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Effect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: A molecular-dynamics study
Accepted Manuscript Effect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: A molecular-dynamics study X.X. Yue, C.T. Liu, S.Y. Pan, A. Inoue, P.K. Liaw, C. Fan PII:
S0921-4526(18)30478-2
DOI:
10.1016/j.physb.2018.07.030
Reference:
PHYSB 310984
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 23 April 2018 Revised Date:
7 July 2018
Accepted Date: 28 July 2018
Please cite this article as: X.X. Yue, C.T. Liu, S.Y. Pan, A. Inoue, P.K. Liaw, C. Fan, Effect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: A molecular-dynamics study, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.07.030. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Effect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: a molecular-dynamics study X.X. Yue,1 C.T. Liu,2 S.Y. Pan,1 A. Inoue,3,4 P.K. Liaw,5 C. Fan,1,a) 1
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School. of Mater. Sci. & Eng. / MIIT Key Lab. of Adv. Metal. & Intermetal. Mater. Tech. , Nanjing University of Science & Technology, Nanjing 210094, P.R. China;
2
Cent. for Adv. Struc. Mater., MBE, College of Sci. & Eng., City University of Hong Kong, Hong Kong, P.R. China; International Institute of Green Materials, Josai International University, Togane 283-8555, Japan;
4
Department of Physics, King Abdulaziz University, Jeddah 22254, Saudi Arabia;
5
Depart. of Mater. Sci. & Eng., the University of Tennessee, Knoxville, USA;
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3
a)
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Author to whom correspondence should be addressed; E-Mail: [email protected];
Abstract
In this paper, the molecular dynamics simulations are utilized to study the cooling rate effect and to understand the relationship among the local atomic structure, free
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volume, and mechanical property in Cu50Zr50 metallic glass. The radial distribution function, bond pair analysis technique, and Voronoi tessellation are performed to characterize the structure evolution and local atomic configurations during the cooling process. The results demonstrate that a faster cooling rate results in a higher glass
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transition temperature and less amount of icosahedra-like clusters. It has been recognized that the concentration of free volumes presents a strong evidence of
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upward trends as the cooling rate increases. The analyses for the free volume and Voronoi polyhedron indicate that icosahedral-like clusters show a lower free volumes as compared with the rest clusters, revealing that well-developed icosahedra-like clusters in Cu50Zr50 make the system densely packed and lower free volume structure. In addition, the simulated alloy obtained at a lower cooling rate exhibits a higher yield strength and elastic modulus, all of which may attribute to the structure with more densely icosahedral-like clusters and less free volumes. Key words: cooling rate; free volume; icosahedra-like; Cu50Zr50
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1. Introduction It is widely known that metallic glasses exhibit many useful properties which cannot be obtained for crystalline alloys. In particular, the excellent mechanical properties make them very promising candidates in engineering applications1, 2. As
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well known, the cooling rate is one of the most important preparation factors for the formation of metallic glasses3-5. A slow cooling rate provides enough time to the system to equilibrate, that is, the atoms have more time to move, which may lead to a crystal structure. However, a faster cooling rate restricts the atomic diffusion and the
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slower atomic movement makes the alloy remain a liquid structure and results in a metallic glass6, 7. Even in fast cooling process, the constituent atoms can be rearranged
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into a short-range ordered and/or medium-range ordered states, resulting in a decrease of free volume fraction. The free volume as "flow defect" is believed to play a significant role in mechanical properties8-10. For example, at the temperature near glass transition temperature, the metallic glass with a high yield stress changes to one with a very low yield stress, depending on the variation of free volumes11. The
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characterization of free volumes is not easy, and hence the influence of the local structure on the free volume at different cooling rates is less concerned in spite of very importance. Therefore, it is necessary to uncover the influence of the atomic structure
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on the free volume in metallic glasses. The short-range orderings (SROs) characterized by different types of polyhedra have been widely accepted12. Recently,
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the free volume on atomic level has been defined and studied13-15, which provides a good way to evaluate the relationship between free volumes and cooling rates, as well as the correlation between the free volume and atomic structure in metallic glasses. Compared with ternary and multi-elemental metallic glasses, the Cu-Zr binary
glassy alloy reduces the complexity of local atomic structures16, and it is easier to understand the local atomic structure. In addition, the Cu-Zr binary alloy is widely regarded as a good candidate for bulk metallic glasses (BMGs)17. In this paper, we firstly consider a simple metallic glass Cu50Zr50 as the model system which can be obtained under different cooling rates by the molecular dynamics (MD) simulations. 2
ACCEPTED MANUSCRIPT The MD has the power and advantage in revealing some information of structural in conjunction with properties, but it is difficult to achieve in laboratory experiments. Based on the radial distribution function (RDF), and bond pair analysis technique, this study analyzes the evolution of structures during the cooling process, and calculates
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the free volume at 300 K. The relations between the free volume and the icosahedra-like clusters is also investigated by analyzing the Voronoi polyhedron. Besides, a simple tensile strain process is performed to show the influence of cooling
2. Computation details The
MD
simulation
was
conducted
by
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rates on mechanical properties.
employing
the
large-scale
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atomic/molecular massively parallel simulator (LAMMPS)18. The EAM potential proposed by Mendelev et al19 was adopted to describe the interactions in the Cu-Zr system. The system was applied with the periodic boundary condition (PBC) for all three-dimensions, and the time step of 0.001 ps was chosen. Nose-Hoover thermostat and barostat were used to control the temperature and pressure, respectively. Firstly,
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the initial box containing 6750 atoms was constructed by the replication of the 250-atom configuration, in which the number of Cu atoms is 3375 and that of Zr atoms is the same as Cu atoms. The simulation was performed at 2000 K that is much
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higher than the experimental melting temperature20. An equilibrated liquid state of Cu50Zr50 was obtained by relaxing 1.0 ns at 2000 K. Followed by quenching the
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model to 300 K at different cooling rates (0.5 × 1011 K/s - 0.5 × 1013 K/s) in the NPT (constant atom number, pressure, and temperature), and then by relaxation at 300 K to get the desired glassy states. For the simulation of mechanical properties, the periodic boundary condition was adopted. The samples obtained at 300 K with different cooling rates were deformed along the z axis at a constant deformation rate of 109 s-1. When strain reached about 15%, we ended the simulation.
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3. Results and discussions
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Figure 1. Volume as a function of temperature under different cooling rates.
Figure 1 shows the volume-temperature plots of the rapidly solidified process when the Cu50Zr50 alloy liquid is cooled at three different quenching rates. The
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variation of the volume occurs continuous and no dramatic drop in the volume is seen upon the whole cooling. The slope of volume versus temperature curves decreases
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below 800 K, indicating the formation of the metallic glass. Since the metallic glass is regarded as a frozen liquid, the variation of the configurational entropy is vanished. Thus, the derivative of entropy with respect to pressure is the derivative of volume with respect to temperature21. The glass transition temperature is dependent on cooling-rate. The higher cooling rate results in a shorter time for the atoms to relax. Thus the falling out of the liquid-state equilibrium occurs earlier, leading to a higher glass transition temperature22. In the present work, the similar tendency is also observed. By dividing the curve into two parts, the intersection is obtained via fitting a straight line to each piece. This intersection is regarded as the glass-transition 4
ACCEPTED MANUSCRIPT temperature23. The glass-transition temperatures at the cooling rate of 0.5 × 1011 K/s,
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0.5 × 1012 K/s, and 0.5 × 1013 K/s are 675 K, 686 K, and 705 K, respectively.
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Figure 2. Total RDF at 300 K obtained under different cooling rates. In general, the RDF, is one of the most important methods to reveal the structure features of a system24. It is often used to describe a structure that is related to the number of finding the center of particle at a given distance from the center of another particle. Figure 2 depicts the g(r)total curves of the model structure at 300 K obtained at
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different cooling rates. For the as-casted samples with different cooling rates, their total RDF curves are very similar. All the curves exhibit a relatively-sharp first peak, a split second peak, and a blunt third peak within the cutoff distance. The first peak is fairly sharp, which is derived from the atomic packing and configuration of the
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nearest neighbor shell, indicating that there exists a strong SRO. It is worth mentioning that the splitting of the second peak is the main characteristic feature in
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the total RDF of a metallic glass25. Thus, quenching the Cu50Zr50 alloy from 2000 K to 300 K at different cooling rates of 0.5 × 1011 K/s - 0.5 × 1013 K/s leads to a metallic glass. The third weakening peak corresponds to the third-nearest neighbors, demonstrating that a little correlation remains. However, it is noted that the influence of cooling rate on the structure cannot be distinguished clearly. Although the RDF just provides the overall description of the structures, it is difficult to describe the details of the local structure. In order to gain a further insight of the atomic structure, the Honeycutt-Anderson (H-A) pair-analysis technique26 was made to determine the detailed structure of simulated systems from the 5
ACCEPTED MANUSCRIPT atomic-bonding view. The first minima of the partial radial distribution function (PRDF) that is not shown here are chosen to be the cutoff distances. The two atoms are taken as the nearest-neighbors and form a bond pair if they are within the cutoff distance. The H-A has been proved successfully to provide the detailed information
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about the microscopic local structure and short-range order in disordered systems, such as liquid and glass27. In this technique, four integers (ijkl) are assigned to classify the pairs of atoms: (1) i is 1 when the two given atoms (a root-pair) are bonded, and i is 2 if the root pairs are not bonded; (2) j is the number of near-neighbor atoms shared
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in common by the two given atoms; (3) k stands for the number of nearest-neighbor bonds among the shared neighbors; (4) l is adopted when the first three integers is not
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sufficient to distinguish the bond geometries. Figure 3 displays the typical bonded
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pairs, the pink balls are the root pair atoms, the blue balls are the near-neighbor atoms.
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Figure 3. Typical bonded pairs: (a)1421; (b)1422; (c)1431; (d)1441; (e)1541; (f)1551; (g)1661;
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Figure 4. Honeycutt-Anderson pair-analysis at 300 K under different cooling rates.
Figure 5. Honeycutt-Anderson pair-analysis during the cooling process: (a) 1551; (b) 1431; (c) 1541; (d) 1422; (e) 1421; (f) 1661; (g) 1441. Figure 4 presents the relative fraction of main bonded pairs in Cu50Zr50 metallic glass at ambient temperature obtained under various cooling rates. It can be noted that the amounts of the ideal icosahedral (ICOS) order (1551) and deformed ICOS order (1541 and 1431) are dominant and the total fraction of them is more than 65%. Nevertheless, the numbers of crystalline type bonded pairs 1422, 1421, 1661, and 7
ACCEPTED MANUSCRIPT 1441 are small percentages. The 1661 and 1441 bond-types are the characteristic bonded pairs for the body-centered-cubic (bcc) crystalline structure. The 1421 and 1422 bond-types exist mainly in the close-packed crystalline structure such as face-centered-cubic (fcc) and hexagonal-close-packed (hcp) clusters. This trend
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implies that icosahedra-like order configurations, including ideal icosahedral types and deformed types, are the main structural units in the amorphous phase, although the crystalline-type-bonded pairs can also exist in the amorphous phase. In addition, the overall icosahedral-like order increases with decreasing the cooling rate due to the
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obvious increase of 1551 at the lower cooling rate. Figure 5 shows the fraction of these bonded pairs as a function of temperature during the rapid solidification process
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with different cooling rates. The fractions of 1541, 1661, and 1441 increase with the decrease of temperature, but they are affected slightly by the cooling rate. When the supercooled liquid transforms into the glass state, the fractions of 1431, 1422, and 1421 decrease slightly as the cooling rate descends. It is worth noting that the slope of the 1551 curve changes obviously near the glass-transition temperature, especially at
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low cooling rates, denoting a substantial increase of locally-favored structures. Recently, the free volume on the atomic level has been defined and studied13-15. This method provides a good tool to study the relationship between the atomic
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structure and free volume in a system. In this method, Nic is taken as the nearest integral number to the corresponding average coordination number (CN) obtained by the Voronoi tessellation analysis. In this paper, Nic of Cu-centered clusters and
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Zr-centered clusters calculated in Cu50Zr50 are 12 and 15, respectively. A cluster with coordination numbers, ni, within a cutoff of rij less than Nic is regarded to have the free volume. Otherwise, there is no free volume. The free volume around every Cu (Zr) atom in the system is defined as: 3
4 1 V fi = ∑ j = ni +1 π ( Rij − rij ) , 3 2 Nic
where Rij represents the distance between the central atom, i, and coordination atom, j, rij is 1.10 times of the first peak position in the partial RDF curve by considering the contribution of the thermal vibration based on Lindemann's melting criterion28,29. 8
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∑
m i =1
V fi , where m is the
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total number in the system30.
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Figure 6. Amount of total free volumes under different cooling rates. Figure 6 displays the total free volume in the system at 300 K under different cooling rates. With increasing of the cooling rate, the free volume presents a strong evidence of upward trends. During the fast cooling process, the annealing rate of the
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free volume will be too slow to maintain an equilibrium state, and the excess free volume will be frozen in, meaning that a larger amount of free volumes is quenched in
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the glassy solid.
Figure 7. Fraction of icosahedral-like, crystal-like, mixed, and other type atoms under different cooling rates.
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Cry-Like 0.40 0.40 0.41
Mixed 0.38 0.38 0.38
Other 0.40 0.40 0.41
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Cooling rate (K/s) 0.5 × 1011 0.5 × 1012 0.5 × 1013
The Voronoi tessellation31 developed by Voronoi has been widely used for the characterization of the short-range order and local atomic environment. This method
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is employed here to explore the relation between free volume and atomic structure. The Voronoi polyhedron (VP) is expresses as, where n3, n4, n5, and
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n6 represent the numbers of triangles, tetragons, pentagons, and hexagons in the Voronoi polyhedron. An i-edged face reflects the local symmetry of the central atom with some nearest-neighbor atoms in a certain direction32. The sum of the Voronoi index denotes the total CN. For example, the Voronoi index (VI) of <0, 0, 12, 0>, called a full icosahedron, describes an icosahedral polyhedron with exactly 12 faces
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with 5 edges each. In this paper, the VP is grouped into four types33: (1) icosahedral-like: <0, 0, 12, 0>, <0, 1, 10, x>, and <0, 2, 8, x>; (2) crystal-like: <0, 4, 4, x> and <0, 5, 2, x>; (3) mixed: <0, 3, 6, x>; (4) other: the remaining VP. Noted that x
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is between 0 and 4. Figure 7 presents the fraction of the four types of polyhedrons. The fraction of the icosahedral-like polyhedron is more than 30%, while the fraction
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of the crystal-like polyhedron is small. The existence of the icosahedral-like polyhedron has been proved by many studies. It has more pentagons and involves a five-fold symmetry that is associated with dense packing in metallic glasses34-36. Table 1 lists the average of free volumes of these four types of polyhedrons. It is noted that the icosahedral-like polyhedron has a smaller free volume, compared with crystal-like, mixed, and other type polyhedrons, illustrating that the structural configuration of pentagons is packed more densely. The slower cooling rate will result in more icosahedral-like polyhedrons, therefore, leading to a lower free volume.
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Figure 8. Stress-strain curves under different cooling rates.
It is well known that the changes of the structure and free volume can affect the
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mechanical property. This paper simply investigates the stress-strain curve of the simulated alloys at 300 K obtained under different cooling rates, as shown in Figure 8. As the cooling rate descends, the sample shows higher yield strength. It is in agreement well with the results of experiments37 and simulations38. In addition, it is obvious that modulus of elasticity increases with decreasing the cooling rate. At the
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low cooling rate, the system is more densely packed, resulting a better resistance
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capacity to elastic deformation.
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Figure 9. Honeycutt-Anderson pair-analysis during tension deformation: (a) 1551; (b) 1431; (c) 1541; (d) 1422; (e) 1421; (f) 1661; (g) 1441.
Figure 10. Variation of ICO-Like fractions during tension deformation.
Figure 9 shows the variation of bond type fractions during deformation. The fractions of crystalline type and 1541 type bonded pairs change slightly within 1%. While the 1551 and 1431 type bonded pairs change in a wider range. The 1551 presents a tendency of decrease and 1431 shows the upward trend. Generally, the 12
ACCEPTED MANUSCRIPT variation of the icosahedral-like bonded pairs first decreases with the strain, and following the content does not change obviously. Figure 10 displays the variation of icosahedral-like polyhedron versus applied strain. One can see that during deformation the fraction of icosahedral-like polyhedron first presents downward trend,
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and consequently saturates during the tension process. The variation of icosahedral-like polyhedron is in accordance with that of the icosahedral-like bonded pairs. Noted, the curves of icosahedral-like polyhedron versus strain tend to have a similar variation pattern under different cooling rates. The results of HA and voronoi
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analysis suggest that icosahedral-like bonded pairs will be weaken and
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icosahedral-like structure will be collapsed during the tension process.
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Figure 11. Variation of free volume fractions during tension deformation. The icosahedral-like polyhedron is densely packed and has a smaller free volume.
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Therefore, the amount of free volume is affected by the content of the icosahedral-like polyhedron. Figure 11 presents the variation of total free volume in system during tension, the curves under different cooling rates display a similar pattern. Compared with the variation of icosahedral-like polyhedron (Figure 10) and free volume (Figure 11), it can be observed that the decrease of icosahedral-like polyhedron is accompanied by the generation of the excess volume. In this process, the icosahedral-like polyhedron becomes more loosely packed structure, leading to the increase of the free volume. Then the amount of free volume reaches a saturated state when the fraction of icosahedral-like polyhedron fluctuates within a certain range. 13
ACCEPTED MANUSCRIPT From these results, it can be concluded that the icosahedral-like polyhedron plays an important role during the tension.
4. Conclusions Based on the RDF, H-A indices, and Voronoi polyhedron, this paper has focused
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on the study of the influence of cooling rate on the local structure, free volume, and mechanical property in Cu50Zr50 metallic glass under different cooling rates (0.5 × 1011 K/s - 0.5 × 1013 K/s) by performing the MD calculations. The main conclusions are summarized below:
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1. The faster cooling rate can cause both a higher glass transition temperature and more abundant of the frozen-in free volume in Cu50Zr50 metallic glass.
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2. Based on the detailed analyses of the H-A indices and Voronoi polyhedron, it has been confirmed that the icosahedra-like clusters are the predominant ones and play a key role in the amount of free volumes in the Cu50Zr50 metallic glass. The relative slow cooling rate makes the atoms have an enough time to move and this induces a more content of icosahedra-like clusters with a five-fold symmetry, all of
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these making the system densely packed with a low free volume. It is worthwhile to note that Voronoi analysis is more sensitive to the cooling rate than the H-A analysis.
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3. The mechanical properties are significantly influenced by atomic structures. The yield strength and elastic modulus are much higher at the lower cooling rate, and
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all of these are due to the different icosahedral-like content induced by the different cooling rates. During the tension deformation, icosahedral-like structure will be collapsed. Therefore, the more loosely packed structure results in the increase of the free volume.
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Acknowledgments The present work was supported by the NSFC (Grant No. 50971057 and 51371099). XXY is thankful for the financial support provided by the MTN Technologies, Inc. CTL is thankful for the financial support provided by City University of Hong Kong
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under the Program of the Structural Materials Development..
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DOI:
10.1016/j.physb.2018.07.030
Reference:
PHYSB 310984
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 23 April 2018 Revised Date:
7 July 2018
Accepted Date: 28 July 2018
Please cite this article as: X.X. Yue, C.T. Liu, S.Y. Pan, A. Inoue, P.K. Liaw, C. Fan, Effect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: A molecular-dynamics study, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/j.physb.2018.07.030. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Effect of cooling rate on structures and mechanical behavior of Cu50Zr50 metallic glass: a molecular-dynamics study X.X. Yue,1 C.T. Liu,2 S.Y. Pan,1 A. Inoue,3,4 P.K. Liaw,5 C. Fan,1,a) 1
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School. of Mater. Sci. & Eng. / MIIT Key Lab. of Adv. Metal. & Intermetal. Mater. Tech. , Nanjing University of Science & Technology, Nanjing 210094, P.R. China;
2
Cent. for Adv. Struc. Mater., MBE, College of Sci. & Eng., City University of Hong Kong, Hong Kong, P.R. China; International Institute of Green Materials, Josai International University, Togane 283-8555, Japan;
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Department of Physics, King Abdulaziz University, Jeddah 22254, Saudi Arabia;
5
Depart. of Mater. Sci. & Eng., the University of Tennessee, Knoxville, USA;
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3
a)
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Author to whom correspondence should be addressed; E-Mail: [email protected];
Abstract
In this paper, the molecular dynamics simulations are utilized to study the cooling rate effect and to understand the relationship among the local atomic structure, free
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volume, and mechanical property in Cu50Zr50 metallic glass. The radial distribution function, bond pair analysis technique, and Voronoi tessellation are performed to characterize the structure evolution and local atomic configurations during the cooling process. The results demonstrate that a faster cooling rate results in a higher glass
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transition temperature and less amount of icosahedra-like clusters. It has been recognized that the concentration of free volumes presents a strong evidence of
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upward trends as the cooling rate increases. The analyses for the free volume and Voronoi polyhedron indicate that icosahedral-like clusters show a lower free volumes as compared with the rest clusters, revealing that well-developed icosahedra-like clusters in Cu50Zr50 make the system densely packed and lower free volume structure. In addition, the simulated alloy obtained at a lower cooling rate exhibits a higher yield strength and elastic modulus, all of which may attribute to the structure with more densely icosahedral-like clusters and less free volumes. Key words: cooling rate; free volume; icosahedra-like; Cu50Zr50
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1. Introduction It is widely known that metallic glasses exhibit many useful properties which cannot be obtained for crystalline alloys. In particular, the excellent mechanical properties make them very promising candidates in engineering applications1, 2. As
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well known, the cooling rate is one of the most important preparation factors for the formation of metallic glasses3-5. A slow cooling rate provides enough time to the system to equilibrate, that is, the atoms have more time to move, which may lead to a crystal structure. However, a faster cooling rate restricts the atomic diffusion and the
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slower atomic movement makes the alloy remain a liquid structure and results in a metallic glass6, 7. Even in fast cooling process, the constituent atoms can be rearranged
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into a short-range ordered and/or medium-range ordered states, resulting in a decrease of free volume fraction. The free volume as "flow defect" is believed to play a significant role in mechanical properties8-10. For example, at the temperature near glass transition temperature, the metallic glass with a high yield stress changes to one with a very low yield stress, depending on the variation of free volumes11. The
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characterization of free volumes is not easy, and hence the influence of the local structure on the free volume at different cooling rates is less concerned in spite of very importance. Therefore, it is necessary to uncover the influence of the atomic structure
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on the free volume in metallic glasses. The short-range orderings (SROs) characterized by different types of polyhedra have been widely accepted12. Recently,
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the free volume on atomic level has been defined and studied13-15, which provides a good way to evaluate the relationship between free volumes and cooling rates, as well as the correlation between the free volume and atomic structure in metallic glasses. Compared with ternary and multi-elemental metallic glasses, the Cu-Zr binary
glassy alloy reduces the complexity of local atomic structures16, and it is easier to understand the local atomic structure. In addition, the Cu-Zr binary alloy is widely regarded as a good candidate for bulk metallic glasses (BMGs)17. In this paper, we firstly consider a simple metallic glass Cu50Zr50 as the model system which can be obtained under different cooling rates by the molecular dynamics (MD) simulations. 2
ACCEPTED MANUSCRIPT The MD has the power and advantage in revealing some information of structural in conjunction with properties, but it is difficult to achieve in laboratory experiments. Based on the radial distribution function (RDF), and bond pair analysis technique, this study analyzes the evolution of structures during the cooling process, and calculates
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the free volume at 300 K. The relations between the free volume and the icosahedra-like clusters is also investigated by analyzing the Voronoi polyhedron. Besides, a simple tensile strain process is performed to show the influence of cooling
2. Computation details The
MD
simulation
was
conducted
by
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rates on mechanical properties.
employing
the
large-scale
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atomic/molecular massively parallel simulator (LAMMPS)18. The EAM potential proposed by Mendelev et al19 was adopted to describe the interactions in the Cu-Zr system. The system was applied with the periodic boundary condition (PBC) for all three-dimensions, and the time step of 0.001 ps was chosen. Nose-Hoover thermostat and barostat were used to control the temperature and pressure, respectively. Firstly,
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the initial box containing 6750 atoms was constructed by the replication of the 250-atom configuration, in which the number of Cu atoms is 3375 and that of Zr atoms is the same as Cu atoms. The simulation was performed at 2000 K that is much
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higher than the experimental melting temperature20. An equilibrated liquid state of Cu50Zr50 was obtained by relaxing 1.0 ns at 2000 K. Followed by quenching the
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model to 300 K at different cooling rates (0.5 × 1011 K/s - 0.5 × 1013 K/s) in the NPT (constant atom number, pressure, and temperature), and then by relaxation at 300 K to get the desired glassy states. For the simulation of mechanical properties, the periodic boundary condition was adopted. The samples obtained at 300 K with different cooling rates were deformed along the z axis at a constant deformation rate of 109 s-1. When strain reached about 15%, we ended the simulation.
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3. Results and discussions
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Figure 1. Volume as a function of temperature under different cooling rates.
Figure 1 shows the volume-temperature plots of the rapidly solidified process when the Cu50Zr50 alloy liquid is cooled at three different quenching rates. The
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variation of the volume occurs continuous and no dramatic drop in the volume is seen upon the whole cooling. The slope of volume versus temperature curves decreases
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below 800 K, indicating the formation of the metallic glass. Since the metallic glass is regarded as a frozen liquid, the variation of the configurational entropy is vanished. Thus, the derivative of entropy with respect to pressure is the derivative of volume with respect to temperature21. The glass transition temperature is dependent on cooling-rate. The higher cooling rate results in a shorter time for the atoms to relax. Thus the falling out of the liquid-state equilibrium occurs earlier, leading to a higher glass transition temperature22. In the present work, the similar tendency is also observed. By dividing the curve into two parts, the intersection is obtained via fitting a straight line to each piece. This intersection is regarded as the glass-transition 4
ACCEPTED MANUSCRIPT temperature23. The glass-transition temperatures at the cooling rate of 0.5 × 1011 K/s,
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0.5 × 1012 K/s, and 0.5 × 1013 K/s are 675 K, 686 K, and 705 K, respectively.
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Figure 2. Total RDF at 300 K obtained under different cooling rates. In general, the RDF, is one of the most important methods to reveal the structure features of a system24. It is often used to describe a structure that is related to the number of finding the center of particle at a given distance from the center of another particle. Figure 2 depicts the g(r)total curves of the model structure at 300 K obtained at
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different cooling rates. For the as-casted samples with different cooling rates, their total RDF curves are very similar. All the curves exhibit a relatively-sharp first peak, a split second peak, and a blunt third peak within the cutoff distance. The first peak is fairly sharp, which is derived from the atomic packing and configuration of the
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nearest neighbor shell, indicating that there exists a strong SRO. It is worth mentioning that the splitting of the second peak is the main characteristic feature in
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the total RDF of a metallic glass25. Thus, quenching the Cu50Zr50 alloy from 2000 K to 300 K at different cooling rates of 0.5 × 1011 K/s - 0.5 × 1013 K/s leads to a metallic glass. The third weakening peak corresponds to the third-nearest neighbors, demonstrating that a little correlation remains. However, it is noted that the influence of cooling rate on the structure cannot be distinguished clearly. Although the RDF just provides the overall description of the structures, it is difficult to describe the details of the local structure. In order to gain a further insight of the atomic structure, the Honeycutt-Anderson (H-A) pair-analysis technique26 was made to determine the detailed structure of simulated systems from the 5
ACCEPTED MANUSCRIPT atomic-bonding view. The first minima of the partial radial distribution function (PRDF) that is not shown here are chosen to be the cutoff distances. The two atoms are taken as the nearest-neighbors and form a bond pair if they are within the cutoff distance. The H-A has been proved successfully to provide the detailed information
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about the microscopic local structure and short-range order in disordered systems, such as liquid and glass27. In this technique, four integers (ijkl) are assigned to classify the pairs of atoms: (1) i is 1 when the two given atoms (a root-pair) are bonded, and i is 2 if the root pairs are not bonded; (2) j is the number of near-neighbor atoms shared
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in common by the two given atoms; (3) k stands for the number of nearest-neighbor bonds among the shared neighbors; (4) l is adopted when the first three integers is not
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sufficient to distinguish the bond geometries. Figure 3 displays the typical bonded
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pairs, the pink balls are the root pair atoms, the blue balls are the near-neighbor atoms.
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Figure 3. Typical bonded pairs: (a)1421; (b)1422; (c)1431; (d)1441; (e)1541; (f)1551; (g)1661;
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Figure 4. Honeycutt-Anderson pair-analysis at 300 K under different cooling rates.
Figure 5. Honeycutt-Anderson pair-analysis during the cooling process: (a) 1551; (b) 1431; (c) 1541; (d) 1422; (e) 1421; (f) 1661; (g) 1441. Figure 4 presents the relative fraction of main bonded pairs in Cu50Zr50 metallic glass at ambient temperature obtained under various cooling rates. It can be noted that the amounts of the ideal icosahedral (ICOS) order (1551) and deformed ICOS order (1541 and 1431) are dominant and the total fraction of them is more than 65%. Nevertheless, the numbers of crystalline type bonded pairs 1422, 1421, 1661, and 7
ACCEPTED MANUSCRIPT 1441 are small percentages. The 1661 and 1441 bond-types are the characteristic bonded pairs for the body-centered-cubic (bcc) crystalline structure. The 1421 and 1422 bond-types exist mainly in the close-packed crystalline structure such as face-centered-cubic (fcc) and hexagonal-close-packed (hcp) clusters. This trend
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implies that icosahedra-like order configurations, including ideal icosahedral types and deformed types, are the main structural units in the amorphous phase, although the crystalline-type-bonded pairs can also exist in the amorphous phase. In addition, the overall icosahedral-like order increases with decreasing the cooling rate due to the
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obvious increase of 1551 at the lower cooling rate. Figure 5 shows the fraction of these bonded pairs as a function of temperature during the rapid solidification process
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with different cooling rates. The fractions of 1541, 1661, and 1441 increase with the decrease of temperature, but they are affected slightly by the cooling rate. When the supercooled liquid transforms into the glass state, the fractions of 1431, 1422, and 1421 decrease slightly as the cooling rate descends. It is worth noting that the slope of the 1551 curve changes obviously near the glass-transition temperature, especially at
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low cooling rates, denoting a substantial increase of locally-favored structures. Recently, the free volume on the atomic level has been defined and studied13-15. This method provides a good tool to study the relationship between the atomic
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structure and free volume in a system. In this method, Nic is taken as the nearest integral number to the corresponding average coordination number (CN) obtained by the Voronoi tessellation analysis. In this paper, Nic of Cu-centered clusters and
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Zr-centered clusters calculated in Cu50Zr50 are 12 and 15, respectively. A cluster with coordination numbers, ni, within a cutoff of rij less than Nic is regarded to have the free volume. Otherwise, there is no free volume. The free volume around every Cu (Zr) atom in the system is defined as: 3
4 1 V fi = ∑ j = ni +1 π ( Rij − rij ) , 3 2 Nic
where Rij represents the distance between the central atom, i, and coordination atom, j, rij is 1.10 times of the first peak position in the partial RDF curve by considering the contribution of the thermal vibration based on Lindemann's melting criterion28,29. 8
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∑
m i =1
V fi , where m is the
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total number in the system30.
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Figure 6. Amount of total free volumes under different cooling rates. Figure 6 displays the total free volume in the system at 300 K under different cooling rates. With increasing of the cooling rate, the free volume presents a strong evidence of upward trends. During the fast cooling process, the annealing rate of the
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free volume will be too slow to maintain an equilibrium state, and the excess free volume will be frozen in, meaning that a larger amount of free volumes is quenched in
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the glassy solid.
Figure 7. Fraction of icosahedral-like, crystal-like, mixed, and other type atoms under different cooling rates.
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ACCEPTED MANUSCRIPT TABLE 1. Average FV (Å3) of ICO-like, crystal-like, mixed, and other type atoms under different cooling rates. ICO-Like 0.32 0.33 0.34
Cry-Like 0.40 0.40 0.41
Mixed 0.38 0.38 0.38
Other 0.40 0.40 0.41
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Cooling rate (K/s) 0.5 × 1011 0.5 × 1012 0.5 × 1013
The Voronoi tessellation31 developed by Voronoi has been widely used for the characterization of the short-range order and local atomic environment. This method
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is employed here to explore the relation between free volume and atomic structure. The Voronoi polyhedron (VP) is expresses as
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n6 represent the numbers of triangles, tetragons, pentagons, and hexagons in the Voronoi polyhedron. An i-edged face reflects the local symmetry of the central atom with some nearest-neighbor atoms in a certain direction32. The sum of the Voronoi index denotes the total CN. For example, the Voronoi index (VI) of <0, 0, 12, 0>, called a full icosahedron, describes an icosahedral polyhedron with exactly 12 faces
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with 5 edges each. In this paper, the VP is grouped into four types33: (1) icosahedral-like: <0, 0, 12, 0>, <0, 1, 10, x>, and <0, 2, 8, x>; (2) crystal-like: <0, 4, 4, x> and <0, 5, 2, x>; (3) mixed: <0, 3, 6, x>; (4) other: the remaining VP. Noted that x
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is between 0 and 4. Figure 7 presents the fraction of the four types of polyhedrons. The fraction of the icosahedral-like polyhedron is more than 30%, while the fraction
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of the crystal-like polyhedron is small. The existence of the icosahedral-like polyhedron has been proved by many studies. It has more pentagons and involves a five-fold symmetry that is associated with dense packing in metallic glasses34-36. Table 1 lists the average of free volumes of these four types of polyhedrons. It is noted that the icosahedral-like polyhedron has a smaller free volume, compared with crystal-like, mixed, and other type polyhedrons, illustrating that the structural configuration of pentagons is packed more densely. The slower cooling rate will result in more icosahedral-like polyhedrons, therefore, leading to a lower free volume.
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Figure 8. Stress-strain curves under different cooling rates.
It is well known that the changes of the structure and free volume can affect the
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mechanical property. This paper simply investigates the stress-strain curve of the simulated alloys at 300 K obtained under different cooling rates, as shown in Figure 8. As the cooling rate descends, the sample shows higher yield strength. It is in agreement well with the results of experiments37 and simulations38. In addition, it is obvious that modulus of elasticity increases with decreasing the cooling rate. At the
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low cooling rate, the system is more densely packed, resulting a better resistance
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capacity to elastic deformation.
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Figure 9. Honeycutt-Anderson pair-analysis during tension deformation: (a) 1551; (b) 1431; (c) 1541; (d) 1422; (e) 1421; (f) 1661; (g) 1441.
Figure 10. Variation of ICO-Like fractions during tension deformation.
Figure 9 shows the variation of bond type fractions during deformation. The fractions of crystalline type and 1541 type bonded pairs change slightly within 1%. While the 1551 and 1431 type bonded pairs change in a wider range. The 1551 presents a tendency of decrease and 1431 shows the upward trend. Generally, the 12
ACCEPTED MANUSCRIPT variation of the icosahedral-like bonded pairs first decreases with the strain, and following the content does not change obviously. Figure 10 displays the variation of icosahedral-like polyhedron versus applied strain. One can see that during deformation the fraction of icosahedral-like polyhedron first presents downward trend,
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and consequently saturates during the tension process. The variation of icosahedral-like polyhedron is in accordance with that of the icosahedral-like bonded pairs. Noted, the curves of icosahedral-like polyhedron versus strain tend to have a similar variation pattern under different cooling rates. The results of HA and voronoi
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analysis suggest that icosahedral-like bonded pairs will be weaken and
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icosahedral-like structure will be collapsed during the tension process.
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Figure 11. Variation of free volume fractions during tension deformation. The icosahedral-like polyhedron is densely packed and has a smaller free volume.
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Therefore, the amount of free volume is affected by the content of the icosahedral-like polyhedron. Figure 11 presents the variation of total free volume in system during tension, the curves under different cooling rates display a similar pattern. Compared with the variation of icosahedral-like polyhedron (Figure 10) and free volume (Figure 11), it can be observed that the decrease of icosahedral-like polyhedron is accompanied by the generation of the excess volume. In this process, the icosahedral-like polyhedron becomes more loosely packed structure, leading to the increase of the free volume. Then the amount of free volume reaches a saturated state when the fraction of icosahedral-like polyhedron fluctuates within a certain range. 13
ACCEPTED MANUSCRIPT From these results, it can be concluded that the icosahedral-like polyhedron plays an important role during the tension.
4. Conclusions Based on the RDF, H-A indices, and Voronoi polyhedron, this paper has focused
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on the study of the influence of cooling rate on the local structure, free volume, and mechanical property in Cu50Zr50 metallic glass under different cooling rates (0.5 × 1011 K/s - 0.5 × 1013 K/s) by performing the MD calculations. The main conclusions are summarized below:
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1. The faster cooling rate can cause both a higher glass transition temperature and more abundant of the frozen-in free volume in Cu50Zr50 metallic glass.
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2. Based on the detailed analyses of the H-A indices and Voronoi polyhedron, it has been confirmed that the icosahedra-like clusters are the predominant ones and play a key role in the amount of free volumes in the Cu50Zr50 metallic glass. The relative slow cooling rate makes the atoms have an enough time to move and this induces a more content of icosahedra-like clusters with a five-fold symmetry, all of
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these making the system densely packed with a low free volume. It is worthwhile to note that Voronoi analysis is more sensitive to the cooling rate than the H-A analysis.
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3. The mechanical properties are significantly influenced by atomic structures. The yield strength and elastic modulus are much higher at the lower cooling rate, and
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all of these are due to the different icosahedral-like content induced by the different cooling rates. During the tension deformation, icosahedral-like structure will be collapsed. Therefore, the more loosely packed structure results in the increase of the free volume.
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Acknowledgments The present work was supported by the NSFC (Grant No. 50971057 and 51371099). XXY is thankful for the financial support provided by the MTN Technologies, Inc. CTL is thankful for the financial support provided by City University of Hong Kong
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under the Program of the Structural Materials Development..
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