<!-- PDF-EXPORT-IGNORE-START --> > [!info] 📄 PDF Version > [Download PDF](./6-Rheology.pdf) <!-- PDF-EXPORT-IGNORE-END --> # Lecture 6: Rheology – Measuring "Softness" and Time-Dependent Response ## 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:** $\gamma(t) = \gamma_0 \sin(\omega t)$ **Measure resulting stress:** $\sigma(t) = \gamma_0 [G' \sin(\omega t) + G" \cos(\omega t)]$ ### 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:** $\tan \delta = \frac{G"}{G'}$ **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 $\sigma(t) = \sigma_0 e^{-t/\tau}$ - **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 | $\|G^*\| = \sqrt{G'^2 + G"^2}$ | | **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) --- **Previous Lecture:** [[5-Viscoelasticity|Lecture 5: Elastic vs Viscous vs Viscoelastic]] **Next Lecture:** [[7-8-Student-Presentations|Lectures 7-8: Student Presentations]] **Course Home:** [[0-README|Course Overview]] > [!significance]- Metadata > Author:: [Vatsal Sanjay](https://vatsalsanjay.com)<br> > Date published:: Jul 19, 2026<br> > Date modified:: Jul 19, 2026 > [!link] Back to main website > [Home](https://comphy-lab.org/), [Team](https://comphy-lab.org/team), [Research](https://comphy-lab.org/research), [Github](https://github.com/comphy-lab) > > 📝 [Edit this page on GitHub](https://github.com/comphy-lab/CoMPhy-Lab-Blogs/blob/main/Lecture-Notes/Intro-Soft-Matter/6-Rheology.md)