## 1. Introduction

## 2. Method and Algorithm in Model Building

_{i}is the molar concentration of the i

^{th}element which satisfies the following equation:

_{i}and set $\partial L(x)/\partial {x}_{i}=0$, one has:

_{i}, for the maximum entropy distribution, is solved from Equation (4):

_{i}as the free space of the particle i for a system with N identical particles in space V. Following the above procedure, it is easy to prove that v

_{i}= V/N under the MaxEnt condition, i.e., the identical particles should have the equal maximum free space V/N within the limitation of maximum entropy. This is a state of uniform spatial distribution of particles. Therefore, the maximum entropy state has the configuration with a uniform particles distribution over the system.

**X**) is the entropy under the system configuration

**X**. Since there is a proportional relationship between entropy and the particle's free space for a closed particle system with a fixed volume V, the contribution of particle i to the system entropy can be written as:

_{i}(

**r**) is the free space of particle i at position

**r**, k is a constant. Derived from Equations (7) and (8), the entropic force f

_{i}(

**r**) on particle i is given by:

**f**(

_{i}**r**) is pointed to the direction of v

_{i}(

**r**) increase. Hence, the MaxEnt configuration can be reached through minimizing f

_{i}(

**r**) or maximizing v

_{i}(

**r**) for each particle in the system.

## 3. Results and Analyses

#### 3.1. Binary Alloys

_{1-x}Cr

_{x}(x = 0.085, 0.111, 0.206) are created to illustrate the MaxEnt algorithm for the atomic structure modeling of MPE alloys in the follows.

_{1-x}Cr

_{x}are illustrated in Figure 1.

**Figure 1.**4 × 4 × 4 atomic structure models of binary alloys Fe

_{1-x}Cr

_{x}. The ratios of solute to solvent atoms in Figure 1a to 1c are 16:173, 21:168, and 39:150, respectively. Cr atoms (black balls) are dispersed in the Fe (transparent balls) solvent matrix.

_{min}and minimum r

_{max}distances for the nearest neighbors among solute atoms. The smaller the value, the more uniform the distribution of the solute atoms. It was suggested that the random packing of hard spheres in three dimensional space has a density limit between 55.5% [24] and 63.4% [25]. The packing density of the atoms’ free-space balls in our built models is a little smaller than the low limit. The main reason can be understood from the fact that there is a BCC lattice-site restriction for the positions of solute atoms in the built models which reduces the packing efficiency to a certain extent. Another reason can be attributed to the small model size.

**Table 1.**Structure analysis of Fe

_{1-x}Cr

_{x}models. The length unit is the lattice constant a

_{0}of FeCr alloys.

Phase | $\overline{{r}_{0}}$ | r_{min} | r_{max} | $\mathrm{\Delta}$r | Density (%) |
---|---|---|---|---|---|

Fe_{0.915}Cr_{0.085} | 2.2361 | 2.2361 | 2.2361 | 0.0000 | 52.6 |

Fe_{0.889}Cr_{0.111} | 2.0684 | 2.0000 | 2.1795 | 0.1795 | 54.3 |

Fe_{0.794}Cr_{0.206} | 1.4705 | 1.4142 | 1.6583 | 0.2441 | 52.0 |

#### 3.2. BCC Multi-Principal-Element Alloys

**Figure 2.**Atomic structure models of four- to eight-element BCC MPE phases by MaxEnt method. (

**a**) 8 × 8 × 8 quaternary phase, (

**b**) 8 × 8 × 8 quinary phase, (

**c**) 8 × 8 × 8 senary phase, (

**d**) 8 × 8 × 8 septenary phase, (

**e**) 8 × 8 × 8 octonary phase.

**Table 2.**Distribution of the shortest distances between the same-element atoms on the nearest neighbor lattice sites in the created BCC and FCC MaxEnt models. The distances for the successive nearest neighbor sites in BCC and FCC lattices are $\sqrt{3}{a}_{0}/2$, a

_{0}, $\sqrt{2}{a}_{0}$,...; and $\sqrt{2}{a}_{0}/2$, a

_{0}, $\sqrt{3/2}{a}_{0}$,..., respectively.

Phase | Cell type | Distance distribution in nearest neighbor sites (%) | |||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

Quaternary phase | BCC | 8.5 | 83.0 | 6.9 | 1.6 | 0.0 | 0.0 |

FCC | 74.3 | 23.8 | 1.9 | 0.0 | 0.0 | 0.0 | |

Quinary phase | BCC | 0.0 | 65.5 | 30.6 | 3.4 | 0.5 | 0.0 |

FCC | 47.3 | 45.0 | 7.5 | 0.2 | 0.0 | 0.0 | |

Senary phase | BCC | 0.0 | 41.4 | 52.4 | 6.0 | 0.2 | 0.0 |

FCC | 17.2 | 64.9 | 17.7 | 0.2 | 0.0 | 0.0 | |

Septenary phase | BCC | 0.0 | 19.2 | 65.3 | 15.0 | 0.3 | 0.2 |

FCC | 3.9 | 50.4 | 44.4 | 1.1 | 0.2 | 0.0 | |

Octonary phase | BCC | 0.0 | 2.5 | 70.9 | 24.3 | 2.1 | 0.2 |

FCC | 0.2 | 27.1 | 69.8 | 2.7 | 0.2 | 0.0 |

#### 3.3. FCC Multi-Principal-Element Alloys

**Figure 3.**Atomic structure models of four- to eight-element FCC MPE phases by MaxEnt method. The graphs from (

**a**) to (

**e**) are the quaternary, quinary, senary, septenary, and octonary phases, respectively.

**Figure 4.**Model of 2 × 2 × 2 quinary FCC MPE phase. There are six atoms in blue color, three atoms in yellow color, and two atoms in magenta color in the first nearest neighborship.

#### 3.4. Applications to the Real Multi-Principal-Element Alloys

_{l}is used in g calculation in this study. The local lattice constant varies due to the lattice distortion in MPE alloys. The calculation method of local lattice constant is illustrated in Figure 6. Parameter g measures the statistical correlation of local lattice constant. The paracrystalline lattice distortion parameters g of these alloys are given in the last column of Table 3. The lattice constant given in this table is the averaged local lattice parameter a

_{l}for all lattice sites in the model. A relatively larger error in the theoretical lattice constant compared to the experimental one of these optimized MPE phases is seen in Table 3. The reason for the problem could be mainly attributed to the empirical interatomic pair-potentials used in the model structure optimization. This problem will be further explored in the discussion section later. The g parameter provides a quantitative evaluation of the lattice distortion in MPE phases, which can be directly measured through x-ray diffraction experiment [32]. There are generally much more serious lattice distortions in BCC MPE alloys than in FCC ones as reflected in Figure 5 and the g parameters in Table 3. The g parameter of BCC MPE alloys is about two to three times higher than that of the FCC ones. The reason for the phenomenon is largely because the atomic radii of the componential elements vary widely in the BCC phases. The degree of lattice distortion in MPE alloy is sensitively depended on its composition as reflected from the calculated g-parameter. There is a close correlation between lattice distortion and the metallic radii [33] of the componential elements. For examples, the FCC phases of FeCoCrNi and CoCrFeMnNi own very small g parameters, which can be explained from the little differences in the metallic radii of their componential elements: Fe 1.26 Å, Co 1.25 Å, Cr 1.28 Å, Ni 1.24 Å, and Mn 1.27 Å. The metallic radius can be served as an effective criterion for the composition design of MPE alloys.

Phase | a (Å) | a_{expt} (Å) | Error | g |
---|---|---|---|---|

FCC FeCoCrNi | 3.84 | 3.56 [27] | 7.9% | 0.0085 |

FCC CoCrFeMnNi | 3.84 | 3.59 [5] | 7.0% | 0.0070 |

BCC AlCoCrFeNi | 3.08 | 2.87 [29] | 7.3% | 0.0210 |

BCC AlCoCrCuFeNi | 3.10 | 2.87 [30] | 8.0% | 0.0150 |

**Figure 5.**Relaxed atomic structure models of four typical MPE alloys. The Al, Fe, Co, Cr, Ni, Mn, and Cu atoms are in red, magenta, green, blue, cyan, yellow, and grey colors respectively.

**Figure 6.**Illustration of the local lattice constant calculation for a deformed cubic cell. The local lattice constant a

_{l}near the yellow atom is calculated by the averaged length of its three neighboring edges a

_{l}=(a1+a2+a3)/3 along the three forward axis-directions.

## 4. Discussion

_{i}(

**r**) is correlated with the local environment at position

**r**. This situation is similar to the charge density calculation ρ(

**r**) in the density-functional theory. However, the structure optimization can be done by maximizing the spacing of same-type particles since v

_{i}(

**r**) is proportional to the distance of their nearest neighbor. As a rough approximation, the entropic force might be expressed in a similar way as the repulsive Coulomb force of same-type charges in the simulation. The Monte Carlo algorithm is used to search for the system configuration with the maximum distance distribution of same-element atoms in this work. The advantage of this method is that it gives a highly efficient implementation of uniform particles distribution by rapid traversing the entire system space, and thus the time-consuming process of particles diffusion can be avoided. In the physics world, substance is the energy in condensed state, and the behavior of energy is determined by entropy. The in-depth study of entropic force is expected to unify all physics forces, including the electromagnetic force [34,35], gravity [20], etc., and finally leads to the establishment of the Unified Field Theory.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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