastic material Viscous Material Stress Stress Stress Strain Strain Strain Raw oscillation data can also be plotted as stress vs. strain. An elliptical shape represents a viscoelastic response. Oscillation Testing The Phase Angle can be used to separate the stress signal into the elastic and viscous components G : Storage Modulus Measure of elasticity, or the ability to store energy G = (Stress/Strain)*cos(δ) G : Loss Modulus Measure of viscosity, or the ability to lose energy G = (Stress/Strain)*sin(δ) Tan Delta Measure of dampening properties Tan (δ) = G /G G*: Complex Modulus Measure of resistance to deformation G*= Stress/Strain η*: Complex Viscosity δ Measure of resistance to flow Storage Modulus η* = Stress/Strain Rate Loss Modulus 9
PDMS Frequency Sweep Comparison Overlay Frequency Sweeps on Solids 10 8 Happy & Unhappy Balls 10 0 Modulus is the same tan δis very different ) ( ' E ² ] m / c n y [ d 10 7 Unhappy ball 10-1 [ ] t a n _ d e l t a ( ) 10 6 Happy ball 10 5 10-1 100 10 1 10 2 10-2 Freq [rad/s] 10
Applications of Polymer Rheology Polymer Characteristics Apart from chemical nature, tacticity and microstructure, polymers have 3 major characteristics which affect processability 1. Average molecular weight 2. Molecular weight distribution 3. Branching (Type and architecture) 11
Long Chain Branching: Improving Processability Long Chain Branching (LCB) provides Shear Thinning, Strain Hardening and Melt Strength Very effective only one per 10000C needed. Difficult to fully characterize length of branches, number per chain and distribution. Common Characterization Techniques Chromatography (SEC) NMR FTIR 12
Influence of Molecular Weight on G and G The G and G curves are shifted to lower frequency (longer time) with increasing molecular weight. 10 6 10 5 Modulus G', G'' [Pa] 10 4 10 3 10 2 10 1 SBR M w [g/mol] G' 130 000 G'' 130 000 G' 430 000 G'' 430 000 G' 230 000 G'' 230 000 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 10 4 10 5 Freq ω [rad/s] Influence of MW on Viscosity The zero shear viscosity increases with increasing molecular weight. TTS is applied to obtain the extended frequency range. Viscosity η* [Pa s] 10 7 10 6 10 5 10 4 10 3 Zero Shear Viscosity η o [Pa s] 10 6 Zero Shear Viscosity Slope 3.08 +/- 0.39 SBR M w [g/mol] 130 000 230 000 320 000 430 000 The high frequency behavior (slope -1) is independent of the molecular weight 10 5 10 2 100000 Molecilar weight M w [Daltons] 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 10 4 10 5 Frequency ω a T [rad/s] 13
Materials Evaluated: Polyolefin Elastomers POE Co-Monomer Density (g/cm 3 ) Long Chain Branching EB-1 Butene 0.901 High EB-2 Butene 0.870 Medium EB-3 Butene 0.870 High EO-1 Octene 0.870 Medium The level of Long Chain Branching (LCB) is difficult to directly measure in POEs Amount and length of short chain branching interferes with NMR Characterize differences in LCB using rheology L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented at Society of Plastics Engineers 2005 ANTEC Boston, MA Conventional Rheology of Polyolefin Elastomers 1,000,000 (b T /αt)η (Poise) Viscosity (poise) 100,000 10,000 EB-1: High LCB EB-2: Med LCB EB-3: High LCB EO-1: Med LCB 1,000 0.01 0.1 1 10 100 1000 Angular Frequency (rad/sec) L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented at Society of Plastics Engineers 2005 ANTEC Boston, MA 14
Conventional Rheology of Polyolefin Elastomers 10 EB-1: High LCB EB-2: Med LCB EB-3: High LCB EO-1: Med LCB Tan Delta 1 0.1 0.01 0.1 1 10 100 1000 Angular Frequency (rad/sec) L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented at Society of Plastics Engineers 2005 ANTEC Boston, MA Extensional Viscosity of Polyolefin Elastomers 3000000-1 ε& = 0.1s 190 C 2500000 ηe (Poise) 2000000 1500000 EB-1: High LCB EB-2: Med LCB EB-3: High LCB EO-1: Med LCB 1000000 500000 0 0 2 4 6 8 10 12 14 16 18 20 Time (s) L. J. Effler,et.al., DuPont Dow Elastomers, Freeport, TX, Presented at Society of Plastics Engineers 2005 ANTEC Boston, MA 15
Linear Viscoelasticity In the Linear Viscoelastic Region, the Stress signal is a sine wave, and meaningful rheology measurements can be made. Linear Viscoelasticity At higher strains, stress becomes non-sinusoidal. Modulus is an approximation. 16
Linear Viscoelasticity Non-Linear Viscoelasticity- Higher Harmonics SAOS γ(t) = τ(t) = γ 0 sin(ωt) τ 0 sin(ωt+δ) LAOS γ(t) = γ 0 sin(ωt) τ(t) = τ 1 sin(ω 1 t+ϕ 1 ) + τ 3 sin(3ω 1 t+ϕ 3 ) + τ 5 sin(5ω 1 t+ϕ 5 ) +... Σ n=1 odd = τ n sin(nω 1 +ϕ n ) 17
Non-Linear Viscoelasticity- Higher Harmonics Sample Response = + + Fundamental 3 rd Harmonic 5 th Harmonic Large Amplitude Oscillatory Shear (LAOS) 1200 10000 8000 1000 8000 6000 4000 800 600 400 6000 4000 2000 200 2000 τ [Pa] 0-2000 -4000-6000 0-200 -400-600 -800 γ [%] τ [Pa] 0-2000 -4000-6000 -8000-1000 PIB, ω=0.1 Hz, γ 0 =1000%, T=140 C -1200 0.0π 0.5π 1.0π 1.5π 2.0π t ω [Pa] -8000-10000 PIB, ω=0.1 Hz, γ 0 =1000%, T=140 C -1200-600 0 600 1200 γ [%] The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly Université Catholique de Louvain (UCL) 18
Large Amplitude Oscillatory Shear (LAOS) 300000 200000 Shear stress (Pa) 100000 0-8 -6-4 -2 0 2 4 6 8-100000 -200000 Branched Linear 5% Branched in Linear -300000 Strain rate (s -1 ) The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly Université Catholique de Louvain (UCL) Long Chain Branching Index The potential of large amplitude oscillatory shear to gain an insight into the long-chain branching structure of polymers ACS Meeting 2008, Florian J. Stadler, Sunil Dhole, Adrien Leygue, Christian Bailly Université Catholique de Louvain (UCL) 19
Long Chain Branching Shear stress (Pa) 75000 50000 25000 High visc. PP Low visc. PP 95% Low visc. PP+5% high visc. PP 50% Low visc. PP+50% high visc. PP 0 Secondary loops not affected by AMW and MWD -25000-50000 -75000 Secondary loops can be associated with linear polymer architecture There has to be a mathematical condition to account for loops -10-5 0 5 10 Shear rate (s-1) Lissajous-Bowditch Plots Ewoldt, R.H; McKinley, G. H. On secondary loops in LAOS via selfintersection of Lissajous-Bowditch Curves. Rheologica Acta 49, no 2. February 12 th, 2010. 213-219. 20
Rheological characterization of EPDMs ML(1+4) at 125 C Burhin, Henri G. Polymer Process Consult Rheological characterization of EPDMs Burhin, Henri G. Polymer Process Consult 21
LAOS testing of EPDMs Burhin, Henri G. Polymer Process Consult Long Chain Branching Index- EPDMs Burhin, Henri G. Polymer Process Consult 22
Dynamic Mechanical Analysis Is DMA Rheology, or Thermal Analysis? Oscillation measurements- E, E, Tan Delta Characterize viscoelastic materials as a function of time, temperature, stress and strain. Glass Transition of PSA measured by DMA 23
Effect of Crosslinking M = MW between c crosslinks 120 160 300 30,000 1500 9000 Temperature Effect of Aging on Elastomer O Rings 24
Conclusions Closed-Die RPA rheometer can match the viscoelastic measurements of the ARES Rheological measurements within the linear visocelastic region can be used to compare MW for non-branched polymers Effects of branching are seen in linear, small-amplitude rheology measurements, especially at low frequency Large Amplitude Oscillatory Shear (LAOS) clearly differentiates linear and branched polymers. DMA measurements also show differences in viscoelastic properties. These can indicate differences in chemistry or molecular structure, but also relate to bulk properties. Thank You The World Leader in Thermal Analysis, Rheology, and Microcalorimetry 25