Oscillatory Rheometry Simulator: Viscoelastic Moduli & Frequency Sweeps

simulator intermediate ~10 min
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G' = 990 Pa, G'' = 99 Pa — elastic-dominated at ω = 10 rad/s

At ω = 10 rad/s with λ = 1 s and G₀ = 1000 Pa (ωλ = 10), the single-mode Maxwell model predicts G' = 990 Pa and G'' = 99 Pa. The material is predominantly elastic with tan δ = 0.1.

Formula

G'(ω) = G₀ × (ωλ)² / (1 + (ωλ)²)
G''(ω) = G₀ × ωλ / (1 + (ωλ)²)
|η*| = √(G'² + G''²) / ω

Probing Viscoelasticity

Every real material exhibits some combination of elastic (solid-like) and viscous (liquid-like) behavior. Oscillatory rheometry reveals this dual nature by subjecting a sample to a small sinusoidal deformation and decomposing the stress response into an in-phase component (stored energy, G') and an out-of-phase component (dissipated energy, G''). A frequency sweep maps the material's behavior across time scales, from fast elastic response to slow viscous flow.

The Maxwell Model

The simplest viscoelastic model places a spring and dashpot in series. At high frequencies (fast deformation), the dashpot has no time to flow and the spring dominates — the material appears elastic with G' approaching the plateau modulus G₀. At low frequencies, the dashpot relaxes all stress and G'' dominates — the material flows. The crossover frequency ω = 1/λ marks the transition and directly reveals the relaxation time. This simulation plots the Maxwell frequency sweep in real time.

Frequency Sweep Fingerprint

A log-log plot of G' and G'' versus frequency is a material fingerprint. Entangled polymer melts show a rubbery plateau in G' spanning decades of frequency. Cross-linked gels show G' flat and above G'' across all frequencies. Dilute solutions show terminal behavior with G' proportional to ω² and G'' proportional to ω. By matching experimental frequency sweeps to model predictions, rheologists extract relaxation spectra that encode the material's microstructural dynamics.

From Lab to Product

Oscillatory rheometry is the workhorse characterization tool in polymer, food, pharmaceutical, and cosmetics industries. A frequency sweep on a new shampoo formulation reveals whether it will feel thick or watery, hold its shape in a bottle, and spread easily on hair. Temperature sweeps detect gelation points in adhesives and curing resins. Strain sweeps map the onset of nonlinearity where microstructure begins to break. This single instrument provides more actionable information about material behavior than any other rheological test.

FAQ

What is oscillatory rheometry?

Oscillatory rheometry applies a sinusoidal strain (or stress) to a material and measures the resulting stress (or strain) response. The in-phase component gives the storage modulus G' (elastic behavior), and the out-of-phase component gives the loss modulus G'' (viscous dissipation). Together they characterize the full viscoelastic response.

What do G' and G'' tell you?

G' (storage modulus) quantifies energy stored elastically per cycle — how solid-like the material is. G'' (loss modulus) quantifies energy dissipated as heat — how liquid-like it is. When G' > G'' the material is gel-like; when G'' > G' it is sol-like. Their ratio tan δ = G''/G' is the loss tangent.

What is the Maxwell model?

The Maxwell model represents a viscoelastic material as a spring (modulus G₀) in series with a dashpot (viscosity η₀ = G₀λ). It predicts a single crossover point where G' = G'' at ω = 1/λ. Real materials often require multiple Maxwell modes (a spectrum of relaxation times) for accurate fitting.

What is the linear viscoelastic regime?

The linear viscoelastic (LVE) regime is the range of strain amplitudes where G' and G'' are independent of strain — the material response is proportional to the applied deformation. Beyond this limit (typically 0.1-10% strain for gels), the microstructure is disrupted and moduli become strain-dependent (nonlinear behavior).

Sources

Other simulations: Rheology & Flow Behavior

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Extensional Flow & Elongational Viscosity
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Thixotropy & Time-Dependent Flow
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Viscosity Models & Shear Response
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Yield Stress & Plastic Flow

Embed

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View source on GitHub