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We study the evolution of step bunches on vicinal surfaces using athermodynamically consistent step-flow model that (i) circumvents thequasistatic approximation that prevails in the literature by accounting for thedynamics of adatom diffusion on terraces and attachment-detachment at steps(referred to as the dynamical effect), and (ii) generalizes the expression ofthe step chemical potential by incorporating the necessary coupling between thediffusion fields on adjacent terraces (referred to as the chemical effect).Having previously shown that these effects can explain the onset of stepbunching without recourse to the inverse Ehrlich-Schwoebel (iES) barrier orother extraneous mechanisms, we are here interested in the evolution of stepbunches beyond the linear-stability regime. In particular, the numericalresolution of the step-flow problem yields a robust power-law coarsening of thesurface profile, with the bunch height growing in time as $H\sim t^{1/2}$ andthe minimal interstep distance as a function of the number of steps in thebunch cell obeying $\ell_{min}\sim N^{-2/3}$. Although these exponents havepreviously been reported, this is the first time such scaling laws are obtainedin the absence of an iES barrier or adatom electromigration. In order tovalidate our simulations, we take the continuum limit of the discrete step-flowsystem, leading to a novel nonlinear evolution equation for the surface height.We investigate the existence of self-similar solutions of this equation andconfirm the 1/2 coarsening exponent obtained numerically for $H$. We highlightthe influence of the combined dynamical-chemical effect and show that it can beinterpreted as an effective iES barrier in the standard BCF theory. Finally, weuse a Pad\'e approximant to derive an analytical expression for the velocity ofsteadily moving step bunches and compare it to numerical simulations.

Scaling laws for step bunching on vicinal surfaces: the role of the dynamical and chemical effects