Short introduction into rheology Basics, testing in rotation, creep and oscillation, extensional rheology

Contents Viscosity Controlled shear rate (CR), shear stress (CS), deformation (CD) Rotational testing - Newtonian and Non-Newtonian flow behavior - Yield stress - Thixotropy Viscoelasticity Structural reasons, modeling Creep & recovery testing - Description with Burgers model - Elastic and viscous share Oscillatory testing - Time sweep e.g. relaxation, gelation, sedimentation - Amplitude sweep Linear viscoelastic range (LVR), stability - Frequency sweep liquid, paste-like or elastic? - Temperature sweep e.g. cross-linking - Cyclic testing stability Extensional Rheology

Shear stress t, deformation g and shear rate g . Sample height: h Deflection: x Deformation: g Shear stress t : force F applied to area A Shear rate = change of deformation per time unit dt d g = . x A F h Direction of force

Typical shear rates Application Shear rate (s-1) Sedimentation 10-6 - 10-4 Phase separation 10-6 - 10-4 Leveling, running 10-1 - 101 Extrusion 100 - 102 Dip Coatings 101 - 102 Chewing 101 - 102 Pumping, stirring 101 - 103 Brushing 101 - 104 Spraying 103 - 104

Absolute and relative viscosity "Resistance to flow" Viscosity can be determined indirectly: torque M * A factor . shear stress Viscosity = = shear rate rotational speed * M factor Absolute viscosity readings with known measuring geometry only!

Relative viscosity Any scale reading S (time, distance, angular deflection) is set into ratio with a known viscosity standard Viscosity of unknown material calculates as follows: Parameters of testing (rotor, speed, filling …) strictly need to be kept constant. Calibration possible for Newtonian Liquids only!

Dynamic and kinematic viscosity (Dynamic) viscosity h [Pas] = shear stress [Pa] = shear rate [1/s] 1 Pas = 1000 mPas 1 mPas = 1 cP (centi Poise) Kinematic viscosity n [mm/s2] = density [kg/m³] 1 mm/s² = 1 cSt (centi Stokes) .

Viscosity of fluids: Measured at 20°C Substance Viscosity Water 1 mPas Milk 5 - 10 mPas Olive oil 100 mPas Engine oil 1000 mPas Honey 10 000 mPas Bitumen 100 000 000 mPas

Measuring flow behaviour Determination of flow behavior as a function of varying shear stress or shear rate Shear Stress  [Pa] Shear Rate  [1/s] Ramp (Thixotropy) Steps (Steady state) . Time t [s]

Newtonian flow behavior Example: Oil - Shear Stress - Viscosity  - Shear Rate . 5 10 15 20 25 30 35 40 45 50 Á [1/s] 100 150 200 250 300 350 400 450 500 ‚ [Pa] 1 ƒ [Pa s] Flow curve . Viscosity curve

Shear thinning flow behavior: Structural reasons Orientation Extension Deformation Dis-aggregation

Flow behavior: Flow curve Linear plot Newtonian Pseudoplastic (shear thinning) Dilatant (shear thickening)

Flow behavior: Viscosity curve Double-logarithmic plot Newtonian Pseudoplastic (shear thinning) Dilatant (shear thickening)

Yield stress 0 / yield point – a model The yield stress 0 is the shear stress  required - to overcome elastic behavior and - obtain viscoelastic flow behavior Shear stress 

Yield stress 0: Determination Controlled deformation (CD) mode: 0: Maximum of the curve shear stress  vs. time t (linear scaling) Controlled rate (CR) ramp: 0: Extrapolation of flow curve to shear rate  = 0 (linear scaling) Controlled stress (CS) ramp: 0: Intersection of tangents in the change in slope of the curve log deformation  vs. log shear stress  .

Yield stress 0: Determination in CD-mode Input: deformation  (constant) Measurement: shear stress  Result: shear stress  = f(time t) Evaluation: Determination of the curve maximum (= yield stress 0) 0.5 1.0 1.5 2.0 2.5 Time t [min] 50 100 150 200 250 Shear Stress ‚ [Pa] Curve discussion : Method t [min] t0 [Pa] --------------------------- Maximum 0.3161 224.9

Yield stress 0: Determination in CR-mode Input: shear rate  (varying) Measurement: shear stress  Result: shear stress  = f(shear rate ) Evaluation: yield stress 0 by Extrapolation of flow curve to shear rate  = 0 using a rheological model . 10 20 30 40 50 60 Á [1/s] 80 100 120 ‚ [Pa] Extrapolation Casson: 0 = 8.808 [Pa] . .

Yield stress 0: Determination in CS-mode Input: shear stress  (increase logarithmic) Measurement: deformation  Result: log deformation  = f(log shear stress ) Evaluation : Transition between the linear regimes (= yield stress 0) 0.1 1.0 10.0 100.0 Shear Stress ‚ [Pa] 0.001 0.010 0.100 1.000 10.000 100.000 Deformation  [-] 0 = 16 Pa

Bingham flow behavior Example: Tooth paste - Shear Stress - Viscosity  - Shear Rate . 5 10 15 20 25 30 35 40 45 50 Á [1/s] 100 150 200 250 300 350 400 450 500 550 ‚ [Pa] ƒ [Pa s] Decrease in h due to yield stress Flow curve . Bingham yield stress: ‚¥ = 29 Pa Viscosity curve

Thixotropy: Structural behavior Time-dependent behavior: Primary particles Agglomerates Network

Thixotropy: Definition and determination Definition of thixotropic flow behaviour: - Decrease of viscosity as a function of time upon shearing, - 100% recovery (= regaining the original structures) as a function of time without shearing. Determination (1) Time Curves - Base-line of intact structure at low shear rate (e.g. CR mode: 1 1/s) or in oscillation (e.g. CD mode: 1% deformation) - Dis-aggregation at constant shear rate (e.g. CR mode: 100 1/s) - Re-aggregating time at low shear rate (e.g. CR mode: 1 1/s) or in oscillation (e.g. CD mode: 1% deformation) (2) Flow Curves - Ramp up, (peak hold,) ramp down at constant temperature. - The hysteresis area in this loop is a measure for the thixotropy.

Thixotropy: Time curve Base-line, dis-aggregation, re-aggregating time

Thixotropy: Flow curve (thixotropy loop) Input: shear rate  - ramp up - (peak hold) - ramp down Measurement: shear stress  Result: viscosity  = f(shear rate , time t) Evaluation: Determination of thixotropic loop area . 50 100 150 200 250 300 350 400 450 500 Shear Rate Á [1/s] Shear Stress ‚ [Pa] Thixotropic loop area .

Viscoelasticity: Structural reasons Entanglement in macromolecules Structure/network of an emulsion

How to model viscoelasticity? Viscous flow Elastic deformation Spring Dash pot .    Voigt/Kelvin- Model   G* Maxwell- Model Burgers-Model

Testing methods for viscoelasticity Method Input Information Shear stress ramp Increasing shear stress Yieldpoint Creep test Const. shear stress Deformation Time curve Const. frequency and Monitoring of const. amplitude chemical reaction Amplitude sweep Stepwise increasing Network stability amplitude Frequency sweep Stepwise increasing Time frequency dependence Temperature curve const. frequency and Temperature const. amplitude dependence

Signals applied by a rheometer . . . (Stepped) Ramp (, ) Jump (, ) (Co-)Sinus (, ) Rotational Testing Creep & Recovery Oscillatory testing

Creep & recovery testing . Shear rate g at low stress Zero shear viscosity h0 Equilibrium compliance Je0 Ratio of viscous and elastic properties Relaxation time l0 Elastic Modulus G0 Mostly elastic sample

Oscillatory testing: Principle =0 (change of direction) =0 (change of direction)

Oscillatory testing: Complex Quantities Complex modulus G* = G’ + i G’’ (i2 = -1) Storage modulus G’ (elastic properties) Loss modulus G’’ (viscous/damping properties ) Loss angle d Loss factor tand = G’’/G’ Complex Viscosity h*= G* / i w Angular frequency w = 2p f G* G” d G’

Amplitude Sweep Material Stability Example: Delicate gel Material Stability Gel strength correlates with the gel's yield point The critical stress from the stress sweep is used as characteristic value. Remember the test is frequency dependent, therefore it is a relative result! LVR

Amplitude Sweep Example: Gels with different carbopol (hydro colloid) content

Frequency Sweep: Frequency and temperature dependence elastic paste flowing

Frequency Sweep Cross-over Material Characterization Paste - Entangled solution (circles) Gel - 3D network (triangles) Note: A Gel is not necessarily “stronger” than a Paste

Time Sweep: Gelation Parameters: f = 0.5 Hz g = 1 % T = 35°C Verlustanteile G" Cross-Over Parameters: f = 0.5 Hz g = 1 % T = 35°C

Curing Storage modulus G’ Loss modulus G”

Test for prediction of temperature stability Brummer et al Oscillation (g , w = const.) Cyclic temperature ramps (-10 ... 50°C, 20 min each) Indicators: G' und G": - G' and G" not affected sample is stable - Changes in G' and G" sample not stable

Test for prediction of temperature stability Brummer et al Example: Cosmetics G´´ [Pa] G´ [Pa] Temp. T [°C] w = konstant Time t [min] Cyclic testing  stable sample

Test for prediction of temperature stability Brummer et al Example: Cosmetics G´´ [Pa] G´ [Pa] Temp. T [°C] w = konstant Time t [min] Cyclic testing  sample not stable

Extensional Rheology HAAKE CaBER 1 - Capillary Breakup Extensional Rheometer - Designed for fluids Extensional behaviour ist relevant for - Processability - Strand formation / stringiness - Time to breakup - Relaxation time - Filling of bottels etc.

Extensional Rheology: HAAKE CaBER 1 - how it works Sample Apparent viscosity Laser micrometer Calculations Result: Apparent extensional viscosity vs. Hencky strain Measurement D=f(t)

Extensional Rheology: Bottle Filling Subtle changes in shampoo formulation caused difference in strand detachment during bottle filling Up-line characterization would prevent costly external washing of poorly-filled bottles

Further Reading A handbook of elementary rheology. H.A. Barnes, University of Wales, Aberystwyth, Dyfed, U.K., 2000 Non-Newtonian flow in the process industries - fundamentals and engineering applications. Chhabra RP, Richardson JF, Butterworth Heinemann, Oxford, 1999 A practical approach to rheology und rheometry G. Schramm, Thermo Haake GmbH, Karlsruhe, 1995 Engineering rheology - Oxford engineering science series vol 52. R.I. Tanner, Oxford University Press, Oxford, 2000

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