Strain-engineering of bimetallic nanomaterials is an important design strategy for developing new catalysts. Herein, we introduce an approach for including strain effects into a recently introduced, density functional theory (DFT)-based alloy stability model. The model predicts adsorption site stabilities in nanoparticles and connects these site stabilities with catalytic reactivity and selectivity. Strain-based dependencies will increase the model’s accuracy for nanoparticles affected by finite-size effects. In addition to the stability of small nanoparticles, strain also influences the heat of adsorption of epitaxially grown metal-on-metal adlayers. In this respect, we successfully benchmark the strain-including alloy stability model with previous experimentally determined trends in the heats of adsorption of Au and Cu adlayers on Pt (111). For these systems, our model predicts stronger bimetallic interactions in the first monolayer than monometallic interactions in the second monolayer. We explicitly quantify the interplay between destabilizing strain effects and the energy gained by forming new metal–metal bonds. While tensile strain in the first Cu monolayer significantly destabilizes the adsorption strength, compressive strain in the first Au monolayer has a minimal impact on the heat of adsorption. Hence, this study introduces and, by comparison with previous experiments, validates an efficient DFT-based approach for strain-engineering the stability, and, in turn, the catalytic performance, of active sites in bimetallic alloys with atomic level resolution.

Rational design of electro- and heterogeneous catalysts has greatly benefited from Density Functional Theory (DFT)-derived screening paradigms.1–4 These paradigms are originally based on scaling relations between reaction intermediates and catalytic descriptors like the adsorption energies of CO*, CH3*, or OH*.5 By integrating these scaling relations into microkinetic models, catalytic figures of merit such as rate or selectivity can be predicted based on rapid and inexpensive adsorption energy calculations of these catalytic descriptors.6–9 The resulting volcano plots have been effectively used to discover new catalyst candidates.6,10–14

Recent efforts to move away from idealized, single-crystalline surfaces toward modeling realistic and dynamic catalyst structures at the nanoscale have focused on integrating the effects of variations in the local coordination environment15–20 and composition21–27 around the adsorption site into scaling relations. As a consequence, descriptors reflecting the inherent stability of metal atoms have emerged.28–35 Examples for such descriptors are the thermodynamic stability of adsorption sites,28–31 surface energies,32 cohesive energies of nanoparticles,33 and the chemical potential of nanoparticles.34,35 These descriptors connect the adsorption site stability with catalytic reactivity and selectivity.29,31

Given the vast materials space of multimetallic alloys, several physics-based28,30,32,33,36 and data-driven37–42 approaches have been constructed to efficiently predict these stability descriptors in multimetallic nanoparticles of diverse sizes, shapes, and compositions. To reduce computational cost, many of these approaches are first parameterized on extended surfaces and subsequently employed to predict the catalytic stability metrics of nanoparticles. Challenges associated with this strategy of building descriptors for the stability of nanoparticles from extended surface calculations arise from quantum- and finite-size effects that are prevalent in small nanoparticles. Jinnouchi et al.38,39 discussed these challenges in their Bayesian regression model for predicting the stability of RhAu nanoparticles. They showed that the Bayesian regression parameters fitted to single crystal surfaces display poor transferability to nanoparticles below 1.5 nm. The model transferability to RhAu nanoparticles smaller than 1.5 nm greatly improves, however, by explicitly including the energies of small nanoclusters within the training set.

These observations illustrate the necessity of precisely delineating nanoparticle size ranges within which models parameterized on extended surfaces can be used to accurately forecast descriptors for the stability of nanoparticles. It is further important to find ways of bridging the gap between models parameterized on extended surfaces and their applicability to nanoparticles. One way of increasing the transferability between extended surfaces and nanoparticles in the finite-size regime is through the inclusion of atomic distance-dependent functional forms. These functional forms can explicitly account for energy changes caused by compression of interatomic distances in nanoparticles in the finite-size regime and can potentially increase the accuracy of the models. In addition to elucidating finite-size effects in nanoparticles, these distant-dependent functional forms can also be applied to understand the impact of external strain on metal–metal energetics. Strain effects are, for instance, naturally encountered on epitaxial metal-on-metal films, wherein the adlayer atom-atom distances are constrained to match the lattice spacing of the underlying substrate, or in bimetallic core-shell nanoparticles.43–46 In addition, external strain is frequently exploited as a design lever for improving the performance of (electro)catalysts.47–49 Hence, functional forms that can account for the effects of external strain with a site-by-site resolution will not only improve the predictive power for descriptors of stability but can also be employed to exquisitely tailor catalytic performance.

In addition to predicting the energetics of metal–metal interactions using computational models, these interactions can also be experimentally measured. Such experiments are essential to benchmarking computational models. Commonly used techniques to derive metal–metal interactions include Temperature-Programmed Desorption (TPD)50 and Single Crystal Adsorption Calorimetry (SCAC).51,52 While TPD works well for bimetallic systems that are immiscible, it cannot be applied to systems forming bulk alloys: Elements intermix before reaching the desorption temperature. SCAC, in contrast, can be performed below bulk-alloy formation temperatures. For example, James et al.53 and Feeley et al.54 successfully used SCAC to measure the adsorption energies of Cu and Au atoms on single-crystalline Pt (111) surfaces. These studies were among the first to report metal adsorption for bulk-alloy forming bimetallic systems. A detailed discussion of these previous experimental studies will be provided later on in the manuscript while comparing the experiments with model predictions. Briefly, in both systems, strain effects appear within the first monolayer because both Cu and Au islands grow pseudo-morphically with interatomic distances matching those of the underlying Pt lattice. While James et al.53 and Feeley et al.54 discussed the interplay between the formation of new metal–metal bonds through Cu and Au adsorption and the effects of strain on the Cu and Au islands, the magnitude of each of these counteracting effects still remains elusive.

Within this work, we use and extend the coordination-based alloy stability model introduced by Roling et al.28,30 to predict and understand metal–metal interactions in nanoparticles and strained metal adlayers in the fcc crystal structure. In the following, we briefly summarize the main elements of the original model.

For monometallic systems, the energy of a metal atom Z having a coordination number n is shown in Eq. (1). Equation (2) represents the energies of a metal atom Z1 having a coordination number of n in a bimetallic system Z1Z2,

EZn=α13Z+i=4nαiZ,
(1)
EZ1Z1Z2n=α13Z1Z1Z2+i=4nαiZ1Z1Z2.
(2)

These energies are partitioned in terms of the bond-associated parameters αnZ and αnZ1Z1Z2, which reflect the energy gained through each new bond formed for monometallic and bimetallic systems, respectively. In both models, the bond-associated parameters for the first three bonds are combined into one parameter, α13Z and α13Z1Z1Z2, while nine additional parameters are used to represent the energy gained through the formation of the fourth to the twelfth bond. Within this formalism, 10 parameters are required to predict the energy of an atom in a monometallic system (Z), while 20 parameters (10 each for αnZ1Z1Z2 and αnZ2Z1Z2) are needed to calculate the energy of an atom in a bimetallic system (Z1Z2). This deconstruction of energies of a metallic atom explicitly includes the first-coordination shell, while it treats compositional variations through a mean field approximation. Thus, αnZ1Z1Z2 parameters for bimetallic systems (Z1Z2) are independent of the ratio of Z1 and Z2 in the system. The bond-associated parameters have been fitted to a limited set of DFT calculations on extended surfaces of fcc crystals for 7 different monometallic and 21 different binary combinations of Ag, Au, Cu, Ir, Pd, Pt, and Rh. These bond-associated parameters successfully predict cohesive energies, adsorption energies of metal atoms, and relative energies for interatomic swaps of 147 atom and truncated-309 atom nanoparticles having diverse shapes and compositions (errors generally within 0.1–0.2 eV). The assumptions involved in formulating the model, like expressing energies of metal atoms in terms of the first coordination shell alone and treating compositional variations through a mean field approximation, are pertinent because of the well-known screening effects of d-electrons in late transition metals.55–57 These screening effects will attenuate any perturbations in structure and composition to the energy of a metal atom beyond its first coordination shell.

The scope of the alloy stability model can be extended by explicitly integrating the effects of strain on metal–metal interaction energies with site-by-site resolution. Quantitative comparisons between model predictions and experimental measurements of metal–metal interaction energies will help benchmark this model extension. In addition, by further reducing the number of calculations needed to obtain the bond-associated parameters, we will be able to more quickly assess the site stabilities, and hence catalytic activities, of new material systems.

In this paper, we first verify the nanoparticle size ranges within which the coordination-based alloy stability model accurately determines the cohesive energies of nanoparticles. We demonstrate that the alloy stability model yields a correlation between cohesive energies and nanoparticle size that resembles the Gibbs–Thomson relation.58 We then introduce functional forms for incorporating the effects of applied strain on αnZ. We use these functional forms for strain together with the alloy stability model to obtain atomic level insights regarding the experimentally determined interactions of Cu and Au atoms with Pt (111) surfaces.53,54 We present a comparison between experimentally determined and model-predicted as well as DFT-calculated monometallic and bimetallic adsorption energies. We further focus on quantifying the interplay between the energy gained through the formation of new metal–metal bonds, and the energy lost as a result of strain induced in the first layer of Cu and Au atoms on Pt (111). Finally, we discuss the importance of adsorption energies measured using SCAC in calibrating and benchmarking computational models that predict unified stability and reactivity trends for bimetallic alloys. Paper II59 focusses on strategies to further reduce the computational cost of parameterizing the coordination-based alloy stability model. We achieve this reduction by using quadratic functional forms to describe the energies of metal atoms as a function of their coordination number and by establishing correlations between the model parameters and the bulk cohesive energies.

We performed first principles periodic DFT calculations using the Quantum ESPRESSO package60 within the Atomic Simulation Environment (ASE).61 We used the revised Perdew-Burke-Ernzerhof (RPBE) exchange–correlation functional,62 with core-states represented by ultrasoft Vanderbilt pseudopotentials,63 to compute self-consistent energies. We performed selected calculations using the PBE,64 BEEF-vdW,65 and PBEsol66 exchange–correlation functionals to facilitate a more rigorous comparison of DFT-derived metal adsorption energies with experiments. We used the Monkhorst–Pack method67 to generate k-point grids on which the Kohn–Sham equations were self-consistently solved in reciprocal space. We used kinetic energy cutoffs for plane waves and charge densities of 500 eV and 5000 eV, respectively. We performed spin-paired calculations of bulk structures, surface slabs, and 147 and 309 atom cuboctahedral (CUBO) nanoparticles, while we calculated spin-polarized total energies for gas phase species, and 13 and 55 atom CUBO nanoparticles. We used the Fermi-Dirac distribution to smear electronic states of bulk structures, nanoparticles, and surface slabs by 0.1 to converge the Kohn–Sham equations. We separated periodic images of surface slabs by at least 12 Å of vacuum. To cancel spurious interactions between periodic images of surface slabs, we applied a dipole correction68 perpendicular to the surface. We separated periodic images of CUBO nanoparticles along the x-, y-, and z-directions by at least 10 Å of vacuum. We converged the total energies to at least 10−5 eV and optimized geometries until the forces were below 0.02 eV/Å. We used the equation of state method on a 12 × 12 × 12 k-point grid to obtain the lattice constants of fcc bulk structures. The lattice constants calculated with RPBE are (all values in Å, experimental values in parentheses69): Au: 4.20 (4.08), Cu: 3.68 (3.61), and Pt: 3.99 (3.92). The calculated lattice constants of Pt with PBE, BEEF-vdW, and PBEsol are 3.97 Å, 3.99 Å, and 3.92 Å, respectively. We calculated the total energies of metal atoms in the gas phase in a 21 Å × 22 Å × 23 Å unit cell with a 1 × 1 × 1 k-point density. We use these gas phase energies as a reference for adsorption energies of metal atoms. We state further computational details such as unit cell sizes, constraints, and k-point densities in the respective in the sections titled Calculation details for Cu and Au island and epitaxial films on Pt (111) as well as Calculation details for diffusion barrier determinations.

We conducted DFT calculations on strained (111), (100), (211), and bulk systems of Au, Cu, and Pt using the same set of differently coordinated structures, as described in detail by Roling et al.28 All calculations on strained systems were performed with atoms constrained to their respectively strained bulk positions, so as to completely decouple the effects of external strain from local geometric relaxations. These calculations form the training set for determining the bond-associated parameters as a function of external strain, αnZ(x). An isotropic strain of ±5% was applied to the (111), (100), (211), and bulk supercells. For Cu, we increased the range of applied strain to ±8% so as to match interatomic distances observed in epitaxial Cu islands on Pt (111) during SCAC studies.

We determined cohesive energies of Pt and Au nanoparticles having 13, 55, 147, and 309 atoms. We performed all calculations on a 1 × 1 × 1 k-point grid. We obtained the total energies for both fully relaxed nanoparticles and for nanoparticles having their atomic coordinates constrained to the bulk positions. In our comparisons between model-predicted and DFT-calculated adsorption energies, we used the DFT-calculated adsorption energies of five-, seven-, eight-, and ninefold coordinated metal atoms in the 147 atom CUBO nanoparticles from Roling et al.28 

We investigated the changes in interatomic distances in Cu and Au islands on Pt (111) as a function of their island size. These calculations involved adsorbing islands consisting of 3–25 atoms on a four-layered, 6 × 6 Pt (111) supercell. The bottom two layers were fixed at their bulk positions with all other atoms allowed to relax. We performed DFT calculations on a 2 × 2 × 1 k-point grid. All atoms of the islands were adsorbed on Pt fcc hollow sites. These islands had shapes ranging from triangular and hexagonal to diamond-like. We also calculated the interaction energies between two islands using DFT. These island–island interactions were computed for two different geometries, namely, triangular islands composed of six atoms and hexagonal islands consisting of seven atoms. The positions of the two islands on the Pt (111) substrate were selected such that they were separated by at least two interatomic distances. We provide images of the structures of these islands in Figs. S8 and S9.

We determined the adsorption energies of complete Cu and Au films on a four-layered 3 × 3 Pt (111) supercell. We performed DFT calculations on a 4 × 4 × 1 k-point grid. The bottom two Pt layers were constrained at their bulk positions with all other atoms allowed to relax. We then calculated the differential adsorption energies of the first and the second monolayer of Cu and Au films on Pt (111) with RPBE. To facilitate a more rigorous comparison with experimentally measured heats of adsorption, we computed the differential adsorption energy for the first layer of Cu and Au on Pt (111) also with PBE,64 BEEF-vdW,65 and PBEsol.66 Given the periodic boundary conditions in plane-wave DFT, interatomic distances of Cu and Au atoms were lattice matched to the Pt (111) substrate, resulting in a tensile strain of 8% for Cu and a compressive strain of 4% for Au. Adsorption energies are reported per atom of the adsorbate layer and are referenced to the corresponding atom in the gas phase. Adsorption energies for Cu and Au films were also determined using the bond-associated parameters of the alloy stability model. Energies of metal atoms of the first Pt layer and the first monolayer of Cu and Au were determined using the bimetallic αnZ1Z1Z2 parameters, while energies of metal atoms in the second monolayer of Cu and Au were computed using the monometallic αnZ parameters.

We calculated the diffusion barriers of Au and Cu on Pt (111) using the machine-learning enhanced nudged elastic band (ML-NEB) method.70 Briefly, the ML-NEB algorithm relies on the climbing image NEB (CI-NEB) method71 (as implemented in ASE61) as a surrogate. A predictive Gaussian process model is built with the DFT-converged initial and final images of the reaction trajectory. The predictive model is then optimized using the CI-NEB. The algorithm chooses the image along the predicted trajectory with the highest uncertainty, calculates it with DFT, and adds it to the training set. The training and testing process is done iteratively by evaluating the images and optimizing the NEBs on the predicted model in a single-image evaluation fashion. This iterative process is performed until the uncertainty of each image along the minimum energy pathway lies below our convergence criteria, i.e., 0.05 eV. After that, the algorithm converges the saddle point at a DFT level, in our case the convergence criteria is set to 0.05 eV/Å. The saddle point is always included in the training set. We considered three different Au and Cu coverages on six-layered 2 × 2 slabs of Pt (111): (1) one adatom (25% coverage), (2) two adatoms (50% coverage), and (3) three adatoms (75% coverage). In (1), the threefold coordinated adatom diffused from one fcc hollow site to a neighboring hcp hollow site, remaining threefold coordinated in its final state. In (2), the fivefold coordinated adatom moved from an fcc hollow site to another fcc hollow site, thereby remaining fivefold coordinated in the final state. Finally, in (3), the sevenfold coordinated adatom moved from an fcc hollow site to another fcc hollow site, remaining sevenfold coordinated in its final state. In the ML-NEB calculations, we used 19 intermediate images between geometry optimized initial and final states. DFT calculations were performed with an 8 × 8 × 1 k-point grid. We obtained initial guesses for images along the reaction coordinate using the idpp interpolation.72 Structures of the initial, transition, and final states are shown in Fig. S6.

Herein, we first investigate the accuracy of the alloy stability model as a function of the nanoparticle size. Second, we derive αnZ(x) parameters that are explicit functions of strain acting on metal–metal bonds to account for bond-distance contractions in nanoparticles and adlayer strain in epitaxial metal-on-metal films. Finally, we compare predictions of metal–metal energetics obtained using the alloy stability model including these strain corrections to calorimetric experiments on bimetallic Cu/Pt and Au/Pt systems.53,54 We quantitatively interpret the hypotheses postulated by the experiments that rationalize changes in metal–metal interaction energies as a function of Cu and Au coverages on Pt (111) substrates. We further illustrate how heats of adsorption of metal atoms depend on an intriguing interplay between destabilizing effects of strain and the stabilizing effects of forming new metal–metal bonds.

Roling et al.28,30 observed that the mean average errors (MAEs) of atom stabilities in 147 atom cuboctahedral (CUBO) nanoparticles predicted by the alloy-stability model are generally about an order of magnitude higher than those in extended surfaces (0.2 eV vs 0.02 eV). We hypothesize that these larger errors are caused by residual quantum- and finite-size effects that may still be prevalent in 147 atom nanoparticles. We probe the electronic and geometric origin of these errors by comparing DFT-derived and model-predicted cohesive energies across a library of Pt and Au CUBO nanoparticles with 13, 55, 147, and 309 atoms for both relaxed nanoparticles and for nanoparticles with metal atoms constrained to their bulk positions. Model-predicted cohesive energies of 13, 55, 147, and 309 atom CUBO nanoparticles are calculated according to Eqs. (3)(6). For a given nanoparticle, we count the number of 5, 7, 8, 9, and 12 coordinated atoms. 12 coordinated atoms in the second layer are corrected using α122ndZ determined by Roling et al.28 We utilize both the constrained and relaxed αnZ parameters that were derived by Roling et al.28 Cohesive energies of larger nanoparticles (up to 17 nm) shown in Fig. 2 are calculated accordingly:

CE13Z=11312EZ5+EZ12+α122ndZ,
(3)
CE55Z=15512EZ5+24EZ7+6EZ8+13EZ12+12α122ndZ,
(4)
CE147Z=114712EZ5+48EZ7+24EZ8+8EZ9+55EZ12+42α122ndZ,
(5)
CE309Z=130912EZ5+72EZ7+54EZ8+24EZ9+147EZ12+92α122ndZ.
(6)

In Fig. 1, we plot the MAEs for relaxed and constrained nanoparticles as a function of their size. The errors associated with constrained nanoparticles, especially the 13 and 55 atom structures, are chiefly connected to electronic effects like quantum-size effects,73,74 whereas the differences between errors observed for relaxed and constrained nanoparticles (black arrows in Fig. 1) are primarily associated with geometric relaxations. These geometric errors correspond to contributions arising from finite-size effects. Such effects occur because intermetallic distances in small nanoparticles are commonly compressed.73–75 Our alloy stability model attains the accuracy observed on extended surfaces for both relaxed and constrained 309 atom nanoparticles (∼2.3 nm in diameter). While cohesive energies of constrained 147 atom nanoparticles (∼1.7 nm in diameter) are well predicted, the relaxed nanoparticles show twice the error. This increased error is caused by finite-size effects due to compressions in metal–metal bond lengths as stated above. Clearly, the model loses its predictive power in the limit of 13 and 55 atoms. These two nanoparticle sizes also see an increased contribution of the error due to electronic effects as compared to 147 and 309 atom nanoparticles. We anticipate our model to break down below the 147-atom limit because the electronic states of metal atoms in 13 and 55 atom nanoparticles are extremely different from the electronic states of extended surfaces in our training set. Based on Fig. 1, we conclude that our model, which is fitted on simple surface slabs, is directly transferable to nanoparticles larger than ∼2 nm with high accuracy. In the section titled Introducing the effects of strain, we will introduce explicit strain-dependent functional forms for αnZ(x) parameters that can potentially mitigate errors associated with finite-size effects.

FIG. 1.

Mean average errors (MAEs) between model-predicted and DFT-derived cohesive energies of relaxed (orange bars) and constrained (blue bars) Pt and Au nanoparticles as a function of their size. Structures of CUBO nanoparticles ranging from 13 atoms (∼0.6 nm) to 309 atoms (∼2.3 nm) are depicted above. The MAEs indicate that our model is highly accurate for nanoparticles larger than 2 nm. The differences between relaxed and constrained nanoparticles (black arrows) reflect the contribution of finite-size effects to the error.

FIG. 1.

Mean average errors (MAEs) between model-predicted and DFT-derived cohesive energies of relaxed (orange bars) and constrained (blue bars) Pt and Au nanoparticles as a function of their size. Structures of CUBO nanoparticles ranging from 13 atoms (∼0.6 nm) to 309 atoms (∼2.3 nm) are depicted above. The MAEs indicate that our model is highly accurate for nanoparticles larger than 2 nm. The differences between relaxed and constrained nanoparticles (black arrows) reflect the contribution of finite-size effects to the error.

Close modal

Cohesive energies of CUBO Pt and Au nanoparticles having diameters ranging from ∼0.6 nm to ∼17 nm are shown in Fig. 2. These cohesive energies are computed using the alloy stability model and Eqs. (3)–(6). Based on Eqs. (7) and (8), we fit a size-dependent function for model-predicted cohesive energies that closely resemble the Gibbs–Thomson equation.35,76 As is intuitively expected, we see that cohesive energies of large nanoparticles asymptotically converge to their corresponding bulk values. Thus, our alloy stability model, which calculates energies with atom-by-atom precision, successfully predicts experimentally observed trends in cohesive energies or chemical potentials of metal atoms as a function of nanoparticle size,

CEnanoparticlePtd=CEbulkPt+0.583d,
(7)
CEnanoparticleAud=CEbulkAu+0.252d.
(8)
FIG. 2.

Model-predicted cohesive energies of CUBO nanoparticles as a function of their nanoparticle size. Predicted cohesive energies for Pt and Au are shown by blue solid and orange solid lines, respectively. Black dashed lines depict fits to the cohesive energies that closely resemble the Gibbs–Thomson relation. Fits for Pt and Au are shown in Eqs. (7) and (8), respectively. DFT-calculated cohesive energies of Pt (blue circles) and Au (orange diamonds) are also marked. As discussed in Fig. 1, the errors between DFT (circles and diamonds) and our model (solid lines) diminish with increasing nanoparticle size.

FIG. 2.

Model-predicted cohesive energies of CUBO nanoparticles as a function of their nanoparticle size. Predicted cohesive energies for Pt and Au are shown by blue solid and orange solid lines, respectively. Black dashed lines depict fits to the cohesive energies that closely resemble the Gibbs–Thomson relation. Fits for Pt and Au are shown in Eqs. (7) and (8), respectively. DFT-calculated cohesive energies of Pt (blue circles) and Au (orange diamonds) are also marked. As discussed in Fig. 1, the errors between DFT (circles and diamonds) and our model (solid lines) diminish with increasing nanoparticle size.

Close modal

In this section, we present αnZ(x) parameters that explicitly include functional forms for an applied isotropic strain, x, in percent change from zero strain. Such strain-dependent functional forms are not only applicable to nanoparticles in the finite-size regime as discussed in the section titled Size-dependent cohesive energies but also relevant toward predicting bimetallic metal–metal interactions for epitaxial islands of metal Z1 grown on substrate Z2. Our training set for determining the αnZ(x) parameters consists of DFT-calculated metal adsorption energies for Cu, Au, and Pt atoms on strained (111), (100), (211), and bulk structures. Initial and final adsorption configurations are equal to those used by Roling et al.28 We apply an isotropic strain ranging from −8% to +8%. The DFT calculations are performed by constraining all atoms to their bulk positions so as to completely decouple the effects of strain from any geometric relaxations.

The 10 αnZ(x) parameters are fitted to 14 DFT-derived adsorption energies using Eq. (9), as discussed for the specific case of 0% strain briefly in the Introduction and extensively in the work of Roling et al.28 The matrix M̃ is also similar in construction to the case with 0% strain. First, we explicitly determine the αnZ(x) parameters at the selected values of strain (e.g., −4.5%, −3%, −1.5%, 0%, 1.5%, and 3% for Pt). We then fit quadratic functions based on Eq. (10) for a given Z (e.g., Pt) and n (e.g., n = 5). These quadratic functions permit efficient interpolations of strain effects across a single continuous variable, x. The coefficients of these functions for bimetallic Cu, Au, and Pt environments are given in Tables S1–S3. Figure 3 illustrates that αnZ(x) parameters calculated using Eq. (10) are in excellent agreement with those determined by solving Eq. (9). These smooth fits further enable accurate interpolations within an external strain of ±8%. We compare DFT-derived and model-predicted adsorption energies of Au, Cu, and Pt atoms on strained surfaces in Fig. 4. Our model is extremely accurate on the training set with an MAE of 0.01 eV,

M̃14x10αx̃10x1=BE(x)14x1Z̃,
(9)
αnZx=αnZx0pnZx2+qnZx+rnZ.
(10)

In the section titled Energetics of metal-metal interactions, we will employ the strain-dependent αnZ(x) parameters determined using Eq. (10) to understand the energetics of metal–metal interactions during the growth of epitaxial Cu and Au islands on Pt (111) substrates. We will compare both absolute and relative metal–metal interactions predicted using the alloy stability model to experimental measurements reported using single crystal adsorption calorimetry. We will use the strain dependent functions in Eq. (10) and Fig. 3 to uncover the fascinating interplay between strain effects that destabilize metal–metal interactions and the formation of new metal–metal bonds that are stabilizing in nature.

FIG. 3.

(a)–(d) Normalized bondassociated parameters αnZ(x)/αnZx0 as a function of applied strain for Au (red), Cu (blue), and Pt (ochre), at n = 1–3, 5, 7, and 9. Quadratic fits for the strained bond-associated parameters are shown by black lines in accordance with αnZx=αnZx0*(pnx2+qnx+rn). The normalized bond-associated parameters are defined as the ratio between strained parameters, αnZ(x), and their corresponding values at zero strain, αnZx0. Quadratic functions for n = 4, 6, 8, 10, 11, and 12 are shown in Fig. S1.

FIG. 3.

(a)–(d) Normalized bondassociated parameters αnZ(x)/αnZx0 as a function of applied strain for Au (red), Cu (blue), and Pt (ochre), at n = 1–3, 5, 7, and 9. Quadratic fits for the strained bond-associated parameters are shown by black lines in accordance with αnZx=αnZx0*(pnx2+qnx+rn). The normalized bond-associated parameters are defined as the ratio between strained parameters, αnZ(x), and their corresponding values at zero strain, αnZx0. Quadratic functions for n = 4, 6, 8, 10, 11, and 12 are shown in Fig. S1.

Close modal
FIG. 4.

Parity plot comparing DFT-derived and model-predicted binding energies of Cu, Au, and Pt atoms having diverse coordination numbers (3-fold to 12-fold) on strained (111), (211), (100), and bulk structures. Model-predictions are obtained using αnZ(x) parameters extracted from quadratic fits shown in Fig. 3 and Fig. S1.

FIG. 4.

Parity plot comparing DFT-derived and model-predicted binding energies of Cu, Au, and Pt atoms having diverse coordination numbers (3-fold to 12-fold) on strained (111), (211), (100), and bulk structures. Model-predictions are obtained using αnZ(x) parameters extracted from quadratic fits shown in Fig. 3 and Fig. S1.

Close modal

In the sections titled Size-dependent cohesive energies and Introducing the effects of strain, we have laid out how strain effects can be integrated into site-stability predictions by the alloy stability model.28,30 We are interested in precisely predicting strain-dependent site stabilities, as they can serve as descriptors for catalytic reactivity.29,31

In the present section, we benchmark the strain-including alloy stability model against experimental data. We do so by combining the model with explicit DFT calculations and compare the results to experimental observations53,54 of single crystal adsorption calorimetry (SCAC) during the growth of Au and Cu films on Pt (111) surfaces. Using this combination, we are able to further understand the experimental observations, in particular, the role of metal adlayer strain in modifying the heat of adsorption. First, we briefly summarize previous experimental findings for these systems.

For the two systems, Au and Cu on Pt (111), Auger electron spectroscopy (AES),77–79 low energy electron diffraction (LEED),79,80 and scanning tunneling microscopy (STM)80,81 indicate a layer-by-layer growth mode for the first monolayer with the adsorbate layer adapting the substrate lattice constant. This lattice matching results in a tensile strain of +8% for Cu and a compressive strain of −4% for Au compared to their respective bulk lattice constants. This initial strain is released by the second Cu layer81 and by the fifth Au layer,79 as indicated by the appearance of the bulk lattice spacing for Cu and Au in LEED analyses. The nucleation of Au and Cu islands is believed to occur at step edges, i.e., imperfections, on the Pt (111) surface. These imperfections provide energetically favorable, high-coordinated adsorption sites, and islands subsequently grow onto adjacent terraces.54,81

Campbell and co-workers53,54 recently employed SCAC in an effort to measure metal-on-metal heats of adsorption for bimetallic systems such as Au/Pt (111) and Cu/Pt (111) that are miscible and form bulk alloys. The heats of adsorption of metal adlayers of such alloy-forming systems are not accessible through conventional temperature-programmed desorption (TPD) measurements since the alloy formation initiates at temperatures lower than the desorption temperature.53,54

The heat of adsorption measured by James et al.53 and Feeley et al.54 for multiple Cu and Au layers on Pt (111) is reproduced in Fig. 5. Throughout this section, we adopt the positive sign convention of microcalorimetry measurements and report all heats of adsorption/adsorption energies (we use these terms interchangeably) obtained from experiments and theory in kJ/mol; higher positive values of the heat of adsorption indicate a stronger adsorption. Model predictions are reported in kJ/mol using the positive sign convention in order to facilitate comparisons with experiments. Since most computational studies report adsorption energies in eV, with more negative values indicating stronger adsorption, we reproduce the theory-only plots in the supplementary material using the negative sign convention and units of eV (Figs. S3–S5, S7, and S10). We briefly compare the experimental observations for the two systems. While James et al.53 observed that the heat of adsorption linearly decreases within the first ML for Cu on the Pt (111) system (shown in red triangles), Feeley et al.54 observe an initial steep increase up to 0.03 ML, then a constant heat of adsorption up to 0.7 ML, and finally a steep decrease in the heat of adsorption up to 1 ML for Au on the Pt (111) system (shown in blue dots). In both cases, the authors attribute the decreasing heat of adsorption within the first monolayer to an increased strain as island sizes grow. The strain on the islands, in turn, induces strain effects on the substrate atoms that can result in island–island repulsion. In the case of copper, it is concluded that the heat of adsorption is immediately affected by strained Cu–Cu bonds,53 while in the case of Au, it is hypothesized that the strain only takes effect after reaching a critical island size at 0.7 ML.54 For subsequent layers, the adsorption energy of Cu continues to decrease until reaching a minimum at roughly two monolayers before finally increasing to the heat of sublimation of bulk Cu. For the Au/Pt system, oscillations in the heat of adsorption were observed even during the second and third layers. These oscillations were attributed to multilayer growth before converging to the bulk heat of sublimation of Au after the fourth monolayer.

FIG. 5.

Heat of adsorption measurements vs Cu/Au coverage on Pt (111) obtained from single crystal adsorption calorimetry measurements during the deposition of the first two monolayers of Cu/Au. Cu is shown in red triangles; Au is shown in dark blue circles. Reproduction of the initial part of the data published in Ref. 53 (Cu) and Ref. 54 (Au).

FIG. 5.

Heat of adsorption measurements vs Cu/Au coverage on Pt (111) obtained from single crystal adsorption calorimetry measurements during the deposition of the first two monolayers of Cu/Au. Cu is shown in red triangles; Au is shown in dark blue circles. Reproduction of the initial part of the data published in Ref. 53 (Cu) and Ref. 54 (Au).

Close modal

For the analysis of the experimental data, we compare it to heats of adsorption obtained from explicit DFT calculations and predictions by the alloy stability model presented above. First, we focus on differences between the adsorption strength of Au/Cu on Pt (111) in the first monolayer and that of Au on Au and Cu on Cu in the second monolayer. Second, we closely examine the different trends observed in the measured heats of adsorption of Au and Cu on Pt (111) within the first monolayer. We focus on quantifying the interplay between strain effects and the formation of new metal–metal bonds on the heats of adsorption for the two different systems. Strain effects tend to destabilize metal–metal interactions, while the formation of new metal-bonds energetically favors metal–metal interactions.

In Figs. 6(a) and 6(b), we show the heats of adsorption of the first and second monolayer of Au and Cu on Pt (111), respectively. We compare energies obtained from experiments, calculated by DFT, and predicted by the alloy stability model (referred to as α in Fig. 6) with and without the inclusion of strain. For the experiments, we selected the measured value at 0.5 ML coverage. We chose this coverage since at this point extended islands will have formed, and the measured interaction will be governed by the bimetallic interactions between high-coordinated Au/Cu adatoms and the Pt (111) substrate. Since the majority of the atoms in extended islands on Pt (111) at an experimental coverage of 0.5 ML are ninefold coordinated, we shall use the heat of adsorption values for which all Cu and Au atoms are ninefold coordinated in our comparison with DFT and the alloy stability model. In the alloy stability model prediction that includes strain, we assume the Au film to be compressively strained by 2% and the Cu film to have a tensile strain of 4%; hence, in both cases, we consider half the strain seen for a full monolayer.

FIG. 6.

Comparison of the heats of adsorption of the first and second monolayer (ML) of (a) Au and (b) Cu on Pt (111) obtained from the experiment53,54 (expt.), calculated by RPBE-based DFT (RPBE) and predicted by the alloy stability model without (α w/o strain) and with (α w/strain) the inclusion of strain. For the first monolayer, we chose the experimental values at 0.5 ML and compared them with values for ninefold coordinated atoms onto which a tensile strain 4% on Cu and a compressive strain of 2% on Au acts. For the second ML, we use average experimental values and 0% strain for both systems.

FIG. 6.

Comparison of the heats of adsorption of the first and second monolayer (ML) of (a) Au and (b) Cu on Pt (111) obtained from the experiment53,54 (expt.), calculated by RPBE-based DFT (RPBE) and predicted by the alloy stability model without (α w/o strain) and with (α w/strain) the inclusion of strain. For the first monolayer, we chose the experimental values at 0.5 ML and compared them with values for ninefold coordinated atoms onto which a tensile strain 4% on Cu and a compressive strain of 2% on Au acts. For the second ML, we use average experimental values and 0% strain for both systems.

Close modal

We observe that the absolute magnitude of the heats of adsorption determined by theory deviates from those acquired in experiments. These differences are routinely noted when comparing experimental and DFT-derived heats of adsorption.82 In Fig. S2, we show a comparison of the experimental values of the heats of adsorption of the first monolayer with those obtained from DFT when using four different functionals (RPBE,62 BEEF-vdW,65 PBE,64 and PBESol66). Such comparisons between carefully performed experimental heat of adsorption measurements of bimetallic systems and DFT calculations and model predictions, respectively, will help us to develop a post-correction scheme for parameters in the alloy stability model to tackle the absolute differences between the heats of adsorption obtained from experiments and theory.

While the absolute values of the measured heats of adsorption differ from those calculated and predicted, we show that the relative trends are well described with both DFT and our alloy stability model: For both Au and Cu on Pt (111), we find that the second layer adsorbs more weakly than the first monolayer. The only outlier is the alloy stability model prediction including 4% tensile strain of the first ML and assuming 0% strain in the second ML of Cu on Pt (111). We will see later on in our analysis that the inclusion of strain effects for Cu has a strong impact on the heat of adsorption, while strain minimally affects the heat of adsorption of the Au/Pt system. The number presented for the second ML of Cu on Pt (111) should hence be seen as an upper bound to the actual heat of adsorption. If there is still some residual strain in the second monolayer or interlayer strain between the first and second monolayer, the reported number could be reduced by as much as 20 kJ/mol and, hence, be lower than the number predicted for the first ML.

Having established the applicability of DFT and the alloy stability model to predict trends in adsorption energies, we now focus on examining the experimental trends observed during the deposition of the first monolayer of Au/Cu on Pt (111). We begin by considering the nucleation of Cu or Au islands on the Pt surface. Experimental studies indicate that the nucleation of islands is initiated at high-coordinated sites located at the bottom of steps. To test this assumption, we compare the adsorption energy of the first adsorbed atom both at the bottom and the top of a step site, as shown in Fig. 7.

FIG. 7.

Top view of atomic models of a (211) step present as a defect in a Pt (111) single crystalline surface. Pt atoms are shown in gray; Au atoms are in yellow. Changes in coordination number of atoms upon the adsorption of an Au atom on a terrace-like site on top of the step (a) and at the bottom of the step edge (b) are indicated in black numbers for Au atoms and in red numbers for Pt atoms. For example, in (a), the adsorbed Au atom changes its coordination number from 0 in the gas phase to 3 at the adsorption site. These changes in coordination numbers are used in combination with the alloy stability model to predict the site stability of the two different adsorption sites.

FIG. 7.

Top view of atomic models of a (211) step present as a defect in a Pt (111) single crystalline surface. Pt atoms are shown in gray; Au atoms are in yellow. Changes in coordination number of atoms upon the adsorption of an Au atom on a terrace-like site on top of the step (a) and at the bottom of the step edge (b) are indicated in black numbers for Au atoms and in red numbers for Pt atoms. For example, in (a), the adsorbed Au atom changes its coordination number from 0 in the gas phase to 3 at the adsorption site. These changes in coordination numbers are used in combination with the alloy stability model to predict the site stability of the two different adsorption sites.

Close modal

In order to employ the alloy stability model, we need to first identify the changes in coordination numbers (CNs) of all involved atoms during the adsorption process. In the following, we use M to interchangeably denote Au and Cu. In the case of the atom adsorbing at the bottom of the step edge, the CN of the adsorbed M atom changes from 0 in the gas phase to 5 at the adsorption site. Of the five coordinating Pt atoms, one changes from 9 to 10, two change from 10 to 11, and another two change from 7 to 8. Accordingly, for the atom adsorbing on a terrace-like position on top of the step, the CN of M changes from 0 to 3 and the CNs of the three coordinating Pt atoms change from 7 to 8 for one atom and from 9 to 10 for two atoms. The corresponding matrices are shown in Table I.

TABLE I.

Matrices in terms of αnZ1Z1Z2 parameters describing the bonds being formed during Au/Cu (denoted as M) adsorption processes at the bottom of a step edge and on a terrace-like position on top of the step edge.

αnMMPtαnPtPtM
n 1–3 10 11 12 1–3 10 11 12 
Step edge 
Terrace 
αnMMPtαnPtPtM
n 1–3 10 11 12 1–3 10 11 12 
Step edge 
Terrace 

We obtain the adsorption energy of the two configurations by multiplying the matrices shown in Table I with the αnZ1Z1Z2 parameters of the alloy stability model for unstrained bimetallic systems30 given in Tables S4–S6. For better comparison to experiments, we adopt the positive sign convention of microcalorimetry and convert the values to kJ/mol as stated above. As shown in Fig. 8, our model yields that adsorbing the first atom at the bottom of a Pt step [Fig. 7(b)] is ∼50 kJ/mol more favorable than adsorbing it on the Pt terrace [Fig. 7(a)] for both Au and Cu. Hence, in agreement with experiments, our model predicts nucleation to be energetically more favorable at high-coordinated step-edge sites.

FIG. 8.

Comparison of the predicted adsorption energy of Au/Cu atoms on a terrace-like site on top and at the bottom of a step defect site on a Pt (111) surface. For both systems, adsorption at the bottom of the step is more stable by ∼50 kJ/mol.

FIG. 8.

Comparison of the predicted adsorption energy of Au/Cu atoms on a terrace-like site on top and at the bottom of a step defect site on a Pt (111) surface. For both systems, adsorption at the bottom of the step is more stable by ∼50 kJ/mol.

Close modal

Next, we focus on the initial increase in the heat of adsorption at coverages below 0.03 ML for the Au/Pt (111) system. In the paper by Feeley et al.,54 they hypothesized that this trend is observed since the initially deposited atoms are quite well dispersed and therefore on average have low coordination numbers. Such low-coordinated atoms are generally less stable than their high-coordinated counterparts.28,30 As more Au is deposited in subsequent calorimetry cycles, the average coordination number of the deposited atoms increases since larger islands are forming. Hence, we expect to see an increase in the stability of the atoms and a concomitant increase in the measured heat of adsorption.

We model this scenario by considering the energetics of hexagonally shaped Au islands of increasing size on the Pt (111) surface. Specifically, we apply the alloy stability model to determine the differential adsorption energy per atom of each additional hexagonal layer formed as illustrated for the first ten hexagons by the color gradient in Fig. 9(a). The elegance of the alloy stability model is that it enables us to predict the energetics of such extended systems, whose assessment would be prohibitively expensive with DFT. Figure 9(b) shows in dark blue dots the evolution of the differential adsorption energy of new atoms added as the island grows in size. As a comparison to our model, we explicitly calculated this differential adsorption energy with DFT for islands with 1, 7, and 19 atoms as well as for a complete monolayer. The results are represented by light blue diamonds in Fig. 9(b). The alloy stability model and explicit DFT agree within 10 kJ/mol. Both DFT and the alloy stability model confirm that the atoms are more strongly adsorbed as the island size grows as has been hypothesized by Feeley et al.54 We see that the magnitude of the differential adsorption energy increases more significantly as the first few Au atoms are adsorbed. This increased stabilization during the initial phase of Au deposition is consistent with the steeply increasing first three measurement points by Feeley et al.54 (Fig. 5) at which atoms are likely still isolated on the surface.

FIG. 9.

(a) Top view of the structure of an extended Pt (111) surface onto which hexagonal islands of Au atoms are adsorbed. Pt atoms are shown in gray; Au atoms are shown in a color gradient from dark blue to pink. In our simulation, we first add the central atom (dark blue) and then sequentially add new hexagonal layers around this central atom as indicated by the color code. Using the alloy stability model, we determine the differential adsorption energy per atom of each new hexagonal outer layer adsorbed. (b) Predicted (alloy stability model, dark blue circles) and calculated (DFT, light blue diamonds) differential adsorption energies per atom of each new hexagon vs the total number of atoms in the island. For comparison, the monolayer (ML) limit calculated by DFT is shown as light blue horizontal bar.

FIG. 9.

(a) Top view of the structure of an extended Pt (111) surface onto which hexagonal islands of Au atoms are adsorbed. Pt atoms are shown in gray; Au atoms are shown in a color gradient from dark blue to pink. In our simulation, we first add the central atom (dark blue) and then sequentially add new hexagonal layers around this central atom as indicated by the color code. Using the alloy stability model, we determine the differential adsorption energy per atom of each new hexagonal outer layer adsorbed. (b) Predicted (alloy stability model, dark blue circles) and calculated (DFT, light blue diamonds) differential adsorption energies per atom of each new hexagon vs the total number of atoms in the island. For comparison, the monolayer (ML) limit calculated by DFT is shown as light blue horizontal bar.

Close modal

We also expect such an initial increase in the heat of adsorption for the first Cu atoms that are deposited on the Pt surface due to the same considerations as for the Au/Pt (111) system. Experimentally, however, such an increase does not appear in the initially measured heat of adsorption.53 We have two hypotheses that might explain the observed difference between Cu on Pt (111) and Au on Pt (111): First, the Cu doses are larger than the Au doses, which can be seen by the larger spacing between the measurement points (Fig. 5). In fact, the first measurement point for the Cu/Pt (111) system is recorded at ∼0.03 ML, a coverage at which the heat of adsorption has already leveled out in the Au/Pt (111) system. Second, the rate of diffusion of Au and Cu adatoms on Pt (111) might differ, leading to different island sizes at comparable coverages. To test the latter hypothesis, we calculated the diffusion barriers of Cu and Au on a 2 × 2 × 6 Pt (111) slab. We considered three different scenarios: (1) an adatom moving from a threefold coordinated position to another threefold coordinated position, (2) an adatom moving from a fivefold coordinated position to another fivefold coordinated position, (3) an adatom moving from a sevenfold coordinated position to another sevenfold coordinated position. A comparison of the resulting diffusion barriers for Au and Cu on Pt (111) is shown in Fig. 10.

FIG. 10.

Calculated diffusion barriers of Au/Cu adatoms with different coordination numbers (CN) on a Pt (111) surface. The notation CN 3 → 3 describes a threefold coordinated Au/Cu atom in an fcc hollow site diffusing to neighboring hcp hollow site, where it is also threefold coordinated. Accordingly, the notations CN 5 → 5 and CN 7 → 7 describe five- and sevenfold coordinated atoms diffusing from an fcc to another fcc hollow site.

FIG. 10.

Calculated diffusion barriers of Au/Cu adatoms with different coordination numbers (CN) on a Pt (111) surface. The notation CN 3 → 3 describes a threefold coordinated Au/Cu atom in an fcc hollow site diffusing to neighboring hcp hollow site, where it is also threefold coordinated. Accordingly, the notations CN 5 → 5 and CN 7 → 7 describe five- and sevenfold coordinated atoms diffusing from an fcc to another fcc hollow site.

Close modal

While the diffusion barriers for threefold coordinated Au and Cu atoms on Pt (111) are comparable, we find that the diffusion barriers for higher-coordinated atoms are ∼20 kJ/mol lower for Cu adatoms. This finding indicates that at room temperature, Cu forms larger islands with a higher average coordination number than Au at similar coverages. We conclude that the islands forming during the first points of Cu deposition likely have already reached the critical island size for which the differential heat of adsorption is governed by high-coordinated atoms.

In the following, we focus on the differences between Au and Cu on Pt (111) in the range between 0.03 ML and 1 ML coverage (see Fig. 5). While the heat of adsorption steadily decreases for the Cu/Pt (111) system, it remains constant for the Au/Pt (111) system up to a coverage of 0.7 ML and then steeply decreases. We want to test the hypothesis put forward by James et al.53 and Feeley et al.54 that increasing strain connected to growing island size causes the observed decrease in the measured heats of adsorption.

First, we used DFT to relax structures of Au and Cu islands of various shapes and sizes (between 3 and 25 adatoms) on 6 × 6 × 4 Pt (111) slabs (see images in Fig. S8). From these relaxed structures, we determined the mean Cu–Cu/Au–Au distances to deduce the strain acting on the metal adlayer bonds. The corresponding strain for the Cu and Au on Pt (111) systems is shown in Fig. 11(a) with the ML strain included for comparison. We indeed observe that the strain acting on the metal adlayer bonds increases as the islands grow in size.

FIG. 11.

(a) Average Au (dark blue circles) and Cu (red triangles) adlayer strain on a Pt (111) surface vs the number of atoms in the island, obtained from relaxed DFT calculations. We determined the average strain by comparing the average Au–Au/Cu–Cu distances in the adlayer and the bulk Au–Au/Cu–Cu distances. As a comparison, the strain of a complete monolayer of Au/Cu on Pt (111) is shown for which the Au/Cu adlayer atoms adopt the lattice constant of Pt resulting in a compressive strain of 4% for Au and a tensile strain of 8% for Cu. (b) Adsorption energy of ninefold coordinated Au (dark blue circles)/Cu (red triangles) atoms in extended Au/Cu islands on Pt (111) vs the average adlayer strain. The light blue and orange lines serve as guides to the eye to evaluate the impact that a 4% compressive and an 8% tensile strain for Au and Cu, respectively, have on the metal adlayer adsorption energy compared to 0% strain.

FIG. 11.

(a) Average Au (dark blue circles) and Cu (red triangles) adlayer strain on a Pt (111) surface vs the number of atoms in the island, obtained from relaxed DFT calculations. We determined the average strain by comparing the average Au–Au/Cu–Cu distances in the adlayer and the bulk Au–Au/Cu–Cu distances. As a comparison, the strain of a complete monolayer of Au/Cu on Pt (111) is shown for which the Au/Cu adlayer atoms adopt the lattice constant of Pt resulting in a compressive strain of 4% for Au and a tensile strain of 8% for Cu. (b) Adsorption energy of ninefold coordinated Au (dark blue circles)/Cu (red triangles) atoms in extended Au/Cu islands on Pt (111) vs the average adlayer strain. The light blue and orange lines serve as guides to the eye to evaluate the impact that a 4% compressive and an 8% tensile strain for Au and Cu, respectively, have on the metal adlayer adsorption energy compared to 0% strain.

Close modal

Second, we use our alloy stability model to determine the strain-dependent adsorption energy of ninefold coordinated Au/Cu atoms on a Pt (111) substrate. We use the αnZ1Z1Z2 parameters determined for the bimetallic alloy combinations, and in addition, we impose the strain dependence that we have found for the corresponding monometallic system, as described in the first part of the manuscript [Eq. (10)]. In doing so, we obtain the strain-dependent Cu and Au adsorption energies on Pt (111) shown in Fig. 11(b). While we do observe that a tensile strain of 8% weakens the adsorption energy of Cu on Pt (111) by ∼10%, the adsorption energy of Au on Pt (111) is minimally impacted by a compressive strain of 4%.

This analysis identifies the differences between the experimentally observed trends for the heats of adsorption of Cu and Au on Pt (111) between 0.03 and 0.7 ML (Fig. 5). While the increasing strain correlates with the heat of adsorption of Cu on Pt (111), it does not affect the heat of adsorption of Au on Pt (111). This strain-dependence of the heat of adsorption of Cu on Pt (111) explains the experimentally observed linear decrease for Cu, as hypothesized by James et al.53 For Au on Pt (111), the apparent strain-independence of the heat of adsorption explains why the measured value remains steady for island sizes between 0.03 ML and 0.7 ML.

Finally, we want to address the observed rapid decrease in the heat of adsorption for Au on Pt (111) between 0.7 ML and 1 ML. Given our previous analysis on the minimal impact of compressive strains of up to 4% on the adsorption energy of Au on Pt (111), it does not seem likely that strain effects that start at 0.7 ML are capable of causing such a rapid drop in the heat of adsorption.

Another hypothesis that has been raised to explain this decrease in heat of adsorption is island–island repulsion by neighboring islands caused by substrate strain acting in opposite directions.53,54,83 To test this hypothesis for the considered systems, we use DFT calculations to compare the adsorption energy per atom for isolated and neighboring islands of Au and Cu on Pt (111). We consider both 6-atom triangular and 7-atom hexagonal islands (see the structures in Fig. S9). The results of this comparison are summarized in Fig. 12. While we observe a 1% weakening of the metal–metal interactions for islands in close vicinity, this effect is considerably less than the ∼6% decrease in heat of adsorption observed experimentally for both Au and Cu on Pt (111). Since the substrate atoms under small islands (6–7 atoms) in close proximity have more degrees of freedom than those under larger islands, their bonds will likely be more strained. Hence, we do not expect the island–island repulsion effect to be more pronounced on larger neighboring islands.

FIG. 12.

Comparison of the average adsorption energy per atom of a single and two neighboring 6-atom triangular and 7-atom hexagonal Au/Cu islands on a Pt (111) surface.

FIG. 12.

Comparison of the average adsorption energy per atom of a single and two neighboring 6-atom triangular and 7-atom hexagonal Au/Cu islands on a Pt (111) surface.

Close modal

A different hypothesis is that a fraction of the deposited atoms initiates the formation of the second layer for Au on Pt (111) already at 0.7 ML, hence before the first layer is fully completed. The alloy stability model including 4% compressive strain supports this picture as it yields adsorption energies of Au upon completion of the first monolayer on Pt (111) of ∼260 kJ/mol, while the adsorption energy of Au in the second monolayer on Au is ∼220 kJ/mol–230 kJ/mol (depending on island size, stronger adsorption for larger islands with higher coordinated atoms). This comparison shows that if the first layer were fully completed before the second layer started forming, one would expect the measured heat of adsorption to suddenly drop by ∼30 kJ/mol as the first monolayer is completed. Experimental results by Feeley et al.,54 however, do not observe such a sudden drop in the heat of adsorption at a coverage of 1 ML. Instead, they see a continuous drop of a total of 25 kJ/mol from 0.7 ML to 1 ML. This observation points to a gradual increase in atoms deposited in the second layer before the first layer is fully completed.

We note here that the situation for the Cu/Pt (111) system might differ from that in the Au/Pt (111) system. For the Cu/Pt (111) system, the first monolayer may be fully completed before the second layer starts growing without showing a discontinuity in the heats of adsorption between the first and the second layer. The alloy stability model including 8% tensile strain yields adsorption energies of Cu upon completion of the first monolayer on Pt of ∼275 kJ/mol, while the adsorption energy of Cu in the initial stages of the second monolayer with 0% strain on Cu is ∼290 kJ/mol. We note that the 290 kJ/mol serves as an upper bound. The true adsorption energy is likely weaker since there might still be interlayer strain between the first and second layer of Cu impacting the heat of adsorption, which is not included in the 0% strain assumption we adopted for the second layer here. Hence, for the Cu/Pt system, we do not necessarily expect to see a discontinuity as the first layer is completed and the second layer starts to grow. Finally, we note that only carefully performed microscopic and surface science experiments will eventually be able to completely resolve the true growth mechanisms of Au and Cu islands on Pt (111).

In summary, we have demonstrated the suitability of our predictive models to describe and explain trends observed in experimental measurements of heats of adsorption of metal adlayers in bimetallic systems that form bulk alloys. Given the delicate nature of these calorimetric studies, our strain-including alloy stability model serves as a complementary approach for rapidly predicting metal–metal energetics for systems that have not yet been accessed experimentally.

First, we deconstructed the relative contributions of the errors arising from quantum-size/electronic and finite-size effects by comparing cohesive energy of relaxed and constrained Pt and Au cuboctahedral nanoparticles having 13, 55, 147, and 309 atoms calculated by DFT and predicted by the alloy stability model.28 We found that errors in cohesive energies associated with quantum-size/electronic effects tend to diminish at 309 atoms (∼2.3 nm in diameter) and that the size dependence of model-predicted cohesive energies closely resembles the Gibbs–Thomson relation.

Second, we integrated quadratic functional forms into the parameters of the alloy stability model, αnZ parameters to account for the effect of external strain on the adsorption site stability. These strain-dependent αnZ(x) parameters can potentially account for the errors caused by finite-size effects and are able to predict adsorption energies of Au, Cu, and Pt atoms on strained surfaces with an MAE as low as 0.01 eV on the training set.

Finally, we demonstrated the applicability of these strain-dependent αnZ parameters toward predicting metal–metal interaction energies during the growth of epitaxial Au and Cu islands on Pt (111) substrates. With this aim, we benchmarked heats of adsorption of metal atoms predicted by the strain-including alloy stability model with experimentally measured values from single crystal adsorption calorimetry by James et al.53 and Feeley et al.54 and quantitatively evaluated the influence of strained metal–metal bonds on the heats of adsorption. First, we considered the metal–metal interaction strengths during the deposition of the first and second monolayer of Au and Cu on Pt (111), respectively. In agreement with experiments, our model yields that bimetallic interactions between Cu and Pt (Au and Pt) in the first monolayer are stronger than monometallic interactions between Cu and Cu (Au and Au) in the second monolayer. We then used our model to closely examine the trends in the experimentally measured heats of adsorption right from the nucleation of islands to the completion of an epitaxial monolayer. In both systems, our model confirms that the nucleation of islands is initiated at high-coordinated adsorption sites located at the bottom of step edges. Subsequently, as the islands start growing onto adjacent terraces, the model reveals a steep initial increase in the heat of adsorption due to the stabilizing effect of higher coordination environments. In experiments, this initial increase is only captured for the Au on Pt (111) system, given the smaller doses of Au as compared with Cu.53,54 Finally, our strain analysis for extended islands revealed a strong impact of tensile strain [as is the case for Cu on Pt (111)] and, depending on the extent of compression, a negligible to moderate influence of compressive strain [as is the case for Au on Pt (111)] on the heats of adsorption. These differences between the effects of tensile and compressive strain in the first monolayer help elucidate the different trends observed experimentally within the first monolayer for the two systems. The increase in tensile strain caused by increasing Cu island size progressively reduces the heat of adsorption of Cu on Pt (111). Conversely, the increasing compressive strain as the Au islands grow larger does not influence the heat of adsorption of Au on Pt (111).

Given the wide applicability of the alloy stability model, our approach for predicting the relative interplay between the destabilizing effects of strain and the stabilizing effect of forming new metal–metal bonds can be transferred beyond the two specific systems considered here. Since our paradigm predicts the stabilities of metal atoms on strained systems, which, in turn, can be correlated with catalytic reactivity and selectivity descriptors, it can be effectively employed to strain-engineer adsorption sites on bimetallic catalysts with a remarkable site-by-site precision.

The supplementary material contains comparisons of heats of adsorption with different DFT functionals, geometries of Cu and Au islands on Pt (111), transition states for Cu and Au diffusion on Pt (111), and bond-associated parameters as a function of external strain.

V.S. and T.S.C. contributed equally to this work.

This research was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, Catalysis Science Program to the SUNCAT Center for Interface Science and Catalysis. V.S. acknowledges financial support from the Alexander von Humboldt Foundation. We acknowledge computational support from the National Energy Research Scientific Computing Center (computer time allocation m2997), a Department of Energy Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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