Decoupling atomic diffusion degeneracy in L1₂ intermetallics

Abstract

In this work, a degenerate diffusion state is identified for common L1₂ intermetallics in well-defined five-frequency model (FFM) and anti-site bridge model (ASB). The relative energy barrier $E_4/E_B^{\text{β}\to \alpha }$ ($E_4$ in FFM and $E_B^{\text{β}\to \alpha }$ in ASB) of the degenerate state serves as a decomposer which not only bifurcates the degenerate pathways, but also determines the most probable diffusion model. The diffusion coefficients obtained from the most probable diffusion model coincides the experimental value. We find the proposed decomposer complies with the Onsager phenomenological theory and atomistic correlation understanding, and is also supported by the relative work $W_{\mathrm{FFM}}/W_{\mathrm{ASB}}$ done by electronic wind force.

GRAPHICAL ABSTRACT

KEYWORDS:

• Diffusion model
• L1₂ intermetallics
• degenerate state
• decoupling rule

1. Introduction

The diffusion of atoms in gases or liquids is generally Brownian, as proposed by Einstein, which is determined by the root-mean-square of total displacements [1]. The diffusion pathways in solids, on the other hand, are considered finite and restricted by the crystallography, thermodynamics, kinetics, and statistics. The pursuit of possible pathways in solid-state diffusion usually takes years or decades [2–4]. Lacking in situ monitoring and a clear physical picture of the atomic movement in solids makes those proposed pathways ambiguous from one to another [5–7].

In past decades, the atomic behaviors (jump frequency, transition states, thermodynamic factors, etc.) associated with each pathway can be accurately determined by density function theory [8–10]. Many stochastic methods, such as kinetic Monte Carlo calculations [11–13], path probability methods [14,15], mean-field methods [16–18], Ritz variational methods [19–21], and Green function methods [22–24] have been developed to compute the diffusion coefficient from calculated atomic mechanisms, and used to validate the proposed diffusion pathways together with experimental data [25]. Analytically, correlation, drag ratio, escape frequency, etc. have been proposed to connect the possible pathways in different diffusion models. In this endeavor, the important role of some particular pathways in the resulting diffusion coefficient has been emphasized. In some cases, bifurcations caused by the same pathway in different diffusion models may result in distinct outcomes. For example, Trinkle et al. found the asymmetrical solute-vacancy diffusion in the first-nearest neighbor leading to quite different drag ratio in hexagonal close packed (HCP) Mg through Green function methods [23]. Most stochastic methods work well for solute diffusion in dilute solid solution phase as its movement is short ranged [23,26]. However, they strongly depended on the input transition state energies, and often be trapped to the pathway with the lowest energy barrier [27,28].

The atomic movement in ordered intermetallics, on the other hand, is not short ranged, since the direct exchange of atomic position of different sublattice will lead to the frustration of ordered structure. A common L1₂ A₃B intermetallics, which is widely existed in superalloys, bimetallic catalysts, high-entropy alloys, etc. [29–31], needs anti-site defects to fulfill the transportation of B atoms. Five-frequency model (FFM) and anti-site bridge model (ASB) are two well-documented diffusion models describing the diffusion pathways of anti-site B atoms in the L1₂ A₃B intermetallics (refer to supplementary material for further details) [32,33]. B atoms can achieve effective migration in these models without breaking the structural integrity. As depicted in Figure 1(a), these two models are not independent due to a shared diffusion state. The initial state of $w_4$ is identical to the initial state of $w_B^{\text{β}\to \alpha }$ marked by the red rectangle vacancy, leading to a blending of the two diffusion models [29]. Consequently, when the vacancy ‘disassociates’ with the anti-site B atom along the $w_3$ pathway to the initial state of $w_4$, it becomes uncertain whether the next jump will follow the FFM pathway ($w_4$) or the ASB pathway ($w_B^{\text{β}\to \alpha }$).

Figure 1. (a) Atomic illustration of FFM and ASB model for L1₂ intermetallics. (b) Calculated diffusion energy barriers $E$ of FFM and ASB in L1₂ Al₃Ti, Ni₃Al and Al₃Li at 0 K.

In this work, a simple yet ingenious physical picture is constructed by considering the same route in these two diffusion models as a degenerate diffusion state. The bifurcation of this degeneracy state thus lead to quite distinct phenomenological Onsager transport coefficients and atomistic correlation relationship [34]. In order to unravel the decomposer of this degenerate state, how it decouples this state and determines the final diffusion coefficient, 11 common L1₂ intermetallics are chosen and calculated using both FFM and ASB models through density function theory. Al₃Ti, Ni₃Al and Al₃Li are discussed in detail to show how electronic wind force untangles this degeneracy.

2. Methods

The factors affecting the selection of B atom diffusion model are studied using first principles calculations using Vienna ab initio simulation package (VASP) [35] with projector augmented wave (PAW) method [36] and the Perdew−Burke−Ernzerhof (PBE) exchange−correlation functional [37]. Monkhorst−Pack method [38] was used to sample the 7 × 7 × 7 k−point mesh, and 2 × 2 × 2 supercell was adopted in L1₂ structure which has sufficient precision compared with the larger 3 × 3 × 3 supercell shown in Figure S1 and Table S1. A cutoff energy for the plane−wave basis was set to 500 eV. Convergence criteria for electronic and ionic relaxations were 0.001 meV and 0.01 eV/Å, respectively, whereas that for static, self−consistent calculations was 0.001 meV. The energy barriers of the atom migration were calculated using climb−image nudged elastic band (CI−NEB) [39] calculations. Alloy Theoretic Automated Toolkit (ATAT) [40] package was used to obtain the effective frequency and quasi−harmonic thermodynamic properties through phonon calculation based on the direct force constant supercell approach [41]. The phonon dispersion results provide information about the atomic diffusion behaviors and the onset of the diffusion processes [42–45], which are available in the supplementary material. All parameters have been tested in our previous work [46].

3. Results and discussion

The diffusion energy barriers $E$ corresponding to the pathways in Figure 1(a) of the studied L1_2­ systems are listed in Table 1. Al₃Ti, Ni₃Al and Al₃Li, as typical examples, are illustrated in Figure 1(b). The calculated diffusion energy barrier data align well with other literature values (Table S1) [47,48]. It is observed that the diffusion energy barrier $E_2$ is the lowest among FFM pathways, especially lower than the degenerated $E_4$. In ASB, the degenerated $E_B^{\text{β}\to \alpha }$ is always larger than $E_B^{\alpha \to \text{β}}$, indicating that the first jump is harder than the second jump due to the introduction of two anti-site defects. Generally speaking, the relative energy barrier of $w_2$ and $w_B^{\text{β}\to \alpha }$ should be able to differentiate the atomic propensity toward to FFM or ASB. As shown in Figure 1(b), Al₃Ti and Al₃Li would choose FFM, while Ni₃Al adopt ASB. Even though there are many works using the energy barriers to select the preferred diffusion pathways [49,50], it is widely acknowledged that energy barrier itself cannot solely determine the diffusion behaviors [51].

Table 1. Calculated diffusion energy barriers $E$, works $W$, pre-factor $D_0$, activation energy $Q$, the values of $W_{\mathrm{FFM}}/W_{\mathrm{ASB}}$ and $E_4/E_B^{\text{β}\to \alpha }$ of all studied L1₂ intermetallics.

		FFM								ASB
		Barrier E (eV)								Barrier E (eV)
System	Preferred model (Larger diffusion coefficient $D$)	$E_0$	$E_1$	$E_2$	$E_3$	$E_4$	Work ($W_{\mathrm{FFM}}$)	$D_0$(cm^2/s)	$Q$(eV)	$E_B^{\text{β}\to \alpha }$	$E_B^{\alpha \to \text{β}}$	Work ($W_{\mathrm{ASB}}$)	$D_0$(cm^2/s)	$Q$(eV)	T range^a (K)	$\frac{W_{\mathrm{FFM}}}{W_{\mathrm{ASB}}}$	$\frac{E_4}{E_B^{\text{β}\to \alpha }}$
Al3Sc	FFM	0.85	0.99	0.13	0.82	0.81	0.06	0.45	4.76	1.39	0.35	0.10	6.66	6.07	300–700	0.59	0.58
Al3Ti	FFM	0.93	0.81	0.70	0.92	1.17	0.03	0.76	2.11	1.70	1.20	0.04	20.4	3.09	500–900	0.59	0.69
Al3Zr	FFM	0.87	1.04	0.52	0.89	0.98	0.02	0.37	4.07	1.90	0.67	0.03	2.86	5.34	500–900	0.58	0.51
Al3Hf	FFM	0.92	1.01	0.77	0.93	1.05	0.02	0.15	2.15	1.65	0.92	0.04	0.38	2.86	500–900	0.56	0.64
Al3Li	ASB	0.62	0.69	0.29	0.59	0.52	0.05	0.17	1.62	0.43	0.37	0.05	0.19	1.57	300–700	1.11	1.21
Al3Mg	ASB	0.46	0.59	0.37	0.46	0.49	0.30	0.04	1.24	0.35	0.38	0.18	0.11	1.17	300–700	1.61	1.40
Al3Zn	ASB	0.41	0.41	0.29	0.45	0.45	0.01	0.03	0.85	0.37	0.27	0.02	0.10	0.83	300–700	0.85	1.20
Al3Cu	ASB	0.35	0.17	0.51	0.36	0.41	0.21	1.39 × 10^−7	1.06	0.33	0.38	0.09	2.95	0.68	300–700	2.27	1.24
Fe3Al	ASB	1.07	1.21	0.64	1.08	0.99	0.75	0.04	3.48	0.80	0.32	0.50	0.16	3.39	800–1600	1.50	1.23
Co3Al	ASB/FFM^b	0.58	0.73	0.23	0.61	0.63	0.65	0.01	1.96	0.64	0.17	0.39	0.07	2.16	800–1600	1.65	0.98
Ni3Al	ASB	1.00	0.96	0.72	1.05	1.04	0.74	0.02	3.00	0.67	0.32	0.54	0.05	2.61	800–1600	1.36	1.56

^a It should be noted that the values of $D_0$ and $Q$ are obtained by fitting $D$over $T$.

^b The calculated $D_{\mathrm{FFM}}$ and $D_{\mathrm{ASB}}$ in Co₃Al are almost the same, so both the two models can be chosen as the preferred model.

A more accurate description of atomic mechanisms on the diffusion coefficient is proposed by Bakker [52], which is determined by two parts: the jump frequency $w$ of the target diffusion atom exchanging with its nearest neighbor (NN) vacancy (effective jump frequencies $w_2$ in FFM and the ${\tilde{w}}_B^{\alpha \to \alpha }$ in ASB, see supplementary material for the definition), and the probability $P$ that the vacancy is at the NN sites of the target diffusion atom. The diffusion coefficient is then obtained by: (1) $D_B^{A_{3}B}=\mathrm{KPw}$(1) where $K$ is a constant for the same L1₂ structure in FFM and ASB, determined by lattice constant and point defect concentrations.

The calculated results of $w_2$ in FFM and ${\tilde{w}}_B^{\alpha \to \alpha }$ in ASB are shown in Figure 2(a–c) and Figure S3. It can be seen that the relative values of $w_2$ and ${\tilde{w}}_B^{\alpha \to \alpha }$ are consistent with the relative value of $E_2$ and $E_B^{\text{β}\to \alpha }$. That is the lower the energy barrier, the higher the internal jump frequency. In this way, the preference of FFM or ASB is the same as that indicated by the energy barrier. As the stochastic methods use energy barrier or jump frequency to compute the diffusion coefficient, they may overweight the contribution of energy barrier and prejudice the lower one against the higher one [27,28]. For example, the ASB pathway proposed by Belova and Murch is first concepted by the direct movement of anti-site B atom in $\alpha$-sublattice (major A sublattice) through a transition state of bridged two anti-site B states. The diffusion coefficient of Ni₃X (X = Al, Ga, Ge) obtained from kinetic Monte Carlo calculations is closer to experimental values than those from the previous FFM mechanism [33]. With almost a decade development, the ASB mechanism is constructed analytically like FFM. However, still many works adopted or found FFM over ASB [32,48,50]. This may originate from the factor that there is a degenerate state between these two mechanisms.

Figure 2. Calculated results of $P$, $w$ and $D$ in Equation (1) of L1₂ Al₃Ti, Ni₃Al and Al₃Li, with available data in the literature [29,53–59].

Figure 2(d–f) and Figure S8 show the calculated probabilities $P_{\mathrm{FFM}}$ and $P_{\mathrm{ASB}}$. The $P$ value of the two models differ greatly in different diffusion systems, and do not have apparent relationship with energy barriers or jump frequency of effective jump for FFM or ASB. For instance, $P_{\mathrm{FFM}}$ is smaller than $P_{\mathrm{ASB}}$ for Al₃Li, while the respective $w_2$ and ${\tilde{w}}_B^{\alpha \to \alpha }$ show reversed trend. Similar phenomenon is also observed in other L1_2­ intermetallic compounds like Al₃Sc, Al₃Mg and Co₃Al as shown in Figure S8. This reversed relationship makes the diffusion mode deduced solely from energy barriers or jump frequency ambiguous. In Al₃Ti and Ni₃Al, the relative value of $P$ follows the same rule with energy barriers or jump frequency, that $P_{\mathrm{FFM}}$ is larger than $P_{\mathrm{ASB}}$. However, the slope trend does not resemble with each other, i.e. the slope is almost the same for $P$ in Ni₃Al, while that differs greatly for $w$.

Based on the $P$ and $w$ values, the resulting diffusion coefficients are calculated analytically and shown in Figure 2(g–i) and Figure S8 together with available experimental values [29,53–59]. The diffusion coefficients of majority A in L1₂ A₃B are also plotted for comparison. Compared with experimental values, $D_B^{A_{3}B}$ obtained from different models show quite distinct behaviors. For example, $D_B^{A_{3}B}$ of Al₃Ti from FFM agree well with experimental ones, while those of Ni₃Al and Al₃Li from ASB are consistent with experiments. This behavior indicates the transportation of B atoms may prefer one model over another, resembling the rate-determined reaction in chemical science. Comparing the results of $w$, $P$ and $D$, we find the relative values and variation trends of $P$ and $D$ are almost the same. This finding indicates that the probability of the vacancy remaining at the NN site of the target diffusion atom could be regarded the manipulator of the ‘rate-determined pathway’.

We now look into the internal connection between $P$ and the degenerate diffusion state. The meaning of $P$ is how likely the vacancy provides new diffusion sites for B atoms to make the intended move ($w_2$ or ${\tilde{w}}_B^{\alpha \to \alpha }$) for FFM or ASB pathways. According to Einstein’s atomic mechanism [1], the probability $P$ could represent the correlation factor as it determines the effective jump in the diffusion process. The degenerate state $w_4$ and $w_B^{\text{β}\to \alpha }$ is the jump of vacancy to the NN of anti-site B. Thus, how many vacancies available for the effective jump of FFM or ASB is associated with how many jumps from degenerate $w_4$ or $w_B^{\text{β}\to \alpha }$, respectively. As a result, $P_{\mathrm{FFM}}$ and $P_{\mathrm{ASB}}$ can be interpreted as the probability amplitude of degenerate diffusion state, while $w_4$ and $w_B^{\text{β}\to \alpha }$ can be used to determine the preferred diffusion pathway. As shown in Table 1, the diffusion energy barriers $E_4$ and $E_B^{\text{β}\to \alpha }$ (as they does not change with temperature like $w_4$ and $w_B^{\text{β}\to \alpha }$) are compared with each other and the preferred diffusion model. When the B atom prefers FFM, the value of $E_4/E_B^{\text{β}\to \alpha }<1$, and when B atom prefers ASB, $E_4/E_B^{\text{β}\to \alpha }>1$.

As the atomic mechanism suggests the deciding role of degenerate $w_4$ or $w_B^{\text{β}\to \alpha }$, its phenomenological understanding deserves further attentions. According to Onsager relationship, the movement direction of the vacancy towards the diffusion atom can be predicted by the drag ratio $L_{\mathrm{BV}}/L_{\mathrm{BB}}$. The definition of $P$ implies that the drag ratio complies with $P$, that how many vacancies are brought to the NN of anti-site B. Accordingly, the degenerate $w_4$ or $w_B^{\text{β}\to \alpha }$ also reflect the effect of drag ratio. Indeed, Trinkle et al. pointed out that during the jump of $w_2$ in FFM, the barycenter of the solute-vacancy pair remains unchanged, so that $w_2$ has less impact on the drag radio than $w_4$ [22].

The intrinsic mechanism of the degenerate pathway and its role in the final (observable) diffusion coefficient is analyzed by the electronic density difference. Figure 3 shows charge density difference contour plots on (010) plane intersecting the vacancy in $\alpha$ sublattice of the initial and final state of $w_2$ in FFM and the first jump $w_B^{\text{β}\to \alpha }$ of ASB. Charge densities in column 1 and 3 represent the end of untanglement and those in column 2 and 4 are the end of effective movement of B atom. It can be seen that the introduction of vacancy near the anti-site B induces charge redistribution between the vacancy and anti-site B atom [60,61]. The partial covalent bonding feature of L1₂ intermetallics, as reflected by the red regions in Figure 3, makes the diffusion of anti-site B atoms directional. The eletrons are accumulated around the anti-site B, while electrons are depleted on the vacancy. However, the amount of electrons accumulated or depleted is different for FFM and ASB models. For example, in Al₃Ti the vacancy has negative value of charge density for FFM model, while it is positive for ASB model. As the Ti atoms are surrounded by positive charge density, it will push Ti into vacancy easily in FFM model, while hard for ASB. On the contrary, in Al₃Li the vacancy has positive charge density in the case of FFM, while slight negative charge density for ASB, leading to the movement of B atoms toward ASB. In Ni₃Al, the charge density distribution, on the other hand, shows similar feature between FFM and ASB, but the distance from the vacancy to the anti-site B is different. Comparing the charge density distribution with $P$ value, it could be deduced that they are in line with each other in decoupling the degenerate pathways.

Figure 3. The charge density difference contour plots of L1₂ Al₃Ti, Ni₃Al and Al₃Li on the $\left(010\right)$ plane.

As mentioned before, the $P$ value is closely related to correlation in FFM and ASB. The electron density is also associated with correlation through the electron wind force [62] proposed by Huntington (details are presented in the supplementary material). As shown in Figure 3, the accumulation and depletion of charge density build a naturally electrostatic potential $\varphi$. The atoms carry electrons will move in that potential. Based on the Huntington and Onsager theory, the correlation factor $f$ can be represented by (2) $f=\frac{D^{\ast }}{D}=\frac{n_B{z}_B|e|\mathrm{\nabla \varphi }}{\mathrm{kT}\mathrm{\nabla }{n}_B}$(2) where $D^{\ast }$ is the diffusion coefficient considering the correlation effect, $D$ is the random diffusion coefficient, $n_B$ is the concentration of diffusion atom (the anti-site B atom in this work), $z_B$ is the effective charge number of B atom, $|e|$ is the absolute value of the electronic charge and $k$ is the Boltzmann’s constant (The derivation of this equation is put in the supplementary material.). From above equation, the key factor in correlation is $\mathrm{\nabla \varphi }$, the potential gradient induced by charge density redistribution. Thus, the steeper the gradient, the larger the correlation factor. As discussed before, the larger charge density difference between the anti-site B and vacancy facilitates the atomic movement and decouples the degenerate pathways. Note that the energy barrier cannot be simply related to the charge density difference, as it is also influenced by volume, long-range interactions, lattice distortion, and so on.

Similar to the $E_4$/$E_B^{\text{β}\to \alpha }$ value (Figure 4(a)), we construct a quantitative description of electron wind force. Thus, the electron wind force induced by electrostatic potential $\varphi$ for the given atom is defined as: (3) $F=|e|z_B\varphi$(3) where $|e|$ is the absolute value of the electronic charge, $z_B$ is the effective charge number of B atom, and $\varphi$ is the electrostatic potential. On the basis of symmetry, we only consider the electric field generated by the NN atoms of the anti-site diffusion atom along the direction of the diffusion pathway (A atom shown in Figure 4(b)). The $|e|$ is estimated through Bader charge analysis [63]. Unfortunately, direct comparison of diffusion driving force may not be adequate since the diffusion is also related to the distance of atom movement. The diffusion distance, as noticed by Janotti et al., causes ‘Larger Atoms Can Move Faster’ in metals [49].

Figure 4. Schematic illustration of (a) the decoupling direction and (b) the electronic work during the diffusion process. (c) Guiding diagram for the diffusion model selection.

Accordingly, we use the work done in the process of atomic diffusion to quantitatively assess the impacts of the charge density differences and displacements in Figure 3, and to differentiate the role of electron wind force in untangling of degenerate pathways. The work is defined as: (4) $W={\int }_{r_i}^{r_f}\mathrm{Fdr}={\int }_{r_i}^{r_f}{k\frac{Q_A}{r^2}|e|z_B\mathrm{dr}=\frac{Q_A|e|z_B}{r}|}_{r_i}^{r_f},$(4) where $r$ is the distance from the A atom, $k$ is the Coulomb constant, $Q_A$ is the charge of the NN atom. It should be noted that the displacement (the white arrow in Figure 3) during diffusion process is equal to $r_f-r_i$. See Table 1 and Table S4 for the value of each parameter in Equation (4). In practice, the more work the diffusion atoms do during diffusion, the harder the diffusion atom to get rid of the surrounding A atoms to complete diffusion. With the work for diffusion, we can quantitatively decouple the degeneracy in diffusion pathways. Comparing $W$ in FFM ($W_{\mathrm{FFM}}$) with $W$ in ASB ($W_{\mathrm{ASB}}$), when $W_{\mathrm{FFM}}/W_{\mathrm{ASB}}>1$, it indicates that more energy is required for diffusion through FFM. As shown in Figure 4(c), the relative value of $E_4/E_B^{\text{β}\to \alpha }$ and $W_{\mathrm{FFM}}/W_{\mathrm{ASB}}$ are consistent with each other in decoupling the degenerate pathway into the most probable diffusion model, which reflect the correlation between the probability $P$ and the charge density differences quantitatively. Among the L1₂ structure intermetallics we studied, Al₃Ti, Al₃Zr, Al₃Hf and Al₃Sc in the red region ($W_{\mathrm{FFM}}/W_{\mathrm{ASB}}<1$and $E_4/E_B^{\text{β}\to \alpha }<1$) choose FFM model for diffusion, while Ni₃Al, Fe₃Al, Al₃Mg and others in the blue region ($W_{\mathrm{FFM}}/W_{\mathrm{ASB}}>1$ and $E_4/E_B^{\text{β}\to \alpha }>1$) will choose ASB for diffusion.

4. Conclusion

In summary, the decisive role of the degenerate $w_4$ and $w_B^{\text{β}\to \alpha }$ on the most probable diffusion model has been uncovered from the perspective of the probability $P$ correlated to the Onsager relationship and the electronic work $W$ done by the electron wind force. This decoupling rule establishes a framework for choosing the exact analytical model of diffusion coefficient in L1₂ intermetallics. Our study is pivotal for unraveling the intricacies of atomic diffusion in solids and addressed the limitations of current diffusion research methodologies. The physical picture provided in this work not only sheds light on the atomistic and phenomenological theory of current state-of-art diffusion models, but also would facilitate the development of long-ranged diffusion model and reliable stochastic methods.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (grant numbers 51931012, 51801236). We are grateful for technical support from the High Performance Computing Center of Central South University.