A new model has been proposed for predicting the hardness of materials.

The results of the research are presented in the journal Physical Review Materials. Hardness is a crucial property of materials that determines their ability to resist deformation and other types of damage (such as dents and scratches) caused by external forces. Typically, hardness is assessed by pressing an indenter into the test sample, with the indenter made from a harder material, usually diamond.

In this case, hardness is defined based on the ratio between the maximum indentation force and the impression left on the sample. Modern industry requires new hard and superhard materials with enhanced mechanical properties compared to traditional materials. One solution to this challenge is the use of advanced computational methods for high-throughput screening of materials with improved characteristics.

“Today, computational methods are sufficiently advanced to accurately predict the structure and properties of various compounds and materials. However, it is essential not only to predict the material's structure but also to accurately calculate its mechanical properties, such as hardness, which are necessary for the experimental synthesis of materials with predefined characteristics.

Existing empirical models for predicting hardness are based on the strength of chemical bonds, the degree of ionicity, the electronegativity of crystals, and the elastic moduli of materials. We have proposed a simple and accurate model based on properties such as the shear modulus of elasticity and the derivative of the bulk modulus of elasticity with respect to pressure. Both properties can be obtained through experiments or atomistic modeling,” said the first author of the study, Faridun Djalolov, a graduate student in the “Materials Science” program at Skoltech.

The importance of using the shear modulus in the hardness model is due to its characteristic dependence on the directions of deformation of the crystalline structure—this allowed us to calculate the spatial dependence of hardness for a range of materials while taking into account the anisotropy of crystal structures. The derivative of the elastic modulus with respect to pressure, derived from the equation of state, enabled us to consider the influence of temperature on hardness.

“We demonstrated that the hardness model works for hard and superhard materials, illustrated by the examples of rhenium diboride (ReB₂) and boron carbide (B₄C). The obtained temperature dependence of hardness aligns well with existing experimental measurements and predictions from machine learning-based models. All quantities in our model can be directly derived from calculations or experiments, making the model suitable for practical applications,” added Alexander Kvashnin, a professor at Skoltech's Energy Transition Project Center, co-author, and scientific supervisor of the work.