High-Throughput Determination of Interdiffusion Coefficients for Co-Cr-Fe-Mn-Ni High-Entropy Alloys

Abstract

In this report, a combination of the diffusion multiple technique and the recently developed pragmatic numerical inverse method was employed for a high-throughput determination of interdiffusivity matrices in Co-Cr-FeMn-Ni high-entropy alloys (HEAs). Firstly, one face-centered cubic (fcc) quinary Co-Cr-Fe-Mn-Ni diffusion multiple at 1373 K was carefully prepared by means of the hot-pressing technique. Based on the composition profiles measured by the field emission electron probe micro analysis (FE-EPMA), the composition-dependent interdiffusivity matrices in quinary Co-Cr-Fe-Mn-Ni system at 1373 K were then efficiently determined using the pragmatic numerical inverse method. The determined interdiffusivities show good agreement with the limited results available in the literature. Moreover, the further comparison with the interdiffusivities in the lower-order systems indicates the sluggish diffusion effect in Co-Cr-Fe-Mn-Ni HEAs, which is however not observed in tracer diffusivities. In order for the convenience in further analysis, a generalized transformation relation among interdiffusivities with different dependent components in multicomponent systems was finally derived.

Appendix

In an N-component system, the interdiffusion flux of a component i under constant molar volume can be expressed by Fick’s first law,

$$\tilde{J}_{i} = - \sum\limits_{j = 1}^{N - 1} {\left( {\tilde{D}_{ij}^{N} \cdot \frac{{\partial c_{j} }}{\partial x}} \right)} \quad i = 1, \ldots ,N - 1$$

(9)

Then, one can obtain

$$\tilde{J}_{i} = - \sum\limits_{j = 1}^{N - 2} {\left( {\tilde{D}_{ij}^{N} \cdot \frac{{\partial c_{j} }}{\partial x}} \right)} - \tilde{D}_{iN - 1}^{N} \cdot \frac{{\partial c_{N - 1} }}{\partial x}\quad i = 1, \ldots ,N - 2$$

(10)

$$\tilde{J}_{N - 1} = - \sum\limits_{j = 1}^{N - 2} {\left( {\tilde{D}_{N - 1j}^{N} \cdot \frac{{\partial c_{j} }}{\partial x}} \right)} - \tilde{D}_{N - 1N - 1}^{N} \cdot \frac{{\partial c_{N - 1} }}{\partial x}$$

(11)

In fact, Eq 9-11 are presented when the component N is treated as the dependent component. If the component N − 1 is selected as the dependent component, one can have

$$\tilde{J}_{i} = - \sum\limits_{j = 1}^{N - 2} {\left( {\tilde{D}_{ij}^{N - 1} \cdot \frac{{\partial c_{j} }}{\partial x}} \right)} - \tilde{D}_{iN}^{N - 1} \cdot \frac{{\partial c_{N} }}{\partial x}\quad i = 1, \ldots ,N - 2$$

(12)

$$\begin{aligned} \tilde{J}_{N - 1} & = 0 - \tilde{J}_{N} - \sum\limits_{i = 1}^{N - 2} {\tilde{J}_{i} } \\ & { = }\sum\limits_{j = 1}^{N - 2} {\left( {\tilde{D}_{Nj}^{N - 1} \cdot \frac{{\partial c_{j} }}{\partial x}} \right)} + \tilde{D}_{NN}^{N - 1} \cdot \frac{{\partial c_{N} }}{\partial x} - \sum\limits_{i = 1}^{N - 2} {\tilde{J}_{i} } \\ \end{aligned}$$

(13)

Considered the relation ($\frac{{\partial c_{N} }}{\partial x} = - \sum\limits_{j = 1}^{N - 1} {(\frac{{\partial c_{j} }}{\partial x})}$), Eq 12 and 13 can be revised as,

$$\tilde{J}_{i} = - \sum\limits_{j = 1}^{N - 2} {\left[ {(\tilde{D}_{ij}^{N - 1} - \tilde{D}_{iN}^{N - 1} ) \cdot \frac{{\partial c_{j} }}{\partial x}} \right]} - \left( { - \tilde{D}_{iN}^{N - 1} } \right) \cdot \frac{{\partial c_{N - 1} }}{\partial x}\quad i = 1, \ldots ,N - 2$$

(14)

$$\tilde{J}_{N - 1} = - \sum\limits_{j = 1}^{N - 2} {\left[ {\left( {\tilde{D}_{NN}^{N - 1} - \tilde{D}_{Nj}^{N - 1} } \right) \cdot \frac{{\partial c_{j} }}{\partial x}} \right]} - \tilde{D}_{NN}^{N - 1} \cdot \frac{{\partial c_{N - 1} }}{\partial x} - \sum\limits_{i = 1}^{N - 2} {\tilde{J}_{i} }$$

(15)

A direct comparison between Eq A6 with Eq A4 leads to,

$$\begin{aligned} \tilde{D}_{ij}^{N - 1} = \tilde{D}_{ij}^{N} - \tilde{D}_{iN - 1}^{N} \quad i,j = 1, \ldots ,N - 2 \hfill \\ \tilde{D}_{iN}^{N - 1} = - \tilde{D}_{iN - 1}^{N} \quad \quad i = 1, \ldots ,N - 2 \hfill \\ \end{aligned}$$

(16)

By substituting Eq 14 into Eq 15, one can obtain

$$\begin{aligned} \tilde{J}_{N - 1} & = - \sum\limits_{j = 1}^{N - 2} {\left[ {\left( {\tilde{D}_{NN}^{N - 1} - \tilde{D}_{Nj}^{N - 1} } \right) \cdot \frac{{\partial c_{j} }}{\partial x}} \right]} - \tilde{D}_{NN}^{N - 1} \cdot \frac{{\partial c_{N - 1} }}{\partial x} - \sum\limits_{i = 1}^{N - 2} {\left\{ {\sum\limits_{j = 1}^{N - 2} {\left[ {\left( {\tilde{D}_{iN}^{N - 1} - \tilde{D}_{ij}^{N - 1} } \right) \cdot \frac{{\partial c_{j} }}{\partial x}} \right]} + \tilde{D}_{iN}^{N - 1} \cdot \frac{{\partial c_{N - 1} }}{\partial x}} \right\}} \\ & = - \sum\limits_{j = 1}^{N - 2} {\left\{ {\left[ {\left( {\tilde{D}_{NN}^{N - 1} - \tilde{D}_{Nj}^{N - 1} } \right) + \sum\limits_{i = 1}^{N - 2} {\left( {\tilde{D}_{iN}^{N - 1} - \tilde{D}} \right)_{ij}^{N - 1} } } \right] \cdot \frac{{\partial c_{j} }}{\partial x}} \right\}} - \left( {\tilde{D}_{NN}^{N - 1} + \sum\limits_{i = 1}^{N - 2} {\tilde{D}_{iN}^{N - 1} } } \right) \cdot \frac{{\partial c_{N - 1} }}{\partial x} \\ \end{aligned}$$

(17)

A direct comparison between Eq 17 with Eq 11 leads to,

$$\begin{aligned} \tilde{D}_{NN}^{N - 1} & = \tilde{D}_{N - 1N - 1}^{N} - \sum\limits_{i = 1}^{N - 2} {\tilde{D}_{iN}^{N - 1} } = \sum\limits_{i = 1}^{N - 1} {\tilde{D}_{iN - 1}^{N} } \\ \tilde{D}_{Nj}^{N - 1} & = \tilde{D}_{NN}^{N - 1} + \sum\limits_{i = 1}^{N - 2} {\left( {\tilde{D}_{iN}^{N - 1} - \tilde{D}_{ij}^{N - 1} } \right)} - \tilde{D}_{N - 1j}^{N} \\ & = \sum\limits_{i = 1}^{N - 1} {\tilde{D}_{iN - 1}^{N} } - \sum\limits_{i = 1}^{N - 2} {\tilde{D}_{ij}^{N} } - \tilde{D}_{N - 1j}^{N} \\ & = \sum\limits_{i = 1}^{N - 1} {\left( {\tilde{D}_{iN - 1}^{N} - \tilde{D}_{ij}^{N} } \right)} \, j = 1, \ldots ,N - 2 \\ \end{aligned}$$

(18)

According to the above mathematical derivations, a generalized transformation relation in an N-component system is achieved.