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We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady-state solutions on the wettability contrast in that case.
We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]d random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξN ∈ [0, 1]d. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.
Motivated by a recent investigation of Millar and McKay [Director orientation of a twisted nematic under the influence of an in-plane magnetic field. Mol. Cryst. Liq. Cryst435, 277/–286/ (2005)], we study the magnetic field twist-Fréedericksz transition for a nematic liquid crystal of positive diamagnetic anisotropy with strong anchoring and pre-twist boundary conditions. Despite the pre-twist, the system still possesses ℤ2 symmetry and a symmetry-breaking pitchfork bifurcation, which occurs at a critical magnetic-field strength that, as we prove, is above the threshold for the classical twist-Fréedericksz transition (which has no pre-twist). It was observed numerically by Millar and McKay that this instability occurs precisely at the point at which the ground-state solution loses its monotonicity (with respect to the position coordinate across the cell gap). We explain this surprising observation using a rigorous phase-space analysis.
It was demonstrated, on general thermodynamic grounds, that, in non-hydrostatically stressed elastic systems, phase and grain interfaces undergo morphological destabilization due to different mechanisms of “mass rearrangement”. Destabilization of free surfaces due to the combined action of mass rearrangement in the presence of electrostatic field has been well known since the end of the 19th century. Currently, morphological instabilities of thin solid films with electro-mechanical interactions have found various applications in physics and engineering. In this paper, we investigate the combined effects of the stress driven rearrangement instabilities and the destabilization due to the electro-mechanical interactions. The paper presents relevant results of theoretical studies for ferroelectric thin films. Theoretical analysis involves highly nonlinear equations allowing analytical methods only for the initial stage of unstable growth. At present, we are unable to explore analytically the most important, deeply nonlinear regimes of growth. To avoid this difficulty, we developed numerical tools facilitating the process of solving and interpreting the results by means of visualization of developing morphologies.
At present, there is a consensus that various Stress Driven Rearrangement Instabilities (SDRI) are the implications of the mathematically rigorous theoretical Gibbs thermodynamics. Many applied researchers and practitioners believe that SDRI are also universal physical phenomena occurring over a large range of length scales and applied topics. There is a multitude of publications claiming experimental observation of the SDRI based phenomena. This opinion is challenged by other highly respected scholars claiming theoretical inconsistencies and multiple experimental counterexamples. Such an uncertainty is too costly for further progress on the SDRI topic. The ultimate goal of our project is to resolve this controversy.
The project includes experimental, theoretical, and numerical studies. Among various plausible manifestations of SDRI, the authors focused only on two most promising for which the validity of the SDRI has already been claimed by other researchers: a) stress driven corrugations of the solid-melt phase interface in macroscopic quantum 4He and b) the dislocation-free Stranski-Krastanov pattern of growth of semiconductor quantum dots. We devised a program and experimental set-ups for testing applicability of the SDRI mechanisms using the same physical systems as before but using implications of the SDRI theory for 2D patterning which have never been tested in the past.
Connections are established between mixing or ergodic properties of maps on the one hand, and the convergence of the iterates of the map, or of the empirical measures of the iterates, to a constant measure-valued map, on the other. The uniqueness of an absolutely continuous ergodic measure can also be verified via the convergence. The technique helps to identify ergodic and mixing pairs and verify the uniqueness in specific examples.
Coarsening of solutions of the integro-differential equation
where Ω ⊂ ℝn, J(·) [ges ] 0, ε > 0 and
f(u) = u3 − u (or similar bistable nonlinear term), is
examined, and compared with results for the Allen–Cahn partial differential equation. Both
equations are used as models of solid phase transitions. In particular, it is shown that when
ε is small enough, solutions of this integro-differential equation do not coarsen, in contrast
to those of the Allen–Cahn equation. The special case J(·) ≡ 1 is explored in detail, giving
insight into the behaviour in the more general case J(·) [ges ] 0. Also, a numerical approximation
method is outlined and used on tests in both one- and two-space dimensions to verify and
illustrate the main result.
The equilibrium configurations of a one-dimensional variational model that
combines terms expressing the bulk energy of a deformable crystal and its
surface energy are studied. After elimination of the displacement, the
problem reduces to the minimization of a nonconvex and nonlocal functional of
a single function, the thickness. Depending on a parameter which strengthens
one of the terms comprising the energy at the expense of the other, it is
shown that this functional may have a stable absolute minimum or only a
minimizing sequence in which the term corresponding to the bulk energy is
forced to zero by the production of a crack in the material.
Surface diffusion within intergrain space is studied theoretically with the help of the Onsageristic approach of irreversible thermodynamics. The grains are treated as isotropic elastic half solids. It is assumed that species of one of the grains are capable to leave their equilibrium positions in the parent solid body and then they migrate within the inter-grain space along the surface of their parent body. Decrease of the total accumulated energy is postulated to be a driving force of the diffusion processes.
We investigate the influence of mass forces (in particular, of gravitation and van der Waals forces) on the critical film thickness of thin films attached to solid substrates and establish corresponding corrections of the earlier published formula Hcril = ΣΜ/τ2 (where Σ is the surface energy, Μ - the shear Modulus, and τ - the mismatch stress). It is assumed that the films’ particles are able to rearrange their relative positions in the lattices, and the equilibrium rearrangement is determined by minimizing the total static energy. Recently, it was demonstrated that morphological stability of interfaces in crystalline solids with the rearrangement is extremely sensitive to the presence of shear stresses. Equilibrium theory of elasticity of pre-stressed solids with the rearrangement of their material particles has already allowed the prediction of the appearance of corrugations in He4 films and to explain the dislocation-free Stranski-Krastanov pattern of epitaxial growth of thin solid films. The explicit asymptotic formulae announced here are especially useful in the case of small mass force, the effects of which can be detectable and even significant for some of the above mentioned phenomena.
We study possible morphologies of epitaxial films atop attractive substrates appearing as a result of competition of misfit stresses, van der Waals forces and surface energy. Corresponding formula for the critical thickness of the dislocation-free Stranski-Krastanov pattern is established for the isotropic deformable films and substrates. If the film thickness exceeds the critical magnitude the layer-by-layer pattern switches to islanding. At the first stage the islands have a shape of striae (i.e. long parallel trenches with periodic spacing). We discuss also i)the circumstances in which surface morphology of the film corresponds to a two-dimensional superlattice of islands rather than a one dimensional lattice of striae and ii)the influence of a buffer inter-layer.
It was demonstrated earlier [1,2] in the framework of equilibrium thermodynamics that the morphological stability of the free boundaries and interfaces in crystals is extremely sensitive to the presence of shear stresses. Relying on that idea we have established the formula H = μσ/τ2 of a critical thickness of solidifying He4 films and of the dislocation-free Stranski-Krastanow growth of epitaxial films (where σ – the coefficient of surface tension, μ - the shear module of the crystal, τ - the external or misfit stress). In this report we present certain facts pertaining to possible patterns of the growing corrugations and introduce the second critical thickness at which a symmetry change in the patterns has to occur.
We discuss the static and quasi-static problems appearing in the theory of morphological instability of interfaces. The approach has allowed to predict the corrugations in He4 films and to explain the dislocation-free Stranski-Krastanow pattern of epitaxial growth of thin solid films with the critical film thickness H = σμ/τ2 (σ is a surface energy, μ- the shear modulus, and τ - the mismatch stress). In this paper we discuss possible morphological patterns of corrugations and their changes which appear in result of the stress driven “rearrangement” destabilization of originally flat interfaces.
In the absence of surface tension and external force fields, the equilibrium between a hydrostatically stressed crystal and its melt is neutral with respect to the perturbations associated with particle transfer from one region of the boundary into another. However, under the action of arbitrary small nonhydrostatic components of the stress field in the elastic crystal, the neutral equilibrium is transformed to an unstable equilibrium . This instability is very general in nature; for example, for it to be seen the liquid media need only to be able to dissolve the solid phase or in some way to assist the transport of particles along the crystal's surface. In contrast, the surface tension, roughly speaking, stabilizes the shape of the interphase boundary but it cannot suppress the instability generated by the nonhydrostatic components of the stress field in the region of sufficiently long perturbations. Until now the basic instability mechanism discussed here seems to have escaped the attention of theorists. This mechanism allows one to look in a completely new way at a broad range of phenomena. We discuss tentative manifestations and role of this instability in low temperature physics, in materials science, in theory of crystal growth, and, in particular, in theory of epitaxy and of the Stranski-Krastanow pattern of growth of thin crystalline films.
It was demonstrated in  that, in the absence of surface tension aflat boundary of non-hydrostatically stressed elastic solids is always unstable with respect to “mass rearrangement”. The physical mechanisms of the rearrangement can be different, for instance, a)melting-freezing or vaporization-sublimation processes at liquid-solid or vapor-solid phase boundaries, b»surface diffusion of particles along free or interfacial boundaries, b)adsorption-desorbtion of the atoms in epitaxial crystal growth, etc… We discuss the role of this instability in the problems of epitaxy and, in particular, the opportunities delivered by this instability for explanation of the recently discovered phenomena of the dislocation-free Stranski-Krastanow pattern of growth . These phenomena cannot be interpreted in the framework of traditional viewpoints since, according to the classical theory, the Stranski-Krastanow pattern is a result of proliferation of the misfit dislocation appearing on the interface “crystalline film-substratum” .
In this paper we examine the influence of capillarity on existence and uniqueness of travelling wave solutions in an isothermal system of van der Waals fluids. Existence and non-uniqueness theorems are proved using phase-space analysis and topological methods.
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