Reference Manual Rheology of Fluids Using the AR2000ex Advanced Rheometer SystemSection 1:Theory of Rheology Part 1:Introduction and General Theory of Rheology Part 2: Viscometric Theory Part 3: Viscoelastic Theory Section 1:Theory of Rheology Part 1:Introduction and General Theory of Rheology Definition Rheology is defined as the study of the flow and deformation of matter. The word was coined in the 1920s by Professor E.C. Bingham at Lafayette College in Indiana. Flow How liquids behave or respond to stress We refer to this as a Viscous response Deformation How solids behave or respond to stress We refer to this as an Elastic response Two Important Material Laws 1. Newtons Law of Viscosity (for Fluids) Equation: o = q * o = Shear Stress, a Force Parameter = Shear Rate, a Velocity Parameter q = Viscosity of the fluid. Viscosity is a material property. For simple, Newtonian fluids, the symbol for viscosity is often . 2. Hookes Law of Elasticity (for Solids) Equation: o = G * o = Shear Stress, a Force Parameter = Shear Strain, a Deformation Parameter G = Modulus of the solid, which is a material property. In tensor form,o = q In tensor form,o = G Constitutive Relations ModulusStrainStress=ity Visrate ShearStresscos =Strain, = x(t) y0 ModulusG = t t= F A y0 x(t) u V y x A z Shear deformation . =A At Shear Deformation Viscosityq = t . Rheometric Testing Two general classifications of testing Viscometric:various test methods in which the material is sheared continuously in order to observe the response of a fully-developed flow field to changes in shear rate, shear stress, temperature or time specialized Transient testing a subset of viscometric testing in which stress or shear rate is applied to the sample and no limitation is applied to the resultant deformation. However, in transient testing we are primarily concerned with the time dependent response to very basic changes in stress or strain Creep/Creep Recovery Stress Relaxation Oscillatory or Dynamic Third specialized mode Transient:a subset of viscometric testing in which stress or shear rate is applied to the sample and no limitation is applied to the resultant deformation. However, in transient testing we are primarily concerned Creep/Creep Recovery Stress Relaxation Rheometric Testing (contd) Two general classifications of testing Oscillatory or Dynamic Testing modes This range of testing capability is what separates our type of rheometers from simple viscometers and is the primary reason for their existence Dynamic testing is more of a perturbation technique in which typically low levels of stress/strain are applied using a frequency-based control of rate.This technique allows the isolation of the elastic and viscous components of the complete viscoelastic response of the fluid to be evaluated. This technique is more sensitive to wide ranges in morphology and molecular-level differences in materials than any other single analytical technique Section 1:Theory of Rheology Part 2: Viscometric Theory Steady Simple Shear Flow (Fluid Flow) Top Plate Velocity = V0; Area = A; Force = F Bottom Plate Velocity = 0 x y H vx = (y/H)*V0 = dvx/dy = V0/H . o = F/A q = o/ . Shear Rate, sec-1 Shear Stress, Pascals Viscosity, Pa-sec This is obtained from solving the equation of motion using the boundary conditions at y = 0 (vx=0) and y = H (vx=V0). See next page. These are the fundamental flow parameters. Shear rate is always a change in velocity with respect to distance. Viscosity: Units The Units of Viscosity are . . . . .SI (International) unit is the Pascal.second (Pa.s) c-g-s unit is the Poise Conversion:10 Poise =1 Pa.s1 cP (centipoise) = 1 m Pa.s (milli-pascal-second) Typical Viscosity Values (Pa-s) Asphalt Binder.. Polymer Melt Molasses .. Liquid Honey Glycerol Olive Oil Water Air . 100,000 1,000 100 10 1 0.01 0.001 0.00001 The range of viscosity values one encounters suggests the use of logarithmic scales when plotting data. Viscous Behavior Ideal Liquid 1687: Isaac Newton addresses liquids and steady shearing flow in Principia The resistance which arises from the lack of slipperiness of the parts of theliquid, other things being equal, is proportional to the velocity with which theparts of the liquid are separated from one another. Dashpot: Mechanical Analogue of Viscous Response Newtons Law ofViscosity Newtons Law:stress = (coefficient of viscosity) shear-rate q o=Viscosity is affected by the following: Material Molecular Weight Molecular Weight Distribution, Branching Temperature Concentration Particle Size Particle Size Distribution Newtonian and Non-Newtonian Fluids Newtonian Fluids- constant proportionality between shear stress and shear-rate Non-Newtonian Fluids - Viscosity is time or shear rate dependentTime :At constant shear-rate, if viscosity Decreases with time Thixotropy Increases with time - Rheopexy Shear-rate : Shear - thinning Shear - thickening General Flow Classifications TimeViscosity (Pa-sec)RheopecticStableThixotropicShear Rate (1/sec)Shear Stress (Pa)DilatantNewtonianPseudoplasticSome dough-like materials exhibit rheopecticity when processed. Coatings and inks can display thixotropy when sheared due to structure breakdown. An example of dilatancy is wet sand on the beach. Most materials used industrially are pseudoplastic. o, Pa ,1/s or o, Pa q, Pa.s ,1/s Ideal Yield Stress (Bingham Yield) Characteristic Diagrams for Newtonian Fluids Characteristic Diagrams for Shear Thinning Fluids ,1/s 105 103 101 10-1 10-610-410-2100102104 o, PaIdeal Yield Stress (Bingham plastic) 105 103 101 10-1 10-6 10-410-2 100 102 104 q, Pa.s ,1/s 105 103 101 10-1 10-1 10-0 10-1102 103 q, Pa.s o, Pa Characteristic Diagrams for Shear Thickening Log q, Pa.s Log ,1/s Non-Newtonian, Time Dependent Fluids time Viscosity Thixotropic Rheopectic Shear Rate = Constant Typical Flow Curve, showing Different Flow RegimesLog Shear Rate (1/s) Log Viscosity (Pa-sec) Newtonian Region: Viscosity is essentially constant. This viscosity is often called the Zero Shear Viscosity, q0, or the Newtonian Viscosity. For undiluted polymers, q0 = K Mw3.4 when Mw > Me, the molecular weight between entanglements. Transition Region to non-Newtonian behavior Power Law Region:o m or, equivalently, q m-1

m is usually 0.15 to 0.6 Second Newtonian Region: Viscosity is essentially constant again. Structure Breakdown under Shear: Increasing Time at Rest Correlations of Shear Rates with Various Processes1 1) Sedimentation 2) Leveling, Sagging 3) Draining under gravity 4) Chewing and swallowing 5) Dip coating 6) Mixing and stirring 7) Pipe flow 8) Spraying and brushing 9) Rubbing 10) Milling pigments in fluid base 11) High Speed coating 23 6 5 89 1.001.00E-51.00E-41.00E-30.01000.100 shear rate (1/s) 10.00100.001000.001.00E41.00E5 log q 1.00E6 117 4 10 Section 1:Theory of Rheology Part 3: Viscoelastic Theory Viscous and Elastic Responses Most materials behave such that they have a combination ofviscous and elastic responses under stress or deformation. Materials behave in the linear manner, as described by Hooke and Newton, only in rare cases where the morphology is simple, or when the deformation or shear stress is on a small scale. Most Materials Ideal SolidIdeal Liquid Viscoelasticity:the Deborah Number Deborah was an Old Testament Prophetess who said : "The Mountains Flow before the Lord" She was the first rheologist -- Everything Flows if you wait long enough! Deborah Number, De - The ratio of a characteristic relaxation time of a material (t) to a characteristic time of the relevant deformation process ( T ). De = t/T Deborah Number Hookean elastic solid - t is infinite Newtonian Viscous Liquid - t is zero Polymer melts processing - t may be a few seconds High DeSolid-like behavior Low De Liquid-like behavior IMPLICATION: Material can appear solid-like because 1) it has a very long characteristic relaxation time or 2) the relevant deformation process is very fastElastic Behavior Ideal Solid 1678: Robert Hooke develop his True Theory of Elasticity The power of any spring is in the same proportion with the tension thereof. o = G Hookes Law ofElasticity Hookes Law: Stress = Modulus Strain Spring: Mechanical Analogue of Elastic Response Simple Shear Deformation (Solids and/or Oscillation) Top Plate Displacement = X0; Area = A; Force = F Bottom Plate Displacement = 0 x y H x = (y/H)*X0 = dxx/dy = X0/H o = F/A G = o/ Shear Strain, unitless Shear Stress, Pascals Modulus, Pa These are the fundamental deformation parameters. Shear strain is always a change in displacement with respect to distance. Remember Liquid Flow Behavior? 0123 F Move piston through dashpot Force (F) depends on velocity (v) and vice versa. Force is independent of position (x) (within reason) For simple fluids, F v1; F = F(x) x Heres Viscoelasticity (Solid+Liquid) F F Viscoelastic Materials: Force depends on both Deformation and Rate of Deformation and vice versa. The material exhibits behavior lying in between classic fluid and classic solid behavior Maxwell Model Kelvin (Voigt) Model Time-Dependent Viscoelastic Behavior T is short [< 1s]T is long [24 hours] Deborah Number [De] =t / T Viscoelastic Modeling for Transient Testing F Maxwell Model The Maxwell model is useful for illustrating stress relaxation behavior. For 1 element: G(t) = Giexp(-t/ti) Where G(t) is the modulus at time, t, following a strain imposition at t = 0. Gi is the spring constant; q = viscosity ti is the Relaxation Time = qi/Gi Log time (sec)Log G(t) (Pa)For N elements in Parallel: G(t) = EGiexp(-t/ti) i = 1 N 1-element stress relaxation curve Integral Form: G(t) = Ge + }Hexp(-t/ti)dln(t) - Storage and Loss for a Viscoelastic Material TENNIS BALL STORAGE (G) LOSS(G) SUPER BALL STORAGE (G) LOSS (G) Viscoelastic Test Methods:Oscillation Dynamic or Oscillatory Testing Oscillatory Stress or Strain sweeps to determine Linear Viscoelastic Region Yield Stress Frequency sweeps to evaluate rate response characteristics Time sweeps to determine kinetic events, such as reactions, curing, stability, or structure buildup/breakdown Temperature ramps or sweeps to determine mechanical properties and transition temperatures SINUSOIDAL DEFORMATION FOR DYNAMIC TESTING TimeResponse, Strain90o oo STRAIN ELASTIC RESPONSE VISCOUS RESPONSE VISCOELASTIC RESPONSE Polymers and many suspensions, emulsions and solutions are viscoelastic materials.Both components viscosity and elasticity are important. Viscoelastic Parameters The Elastic (Storage) Modulus:Measure of elasticity of material.The ability of the material to store energy. G' = (stress*/strain)coso =stress/strain G" = (stress*/strain)sino = stress/strain The Viscous (loss) Modulus: The ability of the material to dissipate energy.Energy lost as heat. The Complex Modulus:Measure of materials overall resistance to deformation. G* = Stress*/Strain G* = G + iGTan o = G"/G' Tan Delta: Measure of material damping - such as vibration or sound damping. Gel Structure in Suspensions Unbroken GelStructure Increasing Oscillation Amplitude/MagnitudeIn Shear Deformation Structure begins to Breakdown Broken Structure Dynamic Stress (Amplitude) Sweeps for Yield Determination (Structure Break-Down) 100.0 0.1000 1.000 10.00osc. stress (Pa)1000010.00100.01000G' (Pa)Ink Samples: Oscillation Stress Sweeps @ 6.28 rad/sDynamic Rheological Parameters ParameterShearElongationUnits Strain = 0 sin(t) = 0 sin(t)--- Stresso = o0sin(t + )t = t0sin(t + )Pa Storage Modulus (Elasticity) G = (o0/0)cosE = (t0/0)cosPa Loss Modulus (Viscous Nature) G = (o0/0)sinE = (t0/0)sinPa Tan G/GE/E--- Complex ModulusG* = (G2+G2)0.5 E* = (E2+E2)0.5Pa Complex Viscosityq* = G*/qE* = E*/Pa-sec Cox-Merz Rule for Linear Polymers: q*() = q() @ = .. Section 2:Rheometer Basics Rheometer construction Geometry selectionBasic operational principles The AR2000ex Rheometer Applied Strain orRotation Measured Torque (Stress) DirectDriveMotorTransducer Separatemotor & transducer Combined motor & transducer DisplacementSensor Measures Strain (angular rotation) and rate of rotation(Shear Rate) Non-Contact Drag Cup Motor Applied Torque (Stress) Static Plate AR2000ex Controlled Stress/Strain CMT Combined Motor-Transducer Five Important Rheometer Specifications Torque range This specification is the most practically important specification for daily use of the rheometer.You cannot expect to get accurate of precise data below the minimum torque specification of the rheometer. Angular Resolution This specification may be important when working with very stiff solid samples, as it relates to how small of a strain can be properly measured or controlled by the rheometer Angular Velocity Range For fluids, the minimum angular velocity relates to the lowest controllable shear rate, when operating in rate controlled mode. When operating in stress controlled mode, the rate is measured passively and can achieve rates on the order of 1e-8 1/s Frequency Range Specified frequency range of stress rheometers (like the AR2000ex) may not be useable on all samples.The inertia effect of the motor on the resolution of phase angle will be compromised at higher frequencies, especially at low viscosities. The AR2000ex allows you to observer the raw phase angle if this number exceeds 160 degrees, the inertial effect is probably not correctable anymore and the data should be suspect. Normal Force The first normal stress difference (N1) can be very important information in the analysis of viscoelastic fluids --- however, very few applied fluids labs use this function and it tends to be more academic/research in nature. The N1 is directly related to extensional viscosity behavior. Specification TA AR1500ex TA AR2000ex TA AR G2 Min. Torque [N.m]: Oscillation Flow 0.1 0.1 0.03 0.1 0.003 0.01 Max. Torque [N.m]150200200 Min. Frequency [Hz]1.00e-061.00e-071.00e-07 Max. Frequency [Hz]100100100 Min. Ang. Velocity CS [rad/s]000 Min. Ang. Velocity CR [rad/s]1.00e-071.00e-081.00e-09 Max. Ang. Velocity CS [rad/s]300300300 Displacement resolution [Rad]0.040.040.025 Air bearing typePorous carbonPorous carbonMagnetic Normal force transducerN/aWheatstone bridge in base Wheatstone bridge in base Min. Normal Force [N]N/a0.0050.005 Max. Normal Force [N]N/a5050 AR-Series Instrument Specifications What does a Rheometer do? Rheometer an instrument that measures both viscosity and viscoelasticity of fluids, semi-solids and solids It provides information about the materials: Viscosity function of shear rate or stress, time & temperature dependence Viscoelastic properties (G, G, tan o) with respect to time, temperature, frequency & stress/strain Transient response (relaxation modulus, creep compliance, creep recovery) How do Rheometers work? From the definition of rheology, the science of flow and deformation of matter or the study of stress (Force / Area) deformation(Strain orStrain rate) relationships. Fundamentally a rotational rheometer will apply or measure the following: 1.Torque (Angular Force) 2.Angular Displacement3.Angular Velocity 1. Torque Torque M is a measure of how much a force F acting on an object causes that object to rotate. The object rotates about an axis, called the pivot point, labeled 'O'. The distance from the pivot point to the point where the force acts is called the moment arm, and is denoted by 'r'.M = r x F sin(u) = r x F, for u = 90 r F Torque Shear Stress In a rheometer, the stress is calculated from the torque. The formula for stress is: o = Ko x M Whereo = Stress (Pa or Dyne/cm2) Ko = Stress ConstantM = torque in mNm or gm.cm The stress constant, Ko, is a geometry dependent factor. 2. Angular Displacement Angular displacement is the angle (distance) that a rotating body goes through s = r u Where: s is the arc-length r is the radius of the circle is the central angle (inradian) swept out by the arc Multiply the angle in degrees by t/180 to determine radians Multiply the angle in radian by 180/t to determine degrees r s u Angular DisplacementShear Strain In a SMT Rheometer the angular displacement is directly applied by a motor. The formula for strain is: = K x u(% = *100) where = Strain K = Strain Constant u = Angular motor deflection (radians) The strain constant, K, is a geometry dependent factor. 3. Angular Velocity Strain Rate =Change of strain per unit time of measurement F s2 s1 s = r u Angular Velocity Shear Rate In a SMT rheometer, the angular speed is directly controlled by the motor). The formula for shear rate is: where= Shear rate K = Strain Constant O = Motor angular velocity in rad/sec. The strain constant, K, is a geometry dependent factor. O = x KModes of Deformation Bending Compressive Rectangular Torsion Tensile DMA Rheology Torsional Shear Strain, = x(t) y0 ModulusG = t t= F A y0 x(t) u V y x A z Shear deformation . =A At Shear Deformation Viscosityq = t . Equation for Modulus Raw rheometerSpecifications Geometric ShapeConstants Constitutive Equation GK xK x M= =ouoRheological Parameter In Spec Describe Correctly Equation for Viscosity qoo= =OK xK x MRaw rheometerSpecifications Geometric ShapeConstants Constitutive Equation Rheological Parameter In Spec Describe Correctly Ranges of Rheometers and DMAs Loss Modulus (E" or G") Storage Modulus (E' or G') Temperature Range of AR/ARES Rheometer Range of DMA/RSA Some Viscoelastic LiquidCharacterization Possible with Shear SandwichTemperature ControlSystems:Fluids AR:40 C 200 C Peltier Plate Peltier Concentric cylinder AR:-20 C 150 C Rheological Testing Basic Rheological Approach 1.Apply Stress/Torque/Force and measure resulting Deformation and/or Deformation Rate(Controlled Force, Controlled Stress) 2. Control Deformation and/or Deformation Rate and measure Force needed (Controlled Strain or Shear Rate) DisplacementSensor Measured Strain or RotationNon-Contact Drag Cup Motor Applied Torque (Stress) Stationary Plate Typical Geometries for Rheological Testing in Shear Parallel Plate Cone and Plate Concentric Cylinders Torsion Rectangular Very Low to MediumViscosity Low to HighViscosity Low Viscosity to Soft Solids Very Soft to Very Rigid Solids The equation of motion and other relationships have been used to determine the appropriate equations to convert machine parameters (torque, angular velocity, and angular displacement) to rheological parameters. Concentric Cylinder Features: Good for testing suspensions withlimited stability Good for testing samples with large particle size Suitable for low viscosity fluids Shear Rate and Shear Stress are reasonably constant with small gaps. Strain Constant: K =2 1-(R1/R2)2 Stress Constant: Ko = 1 2tL(R1)2 R1 = Inner Radius R2 = Outer Radius L = Cylinder immersed height Cone and Plate Strain Constant: K =1 | Stress Constant: Ko = 3 2tR3 Features: Shear Stress and Shear Rate are constant throughout the gap. Suitable for low to medium viscosity fluids Can be used to determine Normal Stress Differences (see next page.) R = Plate Radius | = Cone Angle (radians) Note: If there are particles in the fluid, they should be