# Modeling Hall–Petch Relationships of Alloyed Nanostructured Materials Regarding Bonding Nature at Grain Boundaries

## TL;DR

This paper proposes a unified formula to explain how grain size affects the strength of nanostructured alloys and pure metals.

## Contribution

A thermodynamic model unifying Hall–Petch relationships for both pure and alloyed nanostructured materials is introduced.

## Key findings

- Yield strength increases then decreases for pure and low-melting alloys as grain size shrinks.
- High-melting alloys show a monotonic increase in strength with decreasing grain size.
- Bond weakening and interface energy changes explain the observed trends.

## Abstract

Based on a thermodynamic approach, a unified formula was proposed to describe distinct Hall–Petch relationships (HPRs) of unalloyed nanostructured materials (u-NSs: Fe, Cu, Ni, Pd, and Mo) and alloyed ones with low- or high-melting temperature alloying metals (low-Tm or high-Tm a-NSs: NixMo1−x, FexZr1−x, NixCu1−x, and FexCu1−x). As the grain size decreases to several nanometers, the yield strength first increases and then decreases for u-NSs and low-Tm a-NSs, obeying the inverse HPR (IHPR), while it monotonically increases for high-Tm a-NSs. For the former, the decrease is induced by the reduction in activation energies of interface migration and dislocation gliding, along with the thermally driven decline, lattice expansion, and bond weakening of interface atoms. In the latter case, the monotonic increase or the elimination of IHPR is relevant to the negative interface energy induced by the segregation of alloying atoms at grain boundaries. Our predictions are validated by the available experimental results.