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Rotation of small clusters in sheared metallic glasses
Chemical Physics 386 (2011) 101–104
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Rotation of small clusters in sheared metallic glasses Francesco Delogu Dipartimento di Ingegneria Chimica e Materiali, Università di Cagliari, piazza d’Armi, I-09123 Cagliari, Italy
a r t i c l e
i n f o
Article history: Received 7 May 2011 In final form 18 June 2011 Available online 24 June 2011 Keywords: Metallic glasses Plastic deformation Non-affine displacements Molecular dynamics
a b s t r a c t Molecular dynamics methods were used to simulate the response of a Cu50Ti50 metallic glass to shear deformation. Attention was focused on the atomic displacements taking place during the irreversible rearrangement of local atomic structures. It is shown that the apparently disordered dynamics of such events hides the rigid body rotation of small clusters. Cluster rotation was investigated by evaluating rotation angle, axis and lifetimes. This permitted to point out that relatively large clusters can undergo two or more complete rotations. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Numerical studies have clearly shown that the shear deformation of metallic glasses (MGs) is mediated by relatively high-energy atomic rearrangements, consisting of inelastic shear distortions localized in shear transformation zones (STZs) [1–23]. When STZs operate, about 10–200 atoms [24–35] undergo non-affine atomic displacements [11–20]. Correspondingly, local structures move from a relatively low-energy configuration to another [1–23]. The mobility burst lasts about 40 ps [24–35] and is accompanied by atomic stress drops of about 10–30 GPa [28,29]. The process is governed by an activation barrier roughly 30–80 kJ mol 1 high [1–23] and an activation volume approximately equal to 10 Å3 [24–35]. The regions participating in local rearrangements exhibit lower mechanical stability [30,31,36–39]. Particular structural motifs are more sensitive than others to shear deformation [24–27] and different regimes of STZ operation emerge depending on deformation conditions [32–35]. Although the above mentioned findings provide valuable insight into the elementary processes underlying the MG deformation mechanism, the complexity of the subject leaves a number of fundamental questions open. An example is the connection between local structures and STZs, which is still unsatisfactorily understood despite remarkable recent advances [1,24–27]. The same is true for the degree of correlation between individual atomic displacements during the STZ operation [1]. In fact, most studies use non-affine displacements only to point out STZs, paying scarce attention to their intrinsic dynamics [1]. Yet, characterizing local rearrangements in terms of relative atomic positions can in principle reveal interesting details of their intimate mechanism. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.06.030
Focusing precisely on the relative motion of atoms involved in the operation of STZs, the present work points out the occurrence of synchronous atomic displacements resulting in the rotation of small atomic clusters around an axis parallel to the shear deformation plane. Molecular dynamics methods were used to simulate the shear deformation of a Cu50Ti50 MG. The details of numerical simulations are given below. 2. Molecular dynamics simulations Calculations were performed by employing an embedded-atom method potential [40] extensively used to study the disorder-induced amorphization of Cu–Ti systems [41,42]. Potential parameters were taken from literature [41,42]. Forces were computed within a cut-off radius of about 0.7 nm. Simulations were carried out with number N of atoms, pressure P and temperature T constant [43,44]. Equations of motion were solved with a fifth-order predictor–corrector algorithm [45] and a time step of 2 fs. A Cu50Ti50 MG roughly including 131,000 atoms in a cubic volume with sides about 12.4 nm long was prepared by quenching the melt from 3000 to 20 K at a rate of 0.02 K ns 1 and zero pressure. Periodic boundary conditions (PBCs) were applied along the x, y and z Cartesian directions. Pointed out by the Wendt–Abraham method [46], the glass transition occurred at about 620 K, in accordance with previous work [29]. The obtained MG was relaxed for 0.5 ns. The amorphous character of its structure was clearly demonstrated by the pair distribution function, not shown for brevity. The mechanical response of the MG to shear deformation was studied at 20 K as detailed in the following. Following previous work [30,31,29], rigid reservoirs 1.5 nm thick were created at the top and bottom of the simulation cell along the z Cartesian direction. PBCs were kept along x and y ones. Shear deformation along
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the x Cartesian direction was produced by imposing to reservoirs, kept at constant distance, incremental displacements of about 3.1 10 3 nm every 20 ps. Accordingly, a strain increment de of about 2.5 10 4 was produced every 20 ps, which corresponds to a strain rate of about 1.25 10 2 ns 1. Similar results were obtained at 5 10 3 and 5 10 2 ns 1. No thermostat was applied during deformation, carried out in the average strain e range between 0 and 0.2. Voronoi polyhedra [47] were constructed around each atom to evaluate the atomic volumes, X, and stresses, rat [48]. In turn, the macroscopic stress r was calculated by averaging over all of the rat values. The obtained estimate is about 5% lower than the one given by the virial expression [45]. The number n of nearest neighbors of a given atom was set equal to the number of faces of its Voronoi polyhedron. Atomic positions, stresses, and volumes were averaged over 5-ps long time intervals.
3. Results and discussion The macroscopic shear stress r is shown in Fig. 1 as a function of the macroscopic shear strain e. The curve provides a description of the mechanical response of the MG systems based on macroscopic features. Initially, the MG undergoes a deformation stage characterized by a linear increase of r with e. This stage approximately extends up to a strain e value of 0.04. Yield occurs at a strain e of about 0.07, when the stress r reaches a maximum value roughly equal to 13 GPa. Afterwards, a steady-state flow regime is attained, characterized by r values fluctuating around 8 GPa. The obtained curve is very close to the one exhibited by the same MG at 10 K [29]. The atomic positions were constantly monitored throughout the deformation process to point out local irreversible rearrangements (IRs). The atoms involved in IRs were identified by using the non-affine component d of atomic displacements. The d values were calculated by comparing the atomic positions in consecutive MG configurations separated by a single strain increment de and a relaxation period of 100 ps. Thus, for any given atom, d measures the deviation of the atomic position from the one expected on the basis of an affine deformation [1,11–17,29]. Taking into account that the average amplitude of thermal vibrations amounts to about 0.1 Å, a threshold of 0.5 Å was used to eliminate the contributions to d due to atomic vibrations [29]. Any atom undergoing a non-affine displacement d longer than 0.5 Å was then regarded as involved in IRs. The number NIR of atoms involved in IRs at least once is also shown in Fig. 1 as a function of the macroscopic strain e. It can be seen that NIR increases starting from the very beginning of
Fig. 1. The macroscopic stress shear r and the number NIR of atoms involved in IRs at least once as a function of the macroscopic shear strain e. For illustration purposes, data in the e range between 0.15 and 0.2 are not plotted. Two clusters of atoms involved in IRs, similar to a branched chain polymer (right) and to globular star cluster (left), are also shown.
deformation. Correspondingly, IRs of local structures are also observed in the initial deformation stage characterized by a roughly linear change of the shear stress r with the shear strain e. It follows that the linear change of r with e does not correspond to an elastic deformation regime, usually observed in the case of crystalline systems. Rather, the MG structure exhibits an inelastic deformation even in the initial stages. Following a roughly linear trend, when e equals 0.15 NIR reaches a value of about 5.6 104, which corresponds to a percentage of atoms that have participated in IRs at least once of about 43%. A close look to the NIR curve reveals the irregular NIR increase, characterized by apparently random increments. Irregular are also the shapes exhibited by the atomic arrangements involved in IRs. Two examples of such atomic arrangements are shown in Fig. 1. Although the shape of atomic arrangements undergoing IRs is quite variegated, the arrangements shown are somewhat representative of the two general typologies observed. One of them recalls the branched chain of a polymer, whereas the other is similar to a globular star cluster. As a whole, the displacements of atoms involved in IRs look like part of a general rearrangement process with a relatively disordered dynamics. The direct visualization of displacing atoms also generally gives the idea of a complicate collective motion with no evident correlation between individual displacements. However, a detailed analysis of the dynamics of individual IRs points out that ordered events such as the rigid body rotation of small clusters can also take place. In this work, cluster rotation processes were identified by using three criteria to be simultaneously satisfied: (i) the distances between the cluster atoms must keep constant, except for small fluctuations on the order of ±0.2 Å due to thermal vibrations or ascribable to small local distortions; (ii) the angle between the vectors connecting a given cluster atom with two of its nearest neighbors, both included in the cluster, must also keep approximately constant, with a tolerance of ±3°; (iii) the orientation of the vectors connecting cluster atoms lying approximately on a plane perpendicular to the rotation axis must change of the same angle with respect to the inertial Cartesian reference framework. Here, it is worth noting that tolerating uncertainties in interatomic distances and angles for criteria (i) and (ii) is necessary in the light of the complex dynamics exhibited by IRs. Indeed, the overall displacements of individual atoms originates from a combination of thermal fluctuations and local mechanical forces that do not allow any simple deconvolution of motion. The situation is even more complicated for criterion (iii), where the mutual orientation of the vectors connecting the various pairs of cluster atoms is considered. The difficulty arises from the fact that only a few vectors lie on the approximate plane of the cluster rotation. Therefore, most of vectors point along different directions, which also means that their apparent rotation angle is different from the one of the other vectors. This is the reason for which only a few of them, and in particular, the ones lying approximately on the plane perpendicular to the cluster rotation axis, were considered to check criterion (iii). It must be also noted that there is no need to refer to non-affine displacements to identify rotating clusters. In fact, the rigid body rotation implies the conservation of the cluster symmetry, and then of relative atomic positions. Thus, absolute atomic positions can be used. Along the same line, once identified the clusters exhibiting a rigid body behavior, classical physics relationships permit to evaluate the rotation axis and the rotation angle. The total number nrot of rotating clusters observed during the deformation up to an average strain e of 0.2 is on the order of 2.7 103. The statistical distribution p(Ncl) of the cluster size Ncl is shown in Fig. 2a. The observed clusters are invariably small, with a number Ncl of connected atoms ranging from 3 to 45. The p(Ncl)
F. Delogu / Chemical Physics 386 (2011) 101–104
(a)
(b)
Fig. 2. (a) The statistical distribution p(Ncl) of the cluster size Ncl. (b) The statistical distribution p(srot) of the cluster rotation lifetime srot. Data refer to 1161 clusters including 3 atoms (white), 162 clusters including 10 atoms (light gray) and 38 clusters including 20 atoms (dark gray).
curve has an exponential character. Roughly the 74% of clusters have size Ncl below 13 and only the 2% include more than 30 atoms. The total number of atoms involved in cluster rotation processes at least once is equal to about 4.3 103, i.e. approximately the 7.7% of the atoms involved in IRs. Rotating clusters are observed throughout the deformation process. However, only clusters with size Ncl smaller than 7 rotate in the elastic deformation regime, whereas clusters including more than 24 atoms are observed only in the flow deformation regime, when e is larger than 0.09. Thus, only in the latter deformation stages the complex conditions required for a relatively large cluster to rotate are satisfied. Clusters rotate for relatively small time, or strain, intervals. In the 93% of cases, the interruption of the rotation process is accompanied by the disaggregation of the cluster, with one or more atoms moving away from it. This mechanism operates in all of the clusters with size Ncl smaller than 19. For larger clusters, it is sometimes observed that rotation simply interrupts with no consequence on the cluster integrity. The cluster rotation lifetime srot does not trivially correlate with the cluster size Ncl. This can be easily pointed out by selecting rotating clusters of a given size Ncl and looking at the statistical distribution p(srot) of their srot values. The p(srot) curves for 1161, 162 and 38 clusters including 3, 10 and 20 atoms respectively are shown in Fig. 2b. The distributions p(srot) are quite similar to each other and extend over a relatively large time domain from 0.1 to 1.6 ns. However, about the 75% of clusters rotate for only 0.1 and 0.3 ns, irrespective of their size. Analogous results were obtained in all of the cases in which a rough statistical analysis was possible, i.e. when the number of rotating clusters with the same size was larger than 30. For about the 77% of clusters, the rotation axis is approximately parallel to the y Cartesian direction, the angular deviation being at most of 5°. In the other cases, the rotation axis is randomly oriented. Among the clusters with a rotation axis parallel to the y Cartesian direction, about the 60% exhibit a rotation axis external to the cluster, whereas in the remaining 40% the rotation axis is internal.
103
When the rotation axis lies externally to the cluster, a complete revolution is never observed. In contrast, for clusters with internal rotation axis, even short lifetimes srot can correspond to one or more complete rotation cycles depending on the cluster size Ncl. The statistical distribution p(hrot) of rotation angles hrot, evaluated on the 829 clusters with internal rotation axis, is shown in Fig. 3a. The p(hrot) curve exhibits a relatively irregular shape, indicating that roughly the 96% of clusters rotate for angles hrot smaller than 360°. However, two clusters, including 8 and 37 atoms respectively, were able to rotate for more than two complete rotation cycles. Whereas small clusters generally undergo an irregular, but continuous, rotation, the largest clusters can also exhibit an intermittent rotation regime. For sake of illustration, the cases of two clusters including 6 and 27 atoms respectively are shown in Fig. 3b. Here, the rotation angle hrot is plotted as a function of the number m of strain increments de forming the strain interval in which the selected clusters rotate. The curves in Fig. 3b can be regarded as representative of the behavior of clusters with the same size. To allow a reliable comparison on a relative basis, the clusters were suitably chosen to have roughly the same lifetime srot of about 500 ps. It can be seen that for the cluster including 6 atoms the rotation angle hrot undergoes a steady, though irregular, increase. In contrast, the hrot curve regarding the 27-atom cluster exhibits two marked plateaus starting at m values respectively equal to 10 and 14. The rotation of relatively large clusters is characterized by a relatively complex dynamics. It is accompanied by a number of processes, including atomic displacements due to IRs and atomic coordination changes. An idea of what happens in the case of a cluster including 32 atoms is given in Fig. 4, where the cluster and the surrounding atoms are shown at different times. The cluster rotates of about 60° in roughly 360 ps, with relative atomic positions substantially conserved. This allows the cluster to rotate as a rigid body around a rotation axis that passes close to the cluster center of mass and is almost perfectly parallel to the y Cartesian direction. The atoms surrounding the rotating cluster are involved
(a)
(b)
Fig. 3. (a) The statistical distribution p(hrot) of cluster rotation angles hrot. Data refer to 829 clusters with internal rotation axis. (b) The rotation angle hrot is plotted as a function of the number m of strain increments de during which the selected clusters undergo rotation. Data refer to clusters including 6 and 27 atoms respectively.
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cases even relatively large clusters can undergo two or more rotation cycles. Acknowledgement Financial support has been given by the University of Cagliari. A. Ermini, ExtraInformatica s.r.l., is gratefully acknowledged for his kind assistance. References
Fig. 4. A cluster of 32 atoms exhibiting a rotation angle hrot of about 0°, 23°, 48° and 61° after the time intervals indicated. Light and dark gray have been used for the Cu and Ti cluster atoms respectively. White was used for the atoms surrounding the rotating cluster. Four of them have been marked with numbers to help visualizing the rotation process. A projection on the (y; z) Cartesian plane is shown.
in an IR, which is a necessary condition to have rotation. The atoms involved in the IR and not belonging to the rotating cluster modify their relative positions. Some of them accompany the cluster in its rotation, thus rotating in the same direction. However, their displacements do not satisfy the three previously mentioned criteria. As a consequence, their motion cannot be described as a rigid body rotation, but rather as the simple rearrangement of a local structure. Nevertheless, it is precisely their displacement to assist and permit the cluster rotation. Similar observations can be made for all of the observed rotation processes. The cluster is located in a region participating in an IR and its rotation is accompanied by a collective displacement of surrounding atoms in the same direction of rotation.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
4. Conclusions
[38] [39]
The numerical findings indicate that the deformation is accompanied by IRs. A detailed analysis of the atomic-scale dynamics underlying IRs pointed out the occurrence of ordered rotation processes. In particular, atomic clusters can undergo a rigid body rotation. Rotating clusters are small, including at most 45 atoms, and are always embedded into a region undergoing an IR. Thus, the cluster rotation is governed by the IR dynamics. Depending on the cluster size, rotation can be regular or intermittent. The lifetime of rotation processes is generally short, although in a few
[40] [41] [42] [43] [44] [45] [46] [47] [48]
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Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Rotation of small clusters in sheared metallic glasses Francesco Delogu Dipartimento di Ingegneria Chimica e Materiali, Università di Cagliari, piazza d’Armi, I-09123 Cagliari, Italy
a r t i c l e
i n f o
Article history: Received 7 May 2011 In final form 18 June 2011 Available online 24 June 2011 Keywords: Metallic glasses Plastic deformation Non-affine displacements Molecular dynamics
a b s t r a c t Molecular dynamics methods were used to simulate the response of a Cu50Ti50 metallic glass to shear deformation. Attention was focused on the atomic displacements taking place during the irreversible rearrangement of local atomic structures. It is shown that the apparently disordered dynamics of such events hides the rigid body rotation of small clusters. Cluster rotation was investigated by evaluating rotation angle, axis and lifetimes. This permitted to point out that relatively large clusters can undergo two or more complete rotations. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Numerical studies have clearly shown that the shear deformation of metallic glasses (MGs) is mediated by relatively high-energy atomic rearrangements, consisting of inelastic shear distortions localized in shear transformation zones (STZs) [1–23]. When STZs operate, about 10–200 atoms [24–35] undergo non-affine atomic displacements [11–20]. Correspondingly, local structures move from a relatively low-energy configuration to another [1–23]. The mobility burst lasts about 40 ps [24–35] and is accompanied by atomic stress drops of about 10–30 GPa [28,29]. The process is governed by an activation barrier roughly 30–80 kJ mol 1 high [1–23] and an activation volume approximately equal to 10 Å3 [24–35]. The regions participating in local rearrangements exhibit lower mechanical stability [30,31,36–39]. Particular structural motifs are more sensitive than others to shear deformation [24–27] and different regimes of STZ operation emerge depending on deformation conditions [32–35]. Although the above mentioned findings provide valuable insight into the elementary processes underlying the MG deformation mechanism, the complexity of the subject leaves a number of fundamental questions open. An example is the connection between local structures and STZs, which is still unsatisfactorily understood despite remarkable recent advances [1,24–27]. The same is true for the degree of correlation between individual atomic displacements during the STZ operation [1]. In fact, most studies use non-affine displacements only to point out STZs, paying scarce attention to their intrinsic dynamics [1]. Yet, characterizing local rearrangements in terms of relative atomic positions can in principle reveal interesting details of their intimate mechanism. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.06.030
Focusing precisely on the relative motion of atoms involved in the operation of STZs, the present work points out the occurrence of synchronous atomic displacements resulting in the rotation of small atomic clusters around an axis parallel to the shear deformation plane. Molecular dynamics methods were used to simulate the shear deformation of a Cu50Ti50 MG. The details of numerical simulations are given below. 2. Molecular dynamics simulations Calculations were performed by employing an embedded-atom method potential [40] extensively used to study the disorder-induced amorphization of Cu–Ti systems [41,42]. Potential parameters were taken from literature [41,42]. Forces were computed within a cut-off radius of about 0.7 nm. Simulations were carried out with number N of atoms, pressure P and temperature T constant [43,44]. Equations of motion were solved with a fifth-order predictor–corrector algorithm [45] and a time step of 2 fs. A Cu50Ti50 MG roughly including 131,000 atoms in a cubic volume with sides about 12.4 nm long was prepared by quenching the melt from 3000 to 20 K at a rate of 0.02 K ns 1 and zero pressure. Periodic boundary conditions (PBCs) were applied along the x, y and z Cartesian directions. Pointed out by the Wendt–Abraham method [46], the glass transition occurred at about 620 K, in accordance with previous work [29]. The obtained MG was relaxed for 0.5 ns. The amorphous character of its structure was clearly demonstrated by the pair distribution function, not shown for brevity. The mechanical response of the MG to shear deformation was studied at 20 K as detailed in the following. Following previous work [30,31,29], rigid reservoirs 1.5 nm thick were created at the top and bottom of the simulation cell along the z Cartesian direction. PBCs were kept along x and y ones. Shear deformation along
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the x Cartesian direction was produced by imposing to reservoirs, kept at constant distance, incremental displacements of about 3.1 10 3 nm every 20 ps. Accordingly, a strain increment de of about 2.5 10 4 was produced every 20 ps, which corresponds to a strain rate of about 1.25 10 2 ns 1. Similar results were obtained at 5 10 3 and 5 10 2 ns 1. No thermostat was applied during deformation, carried out in the average strain e range between 0 and 0.2. Voronoi polyhedra [47] were constructed around each atom to evaluate the atomic volumes, X, and stresses, rat [48]. In turn, the macroscopic stress r was calculated by averaging over all of the rat values. The obtained estimate is about 5% lower than the one given by the virial expression [45]. The number n of nearest neighbors of a given atom was set equal to the number of faces of its Voronoi polyhedron. Atomic positions, stresses, and volumes were averaged over 5-ps long time intervals.
3. Results and discussion The macroscopic shear stress r is shown in Fig. 1 as a function of the macroscopic shear strain e. The curve provides a description of the mechanical response of the MG systems based on macroscopic features. Initially, the MG undergoes a deformation stage characterized by a linear increase of r with e. This stage approximately extends up to a strain e value of 0.04. Yield occurs at a strain e of about 0.07, when the stress r reaches a maximum value roughly equal to 13 GPa. Afterwards, a steady-state flow regime is attained, characterized by r values fluctuating around 8 GPa. The obtained curve is very close to the one exhibited by the same MG at 10 K [29]. The atomic positions were constantly monitored throughout the deformation process to point out local irreversible rearrangements (IRs). The atoms involved in IRs were identified by using the non-affine component d of atomic displacements. The d values were calculated by comparing the atomic positions in consecutive MG configurations separated by a single strain increment de and a relaxation period of 100 ps. Thus, for any given atom, d measures the deviation of the atomic position from the one expected on the basis of an affine deformation [1,11–17,29]. Taking into account that the average amplitude of thermal vibrations amounts to about 0.1 Å, a threshold of 0.5 Å was used to eliminate the contributions to d due to atomic vibrations [29]. Any atom undergoing a non-affine displacement d longer than 0.5 Å was then regarded as involved in IRs. The number NIR of atoms involved in IRs at least once is also shown in Fig. 1 as a function of the macroscopic strain e. It can be seen that NIR increases starting from the very beginning of
Fig. 1. The macroscopic stress shear r and the number NIR of atoms involved in IRs at least once as a function of the macroscopic shear strain e. For illustration purposes, data in the e range between 0.15 and 0.2 are not plotted. Two clusters of atoms involved in IRs, similar to a branched chain polymer (right) and to globular star cluster (left), are also shown.
deformation. Correspondingly, IRs of local structures are also observed in the initial deformation stage characterized by a roughly linear change of the shear stress r with the shear strain e. It follows that the linear change of r with e does not correspond to an elastic deformation regime, usually observed in the case of crystalline systems. Rather, the MG structure exhibits an inelastic deformation even in the initial stages. Following a roughly linear trend, when e equals 0.15 NIR reaches a value of about 5.6 104, which corresponds to a percentage of atoms that have participated in IRs at least once of about 43%. A close look to the NIR curve reveals the irregular NIR increase, characterized by apparently random increments. Irregular are also the shapes exhibited by the atomic arrangements involved in IRs. Two examples of such atomic arrangements are shown in Fig. 1. Although the shape of atomic arrangements undergoing IRs is quite variegated, the arrangements shown are somewhat representative of the two general typologies observed. One of them recalls the branched chain of a polymer, whereas the other is similar to a globular star cluster. As a whole, the displacements of atoms involved in IRs look like part of a general rearrangement process with a relatively disordered dynamics. The direct visualization of displacing atoms also generally gives the idea of a complicate collective motion with no evident correlation between individual displacements. However, a detailed analysis of the dynamics of individual IRs points out that ordered events such as the rigid body rotation of small clusters can also take place. In this work, cluster rotation processes were identified by using three criteria to be simultaneously satisfied: (i) the distances between the cluster atoms must keep constant, except for small fluctuations on the order of ±0.2 Å due to thermal vibrations or ascribable to small local distortions; (ii) the angle between the vectors connecting a given cluster atom with two of its nearest neighbors, both included in the cluster, must also keep approximately constant, with a tolerance of ±3°; (iii) the orientation of the vectors connecting cluster atoms lying approximately on a plane perpendicular to the rotation axis must change of the same angle with respect to the inertial Cartesian reference framework. Here, it is worth noting that tolerating uncertainties in interatomic distances and angles for criteria (i) and (ii) is necessary in the light of the complex dynamics exhibited by IRs. Indeed, the overall displacements of individual atoms originates from a combination of thermal fluctuations and local mechanical forces that do not allow any simple deconvolution of motion. The situation is even more complicated for criterion (iii), where the mutual orientation of the vectors connecting the various pairs of cluster atoms is considered. The difficulty arises from the fact that only a few vectors lie on the approximate plane of the cluster rotation. Therefore, most of vectors point along different directions, which also means that their apparent rotation angle is different from the one of the other vectors. This is the reason for which only a few of them, and in particular, the ones lying approximately on the plane perpendicular to the cluster rotation axis, were considered to check criterion (iii). It must be also noted that there is no need to refer to non-affine displacements to identify rotating clusters. In fact, the rigid body rotation implies the conservation of the cluster symmetry, and then of relative atomic positions. Thus, absolute atomic positions can be used. Along the same line, once identified the clusters exhibiting a rigid body behavior, classical physics relationships permit to evaluate the rotation axis and the rotation angle. The total number nrot of rotating clusters observed during the deformation up to an average strain e of 0.2 is on the order of 2.7 103. The statistical distribution p(Ncl) of the cluster size Ncl is shown in Fig. 2a. The observed clusters are invariably small, with a number Ncl of connected atoms ranging from 3 to 45. The p(Ncl)
F. Delogu / Chemical Physics 386 (2011) 101–104
(a)
(b)
Fig. 2. (a) The statistical distribution p(Ncl) of the cluster size Ncl. (b) The statistical distribution p(srot) of the cluster rotation lifetime srot. Data refer to 1161 clusters including 3 atoms (white), 162 clusters including 10 atoms (light gray) and 38 clusters including 20 atoms (dark gray).
curve has an exponential character. Roughly the 74% of clusters have size Ncl below 13 and only the 2% include more than 30 atoms. The total number of atoms involved in cluster rotation processes at least once is equal to about 4.3 103, i.e. approximately the 7.7% of the atoms involved in IRs. Rotating clusters are observed throughout the deformation process. However, only clusters with size Ncl smaller than 7 rotate in the elastic deformation regime, whereas clusters including more than 24 atoms are observed only in the flow deformation regime, when e is larger than 0.09. Thus, only in the latter deformation stages the complex conditions required for a relatively large cluster to rotate are satisfied. Clusters rotate for relatively small time, or strain, intervals. In the 93% of cases, the interruption of the rotation process is accompanied by the disaggregation of the cluster, with one or more atoms moving away from it. This mechanism operates in all of the clusters with size Ncl smaller than 19. For larger clusters, it is sometimes observed that rotation simply interrupts with no consequence on the cluster integrity. The cluster rotation lifetime srot does not trivially correlate with the cluster size Ncl. This can be easily pointed out by selecting rotating clusters of a given size Ncl and looking at the statistical distribution p(srot) of their srot values. The p(srot) curves for 1161, 162 and 38 clusters including 3, 10 and 20 atoms respectively are shown in Fig. 2b. The distributions p(srot) are quite similar to each other and extend over a relatively large time domain from 0.1 to 1.6 ns. However, about the 75% of clusters rotate for only 0.1 and 0.3 ns, irrespective of their size. Analogous results were obtained in all of the cases in which a rough statistical analysis was possible, i.e. when the number of rotating clusters with the same size was larger than 30. For about the 77% of clusters, the rotation axis is approximately parallel to the y Cartesian direction, the angular deviation being at most of 5°. In the other cases, the rotation axis is randomly oriented. Among the clusters with a rotation axis parallel to the y Cartesian direction, about the 60% exhibit a rotation axis external to the cluster, whereas in the remaining 40% the rotation axis is internal.
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When the rotation axis lies externally to the cluster, a complete revolution is never observed. In contrast, for clusters with internal rotation axis, even short lifetimes srot can correspond to one or more complete rotation cycles depending on the cluster size Ncl. The statistical distribution p(hrot) of rotation angles hrot, evaluated on the 829 clusters with internal rotation axis, is shown in Fig. 3a. The p(hrot) curve exhibits a relatively irregular shape, indicating that roughly the 96% of clusters rotate for angles hrot smaller than 360°. However, two clusters, including 8 and 37 atoms respectively, were able to rotate for more than two complete rotation cycles. Whereas small clusters generally undergo an irregular, but continuous, rotation, the largest clusters can also exhibit an intermittent rotation regime. For sake of illustration, the cases of two clusters including 6 and 27 atoms respectively are shown in Fig. 3b. Here, the rotation angle hrot is plotted as a function of the number m of strain increments de forming the strain interval in which the selected clusters rotate. The curves in Fig. 3b can be regarded as representative of the behavior of clusters with the same size. To allow a reliable comparison on a relative basis, the clusters were suitably chosen to have roughly the same lifetime srot of about 500 ps. It can be seen that for the cluster including 6 atoms the rotation angle hrot undergoes a steady, though irregular, increase. In contrast, the hrot curve regarding the 27-atom cluster exhibits two marked plateaus starting at m values respectively equal to 10 and 14. The rotation of relatively large clusters is characterized by a relatively complex dynamics. It is accompanied by a number of processes, including atomic displacements due to IRs and atomic coordination changes. An idea of what happens in the case of a cluster including 32 atoms is given in Fig. 4, where the cluster and the surrounding atoms are shown at different times. The cluster rotates of about 60° in roughly 360 ps, with relative atomic positions substantially conserved. This allows the cluster to rotate as a rigid body around a rotation axis that passes close to the cluster center of mass and is almost perfectly parallel to the y Cartesian direction. The atoms surrounding the rotating cluster are involved
(a)
(b)
Fig. 3. (a) The statistical distribution p(hrot) of cluster rotation angles hrot. Data refer to 829 clusters with internal rotation axis. (b) The rotation angle hrot is plotted as a function of the number m of strain increments de during which the selected clusters undergo rotation. Data refer to clusters including 6 and 27 atoms respectively.
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cases even relatively large clusters can undergo two or more rotation cycles. Acknowledgement Financial support has been given by the University of Cagliari. A. Ermini, ExtraInformatica s.r.l., is gratefully acknowledged for his kind assistance. References
Fig. 4. A cluster of 32 atoms exhibiting a rotation angle hrot of about 0°, 23°, 48° and 61° after the time intervals indicated. Light and dark gray have been used for the Cu and Ti cluster atoms respectively. White was used for the atoms surrounding the rotating cluster. Four of them have been marked with numbers to help visualizing the rotation process. A projection on the (y; z) Cartesian plane is shown.
in an IR, which is a necessary condition to have rotation. The atoms involved in the IR and not belonging to the rotating cluster modify their relative positions. Some of them accompany the cluster in its rotation, thus rotating in the same direction. However, their displacements do not satisfy the three previously mentioned criteria. As a consequence, their motion cannot be described as a rigid body rotation, but rather as the simple rearrangement of a local structure. Nevertheless, it is precisely their displacement to assist and permit the cluster rotation. Similar observations can be made for all of the observed rotation processes. The cluster is located in a region participating in an IR and its rotation is accompanied by a collective displacement of surrounding atoms in the same direction of rotation.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
4. Conclusions
[38] [39]
The numerical findings indicate that the deformation is accompanied by IRs. A detailed analysis of the atomic-scale dynamics underlying IRs pointed out the occurrence of ordered rotation processes. In particular, atomic clusters can undergo a rigid body rotation. Rotating clusters are small, including at most 45 atoms, and are always embedded into a region undergoing an IR. Thus, the cluster rotation is governed by the IR dynamics. Depending on the cluster size, rotation can be regular or intermittent. The lifetime of rotation processes is generally short, although in a few
[40] [41] [42] [43] [44] [45] [46] [47] [48]
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