5 Superposability of Stress 12121+2gsJust for g11(t) = G(t-t1) g1Just for g22(t) = G(t-t2) g2Forg1 + g21+2(t) = 1(t) + 2(t)= G(t-t1) g1 + G(t-t2) g2
6 Boltzmann Principle For infinitesimal strain dgi at time dt'dgitidiFor infinitesimal strain dgi at timefor strain g(t) of arbitrary historyt
7 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.
8 Linear Viscoelasticity (Oscillatory Shear) InputOutputη*: complex viscosity
9 Frequency Defined Test Input: strain (g), frequency (w), and gap (H). Measure: torque (M) and phase angle (d).
10 Frequency SweepCurve of a frequency sweep used to talk about oscillation modes; talk about other ways to run the experiment like time sweeps, temp sweeps; ability to do in both controlled strain and stress mode and advantage of eachthe waveform and why that is important(and unique to us)The amplitude of the perturbation can be freely chosen for each frequency, and dynamic modulus measurement is so far the most common method of linear viscoelastic characterization currently.
11 Stress Relaxation vs. Frequency Spectrum G(t) vs. tG'(ω) vs. ωthe amplitude of the perturbation can be freely chosen for each frequency,A is monodisperse with M<Mc; B is monodisperse with M>>Mc and C is polydisperseLVE response is very sensitive to the molecular structure of the polymers
12 线性粘弹性函数之间的关系 Dynamic Compliance J*(ω) Dynamic Modulus G*(ω) Relaxation G(t)FourierTransformsAlgebraicEquationsFourierTransformsCreepComplianceJ(t)IntegralEquationsLaplaceTransformsLaplaceTransformsRetardation TimeDistributionL(τ)RelaxationTimeDistributionH(τ)IntegralTransforms12Polymeric liquids and networks – Dynamic and rheology. Page 122.
19 旋转流变仪的种类 AR-Series Hybrid-Series Aton Paar ARES Malvern ARES-G2 应变控制型 （SMT）应力控制型（CMT）19Separate Motor and Transducer
20 Torque Measurement is Unaffected by Motor Inertia & Friction 旋转流变仪的种类FRTTorque Measurement is Unaffected by Motor Inertia & FrictionMotor Inertia & friction Involved in Torque MeasurementMotor/ TransducerPrimary Moving ElementsMotorThe SMT architecture is preferable to the CMT architecture because the most massive moving element (the motor) is physically isolated from the torque measurement. The motor is also where the largest motions occur. This is in contrast to the CMT in which the motor and sample torque are conflated because the heavy, large moving element is used to measure (apply) the torque.应力控制型（CMT）应变控制型（SMT）20
21 Strain vs. Stress controlled Strain ControlledGood for oscillatory measurementsGood for fixed shear rate/strain measurements (Stress relaxation)Motors are really good - good for weak materialsVery sensitive torque transducersStress ControlledOK for oscillatory measurementsGood for fixed stress measurementsGood for creep measurementsDrag cup motors often cannot do low stresses wellEC motors often have more inertial effectsOften assumes certain type of material response两种流变仪差别越来越小！
30 Rheological Measurements Flow testsConstant shear rateContinuous stress/rate ramp and downSteady state shear rate sweepFlow temperature rampFlow reversalOscillation testsFrequency sweepTime sweepStrain/stress sweep (LVE)Temperature rampTemperature/Frequency sweep (TTS)Fast SamplingMultiwaveTransient testsStress relaxationCreep & creep recoveryothersElongational testLAOSStrain-Rate Frequency Superposition (SRFS)
31 Slow Relaxation Behavior of Linear Chains Polybutadiene, 40Crelaxation time t ~ M3.4±0.2Delay of orientation/stress relaxation due toentanglement of uncrossable chains
32 Slow Relaxation of Star-branched Chains PBD:Linear Mw=160K6-arm star Ma=77KRelaxation timet ~ exp(0.6Marm/Me)Much stronger delayfor star chaincf. ~ M3.4±0.2for linear chain
33 利用蠕变测试扩展SAOS测试频率Example for the extension of the frequency range using the retardationspectrum obtained from creep-recovery tests (recover time up to 104 s).J Rheol. 2014, 58(3), 565.
34 利用应力松弛测试扩展SAOS测试频率 UHMWPE DFreq SR ARES-G2 Fourier Transforms RelaxationModulusG(t)DynamicModulusG*(ω)
35 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),
36 Oscillation Time Sweep Effect of thermally reduced graphite oxide (TrGO) on the polymerization kinetics of poly(butylene terephthalate)Polymer. 2013, 54 (6), 1603.
37 Multiwave Oscillation The total strain amplitude should not exceed the linear viscoelastic regimeThe test time is the same as the dynamic single point experiment under the fundamental frequency
38 Multiwave Oscillation Evolution of the loss tangent during a curing reaction. The gel point is the point, when tan δ becomes independent of frequency.
39 Oscillation Temperature Ramp Cross-linking kinetics of XLPE
40 Oscillation Temperature Ramp Phase separation temperature of polymer blendsPS/PVMEwith big difference in TgPB/PIwith big discrepancyin viscoelasticityMiscibleMetastablePhase-separatedDynamic temperature s ramp for a 50:50 PS 38K/PVME-23K blendJ Phys Chem B. 2004, 108 (35),
46 Further ReadingsThe Rheology Handbook-For Users of Oscillatory Rheometers ( 3rd ed.)Thomas G. Mezger2013Structure and Rheology of Molten Polymers:From Structure To Flow Behavior and Back AgainJohn M. Dealy , Ronald G. Larson.2006
47 Further ReadingsMelt Rheology and Its Applications in the Plastics IndustryJohn M. Dealy , Jian Wang2013Colloidal Suspension RheologyNorman J. Wagner, Jan Mewis.2012
48 Further Readings Viscoelastic Properties of Polymers (3rd Revised) John D. Ferry1980Rheology: Principles, Measurements, and Applications Ch. W. Macosko1994
49 Rheology Related Journals Journal of RheologyRheologica ActaJournal of Non-Newtonian Fluid MechanicsApplied RheologyKorea-Australia Rheology JournalNihon Reorogi Gakkaishi (Journal of Society of Rheology Japan)MacromoleculesLangmuirSoft MatterPhysical Review LettersPhysical Review EJournal of Chemical Physics
54 Rheology needs a lot of experience Rheology needs a lot of experience. Modern rheometers will give you numbers, no problem, but the question is always whether they are correct. That and the optimization of the parameters to minimize the noise and do what you want to the material (destroy or not destroy a structure) is what sets a good rheologist apart from an inexperienced one.