Stepwise emergence of an intensive physical property from a single-atom to bulk

TL;DR

This study reveals that the strength of silver transitions from single atoms to bulk in a staircase pattern, demonstrating size-dependent emergent properties and potential for materials design based on atomic configurations.

Contribution

It uncovers the stepwise emergence of strength as size increases, linking geometric atomic arrangements to size-dependent property evolution, a novel insight into nanoscale material behavior.

Findings

Strength approaches theoretical limit at single-atom scale

Strength variation occurs in size-dependent staircase steps

Potential for designing materials with tailored properties

Abstract

Intensive or size-invariant physical properties are well known to become size-dependent when the bulk material is reduced to the nanometer scale. Using silver, the present study shows a remarkable emergent characteristic of extensive properties as the size of the system is increased from a single atom to bulk - the property (strength) evolves in a staircase manner, as opposed to the intuitively assumed continuous approach to a saturating bulk value. In other words, the observed variation with size retains the inherent trait of intensive properties in the form of size-independent staircase plateaus. The steps can be explained by simple and necessary geometric configurations that atoms must assume in their initial approach to bulk. Were it not for these observations, it would have been remarkable to have reported that the observed strength in the limit of a single-atom Ag bridge approaches the theoretical limit, being 3-4 orders of magnitude higher than in bulk single crystals. The appearance of steps is acutely sensitive to minute perturbations, and their observation is enabled by pico-level resolution in measured forces and displacements in highly stable atomic sized bridges. However, suitably harnessed, results open the possibility to impart vastly different strengths to the same material based only on its size. This combined with possibility to change the position and height of the steps by mixing elements of different diameters and surface energy, opens a realistic approach to 'materials by design'. Results also provide an example of the evolutionary trace of an emergent physical property in materials.