Browsing by Author "Goriely, Alain"
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- ItemControlled topological transitions in thin film phase separation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hennessy, Mathew G.; Burlakov, Victor M.; Münch, Andreas; Wagner, Barbara; Goriely, AlainIn this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.
- ItemMini-Workshop: Mathematics of Differential Growth, Morphogenesis, and Pattern Selection(Zürich : EMS Publ. House, 2015) Goriely, Alain; Kuhl, Ellen; Menzel, AndreasLiving structures are highly heterogeneous systems that consist of distinct regions made up of characteristic cell types with a specific structural organization. During evolution, development, disease, or environmental adaptation each region may grow at its own characteristic rate. Differential growth creates a balanced interplay between tension and compression and plays a critical role in biological function. In plant physiology, typical every-day examples include the petioles of celery, caladium, or rhubarb with a slower growing compressive outer surface and a faster growing tensile inner core. In developmental biology, differential growth is critical to the organogenesis of various structures including the gut, the heart, and the brain. From a structural point of view, these phenomena are close associated with instabilities, of twisting, looping, folding, and wrinkling. From a mathematical point of view, the governing equations of organogenesis are highly nonlinear and often characterized through multiple bifurcation points. Bifurcation is critical in symmetry breaking, pattern formation, and selection of shape. While biologists are studying differential growth, morphogenesis, and pattern selection merely by observation, our goal in this workshop is to explore, discuss, and advance the fundamental theory of differential growth to characterize morphogenesis and pattern selection by mathematical modeling. This workshop will bring together scientists with similar interests and complementary backgrounds in applied mathematics, mathematical biology, developmental biology, plant biology, dynamical systems, biophysics, biomechanics, and clinical sciences. We will identify common features of growth phenomena in living systems with the overall objectives to establish a unified mathematical theory for growing systems and to identify the necessary mathematical tools to address challenging questions in biology and medicine.
- ItemPropagating topological transformations in thin immiscible bilayer films(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Hennessy, Mathew G.; Burlakov, Victor M.; Münch, Andreas; Wagner, Barbara; Goriely, AlainA physical mechanism for the topological transformation of a two-layer system confined by two substrates is proposed. Initially the two horizontal layers, A and B, are on top of each other, but upon a sufficiently large disturbance, they can rearrange themselves through a spontaneously propagating sectioning to create a sequence of vertical alternating domains ABABAB. This generic topological transformation could be used to control the morphology of fabricated nanocomposites by first creating metastable layered structures and then triggering their transformation. The generality is underscored by formulating conditions for this topological transformation in terms of the interface energies between phases and substrates. The theoretical estimate for the width of the domains is confirmed by simulations of a phase-field model and its thin-film/sharp-interface approximation.