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A metric to gauge local distortion in metallic glasses and supercooled liquids
Available online at www.sciencedirect.com
ScienceDirect Acta Materialia 72 (2014) 229–238 www.elsevier.com/locate/actamat
A metric to gauge local distortion in metallic glasses and supercooled liquids Chen Wu a, Nikos Ch. Karayiannis b, Manuel Laso b, Dongdong Qu a, Qiang Luo c, Jun Shen c,⇑ b
a School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China Institute for Optoelectronics and Microsystems (ISOM) and ETSII, Polytechnic University of Madrid (UPM), Madrid 28006, Spain c School of Materials Science and Engineering, Tongji University, Shanghai 201804, People’s Republic of China
Received 9 February 2014; received in revised form 18 March 2014; accepted 19 March 2014 Available online 23 April 2014
Abstract We propose a new structural descriptor to characterize distortion and short-range order in metallic glasses and supercooled liquids. Through the new metric, salient structural differences in local clusters of identical Voronoi indices can be accurately identified. As a test case, we employ the short-range-order symmetry parameter (SSP) in atomistic configurations of Cu50Zr50 generated through extensive molecular dynamics simulations at various temperatures. Structural analysis reveals the dependence of short-range order on local chemistry and thermal fluctuations. Furthermore, the new descriptor can be directly correlated with the atomic level shear stress, the local potential energy surface and the dynamical heterogeneity. Combined with existing analytical tools, the SSP can provide significant information, at the atomic level, on the structure–property relation in metallic glasses and in general complex particulate systems. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Metallic glasses; Supercooled liquids; Molecular dynamics simulation; Symmetry and distortion of short-range order; Dynamics of atoms
1. Introduction The identification and successive analysis of atomic structure in metallic glasses (MGs) and supercooled liquids (SLs) is a formidable scientific challenge which is attracting constant interest [1–6]. During the last few years, numerous studies [7–18] have provided significant information on this subject. In parallel, groundbreaking atomistic models [19–21] have been proposed to describe the atomic structure in such complex systems. However, key factors that dictate the correlation between atomic structure and properties remain elusive. For example, due to the non-equilibrium nature of MGs, many studies focus on the formation ⇑ Corresponding author.
E-mail address: [email protected] (J. Shen).
process, that is, supercooling of liquids below their melting temperatures. Although spatial dynamic fluctuations play the central role in modern description of supercooled liquids, their dependence on local structure is not yet fully understood. [22] Therefore, a detailed description of the atomic structure of MGs and SLs is highly desirable. Recent studies [6,11] have shown that the local environment in MGs and SLs does possess short-range order (SRO). Nowadays, scientists commonly use the bond orientation order (BOO) parameter [23], the Voronoi analysis [24] and the common neighbor analysis [25–27] methods to characterize SRO. By quantifying the angular distortion of local structure, the BOO parameter identifies the type of perfect order that the local structure resembles. However, radial distortion from perfect order is ubiquitous in glassy systems and the extraction of structural
http://dx.doi.org/10.1016/j.actamat.2014.03.046 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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C. Wu et al. / Acta Materialia 72 (2014) 229–238
information through the BOO parameters is difficult in MGs and supercooled liquids [28]. The Voronoi tessellation method [24] generates the Voronoi polyhedron (VP), which contains the space closer to a given atom than to any other atom. SRO is characterized by the Voronoi indices (n3, n4, n5, n6) where ni stands for the number of i-edged faces. However, many researchers have pointed out that the Voronoi indices do not provide a complete description of SRO [28–30]. Srolovitz et al. [29] showed that, in order to quantify SRO, additional information on the distortion of individual clusters with identical Voronoi indices is required. The common neighbor analysis (CNA) method identifies the structural unit, the so-called n-bipyramid [30], formed by a pair of nearest neighbor atoms and their n common neighbors. Miracle [30] reported that the efficient filling of space can be effectively explored by taking the n-bipyramids as the basic structural units. Furthermore, n-bipyramids can be related to plastic deformation [31]. However, these n-bipyramids might be severely distorted, and the degree of distortion, which varies from site to site, cannot be fully quantified by the CNA method. Often, n-fold bipyramids are denoted as “n-fold symmetry”, although there is no real symmetry in these configurations. Instead, the notation “n-fold symmetry” identifies these bipyramids as n-bipyramids. Nevertheless, this characterization has motivated us to quantify the distortion of n-bipyramids by calculating the degree of rotational symmetry. Lowsymmetry bipyramids can be effectively viewed as highdistortion bipyramids compared to ideal ones. It is well established that the degree of symmetry in SRO strongly influences the glass-forming ability of MGs [32]. The discussion above emphasizes the need to introduce a novel structural indicator which can quantify distortion by measuring the degree of symmetry. Srolovitz et al. [29] analyzed the degree of symmetry in SRO by employing a set of parameters inspired by the atomic level stress theory. This theory, proposed by Egami et al. [33–36], provides a clear physical interpretation of glass formation and mechanical behavior in MGs. However, finding structural descriptors, which can be correlated with the atomic level stresses, remains an open research field [6,37]. Thus, a refined structural analysis tool is required to measure the degree of symmetry in SRO accurately. A combination of the new descriptor with an existing structural indicator (i.e. the Voronoi index) could provide a better understanding of SRO. In this work, we propose the SRO symmetry parameter (SSP) as a way to quantify the degree of symmetry and distortion in local structure, providing information which is untraceable by conventional methods. Section 2 describes the molecular dynamics (MD) simulations used for the generation of the samples in detail and provides the SSP formalism. Results from the structural analysis based on the SSP are presented and discussed in Section 3. The correlation between the SSP and the dynamics of atoms in supercooled liquids is analyzed in Section 4, further demonstrating the usefulness of SSP as a detailed
descriptor of local order. Finally, the main results and conclusions are summarized in Section 5. 2. Simulation details and the SSP 2.1. Molecular dynamics In the present work, atomic structures of Cu50Zr50 were generated through LAMMPS [38] with the embeddedatom method (EAM) potential [39,40]. A cubic simulation box containing 10,000 atoms, with periodic boundary conditions applied in all dimensions, was equilibrated through isothermal–isobaric (NPT) simulations (P = 0 atm, T = 2000 K) for 2 ns. Pressure and temperature oscillations were controlled through a Nose´–Hoover barostat and thermostat, respectively. In the continuation, melt configurations were cooled down to the target temperatures (T = 1000, 800 and 300 K) at a rate of 1012 K s1. In all cases the time integration step was set at 1 fs. According to literature data, the liquidus temperature for Cu50Zr50 MG is approximately 1150 K [41]. 2.2. The SSP In this section, we give a detailed account of the SRO symmetry norm method and the corresponding parameter (SSP). The new descriptor is inspired by the characteristic crystallographic element norm method [42–47]. The SSP of a given atom is defined by the degree of rotational symmetry for the bipyramids extracted from the coordinating polyhedron (CP) of the atom. We consider a configuration of atoms, denoted by X, consisting of the reference atom j (with position vector rj) and its nearest neighbors (with position vectors rk, k e X, in an arbitrary reference frame) determined by the Voronoi tessellation method. To obtain the SSP of atom j, the coordinating polyhedron is first divided into Nsub n-bipyramids, where Nsub is the number of nearest neighbors of atom j. Fig. 1(a) depicts a Cu-centered (0, 1, 10, 2) CP in the simulated Cu50Zr50 MG, while Fig. 1(b) shows a hexagonal bipyramid (6-bipyramid), a pentagonal bipyramid (5-bipyramid) and
Fig. 1. (a) A Cu-centered (0, 1, 10, 2) CP in Cu50Zr50 MG and (b) representative sub-bipyramids extracted from this CP.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
a square bipyramid (4-bipyramid) extracted from the parent (0, 1, 10, 2) CP of Fig. 1(a). In general, all types of polyhedra can be divided into subbipyramids through the Voronoi tessellation. A CP with Voronoi index (n3, n4, n5, n6) can be divided into n3 3-, n4 4-, n5 5- and n6 6-bipyramids. If we suppose that the coordinating polyhedron around j is an ideal, perfect polyhedron with equilateral edges, then each of the n-bipyramids possesses rotational symmetry (an n-fold rotational axis penetrates the axial sites for every n-bipyramid). However, in real systems, deviations from ideality are expected: the n-bipyramids do not possess rotational symmetry due to intrinsic fluctuations and various arrangements of different constituent atoms. Thus, rotating an n-bipyramid, obtained
erXj 6-bipyramid
¼ minR;RSOð3Þ
1 pffiffiffiffiffiffiffiffiffi r 57
ationally, for the n-bipyramid of j, we first randomly choose an initial orientation of the n-fold rotation axis passing through rj. The Ng distinct proper operations are then applied to the n-bipyramid. After each application, the discrepancy is calculated as in Eq. (1). This procedure is repeated by searching for the orientation of the rotation axis in the two-parameter space of azimuthal and polar angles so as to find the minimum discrepancy, which is then set to be erXj sub . This step corresponds to the first minimization in Eq. (1). An illustrative example is given for the 6-bipyramid, where the 6-fold axis defines Ng = 5 different elements ({R = 61, 62, 63, 64 and 65}). Eq. (1) can then be written as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h X7 X5 i 6-bipyramid 6-bipyramid 2 l mini26-bipyramid rk R ri l¼1ðRl 2RÞ k¼1
from the CP in real systems, will not keep the n-bipyramid unchanged. Accordingly, the discrepancy between the rotated and initial n-bipyramids can be used to quantify the degree of symmetry for the n-bipyramid. The symmetry parameter (eXrj sub ) of the n-bipyramid (X_sub) extracted from X can be defined as ( 1 X sub pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er j ¼ minR;RSOð3Þ r N g N X sub rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h XN X sub XN g i l X sub X sub 2 min r R r i2X sub i k l¼1ðRl 2RÞ k¼1 ð1Þ where R is a cyclic group generated by an n-fold rotation operator which describes a rotation of 2p/n rad around an axis with a given orientation. Ng is the number of distinct operations in the cyclic group (the unit element E is omitted) and Rl is the lth element (proper rotation) of the Ng distinct operations. In implementing Eq. (1), Rl is substituted by a three-dimensional orthogonal matrix that performs the proper rotation Rl. NX_sub denotes the number of atoms in the sub-n-bipyramid. r is a normalization factor that renders eXrj sub dimensionless and eliminates the volume effect of the bipyramid. Here, r is the circumference divided by the number of edges in the n-bipyramid. The term rkX sub Rl rXi sub in Eq. (1) corresponds to the discrepancy between the initial atomic coordinates and the transformed ones after the application of Rl. When calculating this term, the point closer to each rXk sub is taken into account according to the second minimization in Eq. (1). Oper-
231
The SSP of the CP around j is defined as . X eXrj ¼ erXj sub N sub
ð2Þ
ð3Þ
N sub
The summation runs over all the Nsub sub-n-bipyramids. If the CP of j is an ideal polyhedron, then, according to Eqs. (1) and (3), the individual erXj sub and the total eXrj will all be zero. Any radial and/or orientational deviations from the ideal polyhedron will produce positive SSP. The larger the SSP, the lower the degree of symmetry the CP possesses. In parallel, the SSP measures the extent to which the polyhedron deviates from an ideal one with the same Voronoi index. Such deviation is induced by the distortion of each individual CP, in the form of non-equilateral bond length or angular difference from the characteristic bond angle of the ideal polyhedron. (Throughout the text, the term “bond” corresponds to the line connecting the central and peripheral atoms in the CP.) Obviously, the SSP provides further quantitative information on the level of distortion for each individual CP. This distortion is ubiquitous in MGs and SLs, and varies from site to site. Such information is not available through a conventional Voronoi analysis method. To summarize, a coordinating polyhedron with large SSP is one with a low degree of symmetry and a large distortion, while a low-SSP CP possesses a high degree of symmetry and a small distortion. Instead of applying the symmetry operations to the whole CP, the SSP considers the degree of symmetry for each sub-n-bipyramid. This definition renders the descriptor applicable to all types of polyhedra. Accordingly, an ideal icosahedron and an ideal
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(0, 1, 10, 2) both correspond to zero SSP, although the former possesses more symmetry elements than the later. This definition is consistent with the fact that the SSP is proposed to quantify the distortion of a CP compared to an ideal polyhedron of the same Voronoi index. Comparing the SSPs among different types of CPs can only give qualitatively meaningful results. In the following, the validity of the SSP method is further demonstrated by comparing the angular distortion of icosahedra as quantified by SSP and by the BOO parameter W6 (Section 3.2). 3. Characterizing atomic structure using the SSN method 3.1. Local atomic structure Initially, atomic configurations of the Cu50Zr50 MG and SL are analyzed through the Voronoi method. Fig. 2 shows the fractions of the top 19 most abundant CPs. A non-uniform distribution can be observed. The six dominant types of Cu-centered CPs are (0, 2, 8, 1), (0, 2, 8, 2), (0, 0, 12, 0), (0, 3, 6, 3), (0, 2, 8, 0) and (0, 3, 6, 4) at all temperatures. The Cu atoms surrounded by these polyhedra take more than 60% of the Cu population. These results are in general consistent with past findings [48–50], except that the currently observed fraction of Cu-centered icosahedra is relatively smaller than the one reported previously [48]. As pointed out by Cheng and Ma [6], the icosahedral fraction is particularly sensitive to the cooling rate. Accordingly, the different amount of Cu-centered icosahedra can be attributed to the different cooling rates employed in MD simulations. Previous studies have reported that the Cu-centered CPs can be chosen as the basic structural units in Cu–Zr systems due to their compactness [8] and important role in controlling properties [51]. Therefore, our analysis focuses exclusively on the Cu-centered CPs. However, the corresponding data for the Zr-centered CPs are also readily available. 3.2. The influence of local chemistry on SSP From Eqs. (1) and (3), it can be seen that, in MGs with more than two constituent elements, the SSP depends on
Fig. 2. Population distribution of the 19 most abundant CPs around Cu and Zr atoms in the simulated Cu50Zr50 alloy at 1000, 800 and 300 K.
the local chemistry of the CP and the thermal fluctuations which cause structural variations [6] in the local cluster. Consequently, in this section, the influencing factors of SSP are analyzed taking these two aspects into account. First, we study the correlation between the local chemistry and SSP. Fig. 3 shows the variation of SSP as Zr increases in various Cu-centered CPs in Cu50Zr50 MG. Only the six dominant types of Cu-centered CPs are shown. To eliminate the effect of temperature-induced structural fluctuations, the SSP for a given type of CP (Voronoi index and chemistry) is obtained as a mean value averaged over all the CPs of this type. Fig. 3 further depicts the probability distribution of the local chemistry for various types of CPs. It is clearly seen that, for (0, 2, 8, 1), (0, 2, 8, 2), (0, 3, 6, 3), (0, 2, 8, 0) and (0, 3, 6, 4), the variation of SSP does not show any obvious regularity as the Zr content changes. However, for icosahedra, the SSP initially increases monotonically with increasing Zr content. It reaches a maximum at Cu3Zr9, then decreases in Cu2Zr10. For other CPs not shown in Fig. 3 there is also no clear correlation between the SSP and local chemistry. Fig. 3 further shows vividly that the icosahedron is significantly less distorted than other CPs. From the perspective of geometry, the SSP can be divided into two parts. One is the radial part resulting from the various bond lengths, the other being the angular part corresponding to the deviations from the characteristic bond angles of the perfect polyhedron. In order to explain the trends in Fig. 3, we studied the individual evolution of the radial and angular parts of the SSP with increasing Zr content. The radial part of the SSP (RSSP) is related to the distribution of bond lengths in the CP. The standard deviation (D) of the bond lengths is plotted vs. the local chemistry for various CPs in Fig. 4(a). For (0, 2, 8, 1), (0, 2, 8, 2), (0, 3, 6, 3) and (0, 3, 6, 4), D increases monotonically with increasing Zr. In contrast, for icosahedra and (0, 2, 8, 0), D first increases, reaches a maximum at Zr9, then decreases at Zr10. To explain these different behaviors, the variance (V) of the bond length (to monitor the changing of V as a single bond change from Cu–Cu to Cu–Zr in the CP, V
Fig. 3. Distribution of the local chemistry (bars) for the six most abundant Cu-centered CPs, together with the evolution of the average SSP as a function of the local chemistry for these CPs (lines and balls) at 300 K.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
Fig. 4. (a) The standard deviation, D, of the bond-length distribution in various CPs with different local chemistries in Cu50Zr50 MG. (b) Histograms showing V1,Cu–Cu, V1,Cu–Zr, V2,Cu–Cu and V2,Cu–Zr for various CPs with different chemistries.
is replaced by N V where N is the coordination number of the CP) was divided into four parts: N V ¼
N X 2 ðLi LÞ ¼ V 1 þ V 2 ¼ V 1;Cu–Cu þ V 1;Cu–Zr þ V 2;Cu–Cu þ V 2;Cu–Zr
ð4Þ
i¼1
V 1;Cu–Cu ¼
NX Cu–Cu
2
ðLi;Cu–Cu LCu–Cu Þ
ð5Þ
i¼1 2
V 2;Cu–Cu ¼ N Cu–Cu ðLCu–Cu LÞ V 1;Cu–Zr ¼
NX Cu–Zr i¼1
Li;Cu–Zr LCu–Zr 2
V 2;Cu–Zr ¼ N Cu–Zr ðLCu–Zr LÞ
2
ð6Þ ð7Þ ð8Þ
where Li, Li,X–Y, L and LX –Y (X, Y = Cu, Zr) are the individual bond length, individual X–Y bond length, overall average bond length and average X–Y bond length in the CP, respectively. NX–Y corresponds to the number of X– Y bonds. V1,X–Y is the variance of X–Y bond length, while V2,X–Y describes the difference between the average X–Y bond length and the (overall) average bond length. The evolution of these four contributions, with varied Zr content, is presented in Fig. 4(b). Here only (0, 2, 8, 1), (0, 2, 8, 2), icosahedra along with (0, 2, 8, 0) are shown. For the first two clusters, D increases monotonically with increasing Zr, while for the last two the maximum deviation is observed at Zr9. Fig. 4(b) shows that the increase in D (or V) from Zr4 to Zr9 is mainly due to the increase in V1 (V1,Cu–Cu + V1,Cu–Zr). With increasing Zr, V1,Cu–Cu
233
decreases and V1,Cu–Zr increases simply because more Cu–Zr bonds are involved in the CP. The reduction of V1,Cu–Cu is more than compensated for by the growth in V1,Cu–Zr. This trend demonstrates that, as the composition changes from Zr4 to Zr5, the Cu–Zr bond length changes more than that of Cu–Cu. The net result is an increase in V1 and D as we move from Zr4 to Zr9. Previous studies [52,53] have shown that, in Cu50Zr50 MG, the Cu–Zr bond length possesses greater variability than that of Cu–Cu. In the present study, we find that, on the microscopic scale, the Cu–Zr bond shows significantly greater variability with varying CP composition. For Zr10, the situation is different: the increasing D for (0, 2, 8, 1) and (0, 2, 8, 2) can be attributed to the rising V2,Cu–Cu. As mentioned earlier, V2,Cu–Cu describes the difference between L and LCu–Cu . With increasing Zr, the difference between L and LCu–Zr decreases. The increased V2,Cu–Cu is a direct consequence of the large difference between the Cu–Cu and Cu–Zr bond lengths. Similar to the effect of increasing probability of comfortable local arrangements to reach icosahedra induced by introducing different atoms (different radii) into the system [2], such diversity in bond lengths enhances the ability to construct clusters with certain topological characteristics. According to our data, the contribution of V2,Cu–Cu in the icosahedra and (0, 2, 8, 0) decreases at Zr10. V2,Cu–Cu of Zr10 (0, 2, 8, 0) obviously vanishes in the absence of peripheral Cu atoms. For icosahedra, V2,Cu–Cu also decreases at Zr10, as there are fewer Cu atoms inside compared to Z9 icosahedra, suggesting that the difference between the bond length of Cu–Cu and Cu–Zr is small in Zr10 icosahedra. That is, the bond lengths in Cu2Zr10 icosahedra are more homogeneous than in other clusters at Zr10. This can also be deduced from the fact that the D or V of icosahedra is considerably smaller than other CPs. To summarize, D in (0, 2, 8, 1), (0, 2, 8, 2), (0, 3, 6, 3) and (0, 3, 6, 4) increases monotonically with increasing Zr due to the cooperative effect of V1,Cu–Cu and V2,Cu–Cu. The decreasing D for Cu2Zr10 icosahedra is due to the homogeneous bond length inside the CP. The angular part of the SSP (ASSP) can be obtained by eliminating the radial contribution from the total SSP. We first artificially set the bond lengths in all the sub-bipyramids to be equal to the average bond length of the initial CP. This equalization is implemented by radially adjusting the position vectors of the shell atoms in the CP. After employing Eq. (1) to the “adjusted” sub-bipyramids, the ASSP is calculated by substituting eXrj sub of the adjusted sub-bipyramids to Eq. (3). Fig. 5 visualizes the ASSP dependence on Zr content in the six dominant types of Cu-centered CPs. Similar to the SSP case in Fig. 3, the variation of the ASSP does not show obvious regularity as the Zr content changes except for icosahedra. In the icosahedra, the ASSP shows exactly the same tendency as the SSP (Fig. 3): it increases monotonically from Cu8Zr4 to Cu3Zr9 and decreases for Cu2Zr10. For comparison, the W6 parameter of the central atoms in the icosahedra is
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C. Wu et al. / Acta Materialia 72 (2014) 229–238
Fig. 5. The ASSP for various clusters with different local chemistries (solid lines). Also shown for comparison are the W6 values of icosahedra with the different chemistries (dashed line).
plotted against Zr content in Fig. 5. The W6 parameter closely matches the ASSP in describing the angular distortion for icosahedra at different chemistries. This consistency further demonstrates the validity of the SSP method. The strong relationship between the ASSP and the Zr content for icosahedra can be associated with the variation of D in the icosahedra during the change in composition. As reported previously, D increases from Cu8Zr4 to Cu3Zr9. Accordingly, the angle between the bonds in the icosahedra
becomes more distorted in order to accommodate diverse bond lengths. Cu2Zr10 icosahedra correspond to more homogeneous bond lengths compared to Cu3Zr9 ones, explaining why ASSP decreases at Cu2Zr10. For other types of CPs, the ASSP shows no correlation with D. As mentioned before, clusters, such as (0, 2, 8, 1) and (0, 2, 8, 2), are remarkably more distorted than icosahedra. The angular distortion is insensitive to D in these CPs. For icosahedra, by combining the data of Figs. 4(a) and 5, it is clear that the ASSP depends non-monotonically on D. For example, the ASSP of Cu2Zr10 icosahedra lies between those of Cu7Zr5 and Cu6Zr6, while the D of Cu2Zr10 icosahedra is larger than that of Cu5Zr7 ones. Accordingly, the dependence of ASSP on D in the icosahedra is substantial. In parallel, it can be drawn from Figs. 3 and 5 that the angular contribution is the dominant part of the total SSP. In summary, we find that the icosahedron is a special type of cluster the degree of distortion of which is sensitive to local chemistry (Fig. 3), in contrast to other local structures. The main factor for this singularity is the angular distortion of the icosahedron, which appears to be sensitive to changes in local composition. With increasing Zr content, the variability of bond lengths increases, which in turn affects the angular distortion in the local cluster
Fig. 6. SSP distribution spectra for various clusters in Cu50Zr50 MG and SLs (T = 800, 1000 K). Also shown are the Gaussian fittings of the distribution spectra.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
significantly, as can be seen in Fig. 5. In contrast, the other cluster types are characterized by angular distortions which are considerably larger than that of the icosahedron (Fig. 5). Accordingly, the variation of the bond length diversity produces an effect which is small compared to the total angular distortion. In practice, this contribution is overshadowed by variation of the angular distortion caused by the temperature-induced fluctuation of the local environment. As a result, and in contrast to the trends observed in the icosahedron, both the ASSP (angular distortion) and SSP (total distortion) do not show a systematic trend as the local composition changes in these types of clusters.
235
(Fig. 3). The distributions are broad, indicating that the atomic structures of CPs with the same Voronoi index and local chemistry can be distinctly different from each other. Fig. 6 also shows Gaussian fittings on the SSP distributions. In Table 1, the fitting parameters (the peak value, b, the width of peak, w, and the coefficient of determination, R2) for various types of clusters are also listed. As seen by the R2 values (Table 1), all distributions are very well described by a Gaussian, which is a direct consequence of random thermal fluctuations. Furthermore, with decreasing temperature, the spectra clearly shift towards the lowSSP side. This trend demonstrates that the clusters become less distorted and more symmetric during cooling. At high temperatures, due to enhanced thermal fluctuations, the peripheral atoms can make spatial excursions of larger amplitude, thus increasing the SSP. Consequently, and not surprisingly, the SSP distribution spectrum shifts towards the large-SSP regime at high temperatures. Furthermore, we try to correlate the SSP with the atomic level stress [29,33,34,36,54] exerted on atoms. The atomic level stress at atom i is
3.3. The influence of thermal fluctuation on SSP After exploring the effect of local chemistry, we further study the influence of thermal fluctuations on SRO by applying the SSP descriptor. Fig. 6 shows the SSP distribution spectra of (0, 2, 8, 1), (0, 2, 8, 2) and (0, 0, 12, 0) with varying local chemistry. For each type of CP, only the top three most abundant chemistries are shown since they account for more than 70% of the total number of that type of CP
Table 1 Gaussian fitting parameters of the six dominant types of Cu-centered CPs with varied chemistry in Cu50Zr50 SLs (T = 1000 and 800 K) and MG. Gaussian fitting at 300 K 2
Gaussian fitting at 800 K 2
Gaussian fitting at 1000 K 2
Correlation coefficients at 800 K
P
b
R
P
b
R
P
b
R
l(p, e)
l(s, e)
(0, 2, 8, 1)
Cu6Zr5 Cu5Zr6 Cu4Zr7 Cu3Zr8 Cu2Zr9
0.2771 0.2748 0.2746 0.2699 0.2753
0.0329 0.0416 0.0394 0.0394 0.0379
0.9621 0.9811 0.9792 0.9814 0.9557
0.2871 0.2893 0.2873 0.2846 0.2855
0.0454 0.0423 0.0430 0.0434 0.0412
0.9526 0.9917 0.9818 0.9778 0.9681
0.2953 0.2933 0.2922 0.2924 0.2916
0.0471 0.0468 0.0446 0.0473 0.0441
0.9464 0.9738 0.9767 0.9958 0.9621
0.1827 0.0567 0.0441 0.1812 0.1127
0.6846 0.611 0.7679 0.8192 0.7477
(0, 2, 8, 2)
Cu7Zr5 Cu6Zr6 Cu5Zr7 Cu4Zr8 Cu3Zr9
0.2654 0.2689 0.2714 0.2709 0.2706
0.0403 0.0462 0.0459 0.0449 0.0451
0.9598 0.974 0.9848 0.9843 0.9788
0.2842 0.2856 0.2869 0.2885 0.2914
0.0484 0.0489 0.0462 0.0462 0.0459
0.9718 0.9631 0.9816 0.9771 0.9685
0.2931 0.2962 0.2978 0.2951 0.2919
0.0516 0.0494 0.0498 0.0478 0.0462
0.9623 0.9763 0.9634 0.9736 0.9484
0.154 0.1127 0.1069 0.0067 0.08
0.7236 0.6807 0.6527 0.7247 0.7561
(0, 0, 12, 0)
Cu7Zr5 Cu6Zr6 Cu5Zr7 Cu4Zr8 Cu3Zr9
0.1848 0.1904 0.2018 0.2087 0.2118
0.0410 0.0510 0.0486 0.0478 0.0497
0.9585 0.9821 0.9817 0.9776 0.9442
0.2068 0.2173 0.218 0.2255 0.2277
0.0490 0.0541 0.0526 0.0541 0.0519
0.9659 0.9731 0.9690 0.9787 0.9533
0.2225 0.2303 0.2318 0.2367 0.2417
0.0572 0.0591 0.0575 0.0544 0.0532
0.9478 0.9786 0.9855 0.9679 0.9707
0.1841 0.0955 0.0274 0.1359 0.1981
0.8136 0.7378 0.7557 0.6695 0.708
(0, 3, 6, 3)
Cu7Zr5 Cu6Zr6 Cu5Zr7 Cu4Zr8 Cu3Zr9
0.2905 0.2875 0.2961 0.2901 0.2896
0.0424 0.0479 0.0519 0.0477 0.0497
0.9692 0.9737 0.9783 0.9816 0.9511
0.2976 0.2937 0.2974 0.2933 0.2924
0.0492 0.0498 0.0545 0.0523 0.0544
0.9541 0.9782 0.9655 0.9731 0.9547
0.3122 0.3028 0.3106 0.3008 0.3109
0.0524 0.0547 0.0575 0.0557 0.0552
0.9624 0.9748 0.9622 0.9488 0.9538
0.0274 0.1359 0.043 0.0364 0.0956
0.7529 0.6672 0.795 0.6714 0.6455
(0, 2, 8, 0)
Cu5Zr5 Cu4Zr6 Cu3Zr7 Cu2Zr8 CuZr9
0.2713 0.2604 0.2588 0.2702 0.2629
0.0399 0.0435 0.0407 0.0411 0.0398
0.9728 0.9823 0.9637 0.9591 0.9676
0.2733 0.2711 0.2654 0.2798 0.2691
0.0452 0.0461 0.0436 0.0439 0.0451
0.9773 0.9504 0.9722 0.9654 0.9539
0.2773 0.2765 0.2709 0.2809 0.2793
0.0477 0.0469 0.0454 0.0463 0.046
0.9591 0.9418 0.9617 0.9811 0.9513
0.1101 0.1894 0.0605 0.0821 0.0182
0.748 0.6345 0.6882 0.7564 0.7302
(0, 3, 6, 4)
Cu8Zr5 Cu7Zr6 Cu6Zr7 Cu5Zr8 Cu4Zr9
0.2615 0.2742 0.2739 0.2782 0.2811
0.0431 0.0496 0.0476 0.0507 0.0521
0.9616 0.9728 0.9535 0.9612 0.9528
0.2738 0.2764 0.2787 0.2796 0.2834
0.0457 0.0519 0.0507 0.0519 0.0538
0.9742 0.9618 0.9519 0.9457 0.9533
0.2795 0.2788 0.2804 0.2803 0.2846
0.0474 0.0526 0.0538 0.0574 0.0566
0.9738 0.9586 0.9417 0.9696 0.9643
0.105 0.191 0.099 0.1731 0.0005
0.685 0.7287 0.8455 0.6918 0.6727
For each type of CP, only the five most abundant chemistries are presented, as they account for more than 90% of this type of CP. Also shown in this table are the correlation coefficients l(p, e) and l(s, e) at 800 K.
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C. Wu et al. / Acta Materialia 72 (2014) 229–238
"
# b
1 1 X @E raij rij a b ¼ mi vi vi ; ða; b ¼ x; y; zÞ ð9Þ Xi 2 j;j–i @rij rij " ! # X X 1X E¼ Fi qj ðrij Þ þ / ðrij Þ ð10Þ 2 j;j–i ij i j;j–i rab i
Here, m and v are mass and velocity, respectively, and rij is the distance between atoms i and j. Xi is the volume of the VP. E is the potential energy in the EAM formulation. For MGs and SLs which are macroscopically isotropic, it is desirable to use two rotational invariants, p and s, p ¼ ðr1 þ r2 þ r3 Þ=3
12 1 1 1 1 2 2 2 s¼ ðr1 r2 Þ þ ðr2 r3 Þ þ ðr3 r1 Þ 3 2 2 2
ð11Þ ð12Þ
where r1, r2 and r3 are the three principal stresses. p is the atomic level pressure, while s is the deviatoric stress or atomic level von Mises shear stress. To correlate the SSP (e) with the atomic level stress, the correlation coefficients l of (p, e) and (s, e) for the J type of cluster (with a number of NJ clusters) are calculated as X lJ ðp; eÞ ¼ ðpi pJ Þðei eJ Þ i2J
X lJ ðs; eÞ ¼ ðsi sJ Þðei eJ Þ i2J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,v #" #, u" X u X t N2 ðp p Þ2 ðe e Þ2 J
ð13Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,v #" #, ffi u" X u X 2 2 t ðs s Þ ðe e Þ N2
ð14Þ
J
i
i2J
i
i2J
i
J
i2J
J
i
J
J
i2J
where pi, si and ei correspond to cluster i (i 2 J ), for various CPs in Cu50Zr50 alloy at 800 K. The results are listed in Table 1. Clearly, for clusters with identical Voronoi indices and chemistries, s and e show strong correlation, while l(p, e) is small. This correlation demonstrates that the clusters with large e are likely to be associated with large s. In other words, the central atoms of the clusters with large distortion (low degree of symmetry) tend to be in a state of large local shear stress. This finding is a validation, from the geometrical perspective of atomic detail, of previous arguments that the local shear stress originates from the deviation of local environment from spherical symmetry [36,54].
In this work, we focus on the short-time atomic dynamics, that is, the b relaxation. To characterize the heterogeneous dynamics of atoms, centered at different clusters, we calculate the propensity of motion [56] for Cu atoms in Cu50Zr50 alloy at 1000 K. To this end, 1000 independent simulations in the isoconfigurational ensemble [56] were conducted. The length of the time interval was chosen as the time at which the nonGaussian parameter [57] reaches the maximum. This time interval is sufficiently long, as it corresponds to the late part of the b relaxation, for the system to reach heterogeneous dynamics and simultaneously to probe the effect of the initial structure on dynamics [48,58]. By using the method reported in previous studies [48,55], Cu atoms are sorted and divided into 20 groups according to their mobility, with 5% Cu atoms in each group. The local structure of each group is categorized according to its Voronoi index. The results are listed in Fig. 7. The most prominent feature of Fig. 7 is that icosahedra play an important role in dynamical heterogeneity. About 33% of the slowest 5% Cu atoms are surrounded by icosahedra. This feature is consistent with previous studies [59–64], and demonstrates the important role of icosahedra and their spatial correlation in the dynamical slowing down. Another important feature of Fig. 7 is the wide distribution of various types of clusters in all groups. For example, the (0, 2, 8, 1) cluster tends to be evenly distributed in all 20 groups. This wide distribution reveals the profound effect of structural heterogeneity on dynamics. To characterize this heterogeneous structure and further correlate it with the DH, Fig. 8 presents the average SSP of Cu atoms centered in various types of clusters in each of the 20 groups obtained above. Albeit with statistical noise, the overall trend is that the average SSP increases systematically from a low- to a high-mobility range. This result demonstrates the strong correlation between the SSP and atomic dynamics, that is, large SSP tends to be accompanied by large mobility.
4. Correlating SSP with the short-time dynamics of atoms in supercooled liquids The dynamics of atoms is strongly influenced by the structure of the corresponding coordination clusters. Due to the heterogeneous structure of these clusters, atoms in SLs show highly heterogeneous mobility. Although this dynamical heterogeneity (DH) in atomic motion plays a crucial role in glass formation, its microscopic origins are still not fully understood [6,22,55]. The atomic mobility and related DH in SLs can be analyzed from the perspective of the local potential energy surface or the atomic level stress [6,36]. Considering the correlation between SSP and atomic level shear stress, it is reasonable to attempt a connection between the new descriptor and particle dynamics.
Fig. 7. Correlation between the atomic propensity of motion and topological structure of Cu-centered CPs in Cu50Zr50 SL at 1000 K. Cu atoms are sorted according to their mobility and evenly divided into 20 groups. From left to right, all groups are arranged in increasing order of atomic mobility. Only the fractions of the six most abundant clusters from Voronoi indices in each group are presented.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
237
Table 2 Correlation between the cluster lifetime and SSP for (0, 2, 8, 1), (0, 2, 8, 2) and (0, 0, 12, 0) clusters with different chemistries in Cu50Zr50 SL at 1000 K. Group ID
Average scaled cluster lifetime (0, 2, 8, 1)
Fig. 8. Color map showing the average SSP for various types of Cucentered CPs in each of the 20 groups obtained in Fig. 7. The groups are arranged vertically in an increasing order of atomic mobility from the bottom to the top. The color shows the SSP value for each data point. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
As mentioned above, for clusters with the same topology and chemistry, the central atoms of the clusters with large SSP bear a large shear stress. Previous studies [36,54,65] have found that, in MGs, the atomic level pressure depends predominantly on the topology and chemistry of local environment. Accordingly, for clusters with specific Voronoi index and chemistry, a large SSP corresponds to a large internal stress on the central atoms, making these atoms “liquid-like”, or ones which are moving faster than the corresponding atoms of low SSP. The same conclusion can be drawn from the perspective of a local potential energy surface (PES). A direct and robust way to correlate local PES with atomic structural motifs is to calculate the cluster lifetime [9,66]. Central atoms of clusters with a long lifetime tend to be arrested in deep energy minima, while short lifetimes correspond to shallow minima [66]. In order to explore the relation between the SSP and atomic dynamics, we further investigate the correlation between the cluster lifetime and the SSP. The lifetime of a cluster is defined as the period during which its Voronoi index remains unchanged [9]. Practically, the lifetime of a cluster with a given Voronoi index, Dt, is calculated as t t0, where t0 and t are the times at which this cluster first appears and then vanishes, respectively, in the MD simulation. Consequently, we studied the correlation between the SSP of the cluster at t0 (denoted the “initial cluster” in the following text) and the lifetime Dt of this cluster. First, the initial clusters were screened out from the whole MD trajectory. Then, for the initial clusters with the same Voronoi index and chemistry, the central atoms were divided into 10 groups of equal size, according to their SSP values. They were labeled in ascending order, with smaller numbered groups corresponding to smaller SSPs. The average lifetime for each group was then calculated, as reported in Table 2. Within statistical fluctuations, the general trend is that, for clusters of specific Voronoi index and chemistry, the lifetime decreases with increasing SSP. This suggests that, for any given cluster type, clusters with large SSP tend to be short-
1 2 3 4 5 6 7 8 9 10
(0, 2, 8, 2)
(0, 0, 12, 0)
Zr6
Zr7
Zr8
Zr6
Zr7
Zr8
Zr6
Zr7
Zr8
2.46 2.11 2.09 1.87 1.88 1.52 1.65 1.42 1.35 1.24
2.48 2.26 2.14 2.17 1.92 1.82 1.65 1.76 1.44 1.32
2.49 2.22 2.09 2.08 1.93 1.87 1.74 1.72 1.44 1.49
2.03 1.97 1.87 1.72 1.77 1.55 1.49 1.34 1.22 1.11
2.07 1.87 1.79 1.72 1.51 1.66 1.45 1.21 1.28 1.10
2.06 1.72 1.79 1.62 1.59 1.49 1.61 1.42 1.38 1.12
4.33 4.18 3.91 3.73 3.76 3.62 3.51 3.53 3.35 3.28
4.29 4.12 3.90 3.99 3.79 3.58 3.45 3.38 3.20 3.32
4.37 4.24 3.98 3.68 3.81 3.73 3.52 3.36 3.39 3.25
Here the scaled cluster lifetime is used, which is obtained by dividing the lifetime (in unit of MD steps) of a cluster by the average cluster lifetime of P the SL. The average lifetime is defined as i pðiÞtðiÞ where p(i) and t(i) are the population fraction and lifetime of the i-type cluster; the summation is over all types of clusters.
lived. In other words, the central atoms of large-SSP clusters are trapped in the shallow local basins of local PES while small-SSP clusters are associated with deep local energy minima. High potential energy barriers have to be overcome for the central atoms of low-SSP clusters to escape the cage. Thus, they reside “trapped” for extended time periods, practically rattling in the cages, before escaping from the cluster, compared to the central atoms of large-SSP clusters. This long cage-rattling process postpones the onset of the diffusive behavior for the central atoms of low-SSPs. Therefore, the central atoms of low-SSP clusters show lower mobility than those of high-SSP clusters. From the analysis above, the degree of symmetry or distortion of local clusters, which is quantified in the present work by the SSP, shows the influence of atomic structure on dynamics vividly. Note that previous studies [48,55,67–69] found that the local topology and chemistry play an important role in the heterogeneous dynamics of SLs. These findings, together with our results, demonstrate the multifaceted relationship between the local atomic structure and the dynamics of supercooled liquids. 5. Conclusions In this work, we have introduced and employed a new descriptor of local structure – the short-range-order symmetry parameter (SSP) method – to study the salient characteristics of short-range order in metallic glasses and supercooled liquids. By calculating the degree of symmetry for each cluster (mapped into symmetry parameter, SSP), we quantify the degree of distortion, both radially and orientationally, of an individual cluster from the ideal one. From the present analysis, the local structure of an icosahedron, as quantified by the new metric, appears to be sensitive to changes in local chemistry. This structural
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characteristic of icosahedra is not encountered in other polyhedral cluster types. The SSP can be correlated with the atomic level shear stress, the local potential energy surface and successively with dynamical properties (atomic mobility). These correlations indicate that the SSP is an important structural identifier that can accurately quantify the structure of SRO and shed light on the microstructure–property relationship. By combing existing structural analysis methods with the newly introduced metric a detailed and accurate description of short-range order in complex atomic systems can be obtained. Acknowledgements This work was financially supported by the National Nature Science Foundation of China under Grant No. 51025415. The authors acknowledge Dr. Daniel Miracle from Air Force Research Laboratory, Ohio, USA, for fruitful discussions. NCK acknowledges support by the Spanish Ministry of Economy and Competitiveness (MINECO) through projects “Ramon y Cajal” (RYC2009-05413), “I3” and MAT2011-24834.
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A metric to gauge local distortion in metallic glasses and supercooled liquids Chen Wu a, Nikos Ch. Karayiannis b, Manuel Laso b, Dongdong Qu a, Qiang Luo c, Jun Shen c,⇑ b
a School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China Institute for Optoelectronics and Microsystems (ISOM) and ETSII, Polytechnic University of Madrid (UPM), Madrid 28006, Spain c School of Materials Science and Engineering, Tongji University, Shanghai 201804, People’s Republic of China
Received 9 February 2014; received in revised form 18 March 2014; accepted 19 March 2014 Available online 23 April 2014
Abstract We propose a new structural descriptor to characterize distortion and short-range order in metallic glasses and supercooled liquids. Through the new metric, salient structural differences in local clusters of identical Voronoi indices can be accurately identified. As a test case, we employ the short-range-order symmetry parameter (SSP) in atomistic configurations of Cu50Zr50 generated through extensive molecular dynamics simulations at various temperatures. Structural analysis reveals the dependence of short-range order on local chemistry and thermal fluctuations. Furthermore, the new descriptor can be directly correlated with the atomic level shear stress, the local potential energy surface and the dynamical heterogeneity. Combined with existing analytical tools, the SSP can provide significant information, at the atomic level, on the structure–property relation in metallic glasses and in general complex particulate systems. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Metallic glasses; Supercooled liquids; Molecular dynamics simulation; Symmetry and distortion of short-range order; Dynamics of atoms
1. Introduction The identification and successive analysis of atomic structure in metallic glasses (MGs) and supercooled liquids (SLs) is a formidable scientific challenge which is attracting constant interest [1–6]. During the last few years, numerous studies [7–18] have provided significant information on this subject. In parallel, groundbreaking atomistic models [19–21] have been proposed to describe the atomic structure in such complex systems. However, key factors that dictate the correlation between atomic structure and properties remain elusive. For example, due to the non-equilibrium nature of MGs, many studies focus on the formation ⇑ Corresponding author.
E-mail address: [email protected] (J. Shen).
process, that is, supercooling of liquids below their melting temperatures. Although spatial dynamic fluctuations play the central role in modern description of supercooled liquids, their dependence on local structure is not yet fully understood. [22] Therefore, a detailed description of the atomic structure of MGs and SLs is highly desirable. Recent studies [6,11] have shown that the local environment in MGs and SLs does possess short-range order (SRO). Nowadays, scientists commonly use the bond orientation order (BOO) parameter [23], the Voronoi analysis [24] and the common neighbor analysis [25–27] methods to characterize SRO. By quantifying the angular distortion of local structure, the BOO parameter identifies the type of perfect order that the local structure resembles. However, radial distortion from perfect order is ubiquitous in glassy systems and the extraction of structural
http://dx.doi.org/10.1016/j.actamat.2014.03.046 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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C. Wu et al. / Acta Materialia 72 (2014) 229–238
information through the BOO parameters is difficult in MGs and supercooled liquids [28]. The Voronoi tessellation method [24] generates the Voronoi polyhedron (VP), which contains the space closer to a given atom than to any other atom. SRO is characterized by the Voronoi indices (n3, n4, n5, n6) where ni stands for the number of i-edged faces. However, many researchers have pointed out that the Voronoi indices do not provide a complete description of SRO [28–30]. Srolovitz et al. [29] showed that, in order to quantify SRO, additional information on the distortion of individual clusters with identical Voronoi indices is required. The common neighbor analysis (CNA) method identifies the structural unit, the so-called n-bipyramid [30], formed by a pair of nearest neighbor atoms and their n common neighbors. Miracle [30] reported that the efficient filling of space can be effectively explored by taking the n-bipyramids as the basic structural units. Furthermore, n-bipyramids can be related to plastic deformation [31]. However, these n-bipyramids might be severely distorted, and the degree of distortion, which varies from site to site, cannot be fully quantified by the CNA method. Often, n-fold bipyramids are denoted as “n-fold symmetry”, although there is no real symmetry in these configurations. Instead, the notation “n-fold symmetry” identifies these bipyramids as n-bipyramids. Nevertheless, this characterization has motivated us to quantify the distortion of n-bipyramids by calculating the degree of rotational symmetry. Lowsymmetry bipyramids can be effectively viewed as highdistortion bipyramids compared to ideal ones. It is well established that the degree of symmetry in SRO strongly influences the glass-forming ability of MGs [32]. The discussion above emphasizes the need to introduce a novel structural indicator which can quantify distortion by measuring the degree of symmetry. Srolovitz et al. [29] analyzed the degree of symmetry in SRO by employing a set of parameters inspired by the atomic level stress theory. This theory, proposed by Egami et al. [33–36], provides a clear physical interpretation of glass formation and mechanical behavior in MGs. However, finding structural descriptors, which can be correlated with the atomic level stresses, remains an open research field [6,37]. Thus, a refined structural analysis tool is required to measure the degree of symmetry in SRO accurately. A combination of the new descriptor with an existing structural indicator (i.e. the Voronoi index) could provide a better understanding of SRO. In this work, we propose the SRO symmetry parameter (SSP) as a way to quantify the degree of symmetry and distortion in local structure, providing information which is untraceable by conventional methods. Section 2 describes the molecular dynamics (MD) simulations used for the generation of the samples in detail and provides the SSP formalism. Results from the structural analysis based on the SSP are presented and discussed in Section 3. The correlation between the SSP and the dynamics of atoms in supercooled liquids is analyzed in Section 4, further demonstrating the usefulness of SSP as a detailed
descriptor of local order. Finally, the main results and conclusions are summarized in Section 5. 2. Simulation details and the SSP 2.1. Molecular dynamics In the present work, atomic structures of Cu50Zr50 were generated through LAMMPS [38] with the embeddedatom method (EAM) potential [39,40]. A cubic simulation box containing 10,000 atoms, with periodic boundary conditions applied in all dimensions, was equilibrated through isothermal–isobaric (NPT) simulations (P = 0 atm, T = 2000 K) for 2 ns. Pressure and temperature oscillations were controlled through a Nose´–Hoover barostat and thermostat, respectively. In the continuation, melt configurations were cooled down to the target temperatures (T = 1000, 800 and 300 K) at a rate of 1012 K s1. In all cases the time integration step was set at 1 fs. According to literature data, the liquidus temperature for Cu50Zr50 MG is approximately 1150 K [41]. 2.2. The SSP In this section, we give a detailed account of the SRO symmetry norm method and the corresponding parameter (SSP). The new descriptor is inspired by the characteristic crystallographic element norm method [42–47]. The SSP of a given atom is defined by the degree of rotational symmetry for the bipyramids extracted from the coordinating polyhedron (CP) of the atom. We consider a configuration of atoms, denoted by X, consisting of the reference atom j (with position vector rj) and its nearest neighbors (with position vectors rk, k e X, in an arbitrary reference frame) determined by the Voronoi tessellation method. To obtain the SSP of atom j, the coordinating polyhedron is first divided into Nsub n-bipyramids, where Nsub is the number of nearest neighbors of atom j. Fig. 1(a) depicts a Cu-centered (0, 1, 10, 2) CP in the simulated Cu50Zr50 MG, while Fig. 1(b) shows a hexagonal bipyramid (6-bipyramid), a pentagonal bipyramid (5-bipyramid) and
Fig. 1. (a) A Cu-centered (0, 1, 10, 2) CP in Cu50Zr50 MG and (b) representative sub-bipyramids extracted from this CP.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
a square bipyramid (4-bipyramid) extracted from the parent (0, 1, 10, 2) CP of Fig. 1(a). In general, all types of polyhedra can be divided into subbipyramids through the Voronoi tessellation. A CP with Voronoi index (n3, n4, n5, n6) can be divided into n3 3-, n4 4-, n5 5- and n6 6-bipyramids. If we suppose that the coordinating polyhedron around j is an ideal, perfect polyhedron with equilateral edges, then each of the n-bipyramids possesses rotational symmetry (an n-fold rotational axis penetrates the axial sites for every n-bipyramid). However, in real systems, deviations from ideality are expected: the n-bipyramids do not possess rotational symmetry due to intrinsic fluctuations and various arrangements of different constituent atoms. Thus, rotating an n-bipyramid, obtained
erXj 6-bipyramid
¼ minR;RSOð3Þ
1 pffiffiffiffiffiffiffiffiffi r 57
ationally, for the n-bipyramid of j, we first randomly choose an initial orientation of the n-fold rotation axis passing through rj. The Ng distinct proper operations are then applied to the n-bipyramid. After each application, the discrepancy is calculated as in Eq. (1). This procedure is repeated by searching for the orientation of the rotation axis in the two-parameter space of azimuthal and polar angles so as to find the minimum discrepancy, which is then set to be erXj sub . This step corresponds to the first minimization in Eq. (1). An illustrative example is given for the 6-bipyramid, where the 6-fold axis defines Ng = 5 different elements ({R = 61, 62, 63, 64 and 65}). Eq. (1) can then be written as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h X7 X5 i 6-bipyramid 6-bipyramid 2 l mini26-bipyramid rk R ri l¼1ðRl 2RÞ k¼1
from the CP in real systems, will not keep the n-bipyramid unchanged. Accordingly, the discrepancy between the rotated and initial n-bipyramids can be used to quantify the degree of symmetry for the n-bipyramid. The symmetry parameter (eXrj sub ) of the n-bipyramid (X_sub) extracted from X can be defined as ( 1 X sub pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi er j ¼ minR;RSOð3Þ r N g N X sub rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h XN X sub XN g i l X sub X sub 2 min r R r i2X sub i k l¼1ðRl 2RÞ k¼1 ð1Þ where R is a cyclic group generated by an n-fold rotation operator which describes a rotation of 2p/n rad around an axis with a given orientation. Ng is the number of distinct operations in the cyclic group (the unit element E is omitted) and Rl is the lth element (proper rotation) of the Ng distinct operations. In implementing Eq. (1), Rl is substituted by a three-dimensional orthogonal matrix that performs the proper rotation Rl. NX_sub denotes the number of atoms in the sub-n-bipyramid. r is a normalization factor that renders eXrj sub dimensionless and eliminates the volume effect of the bipyramid. Here, r is the circumference divided by the number of edges in the n-bipyramid. The term rkX sub Rl rXi sub in Eq. (1) corresponds to the discrepancy between the initial atomic coordinates and the transformed ones after the application of Rl. When calculating this term, the point closer to each rXk sub is taken into account according to the second minimization in Eq. (1). Oper-
231
The SSP of the CP around j is defined as . X eXrj ¼ erXj sub N sub
ð2Þ
ð3Þ
N sub
The summation runs over all the Nsub sub-n-bipyramids. If the CP of j is an ideal polyhedron, then, according to Eqs. (1) and (3), the individual erXj sub and the total eXrj will all be zero. Any radial and/or orientational deviations from the ideal polyhedron will produce positive SSP. The larger the SSP, the lower the degree of symmetry the CP possesses. In parallel, the SSP measures the extent to which the polyhedron deviates from an ideal one with the same Voronoi index. Such deviation is induced by the distortion of each individual CP, in the form of non-equilateral bond length or angular difference from the characteristic bond angle of the ideal polyhedron. (Throughout the text, the term “bond” corresponds to the line connecting the central and peripheral atoms in the CP.) Obviously, the SSP provides further quantitative information on the level of distortion for each individual CP. This distortion is ubiquitous in MGs and SLs, and varies from site to site. Such information is not available through a conventional Voronoi analysis method. To summarize, a coordinating polyhedron with large SSP is one with a low degree of symmetry and a large distortion, while a low-SSP CP possesses a high degree of symmetry and a small distortion. Instead of applying the symmetry operations to the whole CP, the SSP considers the degree of symmetry for each sub-n-bipyramid. This definition renders the descriptor applicable to all types of polyhedra. Accordingly, an ideal icosahedron and an ideal
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(0, 1, 10, 2) both correspond to zero SSP, although the former possesses more symmetry elements than the later. This definition is consistent with the fact that the SSP is proposed to quantify the distortion of a CP compared to an ideal polyhedron of the same Voronoi index. Comparing the SSPs among different types of CPs can only give qualitatively meaningful results. In the following, the validity of the SSP method is further demonstrated by comparing the angular distortion of icosahedra as quantified by SSP and by the BOO parameter W6 (Section 3.2). 3. Characterizing atomic structure using the SSN method 3.1. Local atomic structure Initially, atomic configurations of the Cu50Zr50 MG and SL are analyzed through the Voronoi method. Fig. 2 shows the fractions of the top 19 most abundant CPs. A non-uniform distribution can be observed. The six dominant types of Cu-centered CPs are (0, 2, 8, 1), (0, 2, 8, 2), (0, 0, 12, 0), (0, 3, 6, 3), (0, 2, 8, 0) and (0, 3, 6, 4) at all temperatures. The Cu atoms surrounded by these polyhedra take more than 60% of the Cu population. These results are in general consistent with past findings [48–50], except that the currently observed fraction of Cu-centered icosahedra is relatively smaller than the one reported previously [48]. As pointed out by Cheng and Ma [6], the icosahedral fraction is particularly sensitive to the cooling rate. Accordingly, the different amount of Cu-centered icosahedra can be attributed to the different cooling rates employed in MD simulations. Previous studies have reported that the Cu-centered CPs can be chosen as the basic structural units in Cu–Zr systems due to their compactness [8] and important role in controlling properties [51]. Therefore, our analysis focuses exclusively on the Cu-centered CPs. However, the corresponding data for the Zr-centered CPs are also readily available. 3.2. The influence of local chemistry on SSP From Eqs. (1) and (3), it can be seen that, in MGs with more than two constituent elements, the SSP depends on
Fig. 2. Population distribution of the 19 most abundant CPs around Cu and Zr atoms in the simulated Cu50Zr50 alloy at 1000, 800 and 300 K.
the local chemistry of the CP and the thermal fluctuations which cause structural variations [6] in the local cluster. Consequently, in this section, the influencing factors of SSP are analyzed taking these two aspects into account. First, we study the correlation between the local chemistry and SSP. Fig. 3 shows the variation of SSP as Zr increases in various Cu-centered CPs in Cu50Zr50 MG. Only the six dominant types of Cu-centered CPs are shown. To eliminate the effect of temperature-induced structural fluctuations, the SSP for a given type of CP (Voronoi index and chemistry) is obtained as a mean value averaged over all the CPs of this type. Fig. 3 further depicts the probability distribution of the local chemistry for various types of CPs. It is clearly seen that, for (0, 2, 8, 1), (0, 2, 8, 2), (0, 3, 6, 3), (0, 2, 8, 0) and (0, 3, 6, 4), the variation of SSP does not show any obvious regularity as the Zr content changes. However, for icosahedra, the SSP initially increases monotonically with increasing Zr content. It reaches a maximum at Cu3Zr9, then decreases in Cu2Zr10. For other CPs not shown in Fig. 3 there is also no clear correlation between the SSP and local chemistry. Fig. 3 further shows vividly that the icosahedron is significantly less distorted than other CPs. From the perspective of geometry, the SSP can be divided into two parts. One is the radial part resulting from the various bond lengths, the other being the angular part corresponding to the deviations from the characteristic bond angles of the perfect polyhedron. In order to explain the trends in Fig. 3, we studied the individual evolution of the radial and angular parts of the SSP with increasing Zr content. The radial part of the SSP (RSSP) is related to the distribution of bond lengths in the CP. The standard deviation (D) of the bond lengths is plotted vs. the local chemistry for various CPs in Fig. 4(a). For (0, 2, 8, 1), (0, 2, 8, 2), (0, 3, 6, 3) and (0, 3, 6, 4), D increases monotonically with increasing Zr. In contrast, for icosahedra and (0, 2, 8, 0), D first increases, reaches a maximum at Zr9, then decreases at Zr10. To explain these different behaviors, the variance (V) of the bond length (to monitor the changing of V as a single bond change from Cu–Cu to Cu–Zr in the CP, V
Fig. 3. Distribution of the local chemistry (bars) for the six most abundant Cu-centered CPs, together with the evolution of the average SSP as a function of the local chemistry for these CPs (lines and balls) at 300 K.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
Fig. 4. (a) The standard deviation, D, of the bond-length distribution in various CPs with different local chemistries in Cu50Zr50 MG. (b) Histograms showing V1,Cu–Cu, V1,Cu–Zr, V2,Cu–Cu and V2,Cu–Zr for various CPs with different chemistries.
is replaced by N V where N is the coordination number of the CP) was divided into four parts: N V ¼
N X 2 ðLi LÞ ¼ V 1 þ V 2 ¼ V 1;Cu–Cu þ V 1;Cu–Zr þ V 2;Cu–Cu þ V 2;Cu–Zr
ð4Þ
i¼1
V 1;Cu–Cu ¼
NX Cu–Cu
2
ðLi;Cu–Cu LCu–Cu Þ
ð5Þ
i¼1 2
V 2;Cu–Cu ¼ N Cu–Cu ðLCu–Cu LÞ V 1;Cu–Zr ¼
NX Cu–Zr i¼1
Li;Cu–Zr LCu–Zr 2
V 2;Cu–Zr ¼ N Cu–Zr ðLCu–Zr LÞ
2
ð6Þ ð7Þ ð8Þ
where Li, Li,X–Y, L and LX –Y (X, Y = Cu, Zr) are the individual bond length, individual X–Y bond length, overall average bond length and average X–Y bond length in the CP, respectively. NX–Y corresponds to the number of X– Y bonds. V1,X–Y is the variance of X–Y bond length, while V2,X–Y describes the difference between the average X–Y bond length and the (overall) average bond length. The evolution of these four contributions, with varied Zr content, is presented in Fig. 4(b). Here only (0, 2, 8, 1), (0, 2, 8, 2), icosahedra along with (0, 2, 8, 0) are shown. For the first two clusters, D increases monotonically with increasing Zr, while for the last two the maximum deviation is observed at Zr9. Fig. 4(b) shows that the increase in D (or V) from Zr4 to Zr9 is mainly due to the increase in V1 (V1,Cu–Cu + V1,Cu–Zr). With increasing Zr, V1,Cu–Cu
233
decreases and V1,Cu–Zr increases simply because more Cu–Zr bonds are involved in the CP. The reduction of V1,Cu–Cu is more than compensated for by the growth in V1,Cu–Zr. This trend demonstrates that, as the composition changes from Zr4 to Zr5, the Cu–Zr bond length changes more than that of Cu–Cu. The net result is an increase in V1 and D as we move from Zr4 to Zr9. Previous studies [52,53] have shown that, in Cu50Zr50 MG, the Cu–Zr bond length possesses greater variability than that of Cu–Cu. In the present study, we find that, on the microscopic scale, the Cu–Zr bond shows significantly greater variability with varying CP composition. For Zr10, the situation is different: the increasing D for (0, 2, 8, 1) and (0, 2, 8, 2) can be attributed to the rising V2,Cu–Cu. As mentioned earlier, V2,Cu–Cu describes the difference between L and LCu–Cu . With increasing Zr, the difference between L and LCu–Zr decreases. The increased V2,Cu–Cu is a direct consequence of the large difference between the Cu–Cu and Cu–Zr bond lengths. Similar to the effect of increasing probability of comfortable local arrangements to reach icosahedra induced by introducing different atoms (different radii) into the system [2], such diversity in bond lengths enhances the ability to construct clusters with certain topological characteristics. According to our data, the contribution of V2,Cu–Cu in the icosahedra and (0, 2, 8, 0) decreases at Zr10. V2,Cu–Cu of Zr10 (0, 2, 8, 0) obviously vanishes in the absence of peripheral Cu atoms. For icosahedra, V2,Cu–Cu also decreases at Zr10, as there are fewer Cu atoms inside compared to Z9 icosahedra, suggesting that the difference between the bond length of Cu–Cu and Cu–Zr is small in Zr10 icosahedra. That is, the bond lengths in Cu2Zr10 icosahedra are more homogeneous than in other clusters at Zr10. This can also be deduced from the fact that the D or V of icosahedra is considerably smaller than other CPs. To summarize, D in (0, 2, 8, 1), (0, 2, 8, 2), (0, 3, 6, 3) and (0, 3, 6, 4) increases monotonically with increasing Zr due to the cooperative effect of V1,Cu–Cu and V2,Cu–Cu. The decreasing D for Cu2Zr10 icosahedra is due to the homogeneous bond length inside the CP. The angular part of the SSP (ASSP) can be obtained by eliminating the radial contribution from the total SSP. We first artificially set the bond lengths in all the sub-bipyramids to be equal to the average bond length of the initial CP. This equalization is implemented by radially adjusting the position vectors of the shell atoms in the CP. After employing Eq. (1) to the “adjusted” sub-bipyramids, the ASSP is calculated by substituting eXrj sub of the adjusted sub-bipyramids to Eq. (3). Fig. 5 visualizes the ASSP dependence on Zr content in the six dominant types of Cu-centered CPs. Similar to the SSP case in Fig. 3, the variation of the ASSP does not show obvious regularity as the Zr content changes except for icosahedra. In the icosahedra, the ASSP shows exactly the same tendency as the SSP (Fig. 3): it increases monotonically from Cu8Zr4 to Cu3Zr9 and decreases for Cu2Zr10. For comparison, the W6 parameter of the central atoms in the icosahedra is
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C. Wu et al. / Acta Materialia 72 (2014) 229–238
Fig. 5. The ASSP for various clusters with different local chemistries (solid lines). Also shown for comparison are the W6 values of icosahedra with the different chemistries (dashed line).
plotted against Zr content in Fig. 5. The W6 parameter closely matches the ASSP in describing the angular distortion for icosahedra at different chemistries. This consistency further demonstrates the validity of the SSP method. The strong relationship between the ASSP and the Zr content for icosahedra can be associated with the variation of D in the icosahedra during the change in composition. As reported previously, D increases from Cu8Zr4 to Cu3Zr9. Accordingly, the angle between the bonds in the icosahedra
becomes more distorted in order to accommodate diverse bond lengths. Cu2Zr10 icosahedra correspond to more homogeneous bond lengths compared to Cu3Zr9 ones, explaining why ASSP decreases at Cu2Zr10. For other types of CPs, the ASSP shows no correlation with D. As mentioned before, clusters, such as (0, 2, 8, 1) and (0, 2, 8, 2), are remarkably more distorted than icosahedra. The angular distortion is insensitive to D in these CPs. For icosahedra, by combining the data of Figs. 4(a) and 5, it is clear that the ASSP depends non-monotonically on D. For example, the ASSP of Cu2Zr10 icosahedra lies between those of Cu7Zr5 and Cu6Zr6, while the D of Cu2Zr10 icosahedra is larger than that of Cu5Zr7 ones. Accordingly, the dependence of ASSP on D in the icosahedra is substantial. In parallel, it can be drawn from Figs. 3 and 5 that the angular contribution is the dominant part of the total SSP. In summary, we find that the icosahedron is a special type of cluster the degree of distortion of which is sensitive to local chemistry (Fig. 3), in contrast to other local structures. The main factor for this singularity is the angular distortion of the icosahedron, which appears to be sensitive to changes in local composition. With increasing Zr content, the variability of bond lengths increases, which in turn affects the angular distortion in the local cluster
Fig. 6. SSP distribution spectra for various clusters in Cu50Zr50 MG and SLs (T = 800, 1000 K). Also shown are the Gaussian fittings of the distribution spectra.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
significantly, as can be seen in Fig. 5. In contrast, the other cluster types are characterized by angular distortions which are considerably larger than that of the icosahedron (Fig. 5). Accordingly, the variation of the bond length diversity produces an effect which is small compared to the total angular distortion. In practice, this contribution is overshadowed by variation of the angular distortion caused by the temperature-induced fluctuation of the local environment. As a result, and in contrast to the trends observed in the icosahedron, both the ASSP (angular distortion) and SSP (total distortion) do not show a systematic trend as the local composition changes in these types of clusters.
235
(Fig. 3). The distributions are broad, indicating that the atomic structures of CPs with the same Voronoi index and local chemistry can be distinctly different from each other. Fig. 6 also shows Gaussian fittings on the SSP distributions. In Table 1, the fitting parameters (the peak value, b, the width of peak, w, and the coefficient of determination, R2) for various types of clusters are also listed. As seen by the R2 values (Table 1), all distributions are very well described by a Gaussian, which is a direct consequence of random thermal fluctuations. Furthermore, with decreasing temperature, the spectra clearly shift towards the lowSSP side. This trend demonstrates that the clusters become less distorted and more symmetric during cooling. At high temperatures, due to enhanced thermal fluctuations, the peripheral atoms can make spatial excursions of larger amplitude, thus increasing the SSP. Consequently, and not surprisingly, the SSP distribution spectrum shifts towards the large-SSP regime at high temperatures. Furthermore, we try to correlate the SSP with the atomic level stress [29,33,34,36,54] exerted on atoms. The atomic level stress at atom i is
3.3. The influence of thermal fluctuation on SSP After exploring the effect of local chemistry, we further study the influence of thermal fluctuations on SRO by applying the SSP descriptor. Fig. 6 shows the SSP distribution spectra of (0, 2, 8, 1), (0, 2, 8, 2) and (0, 0, 12, 0) with varying local chemistry. For each type of CP, only the top three most abundant chemistries are shown since they account for more than 70% of the total number of that type of CP
Table 1 Gaussian fitting parameters of the six dominant types of Cu-centered CPs with varied chemistry in Cu50Zr50 SLs (T = 1000 and 800 K) and MG. Gaussian fitting at 300 K 2
Gaussian fitting at 800 K 2
Gaussian fitting at 1000 K 2
Correlation coefficients at 800 K
P
b
R
P
b
R
P
b
R
l(p, e)
l(s, e)
(0, 2, 8, 1)
Cu6Zr5 Cu5Zr6 Cu4Zr7 Cu3Zr8 Cu2Zr9
0.2771 0.2748 0.2746 0.2699 0.2753
0.0329 0.0416 0.0394 0.0394 0.0379
0.9621 0.9811 0.9792 0.9814 0.9557
0.2871 0.2893 0.2873 0.2846 0.2855
0.0454 0.0423 0.0430 0.0434 0.0412
0.9526 0.9917 0.9818 0.9778 0.9681
0.2953 0.2933 0.2922 0.2924 0.2916
0.0471 0.0468 0.0446 0.0473 0.0441
0.9464 0.9738 0.9767 0.9958 0.9621
0.1827 0.0567 0.0441 0.1812 0.1127
0.6846 0.611 0.7679 0.8192 0.7477
(0, 2, 8, 2)
Cu7Zr5 Cu6Zr6 Cu5Zr7 Cu4Zr8 Cu3Zr9
0.2654 0.2689 0.2714 0.2709 0.2706
0.0403 0.0462 0.0459 0.0449 0.0451
0.9598 0.974 0.9848 0.9843 0.9788
0.2842 0.2856 0.2869 0.2885 0.2914
0.0484 0.0489 0.0462 0.0462 0.0459
0.9718 0.9631 0.9816 0.9771 0.9685
0.2931 0.2962 0.2978 0.2951 0.2919
0.0516 0.0494 0.0498 0.0478 0.0462
0.9623 0.9763 0.9634 0.9736 0.9484
0.154 0.1127 0.1069 0.0067 0.08
0.7236 0.6807 0.6527 0.7247 0.7561
(0, 0, 12, 0)
Cu7Zr5 Cu6Zr6 Cu5Zr7 Cu4Zr8 Cu3Zr9
0.1848 0.1904 0.2018 0.2087 0.2118
0.0410 0.0510 0.0486 0.0478 0.0497
0.9585 0.9821 0.9817 0.9776 0.9442
0.2068 0.2173 0.218 0.2255 0.2277
0.0490 0.0541 0.0526 0.0541 0.0519
0.9659 0.9731 0.9690 0.9787 0.9533
0.2225 0.2303 0.2318 0.2367 0.2417
0.0572 0.0591 0.0575 0.0544 0.0532
0.9478 0.9786 0.9855 0.9679 0.9707
0.1841 0.0955 0.0274 0.1359 0.1981
0.8136 0.7378 0.7557 0.6695 0.708
(0, 3, 6, 3)
Cu7Zr5 Cu6Zr6 Cu5Zr7 Cu4Zr8 Cu3Zr9
0.2905 0.2875 0.2961 0.2901 0.2896
0.0424 0.0479 0.0519 0.0477 0.0497
0.9692 0.9737 0.9783 0.9816 0.9511
0.2976 0.2937 0.2974 0.2933 0.2924
0.0492 0.0498 0.0545 0.0523 0.0544
0.9541 0.9782 0.9655 0.9731 0.9547
0.3122 0.3028 0.3106 0.3008 0.3109
0.0524 0.0547 0.0575 0.0557 0.0552
0.9624 0.9748 0.9622 0.9488 0.9538
0.0274 0.1359 0.043 0.0364 0.0956
0.7529 0.6672 0.795 0.6714 0.6455
(0, 2, 8, 0)
Cu5Zr5 Cu4Zr6 Cu3Zr7 Cu2Zr8 CuZr9
0.2713 0.2604 0.2588 0.2702 0.2629
0.0399 0.0435 0.0407 0.0411 0.0398
0.9728 0.9823 0.9637 0.9591 0.9676
0.2733 0.2711 0.2654 0.2798 0.2691
0.0452 0.0461 0.0436 0.0439 0.0451
0.9773 0.9504 0.9722 0.9654 0.9539
0.2773 0.2765 0.2709 0.2809 0.2793
0.0477 0.0469 0.0454 0.0463 0.046
0.9591 0.9418 0.9617 0.9811 0.9513
0.1101 0.1894 0.0605 0.0821 0.0182
0.748 0.6345 0.6882 0.7564 0.7302
(0, 3, 6, 4)
Cu8Zr5 Cu7Zr6 Cu6Zr7 Cu5Zr8 Cu4Zr9
0.2615 0.2742 0.2739 0.2782 0.2811
0.0431 0.0496 0.0476 0.0507 0.0521
0.9616 0.9728 0.9535 0.9612 0.9528
0.2738 0.2764 0.2787 0.2796 0.2834
0.0457 0.0519 0.0507 0.0519 0.0538
0.9742 0.9618 0.9519 0.9457 0.9533
0.2795 0.2788 0.2804 0.2803 0.2846
0.0474 0.0526 0.0538 0.0574 0.0566
0.9738 0.9586 0.9417 0.9696 0.9643
0.105 0.191 0.099 0.1731 0.0005
0.685 0.7287 0.8455 0.6918 0.6727
For each type of CP, only the five most abundant chemistries are presented, as they account for more than 90% of this type of CP. Also shown in this table are the correlation coefficients l(p, e) and l(s, e) at 800 K.
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"
# b
1 1 X @E raij rij a b ¼ mi vi vi ; ða; b ¼ x; y; zÞ ð9Þ Xi 2 j;j–i @rij rij " ! # X X 1X E¼ Fi qj ðrij Þ þ / ðrij Þ ð10Þ 2 j;j–i ij i j;j–i rab i
Here, m and v are mass and velocity, respectively, and rij is the distance between atoms i and j. Xi is the volume of the VP. E is the potential energy in the EAM formulation. For MGs and SLs which are macroscopically isotropic, it is desirable to use two rotational invariants, p and s, p ¼ ðr1 þ r2 þ r3 Þ=3
12 1 1 1 1 2 2 2 s¼ ðr1 r2 Þ þ ðr2 r3 Þ þ ðr3 r1 Þ 3 2 2 2
ð11Þ ð12Þ
where r1, r2 and r3 are the three principal stresses. p is the atomic level pressure, while s is the deviatoric stress or atomic level von Mises shear stress. To correlate the SSP (e) with the atomic level stress, the correlation coefficients l of (p, e) and (s, e) for the J type of cluster (with a number of NJ clusters) are calculated as X lJ ðp; eÞ ¼ ðpi pJ Þðei eJ Þ i2J
X lJ ðs; eÞ ¼ ðsi sJ Þðei eJ Þ i2J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,v #" #, u" X u X t N2 ðp p Þ2 ðe e Þ2 J
ð13Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,v #" #, ffi u" X u X 2 2 t ðs s Þ ðe e Þ N2
ð14Þ
J
i
i2J
i
i2J
i
J
i2J
J
i
J
J
i2J
where pi, si and ei correspond to cluster i (i 2 J ), for various CPs in Cu50Zr50 alloy at 800 K. The results are listed in Table 1. Clearly, for clusters with identical Voronoi indices and chemistries, s and e show strong correlation, while l(p, e) is small. This correlation demonstrates that the clusters with large e are likely to be associated with large s. In other words, the central atoms of the clusters with large distortion (low degree of symmetry) tend to be in a state of large local shear stress. This finding is a validation, from the geometrical perspective of atomic detail, of previous arguments that the local shear stress originates from the deviation of local environment from spherical symmetry [36,54].
In this work, we focus on the short-time atomic dynamics, that is, the b relaxation. To characterize the heterogeneous dynamics of atoms, centered at different clusters, we calculate the propensity of motion [56] for Cu atoms in Cu50Zr50 alloy at 1000 K. To this end, 1000 independent simulations in the isoconfigurational ensemble [56] were conducted. The length of the time interval was chosen as the time at which the nonGaussian parameter [57] reaches the maximum. This time interval is sufficiently long, as it corresponds to the late part of the b relaxation, for the system to reach heterogeneous dynamics and simultaneously to probe the effect of the initial structure on dynamics [48,58]. By using the method reported in previous studies [48,55], Cu atoms are sorted and divided into 20 groups according to their mobility, with 5% Cu atoms in each group. The local structure of each group is categorized according to its Voronoi index. The results are listed in Fig. 7. The most prominent feature of Fig. 7 is that icosahedra play an important role in dynamical heterogeneity. About 33% of the slowest 5% Cu atoms are surrounded by icosahedra. This feature is consistent with previous studies [59–64], and demonstrates the important role of icosahedra and their spatial correlation in the dynamical slowing down. Another important feature of Fig. 7 is the wide distribution of various types of clusters in all groups. For example, the (0, 2, 8, 1) cluster tends to be evenly distributed in all 20 groups. This wide distribution reveals the profound effect of structural heterogeneity on dynamics. To characterize this heterogeneous structure and further correlate it with the DH, Fig. 8 presents the average SSP of Cu atoms centered in various types of clusters in each of the 20 groups obtained above. Albeit with statistical noise, the overall trend is that the average SSP increases systematically from a low- to a high-mobility range. This result demonstrates the strong correlation between the SSP and atomic dynamics, that is, large SSP tends to be accompanied by large mobility.
4. Correlating SSP with the short-time dynamics of atoms in supercooled liquids The dynamics of atoms is strongly influenced by the structure of the corresponding coordination clusters. Due to the heterogeneous structure of these clusters, atoms in SLs show highly heterogeneous mobility. Although this dynamical heterogeneity (DH) in atomic motion plays a crucial role in glass formation, its microscopic origins are still not fully understood [6,22,55]. The atomic mobility and related DH in SLs can be analyzed from the perspective of the local potential energy surface or the atomic level stress [6,36]. Considering the correlation between SSP and atomic level shear stress, it is reasonable to attempt a connection between the new descriptor and particle dynamics.
Fig. 7. Correlation between the atomic propensity of motion and topological structure of Cu-centered CPs in Cu50Zr50 SL at 1000 K. Cu atoms are sorted according to their mobility and evenly divided into 20 groups. From left to right, all groups are arranged in increasing order of atomic mobility. Only the fractions of the six most abundant clusters from Voronoi indices in each group are presented.
C. Wu et al. / Acta Materialia 72 (2014) 229–238
237
Table 2 Correlation between the cluster lifetime and SSP for (0, 2, 8, 1), (0, 2, 8, 2) and (0, 0, 12, 0) clusters with different chemistries in Cu50Zr50 SL at 1000 K. Group ID
Average scaled cluster lifetime (0, 2, 8, 1)
Fig. 8. Color map showing the average SSP for various types of Cucentered CPs in each of the 20 groups obtained in Fig. 7. The groups are arranged vertically in an increasing order of atomic mobility from the bottom to the top. The color shows the SSP value for each data point. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
As mentioned above, for clusters with the same topology and chemistry, the central atoms of the clusters with large SSP bear a large shear stress. Previous studies [36,54,65] have found that, in MGs, the atomic level pressure depends predominantly on the topology and chemistry of local environment. Accordingly, for clusters with specific Voronoi index and chemistry, a large SSP corresponds to a large internal stress on the central atoms, making these atoms “liquid-like”, or ones which are moving faster than the corresponding atoms of low SSP. The same conclusion can be drawn from the perspective of a local potential energy surface (PES). A direct and robust way to correlate local PES with atomic structural motifs is to calculate the cluster lifetime [9,66]. Central atoms of clusters with a long lifetime tend to be arrested in deep energy minima, while short lifetimes correspond to shallow minima [66]. In order to explore the relation between the SSP and atomic dynamics, we further investigate the correlation between the cluster lifetime and the SSP. The lifetime of a cluster is defined as the period during which its Voronoi index remains unchanged [9]. Practically, the lifetime of a cluster with a given Voronoi index, Dt, is calculated as t t0, where t0 and t are the times at which this cluster first appears and then vanishes, respectively, in the MD simulation. Consequently, we studied the correlation between the SSP of the cluster at t0 (denoted the “initial cluster” in the following text) and the lifetime Dt of this cluster. First, the initial clusters were screened out from the whole MD trajectory. Then, for the initial clusters with the same Voronoi index and chemistry, the central atoms were divided into 10 groups of equal size, according to their SSP values. They were labeled in ascending order, with smaller numbered groups corresponding to smaller SSPs. The average lifetime for each group was then calculated, as reported in Table 2. Within statistical fluctuations, the general trend is that, for clusters of specific Voronoi index and chemistry, the lifetime decreases with increasing SSP. This suggests that, for any given cluster type, clusters with large SSP tend to be short-
1 2 3 4 5 6 7 8 9 10
(0, 2, 8, 2)
(0, 0, 12, 0)
Zr6
Zr7
Zr8
Zr6
Zr7
Zr8
Zr6
Zr7
Zr8
2.46 2.11 2.09 1.87 1.88 1.52 1.65 1.42 1.35 1.24
2.48 2.26 2.14 2.17 1.92 1.82 1.65 1.76 1.44 1.32
2.49 2.22 2.09 2.08 1.93 1.87 1.74 1.72 1.44 1.49
2.03 1.97 1.87 1.72 1.77 1.55 1.49 1.34 1.22 1.11
2.07 1.87 1.79 1.72 1.51 1.66 1.45 1.21 1.28 1.10
2.06 1.72 1.79 1.62 1.59 1.49 1.61 1.42 1.38 1.12
4.33 4.18 3.91 3.73 3.76 3.62 3.51 3.53 3.35 3.28
4.29 4.12 3.90 3.99 3.79 3.58 3.45 3.38 3.20 3.32
4.37 4.24 3.98 3.68 3.81 3.73 3.52 3.36 3.39 3.25
Here the scaled cluster lifetime is used, which is obtained by dividing the lifetime (in unit of MD steps) of a cluster by the average cluster lifetime of P the SL. The average lifetime is defined as i pðiÞtðiÞ where p(i) and t(i) are the population fraction and lifetime of the i-type cluster; the summation is over all types of clusters.
lived. In other words, the central atoms of large-SSP clusters are trapped in the shallow local basins of local PES while small-SSP clusters are associated with deep local energy minima. High potential energy barriers have to be overcome for the central atoms of low-SSP clusters to escape the cage. Thus, they reside “trapped” for extended time periods, practically rattling in the cages, before escaping from the cluster, compared to the central atoms of large-SSP clusters. This long cage-rattling process postpones the onset of the diffusive behavior for the central atoms of low-SSPs. Therefore, the central atoms of low-SSP clusters show lower mobility than those of high-SSP clusters. From the analysis above, the degree of symmetry or distortion of local clusters, which is quantified in the present work by the SSP, shows the influence of atomic structure on dynamics vividly. Note that previous studies [48,55,67–69] found that the local topology and chemistry play an important role in the heterogeneous dynamics of SLs. These findings, together with our results, demonstrate the multifaceted relationship between the local atomic structure and the dynamics of supercooled liquids. 5. Conclusions In this work, we have introduced and employed a new descriptor of local structure – the short-range-order symmetry parameter (SSP) method – to study the salient characteristics of short-range order in metallic glasses and supercooled liquids. By calculating the degree of symmetry for each cluster (mapped into symmetry parameter, SSP), we quantify the degree of distortion, both radially and orientationally, of an individual cluster from the ideal one. From the present analysis, the local structure of an icosahedron, as quantified by the new metric, appears to be sensitive to changes in local chemistry. This structural
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characteristic of icosahedra is not encountered in other polyhedral cluster types. The SSP can be correlated with the atomic level shear stress, the local potential energy surface and successively with dynamical properties (atomic mobility). These correlations indicate that the SSP is an important structural identifier that can accurately quantify the structure of SRO and shed light on the microstructure–property relationship. By combing existing structural analysis methods with the newly introduced metric a detailed and accurate description of short-range order in complex atomic systems can be obtained. Acknowledgements This work was financially supported by the National Nature Science Foundation of China under Grant No. 51025415. The authors acknowledge Dr. Daniel Miracle from Air Force Research Laboratory, Ohio, USA, for fruitful discussions. NCK acknowledges support by the Spanish Ministry of Economy and Competitiveness (MINECO) through projects “Ramon y Cajal” (RYC2009-05413), “I3” and MAT2011-24834.
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