Special Issue on Spatial Moment Techniques for Modelling Biological Processes

Over the last two decades, there has been an increasing awareness of, and interest in, the use of spatial moment techniques to provide insight into a range of biological and ecological processes. Models that incorporate spatial moments can be viewed as extensions of mean-field models. These mean-field models often consist of systems of classical ordinary differential equations and partial differential equations, whose derivation, at some point, hinges on the simplifying assumption that individuals in the underlying stochastic process encounter each other at a rate that is proportional to the average abundance of individuals. This assumption has several implications, the most striking of which is that mean-field models essentially neglect any impact of the spatial structure of individuals in the system. Moment dynamics models extend traditional mean-field descriptions by accounting for the dynamics of pairs, triples and higher n-tuples of individuals. This means that moment dynamics models can, to some extent, account for how the spatial structure affects the dynamics of the system in question.

Modeling of Convection and Macrosegregation through Appropriate Consideration of Multiphase/Multiscale Phenomena during Alloy Solidification

Solidification processes are complex in nature, involving multiple phases and several length scales. The properties of solidified products are dictated by the microstructure, the mactostructure, and various defects present in the casting. These, in turn, are governed by the multiphase transport phenomena Occurring at different length scales. In order to control and improve the quality of cast products, it is important to have a thorough understanding of various physical and physicochemical phenomena Occurring at various length scales. preferably through predictive models and controlled experiments. In this context, the modeling of transport phenomena during alloy solidification has evolved over the last few decades due to the complex multiscale nature of the problem. Despite this, a model accounting for all the important length scales directly is computationally prohibitive. Thus, in the past, single-phase continuum models have often been employed with respect to a single length scale to model solidification processing. However, continuous development in understanding the physics of solidification at various length scales oil one hand and the phenomenal growth of computational power oil the other have allowed researchers to use increasingly complex multiphase/multiscale models in recent. times. These models have allowed greater understanding of the coupled micro/macro nature of the process and have made it possible to predict solute segregation and microstructure evolution at different length scales. In this paper, a brief overview of the current status of modeling of convection and macrosegregation in alloy solidification processing is presented.

Characterization of hot deformation behavior of Zr-1Nb-1Sn alloy

The hot deformation behavior of beta-quenched Zr-1 Nb-1Sn was studied in the temperature range 650-1050 degrees C and strain rate range 0.001-100 s(-1) using processing maps. These maps revealed three different domains: a domain of dynamic recovery at temperatures <700 degrees C and at strain rates <3 x 10(-3) s(-1), a domain of dynamic recrystallization in the temperature range 750-950 C-degrees and at strain rates <10(-2) S-1 with a peak at 910 degrees C and 10(-3) S-1 (in alpha + beta phase field), and a domain of large-grain superplasticity in the beta phase field at strain rates <10(-2) s(-1). In order to identify the rate controlling mechanisms involved in these domains, kinetic analysis was carried out to determine the various activation parameters. In addition, the processing maps showed a regime of flow instability spanning both alpha + beta and beta phase fields. The hot deformation behavior of Zr 1Nb-1Sn was compared with that of Zr, Zr-2.5Nb and Zircaloy-2 to bring out the effects of alloy additions. (C) 2006 Elsevier BN. All rights reserved.

Processing Response of Boron Modified Ti-6Al-4V Alloy In (alpha plus beta) Working Regime

Titanium alloys like Ti-6A-4V are the backbone materials for aerospace, energy and chemical industries. Hypoeutectic boron addition to Ti-6Al-4V alloy produces a reduction in as-cast grain size by roughly an order of magnitude resulting in the possibility of avoiding ingot breakdown step and thereby reducing the processing cost. In the present study, ISM processed as-cast boron added Ti-6Al-4V alloy is deformed in (alpha+beta)-phase field, where alpha-lath bending seemed to be the dominating deformation mechanism.

Role of entropy in the stability of cobalt titanates

The standard molar Gibbs free energy of formation of Co2TiO4, CoTiO3,and CoTi2O5 as a function of temperature over an extended range (900 to 1675) K was measured using solid-state electrochemical cells incorporating yttria-stabilized zirconia as the electrolyte, with CoO as reference electrode and appropriate working electrodes. For the formation of the three compounds from their component oxides CoO with rock-salt and TiO2 with rutile structure, the Gibbs free energy changes are given by:Delta(f)G degrees((ox))(Co2TiO4) +/- 104/(J . mol(-1)) = -18865 - 4.108 (T/K)Delta(f)G degrees((ox))(CoTiO3) +/- 56/(J . mol(-1)) = -19627 + 2.542 (T/K) Delta(f)G degrees((ox))(CoTi2O5) +/- 52/(J . mol(-1)) = -6223 - 6.933 (T/K) Accurate values for enthalpy and entropy of formation were derived. The compounds Co2TiO4 with spinel structure and CoTi2O5 with pseudo-brookite structure were found to be entropy stabilized. The relatively high entropy of these compounds arises from the mixing of cations on specific crystallographic sites. The stoichiometry of CoTiO3 was confirmed by inert gas fusion analysis for oxygen. Because of partial oxidation of cobalt in air, the composition corresponding to the compound Co2TiO4 falls inside a two-phase field containing the spinet solid solution Co2TiO4-Co3O4 and CoTiO3. The spinel solid solution becomes progressively enriched in Co3O4 with decreasing temperature. (c) 2010 Elsevier Ltd. All rights reserved.

Effect of interface friction on the mechanics of indentation of a finite layer resting on a rigid substrate

Copper strips of 2.5 mm thickness resting on stainless steel anvils were normally indented by wedges under nominal plane strain conditions. Inflections in the hardness-penetration characteristics were identified. Inflections separate stages where each stage has typical mechanics of deformation. These are arrived at by studying the distortion of 0.125 mm spaced grids inscribed on the deformation plane of the strip. The sensitivity of hardness and deformation mechanics to wedge angle and the interfacial friction between strip and anvil were investigated within the framework of existing slip line field models of indentation of semi-infinite and finite blocks.

Continuous ordering below the tricritical point: A critical test for the kinetic pathway in Fe-12at.% Si alloy

A critical test has been presented to establish the nature of the kinetic pathways for the decomposition of Fe-12 at.% Si alloy below the metastable tricritical point. The results, based on the measurements of saturation magnetization, establish that a congruent ordering from B2 --> D0(3) precedes the development of a B2 + D0(3) two-phase field, consistent with the predictions in 1976 of Allen and Cahn.

Hot-working characteristics of Zircaloy-2 in the temperature range of 650–950°C

The hot-working characteristics of Zircaloy-2 have been studied in the temperature range of 650 to 950°C and in the strain-rate range of 10−3 to 102 s−1 using power dissipation maps which describe the variation of the efficiency of power dissipation, η = 2m /(m + 1) where m is the strain-rate sensitivity of flow stress. The individual domains exhibited by the map have been interpreted and validated by detailed metallographic investigations. Dynamic recrystallization occurs in the temperature range of 730 to 830°C and in the strain-rate range of 10−2 to 2 s−1. The peak efficiency occurs at 800°C and 0.1 s−1 which may be considered as the optimum hot-working parameters in the α-phase field of Zircaloy-2. Superplastic behaviour, characterized by a high efficiency of power dissipation is observed at temperatures greater than 860°C and at strain rates lower than 10−2 s−1. When deformed at 650°C and 10−3 s−1, the primary restoration mechanism is dynamic recovery, while at rates higher than 2s−1, the material exhibits microstructural instabilities in the form of localized shear bands.

Texture Evolution in Boron Modified Ti-6Al-4V Alloy

Owing to their high strength-to-weight ratio, excellent mechanical properties and corrosion resistance, titanium (Ti) and its alloys, especially (alpha+beta) alloys like Ti-6Al-4V is the backbone materials for aerospace, energy, and chemical industries. Trace boron addition (similar to 0.1 wt. %) to the alloy Ti-6Al-4V produces a reduction in as-cast grain size by roughly an order of magnitude resulting in enhanced ductility, higher stiffness, strength and good fracture resistance. Boron addition could also affect the evolution of texture and microstructure in the material. The solidification microstructures of Boron free as well as Boron containing Ti-6Al-4V are found to be almost homogeneous from periphery towards the center of as-cast ingot in terms of both alpha-colony size and distribution. Boron addition substantially reduces alpha-colony size (similar to 50-80 mu m). A gradual change in alpha texture from periphery towards the center has been observed with orientations close to specific texture components suggesting the formation of texture zones. The mechanism of texture evolution can be visualized as a result of variant selection during solidification through (alpha+beta) phase field.

Evolution of Transformation Texture in a Metastable B - Titanium Alloy

Texture evolution in h. c. p. (alpha) phase derived from aging of a differently processed metastable b.c.c. (beta) titanium alloy was investigated. The study was aimed at examining (i) the effect of different b. c. c. cold rolling textures and (ii) the effect of different defect structures on the h. c. p transformation texture. The alloy metastable beta alloy Ti-10V-4.5Fe-1.5Al was rolled at room temperature by unidirectional (UDR) and multi-step cross rolling (MSCR). A piece of the as-rolled materials were subjected to aging in order to derive the h. c. p. (alpha) phase. In the other route, the as-rolled materials were recrystallized and then aged. Textures were measured using X-ray as well as Electron Back Scatter Diffraction. Rolling texture of beta phase, as characterized by the presence of a strong gamma fibre, was found stronger in M S C R compared to UDR, although they were qualitatively similar. The stronger texture of MSCR sample could be attributed to the inhomogeneous deformation taking place in the sample that might contribute to weakening of texture. Upon recrystallization in beta phase field close to beta-transus. the textures qualitatively resembled the corresponding beta deformation textures; however, they got strengthed. The aging of differently beta rolled samples resulted in the product alpha-phase with different textures. The (UDR + Aged) sample had a stronger texture than (MSCR + Aged) sample, which could be due to continuation of defect accumulation in UDR sample, thus providing more potential sites for the nucleation of alpha phase. The trend was reversed in samples recrystallized prior to aging. The (MSCR + Recrystallized + Aged) sample showed stronger texture of alpha phase than the (UDR + Recrystallized + Aged) sample. This could be attributed to extensive defect annihilation in the UDR sample on recrystallization prior to aging. The (MSCR + Aged) sample exhibited more alpha variants when compared to (MSCR + Recrystallized + Aged) sample. This has been attributed to the availability of more potential sites for nucleation of alpha phase in the former. It could be concluded that alpha transformation texture depends mainly on the defect structure of the parent phase.

Effect of microstructure on the creep behaviour of Ti-24Al-11Nb

In this investigation, the influence of microstructure on the high temperature creep behaviour of Ti-24Al-11Nb alloy has been studied. Different microstructures are produced by devising suitable heat treatments from the beta phase field. Creep tests are conducted in the temperature range of 923-1113 K, over a wide stress range at each temperature, employing the impression creep technique. The creep behaviour is found tb be sensitive to the crystallographic texture as well as to the details of microstructure. Best creep resistance is shown when the microstructure contains smaller alpha(2) plates and a lower beta volume fraction. This can be understood in terms of the dislocation barriers offered by alpha(2) beta boundaries and the case of plastic flow in the beta phase at high temperatures.

Characterization of hot deformation behaviour of Zr---2.5Nb---0.5Cu using processing maps

The characteristics of hot deformation of beta-quenched Zr-2.5Nb-0.5Cu in the temperature range 650-1050 degrees C and in the strain rate range 0.001-100 s(-1) have been studied using hot compression testing. For this study, the approach of processing maps has been adopted and their interpretation done using the Dynamic Materials Model. The efficiency of power dissipation given by [2m/(m + 1)], where m is strain rate sensitivity, is plotted as a function of temperature and strain rate to obtain a processing map. The processing map for Zr-2.5Nb-0.5Cu within (alpha + beta) phase field showed a domain of dynamic recrystallization, occurring by shearing of alpha-platelets followed by spheroidization, with a peak efficiency of 48% at 750 degrees C and 0.001 s(-1). The stress-strain curves in this domain had features of continuous flow softening and all these are similar to that in Zr-2.5Nb alloy. In the beta-phase field, a second domain with a peak efficiency of 47% occurred at 1050 degrees C and 0.001 s(-1) and this domain is correlated with the superplasticity of beta-phase. The beta-deformation characteristics of this alloy are similar to that observed in pure beta-zirconium with large grain size. Analysis of flow instabilities using a continuum criterion revealed that the Zr-2.5Nb-0.5Cu exhibits flow localization at temperatures higher than 800 degrees C and strain rates higher than about 30 s(-1) and that the addition of copper to Zr-2.5Nb reduces its susceptibility to flow instability, particularly in the (alpha + beta) phase field.

Microstructure and texture evolution during beta extrusion of boron modified Ti-6A1-4V alloy

Boron addition to conventional titanium alloys below the eutectic limit refines the cast microstructure and improves mechanical properties. The present work explores the influence of hypoeutectic boron addition on the microstructure and texture evolution in Ti-6Al-4V alloy under beta extrusion. The beta extruded microstructure of Ti-6Al-4V is characterized by shear bands parallel to the extrusion direction. In contrast, the extruded Ti-6Al-4V-0.1B alloy shows a regular beta worked microstructure consisting of fine prior beta grains and acicular alpha-lamellae with no signs of the microstructural instability. Crystallographic texture after extrusion was almost identical for the two alloys indicating the similarity in their transformation behavior, which is attributed to complete dynamic recrystallization during beta processing. Microstructural features as well as crystallographic texture indicate dominant grain boundary related deformation processes for the boron modified alloy that leads to homogeneous deformation without instability formation. The absence of shear bands has significant technological importance as far as the secondary processing of boron added alloys in (alpha + beta)-phase field are concerned. (C) 2012 Elsevier B.V. All rights reserved.

The influence of temperature and strain rate on the deformation response and microstructural evolution during hot compression of a titanium alloy Ti-6Al-4V-0.1B

Hot deformation behavior of a hypoeutectic Ti-6Al-4V-0.1B alloy in (alpha + beta) phase field is investigated in the present study with special reference to flow response, kinetics and microstructural evolution. For a comparison, the base alloy Ti-6Al-4V was also studied under identical conditions. Dynamic recovery of alpha phase occurs at low temperatures while softening due to globularization and/or dynamic recrystallization dominates at high temperatures irrespective of boron addition. Microstructural features for both the alloys display bending and kinking of alpha lamellae for near alpha test temperatures. Unlike Ti-6Al-4V, no sign of instability formation was observed in Ti-6Al-4V-0.1B for any deformation condition except for cavitation around TiB particles, due to deformation incompatibility and strain accumulation at the particle-matrix interface. The absence of macroscopic instabilities and early initiation of softening mechanisms as a result of boron addition has been attributed to microstructural features (e.g. refined prior beta grain and alpha colony size, absence of grain boundary alpha layer, presence of TiB particles at prior beta boundaries, etc.) of the respective alloys prior to deformation. (C) 2012 Elsevier B.V. All rights reserved.

On the absence of shear cracking and grain boundary cavitation in secondary tensile regions of Ti-6Al-4V-0.1B alloy during hot (alpha plus beta)-compression

Deformation instabilities, such as shear cracking and grain boundary cavitation, which are observed in the secondary tensile region of Ti-6Al-4V alloy during compressive deformation in the (+)-phase field, do not form in Ti-6Al-4V-0.1B alloy when processed under the same conditions. This has been attributed to the microstructural modifications, e.g. the absence of grain boundary and adjacent grain boundary retained layers and a lower proportion of 90(o)-misoriented -colonies that occur with boron addition.