Melting behavior of (Pd_xPt_1−x)_n nanoclusters confined in single-walled carbon nanotubes: a molecular dynamics investigation on the effects of chirality and diameter of nanotubes, and size and composition of nanoclusters

Abstract

The effects of chirality and nanotube diameter, accompanied with size and concentration of Pd, on the melting behavior of (Pd_xPt_1−x)_n bimetallic nanoclusters (with n = 108 and 256, x_Pd = 0.3, 0.5, and 0.7) confined in SWCNTs with different sizes, diameters, and chiralities were studied by MD simulation. The effect of chirality of the nanotube is significant in such a way that nanoclusters in zigzag nanotubes have higher melting temperature and greater thermodynamic stability. The effect of diameter of the nanotube is negligible and replaces zero energy. Also, the effect of cluster size is considerable, and clusters with small sizes have lower melting temperature, decreased thermodynamic stability, and stronger interactions with the tube wall. Moreover, there is a significant effect of Pd concentration in the nanocluster, and clusters with a higher concentration of Pd have a lower melting temperature and decreased thermodynamic stability. Also, a core–shell (Pt–Pd) structure was observed for (Pd_xPt_1−x)₂₅₆ nanoclusters, which is an intrinsic property and depends only on their size, not on their environment. The other important result is an fcc to hcp-like transition near the melting temperature of the studied nanoclusters, which is not dependent on the cluster size, concentration, or its environment.

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Introduction

Metallic nanoclusters and carbon nanotubes are two categories of important nanomaterials that are currently being explored by experimental and theoretical scientists due to their unique properties that can be utilized for new electronic and photonic devices.^1–6 An important property of metallic nanoclusters is their high chemical activity that is due to the large surface/bulk ratio. This high activity leads to increased thermodynamic instability; however, one solution for this problem is the application of carbon nanotubes as a support for the stabilization and encapsulation of metallic nanoclusters.^7–9 Therefore, it is important for nanotechnology research to investigate the physical and chemical properties of these new carbon–nanotube/metallic–nanocluster composites.

One of the interesting physical phenomena regarding these types of nanocomposites is the melting behavior of nanoclusters surrounded by single-walled carbon nanotubes (SWCNTs). There is a lack of experimental data for melting characteristics of transition metal nanoclusters (TMNCs) inside carbon nanotubes, due to many technical and instrumental limitations. Under these circumstances, theoretical studies using molecular simulation methods can be very helpful. Due to the large number of transition metal atoms and electrons in nanoclusters, the application of quantum-mechanical methods for such studies is too difficult and sometimes impossible. Because melting is a physical phenomenon without any electron transfer, classical molecular dynamic (MD) simulations with appropriate force fields can give reasonable results for such physical properties. Therefore, many theoretical studies have been performed to simulate the melting of TMNCs encapsulated in SWCNTs.

Cheng et al. used a Monte-Carlo (MC) simulation in order to study the melting behavior of icosahedral Pt₅₅ nanoclusters encapsulated in (15,15) and (20,20) SWCNTs.¹⁰ They used tight-binding and Lennard-Jones (LJ) potentials for metal–metal and carbon–metal interactions, respectively, and assumed that the carbon atoms of the SWCNTs were fixed. Their results indicated that the melting-like temperature of the nanocluster is increased by the diameter of nanotube.¹⁰ In another work, Cheng and Lan studied the melting of icosahedral Pd_N (with N = 55, 147, 369, and 561) nanoclusters confined in armchair SWCNTs (with chirality of (15,15), (20,20), (25,25), (30,30), and (35,35)) using MD simulation in order to investigate the effects of cluster size, nanotube diameter, and metal–tube interaction on the melting properties of Pd nanoclusters.¹¹ They used two different metal–carbon potentials; the first was LJ and the second was a Morse-type potential that was calculated by density functional theory (DFT). Their results indicated that the melting temperature of nanoclusters has a linear trend with the inverse diameter of the cluster and is smaller than that of the free nanocluster due to reconstruction of the cluster within the nanotube before melting. Moreover, they discovered that the diameter of the nanotube does not have a considerable effect on the melting temperature of the nanocluster.¹¹ However, these results were not dependent on the carbon–metal interaction type, for which the only difference between LJ and Morse carbon–metal potentials was an increased deformation of the nanotube due to the strong Morse-type potential. This Morse-type potential cannot be exact due to the fact that there are many reports in theoretical studies that emphasize the significant error of DFT in predicting intermolecular potentials except for new modified density functionals.^12,13

Shao et al. studied the melting and freezing of Au_n (with n = 467, 818, 1522, and 2230) nanoclusters confined in armchair SWCNTs (with chirality of (15,15), (19,19), (25,25), and (30,30)) by MD simulation.¹⁴ Their results reveal that: (i) Au nanoclusters have multi-shell structures and order–disorder transformation of atoms in each layer that occurs during phase transition in such a way that melting starts from the innermost layer while freezing starts from the outermost one, and (ii) the Au–nanotube interface plays an important role in the ordered structure of the nanocluster and its stability.¹⁴ Fang et al. used MD to study the melting of Al nanowires within SWCNTs.¹⁵ Their results showed that intriguing phenomena may appear when the diameters of the inner nanowires exceed a threshold value in such a way that melting starts first at the inner layers and then the inner atoms diffuse toward the surface layers. They obtained two melting temperatures, where one was lower than the bulk and another was higher.¹⁵

An important group of TMNCs are those that are bimetallic.¹ There are considerable experimental studies that have reported successful synthesis of Pd–Pt nanoclusters.^16–18 Therefore, a portion of MD simulations for the melting behavior of nanoclusters confined in SWCNTs are performed for bimetallic nanoclusters. Shi et al. used MD simulation to study the melting and freezing of (Au_1−xPt_x)_N (with x = 0.2, 0.4, 0.6, 0.8, and 1.0 and N = 818, 1522, and 2230) nanoparticles confined in armchair SWCNTs (with chirality of (19,19), (25,25), and (30,30)).¹⁹ Their calculations indicate that Pt atoms tend to congregate on the nanotube wall due to the stronger Pt–C interactions in comparison with Au–C ones. As the number of Pt atoms increases, they distribute from the tube walls to its center, while Au atoms distribute in the opposite direction. Also, melting starts from the innermost layer, but freezing starts from the outermost layer,¹⁹ which is an observation that is similar to that obtained by Shao et al.¹⁴ However, in the abovementioned studies, the effect of chirality of the nanotube was not investigated.

Sankaranarayanan et al. used MD simulation to study the melting of Pd–Pt nanoclusters.²⁰ They used MC calculations to produce minimum energy structures as starting points for MD simulations using quantum Sutten–Chen (QSC) many-body potentials. The calculated minimum energy structures indicate a concentration of Pd on the cluster surface and Pt aggregation in its core. This core–shell distribution remains unchanged even at melting conditions. Also, Pd–Pt nanoclusters show two melting stages: the first is surface melting of the Pd-rich shell, and the second is followed by homogeneous melting of the Pt-rich core. Moreover, Pt tends to diffuse to the surface of the cluster after melting in order to reduce its surface energy, and melting temperatures were found to be lower than the macroscopic bulk of Pd and Pt crystals.²⁰ In another work, Sankaranarayanan et al. studied the melting of Pd–Pt, Pd–Rh, and Pd–Cu nanoclusters on the surface of graphite by MD simulation.²¹ They found that melting temperatures decrease when the concentrations of Cu (in Pd–Cu) and Pd (in Pd–Pt, and Pd–Rh) increase. Also, similar to SWCNTs, one of the metal atoms has increased tendencies to migrate to the graphite surface. Moreover, their calculations determined that melting starts from the surface of the nanocluster due to the decreased distance from the graphite surface.²¹ As previously mentioned, the effect of nanotube chirality on the melting properties of bimetallic nanoclusters has not yet been investigated.

In this work, we studied the melting behavior of Pd–Pt nanoclusters confined in SWCNTs in order to determine the effects of cluster size, diameter, and chirality of nanotubes, and the composition of nanoclusters using MD simulation. Finally, the effects of the previously mentioned parameters will be compared and discussed.

Calculation method

MD simulations were used to investigate the melting behavior of (Pd_xPt_1−x)_n bimetallic nanoclusters (with n = 108 and 256, x_Pd = 0.3, 0.5, and 0.7) confined in SWCNTs with different sizes, diameters, and chiralities. In order to analyze the effects of diameter and chirality of SWCNTs on the melting behavior of (Pd_xPt_1−x)_n bimetallic nanoclusters, three different SWCNTs were selected: ((22,22), (30,30), and (38,0)). The SWCNT systems chosen in the present study include two armchair nanotubes and one zigzag nanotube. The nanotubes were carefully chosen in order to address the following fundamental issues. First, in order to understand the diameter effect on the melting behavior of nanoclusters, we selected nanotubes with different diameters ((22,22) and (30,30)). Next, in order to investigate the effect of nanotube chirality, we intentionally chose nanotubes with different chiral architectures and similar diameters ((22,22) and (38,0)).

The Pd–Pt nanocluster is encapsulated in the SWCNT. Each system includes a (Pd_xPt_1−x)_n nanocluster confined in an infinitely long SWCNT (simulated by a box replicated using periodic boundary conditions along the tube axis).

In these simulations, the nanocluster inside the SWCNT was melted by bringing it to a temperature above 1300 K for 1 ns. Then, a slow cooling process started at 1300 K, with a temperature step of 50 K and time step of 1 fs. MD simulations continued until the temperature reached 300 K, and the final structure at 300 K was considered to be the initial configuration for a slow heating process. In this process, each structure was heated from 300 K to 1400 K with temperature steps of 100 K. Near the melting point, the temperature increments were reduced to 10 K. The simulations were carried out for 1 ns of equilibration followed by a production time of 2 ns for generating time-averaged properties.

The MD simulations were carried out in canonical ensemble (NVT). To keep the temperature constant, a Berendsen thermostat²² with a relaxation time of 0.1 ps was applied. The equations of motion were integrated using the Verlet–Leapfrog algorithm with a time step of 1 fs. The analysis of simulated trajectories and the calculations for different properties were performed using the utilities of the DL_POLY 4.03 program.²³

To successfully predict the properties of Pd–Pt nanoclusters, we used the many-body quantum Sutton–Chen potential (QSC) for Pt–Pt, Pd–Pd, and Pd–Pt interactions.²⁰ For the rest of the interactions ((Pd–C) and (Pt–C)), we used the Lennard-Jones 12–6 (L-J) potential.²¹ The L-J parameters for dissimilar atoms are obtained by the Lorentz–Berthelot mixing rules. A static substrate with fixed positions for carbon atoms was used to reduce the computational load. The mentioned force fields give reliable results in comparison with the experimental data, which were confirmed by Sankaranarayanan et al.²⁰ for the application of QSC potential for Pd–Pt bimetallic nanocluster and using the LJ potential for C–Pd and C–Pt interactions.²¹

To determine the melting point with increased accuracy, we calculated the specific heat capacity in a constant volume as follows:²⁰

where N is the number of atoms in the cluster, E_p is the potential energy, k is the Boltzmann constant, and α is the ratio of the standard deviations of kinetic and potential energies. In canonical ensemble with a coupling parameter of 0.1 ps, C_v can be obtained by eqn (2):

Also, in order to examine the nanocluster melting point, we employed the deformation parameter ε(r) as a symptom of phase transition of nanoclusters that is defined as:²¹

where r_i refers to either the position coordinates of the ith atom in an N-atom cluster along one of the three directions, and r_CM is the nanocluster center of mass.

Results and discussion

The results of this work are separately discussed in four sections as follows:

(a) Effect of carbon nanotube chirality

The potential energy versus temperature for the Pd₁₂₈Pt₁₂₈ nanocluster in SWCNTs with chirality of (22,22) and (38,0), which have the same diameters, is presented in Fig. 1. As shown in this figure, the difference between the two mentioned chiralities becomes smaller near the melting point, and this difference is eliminated at higher temperatures. This is a reasonable result due to the fact that increasing temperature provides additional kinetic energy to the metal atoms, which enables them to overcome the different nanocluster–tube interactions in the two mentioned nanotube chiralities. The melting temperatures of the Pd₁₂₈Pt₁₂₈ nanoclusters in the (22,22) and (38,0) nanotubes were 1240 ± 10 K and 1260 ± 10 K, respectively. Therefore, melting of Pd–Pt nanoclusters is easier when it occurs in the armchair nanotubes. In the applied force field, C–Pd and C–Pt pair potentials are the same for both armchair and zigzag nanotubes; however, a different arrangement of C atoms relative to the nanocluster atoms in armchair and zigzag tubes leads to different cluster–tube interactions. Therefore, the effect of chirality is not implemented within a simple pair potential, but rather, it is related to a number of C atoms that directly interact with the metallic nanocluster.

Fig. 1 Potential energy versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0).

Radial distribution function (RDF) plots of the system can be helpful to understand this property. The RDF plots for C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for the Pd₁₂₈Pt₁₂₈ nanocluster confined in both (22,22) and (38,0) SWCNTs at the temperatures of 300 K, melting temperature, and 1500 K are presented in Fig. 2. By comparison between C–Pd and C–Pt pair potentials in Fig. 2, the first important result is the increased C–Pd and C–Pt interactions for the (22,22) chirality in comparison with the (38,0) SWCNTs. The first peak of the RDF plots for (22,22) is higher than that of the (38,0), except for C–Pt in the (22,22) tube. Therefore, the Pd₁₂₈Pt₁₂₈ nanocluster has more interface atoms with the (22,22) nanotube, and short-range repulsive forces are more important in this condition, which leads to more positive potential energy of the nanocluster for the (22,22) tube (see Fig. 1).

Fig. 2 RDF plots of C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for Pd₁₂₈Pt₁₂₈ nanoclusters confined in both (22,22) and (38,0) SWCNTs at temperatures of 300 K, melting temperature, and 1500 K.

The second result of the RDF plots for C–Pd and C–Pt pairs is the increased concentration of Pd atoms near the tube wall at all three temperatures for the (22,22) armchair nanotube; however this difference is not considerable for the (38,0) nanotube. Despite these exceptions, the differences in surface concentrations between Pd and Pt become smaller at higher temperatures, which are expected due to the increased kinetic energy of atoms under these conditions. Moreover, a higher first peak of the C–Pd pair for the (22,22) tube implies a stronger nanotube–nanocluster interaction when the tube possesses armchair chirality. Also, the significant difference between the C–Pt interaction in the (22,22) nanotube and the C–Pd interaction in the (38,0) nanotube at 300 K presented in Fig. 2 reflects the important effect of nanotube chirality on the structural arrangement of bimetallic nanoclusters inside the nanotube at room or lower temperatures.

A comparison between RDF plots for Pd–Pd, Pd–Pt, and Pt–Pt in the (22,22) tube with those of the (38,0) tube shows the more ordered structure of the nanocluster inside the (22,22) SWCNT. This more disordered structure of clusters in the zigzag nanotube can be related to its weaker interaction with SWCNTs, which leads to increased nanocluster stability. This conclusion can be understood by simple thermodynamic logic as follows. Suppose that E_T is the internal energy of the nanotube, and E_C(A) and E_C(Z) are taken as the internal energy of the nanocluster inside the armchair and zigzag tubes, respectively. Also, E_TC(A) and E_TC(Z) are considered to be the interaction energy between the tube and cluster for armchair and zigzag chiralities, respectively. The total internal energy E_tot of the system can be obtained as:

E_tot = E_T + E_C(A) + E_TC(A) = E_T + E_C(Z) + E_TC(Z) (4)

where the conservation of total energy is considered due to the equal number of atoms in both cases. If the atomic coordinates of the nanotube are held fixed during the simulation, E_T is the same for both of the abovementioned systems. Therefore, we can conclude that:

E_C(A) + E_TC(A) = E_C(Z) + E_TC(Z) (5)

On the basis of the RDF plots obtained from simulations, the nanocluster has a stronger interaction with the armchair nanotube, i.e., E_TC(A) < E_TC(Z), which indicates that E_TC(A) is more negative. Hence, in order to maintain the above equality, E_C(A) should be more positive than E_C(Z) or E_C(A) > E_C(Z). This indicates that the nanocluster has more stability inside the zigzag structure, which can appear as a higher melting temperature of nanoclusters under these circumstances. Therefore, it is reasonable that nanoclusters should have a more disordered structure inside the zigzag tube in order to reduce its internal energy. This disordered structure leads to wider peaks in the RDF plot for pair interactions inside the nanocluster body (see Fig. 2). Another interesting result of the RDF plots for C–Pd and C–Pt potentials is the different diffusive behavior of Pt atoms resulting from a change in temperature for armchair and zigzag nanotubes. For the (22,22) tube, Fig. 2 for C–Pd and C–Pt pairs shows the diffusion of both Pd and Pt atoms toward the tube wall in such a way that the first peaks of C–Pd and C–Pt pair densities obey a 300 K < 1500 K < melting temperature trend. This denotes that as an overall view, the diffusion of Pd and Pt atoms towards the tube wall is increased by temperature, but at the melting temperature, this diffusion reaches a maximum value that can be described by the variation of kinetic energy of atoms near the melting temperature. The internal energy and hence kinetic energy of the particles exhibit a great variation near a phase transition such as melting, and thus, it is expected that under these circumstances, Pd and Pt atoms will be able to diffuse out from the center of mass of the cluster towards the nanotube wall.

This prediction can be confirmed by a comparison of nanocluster diffusion coefficients inside the nanotube as a function of temperature, which is presented in Fig. 3. In this figure, the diffusion coefficient of Pd₁₂₈Pt₁₂₈ nanoclusters confined in (22,22) and (38,0) SWCNTs versus temperature exhibits a sudden variation near the melting point, which is in agreement with the results of Feng et al. for the melting of ionic liquid²⁴ and Qiao et al. for the melting of gold nanoparticles.²⁵ However, for the (38,0) nanotube, the first RDF peaks of C–Pd and C–Pt potentials have less sensitivity to the temperature, which reflects the important effect of chirality on the melting behavior of the nanocluster. Moreover, the first RDF peaks of C–Pd in both the (22,22) and (38,0) nanotubes are more than those of C–Pt. Also, the first RDF peaks of Pd–Pd in both the armchair and zigzag nanotubes are more than those of Pd–Pt, which implies a core–shell structure of nanoclusters where Pd atoms tend to aggregate at the surface of the nanocluster, while Pt atoms tend to aggregate in its core. Therefore, the chirality of the SWCNT has a significant effect on the melting behavior of the Pd–Pt nanocluster but it cannot change its core–shell regime.

Fig. 3 The diffusion coefficient versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0).

The isochoric heat capacity (C_v) versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0) is presented in Fig. 4. Corresponding to the potential energy jumping around the melting temperature of the cluster, the heat capacity also exhibits a large and sharp peak at this temperature. Unlike the potential energy, the C_v of the nanocluster is not considerably affected by the chirality of the nanotubes. If each side of eqn (5) is considered as the total energy of the cluster, and C_v is assumed as the variation of this energy relative to temperature, then the variation of total energy of the nanocluster in the armchair and zigzag nanotubes is approximately the same. Also, the deformation parameter versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0) is shown in Fig. 5, which illustrates the deformation parameter increasing with temperature and exhibits the diffusion of both of Pd and Pt atoms outward from the nanocluster center of mass. Near the melting temperature, the deformation parameter sharply changes, and the cluster reaches a liquid-like motion with diffusion of atoms. However, as shown in Fig. 5, variation of the deformation parameter for nanoclusters with temperature does not depend on the chirality of the nanotube. This low sensitivity of the deformation parameter to the chirality of the nanotubes can be described by the same force field parameters for nanoclusters inside the armchair and zigzag tubes, because the deformation parameter depends only on the coordinates of cluster atoms, and its center of mass and is not directly related to cluster–tube interactions.

Fig. 4 C_v versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0).

Fig. 5 Deformation parameter versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0).

Finally, the Wigner parameter (W₆) versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0) is presented in Fig. 6, which shows that the Wigner parameter of the nanocluster undergoes a transition from an fcc to a hcp-like structure near the melting point. Also, this transition does not depend on the chirality of the nanotube. This independence can be attributed to the stronger metal–metal versus metal–tube interactions. For other concentrations of Pd–Pt nanoclusters confined in other nanotubes with different diameter and chirality, the same trend was observed.

Fig. 6 The Wigner parameter (W₆) versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in SWCNTs with chirality of (22,22) and (38,0).

(b) Effect of the carbon nanotube diameter

The potential energy versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in the (22,22) and (30,30) armchair SWCNTs is presented in Fig. 7. Similar to Fig. 1, there is a jump in potential energy around the melting point of the cluster. However, unlike Fig. 1, the melting temperatures of the cluster in the (22,22) and (30,30) nanotubes with different diameters is approximately the same, 1240 ± 10 K. This insensitivity of melting point to the diameter of the nanotube can be explained by the similarity of the RDF for C–Pd pairing of (22,22) and (30,30) nanotubes at 300 K. The RDF plots for C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for Pd₁₂₈Pt₁₂₈ nanoclusters confined in both the (22,22) and (30,30) SWCNTs at 300 K, melting temperature, and 1500 K are presented in Fig. 8. The C–Pd pair density is similar for the (22,22) and (30,30) nanotubes, and this similarity remains at higher temperatures. The same trend is also observed for C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair densities. This is a reasonable result, due to the fact that in both the (22,22) and (30,30) cases, the metal–metal and metal–tube interactions are the same but the number of interactions may be different. Therefore, the different numbers of corresponding interactions may change the zero energy only, which appears in Fig. 7 in such a way that the variation of potential energy by temperature is very similar and leads to the same melting temperature. Moreover, by comparing the RDF plots of Fig. 2 and 8, it is observed that unlike Fig. 2, there is no considerable difference between the first peak of the RDF plots of C–Pd for the (22,22) and (30,30) nanotubes at 300 K in Fig. 8. Similar behavior can be observed for C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt interactions in this figure. This trend is almost reversed at higher temperatures, although C–Pt interaction shows a greater difference between two tubes with different diameters (see Fig. 8). However, similar to Fig. 2, the difference between the first RDF peak for C–Pd and C–Pt is significant, which reflects the stronger C–Pd interaction and is clear from Fig. 8. Moreover, the first RDF peaks of Pd–Pd are higher than those of Pd–Pt, which emphasizes the core–shell structure of the nanotubes, as previously mentioned. Similar to Fig. 3, the diffusion coefficient of Pd₁₂₈Pt₁₂₈ nanoclusters confined in (22,22) and (30,30) SWCNTs versus temperature is shown in Fig. 9. Again, near the melting point of the nanoclusters, its diffusion coefficient is suddenly increased, and the effect of nanotube diameter is seen in the difference between the diffusion coefficient of nanoclusters inside the (22,22) and (30,30) nanotubes with the same chirality and different diameters after the melting point.

Fig. 7 Potential energy versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) armchair SWCNTs.

Fig. 8 RDF plots of C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) armchair SWCNTs at the temperatures of 300 K, melting temperature, and 1500 K.

Fig. 9 The diffusion coefficient versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) armchair SWCNTs.

The C_v versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) SWCNTs is presented in Fig. 10, which shows that the effect of the nanotube diameter on the heat capacity of nanoclusters is not significant. It is noticeable that thermodynamic properties that depend only on the variation of energy, such as heat capacity and entropy, are not different between the two systems, which differ only by the zero of energy. Also, the deformation parameter versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) SWCNTs is shown in Fig. 11. Similar to heat capacity, there is no considerable difference between the deformation parameter of the nanoclusters inside the (22,22) and (30,30) nanotubes. Then, the melting temperature of Pd₁₂₈Pt₁₂₈ nanoclusters is obtained, and it is the same as that which appears in Fig. 7 and 10. Moreover, the Wigner parameter versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) SWCNTs is represented in Fig. 12. Again, similar to the heat capacity and deformation parameter, there are no observed differences between (22,22) and (30,30) nanotubes for Wigner parameters of the nanocluster located inside them. Similar to Fig. 6, an fcc to hcp-like transition near the melting point is observed for Pd₁₂₈Pt₁₂₈ nanoclusters confined in (22,22) and (30,30) nanotubes. Therefore, it can be concluded that when interatomic forces inside the cluster are sufficiently strong, the diameter of the nanotube cannot considerably affect its physical properties. This phenomenon was observed for other sizes of (Pd_xPt_1−x)_n clusters inside other nanotubes with the same chirality and various diameters. Overall, the nanotube diameter has a negligible effect on the melting behavior of the bimetallic nanoclusters confined in them.

Fig. 10 C_v versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) armchair SWCNTs.

Fig. 11 Deformation parameter versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) armchair SWCNTs.

Fig. 12 The Wigner parameter versus temperature for Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) and (30,30) armchair SWCNTs.

(c) Effect of the nanocluster size

The potential energy versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs is presented in Fig. 13. As expected, these nanoclusters exhibit different melting temperatures: 1240 ± 10 K for the larger nanocluster and 1130 ± 10 K for smaller one. It is well-known that smaller nanoclusters have less thermodynamic stability due to the fact that the smaller clusters have a larger surface/bulk ratio, which leads to lower cohesive energy because the surface atoms have less binding energy in comparison with the inner atoms of the cluster. The cohesive energy of a nanocluster can be expressed by:²⁶where E_C(N) and E_b0 are cohesive energies of the N-atom cluster and complete crystalline bulk, respectively. B_a is the rest bond order or actual bond order of the cluster surface, and B_t is the bond order in the perfect crystalline bulk. By decreasing the cluster size, B_a is decreased, due to the larger surface/bulk ratio and lower bond order of the surface atoms, which lead to increased thermodynamic instability of smaller nanoclusters.

Fig. 13 Potential energy versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs.

As the cohesive energy of the nanocluster decreases with its size, its tendency to form nanotubes increases in order to increase its cohesive energy and thermodynamic stability. This argument can be understood from RDF plots of C–Pt and C–Pd pair potentials for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters. The RDF plots of C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters confined in the (22,22) nanotube are shown in Fig. 14. The RDF plots for C–Pd and C–Pt in Fig. 14 show that the first peaks of C–Pd and C–Pt pair densities for Pd₅₄Pt₅₄ clusters is significantly more than those of Pd₁₂₈Pt₁₂₈. This indicates that the former has a stronger interaction with the tube wall. Also, by comparison between RDF plots of C–Pd and C–Pt potentials, the first nearest neighbors of C atoms are mainly Pd, which shows the stronger interaction of Pd with the tube wall. However, the core–shell structure of the cluster is not maintained for the Pd₅₄Pt₅₄ nanoclusters. In Fig. 14, the relative intensity of the RDF peaks of Pd–Pd, Pd–Pt, and Pt–Pt pair densities for Pd₅₄Pt₅₄ nanoclusters is not the same as that of the Pd₁₂₈Pt₁₂₈ nanoclusters; the former exhibits a Pd–Pt > Pt–Pt > Pd–Pd trend, while the latter obeys a Pd–Pd > Pt–Pt > Pd–Pt trend. Therefore, the core–shell structure for Pd₅₄Pt₅₄ nanoclusters is not confirmed. It seems that Pd₅₄Pt₅₄ exhibits increased mixing of Pd and Pt atoms, which is considerably different than that of the Pd₁₂₈Pt₁₂₈ nanocluster with a core–shell arrangement of metallic atoms. In Fig. 15, the diffusion coefficients of Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs versus temperature are presented, and as expected, both of the nanoclusters exhibit a jump in the diffusion coefficient near the melting point. However, the Pd₅₄Pt₅₄ nanocluster is more diffusive due to its smaller size and easier mobility in comparison with the Pd₁₂₈Pt₁₂₈ nanocluster.

Fig. 14 RDF plots of C–Pd, C–Pt, Pd–Pd, Pd–P, and Pt–Pt pair interactions for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs at the temperatures of 300 K, melting temperature, and 1500 K.

Fig. 15 The diffusion coefficient versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs.

The C_v versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs is presented in Fig. 16. As expected, the melting temperature of the Pd₅₄Pt₅₄ nanocluster is higher than that of Pd₁₂₈Pt₁₂₈, which reflects the increased thermodynamic instability of the Pd₅₄Pt₅₄ nanocluster and was discussed above. Also, the same result can be obtained from the plot of the deformation parameter versus temperature, which is presented in Fig. 17 for both Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs. It is noticeable that the deformation parameter at each temperature for Pd₁₂₈Pt₁₂₈ nanoclusters is more than that for Pd₅₄Pt₅₄ nanoclusters due to the fact that the decreased cohesive energy and thermodynamic stability of Pd₅₄Pt₅₄ clusters do not permit its components to be located at large distances from each other. Finally, the Wigner parameter versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs is presented in Fig. 18. Similar to the previous effects (chirality and diameter of the nanotube), both Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters exhibit a fcc to hcp-like transition before melting. This result is in agreement with that of Sankaranarayanan et al. for bare Pd–Pt bimetallic nanoclusters.²⁰ Therefore, it can be concluded that this fcc–hcp structural phase transition is an intrinsic property of the nanocluster that does not depend on its environment.

Fig. 16 C_v versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs.

Fig. 17 Deformation parameter versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs.

Fig. 18 The Wigner parameter versus temperature for Pd₅₄Pt₅₄ and Pd₁₂₈Pt₁₂₈ nanoclusters in (22,22) SWCNTs.

(d) Effect of the nanocluster composition

The potential energy versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs is presented in Fig. 19. The melting temperature is decreased by increasing Pd concentration in such a way that values of 1320 ± 10 K, 1240 ± 10 K, and 1150 ± 10 K are obtained, respectively for x_Pd = 0.3, 0.5, and 0.7. This is a reasonable result because, as mentioned earlier, Pd atoms have a stronger interaction with the nanotube wall. If we use a simple equation such as eqn (5) for two nanoclusters with the same size but different Pd mole fractions x₁ and x₂ (x₁ > x₂), we have:

E_C(x1) + E_TC(x1) = E_C(x2) + E_TC(x2) (7)

Fig. 19 Potential energy versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs.

If the interaction between the tube and cluster for cluster 1 is more than 2, E_TC(x1) is more negative than E_TC(x2), and hence, due to the satisfaction of eqn (7), E_C(x1) should be more positive than E_C(x2) or E_C(x1) > E_C(x2), which indicates that nanoclusters with a higher concentration of Pd have lower thermodynamic stability and melting temperatures. The RDF plots of C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for the (Pd_xPt_1−x)₂₅₆ nanocluster with (x_Pd = 0.3, 0.5, and 0.7) confined in the (22,22) SWCNTs are presented in Fig. 20. As shown in this figure, the first RDF peaks of C–Pd potential for the mentioned mole fractions of Pd obey a 0.7 > 0.3 > 0.5 trend. By comparison of this result with that in Fig. 19, one can conclude that different mole fractions of Pd lead to different nanocluster shapes inside the nanotube. Moreover, in Fig. 20, the first RDF peaks of C–Pt potential exhibit a 0.3 > 0.7 > 0.5 trend, which confirms a complex variation of the cluster shape by its Pd concentration. Also, by comparison of the Pd–Pd, Pd–Pt, and Pt–Pt RDF plots in Fig. 20, one can see a Pt–Pt > Pd–Pd > Pd–Pt trend for all x_Pd = 0.3, 0.5, and 0.7 concentrations, which demonstrates the core–shell structure of these clusters. This indicates that the core–shell arrangement of Pt and Pd atoms does not depend on the composition of the nanocluster, but rather, it depends on the size of the nanocluster, which was mentioned in Section (c) of the Discussion. The increased thermodynamic instability of nanoclusters with a higher Pd content also can be concluded from diffusion coefficients of the mentioned nanoclusters inside the nanotube. The diffusion coefficients versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) confined in the (22,22) SWCNTs are presented in Fig. 21. The diffusion coefficients for the (Pd_xPt_1−x)₂₅₆ nanocluster obey a x_Pd = 0.3 < x_Pd = 0.5 < x_Pd = 0.7 trend, which reflects the increased mobility and decreased melting point of the (Pd_0.7Pt_0.3)₂₅₆ nanocluster.

Fig. 20 RDF plots of C–Pd, C–Pt, Pd–Pd, Pd–Pt, and Pt–Pt pair interactions for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs at the temperatures of 300 K, melting temperature, and 1500 K.

Fig. 21 The diffusion coefficient versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs.

The C_v versus temperature for the (Pd_xPt_1−x)₂₅₆ nanocluster with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs is presented in Fig. 22. The results of this figure are in agreement with those of the Fig. 19, which emphasize a decreased melting point and thermodynamic stability resulting from an increase in Pd concentration. Also, a similar result can be concluded from the plot of the deformation parameter versus temperature. The deformation parameter versus temperature for the (Pd_xPt_1−x)₂₅₆ nanocluster with (x_Pd = 0.3, 0.5, and 0.7) confined in (22,22) SWCNTs is presented in Fig. 23. As mentioned for Fig. 14, a decrease in thermodynamic stability of the nanocluster leads to its lower deformation parameter. Therefore, the (Pd_0.3Pt_0.7)₂₅₆ nanocluster with the highest thermodynamic stability exhibits the highest deformation parameter in comparison with (Pd_0.5Pt_0.5)₂₅₆ and (Pd_0.7Pt_0.3)₂₅₆ nanoclusters. Moreover, the Wigner parameter versus temperature for the (Pd_xPt_1−x)₂₅₆ nanocluster with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs is shown in Fig. 24. It is interesting that for all of the mentioned mole fractions of Pd, an fcc–hcp transition can be seen. Therefore, the fcc–hcp transition near the melting temperature of the Pd–Pt nanoclusters is an intrinsic property that does not depend on the concentration of the nanocluster. Overall, the Pd concentration of the nanoclusters has considerable effect on their melting behavior inside the SWCNTs.

Fig. 22 C_v versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs.

Fig. 23 Deformation parameter versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs.

Fig. 24 The Wigner parameter versus temperature for (Pd_xPt_1−x)₂₅₆ nanoclusters with (x_Pd = 0.3, 0.5, and 0.7) surrounded by (22,22) SWCNTs.

In Fig. 25, a snapshot is shown of simulated Pd₁₂₈Pt₁₂₈ nanoclusters inside SWCNTs with different diameters and chiralities at temperatures of 300 K, melting temperature, and 1500 K. At 300 K, the nanoclusters tend to have a layered structure that is approximately maintained up to melting temperature. However, at temperatures higher than melting, this layered structure is considerably perturbed, which is a reasonable result. This ordered–disordered transition is in agreement with RDF plots of Pd–Pd, Pd–Pt, and Pt–Pt pair potentials for the studied Pd–Pt nanoclusters confined in SWCNTs in such a way that the mentioned RDF peaks become wider, which reflects a liquid-like structure at high temperatures.

Fig. 25 A snapshot of Pd₁₂₈Pt₁₂₈ nanoclusters inside SWCNTs with different diameters and chiralities at the temperatures of 300 K, melting temperature, and 1500 K, simulated in this work.

Conclusion

In this work, the effects of chirality and nanotube diameter, accompanied with size and concentration of the nanocluster, on the melting behavior of (Pd_xPt_1−x)_n nanoclusters (with n = 108 and 256, x_Pd = 0.3, 0.5, and 0.7) surrounded by SWCNTs with different sizes, diameters and chiralities were studied by MD simulation. The important results of this study are as follows:

(1) The core–shell structure of the (Pd_xPt_1−x)_n bimetallic nanocluster is an intrinsic property that does not depend on its concentration or environment. However, it is strongly dependent on the size of the nanocluster.

(2) The effect of nanotube chirality on the melting of (Pd_xPt_1−x)_n bimetallic nanoclusters is significant, in such a way that the nanocluster inside the zigzag nanotube has increased thermodynamic stability, a higher melting temperature, and increased deformation ability. However, the chirality of the nanotube cannot change the core–shell regime of the nanocluster because, as mentioned before, this property does not depend on the environment of the nanocluster.

(3) The effect of nanotube diameter on the melting behavior of (Pd_xPt_1−x)_n bimetallic nanoclusters is negligible. Its effect is simply the change of zero energy, which does not appear in the thermodynamic properties that depend on the variation of the internal energy by temperature.

(4) The effect of the cluster size on the melting behavior of (Pd_xPt_1−x)_n bimetallic nanoclusters confined in a SWCNT is also considerable. Clusters with a smaller size have less cohesive energy, decreased melting temperature, less thermodynamic stability, and stronger interactions with tube walls. Also, the core–shell structure of the nanoclusters may be violated by variation in size.

(5) The Pd concentration of the nanocluster also has a significant effect on the melting properties of (Pd_xPt_1−x)_n bimetallic nanoclusters confined in a SWCNT. As the Pd concentration of the nanocluster increases, its melting temperature and thermodynamic stability decreases. The RDF plots for nanoclusters with different Pd concentration imply that the shape of the nanocluster varies depending upon its Pd concentration.

(6) During the melting of (Pd_xPt_1−x)_n bimetallic nanoclusters, an fcc to hcp-like transition is observed near their melting point. This phenomenon does not depend on the size or concentration of the nanocluster or the properties of its surrounding nanotube, and therefore is a universal property of these nanoclusters. However, additional studies are required to establish the universality of this phenomenon.

These results can be helpful for scientists who are trying to synthesize new nanomaterials and construct new devices in order to improve their efficiency by increased application of nanotechnology for this aim.

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