Τ Thomas G. Mason Department of Chemistry and Biochemistry Department of Physics and Astronomy California NanoSystems Institute Τ γ 26 by Thomas G. Mason All rights reserved. γ (t) τ (t) γ τ Δt 2π t γ Rheology Study of deformation and flow of materials Elastic Solids Viscous Liquids And everything in between Soft Materials Emulsions, Polymers, Glassy Materials Thermally agitated colloidal structures (1 nm - 1 µm) Describe mechanical constitutive relationships: material response Focus on isotropic disordered materials (Math is more complicated for partially or fully ordered materials) 1
Rheology - Outline Basic Definitions: Stress and Strain Common Shear Flows and Simple Examples Equations of Continuity and Momentum Linear Shear Viscoelastic Rheology Introduction to Non-linear Rheology Macroscopic Mechanical Shear Rheometry Example of a Soft Glassy Material: Concentrated Emulsions Affine and Non-Affine Shear Deformations Affine All points in the material deform uniformly Homogeneous flow Non-Affine Different points in the material deform differently Apparent macroscopic strain is not the local strain everywhere Local non-affine motion of constituents in soft materials can have an important impact on their rheology For simplicity, we focus on affine rheology 2
Incompressible Soft Materials Isothermal Compressibility β = 1 V V p T Important for gases, but much smaller for condensed matter Mass Conservation of Fluid Flow: Equation of Continuity ρ t = ( r ρv r ρ is density ) v is velocity Constant density implies: ( r r v ) = (Incompressible) Good approximation for most complex fluids, but not for foams Example of Shear Flow Shear Deformations: Force is applied along surface, not normal to it Unidirectional Simple Shear Flow parallel plates (plane Couette) fixed gap separation: d Shear Strain: γ = Δx/d v Δx(t) Shear Strain Rate: γ = v(t)/d = Δ x (t)/d z y x Strain Tensor γ γ = γ Rate of Strain Tensor γ γ = γ = [ r v r + ( r v r ) ] velocity gradient tensor Convention: x is along shear direction (1-direction) y is normal to shearing surface (2-direction) z is tangential to shearing surface (3-direction) 3
Example of Extensional Flow y Planar Extensional (Elongational) Flow Pure Shear Flow roller 4 roll mill z x extension rate Rate of Strain Tensor ε γ = ε Applying Shear Stresses and Pressures Area of Face: A y x z τ xy = F xy /A, Shear Stress Tensor τ xy τ xz τ = τ yx τ yz τ zx τ zy F xy F xy F xx Simple isotropic pressure: p p = p = pδ p Total Stress Tensor p τ xy τ xz σ = τ pδ = τ yx p τ yz τ zx τ zy p Total description including shear-free flows is more complex (we ve neglected normal stresses τ xx, τ yy, τ zz for now) 4
Momentum Conservation: Navier-Stokes Equation Newton s Law for Fluid Elements Form for Simple Shear Flow t ρr v = [ r ρv r v r ]+[ r σ ]+ ρg r Assume viscous dissipation: σ = pδ +η γ η is the viscosity neglect dilational viscosity A Very Simple Rheological Equation: Navier-Stokes Equation ρ D r v = r p +η r 2 r v + ρg r Dt where D Dt = + v r r t is the convective or substantial derivative operator (derivative operator in reference frame of moving liquid) We ll simplify to scalar equations mostly from here on out Simple Viscous Liquids and Elastic Solids Stress: τ Viscous Liquid. Strain Rate: γ(t) Viscosity resists motion proportional to rate: Newtonian Liquid τ v = η γ Stress: τ Elastic Solid Strain: γ(t) Elasticity resists motion proportional to strain: τ e = Gγ Hookean Solid Both η and G are intensive thermodynamic properties 5
Thermodynamic Definition: Shear Modulus G Helmholtz Free Energy F = F + VG γ 2 2 + O[γ 4 ] Intensive form of Hooke s Law: differentiate once w.r.t. strain τ = 1 V F γ γ= = Gγ γ = is used to eliminate any nonlinear dependence Elastic Shear Modulus: differentiate again G = 1 2 F V γ 2 γ= = φ V d 2 F γ 2 γ= related to curvature of energy well Linear Viscoelastic Response of Soft Materials Soft materials: colloidal components dispersed in a viscous liquid Examples: polymer chains, clay platelets, emulsion droplets Strain response of material is dependent on strength of interactions between components (e.g. polymer chains, clay platelets, droplets) and relaxation time scales of microstructure Stress: τ Strain: γ(t) Response for small strains is neither perfectly viscous nor elastic Mechanical response is called viscoelastic 6
Linear Shear Viscoelastic Rheology Study of small (near-equilibrium) deformation response of materials Key Idea There is only one real function of one real variable (e.g. time t) that describes the relaxation of shear stress Stress Relaxation Modulus: G r (t) There are many equivalent ways of representing the information contained in this one real function G r (t) Preference for a particular representation is based on expt. method This can be very confusing to people new to the field Stress Response to a Step Strain Step the strain: γ(t) = γ u(t) Measure Shear Stress: τ(t) Strain: γ(t) = γ <<1 Define a function, the Stress Relaxation Modulus G r (t): G r (t) τ(t) γ This one function contains all of the information about the equilibrium (linear) stress-strain response of the soft material Simple viscous liquid: Simple elastic solid: G r (t) = η δ(t) G r (t) = G u(t) 7
Strain Response to a Step Stress Step the stress: τ = τ u(t) Measure Strain: γ(t) Define an Equivalent Function, the Creep Compliance, J(t): J(t) = γ(t)/τ In principle, if measured over a large enough dynamic range in J and t, we can perform mathematical manipulations to determine J(t) from G r (t). This is cumbersome and is usually done numerically Simple viscous liquid: Simple elastic solid: J(t) = t/η J(t) = (1/G) u(t) Modeling Viscoelastic Materials Spring and Dashpot Mechanical Models Are Useful For Schematically Representing Relaxation Modes in Soft Materials Hookean Spring Newtonian Dashpot τ(t) = -G γ(t) τ(t) = η γ (t) G η Maxwell combined the ideas behind these equations: τ + η G τ = η γ This is one of the first viscoelastic equations 8
General Linear Viscoelastic Model Use as many springs and dashpots in series or in parallel to model the response function of the material A solution of the set of coupled differential equations and crossover from discrete to continuum notation yields: t τ(t) = G r (t t ) γ ( t )d t The shear stress is related to the convolution integral of the stress relaxation modulus with the strain rate history You have to know the history of applied shear! Check: step strain -> delta function strain rate Equivalently: t τ(t) = M (t t )γ( t )d t Memory function: M (t t ) = G r (t t ) t Frequency Domain Representation: Complex Viscosity What is the Unilateral Fourier Transform of G r (t)? Answer: the complex frequency-dependent viscosity η *() = G r ( t )exp( i t )d t η *() = G r ( t )[cos( t ) isin( t )]d t η *() = [ G r ( t )cos( t )d t ] i[ G r ( t ) sin( t )d t ] in-phase w.r.t. strain rate Complex Viscosity: η () = G r ( t )cos( t )d t η () = G r ( t )sin( t )d t 9 out-of-phase w.r.t. strain rate η *() = η () i η () 9
Frequency Domain: Complex Shear Modulus Another equivalent -domain representation of G r (t) Complex Modulus: G*() = iη*() pick up from time derivative G *() = G ()+ i G () Real Part: G () Storage Modulus Imaginary Part: G () Loss Modulus Complex phasor notation is mathematically convenient But in reality, we measure real functions of real variables Forced Parallel Plate Oscillatory Viscometry γ(t) = γ sin(t) γ (t) = γ cos(t) Measure: Stress τ(t) Controlled Strain Oscillations γ (t) τ (t) γ τ Keep γ small enough that the stress response is also sinusoidal Measured Stress: Δt 2π τ(t) = τ () sin [t + δ()] phase lag: δ = Δt τ(t) = τ () cosδ() γ sint + τ () sinδ() γ cost γ In-phase w.r.t. strain G () = τ () cosδ() γ γ 9 out-of-phase w.r.t. strain G () = τ () sinδ() γ t 1
Forced Parallel Plate Oscillatory Viscometry γ(t) = γ sin(t) γ (t) = γ cos(t) Controlled Strain Oscillations γ (t) τ (t) γ τ Δt 2π t Measure: Stress τ(t) phase lag: Δt t t τ(t) = G r (t t ) γ ( t )d t = G r (t t ) γ cos( t )d t τ(t) = γ G r (s)cos[(t s)]ds s = t t τ(t) = [ G r (s)cos(s)ds] γ cos(t) [ G r (s)sin(s)ds] γ sin(t) In-phase w.r.t strain: G () = η () 9 out-of-phase w.r.t strain: G () = η () Storage & Loss Moduli are Not Independent Functions The complex function G* consists of two real functions G and G But both of these are derived from only one real function G r (t) G *() = G ()+ i G () G *() = i G r ( t )exp( i t )d t So, G and G express the same information about stress relaxation If either G and G are known over a large enough frequency range the other function can be determined by the Kramers-Kronig Relations G '() = 2 d ˆ ˆ G ''( ˆ ) π 2 ˆ 2 G ''() η' = 2 G '( ˆ ) d ˆ π ˆ 2 2 11
Viscoelastic Soft Materials with Low Freq. Relaxation Maxwell Model: combine high freq elasticity and low freq viscosity Spring and dashpot in series (simple 1D model) J(t) G p η e.g. ideal polymer entanglement solution G*() G p Equivalent 1/G p η /G p t = t c crossover time crossover frequency 2πG p /η c = 2π/t c J(t) = (1/G p ) + t/η G () G () G*() = G p {(t c ) 2 /[1+(t c ) 2 ]+ it c /[1+(t c ) 2 ]} Linear Viscoelastic Soft Material with Low Freq. Elasticity Voigt Model: combine low freq elasticity and high freq viscosity Spring and dashpot in parallel (simple 1D model) G p J(t) 1/G p G*() η Equivalent G e η /G p t G e /η J(t) = (1/G p ) [1-exp(-G p t/η )] G*() = G p + iη 12
Brief Introduction to Non-Linear Rheology Sinusoid in does not give a sinusoid out: Harmonics are seen τ τ y Yield stress: τ y Yield strain: γ y γ y γ Most Soft Glassy Materials: γ y <.2 typically Shear Thinning: viscosity decreases as strain rate increases Shear Thickening: viscosity increases as strain rate increases Strain Hardening: stress increases greater than linear strain Measuring Viscoelastic Response: Mechanical Rheometry - Controlled Strain Rheometry - Apply a Strain (Motor) and Measure a Torque (Transducer) Use Geometry to Convert Torque to Stress Cone and Plate Τ Double Wall Couette Τ γ vapor trap prevents evaporation of continuous phase These are among preferred geometries: the strain field is homogeneous throughout the gap γ 13
Controlled Stress vs Controlled Strain Rheometry Controlled Strain: Good for G*() and G r (t), steady shear Best for very weak liquid-like materials Motors are excellent Torque transducers are very sensitive Can be damaged more easily Controlled Stress: Good for J(t), OK for G*(), steady shear Bad for weak materials Drag cup motors can t produce low stresses Feedback strain control is problematic Assumes a certain type of material response Some newer rheometers have both stress and strain control Get the best of both worlds in principle An Example: Viscoelasticity of Soft Glassy Emulsions T.G. Mason & D.A. Weitz PRL 75 251 (1995) Linear regime at low strain: G and G do not depend on γ Yield strain is evident: 1-2 < τ y < 1-1 14
An Example: Viscoelasticity of Soft Glassy Emulsions T.G. Mason & D.A. Weitz PRL 75 251 (1995) Plateau Storage Modulus over many decades in frequency: G p Viscous Loss Modulus has a shallow minimum: G min An Example: Viscoelasticity of Soft Glassy Emulsions G P /(σ / a),g ' ' min /(σ / a) 1 1-1 1-2 1-3 1-4 1-5 1-6 T.G. Mason & D.A. Weitz PRL 75 251 (1995) solid symbols: G p flows elastic solid G p ~ (σ/a) (φ eff - φ MRJ ) Disorderd droplets begin to pack & deform φ MRJ = φ RCP =.64.5.6.7.8.9 1 φ eff 15
A Model and Simulation: Explains Emulsion Elasticity M.-D. Lacasse, G. S. Grest, D. Levine, T. G. Mason, and D. A. Weitz, PRL 76, 3448 (1996). Jamming of disordered objects that interact with a repulsive potential Surface Evolver: -> Effective spring constant Non-Affine Local Droplet Motion in Disordered Glassy Soft Materials Response to Applied Shear is Complex! Summary: Rheology is a convenient macroscopic measurement of stress-strain response of a micro or nano structured soft material Stress-strain relationships are related to the structure and interactions of constituents in the soft material: a thermodynamic property Many equivalent representations exist for linear viscoelasticity We have only begun to scratch the surface of a very deep field Describing non-newtonian fluid dynamics mathematically is still an area of very active research See reading list for references: Bird, Ferry, Russel, Tabor, Larson, 16