Abstract
The Prandtl model is certainly the simplest and most generic microscopic model describing solid friction. It consists of a single, thermalized atom attached to a spring, which is dragged past a sinusoidal potential representing the surface energy corrugation of a counterface. While it was primarily introduced to rationalize how Coulomb’s friction law can arise from small-scale instabilities, Prandtl argued that his model also describes the shear thinning of liquids. Given its success regarding the interpretation of atomic-force-microscopy experiments, surprisingly little attention has been paid to the question how the Prandtl model relates to fluid rheology. Analyzing its Langevin and Brownian dynamics, we show that the Prandtl model produces friction–velocity relationships, which, converted to a dependence of effective (excess) viscosity on shear rate η ( γ ˙ ) , is strikingly similar to the Carreau–Yasuda (CY) relation, which is obeyed by many non-Newtonian liquids. The two dimensionless parameters in the CY relation are found to span a broad range of values. When thermal energy is small compared to the corrugation of the sinusoidal potential, the leading-order γ ˙ 2 corrections to the equilibrium viscosity only matter in the initial part of the cross-over from Stokes friction to the regime, where η obeys approximately a sublinear power law of 1 / γ ˙ .
Highlights
Understanding frictional forces and energy dissipation between two sliding bodies is a central task of tribology
We first describe the Prandtl model in a slightly modified form, that is, the explicitly introduced damping of the mass point does not occur relative to the substrate but within the spring
The three different methods pursued to study the dynamics of the system are described
Summary
Understanding frictional forces and energy dissipation between two sliding bodies is a central task of tribology. The Prandtl model is arguably the simplest and most generic non-linear model explaining why and how energy dissipates microscopically [1,2,3]. It consists of a mass point, which is dragged by a spring of stiffness k past a corrugated potential and subjected to a Stokesian drag force as well as to thermal fluctuations. If the reduced spring stiffness k ≡ k/Vmax k B T is finite, the dependence of the friction force F on velocity v is Stokesian at small v but usually much enhanced compared to that caused by the artificially added damping term
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