*4.3. Assessing Degree of Lattice Distortion in High Entropy Alloys by Parameter δ*

The phase formation rule in HEA is of great importance both scientifically and technologically. The formed phase(s) in HEAs (solid solution, intermetallics and amorphous phase) at certain conditions (alloy composition, preparation method, service environment, etc.) remain unknown for most HEAs [10,13,15,18,42]. Many researchers proposed various criteria to solve this problem, such as the *δ-*Δ*H*mix diagram proposed by Zhang et al. [13], VEC criteria proposed by Guo et al. [15], electronegativity mismatch *D*<sup>c</sup> proposed by Toda-Caraballo et al. [43], etc. Lattice distortion is a crucial factor in HEAs, and it is also very important in determining phase formation. However, the relationship between lattice distortion and phase formation is still not clear. Much research has been devoted to characterizing the degree of lattice distortion, and to further illustrate its correlation with phase formation, such as the *γ* parameter proposed by Wang et al. [44], the *α*<sup>2</sup> parameter proposed by Wang et al. [45], etc. However, it is far from clearly understanding. Further investigation is still required.

It is noticed from Table 2 that in most HE-BMGs, the atomic radius of the constituent elements atomic sizes distribute in a wide range; while for many solid solution forming HEAs, atomic sizes are more concentrated (especially for CuCoCrNiFe [10] and Cantor alloy [11], they both possess FCC structure, meanwhile atomic size difference of the constituent elements are very small). However, this is a qualitative description, and it is somehow ambiguous. As a result, a quantitative exemplification is needed.

Based on Table 2, here we propose a new parameter *δ* to assess the degree of lattice distortion in HEAs. Supposing that a HEA contains N elements, the atomic fractions are c1, c2 ... ... cN, respectively, and the atomic radii are *r*1, *r*<sup>2</sup> ... ... *r*<sup>N</sup> (*r*<sup>1</sup> < *r*<sup>2</sup> < ... ... < *r*N), respectively (data from ref. [46]). Then, the average atomic size is defined as *r*:

$$
\overline{r} = \sum\_{1}^{N} c\_{i} r\_{i} \tag{1}
$$

The lattice distortion parameter *δ* is defined as

$$\delta'=100\sum\_{1}^{N-1}\frac{c\_{i+1}+c\_i}{2}\frac{r\_{i+1}-r\_i}{\tilde{r}}\tag{2}$$

In particular, for equal atomic alloy, *δ* is given as

$$
\delta' = \frac{100}{N} \frac{r\_N - r\_1}{\overline{r}} \tag{3}
$$

According to Formula (2), lattice distortion parameter *δ* for some typical HEAs were calculated and listed in Table 3. For clarity, the relationship between atomic size distribution, lattice distortion parameter *δ* , and phase selection is demonstrated in Figure 4. It is noticed that *δ'* is closely related to phase selection in HEAs: when atomic size difference is relatively small, *δ* is also small (*δ* < 2.2), FCC solid solution would be formed; when the atomic size difference became larger, *δ* increased, FCC + BCC solid solution would tend to form as 2.2 < *δ* < 2.9; with even larger *δ* (2.9 < *δ* < 4.9), BCC solid solution would be formed; amorphous phase would be formed as *δ* exceeds 4.9.

**Figure 4.** Correlation between atomic size distribution, lattice distortion degree parameter *δ'* and phase selection in some typical HEAs [10–12].

The parameter *δ* can be understood from the point of view of dense atomic packing. Since it is correlated with the degree of lattice distortion, when *δ* was small, dense random packing FCC phase formed (its density is about 74% for monolic element); with the increase of lattice distortion, *δ* became larger, looser BCC phase (density of about 68% for monolic element) appeared, and its concentration increased accordingly; when lattice distortion became even serious, lattice collapse and amorphous phase would form eventually. Then, adjusting the lattice distortion or the parameter *δ* of HEAs, such as by similar element substitution or addition, could be helpful in designing high-entropy metallic glasses.

The new parameter *δ* we proposed here is somewhat similar with the *δ* parameter proposed by Zhang et al. [13]; it is also affected by the number, type, and concentration of the elements, while its value is smaller than *δ*, as is demonstrated in Table 3. As compared with *δ*, *δ* is more sensitive to addition/substitution of an ultra large/small atom. Taking alloy 11 and 12 in Table 3 for example, it can be seen that by substituting Hf element with much smaller Be element in the Ti-Zr-Hf-Cu-Ni HEA, *δ* increased from 10.324 to 12.514, the growth rate is 21%; while *δ* increased from 4.977 to 7.065, the growth rate is 42%, much larger than that of *δ*. It indicates that the new parameter *δ* is more sensitive than *δ* in certain circumstance.

Additionally, it is noticed from Formula (3), for equiatomic high entropy alloys, as the number of elements N increased, *δ* decreased and lattice distortion is mitigated accordingly. As a result, it is not beneficial for amorphous phase formation, especially for N > 10. This is in consistent with Cantor's result that an alloy with 16 to 20 elements in equiatomic concentration does not form amorphous phase [11].

**Table 3.** Correlation between atomic size distribution, lattice distortion and phase selection in some typical HEAs.


#### **5. Conclusions**

In this paper, two new high entropy bulk metallic glasses (HE-BMGs) have been successfully fabricated using copper mold casting method, namely Ti20Hf20Cu20Ni20Be20 with a critical diameter of 2 mm and Ti16.7Zr16.7Nb16.7Cu16.7Ni16.7Be16.7 with a critical diameter of 1.5 mm. These two HE-BMGs exhibit high fracture strength over 2300 MPa. The glass forming ability and atomic size distribution characteristics of the HE-BMGs

are discussed, and it is found that atomic radius spans over a wide range in HE-BMGs. Moreover, we propose a new parameter *δ* to assess the degree of lattice distortion in high entropy alloys (HEAs). It emphasizes the difference between atoms with adjacent atomic size, and it is closely related to phase selection in HEAs. When *δ* is relatively small (*δ* < 2.2), FCC solid solution formed; when 2.2 < *δ* < 2.9, FCC + BCC phases formed; when 2.9 < *δ* < 4.9, BCC phase formed; while *δ* > 4.9, amorphous phase would be formed. This new parameter *δ* is beneficial for understanding lattice distortion and phase selection in HEAs. The present work suggests that through adjusting the parameter *δ* by similar element substitution/addition, that is, adjusting the lattice distortion, is an effective way for designing high entropy bulk glassy alloy.

**Author Contributions:** Conceptualization, H.D.; Data curation, H.X.; Formal analysis, H.L.; Funding acquisition, H.D. and K.Y.; Investigation, H.B.; Methodology, H.B.; Project administration, K.Y.; Software, H.X.; Validation, H.X.; Visualization, H.L.; Writing–Original draft, H.D.; Writing–review & editing, H.L., H.B. and K.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Key Basic Research and Development Program (Grant No. 2016YFB00300500), National Natural Science Foundation of China (Grant Nos. 51571127 and 51871129). and Youth Fund of Jiangsu Natural Science Foundation (Grant No. BK20190979).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

**Conflicts of Interest:** The authors declare no competing interests.We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work.

### **References**

