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Atomic structure models of multi-principal-element alloys (or high-entropy alloys) composed of four to eight componential elements in both BCC and FCC lattice structures are built according to the principle of maximum entropy. With the concept of entropic force, the maximum-entropy configurations of these phases are generated through the use of Monte Carlo computer simulation. The efficiency of the maximum-entropy principle in modeling the atomic structure of random solid-solution phases has been demonstrated. The bulk atomic configurations of four real multi-principal-element alloys with four to six element components in either BCC or FCC lattice are studied using these models.

The discovery of high-entropy alloys [

High-entropy alloys generally contain five or more componential elements in nearly equimolar composition [

The principle of maximum entropy (MaxEnt) is a combinatorial theory comprising Shannon’s entropy in information theory [

In this work we build atomic structure models of bulk MPE alloys with BCC and FCC lattices for compositions ranging from four to eight principal elements according to the MaxEnt principle. The entropic force for the particle distribution in a closed system is defined. The MaxEnt configuration is generated by Monte Carlo simulation through entropic force minimization. Based on the built models, the atomic structure features of bulk MPE alloys are analyzed. The optimized lattice structures and lattice distortion parameters of several real quaternary, quinary, and senary MPE alloys are obtained using the models built in this work.

The entropy of an alloy system mainly consists of four parts: electronic entropy, magnetic entropy, vibrational entropy and configurational entropy or entropy of mixing. The electronic entropy is caused by the variation of electron distribution with structure configuration and temperature. The contribution of electronic entropy to the system free energy is usually very small. Alloys with iron, nickel, and cobalt elements may have magnetic entropy. The magnetic entropy describes the change of material magnetization with temperature and magnetic field [

The configurational entropy of multi-component alloy melt is very like the entropy of mixing in ideal gases. Like the mixing entropy of ideal gases, the configurational entropy of _{i}^{th} element which satisfies the following equation:

A Lagrangian equation can be constructed from Equations (1) and (2) according to the MaxEnt principle [

Taking the derivative of _{i}

The _{i}

Noticing that

The free space of a particle is defined as its maximum non-overlapping space with the other particles. We define _{i}_{i} = V/N

According to the MaxEnt principle, every particle is striving for the maximum free space in the system. The driving force for this trend is called as the entropic force. The definition of the entropic force is given by [_{i}_{i}(

Equation (9) indicates that the entropic force _{i}_{i}(_{i}(_{i}(

The atomic structure models of BCC binary random-substitution alloys Fe_{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.

A n × n × n BCC lattice of Fe matrix is created firstly. Then, the Fe atoms at some random sites are replaced by the solute atoms Cr. In the next step, this initial configuration is optimized according to the MaxEnt principle in the following way: The state of the maximum system entropy for this binary phase should be that each of Cr atom approaches to its maximum free space. Let
_{1-x}Cr_{x} are illustrated in

4 × 4 × 4 atomic structure models of binary alloys Fe_{1-x}Cr_{x}. The ratios of solute to solvent atoms in

The created models are carefully examined to assess the degree of compliance with the MaxEnt principle. The results of model structure analysis are given in _{min}_{max}

Structure analysis of Fe_{1-x}Cr_{x} models. The length unit is the lattice constant _{0}

Phase | _{min} |
_{max} |
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 |

Bulk atomic structure models of BCC MPE alloys for compositions from four to eight componential elements were created in this study. These models can be used in the studies for any four- to eight-element MPE alloys through appropriate element substitution. Since there is not a major element to act as solvent in MPE alloys, all atoms from the componential elements were randomly filled up in a cubic box by a n × n × n BCC lattice at the beginning of the model building. Then these atoms undergo distribution optimization according to the MaxEnt principle for the configuration with their maximum free space,

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

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 _{0}_{0}

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 |

FCC MPE phases are formed when the difference of the atomic radii between the componential elements are small. The built 6 × 6 × 6 bulk structure models for four- to eight-element FCC MPE phases in this work are presented in

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

Since the number of atoms in FCC bulk is twice as much as in the BCC phase, it becomes more difficult to separate the same-element atoms. The element distribution analyses given in

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.

Because no direct interatomic interaction was considered in the above modeling work, the built models should subject to further atomic-coordinates optimization [_{l}_{l}

Lattice constant and lattice distortion parameter of the optimized MPE alloys.

Phase | _{expt} |
Error | ||
---|---|---|---|---|

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

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

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

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

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.

Illustration of the local lattice constant calculation for a deformed cubic cell. The local lattice constant _{l}_{l}=(a1+a2+

It could be an impossible task to deduce an explicit expression of entropic force from the definition in Equation (9) because the _{i}(_{i}(

How to accurately modeling the microstructure of the condensed matters with lack of strict periodicity is a long-standing issue in materials research. More difficulty is imposed by the structural description of the random multi-componential systems. Historically, the cluster expansion method [

The advantages for using pair-potentials in the structure optimization of MPE alloys are simplicity, easy creation, and low calculation cost. Almost all the pair-potentials between any two metal elements in the Periodic Table of elements had been created by Chen's lattice inversion method, which is greatly convenient to the studies for the complicated and diverse alloy systems, such as high-entropy alloys. However, these pair-potentials were created from the first-principles total-energy calculation of the relevant unary or binary phases in their ideal stable lattice configurations [

In this paper we have successfully built atomic structure models of multi-principal-element alloys with components ranging from four to eight elements according to the principle of maximum entropy. The principle of maximum entropy predicts a uniform particle distribution in a random-alloy system. With the concept of entropic force, a Monte Carlo method was developed for generating the structure configurations of uniform particle distributions in the models. The lattice geometries of four real MPE alloys are optimized, and their lattice distortion parameters are calculated.

We grateful to Nanxian Chen and Jiang Shen for the interatomic potentials. This work was supported by the National Basic Research Program of China (No. 2011CB606403) and the National Natural Science Foundation of China (No. 50971119, 51071149). POV-Ray code [39] was used for graph rendering.

The authors declare no conflict of interest.

_{3}(Al

_{x}Si

_{1-x}) alloys

_{x}CoCrFeNi (0 ≤ x ≤ 2) high-entropy alloys