Two-dimensional Cellular Pattern Research Articles

Mechanical performance of additively manufactured lightweight cellular solids: Influence of cell pattern and relative density on the printing time and compression behavior

A comprehensive investigation is presented on the Fused Filament Fabrication (FFF) technology’s possibilities to create cellular solids with a broad spectrum of specific stiffness and strength, modifying cell geometry and size, while addressing manufacturing matters such as inherent defects and built time. Thirteen typologies of two-dimensional cellular patterns with different relative densities are examined. Results have allowed conclusions to be drawn regarding the influence of cell type and infill density on mechanical performance. Intra-layer and inter-layer inherent defects identified after manufacturing highlight the importance of optimizing filament trajectories. A reliable comparison of the elastic properties of the cellular patterns as a function of their density is presented. An experimentally validated numerical model is provided for predicting the compression stiffness of the different cell patterns with an average deviation below 5%. The model can reproduce the behavior in the elastic range based on tensile specimen properties and a Normal Stiffness Factor to account for the phenomenon of elastic asymmetry of the FFF printed samples. The wide range of results achieved is experimental confirmation of the potential of FFF cellular solids. Lastly, this investigation provides analytical, numerical, and empirical validated evidence to further design-for-additive manufacturing strategies with cellular solids for designing advanced lightweight structures.

Phase field crystal simulations of the kinetics of Ostwald ripening in two dimensions.

The kinetics of Ostwald ripening of solid domains in the liquid phase of one-component systems in two dimensions is investigated numerically via the phase field crystal model. The simulations, which are performed systematically as a function of volume fraction of the solid phase, show that dynamical scaling is reached during late times, and the growth law is in good agreement with the classical theory of Lifshitz, Slyozov, and Wagner (LSW), i.e., R[over ¯]∼t^{1/3}, an indication that domain growth is mediated by the long-range interdomain diffusion of atoms. In contrast to the LSW theory, however, the domain size distribution is symmetric, and can be fit with a Gaussian. The investigation of the topological domain structure, through the Voronoi tessellation of the domains' centers of mass shows that both the Lewis law and the Aboav-Weaire law of two-dimensional cellular patterns are satisfied, implying that the kinetics proceed such that the conformational entropy of the domain-containing Voronoi cells is maximized. These results are in very good agreement with an earlier experimental study of a phase-separating phospholipid-cholesterol Langmuir film.

Dynamic Transfer of Two-dimensional Cellular Patterns into Fibrin Gels

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Control of Cellular Pattern Formation in the Vertebrate Inner Retina by Homotypic Regulation of Cell-Fate Decisions

The vertebrate retina is composed of cellular arrays that are nonrandom across two-dimensional space. The determinants of these nonrandom two-dimensional cellular patterns in the inner nuclear layer of the retina were investigated using empirical and computational modeling techniques. In normal and experimental models of goldfish retinal growth, the patterns of tyrosine hydroxylase- and serotonin-positive cells indicated that neither cell death nor lateral migration of differentiated cells were dominant mechanisms of cellular pattern formation. A computational model of cellular pattern formation that used a signaling mechanism arising from differentiated cells that inhibited homotypic cell-fate decisions generated accurate simulations of the empirically observed patterns in normal retina. This model also predicted the principal atypical cellular pattern characteristic, a transient cell-type-specific hyperplasia, which was empirically observed in the growing retina subsequent to selective ablation of differentiated retinal cells, either tyrosine hydroxylase positive or serotonin positive. The results support the hypothesis that inhibitory spatiotemporal regulation of homotypic cell-fate decisions is a dominant mechanistic determinant of nonrandom cellular patterns in the vertebrate retina.

Deterministic cellular descendance and its relationship to the branching of plant organ axes

A double-wall map L-system, designated as S(5-5), was developed to simulate the cellular pattern found at the summit of shoot apices of Psilotum nudum. Commencing from a 3-sided autoreproductive founder cell, fives steps of simulation established a basic set of ten different cell types. Continuing the simulation beyond the fifth step revealed that, in addition to the regular production of new 3-sided cells, a group of autoreproductive 5-sided cells came into being. A close correspondence exists between the cells of the two-dimensional simulation and the two-dimensional cellular patterns found on the epidermis of the apices of Psilotum species. The 3-sided cells produced during the simulation correspond to the potentially organogenetic 3-sided cells that can be seen upon the apical surfaces. Successive generations of these 3-sided apical cells (which are actually 4-sided tetrahedral cells when viewed in three dimensions) and their immediate descendants are thought to be selected to organise the successive pairs of apices that bring about the repeated bifurcation of the Psilotum shoots. The 5-sided cells contribute to the cellular "pavements" which separate these pairs of organogenetic centres, each with their 3-sided apical cells. The cellular patterns simulated by the S(5-5) system may also correspond to the cellular patterns found on the surfaces of some other pteridophyte apices, including that of the rhizophores of Selaginella species. Tritiated-thymidine labelling of rhizophore apices revealed a group of nonproliferating cells that was associated with rhizophore bifurcation and which may correspond to a group of pavement cells. Nonproliferating cells, by regulating the siting of new organogenetic centres, may have evolved as an accompaniment to branching events such as the bifurcation of root and organ axes.

Lower bounds for the surface energy of two-dimensional foams.

Amongst the two-dimensional cellular patterns that fill a plane, dry foams at stable equilibrium typify a particular subset for which the total perimeter P of cell boundaries ( i.e., films between bubbles) has a local minimum. For a given set of bubble areas Ai (i=1,..., N), P can be written in the form P=R(SigmaN(i=1) square root Ai)/2, where R is topology dependent. We seek the set of areas Ai and the cluster topology that minimise R, and propose lower bounds for R that set lower bounds for the surface energy of i) individual bubbles, with circular edges meeting at 2pi/3 angles at vertices (Plateau cells), and ii) infinite periodic bubble clusters.

Lewis' law versus Feltham's law in soap froth

The scaling of area and perimeter of n-sided cells versus n in a two-dimensional cellular pattern are investigated. Lewis' law states that 〈< A( n)〉 is linear in n while Feltham's law states that the perimeter 〈 E( n)〉 is linear in n. We have tested these two different predictions on soap froth for large samples, starting with approximately 5000 cells, at different times and different gap thicknesses. We conclude that Feltham's law has a smaller statistical error than Lewis' law. We also compared the results with several two-dimensional Voronoi constructions of random points and perturbed triangular lattice. We found that the error bars of the Lewis's law and Feltham's law are comparable in the Voronoi constructions. Using geometric constraints, we argue and verified by experimental data that there is a universal law obeyed by the slopes and intercepts of these laws, on all samples.

Phase Wave in a Cellular Structure

Phase waves and their stability in a cellular pattern is discussed from a viewpoint of exper­ imentalists. When the cellular pattern exhibits oscillatory instability, rich phenomena can be observed. The prototype experimental system is the electrohydrodynamic convection of a nematic liquid crystal in which a two-dimensional cellular pattern of convection exhibits limit cycle oscilla­ tion. We observe localized target pattern of wave fronts emitted from a self-organized center. The localized structure has a life cycle: it arises from a uniform state, grows and collapses by nucleating defects at its center. They are created by a phase instability through the coupling between the oscillatory mode and the underlying cellular pattern. We experimentally determine coupled complex Ginzburg- Landau equations as well as phase equations derived from them, and we confirm that the equations quantitatively describe the instability. We also discussed mechanism and process of collapse of target which are closely related to a turbulent nature of defects in the cellular pattern.

Phase Wave in a Cellular Structure

Phase waves and their stability in a cellular pattern is discussed from a viewpoint of experimentalists. When the cellular pattern exhibits oscillatory instability, rich phenomena can be observed. The prototype experimental system is the electrohydrodynamic convection of a nematic liquid crystal in which a two-dimensional cellular pattern of convection exhibits limit cycle oscillation. We observe localized target pattern of wave fronts emitted from a self-organized center. The localized structure has a life cycle: it arises from a uniform state, grows and collapses by nucleating defects at its center. They are created by a phase instability through the coupling between the oscillatory mode and the underlying cellular pattern. We experimentally determine coupled complex Ginzburg-Landau equations as well as phase equations derived from them, and we confirm that the equations quantitatively describe the instability. We also discussed mechanism and process of collapse of target which are closely related to a turbulent nature of defects in the cellular pattern.

Complex dynamics of a localized target pattern in electrohydrodynamic convection.

A two-dimensional cellular pattern (lattice) created by electrohydrodynamic convection of a nematic liquid crystal exhibits limit cycle oscillations. We found a localized target pattern of wave fronts emitted from a self-organized oscillating center. The localized structure has a life cycle: it arises from a uniform state, grows, and collapses from nucleating defects at its center. This phenomenon is caused by an interaction between the waves and deformations of the underlying lattice, and is different from that in localized traveling waves.

Computer simulation of cellular pattern growth in two and three dimensions

Abstract Computer simulations of cellular pattern growth (domain growth) have been carried out for two- and three-dimensional vertex model. Geometrical properties beyond the edge-number distribution have been studied for a two-dimensional cellular pattern in the scaling regime. The method of simulation is shown in three dimensions and simulation results are presented for the self-similarity of the cellular pattern growth, the growth law and the topological properties. The results are compared with existing simulations and experiments. It is concluded that the three-dimensional vertex models provide us with efficient simulation models of systems with large numbers of cells.

Vertex Dynamics of Two-Dimensional Cellular Patterns

Time evolution of a two-dimensional cellular structure is studied by computer for a deterministic dissipative model system of interacting vertices. Despite its completely deterministic nature, our model gives rise to fully-developed random cellular structures which abound in nature. Both the growth law and the shape distribution for cells are obtained and are compared with other existing studies.