Rheology of Soft Materials

Thomas G. Mason

Department of Chemistry and Biochemistry Department of Physics and Astronomy California NanoSystems Institute Τ ω Τ ω

γ τ γ γ (t) 0 0 ω τ (t) 2π ω γ ω t Δt © 2006 by Thomas G. Mason All rights reserved.

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)

Rheology of Soft Materials © 2006 Thomas G. Mason

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

Rheology of Soft Materials © 2006 Thomas G. Mason

Affine and Non-Affine Shear Deformations

Affine Non-Affine

All points in the material Different points in the material deform uniformly deform differently

“Homogeneous flow” 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

Rheology of Soft Materials © 2006 Thomas G. Mason

2 Incompressible Soft Materials Isothermal Compressibility 1 ∂V β = − V ∂p T

Important for gases, but much smaller for condensed matter € Mass Conservation of Fluid Flow: Equation of Continuity ∂ρ r r ρ is density = −(∇ ⋅ ρv ) v is velocity ∂ t r Constant density implies: r (Incompressible) ( ∇ ⋅ v ) = 0 € Good approximation for most complex fluids, but not for foams € Rheology of Soft Materials © 2006 Thomas G. Mason

Example of Shear Flow Shear Deformations: Force is applied along surface, not normal to it Unidirectional Simple Shear Flow Strain Tensor parallel plates (plane Couette) 0 γ 0 v y 0 0 γ = γ z x 0 0 0 fixed gap separation: d Δx(t) Rate of Strain Tensor Shear Strain: 0 ˙ 0 γ = Δx/d € γ r r † γ˙ = γ˙ 0 0 = [∇ vr + (∇ vr ) ] Shear Strain Rate: 0 0 0 γ˙ = v(t)/d = Δx˙ (t)/d 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) Rheology of Soft Materials © 2006 Thomas G. Mason

3 Example of Extensional Flow Planar Extensional (Elongational) Flow “Pure Shear Flow” 4 roll mill y roller

z x

extension rate

Rate of Strain Tensor ε ˙ 0 0 ˙ 0 ˙ 0 γ = −ε 0 0 0

Rheology of Soft Materials © 2006 Thomas G. Mason €

Applying Shear Stresses and Pressures

Fxy Area of Face: A Simple isotropic pressure: y − p 0 0 x p = 0 −p 0 = −pδ z Fxx F 0 0 −p τxy = Fxy/A, … xy Shear Stress Tensor Total Stress Tensor

0 τ xy τ xz € −p τ xy τ xz τ = τ yx 0 τ yz σ = τ − pδ = τ yx −p τ yz τ zx τ zy 0 τ zx τ zy −p

Total description including shear-free flows is more complex (we’ve neglected normal stresses , , for now) € € τxx τyy τzz Rheology of Soft Materials © 2006 Thomas G. Mason

4 Momentum Conservation: Navier-Stokes Equation Newton’s Law for Fluid Elements Form for Simple Shear Flow ∂ r r ρvr = −[∇ ⋅ ρvr vr ]+[∇ ⋅σ ]+ ρgr ∂ t

Assume viscous dissipation: σ = −pδ +ηγ˙ η is the viscosity neglect dilational viscosity € A Very Simple Rheological Equation: Navier-Stokes Equation € D r r ρ vr = −∇ p +η∇ 2vr + ρgr Dt r r where D = ∂ +v ⋅∇ is the convective or substantial derivative operator Dt ∂t (derivative operator in reference frame of moving liquid) € €We’ll simplify to scalar equations mostly from here on out Rheology of Soft Materials © 2006 Thomas G. Mason

Simple Viscous Liquids and Elastic Solids

Viscous Liquid . Stress: τ Strain Rate: γ(t)

Viscosity resists motion proportional to rate: τ v = −ηγ˙ “Newtonian Liquid”

Elastic Solid Stress: τ € Strain: γ(t)

Elasticity resists motion proportional to strain: τ e = −Gγ “Hookean Solid” Both η and G are intensive thermodynamic properties Rheology of Soft Materials € © 2006 Thomas G. Mason

5 Thermodynamic Definition: Shear Modulus G

Helmholtz Free Energy γ 2 F = F + VG + O[γ 4 ] 0 2 Intensive form of Hooke’s Law: differentiate once w.r.t. strain

1 ∂F € τ = − = −Gγ V ∂γ γ=0 γ = 0 is used to eliminate any nonlinear dependence

Elastic Shear Modulus: differentiate again € 1 ∂ 2F φ ∂ 2F G = = related to curvature V 2 V 2 of energy well ∂γ γ=0 d ∂γ γ=0

Rheology of Soft Materials © 2006 Thomas G. Mason

€ €

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”

Rheology of Soft Materials © 2006 Thomas G. Mason

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: Gr(t)

There are many equivalent ways of representing the information

contained in this one real function Gr(t)

Preference for a particular representation is based on expt. method This can be very confusing to people new to the field

Rheology of Soft Materials © 2006 Thomas G. Mason

Stress Response to a Step Strain

Step the strain: γ(t) = γ0 u(t) Measure Shear Stress: τ(t) Strain:

γ(t) = γ0 <<1

Define a function, the Stress Relaxation Modulus Gr(t): τ(t) Gr (t) ≡ γ0

This one function contains all of the information about the equilibrium (linear) stress-strain response of the soft material €

Simple viscous liquid: Gr(t) = η δ(t) Simple elastic solid: Gr(t) = G u(t)

Rheology of Soft Materials © 2006 Thomas G. Mason

7 Strain Response to a Step Stress

Step the stress: τ = τ0 u(t) Measure Strain: γ(t)

Define an Equivalent Function, the Creep Compliance, J(t):

J(t) = γ(t)/τ0 In principle, if measured over a large enough dynamic range in J and t, we can perform mathematical manipulations to

determine J(t) from Gr(t). This is cumbersome and is usually done numerically

Simple viscous liquid: J(t) = t/η Simple elastic solid: J(t) = (1/G) u(t)

Rheology of Soft Materials © 2006 Thomas G. Mason

Modeling Viscoelastic Materials

Spring and Dashpot Mechanical Models Are Useful For Schematically Representing Relaxation Modes in Soft Materials

G Hookean Spring τ(t) = -G γ(t)

Newtonian Dashpot τ(t) = −η γ˙ (t) η

Maxwell combined€ the ideas behind these equations: η τ + τ˙ = −ηγ˙ G

This is one of the first viscoelastic equations

Rheology of Soft Materials € © 2006 Thomas G. Mason

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) t G (t t )˙ (t )dt Check: step strain -> τ = − ∫−∞ r − ′ γ ′ ′ delta function strain rate 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! Equivalently: Memory function: t (t) M (t t ′) (t ′) dt ′ ∂Gr (t − t ′) τ = ∫−∞ − γ M (t − t ′) = ∂t ′

Rheology of Soft Materials © 2006 Thomas G. Mason € €

Frequency Domain Representation: Complex Viscosity

What is the Unilateral Fourier Transform of Gr(t)? Answer: the complex frequency-dependent viscosity ∞ η *(ω) = ∫0 Gr (t ′) exp(−iωt ′) dt ′ ∞ η *(ω) = ∫0 Gr (t ′)[cos( ωt ′) − isin(ωt ′ )]dt ′ ∞ ∞ η€* (ω) = [ ∫0 Gr (t ′) cos(ωt ′ )dt ′] − i[ ∫0 Gr (t ′) sin(ωt ′) dt ′ ] in-phase w.r.t. strain rate 90° out-of-phase w.r.t. strain rate € Complex Viscosity: η *(ω) = η ′( ω)− iη ′′ ( ω) € ∞ η ′( ω) = ∫0 Gr (t ′) cos(ωt ′ )dt ′ ∞ €η ′ ′ ( ω) = ∫0 Gr (t ′) sin(ωt ′) dt ′

Rheology of Soft Materials€ © 2006 Thomas G. Mason €

9 Frequency Domain: Complex Shear Modulus

Another equivalent ω-domain representation of Gr(t) Complex Modulus: G*(ω) = iωη*(ω)

pick up from time derivative

G *(ω) = G ′( ω)+ iG ′′ ( ω)

Real Part: G’(ω) “Storage Modulus” € Imaginary Part: G’’(ω) “Loss Modulus”

Complex “phasor notation” is mathematically convenient But in reality, we measure real functions of real variables

Rheology of Soft Materials © 2006 Thomas G. Mason

Forced Parallel Plate Oscillatory Viscometry Controlled Strain Oscillations γ (t) γ 0 τ 0 γ(t) = γ0 sin(ωt) τ (t) γ˙ (t) = γ˙ 0 cos(ωt) 2π ω t Δt

€ Measure: Stress τ(t) phase lag: δ = ωΔt € Keep γ0 small enough that the stress response is also sinusoidal

Measured Stress: τ(t) = τ 0(ω) sin [ωt + δ(ω)]

τ 0 (ω) τ 0 (ω) τ(t) = cosδ(ω) γ0 sinωt + sinδ(ω) γ0 cosωt γ0 γ0 € In-phase w.r.t. strain 90° out-of-phase w.r.t. strain

τ 0 (ω) τ 0 (ω) € G ′( ω) = cosδ(ω) G ′′ ( ω) = sinδ(ω) γ0 γ0 Rheology of Soft Materials © 2006 Thomas G. Mason

€ €

10 Forced Parallel Plate Oscillatory Viscometry Controlled Strain Oscillations γ (t) γ 0 τ 0 γ(t) = γ0 sin(ωt) τ (t) γ˙ (t) = γ˙ 0 cos(ωt) 2π ω t Δt

€ Measure: Stress τ(t) phase lag: ωΔt € (t) t G (t t )˙ (t )dt t G (t t )˙ cos( t )dt τ = − ∫−∞ r − ′ γ ′ ′ = − ∫−∞ r − ′ γ 0 ω ′ ′ ∞ τ(t) = −γ˙ 0 ∫0 Gr (s)cos[ω(t − s)]ds s = t − t ′ ∞ ∞ € τ(t) = −[ ∫0 Gr (s)cos(ωs)ds]γ˙ 0 cos(ωt)−[ ∫0 Gr (s)sin(ωs)ds]γ˙ 0 sin(ωt) € In-phase w.r.t strain: € G ′( ω) = ωη ′′ ( ω) 90° out-of-phase w.r.t strain: G ′′ ( ω) = ωη ′( ω) € Rheology of Soft Materials © 2006 Thomas G. Mason € €

Storage & Loss Moduli are Not Independent Functions The complex function G* consists of two real functions G’ and G’’

G *(ω) = G ′( ω)+ iG ′′ ( ω)

But both of these are derived from only one real function Gr(t) ∞ G *(ω) = iω G (t ′) exp(−iωt ′) dt ′ € ∫0 r 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

2ω ∞ ω G ''(ωˆ ) G '(ω) = ∫ dωˆ π 0 ωˆ ω 2 −ωˆ 2 2ω ∞ G '(ωˆ ) G ''(ω) − η'∞ ω = ∫ dωˆ π 0 ωˆ 2 −ω 2 Rheology of Soft €Materials © 2006 Thomas G. Mason

€

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)

Gp e.g. ideal polymer entanglement solution η0 J(t) G*(ω)

Gp Equivalent

1/Gp

η0/Gp t 2πGp/η0 ω = tc crossover time crossover frequency ωc = 2π/tc

J(t) = (1/Gp) + t/η0 G’(ω) G’’(ω) 2 2 2 G*(ω) = Gp{(ωtc) /[1+(ωtc) ]+ iωtc/[1+(ωtc) ]} Rheology of Soft Materials © 2006 Thomas G. Mason

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)

Gp

J(t) η∞ G*(ω)

1/Gp

Equivalent

Ge

η∞/Gp t Ge/η∞ ω

J(t) = (1/Gp) [1-exp(-Gpt/η∞)] G*(ω) = Gp + iωη∞ Rheology of Soft Materials © 2006 Thomas G. Mason

12 Brief Introduction to Non-Linear Rheology Sinusoid in does not give a sinusoid out: Harmonics are seen

τ Yield stress: τy τ y Yield strain: γy

Most Soft Glassy Materials: γ γy < 0.2 typically γy Shear Thinning: viscosity decreases as strain rate increases

Shear Thickening: viscosity increases as strain rate increases

Strain Hardening: stress increases greater than linear strain

Rheology of Soft Materials © 2006 Thomas G. Mason

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 Double Wall Couette Τ ω Τ ω

γ ω vapor trap prevents evaporation of continuous phase γ ω These are among preferred geometries: the strain field is homogeneous throughout the gap

Rheology of Soft Materials © 2006 Thomas G. Mason

13 Controlled Stress vs Controlled Strain Rheometry

Controlled Strain: Good for G*(ω) and Gr(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*(ω), 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

Rheology of Soft Materials © 2006 Thomas G. Mason

An Example: Viscoelasticity of Soft Glassy Emulsions

T.G. Mason & D.A. Weitz PRL 75 2051 (1995)

Linear regime at low strain: G’ and G’’ do not depend on γ -2 -1 Yield strain is evident: 10 < τy < 10

Rheology of Soft Materials © 2006 Thomas G. Mason

14 ...

An Example: Viscoelasticity of Soft Glassy Emulsions

T.G. Mason & D.A. Weitz PRL 75 2051 (1995)

Plateau Storage Modulus over many decades in frequency: G’p

Viscous Loss Modulus has a shallow minimum: G’’min

Rheology of Soft Materials © 2006 Thomas G. Mason

An Example: Viscoelasticity of Soft Glassy Emulsions

T.G. Mason & D.A. Weitz PRL 75 2051 (1995) 0

) 10

a solid symbols: G’p / -1

σ 10

/( flows elastic “solid” 10-2 min ' ' -3

G 10 ),

a -4 G’p ~ (σ/a) (φeff - φMRJ) / 10

σ Disorderd droplets begin to pack & deform /( 10-5 P φMRJ = φRCP = 0.64 G 10-6 0.5 0.6 0.7 0.8 0.9 1 φeff Rheology of Soft Materials © 2006 Thomas G. Mason

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!

Rheology of Soft Materials © 2006 Thomas G. Mason

Summary: Rheology of Soft Materials

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, …

Rheology of Soft Materials © 2006 Thomas G. Mason

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