Abstract. We study linkages of two scales in the relaxation of an axisymmetric crystal with a facet in evaporation-condensation kinetics. The macroscale evolution is driven by the motion of concentric circular, repulsively interacting line defects (steps) which exchange atoms with the vapor. At the microscale, the step velocity is proportional to the variation of the total step free energy, leading to large systems of differential equations for the step radii. We focus on two step flow models. In one model (called M1) the discrete mobility is simply proportional to the upper-terrace width; in another model (M2) the mobility is altered by an extra geometric factor. By invoking self-similarity at long time, we numerically demonstrate that (i) in M1, discrete slopes follow closely a continuum thermodynamics approach with "natural boundary conditions" at the facet edge; (ii) in contrast, predictions of M2 deviate from results of the above continuum approach; and (iii) this discrepancy can be eliminated via a continuum boundary condition with a geometry-induced jump for top-step collapses. At the macroscale, both step models give rise to free-boundary problems for a second-order, parabolic partial differential equation, which we study via the subgradient formalism. We discuss the interpretation of the facet height as shock and prove convergence of the solution of each discrete scheme to the (weak) entropy solution of a conservation law if steps do not interact.

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a b s t r a c tWe apply classical homogenization to derive macroscopic relaxation laws for crystal surfaces with distinct inhomogeneities at the microscale. The proposed method relies on a formal multiscale expansion in one spatial coordinate. This approach transcends the coarse graining applied previously via Taylor expansions. Our work offers an extension of the static homogenization formulated in a brief report [D. Margetis, Homogenization of reconstructed crystal surfaces: Fick's law of diffusion, Phys. Rev. E 79 (2009) 052601] to account for surface evolution. The starting point is the Burton-Cabrera-Frank (BCF) model for the motion of line defects (steps) separating nanoscale terraces. We enrich this model with sequences of distinct material parameters, i.e., disparate diffusivities of adsorbed atoms (adatoms) across terraces, kinetic sticking rates at step edges, and step energy parameters for elastic-dipole interactions. Multiscale expansions for the adatom concentration and flux are used, with a slow diffusive time scale consistent with the quasi-steady regime for terrace diffusion. The ensuing macroscopic, nonlinear evolution laws incorporate averages of the microscale parameters.

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Purpose To investigate the paths of thoracic epidural catheters in children, this retrospective study was performed. Methods We investigated 73 children aged 4 to 12 (mean ± SD 7.8 ± 2.3) years, who underwent the Nuss procedure for pectus excavatum repair under combined general and epidural anesthesia over a 5-year period at Tokyo Metropolitan Police Hospital. Following induction of general anesthesia, we inserted a radiopaque epidural catheter via the T5/6 or T6/7 interspace and advanced for 5 cm cephalad in the thoracic epidural space. We evaluated the paths of the epidural catheters on plain chest radiographs after surgery. Results The median level for the catheter tip location was T3 (range C6–T7), while the median number of vertebrae crossed by the catheter tips was 2.5. In most children, the catheters advanced straight for the first 2–3 cm (1–1.5 vertebrae) in the thoracic epidural space. However, they continued to advance straight in only 25 children, while they exhibited curved or coiled paths in the remaining 48. The catheter tips were located at higher levels in children with straight epidural catheter paths [median (range) T2 (C6–T4)] than in those with curved or coiled paths after the initial 2–3 cm [median (range) T4 (T2–T7)] (p < 0.0001). Conclusions Our findings indicate that the course of epidural catheters in children is unpredictable after the first 2–3 cm in the thoracic epidural space. Clinicians should be aware of such findings, although further studies are required for confirmation.