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The effects of size and shape on the structural and thermal stability of platinum nanoparticles
Computational Materials Science 169 (2019) 109090
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The effects of size and shape on the structural and thermal stability of platinum nanoparticles ⁎
Gang Wanga,b,c, Yi-Shuang Xub,c, Ping Qiana,b, , Yan-Jing Sua,
T
⁎
a
Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing, Beijing 100083, China Department of Physics, University of Science and Technology Beijing, Beijing 100083, China c TianJin College, University of Science and Technology Beijing, Tianjin 301830, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanoparticle Structural stability Thermal stability
A new EAM potential was obtained by fitting to the experimental data of bulk Pt, and the cohesive energy, surface energy, structure evolution during the heating process and melting point of Pt nanoparticles with different shapes and sizes were studied with the EAM potential. According to the calculation results and compared with other models, a model is developed to describe the cohesive energy and melting point as functions of the particle size and average coordination number.
1. Introduction
NPs.
Platinum (Pt) nanoparticles (NPs) have been widely used as catalysts in the chemical, pharmaceutical and fuel cell industries because of their excellent catalytic activity and chemical stability [1–3]. Various shapes of Pt NPs have been experimentally obtained by shape-controlled synthesis, including tetrahedron [4–6], octahedron [5–7], icosahedron [4,5,7], cube [4–7], cuboctahedron [4–7], truncated octahedron[5,6], sphere[6,7], tetrahexahedron [5,6], decahedron [5], and others. The size, shape, and reaction temperature of NPs have important effects on the catalytic activity [8]. Therefore, it is necessary to study the structural and thermal stability of Pt NPs with different sizes and shapes. The structural and thermal stability of metal NPs have been studied theoretically and experimentally [9–15]. The results show that as the size of NP decreases, the cohesive energy and melting point gradually decrease, and the structural and thermal stability deteriorate. However, these studies are less concerned with the effect of shape on stability, and the conclusions are inconsistent. Huang et al. proposed that the cohesive energies of Pt NPs with different shapes are mainly determined by the surface energy, and the melting point is roughly independent of the particle shape [13]. Qi et al. showed that spherical NP has the largest cohesive energy, the highest melting point, and the most stable structure among the same size NPs [14,15]. Mei et al. posited that the melting point of NPs should be a function of the particle size and surface energy [12]. Therefore, in this paper, we systematically studied the effect of the size and shape on the structural and thermal stability of Pt
2. Simulation methodology
⁎
2.1. Molecular dynamics Seven different shapes were taken into account in this paper: cube (C), tetrahedron (T), octahedron (O), cuboctahedron (CO), truncated octahedron (TO), sphere (SP), and tetrahexahedron enclosed with {2 1 0} facets (TTH). Because of computing limitations, when studying the structural stability of NPs, the size range of Pt NPs is 2–20 nm, which contains approximately 400–400000 atoms. When researching the thermal stability of NPs, the size range of Pt NPs is 2–10 nm, which contains about 400–50000 atoms. To study the effects of the size and shape on the thermal stability of Pt NPs, we calculated the structural evolution during the heating process and the melting point of Pt NPs using molecular dynamics simulation. Throughout the simulation process, the isothermal-isobaric ensemble (NPT), Nose-Hoover thermostat and Berendsen barostat were adopted, and the time step was 1 fs. Starting from 0 K, the system temperature was increased by 20 K within 40 ps, and then the system was equilibrated by 40 ps under a constant temperature. These heating and equilibrium processes were repeated until the NP completely melted. The output value of each physical quantity was obtained from the statistical average value in the last 20 ps of the equilibrium process.
Corresponding authors. E-mail addresses: [email protected] (P. Qian), [email protected] (Y.-J. Su).
https://doi.org/10.1016/j.commatsci.2019.109090 Received 1 March 2019; Received in revised form 23 May 2019; Accepted 17 June 2019 0927-0256/ © 2019 Published by Elsevier B.V.
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Nomenclature NP C T O CO
TO SP TTH ACN CN CE SED
Nanoparticle Cube Tetrahedron Octahedron Cuboctahedron
vacancy migration energy is obtained by the nudged elastic band (NEB) method [25], and the melting point is obtained via the solid liquid coexistence method [26]. Table 2 shows that the results obtained by the present EAM potential are in good agreement with the experimental data. The equation of state (the energy as a function of the nearest neighbour distance) can reflect the interaction between atoms in largescale deformation. Therefore, it is important to the study of the properties related to deformation. The equation of state of Pt calculated with the present EAM potential is shown in Fig. 1. For comparison, the experimental Rose equation [27] are also plotted in Fig. 1. Fig. 1 shows that the calculation results of the present EAM potential agree well with the experimental data. For further verification, the pressure-volume curve is demonstrated in Fig. 2. The pressure-volume relationship calculated by the present EAM potential provides a good agreement with the experimental curve [28]. It shows that the present EAM potential is effective for large-scale deformation of Pt. Linear thermal expansions of Pt calculated with the present EAM potentials in comparison with experimental data [29] are illustrated in Fig. 3, proving that the present EAM potential can well describe the thermal expansion properties of Pt. It can be concluded from the analysis that describing the properties of Pt using the present EAM potential is effective.
2.2. Interatomic interaction potential 2.2.1. Construction of EAM potential The interatomic interaction potential is in the form of EAM potential. In the EAM model [16], the total energy of crystal can be expressed as
Etot =
∑ Fi (ρi ) + i
ρi =
∑
1 2
∑
φij (rij )
i, j (i ≠ j )
f j (rij )
(1)
(2)
j (j ≠ i)
where Fi is the embedding energy of atom i, ρi is the host electron density at atom i, φij is the pair potential between atoms i and j, rij is the distance between atoms i and j, and f j is the atomic electron density contributed by atom j. In this paper, the pair potential function is a polynomial
φ (r ) = (k1 r −12 + k2 r −6 + k3 r −3 + k 4 r −2 + k5 r −1 + k6 + k 7 r + k8 r 6)Ψ ⎛ r − rc ⎞ ⎝ h1 ⎠ ⎜
Truncated octahedron Sphere Tetrahexahedron Average coordination number Coordination number Cohesive energy Surface energy density
⎟
(3)
where Ψ(x ) is a cutoff function defined as [17] 4
Ψ(x ) =
⎧ x 4 x < 0, 1+x ⎨ 0, x ⩾ 0. ⎩
(4)
3. Results and discussion
ki (i = 1, 2, …, 8) , rc and h1 are parameters for fitting. The function of the embedding energy is a piecewise function 5
F (ρ) =
If the NP is an ideal cube, a3 = Na03/4 , which obtains a = (N /4)1/3a0 , where a is the side length of the cube, a 0 is the lattice constant of the bulk Pt, and N is the atomic number of the NP. If the NP is an ideal sphere, πD3 /6 = Na03/4 , which obtains D = (N /2.09)1/3a0 , where D is the diameter of the sphere. The size parameters of NPs with the same atomic numbers but different shapes vary. To facilitate the comparison of NPs with the same atomic numbers but different shapes, we use the value between the above two values as the particle size. The NP size d is uniformly defined as
ρ
⎧ F0 + ∑i = 2 Fi ( ρ − 1)i ρ < 1.15ρe e
⎨ ∑3 Fni ( ρ − 1)i i=0 ρe ⎩
ρ ⩾ 1.15ρe
(5)
where ρe is the equilibrium value of the host electron density, Fi (i = 0, 2, 3, 4, 5) and Fni (i = 0, 1, 2, 3) are adjustable parameters. The function of the atomic electron density is expressed in the form of exponent
r − rc ⎞ f (r ) = p0 exp(−p1 r )Ψ ⎛ , ⎝ h2 ⎠ ⎜
d=
⎟
(6)
where p0 , p1, rc and h2 are fitting parameters. The fitting database of the EAM potential includes the experimental values of the lattice constant, cohesive energy, equation of state, elastic constants, vacancy forming energy and linear thermal expansion. The fitting results of the potential parameters are listed in Table 1. In this work, all of the calculations using the EAM potential are performed by LAMMPS [18].
3
N a0 . 3
(7)
Table 1 Parameters of the EAM potential of Pt. Parameter rc (Å) h1 (Å) h2 (Å) k1 (eV·Å12) k2 (eV·Å6) k3 (eV·Å3) k4 (eV·Å2) k5 (eV·Å) k6 k7 (eV·Å−1) k8 (eV·Å−6) ρe (Å−3)
2.2.2. Validity of the EAM potential To test the validity of the present EAM potential, various properties of Pt calculated with the present EAM potential are compared with the experimental [19–24] data in Table 2. These properties include the lattice constant (a 0 ), cohesive energy (Ec ), elastic constants (Cij ), energy differences between alternative structures and FCC structure (ΔEHCP → FCC and ΔEBCC → FCC ), vacancy formation energy (Evf ), vacancy migration energy (Evm ), stacking fault energy (γsf ), twin boundary energy (γtb ), surface energy density (γ ) and melting point (Tm ). The 2
Value 7.46652 1 0.7 2.35234 × 103 1.14886 × 103 −5.77507 × 102 4.03998 × 102 −1.13056 × 102 1.47760 × 10 −7.65104 × 10−1 1.26828 × 10−6 1.65 × 10−2
Parameter −3
p0 (Å ) p1 (Å−1) F0 (eV) F2 (eV) F3 (eV) F4 (eV) F5 (eV) Fn0 (eV) Fn1 (eV) Fn2 (eV) Fn3 (eV)
Value 7.99899 × 10−2 1.53013 −4.3 3.37821 4.06570 × 10−1 7.93970 × 10−1 −5.34387 × 10−1 −4.30012 −2.06221 × 10−2 3.66205 −3.66231 × 10−1
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Table 2 Properties of Pt calculated with the present EAM potential in comparison with the experimental data. Property
Experiment
EAM
a0 (Å) Ec (eV) C11 (GPa) C12 (GPa) C44 (GPa) ΔEHCP→FCC (eV) ΔEBCC→FCC (eV) Evf (eV) Evm (eV) γsf (mJ/m2) γtb (mJ/m2) γ110 (mJ/m2) γ100 (mJ/m2) γ111 (mJ/m2) Tm (K)
3.92a 5.84a 358b 253.6b 77.4b 0.03c 0.16c 1.51d 1.43d
3.92 5.84 365 258.7 80.4 0.03 0.11 1.64 1.00
322e 161e 2691f 2691f 2691f 2042c
164 83 2405 2258 2107 2000
a b c d e f
Fig. 2. Pressure-volume relationship of Pt calculated with the present EAM potential in comparison with the experimental data.
Ref. [19]. Ref. [20]. Ref. [21]. Ref. [22]. Ref. [23]. Mean value of polycrystalline, Ref. [24].
Fig. 3. Linear thermal expansion of Pt calculated with the present EAM potential in comparison with the experimental data.
Fig. 1. Equations of state of Pt calculated with the present EAM potential in comparison with the experimental data.
3.1. Size and shape dependent structural stability of Pt NPs 3.1.1. Average coordination number The average coordination number (ACN) is a very important parameter of NP, and many of the properties of NP are affected by it [30,31]. If the geometrical structure of NP is determined, the ACN is a definite value. We first calculate the coordination number (CN) of each atom by LAMMPS, and then obtain ACN by taking its average value, as shown in Fig. 4. As shown in Fig. 4, the ACNs of Pt NPs are smaller than the CN of bulk Pt. As the NP size decreases, the ACN of the NP descends. This is because the CNs of the surface atoms of the NPs are less than that of the internal atoms, and the presence of the surface causes the ACNs of the NPs to be less than the CN of the bulk. As the NP size decreases, the proportion of the surface atoms increases, which results in the reduction of the ACN. Fig. 4 also shows that the ACNs of the NPs with the same size but different shapes vary: the ACN of the TO structure is the largest, followed by the O, CO, and SP structures (the ACNs of the three structures are approximately the same), and then the TTH, C and T
Fig. 4. Size dependent average coordination number (ACN) of Pt nanoparticles with different shapes. 3
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structures.
ES = γA = Enp − NEb = −NEcp + NEcb 3 = − 3 (A1 d 2 + A2 d + A3 + B1 d3 (12−ACN) B2), a0
3.1.2. Cohesive energy The cohesive energy (CE) of Pt NPs with different sizes and shapes was calculated with the EAM potential, as shown in Fig. 5(a). Fig. 5(a) demonstrates that the CE decreases with the increase of NP size d . Due to the presence of surface energy, the CE of the surface atom of NP is lower than that of the internal atom, which causes the CE of NP to be smaller than that of bulk. As d reduces, the ratio of the surface atoms number increases, which leads to the decrease of the CE. As shown in Fig. 5(a), the CEs of NPs with the same size but different shapes vary: the CE of the TO structure is the largest, followed by the O, CO, and SP structures (the CEs of these three structures are basically the same), and then the TTH, C and T structures. According to Qi’s generalised bond-energy model [32], the CE of NP (Ecp ) can be written as
Ecp =
1 (Ni E¯ i + Nf E¯ f + Ne E¯e + Nc E¯c ), N
Wei et al. [34] reported that if different methods are used to estimate the surface area of NP, the calculated SED has an opposite sizedependent trend. The SED of NP was found to either increase or decrease with the increase of the NP size [34]. Therefore, it is very important to correctly calculate the SED. Due to the inhomogeneous lattice distortion [37], the surface shape of the NP is no longer the ideal shape, and it is difficult to calculate the surface area theoretically. In this work, the OVITO software package [38] was used to construct the surface grid and calculate A . The SEDs of Pt NPs with different sizes and shapes were computed with the present EAM potential, the results are illustrated in Fig. 6. Fig. 6 shows that γ decreases with the increase of d , which is the same as the previous calculations[39–42]. Moreover, the γ of the NPs with the same size but different shapes vary. The γ of O and T structures enclosed with {1 1 1} facets are the lowest, followed by the TO and CO structures enclosed with {1 1 1} and {1 0 0} facets, then the C structure enclosed with {1 0 0} facets, then the SP structure enclosed with {1 1 1}, {1 0 0} and high-index facets, and finally the TTH structure enclosed with {2 1 0} facets. This result can be explained qualitatively by the SED of bulk. The SEDs of the {1 1 1}, {1 0 0} and {2 1 0} planes in the FCC bulk crystal increase sequentially. The larger the SEDs of the enclosed facets, the higher the SED of NP. However, when d is less than 5 nm, γ of the TO structure is smaller than that of the T structure. This may be related to the lattice distortion of NP. Eq. (12) demonstrates that when the size are the same, the lower the surface energy of the NP, the greater the CE and the more stable the structure. As shown in Figs. 5 and 6, the SP structure has the smallest surface area at the same NP size, but its SED is relatively high (only lower than that of the TTH structure). The competition of SED and surface area makes the CE of the SP structure smaller than that of the TO structure, and larger than that of the C, T and TTH structures. Therefore, the CE of NPs with the same size depends on the competition results of SED and surface area.
(8)
where N , Ni , Nf , Ne , and Nc are the numbers of total, interior, face, edge and corner atoms, respectively; E¯ i , E¯ f , E¯e , and E¯c are the average CEs of interior, face, edge, and corner atoms, respectively. According to the Eqs. (7) and (8), the relationship between Ecp and d can be approximately written in polynomial form [13,33]
Ecp = A +
B C D + 2 + 3, d d d
(9)
where A , B , C and D are the fitting parameters, which vary for different shapes of NPs. As shown in Figs. 4 and 5(a), for Pt NPs with the same size but different shapes, the larger the ACN, the greater the CE. So the CE is not only related to d , but also related to the ACN. This is because the greater the CN of an atom, the more bonds between the atom and its surrounding atoms, the better the structural stability, and the larger the CE. Considering that when d tends to infinity and the ACN tends to 12, the CEs of Pt NPs should be the same as the CE of the bulk Pt. We add an additional term after Eq. (9) to define the CE of Pt NP as follows
Ecp = Ecb +
A1 A A + 22 + 33 + B1 (12−ACN) B2 , d d d
(12)
3.2. Size and shape dependent thermal stability of Pt NPs 3.2.1. Structural evolution during the heating process To clearly observe the structural evolution of Pt NPs during the heating process, we recorded the atomic coordinates of the equilibrium structures at different temperatures. Fig. 7 illustrates the atomistic
(10)
where Ecp is the CE of Pt NP, Ecb is the CE of bulk Pt, and A1, A2 , A3 , B1, and B2 are the fitting parameters. We adopted the Eq. (10) to fit the calculated data, and the fitting method is the least squares method (the least squares method is used in the fitting process later in this paper). The fitting parameters are listed in Table 3, and the curves obtained by fitting are shown in Fig. 5(a). Fig. 5(a) demonstrates that the fitting curves are in good agreement with the calculation results. In Fig. 5(b), we extended d to 100 nm, and predicted the CEs of Pt NPs in the size range of 20–100 nm. Fig. 5 shows that the order of the CEs of Pt NPs with sizes of 20–100 nm is the same as that of 2–20 nm. 3.1.3. Surface energy The specific surface area of NP is very large, which has a significant influence on their physical and chemical properties. The surface energy density (SED) is the most important physical quantity the reflecting surface effect. The SED of NP (γ ) can be expressed as [34–36]
γ=
Enp − NEb A
,
(11) Fig. 5. Size and shape dependent cohesive energy (Ecp ) of Pt nanoparticles. (a) nanoparticle sizes from 2 to 20 nm, (b) nanoparticle sizes from 20 to 100 nm. The symbols are the calculated data, and the lines are the results predicted by the fitted curves.
where Enp is the total energy of NP, N is the total number of atoms, Eb is the energy per atom in bulk, and A is the surface area of NP. According to Eqs. (7), (10) and (11), the surface energy of NP (ES ) can also be expressed as a function of d and ACN. 4
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potential energy, the more likely the atom will migrate under the action of kinetic energy.
Table 3 Fitting parameters of the cohesive energy of Pt nanoparticles. Parameter
Value
A1(eV·nm) A3(eV·nm3) B2
−0.016043 −0.306320 1.000642
Parameter
Value 2
A2 (eV·nm ) B1 (eV)
3.2.2. Melting point The temperature of solid-liquid phase transition can be determined by the curve of the potential energy (Ep ) varying versus the temperature (T), and the temperature at which ∂Ep/ ∂T reaches its maximum is usually defined as the melting point [11,13,43]. Using SP-19861 as an example, the potential energy and its first derivative at different temperatures are calculated, as shown in Fig. 8(a). To compare with other methods for determining the melting point, we also calculated the radial distribution function g(r) at different temperatures. The results are shown in Fig. 8(b). Fig. 8(a) demonstrates that ∂Ep/ ∂T is maximum when the temperature increases to 1920 K. Fig. 8(b) indicates that as the temperature increases, the g(r) peak becomes broader and lower. At 1920 K, the second peak vanishes, meaning that the NP completely melt and become liquid; this temperature is the melting point [44]. Thus, the melting points defined by the radial distribution function and the potential energy temperature curve are the same. In this paper, the melting point is defined by the potential energy temperature curve. Fig. 9(a) shows the normalised melting points of Pt NPs with different shapes and sizes. Fig. 9(a) demonstrates that the melting point of NP (Tmp ) is lower than the melting point of bulk (Tmb ), and the Tmp decrease with the reduction of d . Fig. 9(a) also shows that the melting point of Pt NPs with the same size but different shapes are unequal. The melting point of TO structure is the highest, followed by the O structure, and then the T, SP, CO and TTH structures (the melting points of these structures are basically the same), and finally the C structure. Both experimental and theoretical studies show that the melting point of Pt NPs drops with the decrease of the NP size [9,12,45], which is the same as the calculation results in this paper. The reason for the decrease in the melting point is the increase in the number of atoms with low CN [46]. The existing size-dependent melting point models of NPs can be written in the following basic forms [46]:
0.130266 −0.283995
Fig. 6. The relationships between the surface energy density (γ ) and nanoparticle size (d) of Pt nanoparticles with different shapes.
snapshots of Pt NPs with different shapes and sizes of approximately 4.5 nm at five typical temperatures. Seven different Pt NPs were considered in the simulations, namely a SP structure containing 4897 atoms (SP-4897), a C structure containing 4631 atoms (C-4631), a CO structure containing 5083 atoms (CO-5083), a TO structure containing 4033 atoms (TO-4033), an O structure containing 4579 atoms (O4579), a T structure containing 4495 atoms (T-4495), and a TTH structure containing 3285 atoms (TTH-3285). As shown in Fig. 7, all of the structures except TTH-3285 are stable at room temperature of 300 K. With the increase of the temperature of the NPs, the corner atoms become disordered first, followed by the edge atoms, then the surface atoms, and finally the entire NPs melt completely into spheres. However, the temperatures at which the corner atoms of the SP-4897, C-4631, CO-5083, TO-4033, O-4579, T-4495 and TTH-3285 structures become disordered are not the same. They are respectively 1360 K, 400 K, 1240 K, 1560 K, 1000 K, 560 K, and 280 K. The CNs of the corner atoms of the above seven structures are 6, 3, 5, 6, 4, 3, and 3, respectively. During the heating process, the corner atom with the lowest CN more easily migrates away from its initial position. Both the CO and TO structures are covered by {1 1 1} and {1 0 0} facets. As shown in Fig. 7(c) and (d), the facets parallel to the page are the {1 0 0} facets, which are surrounded by the {1 1 1} facets. As the temperature increases, the {1 0 0} facets become disordered while the {1 1 1} facets remain ordered. Fig. 7(g) shows that the initial TTH structure is covered with the {2 1 0} facets. After the corner atoms are disordered, the {1 1 1} facets are exposed, and then the {1 1 1} facets increase with the temperature. At 1640 K, the {2 1 0} facets disappear, leaving only the {1 1 1} facets. The phenomena can be explained by the SED: the facet with the lowest SED is more stable, so it needs a higher temperature to become disordered. The CNs of the {1 1 1}, {1 0 0}, and {2 1 0} faces are 9, 8, and 6, respectively. In summary, the atoms with lower CNs more easily migrate and make the structure disordered during the heating process. This can be explained by the atomic potential energy. The lower the CN of the atom, the higher the potential energy. During the heating process, the average kinetic energy of each atom is the same at the same temperature, therefore the higher the
X Tmp = Tmb ⎛1 − ⎞. d⎠ ⎝
(13)
There are varying X values in different models. The homogeneous growth model [47], Heterogeneous growth model [48], liquid shell model [49], liquid nucleation growth model [9] and droplet model [50] are all derived from an ideal sphere, and X contains the linear term of SED. If these models are applied to NPs with other shapes, the lower the SED of NPs with the same size, the higher the melting point. However, Figs. 6 and 9(a) show that the order of the surface energy of Pt NPs with various shapes differs significantly from that of the melting point, thus the melting point is difficult to write as a function of the particle size and surface energy. In Qi’s bond energy model [14,15], the X contains a shape factor, which leads to the conclusion that the spherical NPs have the highest melting point in the same size NPs, which is inconsistent with the results in Fig. 9(a). Therefore, the previous size-dependent melting point model of NPs cannot be directly used to calculate the melting point of NPs with different shapes. In the models of Shandiz et al. [30] and Mirjalili et al. [51], the melting point of NP is written as a function of the ACN
Tmp Tmb
=
ACN , ACNb
(14)
where ACNb is the ACN of the atoms in the bulk material. According to the previous analysis, atoms with smaller CNs in NPs are easier to migrate and make the structure disordered. Therefore, we suggest that the melting point of the NPs is related to the particle size and ACN. For NPs with the same size, the smaller the ACN, the lower the melting point. However, in Fig. 9(a), the order of the melting point of Pt NPs with different shapes varies from that of the ACN. This may be due to the shape change of the NPs during the heating process (see Fig. 7), resulting in the ACN different from the initial value. To describe the effect 5
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Fig. 7. Snapshots of Pt nanoparticles with different shapes at five typical temperatures during the heating process. (a) SP-4897, (b) C-4631, (c) CO-5083, (d) TO4033, (e) O-4579, (f) T-4495, and (g) TTH-3285.
of the shape change on the melting point, we introduced a parameter α , called the deformation factor, different shapes have varying α . Although for the NPs with the same shape there is no reason to believe that α is the same for each size, determining α as a function of d does not significantly improve our fitting results, so we roughly consider α as a size-independent constant. Therefore, the melting point of NPs should be a function of d , ACN and α . When d is equal to infinity and ACN is equal to 12, the melting point of Pt NPs should be the same as that of bulk. Thus, we represent the melting point of NPs as follows:
Tmp/ Tmb = 1 +
A1 A A + 22 + 33 − α (12 − ACN ) B , d d d
(15)
where A1, A2 , A3 , B and α are fitting parameters. We adopted Eq. (15) to fit the calculated data. The fitting parameters are provided in Table 4, and the melting points predicted by the fitting parameters are shown in Fig. 9(a) for comparison. The fitting curve agree well with the calculated results. In Fig. 9(b), we extended d to 100 nm and predicted the melting point of NPs with sizes between 10 and 100 nm. Fig. 9 shows that the order of the melting point of Pt NPs with sizes of 10–100 nm is the same as that of 2–10 nm. 6
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Table 4 Fitting parameters of melting point of Pt nanoparticles. Parameter
Value
A1(nm) A3(nm3) αTO αT αTTH αC
−0.627385 −0.592817 0.093612 0.122371 0.134092 0.148397
Parameter 2
A2 (nm ) B αO αSP αCO
Value 0.679604 0.285036 0.105981 0.128845 0.137337
results show that the structural and thermal stability of Pt NPs strongly depended on their sizes and shapes. The stabilities of Pt NPs with the same shape decrease with the reduction of the particle size. The stabilities of Pt NPs with the same size increase with the ACN. With the same size, the TO structure has the best structural and thermal stability, and the T structure has the worst structural stability, and the C structure has the worst thermal stability. The cohesive energy and melting point of Pt NP can be expressed as a function of the particle size and average coordination number. The shape of Pt nanoparticle changes during the melting process, resulting in the change of the average coordination number. To describe the effect of the shape change on the melting point, we presented a deformation factor as a function of the melting point. CRediT authorship contribution statement Gang Wang: Conceptualization, Methodology, Software, Validation, Data curation, Writing - original draft, Writing - review & editing. Yi-Shuang Xu: Validation, Writing - review & editing. Ping Qian: Conceptualization, Methodology, Writing - review & editing, Resources, Supervision. Yan-Jing Su: Conceptualization, Methodology, Writing - review & editing, Resources, Supervision. Fig. 8. (a) Temperature dependent potential energy (Ep) and its first derivative of sphere nanoparticle containing 19,861 atoms (SP-19861). (b) Radial distribution function g(r) of SP-19861 at different temperatures.
Acknowledgements This work was financially supported by the National Key Research and Development Program of China (Grant Nos. 2016YFB0700500 and 2018YFB0704300). References [1] S. Shen, C.-Y. Sun, X.-J. Du, H.-J. Li, Y. Liu, J.-X. Xia, Y.-H. Zhu, J. Wang, Codelivery of platinum drug and siNotch1 with micelleplex for enhanced hepatocellular carcinoma therapy, Biomaterials 70 (2015) 71–83, https://doi.org/10.1016/j. biomaterials.2015.08.026. [2] C. Santoro, S. Rojas-Carbonell, R. Awais, R. Gokhale, M. Kodali, A. Serov, K. Artyushkova, P. Atanassov, Influence of platinum group metal-free catalyst synthesis on microbial fuel cell performance, J. Power Sources 375 (2018) 11–20, https://doi.org/10.1016/j.jpowsour.2017.11.039. [3] G. Cognard, G. Ozouf, C. Beauger, I. Jiménez-Morales, S. Cavaliere, D. Jones, J. Rozière, M. Chatenet, F. Maillard, Pt nanoparticles supported on niobium-doped tin dioxide: impact of the support morphology on Pt utilization and electrocatalytic activity, Electrocatalysis 8 (2017) 51–58, https://doi.org/10.1007/s12678-0160340-z. [4] T.S. Ahmadi, Z.L. Wang, T.C. Green, A. Henglein, M.A. El-Sayed, Shape-controlled synthesis of colloidal platinum nanoparticles, Science 272 (1996) 1924–1925, https://doi.org/10.1126/science.272.5270.1924. [5] N. Tian, Z.-Y. Zhou, S.-G. Sun, Y. Ding, Z.L. Wang, Synthesis of tetrahexahedral platinum nanocrystals with high-index facets and high electro-oxidation activity, Science 316 (2007) 732–735, https://doi.org/10.1126/science.1140484. [6] Z. Peng, H. Yang, Designer platinum nanoparticles: control of shape, composition in alloy, nanostructure and electrocatalytic property, Nano Today 4 (2009) 143–164, https://doi.org/10.1016/j.nantod.2008.10.010. [7] Y. Kang, J.B. Pyo, X. Ye, R.E. Diaz, T.R. Gordon, E.A. Stach, C.B. Murray, Shapecontrolled synthesis of Pt nanocrystals: the role of metal carbonyls, ACS Nano 7 (2013) 645–653, https://doi.org/10.1021/nn3048439. [8] L. Liu, A. Corma, Metal catalysts for heterogeneous catalysis: from single atoms to nanoclusters and nanoparticles, Chem. Rev. (2018), https://doi.org/10.1021/acs. chemrev.7b00776. [9] R.R. Vanfleet, J. Mochel, Thermodynamics of melting and freezing in small particles, Surf. Sci. 341 (1995) 40–50, https://doi.org/10.1016/0039-6028(95)00728-8. [10] T. Bachels, H.-J. Güntherodt, R. Schäfer, Melting of isolated tin nanoparticles, Phys.
Fig. 9. The relationships between the normalised melting point and nanoparticle size of Pt nanoparticles with different shapes. (a) nanoparticle sizes from 2 to 10 nm, (b) nanoparticle sizes from 10 to 100 nm. The symbols are the calculated data, and the lines are the results predicted by the fitting curves.
4. Conclusion In this paper, the structural and thermal stability of Pt NPs with different sizes and shapes were systematically studied by the EAM potential. The cohesive energy, surface energy density, structural evolution during the heating process and melting point were calculated. The 7
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8
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The effects of size and shape on the structural and thermal stability of platinum nanoparticles ⁎
Gang Wanga,b,c, Yi-Shuang Xub,c, Ping Qiana,b, , Yan-Jing Sua,
T
⁎
a
Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing, Beijing 100083, China Department of Physics, University of Science and Technology Beijing, Beijing 100083, China c TianJin College, University of Science and Technology Beijing, Tianjin 301830, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanoparticle Structural stability Thermal stability
A new EAM potential was obtained by fitting to the experimental data of bulk Pt, and the cohesive energy, surface energy, structure evolution during the heating process and melting point of Pt nanoparticles with different shapes and sizes were studied with the EAM potential. According to the calculation results and compared with other models, a model is developed to describe the cohesive energy and melting point as functions of the particle size and average coordination number.
1. Introduction
NPs.
Platinum (Pt) nanoparticles (NPs) have been widely used as catalysts in the chemical, pharmaceutical and fuel cell industries because of their excellent catalytic activity and chemical stability [1–3]. Various shapes of Pt NPs have been experimentally obtained by shape-controlled synthesis, including tetrahedron [4–6], octahedron [5–7], icosahedron [4,5,7], cube [4–7], cuboctahedron [4–7], truncated octahedron[5,6], sphere[6,7], tetrahexahedron [5,6], decahedron [5], and others. The size, shape, and reaction temperature of NPs have important effects on the catalytic activity [8]. Therefore, it is necessary to study the structural and thermal stability of Pt NPs with different sizes and shapes. The structural and thermal stability of metal NPs have been studied theoretically and experimentally [9–15]. The results show that as the size of NP decreases, the cohesive energy and melting point gradually decrease, and the structural and thermal stability deteriorate. However, these studies are less concerned with the effect of shape on stability, and the conclusions are inconsistent. Huang et al. proposed that the cohesive energies of Pt NPs with different shapes are mainly determined by the surface energy, and the melting point is roughly independent of the particle shape [13]. Qi et al. showed that spherical NP has the largest cohesive energy, the highest melting point, and the most stable structure among the same size NPs [14,15]. Mei et al. posited that the melting point of NPs should be a function of the particle size and surface energy [12]. Therefore, in this paper, we systematically studied the effect of the size and shape on the structural and thermal stability of Pt
2. Simulation methodology
⁎
2.1. Molecular dynamics Seven different shapes were taken into account in this paper: cube (C), tetrahedron (T), octahedron (O), cuboctahedron (CO), truncated octahedron (TO), sphere (SP), and tetrahexahedron enclosed with {2 1 0} facets (TTH). Because of computing limitations, when studying the structural stability of NPs, the size range of Pt NPs is 2–20 nm, which contains approximately 400–400000 atoms. When researching the thermal stability of NPs, the size range of Pt NPs is 2–10 nm, which contains about 400–50000 atoms. To study the effects of the size and shape on the thermal stability of Pt NPs, we calculated the structural evolution during the heating process and the melting point of Pt NPs using molecular dynamics simulation. Throughout the simulation process, the isothermal-isobaric ensemble (NPT), Nose-Hoover thermostat and Berendsen barostat were adopted, and the time step was 1 fs. Starting from 0 K, the system temperature was increased by 20 K within 40 ps, and then the system was equilibrated by 40 ps under a constant temperature. These heating and equilibrium processes were repeated until the NP completely melted. The output value of each physical quantity was obtained from the statistical average value in the last 20 ps of the equilibrium process.
Corresponding authors. E-mail addresses: [email protected] (P. Qian), [email protected] (Y.-J. Su).
https://doi.org/10.1016/j.commatsci.2019.109090 Received 1 March 2019; Received in revised form 23 May 2019; Accepted 17 June 2019 0927-0256/ © 2019 Published by Elsevier B.V.
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Nomenclature NP C T O CO
TO SP TTH ACN CN CE SED
Nanoparticle Cube Tetrahedron Octahedron Cuboctahedron
vacancy migration energy is obtained by the nudged elastic band (NEB) method [25], and the melting point is obtained via the solid liquid coexistence method [26]. Table 2 shows that the results obtained by the present EAM potential are in good agreement with the experimental data. The equation of state (the energy as a function of the nearest neighbour distance) can reflect the interaction between atoms in largescale deformation. Therefore, it is important to the study of the properties related to deformation. The equation of state of Pt calculated with the present EAM potential is shown in Fig. 1. For comparison, the experimental Rose equation [27] are also plotted in Fig. 1. Fig. 1 shows that the calculation results of the present EAM potential agree well with the experimental data. For further verification, the pressure-volume curve is demonstrated in Fig. 2. The pressure-volume relationship calculated by the present EAM potential provides a good agreement with the experimental curve [28]. It shows that the present EAM potential is effective for large-scale deformation of Pt. Linear thermal expansions of Pt calculated with the present EAM potentials in comparison with experimental data [29] are illustrated in Fig. 3, proving that the present EAM potential can well describe the thermal expansion properties of Pt. It can be concluded from the analysis that describing the properties of Pt using the present EAM potential is effective.
2.2. Interatomic interaction potential 2.2.1. Construction of EAM potential The interatomic interaction potential is in the form of EAM potential. In the EAM model [16], the total energy of crystal can be expressed as
Etot =
∑ Fi (ρi ) + i
ρi =
∑
1 2
∑
φij (rij )
i, j (i ≠ j )
f j (rij )
(1)
(2)
j (j ≠ i)
where Fi is the embedding energy of atom i, ρi is the host electron density at atom i, φij is the pair potential between atoms i and j, rij is the distance between atoms i and j, and f j is the atomic electron density contributed by atom j. In this paper, the pair potential function is a polynomial
φ (r ) = (k1 r −12 + k2 r −6 + k3 r −3 + k 4 r −2 + k5 r −1 + k6 + k 7 r + k8 r 6)Ψ ⎛ r − rc ⎞ ⎝ h1 ⎠ ⎜
Truncated octahedron Sphere Tetrahexahedron Average coordination number Coordination number Cohesive energy Surface energy density
⎟
(3)
where Ψ(x ) is a cutoff function defined as [17] 4
Ψ(x ) =
⎧ x 4 x < 0, 1+x ⎨ 0, x ⩾ 0. ⎩
(4)
3. Results and discussion
ki (i = 1, 2, …, 8) , rc and h1 are parameters for fitting. The function of the embedding energy is a piecewise function 5
F (ρ) =
If the NP is an ideal cube, a3 = Na03/4 , which obtains a = (N /4)1/3a0 , where a is the side length of the cube, a 0 is the lattice constant of the bulk Pt, and N is the atomic number of the NP. If the NP is an ideal sphere, πD3 /6 = Na03/4 , which obtains D = (N /2.09)1/3a0 , where D is the diameter of the sphere. The size parameters of NPs with the same atomic numbers but different shapes vary. To facilitate the comparison of NPs with the same atomic numbers but different shapes, we use the value between the above two values as the particle size. The NP size d is uniformly defined as
ρ
⎧ F0 + ∑i = 2 Fi ( ρ − 1)i ρ < 1.15ρe e
⎨ ∑3 Fni ( ρ − 1)i i=0 ρe ⎩
ρ ⩾ 1.15ρe
(5)
where ρe is the equilibrium value of the host electron density, Fi (i = 0, 2, 3, 4, 5) and Fni (i = 0, 1, 2, 3) are adjustable parameters. The function of the atomic electron density is expressed in the form of exponent
r − rc ⎞ f (r ) = p0 exp(−p1 r )Ψ ⎛ , ⎝ h2 ⎠ ⎜
d=
⎟
(6)
where p0 , p1, rc and h2 are fitting parameters. The fitting database of the EAM potential includes the experimental values of the lattice constant, cohesive energy, equation of state, elastic constants, vacancy forming energy and linear thermal expansion. The fitting results of the potential parameters are listed in Table 1. In this work, all of the calculations using the EAM potential are performed by LAMMPS [18].
3
N a0 . 3
(7)
Table 1 Parameters of the EAM potential of Pt. Parameter rc (Å) h1 (Å) h2 (Å) k1 (eV·Å12) k2 (eV·Å6) k3 (eV·Å3) k4 (eV·Å2) k5 (eV·Å) k6 k7 (eV·Å−1) k8 (eV·Å−6) ρe (Å−3)
2.2.2. Validity of the EAM potential To test the validity of the present EAM potential, various properties of Pt calculated with the present EAM potential are compared with the experimental [19–24] data in Table 2. These properties include the lattice constant (a 0 ), cohesive energy (Ec ), elastic constants (Cij ), energy differences between alternative structures and FCC structure (ΔEHCP → FCC and ΔEBCC → FCC ), vacancy formation energy (Evf ), vacancy migration energy (Evm ), stacking fault energy (γsf ), twin boundary energy (γtb ), surface energy density (γ ) and melting point (Tm ). The 2
Value 7.46652 1 0.7 2.35234 × 103 1.14886 × 103 −5.77507 × 102 4.03998 × 102 −1.13056 × 102 1.47760 × 10 −7.65104 × 10−1 1.26828 × 10−6 1.65 × 10−2
Parameter −3
p0 (Å ) p1 (Å−1) F0 (eV) F2 (eV) F3 (eV) F4 (eV) F5 (eV) Fn0 (eV) Fn1 (eV) Fn2 (eV) Fn3 (eV)
Value 7.99899 × 10−2 1.53013 −4.3 3.37821 4.06570 × 10−1 7.93970 × 10−1 −5.34387 × 10−1 −4.30012 −2.06221 × 10−2 3.66205 −3.66231 × 10−1
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Table 2 Properties of Pt calculated with the present EAM potential in comparison with the experimental data. Property
Experiment
EAM
a0 (Å) Ec (eV) C11 (GPa) C12 (GPa) C44 (GPa) ΔEHCP→FCC (eV) ΔEBCC→FCC (eV) Evf (eV) Evm (eV) γsf (mJ/m2) γtb (mJ/m2) γ110 (mJ/m2) γ100 (mJ/m2) γ111 (mJ/m2) Tm (K)
3.92a 5.84a 358b 253.6b 77.4b 0.03c 0.16c 1.51d 1.43d
3.92 5.84 365 258.7 80.4 0.03 0.11 1.64 1.00
322e 161e 2691f 2691f 2691f 2042c
164 83 2405 2258 2107 2000
a b c d e f
Fig. 2. Pressure-volume relationship of Pt calculated with the present EAM potential in comparison with the experimental data.
Ref. [19]. Ref. [20]. Ref. [21]. Ref. [22]. Ref. [23]. Mean value of polycrystalline, Ref. [24].
Fig. 3. Linear thermal expansion of Pt calculated with the present EAM potential in comparison with the experimental data.
Fig. 1. Equations of state of Pt calculated with the present EAM potential in comparison with the experimental data.
3.1. Size and shape dependent structural stability of Pt NPs 3.1.1. Average coordination number The average coordination number (ACN) is a very important parameter of NP, and many of the properties of NP are affected by it [30,31]. If the geometrical structure of NP is determined, the ACN is a definite value. We first calculate the coordination number (CN) of each atom by LAMMPS, and then obtain ACN by taking its average value, as shown in Fig. 4. As shown in Fig. 4, the ACNs of Pt NPs are smaller than the CN of bulk Pt. As the NP size decreases, the ACN of the NP descends. This is because the CNs of the surface atoms of the NPs are less than that of the internal atoms, and the presence of the surface causes the ACNs of the NPs to be less than the CN of the bulk. As the NP size decreases, the proportion of the surface atoms increases, which results in the reduction of the ACN. Fig. 4 also shows that the ACNs of the NPs with the same size but different shapes vary: the ACN of the TO structure is the largest, followed by the O, CO, and SP structures (the ACNs of the three structures are approximately the same), and then the TTH, C and T
Fig. 4. Size dependent average coordination number (ACN) of Pt nanoparticles with different shapes. 3
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structures.
ES = γA = Enp − NEb = −NEcp + NEcb 3 = − 3 (A1 d 2 + A2 d + A3 + B1 d3 (12−ACN) B2), a0
3.1.2. Cohesive energy The cohesive energy (CE) of Pt NPs with different sizes and shapes was calculated with the EAM potential, as shown in Fig. 5(a). Fig. 5(a) demonstrates that the CE decreases with the increase of NP size d . Due to the presence of surface energy, the CE of the surface atom of NP is lower than that of the internal atom, which causes the CE of NP to be smaller than that of bulk. As d reduces, the ratio of the surface atoms number increases, which leads to the decrease of the CE. As shown in Fig. 5(a), the CEs of NPs with the same size but different shapes vary: the CE of the TO structure is the largest, followed by the O, CO, and SP structures (the CEs of these three structures are basically the same), and then the TTH, C and T structures. According to Qi’s generalised bond-energy model [32], the CE of NP (Ecp ) can be written as
Ecp =
1 (Ni E¯ i + Nf E¯ f + Ne E¯e + Nc E¯c ), N
Wei et al. [34] reported that if different methods are used to estimate the surface area of NP, the calculated SED has an opposite sizedependent trend. The SED of NP was found to either increase or decrease with the increase of the NP size [34]. Therefore, it is very important to correctly calculate the SED. Due to the inhomogeneous lattice distortion [37], the surface shape of the NP is no longer the ideal shape, and it is difficult to calculate the surface area theoretically. In this work, the OVITO software package [38] was used to construct the surface grid and calculate A . The SEDs of Pt NPs with different sizes and shapes were computed with the present EAM potential, the results are illustrated in Fig. 6. Fig. 6 shows that γ decreases with the increase of d , which is the same as the previous calculations[39–42]. Moreover, the γ of the NPs with the same size but different shapes vary. The γ of O and T structures enclosed with {1 1 1} facets are the lowest, followed by the TO and CO structures enclosed with {1 1 1} and {1 0 0} facets, then the C structure enclosed with {1 0 0} facets, then the SP structure enclosed with {1 1 1}, {1 0 0} and high-index facets, and finally the TTH structure enclosed with {2 1 0} facets. This result can be explained qualitatively by the SED of bulk. The SEDs of the {1 1 1}, {1 0 0} and {2 1 0} planes in the FCC bulk crystal increase sequentially. The larger the SEDs of the enclosed facets, the higher the SED of NP. However, when d is less than 5 nm, γ of the TO structure is smaller than that of the T structure. This may be related to the lattice distortion of NP. Eq. (12) demonstrates that when the size are the same, the lower the surface energy of the NP, the greater the CE and the more stable the structure. As shown in Figs. 5 and 6, the SP structure has the smallest surface area at the same NP size, but its SED is relatively high (only lower than that of the TTH structure). The competition of SED and surface area makes the CE of the SP structure smaller than that of the TO structure, and larger than that of the C, T and TTH structures. Therefore, the CE of NPs with the same size depends on the competition results of SED and surface area.
(8)
where N , Ni , Nf , Ne , and Nc are the numbers of total, interior, face, edge and corner atoms, respectively; E¯ i , E¯ f , E¯e , and E¯c are the average CEs of interior, face, edge, and corner atoms, respectively. According to the Eqs. (7) and (8), the relationship between Ecp and d can be approximately written in polynomial form [13,33]
Ecp = A +
B C D + 2 + 3, d d d
(9)
where A , B , C and D are the fitting parameters, which vary for different shapes of NPs. As shown in Figs. 4 and 5(a), for Pt NPs with the same size but different shapes, the larger the ACN, the greater the CE. So the CE is not only related to d , but also related to the ACN. This is because the greater the CN of an atom, the more bonds between the atom and its surrounding atoms, the better the structural stability, and the larger the CE. Considering that when d tends to infinity and the ACN tends to 12, the CEs of Pt NPs should be the same as the CE of the bulk Pt. We add an additional term after Eq. (9) to define the CE of Pt NP as follows
Ecp = Ecb +
A1 A A + 22 + 33 + B1 (12−ACN) B2 , d d d
(12)
3.2. Size and shape dependent thermal stability of Pt NPs 3.2.1. Structural evolution during the heating process To clearly observe the structural evolution of Pt NPs during the heating process, we recorded the atomic coordinates of the equilibrium structures at different temperatures. Fig. 7 illustrates the atomistic
(10)
where Ecp is the CE of Pt NP, Ecb is the CE of bulk Pt, and A1, A2 , A3 , B1, and B2 are the fitting parameters. We adopted the Eq. (10) to fit the calculated data, and the fitting method is the least squares method (the least squares method is used in the fitting process later in this paper). The fitting parameters are listed in Table 3, and the curves obtained by fitting are shown in Fig. 5(a). Fig. 5(a) demonstrates that the fitting curves are in good agreement with the calculation results. In Fig. 5(b), we extended d to 100 nm, and predicted the CEs of Pt NPs in the size range of 20–100 nm. Fig. 5 shows that the order of the CEs of Pt NPs with sizes of 20–100 nm is the same as that of 2–20 nm. 3.1.3. Surface energy The specific surface area of NP is very large, which has a significant influence on their physical and chemical properties. The surface energy density (SED) is the most important physical quantity the reflecting surface effect. The SED of NP (γ ) can be expressed as [34–36]
γ=
Enp − NEb A
,
(11) Fig. 5. Size and shape dependent cohesive energy (Ecp ) of Pt nanoparticles. (a) nanoparticle sizes from 2 to 20 nm, (b) nanoparticle sizes from 20 to 100 nm. The symbols are the calculated data, and the lines are the results predicted by the fitted curves.
where Enp is the total energy of NP, N is the total number of atoms, Eb is the energy per atom in bulk, and A is the surface area of NP. According to Eqs. (7), (10) and (11), the surface energy of NP (ES ) can also be expressed as a function of d and ACN. 4
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potential energy, the more likely the atom will migrate under the action of kinetic energy.
Table 3 Fitting parameters of the cohesive energy of Pt nanoparticles. Parameter
Value
A1(eV·nm) A3(eV·nm3) B2
−0.016043 −0.306320 1.000642
Parameter
Value 2
A2 (eV·nm ) B1 (eV)
3.2.2. Melting point The temperature of solid-liquid phase transition can be determined by the curve of the potential energy (Ep ) varying versus the temperature (T), and the temperature at which ∂Ep/ ∂T reaches its maximum is usually defined as the melting point [11,13,43]. Using SP-19861 as an example, the potential energy and its first derivative at different temperatures are calculated, as shown in Fig. 8(a). To compare with other methods for determining the melting point, we also calculated the radial distribution function g(r) at different temperatures. The results are shown in Fig. 8(b). Fig. 8(a) demonstrates that ∂Ep/ ∂T is maximum when the temperature increases to 1920 K. Fig. 8(b) indicates that as the temperature increases, the g(r) peak becomes broader and lower. At 1920 K, the second peak vanishes, meaning that the NP completely melt and become liquid; this temperature is the melting point [44]. Thus, the melting points defined by the radial distribution function and the potential energy temperature curve are the same. In this paper, the melting point is defined by the potential energy temperature curve. Fig. 9(a) shows the normalised melting points of Pt NPs with different shapes and sizes. Fig. 9(a) demonstrates that the melting point of NP (Tmp ) is lower than the melting point of bulk (Tmb ), and the Tmp decrease with the reduction of d . Fig. 9(a) also shows that the melting point of Pt NPs with the same size but different shapes are unequal. The melting point of TO structure is the highest, followed by the O structure, and then the T, SP, CO and TTH structures (the melting points of these structures are basically the same), and finally the C structure. Both experimental and theoretical studies show that the melting point of Pt NPs drops with the decrease of the NP size [9,12,45], which is the same as the calculation results in this paper. The reason for the decrease in the melting point is the increase in the number of atoms with low CN [46]. The existing size-dependent melting point models of NPs can be written in the following basic forms [46]:
0.130266 −0.283995
Fig. 6. The relationships between the surface energy density (γ ) and nanoparticle size (d) of Pt nanoparticles with different shapes.
snapshots of Pt NPs with different shapes and sizes of approximately 4.5 nm at five typical temperatures. Seven different Pt NPs were considered in the simulations, namely a SP structure containing 4897 atoms (SP-4897), a C structure containing 4631 atoms (C-4631), a CO structure containing 5083 atoms (CO-5083), a TO structure containing 4033 atoms (TO-4033), an O structure containing 4579 atoms (O4579), a T structure containing 4495 atoms (T-4495), and a TTH structure containing 3285 atoms (TTH-3285). As shown in Fig. 7, all of the structures except TTH-3285 are stable at room temperature of 300 K. With the increase of the temperature of the NPs, the corner atoms become disordered first, followed by the edge atoms, then the surface atoms, and finally the entire NPs melt completely into spheres. However, the temperatures at which the corner atoms of the SP-4897, C-4631, CO-5083, TO-4033, O-4579, T-4495 and TTH-3285 structures become disordered are not the same. They are respectively 1360 K, 400 K, 1240 K, 1560 K, 1000 K, 560 K, and 280 K. The CNs of the corner atoms of the above seven structures are 6, 3, 5, 6, 4, 3, and 3, respectively. During the heating process, the corner atom with the lowest CN more easily migrates away from its initial position. Both the CO and TO structures are covered by {1 1 1} and {1 0 0} facets. As shown in Fig. 7(c) and (d), the facets parallel to the page are the {1 0 0} facets, which are surrounded by the {1 1 1} facets. As the temperature increases, the {1 0 0} facets become disordered while the {1 1 1} facets remain ordered. Fig. 7(g) shows that the initial TTH structure is covered with the {2 1 0} facets. After the corner atoms are disordered, the {1 1 1} facets are exposed, and then the {1 1 1} facets increase with the temperature. At 1640 K, the {2 1 0} facets disappear, leaving only the {1 1 1} facets. The phenomena can be explained by the SED: the facet with the lowest SED is more stable, so it needs a higher temperature to become disordered. The CNs of the {1 1 1}, {1 0 0}, and {2 1 0} faces are 9, 8, and 6, respectively. In summary, the atoms with lower CNs more easily migrate and make the structure disordered during the heating process. This can be explained by the atomic potential energy. The lower the CN of the atom, the higher the potential energy. During the heating process, the average kinetic energy of each atom is the same at the same temperature, therefore the higher the
X Tmp = Tmb ⎛1 − ⎞. d⎠ ⎝
(13)
There are varying X values in different models. The homogeneous growth model [47], Heterogeneous growth model [48], liquid shell model [49], liquid nucleation growth model [9] and droplet model [50] are all derived from an ideal sphere, and X contains the linear term of SED. If these models are applied to NPs with other shapes, the lower the SED of NPs with the same size, the higher the melting point. However, Figs. 6 and 9(a) show that the order of the surface energy of Pt NPs with various shapes differs significantly from that of the melting point, thus the melting point is difficult to write as a function of the particle size and surface energy. In Qi’s bond energy model [14,15], the X contains a shape factor, which leads to the conclusion that the spherical NPs have the highest melting point in the same size NPs, which is inconsistent with the results in Fig. 9(a). Therefore, the previous size-dependent melting point model of NPs cannot be directly used to calculate the melting point of NPs with different shapes. In the models of Shandiz et al. [30] and Mirjalili et al. [51], the melting point of NP is written as a function of the ACN
Tmp Tmb
=
ACN , ACNb
(14)
where ACNb is the ACN of the atoms in the bulk material. According to the previous analysis, atoms with smaller CNs in NPs are easier to migrate and make the structure disordered. Therefore, we suggest that the melting point of the NPs is related to the particle size and ACN. For NPs with the same size, the smaller the ACN, the lower the melting point. However, in Fig. 9(a), the order of the melting point of Pt NPs with different shapes varies from that of the ACN. This may be due to the shape change of the NPs during the heating process (see Fig. 7), resulting in the ACN different from the initial value. To describe the effect 5
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Fig. 7. Snapshots of Pt nanoparticles with different shapes at five typical temperatures during the heating process. (a) SP-4897, (b) C-4631, (c) CO-5083, (d) TO4033, (e) O-4579, (f) T-4495, and (g) TTH-3285.
of the shape change on the melting point, we introduced a parameter α , called the deformation factor, different shapes have varying α . Although for the NPs with the same shape there is no reason to believe that α is the same for each size, determining α as a function of d does not significantly improve our fitting results, so we roughly consider α as a size-independent constant. Therefore, the melting point of NPs should be a function of d , ACN and α . When d is equal to infinity and ACN is equal to 12, the melting point of Pt NPs should be the same as that of bulk. Thus, we represent the melting point of NPs as follows:
Tmp/ Tmb = 1 +
A1 A A + 22 + 33 − α (12 − ACN ) B , d d d
(15)
where A1, A2 , A3 , B and α are fitting parameters. We adopted Eq. (15) to fit the calculated data. The fitting parameters are provided in Table 4, and the melting points predicted by the fitting parameters are shown in Fig. 9(a) for comparison. The fitting curve agree well with the calculated results. In Fig. 9(b), we extended d to 100 nm and predicted the melting point of NPs with sizes between 10 and 100 nm. Fig. 9 shows that the order of the melting point of Pt NPs with sizes of 10–100 nm is the same as that of 2–10 nm. 6
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Table 4 Fitting parameters of melting point of Pt nanoparticles. Parameter
Value
A1(nm) A3(nm3) αTO αT αTTH αC
−0.627385 −0.592817 0.093612 0.122371 0.134092 0.148397
Parameter 2
A2 (nm ) B αO αSP αCO
Value 0.679604 0.285036 0.105981 0.128845 0.137337
results show that the structural and thermal stability of Pt NPs strongly depended on their sizes and shapes. The stabilities of Pt NPs with the same shape decrease with the reduction of the particle size. The stabilities of Pt NPs with the same size increase with the ACN. With the same size, the TO structure has the best structural and thermal stability, and the T structure has the worst structural stability, and the C structure has the worst thermal stability. The cohesive energy and melting point of Pt NP can be expressed as a function of the particle size and average coordination number. The shape of Pt nanoparticle changes during the melting process, resulting in the change of the average coordination number. To describe the effect of the shape change on the melting point, we presented a deformation factor as a function of the melting point. CRediT authorship contribution statement Gang Wang: Conceptualization, Methodology, Software, Validation, Data curation, Writing - original draft, Writing - review & editing. Yi-Shuang Xu: Validation, Writing - review & editing. Ping Qian: Conceptualization, Methodology, Writing - review & editing, Resources, Supervision. Yan-Jing Su: Conceptualization, Methodology, Writing - review & editing, Resources, Supervision. Fig. 8. (a) Temperature dependent potential energy (Ep) and its first derivative of sphere nanoparticle containing 19,861 atoms (SP-19861). (b) Radial distribution function g(r) of SP-19861 at different temperatures.
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Fig. 9. The relationships between the normalised melting point and nanoparticle size of Pt nanoparticles with different shapes. (a) nanoparticle sizes from 2 to 10 nm, (b) nanoparticle sizes from 10 to 100 nm. The symbols are the calculated data, and the lines are the results predicted by the fitting curves.
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