Stochastic Continuum Models for High–Entropy Alloys with Short-range Order

In this subsection, we establish the atomistic model of HEAs in one row without defects. In an HEA, each atom site is randomly occupied by one of the main elements. Denote the set of all these elements by $\Omega$:

$$\Omega=\{e_{1},~{}e_{2},\cdots,~{}e_{m}\}$$

(1)

For each lattice site, a random variable $\omega$ is defined with sample space $\Omega$ and probability measure:

$$\mathbf{P}(e_{k}):=p_{k}\geq 0,\ \ \ \sum_{i=1}^{m}p_{i}=1.$$

(2)

Here $p_{k}$ is the probability of element $e_{k}$ occupying the lattice site. We denote $\omega_{j}$ to be such a random variable at the $j$-th site in the HEA. The location of the $j$-th atom is denoted by $a_{j}$. See Fig. 1 for an illustration.

Consider an elastic displacement field $\{u_{j}\}_{j\in\mathbb{Z}}\subset\mathbb{R}$ on the HEA lattice $\{a_{j}\}_{j\in\mathbb{Z}}$. Suppose that the HEA system is described by pairwise potentials. Without loss of generality, we consider the nearest neighbor interaction. In the HEA system, the pairwise potentials depend on not only the distance between the two atoms but also their elements. Therefore, the interaction energy between atoms $a_{j}$ and $a_{j+1}$ can be written as

$$V\left(h_{j}+u_{j+1}-u_{j},\omega_{j},\omega_{j+1}\right),$$

(3)

where $h_{j}:=h(\omega_{j},\omega_{j+1})$ is the random lattice constant which is the solution of

$$\frac{\,\mathrm{d}V(r,\omega_{j},\omega_{j+1})}{\,\mathrm{d}r}\Big{|}_{r=h_{j}}=0.$$

(4)

Here $r$ is the distance between these two atoms. Hence the total energy of the HEA using the atomistic model is

$$E_{\text{a-el}}=\sum_{j\in\mathbb{Z}}\left[V\left(h_{j}+u_{j+1}-u_{j},\omega_{j},\omega_{j+1}\right)-V\left(h_{j},\omega_{j},\omega_{j+1}\right)\right].$$

(5)

This elastic energy has the following approximate formula:

	$\displaystyle E_{\text{a-el}}=$	$\displaystyle\sum_{j\in\mathbb{Z}}\left[V\left(h_{j}+u_{j+1}-u_{j},\omega_{j},\omega_{j+1}\right)-V\left(h_{j},\omega_{j},\omega_{j+1}\right)\right]$
	$\displaystyle\approx$	$\displaystyle\frac{1}{2}\sum_{j\in\mathbb{Z}}V^{\prime\prime}_{j}h_{j}^{2}\left(\frac{u_{j+1}-u_{j}}{h_{j}}\right)^{2}$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\sum_{j\in\mathbb{Z}}\beta_{j}\left(\frac{u_{j+1}-u_{j}}{h_{j}}\right)^{2}h_{j},$		(6)

where

	$\displaystyle\beta_{j}:=$	$\displaystyle V^{\prime\prime}_{j}h_{j},$		(7)
	$\displaystyle V^{\prime\prime}_{j}:=$	$\displaystyle\frac{\,\mathrm{d}^{2}V(r,\omega_{j},\omega_{j+1})}{\,\mathrm{d}r^{2}}\Big{|}_{r=h_{j}}.$		(8)

Here $\beta_{j}$ can be considered as the elastic modulus on the atomic scale. In fact, from Eq. (6), we have $E_{\text{a-el}}\approx\int_{x\in\mathbb{R}}\frac{1}{2}\beta_{\text{atom}}(x)\left(\frac{\,\mathrm{d}u}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x$, where $\beta_{\text{atom}}(x)\approx\beta_{j}$ for $x\in[a_{j},a_{j+1})$.

In this subsections, we present the atomistic model and assumptions for the short-range order in HEAs. We employ the $\alpha$-mixing coefficients $\alpha_{n}$ [33] to describe the short-range order in HEAs. For the random variable sequence $\{X_{j}\}_{j\in\mathbb{Z}}$, the $\alpha$-mixing coefficient $\alpha_{n}$ is defined as [33]

$$\alpha_{n}=\sup\left\{|P(A\cap B)-P(A)P(B)|:A\in\mathcal{F}_{-\infty}^{k},B\in\mathcal{F}_{k+n}^{+\infty},\ \forall k\in\mathbb{Z}\right\}$$

(12)

where $\mathcal{F}_{a}^{b}$ is the $\sigma$-field generated by $\{X_{a},~{}X_{a+1},~{}\cdots,~{}X_{b}\}$.

Recall that the method of Warren-Cowley pair-correlation parameters [28] is one of the widely used classical methods to describe short-range order in muliti-component systems including HEAs (e.g., [8, 10, 11, 14, 15, 22]): $\alpha_{e_{i}e_{j}}(n):=1-\frac{P_{e_{i}e_{j}}(n)}{p_{j}}$, where $P_{e_{i}e_{j}}(n)$ is the probability of finding an atom of type $e_{j}$ at $a_{n+r}$ given an atom of type $e_{i}$ at $a_{r}$, and $p_{j}$ is the probability of element $e_{j}$ occupying the lattice site defined in Eq. (2). The $\alpha$-mixing coefficients $\alpha_{n}$ of $\{\omega_{j}\}_{j\in\mathbb{Z}}$ in HEAs are stronger than the pair-correlation parameters $\alpha_{e_{i}e_{j}}(n)$ since correlations between groups of atoms are also considered in the definition of the $\alpha$-mixing coefficients:

	$\displaystyle\alpha_{n}=$	$\displaystyle\sup\left\{|P(A\cap B)-P(A)P(B)|:A\in\mathcal{F}_{-\infty}^{k},B\in\mathcal{F}_{k+n}^{+\infty},\ \forall k\in\mathbb{Z}\right\}$
	$\displaystyle\geq$	$\displaystyle\sup_{k}\big{\{}|\mathbf{P}(\omega_{k}=e_{i},\omega_{k+n}=e_{j})-\mathbf{P}(\omega_{k}=e_{i})\mathbf{P}(\omega_{k+n}=e_{j})|\big{\}},\ e_{i},e_{j}\in\Omega$
	$\displaystyle=$	$\displaystyle p_{i}p_{j}|\alpha_{e_{i}e_{j}}(n)|$
	$\displaystyle\geq$	$\displaystyle\lambda|\alpha_{e_{i}e_{j}}(n)|,$

where $\lambda:=\min\{p_{i}^{2}\}$.

In this paper, we consider a short range order in HEAs that is rapidly decaying with atomic distance as in the following assumption:

The rapidly decaying short-range order described in Assumption 1 is consistent with the property of short-range order in alloys and HEAs. In Refs. [34, 35], they showed that short-range order exists in alloys when the temperature is greater than the critical temperature, and it decays quickly with the atomic distance. In Refs. [14, 15, 10], they only considered the first and second nearest-neighbor shell short range orders ($\alpha_{e_{i}e_{j}}(n)$ for $n=1,2$) in HEAs based on the fact that the correlation parameter $\alpha_{e_{i}e_{j}}(n)\to 0$ quickly as $n$ increases and the first and second nearest-neighbor shell short range orders play dominant role in the correlation effect. Moreover, the rapid decay of the $\alpha$-mixing coefficients $\alpha_{n}$ implies rapid decay of the correlation. This is can be proved by a lemma in Ref. [33] (Lemma 2 on Page 365) that the correlation is bounded by the $\alpha$-mixing coefficient.

We will use the following generalized central limit theorem in our derivation. Note that it holds for $\{\omega_{j}\}_{j\in\mathbb{Z}}$ due to the Assumption 1.

Theorem [33, Theorem 27.4] Suppose that random variables $X_{1},X_{2},\cdots$ are stationary with $\alpha$-mixing coefficient $\alpha_{n}=O\left(n^{-5}\right),$ and $\mathbf{E}\left(X_{n}\right)=0$, $\mathbf{E}\left[X_{n}^{12}\right]<\infty$. Let $S_{n}=X_{1}+$ $\cdots+X_{n}$ and $\sigma^{2}={\displaystyle\lim_{n\rightarrow\infty}\mathbf{E}\left[S_{n}^{2}\right]/n}$, where $\sigma$ is positive. Then

$$\frac{S_{n}}{\sigma\sqrt{n}}\stackrel{{\scriptstyle d}}{{\longrightarrow}}\mathcal{N}(0,1),\quad\text{ as }n\rightarrow\infty,$$

(14)

where $\mathcal{N}(0,1)$ is the standard Gaussian distribution, and $\stackrel{{\scriptstyle d}}{{\longrightarrow}}$ is the convergence in distribution.

As described in the previous subsection, we consider the nearest neighbor interaction in this paper. The following lemma is able to give the relationship between the $\alpha$-mixing coefficient of $\{\omega_{j}\}$ and the $\alpha$-mixing coefficient of the pairwise interaction energies associated with $\{(\omega_{j},\omega_{j+1})\}$.

This lemma can be proved by the definition of the $\alpha$-mixing coefficient, and the fact that $\sigma(Y_{j})\subset\sigma(X_{j},X_{j+1})$ and as a result, $\sigma(Y_{k},Y_{k+1},\cdots,Y_{k+n})\subset\sigma(X_{k},X_{k+1},\cdots,X_{k+n+1})$.

Based on Assumption 1, we define the following length to characterize the range of short-range order, based on the nearest neighbor interaction energies associated with $\{(\omega_{j},\omega_{j+1})\}$ (i.e., the bounds between atoms $\{(a_{j},a_{j+1})\}$).

Fig. 2 illustrates this length $H$ of nonzero short range order for the bond between atoms $a_{0}$ and $a_{1}$, i.e., it is independent with those bonds outside this range. Note that $H/2$ is the correlation length in this atomistic model. Here we neglect the perturbation of lattice constant for simplicity in this definition. Our analysis also works for stochastic $H$.

In order to derive a continuum model from the atomistic model, we assume that there are three length scales: the atomic scale, the supercell with size of $H$, and the continuum scale. This is summarized in the following assumption.

Finally in this subsection, we define the continuum limit of the average of random variables, which will be used to obtain the stochastic continuum models.

This definition of continuum limit of average of random variables is different from that in the deterministic case. In addition to the continuum limit of the simple average $A={\displaystyle\lim_{n\to\infty}}\frac{1}{n}\sum_{i=1}^{n}X_{i}=E+\lambda U$, there is also a contribution ${\displaystyle\lim_{n\to\infty}}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}R_{i}=Z_{i}$ to account for the average of variances, which comes from the weakly dependent sequence $\{R_{i}\}_{i=1}^{n}$ and vanishes in the simple average: ${\displaystyle\lim_{n\to\infty}}\frac{1}{n}\sum_{i=1}^{n}R_{i}=0$ due to the generalized central limit theorem Theorem [33, Theorem 27.4] shown above. The above definition guarantees that $\sum_{i=1}^{n}Z_{i}$ and $\sum_{i=1}^{n}R_{i}$ have the same collective behavior, i.e., $\sim\mathcal{N}(0,n\Delta_{R}^{2})$ for large $n$.

For an example, for the i.i.d. random variables $\{X_{i}\}_{i=1}^{n}$ with Gaussian distribution $\mathcal{N}(E,\sigma^{2})$, the averaged sequence given by the above definition are $\{Y_{i}\}_{i=1}^{n}$ that are still i.i.d. with Gaussian distribution $\mathcal{N}(E,\sigma^{2})$, which is the desired result. On the other hand, if we simply use ${\displaystyle\lim_{n\to\infty}}\frac{1}{n}\sum_{i=1}^{n}X_{i}$ to approximate each random variable, then we have $Y_{i}=E$ for all $i$. The information of variance, i.e., the information of randomness, is lost in this continuum limit.

In this subsection, we derive a continuum stochastic elasticity theory in HEAs with short range order from the atomistic model. We start from continuum approximation of the atomic-level stochastic elastic modulus $\{\beta_{j}\}$ defined in Eq. (7). From its definition, $\beta_{j}=\beta(\omega_{j},\omega_{j+1})$ depending on the interaction of the bond between atoms $a_{j}$ and $a_{j+1}$. As discussed in the previous subsection on the interaction energies associated with atomic bonds, these $\beta_{j}$’s have identical distribution by the setting of the system as described in the previous subsection, but they are not necessarily independent due to the existence of the short-range order. The length of range of short-range order of $\{\beta_{j}\}$ is $H$ specified in Definition 1. We also have the following two lemmas that are related to the short-range order of $\{\beta_{j}\}$.

The conclusions in this lemma come directly from Assumption 1, the definition of $\{\beta_{j}\}$ in Eqs. (7) and (8), and Lemma 1.

Now we consider the continuum approximation of $\mathrm{d}\beta(x)$. Starting from $\beta_{i}$, consider $\{\beta_{j}\}$ over a cell with size $H$ defined in Eq. (16) which contains $n_{s}$ atomic sites, i.e., $\beta_{k+i}$, $k=0,1,2,\cdots,n_{s}$. We have

$$\sum_{k=0}^{n_{s}}(\beta_{k+i}-\beta_{i})=(n_{s}+1)\bar{\beta}-(n_{s}+1)\beta_{i}+\sum_{k=0}^{n_{s}}\left(\beta_{k+i}-\mathbf{E}\beta_{k+i}\right),$$

(30)

where

$$\bar{\beta}:=\mathbf{E}\beta_{i}.$$

(31)

Note that $\bar{\beta}=\mathbf{E}\beta_{j}$ for any $j$.

Let $\beta(x)$ be the elastic modulus on the continuum scale. We want to use the average of $\sum_{k=0}^{n_{s}}(\beta_{k+i}-\beta_{i})$ to approximate $\frac{1}{2}\,\mathrm{d}\beta(x)$ over the length $H$. (Note that in the deterministic case, $\sum_{k=0}^{n_{s}}(\beta_{k+i}-\beta_{i})\approx\frac{1}{2}n_{s}(n_{s}+1)\beta^{\prime}(a_{i})\bar{h}$, while $\frac{1}{2}\,\mathrm{d}\beta(x)\approx\frac{1}{2}n_{s}\beta^{\prime}(a_{i})\bar{h}$, i.e., $\frac{1}{n_{s}+1}\sum_{k=0}^{n_{s}}(\beta_{k+i}-\beta_{i})\approx\frac{1}{2}\,\mathrm{d}\beta(x)$.) From Eq. (30), the average of $\sum_{k=0}^{n_{s}}(\beta_{k+i}-\beta_{i})$ equals the average of $(n_{s}+1)\bar{\beta}-(n_{s}+1)\beta_{i}+\sum_{k=0}^{n_{s}}\left(\beta_{k+i}-\mathbf{E}\beta_{k+i}\right)$, which by Definition 2 of the continuum limit of average of random variables and the decay property of the $\alpha$-mixing coefficient $\alpha_{n}$ of $\{\beta_{j}\}$ in Lemma 2, is approximately $\bar{\beta}-\beta_{i}+\mathcal{N}(0,\Delta_{e}^{2})$, where $\Delta_{e}^{2}$ is defined in Eq. (27). Note that the limit $n_{s}\to\infty$ holds due to the condition $H\gg\bar{h}$ in Assumption 2, where $n_{s}=H/\bar{h}$.

To summarize, the continuum limit of the average of $\sum_{k=0}^{n_{s}}(\beta_{k+i}-\beta_{i})$, i.e., $\frac{1}{2}\,\mathrm{d}\beta(x)$ over length $H$ starting from site $a_{i}$, is

$$\frac{1}{2}\,\mathrm{d}\beta(x)\approx\bar{\beta}-\beta_{i}+\mathcal{N}(0,\Delta_{e}^{2}).$$

(32)

This averaged increment defines $\beta_{i+\frac{n_{s}}{2}}$ in the continuum formulation in the middle of the supercell between locations $a_{i}$ (with value $\beta_{i}$) and $a_{i+n_{s}}$ (with value $\beta_{i+n_{s}}$). That is, in the continuum model, the values of $\beta_{i+jn_{s}}$, for integer $j$, are inherited directly from the atomistic model, and the values of $\beta_{i+(j+\frac{1}{2})n_{s}}$ are defined through the averaged increment within a supercell in the atomistic model as given in Eq. (32); see an illustration of the values defined in the continuum model in Fig. 3. The averaged increment $\frac{1}{2}\,\mathrm{d}\beta$ obtained in Eq. (32) serves as a link from atomistic model to continuum model that passes the atomic level short-range order to the continuum model.

Since we are approximating $\,\mathrm{d}\beta$ over the length of $H$ scale, setting $\,\mathrm{d}x=H$ and $\,\mathrm{d}B_{x}=B_{x+H}-B_{x}$, where $B_{x}$ is the Brownian motion, we have the approximation at $x=a_{i}$:

$$\frac{1}{2}\,\mathrm{d}\beta(x)=\frac{\bar{\beta}-\beta(x)}{H}\,\mathrm{d}x+\frac{\Delta_{e}}{\sqrt{H}}\,\mathrm{d}B_{x}.$$

(33)

This is the Ornstein–Uhlenbeck (OU) process [29, 30, 31, 32]. The OU process has been employed phenomenologically in Ref. [17] to model the short-range order in HEAs, and here we provide a rigorous derivation from stochastic atomistic model with parameters directly from the atomistic model.

The continuum model in Eq. (33) was obtained based on the approximation of $\mathrm{d}\beta(x)$ over a supercell with size $H$, which is defined in Eq. (16). This continuum formulation strongly depends on $H$, and here we discuss more on why this length is appropriate for the continuum approximation of $\mathrm{d}\beta(x)$. First, since we want to have a well-defined continuum limit, the length $H_{\beta}$ over which $\mathrm{d}\beta(x)$ is obtained has to be no less than $H$, so that the variance $\Delta_{e}^{2}$ defined in (27) does not change when $H_{\beta}$ is further increased. Moreover, on the continuum level, $\beta(x)$ satisfies Eq. (33), which is an OU process whose correlation length is $H/2$ (given in Eq. (37) below). This correlation length in the continuum model agrees with that in the atomistic model. On the other hand, if $\mathrm{d}\beta(x)$ is approximated by the average over a length $H_{\beta}>H$, we will have parameter $H_{\beta}$ instead of $H$ in the continuum model in Eq. (33), and as a result, the correlation length in the continuum model will be $H_{\beta}/2$, which is strictly greater than the correlation length $H/2$ in the atomistic model. Therefore, $H$ is the appropriate length for the continuum approximation of $\mathrm{d}\beta(x)$ from atomistic model.

From Eq. (33) and the solution formula of the OU process [29, 30, 31, 32], the stochastic elastic modulus $\beta(x)$ is

$$\beta(x)=\bar{\beta}+\Delta_{e}Y_{x},$$

(34)

where $\{Y_{x}\}_{x\in\mathbb{R}}$ is an OU process:

$$Y_{x}=\int_{-\infty}^{x}\frac{2}{\sqrt{H}}e^{\frac{2}{H}(s-x)}\,\mathrm{d}B_{s}.$$

(35)

For each point $x$, $\beta(x)$ is a Gaussian:

$$\beta(x)\sim\mathcal{N}\left(\bar{\beta},\Delta_{e}^{2}\right),$$

(36)

and the correlation of $\beta(x)$ at two points $x_{1}$ and $x_{2}$ is

$$\operatorname{Cov}(\beta(x_{1}),\beta(x_{2}))=\Delta_{e}^{2}e^{-\frac{2|x_{1}-x_{2}|}{H}}.$$

(37)

Therefore, $\Delta_{e}$ indicates the amplitude of randomness at each lattice site, and $H/2$ is the correlation length. These agree with the definitions of $\Delta_{e}^{2}$ in (27) (which is the average of variances at $n_{s}$ lattice sites) and $H$ in (16) in the atomistic model.

With this continuum stochastic elastic modulus, the stochastic elastic energy $E_{\text{elas}}$ satisfies

$$\,\mathrm{d}E_{\text{elas}}=\frac{1}{2}\beta(x)\left(\frac{\,\mathrm{d}u}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x.$$

(38)

When the range of the short-range order, i.e., the correlation length, $H\to+\infty$, by Eq. (37), $\mathbf{E}\left([\beta(x_{1})-\beta(x_{2})]^{2}\right)=2\Delta_{e}^{2}\left(1-e^{-\frac{2|x_{1}-x_{2}|}{H}}\right)\to 0$ for any $x_{1}$ and $x_{2}$, which is the case of uniform randomness. When $H\to 0$, by Eq. (37), $\operatorname{Cov}(\beta(x_{1}),\beta(x_{2}))\to 0$ for $x_{1}\neq x_{2}$, which means that $\beta(x_{1})$ and $\beta(x_{2})$ are independent for any $x_{1}\neq x_{2}$. Especially, in the regime of $H\to 0$ with $\beta(x_{1})$ and $\beta(x_{2})$ being independent for different lattice sites $x_{1}\neq x_{2}$, the right-hand side of Eq. (33) dominates, and Eq. (33) becomes $\beta(x)\,\mathrm{d}x=\bar{\beta}\,\mathrm{d}x+\Delta_{e}\sqrt{\bar{h}}\,\mathrm{d}B_{x}$, where $\bar{h}$ is the average lattice constant which is the smallest distance between atomic sites. In this independent case, we have

$$\,\mathrm{d}E_{\text{elas}}=\frac{1}{2}\left(\frac{\,\mathrm{d}u}{\,\mathrm{d}x}\right)^{2}\left(\bar{\beta}\,\mathrm{d}x+\Delta_{e}\sqrt{\bar{h}}\,\mathrm{d}B_{x}\right).$$

(39)

This agrees with the energy formulation derived in Ref. [27] ((5.14) there) under the assumption of independent randomness, i.e., without short-range order.

In the continuum formulation of the elastic energy $E_{\text{elas}}$ given in Eq. (38), both the elastic modulus $\beta(x)$ and the displacement gradient $\frac{\,\mathrm{d}u}{\,\mathrm{d}x}$ contain the effect of randomness. When the effect of randomness is small, we have

	$\displaystyle E_{\text{elas}}=$	$\displaystyle\int_{\mathbb{R}}\frac{1}{2}\beta(x)\left(\frac{\,\mathrm{d}u}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\frac{1}{2}(\bar{\beta}+\delta\beta(x))\left(\frac{\,\mathrm{d}\bar{u}}{\,\mathrm{d}x}+\frac{\,\mathrm{d}\delta u}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x\,$
	$\displaystyle\approx$	$\displaystyle\int_{\mathbb{R}}\frac{1}{2}\bar{\beta}\left(\frac{\,\mathrm{d}\bar{u}}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x+\int_{\mathbb{R}}\bar{\beta}\frac{\,\mathrm{d}\bar{u}}{\,\mathrm{d}x}\frac{\,\mathrm{d}\delta u}{\,\mathrm{d}x}\,\mathrm{d}x+\int_{\mathbb{R}}\frac{1}{2}\delta\beta(x)\left(\frac{\,\mathrm{d}\bar{u}}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x,$		(40)

where $\bar{u}(x)=\mathbf{E}u(x)$. Here the first term is the $O(1)$ elastic energy, the second term is the leading order perturbation due to the randomness in displacement (i.e., randomness in lattice constant), and the third term is leading order perturbation due to the randomness effect in elastic modulus.

The Peierls-Nabarro models [25, 26, 37, 36, 38] are continuum models for dislocations that incorporate the atomistic structure. In the classical Peierls–Nabarro model for a dislocation in a crystal with a single type atoms, the slip plane of the dislocation separates the entire system into two continuums described by linear elasticity theory, and the interaction across the slip plane is modeled by a nonlinear potential (the $\gamma$-surface) that comes from the atomic interaction [37]. This continuum description enables nonlinear interaction within the dislocation core where the atomic structure is heavily distorted. In the Peierls-Nabarro model for the two-layer system, each layer is a continuum governed by linear elasticity, and there is a nonlinear interaction between the two layers; see Fig. 4(b) for an illustration of the atomic structure.

Consider an edge dislocation in a bilayer system as illustrated in Fig. 4(b). The Burger vector of the dislocation is $\bm{b}=(\bar{h},0)$. The displacements in the $x$ direction (along the layer) of the top and bottom layers are $u^{+}(x)$ and $u^{-}(x)$, respectively. The disregistry (relative displacement) between the two layers is

$$\phi(x):=u^{+}(x)-u^{-}(x).$$

(41)

For this edge dislocation, $\phi(x)$ satisfies the boundary condition

$$\lim_{x\to-\infty}\phi(x)=0,~{}\lim_{x\to+\infty}\phi(x)=\bar{h},$$

(42)

where $\bar{h}$ is the lattice constant; see Fig. 4(a).

In the Peierls-Nabarro model, the total energy is written as the sum of the elastic energy and the misfit energy. The elastic energy is the energy in the two continuums separated by the slip plane, and here in the bilayer system it is the intra-layer elastic energy. The misfit energy in the bilayer system is the inter-layer energy, whose density is the $\gamma$-surface depending on the disregistry $\phi$ [37], denoted by $\bar{\gamma}(\phi)$. Under the assumption that $u^{+}(x)=-u^{-}(x)$, the elastic energy density can also be written based on $\phi$. That is,

	$\displaystyle E_{\text{PN}}$	$\displaystyle=E_{\text{elas}}[\phi]+E_{\text{mis}}[\phi],$		(43)
	$\displaystyle E_{\text{elas}}[\phi]$	$\displaystyle=\int_{\mathbb{R}}\frac{1}{4}\bar{\beta}\left(\frac{\,\mathrm{d}\phi}{\,\mathrm{d}x}\right)^{2}\,\mathrm{d}x,$		(44)
	$\displaystyle E_{\text{mis}}[\phi]$	$\displaystyle=\int_{\mathbb{R}}\bar{\gamma}(\phi)\,\mathrm{d}x.$		(45)

Here $E_{\text{PN}}$ is the total energy in the classical Peierls-Nabarro model, $E_{\text{elas}}[\phi]$ and $E_{\text{mis}}[\phi]$ are the elastic energy and misfit energy, respectively, and $\bar{\beta}$ is the elastic constant. The $\gamma$-surface $\bar{\gamma}(\phi)$ can be calculated from the atomistic model as the energy increment when the perfect lattice system has a uniform shift $\phi$ between the two layers [37].

Note that a rigorous derivation from atomistic model to the classical Peierls-Nabarro model for the dislocation in a bilayer system with the same type of atoms has been presented in Ref. [39].

The atomic structure of a perfect bilayer HEA is shown in Fig. 5(a). Similarly to the single layer HEA discussed in Sec. 2.1, we denote $j$-th atom at upper (or bottom) layer as $a_{j}^{+}$ (or $a_{j}^{-}$) and the random variable to describe the element at $a_{j}^{+}$ (or $a_{j}^{-}$) as $\omega_{j}^{+}$ (or $\omega_{j}^{-}$). Each random variable $\omega_{j}^{+}$ or $\omega_{j}^{-}$ has the distribution in Eq. (2). The averaged locations of atoms of the bilayer system form a triangular lattice.

We consider the pairwise potential with nearest neighbor interaction. For the upper layer, the random lattice constant $h(\omega_{j}^{+},\omega_{j+1}^{+})$ due to the intra-layer interaction is denoted as $h^{+}_{j}$, and similarly $h(\omega_{j}^{-},\omega_{j+1}^{-})$ as $h^{-}_{j}$ in the lower layer. The distance between neighboring inter-layer atoms is $h(\omega_{j}^{+},\omega_{j}^{-})$ or $h(\omega_{j}^{+},\omega_{j-1}^{-})$. Following Sec. 2.1, the potential for the intra-layer atomic interaction is $V\left(h_{j}^{\pm}+u_{j+1}^{\pm}-u_{j}^{\pm},\omega_{j}^{\pm},\omega_{j+1}^{\pm}\right)$, and the inter-layer atomic interaction is denoted as $U\left(h(\omega_{j}^{+},\omega_{j}^{-}),\omega_{j}^{+},\omega_{j}^{-}\right)$ or $U\left(h(\omega_{j}^{+},\omega_{j-1}^{-}),\omega_{j}^{+},\omega_{j-1}^{-}\right)$. We assume that $\mathbf{E}(h^{+}_{j})=\mathbf{E}(h^{-}_{j})=\mathbf{E}(h(\omega_{j}^{+},\omega_{j}^{-}))=\mathbf{E}(h(\omega_{j}^{+},\omega_{j-1}^{-}))=\bar{h}$. Note that the inter-layer interaction potential and the average inter-atomic distance across the two layers may be different from those within each layer, and this does not lead to essential difference in the derivation of the continuum model.

We consider an edge dislocation with Burgers vector $\bm{b}=(\bar{h},0)$ in bilayer HEA as shown in Fig. 5(b). The displacement field at the lattice sites $\{a^{+}_{j},a^{-}_{j}\}_{j\in\mathbb{Z}}$ is $\{u^{+}_{j},u^{-}_{j}\}_{j\in\mathbb{Z}}$, which satisfies

$$\lim_{j\to-\infty}u^{+}_{j}-u^{-}_{j}=0,~{}\lim_{j\to+\infty}u^{+}_{j}-u^{-}_{j}=\bar{h}.$$

(46)

In this paper, we focus on the models for HEAs with short-range order. Similarly to the $\alpha$-mixing coefficients and Assumption 1 for a single layer HEA in Sec. 2.2, we first generalize the definition of $\alpha$-mixing coefficients to two dimensions based on the bilayer HEAs, and then assume a rapidly decaying property of it.