Dynamic Mechanical Analysis (DMA): Storage Modulus E', Loss Modulus E'' & Tan Delta

DMA is the most powerful thermal characterisation tool for viscoelastic materials — it simultaneously measures stiffness, damping, and thermal transitions as a function of frequency and temperature. It detects Tg with 10× more sensitivity than DSC, maps secondary relaxations that predict creep and fatigue, and evaluates rubber vulcanisation completeness via crosslink density. Automotive OEMs (Tata, Mahindra, Maruti), tyre manufacturers (MRF, Apollo), and medical device companies require DMA data for material qualification.

• Derive the complex modulus E* = E' + iE'' from stress-strain phase shift.
• Interpret DMA thermograms: locate Tg from tan δ peak, E' onset, and E'' peak.
• Explain time-temperature superposition (TTS) and WLF equation for master curves.
• Calculate crosslink density from rubber plateau modulus using rubber elasticity theory.
• Identify ASTM E1640 and ISO 6721 DMA test standards.

3.1 Viscoelastic Response & Complex Modulus

Under sinusoidal loading at frequency ω, a viscoelastic material responds with stress lagging strain by phase angle δ:

$$\sigma(t) = \sigma_0 \sin(\omega t + \delta)$$

The complex modulus E* separates into:

$$E^* = E' + iE''$$

3.2 Reading a DMA Thermogram

As temperature increases from glassy to rubbery region:

Three methods to determine Tg from DMA:

1. Onset of E' drop — gives highest Tg value (~15°C above DSC Tg)
2. Peak of E'' — intermediate value (~5–10°C above DSC Tg)
3. Peak of tan δ — most commonly reported; ~10–20°C above DSC Tg

3.3 Crosslink Density from Rubbery Plateau (Rubber Elasticity Theory)

In the rubbery plateau above Tg, the equilibrium modulus E' is related to crosslink density ν_c:

$$E'_r = 3 \nu_c R T$$

Where: E'_r = storage modulus in rubbery plateau (Pa), R = 8.314 J/(mol·K), T = temperature (K), ν_c = crosslink density (mol/m³).

$$\nu_c = \frac{E'_r}{3RT}$$

3.4 Time-Temperature Superposition (TTS)

For thermorheologically simple materials, frequency and temperature are interchangeable via the WLF equation:

$$\log a_T = \frac{-C_1(T - T_{ref})}{C_2 + (T - T_{ref})}$$

Where C₁ ≈ 17.44 and C₂ ≈ 51.6 K for Tg as reference temperature. This allows master curve construction from short-time DMA data to predict long-time creep behaviour.

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Problem: A vulcanised SBR compound shows E'_r = 2.1 MPa at 60°C (333 K) in the rubbery plateau. Calculate crosslink density ν_c.

$$\nu_c = \frac{E'_r}{3RT} = \frac{2.1 \times 10^6 \text{ Pa}}{3 \times 8.314 \text{ J/(mol·K)} \times 333 \text{ K}}$$ $$\nu_c = \frac{2.1 \times 10^6}{8.306 \times 10^3} = \textbf{253 mol/m}^3$$

Interpretation: ν_c = 253 mol/m³ — this crosslink density corresponds to a well-cured tyre compound. Under-cured rubber shows lower E'_r (and lower ν_c), while over-cured shows higher E'_r but reduced elongation at break.

Apollo Tyres (Gurgaon) uses DMA routinely to characterise tyre tread compound tan δ at 0°C (wet grip index) and tan δ at 60°C (rolling resistance index). Lower tan δ at 60°C is the key target for fuel-efficient "green tyres" — CEAT's "SecuraDrive" line achieved 15% rolling resistance reduction through silica-silane compound DMA optimisation.

Mahindra & Mahindra NVH (Noise, Vibration, Harshness) labs in Pune use DMA to qualify EPDM door seals and NBR engine mounts — tan δ at service temperature predicts vibration damping performance across the Bolero and Scorpio SUV temperature operating range (−20°C to +80°C).

• E' (Storage Modulus): Elastic, in-phase response — energy stored per cycle.
• E'' (Loss Modulus): Viscous, out-of-phase response — energy dissipated per cycle.
• tan δ = E''/E': Loss factor (damping coefficient); peak marks Tg.
• Tg from DMA: Peak of tan δ is ~10–20°C higher than DSC Tg — important in reporting.
• Crosslink density ν_c: Calculated from rubbery plateau E' using rubber elasticity theory.
• TTS (Time-Temperature Superposition): Constructs master curves — predicts long-time creep from short-time DMA.

1. ASTM E1640 — Assignment of glass transition temperatures by DMA
2. ISO 6721-1 — Plastics — Determination of dynamic mechanical properties — General principles
3. ISO 6721-5 — Flexural vibration — Non-resonance method
4. ASTM D4065 — Determining and reporting dynamic mechanical properties of plastics
5. ISO 48-2 — Rubber, vulcanised or thermoplastic — Determination of hardness

1. A polymer shows E' = 3.2 GPa at −40°C and E' = 1.8 MPa at +80°C. Calculate tan δ if E'' = 520 MPa at Tg (−5°C). What physical transition is occurring?
2. Using rubber elasticity theory, calculate E'_r at 25°C for a crosslinked network with ν_c = 180 mol/m³.
3. Explain why DMA detects Tg at a higher temperature than DSC for the same polymer sample.

Q1. Storage modulus E' represents:

• A) Elastic energy stored per cycle (in-phase component)

Q2. Tg from DMA is typically determined as:

• C) Peak of tan δ — 10–20°C higher than DSC Tg

Q3. Crosslink density ν_c is calculated from:

• B) Rubbery plateau storage modulus E'_r using ν_c = E'_r / 3RT

Q4. Time-Temperature Superposition (TTS) is used to:

• C) Predict long-time creep behaviour from short-time DMA frequency sweeps

Q5. Which ASTM standard governs DMA glass transition temperature assignment?

• A) ASTM E1640