Giovanni Zanzotto

Crystal symmetry and the reversibility of martensitic transformations

Martensitic transformations are diffusionless, solid-to-solid phase transitions, and have been observed in metals, alloys, ceramics and proteins. They are characterized by a rapid change of crystal structure, accompanied by the development of a rich microstructure. Martensitic transformations can be irreversible, as seen in steels upon quenching, or they can be reversible, such as those observed in shape-memory alloys. In the latter case, the microstructures formed on cooling are easily manipulated by loads and disappear upon reheating. Here, using mathematical theory and numerical simulation, we explain these sharp differences in behaviour on the basis of the change in crystal symmetry during the transition. We find that a necessary condition for reversibility is that the symmetry groups of the parent and product phases be included in a common finite symmetry group. In these cases, the energy barrier to lattice-invariant shear is generically higher than that pertaining to the phase change and, consequently, transformations of this type can occur with virtually no plasticity. Irreversibility is inevitable in all other martensitic transformations, where the energy barrier to plastic deformation (via lattice-invariant shears, as in twinning or slip) is no higher than the barrier to the phase change itself. Various experimental observations confirm the importance of the symmetry of the stable states in determining the macroscopic reversibility of martensitic transformations.

Continuum Models for Phase Transitions and Twinning in Crystals

Introduction Preliminaries Simple Lattices Weak-Transition Neighborhoods Subgroups, Cosets and variants Nonlinear Elasticity of Crystals Bifurcation Patterns Mechanical Twinning Transformation Twins Microstructures

Ericksen's bar revisited : Energy wiggles

This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy. According to Ericksen, for the bar in a hard device, the piecewise constant functions delivering the global minimum of the energy can have an arbitrary numberN of discontinuities in strain (phase-boundaries). Following some previous work in this area, we regularize the problem in order to resolve this degeneracy. We add two non-local terms to the energy density: one depends on the high (second) derivatives of the displacement, the other contains low (zero) derivatives. The low-derivative term (scaled with a constant β) introduces a strong non-locality, and simulates a three-dimensional interaction with the loading device, forcing the formation of layered microstructures in the process of energy minimization. The high-derivative (strain-gradient) term (scaled with a different constant α), represents a surface energy contribution which penalizes the formation of phase interfaces and prevents the infinite refinement of microstructures. In our description we consider the positions of interfaces as variables. This singles out in a natural way an infinite number of finite-dimensional subspaces, where all the essential nonlinearity is concentrated. In this way we can calculate explicitly the local minimizers (metastable states) and their energy, which turns out to be a multi-valued function of the interface positions and the imposed overall straind. Our approach thus gives an explicit framework for the study of the rich variety of finite-scale equilibrium microstructures for the bar and their stability properties. This allows for the study of a number of properties of phase transitions in solids; in particular their hysteretic behavior. Among our goals is the investigation of the phase diagram of the system, described by the functionN(d, α, β) giving the number of phase-boundaries in the absolute minimizer. We observe the somewhat counterintuitive effect that the energy at the global minimum, as a function of the overall strain, generically develops non-smooth oscillations (wiggles).

On the material symmetry group of elastic crystals and the Born Rule

The aim of this paper is to investigate the extent to which non-linear elasticity theory can be used for describing the behavior of crystalline solids. The results we obtain show that some strict and definite boundaries must be set to the possibility of actually doing so. Specifically, we address here a twofold problem: On one hand, we pose the question of the validity of the socalled “Born Rule”, a fundamental hypothesis due to Cauchy and, in a weaker form, to Born, by means of which continuum theories of crystal mechanics are formulated. On the other hand, we explore the possibility of developing an effective elastic model independently of the Rule, in those cases in which the Rule itself does not work. Our results are based on a close study of the implications of the phenomenon of mechanical twinning with regard to the symmetry properties of the energy function of an elastic crystal. These are summarized by the choice of a “material symmetry group” G: The main experimental features of twinning lead one to consider a class of “twinning subgroups” of G that are particular “reflection groups”, in fact, particular linear representations, possibly unfaithful, of abstract “Coxeter groups”. In the “generic” case, the properties of such groups prevent the Born Rule from holding. Only when some very special non-generic conditions are met by the twinning modes of a crystalline substance is the Rule valid; the development of an elastic model is then possible by following a well-known procedure. The analysis of relevant experimental data confirms that, while basically all crystals exhibit twins, most of them do exhibit generic twinning modes for which the hypothesis of Born is violated. We also show that in such generic cases any tentative thermoelastic approach developed independently of the Rule does not give physically sound results and thus cannot be usefully adopted, because some quite undesirable conclusions regarding the symmetry of the energy can be drawn that definitely make elasticity inadequate for our purposes. Experimental data point out nonetheless two quite remarkable classes of “nongeneric” materials for which the Born Rule is never violated, and to which an elastic model safely applies.

Sexually Antagonistic Selection in Human Male Homosexuality

Several lines of evidence indicate the existence of genetic factors influencing male homosexuality and bisexuality. In spite of its relatively low frequency, the stable permanence in all human populations of this apparently detrimental trait constitutes a puzzling ‘Darwinian paradox’. Furthermore, several studies have pointed out relevant asymmetries in the distribution of both male homosexuality and of female fecundity in the parental lines of homosexual vs. heterosexual males. A number of hypotheses have attempted to give an evolutionary explanation for the long-standing persistence of this trait, and for its asymmetric distribution in family lines; however a satisfactory understanding of the population genetics of male homosexuality is lacking at present. We perform a systematic mathematical analysis of the propagation and equilibrium of the putative genetic factors for male homosexuality in the population, based on the selection equation for one or two diallelic loci and Bayesian statistics for pedigree investigation. We show that only the two-locus genetic model with at least one locus on the X chromosome, and in which gene expression is sexually antagonistic (increasing female fitness but decreasing male fitness), accounts for all known empirical data. Our results help clarify the basic evolutionary dynamics of male homosexuality, establishing this as a clearly ascertained sexually antagonistic human trait.

Hysteresis in discrete systems of possibly interacting elements with a double-well energy

SummaryWe study in this paper the hysteretic behavior of a discrete system constituted by a finite number of elements (“snap-springs”) whose energy has two parabolic wells. The guideline idea is that, in many circumstances, hysteresis can be due to the presence of relative minimizers of a potential (“metastable states”) in which the system might get locked during its quasistatic evolution. A careful investigation is thus carried out of the relative minimizers of the total energy of our system of snap-springs under imposed total displacement, and of the barriers separating them. This is done both in the case of noninteracting elements and in the case in which some interaction is present that gives rise in the energy to an extra “coherence” term of special form. The results allow discussion of various hysteretic phenomena, also in the presence of vibrational motion of the elements. This study of a simple but suggestive discrete system will hopefully prove itself of help in understanding the implications regarding hysteresis of certain continuum theories recently proposed to model phase transitions in the solid state, in which the energy density is assumed, as here, to be biparabolic, and in which the coherence energy term we adopt arises in a natural way when equilibria involving mixtures of kinematically noncoherent phases are considered. In these cases the optimal microstructures are known to be layered, and physically this gives a good basis to our discrete calculation.

Generic and non-generic cubic-to-monoclinic transitions and their twins

Abstract The analysis of microstructure, for instance in shape-memory alloys, is largely based on the knowledge of the twins that arise from a symmetry-breaking phase transition. Such ‘transformation twins’ can be determined by analysing the kinematic compatibility conditions between the variants of the low-symmetry product phase. In this paper we report on the remarkable properties of twinning in the two cubic-to-monoclinic transitions that are possible. Not all these twins are conventional of Type 1 or Type 2. Indeed, our aim is to underline a peculiarity of the cubic-to-monoclinic transitions: if the lattice parameters of the monoclinic phase satisfy certain relations that we give explicitly, there exist many more twins than in the generic case. This is important because an abundance of twins implies that the material can form, in connection with the phase change, a wider variety of microstructures to accommodate imposed displacements or loads. In turn, this should noticeably influence the macroscopic behavior of the crystal, for instance enhancing its memory characteristics. This theoretical analysis may thus assist in the search of new and interesting materials. We propose to the experimental community toinvestigate the non-generic twins here described, and to check whether the non-generic monoclinicmaterials do indeed exhibit the remarkable properties that are expected due to the existence of the extra twinning modes.

One-dimensional quasistatic nonisothermal evolution of shape-memory material inside the hysteresis loop

We study in this paper the quasistatic nonisothermal processes of a onedimensional bar consisting of a two-phase shape-memory material. The system of p.d.e.s governing the evolution of the bar is obtained by means of a temperature-dependent hysteretic stress-strain law that we formulate as a “plasticity” criterion and a hysteresis operator. The constitutive theory is developed here on the basis of the mixture approach proposed by Mtiller [t] and of a natural extrapolation of the isothermal experimental data regarding the behavior of the material inside the hysteresis loop recently described by Miiller and Xu [2]. Numerical simulations are presented for three initial and boundary-value problems of interest with regard to uniaxial stretching experimental tests.

A crystallographic approach to structural transitions in icosahedral viruses

Viruses with icosahedral capsids, which form the largest class of all viruses and contain a number of important human pathogens, can be modelled via suitable icosahedrally invariant finite subsets of icosahedral 3D quasicrystals. We combine concepts from the theory of 3D quasicrystals, and from the theory of structural phase transformations in crystalline solids, to give a framework for the study of the structural transitions occurring in icosahedral viral capsids during maturation or infection. As 3D quasicrystals are in a one-to-one correspondence with suitable subsets of 6D icosahedral Bravais lattices, we study systematically the 6D-analogs of the classical Bain deformations in 3D, characterized by minimal symmetry loss at intermediate configurations, and use this information to infer putative viral-capsid transition paths in 3D via the cut-and-project method used for the construction of quasicrystals. We apply our approach to the Cowpea Chlorotic Mottle virus (CCMV) and show that the putative transition path between the experimentally observed initial and final CCMV structures is most likely to preserve one threefold axis. Our procedure suggests a general method for the investigation and prediction of symmetry constraints on the capsids of icosahedral viruses during structural transitions, and thus provides insights into the mechanisms underlying structural transitions of these pathogens.

Elastic crystals with a triple point

Abstract The peculiar behavior of active crystals is due to the presence of evolving phase mixtures the variety of which depends on the number of coexisting phases and the multiplicity of symmetry-related variants. According to Gibbs’ phase rule, the number of phases in a single-component crystal is maximal at a triple point in the p – T phase diagram. In the vicinity of this special point the number of metastable twinned microstructures will also be the highest—a desired effect for improving performance of smart materials. To illustrate the complexity of the energy landscape in the neighborhood of a triple point, and to produce a workable example for numerical simulations, in this paper we construct a generic Landau strain–energy function for a crystal with the coexisting tetragonal (t), orthorhombic (o), and monoclinic (m) phases. As a guideline, we utilize the experimental observations and crystallographic data on the t–o–m transformations of zirconia (ZrO 2 ), a major toughening agent for ceramics. After studying the kinematics of the t–o–m phase transformations, we re-evaluate the available experimental data on zirconia polymorphs, and propose a new mechanism for the technologically important t–m transition. In particular, our proposal entails the softening of a different tetragonal modulus from the one previously considered in the literature. We derive the simplest expression for the energy function for a t–o–m crystal with a triple point as the lowest-order polynomial in the relevant strain components, exhibiting the complete set of wells associated with the t–o–m phases and their symmetry-related variants. By adding the potential of a hydrostatic loading, we study the p – T phase diagram and the energy landscape of our crystal in the vicinity of the t–o–m triple point. We show that the simplest assumptions concerning the order-parameter coupling and the temperature dependence of the Landau coefficients produce a phase diagram that is in good qualitative agreement with the experimental diagram of ZrO 2 .