Physical chemistry

Rheology

The science of how soft matter flows and deforms: viscosity and elasticity combined, shear-thinning blood and ketchup, viscoelastic memory, and the microscopic origins — entanglement, crowding, structure — of non-Newtonian behaviour.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionIntuition: solids on short times, liquids on long ones

Silly Putty bounces like rubber but flows into a puddle overnight. Cornstarch in water runs off your finger yet cracks under a quick punch — try walking on it and you can. Neither is truly solid nor liquid: whether the material “has time” to flow depends on how fast you probe it. Rheology is the physics of this in-between world, where a single material carries both a spring and a dashpot, and where microscopic structure — entangled chains, crowded particles, transient networks — decides which personality shows up.

Four canonical behaviours on one axes pair: Newtonian (τ = ηγ̇), shear-thinning (τ = Kγ̇ⁿ, n = 0.5), shear-thickening (n = 1.5), and Bingham plastic with its yield stress. The local ratio τ/γ̇ is the apparent viscosity.

SchoolSchool: viscosity, and when ketchup misbehaves

Definition: Shear stress, shear rate, viscosity

Slide the top plate of a fluid layer of thickness h at speed v while the bottom stays put: the shear rate is γ̇ = v/h (s⁻¹) and the shear stress τ (Pa) is the force per area needed to keep it sliding. Their ratio is the viscosity η = τ/γ̇ (Pa·s). Water near room temperature has η ≈ 1 mPa·s; honey ≈ 10 Pa·s; a Newtonian fluid keeps η constant while a non-Newtonian fluid lets η depend on γ̇, time, or stress.

Three everyday offenders against Newton’s law: ketchup and blood shear-thin — particles or cells align and unclog as γ̇ rises, so they pour slowly then suddenly fast; wet cornstarch shear-thickens as particles jam under sudden load; toothpaste and mayonnaise have a yield stress — below it they behave as solids and stay on the brush, above it they flow. A yield stress is the difference between a sauce that clings to food and one that pools at the bottom of the plate.

Canonical flow behaviours
Type	Law	Example
Newtonian	τ = ηγ̇, η constant	water, oils
Shear-thinning	τ = Kγ̇ⁿ, n < 1	ketchup, blood, paint
Shear-thickening	τ = Kγ̇ⁿ, n > 1	cornstarch slurry
Bingham plastic	τ = τ_y + η_pγ̇	toothpaste, mayonnaise
Thixotropic	η falls with time under shear	yogurt, drilling mud

UndergraduateUniversity: models, dimensionless numbers, viscoelasticity

Shear-thinning power-law fluids are captured by the Ostwald–de Waele law τ = Kγ̇ⁿ, but the exponent is only local. Better models bound the viscosity at both ends: Cross and Carreau fluids keep a Newtonian plateau η₀ at low shear and a high-shear tail η ~ γ̇^(m−1). Whether a viscoelastic fluid has “time” to relax during an experiment is decided by the Deborah number De = λ/T_obs, the ratio of the material’s relaxation time to the observation time: De ≪ 1 flows, De ≫ 1 snaps back, De ≈ 1 is where the interesting physics lives — putty bouncing (short T) versus spreading (long T).

\tau=K\dot\gamma^{n}\qquad \eta_{\mathrm{app}}=\frac{\tau}{\dot\gamma}=K\dot\gamma^{\,n-1}\qquad \mathrm{De}=\frac{\lambda}{T_{\mathrm{obs}}}

Viscoelastic fluids carry a fading memory of their deformation. The simplest one-mode description is the Maxwell element — spring and dashpot in series — giving stress relaxation G(t) = G₀e^(−t/λ) and, under oscillation at frequency ω, an elastic storage modulus G′ = G₀(ωλ)²/(1 + (ωλ)²) and a viscous loss modulus G″ = G₀ωλ/(1 + (ωλ)²). They cross at ωλ = 1: below the crossover the material dissipates like a liquid, above it stores like a solid. Real soft materials are multi-mode, so the crossover broadens into a spectrum — linear viscoelasticity measures that spectrum by sweeping ω.

G'(\omega)=\frac{G_0(\omega\lambda)^2}{1+(\omega\lambda)^2}\qquad G''(\omega)=\frac{G_0\,\omega\lambda}{1+(\omega\lambda)^2}\qquad \tan\delta=\frac{G''}{G'}

Example: Reading a flow curve

A paint obeys τ = Kγ̇ⁿ with K = 40 Pa·sⁿ and n = 0.4. What is its apparent viscosity at γ̇ = 1 s⁻¹ (brushing starts) and at γ̇ = 1000 s⁻¹ (vigorous brushing)? Why is this combination desirable for paint?

Solution

η_app = Kγ̇^(n−1): at γ̇ = 1, η = 40 Pa·s; at γ̇ = 1000, η = 40 × 1000^(−0.6) = 40 × 10^(−1.8) ≈ 0.63 Pa·s. High low-shear viscosity stops drips and sagging on a wall; the ~60-fold drop under brushing lets the paint spread thin and level out — shear-thinning is the formulation goal, not a defect.

AdvancedAdvanced: where the flow curve comes from

Non-Newtonian behaviour is structure responding to stress. In polymer melts, chains are entangled like a bowl of noodles; reptation theory pictures each chain slithering along its own tube, and predicts the viscosity scaling η ∝ M^3.4 seen experimentally — with M the molar mass — one of the triumphs of soft-matter theory. In concentrated suspensions, viscosity diverges as particles approach random close packing, roughly as Krieger–Dougherty η = η_s(1 − φ/φ_m)^(−[η]φ_m); near φ_m a suspension ceases to be a liquid at all. Shear-thickening suspensions add a twist: above a critical stress, hydrocluster or frictional contacts switch on and viscosity jumps.

\eta=\eta_s\Big(1-\frac{\phi}{\phi_m}\Big)^{-[\eta]\phi_m}\qquad N_1=\tau_{xx}-\tau_{yy}>0

Viscoelasticity adds elastic stresses shear cannot absorb: the first normal-stress difference N₁ makes polymer solutions climb a rotating rod (Weissenberg effect) and swell on leaving a die. Constitutive equations upgrade Maxwell by adding convected time derivatives — the stress tensor must be carried and rotated with the flow — giving the Oldroyd-B and Giesekus models that predict extensional thickening (the reason a honey thread resists being pulled apart far more than its shear viscosity suggests). Mapping which model fits which microstructure is the ongoing project of constitutive rheology.

ResearchResearch frontier

References

On the formulation of rheological equations of state · J. G. Oldroyd, 1950
Reviewing the roots of continuum formulations: molecular-level free-energy functionals and generalised moments of macromolecules · R. H. Ewoldt, 2017
The Physics of a Shear Thickening Suspension · M. Wyart, M. E. Cates, 2014