Impurity-induced van der Waals transition during decohesion

Current understanding of the mechanisms leading to me- chanical failure of materials is far from complete. This has its origin in the immense complexity associated with an event such as fracture in which many phenomena operate at varying length scales. While dislocation nucleation and glide often play an important role, it is the propagation of cracks that ultimately leads to fracture. Crack growth results in the separation of atomic planes either along grain boundaries or through grains. Hence efforts have been made to relate the atomic-level description of bond breaking to fracture resis- We present a first-principles thermodynamic model to study the effect of impurities at constant chemical potential on the traction curve of a uniformly decohering solid. We apply it to decohesion of fcc aluminum along a pair of ~111! planes accounting for the presence of hydrogen or oxygen atoms in the decohering region at constant chemical poten- tial. We find that while the tractions at constant impurity concentration have a form similar to those of previously ex- plored systems, 1 a constant impurity chemical potential can lead to a discontinuity in the relation between decohesion distance and stress. This first-order transition in the stress- displacement diagram, reminiscent of a van der Waals tran- sition, leads to an abrupt drop in the maximal cohesive stress once a critical impurity chemical potential is exceded. In- stead of a gradual change in the resistance to crack growth with increasing impurity chemical potential, this result indi- cates that an abrupt change in crack growth mechanism should occur at a characteristic impurity chemical potential. The traction curve for decohesion can be obtained as the derivative of the energy of a solid as two slabs of bulk ma- terial are uniformly separated along a pair of adjacent atomic planes. This stress-separation relation then serves as a first- principles input to describe the resistance to crack growth of the cohesive zone in continuum simulations of fracture. 3,4 Impurity atoms can segregate between the decohering atomic planes. Calculating the energy of decohesion as a function of slab separation from first principles is straightforward if the impurity concentration and arrangement between the separat- ing atomic planes is kept fixed. 5-7 Obtaining the traction curve at constant impurity chemical potential though requires additional thermodynamic considerations. Thermodynamically, the decohering region can be consid- ered as a subregion of the solid characterized by excess ex- tensive quantities 8 as in the standard thermodynamic descrip- tion of surface properties. The excess internal energy for the decohering region per unit area ~of the separating atomic planes! u is related to the other excess extensive quantities according to