Zbigniew Domański

Application of Control Volume Method Using the Voronoi Tessellation in Numerical Modelling of Solidification Process

The paper presents the method to analyse the thermal processes occurring in the cast composite solidification. The cast is formed by a bundle of parallel fibres randomly immersed in a host metal matrix. The heat is transferred from the metal matrix and absorbed by the fibres. The objective of this paper is to evaluate the volumetric fraction of the fibres for which the solidification of the metal matrix occurs only due to the presence of fibres playing a role of internal chills. Our method is to compute Voronoi diagrams with Voronoi regions representing the geometric location of the fibres in the metal matrix and to use these regions as control volumes within a variant of the Control Volume Method.

Avalanche of bifurcations and hysteresis in a model of cellular differentiation

Cellular differentiation in a developing organism is studied via a discrete bistable reaction-diffusion model. A system of undifferentiated cells is allowed to receive an inductive signal emanating from its environment. Depending on the form of the nonlinear reaction kinetics, this signal can trigger a series of bifurcations in the system. Differentiation starts at the surface where the signal is received and either cells change type up to a given distance or, under other conditions, the differentiation process propagates throughout the whole domain. When the signal diminishes, hysteresis is observed.

Statistics of Critical Avalanches in Vertical Nanopillar Arrays

Nanopillar arrays are encountered in numerous areas of nanotechnology such as bio-medical and chemical sensing, nanoscale electronics, photovoltaics or thermoelectrics. Especially arrays of nanopillars subjected to uniaxial microcompression reveal the potential applicability of nanopillars as components for the fabrication of electro-mechanical sense devices. Thus, it is worth to analyze the failure progress in such systems of pillars. Under the growing load pillars destruction forms an avalanche and when the load exceeds a certain critical value the avalanche becomes self-sustained until the system is completely destroyed. In this work we have explored the distributions of such catastrophic avalanches appearing in overloaded systems. Specifically, we analyze the relations between the size of an avalanche being the numbers of instantaneously crushed pillars and the size of the corresponding array of nanopillars using different load transfer protocols.

Geometry-Induced Transport Properties of Two Dimensional Networks

This work analyses, in a general way, how the geometry of a network influences the transport of a hypothetical fluid through the network’s channels. Here, it is the geometry of the network that matters even though the network and fluid bear broad interpretations ranging from a liquid passing through channel space of a filter, electrons moving inside circuits, bits flying between servers to a suburban highways crowded by cars. The geometrical properties of networks have attracted much attention due to the progress in the field of computer science, mathematical biology, statistical physics and technology. A lot of systems operate as a two-dimensional network and numerous devices are constructed in a planar fashion. Examples are grids of processors, radar arrays, wireless sensor networks, as well as a wide range of micromechanical devices. Especially, the microfluidic systems are built with the use of methods borrowed from the semiconductor industry. Such systems generally employ the fabrication of highly ordered microscale structures. Also a migration of voids in almost jammed granulates in an example worth to mention in this context since the void-position rearrangement resembles the sliding block puzzles. Theoretical models related to a given problem are useful if they help researches to explain observed facts and enable them to predict the system’s behaviour beyond the experiments already conducted. The complexity of a real system frequently prevents constructing a model, in which all the observed characteristics can be accurately captured. Instead of constructing a model to acquire all the details, and in consequence building the model which is complicated and analytically untreatable, it is possible to formulate a rather rude, but statistically correct, description of the transport phenomena which obeys averaged characteristics. The premise of statistical modelling of a network flow phenomena is the graph theory with the fundamental equivalence between the maximum flow and minimal cost circulation and the cost-capacity scaling. Thus, the populations of transportingnetwork, appropriate for such statistical analysis, and based on graph theory may provide valuable information about the effectiveness of the network topology.

The tricritical point in the quantum Ising S = 1 spin glass with biaxial crystal-field effects

We study a quantum generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with biaxial crystal-field effects described by two uniaxial anisotropy parameters Dx and Dy. For spin dimensionality S = 1 we report an analytical and numerical analysis in the (T, Dx, Dy) parameter space (with T being the temperature). For D ≡ Dx = Dy the model effectively becomes classical and identical with the crystal-field-split spin-glass Ising model (introduced by Ghatak and Sherrington) showing a discontinuous phase transition to the spin-glass phase on a portion of the T-D line.

The Model Coupling Liquid Bridge Between Ellipsoidal Grains

Granular materials are the subject of scientific studies due to their unusual physical properties which differ significantly from solid and liquid states of matter. In this paper we are interested in the role played by humidity on static and dynamic properties of systems consisting of non-spherical grains. The addition of a liquid to the material adds an attractive force to the system and then, increased its stability. Quantitative description of wetting thermodynamics is sensitive not only to the contact angle between solid and the liquid but also to the to the shape of grains and thus we analyze ellipsoidal grains and we assume that liquid spreads uniformly over the whole grain’s surface. We consider a model grain’s surface consisting of asperities of equal size uniformly distributed over the grain’s surface. We also suppose that each asperity may be either totally filled with liquid or stay empty. Thus, in our approach we consider two regimes of the inter-grain adhesive force versus volume of the wetting layer. For very small amount of liquid, the capillary force comes from the fluid accumulated around a small number of asperities at which two neighbouring grains are in contact. If the fluid wets the surface of the grains then all asperities are filled and inter-grain adhesive energy is determined mainly by the macroscopic curvature of the grain, and the surface roughness does not play a crucial role. In this case, the distribution of values of inter-grain energies is determined only by macroscopic quantities, i.e. the geometry of grains and material characteristics. Using toroidal approximation for the shape of liquid bridges and some simple probabilistic arguments we analyze the influence of amount of liquid on mutual grain - grain orientation. We found two energetically favorable orientations: one with mutually parallel and second with perpendicular axes of contacted grains.

The asymmetric Hubbard model on a two-dimensional cluster

An interaction between two heavy electrons mediated by two light electrons is investigated within the asymmetric Hubbard model on a square cluster with sites. The pair correlation function describing the probabilities of all relative positions of the heavy electrons is calculated by an approximative method. An effective attraction is found to exist in a certain range of model parameters.

Interaction between heavy electrons mediated by itinerant electrons in the asymmetric Hubbard model

Abstract An interaction between two heavy electrons mediated by two light electrons is investigated within the asymmetric Hubbard model on finite systems. The probability of finding one heavy electron at site l when the other occupies site k is calculated by an approximative method. An effective attraction is found to exist for a certain set of model parameters.

Distribution of the Distance Between Receptors of Ordered Micropatterned Substrates

We study the statistics of equally spaced pairs of receptors on a family of ordered flat microsubstrates whose adhesive centers form regular tessellations. We establish relationship between the symmetry of substrates and the probability density of the end-to-end polymer separation in terms of the so-called Manhattan distance.

Estimation of Susceptibility to Hot Tearing in Solidifying Casting

Hot tearing, also called hot cracking, is a serious defect that appears during the solidification of an alloy. Due to the low recurrence of the phenomena occurring during alloy solidification, such as the evolution of grained structure or stress redistributions, the casting’s susceptibility to hot tearing can be estimated only in an approximate way. Predicting the appearance of hot tears in alloys is thus an important issue in industrial practice. This work concerns with a new criterion for hot tearing evaluation in castings. An algorithm for the computer simulations of the phenomena accompanying the casting formation is introduced and discussed.