Rheology and Viscoelasticity

Maxwell model relaxation time

The Maxwell model describes a linear viscoelastic fluid as a Hookean spring of modulus E (or G) and a Newtonian dashpot of viscosity \eta connected in series. Because the two elements carry the same stress, the total strain is the sum of the spring strain and the dashpot strain. The single characteristic time of the model is the relaxation time \tau = \eta/E, the ratio of viscosity to modulus, which sets the time scale over which stress decays.

Under a sudden strain \varepsilon_0 that is then held constant, the stress relaxes exponentially: \sigma(t) = E\varepsilon_0 e^{-t/\tau}. At t = \tau the stress has fallen to 1/e (about 37%) of its initial value. When the observation time is much shorter than \tau, the material responds as an elastic solid; when it is much longer, the material flows viscously.

Under constant stress \sigma_0, the same model gives a creep strain \varepsilon(t) = \sigma_0(1/E + t/\eta), so the creep compliance is J(t) = 1/E + t/\eta. The spring strain \sigma_0/E appears instantly, and the dashpot strain grows linearly with time.

• Relaxation time — defined as \eta/E in the Maxwell model, not E/\eta. A larger viscosity means slower relaxation.
• Unit conversion — 1 poise = 0.1 Pa·s and 1 dyn cm^{-2} = 0.1 Pa. Poise divided by dyn cm^{-2} gives seconds, so the ratio can be used directly as a time after unit conversion to days by dividing by 86400.
• Series versus parallel — in series (Maxwell) the strains add; in parallel (Kelvin-Voigt) the stresses add.

Example: for \eta = 10^{9} Pa·s and E = 10^{7} Pa, \tau = 100 s. After 100 s the stress is e^{-1} of its initial value.

Creep and stress relaxation

Two standard step experiments characterize viscoelastic materials. In creep, a constant stress is applied and the strain is recorded. In stress relaxation, a constant strain is imposed and the stress decays. A purely elastic solid responds with an instantaneous strain that stays constant under load and is fully recovered on unloading. A purely viscous liquid responds with a strain that grows linearly with time and does not recover. A viscoelastic material shows an instantaneous elastic jump, followed by time-dependent creep and partial recovery after unloading, which combines both behaviours.

For a Kelvin-Voigt element (spring and dashpot in parallel) under constant stress \sigma_0, the strain approaches its equilibrium value exponentially: \varepsilon(t) = (\sigma_0/E)(1 - e^{-t/\tau}), where the retardation time is \tau = \eta/E. The strain approaches \sigma_0/E asymptotically and never exceeds it.

In a deformed sample, stress relaxation dissipates energy: the excess free energy stored in the network is released as chains rearrange, so the stress falls under fixed strain. The relaxation mechanism depends on the temperature regime:

• Glassy region (below T_g) — only local motions are active, so the response depends on local structure and is largely independent of molecular weight.
• Region around T_g — segmental motion of the chains dominates, and the modulus falls steeply.
• Rubbery regime — large-scale chain rearrangement occurs, the excess free energy is dissipated, and the terminal relaxation time depends strongly on molecular weight.

Shear flow curves of materials

Steady shear flow curves distinguish materials by how the shear stress \sigma and the viscosity \eta = \sigma/\dot{\gamma} depend on the shear rate \dot{\gamma}. A Newtonian liquid obeys \sigma = \eta_0\dot{\gamma}. Its stress curve is a straight line through the origin, and its viscosity is constant at all shear rates.

A solid-like material with yielding does not flow until the stress exceeds a yield stress \sigma_y. In the idealized (Bingham) limit, the stress is \sigma = \sigma_y + \eta_p\dot{\gamma}, so the apparent viscosity \eta = \sigma_y/\dot{\gamma} + \eta_p diverges as \dot{\gamma} \to 0 and decreases as the shear rate rises. The Herschel-Bulkley model generalizes this as \sigma = \sigma_y + K\dot{\gamma}^n.

Polymer melts usually show shear thinning. The viscosity stays at its zero-shear value \eta_0 at low shear rate and then decreases. Consequently the stress curve starts linearly with slope \eta_0 and then bends downward as the slope decreases.

• Yield stress — the threshold stress below which no steady flow occurs.
• Shear thinning — viscosity decreases with shear rate; the stress curve is concave downward.
• Shear thickening — viscosity increases with shear rate; the stress curve is concave upward.

Example: a Newtonian liquid with \eta_0 = 2 Pa·s gives \sigma = 2 Pa at \dot{\gamma} = 1 s^{-1} and \sigma = 20 Pa at \dot{\gamma} = 10 s^{-1}. A viscosity that is flat over all shear rates identifies Newtonian behaviour.

Entropic elasticity of rubber

Above the glass transition, a crosslinked elastomer stores its elastic response mainly as entropy rather than internal energy. For an ideal network, the internal energy changes little on stretching, so the retractive force is f = -T\left(\partial S/\partial L\right)_{T}. Stretching reduces the number of chain conformations, so the entropy falls and the network pulls back. The retractive force is therefore proportional to the absolute temperature at a given extension.

The consequence is that, at constant applied force, raising the temperature increases the retractive tendency and the specimen contracts along the loading direction. This is the reverse of ordinary solids, which expand when heated. Thermal contraction under load is a diagnostic sign of entropic elasticity. The statement holds between the glass transition and the melting point, where the network is intact.