Master equation approach to friction at the mesoscale

Contact Dependence and Velocity Crossover in Friction between Microscopic Solid/Solid Contacts

Friction at the nanoscale differs markedly from that between surfaces of macroscopic extent. Characteristically, the velocity dependence of friction between apparent solid/solid contacts can strongly deviate from the classically assumed velocity independence. Here, we show that a nondestructive friction between solid tips with radius on the scale of hundreds of nanometers and solid hydrophobic self-assembled monolayers has a strong velocity dependence. Specifically, using laterally oscillating quartz tuning forks, we observe a linear scaling in the velocity at the lowest accessed velocities, typically hundreds of micrometers per second, crossing over into a logarithmic velocity dependence. This crossover is consistent with a general multicontact friction model that includes thermally activated breaking of the contacts at subnanometric elongation. We find as well a strong dependence of the friction on the dimensions of the frictional probe.

Slow wave propagation in soft adhesive interfaces

Stick-slip in sliding of soft adhesive surfaces has long been associated with the propagation of Schallamach waves, a type of slow surface wave. Recently it was demonstrated using in situ experiments that two other kinds of slow waves-separation pulses and slip pulses-also mediate stick-slip (Viswanathan et al., Soft Matter, 2016, 12, 5265-5275). While separation pulses, like Schallamach waves, involve local interface detachment, slip pulses are moving stress fronts with no detachment. Here, we present a theoretical analysis of the propagation of these three waves in a linear elastodynamics framework. Different boundary conditions apply depending on whether or not local interface detachment occurs. It is shown that the interface dynamics accompanying slow waves is governed by a system of integral equations. Closed-form analytical expressions are obtained for the interfacial pressure, shear stress, displacements and velocities. Separation pulses and Schallamach waves emerge naturally as wave solutions of the integral equations, with oppositely oriented directions of propagation. Wave propagation is found to be stable in the stress regime where linearized elasticity is a physically valid approximation. Interestingly, the analysis reveals that slow traveling wave solutions are not possible in a Coulomb friction framework for slip pulses. The theory provides a unified picture of stick-slip dynamics and slow wave propagation in adhesive contacts, consistent with experimental observations.

Friction and nonlinear dynamics

The nonlinear dynamics associated with sliding friction forms a broad interdisciplinary research field that involves complex dynamical processes and patterns covering a broad range of time and length scales. Progress in experimental techniques and computational resources has stimulated the development of more refined and accurate mathematical and numerical models, capable of capturing many of the essentially nonlinear phenomena involved in friction.

Speed of fast and slow rupture fronts along frictional interfaces

The transition from stick to slip at a dry frictional interface occurs through the breaking of microjunctions between the two contacting surfaces. Typically, interactions between junctions through the bulk lead to rupture fronts propagating from weak and/or highly stressed regions, whose junctions break first. Experiments find rupture fronts ranging from quasistatic fronts, via fronts much slower than elastic wave speeds, to fronts faster than the shear wave speed. The mechanisms behind and selection between these fronts are still imperfectly understood. Here we perform simulations in an elastic two-dimensional spring-block model where the frictional interaction between each interfacial block and the substrate arises from a set of junctions modeled explicitly. We find that material slip speed and rupture front speed are proportional across the full range of front speeds we observe. We revisit a mechanism for slow slip in the model and demonstrate that fast slip and fast fronts have a different, inertial origin. We highlight the long transients in front speed even along homogeneous interfaces, and we study how both the local shear to normal stress ratio and the local strength are involved in the selection of front type and front speed. Last, we introduce an experimentally accessible integrated measure of block slip history, the Gini coefficient, and demonstrate that in the model it is a good predictor of the history-dependent local static friction coefficient of the interface. These results will contribute both to building a physically based classification of the various types of fronts and to identifying the important mechanisms involved in the selection of their propagation speed.

Aftershocks in a frictional earthquake model

Inspired by spring-block models, we elaborate a "minimal" physical model of earthquakes which reproduces two main empirical seismological laws, the Gutenberg-Richter law and the Omori aftershock law. Our point is to demonstrate that the simultaneous incorporation of aging of contacts in the sliding interface and of elasticity of the sliding plates constitutes the minimal ingredients to account for both laws within the same frictional model.

Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces

The failure of the population of microjunctions forming the frictional interface between two solids is central to fields ranging from biomechanics to seismology. This failure is mediated by the propagation along the interface of various types of rupture fronts, covering a wide range of velocities. Among them are the so-called slow fronts, which are recently discovered fronts much slower than the materials' sound speeds. Despite intense modeling activity, the mechanisms underlying slow fronts remain elusive. Here, we introduce a multiscale model capable of reproducing both the transition from fast to slow fronts in a single rupture event and the short-time slip dynamics observed in recent experiments. We identify slow slip immediately following the arrest of a fast front as a phenomenon sufficient for the front to propagate further at a much slower pace. Whether slow fronts are actually observed is controlled both by the interfacial stresses and by the width of the local distribution of forces among microjunctions. Our results show that slow fronts are qualitatively different from faster fronts. Because the transition from fast to slow fronts is potentially as generic as slow slip, we anticipate that it might occur in the wide range of systems in which slow slip has been reported, including seismic faults.

History-dependent friction and slow slip from time-dependent microscopic junction laws studied in a statistical framework

To study how macroscopic friction phenomena originate from microscopic junction laws, we introduce a general statistical framework describing the collective behavior of a large number of individual microjunctions forming a macroscopic frictional interface. Each microjunction can switch in time between two states: a pinned state characterized by a displacement-dependent force and a slipping state characterized by a time-dependent force. Instead of tracking each microjunction individually, the state of the interface is described by two coupled distributions for (i) the stretching of pinned junctions and (ii) the time spent in the slipping state. This framework allows for a whole family of microjunction behavior laws, and we show how it represents an overarching structure for many existing models found in the friction literature. We then use this framework to pinpoint the effects of the time scale that controls the duration of the slipping state. First, we show that the model reproduces a series of friction phenomena already observed experimentally. The macroscopic steady-state friction force is velocity dependent, either monotonic (strengthening or weakening) or nonmonotonic (weakening-strengthening), depending on the microscopic behavior of individual junctions. In addition, slow slip, which has been reported in a wide variety of systems, spontaneously occurs in the model if the friction contribution from junctions in the slipping state is time weakening. Next, we show that the model predicts a nontrivial history dependence of the macroscopic static friction force. In particular, the static friction coefficient at the onset of sliding is shown to increase with increasing deceleration during the final phases of the preceding sliding event. We suggest that this form of history dependence of static friction should be investigated in experiments, and we provide the acceleration range in which this effect is expected to be experimentally observable.

Size scaling of static friction

Sliding friction across a thin soft lubricant film typically occurs by stick slip, the lubricant fully solidifying at stick, yielding and flowing at slip. The static friction force per unit area preceding slip is known from molecular dynamics (MD) simulations to decrease with increasing contact area. That makes the large-size fate of stick slip unclear and unknown; its possible vanishing is important as it would herald smooth sliding with a dramatic drop of kinetic friction at large size. Here we formulate a scaling law of the static friction force, which for a soft lubricant is predicted to decrease as f(m)+Δf/A(γ) for increasing contact area A, with γ>0. Our main finding is that the value of f(m), controlling the survival of stick slip at large size, can be evaluated by simulations of comparably small size. MD simulations of soft lubricant sliding are presented, which verify this theory.

Thermal activation in atomic friction: revisiting the theoretical analysis

The effect of thermal activation on atomic-scale friction is often described in the framework of the Prandtl-Tomlinson model. Accurate use of this model relies on parameters that describe the shape of the corrugation potential β and the transition attempt frequency f(0). We show that the commonly used form of β for a sinusoidal corrugation potential can lead to underestimation of friction, and that the attempt frequency is not, as is usually assumed, a constant value, but rather varies as the energy landscape evolves. We partially resolve these issues by demonstrating that numerical results can be captured by a model with a fitted β and using harmonic transition state theory to develop a variable form of the attempt frequency. We incorporate these developments into a more accurate and generally applicable expression relating friction to temperature and velocity. Finally, by using a master equation approach, we verify the improved analytical model is accurate in its expected regime of validity.

Crack in the frictional interface as a solitary wave

We introduce and investigate a multiscale model for the propagation of rupture fronts in friction. Taking advantage of the correlation length for the motion of individual contacts in elastic theory, we introduce collective contacts which can be characterized by a master equation approach. The problem of the dynamics of a chain of those effective contacts under stress is studied. We show that it can be reduced to an analog of the Frenkel-Kontorova model. In some limits this allows us to derive analytical solutions for kinks describing the rupture fronts. Numerical simulations are used to study more complex cases.

Surface roughness and dry friction

Persson's multiscale contact mechanics theory combined with a multiscale Brillouin-Prandtl-Tomlinson model is used to show that on the basis of these models "dry friction" [i.e., kinetic friction that remains at exceedingly small velocities (but still above the creep range) close to its value at higher velocities] should almost always occur for self-affine surfaces when the dominant interaction between two surfaces in contact is due to interatomic hard core repulsion, except for extremely smooth surfaces (i.e., surfaces with a Hurst index very close to 1).

Dependence of kinetic friction on velocity: master equation approach

We investigate the velocity dependence of kinetic friction with a model that makes minimal assumptions on the actual mechanism of friction so that it can be applied at many scales, provided the system involves multicontact friction. Using a recently developed master equation approach, we investigate the influence of two concurrent processes. First, at a nonzero temperature, thermal fluctuations allow an activated breaking of contacts that are still below the threshold. As a result, the friction force monotonically increases with velocity. Second, the aging of contacts leads to a decrease of the friction force with velocity. Aging effects include two aspects: the delay in contact formation and aging of a contact itself, i.e., the change of its characteristics with the duration of stationary contact. All these processes are considered simultaneously with the master equation approach, giving a complete dependence of the kinetic friction force on the driving velocity and system temperature, provided the interface parameters are known.