Atomistic Simulation of Decomposition Processes in AgâCu Hollow

J. Phys. Chem. C 2008, 112, 11135–11143

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Atomistic Simulation of Decomposition Processes in Ag-Cu Hollow Nanocubes Francesco Delogu† Dipartimento di Ingegneria Chimica e Materiali, UniVersita` degli Studi di Cagliari, piazza d’Armi, I-09123 Cagliari, Italy ReceiVed: January 7, 2008; ReVised Manuscript ReceiVed: April 21, 2008

Molecular dynamics simulations have been employed to study the thermal behavior of an alloyed Ag-Cu hollow cube with a side about 8 nm long and walls about 2 nm thick. Gradual heating determines at about 300 K a partial decomposition with the formation of Ag- and Cu-rich crystalline domains. The interplay between highly cooperative atomic displacements and thermodynamic factors produce a remarkable selforganization which results in the formation of a hollow cube with crystalline facets mostly occupied by Agrich domains, i.e. by the species with the lowest surface energy. Further heating results in the collapse and the final melting of the whole structure. I. Introduction Nanometer-sized systems are known to exhibit an impressive suite of physical and chemical properties unobtainable in bulk counterparts and originating from the intimate coupling between effects related to size and geometry.1–3 These can be connected in turn with two fundamental aspects.1–3 First, the fraction of atoms with coordination number smaller than the one attained in equilibrated bulks is no longer negligible and phenomena in which surfaces are typically involved, e.g., self-assembly and catalysis, are consequently affected.1–3 Second, the number of atoms included in nanometer-sized systems can be remarkably far from the thermodynamic limit.1–4 Correspondingly, classical equilibrium thermodynamics can be frustrated and processes generally highly improbable can well take place.1–4 Intertwined to such factors are the stability of nanometersized solids and their capability of withstanding external perturbations. These aspects assume a particular importance for the metastable phases formed by immiscible elements, for which the equilibrium phase diagram indicates a terminal solubility limited to very small atomic percentages.5 Massive crystalline alloys can be synthesized by imposing suitable kinetic constraints6–8 to bypass the tendency to decomposition due to the positive enthalpy of mixing.5 The obtained systems, yet metastable in thermodynamic sense, are stable on relatively long time scales.6–8 Such apparent stability can be however questioned for nanometer-sized structures where local instabilities, due for example to compositional fluctuations, can have unexpected amplification. The deep connection between structural evolution and particle geometry for a nanometer-sized hollow cube consisting of an equal number of randomly mixed Ag and Cu atoms is here investigated by classical molecular dynamics (MD) methods. Ag-Cu is a classical immiscible system,5 but it should be regarded here as a model system and the study as a qualitative one. Hollow cubes can be however prepared Via refined solution methods9–17 and the questions addressed in this work are in principle amenable to experimental investigation. Nevertheless, no hollow cube of immiscible elements has been synthesized yet. In addition, experimentally prepared hollow cubes have in general size in the range between 20 and 100 nm. Unfortunately, these length scales are out of † E-mail: [email protected].

range of routine MD simulations. The Ag-Cu hollow cube with sides about 8 nm long studied in the present work must be therefore regarded as a model system. II. Numerical Simulations Nanometer-sized hollow cubes were generated by assembling six crystalline domains shaped as truncated square pyramids approximately 8 nm long and 2 nm thick. Partial overlaps between neighboring domains were eliminated by removing one of any two atoms located at distances smaller than 2 Å. Each domain was independently grown starting from a point in threedimensional space randomly selected within suitable ranges of Cartesian coordinates. Such choice was motivated by the evidence that (i) templating surfaces and walls of hollow structures respectively employed and produced in experiments are epitaxially correlated and (ii) walls nucleate on random surface sites.10–13 It follows that grain boundary (GB) regions generated at assembly by neighboring walls are generally different from each other.10–13 Differences are further enhanced by relaxation and thermal disordering processes. The final six cF4 face-centered cubic (fcc) lattices formed a hollow cube of about 31500 atoms. Hollow cubes with (100) and (111) crystallographic facets were studied. However, the behavior of hollow cubes is not qualitatively affected by surface topology and the same results are obtained with (100) and (111) crystallographic facets at temperatures differing only of 2 or 3 K. Only the case of cubes with (100) facets will be therefore discussed in detail for sake of brevity. A semiempirical potential based on the second-moment approximation of a tight-binding (TB) Hamiltonian was used to reproduce interatomic forces.18 The cohesive energy depends

on the distance r0,Rβ between nearest neighbors of species R and β at 0 K, on the distance rij ) |ri - rj| between atoms i and j and on characteristic potential parameters ARβ, ξRβ, pRβ, and qRβ for pure and cross interactions between R and β species.18

10.1021/jp800112z CCC: $40.75  2008 American Chemical Society Published on Web 07/03/2008

11136 J. Phys. Chem. C, Vol. 112, No. 30, 2008

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nj(τ) )

Figure 1. A cross section about 0.5 nm thick of the hollow cube with (100) crystallographic facets equilibrated at 200 K. Thick lines are superposed to point out the arrangement of lateral walls. Ag and Cu atoms are shown in light and dark gray respectively.

Interactions were computed within a cutoff radius extending approximately to seventh neighbors. Potential parameters, taken from literature,18,19 define an enthalpy of mixing of about + 13 kJ mol-1, which is larger than the experimental value for a bulk chemically disordered Ag50Cu50 fcc solid solution.5 It is however worth noting that the difference between predicted and observed values is unimportant in view of the qualitative character of the present study. A larger positive enthalpy of mixing is even more desirable because, enhancing the thermodynamic tendency to system decomposition, it allows a significant decrease of simulation times. Equations of motion were solved with a fifth-order predictorcorrector algorithm20 and a time step of 2 fs. Calculations were carried out in the NVT ensemble with number of atoms N, volume V and temperature T constant.20,21 The initial atomic configuration was relaxed at 200 K for 300 ps. This produced only a few short-range atomic displacements aimed at accommodating local atomic strains. A cross section of the structure equilibrated at 200 K is reported in Figure 1. Starting from 200 K, the temperature was raised of 2 K every 40 ps. A general progressive increase of the amplitude of atomic vibrations around equilibrium lattice sites correspondingly occurred. Being atomic dynamics affected by degree of saturation and geometry of coordination shells, each atom underwent an actual increase dependent on the local arrangement of its nearest neighbors.22,23 Species located at edges, surfaces, GBs and in the remaining bulk-like regions of the hollow cube exhibited then different thermal responses.24 Whenever necessary, surface species were defined as the atoms with a coordination number smaller than 10 in order to include both edge and plane surface species. Atomic mobility was quantified by means of the mean square (ms) atomic displacement 〈∆r2〉.20 The self-part of the van Hove function Gs(l,∆t) and the non-Gaussian parameter R2(∆t) were used to gain further information on the system dynamics in case of collective atomic rearrangements.25,26 The former measures the probability for atoms to cover a distance l in a time interval ∆t, whereas the latter quantifies the deviation of Gs(l,∆t) from a Gaussian distribution. A characteristic time scale τ, always on the order of 20-25 ps, was identified at each temperature. Collective atomic motion was further characterized by evaluating the average number of atoms participating in cooperative displacements in a time period τ25,27

∑ nPn(τ) ∑ nPn(τ)

(2)

where Pn(τ) represents the probability of finding a group of n collectively rearranging atoms. These were identified on the basis of their relatively high mobility and capability of remaining neighbors after motion has occurred.25,27 A given atom i was defined mobile when the condition 0.35rnn < |ri(τ) - ri(0)| < 0.86rmin was satisfied, with ri(t) being the ith atom position at time t, rnn the distance of nearest neighbors indicated by the global pair correlation function (PCF)20 and rmin the distance at which the global PCF has the first minimum at the temperature considered. The condition min [|ri(τ) - rj(0)|, |rj(τ) - ri(0)|] < 0.43 rnn was instead employed to identify mobile atoms i and j remaining neighbors in correlated displacements.27 The numerical coefficients used effectively separate cooperative motion from thermal vibrations and vacancy-mediated atomic exchanges.27 The analysis of the atomic dynamics provides similar results when slightly different values are used. Demixing phenomena mediated by atomic displacements were monitored by evaluating the chemical short-range order (CSRO) parameter28–30

Ω)

[ 1 -c c N + 1 -c c N ](N + N ) β R

R β

R R

β -1 -1 β

(3)

where c is the fraction of R atoms and NRβ (NβR) is the average number of nearest neighbors of species β (R) around R (β) atoms. Segregated, randomly mixed and short-range ordered systems can be thus conveniently identified.28–30 For such ideal structures Ω takes indeed respectively the values -1, 0, and 1. MD simulations were also performed on a single-crystal cube with sides about 8 nm long and on a chemically disordered Ag50Cu50 bulk system closed by periodic boundary conditions.20 Hollow cubes of Ag and Cu were also considered. The results concerning these systems will be reported only when useful for sake of comparison. Calculations were also carried out on Ag50Cu50 hollow cubes with wall thickness of about 2 nm and sides about 6, 10 and 12 nm long as well as on Ag50Cu50 hollow cubes with sides 8 nm long but wall thickness of about 1 and 3 nm. The energies γ of (100) and (111) surfaces of Ag, Cu and Ag50Cu50 alloys were evaluated in simulations carried out in the NPT ensemble with N, pressure P and T constant on a semicrystal according to a procedure described in detail elsewhere.31 Systems of about 17100 atoms arranged in stacking sequences of 19 (100) or (111) atomic planes were used. III. Low-Temperature Thermal Behavior According to general expectations, the progressive temperature rise induces a corresponding increase of atomic mobility. As shown in Figure 2, Ag and Cu ms displacements 〈∆r2〉Ag and 〈∆r2〉Cu display a linear dependence on temperature T. The comparison with analogous data for Ag and Cu atoms in bulks, also reported in Figure 2, points out that hollow cube atoms possess a significantly larger mobility. This must be ascribed to both structural and thermodynamic factors. Structural factors can be properly pointed out by taking into account that ms displacements shown in Figure 2 are mean values obtained by averaging over all the hollow cube atoms. Whereas bulk atoms are equivalent, atomic species in nanometer-sized systems differ in number and arrangement of nearest neighbors. In the hollow cube case, atoms located respectively at external edges, internal and external surfaces, GBs and remaining bulk-like regions form

Atomistic Simulation of Decomposition Processes

Figure 2. The atomic ms displacements 〈∆r2〉Ag and 〈∆r2〉Cu of Ag and Cu species as a function of temperature T. Data refer to the cube with (100) crystallographic facets (full symbols). Analogous data for Ag and Cu atoms in bulk systems are also reported for sake of comparison (open symbols).

J. Phys. Chem. C, Vol. 112, No. 30, 2008 11137

Figure 4. CSRO parameter Ω as a function of temperature T. Data refer to the cube with (100) crystallographic facets.

Figure 3. Ms displacement 〈∆r2〉 characteristic of external edge, surface, GB and bulk-like Cu atoms as a function of the average potential energy u of each set. Data refer to the relaxed configuration of the cube with (100) crystallographic facets at 200 K. The ms displacement of a bulk atom is also quoted (open symbol), partially hidden by the point of hollow cube bulk-like atoms. The best-fitted line is shown.

four different sets with characteristic ms displacement and potential energy. The ms displacements of Cu atoms at 200 K are shown in Figure 3 as a function of their average potential energy u. A roughly linear relationship is observed. It appears that edge atoms are more mobile than surface ones and both of them are significantly more mobile than GB and bulk-like species. The larger atomic mobility in the hollow cube essentially results thus from contributions given by edge and surface atoms, i.e. by the species with the lowest coordination numbers. Thermodynamic factors have a quite secondary role respect to structural ones. Their influence can be however inferred by noting that the ms displacements of both Ag and Cu bulk-like atoms are 5-10% larger than ms displacements of Ag and Cu atoms in pure Ag and Cu bulk phases, but approximately the same of Ag and Cu atoms in disordered Ag50Cu50 bulks. This mobility enhancement can be tentatively connected with the unfavorable energy of cross Ag-Cu interactions and then with the positive enthalpy of mixing exhibited by the Ag-Cu system, which induces a tendency toward spontaneous decomposition to form segregated Ag- and Cu-rich domains. According to the CSRO parameter Ω values reported in Figure 4 as a function of temperature T, the first clues to system decomposition are detected at about 264 K. Around such

Figure 5. Two snapshots of the configuration attained by atomic species at 292 K. A lateral view of the hollow cube is shown in the upper part, whereas a cross section is reported in the lower part. Ag and Cu atoms are in light and dark gray respectively. Data refer to the cube with (100) crystallographic facets.

temperature Ω starts indeed a gradual decrease from 0 to negative values, pointing out an increase of segregation. The rate of Ω decrease keeps relatively low up to about 284 K, above which it undergoes a sudden rise. Between 284 and 290 K Ω drops approximately from -0.15 to -0.81, providing a clear indication of the occurrence of system decomposition. At 292 K the cube is characterized by considerable segregation, with an Ag-rich phase separated by a Cu-rich one. If segregation directly follows from the known equilibrium phase diagram and can be then expected5, the capability of selforganization displayed by the hollow structure undergoing decomposition is instead surprising. Two snapshots of the configuration attained by atomic species at 292 K are shown in Figure 5. It can be seen that the Ag-rich phase is selectively located at the center of cube faces, whereas the Cu-rich phase occupies the remaining portions of the hollow structure.


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TABLE 1: Surface Energy γ for (100) and (111) Ag, Cu, and Ag50Cu50 Surfaces Predicted by the TB Potential Parameters for Ag-Ag, Cu-Cu, and Ag-Cu Employed in the Present Work, Where Data are expressed in J m-2 Ag Cu Ag50Cu50

(100)

(111)

0.98 1.53 1.34

0.86 1.37 1.21

The difference between Ag and Cu surface energies γ can be tentatively invoked as a rationale for such behavior involving both external and internal sides. The γ values predicted for (100) and (111) surfaces of Ag and Cu species by the TB potential employed in the present work are reported in Table 1 together with the corresponding values for disordered Ag50Cu50 alloy semicrystals. The energy of Ag surfaces is always lower than the one of Cu and Ag50Cu50 surfaces.5,32 The fact that after decomposition the hollow cube walls consist to a large extent of Ag-rich domains suggests that demixing promotes the formation of the system with the lowest possible surface energy γ. A decomposed system with walls formed by the Cu-rich phase would indeed have a surface energy larger than the initial one. In such case walls would be formed by the species with the highest γ values, whereas the Ag-rich phase with the lowest surface energy would be located at edges, i.e. in a remarkably unfavorable position. It must be also noted that the tendency of Ag atoms to segregate at the surface of Ag-Cu nanometersized systems is known.33,34 In the case of the single-crystal cube a remarkable surface segregation of Ag atoms has been also observed at temperatures around 300 K. Being free surfaces absent, the bulk system still undergoes decomposition, but at temperatures remarkably higher. A degree of segregation quantified by a CSRO parameter Ω value of about -0.78 is reached only at about 340 K. The hollow structure attained at 292 K is characterized by the coexistence of Ag- and Cu-rich phases with a stoichiometry of about Ag90Cu10 and Ag10Cu90 respectively. A total separation of Ag and Cu species has not been therefore obtained. Actually, any further temperature increase within a range of about 300 or 400 K determines only a very slow decrease of Ω values from -0.81 to -1. A complete decomposition requires therefore very long times. For example, at temperatures as high as 750 K Ω still amounts to only -0.89. Compared with the very short times of the Ω drop from -0.15 to -0.81, this suggests that a drastic change in atomic displacement mechanism has occurred. IV. Decomposition Mechanism The hollow cube decomposition is a gradual process starting at about 240 K and progressively accelerating up to about 284 K, when a fast sequence of rearrangements produces a significant phase separation. Three consecutive stages can be roughly identified. In stage I exchanges involving edge and surface atoms promote the formation of the first nuclei of Ag- and Cu-rich phase domains. In stage II also atomic species in bulk-like regions become involved in local demixing events, which permits a rapid formation of extended phase domains. Stage III is finally characterized by low atomic mobility and further demixing requires high temperatures. The transition between consecutive stages is accompanied by significant changes in atomic displacement mechanisms which are described in some detail in the following. IV.1. Stage I. Atomic mobility in stage I is intimately connected with the dynamics of edge and surface species. Already at temperatures of about 240 K edge atoms perform

Figure 6. Numbers NAg,is and NCu,is of Ag and Cu surface islands as a function of temperature T. The inset quotes the fraction RAg,s of Ag surface atoms as a function of temperature T. The transition temperature TI-II of 258 K between two mechanistic regimes is indicated.

relatively long-range displacements with an apparently stochastic dynamics. Not only they move along the hollow cube external edges, but also explore neighboring surfaces like adsorbed species undergoing a two-dimensional surface diffusion. Under such circumstances, edge atoms occasionally exchange their position with atoms of the underlying surface. Atoms originally located at external edges become thus surface atoms. Conversely, a surface atom transforms into an adsorbed species and diffuses on the surface. These exchanges are localized events involving two species, with no evident indication of collective character. Numerical findings point out that exchanges are aimed at establishing the highest possible number of favorable interactions, each atom attempting to increase the number of like species in its coordination shell. As a consequence, exchanges hardly involve the same chemical species. For example, adsorbed Ag atoms exchange their positions with surface Cu atoms and not with Ag ones. The slow replacement of surface atoms determines the gradual formation of small pure Ag and Cu islands. The numbers Nis of surface domains were roughly estimated by considering the surface regions in which a given surface atom has at least five nearest neighbors of its same species also located at the surface. Up to a temperature of about 256 K Ag and Cu surface islands have roughly the same probability to form, as shown by the numbers NAg,is and NCu,is of Ag and Cu islands reported in Figure 6 as a function of temperature T. However, starting from 258 K NAg,is and NCu,is respectively increases and decreases, indicating that Ag surface islands are formed preferentially. At the same time the fractions RAg,s and RCu,s of Ag and Cu surface atoms, initially almost coincident, also change. In particular, the inset of Figure 6 shows that RAg,s increases with temperature T. Of course, the progressive decrease of RCu,s necessarily determines the growth and coalescence of Ag islands. A decrease of NAg,is is therefore expected to take place at higher temperatures. The aforementioned evidence suggest that a change in the mechanism of atomic displacement occurs at the temperature of 258 K and that it can be regarded as the transition temperature TI-II between stages I and II. Island formation during stage I deserves further comments. Relatively long simulations carried out at constant temperature show that Ag and Cu surface islands appear as well at 250 K and both NAg,is and NCu,is attain progressively roughly the same values reported in Figure 6 despite these were obtained at different temperatures. Also, when the system is left free to evolve at 262 K, NAg,is and NCu,is respectively increases and



Figure 8. Group of cooperatively rearranging atoms. Data refer to a hollow cube with (100) crystallographic facets at 284 K. Figure 7. The average number nj(τ) of atoms participating to collective displacements as a function of temperature T. The inset shows the probability Pn(τ) of finding a group of n mobile atoms in a time period τ at 284 K.

decreases. Of course, longer simulation times are necessary. It seems therefore that the system evolution is governed by an underlying instability which would determine the occurrence of decomposition also at relatively low temperatures. The temperature rise simply promotes the enhancement of atomic mobility thus shortening demixing time scales. Additional simulations also revealed that the temperature TI-II at which the change of mechanism apparently takes place is correlated to both NAg,is and NCu,is. When Ag and Cu surface islands are artificially created by suitably transforming Ag species into Cu ones and ViceVersa, TI-II changes. In particular, any NAg,is increase determines a TI-II decrease, whereas TI-II increases as NAg,is is decreased. No systematic investigation has been however carried out. IV.2. Stage II. The change of mechanism suggested by the preferential formation of Ag surface islands above 258 K is related to the involvement of bulk-like species in atomic rearrangements. This permits indeed a significant increase of the total number of Ag surface atoms. Atomic displacements from bulk-like regions to surface ones and Vice Versa imply in turn a definite modification of atomic mobility regimes. While individual localized exchanges can easily take place as far as only surface species are involved, more complex rearrangements often require cooperative processes. The self-part of van Hove function Gs(l,∆t) was then employed to characterize the Ag and Cu atomic mobility in stage II. For both of them Gs(l,∆t) exhibits the growth of a second peak approximately at the average distance rnn,RR of nearest neighbors indicated by the partial RR PCFs. This peak attains a maximum value after a time period τ of about 25 ps which is roughly the same for the two species. At the same τ value the non-Gaussian parameter R2(∆t) also displays a maximum. Following previous work,25,26 it can be inferred that cooperative diffusion processes are operating. The number of atoms participating in collective displacements on time scales measured by τ also suggests that 258 K can be considered as a sort of transition temperature. The average number nj(τ) of such atoms is reported in Figure 7 as a function of temperature T. It can be seen that nj(τ) takes very small values below 256 K and progressively increases above. Accordingly, individual atomic displacements taking place in stage I are gradually replaced by collective rearrangements in stage II. The probability Pn(τ) of finding a cluster of n mobile atoms in a time period τ is shown in the inset of Figure 7. Its shape is characteristic and is not affected by temperature. The average chemical composition of mobile atom clusters, i.e. the relative number of Ag and Cu species participating to correlated

Figure 9. Neperian logarithm of mobile cluster length L, ln L, as a function of the Neperian logarithm of the cluster size n, ln n. Data refer a hollow cube with (100) crystallographic facets at 284 K. The best-fitted line is also shown.

displacements, is also left unaffected by temperature rises. Irrespective of temperature, a cluster of n mobile atoms is found to consist on the average of about 0.49 n Ag and 0.51 n Cu atoms. In collective displacements Ag and Cu exhibit therefore roughly the same atomic mobility. A cluster of mobile atoms is shown in Figure 8 for sake of illustration. Atomic species in clusters with relatively large size have a relatively disordered spatial distribution resembling the one of fractal aggregates. The dimensionality of clusters with n larger than 10 was quantified by estimating their average fractal dimension df. To this end, in any cluster considered the two more distant atoms were identified and their distance evaluated. This defined the so-called cluster length L.35 The Neperian logarithm of L, ln L, was then plotted as a function of the Neperian logarithm of n, ln n. The obtainment of roughly linear plots with slope ν like the one reported in Figure 9 indicates that L∝nν. Evaluated ν by means of linear best-fitting procedures, the average fractal dimension df was finally worked out by the relationship ν ≈ df-1.35 The df value obtained, amounting to about 1.81, indicates that mobile atom clusters tend to a two-dimensional occupancy of space. This is probably due to the fact that most clusters include surface atoms and are located in cube walls between free surfaces, which necessarily implies at least for the larger ones a pseudo-two-dimensional atomic arrangement. The formation of mobile atom clusters and the cooperative character underlying atomic mobility in stage II determine a relatively fast evolution of the hollow cube structure. Such evolution has been already quantified by means of the CSRO parameter Ω, the values of which have been reported in Figure 4. Another useful quantity is the fraction f of Ag atoms contained in hollow cube walls. More specifically, f was evaluated over the total number of atoms included in six square parallelepiped volume regions corresponding to the volume spanned by internal cube surfaces when moved along their normal toward the external surfaces. One of such domains is shown in Figure 10. This figure also reports f as a function of temperature T. It can be seen that f undergoes a rapid increase between 284 and 290


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Figure 10. Fraction f of Ag atoms contained in hollow cube walls as a function of temperature T. The inset shows the regions of the hollow cube occupied by Ag-rich domains.

K, attaining a final asymptotic value roughly equal to 0.93, which confirms the conclusions already drawn on the basis of Figure 5. IV.3. Stage III. The attainment of Ω and f values of about -0.81 and 0.93 respectively coincides with a drop in atomic mobility. The segregation of Ag and Cu atomic species determines then another drastic change in the mechanism of atomic displacement. Once the system has decomposed cooperative rearrangements are rapidly replaced by usual thermally activated short-range diffusion events. These are mostly restricted to edges and surfaces until temperatures as high as 750 K are reached. Ms displacements of both Ag and Cu atoms are smaller than during stage I, pointing out a sort of dynamical arrest consequent to demixing. Reducing the thermodynamic metastability, the partial system decomposition has also reduced the atomic mobility. V. High-Temperature Thermal Behavior The self-organized decomposed structure obtained at about 300 K must not be regarded as the final hollow cube configuration, but, on the contrary, as an intermediate structure with an inherent degree of instability. Its potential energy is indeed significantly larger than the one possessed by the most stable possible configuration, characterized by minimum areas of free surfaces and Ag-Cu interfaces. To achieve these characteristics, the final structure must not possess internal cavities and its chemical decomposition must be complete. In light of its structure, the decomposed cube formed at about 300 K must therefore necessarily correspond to a local minimum in potential energy. On such basis it is reasonable to expect that it could undergo further structural evolution as the temperature is raised. Numerical findings indicate that no significant dynamical event takes place at temperatures below 750 K except for a slow increase of the frequency of individual displacement processes involving both edge and surface atoms. These displacements promote further demixing according to a mechanistic scenario similar to the one operating during stage I, producing an increase of the external free surface area of Ag-rich domains. Around 850 K the slow system evolution is replaced by a relatively fast one resulting in both structural and geometrical modifications. The combined structural and chemical transformations were described and characterized in detail by evaluating the total system volume sys, including the system internal cavity, and the volume in of the cavity itself. The former was roughly estimated by defining a rigid cubic box of known size including the hollow cube and filling it virtually with smaller cubes of side 1 Å long under the condition that virtual cubes must not include or “touch” Ag and Cu atoms. These were considered as rigid spheres of radius defined by the average distance of nearest neighbors in pure Ag and Cu bulks at the temperature considered. The difference between the rigid box volume and the one filled with small cubes quantified the excluded volume

Figure 11. Volumes Vsys and Vin quoted as a function of the temperature T.

Figure 12. Lateral cross-sectional view of the atomic configurations attained by the system at 884 K. Ag and Cu atoms are in light and dark gray respectively.

occupied by the hollow structure. The volume in of the internal cavity was estimated by evaluating the number of small cubes it is able to contain. The volumes Vsys and Vin are quoted in Figure 11 as a function of temperature T. They show approximately constant values up to a temperature of about 850 K, when both undergo a slow decrease due to a structural collapse. Around 882 K Vsys undergoes a drop to the volume simply occupied by Ag and Cu atoms, whereas Vin exhibits a definite upward jump to such value. The walls of the hollow structure have correspondingly broken and the internal cavity disappeared, so that the excluded volume is represented by the one of atomic species. The fracture produced in lateral walls is shown in Figure 12, which also permits to appreciate the shape change occurred. The analysis of atomic trajectories within the temperature range between 800 and 900 K indicates that the wall fracture is preceded by a gradual migration of Cu atoms toward the cube vertices and their replacement by Ag species. This determines an increase of the surface area of Ag-rich domains, two of which become finally directly connected. The connection of neighboring Agrich domains, together with the distortion produced in the hollow structure by combined atomic rearrangements, promotes then the fracture process along the edge. The atomic dynamics immediately before and after this event is relatively disordered and trajectories resemble the ones generally observed in liquid phases. It should be however noted that such liquid-like rearrangements are extremely localized and that the whole structure should be still regarded as a solid. As the temperature increases the structural evolution proceeds and further connections between Ag-rich domains occur. The total fraction Rs of surface species, quoted in Figure 13 as a function of the temperature T, undergoes an irregular decrease


Figure 13. Total fraction Rs of surface species as a function of the temperature T.

Figure 14. System atomic configuration at 925 K. Ag and Cu atoms are in light and dark gray respectively.

starting from the value for the initial hollow cube and probably tending to the smallest possible one of about 0.63 characteristic of a full sphere. The slow transformation into a sphere containing two large decomposed Ag and Cu domains separated by a single interface is however pre-empted by surface premelting, marked by a discontinuity in the increase of ms displacements of surface species with the temperature T. Premelting is detected at about 920 K. Above this value, the structural evolution becomes faster. Rather than a complete decomposition, it induces however a mixing of surface species being Ag and Cu atoms miscible in the liquid phase. At the same time, a progressive shape modification is observed. Vertices and corners disappear, transforming the hollow structure into an irregular full ellipsoid. The structure obtained at 925 K is shown in Figure 14 for sake of illustration. The predominance at the ellipsoid surface of Ag atoms over Cu ones is evident. The system finally melts at about 940 K and already at 944 K the mixing of atomic species is complete as a consequence of the relatively fast dynamics of the liquid phase. VI. Effects of Side Length and Wall Thickness on the Decomposition Mechanism The three-stage mechanism of phase separation discussed in detail for a Ag50Cu50 hollow cube with sides about 8 nm long and walls about 2 nm thick is intrinsically related to the characteristic length scales of the system. These consist essentially of the cube side length and of the wall thickness. The former has a critical influence on the extension of surface that edge atoms can actually explore when diffusing on the plane surfaces like adsorbed species. The extension of surface area explored is determined by the probability that a diffusing edge atom could exchange its position with an underlying surface species. Such probability measures the number of diffusing jumps undergone by the edge atom before the position exchange and is connected with the surface topology and the temperature. Being the two latter factors constant, the increase of the side length undermines the capability of edge atoms to explore the

J. Phys. Chem. C, Vol. 112, No. 30, 2008 11141 whole surface. In particular, as the side length increases beyond a certain threshold an increasing portion of the central region of the facets is expected to remain outside the exploration range of edge atoms. It follows that, under such conditions, the surface species in the central region will not undergo exchanges with edge atoms. Therefore, the first stage of the mechanism, i.e. the one connected with the formation of Ag- and Cu-rich surface domains, will be correspondingly critically affected. The latter length scale, i.e. the wall thickness, can instead affect the stability of the lateral walls and the capability of Ag bulk-like atoms to attain the surface Via collective rearrangements. On the one hand, excessively thin walls are in fact expected to be unable to withstand the migration of Ag species toward the surface. Holes can thus appear in the cube walls which would definitely undermine the system mechanical stability. On the other hand, excessively thick walls are expected to hinder the diffusion of Ag atoms at the surface, which would affect the second stage of the decomposition mechanism. Aimed at exploring the effects of both side length and wall thickness, additional simulations were carried out. The results obtained from these simulations will not be here detailed as in the case of the hollow cube with sides 8 nm long and walls 2 nm thick previously discussed. Rather, the differences between the various cases will be specifically highlighted. VI.1. Side Length. A side about 6 nm long defines a hollow structure with a cavity of about 8 nm3 in volume and surface areas exposed per single internal and external faces roughly equal to 4 and 36 nm2 respectively. The fraction of edge atoms with respect to the total number of surface species is equal to about 0.32. It is then larger than for the 8 nm cube, for which such fraction amounts to about 0.24. Under such conditions, the frequency of exchange processes involving edge and surface species is significantly larger and the formation of Ag- and Curich surface island is correspondingly affected. More specifically, Ag islands begin to form preferentially at lower temperatures. At about 252 K, the cube external faces are already mostly formed by Ag atoms. It is also worth noting that at such low temperature collective rearrangements have not started yet. The modification of the surface composition, enriched in Ag, is due to the frequency of surface exchange events and to the involvement of Ag and Cu atoms located under the exposed layers at the edges. The edge regions of the 6 nm cube become indeed depleted of Ag atoms at temperatures lower than the 8 nm cube. According to a rough description of the operating mechanism, the GB regions immediately below the surface at edges provide the necessary Ag atoms to modify the surface composition. These Ag atoms are able to reach the surface not as a consequence of collective rearrangements, but as a result of individual displacements on the surface and occasional position exchange with surface species. No other significant difference with the case of the 8 nm hollow cube is observed from a mechanistic point of view. It is however worth noting that the segregation of Ag- and Cu-rich phases at surface and GB regions, although similar to the one illustrated by Figure 5, is much less defined. The area of internal surfaces, in particular, is so small that Ag and Cu atoms appear only weakly segregated even at temperatures as high as 300 K. When the sides are 10 nm long, the thermal response of the hollow cube is strictly similar to the one exhibited by the 8 nm system. The phase separation process takes place Via a threestage mechanism involving individual displacements of surface species, collective rearrangements of surface and bulk-like atoms and finally thermally activated diffusion. No significant difference between the behavior of 8 and 10 nm cubes is observed,


Delogu

Figure 15. The central region of an external surface at 272 K. Data refer to a hollow cube with sides about 12 nm long and walls about 2 nm thick. Ag and Cu atoms are in light and dark gray respectively.

except for the temperature TI-II at which the transition from the first to the second stage mobility regimes occurs. More specifically, in the case of the 10 nm cube TI-II amounts roughly to 262 K, a value about 4 K larger than the one for the 8 nm cube. In the light of the connection of TI-II with the number NAg,is of Ag surface islands and then the fraction f of Ag atoms contained in hollow cube walls, an increase of TI-II suggests the formation of a smaller number of Ag surface islands. Better, it suggests a decrease of the surface density FAg,is of Ag islands, quantified by the ratio between NAg,is and the surface area. Only this quantity permits to suitably take into account the variation of NAg,is with the system size. Such decrease is actually observed and the FAg,is values for the 10 nm cube are about the 8% smaller than the ones for the 8 nm system. In addition, also the values of the fraction f of Ag atoms contained in hollow cube walls are smaller for the 10 nm cube. The visualization of the cube surface allows finally the identification of a central portion, with approximate radius of 4 Å, in which Ag and Cu surface atoms are still mixed together. Actually, no exchange between edge and surface atoms has there occurred because no edge atoms has been able to reach such region starting from its original position. The observations above are valid also for the hollow cube with sides 12 nm long. Its thermal response is again analogous to the one discussed in detail for the 8 nm cube. In this case, however, the existence of a central portion in the external surfaces not involved in individual exchange processes is much clearer. First, the transition from stage I to stage II of decomposition mechanism takes place at about 270 K, i.e. Twelve K above the TI-II value observed in the case of the 8 nm cube. Second, the density FAg,is of Ag surface islands is about the 13% smaller than the one for the 8 nm system. Third, a region of about 1.5 nm in radius can be identified at the center of the cube external surfaces in which Ag and Cu atoms are still randomly mixed. One of such regions, clearly not involved in surface exchange processes between edge and surface atoms, is shown in Figure 15 for sake of illustration. The whole body of evidence discussed above points out that the details of the phase separation process observed are clearly affected by the side length of the hollow cube. The competition between edge atom displacement and surface exchange processes defines in particular an optimum size for the hollow cube in connection with the capability of edge atoms of exploring the whole surface before remaining involved in exchange processes with the underlying surface species. It is however worth noting that the occurrence of the peculiar segregation of Ag- and Cu-rich phases observed does not seem to be prevented at relatively large system size. The formation of well separated

Figure 16. The atomic configuration of an external surface at 266 K showing a hole. Data refer to a hollow cube with sides about 8 nm long and walls about 1 nm thick. Ag and Cu atoms are in light and dark gray respectively.

Ag- and Cu-rich domains respectively at the center of external surfaces and at the edges of the hollow cube is however somewhat undermined. VI.2. Wall Thickness. The influence of the wall thickness on the system stability is, in a sense, simpler to understand and to point out. The hollow cube with sides about 8 nm long and walls about 3 nm thick has a thermal response substantially equal to the one of the 8 nm cube with walls about 2 nm thick. However, the fraction f of Ag atoms contained in hollow cube walls after the phase separation is significantly smaller. For example, at 300 K the f value is roughly equal to 0.81 for the cube with walls about 3 nm thick, whereas it approximately amounts to 0.93 for the cube with walls about 2 nm thick. The reasons for such difference lie in the different mobility of surface and bulk-like species. In particular, the wall thickness increase hinders the attainment of high degrees of phase separation because of the longer distance that Ag bulk-like atoms must cover to reach the surface. It is therefore expected that the fraction f of Ag atoms contained in hollow cube walls decreases with the wall thickness. The case of the 8 nm hollow cube with walls about 1 nm thick is remarkably different from all the other ones. The dynamics of atomic species is initially similar, with edge atoms undergoing individual displacements and surface exchange processes. As soon as bulk-like species become involved in cooperative rearrangements at about 258 K, the migration of Ag atoms at the surface induces the formation of one or more holes in the lateral walls such as the one shown in Figure 16. The system follows successively a completely different evolution, with a decomposition process determining the usual surface segregation of Ag atoms. In one of the two cases investigated, the temperature rise determined however also an enlargement of the holes in the lateral walls and the progressive collapse of the structure in a way similar to that observed for the 8 nm cube with walls about 2 nm thick at high temperatures. Hollow systems with thinner walls are expected to undergo a structural collapse even at lower temperatures. In the light of the evidence discussed above, the hollow cube appears as a nanometer-sized system the behavior of which is affected by evident size effects. Not only the length of the cube sides is important, but also the thickness of its walls. It is finally worth noting that the results described above provide an a posteriori justification for having focused the attention on a hollow cube with sides about 8 nm long and walls about 2 nm



thick. Such system is the best one to point out the particular phase separation process investigated in the present work.

Acknowledgment. Financial support has been provided by the University of Cagliari. A. Ermini, ExtraInformatica srl, is gratefully acknowledged for his technical support.

VII. Conclusions

References and Notes

The Ag-Cu hollow cube studied in this work exhibited a relatively complex structural and chemical evolution under gradual heating conditions. The observed behavior originates from two general evidence connected to the particular structure and geometry. First, atoms located at edges, free plane surfaces, GBs and bulk-like regions possess different coordination numbers and their thermal response is correspondingly different. Second, the system is characterized by an intrinsic metastability due to the immiscibility of Ag and Cu elements in the solid state. All these factors concur in enhancing the atomic mobility respect to alloyed bulk systems or even to hollow cubes of pure Ag and Cu of similar size. The larger mobility is necessarily oriented to promote the system decomposition, which lowers the total potential energy of the system Via the creation of segregated Ag and Cu domains. As soon as allowed by thermal motion, decomposition actually occurs in different mechanistic stages. Atomic displacements do not produce immediately a complete separation of Ag and Cu phases. The intermediate structure resulting from the partial demixing is characterized by a high degree of self-organization, probably ruled and promoted by surface energies of Ag, Cu and Ag-Cu coherent domains. Ag-rich domains occupy in fact in the central parts of hollow cube walls, whereas the Cu-rich phase forms a continuous framework along the edges. The largest fraction of free plane surfaces consists thus of the crystalline phase with the lowest surface energy. Under such conditions, the hollow structure is in a local minimum of potential energy. A temperature rise above 850 K destabilizes such metastable structure and determines the final evolution toward a completely decomposed structure with no internal cavity, which permits to further lower the total potential energy by eliminating the contribution of internal surfaces. The accomplishment of such structural condition is however prevented by premelting and melting processes, which on the contrary induce a remixing of Ag and Cu species in the liquid phase. It is here worth noting that the interesting process of shape change and chemical decomposition observed below the premelting point can be in principle followed in long simulations at constant temperature, provided that such temperature is large enough to allow atomic rearrangements. Such investigation could provide valuable details on the mechanistic scenario governing the phase separation and the faceting in nanometersized solids.

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