Phase Diagram of Continuous Binary Nanoalloys: Size, Shape, and Segregation Effects

The phase diagrams of continuous binary nanoalloys are important in providing guidance for material designs and industrial applications. However, experimental determination of the nano-phase diagram is scarce since calorimetric measurements remain quite challenging at the nanoscale. Based on the size-dependent cohesive energy model, we developed a unified nano-thermodynamic model to investigate the effects of the size, shape, and segregation on the phase diagrams of continuous binary nanoalloys. The liquidus/solidus dropped in temperature, two-phase zone was narrowed, and the degree of surface segregation decreased with decrease in the size or increase in the shape factor. The congruent melting point of Cu-Au nanoalloys with and without segregation is linearly shifted to higher Au component and lower temperature with decreasing size or increasing shape factor. By reviewing surface segregated element of different binary nanoalloys, two segregation rules based on the solid surface energy and atomic size have been identified. Moreover, the established model can be employed to describe other physicochemical properties of nanoalloys, e.g. the cohesive energy, catalytic activation energy, and order-disorder transition temperature, and the validity is supported by available other theoretical prediction, experimental data and molecular dynamic simulations results. This will help the experimentalists by guiding them in their attempts to design bimetallic nanocrystals with the desired properties.

two new segregation rules to determine the nature of the segregated element at the surface of bimetallic nanoalloys 1,20,24,25 . However, the results and discussions in the work of Guisbiers are quite questionable 26 , because their employed values of surface energy were calculated by the full charge density (FCD) method with the generalized gradient approximation (GGA). Note that it has been shown that GGA needs to be corrected due to the exclusion of surface electron self-interactions 27 . For example, the surface energies of solid Au, Cu, and Ni (111) were determined to be 1.28, 1.95, and 2.01 J/m 2 by GGA 27 , which are 15%, − 9% and 18% different from the corresponding experimental data of 1.50 (or 1.51), 1.79 (or 1.83), and 2.45 (or 2.38) J/m 2 and other theoretical results of 1.52, 1.83, and 2.44 J/m 2 27 . Moreover, in FCD calculations with GGA, there are often exceptions where the most close-packed surface does not have the lowest surface energy values or there is a weak orientation dependence 27 . The authors of reference 1 used the above questionable data, resulting in the prediction of surface Ni segregation in Cu-Ni system, contrary to the fact that Cu segregates to the surface in this system 28,29 . As a result, the segregation rules proposed by Guisbiers are self-contradictory and incorrect 26 . Therefore, it is necessary to develop a reasonable model to investigate the effects of size, shape, and segregation on the phase diagrams of continuous binary nanoalloys and propose logical segregation rules to predict the nature of the segregated element.

Formula
When a binary system is in thermodynamic equilibrium, the chemical potentials of component A (or B) in the solid phase and liquid phase are equal. In this regard, for a regular solution, the solidus and liquidus of continuous binary alloys have been deduced as 22 , , R denotes the ideal gas constant and x is the composition where the superscripts s and l denote the solid and liquid phases at given temperature T. T m , H m , and Ω are the melting temperature, melting enthalpy, and interaction parameter, respectively. The model has successfully predicted the bulk binary regular solution phase diagrams of metals 1,20,22,25 .
The surface segregation is a critical issue affecting the surface properties and electrocatalysts of binary alloys. The surface segregation of bulk alloys has been well studied experimentally and theoretically [28][29][30] . According to semi-empirical theories, surface segregation is caused by the difference of surface energy between two components and lattice strain energy arising from lattice mismatch. Thus, Tomanek et  where x core solidus is the bulk solidus composition. The heat of segregation Q seg is the work involved in exchanging a surface atom and a bulk atom.
To calculate the phase diagrams at the nanoscale, the size-and shape-dependent thermodynamic parameters are required. On the basis of Lindemann's criterion of melting, the melting temperature is linear to the force constant of the lattice vibration where the latter can be expressed by the cohesive energy E c , i.e. E c ≈ m 1 T m with m 1 being a constant 31 . Similar linear relations between the melting enthalpy or the heat of segregation and cohesive energy also exist for metals, namely E c ≈ m 2 H m and E c ≈ m 3 Q seg where m 2 is a constant and m 3 is a material parameter related to the structure 1,20,25,31 . When the nanocrystals have the same structure as the corresponding bulk, m 1 , m 2 , and m 3 are size-independent, and thus the above three relationship can be extended to the range of nanoscale with the same forms. As a result, T m , H m , and Q seg have the same size dependence as the cohesive energy 31 , namely where X denotes E c , T m , H m , or Q seg . S 0 is the bulk solid-vapor transition entropy of crystal. D 0 is a critical size at which all atoms of crystal are located on its surface, which can be determined 31 as D 0 = 2(3 − d)h with d and h being the dimensionality and the nearest atomic distance. d = 0, 1, and 2 for nanoparticle, nanowire, and thin film 31 . The shape factor λ, describing the shape effect on the ratio of surface atoms to the total atoms, has been determined as 32 , where η S denotes surface packing density, A and V are the surface area and volume of nanoparticles, and the subscripts 1 and 2 denote nanoparticles with a spherical shape and with other shape, respectively. Obviously, the λ value for spherical nanoparticle is equal to one according to Eq. (4) and λ for other polyhedral particles are calculated and listed in Table 1.
Scientific RepoRts | 7:41990 | DOI: 10.1038/srep41990 As shown in Fig. 1, the size-and shape-dependent melting temperatures T m (D,λ) of Au and Ni nanoparticles have been plotted in terms of Eq. (3) together with the necessary parameters listed in Tables 1 and 2. It is found that the melting temperature of nanoparticles decreases with decrease in the size or increase in the shape factor and the shape effect on T m (D,λ) becomes evident at small size. Available MD simulations results [34][35][36][37][38][39] and experimental data 39,40 are also listed for comparisons. Obviously, the model predictions agree with the corresponding experimental and MD simulations results, which indicates the validity and accuracy of Eq. (4) in determining the shape factor.
It is known that Ω = ZN a [ε AB − (ε AA + ε BB )/2] where Z denotes the coordinate number of an atom, N a is the Avogadro's constant, and ε denotes the bond energy 22 . The size dependence of interaction energy should also be the same as that of the cohesive energy, since both of them are proportional to the bond energy [20][21][22] . Combining Eq. (3) with the consideration of the composition effect, a unified model to describe size-, shape-, and composition-dependent interaction energy Ω(x,D,λ) can be deduced as 21

Au Cu Ni
T m (K) 33 As a first-order approximation, the composition-dependent bulk vaporization entropy S 0 (x) and critical size D 0 (x) can be obtained by the Fox equation 42 , In terms of Eqs (1)- (7), size-and shape-dependent continuous binary nano-phase diagrams with and without surface segregation can be calculated. Ni-Cu and Cu-Au systems have been selected as typical examples due to their representative phase diagram shapes.  (1) and (3)-(7) at two sizes of 10 and 4 nm, where the bulk phase diagram is also calculated and listed for comparison. It can be found that both the solidus and the liquidus drop with decrease in the size and the two-phase zone diminish gradually because the difference between Ω s (D) and Ω l (D) decreases with the size. When the size decreases to a critical size D*, the regular solution deteriorates into the ideal solution and the two-phase zone vanishes where the liquid phase and the solid phase are indistinguishable in structure and thus the phase numbers transform from two to one, as shown in Fig. 2f. Note that  Table 1, and thus the higher shape factor, the narrower two-phase zone and the larger critical size D*. Moreover, the melting temperature of nanoalloys also decreases with the increasing shape factor since the physical origin for the depression of melting temperature is thought of as the enormous ratio of the number of surface-to-volume atoms 43 , where the surface-to-volume ratio is related to the shape. By comparing the phase diagrams of different Ni-Cu polyhedral nanoalloys as shown in Fig. 2a-f, it can be found that the dodecahedron presents the higher solidus and liquidus than others at the same size, which implies that the most stable shape investigated theoretically of nanoalloys is dodecahedron. Nevertheless, the predictions can differ from the shapes observed experimentally due to the critical role played by defects and adsorbed species on the surface of nanoparticles 44 .

Result & Discussion
Because there are no systematic experimental investigations or computer simulations on the phase diagrams of binary nanoalloys over the whole composition range at the selected size (4 or 10 nm), no comparison between our model prediction and experimental data or computer simulation results are made in Fig. 2. However, some MD simulations have been carried out on the size-dependent melting temperature T m (D) of Ni-Cu nanoalloys at certain compositions, i.e. Ni 0.8 Cu 0.2 , Ni 0.75 Cu 0.25 , Ni 0.5 Cu 0.5 and Ni 0.2 Cu 0. 8 14,16 . Figure 3 compares the T m (D) functions of these four Ni-Cu nanoalloys between our model predictions and the corresponding MD simulation results 14,16 , where good agreements can be found. Similar to the depression of T m (D) for pure elements as shown in Fig. 1, the melting temperature of nanoalloys also decreases with the decreasing size at a given composition and the drop becomes dramatic at D < 5 nm. Furthermore, as shown in Fig. 3, the melting temperature of Ni-Cu nanoalloys ascends with decrease in the composition of Cu at the same size.
Different from the phase diagram of Ni-Cu where both the solidus and liquidus vary monotonically with increasing x B , there is an intersection between the solidus and liquidus curves in the phase diagram of Cu-Au. Note that the congruent melting point is lower than the melting temperature of either Cu or Au, implying the greater stability of the liquid solution than the solid one. Figure 4 presents the predicted nano-phase diagrams of Cu-Au alloy with different shapes in terms of Eqs (1) and (3)-(7) at two sizes of 10 and 4 nm. The bulk phase diagram is also calculated and listed for comparison. It can be found that the reduction of two-phase zone becomes less distinguishable with the decreasing size than that in Ni-Cu nanoalloys, originating from the small difference of S 0 between Cu and Au in comparison with that between Cu and Ni since S 0 is the only variable parameter in Eqs (3) and (5)-(7) at certain composition, size and shape. The solidus and liquidus of bulk phase diagram of Cu-Au meet at the congruent melting point, which occurs at 52%Au composition and 1165 K. As shown in Fig. 4, the congruent melting point of the nano-phase diagrams is linearly shifted towards higher Au component and lower temperature with decreasing size. It is reasonable and understandable: Au possesses higher S 0 value than Cu, and the size dependences of solidus and liquidus with higher Au component are stronger than those with lower Au component. Moreover, the higher shape factor, the larger slope of the arrow (k 1 < k 2 < k 3 < k 4 < k 5 < k 6 ), namely, the greater effect of shape factor on the shift of the congruent melting point.
Continuous binary nanoalloy with well-controlled size and shape is a remarkable catalyst, which has low catalytic activation energy and thus high rate of reaction 3,4 . Recently, Wang et al. have proposed that Au 0.5 Ni 0.5 nanoalloys exhibited the lowest catalytic activation energy in comparison with Au 0.9 Ni 0.1 , Au 0.8 Ni 0.2 , Au 0.2 Ni 0.8 , and Au 0.1 Ni 0.9 at similar size for a reaction of hydrogen generation 45 . It has been reported that size-and shape-dependent catalytic activation energy of nanocrystals is directly proportional to its melting temperature 19,32 . Since the shape of phase diagram of Au-Ni alloy is similar to that of Cu-Au alloy and the composition of Au 0.5 Ni 0.5 is more closer to that of the congruent melting point (43%Ni composition) than other Au 1 − x Ni x alloys with x = 0.1, 0.2, 0.8 and 0.9, Au 0.5 Ni 0.5 nanoalloy should have the lowest melting temperature and catalytic activation energy among these five Au-Ni nanoalloys, which agree with the experimental results of Wang 45 . This agreement also suggests that our model is valuable for the design of catalyst of nanoalloys.
The cohesive energy of solid is an important physical quantity to account for the binding strength of the crystal and estimate the stability of nanoalloys with different sizes and shapes. Similar to the pure elements, the cohesive energy of nanoalloys should also be directly proportional to its melting temperature when the structure remains the same 31 ,  46,47 . Note that the disc-shaped nanocrystal can be assumed to have a quasi-dimensionality of d = 1 19 and the shape factor of disc-shaped nanocrystal can be determined as λ = (L + D/2)/L in terms of Eq. (4). Since T m (D,λ,x) value of nanoalloys can be obtained through our established nano-phase diagram, E c (D,λ,x) of nanoalloys can be calculated in terms of Eq. (8). Figure 5a compares the E c (D) function of disc-shaped Cu 0.7 Au 0.3 nanoalloys between the model prediction with d = 1 and λ = (L + D/2)/L ≈ 2.25 and other theoretical estimation 46,47 , where an agreement between model prediction and one data point can be found. The cohesive energy decreases with decrease in the size, which reflects the instability of nanoalloys in comparison with the corresponding bulk crystals. This trend is expected since the surface/ volume ratio increases with decreasing size while the surface atoms have lower coordination number and higher energetic state, and consequently the low stability of nanoalloys 48 .
Not only the cohesive energy but also the order-disorder transition temperature T o (D) of nanoalloys can be determined through our established nano-phase diagram. It has been reported that the size dependence of T o (D) is directly proportional to the root-mean-square of that of T m (D) 49 , i.e. As shown in Fig. 6, solidus(core)/liquidus(core) and solidus(surface) of Ni-Cu and Cu-Au nanoalloys with and without segregation are plotted for the most stable shape (dodecahedron), basal shape (sphere), and the least stable shape (tetrahedron). It is obvious that Cu and Au always segregate to the surfaces of Ni-Cu and Cu-Au nanoalloys, which agrees with the corresponding experimental results 28,29 . Red and cyan arrows in Fig. 6d-f are used to highlight the size and shape effects on the congruent melting point without and with segregation, respectively. It can be found that the congruent melting point in the nano-phase diagrams always shifts towards higher Au composition and lower temperature where the slope of cyan arrow is always smaller than that of red arrow at a given shape of nanoalloy (k' 1 < k 1 , k' 4 < k 4 and k' 6 < k 6 ), i.e. the occurrence of surface segregation causes the composition change to slow down at the congruent melting point. Moreover, the higher shape factor, the greater slope of the cyan arrow (k' 1 < k' 4 < k' 6 ), which indicates that the degree of segregation decreases with decrease in the size or increase in the shape factor. This changing trend agrees with the MD simulation results 29 . Based on elemental variables related to the work functions and electron densities of the elements, Hamilton concluded that the surface energy difference was the dominant driving force while the strain energy and the heat of solution played a minor role in determining segregation behaviour for most binary alloys 51 . Since the surface energy is also descended with decreasing size 27 , the drop in driving force for surface segregation is expected.
To predict the nature of the segregated element, preferentially found at the surface of the binary nanoalloys, we identified two segregation rules based on the solid surface energy γ s and atomic size h: The first rule says that if the surface energy of element A is larger than the element B, then element B will segregate to the surface; When the surface energy difference between two elements is less than ~10% of the highest surface energy, then the element with the largest atomic size segregates to the surface to release the strain energy, this is the second rule. These two rules have been summarized in Table 3 to explain the surface segregation of different binary nanoalloys. For example, since γ s of Ag is obviously smaller than those of Au, Co, Cu, Ni and Pd 27 , the segregated element in Au-Ag, Ag-Co, Ag-Cu, Ag-Ni and Ag-Pd alloys is Ag according to our first rule, in agreements with the corresponding experimental results 9 . Similar conditions occur in Au-Cu, Au-Ni, Au-Pd, Au-Pt, Cu-Ni, Pd-Ni and Pd-Pt alloys. While for Fe-Ni and Pt-Ni alloys with h(Fe) < h(Ni) < h(Pt) 52 , Ni and Pt respectively segregate to the surfaces according to our second rule, since the differences of γ s among Fe, Ni and Pt are smaller than 5% 27 .
Remarkably, these two segregation rules are applicable when the binary alloys are not affected by external factors, such as heat treatment, adsorbents, substrates, and other inducements 30,53 , since these factors can drastically alter the surface energy of nanoalloys and thus directly modify the surface segregation. For example, Au with the lower surface energy segregates to the surface of Cu-Au nanoalloys. Nevertheless, the sample exposed to O 2 would lead to a preference of Au to stay in the interior of Cu-Au nanoalloys, caused by the energetically much more favorable O binding to Cu than to Au atoms 29 .

Concluding Remarks
A unified thermodynamic model based on the size-dependent cohesive energy model has been developed to predict the size, shape and segregation effects on phase diagrams of continuous binary nanoalloys. For Cu-Au nanoalloys with segregation, the congruent melting point is linearly shifted to higher Au composition and lower  . Solidus(core)/liquidus(core) and solidus(surface) of dodecahedral, spherical, and tetrahedral Ni-Cu (a-c) and Cu-Au (d-f) nanoalloys without and with surface segregation at bulk, 10, and 4 nm, where the red and cyan arrows are only to guide the eyes and highlight the size and shape effects on the congruent melting point, respectively.
temperature and the degree of segregation decreases with decrease in the size or increase in the shape factor. Moreover, it is found that surface segregated elements are Cu and Au in Ni-Cu and Cu-Au nanoalloys for all the shapes investigated. Two segregation rules based on the solid surface energy and atomic size have been developed to predict the segregated elements when the binary alloys are not affected by external factors, and these rules agree with experimental measurements. Finally, the established nano-phase diagrams can be employed to describe the physicochemical of nanoalloys, and the validity is supported by available theoretical predictions, experimental data, and MD simulations results.