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Thermally-activated atomic rearrangements in elastically deformed metallic glasses
Materials Chemistry and Physics 126 (2011) 152–155
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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Thermally-activated atomic rearrangements in elastically deformed 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 25 August 2010 Received in revised form 22 October 2010 Accepted 23 November 2010 Keywords: Glasses Metals Molecular dynamics Deformation Elastic properties
a b s t r a c t This study investigates by molecular dynamics the response of model metallic glasses to artificial atom displacements at increasing deformation degrees. Glasses are shown to contain unstable regions that spontaneously rearrange under deformation. The relative stability of such regions was measured by the length of artificial atomic displacements inducing a rearrangement. Such length scales with the amplitude of atomic thermal vibrations, which suggests for deformation-induced rearrangements a thermal activation mechanism. © 2010 Elsevier B.V. All rights reserved.
1. Introduction It is well known that the lack of translational symmetry prevents in metallic glasses (MGs) the occurrence of dislocation-mediated deformation mechanisms characteristic of crystalline phases [1–5]. In contrast, deformation takes place via relatively high-energy rearrangements localized in shear transformation zones (STZs) [3–22]. These consist of groups of atoms that undergo a transition between two relatively low-energy configurations separated by an activation barrier [3–22]. The cooperation of STZs in strained regions underlies the undesired tendency of MGs to localize shear into bands [3–5]. For this reason, numerous studies aimed at elaborating strategies for controlling strain localization have focused precisely on STZs and their properties [6–28]. In spite of this, the conceptual framework is still quite fragmentary on various basic questions, including the relationship between STZs and local structural arrangements [4,5]. Regarding this latter feature, it must be noted that local atomic arrangements exhibit significantly different degrees of mechanical stability, which allow them to evolve as a consequence of both atomic thermal motion and mechanical solicitations [23–28]. Molecular dynamics (MDs) simulations have clearly shown that shear deformation is able to activate the irreversible rearrangement (IR) of small groups of atoms, which can be tentatively identified with STZs [6–28]. However, IRs can be also triggered by the artifi-
∗ Tel.: +39 070 675 50 73; fax: +39 070 675 50 67. E-mail address: [email protected] 0254-0584/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2010.11.047
cial displacement of individual atoms, which provides a method to measure the relative stability of local atomic arrangements [28]. This work aims precisely at applying such method to show its potential in connecting the occurrence of shear-induced IRs with the relative stability of the atomic arrangements involved. 2. Computational outline MD computations were employed to study the response of model Ni50 Zr50 MGs to elastic shear deformation at different temperatures. A semi-empirical tight-binding force scheme based on the second-moment approximation to the electronic density of states was selected to reproduce the interatomic interactions [29]. Accordingly, the cohesive energy was expressed as
⎧ ⎪ ⎪ Nˇ ⎨
−p
E= ˛
i˛ =1
⎪ ⎪ ⎩
˛ˇ
A˛ˇ e ˇ
ij ˛ˇ −1 d˛ˇ r
N˛
jˇ =1
⎡ ij ⎤1/2 ⎫ r ⎪ ˛ˇ ⎪ Nˇ −2q˛ˇ −1 ⎬ d˛ˇ ⎥ ⎢ 2 −⎣ ˛ˇ e ⎦ ⎪ ⎪ ˇ jˇ =1 ⎭ (1)
ij
j
where r˛ˇ = r˛i − rˇ is the distance between two particles and the indexes j˛ and jˇ run over all the particles. The parameters A˛ˇ , ˛ˇ , p˛ˇ and q˛ˇ quantify the interatomic potential between ˛ and ˇ species. The term d˛ˇ represents the nearest-neighbors distance at 0 K. The first member on the right hand side of Eq. (1) expresses the repulsive part of the potential as a Born–Mayer pairwise interaction, while the second member expresses the attractive
F. Delogu / Materials Chemistry and Physics 126 (2011) 152–155 Table 1 The values of the TB potential parameters for the pure Ni–Ni and Zr–Zr interactions as well as for the cross Ni–Zr ones. Interaction
A˛ˇ , kJ mol−1
˛ˇ , kJ mol−1
p˛ˇ
q˛ˇ
d˛ˇ , Å
Ni–Ni Ni–Zr Zr–Zr
13.1474 20.8959 15.5803
169.4056 206.3545 225.7455
10 8.36 9.3
2.7 2.23 2.1
2.491 2.761 3.18
part in the framework of the second-moment approximation of the tight-binding band energy [29,30]. Interactions were computed for ij distances r˛ˇ within a spherical cutoff radius rc of about 0.7 nm, approximately corresponding to the seventh shell of neighbors. Potential parameters were taken from the literature [29,30] and their values are shown in Table 1. The selected parameter values, in connection with the chosen cutoff distance, permit the reproduction to a fairly good extent of the most important thermodynamic quantities as well as of the elastic ones [29,30]. In particular, the extension of the cutoff distance with respect to previous work allows improving the accuracy with which elastic constants of elements and their crystalline alloys are reproduced. In turn, this also assures dealing with improved computations of the mechanical properties of amorphous phases. Of course, despite the generally good performances of the TB potential, the results obtained in the present work should be regarded as good approximations at best. By no means, the numerical evidences can be interpreted as a quantitative estimation of mechanical properties of Ni50 Zr50 MGs. However, the work provides a wealth of information of qualitative value quite independent of the potential employed. All of this is perfectly in line with the specifically qualitative purpose of the research carried out. Calculations were performed with number of atoms, pressure and temperature constant [31–33]. Equations of motion were solved with a fifth-order predictor–corrector algorithm [33] and a time step of 2 fs. Seven Ni50 Zr50 MGs including 131,072 atoms in a volume of about 13.3 nm × 13.3 nm × 13.3 nm were created by quenching a melt from 2000 K to a temperature Tq between 200 and 800 K at a rate of 100 K ps−1 and zero pressure. Periodic boundary conditions (PBCs) were applied along the three Cartesian directions. According to the Wendt–Abraham method [34], the glass transition occurred at about 1010 K. After a relaxation of 1 ns, the Nosè thermostat was removed avoid perturbing the MG dynamics with its stochastic action. Indeed, it is worth remembering that the Nosè thermostat couples local temperature fluctuations with a fictitious mass that governs the thermal inertia of the simulated system [32]. Equations for the evolution of the fictitious mass are deterministic and timereversible [32]. Despite this, due to the description of the fictitious mass dynamics by a second-order differential equation, heat may flow in and out the system according to an oscillatory fashion [32]. Depending on the general computational conditions imposed, such fluctuations can affect the local atomic behavior with quasi-random effects [32], which could be deleterious for the identification of local rearrangements. After the thermostat removal, the initial unstrained configuration for successive computations was finally created by further relaxing the MGs for 2 ns. Following previous work [28], the relative stability of local structures was roughly measured by instantaneously displacing an atom from its position and studying the system response in the following 20 ps. A single atom was displaced time by time. Moves were performed along 30 random directions for a length lk = klm /60, where lm is the distance between the atom and the side of its Voronoi cell [35] and k is an integer between 1 and 60. Then, any given atom can undergo up to 1800 independent moves.
153
IRs were identified by comparing the initial and final 5-ps averaged positions of the atoms distant less than r3 from the displaced atom, being r3 the distance of the third minimum of the MG global pair correlation function [33]. An atom was considered involved in an IR when its initial and final positions differed at least twice the average amplitude ı of thermal vibrations. This method yields results in substantial agreement with the ones obtained by identifying IRs as localized non-affine deformations [11], with differences in the number of rearranging atoms of about ±2%. Local stability was analyzed in both unstrained and strained configurations. To generate the latter ones, two rigid reservoirs 1.5 nm thick were created at the top and bottom of the unstrained configuration along the z Cartesian direction [21,28], leaving PBCs along the others. Then, reservoirs were incrementally displaced of 0.02 nm every 50 ps while kept at constant distance. The average strain ε was increased in the elastic range up to 0.04. Before discussing the results, it is worth noting that relating local stability to IRs requires that the number of unstable regions is large enough to have a significant statistics and the strain at which IRs occur is accurately identified. To satisfy both these conditions, an unusually high quenching rate was selected to maximize the number of unstable regions, whereas the chosen deformation rate was the lowest possible one allowing calculation feasibility. Of course, under these circumstances only qualitative indications on the MG response to shear can be obtained, with scarce relevance for experimental data. Despite this, the numerical findings could as well provide interesting clues to address a more accurate future work.
3. Results and discussion The case of the MG quenched at 200 K exemplifies the general MG behavior. Individual atom displacements perturb the initial configuration with a strength scaling with k. Only about the 6% of moves induce IRs, which are seen to involve 8–45 atoms and to last 1–4 ps in agreement with recent work [36]. About the 65% of atoms participating in the IR of an unstable region are in turn able to induce an IR in the same region. In roughly the 85% of cases, the atomic positions after IRs induced by different atoms differ less than the average amplitude ı of atomic thermal vibrations. For most of atoms belonging to unstable regions, the minimum k values activating an IR along the different directions are quite close to each other, which can be taken as an indirect indication of the relative isotropy of individual Voronoi cells. Thus, representative average km values were evaluated. The km values of the atoms belonging to a few unstable regions in the unstrained configuration are shown in Fig. 1a. Each unstable region is characterized by a spectrum of km values, which clearly indicates that different unstable regions can exhibit very different stability. For atoms activating IRs, the km value correlates with their 5-ps averaged potential energy u, calculated before the artificial displacement by summing over the neighbors within the potential cut-off radius and normalizing to their number. In particular, the inset in Fig. 1a shows that the logarithm of km , lnkm , scales linearly with the ratio u/U, where U is the average potential energy of the MG. Thus, the higher the potential energy u of the displaced atom, the smaller the km value triggering the IR. For any given unstable region, the smallest km value, km,thr , represents the threshold below which no IR takes place. Then, km,thr is a measure of local stability and can be used to monitor the response of unstable regions to shear deformation. Data in Fig. 1b, which refer to three representative regions of significantly different stability, indicate that km,thr undergoes a linear decrease with the square of the average strain ε, ε2 . It follows that deformation facilitates IRs. A similar quadratic dependence on ε is shown by the average potential energy u¯ of the atoms located in unstable regions,
154
F. Delogu / Materials Chemistry and Physics 126 (2011) 152–155
Fig. 2. (a) The number nIR of atoms involved in spontaneous IRs and (b) the threshold km,thr values for unstable regions participating in spontaneous IRs as a function of the average strain ε. Data refer to MGs quenched at the temperatures Tq indicated. Horizontal lines in (b) are a guide to the eyes.
Fig. 1. (a) The km values for the atoms belonging to three unstable regions in the unstrained configuration. The threshold km,thr is indicated by the highest bar. The inset shows the logarithm of km , lnkm , as a function of the ratio between the potential energy u of individual atoms in unstable regions and the average potential energy U of the system. The best-fitted line is also shown. (b) The threshold km,thr for three regions of different stability as a function of the function of the square of average strain ε, ε2 . The horizontal dotted line is a guide to the eyes. (c) The ratio u¯ c /u¯ c,0 for the atoms with the smallest and the largest km values as a function of the square of ¯ u¯ 0 between potential energies averaged over all the average strain ε, ε2 . The ratio u/ atoms in unstable regions is also shown. Data refer to the MG quenched at 200 K.
as shown in Fig. 1c by the linear plot of the u¯ estimates normalized to the initial values u¯ 0 . However, the response to deformation of individual atoms in unstable regions is not homogeneous. To specifically study this feature, all of the atoms belonging to unstable regions in unstrained and strained configurations were divided into classes based on their km values. Then, the average potential energy u¯ c per class was evaluated and normalized to its initial value u¯ c,0 to facilitate the comparison. The data referring to the 10% of atoms with the smallest km and to the 10% of atoms with the largest km are also shown in Fig. 1c. All of the data sets arrange according to linear trends. The u¯ c /u¯ c,0 data sets corresponding to the atoms with the smallest and the largest km values lie respectively above and below ¯ u¯ 0 . Thus, the atoms with high potential energy u and small the u/ km values are the most affected by deformation. It follows that precisely such atoms must be expected to play a role in the activation of IRs during deformation. Regarding this point, another useful indication comes from the km,thr plots in Fig. 1b. Such plots terminate at different deformation degrees, namely when the considered unstable regions disappears as a consequence of a deformation-induced IR. Yet, they exhibit approximately the same final km,thr value. Then, it seems that disappearing unstable regions share similar km,thr values irrespective of the deformation degree. This possibility was investigated by focusing on IRs induced by deformation, hereafter referred to as spontaneous IRs. The number nIR of atoms involved in spontaneous IRs is shown in Fig. 2a as a function of the average strain ε for MGs quenched at different temperatures Tq . As obtained in previous work [37,38], nIR increases
linearly with ε. Instead, at any given strain nIR , nIR decreases with Tq . Spontaneous IRs can be ascribed to the increasing destabilization of unstable regions as deformation proceeds, accompanied by the decrease of km,thr with the average strain ε. The km,thr values, km,thr,IR , pertaining to unstable regions at the onset of their spontaneous IR are shown in Fig. 2b as a function of ε for two MGs at different temperatures Tq . It can be seen that, for a given MG, data are scattered around and above roughly horizontal lines, which define an average k¯ m,thr,IR value common to all of the spontaneous IRs activated at different deformation degrees. Both k¯ m,thr,IR and the amplitude of data scattering increase with the MG quenching temperature Tq . All of the above mentioned findings suggest that spontaneous IRs (i) onset at a local instability level that is roughly the same for different unstable regions at different strain, and (ii) exhibit an activation mechanism somewhat dependent on thermal conditions. In addition, calculations indicate that most unstable regions rearrange first, and that the sequence of spontaneous IRs is determined by the rate at which the km,thr value of an unstable region decreases with ε down to about k¯ m,thr,IR . In addition, computations indicate that spontaneous IRs must be regarded as cooperative rearrangements in which the whole set of atoms involved in any given unstable region rearrange on a very short time scale, on the order of 1–4 ps. A few successive atomic configurations of an unstable region are given in Fig. 3 to illustrate the evolution of an incoming IR. It can be seen that atomic displacements take place collectively, with very small distances being covered. Also, it is worth noting that the final positions do not make the stable region formed suitably distinguishable from the initial unstable one. Therefore, once again simulations point out that simple structural analyses are not useful to successfully discriminate between atomic arrangements of different relative stability. To throw some light on this latter point, the k¯ m,thr,IR values were compared with the average amplitude ı of atomic thermal vibrations. This quantity was evaluated by averaging over the all of the atoms in the unstrained configurations. The obtained ı values exhibit the expected square-root dependence on temperature [39], changing roughly from 0.011 to 0.041 nm in the temperature range between 200 and 800 K. The k¯ m,thr,IR data are shown in Fig. 4 as a function of ı. A roughly linear increase of k¯ m,thr,IR with ı is observed. It is now worth noting that the k¯ m,thr,IR values correspond to lengths lk varying approximately from 0.026 to 0.059 nm.
F. Delogu / Materials Chemistry and Physics 126 (2011) 152–155
155
identified. k¯ m,thr,IR increases with the temperature, scaling with the average amplitude of atomic thermal vibrations. Furthermore, the displacement length lk corresponding to k¯ m,thr,IR is comparable with the atomic vibration amplitude ı. This suggests that an unstable region undergoes a spontaneous IR when ı of is on the same order than the artificial displacement length lk needed for activating the IR. Finally, it must be noted that the regions with a degree of stability incompatible with the quenching temperature, i.e. with km,thr smaller than k¯ m,thr,IR at such temperature, are destined to rapidly decade. This is exactly what happens during the 2-ns relaxation following quenching, with various unstable regions rearranging and disappearing due to IRs triggered by simple thermal motion. At any given quenching temperature, only the unstable regions with km,thr larger than k¯ m,thr,IR survive. This also explains the decrease of the number of IRs as the final quenching temperature increases. Fig. 3. Atomic configurations describing the spontaneous IR of an unstable region. Data refer to the case of a MG quenched to 300 K at an average strain ε equal to 0.03. Ni and Zr atoms are shown in light and dark gray, respectively.
Acknowledgements 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. The average threshold k¯ m,thr,IR exhibited by unstable regions involved in spontaneous IRs as a function of the average amplitude ı of atomic thermal vibrations. The best-fitted line is also shown.
In other words, they are on the same order of the amplitudes of atomic thermal vibrations. Therefore, being the average threshold k¯ m,thr,IR a measure of ultimate local stability, it can be inferred that unstable regions participate in spontaneous IRs when the activation of such IRs is compatible with thermal motion. 4. Conclusions In summary, the method of artificial atom displacements permits to identify unstable regions prone to IRs. The minimum displacement length lk allowing the IRs to occur, corresponding to a km value, varies from atom to atom, even for the same unstable region. For any given unstable region, the smallest km value defines the threshold km,thr below which no IR can take place. The higher the km,thr value, the more stable the region. Unstable regions exhibit different stability degrees, i.e. different km,thr values. Their stability diminishes with deformation, as pointed out by the decrease of km,thr with strain ε. In general, the smaller the initial km,thr value, the smaller the strain at which its IR take place. Therefore, most unstable regions rearrange first. The study of spontaneous IRs indicates that the threshold km,thr,IR value at the onset of IRs is roughly the same for all of the unstable regions irrespective of the strain level at which the spontaneous IR occurs. Thus, a representative average value k¯ m,thr,IR can be
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Contents lists available at ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Thermally-activated atomic rearrangements in elastically deformed 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 25 August 2010 Received in revised form 22 October 2010 Accepted 23 November 2010 Keywords: Glasses Metals Molecular dynamics Deformation Elastic properties
a b s t r a c t This study investigates by molecular dynamics the response of model metallic glasses to artificial atom displacements at increasing deformation degrees. Glasses are shown to contain unstable regions that spontaneously rearrange under deformation. The relative stability of such regions was measured by the length of artificial atomic displacements inducing a rearrangement. Such length scales with the amplitude of atomic thermal vibrations, which suggests for deformation-induced rearrangements a thermal activation mechanism. © 2010 Elsevier B.V. All rights reserved.
1. Introduction It is well known that the lack of translational symmetry prevents in metallic glasses (MGs) the occurrence of dislocation-mediated deformation mechanisms characteristic of crystalline phases [1–5]. In contrast, deformation takes place via relatively high-energy rearrangements localized in shear transformation zones (STZs) [3–22]. These consist of groups of atoms that undergo a transition between two relatively low-energy configurations separated by an activation barrier [3–22]. The cooperation of STZs in strained regions underlies the undesired tendency of MGs to localize shear into bands [3–5]. For this reason, numerous studies aimed at elaborating strategies for controlling strain localization have focused precisely on STZs and their properties [6–28]. In spite of this, the conceptual framework is still quite fragmentary on various basic questions, including the relationship between STZs and local structural arrangements [4,5]. Regarding this latter feature, it must be noted that local atomic arrangements exhibit significantly different degrees of mechanical stability, which allow them to evolve as a consequence of both atomic thermal motion and mechanical solicitations [23–28]. Molecular dynamics (MDs) simulations have clearly shown that shear deformation is able to activate the irreversible rearrangement (IR) of small groups of atoms, which can be tentatively identified with STZs [6–28]. However, IRs can be also triggered by the artifi-
∗ Tel.: +39 070 675 50 73; fax: +39 070 675 50 67. E-mail address: [email protected] 0254-0584/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2010.11.047
cial displacement of individual atoms, which provides a method to measure the relative stability of local atomic arrangements [28]. This work aims precisely at applying such method to show its potential in connecting the occurrence of shear-induced IRs with the relative stability of the atomic arrangements involved. 2. Computational outline MD computations were employed to study the response of model Ni50 Zr50 MGs to elastic shear deformation at different temperatures. A semi-empirical tight-binding force scheme based on the second-moment approximation to the electronic density of states was selected to reproduce the interatomic interactions [29]. Accordingly, the cohesive energy was expressed as
⎧ ⎪ ⎪ Nˇ ⎨
−p
E= ˛
i˛ =1
⎪ ⎪ ⎩
˛ˇ
A˛ˇ e ˇ
ij ˛ˇ −1 d˛ˇ r
N˛
jˇ =1
⎡ ij ⎤1/2 ⎫ r ⎪ ˛ˇ ⎪ Nˇ −2q˛ˇ −1 ⎬ d˛ˇ ⎥ ⎢ 2 −⎣ ˛ˇ e ⎦ ⎪ ⎪ ˇ jˇ =1 ⎭ (1)
ij
j
where r˛ˇ = r˛i − rˇ is the distance between two particles and the indexes j˛ and jˇ run over all the particles. The parameters A˛ˇ , ˛ˇ , p˛ˇ and q˛ˇ quantify the interatomic potential between ˛ and ˇ species. The term d˛ˇ represents the nearest-neighbors distance at 0 K. The first member on the right hand side of Eq. (1) expresses the repulsive part of the potential as a Born–Mayer pairwise interaction, while the second member expresses the attractive
F. Delogu / Materials Chemistry and Physics 126 (2011) 152–155 Table 1 The values of the TB potential parameters for the pure Ni–Ni and Zr–Zr interactions as well as for the cross Ni–Zr ones. Interaction
A˛ˇ , kJ mol−1
˛ˇ , kJ mol−1
p˛ˇ
q˛ˇ
d˛ˇ , Å
Ni–Ni Ni–Zr Zr–Zr
13.1474 20.8959 15.5803
169.4056 206.3545 225.7455
10 8.36 9.3
2.7 2.23 2.1
2.491 2.761 3.18
part in the framework of the second-moment approximation of the tight-binding band energy [29,30]. Interactions were computed for ij distances r˛ˇ within a spherical cutoff radius rc of about 0.7 nm, approximately corresponding to the seventh shell of neighbors. Potential parameters were taken from the literature [29,30] and their values are shown in Table 1. The selected parameter values, in connection with the chosen cutoff distance, permit the reproduction to a fairly good extent of the most important thermodynamic quantities as well as of the elastic ones [29,30]. In particular, the extension of the cutoff distance with respect to previous work allows improving the accuracy with which elastic constants of elements and their crystalline alloys are reproduced. In turn, this also assures dealing with improved computations of the mechanical properties of amorphous phases. Of course, despite the generally good performances of the TB potential, the results obtained in the present work should be regarded as good approximations at best. By no means, the numerical evidences can be interpreted as a quantitative estimation of mechanical properties of Ni50 Zr50 MGs. However, the work provides a wealth of information of qualitative value quite independent of the potential employed. All of this is perfectly in line with the specifically qualitative purpose of the research carried out. Calculations were performed with number of atoms, pressure and temperature constant [31–33]. Equations of motion were solved with a fifth-order predictor–corrector algorithm [33] and a time step of 2 fs. Seven Ni50 Zr50 MGs including 131,072 atoms in a volume of about 13.3 nm × 13.3 nm × 13.3 nm were created by quenching a melt from 2000 K to a temperature Tq between 200 and 800 K at a rate of 100 K ps−1 and zero pressure. Periodic boundary conditions (PBCs) were applied along the three Cartesian directions. According to the Wendt–Abraham method [34], the glass transition occurred at about 1010 K. After a relaxation of 1 ns, the Nosè thermostat was removed avoid perturbing the MG dynamics with its stochastic action. Indeed, it is worth remembering that the Nosè thermostat couples local temperature fluctuations with a fictitious mass that governs the thermal inertia of the simulated system [32]. Equations for the evolution of the fictitious mass are deterministic and timereversible [32]. Despite this, due to the description of the fictitious mass dynamics by a second-order differential equation, heat may flow in and out the system according to an oscillatory fashion [32]. Depending on the general computational conditions imposed, such fluctuations can affect the local atomic behavior with quasi-random effects [32], which could be deleterious for the identification of local rearrangements. After the thermostat removal, the initial unstrained configuration for successive computations was finally created by further relaxing the MGs for 2 ns. Following previous work [28], the relative stability of local structures was roughly measured by instantaneously displacing an atom from its position and studying the system response in the following 20 ps. A single atom was displaced time by time. Moves were performed along 30 random directions for a length lk = klm /60, where lm is the distance between the atom and the side of its Voronoi cell [35] and k is an integer between 1 and 60. Then, any given atom can undergo up to 1800 independent moves.
153
IRs were identified by comparing the initial and final 5-ps averaged positions of the atoms distant less than r3 from the displaced atom, being r3 the distance of the third minimum of the MG global pair correlation function [33]. An atom was considered involved in an IR when its initial and final positions differed at least twice the average amplitude ı of thermal vibrations. This method yields results in substantial agreement with the ones obtained by identifying IRs as localized non-affine deformations [11], with differences in the number of rearranging atoms of about ±2%. Local stability was analyzed in both unstrained and strained configurations. To generate the latter ones, two rigid reservoirs 1.5 nm thick were created at the top and bottom of the unstrained configuration along the z Cartesian direction [21,28], leaving PBCs along the others. Then, reservoirs were incrementally displaced of 0.02 nm every 50 ps while kept at constant distance. The average strain ε was increased in the elastic range up to 0.04. Before discussing the results, it is worth noting that relating local stability to IRs requires that the number of unstable regions is large enough to have a significant statistics and the strain at which IRs occur is accurately identified. To satisfy both these conditions, an unusually high quenching rate was selected to maximize the number of unstable regions, whereas the chosen deformation rate was the lowest possible one allowing calculation feasibility. Of course, under these circumstances only qualitative indications on the MG response to shear can be obtained, with scarce relevance for experimental data. Despite this, the numerical findings could as well provide interesting clues to address a more accurate future work.
3. Results and discussion The case of the MG quenched at 200 K exemplifies the general MG behavior. Individual atom displacements perturb the initial configuration with a strength scaling with k. Only about the 6% of moves induce IRs, which are seen to involve 8–45 atoms and to last 1–4 ps in agreement with recent work [36]. About the 65% of atoms participating in the IR of an unstable region are in turn able to induce an IR in the same region. In roughly the 85% of cases, the atomic positions after IRs induced by different atoms differ less than the average amplitude ı of atomic thermal vibrations. For most of atoms belonging to unstable regions, the minimum k values activating an IR along the different directions are quite close to each other, which can be taken as an indirect indication of the relative isotropy of individual Voronoi cells. Thus, representative average km values were evaluated. The km values of the atoms belonging to a few unstable regions in the unstrained configuration are shown in Fig. 1a. Each unstable region is characterized by a spectrum of km values, which clearly indicates that different unstable regions can exhibit very different stability. For atoms activating IRs, the km value correlates with their 5-ps averaged potential energy u, calculated before the artificial displacement by summing over the neighbors within the potential cut-off radius and normalizing to their number. In particular, the inset in Fig. 1a shows that the logarithm of km , lnkm , scales linearly with the ratio u/U, where U is the average potential energy of the MG. Thus, the higher the potential energy u of the displaced atom, the smaller the km value triggering the IR. For any given unstable region, the smallest km value, km,thr , represents the threshold below which no IR takes place. Then, km,thr is a measure of local stability and can be used to monitor the response of unstable regions to shear deformation. Data in Fig. 1b, which refer to three representative regions of significantly different stability, indicate that km,thr undergoes a linear decrease with the square of the average strain ε, ε2 . It follows that deformation facilitates IRs. A similar quadratic dependence on ε is shown by the average potential energy u¯ of the atoms located in unstable regions,
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Fig. 2. (a) The number nIR of atoms involved in spontaneous IRs and (b) the threshold km,thr values for unstable regions participating in spontaneous IRs as a function of the average strain ε. Data refer to MGs quenched at the temperatures Tq indicated. Horizontal lines in (b) are a guide to the eyes.
Fig. 1. (a) The km values for the atoms belonging to three unstable regions in the unstrained configuration. The threshold km,thr is indicated by the highest bar. The inset shows the logarithm of km , lnkm , as a function of the ratio between the potential energy u of individual atoms in unstable regions and the average potential energy U of the system. The best-fitted line is also shown. (b) The threshold km,thr for three regions of different stability as a function of the function of the square of average strain ε, ε2 . The horizontal dotted line is a guide to the eyes. (c) The ratio u¯ c /u¯ c,0 for the atoms with the smallest and the largest km values as a function of the square of ¯ u¯ 0 between potential energies averaged over all the average strain ε, ε2 . The ratio u/ atoms in unstable regions is also shown. Data refer to the MG quenched at 200 K.
as shown in Fig. 1c by the linear plot of the u¯ estimates normalized to the initial values u¯ 0 . However, the response to deformation of individual atoms in unstable regions is not homogeneous. To specifically study this feature, all of the atoms belonging to unstable regions in unstrained and strained configurations were divided into classes based on their km values. Then, the average potential energy u¯ c per class was evaluated and normalized to its initial value u¯ c,0 to facilitate the comparison. The data referring to the 10% of atoms with the smallest km and to the 10% of atoms with the largest km are also shown in Fig. 1c. All of the data sets arrange according to linear trends. The u¯ c /u¯ c,0 data sets corresponding to the atoms with the smallest and the largest km values lie respectively above and below ¯ u¯ 0 . Thus, the atoms with high potential energy u and small the u/ km values are the most affected by deformation. It follows that precisely such atoms must be expected to play a role in the activation of IRs during deformation. Regarding this point, another useful indication comes from the km,thr plots in Fig. 1b. Such plots terminate at different deformation degrees, namely when the considered unstable regions disappears as a consequence of a deformation-induced IR. Yet, they exhibit approximately the same final km,thr value. Then, it seems that disappearing unstable regions share similar km,thr values irrespective of the deformation degree. This possibility was investigated by focusing on IRs induced by deformation, hereafter referred to as spontaneous IRs. The number nIR of atoms involved in spontaneous IRs is shown in Fig. 2a as a function of the average strain ε for MGs quenched at different temperatures Tq . As obtained in previous work [37,38], nIR increases
linearly with ε. Instead, at any given strain nIR , nIR decreases with Tq . Spontaneous IRs can be ascribed to the increasing destabilization of unstable regions as deformation proceeds, accompanied by the decrease of km,thr with the average strain ε. The km,thr values, km,thr,IR , pertaining to unstable regions at the onset of their spontaneous IR are shown in Fig. 2b as a function of ε for two MGs at different temperatures Tq . It can be seen that, for a given MG, data are scattered around and above roughly horizontal lines, which define an average k¯ m,thr,IR value common to all of the spontaneous IRs activated at different deformation degrees. Both k¯ m,thr,IR and the amplitude of data scattering increase with the MG quenching temperature Tq . All of the above mentioned findings suggest that spontaneous IRs (i) onset at a local instability level that is roughly the same for different unstable regions at different strain, and (ii) exhibit an activation mechanism somewhat dependent on thermal conditions. In addition, calculations indicate that most unstable regions rearrange first, and that the sequence of spontaneous IRs is determined by the rate at which the km,thr value of an unstable region decreases with ε down to about k¯ m,thr,IR . In addition, computations indicate that spontaneous IRs must be regarded as cooperative rearrangements in which the whole set of atoms involved in any given unstable region rearrange on a very short time scale, on the order of 1–4 ps. A few successive atomic configurations of an unstable region are given in Fig. 3 to illustrate the evolution of an incoming IR. It can be seen that atomic displacements take place collectively, with very small distances being covered. Also, it is worth noting that the final positions do not make the stable region formed suitably distinguishable from the initial unstable one. Therefore, once again simulations point out that simple structural analyses are not useful to successfully discriminate between atomic arrangements of different relative stability. To throw some light on this latter point, the k¯ m,thr,IR values were compared with the average amplitude ı of atomic thermal vibrations. This quantity was evaluated by averaging over the all of the atoms in the unstrained configurations. The obtained ı values exhibit the expected square-root dependence on temperature [39], changing roughly from 0.011 to 0.041 nm in the temperature range between 200 and 800 K. The k¯ m,thr,IR data are shown in Fig. 4 as a function of ı. A roughly linear increase of k¯ m,thr,IR with ı is observed. It is now worth noting that the k¯ m,thr,IR values correspond to lengths lk varying approximately from 0.026 to 0.059 nm.
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identified. k¯ m,thr,IR increases with the temperature, scaling with the average amplitude of atomic thermal vibrations. Furthermore, the displacement length lk corresponding to k¯ m,thr,IR is comparable with the atomic vibration amplitude ı. This suggests that an unstable region undergoes a spontaneous IR when ı of is on the same order than the artificial displacement length lk needed for activating the IR. Finally, it must be noted that the regions with a degree of stability incompatible with the quenching temperature, i.e. with km,thr smaller than k¯ m,thr,IR at such temperature, are destined to rapidly decade. This is exactly what happens during the 2-ns relaxation following quenching, with various unstable regions rearranging and disappearing due to IRs triggered by simple thermal motion. At any given quenching temperature, only the unstable regions with km,thr larger than k¯ m,thr,IR survive. This also explains the decrease of the number of IRs as the final quenching temperature increases. Fig. 3. Atomic configurations describing the spontaneous IR of an unstable region. Data refer to the case of a MG quenched to 300 K at an average strain ε equal to 0.03. Ni and Zr atoms are shown in light and dark gray, respectively.
Acknowledgements 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. The average threshold k¯ m,thr,IR exhibited by unstable regions involved in spontaneous IRs as a function of the average amplitude ı of atomic thermal vibrations. The best-fitted line is also shown.
In other words, they are on the same order of the amplitudes of atomic thermal vibrations. Therefore, being the average threshold k¯ m,thr,IR a measure of ultimate local stability, it can be inferred that unstable regions participate in spontaneous IRs when the activation of such IRs is compatible with thermal motion. 4. Conclusions In summary, the method of artificial atom displacements permits to identify unstable regions prone to IRs. The minimum displacement length lk allowing the IRs to occur, corresponding to a km value, varies from atom to atom, even for the same unstable region. For any given unstable region, the smallest km value defines the threshold km,thr below which no IR can take place. The higher the km,thr value, the more stable the region. Unstable regions exhibit different stability degrees, i.e. different km,thr values. Their stability diminishes with deformation, as pointed out by the decrease of km,thr with strain ε. In general, the smaller the initial km,thr value, the smaller the strain at which its IR take place. Therefore, most unstable regions rearrange first. The study of spontaneous IRs indicates that the threshold km,thr,IR value at the onset of IRs is roughly the same for all of the unstable regions irrespective of the strain level at which the spontaneous IR occurs. Thus, a representative average value k¯ m,thr,IR can be
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