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of Soft Materials

Thomas G. Mason

Department of and Biochemistry Department of 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 and flow of materials

Elastic Viscous And everything in between… “Soft Materials” , , 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: and

Common Flows and Simple Examples

Equations of Continuity and

Linear Shear Viscoelastic Rheology

Introduction to Non-linear Rheology

Macroscopic Mechanical Shear

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 1 ∂V β = − V ∂p T

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

Example of Shear Deformations: is applied along surface, not normal to it Unidirectional Simple Shear Flow Strain 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 :   0 0 0 γ˙ = v(t)/d = Δx˙ (t)/d velocity 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 “ 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

Fxy Area of Face: A Simple isotropic : y − p 0 0    x p = 0 −p 0 = −pδ z Fxx     F  0 0 −p τxy = Fxy/A, … xy 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 ’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 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 ) € €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 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: G

Helmholtz Free 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 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: chains, clay , 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

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 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 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- 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 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 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 (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 have both stress and strain control Get the best of both worlds in principle

Rheology of Soft Materials © 2006 Thomas G. Mason

An Example: 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). “” 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 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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