According to the analytic nearest-neighbor EAM model proposed by Johnson for fcc pure metals [56], both the electron density 𝑓(𝑟) and the two-body potential Φ(𝑟) can be expressed as decreasing exponential functions requiring two parameters each12:
𝑓(𝑟) =𝑓𝑒exp [ −𝛽 (𝑟 𝑟𝑒 −1 )] , 𝑟𝑒= 𝑎 √ 2 Φ(𝑟) = Φ𝑒exp [ −𝛾 (𝑟 𝑟𝑒 −1 )] (3.76)
In addition the embedding function𝐹(𝜌) is obtained from the zero-temperature equation of state proposed by Rose et al. [58]13:
𝐹(𝜌) =−𝐸𝑐 [ 1−𝛼 𝛽ln (𝜌 𝜌𝑒 )] ( 𝜌 𝜌𝑒 )𝛼𝛽 −6Φ𝑒 (𝜌 𝜌𝑒 )𝛾𝛽 , 𝜌𝑒= 12𝑓𝑒 (3.77)
The model parameters (𝑓𝑒, Φ𝑒, 𝛼, 𝛽, and𝛾) are determined by fitting the lattice constant𝑎 or atomic volume 𝑉, the cohesive energy𝐸𝑐, the unrelaxed vacancy-formation energy𝐸𝑈 𝐹, the bulk modulus:
𝐵= 𝐶11+ 2𝐶12
3 (3.78)
and the Voigt-average shear modulus (all these quantities at room temperature):
𝐺= 𝐶11−𝐶12+ 3𝐶44
5 (3.79)
This potential does not include additional parameters to fit alloy properties, but the calculated heat of solution for binary alloys has been shown to be consistent with experimental data [56]. The model parameters for the elements used in this work (Cu, Ni, Pd, Au, Ag) are listed in Table 3.2.
12In prior EAM calculations the electron density 𝑓(𝑟) has been assumed to be well represented by spherically averaged free-atom densities calculated from Hartree-Fock theory. According to Johnson [76], when these Hartree- Fock electronic densities are plotted they can be approximated by a single exponential function. This is therefore the functionality assumed by this author.
13Based upon the compilation of many first-principle calculations, Rose et al. showed that for a broad range of materials the cohesive energy𝐸𝑐as a function of nearest-neighbor distance𝑟𝑒is well approximated by an EOS.
element 𝑓𝑒 [-] Φ𝑒[eV] 𝛼[-] 𝛽 [-] 𝛾 [-] Cu 0.30 0.59 5.09 5.85 8.00 Ni 0.41 0.74 4.98 6.41 8.86 Pd 0.27 0.65 6.42 5.91 8.23 Au 0.23 0.65 6.37 6.67 8.20 Ag 0.17 0.48 5.92 5.96 8.26
Table 3.2: EAM parameters
The information provided above allows the determination of the following functions in equation (3.38): 𝐹𝐴(𝜌),𝐹𝐵(𝜌),𝑓𝐴(𝑟),𝑓𝐵(𝑟), Φ𝐴𝐴(𝑟), Φ𝐵𝐵(𝑟). Johnson builds the remaining Φ𝐴𝐵(𝑟) –which describes the pair interaction between atoms of different species– from the monoatomic functions [56]:
Φ𝐴𝐵(𝑟) =1 2 [𝑓𝐵(𝑟) 𝑓𝐴(𝑟)Φ 𝐴𝐴(𝑟) +𝑓𝐴(𝑟) 𝑓𝐵(𝑟)Φ 𝐵𝐵(𝑟) ] (3.80)
Chapter 4
Variational Coupling for
Non-Equilibrium
Mechano-Chemical Problems
The equilibrium formulation presented in the last chapter allowed us to characterize the effect of concentration on (equilibrium) material properties. However, our final goal is to simulate kinetic processes such as the time evolution of impurity segregation towards grain boundaries. To this end, we present in this chapter an extension of the variational formulation proposed by Yang et al. [5] for mechano-chemical coupled problems.
The main difference between the work done by Yang et al. [5] and the framework we are about to present is that the latter is discrete from the onset. Therefore, instead of discretizing the govern- ing equations describing the behavior of continuum media, we start directly from a discrete system without any reference to the continuum. This seems a more natural starting point for the atom- istic samples considered so far, which are intrinsically discrete. Crystalline solids are of important practical interest in material science, and the anisotropic nature of their properties arises from the geometry and connectivity of the intermolecular bonds in the lattice. We will model these lattices as inherently discrete objects. In particular, we think this could yield a more detailed geometric understanding of how the system evolves in the presence of grain boundaries and other defects.
The formulation that follows relies on Discrete Exterior Calculus (DEC) [84]. This theory de- velopsab initio a calculus on discrete manifolds that parallels the calculus on smooth manifolds of arbitrary finite dimension. In our case we make use ofsimplicial complexes. Familiar examples of
simplicial complexes are meshes of triangles embedded in ℛ3 and tetrahedral meshes occupying a portion ofℛ3. Given a set of lattice points, in this work we construct a simplicial triangulation by recourse to Delaunay triangulation techniques.
In this chapter we show the existence of a joint potential function whose Euler-Lagrange equations yield the equilibrium equation, the kinetic relations and the conservation of mass of different species within a binary alloy. We restrict ourselves to elastic solids undergoing diffusional processes since we do not assume any flow, hardening rules or viscosity law in our work. Moreover, the only kinetic relation that we introducea priori is the generalized Fick’s law of diffusion.
As we shall see, the variational structure determines the coupling between mechanics and diffusion in a unique way. Thus, there is no need to postulate additional constitutive relations relating chemical potentials with concentrations or the state of stress of the system.
4.1
Balance Laws
The following equations describe the mechano-chemical coupled problem for a binary system with no prescribed external fluxes or forces:
1. Linear momentum balance1,
∇ ⋅𝜎=0 in𝑉 (4.1)
2. A-component mass balance2,
˙
𝑥𝐴=−∇ ⋅J𝐴 in𝑉 (4.2)
3. Equation of state,
𝐹 =𝐹(E, 𝑥𝐴) in𝑉 (4.3)
1The very different time scales for elastic and chemical equilibrium justify the assumption of elastic equilibrium at all times [85]
2Since we are not considering chemical reactions, and by definition ∂𝐹
∂𝑥𝑘
=𝜇𝑘, 𝑥𝐴+𝑥𝐵= 1
then𝜇𝐵 =−𝜇𝐴. As we’ll see, the atomic fluxJ𝑘 is proportional to the gradient of chemical potential∇𝜇𝑘. Then there is no need to solve for the B-component balance provided that the number of lattice sites𝑁 remains constant.
where𝜎is the Cauchy stress tensor,𝑥𝐴 is the atomic fraction of the𝐴-component,J𝐴is the atomic flux of that species, 𝐹 is the Helmholtz free energy,Eis an appropriate deformation measure, and 𝑉 is the volume occupied by the system in the current configuration.