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MD study on topologically close-packed and configuration entropy of Mg40Al60 metallic glasses under rapid solidification
Journal of Non-Crystalline Solids 522 (2019) 119578
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MD study on topologically close-packed and configuration entropy of Mg40Al60 metallic glasses under rapid solidification ⁎
T
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Bang-yi Yua, Yong-chao Lianga, , Ze-an Tiana, , Yue-hong Zhanga, Quan Xiea, Ting-hong Gaoa, Yun-fei Mob a b
School of Big Data and Information Engineering, Guizhou University, Guiyang 550025, China School of Electronic and Communication Engineering, Changsha University, Changsha 410003, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Molecular dynamics simulation Cooling rates Rapid solidification Topologically close-packed cluster Configuration entropy
Herein, the effects of cooling rate during rapid solidification of Mg40Al60 metallic glasses are studied using the molecular dynamics (MD) simulation. The system energy, pair distribution function (PDF), local structures, configuration entropy, and three-dimensional visualization are used to systematically analyze the evolution of microstructure with temperature. Results indicate that the lower the cooling rate, the lower the average atomic energy. With the decrease of cooling rate, increases the height difference between two splitting subpeaks of the second peak on PDF curves; while decreases the onset temperature of glass transition. The effect of the cooling rate on the topologically close-packed (TCP) clusters is similar to the local five-fold structures of S555: the lower the cooling rate, the more the amount of TCP clusters, and the lower the configuration entropy. In addition, most TCP clusters are centered the smaller Al atoms.
PACS: 61.20.Ja 61.25.Mv 64.70.pe 71.15.Pd
1. Introduction Formation of amorphous alloys is a dynamic process, and its atomic arrangement is closely related to thermal historic conditions, such as cooling rates and initial conditions that affect both phase transition and microstructure [1]. Rapid cooling may result in some novel structural configuration and outstanding material properties [2]. In other words, differences can be found in the microstructures and material properties of a metal melt during the rapid solidification process at various cooling rates. Therefore, It is important to profoundly understand the microstructural evolutions of liquid metal at different thermal historic conditions under rapid solidification. In recent years, microstructure of pure metals and binary alloys obtained by rapid cooling has been investigated extensively. MD simulation about silver at six cooling rates revealed that an amorphous solid can be obtained with the cooling rate γ ≥ 1.0 × 1013 K/s; as cooling rate ranging within 5.0 × 1012 − 1.38 × 1011 K/s, the coexistence structures of the metastable hcp with stable fcc configurations can be formed with phase separating or layering structures [3]. The rapid solidification experiment at different cooling rates found that cooling rate effectively tailored the microstructure and characteristic
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temperatures of NieTi alloy [4]. For binary TiAl alloys there is also a strong dependency of the microstructure on the Al content and cooling rates [5]. MD simulation for pure Zr concluded that the critical cooling rate of vitrification was approximately 5.0 × 1013 K/s, and a rather perfect bcc phase can be obtained at 1012 K/s [6]. MD investigation upon the correlation between the phase selection of crystal or glass and cooling rate of pure copper demonstrated that the crystalline fractions largely fluctuated along with the cooling rates ranging in 6.3 × 1011−16.6 × 1011 K/s [7]. For good glass former Cu50Zr50 MD simulation shows that a faster cooling rate will resulte in a less amount of icosahedra-like clusters [8], which may shed great guiding significance in comprehending the relation between rapid solidification and cooling rate of such binary alloys. As a type of magnesium-based alloy composed mainly of Al with less Mg or other metal elements for enhancing its hardness [9], Magnesiumaluminum (MgeAl) alloys not only have been widely applied in industry, but also are an important model system for studying materials science and condensed matter physics, playing an important role in the development of amorphous alloy materials [10]. From this viewpoint, herein, MD simulation approach is adopted to explore the influence of cooling rate on the microstructures of Mg40Al60 liquid alloy during the
Corresponding authors. E-mail addresses: [email protected] (Y.-c. Liang), [email protected] (Z.-a. Tian).
https://doi.org/10.1016/j.jnoncrysol.2019.119578 Received 22 March 2019; Received in revised form 7 July 2019; Accepted 15 July 2019 0022-3093/ © 2019 Elsevier B.V. All rights reserved.
Journal of Non-Crystalline Solids 522 (2019) 119578
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of Tg and E of the system. The slower the cooling rate is, the system is more close to an equilibrium state at a certain temperature, and hence the harder to achieve a nonequilibrium transition; therefore, Tg of the system decreased with a decrease in cooling rate. However, in practice, there is a weak dependence of Tg on the cooling rate; even if the cooling rate changes by an order of magnitude, the Tg value changes by only approximately 10 K. The glass transition phenomenon, wherein Tg reflected the nature of the liquid to some extent, showed that the nature and behavior of the amorphous state changed with time [21]. This fact is significant for exploring the microstructural characteristics of the binary alloys.
rapid solidification process. Alloy structure evolution is also profoundly analyzed and investigated from different viewpoints based on a series of effective means, such as the energy-temperature (E-T) relationship [11], pair distribution function (PDF or g(r)) [12], Honeycutt-Andersen (HA) bond-type index [13], the largest standard cluster analysis (LSCA) [14], TCP structures [15], configuration entropy [16] and three-dimensional (3D) visualization. The microscopic structural characterization in Mg40Al60 alloy is discussed in detail. 2. Simulation details MD simulation was performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) [17]. A total of 10,000 atoms, i.e., 4000 Mg atoms and 6000 Al atoms, were randomly placed in a cubic box subsequently operated according to a periodic boundary condition. In this case, the interaction potential between the atoms was indicated by an embedded-atom model (EAM) developed and optimized by Mendelev et al. [18]. The time step for the system operation was set at 10−15 s. The relevant analog computation started at 2000 K, much higher the equilibrium melting point of the MgeAl alloy (no > 1300 K) [19]. The isothermal relaxation lasted for 1,000,000 steps drives the system to an equilibrium state. The system was then cooled at different cooling rates of 1010, 1011, 1012, and 1013 K/s to the proposed temperature of 300 K. In such a rapid temperature reduction, the structural configuration data, i.e., moving speed and space coordinates of all the atoms, should be recorded at an interval of 10 K to acquire the microcosmic information of the system at the corresponding temperature. Then, the E-T curve, g(r) curve, bond-type index, LSCA, TCP structures, configuration entropy, and 3D visualization were adopted to analyze the structural configurations of the system and investigate the formation and evolutionary rules of the alloy microstructures at different temperatures and cooling rates.
3.2. Difference in medium-range order While E-T graph merely reflects some simple statistical information about the microstructure, the PDF or g(r) curve is an important method for describing the overall structural variation characteristics of the system. It refers to the probability of other atoms to appear in a spherical shell with a certain distance from a selected central atom. It is used to calculate the average value of all the atomic distribution probabilities. Being extremely objective and extensively applicable, PDF has been widely applied to investigate the structural characteristics in the field of materials science. Fig. 2 shows the g(r) curves of Mg40Al60 alloy at 300 K obtained at four cooling rates. No distinct differences were found in the first peaks. All the second peak of the g(r) curves clearly split, indicating the formation of metallic glass. Nevertheless, the two subpeaks split from the second peak of the g(r) curves have different relative height, which increases with the decrease of cooling rate; that is 0.228, 0.246, 0.263, 0.282 at 1013, 1012, 1011, and 1010 K/s, respectively. Therefore the structure is different in medium-range order. Fig. 3 shows the g(r) curves during rapid solidification process of the Mg40Al60 alloy at four cooling rates. The four groups of curves clearly have the same change trend. With temperature decrease, the first peak of the g(r) curves became sharp; whereas the second peak exhibit becomes two shoulders as T = 530, 540, 550, and 560 K at 1010, 1011, 1012, and 1013 K/s, respectively. The distance between atoms indicated by the shoulders is more than the nearest neighbors, so that some medium-range order formed in the system. Generally onset temperature of the occurrence of the split second peak on g(r) curve is defined as Tg. The value change of Tg with cooling rates indicates the cooling rate dependence of microstructures.
3. Results and discussion 3.1. Evolution of energy As a simple but effective analysis, the varying of average atomic energy (E) [20] can truly and objectively reflect the preliminary feature of rapid solidification. Fig. 1 shows the relation between E and T of the system at different cooling rates. At four cooling rates, the average atomic energy gently declines without sudden changes, and hence amorphous structures should be formed; at a certain temperature the E values are very close to each other, and at high temperatures evolve in a similar tendency, which reveals that the cooling rate exerted an limited influences. The glass transition temperature (Tg) can be evaluated as approximately 560, 550, 540, and 530 K, at 1013, 1012, 1011, and 1010 K/s, respectively. Therefore the lower the cooling rate, the lower value
3.3. Short-range order The PDF describes the distribution of atom pairs in a system, and hence reveals that feature of short-, medium-, and long-range orders. However, it fails to explicitly reflect the geometrical features of atomic arrangement. The microstructure (short-range order) composed of
Fig. 1. Average atomic energy of Mg40Al60 alloy changes with T at different cooling rates.
Fig. 2. g(r) of Mg40Al60 alloy at 300 K obtained at four cooling rates. 2
Journal of Non-Crystalline Solids 522 (2019) 119578
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Fig. 3. g(r) changes with T at four cooling rates.
neighboring atoms may be different even if the first peaks seem no changed at different cooling rate. Hererin, the structure composed of a reference pair of atoms and their common near neighbors (CNNs) is called a common neighbor sub-cluster (CNS) [22], because such structure reflects the topological feature shared by a group of atoms that are a part of a cluster around a center atom. CNS was proven to be successful in describing the local configurations of the liquid, amorphous, and crystal structures. To characterize the bond-type objects, S555, S544, S433, S421, S422, S444, and S666 are consistent with common 1551, 1541, 1431, 1421, 1422, 1441, and 1661 of the HA bond-type index. Generally, S555, S544, and S433 are closely related to amorphous structure; whereas S421, S422, S444, and S666 are related to crystalline structures. Fig. 4 shows the variations of primary CNSs during rapid solidification process of the liquid Mg40Al60 alloy. At 300 K, S555 is always predominant with the maximum values of 36.79%, 40.04%, 43.86%, and 46.91% at the cooling rates of 1013, 1012, 1011, and 1010 K/s, respectively (Fig. 4(a)). Moreover, the upward trend of the four curves appeared as a plateau after 560, 550, 540, and 530 K, respectively. The relative numbers of the S544, S433, S444, and S666 first increase, and then decline, as shown in Figs. 4(b)-(e). They also reached maximum values of 17%, 20%, 7.5%, and 10% at approximately 850, 1350, 800, and 600 K, respectively. In liquid and super-cooled liquid, the four CNSs comprise various local structures; subsequently, they began to reduce and exhibited differences. The lower the cooling rate, the smaller the proportion occupied by the four CNSs. The relative numbers of the crystalline S421 and S422 start to gently decline at T < Tg, and then gradually increase, as shown Figs. 4(f)-(g). The relative numbers of S421 and S422 at a cooling rate of 1013 K/s can reach 3% and 7% at 300 K, respectively; but < 4% and 8% at 2000 K. Only the amorphous S555 remarkably increases, and apparently far greater than that of the other six bond types. At four cooling rates, the amorphous structures that eventually formed in the system were mainly ascribed to the massive S555.
describe the different basic clusters formed by an atom with its nearest neighbors. Investigations on the atomic configurations and structural characterization of the cluster are significant in broadening the subject fields of materials science and condensed matter physics. Structural analysis is critical for measuring material properties and functions. The applications of particle coordinate quantization and local structure identification for disordered systems based on 3D are still in the phase of development. Herein, LSCA [23] was adopted to effectively quantify the 3D structures according to the spatial distribution and characterize various cluster structures independent of any preset parameters and beyond the range of the nearest neighbors. A cluster is composed of all CNSs that share the same atom; if all CNSs in such a cluster are free from common-neighbor sub-ring (CNSR) and multi-bond point, it is called a standard cluster (SC); and the largest SC around a certain atom is unique, called a LSC (the largest standard cluster). In the system under consideration, all LSCs can be identified and described by a group of binary tuples that is compose of a integer and a notation of CNS [22], where the integer represents the number of this CNS in the LSC. Therefore, the icosahedra (ico), hexagonal close-packed (hcp), and body-centered cubic (bcc) LSCs are successively denoted as [12-S555], [6-S421, 6-S422], and [6-S444, 8-S666], respectively. Atoms can be entirely classified according to the LSC types around them. For example, the centers of the fcc, hcp, and bcc LSCs are referred to as the fcc, hcp, and bcc atoms, respectively. Considering a local structure unique to every atom, the radius (r) of the LSC can change, and all the LSCs can be determined by different cutoff distances (rc) [24]. Such a method of LSC overcomes the deficiencies in the structural characterization of the previous clustering scale approach, such as the structural uncertainties caused by the dependence on a preset parameter. By characterizing an internal structure of the atomic clusters in a general disordered system, LSC is available to effectively improve our understanding on the structure-property relation. TCP LSCs are compact-structured clusters formed by n4-S444, n5S555, and n6-S666 CNSs containing at least one CNS of S555. TCP structure follows the Euler's theorem, i.e., face number (NF) + vertex number (NV) − edge number (NE) = 2; that is, an equation set of {n4 + n5 + n6 = CN, 2n4 + n5 = 12, n5 > 0} is satisfied. Therefore, the topological feature of a TCP LSC can be described by a TCP-index of “Ln”, where “L” is Z, A, B, C, D, or E, indicating n4 = 0, 1, 2, 3, 4, 5; and
3.4. Medium-range order Although CNSs can be successfully used to describe the bonding relation between an atom and its nearest neighbors, they still cannot 3
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Fig. 4. Relative number of various CNSs with T at different cooling rates.
Fig. 5. Schematic diagram of 5 TCP LSCs.
n refers to CN. In the case of n4 = 6 (where n4 = 6 means label “F”), the TCP-index is still valid for the topology; for example, the bcc unit of [6S444, 8-S666] can be denoted as F14, although it is a non-TCP LSC owing to the absence of S555. All the Frank-Kasper clusters (only formed by S555 and S666) [25] are apparently TCP LSCs. Four kinds of Kasper polyhedrons are denoted as K12 (120120), K14 (14 0 12 2), K15 (15 0 12 3), and K16 (16 0 12 4) by a CTIM method [26]. In line with the concepts of the abovementioned terms, Z12 (ico), Z14, Z15, and Z16 may be selected for replacing K12, K14, K15, and K16, respectively, to characterize the cluster structures. As shown in the experiment, the TCP LSCs, as a group of compact local structures, are ubiquitous in metallic glass [27]. Fig. 5 shows 5 common TCP LSCs. As shown in Fig. 6, with the decrease of temperature the sum number of all TCP LSCs increases independent of cooling rate at T > Tg; that is, at different (four) cooling rates at a certain temperature the TCP total number is almost the same value; and the lower the temperature, the higher the increase rate. When T is approaching to Tg, the increase rate of the TCP total number start to depend on cooling rate: the lower the cooling rate, the higher the increase rate. With temperature further decrease, the increase rate decreases, and inclines to be constant at a state of plateau that appears T < Tg. Comparing the relative number of S555 (Fig. 4) with that of TCP LSCs (Fig. 6), one can
Fig. 6. Numbers of the TCP LSCs with T at four cooling rates.
conclude that the variation of TCP LSCs is consistent with that of S555. Table 1 lists the details of primary TCP LSCs (Z12, A13, Z14, Z15, and Z16) in Mg40Al60 alloys at 300 K under four cooling rates. The number of Z12 LSC is apparently more than others under all cooling rates, and decreases from 41.80% to 24.27% of all TCP LSCs as cooling 4
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based on LSCs. The LSCs play a major role in analyzing and quantifying the structural evolution in a liquid-solid transition. As known, the configuration entropies of all the cooling processes reduced with a decline in temperature. At T > Tg, the variation curve of configuration entropy of the system with temperature is almost the same under four cooling rates; while suddenly changed at Tg. In other words, the four overlapped curves separate at T < Tg. As aforementioned, Z12 atoms play a critical role at lower cooling rates during rapid solidification, and the lower the cooling rate, the more the number of Z12 atoms; and the kinds of LSCs that the formation of more ordered local structures would decrease the configuration entropy. Therefore, the number of Z12 atoms and configuration entropies change contrarily (see Figs. 6 and 7). Perhaps, the study between the amorphous structure and entropy is one of the approaches making a breakthrough in recognizing the essence of amorphization [30,31].
Table 1 Details for primary TCP LSCs at 300 K for four cooling rates.
Number of Z12 Number of A13 Number of Z14 Number of Z15 Number of Z16 Total of TCP LSCs Proportion of Z12 (%) System energy (eV) Energy of Z12 (eV)
1010 K/s
1011 K/s
1012 K/s
1013 K/s
991 228 137 127 101 2371 41.80 −2.52 −3.24
788 229 100 121 66 2149 36.67 −2.52 −3.22
558 198 96 67 49 1845 30.24 −2.51 −3.21
400 218 69 51 34 1648 24.27 −2.51 −3.16
TCP LSCs is the researching focus in this paper. And Z12 is the largest number of TCP LSCs. So two of them and their proportion are bolded.
rate increase from 1010 K/s to 1013 K/s. The quantities of A13, Z14, Z15, and Z16 also decrease with the increase of cooling rate. The average atomic energy of the system and Z12 atoms slightly increases with the increase of cooling rate. It signified that the amorphous system structure is more stable obtained at a low cooling rate. With lower average energy, Z12 atoms have a substantial influence on the stability of Mg40Al60 amorphous alloys.
3.6. Visualization analysis 3D visualization [32] was utilized herein to further analyze the structures. Its microstructural configurations then visually distinguished the influence of the cooling rate on the solidified structure, so as to have a deeper understanding of the atomic arrangement and the major cluster distributions of Mg40Al60 alloy. Fig. 8 specifically shows the distribution of five TCP atoms in the system at 800 (> Tg), 500 (~ Tg), and 300 K (< Tg) obtained at four cooling rates. As intuitively indicated by the 3D distribution diagram, the number of Z12 (red) atoms remarkably increases with the decrease of temperature and reached the maximum at 300 K. Based on the number of Mg and Al atoms shown in the pictures in the last column of Fig. 8, we can get the number ratio of Mg to Al atoms, being 27.18% (165 / 607), 27.54% (209 / 759), 27.84% (284 / 1020), and 27.74% (344 / 1240) at 1013, 1012, 1011, 1010 K/s, respectively. Therefore, among all TCP atoms the proportion of Mg is basically 20% (= 165 / (165 + 607) as a example), almost unchanged with cooling rate. The proportion of the Al-centered TCP LSCs was absolutely predominant and more than its concentration in Mg40Al60 alloys. This bias is consistent with the fact that the Al-centered ICO LSCs is geometrically favored, because the radius of Mg atom is 1.60 Å, and Al atom 1.43 Å. Fig. 9 shows the specific numbers of Z12, A13, Z14, Z15, and Z16 atoms nearby Tg at different cooling rates. The number of Z12 is always much more than others in all the rapid cooling processes. In particular, in the system at 300 K it is 400, 558, 788, and 991 under 1013, 1012, 1011, and 1010 K/s respectively. Therefore, Z12 atoms increase with the decrease of cooling rate. In other words, the formation of Z12 LSCs plays a vital role in glass forming and is conspicuously influenced by the cooling rate; while A13, Z14, Z15 and Z16 atoms only change a little at four cooling rates.
3.5. Configuration entropy Entropy is a state function of a real physical system, which is related to the chaos of a system. Several empirical rules are put forward during the long-term exploration of the glass-forming systems and used to express the uniformity of any kind of energy distribution in the space. The more uniform the energy distribution, the bigger the entropy. Entropy can directly reflect the uniformity of a system as a measure of the ordered degree. The smaller the entropy of a system is, the more orderly the system. According to the rule of thermodynamics, the entropy of liquid with an disordered structure is larger than that of crystal with an ordered structure. The analysis method of configuration entropy [28,29] is adopted to better understand the changes in the structure. The configuration entropy is here defined as s = −sum(ρilogρi), where ρi is the proportion (< 1.0) of the i-th kind of structure. For fcc, hcp, and bcc crystals that are composed of only one LSC, the configuration entropy is zero, i.e., s = 0. Obviously the more the kinds of structures, the smaller the value of ρi, and the larger the value of s. The configuration entropy model can effectively establish the relationship between order degree and local structures especially for liquids and amorphous alloys. As an important structural analysis, it has a profound meaning to the studies based on the material amorphousness and randomness. Fig. 7 shows the relation between T and the configuration entropy
4. Conclusions By employing classical MD simulation, this study systematically studied the evolution of microstructure in Mg40Al60 alloys obtained by rapid solidification under four cooling rates of 1013, 1012, 1011, and 1010 K/s. The following conclusions are obtained: (1) Mg40Al60 is a good glass former, vitrified even if cooling rate is low to 1010 K/s. The cooling rate has a crucial influence on the microstructural characteristics of Mg40Al60 alloy during rapid solidification. (2) The analysis upon CNS (S555), TCP clusters, and configuration entropy reveals Tg decreases with the decrease of cooling rate, being approximately 560, 550, 540, and 530 K at 1013, 1012, 1011, and 1010 K/s, respectively. (3) A lower cooling rate benefits the formation of a more stable Mg40Al60 glass, and hence benefits the formation of TCP clusters, reduces the configuration entropy.
Fig. 7. Configuration entropy based on LSCs with T at four cooling rates. 5
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Fig. 8. The distribution of Z12, A13, Z14, Z15, and Z16 atoms at 800, 500, and 300 K obtained at four cooling rates. The pictures in the last column is used to distinguish the difference between Mg and Al atoms of TCP atoms at 300 K.
Acknowledgement This work has been supported by the National Natural Science Foundation of China (Grant No.61264004, 51661005 and 51671004), the Theoretical Physics Special Project of National Natural Science Foundation of China (Grant No.11747123), the Natural Science Foundation of Hunan Province of China (Grant No.2018JJ3560), the Project of Science and Technology Plan of Changsha (Grant No. kc1809022), Sci-Tech Cooperation Program of Guizhou Province of China (Grant No. [2017]5788-81). References [1] C.D. Cao, Z. Sun, X.J. Bai, L.B. Duan, J.B. Zheng, F. Wang, Metastable phase diagrams of Cu-based alloy systems with a miscibility gap in undercooled state, J. Mater. Sci. 46 (2011) 6203–6212, https://doi.org/10.1007/s10853-011-5612-7. [2] D.B. Liu, Y.C. Liu, Y. Huang, R. Song, M.F. Chen, Effects of solidification cooling rate on the corrosion resistance of Mg-Zn-Ca alloy, Prog. Nat. Sci-Mater. 24 (2014) 452–457, https://doi.org/10.1016/j.pnsc.2014.08.002. [3] Z.A. Tian, R.S. Liu, H.R. Liu, C.X. Zheng, Z.Y. Hou, P. Peng, Molecular dynamics simulation for cooling rate dependence of solidification microstructures of silver, J. Non-Cryst. Solids 354 (2008) 3705–3712, https://doi.org/10.1016/j.jnoncrysol. 2008.04.006. [4] G.J. Pan, C. Balagna, L. Martino, S. Spriano, Microstructure and transformation temperatures in rapid solidified Ni-Ti alloys, part I: the effect of cooling rate, J. Alloys Compd. 589 (2014) 628–632, https://doi.org/10.1016/j.jallcom.2013.10. 130. [5] C. Kenel, C. Leinenbach, Influence of cooling rate on microstructure formation during rapid solidification of binary TiAl alloys, J. Alloys Compd. 637 (2015)
Fig. 9. Numbers of Z12, A13, Z14, Z15, and Z16 atoms nearby Tg at four cooling rates.
(4) Visualization unveils that both TCP clusters and two kinds of atoms distribute homogeneously in the space, and the Al-centered TCP clusters is absolutely predominant.
Declaration of Competing Interest The authors declare no conflict of interest. 6
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MD study on topologically close-packed and configuration entropy of Mg40Al60 metallic glasses under rapid solidification ⁎
T
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Bang-yi Yua, Yong-chao Lianga, , Ze-an Tiana, , Yue-hong Zhanga, Quan Xiea, Ting-hong Gaoa, Yun-fei Mob a b
School of Big Data and Information Engineering, Guizhou University, Guiyang 550025, China School of Electronic and Communication Engineering, Changsha University, Changsha 410003, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Molecular dynamics simulation Cooling rates Rapid solidification Topologically close-packed cluster Configuration entropy
Herein, the effects of cooling rate during rapid solidification of Mg40Al60 metallic glasses are studied using the molecular dynamics (MD) simulation. The system energy, pair distribution function (PDF), local structures, configuration entropy, and three-dimensional visualization are used to systematically analyze the evolution of microstructure with temperature. Results indicate that the lower the cooling rate, the lower the average atomic energy. With the decrease of cooling rate, increases the height difference between two splitting subpeaks of the second peak on PDF curves; while decreases the onset temperature of glass transition. The effect of the cooling rate on the topologically close-packed (TCP) clusters is similar to the local five-fold structures of S555: the lower the cooling rate, the more the amount of TCP clusters, and the lower the configuration entropy. In addition, most TCP clusters are centered the smaller Al atoms.
PACS: 61.20.Ja 61.25.Mv 64.70.pe 71.15.Pd
1. Introduction Formation of amorphous alloys is a dynamic process, and its atomic arrangement is closely related to thermal historic conditions, such as cooling rates and initial conditions that affect both phase transition and microstructure [1]. Rapid cooling may result in some novel structural configuration and outstanding material properties [2]. In other words, differences can be found in the microstructures and material properties of a metal melt during the rapid solidification process at various cooling rates. Therefore, It is important to profoundly understand the microstructural evolutions of liquid metal at different thermal historic conditions under rapid solidification. In recent years, microstructure of pure metals and binary alloys obtained by rapid cooling has been investigated extensively. MD simulation about silver at six cooling rates revealed that an amorphous solid can be obtained with the cooling rate γ ≥ 1.0 × 1013 K/s; as cooling rate ranging within 5.0 × 1012 − 1.38 × 1011 K/s, the coexistence structures of the metastable hcp with stable fcc configurations can be formed with phase separating or layering structures [3]. The rapid solidification experiment at different cooling rates found that cooling rate effectively tailored the microstructure and characteristic
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temperatures of NieTi alloy [4]. For binary TiAl alloys there is also a strong dependency of the microstructure on the Al content and cooling rates [5]. MD simulation for pure Zr concluded that the critical cooling rate of vitrification was approximately 5.0 × 1013 K/s, and a rather perfect bcc phase can be obtained at 1012 K/s [6]. MD investigation upon the correlation between the phase selection of crystal or glass and cooling rate of pure copper demonstrated that the crystalline fractions largely fluctuated along with the cooling rates ranging in 6.3 × 1011−16.6 × 1011 K/s [7]. For good glass former Cu50Zr50 MD simulation shows that a faster cooling rate will resulte in a less amount of icosahedra-like clusters [8], which may shed great guiding significance in comprehending the relation between rapid solidification and cooling rate of such binary alloys. As a type of magnesium-based alloy composed mainly of Al with less Mg or other metal elements for enhancing its hardness [9], Magnesiumaluminum (MgeAl) alloys not only have been widely applied in industry, but also are an important model system for studying materials science and condensed matter physics, playing an important role in the development of amorphous alloy materials [10]. From this viewpoint, herein, MD simulation approach is adopted to explore the influence of cooling rate on the microstructures of Mg40Al60 liquid alloy during the
Corresponding authors. E-mail addresses: [email protected] (Y.-c. Liang), [email protected] (Z.-a. Tian).
https://doi.org/10.1016/j.jnoncrysol.2019.119578 Received 22 March 2019; Received in revised form 7 July 2019; Accepted 15 July 2019 0022-3093/ © 2019 Elsevier B.V. All rights reserved.
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of Tg and E of the system. The slower the cooling rate is, the system is more close to an equilibrium state at a certain temperature, and hence the harder to achieve a nonequilibrium transition; therefore, Tg of the system decreased with a decrease in cooling rate. However, in practice, there is a weak dependence of Tg on the cooling rate; even if the cooling rate changes by an order of magnitude, the Tg value changes by only approximately 10 K. The glass transition phenomenon, wherein Tg reflected the nature of the liquid to some extent, showed that the nature and behavior of the amorphous state changed with time [21]. This fact is significant for exploring the microstructural characteristics of the binary alloys.
rapid solidification process. Alloy structure evolution is also profoundly analyzed and investigated from different viewpoints based on a series of effective means, such as the energy-temperature (E-T) relationship [11], pair distribution function (PDF or g(r)) [12], Honeycutt-Andersen (HA) bond-type index [13], the largest standard cluster analysis (LSCA) [14], TCP structures [15], configuration entropy [16] and three-dimensional (3D) visualization. The microscopic structural characterization in Mg40Al60 alloy is discussed in detail. 2. Simulation details MD simulation was performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) [17]. A total of 10,000 atoms, i.e., 4000 Mg atoms and 6000 Al atoms, were randomly placed in a cubic box subsequently operated according to a periodic boundary condition. In this case, the interaction potential between the atoms was indicated by an embedded-atom model (EAM) developed and optimized by Mendelev et al. [18]. The time step for the system operation was set at 10−15 s. The relevant analog computation started at 2000 K, much higher the equilibrium melting point of the MgeAl alloy (no > 1300 K) [19]. The isothermal relaxation lasted for 1,000,000 steps drives the system to an equilibrium state. The system was then cooled at different cooling rates of 1010, 1011, 1012, and 1013 K/s to the proposed temperature of 300 K. In such a rapid temperature reduction, the structural configuration data, i.e., moving speed and space coordinates of all the atoms, should be recorded at an interval of 10 K to acquire the microcosmic information of the system at the corresponding temperature. Then, the E-T curve, g(r) curve, bond-type index, LSCA, TCP structures, configuration entropy, and 3D visualization were adopted to analyze the structural configurations of the system and investigate the formation and evolutionary rules of the alloy microstructures at different temperatures and cooling rates.
3.2. Difference in medium-range order While E-T graph merely reflects some simple statistical information about the microstructure, the PDF or g(r) curve is an important method for describing the overall structural variation characteristics of the system. It refers to the probability of other atoms to appear in a spherical shell with a certain distance from a selected central atom. It is used to calculate the average value of all the atomic distribution probabilities. Being extremely objective and extensively applicable, PDF has been widely applied to investigate the structural characteristics in the field of materials science. Fig. 2 shows the g(r) curves of Mg40Al60 alloy at 300 K obtained at four cooling rates. No distinct differences were found in the first peaks. All the second peak of the g(r) curves clearly split, indicating the formation of metallic glass. Nevertheless, the two subpeaks split from the second peak of the g(r) curves have different relative height, which increases with the decrease of cooling rate; that is 0.228, 0.246, 0.263, 0.282 at 1013, 1012, 1011, and 1010 K/s, respectively. Therefore the structure is different in medium-range order. Fig. 3 shows the g(r) curves during rapid solidification process of the Mg40Al60 alloy at four cooling rates. The four groups of curves clearly have the same change trend. With temperature decrease, the first peak of the g(r) curves became sharp; whereas the second peak exhibit becomes two shoulders as T = 530, 540, 550, and 560 K at 1010, 1011, 1012, and 1013 K/s, respectively. The distance between atoms indicated by the shoulders is more than the nearest neighbors, so that some medium-range order formed in the system. Generally onset temperature of the occurrence of the split second peak on g(r) curve is defined as Tg. The value change of Tg with cooling rates indicates the cooling rate dependence of microstructures.
3. Results and discussion 3.1. Evolution of energy As a simple but effective analysis, the varying of average atomic energy (E) [20] can truly and objectively reflect the preliminary feature of rapid solidification. Fig. 1 shows the relation between E and T of the system at different cooling rates. At four cooling rates, the average atomic energy gently declines without sudden changes, and hence amorphous structures should be formed; at a certain temperature the E values are very close to each other, and at high temperatures evolve in a similar tendency, which reveals that the cooling rate exerted an limited influences. The glass transition temperature (Tg) can be evaluated as approximately 560, 550, 540, and 530 K, at 1013, 1012, 1011, and 1010 K/s, respectively. Therefore the lower the cooling rate, the lower value
3.3. Short-range order The PDF describes the distribution of atom pairs in a system, and hence reveals that feature of short-, medium-, and long-range orders. However, it fails to explicitly reflect the geometrical features of atomic arrangement. The microstructure (short-range order) composed of
Fig. 1. Average atomic energy of Mg40Al60 alloy changes with T at different cooling rates.
Fig. 2. g(r) of Mg40Al60 alloy at 300 K obtained at four cooling rates. 2
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Fig. 3. g(r) changes with T at four cooling rates.
neighboring atoms may be different even if the first peaks seem no changed at different cooling rate. Hererin, the structure composed of a reference pair of atoms and their common near neighbors (CNNs) is called a common neighbor sub-cluster (CNS) [22], because such structure reflects the topological feature shared by a group of atoms that are a part of a cluster around a center atom. CNS was proven to be successful in describing the local configurations of the liquid, amorphous, and crystal structures. To characterize the bond-type objects, S555, S544, S433, S421, S422, S444, and S666 are consistent with common 1551, 1541, 1431, 1421, 1422, 1441, and 1661 of the HA bond-type index. Generally, S555, S544, and S433 are closely related to amorphous structure; whereas S421, S422, S444, and S666 are related to crystalline structures. Fig. 4 shows the variations of primary CNSs during rapid solidification process of the liquid Mg40Al60 alloy. At 300 K, S555 is always predominant with the maximum values of 36.79%, 40.04%, 43.86%, and 46.91% at the cooling rates of 1013, 1012, 1011, and 1010 K/s, respectively (Fig. 4(a)). Moreover, the upward trend of the four curves appeared as a plateau after 560, 550, 540, and 530 K, respectively. The relative numbers of the S544, S433, S444, and S666 first increase, and then decline, as shown in Figs. 4(b)-(e). They also reached maximum values of 17%, 20%, 7.5%, and 10% at approximately 850, 1350, 800, and 600 K, respectively. In liquid and super-cooled liquid, the four CNSs comprise various local structures; subsequently, they began to reduce and exhibited differences. The lower the cooling rate, the smaller the proportion occupied by the four CNSs. The relative numbers of the crystalline S421 and S422 start to gently decline at T < Tg, and then gradually increase, as shown Figs. 4(f)-(g). The relative numbers of S421 and S422 at a cooling rate of 1013 K/s can reach 3% and 7% at 300 K, respectively; but < 4% and 8% at 2000 K. Only the amorphous S555 remarkably increases, and apparently far greater than that of the other six bond types. At four cooling rates, the amorphous structures that eventually formed in the system were mainly ascribed to the massive S555.
describe the different basic clusters formed by an atom with its nearest neighbors. Investigations on the atomic configurations and structural characterization of the cluster are significant in broadening the subject fields of materials science and condensed matter physics. Structural analysis is critical for measuring material properties and functions. The applications of particle coordinate quantization and local structure identification for disordered systems based on 3D are still in the phase of development. Herein, LSCA [23] was adopted to effectively quantify the 3D structures according to the spatial distribution and characterize various cluster structures independent of any preset parameters and beyond the range of the nearest neighbors. A cluster is composed of all CNSs that share the same atom; if all CNSs in such a cluster are free from common-neighbor sub-ring (CNSR) and multi-bond point, it is called a standard cluster (SC); and the largest SC around a certain atom is unique, called a LSC (the largest standard cluster). In the system under consideration, all LSCs can be identified and described by a group of binary tuples that is compose of a integer and a notation of CNS [22], where the integer represents the number of this CNS in the LSC. Therefore, the icosahedra (ico), hexagonal close-packed (hcp), and body-centered cubic (bcc) LSCs are successively denoted as [12-S555], [6-S421, 6-S422], and [6-S444, 8-S666], respectively. Atoms can be entirely classified according to the LSC types around them. For example, the centers of the fcc, hcp, and bcc LSCs are referred to as the fcc, hcp, and bcc atoms, respectively. Considering a local structure unique to every atom, the radius (r) of the LSC can change, and all the LSCs can be determined by different cutoff distances (rc) [24]. Such a method of LSC overcomes the deficiencies in the structural characterization of the previous clustering scale approach, such as the structural uncertainties caused by the dependence on a preset parameter. By characterizing an internal structure of the atomic clusters in a general disordered system, LSC is available to effectively improve our understanding on the structure-property relation. TCP LSCs are compact-structured clusters formed by n4-S444, n5S555, and n6-S666 CNSs containing at least one CNS of S555. TCP structure follows the Euler's theorem, i.e., face number (NF) + vertex number (NV) − edge number (NE) = 2; that is, an equation set of {n4 + n5 + n6 = CN, 2n4 + n5 = 12, n5 > 0} is satisfied. Therefore, the topological feature of a TCP LSC can be described by a TCP-index of “Ln”, where “L” is Z, A, B, C, D, or E, indicating n4 = 0, 1, 2, 3, 4, 5; and
3.4. Medium-range order Although CNSs can be successfully used to describe the bonding relation between an atom and its nearest neighbors, they still cannot 3
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Fig. 4. Relative number of various CNSs with T at different cooling rates.
Fig. 5. Schematic diagram of 5 TCP LSCs.
n refers to CN. In the case of n4 = 6 (where n4 = 6 means label “F”), the TCP-index is still valid for the topology; for example, the bcc unit of [6S444, 8-S666] can be denoted as F14, although it is a non-TCP LSC owing to the absence of S555. All the Frank-Kasper clusters (only formed by S555 and S666) [25] are apparently TCP LSCs. Four kinds of Kasper polyhedrons are denoted as K12 (120120), K14 (14 0 12 2), K15 (15 0 12 3), and K16 (16 0 12 4) by a CTIM method [26]. In line with the concepts of the abovementioned terms, Z12 (ico), Z14, Z15, and Z16 may be selected for replacing K12, K14, K15, and K16, respectively, to characterize the cluster structures. As shown in the experiment, the TCP LSCs, as a group of compact local structures, are ubiquitous in metallic glass [27]. Fig. 5 shows 5 common TCP LSCs. As shown in Fig. 6, with the decrease of temperature the sum number of all TCP LSCs increases independent of cooling rate at T > Tg; that is, at different (four) cooling rates at a certain temperature the TCP total number is almost the same value; and the lower the temperature, the higher the increase rate. When T is approaching to Tg, the increase rate of the TCP total number start to depend on cooling rate: the lower the cooling rate, the higher the increase rate. With temperature further decrease, the increase rate decreases, and inclines to be constant at a state of plateau that appears T < Tg. Comparing the relative number of S555 (Fig. 4) with that of TCP LSCs (Fig. 6), one can
Fig. 6. Numbers of the TCP LSCs with T at four cooling rates.
conclude that the variation of TCP LSCs is consistent with that of S555. Table 1 lists the details of primary TCP LSCs (Z12, A13, Z14, Z15, and Z16) in Mg40Al60 alloys at 300 K under four cooling rates. The number of Z12 LSC is apparently more than others under all cooling rates, and decreases from 41.80% to 24.27% of all TCP LSCs as cooling 4
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based on LSCs. The LSCs play a major role in analyzing and quantifying the structural evolution in a liquid-solid transition. As known, the configuration entropies of all the cooling processes reduced with a decline in temperature. At T > Tg, the variation curve of configuration entropy of the system with temperature is almost the same under four cooling rates; while suddenly changed at Tg. In other words, the four overlapped curves separate at T < Tg. As aforementioned, Z12 atoms play a critical role at lower cooling rates during rapid solidification, and the lower the cooling rate, the more the number of Z12 atoms; and the kinds of LSCs that the formation of more ordered local structures would decrease the configuration entropy. Therefore, the number of Z12 atoms and configuration entropies change contrarily (see Figs. 6 and 7). Perhaps, the study between the amorphous structure and entropy is one of the approaches making a breakthrough in recognizing the essence of amorphization [30,31].
Table 1 Details for primary TCP LSCs at 300 K for four cooling rates.
Number of Z12 Number of A13 Number of Z14 Number of Z15 Number of Z16 Total of TCP LSCs Proportion of Z12 (%) System energy (eV) Energy of Z12 (eV)
1010 K/s
1011 K/s
1012 K/s
1013 K/s
991 228 137 127 101 2371 41.80 −2.52 −3.24
788 229 100 121 66 2149 36.67 −2.52 −3.22
558 198 96 67 49 1845 30.24 −2.51 −3.21
400 218 69 51 34 1648 24.27 −2.51 −3.16
TCP LSCs is the researching focus in this paper. And Z12 is the largest number of TCP LSCs. So two of them and their proportion are bolded.
rate increase from 1010 K/s to 1013 K/s. The quantities of A13, Z14, Z15, and Z16 also decrease with the increase of cooling rate. The average atomic energy of the system and Z12 atoms slightly increases with the increase of cooling rate. It signified that the amorphous system structure is more stable obtained at a low cooling rate. With lower average energy, Z12 atoms have a substantial influence on the stability of Mg40Al60 amorphous alloys.
3.6. Visualization analysis 3D visualization [32] was utilized herein to further analyze the structures. Its microstructural configurations then visually distinguished the influence of the cooling rate on the solidified structure, so as to have a deeper understanding of the atomic arrangement and the major cluster distributions of Mg40Al60 alloy. Fig. 8 specifically shows the distribution of five TCP atoms in the system at 800 (> Tg), 500 (~ Tg), and 300 K (< Tg) obtained at four cooling rates. As intuitively indicated by the 3D distribution diagram, the number of Z12 (red) atoms remarkably increases with the decrease of temperature and reached the maximum at 300 K. Based on the number of Mg and Al atoms shown in the pictures in the last column of Fig. 8, we can get the number ratio of Mg to Al atoms, being 27.18% (165 / 607), 27.54% (209 / 759), 27.84% (284 / 1020), and 27.74% (344 / 1240) at 1013, 1012, 1011, 1010 K/s, respectively. Therefore, among all TCP atoms the proportion of Mg is basically 20% (= 165 / (165 + 607) as a example), almost unchanged with cooling rate. The proportion of the Al-centered TCP LSCs was absolutely predominant and more than its concentration in Mg40Al60 alloys. This bias is consistent with the fact that the Al-centered ICO LSCs is geometrically favored, because the radius of Mg atom is 1.60 Å, and Al atom 1.43 Å. Fig. 9 shows the specific numbers of Z12, A13, Z14, Z15, and Z16 atoms nearby Tg at different cooling rates. The number of Z12 is always much more than others in all the rapid cooling processes. In particular, in the system at 300 K it is 400, 558, 788, and 991 under 1013, 1012, 1011, and 1010 K/s respectively. Therefore, Z12 atoms increase with the decrease of cooling rate. In other words, the formation of Z12 LSCs plays a vital role in glass forming and is conspicuously influenced by the cooling rate; while A13, Z14, Z15 and Z16 atoms only change a little at four cooling rates.
3.5. Configuration entropy Entropy is a state function of a real physical system, which is related to the chaos of a system. Several empirical rules are put forward during the long-term exploration of the glass-forming systems and used to express the uniformity of any kind of energy distribution in the space. The more uniform the energy distribution, the bigger the entropy. Entropy can directly reflect the uniformity of a system as a measure of the ordered degree. The smaller the entropy of a system is, the more orderly the system. According to the rule of thermodynamics, the entropy of liquid with an disordered structure is larger than that of crystal with an ordered structure. The analysis method of configuration entropy [28,29] is adopted to better understand the changes in the structure. The configuration entropy is here defined as s = −sum(ρilogρi), where ρi is the proportion (< 1.0) of the i-th kind of structure. For fcc, hcp, and bcc crystals that are composed of only one LSC, the configuration entropy is zero, i.e., s = 0. Obviously the more the kinds of structures, the smaller the value of ρi, and the larger the value of s. The configuration entropy model can effectively establish the relationship between order degree and local structures especially for liquids and amorphous alloys. As an important structural analysis, it has a profound meaning to the studies based on the material amorphousness and randomness. Fig. 7 shows the relation between T and the configuration entropy
4. Conclusions By employing classical MD simulation, this study systematically studied the evolution of microstructure in Mg40Al60 alloys obtained by rapid solidification under four cooling rates of 1013, 1012, 1011, and 1010 K/s. The following conclusions are obtained: (1) Mg40Al60 is a good glass former, vitrified even if cooling rate is low to 1010 K/s. The cooling rate has a crucial influence on the microstructural characteristics of Mg40Al60 alloy during rapid solidification. (2) The analysis upon CNS (S555), TCP clusters, and configuration entropy reveals Tg decreases with the decrease of cooling rate, being approximately 560, 550, 540, and 530 K at 1013, 1012, 1011, and 1010 K/s, respectively. (3) A lower cooling rate benefits the formation of a more stable Mg40Al60 glass, and hence benefits the formation of TCP clusters, reduces the configuration entropy.
Fig. 7. Configuration entropy based on LSCs with T at four cooling rates. 5
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Fig. 8. The distribution of Z12, A13, Z14, Z15, and Z16 atoms at 800, 500, and 300 K obtained at four cooling rates. The pictures in the last column is used to distinguish the difference between Mg and Al atoms of TCP atoms at 300 K.
Acknowledgement This work has been supported by the National Natural Science Foundation of China (Grant No.61264004, 51661005 and 51671004), the Theoretical Physics Special Project of National Natural Science Foundation of China (Grant No.11747123), the Natural Science Foundation of Hunan Province of China (Grant No.2018JJ3560), the Project of Science and Technology Plan of Changsha (Grant No. kc1809022), Sci-Tech Cooperation Program of Guizhou Province of China (Grant No. [2017]5788-81). References [1] C.D. Cao, Z. Sun, X.J. Bai, L.B. Duan, J.B. Zheng, F. Wang, Metastable phase diagrams of Cu-based alloy systems with a miscibility gap in undercooled state, J. Mater. Sci. 46 (2011) 6203–6212, https://doi.org/10.1007/s10853-011-5612-7. [2] D.B. Liu, Y.C. Liu, Y. Huang, R. Song, M.F. Chen, Effects of solidification cooling rate on the corrosion resistance of Mg-Zn-Ca alloy, Prog. Nat. Sci-Mater. 24 (2014) 452–457, https://doi.org/10.1016/j.pnsc.2014.08.002. [3] Z.A. Tian, R.S. Liu, H.R. Liu, C.X. Zheng, Z.Y. Hou, P. Peng, Molecular dynamics simulation for cooling rate dependence of solidification microstructures of silver, J. Non-Cryst. Solids 354 (2008) 3705–3712, https://doi.org/10.1016/j.jnoncrysol. 2008.04.006. [4] G.J. Pan, C. Balagna, L. Martino, S. Spriano, Microstructure and transformation temperatures in rapid solidified Ni-Ti alloys, part I: the effect of cooling rate, J. Alloys Compd. 589 (2014) 628–632, https://doi.org/10.1016/j.jallcom.2013.10. 130. [5] C. Kenel, C. Leinenbach, Influence of cooling rate on microstructure formation during rapid solidification of binary TiAl alloys, J. Alloys Compd. 637 (2015)
Fig. 9. Numbers of Z12, A13, Z14, Z15, and Z16 atoms nearby Tg at four cooling rates.
(4) Visualization unveils that both TCP clusters and two kinds of atoms distribute homogeneously in the space, and the Al-centered TCP clusters is absolutely predominant.
Declaration of Competing Interest The authors declare no conflict of interest. 6
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