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Lecture 6: Rheology – Measuring “Softness” and Time-Dependent Response

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Key Topics

What is Rheology?

Definition: The study of flow and deformation of matter

Central Question: How do we quantify the softness or flow behavior of a material?

Focus:

• Experimental rheology
• Storage and loss moduli (G’, G”)
• Viscosity as function of shear rate or frequency
• Relaxation timescales

Storage and Loss Moduli

Linear Viscoelastic Characterization

Method: Small oscillatory deformations

Apply oscillating strain:

Measure resulting stress:

Storage Modulus (G’)

Physical Meaning:

• In-phase stress response
• Energy stored like a spring
• Measure of material stiffness
• Elastic component

When G’ dominates:

• Material behaves solid-like
• Most energy stored and recovered
• Little energy dissipation

Loss Modulus (G”)

Physical Meaning:

• Out-of-phase stress response
• Energy dissipated as heat like a dashpot
• Viscous component
• Energy lost per cycle

When G” dominates:

• Material behaves fluid-like
• Energy dissipated as heat
• Flow-dominated behavior

Phase Angle and Loss Tangent

Phase Angle (δ):

• Lag between stress and strain
• δ = 0° → Purely elastic (all energy stored)
• δ = 90° → Purely viscous (all energy lost)

Loss Tangent:

Interpretation:

• tan δ >> 1: Very damped, fluid-like
• tan δ << 1: Elastic, solid-like
• tan δ ~ 1: Balanced viscoelastic behavior

Rheometry Techniques

1. Rotational (Shear) Rheometer

Common Configuration:

• Material sandwiched between plates
• Or cone-and-plate geometry
• Apply controlled strain or stress

Measurements:

Steady Shear:

• Measure viscosity: η = (shear stress)/(strain rate)
• Flow curves
• Shear-thinning or thickening behavior

Oscillatory Shear:

• Measure G’ and G”
• Frequency sweeps
• Amplitude sweeps

Cone-Plate Geometry:

• Advantage: Uniform shear across radius
• Precise measurements
• Common in research

2. Creep Test

Procedure:

1. Apply constant stress
2. Watch strain vs time

Results by Material Type:

Elastic Solid:

• Immediate strain
• Then no further creep
• Constant strain (if truly Hookean)

Viscous Fluid:

• Strain increases linearly forever
• No asymptote
• Slope = 1/η

Viscoelastic Material:

• Kelvin-Voigt behavior: Immediate elastic jump + slow creep → asymptote
• Maxwell behavior: Indefinite creep (has liquid component)

Information Gained:

• Directly measures compliance (how “soft”)
• Time-dependent response
• Viscosity at long times

3. Stress Relaxation Test

Procedure:

1. Apply quick fixed strain
2. Hold constant
3. Measure stress decay over time

Results by Material Type:

Purely Elastic:

• Constant stress (no decay)

Viscoelastic:

• Maxwell fluid: Decays to zero eventually
• With permanent network: Decays to plateau

Information Gained:

• Relaxation time τ
• Material’s internal timescale
• Separates elastic (short-term) and viscous (long-term)

4. Dynamic Mechanical Analysis (DMA)

Application: Solid samples (polymer bars, films)

Method:

• Oscillatory loading
• Often in tension or bending
• Similar concept to rotational rheometry

Advantages:

• For materials that can’t flow into gap
• Temperature sweeps
• Glass transition measurements

5. Other Methods

Capillary Viscometers:

• For simple fluids
• Measure flow through tube
• Use Poiseuille’s law
• Older method but still useful

Interpreting G’ and G”

Practical Meanings

G’ >> G” (Storage dominates):

• Material is gel-like or solid-like
• Can support deforming load elastically
• Barely flows
• Example: Firm gel, rubber

G” >> G’ (Loss dominates):

• Material is liquid-like
• Will flow and not hold shape
• Most energy dissipated
• Example: Viscous liquid, honey

G’ ≈ G” (Crossover):

• Critical point
• Marks characteristic timescale
• Material transitions from solid to liquid behavior
• Important for gel point determination

Frequency Dependence

Many Soft Materials:

• G’ < G” at low frequencies (fluid-like, long times)
• G’ > G” at high frequencies (solid-like, short times)
• Crossover at ω ≈ 1/τ (relaxation time)

Physical Interpretation:

• High frequency (fast deformation): No time to relax → solid
• Low frequency (slow deformation): Plenty of time to relax → liquid

Example Plot:

• X-axis: Frequency (ω)
• Y-axis: G’, G” (log scale)
• Show crossover point
• Identify relaxation time from crossover

Quantifying Softness

For Solids: Modulus

Young’s Modulus or Shear Modulus:

Measure of Stiffness:

• High modulus → stiff → “hard”
• Low modulus → compliant → “soft”

Examples:

• Hard plastic: E ~ 3 GPa
• Rubber: E ~ 1 MPa (1000× softer)
• Soft silicone gel: E ~ 50 kPa (60× softer than rubber)
• Brain tissue: E ~ 1 kPa (extremely soft)

Biological Relevance:

• Tissue engineering cares deeply about modulus
• Cells respond to substrate stiffness
• Preview of Engler et al. paper (Lectures 7-8)

For Fluids: Viscosity

Measure of Flow Resistance:

• Low viscosity → flows easily
• High viscosity → flows slowly

Examples (at room temperature):

• Water: η ~ 1 mPa·s (very fluid)
• Blood: η ~ 3-4 mPa·s
• Olive oil: η ~ 100 mPa·s
• Honey: η ~ 10,000 mPa·s = 10 Pa·s
• Glycerol: η ~ 1000 mPa·s = 1 Pa·s

Non-Newtonian Behavior

Shear-Thinning:

• Viscosity decreases with shear rate
• Common in polymer solutions, paints
• Example: Ketchup (shaking lowers viscosity)

Shear-Thickening:

• Viscosity increases with shear rate
• Example: Cornstarch suspension (oobleck)
• Acts solid under impact
• Preview of Waitukaitis & Jaeger paper

Rheometer Detection:

• Flow curve: Stress vs shear rate
• Slope changes reveal non-Newtonian behavior

Frequency-Dependent Softness

Key Insight: Softness can depend on measurement frequency!

Example: Silly Putty

• Measured slowly (low ω): Low effective modulus (soft, flows)
• Measured quickly (high ω): High effective modulus (stiff, bounces)

Practical Implication: Must specify conditions when reporting “softness”

Examples and Case Studies

Laboratory Demonstration

Rheometer Reading:

• Show real data from rheometer
• Oscillatory test on known material
• Point out G’, G” values
• Discuss crossover frequency

Tabletop Viscoelastic:

• Gelatin dessert
• Quick oscillation (tapping) vs slow deformation (letting sag)
• Relate to G’ and G”

Jello vs Silly Putty

Jello at Room Temperature:

• Jiggles (G’ low, near solid/liquid boundary)
• Breaks if strained too far
• Gel with low modulus
• G’ > G” but both relatively small

Silly Putty:

• Frequency-dependent as discussed previously
• Classic example of viscoelasticity

Polymer Dynamic Mechanical Analysis

Example: Polyvinyl chloride (PVC)

Below Glass Transition (~60°C):

• G’ >> G” (glassy solid)
• Very stiff
• Brittle

Above Glass Transition:

• G” >> G’ (rubbery flow)
• Much softer
• Can flow

Demonstrates:

• Temperature plays similar role to frequency
• Time-temperature superposition

Data Example: Frequency Sweep

Generic Polymer Melt or Colloidal Paste:

Low ω:

• G” > G’ (fluid-like)
• Material has time to flow

High ω:

• G’ > G” (solid-like)
• No time to relax

Crossover at ω₀:

• Mark: ω₀ = 1/τ
• Identifies relaxation time

Shear-Thinning Example

Ketchup Viscosity:

• High viscosity at low shear (doesn’t flow from bottle)
• Low viscosity at high shear (flows when shaken/squeezed)
• Flow curve shows decreasing viscosity with shear rate

Why: Structure breaks down under shear

Discussion Questions

1. Making Measurements

Scenario: New synthesized hydrogel

Question: Want to know if it’s solid-like or liquid-like. What experiment?

Answer:

• Oscillatory shear test at relevant frequency
• Or simple poke test: rebounds (elastic) vs slowly indents (viscous)
• Look at G’ vs G”

2. Interpreting Moduli

Data: Material has G’ = 1000 Pa, G” = 200 Pa at 1 Hz

Question: How would it behave if deformed rhythmically at 1 Hz?

Answer:

• Mostly elastic response
• Will return ~80% of energy
• Only 20% lost
• Will hold shape and only slowly relax

Follow-up: What if frequency increases where G” rises and G’ drops?

3. Measuring Viscosity Without High-End Rheometer

Question: How to measure honey viscosity?

Possible Answers:

• Timed flow experiment (drip and use gravity + Poiseuille’s law)
• Falling ball viscometer (Stoke’s law)
  • Drop marble in honey
  • Measure terminal speed
  • Compute viscosity

Engages: Creative thinking about measurements

4. Relaxation Time Thought Experiment

Comparison: Cold honey vs warm honey

Question: How would relaxation times differ?

Answer:

• Warm honey: Lower viscosity → faster relaxation (low τ)
• Cold honey: Higher viscosity → slower relaxation (high τ)
• At room temp: Flows quickly
• At fridge temp: Almost solid-like (τ very long)

Insight: “Colder = more solid-like” because molecular motion slows

5. Rheology in Everyday Life

Ketchup Question: Why does shaking bottle make it easier to pour?

Answer:

• Shear-thinning behavior
• Structure breaks down
• Viscosity decreases
• Easier flow

Milkshake Question: Why add polymers (like carboxymethyl cellulose) to make thicker?

Answer:

• Increase zero-shear viscosity
• Suspension stays uniform (slower creaming)
• Gives smoother “body”
• Better mouthfeel

Key Concepts Summary

Rheological Properties

Property	Symbol	Units	Meaning
Storage Modulus	G’	Pa	Elastic stiffness, energy stored
Loss Modulus	G”	Pa	Viscous dissipation, energy lost
Complex Modulus	G*	Pa
Loss Tangent	tan δ	-	G”/G’ (damping ratio)
Viscosity	η	Pa·s	Flow resistance

Experimental Methods Summary

Method	Measures	Best For
Oscillatory Shear	G’, G”	Viscoelastic characterization
Steady Shear	η(γ̇)	Flow curves, non-Newtonian
Creep	J(t)	Compliance, long-time behavior
Stress Relaxation	G(t)	Relaxation time, memory
DMA	E’, E”	Solid samples, temperature sweeps

Practical Insights

Why Rheological Properties Matter

Material Processing:

• Polymer melt needs right viscosity to mold
• Cured rubber needs right modulus for application

Formulation Stability:

• Mayonnaise with higher G’ holds shape on sandwich
• Paint with proper shear-thinning flows on brush but not off wall

Product Performance:

• Tire rubber needs damping (G”) for ride comfort
• But stiffness (G’) for handling

Biological Systems:

• Mucus rheology affects clearance
• Blood viscosity affects circulation
• Cell mechanics affect function

Connecting Lab Numbers to Tactile Impressions

Why peanut butter doesn’t flow from jar:

• High zero-shear viscosity
• Often has yield stress
• Requires finite stress to initiate flow

Why shampoo thins when squeezed:

• Shear-thinning behavior
• Polymer network breaks down temporarily

Preparation for Next Lecture

Next lectures: Student Presentations (Lectures 7-8, 2-hour session)

Prepare:

• Read selected paper carefully
• Understand key concepts and methods
• Prepare 10-12 minute presentation
• Be ready for questions from peers

Papers span:

• Historical foundations (Einstein, Onsager, de Gennes)
• Novel materials (auxetics, soft robots)
• Phase transitions (jamming, active matter)
• Applications (mechanobiology, shear thickening)

References

1. “Storage modulus (G’) and loss modulus (G”) for beginners” - Rheology Lab
2. “Basics of Dynamic Mechanical Analysis (DMA)” - Anton Paar Wiki
3. Macosko, C.W. “Rheology: Principles, Measurements, and Applications” (1994)
4. Barnes, H.A. “A Handbook of Elementary Rheology” (2000)

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Previous Lecture: Lecture 5: Elastic vs Viscous vs Viscoelastic Next Lecture: Lectures 7-8: Student Presentations Course Home: Course Overview

Metadata

Author:: Vatsal Sanjay
Date published:: Jul 19, 2026
Date modified:: Jul 19, 2026

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