Linear Viscoelasticity Viscoelastic relaxation modulus of flexible linear

Linear Viscoelasticity Viscoelastic relaxation modulus of flexible linear polymers. 3 Polym J. 2009, 41(11), 929.

Stress Relaxation (Transient Test) strain unit area = G(t) unit height 4

Superposability of Stress Just for g 1 1(t) = G(t-t 1) 1 2 1 t t 2 Just for 2 2(t) = G(t-t 2) 2 1 t 1 2 t 2 1+2 t For 1 + 2 1+2(t) = 1(t) + 2(t) = G(t-t 1) 1 + G(t-t 2) 2 5

Boltzmann Principle For infinitesimal strain d i at time�� d i t for strain (t) of arbitrary history dt' d i t ti t 6

Boltzmann Superposition Principle The principle of linear superposition of stresses and/or deformations： • The response to any event is linear； • All consequent events lead to independent responses. The material reacts to the next action as if no former action took place! Rheology: Concepts, methods and applications. Page 61. 7

Linear Viscoelasticity (Oscillatory Shear) Input Output η*: complex viscosity 8

Frequency Defined Test Input: strain ( ), frequency (w), and gap (H). Measure: torque (M) and phase angle (d). 9

Frequency Sweep The amplitude of the perturbation can be freely chosen for each frequency, and dynamic modulus measurement is so far the most common method of 10 linear viscoelastic characterization currently.

Stress Relaxation vs. Frequency Spectrum G(t) vs. t G'(ω) vs. ω A is monodisperse with M<Mc; B is monodisperse with M>>Mc and C is polydisperse LVE response is very sensitive to the molecular structure of the polymers 11

Dynamic Compliance J*(ω) Algebraic Equations Fourier Transforms Creep Compliance J(t) Fourier Transforms Integral Equations Laplace Transforms Retardation Time Distribution L(τ) Dynamic Modulus G*(ω) Relaxation Modulus G(t) Laplace Transforms Integral Transforms Relaxation Time Distribution H(τ) Polymeric liquids and networks – Dynamic and rheology. Page 122. 12

Time Temperature Superposition (TTS) WLF (Williams Landel Ferry) equation 13

Time Temperature Superposition (TTS) Thermorheologically simple Master curve of the linear viscoelastic moduli J Rheol. 2011, 55(5), 987. 14

Creep – Creep Recovery Recoverable Non Recoverable Principle of a creep recovery experiment J Rheol. 2014, 58(3), 565.

Dynamic Compliance J*(ω) Algebraic Equations Fourier Transforms Creep Compliance J(t) Laplace Transforms Dynamic Modulus G*(ω) Fourier Transforms Integral Equations Relaxation Modulus G(t) Laplace Transforms Retardation Relaxation Integral Time Transforms Distribution L(τ) H(τ) Polymeric liquids and networks – Dynamic and rheology. Page 122. 16

Prog Polym Sci. 2001, 26(6), 895. 17

ARES-G 2 应变控制型 （SMT） Separate Motor and Transducer AR-Series Hybrid-Series Aton Paar Malvern 应力控制型 （CMT） 19

FRT Torque Measurement is Unaffected by Motor Inertia & Friction Motor/ Transducer Primary Moving Elements Motor Inertia & friction Involved in Torque Measurement Motor 应变控制型 （SMT） 应力控制型 （CMT） 20 20

Strain vs. Stress controlled Strain Controlled • • Good for oscillatory measurements Good for fixed shear rate/strain measurements (Stress relaxation) • Motors are really good for weak materials • Very sensitive torque transducers 两种流变仪差别越来越小！ Stress Controlled • • OK for oscillatory measurements • • Good for creep measurements • EC motors often have more inertial effects • Often assumes certain type of material response Good for fixed stress measurements Drag cup motors often cannot do low stresses well

l Torque range （扭矩范围） l Angular Resolution （角位移分辨率） l Angular Velocity Range （角位移速率范围） l Frequency Range （可测频率范围） l Normal Force （法向力范围） l Motor type （驱动马达类型）

一个周期内得到时间间隔为Δt的N个点 From the time into the frequency domain Discrete Fourier transformation (DFT)

Extensional Viscosity Fixture (EVF)

Rheological Measurements n Oscillation tests Ø Ø Ø Frequency sweep Time sweep Strain/stress sweep (LVE) Temperature ramp Temperature/Frequency sweep (TTS) Ø Fast Sampling Ø Multiwave Ø LAOS Ø Strain-Rate Frequency Superposition (SRFS) n Flow tests Ø Constant shear rate Ø Continuous stress/rate ramp and down Ø Steady state shear rate sweep Ø Flow temperature ramp Ø Flow reversal n Transient tests Ø Stress relaxation Ø Creep & creep recovery n others Ø Elongational test

Slow Relaxation Behavior of Linear Chains Polybutadiene, 40 C relaxation time t ~ M 3. 4± 0. 2 Delay of orientation/stress relaxation due to entanglement of uncrossable chains 31

Slow Relaxation of Star branched Chains PBD: Linear Mw=160 K 6 -arm star Ma=77 K Relaxation time ~ exp(0. 6 Marm/Me) Much stronger del for star chain cf. ~ M 3. 4± 0. 2 for linear cha 32

利用蠕变测试扩展SAOS测试频率 Example for the extension of the frequency range using the retardation spectrum obtained from creep-recovery tests (recover time up to 104 s). J Rheol. 2014, 58(3), 565.

利用应力松弛测试扩展SAOS测试频率 UHMWPE DFreq SR ARES G 2 Fourier Relaxation Transforms Dynamic Modulus G(t) G*(ω)

Oscillation Time Sweep Re-entanglement kinetics of freeze-dried polymers (a) Buildup of modulus in polystyrene samples with time. (b) Equilibrium entanglement time of samples freeze dried from solutions with different original concentrations. Macromolecules. 2012, 45 (16), 6648.

Oscillation Time Sweep Effect of thermally reduced graphite oxide (Tr. GO) on the polymerization kinetics of poly(butylene terephthalate) Polymer. 2013, 54 (6), 1603.

Multiwave Oscillation Ø The total strain amplitude should not exceed the linear viscoelastic regime Ø The test time is the same as the dynamic single point experiment under the fundamental frequency

Multiwave Oscillation Evolution of the loss tangent during a curing reaction. The gel point is the point, when tan δ becomes independent of frequency.

Oscillation Temperature Ramp Cross-linking kinetics of XLPE

Oscillation Temperature Ramp Phase separation temperature of polymer blends PS/PVME with big difference in Tg PB/PI with big discrepancy in viscoelasticity Miscible Metastable Phase separated Dynamic temperature s ramp for a 50: 50 PS 38 K/PVME 23 K blend J Phys Chem B. 2004, 108 (35), 13220.

Steady Shear Stress/Rate Sweep Physics Today. 2009, 62(10), 27.

Shear Reversal Results of flow reversal studies of a 4. 80 wt % PP/clay hybrid nanocomposite. Macromolecules. 2001, 34 (6), 1864.

Elongational Test 1 Polylactide with long-chain branched structure Strain hardening coefficient: Ind Eng Chem Res. 2014, 53(3), 1150.

Elongational Test 2 (a) Chewing and (b) bubble gum behavior during start-up of uniaxial extension J Rheol. 2014, 58(4), 821.

Prog Polym Sci. 2001, 26(6), 895. 45

The Rheology Handbook-For Users of Oscillatory Rheometers ( 3 rd ed. ) Thomas G. Mezger 2013 Structure and Rheology of Molten Polymers: From Structure To Flow Behavior and Back Again John M. Dealy , Ronald G. Larson. 2006

Melt Rheology and Its Applications in the Plastics Industry John M. Dealy , Jian Wang 2013 Colloidal Suspension Rheology Norman J. Wagner, Jan Mewis. 2012

Viscoelastic Properties of Polymers (3 rd Revised) John D. Ferry 1980 Rheology: Principles, Measurements, and Applications Ch. W. Macosko 1994

Ø Ø Ø Journal of Rheology Rheologica Acta Ø Ø Ø Macromolecules Langmuir Journal of Non-Newtonian Fluid Mechanics Applied Rheology Korea-Australia Rheology Journal Nihon Reorogi Gakkaishi (Journal of Society of Rheology Japan) Soft Matter Physical Review Letters Physical Review E Journal of Chemical Physics

Rheology needs a lot of expe ri ence. Modern rheome ters will give you num bers, no prob lem, but the ques tion is always whether they are cor rect. That and the opti miza tion of the para me ters to min i mize the noise and do what you want to the mate r ial (destroy or not destroy a struc ture) is what sets a good rhe ol o gist apart from an inexperienced one. 54