Symmetry Breaking Resulting from Long-Range Interactions in Out of Equilibrium Systems: Elastic Properties of Irradiated AgCu

This work presents a consistent formulation of the phase-field approach to model the behavior of nonmiscible alloys under irradiation which includes elastic strain fields, an example of a long-range interaction. Simulations show that the spatial isotropy that is characteristic of radiation-induced patterns breaks down as a result of the elastic strain energy. The consequence of this is the emergence of superlattice structures under irradiation liable to modify macroscopic material properties. This approach is assessed against the experimental study of a AgCu alloy under irradiation: we compare our simulation results to measured solubility limits and Young moduli.

Figure 1

Calculated steady state modulations of the silver concentration in ${\mathrm{Ag}}_{0.42}{\mathrm{Cu}}_{0.58}$ samples irradiated with 1.6 MeV Kr ions at $T=443\mathrm{K}$ (flux of $6\times {10}^{11}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$). These microstructures are computed for different contributions of the elastic strain energy [(a) $W_0=0$ and $W_1=0$ (no contribution), (b) $W_0\ne 0$ and $W_1=0$ (zero-order expansion of elastic strain energy), (c) $W_0\ne 0$ and $W_1\ne 0$ (first-order expansion of elastic strain energy)] and exhibit quite different characteristics. The formation of ordered superlattices of precipitates occurs when an elastic interaction is modeled, as illustrated by the Fourier transforms [(d) $W_0=0$ and $W_1=0$, (e) $W_0\ne 0$ and $W_1=0$, (f) $W_0\ne 0$ and $W_1\ne 0$].

Figure 2

Comparison between experimental solubility limits (cyan triangles) of Ag in irradiated ${\mathrm{Ag}}_{0.42}{\mathrm{Cu}}_{0.58}$ thin films at different temperatures and computed from Eq. (1) [$W_0=W_1=0$, red squares; $W_0\ne 0$ and $W_1=0$, full green circles; $W_0\ne 0$ and $W_1\ne 0$, blue stars].

Figure 3

Comparison between calculated and experimental irradiation temperature dependent Young moduli. Two simulation results are reported, both of which were obtained from the minimization of $\mathcal{L}[\theta ]$. The first (red line) concerns calculations for which only the chemical potential and ballistic mixing contributions to the Lyapounov were taken into account. The second (blue stars) concerns calculations for which chemical potential, ballistic mixing, and elastic contributions were modeled. The experimental data are obtained from nanoindentation experiments on irradiated ${\mathrm{Ag}}_{0.42}{\mathrm{Cu}}_{0.58}$ thin films (cyan triangles).