TTSP

Time-Temperature Superposition Principle (TTSP) is a concept in rheology and materials science that describes how the viscoelastic response of thermorheologically simple materials can be made to collapse onto a single master curve by shifting data along the time or frequency axis in response to temperature changes. When a material obeys TTSP, its relaxation spectrum at different temperatures can be superposed by applying a temperature-dependent horizontal shift factor a_T, so that measurements taken at various temperatures align with those at a chosen reference temperature T_ref.

In practice, TTSP is applied to quantities such as dynamic moduli G*(ω, T) and creep compliance J(t, T). The master curve is constructed by plotting the response versus log ω or log t and horizontally shifting curves by log a_T until they coincide with the reference curve. The relation between the temperatures and the shift is often written as G*(ω, T) = G*(ω a_T, T_ref) or J(t, T) = J(t a_T, T_ref). The shift factor a_T is a function of temperature and is determined from the data or by a constitutive equation.

The most common form for a_T near the glass transition is the Williams-Landel-Ferry (WLF) equation: log a_T = -C1 (T - T_ref) / (C2 + (T - T_ref)), where C1 and C2 are material constants. For higher temperatures or different regimes, an Arrhenius form log a_T = (E_a/R)(1/T - 1/T_ref) may be used. TTSP requires thermorheological simplicity; if temperature alters the relaxation spectrum in a nonuniform way, the principle fails.

TTSP is widely used to build master curves for polymers, rubbers, and related materials, enabling long-term predictions of viscoelastic behavior from short-time measurements and informing material design and processing. See also: WLF equation, thermorheological simplicity.