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Nanoparticles have been used widely in various fields, and their size and shape greatly affect the functional properties. Therefore, controlling the morphology of the particles is important, and evaluation of the surface energy is indispensable for that purpose. In this study, the surface energy of nanoparticles was evaluated by numerical simulation and formulated in a polynomial equation. First, molecular dynamics simulations were carried out for variously shaped polyhedral nanoparticles. A cube and an octahedron were introduced as reference shapes, and truncated hexahedrons and truncated octahedrons were created by cutting out their vertices. The surface energy was plotted for various polyhedrons. The lowest energy was observed in an octahedron because of the stability of the (111) plane, and the highest energy was observed in a cube because of the relatively higher energy of the (100) plane. Then, the surface energy was formulated in a polynomial equation, in which the parameters obtained by the molecular-dynamics simulations were introduced. As a result, stability of the octahedron and relative instability of the cube were fairly captured by the proposed polynomial equation, while a slight underestimation was inevitable. Finally, the parameters were revised to continuous numbers to extend the application range. Consequently, an application for various materials, such as a cube having equivalent stability to an octahedron, was demonstrated by imposing rather exaggerated parameters.

A small particle with a nano-meter order diameter is called a “nanoparticle”, and it has been used in a wide range of fields such as electronics, photonics, environmental engineering, medical sciences, and pharmaceuticals [

Concerning the evaluation of surface energy, a phase-field method [

Face-centered-cubic (fcc) metals are considered in this study. In the fcc crystal, (111) planes are the most stable, and this is followed by (100) planes, and other planes are relatively unstable. As such, only these two planes were focused on in this study. One of the typical polyhedral shapes is a cube, which has six (100) planes, 12 edges, and eight vertices. Another preferable shape is an octahedron, which comprises eight (111) planes, 12 edges, and six vertices. By setting these two shape types as the standard shapes for the current study, various polyhedrons were created as shown in

squares as shown in

All these polyhedrons comprise either or both the (100) and (111) planes. In addition to these shapes, a regular tetrahedron and an asymmetric truncated octahedron illustrated in

In these polyhedrons, cube, regular octahedron, and regular tetrahedron shapes are termed regular polyhedrons or Platonic solids, in which all faces are identical regular polygons, and all edges and vertices have the same characteristics. In addition, a cuboctahedron and Kelvin cell are termed semi-regular polygons or Archimedean solids. The number of faces, edges, and vertices as well as the Miller indexes of the relevant faces are listed in

To parameterize the size and shape of the polyhedrons, the following variables were introduced: l_{c}, the edge length of the original cube; c, the cutting length from the vertex of the cube along the edge; l_{o}, the edge length of the original octahedron; and d, the cutting length from the vertex of the octahedron along the edge. The ranges of c and d are limited as 0 ≤ c ≤ l c / 2 , and 0 ≤ d ≤ l o / 2 , respectively. The cuboctahedron is generated from the cube when c = l_{c}/2 and from the octahedron when d = l_{o}/2. The Kelvin cell is formed from the octahedron when d = l_{o}/3. Here, the truncated octahedrons can be generated from a cube by setting the value of c as l c / 2 ≤ c ≤ l c . However, this process induces the overlap of the cut parts, which hinders intuitive understanding. Nonetheless, it is useful for simplifying the parameters. Therefore, in this paper, the parameters l_{c} and c are used to identify the shape and size of a polyhedron. The cutting lengths from alternative vertices are set as c_{1} and c_{2} (the diagonal vertices in the original square face have the same index) under a limitation of l c ≤ ( c 1 + c 2 ) ≤ 2 l c . Finally, the size of a regular tetrahedron is simply expressed by the edge length, l_{t}.

The length, l, irrespective of the subscript, is expressed by the number of atoms along the edge and is denoted by integer n. Cutting parameter c is also

Name | Abbreviation | Faces | Edges | Vertices |
---|---|---|---|---|

Cube | Cube | 6 (100) | 12 (100)-(100) | 8 (100)-(100)-(100) |

Truncated hexahedron | trHexa | 6 (100) 8 (111) | 12 (100)-(100) 24 (111)-(100) | 24 (100)-(100)-(111) |

Cuboctahedron | Cubocta | 6 (100) 8 (111) | 24 (111)-(100) | 12 (100)-(100)-(111)-(111) |

Truncated octahedron | trOcta | 6 (100) 8 (111) | 24 (111)-(100) 12 (111)-(111) | 24 (100)-(111)-(111) |

Octahedron | Octa | 8 (111) | 12 (111)-(111) | 6 (111)-(111)-(111)-(111) |

Tetrahedron | Tetra | 4 (111) | 6 (111)-(111)’ | 4 (111)-(111)-(111) |

Asymmetric truncated octahedron | astrOcta | 6 (100) 8 (111) | 24 (111)-(100) 12 (111)-(111) | 24 (100)-(111)-(111) |

expressed by the integer number of atomic layers. The models are identified by the abbreviated notation for their shape, as listed in _{c} = 12 L_{a} (L_{a}: lattice constant), and “trOcta (n12c6)” denotes the truncated octahedron made by cutting six atomic layers from the vertices of Cube (n12).

The surface energy of nanoparticles can be directly calculated using an atomistic model assuming a specific interatomic potential function. In this study, the Lennard–Jones potential function was applied. A quantitative evaluation of a specific material is sensitive to the choice of potential function, but we applied a simple two-body function because our purpose was to present a general methodology. The atoms were arranged on the lattice points of the fcc structure in a cubic space so that the (100), (010), and (001) planes are on the y – z , z – x and x – y planes, respectively. Then, an arbitrary polyhedral shape was formed by removing the atoms out of the target range. For the truncated octahedron, a regular octahedron was made first by the above-mentioned procedure, and then, the atoms in the cutting area were deleted. The initial position of the atoms was equilibrated by the MD method for the 10,000 time-steps, and the surface energy and other properties were calculated. The fundamental equations involved are as follows.

Newton’s equation of motion:

m i r ¨ i = F i = − ∑ j ≠ i d ϕ i j d r r i j r i j . (1)

Interatomic potential function (Lennard-Jones type):

ϕ i j = 4 ε ( ( σ r i j ) 12 − ( σ r i j ) 6 ) . (2)

Here, m_{i} is mass of the i-th atom, r_{i} is the position vector of the i-th atom, and F_{i} is the force acting on the i-th atom, which is defined as the position derivative of interatomic potential function f_{ij}. The parameters ε and σ depend on the material, and a nondimensional analysis is demonstrated by standardizing the energy and length based on ε and σ, respectively.

Common-neighbor analysis (CNA) [

Surface energy γ is defined as the increase in energy due to the existence of a surface and is expressed as the difference in the energy between the atoms on a surface and the atoms in bulk where the energy is unaffected by the surface. Here, the value of γ is calculated as the energy per atom in the atomistic model, while surface energy in general is typically defined as a value per area. In a particle model, if the atoms are observably divided into surface and bulk atoms, the surface energy can be calculated as follow:

γ = ∑ i ∈ surface ( e i s − e b ) S , (3)

where e i s and e^{b} are the energy of the atoms on surface and in bulk, respectively, and S is the surface area. However, establishing a definition of the surface atom is difficult because the influence of the surface acts across more than one atomic layer. Now, assuming the internal atoms are unaffected by the surface and have the same energy as e^{b}, Equation (3) can be calculated as follows:

γ = ∑ i ∈ particle ( e i − e b ) S , (4)

where the summation in taken over all atoms in the model without the need for distinguishing the surface and bulk atoms, and the surface energy finally corresponds to the difference in energy between the particle model and bulk model [^{b} is obtained by a separate simulation using a fully bulk model with periodic boundary condition.

Another difficulty is defining the surface area of the particle. In this study, the particle model was created based on a polyhedral shape, and the surface area of

No. | Color | Type | Property | No. | Color | Type | Property |
---|---|---|---|---|---|---|---|

1 | ● | bulk | fcc | 5 | ● | edge | (100)-(111) |

2 | ● | bulk | hcp | 6 | ● | edge | (111)-(111) |

3 | ● | surface | (111) | 7 | ● | edge | (100)-(100) |

4 | ● | surface | (100) | 8 | ● | others |

the initial polyhedron was applied by neglecting a slight volume change and roughing of the surface occurred during relaxation calculation.

Snapshots of the MD models after the relaxation steps are shown in

the 2000^{th} time-step. The energy of the entire particle was slightly low in the Kelvin cell, and the truncated hexahedron and cuboctahedron were almost the same as shown in

Various polyhedron models were also created, and MD simulations were carried out. The configurations of the atoms are shown in

(111)-(111) edges in the octahedron; i.e., the angle between the faces in the tetrahedron is much sharper than that of the octahedron. The actual Miller indices are (111) and ( 1 ¯ 11 ) or ( 1 1 ¯ 1 ) in the octahedron, and (111) and ( 1 ¯ 1 ¯ 1 ) in the tetrahedron.

The total energy averaged across all atoms in a particle, U, and the average of the atoms on the surface, e, are plotted in relation to the number of atoms in a particle, N, in

According to the MD simulations in the previous section, the surface energy γ of nanoparticles is assumed to be approximated by the following equation:

γ = 1 S ∑ k ∈ D k N k ( e k − e 0 ) (5)

Here, D_{k} is the element constituting the surface, e.g., the (111) face and (100)-(111) edge, e_{k} is the energy per atom depending on the element, e_{0} is the average energy in the bulk region, N_{k} is the number of atoms in D_{k}, and S is the surface area. The values of e_{k} were obtained by the MD simulations. For example, the value for {k: (111) face} was evaluated from the calculation for a regular octahedron (n28) by taking the average value of the (111) atoms in the model. The energy values varied depending on the shape and size of the MD models, even those with the same face type. Here, the values for the model having the largest area or longest edges were selected because these tend to converge as the size becomes larger. The values and relevant polyhedron models are listed in

The calculated surface energy is plotted in

Element | Value | Model | Element | Value | Model |
---|---|---|---|---|---|

(111) face | −5.709 | octa n28 | vertex 111-111-111-111 | −2.377 | octa n28 |

(100) face | −5.352 | cube n16 | vertex 111-111-100-100 | −3.074 | cubocta n15 |

111-100 edge | −4.462 | cubocta n15 | vertex 111-111-100 | −3.704 | Kelvin n17 |

111-111 edge | −4.301 | trOcta n17c22 | vertex 111-100-100 | −3.300 | trHexa n17c22 |

100-100 edge | −3.398 | trHexa n17c06 | vertex 100-100-100 | −2.107 | Cube n16 |

and octahedrons presented in

The calculation in the previous section was based on atomic arrangement, and the edge and cutting length were represented by integer numbers. For a more systematic investigation, the lengths were revised to continuous real numbers. Then, the cutting length from the cube vertex was standardized by the edge length L of the original cube, i.e., c = 0 is the cube, c = 0.5 is the cuboctahedron, c = 0.67 is the Kelvin cell, and c = 1 is the regular octahedron.

The energy per atom was also modified to the energy per area for the faces, energy per length for the edges, and energy per point for the vertices. Corresponding values were obtained by considering the planar and linear density of each face or edge. This enabled the evaluation of the size dependency of the surface energy. Then, size parameters l and c for different shapes could be defined under a constant volume, by which we could compare the difference in shape of the surface energy under a constant-volume condition.

The surface energy is dominated by both the surface area and edge length as

well as by the crystallographic orientation of the faces and edges. Surface energies were calculated for various volumes and plotted in

In the previous sections, the parameters obtained by MD simulation were used, and they were based on the Lennard–Jones potential, for which the fcc structure and the (111) close-packed plane are stable. In this section, the applicability of the present method to different material systems is verified. The parameters of the constituent faces and edges were virtually varied, and the surface energy of the polyhedral particles was calculated. The parameters of the surface energy for the (100) and (111) faces and (111)-(111) and (100)-(100) edges were virtually varied. The calculated surface energies for V = 500 and 3000 are shown in _{inv}”, the energies for the (111) and (100) faces were replaced so that the (100) plane has a smaller value than the (111) plane. Similarly, the energies for the (111)-(111) and (100)-(100) edges were replaced for “E_{inv}”. The overall slope of the surface energy became smaller, but the energy for the cube and octahedron was not reversed by these operations. Therefore, rather exaggerated

values were provided, and the results are shown by the curves noted as “ext”. As a result, the surface energy decreased for the cube and increased for the octahedron, and the values for both shapes reach mostly equivalent values. This means that the stability of various shapes can be explained by providing individual face and edge energies, and it is concluded that the applicability of the proposed model to various material particles is presented.

In this study, the surface energy of nanoparticles with polyhedral shape was calculated by the MD method and formulated as a polynomial equation. In the MD simulation, a cube and an octahedron were introduced as reference shapes, and various polyhedrons were created by cutting out the vertices. Owing to the stability of the (111) plane, octahedron showed the lowest energy, and the cube had the highest because of the relatively higher energy of the (100) plane. This tendency was fairly captured by the proposed polynomial equation by applying atomic energy despite of a slightly inevitable underestimation. The influence across a few atomic layers should be taken into account for a more precise evaluation. To extend the application range of the proposed equation, the parameters were revised to continuous numbers. The calculated results indicated a proper tendency for fcc particle. Additionally, it was shown that a stability of cubic particles could be obtained by imposing rather exaggerated parameters. This result indicates the possibility of the proposed method being applied to a broad range of materials.

The authors declare no conflicts of interest regarding the publication of this paper.

Uehara, T. and Fujiwara, J. (2020) Numerical Evaluation of the Surface Energy of Polyhedral Nanoparticles. Materials Sciences and Applications, 11, 837-850. https://doi.org/10.4236/msa.2020.1112055