Predicting Nonequilibrium Patterns beyond Thermodynamic Concepts: Application to Radiation-Induced Microstructures

In this work, we derive an analytical model to predict the appearance of all possible radiation-induced steady states and their associated microstructures in immiscible $A_{\overline{c}}{B}_{1-\overline{c}}$ alloys, an example of a nonequilibrium dynamical system. This model is assessed against numerical simulations and experimental results which show that different microstructures characterized by the patterning of $A$-rich precipitates can emerge under irradiation. We demonstrate that the steady-state microstructure is governed by irradiation conditions and also by the average initial concentration of the alloy $\overline{c}$. Such a dependence offers new leverage for tailoring materials with specific microstructures overcoming limitations imposed by the equilibrium thermodynamic phase diagram.

Figure 1

Ag isoconcentrations resulting from the 3D numerical resolution of Eq. (2) in a ${\mathrm{Ag}}_{\overline{c}}{\mathrm{Cu}}_{1-\overline{c}}$ alloy irradiated with 1 MeV Kr ions at 330 K with a flux of $\varphi ={10}^{11}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. Distinct periodic microstructures emerge depending upon the average Ag concentration $\overline{c}$ [(a) 1D stripes ($S$), (c) cylinders forming hexagonal patterns ($C$), (e) 3D body-centered cubic structures ($\overline{c}=0.65$) ($B$), (g) solid solution ($\overline{c}=0.75$) (SS)]. Microstructures associated with the coexistence of such domains are also plotted [(b) $S+C$, (d) $C+B$, (f) $B+\mathrm{SS}$].

Figure 2

Theoretical structure diagram displaying all possible radiation-induced microstructures in AgCu. Microstructures presented in Fig. 1 [$D(q_0)=-0.3$] resulting from the numerical resolution of Eq. (2) [$\mathrm{\Delta }(T,\varphi )=0.0151$, i.e., $D^{\text{ball}}(\varphi )=0.01D^{\text{therm}}(T,\varphi )$, and $R=2$] are also plotted (colored dots).

Figure 3

(a) Values of $A_s^j-\overline{\theta }$ resulting from the minimization of $\mathcal{U}$ in the pattern regime (magenta diamonds) and obtained from the numerical resolution of Eq. (2) (cyan dots) [$D(q_0)=-0.30$]. The absolute value of the relative error between the analytical and numerical $A_s$ values is lower than 5%, justifying the one-mode approximation. (b) Measured (red stars) and analytical (blue line) values of $A_s^C$ plotted as a function of $D(q_0)$ for $\overline{\theta }=-0.2$ in the pattern regime.