Free Energy Differences between Ag−Cu Nanophases with Different

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J. Phys. Chem. C 2010, 114, 19946–19951

Free Energy Differences between Ag-Cu Nanophases with Different Chemical Order Francesco Delogu* Dipartimento di Ingegneria Chimica e Materiali, UniVersita` degli Studi di Cagliari, piazza d’Armi, I-09123 Cagliari, Italy ReceiVed: August 25, 2010; ReVised Manuscript ReceiVed: October 22, 2010

Molecular dynamics simulations have been employed to study the thermodynamic stability of nanometersized Ag50Cu50 spherical particles with radius in the range between 1 and 10 nm. Four different structural arrangements of Ag and Cu atoms were considered, namely an unmixed phase in which Ag and Cu are separated by a coherent interface, two core-shell systems with Ag shells and Cu cores and vice versa and a perfect random solid solution. All of these cases share the same crystallographic arrangement of atoms but a different chemical order. Free-energy differences between the unmixed phase and the other ones were estimated according to the Bennett’s method. It is shown that free-energy differences scale inversely with the particle radius. The results obtained also provide information on fundamental thermodynamic properties such as surface and interface free energies. I. Introduction Nanometer-sized systems exhibit an impressive suite of unusual physical and chemical properties originated from the intimate coupling of size and shape effects that cannot be found in corresponding bulk phases.1-3 Size and shape control two correlated fundamental features, namely the total number of atoms in the system and the fraction of surface atoms,1-3 which are characterized by coordination numbers smaller than the ones typically attained in the bulk.1-3 When size is on the order of a few nanometers, the fraction of surface atoms is no longer negligible and all of the processes related to surface activity are correspondingly enhanced.1-3 In addition, the total number of atoms can be significantly far from thermodynamic limit.1-4 Under such conditions, unexpected behaviors can arise.1-4 In most cases, such unexpectedness can be motivated by our limited knowledge about the influence of surface effects on thermodynamic properties.1-4 It is worth noting that it is not a formal limitation, being the formalism of classical thermodynamics perfectly suited to deal with surface effects by including terms accounting for surface free energy and its dependence on curvature.4,5 Rather, it is a practical limitation in our capability of obtaining experimental measurements or numerical estimates of the thermodynamic properties of individual nanometer-sized systems.6 This is particularly true for free energy, the knowledge of which is crucial to understand phase equilibria and thermodynamic stability.4,5,7 Unfortunately, it is not amenable, neither to direct experimental quantification4,5 nor to simple numerical evaluation,7 even in the case of bulk phases. Yet, its importance motivated considerable research.7 Computational approaches include thermodynamic integration,8 nonequilibrium estimates,9 adiabatic switching,10 and parameter-hopping,11 and generally involve the calculation of free-energy differences. Among socalled free-energy perturbation methods, umbrella sampling12 and Bennett’s acceptance ratio13-15 are the most popular ones. Bennett’s method considers two different equilibrium states defined on a phase space by the energy functions U0(q b) and * To whom correspondence [email protected].

should

be

addressed.

E-mail:

U1(q b) and characterized by a free-energy difference ∆F,4,5,7,13-15 expressed by the logarithm of the ratio between the partition b) and U1(q b).4,5,13-15 A work is functions associated with U0(q b) with U1(q b) and associated with the process of changing U0(q vice versa at constant temperature.13-15 A distribution of work estimates in either direction can be obtained by sampling the states’ trajectory in the phase space.13-15 Provided that switching between states is infinitely fast, the distributions simply corb) - U0(q b) canonically respond to the difference ∆U ) U1(q sampled from the initial state.13-15 In recent times, the Bennett’s method has been clearly shown to have considerable potential in the reliable evaluation of freeenergy differences, even for small sample sizes.16-18 Starting from such premise, the present work aims at demonstrating the capabilities and the usefulness of the Bennett’s acceptance ratio method in the case of nanometer-sized systems. Here, attention is focused on the behavior of Ag-Cu particles with radius in the range between 1 and 10 nm. The choice of dealing with Ag-Cu systems is motivated by the expertise developed in previous work,19 where Bennett’s method was used to estimate free-energy differences between Ag-Cu solid solutions with a different degree of chemical order relative to a completely unmixed system. An advantage of the Ag-Cu system over other possible ones is that both elements and solid solutions share a face-centered cubic (fcc) crystalline lattice, which is an important requisite to apply the Bennett’s method.13-19 Furthermore, the immiscibility of the elements in the solid phase, due to a markedly positive value of the enthalpy of mixing,19 permits to identify in completely unmixed systems the necessary reference state to which free-energy differences must be referred. In addition, the interatomic potential employed was suitably parametrized to specifically reproduce thermodynamic quantities such as heat of mixing and latent heats of transition.19 In this work, particles with Ag50Cu50 stoichiometry and three different degrees of chemical order will be considered, namely a complete random solid solution and two completely segregated core-shell systems with Ag shells and Cu cores or vice versa. Their free-energy differences will be referred to the one between completely mixed and completely unmixed bulk phases.19

10.1021/jp108044z  2010 American Chemical Society Published on Web 11/09/2010

Free Energy Differences between Ag-Cu Nanophases

J. Phys. Chem. C, Vol. 114, No. 47, 2010 19947 TABLE 1: Total Number N of Atoms in the Ag50Cu50 Particles with Different Chemical Order

Computational details are given in the following. II. Numerical Simulations Interatomic forces were reproduced by a semiempirical tightbinding (TB) potential based on the second-moment approximation to the electronic state density.20,21 The cohesive energy is equal to NR

E)

{

Nβ

∑ ∑ ∑ ∑ ARβe R

iR)1

-pRβ(

β

jβ)1

[

rijRβ

-1)

dRβ

-

Nβ

∑ ∑ ξRβ2 e-p β

jβ)1

Rβ(

rijRβ dRβ

]} 1/2

-1)

R (nm)

J

A

C

M

1 2 3 4 5 6 7 8 9 10

301 2404 8102 19 214 37 531 64 849 102 978 153 715 218 865 300 226

302 2408 8106 19 213 37 528 64 854 102 987 153 722 218 871 300 229

300 2411 8112 19 219 37 527 64 855 102 989 153 723 218 868 300 231

301 2400 8108 19 218 37 532 64 850 102 976 153 718 218 865 300 225

(1)

where the indexes iR and jβ run over all the NR and Nβ atoms of species R and β, being rRβ ij ) |riR - rjβ| the distance between two atoms. The parameters ARβ, ξRβ, pRβ, and qRβ quantify the potential energy for R and β species. The term dRβ represents the equilibrium distance of nearest neighbors at 0 K. The first member on the right-hand side expresses the repulsive part of the potential as a Born-Mayer pairwise interaction, whereas the second member describes the attractive part in the framework of the second-moment approximation of the TB band energy.20,21 Interactions were computed within a spherical cutoff radius rc of 0.673 nm, approximately corresponding to the seventh shell of neighbors. The potential parameter values were taken from literature.19,20 The TB potential has a considerable capability of reproducing structural, thermodynamic and mechanical properties of both massive and nanometer-sized transition metals as well as of their alloys.20-23 However, it must be noted that the present work has only qualitative purposes. Accordingly, the obtained results should be considered as good approximations at best. The nanometer-sized Ag-Cu particles were created starting from large bulk phases containing about 256 000 atoms arranged in 40 × 40 × 40 cF4 fcc elementary cells, relaxed at 298 K with number N of Ag atoms, pressure P, and temperature T constant.24,25 Periodic boundary conditions were applied along the three Cartesian directions.26 Equations of motion were solved with a fifth-order predictor-corrector algorithm26 and a time step of 2 fs. Two different bulk phases were considered, namely a solid solution in which Ag and Cu atoms are randomly distributed over the fcc lattice sites and pure elemental Ag and Cu crystals. An additional bulk system formed by two large domains of pure Ag and Cu separated by a coherent interface was also used to simulate the unmixed bulk phase. These bulk lattices were used to generate the desired particles. To such aim, spherical regions of radius R roughly between 1 and 10 nm were selected at the center of the bulk phase. The selected regions were isolated from the surrounding by linearly decreasing to zero in 50 ps the ARβ and ξRβ potential parameters for the interactions between atoms inside and outside the regions. This was done for each type of bulk lattice, which resulted in the creation of 40 different particles. Of these, 20 initially include only Ag or Cu atoms. These particles were used to generate Ag-Cu particles with Ag and Cu forming respectively the surface shell and the bulklike core of the particle, or vice versa. The surface shell of Ag (Cu) was created by simply replacing the Cu (Ag) atoms at, and close, to the particle surface with Ag (Cu) ones until the desired Ag50Cu50 composition was reached. The replacement was carried out by gradually changing the ARβ, ξRβ, pRβ, qRβ,

Figure 1. Cross section about 0.5 nm thick of the Ag50Cu50 J, A, C, and M particles with radius R equal to about 2 nm. Ag and Cu atoms are shown in light and dark gray, respectively.

and dRβ potential parameters according to linear trends in 1 ns. This permits avoiding deleterious volume fluctuations due to the sudden substitution of Ag (Cu) atoms with Cu (Ag) ones and favors the obtainment of relaxed unsupported particles. At the end of the replacement process, the particles were relaxed for at least 1 ns to ensure the necessary equilibration for the evaluation of free-energy differences. It is worth noting that the particles generated starting from the unmixed bulk phase are so-called janus particles, that is particles consisting of two different chemical domains separated by an interface. All of the free-energy difference computations were carried out by using the janus particles as the reference systems. For simplicity, the particles with janus structure, Ag surface shell, Cu surface shell and perfectly mixed Ag and Cu will be hereafter referred to as J, A, C, and M particles, respectively. A few details on the various particles are given in Table 1. Examples of particles with different chemical order are shown in Figure 1. With reference to the canonical statistical ensemble with number N of atoms, volume V and temperature T constant, the Helmholtz free energy F can be written in terms of the configurational integral Q as

F ) -kBTln Q where kB is the Boltzmann constant and

(2)

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Q)

∫Ω exp

[

-

]

U(q b) dq b kBT

Delogu

(3)

produce a pair of potential energy difference distributions h0(∆) and h1(∆). Now, two functions are defined, namely

Integration of eq 3 extends over the volume Ω of the coordinate space occupied by the N atoms. The free-energy difference between two states 0 and 1 can be written as

1 f0 ) - ∆ + ln h0(∆) 2

(8a)

1 f1 ) - ∆ + ln h1(∆) 2

(8b)

and

∆F ) F1 - F0 ) kBTln

Q1 Q0

(4)

with Q1 and Q0 integrated over the same volume of the phase space. Now, it is possible to define the energy difference distribution functions h0(∆) and h1(∆) respectively as

h0(∆) )

1 Q0

∫Ω δ[U0(qb) - U1(qb) - ∆] exp

]

[

-

b) U0(q dq b ) 〈δ[U0(q b) - U1(q b) - ∆]〉0 (5a) kBT and

h1(∆) )

1 Q1

∫Ω δ[U0(qb) - U1(qb) - ∆] exp

]

[

-

U1(q b) dq b ) 〈δ[U0(q b) - U1(q b) - ∆]〉1 kBT

(5b)

b) - U1(q b) - ∆] is a Dirac’s delta whereas U0(q b) Here, δ[U0(q b) correspond to the potential energies of states 0 and and U1(q b) - U1(q b) - ∆]〉 represents the average over 1. Instead, 〈δ[U0(q b) - U1(q b) - ∆ for the canonical ensemble of the quantity U0(q states 0 and 1. It follows that

( )

Q1 ∆ h0(∆) ) exp Q0 kBT h1(∆)

(6)

Being the right-hand side of eq 6 independent of ∆, a suitable choice of it permits to obtain h0(∆) ) h1(∆), so that

( )

Q1 ∆F ) exp Q0 kBT

(7)

Therefore, it is sufficient to interpolate the numerical functions describing h0(∆) and h1(∆) to find the intersection value ∆ assuring the validity of eq 7. In practice, two atomic configurations with the same atomic positions but different chemical order are considered.19 The two b) and configurations exhibit different potential energies U0(q b). One of the configurations is taken as the reference one U1(q that governs the evolution of atomic positions by molecular dynamics in the canonical ensemble. The other configuration is only regarded as a mask configuration with different chemical order that shares the same atomic positions with the reference one. At any instant, the potential energy difference ∆ between reference and mask configurations is evaluated. Two independent simulations are carried out so that both atomic configurations with different chemical order are considered alternately as reference and mask. Provided that a suitable sampling of the phase space is performed, the two independent simulations

Then, the two functions f0(∆) and f1(∆) are plotted as a function of the potential energy difference ∆. Depending on the phase space sampling accuracy, f0(∆) and f1(∆) are expected to exhibit roughly parallel trends separated by the free-energy difference ∆F, which can be easily determined by both statistical and graphical tools. Before proceeding with the description and discussion of the results obtained, it is worth noting that the Bennett’s method was originally developed for Monte Carlo simulations.13-18 However, previous work has clearly shown that reliable results can be also obtained by using molecular dynamics.19 Also, it must be noted that the Bennett’s method should be carried out in the canonical statistical ensemble.13-18 In particular, constancy of volume is an important requisite because both the reference and mask configurations must share the same volume.13-18 Of course, this is a condition that cannot be strictly met in the case of unsupported nanometer-sized particles. Nevertheless, sharing the same average chemical composition, the particles also exhibit very close volumes. For this reason, the numerical findings can be considered as only negligibly affected by volume differences between different particles. The goodness of potential energy distributions h0(∆) and h1(∆) supports this inference. The same is true for the adequateness of sampling of the phase space configuration. Of course, sampling is the crucial point for the Bennett’s method as well as for any other method addressing the problem of evaluating relative free-energy differences.7-15 It is well-known that neither a rich Monte Carlo exploration nor a long molecular dynamics reconstruction can warrant the necessary sampling of the phase space trajectory of the system. More specifically, the system evolution not necessarily involves the global minimum configurations and the closest isomers. Also, specific aspects related to local chemical ordering may not be suitably sampled. In front of this, it must be noted that the relative smallness of the systems investigated represents a favorable circumstance. In fact, the evaluation of free-energy differences is expected to improve as the system size decreases, that is as the phase space trajectory becomes increasingly less rich.7-15 Taking into account all of the abovementioned observations, it should be kept in mind that the results presented here must be considered at best as good approximations of the real situation. This perfectly agrees with the qualitative and methodological character of the present study. III. Free-Energy Differences Free-energy differences between unmixed systems and other ones with different degree of chemical order were computed for any given system size, including bulk phases. In each case, a pair of distributions h0(∆) and h1(∆) of free-energy differences was obtained. From these, the correlated f0(∆) and f1(∆) functions were evaluated. Typical f0(∆) and f1(∆) data sets are shown in Figure 2 for 5 nm particles with J and C structures. It can be seen that the points belonging to the two data sets

Free Energy Differences between Ag-Cu Nanophases

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Figure 2. f0(∆) and f1(∆) functions worked out from the distributions h0(∆) (black, lowest bars) and h1(∆) (red, highest bars) of free-energy differences between J and C structures for the particle with radius R equal to about 5 nm. For simplicity, both potential energy difference ∆U and the functions f0(∆) and f1(∆) are normalized to kBT.

clearly define two superposed curves. Curves keep an approximately constant distance for all of the range of potential energy difference ∆ explored. Therefore, it is quite easy to work out a reliable free-energy difference ∆F value, which roughly amounts in the present case to 1.18 kB T, corresponding to about 2.94 kJ mol-1. Similar results are obtained for all of the systems. In all of the cases, f0(∆) and f1(∆) functions are well superposed and keep at roughly constant distance. Being simulation length equal, superposition, and distance constancy become increasingly less satisfactory as the particle size decreases. However, the lengthening of computations produces an improvement of the abovementioned features that is directly proportional to the simulation length. Therefore, it was sufficient to prolong calculations to obtain satisfactory f0(∆) and f1(∆) data sets even for 1 nm particles. Simulation times as long as 20 ns were attained in the most difficult cases. Of course, such long simulation times were made possible only by the significant decrease of the number N of atoms with the reduction of the particle radius R. The free-energy differences ∆F between J and A, ∆FAJ ) FA - FJ, J and C, ∆FCJ ) FC - FJ, and J and M, ∆FMJ ) FM - FJ, structures are shown in part a of Figure 3 as a function of the particle radius R and in part b of Figure 3 as a function of its reciprocal, R-1. The data regarding the case of bulk J and M phases is also quoted, assuming for them an infinite size corresponding to R-1 equal to 0. The most striking evidence is that the points of the different data sets in part b of Figure 3 invariably arrange according to roughly linear trends, indicating that the free-energy difference ∆F has a simple power-law dependence on the particle radius R. Here, it must be noted that all of the ∆FAJ, ∆FCJ, and ∆FMJ values quoted in Figure 3 are positive. Accordingly, for any given particle size R, the free energy FJ of J structures must be larger, in absolute value, than the corresponding quantities FA, FC, and FM, and must be negative. It follows that, on an absolute scale, the arrangement of Ag and Cu atoms characteristic of janus particles always exhibits the lowest free energy, at least in the particle size range explored in simulations. Therefore, on a thermodynamic basis, the J structure is favored over A, C, and M ones. The data shown in Figure 3 indicate that the most stable structural arrangement of Ag and Cu atoms relative to the janus structure J is the one of A particles, with an Ag shell at the surface and a Cu core. The second more stable arrangement relative to the janus structure J is shown by C particles, in which the surface shell is formed by Cu and the inner core by Ag. However, it must be noted that the M particles, consisting of the equiatomic Ag-Cu solid solution become more stable than

Figure 3. Free-energy differences ∆F between J and A, ∆FAJ ) FA - FJ (0), J and C, ∆FCJ ) FC - FJ (O), and J and M, ∆FMJ ) FM FJ (4), structures as a function of the particle radius R (a) and of the reciprocal of particle radius R, R-1 (b). An infinite size corresponding to R-1 equal to 0 is assumed for the case of bulk phases. Best-fitted lines are also shown.

C particles relative to the janus structure J as the particle radius R becomes smaller than about 2 nm. Elsewhere, the solid solution is the less stable phase. Regarding the relative stability of J, A, C, and M structures, it is also worth noting that the linear plot of ∆FAJ in part b of Figure 3 crosses the abscissa axis at R values very close to 0. In such size regime, specific size effects are generally expected to arise, which could irregularly modify the smooth size variation of the ∆FAJ, ∆FCJ, and ∆FMJ free-energy differences. In particular, a variety of experimental and numerical evidence indicate that the A structure is the preferred one in very small Ag-Cu clusters and affine systems.27-40 As the system size is increased, a transition to the janus structure is observed. However, it is not clear whether the surface of the Cu domain is covered by a thin layer of Ag atoms or not.27-40 In no case C or M structures are observed. With reference to the above-mentioned observations, the numerical evidence discussed in the present study seem to support a contradictory scenario, with the J structure always more stable than the A one. Actually, this is only true for the particle radius R range between 1 and 10 nm explored. In fact, the present study does not allow to exclude that the A structure could become the most stable one for small Ag-Cu clusters with radius R below 1 nm. Therefore, at present the numerical findings obtained by the application of the Bennett’s method can be considered in line with the predictions of ab initio and classical numerical simulations as well as with the available experimental data.27-40 Also, no conclusion can be drawn regarding the relative stability of the three-shell configuration with an Ag core, a Cu intermediate shell and an Ag surface layer.28 The ∆F estimates allow to gain indirect information on various thermodynamic quantities. Before doing this, it is necessary to write the thermodynamic expressions for the Helmholtz free energy F of individual J, A, C, and M particles. For nanoparticles with J structure, the expression is

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Delogu

2 FJ ) 2πR2γs,Ag + 2πR2γs,Cu + πR3FAgFAg + 3 2 3 πR FCuFCu + πR2γi,AgCu 3

∆Fsum ) ∆FAJ + ∆FCJ ) 2π(4r2 - R2)γi,AgCu

(18) (9)

where γs,Ag and γs,Cu are the free energies of Ag and Cu surfaces, FAg and FCu the atomic densities of Ag and Cu, FAg, and FCu the free energies of Ag and Cu, and γi,AgCu the free energy of Ag-Cu interfaces. Taking into account that at 298 K FAg and FCu are both equal to 0 by definition, the free energy of nanoparticles with J structure becomes

FJ ) 2πR2γs,Ag + 2πR2γs,Cu + πR2γi,AgCu

(10)

Along the same line, the following expressions were worked out for the free energy of nanoparticles with A, C, and M structure:

FA ) 4πR γs,Ag + 4πr γi,AgCu 2

2

(11)

FC ) 4πR γs,Cu + 4πr γi,AgCu

(12)

4 FM ) 4πR2γs,AgCu + πR3FAgCuFAgCu 3

(13)

2

2

In eqs 11 and 13, r indicates the radius of the spherical domain including the Ag or the Cu atoms. Because of the constraint of equiatomic composition, for A and C particles r is respectively equal to R (1 + FAg/FCu)-(1)/(3) and R (1 + FCu/ FAg)-(1)/(3). Being FAg and FCu equal to about 0.097 and 0.141 mol cm-3 respectively,41 in the two cases r corresponds to about 0.84 R and 0.74 R. In eq 13, γs,AgCu represents the surface free energy of the Ag50Cu50 solid solution, FAgCu its atomic density and FAgCu its Helmholtz free energy. Now, it is possible to combine eqs 10-13 to obtain the expressions for the free-energy differences ∆F between A, C, and M structures and the J one. These are equal to:

∆FAJ ) FA - FJ ) 2πR2(γs,Ag - γs,Cu) + π(4r2 - R2)γi,AgCu

(14)

∆FCJ ) 2πR2(γs,Cu - γs,Ag) + π(4r2 - R2)γi,AgCu

(15) 4 ∆FMJ ) 4πR2γs,AgCu + πR3FAgCuFAgCu - 2πR2 γs,Ag + 3 1 γs,Cu + γi,AgCu (16) 2

(

)

The free-energy difference between nanoparticles with A and C structures can be also written as

∆FAC ) ∆FAJ - ∆FCJ ) 4πR2(γs,Ag - γs,Cu)

(17)

Conversely, the sum of eqs 14 and 15 yields the quantity

which can be used to evaluate the Ag-Cu interface free energy γi,AgCu. It is worth noting that all of the free-energy differences can be compared only when normalized to the total number N of atoms in the particles. Simple algebraic manipulations indicate that N is roughly equal to (2)/(3) π R3 (FAg + FCu) . It follows that the free energy ∆F estimates obtained from simulations must be referred to the following expressions:

{

[( ] } [( ] }

∆FAJ ) 3 (γs,Ag - γs,Cu) +

FAg 1 4 2 FAg + FCu

)

2/3

-

1 γi,AgCu (FAg + FCu)-1R-1

{

∆FCJ ) 3 (γs,Cu - γs,Ag) +

FCu 1 4 2 FAg + FCu

)

2/3

-

1 γi,AgCu (FAg + FCu)-1R-1

∆FMJ ) 2

(19)

FAgCu 3 F + 2γs,AgCu - γs,Ag FAg + FCu AgCu 4 1 γs,Cu - γi,AgCu (FAg + FCu)-1R-1 2

(20)

(

)

(21)

It can be seen that the thermodynamic expressions above predict for all of the different free-energy differences ∆F a linear dependence on the reciprocal of the particle radius R, R-1. Therefore, the linear arrangements obtained by plotting ∆F as a function of R-1 as in part b of Figure 3 can be rationalized on a very simple thermodynamic basis. In addition, eqs 19-21 can be exploited to fit the linear trends in Figure 3 with the aim of obtaining information on the values of surface and interface free energies γs,Ag, γs,Cu, γs,AgCu, and γi,AgCu. In this regard, it must be noted that the Helmholtz free energy FAgCu of the mixed phase corresponds to the ∆FMJ estimate for bulk phases. Then, the value of eq 20 at R-1 ) 0 should be equal to the ∆FMJ value for bulk systems. Such quantity amounts roughly to 9.1 kJ mol-1, in agreement with previous work.19 This permits to further check the reliability of the method employed as well as of the simulations carried out. According to the linear best-fitting of the different data sets in part b of Figure 3, the slopes associated with eqs 19-21 are equal to 0.85, 14.71, and -0.91 kJ nm mol-1, respectively. On the basis of the knowledge of the quantities FAgCu, FAgCu, FAg, and FCu, the above-mentioned values permitted to work out an estimate for the surface and interface free energies γs,Ag, γs,Cu, γs,AgCu, and γi,AgCu. The obtained values roughly amount to 0.93, 1.48, 1.32, and 1.03 J m-2, respectively. These values compare relatively well with the corresponding ones obtained in ab initio simulations.42,43 For example, the surface free energy γs,Ag of Ag has been shown to range between 1.14 and 1.42 J m-2 for plane surfaces of different crystallographic structure.43 Analogously, the surface free energy γs,Cu of Cu has been shown to range approximately from 1.73 to 2.04 J m-2.42,43 Although ab initio estimates are generally larger than the ones obtained by the TB potential used in this work, it must be noted that this aspect is common to all of the other semiempirical potentials such as the ones deriving from the Finnis-

Free Energy Differences between Ag-Cu Nanophases Sinclair (FS) or the embedded-atom method (EAM) approach.42,43 Conversely, the values obtained by using the FS, EAM, or the TB potential are quite close to each other.44-46 Regarding plane Ag-Cu surfaces, the TB potential indicates a value of about 1.13 to 1.42 for different crystallographic facets.44-46 Instead, in the case of (001) Ag-Cu interfaces, previous classical calculations provided a γs,AgCu value ranging between 1.38 and 1.89 J m-2 for coherent and incoherent cases, respectively.19 It follows that the γs,Ag, γs,Cu, γs,AgCu, and γi,AgCu values obtained in the present work are in line with previous estimates. In this regard, at least two further observations are necessary. First, it must be expected that the surface energies exhibit smooth size effects. In the present case, such effects are quite small and the surface energy values do not show any definite trend as a function of the particle radius R. However, it is worth noting that the evaluation of surface energies often requires dedicated simulations, which must be suitably prepared. This is not the case, since the surface energy values have been worked out as average quantities from a best-fitting of data involving all of the size range explored. Second, it is reasonable to define, and then evaluate, a surface energy only for particles with atoms in their interior that exhibit a bulklike behavior. In fact, when such condition is not satisfied, particles become increasingly affected by so-called specific size effects related to geometric and electronic features.47 Under such circumstances, thermodynamic quantities are generally ill-defined.1,2,48 As a final comment, it must be noted that the linearity of the plots in part b of Figure 3 suggests that the surface and interface free energies γs,Ag, γs,Cu, γs,AgCu, and γi,AgCu are only scarcely dependent on the surface curvature. Otherwise, the points obtained at different particle radii R would have arranged according to curved trends. Also, it should be noted that minor effects could have been masked by the uncertainties in the γs,Ag, γs,Cu, γs,AgCu, and γi,AgCu values. These would have been further reduced by prolonging calculations for longer time intervals. However, the purpose of this work is not providing absolute values of the above-mentioned quantities, but rather demonstrating the reliability of the Bennett’s method when applied to the case of nanometer-sized systems. IV. Conclusions The Bennett’s method was profitably applied to the evaluation of free-energy differences between nanometer-sized Ag-Cu phases exhibiting different chemical order. Four different phases were considered, namely a completely unmixed system in which Ag and Cu are separated by a coherent interface, two core-shell arrangements in which shell and core consist alternately of Ag and Cu, and a perfect random solid solution. Free-energy differences were evaluated with reference to the completely unmixed state. Numerical findings indicate that this latter is always the most stable state in the particle size range explored by simulation. Instead, Ag-Cu core-shell particles are the most stable ones relative to the unmixed structure. Conversely, the solid solution is the less stable phase, provided that the particle radius is larger than about 2 nm. At smaller values, it is the Cu-Ag core-shell particle to exhibit the minor thermodynamic stability relative to the unmixed structure. Free-energy differences are shown to exhibit a linear dependence on the reciprocal of the particle radius. Together with the knowledge of a few physical properties and an explicit formulation of the freeenergy difference expressions, the best-fitting of these linear trends permitted to estimate various thermodynamic quantities. In particular, it was possible to estimate the surface free energies of Ag, Cu, and Ag50Cu50 solid solution as well as the free energy of the

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