Arrhenius-Law Viscosity

• Arrhenius-law viscosity is defined by a linear ln(η) versus 1/T relation, indicating a temperature-independent activation energy for viscous flow.
• This model is applied to strong liquids such as silica and optical fibres, serving as a benchmark to diagnose deviations like super-Arrhenius behavior.
• Practical insights include its use in processing diagnostics and simulation benchmarks, linking microscopic dynamics to macroscopic viscous properties.

Arrhenius-law viscosity is the temperature dependence of viscosity in which the logarithm of the viscosity is linear in inverse temperature, corresponding to a temperature-independent activation energy for viscous flow. In its standard forms, $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right)$$ or, in molar units, $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$; equivalently, $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$. Within this framework, viscous transport is treated as an activated process with a constant barrier, and deviations from linearity in an Arrhenius plot, $\ln\eta$ versus $1/T$, signal a breakdown of that assumption. The concept is central in the analysis of strong liquids such as silica, but it also functions as a reference model against which super-Arrhenius, Vogel–Fulcher–Tammann, power-law, and other non-Arrhenius behaviors are diagnosed in supercooled liquids, metallic melts, model glass formers, and even in operationally defined magnetic viscosity (Shao et al., 2013, Junior et al., 2020, Díaz-Méndez et al., 2011).

1. Definition, equations, and diagnostic criteria

The classical Arrhenius description of viscosity is expressed as

$$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$

or, in molar units,

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$

The corresponding linearized forms are

$$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$

A straight line in $\ln\eta$ versus $1/T$ is therefore the defining graphical signature of Arrhenius-law viscosity. A complementary local diagnostic is the temperature-dependent activation energy

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$0

which is constant only in the Arrhenius regime (Shao et al., 2013, Junior et al., 2020).

In systems where the relaxation observable is not a mechanical shear viscosity but an analogous relaxation measure, the Arrhenius construction is used operationally. In the dipolar–anisotropic Heisenberg model, for example, the relaxation time is written as $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$1 in dimensionless units, and Arrhenius scaling is tested through the collapse variable $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$2. In that context, the “magnetic viscosity” is defined as

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$3

and Arrhenius behavior means that $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$4 collapses as a function of $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$5 and that the long-time relaxation can be mapped to an effective barrier distribution through the scaled derivative (Díaz-Méndez et al., 2011).

2. Physical assumptions and microscopic interpretations

The Arrhenius law encodes a specific physical picture: viscous relaxation is governed by a single, temperature-independent activation barrier, with weakly temperature-dependent short-range spatio-temporal correlations and effectively Markovian dynamics. In the framework analyzed by Rosa Jr. et al., this is the “strong” limit, exemplified by silica, for which viscosity is nearly linear in inverse temperature over broad ranges. The same formalism makes explicit that once $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$6 ceases to be constant, the liquid exits the Arrhenius regime and enters a fragile, super-Arrhenius one characterized by growing correlation ranges and non-Markovianity (Junior et al., 2020).

Molecular-dynamics work on the one-component Lennard-Jones liquid shows that the descriptive success of an Arrhenius fit does not by itself establish an activated barrier-crossing mechanism. At fixed pressure, $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$7 versus $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$8 is linear over significant intervals covering most of the liquid phase and part of the supercritical region, and the fitted form $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$9 is accurate over those windows. Yet the extracted $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$0 is reported to be comparable to or even lower than thermal energies over the explored range, which undermines the usual requirement of a clear time-scale separation for genuine Eyring- or Kramers-type activated events. In that system, the free-volume description is more directly tied to local structural parameters, while Arrhenius behavior remains an effective parametrization (Rizk et al., 2022).

A different microscopic route appears in Buchenau’s asymmetry model of highly viscous flow. There, thermally activated jumps of a nanoscale inner core from a practically undistorted state to strongly distorted metastable states are coupled to an elastic matrix through Eshelby backstress. Viscosity arises from those distorted structures that do not jump back but instead become new ground states when the surrounding viscoelastic matrix relaxes. The model is naturally formulated through the shear compliance

$\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$1

with a cutoff time $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$2 separating reversible retardation from irreversible flow, and a consistency relation $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$3 between the Maxwell time and the cutoff. In this formulation, Arrhenius viscosity emerges when $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$4 is Arrhenius and the temperature dependence of $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$5, $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$6, and the spectrum shape is weak (Buchenau, 2011).

3. Strong-liquid realization in silica and optical fibres

A concrete high-temperature realization of Arrhenius-law viscosity is provided by silica optical fibres monitored with regenerated fibre Bragg gratings. In that study, standard SMF-28 fibre containing a GeO$\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$7 $\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$8 mol% core was preloaded with H$\ln\eta=\ln\eta_0+\frac{E_a}{R}\frac{1}{T}$9, regenerated by annealing near $\ln\eta$0, and further post-annealed near $\ln\eta$1 to stabilize the glass structure. Under a constant $\ln\eta$2 g axial load and a fibre cross-sectional area $\ln\eta$3, the tensile stress was $\ln\eta$4. The Bragg condition $\ln\eta$5 and the strain relation $\ln\eta$6 with $\ln\eta$7 lead to the optical viscosity formula

$\ln\eta$8

which avoids the need to calibrate heated length and is particularly advantageous when physical elongations are too small for direct microscopy (Shao et al., 2013).

Across $\ln\eta$9–$1/T$0, measured in $1/T$1 steps, the Bragg wavelength shifts were linear in time at constant temperature, and the resulting $1/T$2 versus $1/T$3 plots were linear across the full range. The extracted activation energy for viscous flow was $1/T$4, and adding a temperature-dependent prefactor left the value essentially unchanged at $1/T$5–$1/T$6. Representative values derived from the wavelength-shift rate were $1/T$7 at $1/T$8 and $1/T$9 at $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$0, corresponding to an annealing point $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$1 for $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$2 and a strain point $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$3 for $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$4 (Shao et al., 2013).

The same study emphasizes that Arrhenius-law viscosity in silica fibres is conditioned by structural state. The measured $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$5 is lower than the idealized $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$6 sometimes quoted for bulk silica, but it is consistent with other literature values in the $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$7–$$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$8 range. The discrepancy is attributed to the higher fictive temperature of fibres, $$\eta(T)=\eta_0 \exp\!\left(\frac{E_a}{k_B T}\right),$$9–$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$0 versus $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$1–$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$2 for bulk silica, and to the importance of equilibration time. Pre-annealing and regeneration relax frozen-in strains and improve measurement fidelity, but they reduce tensile strength from $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$3–$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$4 to $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$5–$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$6, making Arrhenius-law viscosity in silica simultaneously a transport law and a processing-state diagnostic (Shao et al., 2013).

4. Fragility, supercooling, and the breakdown of Arrhenius behavior

The canonical departure from Arrhenius-law viscosity in glass-forming liquids is super-Arrhenius growth. In the nonadditive stochastic framework studied by Rosa Jr. et al., the viscosity is written as

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$7

with the Arrhenius limit recovered at $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$8. For $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right).$9, the effective activation energy

$$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$0

increases on cooling and diverges at $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$1. In that picture, super-Arrhenius viscosity is associated with extended spatio-temporal correlations and non-Markovian dynamics, whereas a fragile-to-strong transition corresponds to a change in the range of those correlations. The reported fits include selenium melt with $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$2, silica in the Arrhenius limit with $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$3 consistent with the literature value $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$4, and GeSe$$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$5 with a crossover from $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$6 to $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$7 (Junior et al., 2020).

Deeply supercooled water provides an experimentally resolved case in which Arrhenius-law viscosity fails throughout the accessible range. Measurements down to $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$8 and up to $$\ln \eta=\ln \eta_0+\frac{E_a}{R}\frac{1}{T}, \qquad \log_{10}\eta=\log_{10}\eta_0+\frac{E_a}{2.303\,R}\frac{1}{T}.$$9 show pronounced curvature in Arrhenius plots at all pressures. At ambient pressure, the viscosity is well fit by the Speedy–Angell form

$\ln\eta$0

with $\ln\eta$1, $\ln\eta$2, and $\ln\eta$3, while for $\ln\eta$4 a Vogel–Fulcher–Tammann description becomes progressively superior. At $\ln\eta$5, the viscosity decreases by more than $\ln\eta$6 between $\ln\eta$7 and $\ln\eta$8, and the quantity $\ln\eta$9 increases on cooling, signaling Stokes–Einstein violation. No fragile-to-strong Arrhenius crossover is identified within the measured domain (Mussa et al., 2023).

Departures from Arrhenius-law viscosity are not confined to deeply supercooled states. A broad survey of metallic, molecular, and network-forming liquids above their liquidus or melting temperatures found that in every liquid studied the local Arrhenius slope $1/T$0 increases as temperature decreases, so viscosity rises more strongly than predicted by a single constant barrier even far above $1/T$1. That work proposes a partial collapse of the form

$1/T$2

with Arrhenius behavior as the special case $1/T$3, and reports that the measured viscosities remain above the scale $1/T$4, with $1/T$5 the number density and $1/T$6 Planck’s constant (Xue et al., 2021).

5. Simulation benchmarks and operational extensions

In atomistic model liquids, Arrhenius-law viscosity serves both as a fit ansatz and as a diagnostic of dynamical regime. For the one-component Lennard-Jones fluid, Green–Kubo calculations show that Arrhenius fits are valid, within statistical error bars, over substantial temperature intervals at fixed pressures from $1/T$7 to $1/T$8 in reduced units. The activation energy $1/T$9 is approximately pressure-independent at low pressure and increases for $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$00, while the prefactor $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$01 increases with pressure. Within the same domain, the Stokes–Einstein relation holds well, the hydrodynamic radius varies by less than $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$02 with temperature, and the reported proportionality $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$03 links transport directly to the first peak position of the radial distribution function (Rizk et al., 2022).

The supercooled Kob–Andersen binary Lennard-Jones liquid shows how an apparently Arrhenius exponential can become non-Arrhenius once the barrier itself becomes temperature-dependent. In the liquid–liquid interface model, the viscosity is written as

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$04

where the interfacial free energy $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$05 is extracted from curvature-weighted interface geometry and pressure contrast. Above the estimated liquid–liquid critical point, $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$06 and $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$07, the system behaves approximately Arrhenius, but below that regime $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$08 rises from $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$09 at $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$10 to $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$11 at $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$12, and the viscosity increases by $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$13 orders of magnitude. The formal exponential therefore remains, but the effective activation energy is no longer constant, which makes the behavior super-Arrhenius (Brickley et al., 17 Feb 2026).

The Arrhenius formalism also extends to magnetic relaxation under the label “magnetic viscosity.” In the $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$14 dipolar–anisotropic Heisenberg model with $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$15 and anisotropies $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$16 and $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$17, Arrhenius scaling of the out-of-plane magnetization holds only above a dynamical crossover temperature $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$18. For $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$19, scaling with $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$20 collapses the data only for $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$21; below that, the relaxation time follows a Vogel–Fulcher–Tammann law with fitted parameters $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$22, $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$23, and $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$24, while the best curve collapse is obtained with $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$25 and $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$26. For $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$27, Arrhenius scaling persists down to $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$28, but at and below that temperature the system freezes within accessible Monte Carlo times. In both regimes, the logarithmic derivative $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$29 yields an effective barrier distribution $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$30, which narrows at higher temperatures and broadens and shifts to higher energies as collective domain rearrangements take over (Díaz-Méndez et al., 2011).

6. Generalizations, alternative formulations, and open issues

Several recent formulations preserve the Arrhenius law as a limiting case while modifying its microscopic meaning. In the Dual Model of Liquids, viscosity is written as $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$31, with momentum transport carried by wave-packets moving through a dual medium of transient solid-like clusters and amorphous matrix. The model recovers an Arrhenius-like temperature dependence through the temperature evolution of $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$32, $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$33, and $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$34, and it links viscosity to the low-frequency mechano-thermal effect through

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$35

For water at $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$36, the order-of-magnitude estimate reported from mesoscopic parameters gives $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$37 between $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$38 and $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$39, compared with a measured dynamic viscosity of $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$40 (Peluso, 2024).

A mathematically different generalization is the fractional Arrhenius law obtained from a Riemann–Liouville fractional derivative version of the Van’t Hoff equation. For viscosity, the resulting expression is

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$41

with $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$42. The classical Arrhenius law is recovered at $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$43, while $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$44 generates curvature in $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$45 versus $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$46 through a nonzero second derivative, providing a compact way to model smooth nonlinearity without introducing a finite-temperature divergence (Lemes et al., 2016).

The CPA + Constraint model likewise embeds Arrhenius-law viscosity as a limit rather than a universal rule. Its fitting form,

$\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$47

reduces at high temperature and weak constraint to $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$48, so that $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$49, and at $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$50 it becomes a Vogel–Fulcher–Tammann relation with $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$51. On the datasets analyzed in that work—ortho-terphenyl and glycerol–water mixtures—the formulation achieves statistical parity with VFT fits over more than $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$52 orders of magnitude, which the authors interpret as evidence that residual curvature in simpler Arrhenius fits carries physical information about constraint growth (Gavant et al., 20 Nov 2025).

Across these frameworks, a common conclusion emerges. Arrhenius-law viscosity is most defensible when a single effective barrier dominates and its temperature dependence is weak; this condition is approximated in strong liquids such as silica, in selected high-temperature windows of simple liquids, and in some operational relaxation problems. Its breakdown is linked, depending on the system, to growing cooperative rearrangements, evolving barrier distributions, liquid–liquid interfacial structure, constraint buildup, or altered correlation ranges. A plausible implication is that Arrhenius-law viscosity is less a universal law than a limiting regime whose validity must be established by linearity of $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$53 versus $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$54, constancy of $\eta(T)=A \exp\!\left(\frac{E_a}{R T}\right)$55, and consistency with the underlying structural and dynamical observables (Junior et al., 2020, Xue et al., 2021).