Novel mechanism for order patterning in alloys driven by irradiation

Kinetic Monte Carlo simulations have been performed to investigate the evolution of ordered domains in model alloys under irradiation. The alloys investigated were equiatomic binary alloys on a simple square lattice with first and second nearest-neighbor interactions, chosen so that a $2\times 2$ ordered structure is the equilibrium phase below a critical order-disorder transition temperature $T_{\mathrm{c}}$. The ratio of second to first nearest-neighbor interactions $R$ was varied from 0 to 0.45 to explore the effect of the thermodynamic frustrations induced by the proximity of the $2\times 1$ phase boundary, which occurs at $R=0.5$ for $T=0$. The atomic mixing produced by nuclear collisions was modeled by forcing the ballistic exchange of pairs of atoms at a controlled rate ${\mathrm{\Gamma }}_{\mathrm{b}}$. This disordering process competed with thermodynamic reordering, resulting in nonequilibrium steady states. Two trivial steady states were found, a disordered state at high ${\mathrm{\Gamma }}_{\mathrm{b}}$ and low $T$, and a long-range ordered state at low ${\mathrm{\Gamma }}_{\mathrm{b}}$ and low $T$. In the $R=0.45$ alloy, however, a third steady state was identified at intermediate ${\mathrm{\Gamma }}_{\mathrm{b}}$ and $T$ values, where multiple long-range ordered domains coexisted dynamically. It is shown that this state of patterning of order resulted from the coupling of the thermodynamic frustrations present in that alloy with the disorder introduced by irradiation. The practical relevance of this novel mechanism for patterning of order under irradiation is discussed in the context of recent observations of domain coexistence in irradiated ${\mathrm{Cu}}_3\mathrm{Au}$.

Figure 1

Transmission electron microscopy imaging of ordered domains in $\mathrm{L}{1}_2$ ordered ${\mathrm{Cu}}_3\mathrm{Au}$ irradiated with 500 keV Ne ions at $350{}^{\circ }\mathrm{C}$ at a dose rate of $1.1\times {10}^{-4}$ displacement per atom per second (dpa/s) (dark field image formed using $\mathrm{L}{1}_2$ superlattice reflection). Notice the presence of many, small new domains, which nucleated within the initial single-domain ordered state.

Figure 2

Ground state diagram for the 2D square lattice with first and second nearest-neighbor interactions [25]. The four small open circles correspond to the systems studied in this paper.

Figure 3

SRO $\eta$ maps at steady state for the $R=0$ alloy at $T=0.8T_{\mathrm{c}}$ and for ballistic frequency ${\mathrm{\Gamma }}_{\mathrm{b}}=0,6\times {10}^7,2\times {10}^8$, and $1\times {10}^9{\mathrm{s}}^{-1}$, from left to right. The color scale ranges from yellow (light gray) for $\eta =1$ to red (dark gray) for $\eta =0$.

Figure 4

SRO $\eta$ maps at steady state for the $R=0.45$ alloy at $T=0.8T_{\mathrm{c}}$ and for ballistic frequency ${\mathrm{\Gamma }}_{\mathrm{b}}=0,1.2\times {10}^3,4.15\times {10}^3$, and $3\times {10}^4{\mathrm{s}}^{-1}$, from left to right. The color scale ranges from yellow (light gray) for $\eta =1$ to red (dark gray) for $\eta =0$.

Figure 5

Temporal evolution of the SRO parameter $\eta$ and the LRO parameter $S$ for $R=0.45$ alloy, $T=0.8T_{\mathrm{c}}$, and ${\mathrm{\Gamma }}_{\mathrm{b}}=3\times {10}^3{\mathrm{s}}^{-1}$. Systems initially ordered and initially disordered reached the same steady state. As expected, the SRO reaches steady state at shorter times than the LRO, and fluctuations are larger for the LRO evolution than for the SRO one.

Figure 6

For alloy $R=0$, steady state values of (a) LRO parameter $S$, (b) absolute net magnetization $\left|M\right|$, and (c) SRO parameter $\eta$ as function of temperature and replacement rate ${\mathrm{\Gamma }}_{\mathrm{b}}$. Initial condition is fully ordered $2\times 2$ state. Maps are constructed from 540 ($T, {\mathrm{\Gamma }}_{\mathrm{b}}$) points. White contour lines are added to highlight $20\%$ decreases in the parameters. Overall phase diagram is displayed in (d) using criteria given in text.

Figure 7

For alloy $R=0.45$, steady-state values of (a) LRO parameter $S$, (b) absolute net magnetization $\left|M\right|$, and (c) SRO parameter $\eta$ as function of temperature and replacement rate ${\mathrm{\Gamma }}_{\mathrm{b}}$. Initial condition is fully ordered $2\times 2$ state. Maps are constructed from 340 ($T, {\mathrm{\Gamma }}_{\mathrm{b}}$) points. White contour lines are added to highlight $20\%$ decreases in the parameters. Overall phase diagram is displayed in (d) using criteria given in text.

Figure 8

Various configurations of antisites and disorder on the $2\times 2$ square lattice, here decomposed into $\alpha$ and $\beta$ sublattice sites, represented by circles and squares, respectively, with $A$ atoms and $B$ atoms shown as black and white symbols, respectively. (a) Isolated $A_{\beta }$ antisite, (b) pair of first nearest-neighbor $A_{\beta }-B_{\alpha }$ antisites, and (c) (10) APB.