www.nature.com/scientificreports OPEN Network topological model of reconstructive solid-state transformations Received: 19 December 2018 Vladislav A. Blatov 1,2,3, Andrey A. Golov2,3, Changhao Yang1, Qingfeng Zeng 1,4 Accepted: 2 April 2019 & Artem A. Kabanov 2,3 Published: xx xx xxxx Reconstructive solid-state transformations are followed by signifcant changes in the system of chemical bonds, i.e. in the topology of the substance. Understanding these mechanisms at the atomic level is crucial for proper explanation and prediction of chemical reactions and phase transitions in solids and, ultimately, for the design of new materials. Modeling of solid-state transitions by geometrical, molecular dynamics or quantum-mechanical methods does not account for topological transformations. As a result, the chemical nature of the transformation processes are overlooked, which limits the predictive power of the models. We propose a universal model based on network representation of extended structures, which treats any reorganization in the solid state as a network transformation. We demonstrate this approach rationalizes the confguration space of the solid system and enables prediction of new phases that are closely related to already known phases. Some new phases and unclear transition pathways are discovered in example systems including elementary substances, ionic compounds and molecular crystals. Te search for new materials requires a deep understanding of reconstructive processes, which occur at the atomic level and lead to signifcant changes in the architecture of the substance. With respect to chemical reac- tions between molecules, such reconstructive processes are always followed by essential reorganization of the solid system, which ofen results in destruction of the solid system. ‘Single crystal -to- single crystal’ transforma- tions have attracted special attention in the last few years1. In this scenario, the substance preserves its crystallin- ity and the transformation is ofen followed by reversible or irreversible chemical reactions between molecules. Single crystal-to-single crystal transformations typically occur in organic or metal-organic compounds, in which the mechanism of transformation usually mimics the reaction of isolated molecules. Te displacement of atoms during the transition in inorganic solids is less obvious, and special methods are required to model and under- stand these processes. In the recent Materials Genome Initiative project2, the study of reconstructive phase transi- tions was approached as the identifcation, mutation and recombination of the underlying architectures (genes) of complex materials systems. Such computational approaches could further accelerate identifcation and screening of the genes within a material, and complement high-throughput (combinatorial) experiments. Since these genes are expected to essentially determine the properties of the material, computational approaches may provide a cost-efective, high-throughput method to shorten the time-to-market for novel materials. Indeed, the task of developing rational models for reconstructive phase transitions goes far beyond the specifc felds of solid-state chemistry and physics. Modeling of solid-state transitions at the atomic level is now possible thanks to the rapid development of density functional theory (DFT) methods over the last 10–15 years. However, several serious obstacles remain, as there are an infnite number of possible transition pathways, and direct scanning of all possible atomic archi- tectures (also termed confguration space, CS) requires huge computational efort. Terefore, the development of specialized methods that can rapidly search for the most energetically favorable pathways is crucial. Te frst attempts to describe reconstructive phase transitions at the atomic level were undertaken long before the DFT era. 1School of Materials Science and Engineering, Northwestern Polytechnical University, Youyi West Rd. 127, Xi’an, 710072, PR China. 2Samara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Molodogvardeyskaya St. 244, Samara, 443100, Russia. 3Samara Center for Theoretical Materials Science (SCTMS), Samara University, Ac. Pavlov St. 1, Samara, 443011, Russia. 4MSEA International Institute for Materials Genome, Jinxiu Rd. 1, Gu’an, 065500, PR China. Correspondence and requests for materials should be addressed to V.A.B. (email: [email protected]) or Q.Z. (email: [email protected]) SCIENTIFIC REPORTS | (2019) 9:6007 | https://doi.org/10.1038/s41598-019-42483-5 1 www.nature.com/scientificreports/ www.nature.com/scientificreports Figure 1. An area of confguration space with four stable network regions (A–D): (lef) a simplifed representation with a linear boundary that is formed by two narrow metastable network regions (S1 and S2); (right) transition pathways from the subnets/supernets to known phases (A and B) and unknown phases (C and D). Mechanisms were suggested for phase transitions in metallic systems between face-centered cubic and hexagonal close packings or body-centered cubic and face-centered cubic structures (martensitic transformations), in ionic crystals between rock salt and zinc blende, and in covalent crystals between diamond and lonsdaleite, among oth- ers3–5. However, no general scheme was proposed until the 1990’s, when the geometrically shortest pathways were identifed manually. Subsequently, geometrical approaches based on group-subgroup relations were developed6 and the DFT modeling within the solid-state nudged elastic band (SSNEB) method was devised7. All of the aforementioned approaches only consider the movement of atoms during the transition and do not account for the fact that reconstructive phase transition leads to changes in crystal structure connectivity, when some interatomic contacts are broken and some new contacts are created. Since connectivity is a topolog- ical feature of solid structures, then some topological properties should change during a transition. Topological approaches for the study of phase transition mechanisms have been in development since the 1990’s, though these approaches initially described the crystal in terms of abstract 3-periodic 2D surfaces8, not in terms of atoms and bonds. Although several phase transitions have been explored using topological approaches, these models have not been widely propagated due to their complexity and a lack of computational tools. Indeed, the limited availability of computational tools for topological approaches is likely related to the fact that no strict universal algorithm has been proposed for the method and each phase transition must be analyzed in a unique manner. In 2007, we proposed another topological approach for the interpretation of reconstructive phase transitions, which was based on the interrelations between periodic nets and their subnets9. However, this approach required improvement as the net-subnet relations did not allow us to fnd all possible transition pathways or to easily deter- mine the geometry of the transition state. In this study, we present an extension of this approach to a universal topological model of solid-state transformations and provide some examples of the application of this model. We restrict ourselves to phase transitions; no chemical reactions in the solid state are considered and the chemical composition remains the same during the transition. Network Topological Model of Confguration Space We consider the confguration space of a crystal system in terms of the network approach. Let us try to assign a particular topology to each point of CS. Obviously, there are CS regions, where the connectivity can be unambigu- ously determined (i.e., stable network regions), as well as intermediate regions, where several topologies can coex- ist (Fig. 1 lef). Te existence of several topologies means diferent nets can be determined for the same atomic confguration depending on the bonding criteria. In particular, some interatomic contacts can be considered as additional bonds or some existing bonds can be ignored, resulting in a supernet or subnet, respectively. Such con- fgurations can be treated as metastable network regions and ofen provide elongated interatomic contacts, which can be considered or ignored as bonds; it is the confgurations that correspond to the subnets or supernets of the nets from stable network regions. Strictly speaking, the boundaries between network regions are fuzzy, since determination of the edges of the nets is somewhat ambiguous; in our model we simplify the CS structure and shrink the metastable network regions to the lines between the stable regions (Fig. 1 lef). We propose a rational description of the CS within a general network topological model, which is based on representation of the crystal structure within each region as a periodic graph (net) and the adjacency of the CS regions as the subnet-supernet relations9. Te model allows us to formulate relatively general statements that govern reconstructive phase transitions and essentially facilitate subsequent, more detailed, quantitative analysis by DFT methods. Tese statements are: (i) Any stable CS region is described by a unique network topology; the stable regions with diferent topolo- gies do not cross each other. (ii) Any reconstructive phase transition is a path, which crosses at least one stable region boundary. In the case of transition A → B, when nets A and B are in a direct supernet-subnet relation, the nets immediately transform to each other on the boundary.