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

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.

Figure 1

Number of constraints per atom $n_c$ as a function of the $\mathrm{Ca}/\mathrm{Si}$ molar ratio. The inset shows the contributions of the radial (BS) and angular (BB) constraints. Lines are linear fits in flexible and stressed-rigid domains and serve as guides for the eye.

Figure 2

Computed hardness of C-S-H (white squares) [29] and comparison with the values predicted by Mauro’s model (green diamonds) using Eq. (1), and by the present model using Eq. (2). Colored areas are guides for the eye, to distinguish the predictions of the two models in the $\mathrm{Ca}/\mathrm{Si}>1.5$ domain.

Figure 3

Predictions of the number of constraints per atom (top) and of the hardness (bottom) as a function of the composition of C-S-H. The red and blue lines are respective predictions of the cohesive limit of the system and of the isostatic compositions. The green, blue, and red points in the top figure represent the compositions of three selected computed samples being, respectively, flexible, isostatic, and rigid.

Figure 4

Experimental values of the hardness of C-S-H (squares) [29], glassy silica (right triangle) [46], quartz (bottom triangle) [46], soda-lime-borates glasses (diamonds) [13], Ge-Se glasses (up triangles) [47], and Ge-Sb-Se (circles) [48] as a function of the number of angular constraints per atom. Fitting parameters are $(dH/d{n}_{\mathrm{BB}})=6.5$, 9.1, 1.3 (which appear to scale with the bond energy, i.e., $\mathrm{B}-\mathrm{O}>\mathrm{Si}-\mathrm{O}/\mathrm{Ca}-\mathrm{O}>\mathrm{Ge}-\mathrm{Se}$) and $n_{\mathrm{BB}}^{\text{crit}}=0.33$, 1.2, 0.75 for silicates, borates, and chalcogenides, respectively. Lines are guides for the eye.