Knowledge Extraction from DMTA Data

DMTA measurements encode rich material information beyond simple modulus values. This page describes the physical quantities that can be extracted from fitted DMTA models in RheoJAX.

Glass Transition Temperature \(T_g\)

The glass transition is the most commonly extracted quantity from DMTA data. Three conventions exist:

Method

Definition

Notes

\(\tan(\delta)\) peak

Temperature at max \(E''/E'\)

ISO 6721-11; highest \(T_g\) value

\(E''\) peak

Temperature at max \(E''\)

ASTM D4065 preferred; closest to DSC \(T_g\)

\(E'\) onset

Inflection point of \(E'(T)\) drop

Lowest \(T_g\); onset of softening

Important

Typical ordering: \(T_g^{E'\text{onset}} < T_g^{E''\text{peak}} < T_g^{\tan\delta\text{peak}}\) with 5–15 °C difference between methods.

Relaxation Spectrum \(H(\tau)\)

The relaxation spectrum is a material property independent of deformation mode. Several RheoJAX models give direct access:

  • Generalized Maxwell: Discrete spectrum \(\{(G_i, \tau_i)\}\) (Prony series)

  • Fractional models: Continuous power-law spectrum \(H(\tau) \sim \tau^{-\alpha}\) where \(\alpha\) is the fractional order

  • VLB / HVM: Molecular-level relaxation from bond exchange kinetics

# After fitting a GMM (see Workflow 2 in dmta_workflows),
# extract discrete spectrum:

# With modulus_type='tensile': parameters are E_i (Pa)
for i in range(1, gmm._n_modes + 1):
    E_i = gmm.parameters.get_value(f"E_{i}")
    tau_i = gmm.parameters.get_value(f"tau_{i}")

# With modulus_type='shear' + deformation_mode='tension':
# parameters remain in G-space (G_i, Pa)
for i in range(1, gmm_g._n_modes + 1):
    G_i = gmm_g.parameters.get_value(f"G_{i}")

See also

DMTA Workflows Workflow 2 for complete GMM fitting.

Note

When using modulus_type='tensile', parameters are named E_inf, E_1, …, E_N and represent tensile moduli directly. When using the default modulus_type='shear' with deformation_mode='tension', the input E* is converted to G* internally, and parameters remain in G-space (G_inf, G_1, …).

Loss Tangent \(\tan\delta\) Analysis

The loss tangent \(\tan\delta = E''/E'\) is the most information-rich single quantity from DMTA:

  • Peak height: Related to the damping capacity. A tall, narrow peak signals a single dominant relaxation mechanism; a broad, low peak indicates a wide distribution (blends, copolymers, filled systems).

  • Peak width at half-height: Quantifies the breadth of the glass transition. For a single relaxation mechanism (Debye peak in \(E''(\omega)\)), FWHM \(\approx\) 1.14 decades; real polymers typically show 2–4 decades.

  • Peak asymmetry: A high-frequency shoulder on \(\tan\delta\) indicates sub-\(T_g\) (beta) relaxations from local chain motions.

import numpy as np

# From fitted model predictions
E_pred = model.predict(omega, test_mode='oscillation')
E_prime = E_pred.real
E_double_prime = E_pred.imag
tan_delta = E_double_prime / E_prime

# Find Tg (tan delta peak)
idx_peak = np.argmax(tan_delta)
omega_peak = omega[idx_peak]
tau_peak = 1.0 / omega_peak  # characteristic relaxation time

Storage Modulus Crossover and Modulus Drop

Two key features of the \(E'(T)\) or \(E'(\omega)\) curve:

  • Modulus drop ratio \(E'_{\text{glassy}} / E'_{\text{rubbery}}\) indicates the degree of chain mobility change through \(T_g\). Typical values: 100–1000 for amorphous polymers, 10–100 for semi-crystalline (crystallites constrain rubbery plateau).

  • :math:`E’ = E’’` crossover frequency \(\omega_c\) gives a characteristic relaxation time \(\tau_c = 1/\omega_c\). For a Maxwell element, this is exactly \(\tau\).

Molecular Weight from Plateau Modulus

For entangled polymers above \(T_g\), the entanglement molecular weight \(M_e\) is directly accessible:

\[M_e = \frac{\rho R T}{G_N^0} = \frac{\rho R T}{E_\infty / [2(1+\nu)]}\]

where \(\rho\) is the polymer density, \(R\) the gas constant, and \(T\) the absolute temperature. This requires a clear rubbery plateau in the DMTA data (i.e., the material must be entangled and above \(T_g\)).

Plateau Modulus \(E_\infty\) / \(G_N^0\)

The equilibrium (rubbery plateau) modulus encodes crosslink density:

\[G_N^0 = \nu_e k_B T \quad \Rightarrow \quad \nu_e = \frac{G_N^0}{k_B T} = \frac{E_\infty}{2(1+\nu)\,k_B T}\]

where \(\nu_e\) is the crosslink density.

  • Zener / Fractional Zener: \(G_e\) parameter = \(G_N^0\)

  • VLB models: \(G_0\) parameter (sum of all network moduli)

Fractional Order \(\alpha\) and Cooperativity

The fractional order from springpot-based models characterises the breadth of the relaxation distribution:

  • \(\alpha \to 0\): Purely elastic (Hookean spring)

  • \(\alpha \approx 0.5\): Critical gel / moderate cooperativity

  • \(\alpha \to 1\): Purely viscous (Newtonian dashpot)

The \(\alpha\) value also correlates with the cooperativity of segmental motion at \(T_g\):

  • \(\alpha \approx 0.3\): Highly cooperative, rigid-chain polymers (e.g., polycarbonate, polysulfone) — broad glass transition

  • \(\alpha \approx 0.5\): Moderate cooperativity, flexible-chain polymers (e.g., polyisoprene, PDMS)

  • \(\alpha \approx 0.7\): Low cooperativity, highly plasticised or low-\(M_w\) systems — narrow, single-mechanism relaxation

This provides a physics-based alternative to the empirical Kohlrausch–Williams–Watts (KWW) stretched exponential parameter \(\beta_{\text{KWW}} \approx \alpha\) (the Mittag-Leffler function \(E_\alpha(-(t/\tau)^\alpha)\) approximates \(\exp(-(t/\tau)^\beta)\) with \(\beta = \alpha\); see Metzler & Klafter, J. Non-Cryst. Solids 305, 81–87, 2002).

See also

DMTA Workflows Workflow 3 for FZSS fitting with Bayesian uncertainty on \(\alpha\).

WLF / Arrhenius Activation Energy

Temperature-dependent DMTA data (master curves) yield activation energy:

  • WLF: \(\log a_T = -C_1(T - T_r) / (C_2 + T - T_r)\)

  • Arrhenius: \(E_a = 2.303\,R\,C_1 C_2\) (from WLF at \(T > T_g + 100\))

  • HVM / HVNM: Direct \(E_a\) parameter from TST kinetics

See also

DMTA Workflows Workflow 2 for WLF extraction from multi-T DMTA, and DMTA Measurement Protocols for recommended TTS pipeline steps.

Network Topology (Vitrimer Models)

HVM and HVNM models extract topology-specific parameters from DMTA data:

  • Permanent crosslinks \(G_P\): Covalent network modulus

  • Exchangeable bonds \(G_E\): Vitrimer BER-active modulus

  • Bond exchange rate \(k_{\text{BER}}\): Arrhenius with \(E_a, \nu_0\)

  • Topology freezing \(T_v\): Where \(\tau_{\text{BER}} \sim\) experimental timescale

See also