Scaling of Nanoalloy Phase Transitions: Elucidating the Distinct Role of Surface Sites

Abstract

Nano-size-induced shifts in alloy phase-separation critical temperatures ($T_C^{nano}$) are investigated by introducing an atomistic concept of site-specific contributions to the shift (SSCS) associated with different atomic coordination environments in cuboctahedral and truncated-octahedral nanoparticles (NPs). $T_C^{nano}$ previously computed using the Free-energy Concentration Expansion Method (FCEM) for the transformation of three small quasi-Janus Pd-Ir NPs into mixed nanophases are extended here to a substantially broader set of 22 NP sizes, ranging from 147 to 49,049 atoms. This dataset provides the basis for the present modeling. The main objective is to elucidate the deviations of the critical-temperature shifts in small NPs from the finite-size-scaling (FSS) inverse-size power law previously proposed on the basis of non-atomistic thermodynamic modeling. Within the SSCS approach, these deviations are described in terms of contributions from face, edge, and vertex sites. The contributions are proportional to the fractions of the corresponding surface-site types and can be approximated by terms proportional to $n^{-1}$, $n^{-2}$, and $n^{-3}$, respectively, where $n$ is the number of nested atomic shells. The SSCS approach can also be applied to other phase transitions in nanoparticles of various shapes and sizes.

1. Introduction

Studies on the finite-size scaling (FSS) of the solid–solid phase transition temperature in immiscible alloy nanoparticles (NPs, “nanoalloys”) are quite rare in the literature. Some general principles of the FSS theory [1,2] are first outlined for macroscopic systems undergoing second-order phase transitions. Thus, thermodynamic and correlation functions exhibit singularities at a critical point $T_C$, and the “correlation length” $\xi$ diverges according to $t^{-\nu }$, with $t (\equiv 1-{\displaystyle \frac{T}{T_C}})$ characterizing the relative deviation of the temperature from the critical value, and $\nu$ denoting a “critical exponent”. The situation changes considerably in finite-size systems, such as NPs, for which $\xi$ grows up to the NP size ($n$) with increasing temperature [1]. Thus, due to the constrained $\xi$, a “pseudo” critical transition is expected at $T_C^{nano}$ (shifted below $T_C$), so that the shift $t_C\equiv 1-{\displaystyle \frac{T_C^{nano}}{T_C}}$ obeys $t_C\propto n^{-1/\nu }$, where $\nu$ differs from the bulk value because of free boundary conditions involving low-coordinated surface sites. According to our previous computations and analysis of the FSS based on the Free-energy Concentration Expansion Method (FCEM) [3], $\nu =1$ for nano-phase separation transitions [4]. Such an inverse relationship between transition temperature shift and the nanoparticle size was claimed in several thermodynamic models [5,6,7]. It can be attributed to the surface/volume energy ratio, which is presumed to vary proportionally to the ratio of the number of surface atoms to the total number of atoms (e.g., for first-order melting transitions in spherical particles as reviewed by Calvo [8] and by Wilde [9]). Recent theoretical and experimental studies further demonstrate that particle size, morphology, composition, and surface energetics affect the phase stability and atomic ordering of metallic alloy nanoparticles, including their order–disorder transformations [10,11].

The present study is aimed at elucidating the critical-temperature shift size dependence by considering the heterogeneity of the nanoalloy surface consisting of atomic sites with different coordinations in cuboctahedrons and truncated octahedrons. As emphasized below, the edge and vertex site fractions do not scale inversely with the NP size, especially for relatively small NPs (belonging to the so called “non-scalable regime” [8]). Hence, this study introduces a revised model based on the concept of site-specific contributions to the $T_C^{nano}$ shift (SSCS) for handling FCEM calculated deviations from the regular inverse FSS relationship. While most computational alloy NP research focused on properties, such as the compositional structure of small and medium-sized particles, typically consisting of tens to thousands of atoms [12,13], the newly developed approach covers the relatively unexplored scaling behavior in similar NP sizes. It belongs to the class of predictive atomistic thermodynamic studies without trying to reproduce specific experimental measurements.

2. Methodology

The FCEM [3,14] was originally derived in the canonical ensemble for modeling elemental segregation in near-surface layers of semi-infinite alloys [3]. Later it was applied to systems consisting of a large number of uniform NPs that can exchange constituent atoms without changing size or shape, allowing NP site concentrations to be treated as quasi-continuous. The FCEM has been validated by Monte Carlo simulations [15], and shows fair agreement with density functional calculations [16].

For an $A-B$ alloy, the FCEM expression is obtained by expanding the free energy in powers of constituent concentrations [17],

$$F=kT\sum _p{N}_p\left(c_p^A\mathit{ln}{c}_p^A+c_p^B\mathit{ln}{c}_p^B\right)+\sum _{p\le q}{N}_{pq}\left({\displaystyle \frac{1}{2}}{w}_{pq}^{A-A}\left(c_p^A+c_q^A\right)+{\displaystyle \frac{1}{2}}{w}_{pq}^{B-B}\left(c_p^B+c_q^B\right)\right)$$

$$-V\sum _{p\le q}{N}_{pq}\left(c_p^A{c}_q^B+c_p^B{c}_q^A\right)-kT\mathit{ln}\left(\mathit{cos}h\left({\displaystyle \frac{2V}{kT}}\right)\right)\sum _{p\le q}{N}_{pq}{c}_p^A{c}_p^B{c}_q^A{c}_q^B.$$

When applied to NPs with geometrically nonequivalent atomic groups, numerical minimization of $F$ (e.g., using MATLAB) yields equilibrium concentrations $c_p^A$ and $c_p^B$ for each group p. Geometric input parameters include the number of atoms in each group, $N_p$, and the number of nearest neighbor (NN) pairs, $N_{pq}$, between groups $p$ and $q$. The first sum represents the configurational entropy. The second sum involves homoatomic NN pair interactions, $w_{pq}^{A-A}$ and $w_{pq}^{B-B}$. The third term includes the relatively small heteroatomic effective pair interactions (EPIs), $V$, between constituents (EPI considered here to be independent of group numbers). The last term accounts for short-range order contributions.

Determination of $T_C^{nano}$: The general procedure follows our earlier work [18]. For each temperature, the free energy was numerically minimized with respect to the independent site-concentration variables, subject to the constraint of fixed overall composition. To reduce the possibility of retaining a metastable solution, calculations were performed using both ascending and descending temperature sequences. In each sequence, the converged equilibrium solution at the preceding temperature was used as the initial condition for the subsequent minimization. When multiple stationary solutions were obtained, the solution with the lowest free energy was identified as the equilibrium state. Constrained nonlinear minimization was performed using the MATLAB fmincon solver, with OptimalityTolerance set to ${10}^{-6}$. Temperature was initially scanned in increments of 10 K and subsequently in increments of 5 K in the vicinity of the separation transition. The value of $T_C^{nano}$ corresponds to the apex of the miscibility gap. Accordingly, its numerical uncertainty was conservatively estimated as ±5 K.

In the “bond proportion model” (BPM) [19], the vibrational entropy effect in substitutional binary bulk alloys is expressed by considering the total effective pair interaction [20,21],

$$V=V_{chem}+T{V}_{vib},$$

where $V_{chem}\equiv {\displaystyle \frac{1}{2}}\left(w_{bulk}^{Pd-Pd}+w_{bulk}^{Ir-Ir}-2w_{bulk}^{Pd-Ir}\right)$. $V_{chem}$ equals $-37 meV$ (indicating separation/demixing tendency) as obtained from DFT-computed mixing enthalpy [22]. $V_{vib}$, the vibrational EPI, is closely related to the vibrational entropy change and was found to be $V_{vib}=0.0053 meV/K$ by fitting the FCEM/BPM computed critical temperature $T_c$ to the experimental $T_c^{exp}$ [18]. Bulk elemental bond energies, derived by eliminating free-atom electronic-relaxation contributions from experimental cohesive energies [23], are $w_{bulk}^{Pd-Pd}=-867 meV$ and $w_{bulk}^{Ir-Ir}=-1375 meV$. The Coordination dependence of near-surface Bond-Energy Variations (CBEV) [17] was extracted from reported pure metal surface-energy anisotropy that was computed [24] by DFT. This two-layer model estimates intra-surface and inter-surface/subsurface pair-bond strengthening, based on elemental surface energies and cohesive energies.

Despite the limitations of FCEM as a mean-field theory, such as focusing on dominant phases while ignoring possible NP configurational fluctuations [25], it has the advantage of separating and elucidating different contributions to the thermal stability of nanophases, including vibrational entropy and short-range order contributions. Furthermore, to enhance computational efficiency, especially for larger NPs, geometrically equivalent sites are grouped together. According to the computations, the surface of the cuboctahedral NP contains nearly pure Pd in the temperature range of interest in this study, i.e., up to and slightly above $T_C^{nano}$. Therefore, the $O_h$ surface sites are grouped according to only four coordinations (vertices, edges, (100) and (111) faces). In the core, geometrically equivalent sites are grouped according to the coarse-grained layer model (CGLM) [26], reducing the number of concentration variables by grouping atomic sites within layers perpendicular to a chosen axis. Distinct concentration variables correspond to core sites in two locations: the inner core and the subsurface.

3. Results and Discussion

3.1. The FCEM-Computed $T_C^{nano}$ Data

The nanophase separation diagrams of Pd-Ir fcc cuboctahedrons, based on $T^{nano}$ values, were computed for the [111] separation axis by the FCEM/CBEV/BPM/CGLM combination. The complete dataset, denoted range I, comprises all 22 cuboctahedrons with $147\le N\le \mathrm{49,049}$, where $N$ is the total number of atoms in the nanoparticle. Range II comprises the three smallest nanoparticles, with $147\le N\le 561$, whereas range III comprises the remaining nanoparticles, with $923\le N\le \mathrm{49,049}$. The FCEM results for range I were previously reported in Ref. [18] and are reused here. The critical temperatures for the larger cuboctahedrons in range III were newly calculated in the present study, as was the SSCS analysis, including the determination of the coordination-specific coefficients and the comparison of scaling models over the combined range I. As shown in Figure 1, the gap in the mixed nanophase (“miscibility gap”) is lower as compared to the alloy bulk gap and moves to lower temperatures for smaller NP sizes. This destabilization of the core-separated quasi-Janus (QJ) configurations is attributed to the NP size reduction limiting the correlation length (and accompanied by a growing fraction of surface atoms). When the number of surface atoms becomes negligible compared to the total number of atoms in rather large NPs, $T_C^{nano}$ closely approaches the bulk critical temperature, $T_C$ (Figure 1).

3.2. Assessment and Application of the SSCS Model

Unlike relatively large nanoparticles, which exhibit finite-size scaling with a critical exponent of $\nu =1$ [4,28], the smaller nanoparticles considered in our previous study showed a stronger size dependence of $t_C$ ($\nu =0.44$ in the range 147–561 atoms) [18]. This discrepancy can be attributed to the limitations of the simple inverse relationship model $t_C\propto n^{-{\displaystyle \frac{1}{\nu }}}$, which is based on the surface-to-volume energy ratio and becomes less accurate for small NPs. Beyond face sites, significant fractions of edge and vertex sites must be considered (Figure 2a), contributing to the increased slope and deviations from linearity observed in Figure 2b. The fitted value of $\nu$ gradually increases and approaches 1 for larger NPs, e.g., $\nu \approx 0.75$ in the range of 2057–49,049 atoms.

Specifically, considering the fractions $f_s$ of the different surface-site types and their site energies $E_s$, the relative shift in the critical temperature can be expressed as

$$t_C=const.{\sum }_s{\displaystyle \frac{E_s}{E_b}}{f}_s,$$

(1)

where $E_b$ is the bulk-site energy. Accordingly, the SSCS model is introduced through the expression

$$t_C={\sum }_s{k}_s{f}_s.$$

(2)

Here, the scaling coefficients $k_s$ for sites $s$ obey the ratio ${\displaystyle \frac{k_s}{k_{s^{\prime }}}}={\displaystyle \frac{E_s}{E_{s^{\prime }}}}$, whereas the surface-site fractions depend on the NP structure.

Equation (2) represents a first-order additive approximation to the effect of the nanoparticle surface on the critical-temperature shift. In conventional finite-size descriptions, $t_C$ is commonly related to the surface-to-volume ratio, $S/V$, implicitly treating the contributions of individual surface sites as additive. The SSCS model refines this approximation by separating the surface into classes of sites with distinct local coordinations. Within each class, the contribution of an individual site is represented by the coefficient $k_s$, whereas the number of such sites enters through its fraction $f_s={\displaystyle \frac{N_s}{N}}$. Here, $N_s$ denotes the number of surface sites of type $s$. The total shift is consequently expressed as the sum of the contributions from all site classes. This linear form neglects higher-order coupling terms between different surface-site classes and is expected to be most appropriate when their effects on the transition-temperature shift can be treated as local and approximately independent.

The coefficients $k_s$ are assumed to be independent of nanoparticle size because they characterize the contribution associated with a given local coordination and atomic environment, whereas the size dependence is introduced explicitly through the site fractions $f_s$. The surface-site fractions depend on the nanoparticle geometry and on the number of nested atomic shells, $n$. For example, for a cuboctahedral nanoparticle consisting of $n$ nested shells, both $N$ and $N_s$ are explicit functions of $n$; the corresponding expressions, together with those for other nanoparticle geometries, are provided in Appendix A. For self-similar faceted nanoparticles, the numbers of face, edge, and vertex sites scale approximately as $n^2$, $n$, and $n^0$, respectively, while the total number of atoms scales as $n^3$. Their contributions therefore behave approximately as $n^{-1}$, $n^{-2}$, and $n^{-3}$, making the SSCS expression analogous to an inverse-size expansion. This approximation may become less accurate when local site energetics vary substantially with nanoparticle size, composition, shape, surface reconstruction, strain, or interactions between neighboring site classes. The fitted coefficients should therefore be regarded as system- and transition-specific effective parameters rather than universal constants.

In the case of $O_h$, having $s=\left(111\right)$, $\left(100\right)$, edge and vertex, with coordinations 9,8,7,5, respectively, only three of the four fractions are independent since $f_{\left(111\right)}={\displaystyle \frac{2}{3}}{f}_{\left(100\right)}-{\displaystyle \frac{1}{6}}{f}_{edge}$ (according to

Table A1. Total number of atoms $\left(N\right)$ and the numbers at different surface sites $\left(N_s\right)$ in $O_h$ and TO nanoparticles in terms of the number of nested shells $\left(n\right)$ [36].

Shape	${\mathit{O}}_{\mathit{h}}$	TO
$N$	${\displaystyle \frac{10}{3}}{n}^3+5n^2+{\displaystyle \frac{11}{3}}n+1$	$16n^3+15n^2+6n+1$
$N_{\left(111\right)}$	$4n^2-12n+8$	$24n^2-24n+8$
$N_{\left(100\right)}$	$6n^2-12n+6$	$6n^2-12n+6$
$N_{edge}$	$24n-24$	$36n-36$
$N_{vert}$	$12$	$24$

). Thus, the SSCS effect can be expressed by

$$t_C=k_{faces}^{\prime }{f}_{\left(100\right)}+k_{edge}^{\prime }{f}_{edge}+k_{vert}^{\prime }{f}_{vert},$$

(3)

where the $k_s^{\prime }$s, are linear combinations of the $k_s$s (given in Appendix A), and the first term represents the overall contribution of the two face types. The three coefficients fitted to the FCEM/CBEV/BPM/CGLM data for cuboctahedrons in range I are given in

Table 1. Fitted SSCS coefficients for $O_h$.

Coefficients	Fitting Range I (147–49,049 Atoms)	Fitting Range II (147–561 Atoms)	Fitting Range III (923–49,049 Atoms)
$k_{faces}^{\prime }$	0.25	0.27	0.24
$k_{edge}^{\prime }$	0.43	0.40	0.47
$k_{vert}^{\prime }$	4.15	4.25	3.86
RMSD ($T_C^{nano}$), $K$	0.74	1.35	4.26

Data provenance: The three FCEM critical temperatures in range II were previously reported in Ref. [18]. The range III FCEM data, the SSCS fits, and all coefficients reported in this table were obtained in the present study. RMSD was assessed over the complete range I, including fitted and out-of-fit-range data.

, together with coefficients obtained from fits over the two narrower ranges II and III. The full-range fit yields the smallest root mean square deviation (RMSD). The three FCEM-computed transition temperatures for the small nanoparticles in range II are sufficient to determine the three SSCS coefficients; however, agreement with these calibration points does not constitute independent validation. To assess the predictive capability of this parameterization, the resulting coefficients were extrapolated, without refitting, to the larger nanoparticles in range III, whose critical temperatures were not used in determining the coefficients. As shown in Figure 3, the extrapolated SSCS predictions agree closely with the FCEM/CBEV/BPM/CGLM results over range III and reproduce the change in slope across the full range of nanoparticle sizes.

A complementary assessment was performed by fitting the coefficients to the larger nanoparticles in range III and extrapolating the resulting parameterization to the small nanoparticles in range II (Figure 4). The coefficients obtained from the full-range and restricted-range fits are similar, and the extrapolated predictions agree well with the FCEM/CBEV/BPM/CGLM results outside the respective fitting ranges. The RMSD values reported in Table 1 are evaluated over the complete range I and therefore represent full-range assessments that include both the fitted subset and the predictions outside the fitting range. Because all the critical temperatures were generated within the same FCEM framework, these extrapolations should not be regarded as validation against a fully independent dataset. Nevertheless, the agreement outside the respective fitting ranges indicates that the SSCS model captures the underlying size dependence rather than merely reproducing the data used to determine its coefficients.

These results contrast with extrapolations done by employing the logarithmic scaling correction [29,30], namely $n\propto t_C^{-\nu }{\left|ln{t}_C\right|}^{-{\nu }^{\prime }}$. In particular, the extrapolated values based on fitting to the FCEM/CBEV/BPM/CGLM data in both ranges II and III provide unsatisfactory extrapolations. Namely, fitting to range II gives an erroneous slope in range III (Figure 3), and fitting to range III significantly deviates from the data for the three smallest nanoparticles in range II (Figure 4).

To determine whether the SSCS model is applicable to nanoparticles with different faceted shapes, in the present study FCEM/CBEV/BPM/CGLM computed $T_C^{nano}$ for 201- and 586-atom truncated octahedrons (TOs), which were combined with the $O_h\left(\mathrm{147,309,561}\right)$ previous data [18] in order to fit all $k_s$ coefficients (Equation (2)), including the common $k_{\left(111\right)}$, $k_{\left(100\right)}$, $k_{edge}$ and the individual $k_{vert,O_h}$ for 5-coordinated vertexes and $k_{vert,TO}$ for 6-coordinated vertexes. Solution of the corresponding 5 linear SSCS equations provides the coefficients shown in

Table 2. SSCS coefficients $k_s$ fitted to $O_h\left(\mathrm{147,309,561}\right)$ and $TO\left(\mathrm{201,586}\right)$.

# of Broken Bonds	Surface Sites	Coefficient
3	(111)	0.15
4	(100)	0.17
5	Edge	0.42
6	TO vertex	2.54
7	$O_h$ vertex	4.25

. The obtained value of $O_h$ vertex coefficient is the same as that given in Table 1 for range II. Furthermore, $k_{vert,TO}$ occurs closely to the arithmetic mean of $k_{edge}$ and $k_{vert,O_h}$, whereas the ratio ${\displaystyle \frac{k_{\left(111\right)}}{k_{\left(100\right)}}}=0.90$ (Table 2) accurately agrees with the surface-site energy ratio 0.89 simply determined from the surface energy ratio $0.77$ measured by STM for truncated octahedron Pd nanoparticles equilibrated at 723 K [31]. This ${\displaystyle \frac{k_{\left(111\right)}}{k_{\left(100\right)}}}$ ratio is consistent with the condition of shape-independent scaling in rather large nanoparticles (discussed below). As expected, the fitted coefficients $k_s$, which characterize the contributions of the individual site types to $t_C$, increase with the number of missing bonds (Table 2). However, the relation is not linear, unlike in the simple case of surface “bond breaking” energetics [32].

The fractions of the critical-temperature shift, $F_s={\displaystyle \frac{k_s{f}_s}{t_C}}$, originating respectively from faces, edges, and vertexes in cuboctahedrons, are displayed in Figure 5a, and the site fractions at the surface alone are shown in Figure 5b. Because of the decrease in face site fraction in smaller NPs, $F_s$ decreases for faces and increases for vertexes (their number is 12 in all $O_h$). $F_{edge}$ vs. $n$ reaches a maximum since in smaller nanoparticles the vertexes dominate, whereas in larger ones the face fraction dominates (Figure 5).

The feasibility of more transparent fitting models based on simple inverse powers of $n$, which approximately reflect the size dependence of the fractions of differently coordinated surface sites, was examined using the alternative expressions listed in

Table 3. Comparison of six models used to represent the FCEM-computed $t_C$ values over the complete range I. AICc and ΔAICc account for the different numbers of adjustable coefficients.

Model	${\mathit{t}}_{\mathit{C}}$ Expression	Number of Adjustable Coefficients, $\mathit{p}$	Range I Parameters	RMSD of Predicted ${\mathit{T}}_{\mathit{C}}^{\mathit{n}\mathit{a}\mathit{n}\mathit{o}}$ (K)	AICc	ΔAICc
1	$a{n}^{-1}$	1	$a=1.11$	$80$	197	204
2	$a{n}^{-2}$	1	$a=4.91$	$30$	154	161
3	$a{n}^{-3}$	1	$a=15.9$	$84$	200	207
4	$a{n}^{-1}+b{n}^{-2}$	2	$a=0.32$ $b=3.69$	$5.2$	80	87
5	$a{n}^{-1}+b{n}^{-2}+c{n}^{-3}$	3	$a=0.41$ $b=2.50$ $c=2.86$	$0.68$	−7	0
6	SSCS-based $\sum _s{k}_s^{\prime }{f}_s$	3	$k_s^{\prime }$, Table 1	$0.74$	−3	4

. Because these models contain different numbers of adjustable coefficients, comparison based solely on the root mean square deviation (RMSD) may favor models with more fitting parameters. Therefore, the models fitted over the complete range I were additionally compared using the corrected Akaike information criterion ($AI{C}_c$) [33,34]:

$$AI{C}_c=m\mathrm{l}\mathrm{n}\left({\displaystyle \frac{RSS}{m}}\right)+2K+{\displaystyle \frac{2K\left(K+1\right)}{m-K-1}},$$

(4)

where $m=22$ is the number of FCEM-computed critical temperatures in range I, RSS is the residual sum of squares, and $K=p+1$, where $p$ is the number of adjustable coefficients in the model. The relative support for each model was assessed using

$$\Delta AI{C}_{c_i}=AI{C}_{c_i}-AI{C}_{min},$$

(5)

so that the model with the lowest $AI{C}_c$ has $\Delta AI{C}_{c_i}=0$. AICc was evaluated only for the fits over the complete range I. The three data points in range II are insufficient for a meaningful AICc comparison of models containing up to three adjustable coefficients; in particular, the three-parameter Models 5 and 6 are exactly determined in that range. The range II parameterizations were therefore used only for extrapolation assessment and not for $AI{C}_c$-based model selection.

Among the one-parameter expressions, Model 2, based on $1/n^2$, provides the smallest RMSD. This behavior is consistent with the previously reported effective power-law dependence for the relatively small nanoparticles [18]. For larger nanoparticles, however, the leading $n^{-1}$ contribution becomes increasingly important because the fraction of face sites dominates over those of edge and vertex sites.

Combining the $n^{-1}$ and $n^{-2}$ terms in Model 4 substantially reduces the RMSD relative to the one-parameter models. Adding the $n^{-3}$ term in Model 5 further reduces the RMSD to below 1 K. These three inverse-size terms approximately correspond to the fractions of face, edge, and vertex sites, respectively: their numbers scale approximately as $n^2$, $n$, and $n^0$, whereas the total number of atoms scales as $n^3$.

The parameter-count-adjusted comparison supports Models 5 and 6 over the simpler inverse-power expressions. Model 5 gives the lowest AICc and is therefore statistically preferred for the fit over the complete range I, whereas the physically based SSCS Model 6 gives the second-lowest AICc. Both models contain three adjustable coefficients and yield RMSD values below 1 K when fitted over range I. Model 5 is a purely empirical inverse-size expansion, whereas the SSCS model assigns direct physical meanings to its terms through the fractions of face, edge, and vertex sites.

The restricted-range assessment provides a complementary result. When the coefficients are determined using only the three small-nanoparticle data points in range II and the resulting parameterizations are assessed over the complete range I, Model 6 gives an RMSD of 1.4 K, compared with 4.0 K for Model 5. Because these full-range RMSD values include both the fitted range II points and the extrapolated range III results, they do not represent strictly independent test-set errors. Nevertheless, the smaller RMSD of Model 6 indicates that its range II parameterization extrapolates more accurately across the full particle-size range.

Thus, Model 5 provides the statistically best fit when all range I data are used for parameter determination, whereas the SSCS model combines nearly comparable full-range accuracy with better restricted-range extrapolation and a direct physical interpretation. The similar performance of the two models is consistent with their common approximate $n^{-1}$, $n^{-2}$, and $n^{-3}$ structure; however, the SSCS coefficients arise from the explicit geometric relations between the different surface-site fractions rather than from an unconstrained inverse-size polynomial.

A similar inverse-quadratic dependence of binding energies and melting temperatures was reported from DFT calculations for small nanoparticles of the semiconductors C, Si, and Ge and the metals Sn and Pb, and it was attributed to variations in surface-atom bonding [35]. In the present Pd–Ir analysis, the approximate $n^{-2}$ contribution is especially important for relatively small nanoparticles, although here it is associated primarily with the appreciable fraction of edge sites. For larger nanoparticles, the $n^{-1}$ face contribution becomes dominant, recovering the conventional inverse-size behavior.

Truncated octahedrons: The calculated scaling coefficients (Table 2) were utilized to compute critical temperatures for truncated octahedrons beyond 201- and 586-atom NPs by extrapolating the SSCS Equation (2). As illustrated in Figure 6, there is very good agreement with the FCEM/CBEV/BPM/CGLM results, indicating general applicability of the scaling coefficients $k_s$. For general applicability to different nanoparticle shapes, it is convenient to replace the shape-specific size-characteristic $n$ in the SSCS expression by the “effective linear size” (ELS), defined for any shape, e.g., as $L\equiv \sqrt[3]{N}$. When the data is presented against $n$, the scaling plots are mutually displaced, as indicated in the inset of Figure 7. When the FCEM/CBEV/BPM/CGLM computed $T_C^{nano}$ data for cuboctahedrons and truncated octahedrons are plotted against $L$ (Figure 7), despite different site fractions the resulting scaling plots exhibit a remarkable level of overlap. This result, together with the previous findings, suggests that the SSCS model may be applicable more broadly, beyond the two faceted fcc nanoparticle geometries considered here. The former result can be tentatively attributed to a kind of “geometric compensation”, namely, while truncated octahedrons have twice as many vertexes (24) as cuboctahedrons (12), the vertex coordination of the TO (6) is larger than that of the $O_h$ vertex (5). This larger coordination that leads to the TO vertex coefficient being approximately half the magnitude of the $O_h$ vertex coefficient, ${\displaystyle \frac{k_{vert,TO}}{k_{vert,O_h}}}=0.60$, is compensated by the higher number of vertex sites.

According to Table A1, in the case of very large $O_h$s, $n\approx \sqrt[3]{{\displaystyle \frac{3}{10}}}L$, and for TOs, $n\approx \sqrt[3]{{\displaystyle \frac{1}{16}}}L$. Using the Appendix A formulas (A7A8) for such $n$, the following relationships are obtained,

$$t_C^{\left(O_h\right)}\approx \sqrt[3]{{\displaystyle \frac{18}{25}}}\left(2k_{\left(111\right)}+3k_{\left(100\right)}\right){\displaystyle \frac{1}{L}}.$$

(6)

$$t_C^{\left(TO\right)}\approx \sqrt[3]{{\displaystyle \frac{27}{32}}}\left(4k_{\left(111\right)}+k_{\left(100\right)}\right){\displaystyle \frac{1}{L}}.$$

(7)

Irrespective of the dissimilar site fractions present in the octahedral and truncated octahedral nanoparticles, these shifts coincide, namely, $t_C^{\left(O_h\right)}=t_C^{\left(TO\right)}={\displaystyle \frac{0.73}{L}}$, when ${\displaystyle \frac{k_{\left(111\right)}}{k_{\left(100\right)}}}=0.88$. As noted above, this ratio is very close to the fitted coefficient ratio (Table 2), as well as to the reported STM data [31], indicating that in the case of faceted spherical $O_h$ and TO large nanoparticles (beyond the size range of the $T_C^{nano}$ input data), the role of shape in the scaling behavior is expected to be negligible too.

Although a similar behavior is obtained for the two faceted fcc nanoparticle geometries considered here, this agreement alone is insufficient to establish universality. The present conclusions are therefore restricted to the investigated alloy system, and geometries. Calculations for additional alloy systems, crystal structures, and morphologies are required to determine the broader generality of the observed trends.

4. Conclusions

The introduced model elucidates for relatively small NPs the deviation of the shift in critical transition temperatures $\left(T_C^{nano}\right)$ from the simple inverse relationship with the size claimed before. For this goal, nanophase separation diagrams (miscibility gaps) of $Pd-Ir$ fcc cuboctahedrons $O_h\left(923-\mathrm{49,049}\right)$ were first computed for the [111] separation axis in the framework of the FCEM/CBEV/BPM/CGLM approach. The miscibility gaps, including the critical points, namely the temperatures of transitions between quasi-Janus and mixed configurations, are gradually lower for smaller sizes compared to the alloy bulk gap. The proposed concept regarding site-specific contributions to the $T_C^{nano}$ shift (SSCS), considering all surface-site types, namely, (111) and (100) faces, edges, and vertices, is a generalization of the more common approach considering the ratio of the overall surface to total nanoparticle energy as the origin of the critical-temperature shift. The present linear SSCS expression is a first-order approximation in which the effective contribution of each coordination class is assumed to be size-independent and additive; extensions may be required when surface reconstruction, strain, compositional changes, or coupling between different site classes become significant. The SSCS expression is supported by accurate agreement of $T_C^{nano}$ with the FCEM/CBEV/BPM/CGLM data for both cuboctahedrons and truncated octahedrons, indicating a certain degree of model transferability between the two nanoparticle shapes. For very large nanoparticles, the impact of edges and vertices on the shift in critical temperature is negligible, and the inverse-size dependence should hold irrespective of shape. The proposed model and the corresponding expression advance our understanding of nanoscaling in relatively small crystalline nanoparticles and provide a promising framework for investigating other nanoparticle shapes and phase transitions. One challenge to look at is the extension of the current approach to melting transitions in alloy nanoparticles.