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Molecular dynamics simulations of the preparation and deformation of nanocrystalline copper
Acta Materialia 52 (2004) 5105–5114 www.actamat-journals.com
Molecular dynamics simulations of the preparation and deformation of nanocrystalline copper Y.W. Zhang a
a,*
, P. Liu b, C. Lu
b
Department of Materials Science, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Institute of High Performance Computing, Singapore Received 26 May 2004; received in revised form 9 July 2004; accepted 13 July 2004 Available online 13 August 2004
Abstract The molecular dynamics method is used here to simulate: (1) the preparation of full-density nanostructured copper by compacting copper nanoparticles and (2) the deformation behaviors of the nanostructured copper under compression. It is found that the packing arrangement, the size of the nanoparticles and the compaction temperature, affect the deformation behaviors of the nanostructured copper. Our simulation results also show that the synergy of the rotation and mass shedding of grains and the thickening and sliding of grain boundaries, prevents the formation of voids and cracks in the nanostructured copper under compression. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocrystalline materials; Nanoparticles consolidation; Molecular dynamics; Plastic deformation; Copper
1. Introduction Nanocrystalline metals are of scientific interest and technological importance [1–12]. At the present stage, fully dense, impurity-free and bulk nanostructured crystalline materials are difficult to fabricate due to the limitation of the present synthesis techniques. It has been shown that loading rates, temperatures, grain sizes, porosity and impurities significantly affect the strength and ductility of nanostructured crystalline materials. Consequently, there are many conflicting experimental results on the strength and ductility of nanocrystalline metals due to the different fabrication techniques. For example, on the one hand, nanocrystalline materials exhibit superplasticity [8,9]; while on the other, nanocrystalline materials also show brittleness [4,11,12]. Molecular dynamics simulations have also been used to unveil the deformation mechanisms of nanocrystalline materials [13–21]. Detailed activities of dislocations *
Corresponding author. Tel.: +65 687 415 42; fax: +65 677 636 04. E-mail address: [email protected] (Y.W. Zhang).
and grain boundaries during the deformation of nanocrystalline materials may be examined through the atomistic simulations and are quantified to a degree that is impossible to determine experimentally. It is now recognized, through both experimental and modeling efforts, that for nanocrystalline materials, several deformation mechanisms may prevail: (1) dislocation slip, (2) diffusion through dislocations, (3) grain boundary sliding, (4) diffusion through grain boundaries, (5) grain rotations, (6) void nucleation, growth and migration and (7) crack nucleation and propagation. Due to the complexities of the nanostructures and deformation mechanisms, the true mechanical behaviors of fully compact, impurity-free and bulk nanocrystalline materials have not yet been fully understood. The formation and deformation of nanocrystalline materials through the compaction of nanoparticles has been studied experimentally by various researchers [22–25]. But due to the high porosity and high impurity, both the strength and ductility of the synthesized materials are low. In addition, due to the recrystallization, the average grain size is much larger than the average
1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.07.018
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nanoparticle size. As a result, the intrinsic mechanical behaviors of fully dense and impurity-free nanocrystalline materials with an average grain size of less than 20 nm, obtained through nanoparticle compaction, are still largely unknown. In this study, molecular dynamics simulations are performed to model the preparation of fully dense and impurity-free nanocrystalline copper through compaction of copper nanoparticles. During the compaction, different temperatures, packing arrangements and sizes of copper nanoparticles are used. The prepared nanocrystalline materials are used to perform compression tests at near 0 K to avoid atomic diffusion. It is found that the compaction temperature, the packing arrangement and the size of the nanoparticles affect the deformation behaviors of the nanostructured crystals. The simulations also show that the synergetic accommodation of the rotation and mass shedding of grains, and the thickening and sliding of grain boundaries, prevents the formation of voids and cracks in nanocrystalline copper.
2. Formulations An embedded atom potential of copper [26] is used in the present simulations. The potential has the following form: X 1=2 1 X X Etot ¼ qi þ V ij ; ð1Þ 2 i jði6¼jÞ i where qi is the second moment of the density of states of atom i and is written as X qi ¼ /ij : ð2Þ jði6¼jÞ
Both Vij and /ij are the functions only of the interatomic distance and are obtained by fitting physical properties of copper [26]. The atomic level stress components associated with an atom, rab i , can be calculated by using the potential, " # X 0 raij rbij 1 X 0 1=2 ab ri ¼ V qi / ; ð3Þ 2Xi j rij j where Xi is the atomic volume at the reference state, V 0 and / 0 are the derivatives of V and /, respectively, with respect to the interatomic distance rij. Simulations start either from an ideally packed unisize copper nanoparticle array or from a randomly packed arrangement with varied copper nanoparticle sizes. The particle orientations are chosen randomly. For ideally packed particles arrays, three packing arrangements: face-centered cubic (FCC), body-centered cubic (BCC) and simple cubic (SC) are used. The numbers of nanoparticles used for the FCC, BCC and SC packings are 108, 54 and 64, respectively. Three
nanoparticle sizes: 3.25, 6.22 and 8.40 nm are used. For randomly packed arrangements, the number of nanoparticles is always 70 and three average particle sizes: 3.49, 5.52 and 7.33 nm are used. Three temperatures: 300, 700 and 1200 K are used in the compacting processes for each configuration. The packed particles are compacted to the volume equal to that occupied by the single crystal at the corresponding temperatures. Subsequently, the temperature and the average hydrostatic stress are brought to a low level; the temperature is approximately 0 K and the hydrostatic stress approximately 0.1 GPa. (In this paper, positive stress indicates compression.) Finally the pressure is applied along the x-direction while keeping the average normal stresses along the y- and z-directions at about 0.1 GPa. It is noted that although the compactifications are done at different temperatures, all the uniaxial compressions are performed at the same temperature, that is, at 0 K. A typical example of the preparation and deformation process (using FCC packing, with a particle size of 6.22 nm and at a temperature of 700 K) is shown in Fig. 1. In the simulations, the number of atoms for the smallest case (with BCC packing at a particle size of 2.54 nm) is 86,454 and that for the largest case (with FCC packing at a particle size of 8.40 nm) is 2,832,516. The periodical boundary conditions are used in both compaction and compression, and the applied strain rate is approximately 6.3 · 109/s. The prepared nanomaterials are analyzed by calculating compact density, radial distribution functions and local atomic orders [27,28]. The degree of crystallinity (DOC), which is defined as the ratio of the number of atoms in a FCC environment to the total number of atoms in the nanocrystal, can be obtained by local atomic order analysis [27,28]. The current calculations show that the fraction of pores is very small after compaction and may be neglected. The nanocrystalline copper samples prepared can be taken to be fully dense and impurity-free.
3. Results and discussions The simulation results indicate that the deformation behaviors of the nanocrystalline materials are closely related to their nanostructures, which in turn are determined by the compaction temperatures, the packing arrangement and the size of the nanoparticles. Several features associated with the compaction process can be obtained from our simulations and involve the affects of: (1) particle sizes, (2) packing arrangements and (3) temperature. The larger the particle size, the higher the DOC of the nanocrystalline materials (the less the disordered portion, including defects and grain boundaries). For example, for nanocrystals prepared at 700 K and with FCC
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Fig. 1. The images show the modeling sequence of the preparation and deformation of nanocrystalline copper under the condition of FCC packing, T = 700 K and a particle size of 3.25 nm: (a) The initial configuration of 108 packed copper nanoparticles (in different colors), (b) after the hydrostatic compaction, (c) after stress relaxation and reduction of temperature and (d) after a uniaxial compression.
packing, the DOC at the nanoparticle diameters of 3.25, 6.22 and 8.40 nm are 0.29, 0.68 and 0.79, respectively. The reason is predominantly due to the DOC before compaction: the smaller the particles, the higher the fraction of surface to bulk atom ratio. Fig. 2(a)–(d) shows the snapshots of a typical particle in the randomly packed arrangement during the compaction. The average particle diameter is 5.52 nm. Fig. 3(a)–(d) shows the compaction of a typical particle in the same packing arrangement but with an average particle diameter of 3.49 nm. It is found that for the large particle, severe deformation mainly concentrates on its surface or subsurface, and the inner core largely remains crystalline; while for the small particle, the whole particle suffers severe deformation as shown Fig. 3. Hence at a same squashing level, the plastic deformation of small particles is much more pronounced than that of large particles. The packing arrangements affect the DOC of compacted copper. For example, the DOC of the nanocrystalline copper prepared at 700 K concerning a particle size of 6.22 nm is 0.68 for FCC, 0.65 for BCC and 0.48 for SC packings. Therefore, the FCC packing gives the highest DOC while the SC packing gives the lowest DOC among the three ideal packing arrangements. The
underlying reasons are due to the porosity level and the pore shape before compaction. For the SC packing arrangement, the porosity level is highest. If a full compaction is reached, the initial nearly-spherical particle must be squashed into a nearly cubic shape. This can be clearly seen from Fig. 4(a) and (b), which shows the squashing process of a particle in the SC packing array. For the FCC and BCC packing arrangements, the packing density is higher than that of the SC packing arrangement. In addition, when a full compaction is reached, the initial nearly-spherical particle is squashed into a shape which is still close to a sphere. This can be seen from Fig. 4(c)–(f) for a particle compacted in the BCC and FCC arrays, respectively. Therefore, the deformation level for the FCC and BCC packing arrangements is less severe than that for the SC packing arrangement. Between FCC and BCC, since the FCC porosity level before compaction is lower, therefore the deformation level should be also lower as well. This is consistent with the DOC order for the three packing conditions. For the randomly packed array with an average particle diameter of 5.52 nm compacted at 700 K, the DOC is about 0.61, which is close to the BCC packing, but higher than the SC packing and lower than the FCC packing. This is expected since the
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Fig. 2. Compact deformation of a typical particle in randomly packed arrangement with an average particle diameter of 5.52 nm at 700 K. (a) At 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction (the final state of compaction with a volume equal to that of the copper single crystal). The percentages here and in subsequent figures refer to the ratio of the reduction of the sample side length vs. the reduction of the final sample side length at the end of the compaction.
Fig. 3. Compact deformation of a typical particle in the randomly packed arrangement with an average particle diameter of 3.49 nm at 700 K. (a) at 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction.
porosity of the randomly packed array should be generally lower than that of the SC array, but higher than that of the FCC array.
The nanostructured copper shows a higher DOC when prepared at relatively low temperatures (300 and 700 K), and a lower DOC when prepared at high temperatures (1200 K). For example, for the samples prepared under the FCC packing and at a particle size of 6.22 nm, the DOC is 0.63 for 1200 K, 0.68 for 700 K and 0.67 for 300 K. For the samples prepared under the random-packing and at an average particle diameter of 5.52 nm, the DOC is 0.54 for 1200 K, 0.61 for 700 K and 0.63 for 300 K. The compaction processes for a typical particle in the above random-packing cases are shown in Fig. 5(a)–(d) for 300 K, Fig. 2(a)–(d) for 700 K and Fig. 6(a)–(d) for 1200 K, respectively. It is clearly seen from Figs. 2, 5 and 6 that the deformation is lest severe for 300 K while most severe for 1200 K. Evidently the nanoparticles have been severely damaged at the higher temperature compaction and the deformed particles show a large portion of amorphous structures (see Fig. 6). Since the packed particles are compacted to the volume equal to that occupied by the copper single crystal, the maximum compacting pressures will be different due to different packing arrangements, geometries and temperatures. Fig. 7 shows the variation of the average hydrostatic pressure with the compacting time during compaction and unloading processes for the randomly
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Fig. 4. Compact deformation of a typical particle with a particle diameter of 3.25 nm at 700 K in different packing arrangements. (a) at 25% compaction and (b) at 100% compaction in the SC packing; (c) at 25% compaction and (d) at 100% compaction in the BCC packing; and (e) at 25% compaction and (f) at 100% compaction in the FCC packing.
packed particle arrangement with three average particle diameters: 3.49, 5.52 and 7.33 nm. It appears that the larger the average particle size, the higher the pressure required to form a dense nanostructured sample. The maximum pressures for the three cases are approximately 10.8, 12.5 and 13.6 GPa. The prepared nanocrystalline copper samples are used to perform uniaxial compression along the x-direction. The atomic stress is calculated using the atomic potential [29]. The stresses reported are the stresses averaged over all atoms in the simulation cells. The applied strain is defined as (L0L)/L0, where L0 and L are the lengths of the sample along the x-direction before
and after compression (here, a positive strain value implies compressive strain). During compression, all of the samples exhibited a rapid softening after yielding when prepared at 300 and 700 K. Consequently, the yielding strength is very close to the maximum compressive stress. A typical example of the average stress Ærxxæ vs. the external applied strain curves for the three particle sizes, 3.25, 6.22 and 8.40 nm, under 700 K and SC packing are shown in Fig. 8. It is interesting to note that the maximum yield strength occurs at the particle size of 6.22 nm. In this case, there is no correlation between the DOC before the compression and the yield strength during compression. For the randomly packed particle
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Fig. 5. Compact deformation of a typical particle in randomly packed arrangement with an average particle diameter of 5.52 nm at 300 K. (a) at 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction.
Fig. 6. Compact deformation of a typical particle in randomly packed arrangement with an average particle diameter of 5.52 nm at 1200 K. (a) At 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction.
15
D=3.49nm D=5.52nm D=7.33nm
10
5
0
0
0. 5
1
1. 5
2
arrangement, the average stress Ærxxæ vs. the external applied strain curves for the three particle sizes, 3.49, 5.52 and 7.33 nm, at a compaction temperature of 700 K are shown in Fig. 9. It is also interesting to note that the maximum yielding strength occurs at the particle size of 5.52 nm. These simulation results suggest that there may be a transition of the Hall–Petch to the inverse Hall–Petch regimes. For the samples prepared at 300 and 700 K, the graphs of the average stress Ærxxæ vs. the applied strain curves show little difference as shown in Fig. 10. However, at 1200 K, the stress vs. strain reveals a roughly elastic-perfectly-plastic deformation, except that the stress oscillates with an increase in applied strain. Its yield strength is much lower than its lower temperature counterparts. A same trend is also observed for the randomly packed particle arrangement as shown in Fig. 11. The reason for the reduction of yield strength at high temperatures can be deduced from Figs. 2, 5 and 6.
The Average Stress <σxx> (GPa)
5 D=3.25nm D=6.22nm D=8.40nm
3 2 SC packing T=700K
1 0 0
0.1
0.2
0.3
The Applied Strain Fig. 8. The average stress Ærxxæ vs. the externally applied strain at three different particle sizes at 700 K and with a SC packing arrangement.
3 2 Random packing Compaction temperature 700K
1
0
0.1
0.2
0.3
The Applied Strain
Fig. 9. The average stress Ærxxæ vs. the externally applied strain at three different particle sizes at 700 K and with a randomly packed arrangement.
5 The Average Stress <σxx> (GPa)
Fig. 7. Variation of the average hydrostatic pressure Ærmæ with the compacting time for the randomly packed particle arrangement with three different average particle diameters: 3.49, 5.52 and 7.33 nm.
D=3.49nm D=5.52nm D=7.33nm
4
0
The Compaction Time (ns)
4
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5
Random packing Compaction temperature 700K
The Average Stress <σxx> (GPa)
The Average Pressure < σm> (GPa)
Y.W. Zhang et al. / Acta Materialia 52 (2004) 5105–5114
T=300K T=700K T=1200K
4 3 2 1 0 0
SC packing Particle size: 6.22nm
0.1
0.2
0.3
The Applied Strain Fig. 10. The average stress Ærxxæ vs. the externally applied strain at different preparation temperatures with a SC packing and a particle size of 6.22 nm.
During compaction at the high temperature, the nanoparticles have been severely damaged and the deformed particles show a large portion of amorphous structures (see Fig. 6). Therefore during compression, the deformation mainly occurs in amorphous (non-crystalline) regions. In these cases, the yield strength is proportionally related to the DOC after compactions, that is, the higher the DOC, the higher the yield strength. The calculation results show that the stress Ærxxæ vs. the applied strain curves are roughly the same for the FCC and BCC packings, but different from that of the SC packing. This can be clearly seen in Fig. 12 for 700 K with a particle diameter of 6.22 nm. The yield strength of SC packing is much higher than that of FCC and BCC packings. The reason for the difference is likely due to the different deformation patterns during the compaction. For the FCC and BCC packings, the initial
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The Average Stress <σxx> (GPa)
5 T=30 0K T=70 0K T=12 00K
4 3 2
Random packing Average particle size 5.52nm
1 0
0
0.1
0.2
0.3
The Applied Strain
Fig. 11. The average stress Ærxxæ vs. the externally applied strain at different preparation temperatures with a randomly packed arrangement and a particle size of 5.52 nm.
SC FCC BCC
4
6
(a)
3
0.074 0.148 0.222 0.296
2 1 0 0
4
Particle size: 6.22nm T=70 0K
0.1
0.2
BCC packing Particle size: 6.22nm T=700K
RDF
The Average Stress <σxx> (GPa)
5
During the compression of a sample obtained through compacting a BCC array at 700 K and with a particle diameter of 6.22 nm, the radial distribution functions (RDFs) are shown in Fig. 13(a) for applied strain levels at 0.074, 0.148, 0.222 and 0.296. It may be observed that with an increase in the applied strain level, the peaks in the RDFs become broader, indicating a lower DOC. This is confirmed by a local bond order analysis. A typical example of such analysis results is given in Fig. 13(b) at a compaction temperature of 700 K and a particle size of 6.22 nm. It can be seen that the FCC fraction gradually decreases, the HCP fraction remains roughly at the same level, while the fraction of the disordered portion increases with an increase in the strain level. Detailed examination of the nanostructures shows that during compression of the nanocrystalline copper samples, the grains shed atoms. As a consequence, grains are downsized and the grain boundary zones are thickened. The synergic accommodation of grain boundary thickening and sliding, and the grain mass shedding and rotating, prevent the formation of voids and cracks as shown in Fig. 14.
2
0.3
The Applied Strain Fig. 12. The average stress Ærxxæ vs. the externally applied strain under SC, FCC and BCC packing arrangements at 700 K and a particle diameter of 6.22 nm.
0.75
1
1.25
1.5
r/a0 (b)
1 FCC HCP Other (GBs)
0.8
Fraction
porosity is low, relatively small deformation and pressure is required to form dense nanostructured copper as shown in Fig. 4(c)–(f). For the SC packing, the initial porosity is high, the particles need to be squashed more to form a compact sample (see Fig. 4(a) and (b)). As a consequence, the deformation degree of nanoparticles under SC packing is higher than that under FCC and BCC packing arrangements. Thus dislocation pileups and twining density in grains under SC packing after compaction will be high. The high dislocation pileup level and twinning density during compaction contribute to the high yield stress for SC packing. For nanoparticles with random packing, different packing arrangements occur at different locations, making the compacted nanocrystalline materials have highly nonuniform nanostructures and deformation behaviors. But the overall deformation behaviors under randompacking should be fallen between the FCC and SC packings.
0 0.5
BCC packing Particle size: 6.22nm T=70 0K
0.6 0.4 0.2 0
0
0.1
0.2
0.3
The Applied Strain Fig. 13. (a) The radial distribution functions at four compressive strain levels for nanocrystalline copper prepared at 700 K, with BCC packing and a particle size of 6.22 nm; (b) The variation of the fractions of atoms in the FCC, HCP and disordered environments at various compressive strain levels.
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Fig. 14. The deformation is accommodated by the mass shedding and rotation of grains and the thickening and sliding of grain boundaries: (a) an external strain level of 0.074, (b) an external strain level of 0.148, (c) an external strain level of 0.222, and (d) a strain level of 0.296. The same atoms during the compressive deformation are shown in the four images.
Due to the high loading rate and low deformation temperature, the calculated stress levels are high compared to experimental results. Also the nanocrystalline materials obtained here are fully dense and impurityfree, which may also contribute to the high stress levels. In addition, due to the high loading rate and low temperature, the diffusion-induced deformation is largely excluded. Under these conditions, it is commonly believed that two deformation mechanisms prevail: (1) the dislocation mediated inner grain plasticity and (2) the grain boundary sliding induced plasticity [2,3, 16,22]. In the present study, the dislocation-mediated deformation plays a minor role in accommodating the large plastic deformation, while surprisingly, grain mass shedding plays an important role. The fraction of grain boundaries with disordered atomic structures is formed at the expense of grain mass shedding during compression. The mass shedding from grains, together with grain rotations reduce the stress concentration due to the grain boundary sliding, and therefore avoid the formation of voids and cracks during the compressive deformation.
4. Conclusions Several conclusions can be drawn from the simulations: (1) The packing arrangements and sizes of
compacted nanoparticles and the preparation temperature affect the mechanical behaviors of the prepared nanocrystalline materials, however, there is no unique relationship between the degree of crystallinity and the yield strength of the nanocrystalline materials. (2) During the compression, a rapid softening occurs when the samples are prepared at lower temperatures, while elastic-perfectly-plastic deformation can be obtained at high preparation temperatures. (3) An increase in the externally applied strain, results in a decrease in the degree of crystallinity due to the mass shedding of the grains and thickening of grain boundaries. (4) During compressive deformation, the synergy of mass shedding and rotation of grains and the thickening and sliding of grain boundaries, avoid the formation of voids and cracks.
References [1] [2] [3] [4] [5] [6]
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[8] Lu L, Sui ML, Lu K. Science 2000;287:1463. [9] McFadden SX, Mishra RS, Valiez RZ, Zhilyaev AP, Mukherjee AK. Nature 1999;398:684. [10] Chen J, Wang W, Qian LH, Lu K. Scripta Mater 2003;49:645. [11] Ma E. Scripta Mater 2003;49:663. [12] Kumar KS, Suresh S, Chisholm MF, Horton JA, Wang P. Acta Mater 2003;51:387. [13] Van Swygenhoven H, Spaczer M, Caro A, Farkas D. Phys Rev B 1999;60:22. [14] Derlet PM, Hasnaoui A, Van Swygenhoven H. Scripta Mater 2003;49:629. [15] Van Swygenhoven H, Caro A. Phys Rev B 1998;58:11246. [16] Schiotz J, Di Tolla FD, Jacobsen KW. Nature 1998;391:561. [17] Schiotz J, Jacobsen KW. Science 2003;301:1357. [18] Schiotz J, Vegge T, Di Tolla FD, Jacobsen KW. Phys Rev B 1999;60:11971.
[19] Yip S. Nature 1998;391:532. [20] Yamakov V, Wolf D, Salazar M, Phillpot SR, Gleiter H. Acta Mater 2001;49:2713. [21] Yamakov V, Wolf D, Phillpot SR, Mukherjee AK. Nature Mater 2002;1:1. [22] Nieman GW, Weertman JR, Siegel RW. J Mater Res 1991;6: 1012. [23] Sanders PG, Witney AB, Weertman JR, Valiev RZ, Siegel RW. Mater Sci Eng A 1995;204:7. [24] Tellkamp VL, Melmed A, Lavernia EJ. Metall Mater Trans A 2001;32:2335. [25] Ragulya AV, Skorokhod VV. Nanostruct Mater 1995;5:835. [26] Finnis MW, Sinclair JE. Phil Mag A 1984;50:45. [27] Honeycutt JD, Andersen HC. J Phys Chem 1987;91:4950. [28] Clarke AS, Jonsson H. Phys Rev E 1993;47:3975. [29] Liu P, Zhang YW, Lu C. J Appl Phys 2003;94:6350.
Molecular dynamics simulations of the preparation and deformation of nanocrystalline copper Y.W. Zhang a
a,*
, P. Liu b, C. Lu
b
Department of Materials Science, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Institute of High Performance Computing, Singapore Received 26 May 2004; received in revised form 9 July 2004; accepted 13 July 2004 Available online 13 August 2004
Abstract The molecular dynamics method is used here to simulate: (1) the preparation of full-density nanostructured copper by compacting copper nanoparticles and (2) the deformation behaviors of the nanostructured copper under compression. It is found that the packing arrangement, the size of the nanoparticles and the compaction temperature, affect the deformation behaviors of the nanostructured copper. Our simulation results also show that the synergy of the rotation and mass shedding of grains and the thickening and sliding of grain boundaries, prevents the formation of voids and cracks in the nanostructured copper under compression. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocrystalline materials; Nanoparticles consolidation; Molecular dynamics; Plastic deformation; Copper
1. Introduction Nanocrystalline metals are of scientific interest and technological importance [1–12]. At the present stage, fully dense, impurity-free and bulk nanostructured crystalline materials are difficult to fabricate due to the limitation of the present synthesis techniques. It has been shown that loading rates, temperatures, grain sizes, porosity and impurities significantly affect the strength and ductility of nanostructured crystalline materials. Consequently, there are many conflicting experimental results on the strength and ductility of nanocrystalline metals due to the different fabrication techniques. For example, on the one hand, nanocrystalline materials exhibit superplasticity [8,9]; while on the other, nanocrystalline materials also show brittleness [4,11,12]. Molecular dynamics simulations have also been used to unveil the deformation mechanisms of nanocrystalline materials [13–21]. Detailed activities of dislocations *
Corresponding author. Tel.: +65 687 415 42; fax: +65 677 636 04. E-mail address: [email protected] (Y.W. Zhang).
and grain boundaries during the deformation of nanocrystalline materials may be examined through the atomistic simulations and are quantified to a degree that is impossible to determine experimentally. It is now recognized, through both experimental and modeling efforts, that for nanocrystalline materials, several deformation mechanisms may prevail: (1) dislocation slip, (2) diffusion through dislocations, (3) grain boundary sliding, (4) diffusion through grain boundaries, (5) grain rotations, (6) void nucleation, growth and migration and (7) crack nucleation and propagation. Due to the complexities of the nanostructures and deformation mechanisms, the true mechanical behaviors of fully compact, impurity-free and bulk nanocrystalline materials have not yet been fully understood. The formation and deformation of nanocrystalline materials through the compaction of nanoparticles has been studied experimentally by various researchers [22–25]. But due to the high porosity and high impurity, both the strength and ductility of the synthesized materials are low. In addition, due to the recrystallization, the average grain size is much larger than the average
1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.07.018
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nanoparticle size. As a result, the intrinsic mechanical behaviors of fully dense and impurity-free nanocrystalline materials with an average grain size of less than 20 nm, obtained through nanoparticle compaction, are still largely unknown. In this study, molecular dynamics simulations are performed to model the preparation of fully dense and impurity-free nanocrystalline copper through compaction of copper nanoparticles. During the compaction, different temperatures, packing arrangements and sizes of copper nanoparticles are used. The prepared nanocrystalline materials are used to perform compression tests at near 0 K to avoid atomic diffusion. It is found that the compaction temperature, the packing arrangement and the size of the nanoparticles affect the deformation behaviors of the nanostructured crystals. The simulations also show that the synergetic accommodation of the rotation and mass shedding of grains, and the thickening and sliding of grain boundaries, prevents the formation of voids and cracks in nanocrystalline copper.
2. Formulations An embedded atom potential of copper [26] is used in the present simulations. The potential has the following form: X 1=2 1 X X Etot ¼ qi þ V ij ; ð1Þ 2 i jði6¼jÞ i where qi is the second moment of the density of states of atom i and is written as X qi ¼ /ij : ð2Þ jði6¼jÞ
Both Vij and /ij are the functions only of the interatomic distance and are obtained by fitting physical properties of copper [26]. The atomic level stress components associated with an atom, rab i , can be calculated by using the potential, " # X 0 raij rbij 1 X 0 1=2 ab ri ¼ V qi / ; ð3Þ 2Xi j rij j where Xi is the atomic volume at the reference state, V 0 and / 0 are the derivatives of V and /, respectively, with respect to the interatomic distance rij. Simulations start either from an ideally packed unisize copper nanoparticle array or from a randomly packed arrangement with varied copper nanoparticle sizes. The particle orientations are chosen randomly. For ideally packed particles arrays, three packing arrangements: face-centered cubic (FCC), body-centered cubic (BCC) and simple cubic (SC) are used. The numbers of nanoparticles used for the FCC, BCC and SC packings are 108, 54 and 64, respectively. Three
nanoparticle sizes: 3.25, 6.22 and 8.40 nm are used. For randomly packed arrangements, the number of nanoparticles is always 70 and three average particle sizes: 3.49, 5.52 and 7.33 nm are used. Three temperatures: 300, 700 and 1200 K are used in the compacting processes for each configuration. The packed particles are compacted to the volume equal to that occupied by the single crystal at the corresponding temperatures. Subsequently, the temperature and the average hydrostatic stress are brought to a low level; the temperature is approximately 0 K and the hydrostatic stress approximately 0.1 GPa. (In this paper, positive stress indicates compression.) Finally the pressure is applied along the x-direction while keeping the average normal stresses along the y- and z-directions at about 0.1 GPa. It is noted that although the compactifications are done at different temperatures, all the uniaxial compressions are performed at the same temperature, that is, at 0 K. A typical example of the preparation and deformation process (using FCC packing, with a particle size of 6.22 nm and at a temperature of 700 K) is shown in Fig. 1. In the simulations, the number of atoms for the smallest case (with BCC packing at a particle size of 2.54 nm) is 86,454 and that for the largest case (with FCC packing at a particle size of 8.40 nm) is 2,832,516. The periodical boundary conditions are used in both compaction and compression, and the applied strain rate is approximately 6.3 · 109/s. The prepared nanomaterials are analyzed by calculating compact density, radial distribution functions and local atomic orders [27,28]. The degree of crystallinity (DOC), which is defined as the ratio of the number of atoms in a FCC environment to the total number of atoms in the nanocrystal, can be obtained by local atomic order analysis [27,28]. The current calculations show that the fraction of pores is very small after compaction and may be neglected. The nanocrystalline copper samples prepared can be taken to be fully dense and impurity-free.
3. Results and discussions The simulation results indicate that the deformation behaviors of the nanocrystalline materials are closely related to their nanostructures, which in turn are determined by the compaction temperatures, the packing arrangement and the size of the nanoparticles. Several features associated with the compaction process can be obtained from our simulations and involve the affects of: (1) particle sizes, (2) packing arrangements and (3) temperature. The larger the particle size, the higher the DOC of the nanocrystalline materials (the less the disordered portion, including defects and grain boundaries). For example, for nanocrystals prepared at 700 K and with FCC
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Fig. 1. The images show the modeling sequence of the preparation and deformation of nanocrystalline copper under the condition of FCC packing, T = 700 K and a particle size of 3.25 nm: (a) The initial configuration of 108 packed copper nanoparticles (in different colors), (b) after the hydrostatic compaction, (c) after stress relaxation and reduction of temperature and (d) after a uniaxial compression.
packing, the DOC at the nanoparticle diameters of 3.25, 6.22 and 8.40 nm are 0.29, 0.68 and 0.79, respectively. The reason is predominantly due to the DOC before compaction: the smaller the particles, the higher the fraction of surface to bulk atom ratio. Fig. 2(a)–(d) shows the snapshots of a typical particle in the randomly packed arrangement during the compaction. The average particle diameter is 5.52 nm. Fig. 3(a)–(d) shows the compaction of a typical particle in the same packing arrangement but with an average particle diameter of 3.49 nm. It is found that for the large particle, severe deformation mainly concentrates on its surface or subsurface, and the inner core largely remains crystalline; while for the small particle, the whole particle suffers severe deformation as shown Fig. 3. Hence at a same squashing level, the plastic deformation of small particles is much more pronounced than that of large particles. The packing arrangements affect the DOC of compacted copper. For example, the DOC of the nanocrystalline copper prepared at 700 K concerning a particle size of 6.22 nm is 0.68 for FCC, 0.65 for BCC and 0.48 for SC packings. Therefore, the FCC packing gives the highest DOC while the SC packing gives the lowest DOC among the three ideal packing arrangements. The
underlying reasons are due to the porosity level and the pore shape before compaction. For the SC packing arrangement, the porosity level is highest. If a full compaction is reached, the initial nearly-spherical particle must be squashed into a nearly cubic shape. This can be clearly seen from Fig. 4(a) and (b), which shows the squashing process of a particle in the SC packing array. For the FCC and BCC packing arrangements, the packing density is higher than that of the SC packing arrangement. In addition, when a full compaction is reached, the initial nearly-spherical particle is squashed into a shape which is still close to a sphere. This can be seen from Fig. 4(c)–(f) for a particle compacted in the BCC and FCC arrays, respectively. Therefore, the deformation level for the FCC and BCC packing arrangements is less severe than that for the SC packing arrangement. Between FCC and BCC, since the FCC porosity level before compaction is lower, therefore the deformation level should be also lower as well. This is consistent with the DOC order for the three packing conditions. For the randomly packed array with an average particle diameter of 5.52 nm compacted at 700 K, the DOC is about 0.61, which is close to the BCC packing, but higher than the SC packing and lower than the FCC packing. This is expected since the
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Fig. 2. Compact deformation of a typical particle in randomly packed arrangement with an average particle diameter of 5.52 nm at 700 K. (a) At 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction (the final state of compaction with a volume equal to that of the copper single crystal). The percentages here and in subsequent figures refer to the ratio of the reduction of the sample side length vs. the reduction of the final sample side length at the end of the compaction.
Fig. 3. Compact deformation of a typical particle in the randomly packed arrangement with an average particle diameter of 3.49 nm at 700 K. (a) at 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction.
porosity of the randomly packed array should be generally lower than that of the SC array, but higher than that of the FCC array.
The nanostructured copper shows a higher DOC when prepared at relatively low temperatures (300 and 700 K), and a lower DOC when prepared at high temperatures (1200 K). For example, for the samples prepared under the FCC packing and at a particle size of 6.22 nm, the DOC is 0.63 for 1200 K, 0.68 for 700 K and 0.67 for 300 K. For the samples prepared under the random-packing and at an average particle diameter of 5.52 nm, the DOC is 0.54 for 1200 K, 0.61 for 700 K and 0.63 for 300 K. The compaction processes for a typical particle in the above random-packing cases are shown in Fig. 5(a)–(d) for 300 K, Fig. 2(a)–(d) for 700 K and Fig. 6(a)–(d) for 1200 K, respectively. It is clearly seen from Figs. 2, 5 and 6 that the deformation is lest severe for 300 K while most severe for 1200 K. Evidently the nanoparticles have been severely damaged at the higher temperature compaction and the deformed particles show a large portion of amorphous structures (see Fig. 6). Since the packed particles are compacted to the volume equal to that occupied by the copper single crystal, the maximum compacting pressures will be different due to different packing arrangements, geometries and temperatures. Fig. 7 shows the variation of the average hydrostatic pressure with the compacting time during compaction and unloading processes for the randomly
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Fig. 4. Compact deformation of a typical particle with a particle diameter of 3.25 nm at 700 K in different packing arrangements. (a) at 25% compaction and (b) at 100% compaction in the SC packing; (c) at 25% compaction and (d) at 100% compaction in the BCC packing; and (e) at 25% compaction and (f) at 100% compaction in the FCC packing.
packed particle arrangement with three average particle diameters: 3.49, 5.52 and 7.33 nm. It appears that the larger the average particle size, the higher the pressure required to form a dense nanostructured sample. The maximum pressures for the three cases are approximately 10.8, 12.5 and 13.6 GPa. The prepared nanocrystalline copper samples are used to perform uniaxial compression along the x-direction. The atomic stress is calculated using the atomic potential [29]. The stresses reported are the stresses averaged over all atoms in the simulation cells. The applied strain is defined as (L0L)/L0, where L0 and L are the lengths of the sample along the x-direction before
and after compression (here, a positive strain value implies compressive strain). During compression, all of the samples exhibited a rapid softening after yielding when prepared at 300 and 700 K. Consequently, the yielding strength is very close to the maximum compressive stress. A typical example of the average stress Ærxxæ vs. the external applied strain curves for the three particle sizes, 3.25, 6.22 and 8.40 nm, under 700 K and SC packing are shown in Fig. 8. It is interesting to note that the maximum yield strength occurs at the particle size of 6.22 nm. In this case, there is no correlation between the DOC before the compression and the yield strength during compression. For the randomly packed particle
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Fig. 5. Compact deformation of a typical particle in randomly packed arrangement with an average particle diameter of 5.52 nm at 300 K. (a) at 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction.
Fig. 6. Compact deformation of a typical particle in randomly packed arrangement with an average particle diameter of 5.52 nm at 1200 K. (a) At 25% compaction, (b) at 50% compaction, (c) at 75% compaction, and (d) at 100% compaction.
15
D=3.49nm D=5.52nm D=7.33nm
10
5
0
0
0. 5
1
1. 5
2
arrangement, the average stress Ærxxæ vs. the external applied strain curves for the three particle sizes, 3.49, 5.52 and 7.33 nm, at a compaction temperature of 700 K are shown in Fig. 9. It is also interesting to note that the maximum yielding strength occurs at the particle size of 5.52 nm. These simulation results suggest that there may be a transition of the Hall–Petch to the inverse Hall–Petch regimes. For the samples prepared at 300 and 700 K, the graphs of the average stress Ærxxæ vs. the applied strain curves show little difference as shown in Fig. 10. However, at 1200 K, the stress vs. strain reveals a roughly elastic-perfectly-plastic deformation, except that the stress oscillates with an increase in applied strain. Its yield strength is much lower than its lower temperature counterparts. A same trend is also observed for the randomly packed particle arrangement as shown in Fig. 11. The reason for the reduction of yield strength at high temperatures can be deduced from Figs. 2, 5 and 6.
The Average Stress <σxx> (GPa)
5 D=3.25nm D=6.22nm D=8.40nm
3 2 SC packing T=700K
1 0 0
0.1
0.2
0.3
The Applied Strain Fig. 8. The average stress Ærxxæ vs. the externally applied strain at three different particle sizes at 700 K and with a SC packing arrangement.
3 2 Random packing Compaction temperature 700K
1
0
0.1
0.2
0.3
The Applied Strain
Fig. 9. The average stress Ærxxæ vs. the externally applied strain at three different particle sizes at 700 K and with a randomly packed arrangement.
5 The Average Stress <σxx> (GPa)
Fig. 7. Variation of the average hydrostatic pressure Ærmæ with the compacting time for the randomly packed particle arrangement with three different average particle diameters: 3.49, 5.52 and 7.33 nm.
D=3.49nm D=5.52nm D=7.33nm
4
0
The Compaction Time (ns)
4
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5
Random packing Compaction temperature 700K
The Average Stress <σxx> (GPa)
The Average Pressure < σm> (GPa)
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T=300K T=700K T=1200K
4 3 2 1 0 0
SC packing Particle size: 6.22nm
0.1
0.2
0.3
The Applied Strain Fig. 10. The average stress Ærxxæ vs. the externally applied strain at different preparation temperatures with a SC packing and a particle size of 6.22 nm.
During compaction at the high temperature, the nanoparticles have been severely damaged and the deformed particles show a large portion of amorphous structures (see Fig. 6). Therefore during compression, the deformation mainly occurs in amorphous (non-crystalline) regions. In these cases, the yield strength is proportionally related to the DOC after compactions, that is, the higher the DOC, the higher the yield strength. The calculation results show that the stress Ærxxæ vs. the applied strain curves are roughly the same for the FCC and BCC packings, but different from that of the SC packing. This can be clearly seen in Fig. 12 for 700 K with a particle diameter of 6.22 nm. The yield strength of SC packing is much higher than that of FCC and BCC packings. The reason for the difference is likely due to the different deformation patterns during the compaction. For the FCC and BCC packings, the initial
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The Average Stress <σxx> (GPa)
5 T=30 0K T=70 0K T=12 00K
4 3 2
Random packing Average particle size 5.52nm
1 0
0
0.1
0.2
0.3
The Applied Strain
Fig. 11. The average stress Ærxxæ vs. the externally applied strain at different preparation temperatures with a randomly packed arrangement and a particle size of 5.52 nm.
SC FCC BCC
4
6
(a)
3
0.074 0.148 0.222 0.296
2 1 0 0
4
Particle size: 6.22nm T=70 0K
0.1
0.2
BCC packing Particle size: 6.22nm T=700K
RDF
The Average Stress <σxx> (GPa)
5
During the compression of a sample obtained through compacting a BCC array at 700 K and with a particle diameter of 6.22 nm, the radial distribution functions (RDFs) are shown in Fig. 13(a) for applied strain levels at 0.074, 0.148, 0.222 and 0.296. It may be observed that with an increase in the applied strain level, the peaks in the RDFs become broader, indicating a lower DOC. This is confirmed by a local bond order analysis. A typical example of such analysis results is given in Fig. 13(b) at a compaction temperature of 700 K and a particle size of 6.22 nm. It can be seen that the FCC fraction gradually decreases, the HCP fraction remains roughly at the same level, while the fraction of the disordered portion increases with an increase in the strain level. Detailed examination of the nanostructures shows that during compression of the nanocrystalline copper samples, the grains shed atoms. As a consequence, grains are downsized and the grain boundary zones are thickened. The synergic accommodation of grain boundary thickening and sliding, and the grain mass shedding and rotating, prevent the formation of voids and cracks as shown in Fig. 14.
2
0.3
The Applied Strain Fig. 12. The average stress Ærxxæ vs. the externally applied strain under SC, FCC and BCC packing arrangements at 700 K and a particle diameter of 6.22 nm.
0.75
1
1.25
1.5
r/a0 (b)
1 FCC HCP Other (GBs)
0.8
Fraction
porosity is low, relatively small deformation and pressure is required to form dense nanostructured copper as shown in Fig. 4(c)–(f). For the SC packing, the initial porosity is high, the particles need to be squashed more to form a compact sample (see Fig. 4(a) and (b)). As a consequence, the deformation degree of nanoparticles under SC packing is higher than that under FCC and BCC packing arrangements. Thus dislocation pileups and twining density in grains under SC packing after compaction will be high. The high dislocation pileup level and twinning density during compaction contribute to the high yield stress for SC packing. For nanoparticles with random packing, different packing arrangements occur at different locations, making the compacted nanocrystalline materials have highly nonuniform nanostructures and deformation behaviors. But the overall deformation behaviors under randompacking should be fallen between the FCC and SC packings.
0 0.5
BCC packing Particle size: 6.22nm T=70 0K
0.6 0.4 0.2 0
0
0.1
0.2
0.3
The Applied Strain Fig. 13. (a) The radial distribution functions at four compressive strain levels for nanocrystalline copper prepared at 700 K, with BCC packing and a particle size of 6.22 nm; (b) The variation of the fractions of atoms in the FCC, HCP and disordered environments at various compressive strain levels.
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Fig. 14. The deformation is accommodated by the mass shedding and rotation of grains and the thickening and sliding of grain boundaries: (a) an external strain level of 0.074, (b) an external strain level of 0.148, (c) an external strain level of 0.222, and (d) a strain level of 0.296. The same atoms during the compressive deformation are shown in the four images.
Due to the high loading rate and low deformation temperature, the calculated stress levels are high compared to experimental results. Also the nanocrystalline materials obtained here are fully dense and impurityfree, which may also contribute to the high stress levels. In addition, due to the high loading rate and low temperature, the diffusion-induced deformation is largely excluded. Under these conditions, it is commonly believed that two deformation mechanisms prevail: (1) the dislocation mediated inner grain plasticity and (2) the grain boundary sliding induced plasticity [2,3, 16,22]. In the present study, the dislocation-mediated deformation plays a minor role in accommodating the large plastic deformation, while surprisingly, grain mass shedding plays an important role. The fraction of grain boundaries with disordered atomic structures is formed at the expense of grain mass shedding during compression. The mass shedding from grains, together with grain rotations reduce the stress concentration due to the grain boundary sliding, and therefore avoid the formation of voids and cracks during the compressive deformation.
4. Conclusions Several conclusions can be drawn from the simulations: (1) The packing arrangements and sizes of
compacted nanoparticles and the preparation temperature affect the mechanical behaviors of the prepared nanocrystalline materials, however, there is no unique relationship between the degree of crystallinity and the yield strength of the nanocrystalline materials. (2) During the compression, a rapid softening occurs when the samples are prepared at lower temperatures, while elastic-perfectly-plastic deformation can be obtained at high preparation temperatures. (3) An increase in the externally applied strain, results in a decrease in the degree of crystallinity due to the mass shedding of the grains and thickening of grain boundaries. (4) During compressive deformation, the synergy of mass shedding and rotation of grains and the thickening and sliding of grain boundaries, avoid the formation of voids and cracks.
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