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Fundamental mechanism of BCC-FCC phase transition from a constructed PdCu potential through molecular dynamics simulation
Computational Materials Science 159 (2019) 440–447
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Fundamental mechanism of BCC-FCC phase transition from a constructed PdCu potential through molecular dynamics simulation ⁎
⁎
W. Weia, L.C. Liua, H.R. Gonga, , M. Songa, , M.L. Changb, D.C. Chenb, a b
T
⁎
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, China School of Materials Science and Energy Engineering, Foshan University, Foshan 528000, Guangdong, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Pd-Cu alloy Phase transformation First principles calculation Molecular dynamic simulation
An n-body PdCu potential is constructed under the framework of the embedded-atom method and is realistic to reproduce phase stability of PdCu phases. Based on this PdCu potential, two interface models with the Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) relationships are established to reveal the kinetics and thermodynamics of the BCC-FCC phase transition through molecular dynamics simulation. It is found that the BCC → FCC phase transition should include the stages of nucleation, growth, and adjustment, and that the slip of edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes fundamentally brings about the nucleation of the FCC phase. Simulations also reveal that the lower energy difference between interface energy and surface energy could serve as the driving force of the nucleation of the FCC phase and intrinsically bring about the BCC → FCC phase transition in the interface with the NW orientation. The different behaviors of the BCC-FCC interface models with the NW and KS relationships are discussed in terms of kinetics and thermodynamics, which could provide a deep understanding of the BCC-FCC phase transition.
1. Introduction The BCC-FCC phase transition has been recognized as the most commonly encountered type of solid-state transformation in pure metals and alloys [1]. Fundamentally, there are already four principal mechanisms in the literature to understand this kind of phase transition [2–5]. In 1924, Bain proposed that the BCC-FCC transformation would be realized through a simple tetragonal deformation (i.e., the change of c/a) and the two phases should follow the orientation relationship of (0 0 1)FCC||(0 0 1)BCC and [1 0 0]FCC||[1 1 0]BCC [2]. Afterwards, another transition path by Zener suggested that the BCC phase would transform to the FCC phase through the shear of (1 1 0)[1¯ 1 0] with the softening of shear mode [3]. In addition, two other mechanisms of the FCC-BCC phase transition were proposed and named as NishiyamaWassermann (NW) and Kurdjumov-Sachs (KS) relationships [4,5], i.e., (1 1 1)FCC||(1 1 0)BCC and [1¯ 1¯ 2]FCC||[1¯ 1 0]BCC for the NW orientation, while (1 1 1)FCC||(1 1 0)BCC and [1 1¯ 0]FCC||[1 1¯ 1]BCC for the KS relationship. The binary Pd-Cu system is well regarded as one of the most promising membranes for hydrogen separation and purification due to its excellent hydrogen selectivity and permeability, superior resistance against poisoning, high thermal stability, nice mechanical properties, and relatively low price, etc. [6–10]. According to the binary phase
⁎
diagram of Pd-Cu [6,9], the face-centered-cubic (FCC) phase could be produced within the entire composition range, whereas the body-centered-cubic (BCC) ordered phase can be only fabricated when the Pd composition is located at the range of 30–50 at.% [8,9,11,12]. Surprisingly, several studies in the literature reveal that the BCC PdCu phase has much higher hydrogen permeability and would be more preferred than its FCC counterpart [12–15]. The BCC-FCC phase transition of PdCu has been extensively studied in the literature [6–9,12,14,16–18]. For instance, Ellis and Mohanty experimentally observed the strain-induced BCC-FCC phase transition of the Pd0.4Cu0.6 phase [17]. Yuan et al. found that the BCC → FCC transition of Pd48Cu52 happened in the temperature range of 723–873 K [16]. Moreover, theoretical calculations were used to find out the effects of Nb and Cr addition on the Bain path of the BCC-FCC phase transition of Pd50Cu50 [19]. Regarding the detailed mechanism of the BCC-FCC phase transition of PdCu, however, there is not any report so far in the literature. It is well known that molecular dynamics (MD) simulation based on an n-body potential has been universally used to predict various phase transitions of metal alloys [20–22]. The present study is therefore aimed to find out the intrinsic mechanism of the BCC-FCC phase transition of PdCu by means of MD simulation. First of all, a realistic n-body potential of PdCu is constructed in the present study, as the three PdCu
Corresponding authors. E-mail addresses: [email protected] (H.R. Gong), [email protected] (M. Song), [email protected] (D.C. Chen).
https://doi.org/10.1016/j.commatsci.2018.12.037 Received 21 November 2018; Received in revised form 17 December 2018; Accepted 17 December 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
Computational Materials Science 159 (2019) 440–447
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potentials in the literature cannot be used to reveal the BCC-FCC phase transition. Two BCC-FCC interface models are then established to reveal the kinetics and thermodynamics of the BCC-FCC phase transition of PdCu through MD simulation with the newly-constructed PdCu potential. It will be shown for the first time that the BCC → FCC phase transition of PdCu should include the stages of nucleation, growth, and adjustment, and that the FCC phase nucleates at the interface area by means of the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. In addition, the lower energy difference between interface energy and surface energy intrinsically serve as the driving force of the nucleation of the FCC phase and therefore bring about the BCC → FCC phase transition.
Table 1 Fitted parameters of Pd-Pd and Cu-Cu potentials. The cutoff distances, rs and rc, are also listed.
The embedded atom method (EAM) has been well regarded as one of the most realistic n-body potentials, and has been widely used to reproduce physical properties of transition metals and alloys [23–25]. For the binary Pd-Cu system, there are already three constructed EAM potentials in the literature [23,24,26]. Unfortunately, it will be shown in detail later that these EAM potentials have all failed to reflect the relative stability of BCC and FCC Pd50Cu50 phases at 0 K and are therefore unable to describe the BCC-FCC phase transition of the PdCu system. In the present study, a new EAM potential is thus constructed, in order to reveal the intrinsic features of the BCC-FCC phase transition of PdCu. After a series of testing, the EAM forms by Cai et al. [24] are selected for the Pd-Pd and Cu-Cu potentials, while a simplified mathematical term proposed by Gong et al. [27] is chosen for the Pd-Cu cross potential. According to the EAM approach, the total energy of N atoms in the system is depicted as:
Etot
∑
φij (rij ),
n
ρi =
c d e h
⎜
⎟
Calculated
Experiment
Calculated
−3.91a 3.89e 1.95c 2.341c 1.761c 0.712c 1.4 h
−3.91 3.89 1.9646 2.3308 1.7814 0.7182 1.4000
−3.54a 3.615b 1.4203c 1.7c 1.225c 0.758c 1.3d
−3.54 3.615 1.3835 1.7001 1.2252 0.7581 1.2999
Ref. [29]. Ref. [30]. Ref. [31]. Ref. [32]. Ref. [33]. Ref. [34].
φPdCu (r ) =
(4)
⎜
Experiment
φPdCu (r ) = [ZPd (r ) + ZCu (r )]/ r ,
⎟
r r φ (r ) = −α ⎡1 + β ⎛ − 1⎞ ⎤ exp ⎡−β ⎛ − 1⎞ ⎤, ⎢ ⎥ ⎢ r r ⎝ a ⎠⎦ ⎠⎥ ⎣ ⎣ ⎝ a ⎦
Cu
intrinsic properties of Pd and Cu [29–34]. The Pd-Cu cross potential is then constructed by means of the derived Pd-Pd and Cu-Cu potentials. As related before, three EAM potentials of Pd-Cu have already been derived in the literature, and the mathematical forms of the Pd-Cu cross potentials are as follows [23,24,26]:
(3)
⎜
11.13548 0.70277 3.52097 1.65393 0.67199 3.69625 4.10247
b
∑ f (rij),
r f (r ) = fe exp ⎡−χ ⎛ − 1⎞ ⎤, ⎢ ⎝ re ⎠⎥ ⎣ ⎦
17.74997 1.42485 0.43287 0.32751 0.28064 3.97743 4.41455
a
(2)
j≠i
Χ α (eV) Β ra (Å) F1 (eV) rs (Å) rc (Å)
Ec (eV/atom) a (Å) B (Mbar) C11 (Mbar) C12 (Mbar) C44 (Mbar) Evf (eV)
n
ρ ρ ⎤ ρ ⎡ F (ρ) = −F0 ⎢1 − ln ⎛⎜ ⎞⎟ ⎥ ⎜⎛ ⎟⎞ + F1 ⎜⎛ ⎟⎞, ρ ρ ρ ⎝ e ⎠ ⎦⎝ e ⎠ ⎝ e⎠ ⎣
Cu-Cu
Pd
(1)
i, j (i ≠ j )
Pd-Pd
Table 2 Comparison between calculated and experimental results of cohesive energy Ec (eV/atom), lattice constant a (Å), elastic constants (Mbar), and vacancy formation energy Evf (eV) for Pd and Cu.
2. Construction of an n-body potential
1 = ∑ Fi (ρi ) + 2 i
Parameter
(6)
f Pd (r ) 1 ⎡ f Cu (r ) Pd φ (r ) + Cu φCu (r ) ⎤, Pd ⎥ ⎢ 2 ⎣ f (r ) f (r ) ⎦
(7)
where ZPd (r ) and ZCu (r ) are effective charges of Pd and Cu, respectively. Accordingly, Table 3 shows the calculated lattice constants and cohesive energies of BCC (B2) and FCC (L10) Pd50Cu50 phases from the above three EAM potentials of PdCu in the literature [23,24,26]. One could observe from Table 3 that the cohesive energy of the FCC
⎟
(5)
where Etot is the total energy, F(ρ) is the embedding energy, ρi is the total electron density at atom i due to all other atoms, φ(rij) is the pair potential between atoms numbered i and j, and f(rij) is the electron density for atom i contributed by atom j. In Eq. (2), F0, ρe, and n are three constants. F0 = Ec − Evf, in which Ec and Evf are cohesive energy and vacancy formation energy, respectively. The parameter ρe represents the host electron density at an equilibrium state. F1 is an adjustable parameter. In Eq. (4), re is an equilibrium first-neighbor distance, and fe is a scaling factor determined by the relationship of fe = Ec/Ω, where Ω is the atomic volume. The cutoff function is employed from the polynomial derived by Guellil et al. [28], and the cutoff distances are set to be between the second- and third-neighbor distances. It should be noted that there are five adjustable parameters (χ, α, β, ra, F1) to be fitted for both Pd-Pd and Cu-Cu potentials. Consequently, Table 1 lists the derived five parameters and the chosen cutoff distances of Pd-Pd and Cu-Cu potentials. In addition, Table 2 summarizes the calculated and experimental physical properties of pure Pd and Cu. It can be seen clearly that the constructed Pd-Pd and Cu-Cu potentials are relevant to reproduce some
Table 3 Comparison of lattice constants (a) and cohesive energies (Ec) of Pd50Cu50 phases with the BCC and FCC structures from the present PdCu potential as well as the other three EAM potentials in the literature [23,24,26]. Also listed are the fitting parameters of the present PdCu potential, i.e., A, m (Å), and n (Å). References
a (Å) BCC
The present potential 2.9185 Exp. [39] 2.958 Ab initio [35,36] EAM potential [26] 3.011 EAM potential [24] 2.988 EAM potential [23] 2.958 Fitting parameters A 3.7907 m (Å)
441
Ec (eV/atom) FCC
BCC
FCC
3.76 3.749
−3.911
−3.869
−3.891 3.777 −3.711 3.764 −4.093 3.704 −4.183 0.6761 n (Å) 1.7424
−3.819 −3.734 −4.17 −4.285
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Pd50Cu50 phase from the above potentials [23,24,26] is lower than its corresponding BCC structure, which should be contrary to the experimentally observed B2 structure (rather than FCC structure) at low temperature as well as the energy sequence of BCC (B2) and FCC (L10) Pd50Cu50 phases from first principles calculations [35,36]. Such a disagreement implies that the above three EAM potentials of PdCu in the literature [23,24,26] should be irrelevant to correctly reproduce the relative stability of BCC and FCC Pd50Cu50 phases, and would be then unable to reflect the BCC-FCC phase transition in the binary PdCu system. In the present study, another mathematical form is therefore selected for the Pd-Cu cross potential by means of a linear combination of the Pd-Pd and Cu-Cu potentials [27]:
φPdCu (r ) = A [φCu (r + m) + φPd (r + n)],
Fig. 1. Constructed BCC-FCC interface models with Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) orientations. For the NW orientation, the xaxis is [0 0 1]BCC and [1 1¯ 0]FCC, the y-axisis[1 1 0]BCC and[1 1 1]FCC, and the z-axis is [1¯ 1 0]BCC and [1¯ 1¯ 2]FCC. For the KS relationship, the x-axis is [1 1¯ 1]BCC and [1 1¯ 0]FCC, the y-axisis[1 1 0]BCC and[1 1 1]FCC, and the zaxis is [1¯ 1 2]BCC and [1¯ 1¯ 2]FCC. The shadow plane refers to the FCC/BCC interface, and τ denotes the shear direction.
(8)
where A, m, and n are three parameters to be fitted. It should be noted that this formula is a modified form of the original cross potential proposed by Chen et al. [37] and Heino [38]. After the fitting process, Table 3 lists the derived potential parameters of the Pd-Cu cross potential as well as the reproduced cohesive energies and lattice constants of BCC (B2) and FCC (L10) Pd50Cu50 phases. It can be seen that the calculated lattice constants of BCC and FCC Pd50Cu50 phases are 2.919 and 3.76 Å, respectively, which match well with the corresponding values of 2.958 and 3.749 Å from experiments [39] and first principles calculations [19,35,36,39], respectively. Moreover, the cohesive energies of BCC and FCC Pd50Cu50 phases from the present Pd-Cu potential are −3.911 and −3.869 eV/atom, respectively, which are in good agreement with the corresponding values of −3.891 and −3.819 eV/atom from first principles calculations [36]. In other words, the BCC structure of Pd50Cu50 is energetically more favorable with lower cohesive energy than its FCC counterpart, which is in excellent with the corresponding observations from experiments and first principles calculations [19,35,36,39]. These nice agreements suggest that the newly constructed Pd-Cu potential should be realistic to reproduce the phase stability of BCC and FCC Pd50Cu50 phases, which is quite different from the performance of the three EAM potentials of Pd-Cu in the literature as related before [23,24,26]. It will be shown in the following simulations that this derived Pd-Cu potential is also relevant to reveal the intrinsic features of the BCC-FCC phase transition of PdCu.
atom layers in the x (y, z) direction. The x, y, and z axes of the interface [1 1¯ 1]BCC||[1 1¯ 0]FCC, with the KS orientation are [1 1 0]BCC||[1 1 1]FCC, and [1¯ 1 2]BCC||[1¯ 1¯ 2]FCC, respectively. It should be pointed out that the lattice constant (3.76 Å) of the substrate (FCC Pd50Cu50) derived from the present EAM potential shown in Table 3 is used in the two interface models, and there are 123,200 and 202,888 atoms for the two BCC-FCC interfaces with the NW and KS relationships, respectively. Accordingly, Fig. 1 shows the constructed BCC-FCC interface models with the NW and KS orientations. In the above interface settings, the interface mismatch between BCC and FCC planes are minimized to be 0.588% and 0.385% for the interfaces with the NW and KS orientations, respectively, and such a small mismatch should have a negligible effect on simulation results. The constructed BCC-FCC interface models with the NW and KS orientations are first relaxed in the canonical ensemble (NVT) at 10 K for 20 ps and then in the isothermal-isobaric (NPT) ensemble at 10 K for 20 ps. After the relaxation, each interface is then under a shear loading process in the isothermal-isobaric (NPT) ensemble for 350 ps with a strain rate of γ̇ = 1 × 109/s, and the shear direction τ is shown clearly in Fig. 1. It should be pointed out that after a series of tests of deformation directions, only the shear direction (τ) displayed in Fig. 1 could bring about the FCC-BCC phase transition of PdCu. For the loading, a uniform stretch is considered and a time step of 0.001 ps is selected. A Gaussian thermostat is employed for the temperature control. All simulation models are performed by three dimensional periodic boundary conditions. The atomic positions of the interface after the simulation are visualized by means of the OVITO software [46], and the crystal structure is identified by the method of polyhedral template matching (PTM) [47].
3. Simulation methods The present MD simulation is performed by means of Large-scale Atomic/Molecular Massively Parallel Simulator package (LAMMPS) [40] with the newly constructed EAM potential of PdCu. Two interface bilayer models between B2 (BCC) and L10 (FCC) Pd50Cu50 phases are intentionally established with the well-known Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) relationships. Specifically, the NW orientation of the interface is described by (1 1 1)FCC||(1 1 0)BCC and [1¯ 1¯ 2]FCC||[1¯ 1 0]BCC, and the KS relationship is expressed by (1 1 1)FCC||(1 1 0)BCC and [1 1¯ 0]FCC||[1 1¯ 1]BCC. It should be noted that the PdCu phase with equal atomic compositions of Pd and Cu is purposely chosen in the present study due to its good hydrogen permeability [8,15,41], and that the interface orientation of (1 1 1)FCC||(1 1 0)BCC with the NW and KS relationships is usually observed in other systems [4,5,42–45]. To form the BCC-FCC interface with the NW relationship, a BCC Pd50Cu50 phase with 80 (40, 40) atom layers in the x (y, z) direction is put at the top of an FCC Pd50Cu50 phase with 44 (45, 36) atom layers in the x (y, z) direction, and the y direction is perpendicular to the interface. For the NW orientation, the x-axis is [0 0 1]BCC and [1 1¯ 0]FCC, the y-axisis[1 1 0]BCC and[1 1 1]FCC, and the z-axis is [1¯ 1 0]BCC and [1¯ 1¯ 2]FCC. Similarly, the BCC-FCC interface of Pd50Cu50 with the KS orientation has the BCC phase with 42 (46, 36) atom layers in the x(y, z) direction, which are added at the top of the FCC phase with 40 (45, 56)
4. Results and discussion 4.1. Kinetics of BCC-FCC phase transition As related before, the BCC-FCC phase transition of PdCu has been observed both experimentally and theoretically, while the mechanism of such a transition has not been reported so far in the literature [6–9,12,14,16–18]. The present MD simulation is therefore used for the first time to reveal the detailed kinetics of the BCC → FCC phase transition of PdCu with the NW orientation, and this phase transition should include the stages of nucleation, growth, and adjustment. It is found that the FCC phase first nucleates at the interface area by means of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes due to lattice mismatch, then grows through the slip of the newly-formed screw 442
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Fig. 2. Snapshots of the BCC-FCC interface model with the NW relationship after the shear at (a) 0 ps, (b) 240 ps, and (c) 280 ps. In the simulation, the blue, green and red atoms represent the BCC, FCC and HCP atoms, respectively. (d) is the structural change of the BCC lattice during the shear process as a function of shear (ε). ε = tan θ, in which θ is the shear angle. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes, and is finally adjusted to a standard FCC lattice by volume expansion. On the contrary, the BCC-FCC phase transition could not be realized in the interface with the KS relationship. First of all, the snapshots of the BCC-FCC interface model with the NW relationship are detected by the method of polyhedral template matching (PTM) [47], and are shown as typical examples in Fig. 2a, b, and c during the process of shear. In addition, Fig. 2d displays the structural change of the BCC lattice during the shear process as a function of shear (ε). It can be seen clearly that in the BCC-FCC interface model with the NW relationship, the BCC lattice has changed gradually to the FCC lattice with the increase of the shear angle, and that the FCC lattice could keep its crystal structure. For the BCC-FCC interface model with the KS relationship, however, no phase transition could be observed, as the crystal structures of both BCC and FCC lattices do not change during the whole shear process (figures not shown). To find out the detailed micro-structural change during the phase transition, Fig. 3 shows the snapshots of the (1 0 0)BCC and (4 4 1¯)FCC planes in the BCC-FCC interface with the NW relationship before and after shear at 120 ps. It can be discerned obviously from Fig. 3a that in the interface area before shear, ten layers of (0 1 1¯)BCC planes exactly match eleven layers of (1¯ 1 1¯)FCC planes, which would therefore bring about the appearance of an edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. After shear at 120 ps, one could see from Fig. 3b that these dislocations have moved into the BCC lattice through the slip process and some atoms in the original BCC lattice have transformed to the FCC phase. The above points suggest that the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes should induce the BCC → FCC phase transition. It should be noted that various shear directions have been tried and only the shear direction shown in Fig. 1 can bring about the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on
(0 1 1)BCC planes and the subsequent BCC-FCC phase transition. Topological models of the BCC-FCC interface are then used for the above discovered orientations of the interface, in order to clearly reveal the process of the nucleation and growth of the formed FCC phase in the interface area [48,49], as the snapshots from MD simulation are unable to do so due to the vibration of the atoms. Consequently, Fig. 4 displays the topological models of the BCC-FCC interface with the NW relationship before and after the slip of the edge dislocation of 1/ 6aBCC[0 1¯ 1] on (0 1 1)BCC planes. As shown in Fig. 4a, an edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes has been formed in the interface area before shear due to the lattice mismatch. After the first slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes in Fig. 4b, the slipped (0 1 1)BCC plane left to the slip plane could be regarded as the first layer (i.e., nucleation) of the newly-formed FCC phase, and a screw dislocation between the newly-formed FCC phase and the original BCC phase will be generated by the movement of the edge dislocation. That is to say, the Burgers vector of the edge dislocation is parallel to the glide direction, and the slip of the edge dislocation induces the nucleation of the FCC phase. To further express the formation of the other layers of the FCC phase, Fig. 5a displays the topological models of the BCC part of Fig. 4b after the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes, while Fig. 5b, c, and d show subsequently the first, second, and third slips of the newly-formed screw dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes between the BCC and newly-generated FCC closepacked planes, respectively. It can be seen clearly that one FCC layer could be formed after each slip of the screw dislocation and the stacking of -A-B-A-B- in the (0 1 1) planes of the BCC structure would finally transform to the stacking of -A-B-C-A-B-C- in the (1 1 1) planes of the FCC structure. Such a process could be thus regarded as the growth of the FCC phase by means of the slip of the screw dislocation of 1/ 443
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Fig. 3. Snapshots of the (1 0 0)BCC or (4 4 1¯)FCC plane in the BCC-FCC interface with the NW relationship (a) before shear and (b) after shear at 120 ps. The blue and green symbols represent the BCC and FCC phases, respectively. The red symbols refer to the undiscerned BCC or FCC atoms due to interface stress. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6aBCC[0 1¯ 1] on (0 1 1)BCC planes. We now turn to investigate the adjustment of the newly-transformed FCC phase. As displayed in Fig. 6a, the transformed FCC phase of PdCu just after shear has the original lattice spacing of the BCC (0 1 1) plane, i.e., abcc (2.919 Å) and √2abcc (4.127 Å) with an interlayer distance of √2/2abcc (2.064 Å). As shown in Fig. 6b, however, the standard FCC phase of PdCu should have the lattice spacing of the FCC (1 1¯ 1) plane, i.e., √2/2afcc (2.659 Å) and √6/2afcc (4.605 Å) with an interlayer distance of √3/3afcc (2.171 Å). The above comparison implies that the transformed FCC phase just after shear in Fig. 6a should be adjusted to the standard FCC phase in Fig. 6b, and that a volume expansion of 6.91% would take place during the above adjustment. It is of interest to discuss a little bit about the process of the BCC → FCC phase transition in the PdCu interface with the NW relationship. The present MD simulation reveals that the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes intrinsically induces the
nucleation of the new FCC phase. It should be pointed out that such a kinetics of BCC-FCC phase transition is observed for the first time and could be generalized to other systems as well. This kind of edge dislocations is originated from the interface mismatch in the present study, and could be also formed from other possibilities in actual samples. Experimental investigations are therefore welcome to have further studies on the BCC-FCC phase transition by means of the edge dislocations. As to the BCC-FCC interface with the KS orientation, it should be noted that some dislocations have been also formed in the interface area due to the lattice mismatch. However, these dislocations are either perpendicular to the slip direction, or not in the close-packed plane (slip plane) of the BCC phase (figures not shown). These dislocations could not be therefore moved efficiently through the slip process to trigger the BCC-FCC phase transition, which is quite different from the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes in the BCC-
Fig. 4. Topological models of the BCC-FCC interface with the NW relationship (a) before and (b) after the first slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. The circled and squared symbols represent the first and second layers of the (1 0 0)BCC or ∼(4 4 1¯)FCC plane, respectively. The filled blue symbols denote the FCC phase. The open and red-filled symbols refer to the BCC phase before and after the first slip, respectively. SP stands for the slip plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 444
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Fig. 5. Topological models of the BCC phase in the BCC-FCC interface with the NW relationship after (a) the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes (red symbols), and the (b) first (green symbols), (c) second (blue symbols), and (d) third (yellow symbols) slips of the newly-formed screw dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. The circled and squared symbols represent the first and second layers of the (1 0 0)BCC plane, respectively. SP stands for the slip plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Topological models of the transformed FCC (1 1¯ 1) planes after shear (a) before and (b) after the adjustment. The red and blue symbols represent the first and second layer of the FCC (1 1¯ 1) planes, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 445
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the area of the interface. It should be pointed out that interface energy refers to the energy needed for the formation of the interface from bulk materials, and is commonly adopted as a kind of criterion to express interface stability. Accordingly, the calculated interface energies of the BCC-FCC interface with the NW and KS relationships are listed in Table 4. One can observe clearly that the interface with the NW relationship should be energetically more stable with much smaller γint value (0.1284 J/m2) than the interface with the KS relationship (γint = 1.0593 J/m2). The above obtained surface energies and interface energies are then used to calculate the Δ values by means of Eqs. (9) and (10). Consequently, Table 4 displays the derived driving force (ΔFCC and ΔBCC) of the nucleation of the FCC and BCC phases in the (1 1 0)BCC||(1 1 1)FCC interface model with the NW and KS relationships. It can be discerned from Table 4 that the FCC phase in the interface with the NW orientation has a negative ΔFCC value of −0.1076 J/m2, which seems much lower than the other three positive values of Δ. Such a feature suggests that the FCC phase, rather than the BCC phase, should be energetically more possible to nucleate at the interface with the NW orientation, and that both BCC and FCC phases would be thermodynamically unfavorable to nucleate at the interface with the KS relationship due to their much big and positive Δ values. For the interface with the NW orientation, one could also deduce from Table 4 that the negative ΔFCC value of the FCC phase should be mainly due to the low interface energy (0.1284 J/m2) as well as the smaller surface energy of the FCC (1 1 1) surface (0.9206 J/m2). On the other hand, the much higher interface energy (1.0593 J/m2) should principally induce the big and positive Δ values of both BCC and FCC phases of the interface with the KS relationship. That is to say, the lower energy difference between interface energy and surface energy intrinsically serve as the driving force of the nucleation of the FCC phase and therefore bring about the BCC → FCC phase transition in the BCCFCC interface with the NW orientation. On the contrary, it is the higher interface energy which fundamentally impedes the BCC-FCC phase transition in the BCC-FCC interface with the KS relationship. All the above statements could give, from the point of view of thermodynamics, a probably reasonable explanation to the different behaviors of the BCC-FCC phase transition in the BCC-FCC interface with the NW and KS relationships.
FCC interface with the NW orientation as related before. In other words, the type of dislocations formed in the interface should fundamentally cause the absence of the BCC-FCC phase transition in the interface with the KS orientation. 4.2. Thermodynamics of BCC-FCC phase transition So far, the detailed process of the BCC-FCC phase transition of PdCu has been successfully revealed from the present MD simulation based on the newly-constructed PdCu potential. The next issue of the present study is to find out the fundamental reason why the BCC-FCC phase transition could happen in the BCC-FCC interface with the NW orientation, instead of the interface with the KS relationship. The driving force of the nucleation of the BCC-FCC phase transition is first defined in terms of the energy difference between surface energy and interface energy. The derived results are therefore discussed to provide a deep understanding of the BCC-FCC phase transition from the point of view of energetics. Thermodynamically, the nucleation of the FCC or BCC phase in the BCC-FCC interface area is governed by the relative energy of these two phases and their interface [50]. A parameter (Δ) is thus proposed to express the driving force of the nucleation of the new FCC or BCC phase in the BCC-FCC interface, and is defined as the energy difference between surface energies and interface energy with the following forms:
ΔFCC = γFCC + γint − γBCC ,
(9)
ΔBCC = γBCC + γint − γFCC ,
(10)
where ΔFCC and ΔBCC are regarded as the driving forces of the nucleation of the new FCC and BCC phases in the (1 1 0)BCC||(1 1 1)FCC interface model, respectively; γBCC and γFCC are surface energies of BCC (1 1 0) and FCC (1 1 1) surfaces, respectively; γint is the interface energy. Physically, a negative value of Δ signifies that this phase is energetically favorable to nucleate in the interface area, while a more positive value of Δ refers to the more unlike nucleation of this phase from the point of view of energetics. To derive the values of Δ of the FCC and BCC phases in the BCC-FCC interfaces with the NW and KS relationships, the surface energies (γBCC and γFCC) and interface energy (γint) should be calculated beforehand. Intrinsically, the surface energy is defined as the energy difference of the surface with respect to its corresponding bulk. After the calculation, the surface energies of FCC (1 1 1) and BCC (1 1 0) surfaces of PdCu are obtained and shown in Table 4. It is of interest to see that the surface energy (0.9206 J/m2) of the FCC (1 1 1) surface of PdCu is smaller than that (1.1566 J/m2) of the BCC (1 1 0) surface. The interface energy (γint) is obtained according to the following form [51,52]:
γint =
Eint − (EFCC + EBCC ) , A
5. Conclusions An EAM potential has been constructed for the PdCu system and is able to correctly reproduce the phase stability of BCC and FCC Pd50Cu50 phases, which should be quite different from the performance of the three PdCu potentials in the literature. Based on this constructed potential, molecular dynamics simulation reveals that in the BCC-FCC interface with the NW relationship, the FCC phase nucleates at the interface area by means of the slip of the edge dislocation of 1/ 6aBCC[0 1¯ 1] on (0 1 1)BCC planes, grows through the slip of the newlygenerated screw dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes, and is finally adjusted to a standard FCC lattice through volume expansion. On the contrary, the BCC-FCC phase transition could not be realized in the interface with the KS relationship due to the type of the formed dislocations. Simulation also shows that the lower energy difference between interface energy and surface energy intrinsically serve as the driving force of the nucleation of the FCC phase and therefore bring about the BCC → FCC phase transition in the BCC-FCC interface with the NW orientation, while the higher interface energy fundamentally impedes the BCC-FCC phase transition in the BCC-FCC interface with the KS relationship.
(11)
where Eint is the total energy of interface model, EFCC and EBCC are total energies of the corresponding FCC and BCC bulks, respectively, and A is Table 4 The driving force (ΔFCC and ΔBCC) of the nucleation of the FCC and BCC phases in the (1 1 0)BCC||(1 1 1)FCC interface model with the NW and KS relationships. γint is the interface energy. γBCC and γFCC are surface energies of BCC (1 1 0) and and FCC (1 1 1) surfaces, respectively. ΔFCC = γFCC + γint − γBCC ΔBCC = γBCC + γint − γFCC.
ΔFCC (J/m2) ΔBCC (J/m2) γint (J/m2) γBCC (J/m2) γFCC (J/m2)
Interface with NW relationship
Interface with KS relationship
−0.1076 0.3644 0.1284
0.8233 1.2953 1.0593
BCC (1 1 0) surface
FCC (1 1 1) surface
1.1566 0.9206
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CRediT authorship contribution statement
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W. Wei: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Visualization, Writing - original draft, Writing - review & editing. L.C. Liu: Formal analysis, Writing original draft. H.R. Gong: Conceptualization, Formal analysis, Funding acquisition, Investigation, Project administration, Resources, Supervision, Validation, Writing - original draft, Writing - review & editing. M. Song: Project administration, Validation, Writing - review & editing. M.L. Chang: Writing - review & editing. D.C. Chen: Funding acquisition, Writing - review & editing. Acknowledgement This work was supported by Natural Science Foundation of Hunan Province, China (No. 2018JJ2508), and Project supported by State Key Laboratory of Powder Metallurgy of Central South University, China, Key Project of Department of Education of Guangdong Province, China (2016GCZX008), and Project of Engineering Research Center of Foshan, China (20172010018). References [1] P. Toledano, V. Dmitriev, Reconstructive Phase Transitions: In Crystals and Quasicrystals, World Scientific, 1996 (chapter 3). [2] E.C. Bain, N.Y. Dunkirk, Trans. AIME 70 (1924) 25. [3] C. Zener, Phys. Rev. 71 (1947) 846. [4] Z. Nishiyama, M.E. Fine, M. Meshii, C.M. Wayman, Martensitic Transformation, Academic Press, New York, 1978 (Mater. Sci. Technol.). [5] G.V. Kurdjumov, G. Sachs, Ann. Phys. 64 (1930) 325. [6] B.D. Morreale, M.V. Ciocco, B.H. Howard, R.P. Killmeyer, A.V. Cugini, R.M. Enick, J. Membr. Sci. 241 (2004) 219. [7] P. Kamakoti, D.S. Sholl, Phys. Rev. B 71 (2005) 014301. [8] P. Kamakoti, D.S. Sholl, J. Membr. Sci. 225 (2003) 145. [9] S. Nayebossadri, J. Speight, D. Book, J. Membr. Sci. 451 (2014) 216. [10] L.C. Liu, J.W. Wang, J. Qian, Y.H. He, H.R. Gong, C.P. Liang, S.F. Zou, J. Membr. Sci. 550 (2018) 230. [11] C. Decaux, R. Ngameni, D. Solas, S. Grigoriev, P. Millet, Int. J. Hydrogen Energy 35 (2010) 4883. [12] L.X. Yuan, A. Goldbach, H.Y. Xu, J. Phys. Chem. B 112 (2008) 12692. [13] M.C. Gao, L.Z. Ouyang, Ö.N. Doğan, J. Alloys Compd. 574 (2013) 368. [14] P. Kamakoti, B.D. Morreale, M.V. Ciocco, B.H. Howard, R.P. Killmeyer, A.V. Cugini,
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Fundamental mechanism of BCC-FCC phase transition from a constructed PdCu potential through molecular dynamics simulation ⁎
⁎
W. Weia, L.C. Liua, H.R. Gonga, , M. Songa, , M.L. Changb, D.C. Chenb, a b
T
⁎
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, China School of Materials Science and Energy Engineering, Foshan University, Foshan 528000, Guangdong, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Pd-Cu alloy Phase transformation First principles calculation Molecular dynamic simulation
An n-body PdCu potential is constructed under the framework of the embedded-atom method and is realistic to reproduce phase stability of PdCu phases. Based on this PdCu potential, two interface models with the Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) relationships are established to reveal the kinetics and thermodynamics of the BCC-FCC phase transition through molecular dynamics simulation. It is found that the BCC → FCC phase transition should include the stages of nucleation, growth, and adjustment, and that the slip of edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes fundamentally brings about the nucleation of the FCC phase. Simulations also reveal that the lower energy difference between interface energy and surface energy could serve as the driving force of the nucleation of the FCC phase and intrinsically bring about the BCC → FCC phase transition in the interface with the NW orientation. The different behaviors of the BCC-FCC interface models with the NW and KS relationships are discussed in terms of kinetics and thermodynamics, which could provide a deep understanding of the BCC-FCC phase transition.
1. Introduction The BCC-FCC phase transition has been recognized as the most commonly encountered type of solid-state transformation in pure metals and alloys [1]. Fundamentally, there are already four principal mechanisms in the literature to understand this kind of phase transition [2–5]. In 1924, Bain proposed that the BCC-FCC transformation would be realized through a simple tetragonal deformation (i.e., the change of c/a) and the two phases should follow the orientation relationship of (0 0 1)FCC||(0 0 1)BCC and [1 0 0]FCC||[1 1 0]BCC [2]. Afterwards, another transition path by Zener suggested that the BCC phase would transform to the FCC phase through the shear of (1 1 0)[1¯ 1 0] with the softening of shear mode [3]. In addition, two other mechanisms of the FCC-BCC phase transition were proposed and named as NishiyamaWassermann (NW) and Kurdjumov-Sachs (KS) relationships [4,5], i.e., (1 1 1)FCC||(1 1 0)BCC and [1¯ 1¯ 2]FCC||[1¯ 1 0]BCC for the NW orientation, while (1 1 1)FCC||(1 1 0)BCC and [1 1¯ 0]FCC||[1 1¯ 1]BCC for the KS relationship. The binary Pd-Cu system is well regarded as one of the most promising membranes for hydrogen separation and purification due to its excellent hydrogen selectivity and permeability, superior resistance against poisoning, high thermal stability, nice mechanical properties, and relatively low price, etc. [6–10]. According to the binary phase
⁎
diagram of Pd-Cu [6,9], the face-centered-cubic (FCC) phase could be produced within the entire composition range, whereas the body-centered-cubic (BCC) ordered phase can be only fabricated when the Pd composition is located at the range of 30–50 at.% [8,9,11,12]. Surprisingly, several studies in the literature reveal that the BCC PdCu phase has much higher hydrogen permeability and would be more preferred than its FCC counterpart [12–15]. The BCC-FCC phase transition of PdCu has been extensively studied in the literature [6–9,12,14,16–18]. For instance, Ellis and Mohanty experimentally observed the strain-induced BCC-FCC phase transition of the Pd0.4Cu0.6 phase [17]. Yuan et al. found that the BCC → FCC transition of Pd48Cu52 happened in the temperature range of 723–873 K [16]. Moreover, theoretical calculations were used to find out the effects of Nb and Cr addition on the Bain path of the BCC-FCC phase transition of Pd50Cu50 [19]. Regarding the detailed mechanism of the BCC-FCC phase transition of PdCu, however, there is not any report so far in the literature. It is well known that molecular dynamics (MD) simulation based on an n-body potential has been universally used to predict various phase transitions of metal alloys [20–22]. The present study is therefore aimed to find out the intrinsic mechanism of the BCC-FCC phase transition of PdCu by means of MD simulation. First of all, a realistic n-body potential of PdCu is constructed in the present study, as the three PdCu
Corresponding authors. E-mail addresses: [email protected] (H.R. Gong), [email protected] (M. Song), [email protected] (D.C. Chen).
https://doi.org/10.1016/j.commatsci.2018.12.037 Received 21 November 2018; Received in revised form 17 December 2018; Accepted 17 December 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
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potentials in the literature cannot be used to reveal the BCC-FCC phase transition. Two BCC-FCC interface models are then established to reveal the kinetics and thermodynamics of the BCC-FCC phase transition of PdCu through MD simulation with the newly-constructed PdCu potential. It will be shown for the first time that the BCC → FCC phase transition of PdCu should include the stages of nucleation, growth, and adjustment, and that the FCC phase nucleates at the interface area by means of the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. In addition, the lower energy difference between interface energy and surface energy intrinsically serve as the driving force of the nucleation of the FCC phase and therefore bring about the BCC → FCC phase transition.
Table 1 Fitted parameters of Pd-Pd and Cu-Cu potentials. The cutoff distances, rs and rc, are also listed.
The embedded atom method (EAM) has been well regarded as one of the most realistic n-body potentials, and has been widely used to reproduce physical properties of transition metals and alloys [23–25]. For the binary Pd-Cu system, there are already three constructed EAM potentials in the literature [23,24,26]. Unfortunately, it will be shown in detail later that these EAM potentials have all failed to reflect the relative stability of BCC and FCC Pd50Cu50 phases at 0 K and are therefore unable to describe the BCC-FCC phase transition of the PdCu system. In the present study, a new EAM potential is thus constructed, in order to reveal the intrinsic features of the BCC-FCC phase transition of PdCu. After a series of testing, the EAM forms by Cai et al. [24] are selected for the Pd-Pd and Cu-Cu potentials, while a simplified mathematical term proposed by Gong et al. [27] is chosen for the Pd-Cu cross potential. According to the EAM approach, the total energy of N atoms in the system is depicted as:
Etot
∑
φij (rij ),
n
ρi =
c d e h
⎜
⎟
Calculated
Experiment
Calculated
−3.91a 3.89e 1.95c 2.341c 1.761c 0.712c 1.4 h
−3.91 3.89 1.9646 2.3308 1.7814 0.7182 1.4000
−3.54a 3.615b 1.4203c 1.7c 1.225c 0.758c 1.3d
−3.54 3.615 1.3835 1.7001 1.2252 0.7581 1.2999
Ref. [29]. Ref. [30]. Ref. [31]. Ref. [32]. Ref. [33]. Ref. [34].
φPdCu (r ) =
(4)
⎜
Experiment
φPdCu (r ) = [ZPd (r ) + ZCu (r )]/ r ,
⎟
r r φ (r ) = −α ⎡1 + β ⎛ − 1⎞ ⎤ exp ⎡−β ⎛ − 1⎞ ⎤, ⎢ ⎥ ⎢ r r ⎝ a ⎠⎦ ⎠⎥ ⎣ ⎣ ⎝ a ⎦
Cu
intrinsic properties of Pd and Cu [29–34]. The Pd-Cu cross potential is then constructed by means of the derived Pd-Pd and Cu-Cu potentials. As related before, three EAM potentials of Pd-Cu have already been derived in the literature, and the mathematical forms of the Pd-Cu cross potentials are as follows [23,24,26]:
(3)
⎜
11.13548 0.70277 3.52097 1.65393 0.67199 3.69625 4.10247
b
∑ f (rij),
r f (r ) = fe exp ⎡−χ ⎛ − 1⎞ ⎤, ⎢ ⎝ re ⎠⎥ ⎣ ⎦
17.74997 1.42485 0.43287 0.32751 0.28064 3.97743 4.41455
a
(2)
j≠i
Χ α (eV) Β ra (Å) F1 (eV) rs (Å) rc (Å)
Ec (eV/atom) a (Å) B (Mbar) C11 (Mbar) C12 (Mbar) C44 (Mbar) Evf (eV)
n
ρ ρ ⎤ ρ ⎡ F (ρ) = −F0 ⎢1 − ln ⎛⎜ ⎞⎟ ⎥ ⎜⎛ ⎟⎞ + F1 ⎜⎛ ⎟⎞, ρ ρ ρ ⎝ e ⎠ ⎦⎝ e ⎠ ⎝ e⎠ ⎣
Cu-Cu
Pd
(1)
i, j (i ≠ j )
Pd-Pd
Table 2 Comparison between calculated and experimental results of cohesive energy Ec (eV/atom), lattice constant a (Å), elastic constants (Mbar), and vacancy formation energy Evf (eV) for Pd and Cu.
2. Construction of an n-body potential
1 = ∑ Fi (ρi ) + 2 i
Parameter
(6)
f Pd (r ) 1 ⎡ f Cu (r ) Pd φ (r ) + Cu φCu (r ) ⎤, Pd ⎥ ⎢ 2 ⎣ f (r ) f (r ) ⎦
(7)
where ZPd (r ) and ZCu (r ) are effective charges of Pd and Cu, respectively. Accordingly, Table 3 shows the calculated lattice constants and cohesive energies of BCC (B2) and FCC (L10) Pd50Cu50 phases from the above three EAM potentials of PdCu in the literature [23,24,26]. One could observe from Table 3 that the cohesive energy of the FCC
⎟
(5)
where Etot is the total energy, F(ρ) is the embedding energy, ρi is the total electron density at atom i due to all other atoms, φ(rij) is the pair potential between atoms numbered i and j, and f(rij) is the electron density for atom i contributed by atom j. In Eq. (2), F0, ρe, and n are three constants. F0 = Ec − Evf, in which Ec and Evf are cohesive energy and vacancy formation energy, respectively. The parameter ρe represents the host electron density at an equilibrium state. F1 is an adjustable parameter. In Eq. (4), re is an equilibrium first-neighbor distance, and fe is a scaling factor determined by the relationship of fe = Ec/Ω, where Ω is the atomic volume. The cutoff function is employed from the polynomial derived by Guellil et al. [28], and the cutoff distances are set to be between the second- and third-neighbor distances. It should be noted that there are five adjustable parameters (χ, α, β, ra, F1) to be fitted for both Pd-Pd and Cu-Cu potentials. Consequently, Table 1 lists the derived five parameters and the chosen cutoff distances of Pd-Pd and Cu-Cu potentials. In addition, Table 2 summarizes the calculated and experimental physical properties of pure Pd and Cu. It can be seen clearly that the constructed Pd-Pd and Cu-Cu potentials are relevant to reproduce some
Table 3 Comparison of lattice constants (a) and cohesive energies (Ec) of Pd50Cu50 phases with the BCC and FCC structures from the present PdCu potential as well as the other three EAM potentials in the literature [23,24,26]. Also listed are the fitting parameters of the present PdCu potential, i.e., A, m (Å), and n (Å). References
a (Å) BCC
The present potential 2.9185 Exp. [39] 2.958 Ab initio [35,36] EAM potential [26] 3.011 EAM potential [24] 2.988 EAM potential [23] 2.958 Fitting parameters A 3.7907 m (Å)
441
Ec (eV/atom) FCC
BCC
FCC
3.76 3.749
−3.911
−3.869
−3.891 3.777 −3.711 3.764 −4.093 3.704 −4.183 0.6761 n (Å) 1.7424
−3.819 −3.734 −4.17 −4.285
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Pd50Cu50 phase from the above potentials [23,24,26] is lower than its corresponding BCC structure, which should be contrary to the experimentally observed B2 structure (rather than FCC structure) at low temperature as well as the energy sequence of BCC (B2) and FCC (L10) Pd50Cu50 phases from first principles calculations [35,36]. Such a disagreement implies that the above three EAM potentials of PdCu in the literature [23,24,26] should be irrelevant to correctly reproduce the relative stability of BCC and FCC Pd50Cu50 phases, and would be then unable to reflect the BCC-FCC phase transition in the binary PdCu system. In the present study, another mathematical form is therefore selected for the Pd-Cu cross potential by means of a linear combination of the Pd-Pd and Cu-Cu potentials [27]:
φPdCu (r ) = A [φCu (r + m) + φPd (r + n)],
Fig. 1. Constructed BCC-FCC interface models with Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) orientations. For the NW orientation, the xaxis is [0 0 1]BCC and [1 1¯ 0]FCC, the y-axisis[1 1 0]BCC and[1 1 1]FCC, and the z-axis is [1¯ 1 0]BCC and [1¯ 1¯ 2]FCC. For the KS relationship, the x-axis is [1 1¯ 1]BCC and [1 1¯ 0]FCC, the y-axisis[1 1 0]BCC and[1 1 1]FCC, and the zaxis is [1¯ 1 2]BCC and [1¯ 1¯ 2]FCC. The shadow plane refers to the FCC/BCC interface, and τ denotes the shear direction.
(8)
where A, m, and n are three parameters to be fitted. It should be noted that this formula is a modified form of the original cross potential proposed by Chen et al. [37] and Heino [38]. After the fitting process, Table 3 lists the derived potential parameters of the Pd-Cu cross potential as well as the reproduced cohesive energies and lattice constants of BCC (B2) and FCC (L10) Pd50Cu50 phases. It can be seen that the calculated lattice constants of BCC and FCC Pd50Cu50 phases are 2.919 and 3.76 Å, respectively, which match well with the corresponding values of 2.958 and 3.749 Å from experiments [39] and first principles calculations [19,35,36,39], respectively. Moreover, the cohesive energies of BCC and FCC Pd50Cu50 phases from the present Pd-Cu potential are −3.911 and −3.869 eV/atom, respectively, which are in good agreement with the corresponding values of −3.891 and −3.819 eV/atom from first principles calculations [36]. In other words, the BCC structure of Pd50Cu50 is energetically more favorable with lower cohesive energy than its FCC counterpart, which is in excellent with the corresponding observations from experiments and first principles calculations [19,35,36,39]. These nice agreements suggest that the newly constructed Pd-Cu potential should be realistic to reproduce the phase stability of BCC and FCC Pd50Cu50 phases, which is quite different from the performance of the three EAM potentials of Pd-Cu in the literature as related before [23,24,26]. It will be shown in the following simulations that this derived Pd-Cu potential is also relevant to reveal the intrinsic features of the BCC-FCC phase transition of PdCu.
atom layers in the x (y, z) direction. The x, y, and z axes of the interface [1 1¯ 1]BCC||[1 1¯ 0]FCC, with the KS orientation are [1 1 0]BCC||[1 1 1]FCC, and [1¯ 1 2]BCC||[1¯ 1¯ 2]FCC, respectively. It should be pointed out that the lattice constant (3.76 Å) of the substrate (FCC Pd50Cu50) derived from the present EAM potential shown in Table 3 is used in the two interface models, and there are 123,200 and 202,888 atoms for the two BCC-FCC interfaces with the NW and KS relationships, respectively. Accordingly, Fig. 1 shows the constructed BCC-FCC interface models with the NW and KS orientations. In the above interface settings, the interface mismatch between BCC and FCC planes are minimized to be 0.588% and 0.385% for the interfaces with the NW and KS orientations, respectively, and such a small mismatch should have a negligible effect on simulation results. The constructed BCC-FCC interface models with the NW and KS orientations are first relaxed in the canonical ensemble (NVT) at 10 K for 20 ps and then in the isothermal-isobaric (NPT) ensemble at 10 K for 20 ps. After the relaxation, each interface is then under a shear loading process in the isothermal-isobaric (NPT) ensemble for 350 ps with a strain rate of γ̇ = 1 × 109/s, and the shear direction τ is shown clearly in Fig. 1. It should be pointed out that after a series of tests of deformation directions, only the shear direction (τ) displayed in Fig. 1 could bring about the FCC-BCC phase transition of PdCu. For the loading, a uniform stretch is considered and a time step of 0.001 ps is selected. A Gaussian thermostat is employed for the temperature control. All simulation models are performed by three dimensional periodic boundary conditions. The atomic positions of the interface after the simulation are visualized by means of the OVITO software [46], and the crystal structure is identified by the method of polyhedral template matching (PTM) [47].
3. Simulation methods The present MD simulation is performed by means of Large-scale Atomic/Molecular Massively Parallel Simulator package (LAMMPS) [40] with the newly constructed EAM potential of PdCu. Two interface bilayer models between B2 (BCC) and L10 (FCC) Pd50Cu50 phases are intentionally established with the well-known Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS) relationships. Specifically, the NW orientation of the interface is described by (1 1 1)FCC||(1 1 0)BCC and [1¯ 1¯ 2]FCC||[1¯ 1 0]BCC, and the KS relationship is expressed by (1 1 1)FCC||(1 1 0)BCC and [1 1¯ 0]FCC||[1 1¯ 1]BCC. It should be noted that the PdCu phase with equal atomic compositions of Pd and Cu is purposely chosen in the present study due to its good hydrogen permeability [8,15,41], and that the interface orientation of (1 1 1)FCC||(1 1 0)BCC with the NW and KS relationships is usually observed in other systems [4,5,42–45]. To form the BCC-FCC interface with the NW relationship, a BCC Pd50Cu50 phase with 80 (40, 40) atom layers in the x (y, z) direction is put at the top of an FCC Pd50Cu50 phase with 44 (45, 36) atom layers in the x (y, z) direction, and the y direction is perpendicular to the interface. For the NW orientation, the x-axis is [0 0 1]BCC and [1 1¯ 0]FCC, the y-axisis[1 1 0]BCC and[1 1 1]FCC, and the z-axis is [1¯ 1 0]BCC and [1¯ 1¯ 2]FCC. Similarly, the BCC-FCC interface of Pd50Cu50 with the KS orientation has the BCC phase with 42 (46, 36) atom layers in the x(y, z) direction, which are added at the top of the FCC phase with 40 (45, 56)
4. Results and discussion 4.1. Kinetics of BCC-FCC phase transition As related before, the BCC-FCC phase transition of PdCu has been observed both experimentally and theoretically, while the mechanism of such a transition has not been reported so far in the literature [6–9,12,14,16–18]. The present MD simulation is therefore used for the first time to reveal the detailed kinetics of the BCC → FCC phase transition of PdCu with the NW orientation, and this phase transition should include the stages of nucleation, growth, and adjustment. It is found that the FCC phase first nucleates at the interface area by means of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes due to lattice mismatch, then grows through the slip of the newly-formed screw 442
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Fig. 2. Snapshots of the BCC-FCC interface model with the NW relationship after the shear at (a) 0 ps, (b) 240 ps, and (c) 280 ps. In the simulation, the blue, green and red atoms represent the BCC, FCC and HCP atoms, respectively. (d) is the structural change of the BCC lattice during the shear process as a function of shear (ε). ε = tan θ, in which θ is the shear angle. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes, and is finally adjusted to a standard FCC lattice by volume expansion. On the contrary, the BCC-FCC phase transition could not be realized in the interface with the KS relationship. First of all, the snapshots of the BCC-FCC interface model with the NW relationship are detected by the method of polyhedral template matching (PTM) [47], and are shown as typical examples in Fig. 2a, b, and c during the process of shear. In addition, Fig. 2d displays the structural change of the BCC lattice during the shear process as a function of shear (ε). It can be seen clearly that in the BCC-FCC interface model with the NW relationship, the BCC lattice has changed gradually to the FCC lattice with the increase of the shear angle, and that the FCC lattice could keep its crystal structure. For the BCC-FCC interface model with the KS relationship, however, no phase transition could be observed, as the crystal structures of both BCC and FCC lattices do not change during the whole shear process (figures not shown). To find out the detailed micro-structural change during the phase transition, Fig. 3 shows the snapshots of the (1 0 0)BCC and (4 4 1¯)FCC planes in the BCC-FCC interface with the NW relationship before and after shear at 120 ps. It can be discerned obviously from Fig. 3a that in the interface area before shear, ten layers of (0 1 1¯)BCC planes exactly match eleven layers of (1¯ 1 1¯)FCC planes, which would therefore bring about the appearance of an edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. After shear at 120 ps, one could see from Fig. 3b that these dislocations have moved into the BCC lattice through the slip process and some atoms in the original BCC lattice have transformed to the FCC phase. The above points suggest that the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes should induce the BCC → FCC phase transition. It should be noted that various shear directions have been tried and only the shear direction shown in Fig. 1 can bring about the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on
(0 1 1)BCC planes and the subsequent BCC-FCC phase transition. Topological models of the BCC-FCC interface are then used for the above discovered orientations of the interface, in order to clearly reveal the process of the nucleation and growth of the formed FCC phase in the interface area [48,49], as the snapshots from MD simulation are unable to do so due to the vibration of the atoms. Consequently, Fig. 4 displays the topological models of the BCC-FCC interface with the NW relationship before and after the slip of the edge dislocation of 1/ 6aBCC[0 1¯ 1] on (0 1 1)BCC planes. As shown in Fig. 4a, an edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes has been formed in the interface area before shear due to the lattice mismatch. After the first slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes in Fig. 4b, the slipped (0 1 1)BCC plane left to the slip plane could be regarded as the first layer (i.e., nucleation) of the newly-formed FCC phase, and a screw dislocation between the newly-formed FCC phase and the original BCC phase will be generated by the movement of the edge dislocation. That is to say, the Burgers vector of the edge dislocation is parallel to the glide direction, and the slip of the edge dislocation induces the nucleation of the FCC phase. To further express the formation of the other layers of the FCC phase, Fig. 5a displays the topological models of the BCC part of Fig. 4b after the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes, while Fig. 5b, c, and d show subsequently the first, second, and third slips of the newly-formed screw dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes between the BCC and newly-generated FCC closepacked planes, respectively. It can be seen clearly that one FCC layer could be formed after each slip of the screw dislocation and the stacking of -A-B-A-B- in the (0 1 1) planes of the BCC structure would finally transform to the stacking of -A-B-C-A-B-C- in the (1 1 1) planes of the FCC structure. Such a process could be thus regarded as the growth of the FCC phase by means of the slip of the screw dislocation of 1/ 443
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Fig. 3. Snapshots of the (1 0 0)BCC or (4 4 1¯)FCC plane in the BCC-FCC interface with the NW relationship (a) before shear and (b) after shear at 120 ps. The blue and green symbols represent the BCC and FCC phases, respectively. The red symbols refer to the undiscerned BCC or FCC atoms due to interface stress. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6aBCC[0 1¯ 1] on (0 1 1)BCC planes. We now turn to investigate the adjustment of the newly-transformed FCC phase. As displayed in Fig. 6a, the transformed FCC phase of PdCu just after shear has the original lattice spacing of the BCC (0 1 1) plane, i.e., abcc (2.919 Å) and √2abcc (4.127 Å) with an interlayer distance of √2/2abcc (2.064 Å). As shown in Fig. 6b, however, the standard FCC phase of PdCu should have the lattice spacing of the FCC (1 1¯ 1) plane, i.e., √2/2afcc (2.659 Å) and √6/2afcc (4.605 Å) with an interlayer distance of √3/3afcc (2.171 Å). The above comparison implies that the transformed FCC phase just after shear in Fig. 6a should be adjusted to the standard FCC phase in Fig. 6b, and that a volume expansion of 6.91% would take place during the above adjustment. It is of interest to discuss a little bit about the process of the BCC → FCC phase transition in the PdCu interface with the NW relationship. The present MD simulation reveals that the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes intrinsically induces the
nucleation of the new FCC phase. It should be pointed out that such a kinetics of BCC-FCC phase transition is observed for the first time and could be generalized to other systems as well. This kind of edge dislocations is originated from the interface mismatch in the present study, and could be also formed from other possibilities in actual samples. Experimental investigations are therefore welcome to have further studies on the BCC-FCC phase transition by means of the edge dislocations. As to the BCC-FCC interface with the KS orientation, it should be noted that some dislocations have been also formed in the interface area due to the lattice mismatch. However, these dislocations are either perpendicular to the slip direction, or not in the close-packed plane (slip plane) of the BCC phase (figures not shown). These dislocations could not be therefore moved efficiently through the slip process to trigger the BCC-FCC phase transition, which is quite different from the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes in the BCC-
Fig. 4. Topological models of the BCC-FCC interface with the NW relationship (a) before and (b) after the first slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. The circled and squared symbols represent the first and second layers of the (1 0 0)BCC or ∼(4 4 1¯)FCC plane, respectively. The filled blue symbols denote the FCC phase. The open and red-filled symbols refer to the BCC phase before and after the first slip, respectively. SP stands for the slip plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 444
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Fig. 5. Topological models of the BCC phase in the BCC-FCC interface with the NW relationship after (a) the slip of the edge dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes (red symbols), and the (b) first (green symbols), (c) second (blue symbols), and (d) third (yellow symbols) slips of the newly-formed screw dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes. The circled and squared symbols represent the first and second layers of the (1 0 0)BCC plane, respectively. SP stands for the slip plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Topological models of the transformed FCC (1 1¯ 1) planes after shear (a) before and (b) after the adjustment. The red and blue symbols represent the first and second layer of the FCC (1 1¯ 1) planes, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 445
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the area of the interface. It should be pointed out that interface energy refers to the energy needed for the formation of the interface from bulk materials, and is commonly adopted as a kind of criterion to express interface stability. Accordingly, the calculated interface energies of the BCC-FCC interface with the NW and KS relationships are listed in Table 4. One can observe clearly that the interface with the NW relationship should be energetically more stable with much smaller γint value (0.1284 J/m2) than the interface with the KS relationship (γint = 1.0593 J/m2). The above obtained surface energies and interface energies are then used to calculate the Δ values by means of Eqs. (9) and (10). Consequently, Table 4 displays the derived driving force (ΔFCC and ΔBCC) of the nucleation of the FCC and BCC phases in the (1 1 0)BCC||(1 1 1)FCC interface model with the NW and KS relationships. It can be discerned from Table 4 that the FCC phase in the interface with the NW orientation has a negative ΔFCC value of −0.1076 J/m2, which seems much lower than the other three positive values of Δ. Such a feature suggests that the FCC phase, rather than the BCC phase, should be energetically more possible to nucleate at the interface with the NW orientation, and that both BCC and FCC phases would be thermodynamically unfavorable to nucleate at the interface with the KS relationship due to their much big and positive Δ values. For the interface with the NW orientation, one could also deduce from Table 4 that the negative ΔFCC value of the FCC phase should be mainly due to the low interface energy (0.1284 J/m2) as well as the smaller surface energy of the FCC (1 1 1) surface (0.9206 J/m2). On the other hand, the much higher interface energy (1.0593 J/m2) should principally induce the big and positive Δ values of both BCC and FCC phases of the interface with the KS relationship. That is to say, the lower energy difference between interface energy and surface energy intrinsically serve as the driving force of the nucleation of the FCC phase and therefore bring about the BCC → FCC phase transition in the BCCFCC interface with the NW orientation. On the contrary, it is the higher interface energy which fundamentally impedes the BCC-FCC phase transition in the BCC-FCC interface with the KS relationship. All the above statements could give, from the point of view of thermodynamics, a probably reasonable explanation to the different behaviors of the BCC-FCC phase transition in the BCC-FCC interface with the NW and KS relationships.
FCC interface with the NW orientation as related before. In other words, the type of dislocations formed in the interface should fundamentally cause the absence of the BCC-FCC phase transition in the interface with the KS orientation. 4.2. Thermodynamics of BCC-FCC phase transition So far, the detailed process of the BCC-FCC phase transition of PdCu has been successfully revealed from the present MD simulation based on the newly-constructed PdCu potential. The next issue of the present study is to find out the fundamental reason why the BCC-FCC phase transition could happen in the BCC-FCC interface with the NW orientation, instead of the interface with the KS relationship. The driving force of the nucleation of the BCC-FCC phase transition is first defined in terms of the energy difference between surface energy and interface energy. The derived results are therefore discussed to provide a deep understanding of the BCC-FCC phase transition from the point of view of energetics. Thermodynamically, the nucleation of the FCC or BCC phase in the BCC-FCC interface area is governed by the relative energy of these two phases and their interface [50]. A parameter (Δ) is thus proposed to express the driving force of the nucleation of the new FCC or BCC phase in the BCC-FCC interface, and is defined as the energy difference between surface energies and interface energy with the following forms:
ΔFCC = γFCC + γint − γBCC ,
(9)
ΔBCC = γBCC + γint − γFCC ,
(10)
where ΔFCC and ΔBCC are regarded as the driving forces of the nucleation of the new FCC and BCC phases in the (1 1 0)BCC||(1 1 1)FCC interface model, respectively; γBCC and γFCC are surface energies of BCC (1 1 0) and FCC (1 1 1) surfaces, respectively; γint is the interface energy. Physically, a negative value of Δ signifies that this phase is energetically favorable to nucleate in the interface area, while a more positive value of Δ refers to the more unlike nucleation of this phase from the point of view of energetics. To derive the values of Δ of the FCC and BCC phases in the BCC-FCC interfaces with the NW and KS relationships, the surface energies (γBCC and γFCC) and interface energy (γint) should be calculated beforehand. Intrinsically, the surface energy is defined as the energy difference of the surface with respect to its corresponding bulk. After the calculation, the surface energies of FCC (1 1 1) and BCC (1 1 0) surfaces of PdCu are obtained and shown in Table 4. It is of interest to see that the surface energy (0.9206 J/m2) of the FCC (1 1 1) surface of PdCu is smaller than that (1.1566 J/m2) of the BCC (1 1 0) surface. The interface energy (γint) is obtained according to the following form [51,52]:
γint =
Eint − (EFCC + EBCC ) , A
5. Conclusions An EAM potential has been constructed for the PdCu system and is able to correctly reproduce the phase stability of BCC and FCC Pd50Cu50 phases, which should be quite different from the performance of the three PdCu potentials in the literature. Based on this constructed potential, molecular dynamics simulation reveals that in the BCC-FCC interface with the NW relationship, the FCC phase nucleates at the interface area by means of the slip of the edge dislocation of 1/ 6aBCC[0 1¯ 1] on (0 1 1)BCC planes, grows through the slip of the newlygenerated screw dislocation of 1/6aBCC[0 1¯ 1] on (0 1 1)BCC planes, and is finally adjusted to a standard FCC lattice through volume expansion. On the contrary, the BCC-FCC phase transition could not be realized in the interface with the KS relationship due to the type of the formed dislocations. Simulation also shows that the lower energy difference between interface energy and surface energy intrinsically serve as the driving force of the nucleation of the FCC phase and therefore bring about the BCC → FCC phase transition in the BCC-FCC interface with the NW orientation, while the higher interface energy fundamentally impedes the BCC-FCC phase transition in the BCC-FCC interface with the KS relationship.
(11)
where Eint is the total energy of interface model, EFCC and EBCC are total energies of the corresponding FCC and BCC bulks, respectively, and A is Table 4 The driving force (ΔFCC and ΔBCC) of the nucleation of the FCC and BCC phases in the (1 1 0)BCC||(1 1 1)FCC interface model with the NW and KS relationships. γint is the interface energy. γBCC and γFCC are surface energies of BCC (1 1 0) and and FCC (1 1 1) surfaces, respectively. ΔFCC = γFCC + γint − γBCC ΔBCC = γBCC + γint − γFCC.
ΔFCC (J/m2) ΔBCC (J/m2) γint (J/m2) γBCC (J/m2) γFCC (J/m2)
Interface with NW relationship
Interface with KS relationship
−0.1076 0.3644 0.1284
0.8233 1.2953 1.0593
BCC (1 1 0) surface
FCC (1 1 1) surface
1.1566 0.9206
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CRediT authorship contribution statement
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