Kinetics of ammonia synthesis at non-equilibrium conditions

Abstract

A recent theoretical model (Blumenfeld, Phys. Rev. Lett. 76 (1996) 3703) is described for modes I and III crack propagation dynamics in noncrystalline materials on mesoscopic lengthscales. Fracture has been one of the longest standing problems in physics and materials science, and despite much effort, several fundamental issues have stubbornly resisted resolution:(i) Running cracks reach a steady-state velocity of roughly half the shear wave speed, while theoretical predictions based on energetics are twice as high. The discrepancy originates from dynamics, but a consistent dynamical model has been slow to emerge.(ii) There is little understanding of the mechanisms for crack initiation and arrest and the hysteresis between them. Lattice trapping, although relevant on the atomic scale, cannot explain this phenomenon on mesoscopic and macroscopic scales.(iii) Another intriguing phenomenon is appearance of velocity periodic oscillations in some materials and the relation between this and material properties.(iv) As a result of the above issues, there is currently no consensus on the form of the equations of motion that govern mesoscale fracture dynamics.Whether explicitly or implicitly, most traditional models use quasi-static and near-equilibrium concepts to analyse the dynamics of propagation. It is argued here that such approaches are bound to fail. Two reasons are responsible for this and consequently for the dire understanding of this problem: First, most fast fracture processes are usually restricted to post-mortem measurements of the already fractured system, while the process itself is too fast to capture. Only recently there emerged experiments where the dynamic process is continuously monitored. Second, it is strongly contended here that the fracture phenomenon is governed by different mechanisms on different length-scales, a crucial aspect that has not received sufficient attention. In ideally brittle propagation, the crack is atomically sharp and therefore atomic potentials are important (5–10 Å). Anharmonicity plays a significant role on this scale due to large local strains at the crack tip, which gives rise to a strong nonlinear behaviour. On large scales (>μm), continuum linear elasticity describes quite well the stress field and the far-away elastic energetics. This is exactly because cracks propagate slower than the bulk speed of sound, which allows the bulk stress to relax to its static value in the frame of the moving crack. Ultimately, this is the reason why contour integral calculations of energy influx into the crack tip are valid as long as the contours are taken well away from the tip. Between the atomic and the continuous scales there are at least two more relevant length-scales: One is that of the cohesive zone, which is the region where the continuous stress field description breaks down due to the discreteness of the lattice. It is of the order of several lattice constants and about one order of magnitude above the atomic scale (∼10–50 Å). The fourth length-scale, and the one we focus on here, is that defined by the sizes of the nano- and microcracks that form dynamically in front of the propagating tip. Traditional continuum theory cannot be used on this scale due to the strong inhomogeneity. Smoothing the disorder by wishful homogeneization methods does not work for reasons to be detailed in this presentation. A strongly disordered region ahead of the crack is indeed observed experimentally (processing zone). We suggest here that on this `dynamic scale' the local stress field ahead of the crack front relaxes very slowly, which gives rise to a supersonic-like local behaviour even though macroscopically the crack propagates slower than the shear wave speed. This leads to shielding of the tip from the far field energy equilibration and therefore to a far-from-equilibrium process. The mesoscale-dominated dynamics do not invalidate the long range continuum quasi-static calculations, as long as the latter are applied not to the bare crack tip, but rather to the tip `dressed' by the processing zone.Starting from the idea that the tip responds to the local stress, an equation of motion is derived from first principles. The resulting dynamic equation is solved exactly and analysed. A rich propagation is found: the crack either propagates at a steady state speed, which can be predicted from material properties, or the speed oscillates periodically. Which mode is chosen depends on one material parameter. Possible sources of noise are discussed next and it is shown that noise can strongly modify the dynamics into: quasi-periodic propagation, intermittent propagation, or a range of noise-driven steady states. The analysis of these behaviours is outlined and future directions are suggested.