An Atomistic-Based Hierarchical Multiscale Examination of Age Hardening in an Al-Cu Alloy

A large class of modern structural alloys derives its strength from precipitation hardening. Precipitates obstruct the motion of dislocations and thereby increase alloy strength. This paper examines the process using an atomistic-based hierarchical multiscale modeling framework. Atomistic modeling is employed to (1) compute solute-dislocation interaction energies for input into a semi-analytic solute hardening model and (2) evaluate precipitate strengths for use in dislocation line tension simulations. The precipitate microstructure in the dislocation line tension simulations is obtained from simple analytic precipitation kinetics relations. Fitting only the rate constants in the precipitation kinetics model, the macroscopic strength predictions of the hierarchical multiscale model are found to correspond reasonably well with experiments. By analyzing the potential sources of discrepancy between the model’s macroscopic predictions and experiments, this work illuminates the importance of specific atomic-scale processes and highlights important challenges that remain before truly predictive mechanism-based plasticity modeling can be realized.

Appendices

Appendix A: Atomistic Modeling of Precipitation Hardening

The atomistic simulations were carried out using the freely available open source LAMMPS code.[93] The code was modified to use a recently developed Al-Cu empirical potential developed by Apostol and Mishin.[57] The potential is an angular-dependent extension of the embedded atom method (EAM)[94,95] where the potential energy of the system is

$$E_{\rm{tot}}=\frac{1}{2} \sum_{i,j (j \neq i)} \Upphi_{ij} (r_{ij}) + \sum_{i} F_i (\bar{\rho_i}) + \frac{1}{2} \sum_{i,\alpha} (\mu_i^\alpha)^2 + \frac{1}{2} \sum_{i,\alpha,\beta} (\lambda_i^{\alpha\beta})^2 -\frac{1}{6} \sum_{i} v_i^2.$$

(A1)

Indices i and j enumerate atoms and the superscripts α, β = 1, 2, 3 refer to the Cartesian directions. The first two terms, taken together, represent regular EAM contributions, where $\Upphi_{ij}$ is the pair-interaction potential and F _i is the embedding energy of atom i with $\bar{\rho_i}$ loosely representing the electron density. The three remaining terms represent the angular dependence through dipole and quadrupole distortions. The details of the potential can be found in Apostol and Mishin[57] and Mishin et al.[95] We validated our implementation by comparing relaxed cohesive energies, vacancy formation energies, surfaces energies, heats of solution, and formation energy for the θ″ (Al₂Cu) phase with independent values provided by Apostol and Mishin.[57] The simulation cell (Figure A1) consisted of an FCC Al lattice, bounded by $\left(\bar110\right), \left(111\right),$ and $\left(11\bar2\right)$ faces in X, Y, and Z directions, respectively. Periodic boundary conditions were applied in the X and Z directions, and the Y surfaces were used to apply the load. Starting from a perfect lattice, a precipitate was created by simply changing selected atom types to Cu. The continuum displacement field of an edge dislocation was used to create a dislocation in the center of the simulation cell (X = Y = 0) with a line direction parallel to the Z axis and the burgers vector $\vec b=1/2[\bar110].$

Fig. A1

(a) Simulation cell with an edge dislocation and GP zone, (b) 3 possible GP zone orientations with respect to the Burgers vector of the dislocation. Type A is referred to as a 0 deg interaction while types B and C are considered equivalent and refered to as 60 deg interactions

Because GP zones and θ″ precipitates only form on {100}-type planes, there are only three physically relevant orientations by which they interact with edge dislocations, two of which are crystallographically equivalent. The Burgers vector of the edge dislocation will either be at an angle of 60 deg from the precipitate plane or be parallel to it (Figure 1). Thus, we will refer to dislocation–precipitate interactions as either being 60 or being 0 deg in nature. As a simplification, we assume that GP zones are monolayers of Cu atoms with a 100 pct Cu content, while θ′′ precipitates are two Cu layers separated by three Al layers.[14,16]

The minimum cell size was approximately 34 × 42 × 16 nm³ and contained approximately 1.4 million atoms. For the simulations with large GP zones and θ′′ precipitates, a larger cell size was used (with largest cell size being approximately 68 × 44 × 64 nm³ consisting of 11 million atoms). In all cases, box size studies were performed to verify that simulation cell boundary forces were negligible in the X and Y directions. The effect of cell size in the Z direction will be discussed later; however, we note that the cell sizes in the Z direction that we explored are consistent with experimental observations of GP zone and θ″ spacing.[17,28]

Prior to loading, NPT dynamics were performed for 50 ps to relax the system and alleviate out-of-balance forces and net stresses. The system was then loaded in shear by subjecting the atoms near the top and bottom Y surfaces to a constant traction in the X direction given by

$$f_{\rm{top}}= \frac{\tau_{xy} A_{xz}}{N}, \qquad f_{\rm{bottom}}= -\frac{\tau_{xy} A_{xz}}{N}$$

(A2)

for top and bottom Y surfaces, respectively, where τ _xy is applied shear stress, A _xz is the surface area, and N is the number of atoms on the respective surfaces. The applied shear stress was increased quasi-statically until the dislocation breaks free from the precipitate. At each load step of 5 MPa, non-linear conjugate gradient was performed to minimize the out-of-balance forces to less than 10⁻⁸ eV/Å.

Appendix B: Analytic Relations for Critical Resolved Shear Strength of the Precipitate, τ_c

Assuming a fixed dislocation line tension, ϕ_c is an intrinsic measure of the strength of the precipitate. Using a simple line tension model and ignoring dislocation self-interactions, the critical stress required for a dislocation to shear a precipitate can be written as

$$\tau_{\rm c}=\frac{2T}{bL} \cos{(\phi_{\rm c}/2)}$$

(B1)

with T being the line tension of the dislocation. Note that this relation suggests τ_c scales as 1/L.

In the limit of an impenetrable precipitate, the dislocation will bow into a configuration with neighboring segments on each side of the precipitate perpendicular to the dislocations’ original glide direction, i.e., ϕ_c = 0. In this situation, the dislocation segments will collapse via the Orowan looping mechanism,[49] with a critical stress of

$$\tau_{\rm c}^{\rm{Orowan}}=\frac{2T}{bL}$$

(B2)

From continuum elasticity theory, T ≈ Gb ²/2, where G is the shear modulus of the matrix material. This leads to $\tau_{\rm c}^{\rm{Orowan}}=\frac{Gb}{L}$ and $\tau_{\rm c}=\frac{Gb}{L} \cos{(\phi_{\rm c}/2)}=\tau_{\rm c}^{\rm{Orowan}}\cos{(\phi_{\rm c}/2)}$ for Orowan looping and shearing, respectively. A simple modification is often used to extend these relations into a domain where D is not negligible with respect to L.[26]

$$\tau_{\rm c}^{\rm{BH}}=\frac{Gb}{L-D_{\rm p}} \cos{(\phi_{\rm c}/2)},$$

(B3)

with L − D _p representing the distance which is available between neighboring precipitate edges for the dislocation to bow-out and D _p being the projected cross-sectional diameter of the precipitate along the dislocation line. Because the above expressions ignore dislocation self-interactions, they overestimate precipitate strengthening. Specifically, the attraction between the two segments of dislocation on each side of the precipitate will promote a more acute bowing angle as well as Orowan looping before ϕ_c = 0 deg. A semi-empirical expression incorporating self-interaction effects has been given by Scattergood and Bacon[50] by fitting to numerical results,

$$\tau_{\rm c}^{\rm{SB}}=\frac{Gb}{2\pi AL}\left[ln(D^{-1}+L^{-1})^{-1}+B\right]$$

(B4)

where B is a fitting parameter and A = 1 when the dislocation is initially pure edge and (1 − ν) when pure screw with ν being the Poisson’s ratio.

Appendix C: Derivation of Solute Hardening

The model is built upon an interaction energy function between a solute atom and a dislocation, U(x _i , y _j ), with x _i and y _j describing the position of the lattice site of the solute atom with respect to a dislocation with a line direction along the z-axis. Using U(x _i , y _j ), which is computed from atomistic simulation, the change in energy associated with the movement of a straight dislocation in a field of random solutes a distance w in the direction x can be written as

$$\Updelta \hat{E_{\rm p}}(w)=\sum_{ij} n_{ij}\left(U(x_i-w,y_j)-U(x_i,y_j)\right)$$

(C1)

where n _ij is an independent random variable representing the number of solute atoms at position (x _i , y _j ) along the dislocation line. Considering that the solute locations are uncorrelated, the typical change in energy associated with the movement of a straight segment of dislocation of length, ξ, can be written using a correlation function

$$\chi(w,y_j)=\sum_{i} U(x_i-w,y_j)U(x_i,y_j)/\sum_{i} U^2(x_i,y_j),$$

(C2)

as

$$\Updelta E_{\rm p} (\xi,w)= -\sqrt{\frac{c_{\rm s}\xi}{\sqrt{3}b}\left[2 \sum_{i,j} \{1-\chi(w,y_j)\} U^2(x_i,y_j) \right]}$$

(C3)

with c _s being the solute concentration. For a finite w, only a finite region near the dislocation segment needs to be considered to compute $\Updelta E_{\rm p}.$

The heterogeneity of the energy landscape experienced by the dislocation promotes a curved dislocation line. In the model, the curvature of the dislocation line is approximated as a series of straight dislocation segments of length ξ that exist at discrete increments of w in the glide direction. The extra line energy due to a segment of dislocation length ξ gliding a distance w away from its neighboring segments can be written as

$$\Updelta E_{\rm{el}}(\xi,w)=\Upgamma \frac{w^2 L}{4\xi^2}$$

(C4)

with $\Upgamma$ representing the line energy.

The total energy change associated with the glide of a segment of dislocation from an originally straight dislocation is the sum of its change in line energy and change in interaction energy.

$$\Updelta E_{\rm{tot}}(\xi,w)=\Updelta E_{\rm{el}}+\frac{\Updelta E_{\rm p} L}{2\xi}$$

(C5)

By minimizing this equation in terms of ξ and w, characteristic length scales ξ_c and w _c can be obtained.

Considering that the strength-controlling energy barriers to dislocation glide result from segments of dislocation length ξ_c that must move from a typical favorable interaction energy fluctuation past a typical unfavorable interaction energy fluctuation over a distance of w _c while regaining the line energy $\Updelta E_{\rm el}(\xi_{\rm c},w_{\rm c}),$ the typical energy barrier to the glide of a dislocation segment is

$$\Updelta E_{\rm b}(\xi_{\rm c},w_{\rm c})=\sqrt{2}\Updelta E_{\rm p}(\xi_{\rm c},w_{\rm c})-\Updelta E_{\rm el}(\xi_{\rm c},w_{\rm c})$$

(C6)

Finally, assuming a sinusoidal energy profile with a peak of $\Updelta E_{\rm b}$ over a distance of w _c provides the zero-temperature shear stress needed to move a typical segment of dislocation forward,

$$\tau_{\rm{ys}}=\frac{\pi}{2}\frac{\Updelta E_{\rm b}}{b \xi_{\rm c} w_{\rm c}}$$

(C7)

Appendix D: Hardening Contributions

The strength of the Al-Cu alloy will increase as Cu atoms transfer from solute to GP zones if $\frac{{\partial \tau _{{{\text{yGP}}}} }}{{\partial c_{{{\text{GP}}}} }} > \frac{{\partial \tau _{{{\text{ys}}}} }}{{\partial c_{{\text{s}}} }}$ with c = c _GP + c _s where c _GP represents the concentration of Cu belonging to GP zones and τ_yGP represents the strengthening due to GP zones. While $\frac{\partial \tau_{\rm{ys}}}{\partial c_{\rm s}}$ can be explicitly expressed in terms of c _s, i.e., τ_ys∝ c ^2/3_s (Section III and,[54] $\frac{\partial \tau_{\rm{yGP}}}{\partial c_{\rm{GP}}}$ must be approximated due to its non-analytic dependence on GP zone diameter. For the Al-4 wt pct Cu alloy aged at 383 K (110 °C), τ_yGP(c _GP) can be described reasonably well with a simple power law

$$\tau_{\rm{yGP}} = \tau_{\rm{yGP}}^0 c_{\rm{GP}}^{3/7}$$

(D1)

with the fitting constant τ ⁰_yGP  = 1400 MPa. Considering this relation and τ_ys/c ^2/3_s  = 1138 MPa (from Section III), the inequality, $\frac{\partial \tau_{\rm{yGP}}}{\partial c_{\rm{GP}}} > \frac{\partial \tau_{\rm{ys}}}{\partial c_{\rm s}},$ implies that strength will increase as Cu atoms transfer from solute to GP zones if c _GP < 0.663 c ^7/12_s .

Appendix E: Derivation of Plastic Strain Rate in the Presence of Thermal Activation

First, the rate at which the controlling thermally activated events occur is assumed to follow an Arrhenius relation

$$\tilde{\nu}=\tilde{\nu_0} exp \left( \frac{-\Updelta E}{k_{\rm B} T} \right).$$

(E1)

The exponential prefactor, $\tilde{\nu_0},$ is assumed to be a constant and the energy barrier, $\Updelta E,$ is assumed to only depend upon the ratio of the applied load, τ, over the load at which the thermally activated events occur instantaneously², τ_inst. Assuming that the shape of the energy barrier, with respect to the reaction coordinate, is sinusoidal, the functional form of $\Updelta E$ can be approximated to the first order as

$$\Updelta E=\Updelta E_0 \left( 1-\frac{\tau}{\tau_{\rm{inst}}} \right)^{3/2},$$

(E2)

Finally, the rate at which the controlling thermally activated events occur, ν, is assumed proportional to the plastic strain rate, $\dot{\gamma}.$

Assembling the above assumptions and Eqs. [E1] and [E2], one can construct a popular expression for relating the applied load τ to the plastic strain rate $\dot{\gamma}$ in the presence of thermal activation,[54,55]

$$\frac{\tau}{\tau_{\rm{inst}}}=1-\left(\frac{k_{\rm B} T}{\Updelta E_0}\hbox {ln}\frac{\dot{\gamma_0}}{\dot{\gamma}}\right)^{2/3}.$$

(E3)