Rigidity Transition in Materials: Hardness is Driven by Weak Atomic Constraints

Abstract

Understanding the composition dependence of the hardness in materials is of primary importance for infrastructures and handled devices. Stimulated by the need for stronger protective screens, topological constraint theory has recently been used to predict the hardness in glasses. Herein, we report that the concept of rigidity transition can be extended to a broader range of materials than just glass. We show that hardness depends linearly on the number of angular constraints, which, compared to radial interactions, constitute the weaker ones acting between the atoms. This leads to a predictive model for hardness, generally applicable to any crystalline or glassy material. DOI: 10.1103/PhysRevLett.114.125502 PACS numbers: 62.20.Qp, 31.15.xv, 61.43.−j Rigidity theory [1-4], or topological constraint theory, is a powerful tool for capturing the atomic topology of glasses by reducing their network to mechanical trusses [5]. Following this mechanical analogy, a glass can be flexible, stressed-rigid, or isostatic, if the number of constraints per atom n c , comprising radial bond stretching (BS) and angular bond bending (BB), is lower, higher, or equal to three, the number of degrees of freedom per atom, respectively. Flexible networks show internal degrees of freedom, the floppy modes [6], which allow for local deformations; whereas stressed-rigid ones are completely locked by their high connectivity. In between, by being rigid but free of eigenstress [7], compositions exhibiting an isostatic behavior show some remarkable properties, such as a space-filling tendency [8], very weak aging [9], and anomalous behaviors, such as maximal fracture toughness [10].One of the major successes of this approach is the design of the Gorilla© Glass 3 by Corning©, which was created by atomic-scale modeling before anything had been melted in the lab [11,12]. Indeed, by capturing the chemical details of glasses that are relevant to macroscopic properties while filtering out those that are not, rigidity theory [13][14][15] has been used to predict the composition dependence of hardness, H, which characterizes resistance to permanent deformations under a load. Hence, topological constraint theory is a promising tool for designing stronger materials, which has recently been identified as a "grand challenge" for the future [16][17][18]. Topological constraint theory has also been applied to studying the folding of proteins [19,20]. However, it is still unknown whether it could be applied to a larger range of materials and, consequently, used to predict mechanical properties like hardness from the mere knowledge of composition.Recently, relying on molecular dynamics (MD) simulations, we showed that rigidity concepts could be applied to calcium-silicate-hydrate, ðCaOÞ x ðSiO 2 Þ 1−x−y ðH 2 OÞ y , or C-S-H, the binding phase of concrete [21]. From a topological point of view, C-S-H is a complex material as (1) it contains several chemical components, (2) its structure is anisotropic, inhomogeneous, and partially crystalline [22][23][24][...

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