View source for Multistage Modeling of Zn Coated Steel

===ICME overview of Zn Coated Steel===
====Problem description====
	Steel grades are becoming increasingly complex, especially with the development of third generation advanced high strength steels (AHSSs), to allow for a reduction in vehicular weight which leads to improvement in fuel efficiency. The complexity of these steel grades is owed to the addition of alloying elements such as Mn, Si, Cr, V, etc, and heat treatments that cause an increase in strength and ductility. The alloying elements diffuse to the surface of the steel during annealing of the steel strip, prior to dipping in the Zn bath, and preferentially oxidize. These oxides negatively influence the wettability of the Zn to the substrate by hindering the reaction between Fe and the dissolved Al in the Zn bath. The Fe and Al preferentially react first to form a thin layer of Fe2Al5 that is on the order of approximately 100 nm. This layer is also called the inhibition layer as it inhibits the premature interaction between Fe and Zn and the formation of unfavorable Fe-Zn phases at the interface. Once the Zn adheres to the substrate, its thickness is regulated using air knives. At this point in time, if the steel requires no further processing in order to preserve the luster of the coating, then it is called a galvanized steel. However, if industry requires the substrate to be coated with a particular Fe-Zn phase to improve weldability, paintability, formability, and adherence, then the steel sheet is passed through a post annealing furnace where the Fe and Zn diffuse into one another and form different phases. A steel with this type of Zn coating is called a galvannealed steel. A Zn coated steel undergoes through multitude of processes before a finished product is obtained. As such, this indicates that the history effects on the substrate are of vital importance. For the present case. only discuss one particular stage of the galvannealing process that involves the post annealing of the Zn coating and expand upon it using the integrated computational materials engineering (ICME) ideology.

====Downscaling====
To know the lowest relevant length scale, one needs to downscale first starting from the desired finished part. In our current case, we start off with a stamped galvannealed steel part shown in the top right corner of the multiscale modeling slide, illustrated in Figure 1. The part was stamped from a galvannealed steel sheet shown in the top left of the figure. To stamp this sheet, one requires the knowledge of whether the Zn coating is going to crack under the applied load and therefore, requires knowledge of which side of the Zn coating is going to experience the maximum stress concentration and crack first. To be able to answer the aforementioned question, a thorough knowledge of the phases present in the Zn coating and their properties is required. The problem can be further understood and downscaled by realizing that the phases are formed by the interdiffusion of Fe and Zn at the microscale. Furthermore, for the diffusion to happen, the Zn first needs to adhere to the steel substrate which requires the comprehension of the interaction between layers of Fe and Zn atoms at the atomistic scale. Finally, to achieve the knowledge of how a multitude of atoms interact, an understanding of the inter-atomic potential between single atoms of Fe and Zn is required at the electronics principle scale. Information is passed between each length scale and to the macroscale ISV continuum model which will be used to simulate the behavior of a steel sheet when stamped. 

====Upscaling====

====Electronic Scale====
	Starting at the lowest length scale, we can employ Density Functional Theory (DFT), which is a method that utilizes the quantum mechanical modeling to analyze the ground-state properties of materials [ ]. The simulation based on DFT basically describes the behavior of a diatomic system of atoms that interact with one another. Some of the material properties that can be found using DFT are the lattice parameters, bulk modulus, elastic modulus, equilibrium energy, generalized stacking fault energies and the surface absorption energies. In the absence of material potentials due to lack of experimental basis, some of these parameters can be passed on to the next length scale at the atomistics level to be incorporated in molecular dynamics simulations. The interatomic potentials needed for Fe and Zn were determined in the literature [ ] using different methods that incorporated the Density Functional Theory. The pairwise potential between two atoms of Zn was modeled using the Morse potential [ ], the interaction between BCC Fe atoms was modeled using the potential function of Embedded Atom Method of Finnis and Sinclair (EAM/FS) [ ] and the Fe-Zn pair potential was calculated using the energy function [ ] below:

The software package available to be used to run DFT calculation is Vienna ab initio Software Package (VASP) [ , , , ] which can provide us with the equilibrium lattice constant, the equilibrium energy and the bulk modulus. We will pass the theoretical bulk modulus calculated from the software up to the macroscale continuum model and hand off the equilibrium lattice constants and equilibrium energy as a bridge between electronic scale to the atomistic scale. 


====Atomistics/Nanoscale Scale====
	At the atomistic scale, molecular dynamics will be used to simulate movement of a multitude of atoms based on Newton’s equation of motion as they are allowed to interact for a period of time based on the interatomic potentials between atoms that is found from the electronic scale. The methodology that is going to be used to simulate molecular dynamics is the Modified Embedded Atom Method (MEAM) [ ]. This method describes the interatomic potential of an atom based on its nearest neighbor atoms that depends on their distance and bond angle along with the embedding energy that is a function of electron cloud density to simulate metallic bonding. Consulting the literature revealed that molecular dynamics simulations have been performed using LAMMPS (Large Scale Atomic/Molecular Massively Parallel Simulator) to determine the energy absorbed at the onset of failure of the Zn coated Fe under applied shear load at various temperatures, thicknesses and shear rates [6]. The deed was accomplished by placing a few layers of Zn atom on top of a block of BCC Fe and applying a shear force at constant rate while holding the Fe block at a fixed position. The displacement of the Zn atoms above the Fe atoms was measured until it reached the lattice constant of Fe and slipped to a new position, which was equivalent to failure of Zn coating. The corresponding value of the energy of the system at the onset of failure provides the shear energy absorbed and, hence, required for delamination of the Zn coating. Another methodology reported in the literature utilizes a calculable parameter, work of adhesion, W¬Ad, to determine the onset of failure (or delamination) [ ]. This parameter determines the work required at the interface to separate Phase A from Phase B per unit phase area and is defined as the following,

where γ_A^S and γ_B^S are the surface energy of phase A and B, respectively, γ_(A-B)^Interaction is the interaction energy between A and B, and γ_(A-B)^Mismatch is the strain energy caused by the mismatch between A and B at their interface. The surface energy is a parameter that can be easily calculated from the electronic scale and passed up to the current scale. The mismatch energy at the interface can be estimated using a simple equation that requires the surface energy of Phase A and B only [ ]. The interaction energy, however, can be evaluated by consulting the work of Song and Sloof [ ] to use the following equation. 

where C_(A_i)^S is the surface fraction of element A_i in solid solution phase A, ΔH_(A_i  in B_i)^Interface is the interface enthalpy of one mole of element A_i in an infinitely large reservoir of element B_j and V_(A_i )  is the atom volume of A_i. Both of the methodologies showed promising results and therefore, can be used to evaluate useful data. The work of adhesion can be bridged off to the macroscale continuum model where it will come into play at the macroscale to simulate the behavior of interfaces of different phases under the influence of load and stress. The information that needs to be bridged to next length scale is the flux of the material present across the interface that will be responsible for a concentration gradient to develop and enable diffusion of atoms. 

====Microscale/Mesoscale====
	At the micro-mesoscale, the Zn coating is fully adhered to the Fe substrate that has been removed from the molten Zn bath. The substrate enters the post annealing furnace where the increase in temperature causes an increase in the equilibrium energy of the atoms, thereby lowering the activation energy required for the interdiffusion of Fe and Zn to form multitude of phases. The diffusion of each element depends upon the diffusion flux, which is the amount of element present per unit area per unit time, to determine the direction of the flow of material and the change in the material concentration with time. These phenomenon are governed by Fick’s first and second laws of diffusion where the diffusion coefficient, D, is evaluated. This parameter is passed on as an internal state variable to the continuum model. The diffusibility of Fe can then be calculated by presenting a solution [ ] found in the literature to Fick’s second law using the following equation,

where c_1 and c_2 are the mole-fractions of Fe in the steel and in the coating, respectively, x is the distance from the Mantano-plane (x = 0 equated to the interface substrate/coating), t is the duration of the galvannealing treatment, and D_eff (Fe) is the desired effective diffusion coefficient of Fe in the Zn coating during the galvannealing treatment. 


====Macroscale - 1====
	The formation of phases and their growth simultaneously occur. The growth rate of each phase depends upon its equilibrium with the adjacent phase and the growth rate kinetics at an elevated temperature. The process occurs throughout the coating thickness, and involves movement of phase boundaries and as such, requires a moving boundary model to capture the entire phenomenon. One such moving boundary model used in the literature is DICTRA [ ], where the boundary migration rate is determined by the rate of diffusion to and from the interface to conserve the number of moles of a component. Another model that deals with a moving boundary problem is a phase-field model that can be successfully handled using diffuse-interface model.  Such models are constructed by assuming that the free energy of a non-uniform system depends upon the phase field variable ϕ and its gradient ∇ϕ, over a volume, V, described by equation 5 [ ] ,

The model can determine the mobility of each phase and the growth rate kinetics can be validated against experimental data available. This information, will also be passed on to the ISV continuum model whereas the volume fraction of phases in their final form and the hardness of each phase will be calculated and handed off to the next modeling step but at the same length scale. 

====Macroscale - 2====
	Once the width, elastic modulus, and hardness of each phase is known, the information can be paired with the work of adhesion between the interfaces of each phase, found at the atomistic length scale. The bulk of information collected is then utilized in a crystal plasticity [ ] finite element method, using ABAQUS as the software package, to run a simulation of a plate undergoing a bending test that would induce complex stress states in the cross section of the sheet. Modeling of similar nature has been performed in the literature using ABAQUS as a 2D plane strain composite layer system consisting of an elastic perfectly plastic steel substrate and homogeneous, isotropic linear elastic Fe-Zn phases [ ]. The parameters inserted in the code, such as Young’s Modulus, Vickers hardness, Thermal expansion coefficient, and the thickness of each phase, were found from literature and experimental observations. For our case, the parameters provided to the ABAQUS software will be obtained from the simulations performed at the lower length scale, which can be then validated against the literature results. The useful information extracted from the ABAQUS simulation to produce conclusive results included the critical stress intensity factor, critical bending angle, and the critical far field stress. The results of these simulations would allow us to determine which phase in the zinc coating experiences the greatest stress and likelihood of initiating and propagating a crack. The valuable information collected is then passed on towards the ISV continuum model.

====References====
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