Article pubs.acs.org/JPCC
Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX
Melting of Nanocrystalline Gold Jialin Liu,† Xiaofeng Fan,*,† Yunfeng Shi,§ David J. Singh,∥,⊥ and Weitao Zheng†,‡ †
Key Laboratory of Automobile Materials (Jilin University), Ministry of Education, and College of Materials Science and Engineering, ‡State Key Laboratory of Automotive Simulation and Control, and ⊥College of Materials Science and Engineering, Jilin University, Changchun 130012, China § Department of Material Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, United States ∥ Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211-7010, United States
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S Supporting Information *
ABSTRACT: We report atomistic simulations of the melting of nanocrystalline gold with mean grain sizes from 1.7 to 23 nm. Analysis of the structural changes near melting point at the atomic scale confirms that in the melting process, the solid−liquid interface sweeps rapidly from grain boundary into inner grain as the temperature increases. We find a linear relation between the melting point and the reciprocal grain size for the larger grain size samples, above 7.7 nm, similar to the observations in nanoparticles. However, based on the critical Lindemann ratio, the grain boundaries in these cases should be liquefied at very low temperature (less than 800 K for grain size of 9.7 nm). At a small grain size, this relation between grain size and melting temperature was broken. In particular, at grain sizes below 4 nm, the melting point was found to be approximately constant. It was proposed that the growth and/or merging of grains at low temperature far from melting was contributed to this observation.
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INTRODUCTION Melting of crystalline solids is not just a simple phenomenon of changing a state of matter but a complex phase transformation process. Insights into melting have come more slowly than in the inverse transformation, solidification.1 This is due in part to the difficulty of microscopic observation of the initial stages of melting in solids particularly in nanostructured materials.2,3 Supercooled liquids are readily accessible, and nucleation and growth of solid phases can be readily observed in many cases, whereas superheating solids and observing nucleation of melting is much more difficult.4,5 It is well-known that particle size has a strong effect on melting temperatures and extensive experimental research has been reported.2,6−8 The melting temperature follows a monotonic decrease with decreasing particle size, especially at the nanoscale.2 In fact, the size dependence of melting temperature in nanoparticles and the related thermodynamic properties have been studied extensively, and the strong modifications that are possible at the nanoscale are an important area of nanoscience.3,9,10 However, size-dependent properties in nanocrystalline materials, specifically melting temperatures and processes, are not well elucidated in spite of the importance of nanostructured bulk materials in various areas, ranging from thermoelectrics to structural materials.11,12 Complications in nanocrystalline materials include the fact that for temperatures below the bulk melting point, melting is a transient phenomenon, which is followed irreversibly by recrystallization and grain growth, and the fact that unlike individual nanoparticles, the nano© XXXX American Chemical Society
crystalline bulk provides a constraining matrix, so that the liquid portion may be confined in grain boundary (GB) regions.4 However, recent developments especially in ultrafast science, where X-ray lasers can provide detailed time-resolved structural characterizations, for example, through diffuse scattering, provide motivation for revisiting these issues.13−15 As discussed below, we find a similar behavior for nanocrystalline Au to previously reported results for nanoparticles above a critical size of ∼8 nm, but below this size, qualitatively different results are found, indicating stabilization of the intergrain regions as compared to nanocrystal surfaces. Normally, melting is initiated at the surface and the solid− liquid interface sweeps rapidly through the solid at the melting temperature.2,16,17 This is because the surface atoms are less effectively coordinated and are therefore more mobile as the temperature is increased. These surface atoms will then enter the liquid state at a temperature lower than the bulk melting temperature.18−20 When the surface becomes an important part of the volume, as in nanoparticles and nanowires,21−23 melting point depression may then appear, and this is in fact observed in experiments.24−27 Similarly, it is reasonable to suppose that melting in nanocrystalline materials starts at the grain boundaries because the atoms at the GB are less well bonded. Therefore, melting temperature in nanocrystalline Received: October 17, 2018 Revised: December 15, 2018 Published: December 18, 2018 A
DOI: 10.1021/acs.jpcc.8b10149 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Figure 1. Atomic configurations of nanocrystalline gold with a mean grain size of 14 nm at (a) 200 K and atomic structures of its unmelting grain in the melting process with (b) 840, (c) 1200, and (d)1240 K, and atomic structures of single-crystalline gold in the melting process with (e) 1372 and (f) 1374 K. Green, yellow, and red denote the atoms in nanograins with fcc, stacking faults, and grain boundaries, respectively. The simulations of nanocrystalline with a grain size of 14 nm and single crystal are in cubic boxes with the sizes of 30 and 10 nm, respectively.
used to effectively describe the interaction between metallic atoms. Here, the EAM potential of Foiles et al.39 (see Appendix I in the Supporting Information) was adopted to explore the melting of nanocrystalline gold. The three-dimensional nanocrystalline gold was constructed by the Voronoi method implanted in the Atomsk software.40 The grains were randomly positioned and periodic boundary conditions were imposed. In order to prevent any atom from being positioned too close to another, the extra atoms were removed automatically. Then, the systems were relaxed to obtain the equilibrium state with energy minimum. Prior to annealing at elevated temperatures, an equilibrium process at 200 K was used to anneal with the sample under isobaric− isothermal conditions (NPT ensemble) with atmospheric pressure in order to equilibrate atoms at grain boundaries. Figure 1a shows the nanocrystalline sample with a mean grain size of 14 nm. The results are visualized with the OVITO software.41 The mean grain sizes of samples investigated ranged from 1.7 to 23 nm. There are several optional methods to obtain the melting points, such as two-phase, Gibbs free energy and heat-untilmelting. Here, the heat-until-melting method was chosen to analyze the melting processes. It may be the best way to track the change of structures in the heating process, though superheating is unavoidable. The Newton equation of motion was solved by the velocity algorithm with a time step of 1.0 fs. Starting with the well-equilibrated structure at 200 K, the temperature of the system was increased linearly with a heat rate of 0.8 K/ps from 200 to 1400 K. The melting point is defined as the temperature point at which a solid becomes liquid. It is a macroscopic concept, and in a single crystal, it is unambiguous. However, for the nanoscale or nanostructure systems, melting is usually a process over a range of temperatures in which the solid and liquid characteristics coexist. In this work, the melting temperature is defined by the maximum value of heat capacity at which the phase transition from solid to liquid is in progress. The heat capacity under isobaric condition is the first derivative of enthalpy by the formula, Cp = dH/dT, where H
solids should show a qualitatively similar rule to that in nanoparticles. Some models based on the GB energy have been proposed.28,29 The related experimental results are still limited,26,30 while molecular dynamics (MD) simulations have taken an important role to understand the size-related phenomena in nanocrystalline materials.6,9,18,31 Previous studies have provided evidence that melting starts from the GB to the inner grain.32,33 However, the melting temperature does not follow the simple rule that melting point has a linear dependence with the reciprocal particle size,2,34 as has been discussed for isolated nanoparticles.22,35 Results on nanocrystalline Ag indicated that the melting temperature almost remains a constant when the grain size is less than 3 nm.34 Noori et al. reported that for Al nanocrystals with a grain size of more than 14 nm, the melting point is constant and similar to that of perfect crystals.36 Results on Cu bicrystals imply that the melting point is connected with types and angles of grain boundaries.33,37 Here, we explore the process and trends for metal nanocrystal melting by analyzing the melting of nanocrystalline Au with a mean grain size from 1.7 to 23 nm with MD simulations. We analyze the change of atomic structures near the melting point by different techniques, such as polyhedral template matching, and use of the Lindemann index. We find that the melting of nanocrystals with a small grain size is much different from the melting of isolated nanoparticles.
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COMPUTATIONAL METHODS It is necessary that simulations of nanocrystals enable consideration of the statistic average of grain size, and thus, we need to consider the interaction among many atoms. For example, the number of grains in our nanocrystal samples is not less than 15. Then, a nanocrystal sample with an average grain size of 23 nm must have approximately 4.5 × 107 atoms. In this work, we performed the simulations with standard classical MD methods as implemented in a large-scale atomic/ molecular massively parallel simulator.38 The embedded atom method (EAM) potential to express the cohesive energy with a pairwise potential and a many-body embedding energy can be B
DOI: 10.1021/acs.jpcc.8b10149 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Figure 2. Melting process of nanocrystalline gold with the grain size 9.7 nm (a−e) by common neighbor analysis at different temperatures, 680, 1000, 1120, 1160, and 1200 K (white and dark blue represent the atoms of grain boundaries and inner grain, respectively), (f−j) by polyhedral template matching at different temperatures (green, red, blue, yellow, violet, and light blue are the atoms of fcc, hcp, bcc, ico, simple cubic, and amorphous, respectively), and (k−o) by displacement magnitude at different temperatures (the atoms with values of displacement magnitude in the range of 0.6−1.38 are shown with RGB colors, and red and blue represent the atoms with lager displacement and smaller displacement, respectively. The thickness of slice is 1 nm along z-axis). The simulation is in a cubic box with the size of 30 nm.
Figure 3. (a) Average enthalpy per atom as a function of temperature for nanocrystalline gold with mean grain sizes from 4 to 23 nm and (b) heat capacity Cp as a function of temperature for melting of nanocrystalline with a grain size of 8 nm (red dashed line) and 23 nm (blue dashed line).
active and the structure deviates from fcc. At the GB, because of the high amplitude of thermal vibration, the atoms are more active than that in grain interiors. However, the activated atoms are randomly distributed and the localized melting is not clearly demonstrated. From polyhedral template matching and displacement magnitude matching analysis, the amplitude of an atom’s deviation from its equilibrium position can indicate the localized melting process. As the temperature is increased, the melted region in the GB becomes wider and wider and gradually extends into the interior of grains to result in the fullphase transition. It is obvious that the melting of a nanocrystalline material is different from that of a single crystal. For single crystals, the melting temperature is a point at which the volume changes discontinuously with the change of other thermodynamic properties, such as potential, as shown in Figure S1. However, in a nanocrystalline material, the thermodynamic properties, such as the enthalpy, do not show such sharp behavior with increasing temperature near the melting point, though there is a change of slope with the temperature as an independent variable. The enthalpy as a function of temperature for the nanocrystalline materials with different mean grain sizes from
is the enthalpy of nanocrystalline system. We can obtain the change of heat capacity by analyzing the change of enthalpy with the increase of temperature.
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RESULTS AND DISCUSSION Lindemann index and Born instability criteria are both typical indicators to describe the melting from solid to liquid.31 For a single crystal, before melting occurs, the superheated crystal will result in the localized random particles that simultaneously satisfy both instability criteria. As shown in Figure 1e,f, a sudden loss of order in atomic motion appears in a very short range of temperature (∼5 K) near the melting point. For nanocrystalline material, the melt nucleation will happen primarily at GB. As shown in Figure 1, for the melting process of nanocrystalline gold with a mean grain size of 14 nm, the melting originates from the grain boundaries to the inner grain, similar to the other simulations for Ag and Al.36 In Figure 2, the localized changes of structures near the melting point are shown by common neighbor analysis, polyhedral template matching, and displacement magnitude matching analysis. From common neighbor analysis, with the increase of temperature, the atoms in the inner grain become C
DOI: 10.1021/acs.jpcc.8b10149 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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Figure 4. (a) Melting points as a function of the reciprocal grain size for nanocrystalline (black square) and nanoparticle gold (red dot), (b) fwhm of heat capacity peak for different grain sizes, and the Lindemann ratio and fraction of fcc atoms as a function of temperature for (c) single-crystal gold and (d) nanocrystal with a grain size of 9.7 nm.
1.7 to 23 nm is shown in Figure 3a. For each curve, there is a linear relationship between enthalpy and temperature for the temperature far away from the melting point. Near the melting point, the slope initially increases and then decreases as the temperature is increased. A similar relationship is found between volume and temperature in Figure S2a. Heat capacity as the first derivative of enthalpy can be used to reveal the region of melting process with solid−liquid coexistence by the full width at half-maximum (fwhm) with the peak value as the melting point. From Figure 3b, the fwhms of nanocrystalline samples with a grain size of 8 and 23 nm are much larger than that of single-crystal Au in Figure S1. The melting point (Tm) of the nanoparticle decreases following the decrease of particle size or diameter (d), as has been well established in prior studies, including for Au decreasing to at least 1.5 nm.8 With a liquid-drop model or other similar model, the difference of melting point between the nanoparticle and a single crystal can be expressed as ΔTm/ Tmb = β/d, where Tmb is the melting point of the single crystal and the parameter β is related to the surface energy of the nanoparticle. As shown in Figure 4a, both the melting point of the nanoparticle and the reciprocal particle size follow the simple linear relation. If we consider the nanocrystal as the aggregation of nanoparticles, the change of melting point will be related with the interface energy or GB energy. It is known that the interface energy in a GB is less than the surface energy because of the atoms in the surface of a nanoparticle. Therefore, for the curve of ΔTm/Tmb as the function of d−1, the curve slope of nanoparticles should be larger than that of nanocrystals. It means the melting point of a nanocrystal is higher than that of a nanoparticle with the same grain size, as shown in Figure 4a. It is noticed that the melting point calculated from the enthalpy is similar to that calculated from volume (Figure S2b). However, we find that the curve for the nanocrystalline material does not follow the single linear curve with the change of grain size. With the decrease of grain size to 7.7 nm, the curve begins to deviate from the linear relation. With the grain size less than 4 nm, the melting point is almost
constant. A similar phenomenon of melting temperature independent of grain size on Ag was observed when the grain size is less than 3 nm.34 It is also noticed that the fwhm has an increased trend with grain size increasing for larger grain size. For grain size less than 7.7 nm, the value of fwhm is around 68 K and does not have an obvious increasing or decreasing trend. It is usually considered that the nanocrystal is composed of both GB phase and grain inner phase embedded by GB for analyzing the mechanical properties of the nanocrystal. For the nanocrystal with a larger grain size, the grain phase is the dominant part. As in the nanoparticles, the melting initiates from the GB, which can be viewed as the “surface” of the grain, and the melting point is correlated with the grain size. Following the decrease of grain size, the GB network will occupy the dominated position. Xiao et al. proposed that this is the main reason due to which the melting point of the Ag nanocrystal with the grain size less than 3 nm almost remains a constant.34 There may be a more in-depth reason for this special phenomenon and we will explore the change of atomic structures near the melting point in the next part. The Lindemann index is a simple but effective measure of a thermally driven disorder in the melting process. It is calculated by the average of local Lindemann ratio with the formula δL =
1 ∑ δi N i
(1)
where δi is the local Lindemann ratio and is equal to the rootmean-square displacement of an atom from its equilibrium position, divided by the average nearest-neighbor distance. For the single crystal in Figure 4c, with the critical Lindemann ratio31 δL* = 0.22, the melting point is 1350 K. From the peak of heat capacity, the melting point is 1370 K. At this point, the ratio of fcc atoms decreases sharply with an increasing temperature. It is a little higher than the experimental melting temperature of 1337 K. From the fwhm value (about 25 K) of heat capacity peak, the critical Lindemann ratio δ*L = 0.22 D
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Figure 5. (a) Probability distribution P(εxx) of atoms as a function of Green−Lagrangian atomic strains (εxx) for nanocrystal gold with a mean grain size of 23 nm under different temperatures and (b) the maximal value of probability distribution P(εxx) as a function of temperature for nanocrystal gold with a grain size of 4, 9.7, and 23 nm.
indicates the beginning of the melting process. Figure 4d shows the evolution of the Lindemann ratio value with temperature for the nanocrystal with a grain size of 9.7 nm. For the melting point at 1134 K defined by the heat capacity peak, the ratio of fcc atoms decreases sharply and the Lindemann ratio increases sharply, similar to the observation in a single crystal. From the fwhm value (about 95 K) of heat capacity peak, the temperature of 1086 K should be the beginning of the melting process. However, in this temperature, the Lindemann ratio is far more larger than δL* = 0.22. The critical Lindemann ratio is corresponding to the temperature 760 K. Combining with the results in Figure 2, it is indicated that the GB is very easy to liquefy with the increase of temperature far below the melting temperature of inner grains in the nanocrystal. This is a limiting factor for experiments, as at this temperature, rapid diffusion in grain boundaries and Ostwald ripening that changes or destroys the nanostructure may occur. Atomic Green−Lagrangian strain (εxx) is a useful indicator for the changes of local displacements of atoms,42 similar to the local Lindemann ratio. It can be derived from an atomic gradient tensor. For each atom, the local deformation gradient tensor can be calculated from the relative displacements of these atom neighbors within the given cutoff radius. The Green−Lagrangian strain is defined in detail in Appendix II in the Supporting Information. In Figure S3, the space distributions of atomic strain for a nanocrystal with a size of 23 nm under different temperatures are shown. It is clear that the large atomic strain is distributed at the GB. In Figure 5a, the probability distribution P(εxx) of atomic strain for a nanocrystal with a grain size of 23 nm under different temperatures are shown. The strain at the peak of P(εxx) is larger than zero, indicating the thermal expansion with uplifted temperature. Following the increase of temperature, the peak value of P(εxx) decreases and the fwhm of the peak in the curve of probability distribution increases. This demonstrates the typical distributions of thermal movement with temperature. In Figure 5b, the peak values as the function of temperature for a grain size of 4, 9.7 and 23 nm are shown. At the melting point defined by the heat capacity peak, each curve has an obvious inflection point for a sharp decrease of the peak value. However, even in the nanocrystal with a grain size of 4 nm, there is no implication that the melting point is dominated by GB phase. The nanocrystalline material is composed of grain atoms and GB atoms. As shown in Figure 2, the grain boundaries become wider and wider with increased temperature. In Figure 6, we
Figure 6. Fraction of GB atoms as a function of temperature for nanocrystal gold with different grain sizes from 1.7 to 23 nm.
analyze the number of GB atoms as a function of temperature for different grain sizes. For a large grain size, it is clear that the number of GB atoms increases with the increase of temperature. However, it is interesting that at the grain sizes less than 4 nm, the number of GB atoms decreases with the increase of temperature at a lower temperature far away from the melting point. This implies that some grain sizes are merged at low temperature. From the Lindemann index, we know that the atoms at GB are very active and even can diffuse along GB at a lower temperature. This makes the merging and growth of grains possible, as in this way, the free energy of the system can decrease. Especially for a small grain size, the large GB network makes the dynamic barrier of merging of grains lower. In Figures 7 and 8, we analyze the evolution of grains for nanocrystals with grain sizes 1.7 and 2.5 nm in detail. In Figure 7, the size of each grain is obtained as follows. First, the core of each grain is identified by the technique of atoms with all their nearest neighbors as perfect fcc. Then, a cluster analysis is performed on these core atoms to differentiate each cluster (i.e., each grain). The grain size can be calculated from the number of core atoms in the cluster, by assuming a cubic grain. Finally, a length of 0.8156 nm for the skin of the grains is added to the calculated grain size. The distribution of grain size with this analysis method matches very well with the inputted grain sizes from Voronoi method. From Figure 7, it is typical that the grains grow and/or merge at low temperature far away from the melting point, while the sizes of some grains decrease and even some disappear in this process. In Figure 8, the distribution of grain size by atomic weight is shown. It is clear that the atomic weight of larger grain size increases with the increase of temperature, whereas the atomic weight of smaller grain size decreases. The distribution with the largest grain size E
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Figure 7. Grain growth and/or merging of nanocrystal gold with a (a−d) grain size of 1.7 nm under different low temperatures 224, 320, 440, and 560 K and (e−h) grain size of 2.5 nm under different low temperatures 224, 273, 344, and 392 K. Blue, red, and yellow represent the inner grain, stacking fault, and surface mesh, respectively.
Figure 8. (a) Grain size distribution in terms of atomic weight for nanocrystal gold with a (a) grain size of 1.7 nm under different low temperatures 224, 320, 440, and 560 K and (b) grain size of 2.5 nm under different low temperatures 224, 273, 344, and 392 K, respectively.
of melting point in nanocrystallines is similar to that in nanoparticles. The results indicated there were obvious different rules of melting and grain size effect at a small grain size less than 7.7 nm. For the nanocrystal with a larger grain size more than 7.7 nm, the melting process is consistent to the previous proposal that the melting starts from GB and the solid−liquid interface sweeps rapidly from GB into the inner grain with the uplifted temperature. It is surprised that the GB is liquefied at very low temperature far away from the corresponding melting point based on the critical Lindemann ratio. For example, the GB is liquefied at 760 K for a grain size of 9.7 nm. It was found that the relation between melting point and grain size began to deviate from a linear relation with a decrease of grain size to 7.7 nm and melting point became a constant (∼1070 K) when the grain size was less than 4 nm. By analyzing the evolution of local grain with increased temperature, the growth and/or merging of grains was found to begin at very low temperature far away from the melting point. It is hoped that these results for nanocrystalline Au will stimulate the further experimental research especially in the context of ultrafast experiments.
is located at approximately 3.75 nm and its atomic weight becomes larger and larger with the uplifted temperature. This may be the reason that the melting point almost remains a constant for the grain size less than 4 nm. The competition between annealing and heating rate could have a substantial effect on the details of the process. If given sufficient time at some temperature far lower than the melting point, Ostwald ripening will occur. In Figure S4, the heating of nanocrystal gold with the grain size of 1.7 nm under 400 K clearly demonstrates the growth and/or merging of grains. At the limited heat rate, the melting point will have a weak shift because of the effect of superheating. With the decrease of heating rate, the melting point will have a small decrease, as shown in Figure S5a. Under different heating rates, the microscopic distribution of grain sizes will be less different (Figure S5). From the statistical view, at low temperature (region of grain growth), the fraction of GB decreases following the decrease of heating rate (Figure S5b). Therefore, low heating rate at low temperature will promote the grain growth. At the proper large size of simulation cell, the cell size does not make any obvious effect on the simulated results (Figure S6).
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ASSOCIATED CONTENT
S Supporting Information *
CONCLUSIONS We have investigated the melting process and grain size effect of nanocrystal gold and atomic mechanism of phase transition under heating using MD simulations with grain sizes from 1.7 to 23 nm. It is usually proposed that the grain size dependence
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b10149. EAM potential with the parameters about gold; definition of Green−Lagrangian strain tensor; melting F
DOI: 10.1021/acs.jpcc.8b10149 J. Phys. Chem. C XXXX, XXX, XXX−XXX
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of single-crystal gold; melting point from volume; melting process of nanocrystalline gold with a grain size of 23 nm common neighbor analysis; grain growth and/or merging of nanocrystal gold with a grain size of 1.7 nm under 400 K; heat rate effect in the melting of nanocrystalline gold with a mean grain size of 2.5 nm; and size effect of the simulation cell (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail: xff
[email protected]. Phone: +86-159-4301-3494. ORCID
Xiaofeng Fan: 0000-0001-6288-4866 Yunfeng Shi: 0000-0003-1700-6049 David J. Singh: 0000-0001-7750-1485 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The research was supported by the National Key R&D Program of China (grant no. 2016YFA0200400) and the National Natural Science Foundation of China (grant nos. 11504123 and 51627805). Work at the University of Missouri was supported by the Department of Energy, Basic Energy Sciences, award DE-SC0019114.
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