A universal symmetry criterion for the design of high performance ferroic materials

Abstract

The symmetry of a crystal has profound effects on its physical properties and so does symmetry-breaking on the characteristics of a phase transition from one crystal structure to another. For an important class of smart materials, the ferroics, their functionality and performance are associated with cycles of transitions from multiple structural states of one phase to those of the other. Using group and graph theories we construct phase transition graph (PTG) and show that both the functionality and performance of ferroics are dictated by the topology of their PTGs. In particular, we demonstrate how the giant piezoelectricity in ferroelectrics and the functional fatigue in shape memory alloys (SMAs) are related to their unique PTG topological features. Using PTG topology as a guide, we evaluate systematically new systems potentially having giant piezoelectricities and giant electro-and magneto-strictions and discuss the design strategies for high performance SMAs with much improved functional fatigue resistance.

The transformation matrices (distortion, orientation, correspondence), their continuous forms and their variants

“…One part of the irreversibility comes from the accumulation of defects (dislocations) generated by a series of transformations, which has been described mathematically with a 'global' group (Bhattacharya et al, 2004) and Cayley graphs (Gao et al, 2017). One part of the irreversibility comes from the accumulation of defects (dislocations) generated by a series of transformations, which has been described mathematically with a 'global' group (Bhattacharya et al, 2004) and Cayley graphs (Gao et al, 2017).…”

“…The distortion matrix F c is a lattice-preserving strain; it belongs to the 'global group' (Bhattacharya et al, 2004;Gao et al, 2016Gao et al, , 2017. The distortion leaves the lattice globally invariant, and all the geometric symmetry elements are put in coincidence.…”

“…The set of accumulated deformations (and defects) during thermal cycling was identified as a group called the 'global group' by Bhattacharya et al (2004). This approach has been followed by Gao et al (2017) who introduced geometric representations (Cayley graphs) of the global group of lattice deformations. For example, the global group of a simple square 'phase' is made of the LIS matrices generated by the simple strain matrix F c given in Section 7.1 and by the square symmetries; it is the special linear group SLð2; ZÞ of matrices of determinant 1 made of integer coefficients.…”

A Revisit to the Notation of Martensitic Crystallography

“…In the literature, the so-called phase transition graph (PTG) is introduced to capture the pathway connectivity during martensitic transformations [31]. In a PTG, each vertex corresponds to a structural state, while each edge (connecting two vertices) corresponds to a transformation pathway (between two structural states).…”

“…However, in 2004 the intersection of the two branches was 2 of 12 indicated in the work of Bhattacharya et al [30] on the investigation of the reversibility of martensitic transformations. Following this work, a graph theory approach was developed to systematically analyze the transformation pathway connectivity associated with the symmetry breaking processes, which provided a general understanding of the crystallographic coupling between structural phase transformations and transformation-induced defects [31,32]. However, the mathematical connection between the two branches is still not clearly identified, partially due to the lack of a systematic notation in common.…”