This work provides a ground for a quantitative interpretation of experiments on step bunching during sublimation of crystals with a pronounced Ehrlich-Schwoebel (ES) barrier in the regime of weak desorption. A strong step bunching instability takes place when the kinetic length d d = Ds/K d is larger than the average distance l between the steps on the vicinal surface; here Ds is the surface diffusion coefficient and K d is the step kinetic coefficient. In the opposite limit d d ≪ l the instability is weak and step bunching can occur only when the magnitude of step-step repulsion is small. The central result are power law relations of the form L ∼ H α , lmin ∼ H −γ between the width L, the height H, and the minimum interstep distance lmin of a bunch. These relations are obtained from a continuum evolution equation for the surface profile, which is derived from the discrete step dynamical equations for the case d d ≫ l. The analysis of the continuum equation reveals the existence of two types of stationary bunch profiles with different scaling properties. Through comparison with numerical simulations of the discrete step equations, we establish the value γ = 2/(n + 1) for the scaling exponent of lmin in terms of the exponent n of the repulsive step-step interaction, and provide an exact expression for the prefactor in terms of the energetic and kinetic parameters of the system. For the bunch width L we observe significant deviations from the expected scaling with exponent γ = 1 − 1/α, which are attributed to the pronounced asymmetry between the leading and the trailing edges of the bunch, and the fact that bunches move. Through a mathematical equivalence on the level of the discrete step equations as well as on the continuum level, our results carry over to the problems of step bunching induced by growth with a strong inverse ES effect, and by electromigration in the attachment/detachment limited regime. Thus our work provides support for the existence of universality classes of step bunching instabilities [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002)], but some aspects of the universality scenario need to be revised.
The surface properties of human meibomian lipids (MGS), the major constituent of the tear film (TF) lipid layer, are of key importance for TF stability. The dynamic interfacial properties of films by MGS from normal eyes (nMGS) and eyes with meibomian gland dysfunction (dMGS) were studied using a Langmuir surface balance. The behavior of the samples during dynamic area changes was evaluated by surface pressure-area isotherms and isocycles. The surface dilatational rheology of the films was examined in the frequency range 10(-5) to 1 Hz by the stress-relaxation method. A significant difference was found, with dMGS showing slow viscosity-dominated relaxation at 10(-4) to 10(-3) Hz, whereas nMGS remained predominantly elastic over the whole range. A Cole-Cole plot revealed two characteristic processes contributing to the relaxation, fast (on the scale of characteristic time τ < 5 s) and slow (τ > 100 s), the latter prevailing in dMGS films. Brewster angle microscopy revealed better spreading of nMGS at the air-water interface, whereas dMGS layers were non-uniform and patchy. The distinctions in the interfacial properties of the films in vitro correlated with the accelerated degradation of meibum layer pattern at the air-tear interface and with the decreased stability of TF in vivo. These results, and also recent findings on the modest capability of meibum to suppress the evaporation of the aqueous subphase, suggest the need for a re-evaluation of the role of MGS. The probable key function of meibomian lipids might be to form viscoelastic films capable of opposing dilation of the air-tear interface. The impact of temperature on the meibum surface properties is discussed in terms of its possible effect on the normal structure of the film.
We propose a unified treatment of the step bunching instability during epitaxial growth. The scaling properties of the self-organized surface pattern are shown to depend on a single parameter, the leading power in the expansion of the biased diffusion current in powers of the local surface slope. We demonstrate the existence of universality classes for the self-organized patterning appearing in models and experiments.
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