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Nanoscale patterning of composition and chemical order induced by displacement cascades in irradiated alloys
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 242 (2006) 265–269 www.elsevier.com/locate/nimb
Nanoscale patterning of composition and chemical order induced by displacement cascades in irradiated alloys Jia Ye
a,*
, Pascal Bellon a, Rau´l A. Enrique
b
a
b
Department of Materials Science and Engineering, Urbana, IL 61801, United States Department of Materials Science and Engineering, Ann-Arbor, MI 48103, United States Available online 8 November 2005
Abstract When irradiation leads to the production of dense displacement cascades, it is predicted at the composition and the chemical order in metallic alloys may undergo spontaneous formation of nanoscale patterns. These predictions, which are based on continuum kinetic models and atomistic simulations, are reviewed and compared with available experimental data. If these predictions were confirmed, ion beam processing would offer a route for the direct synthesis of nanoscale patterns of composition and degree of order, with tunable length. Ó 2005 Elsevier B.V. All rights reserved. PACS: 61.80.Az; 05.65.+b Keywords: Patterning; Decomposition; Order–disorder transitions; Nanostructures; Displacement cascades; Kinetic Monte Carlo simulations
1. Introduction When materials are subjected to irradiation, nuclear collisions with large enough recoil energies result in the formation of displacement cascades [1]. It has been well known for many years that the rate of introduction and the size of these cascades, as well as the point defect evolution during the cascade lifetime, play a determinant role in microstructural evolutions under irradiation, such as segregation and precipitation on sinks, or crystal-to-amorphous transitions. Furthermore, the presence of displacement cascades introduces a bias in point defect formation that may lead to self-organization of defect clusters, such as dislocation loops, voids and gas bubbles [2]. Recently, however, we have shown that displacement cascades can also lead to the spontaneous self-organization of composition [3,4] and of chemical order [5]. Analytical models and
*
Corresponding author. Tel.: +1 217 333 1441; fax: +1 217 333 2736. E-mail address: [email protected] (J. Ye).
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.08.198
atomistic computer simulations reveal that these self-organization reactions find their origin in two length scales introduced by displacement cascades, the cascade size L and the average relocation distance of atoms within the cascade, R. For metallic alloys with moderately positive heats of mixing, DHm 6 10 kJ/g atom, molecular dynamics (MD) simulations indicate that the forced atomic mixing in these cascades is nearly random [1,6] and relocation distances follow an exponential decay [7]. For metallic alloys with negative heats of mixing, MD simulations indicate that any pre-existing long-range order is destroyed in the core of the cascades, thus producing disordered zones [8–12]. Disordered zones have been extensively studied by transmission electron microscopy (see [13] for a review). Under sustained irradiation conditions, the atomic mixing and the chemical disordering produced by displacement cascades may compete with the annealing promoted by the thermally activated migration of point defects. We briefly review in the following sections how this competition may lead to patterning of composition and chemical order in A–B binary alloys when the length scales of the cascade, L and R, exceed some critical values.
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2. Compositional patterning induced by the relocation distance Two decades ago, Martin [14] introduced a model to evaluate phase stability in alloys under irradiation, using a kinetic description that takes into account the presence of several dynamics. In MartinÕs model, forced atomic exchanges proceed by ballistic, i.e. random, nearest neighbor exchanges and the cascade size is not explicitly taken into account. These assumptions are reasonably well satisfied in the case of high-energy electron or light-ion irradiation [1]. In MartinÕs model, an alloy with positive heat of mixing should reach a homogeneous steady state when the irradiation temperature is low enough for the forced mixing to dominate over the thermal decomposition. As the irradiation temperature is increased, one crosses a dynamical phase transition, at which point the steady state microstructure is comprised of two macroscopic phases. Early kinetic Monte Carlo (KMC) simulations [15] and mean-field modeling [16] indicated however, that the presence of medium or long-range ballistic relocations could lead to the stabilization of compositional patterning under irradiation. For a binary alloy undergoing phase separation, we introduced a one-dimensional continuum kinetic model for the time evolution of the composition field, where forced atomic relocations are distributed according to an exponential decay with a characteristic length R [3]. This contribution introduces a non-local term in the kinetic equation. The local stability of a homogeneous steady state was studied by a linear stability analysis, and the global stability of composition profiles was evaluated by minimizing an effective free energy. Both analyses indicated that beyond a threshold value Rc, the stable steady state be comprised of compositional patterns with a characteristic size directly related to R. This prediction was confirmed by atomistic kinetic Monte Carlo simulations [17]. A dynamical phase diagram was built, see Fig. 1, which yields the domain of stability of steady states in the (R, c) space, where c = C/M is the ratio of the ballistic jump frequency to the thermal atomic mobility. The dynamical stabilization of patterns can be seen as a compromise resulting from the competition of two dynamics of similar strength but with different characteristic scales. Indeed, decomposition is more effective than forced mixing at small scale since a few atomic jumps are sufficient to maintain locally decomposition. At larger scales, however, the forced mixing is predominant, and, from a coarse-grained perspective, the alloy remains homogeneous. In the case of alloys that would phase separate at equilibrium into a solid solution and a chemically ordered phase, e.g. Ni-rich Ni–Al alloys, we have extended the above treatment [4]. Here again, it is predicted that beyond a threshold relocation range Rc, irradiation may stabilize compositional patterns. The coupling between the composition and the chemical order fields leads to patterns comprised of B-enriched ordered precipitates in a solid solution matrix.
Fig. 1. Dynamical steady-state phase diagram as a function of the average relocation distance for the ballistic jumps, R and the reduced irradiation intensity c. Solid lines c1, c2 are the transition lines predicted from the minimization of an effective free energy functional for an alloy with CB = 50%, and insets are (1 1 1) cuts of atomistic KMC simulations. A and C are coefficients entering the free energy functional (from [3]).
3. Patterning of chemical order induced by the cascade size In order to isolate the effect of the cascade size L, let us now assume that the ballistic mixing is short range, i.e. proceeds by exchanges between nearest neighbors in the atomistic KMC simulations, or R ! 0 in the continuum formulation. The cascade size is measured through the number of nearest neighbor pairs that are exchanged at once in a spherical volume. For simplicity, the fraction of pairs exchanged in this volume is kept constant at 80%. A cascade size of b = 4000 thus corresponds to a volume containing 10,000 atoms. For the parameters used in the KMC simulations (see [5] for details), the critical cascade size for order patterning is bc 100 in the present case, which would correspond to L 2 nm for the atomic density of Ni3Al. No significant direct effect of L is expected on the composition field, as the cascade effect reduces mainly to modifications of the correlation factors during atomic diffusion. KMC simulations performed on the model alloy with a positive heat of mixing used in Section 2 have confirmed the absence of compositional patterning, even for large cascade sizes. In the case of an ordering alloy, however, a direct effect of L is possible. Indeed, consider a long-range ordered phase; the disordered zone produced by a displacement cascade may re-order either from its interface or from within its core. In the latter case, this re-ordering will lead to the presence of antiphase domains, as more than one variant will be available for re-ordering. Under appropriate conditions, this sustained introduction of antiphase domains may destabilize an initially longrange ordered state, resulting in an ordered state with a finite domain size. This phenomenon has indeed been observed in KMC simulations [5] for a generic alloy on a face centered cubic lattice, which forms L12 and L10
J. Ye et al. / Nucl. Instr. and Meth. in Phys. Res. B 242 (2006) 265–269
ordered structures at B concentrations near 25% and 50%, respectively. We first consider a stoichiometric L12 alloy, i.e. cB = 0.25. The resulting dynamical phase diagram is shown in Fig. 2. Three possible steady states are observed: long-range ordered (LRO), disordered (D), or ordered but with finite domain sizes (OP, which stands for patterning of chemical order). The transition from LRO to non-LRO corresponds to the transition from one macroscopic L12 variant in the LRO state, to four equi-probable variants in the non-LRO state. The existence of the patterning state is derived from the best fit of the spherically averaged structure factor. In the absence of patterning, the best fit is obtained with a Lorentzian function, which corresponds to an exponential distribution of order fluctuations in real space. In the presence of patterns, however, the existence of well-defined interfaces must lead to a 1/q4 decay of the intensity at large q wave vectors, according to PorodÕs law [18]. We thus choose to fit the structure factor with a squared Lorentzian as well as, since this function is compatible with PorodÕs law at large q values. In the absence of irradiation and for small cascade sizes, it is found that the Lorentzian fit is better, and we thus conclude that no patterning is taking place. For cascade sizes larger than a critical value Lc, however, the squared Lorentzian fit is better than the one obtained with a Lorentzian, in agreement with direct visualization of ordered domains in real space. For L > Lc, unfortunately, it is not possible to determine easily the transition from the disordered state to the patterned state. For that purpose, we extended the Cook–Chen–Cohen method (CCC) [19], which is based on the evolution of the intensity of the structure factor at the superlattice position at equilibrium as one approaches an LRO transition from the disordered state. The Cook–Chen–Cohen method makes it possible to identify an ordering instability temperature. For equilibrium first-order transitions, this instability temperature is lower than the order–disorder transition temperature,
Fig. 2. Dynamical phase diagram for the L12 structure at T = 0.09 eV. The threshold for patterning of order is b 100. For a given b, the cascade size L is obtained using Ni3Al atomic density (from [5]).
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below which one ordering variant becomes macroscopic. Transposing this property into the present situation, where the disordering jump frequency C is progressively increased to change the stable steady state from ordered to disordered, we thus expect that ordering instability will take place at a value Cs smaller than Cc, the critical value for the LRO to non-LRO transition. As seen in Fig. 2, for small cascade sizes, 1 6 b 6 100, this inequality is indeed fulfilled. For larger cascade size, however, the two transitions take place in reverse order: Cs is predicted to be larger than Cc. It is both remarkable and satisfactory to note that this reversal is observed for the same cascade size where a squared Lorentzian function fits better the structure factor than a Lorentzian. The two criteria, one for identifying the existence of patterns, and the second for assessing the transition from disorder to patterns are thus fully consistent. For non-stoichiometric alloys, KMC simulations indicate that order patterning can lead to compositional patterning, which is probably driven by the segregation of solvent atoms at antiphase boundaries [5]. We finally turn to the L10, by considering the case cB = 50%. Despite the differences in the number and nature of variants in the two ordered structures, their respective dynamical phase diagrams are very similar. Fig. 3 shows a typical microstructure in the patterning state, where six variants coexist. Analytical modeling based on kinetic equations describing the evolution of interfacial areas and volume fractions of variants has in fact shown that patterning of order should exist in general, as long as re-ordering of the cascade disordered zones proceed in two steps, with a transient formation of new domains. An important result
Fig. 3. Maps of B atoms in a (1 0 0) plan for Cb = 50% at steady state in the patterning regime. The B atoms are coded so as to reveal the six ordering variants of the L10 structure. Note the existence of well-ordered domains of finite size.
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of the modeling is that the boundary between LRO and patterning is predicted to follow a power law with a 3/2 exponent [5], in remarkable agreement with the KMC results for both L12 and L10 phases. Last, preliminary results for B-concentrations intermediate between 25% and 50% indicate that it is possible to achieve a patterning of chemical order that contains both L12 and L10 domains of finite size, thus resulting in patterning of both composition and chemical order. 4. Discussion The results obtained by analytical models and simulations strongly indicate that finite size relocation and large cascade sizes can lead to patterning of composition and degree of order, respectively. It is interesting to notice the complementary roles played by R and Lc: each characteristic length can directly stabilize patterns of only one field, but dual patterning is possible when the alloyÕs microstructure involves coupled composition and degree of order. We note that Heinig and coworkers have also established that finite range atomic relocations can lead to compositional patterning [20]. One difference, though, is that their treatment does not predict the existence of a critical relocation range for patterning to take place. We stress that the above analytical and simulation results have been obtained using several simplifications, an important one being that the evolution of the composition and the order fields is decoupled from, or only weakly coupled to, the evolution of structural defects, such vacancies, interstitials, dislocations and grain boundaries. The effect of stress fields has also been ignored. While this last assumption can be justified for L12 precipitates lattice matched to a solid solution, it is understood that a full treatment of the L10 phase, which is tetragonal, requires including stress effects. This is left for future work. From an experimental viewpoint, there are only few works reporting the formation of patterns in irradiated materials. This scarcity is due to two reasons: these patterns exist only in certain regions of the control parameter space, and these regions can easily be overlooked; the scale of the patterns is usually in the nanometer range, and thus pattern detection requires advanced characterization techniques. In the case of irradiated alloys, as R ranges from ˚ and Lc ranges from a few to 10 nm, only nano1 to a few A meter-scale patterns should be observed. The clear identification of compositional patterns has nevertheless been obtained in Cu–Ag and Cu–Co thin films irradiated with 1 MeV Kr+ ions [21], and saturation of precipitate coarsening has been observed during irradiation of Cu–Co thin films [22]. Composition and order patterning have been reported in Ni–Al alloys irradiated with Ni+ ions at temperatures T 550 °C [23,24]. An important theoretical and practical property that has not yet been clearly established experimentally is the predicted scaling of the pattern length with the intensity of the driving force and the fact that the largest possible pattern length is bounded by the
characteristic length of the external forcing. As Rc and Lc, have typical values of a few Angstroms and a few nanometers, respectively, patterns stabilized by irradiation could thus be used to synthesize nanoscale microstructures. These predictions can be tested by systematic experiments. 5. Conclusion A comprehensive description of the effects of the characteristic lengths introduced by displacement cascades is given. Cascade size and relocation distances in irradiated alloys can lead to the formation of compositional and order patterning. These patterning reactions can be understood by a competition between force and thermally activated dynamics that operate at different length scales. These patterning reactions should make it possible to directly synthesize nanocomposite materials with tunable length scales. Acknowledgments This material is based upon work partly supported by the US Department of Energy, Division of Materials Sciences under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign, by the Materials Computation Center at the University of Illinois, National Science Foundation, under Grants DMR 9976550 and DMR 03-25939, and by the University of Illinois Campus Research Board. We also thank the FS-MRL Center For Computation for its assistance. References [1] R.S. Averback, T. Diaz de la Rubia, Solid State Phys. 51 (1997) 281. [2] W. Ja¨ger, H. Trinkhaus, J. Nucl. Mater. 205 (1993) 394. [3] R.A. Enrique, P. Bellon, Phys. Rev. Lett. 84 (2000) 2885. [4] J.-W. Liu, P. Bellon, Phys. Rev. B 66 (2002) 020303(R). [5] J. Ye, P. Bellon P, Phys. Rev. B 70 (2004) 094104. [6] T.J. Colla, H.M. Urbassek, K. Nordlund, R.S. Averback, Phys. Rev. B 63 (2000) 104206. [7] R.A. Enrique, K. Nordlund, R.S. Averback, P. Bellon, J. Appl. Phys. 93 (2003) 2917. [8] T. Diaz de la Rubia, A. Caro, M. Spaczer, Phys. Rev. B 47 (1993) 11483. [9] M. Spaczer, A. Caro, M. Victoria, T. Diaz de la Rubia, Phys. Rev. B 50 (1994) 13204. [10] F. Gao, D.J. Bacon, Philos. Mag. A 71 (1995) 43. [11] F. Gao, D.J. Bacon, Philos. Mag. A 71 (1995) 65. [12] N.V. Doan, R. Vascon, Nucl. Instr. and Meth. B 135 (1998) 207. [13] M.L. Jenkins, C.A. English, J. Nucl. Mater. 108–109 (1982) 46. [14] G. Martin, Phys. Rev. B 30 (1984) 1424. [15] F. Haider, Habilitation Thesis, University of Goettingen, 1995. [16] V.G. Vaks, V.V. Kamyshenko, Phys. Lett. A 177 (1993) 269. [17] R.A. Enrique, P. Bellon, Phys. Rev. B 63 (2001) 134111. [18] A. Guinier, X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, Dover Publications, New York, 1994, p. 336. [19] H. Chen, J.B. Cohen, Acta Metall. 27 (1979) 603.
J. Ye et al. / Nucl. Instr. and Meth. in Phys. Res. B 242 (2006) 265–269 [20] K.-H. Heinig, T. Mu¨ller, B. Schmidt, M. Strobel, W. Mo¨ller, Appl. Phys. A 77 (2003) 17. [21] R.A. Enrique, P. Bellon, Appl. Phys. Lett. 78 (2001) 4178. [22] P. Krasnochtchekov, R.S. Averback, P. Bellon, this Conference (IBMM 2004).
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[23] R.S. Nelson, J.A. Hudson, D.J. Mazey, J. Nucl. Mater. 44 (1972) 318. [24] G. Schmitz, J.C. Ewert, F. Harbsmeier, M. Uhrmacher, F. Haider, Phys. Rev. B 63 (2001) 224113.
Nuclear Instruments and Methods in Physics Research B 242 (2006) 265–269 www.elsevier.com/locate/nimb
Nanoscale patterning of composition and chemical order induced by displacement cascades in irradiated alloys Jia Ye
a,*
, Pascal Bellon a, Rau´l A. Enrique
b
a
b
Department of Materials Science and Engineering, Urbana, IL 61801, United States Department of Materials Science and Engineering, Ann-Arbor, MI 48103, United States Available online 8 November 2005
Abstract When irradiation leads to the production of dense displacement cascades, it is predicted at the composition and the chemical order in metallic alloys may undergo spontaneous formation of nanoscale patterns. These predictions, which are based on continuum kinetic models and atomistic simulations, are reviewed and compared with available experimental data. If these predictions were confirmed, ion beam processing would offer a route for the direct synthesis of nanoscale patterns of composition and degree of order, with tunable length. Ó 2005 Elsevier B.V. All rights reserved. PACS: 61.80.Az; 05.65.+b Keywords: Patterning; Decomposition; Order–disorder transitions; Nanostructures; Displacement cascades; Kinetic Monte Carlo simulations
1. Introduction When materials are subjected to irradiation, nuclear collisions with large enough recoil energies result in the formation of displacement cascades [1]. It has been well known for many years that the rate of introduction and the size of these cascades, as well as the point defect evolution during the cascade lifetime, play a determinant role in microstructural evolutions under irradiation, such as segregation and precipitation on sinks, or crystal-to-amorphous transitions. Furthermore, the presence of displacement cascades introduces a bias in point defect formation that may lead to self-organization of defect clusters, such as dislocation loops, voids and gas bubbles [2]. Recently, however, we have shown that displacement cascades can also lead to the spontaneous self-organization of composition [3,4] and of chemical order [5]. Analytical models and
*
Corresponding author. Tel.: +1 217 333 1441; fax: +1 217 333 2736. E-mail address: [email protected] (J. Ye).
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.08.198
atomistic computer simulations reveal that these self-organization reactions find their origin in two length scales introduced by displacement cascades, the cascade size L and the average relocation distance of atoms within the cascade, R. For metallic alloys with moderately positive heats of mixing, DHm 6 10 kJ/g atom, molecular dynamics (MD) simulations indicate that the forced atomic mixing in these cascades is nearly random [1,6] and relocation distances follow an exponential decay [7]. For metallic alloys with negative heats of mixing, MD simulations indicate that any pre-existing long-range order is destroyed in the core of the cascades, thus producing disordered zones [8–12]. Disordered zones have been extensively studied by transmission electron microscopy (see [13] for a review). Under sustained irradiation conditions, the atomic mixing and the chemical disordering produced by displacement cascades may compete with the annealing promoted by the thermally activated migration of point defects. We briefly review in the following sections how this competition may lead to patterning of composition and chemical order in A–B binary alloys when the length scales of the cascade, L and R, exceed some critical values.
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J. Ye et al. / Nucl. Instr. and Meth. in Phys. Res. B 242 (2006) 265–269
2. Compositional patterning induced by the relocation distance Two decades ago, Martin [14] introduced a model to evaluate phase stability in alloys under irradiation, using a kinetic description that takes into account the presence of several dynamics. In MartinÕs model, forced atomic exchanges proceed by ballistic, i.e. random, nearest neighbor exchanges and the cascade size is not explicitly taken into account. These assumptions are reasonably well satisfied in the case of high-energy electron or light-ion irradiation [1]. In MartinÕs model, an alloy with positive heat of mixing should reach a homogeneous steady state when the irradiation temperature is low enough for the forced mixing to dominate over the thermal decomposition. As the irradiation temperature is increased, one crosses a dynamical phase transition, at which point the steady state microstructure is comprised of two macroscopic phases. Early kinetic Monte Carlo (KMC) simulations [15] and mean-field modeling [16] indicated however, that the presence of medium or long-range ballistic relocations could lead to the stabilization of compositional patterning under irradiation. For a binary alloy undergoing phase separation, we introduced a one-dimensional continuum kinetic model for the time evolution of the composition field, where forced atomic relocations are distributed according to an exponential decay with a characteristic length R [3]. This contribution introduces a non-local term in the kinetic equation. The local stability of a homogeneous steady state was studied by a linear stability analysis, and the global stability of composition profiles was evaluated by minimizing an effective free energy. Both analyses indicated that beyond a threshold value Rc, the stable steady state be comprised of compositional patterns with a characteristic size directly related to R. This prediction was confirmed by atomistic kinetic Monte Carlo simulations [17]. A dynamical phase diagram was built, see Fig. 1, which yields the domain of stability of steady states in the (R, c) space, where c = C/M is the ratio of the ballistic jump frequency to the thermal atomic mobility. The dynamical stabilization of patterns can be seen as a compromise resulting from the competition of two dynamics of similar strength but with different characteristic scales. Indeed, decomposition is more effective than forced mixing at small scale since a few atomic jumps are sufficient to maintain locally decomposition. At larger scales, however, the forced mixing is predominant, and, from a coarse-grained perspective, the alloy remains homogeneous. In the case of alloys that would phase separate at equilibrium into a solid solution and a chemically ordered phase, e.g. Ni-rich Ni–Al alloys, we have extended the above treatment [4]. Here again, it is predicted that beyond a threshold relocation range Rc, irradiation may stabilize compositional patterns. The coupling between the composition and the chemical order fields leads to patterns comprised of B-enriched ordered precipitates in a solid solution matrix.
Fig. 1. Dynamical steady-state phase diagram as a function of the average relocation distance for the ballistic jumps, R and the reduced irradiation intensity c. Solid lines c1, c2 are the transition lines predicted from the minimization of an effective free energy functional for an alloy with CB = 50%, and insets are (1 1 1) cuts of atomistic KMC simulations. A and C are coefficients entering the free energy functional (from [3]).
3. Patterning of chemical order induced by the cascade size In order to isolate the effect of the cascade size L, let us now assume that the ballistic mixing is short range, i.e. proceeds by exchanges between nearest neighbors in the atomistic KMC simulations, or R ! 0 in the continuum formulation. The cascade size is measured through the number of nearest neighbor pairs that are exchanged at once in a spherical volume. For simplicity, the fraction of pairs exchanged in this volume is kept constant at 80%. A cascade size of b = 4000 thus corresponds to a volume containing 10,000 atoms. For the parameters used in the KMC simulations (see [5] for details), the critical cascade size for order patterning is bc 100 in the present case, which would correspond to L 2 nm for the atomic density of Ni3Al. No significant direct effect of L is expected on the composition field, as the cascade effect reduces mainly to modifications of the correlation factors during atomic diffusion. KMC simulations performed on the model alloy with a positive heat of mixing used in Section 2 have confirmed the absence of compositional patterning, even for large cascade sizes. In the case of an ordering alloy, however, a direct effect of L is possible. Indeed, consider a long-range ordered phase; the disordered zone produced by a displacement cascade may re-order either from its interface or from within its core. In the latter case, this re-ordering will lead to the presence of antiphase domains, as more than one variant will be available for re-ordering. Under appropriate conditions, this sustained introduction of antiphase domains may destabilize an initially longrange ordered state, resulting in an ordered state with a finite domain size. This phenomenon has indeed been observed in KMC simulations [5] for a generic alloy on a face centered cubic lattice, which forms L12 and L10
J. Ye et al. / Nucl. Instr. and Meth. in Phys. Res. B 242 (2006) 265–269
ordered structures at B concentrations near 25% and 50%, respectively. We first consider a stoichiometric L12 alloy, i.e. cB = 0.25. The resulting dynamical phase diagram is shown in Fig. 2. Three possible steady states are observed: long-range ordered (LRO), disordered (D), or ordered but with finite domain sizes (OP, which stands for patterning of chemical order). The transition from LRO to non-LRO corresponds to the transition from one macroscopic L12 variant in the LRO state, to four equi-probable variants in the non-LRO state. The existence of the patterning state is derived from the best fit of the spherically averaged structure factor. In the absence of patterning, the best fit is obtained with a Lorentzian function, which corresponds to an exponential distribution of order fluctuations in real space. In the presence of patterns, however, the existence of well-defined interfaces must lead to a 1/q4 decay of the intensity at large q wave vectors, according to PorodÕs law [18]. We thus choose to fit the structure factor with a squared Lorentzian as well as, since this function is compatible with PorodÕs law at large q values. In the absence of irradiation and for small cascade sizes, it is found that the Lorentzian fit is better, and we thus conclude that no patterning is taking place. For cascade sizes larger than a critical value Lc, however, the squared Lorentzian fit is better than the one obtained with a Lorentzian, in agreement with direct visualization of ordered domains in real space. For L > Lc, unfortunately, it is not possible to determine easily the transition from the disordered state to the patterned state. For that purpose, we extended the Cook–Chen–Cohen method (CCC) [19], which is based on the evolution of the intensity of the structure factor at the superlattice position at equilibrium as one approaches an LRO transition from the disordered state. The Cook–Chen–Cohen method makes it possible to identify an ordering instability temperature. For equilibrium first-order transitions, this instability temperature is lower than the order–disorder transition temperature,
Fig. 2. Dynamical phase diagram for the L12 structure at T = 0.09 eV. The threshold for patterning of order is b 100. For a given b, the cascade size L is obtained using Ni3Al atomic density (from [5]).
267
below which one ordering variant becomes macroscopic. Transposing this property into the present situation, where the disordering jump frequency C is progressively increased to change the stable steady state from ordered to disordered, we thus expect that ordering instability will take place at a value Cs smaller than Cc, the critical value for the LRO to non-LRO transition. As seen in Fig. 2, for small cascade sizes, 1 6 b 6 100, this inequality is indeed fulfilled. For larger cascade size, however, the two transitions take place in reverse order: Cs is predicted to be larger than Cc. It is both remarkable and satisfactory to note that this reversal is observed for the same cascade size where a squared Lorentzian function fits better the structure factor than a Lorentzian. The two criteria, one for identifying the existence of patterns, and the second for assessing the transition from disorder to patterns are thus fully consistent. For non-stoichiometric alloys, KMC simulations indicate that order patterning can lead to compositional patterning, which is probably driven by the segregation of solvent atoms at antiphase boundaries [5]. We finally turn to the L10, by considering the case cB = 50%. Despite the differences in the number and nature of variants in the two ordered structures, their respective dynamical phase diagrams are very similar. Fig. 3 shows a typical microstructure in the patterning state, where six variants coexist. Analytical modeling based on kinetic equations describing the evolution of interfacial areas and volume fractions of variants has in fact shown that patterning of order should exist in general, as long as re-ordering of the cascade disordered zones proceed in two steps, with a transient formation of new domains. An important result
Fig. 3. Maps of B atoms in a (1 0 0) plan for Cb = 50% at steady state in the patterning regime. The B atoms are coded so as to reveal the six ordering variants of the L10 structure. Note the existence of well-ordered domains of finite size.
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of the modeling is that the boundary between LRO and patterning is predicted to follow a power law with a 3/2 exponent [5], in remarkable agreement with the KMC results for both L12 and L10 phases. Last, preliminary results for B-concentrations intermediate between 25% and 50% indicate that it is possible to achieve a patterning of chemical order that contains both L12 and L10 domains of finite size, thus resulting in patterning of both composition and chemical order. 4. Discussion The results obtained by analytical models and simulations strongly indicate that finite size relocation and large cascade sizes can lead to patterning of composition and degree of order, respectively. It is interesting to notice the complementary roles played by R and Lc: each characteristic length can directly stabilize patterns of only one field, but dual patterning is possible when the alloyÕs microstructure involves coupled composition and degree of order. We note that Heinig and coworkers have also established that finite range atomic relocations can lead to compositional patterning [20]. One difference, though, is that their treatment does not predict the existence of a critical relocation range for patterning to take place. We stress that the above analytical and simulation results have been obtained using several simplifications, an important one being that the evolution of the composition and the order fields is decoupled from, or only weakly coupled to, the evolution of structural defects, such vacancies, interstitials, dislocations and grain boundaries. The effect of stress fields has also been ignored. While this last assumption can be justified for L12 precipitates lattice matched to a solid solution, it is understood that a full treatment of the L10 phase, which is tetragonal, requires including stress effects. This is left for future work. From an experimental viewpoint, there are only few works reporting the formation of patterns in irradiated materials. This scarcity is due to two reasons: these patterns exist only in certain regions of the control parameter space, and these regions can easily be overlooked; the scale of the patterns is usually in the nanometer range, and thus pattern detection requires advanced characterization techniques. In the case of irradiated alloys, as R ranges from ˚ and Lc ranges from a few to 10 nm, only nano1 to a few A meter-scale patterns should be observed. The clear identification of compositional patterns has nevertheless been obtained in Cu–Ag and Cu–Co thin films irradiated with 1 MeV Kr+ ions [21], and saturation of precipitate coarsening has been observed during irradiation of Cu–Co thin films [22]. Composition and order patterning have been reported in Ni–Al alloys irradiated with Ni+ ions at temperatures T 550 °C [23,24]. An important theoretical and practical property that has not yet been clearly established experimentally is the predicted scaling of the pattern length with the intensity of the driving force and the fact that the largest possible pattern length is bounded by the
characteristic length of the external forcing. As Rc and Lc, have typical values of a few Angstroms and a few nanometers, respectively, patterns stabilized by irradiation could thus be used to synthesize nanoscale microstructures. These predictions can be tested by systematic experiments. 5. Conclusion A comprehensive description of the effects of the characteristic lengths introduced by displacement cascades is given. Cascade size and relocation distances in irradiated alloys can lead to the formation of compositional and order patterning. These patterning reactions can be understood by a competition between force and thermally activated dynamics that operate at different length scales. These patterning reactions should make it possible to directly synthesize nanocomposite materials with tunable length scales. Acknowledgments This material is based upon work partly supported by the US Department of Energy, Division of Materials Sciences under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign, by the Materials Computation Center at the University of Illinois, National Science Foundation, under Grants DMR 9976550 and DMR 03-25939, and by the University of Illinois Campus Research Board. We also thank the FS-MRL Center For Computation for its assistance. References [1] R.S. Averback, T. Diaz de la Rubia, Solid State Phys. 51 (1997) 281. [2] W. Ja¨ger, H. Trinkhaus, J. Nucl. Mater. 205 (1993) 394. [3] R.A. Enrique, P. Bellon, Phys. Rev. Lett. 84 (2000) 2885. [4] J.-W. Liu, P. Bellon, Phys. Rev. B 66 (2002) 020303(R). [5] J. Ye, P. Bellon P, Phys. Rev. B 70 (2004) 094104. [6] T.J. Colla, H.M. Urbassek, K. Nordlund, R.S. Averback, Phys. Rev. B 63 (2000) 104206. [7] R.A. Enrique, K. Nordlund, R.S. Averback, P. Bellon, J. Appl. Phys. 93 (2003) 2917. [8] T. Diaz de la Rubia, A. Caro, M. Spaczer, Phys. Rev. B 47 (1993) 11483. [9] M. Spaczer, A. Caro, M. Victoria, T. Diaz de la Rubia, Phys. Rev. B 50 (1994) 13204. [10] F. Gao, D.J. Bacon, Philos. Mag. A 71 (1995) 43. [11] F. Gao, D.J. Bacon, Philos. Mag. A 71 (1995) 65. [12] N.V. Doan, R. Vascon, Nucl. Instr. and Meth. B 135 (1998) 207. [13] M.L. Jenkins, C.A. English, J. Nucl. Mater. 108–109 (1982) 46. [14] G. Martin, Phys. Rev. B 30 (1984) 1424. [15] F. Haider, Habilitation Thesis, University of Goettingen, 1995. [16] V.G. Vaks, V.V. Kamyshenko, Phys. Lett. A 177 (1993) 269. [17] R.A. Enrique, P. Bellon, Phys. Rev. B 63 (2001) 134111. [18] A. Guinier, X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies, Dover Publications, New York, 1994, p. 336. [19] H. Chen, J.B. Cohen, Acta Metall. 27 (1979) 603.
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