The influence of complex thermal treatment on mechanical properties of amorphous materials
TThe influence of complex thermal treatment on mechanicalproperties of amorphous materials
Qing-Long Liu and Nikolai V. Priezjev , National Research University Higher School of Economics, Moscow 101000, Russia and Department of Mechanical and Materials Engineering,Wright State University, Dayton, OH 45435 (Dated: November 7, 2018)
Abstract
We study the effect of periodic, spatially uniform temperature variation on mechanical propertiesand structural relaxation of amorphous alloys using molecular dynamics simulations. The disor-dered material is modeled via a non-additive binary mixture, which is annealed from the liquid tothe glassy state with various cooling rates and then either aged at constant temperature or sub-jected to thermal treatment. We found that in comparison to aged samples, thermal cycling withrespect to a reference temperature of approximately half the glass transition temperature leads tomore relaxed states with lower levels of potential energy. The largest energy decrease was observedfor rapidly quenched glasses cycled with the thermal amplitude slightly smaller than the referencetemperature. Following the thermal treatment, the mechanical properties were probed via uniaxialtensile strain at the reference temperature and constant pressure. The numerical results indicatean inverse correlation between the levels of potential energy and values of the elastic modulus andyield stress as a function of the thermal amplitude.Keywords: glasses, deformation, temperature, yield stress, molecular dynamics simulations a r X i v : . [ c ond - m a t . s o f t ] N ov . INTRODUCTION Due to their disordered structure, metallic glasses are known to possess high strength andelastic limit as well as high resistance to corrosion, which makes them potentially suitablefor various structural and biomedical applications [1, 2]. These amorphous alloys, how-ever, suffer from lack of ductility and typically exhibit brittle fracture due to shear bandformation, especially in aged samples [3]. Commonly used methods to rejuvenate glassesand to improve plasticity, such as shot-peening [4], cold rolling and drawing [5], as well ashigh-pressure torsion [6], typically cause severe plastic deformation. By contrast, it wasrecently shown that structural rejuvenation in metallic glasses can be induced by temporar-ily heating samples above T g and subsequently quenching with a suitably fast cooling ratefor recovery annealing [7, 8]. It was later found that for some alloys, thermal rejuvenationcan be enhanced by applying external pressure [9, 10]. Alternatively, rejuvenated states inmetallic glasses can be accessed via cryogenic thermal cycling below T g , which promoteslocal structural transformations due to spatially non-uniform thermal expansion [11, 12].The influence of thermal expansion heterogeneity on rejuvenation of metallic glasses waspredicted to be important at sufficiently large length scales as compared to scales accessibleto atomistic simulations [13]. However, the microscopic details of the thermal processing aswell as the degree of rejuvenation or relaxation that can be achieved by applying multiplecycles remain to be clarified.During the last decade, the dynamic response of amorphous materials to periodic sheardeformation was extensively studied using atomistic simulations and experimental measure-ments [14–28]. Notably, it was shown that thermal aging process in rapidly quenched glassesis facilitated by repetitive subyield cycling that leads to progressively lower levels of poten-tial energy, and the effect is more pronounced at larger strain amplitudes [25, 26, 28]. Morerecently, it was found that relaxed states can be attained by repeatedly heating and coolingbinary glasses at constant pressure with various thermal amplitudes below the glass transi-tion [29, 30]. In particular, the results of numerical simulations have shown that the largestdecrease of the potential energy and the increase in the yield stress occur for rapidly cooledglasses with the thermal amplitude not far below the glass transition temperature [29]. Itwas later found that after hundreds of thermal cycles with respect to a very low referencetemperature, the glasses evolve into steady states, where particle dynamics becomes nearly2eversible after each cycle, similar to the so-called limit cycles observed during athermal pe-riodic shear of amorphous materials [17, 19, 30]. However, the dependence of the potentialenergy and mechanical properties on the preparation history, period and number of thermaloscillations, thermal amplitude, and reference temperature remains unexplored.In this paper, molecular dynamics simulations are performed to examine the effect ofthermal cycling on potential energy states and mechanical properties of binary glasses. Thethermal oscillations with relatively small period are imposed with respect to a referencetemperature of about half the glass transition temperature. It will be shown that regardlessof preparation history, the thermal treatment of one hundred cycles always leads to relaxedstates in a wide range of thermal amplitudes. Subsequent tensile loading of aged and ther-mally cycled glasses reveals that the yield stress and the elastic modulus acquire maxima atthe thermal amplitude that corresponds to the most relaxed states.The paper is structured as follows. The description of molecular dynamics simulationsand thermomechanical processing protocols are provided in the next section. The analysis ofthe potential energy series, particle displacements, and mechanical properties are presentedin section III. Brief conclusions are given in the last section. II. MOLECULAR DYNAMICS SIMULATIONS
The metallic glass is represented by the Kob and Andersen (KA) binary mixture model,which was originally developed to study the amorphous metal alloy Ni P [31, 32]. In thismodel, there are two types of atoms, A and B , with strongly non-additive cross interactionsthat prevent crystallization upon cooling below the glass transition temperature [31]. Morespecifically, the interaction between two atoms α, β = A, B is described by the truncatedLennard-Jones (LJ) potential: V αβ ( r ) = 4 ε αβ (cid:104)(cid:16) σ αβ r (cid:17) − (cid:16) σ αβ r (cid:17) (cid:105) , (1)with the following parametrization ε AA = 1 . ε AB = 1 . ε BB = 0 . σ AB = 0 . σ BB = 0 . m A = m B [31]. The LJ potential is truncated at the cutoff radius r c, αβ = 2 . σ αβ toalleviate the computational burden. The system is composed of 48000 A atoms and 12000 B atoms, with the total number of atoms of 60000. In what follows, all physical quantitiesare reported in terms of the reduced LJ units of length, mass, energy, and time: σ = σ AA ,3 = m A , ε = ε AA , and τ = σ (cid:112) m/ε , respectively. The MD simulations were carried outusing the LAMMPS parallel code with the time step (cid:52) t MD = 0 . τ [34].The simulations were performed in several stages. The system was first thoroughly equi-librated at the temperature of 0 . ε/k B , which is above the glass transition of the KA model, T g ≈ . ε/k B [31]. Here, k B denotes the Boltzmann constant. The temperature is con-trolled via the Nos´e-Hoover thermostat, and the periodic boundary conditions are imposedalong all three dimensions [33]. After the equilibration procedure, the samples were annealedat constant pressure to the temperature T LJ = 0 . ε/k B with the cooling rates 10 − ε/k B τ ,10 − ε/k B τ , 10 − ε/k B τ , and 10 − ε/k B τ . The snapshot of the binary glass annealed with thecooling rate of 10 − ε/k B τ to the temperature of T LJ = 0 . ε/k B is illustrated in Fig. 1. Next,the system was repeatedly heated and cooled with the thermal amplitude ∆ T LJ during 100cycles with the period T = 2000 τ . The thermal oscillations were imposed at constant pres-sure, P = 0, with respect to the reference temperature T LJ = 0 . ε/k B . After the thermaltreatment, the samples were strained along the ˆ x direction at constant pressure with thestrain rate ˙ ε xx = 10 − τ − . At each stage, the potential energy, pressure components, sys-tem dimensions, and atomic configurations were periodically saved for the post-processinganalysis. III. RESULTS
As discussed in Sec. II, the binary mixture was initially equilibrated at the temperatureof 0 . ε/k B and zero pressure, and then annealed below the glass transition point to thetemperature of 0 . ε/k B with different cooling rates. The starting temperature of 0 . ε/k B was chosen to be not far above the glass transition temperature T g ≈ . ε/k B in orderto reduce the annealing time during slow cooling and to avoid significant deformation ofthe simulation domain from a cubic box, since all system dimensions were allowed to varyindependently at constant pressure. After the glasses were annealed with the cooling rates10 − ε/k B τ , 10 − ε/k B τ , 10 − ε/k B τ , and 10 − ε/k B τ , the simulations proceeded at T LJ =0 . ε/k B and P = 0 during the time interval 2 × τ . The results for the potential energyper atom and the average glass density during the aging process are reported in Fig. 2 forthe indicated values of the cooling rate.It can be observed from Fig. 2 that upon cooling with slower rates, the potential energy4evels become deeper, as the system visits a larger number of minima in the potential energylandscape in the vicinity of the glass transition temperature. During the aging process at T LJ = 0 . ε/k B and P = 0, shown in Fig. 2, the potential energy of more rapidly quenchedglasses decays significantly during the time interval of 2 × τ . A similar decrease of thepotential energy during aging below T g at constant volume during 10 τ was reported inthe previous MD study, where it was also shown that the system dynamics, as measuredby the the decay of the two-time intermediate scattering function, becomes progressivelyslower [35]. We also comment that the potential energy levels reported in the recent studywith a similar setup, except that the reference temperature is T LJ = 0 . ε/k B , remainednearly constant during the time interval 10 τ and relatively low, i.e. , U (cid:46) − . ε [29]. Asshown in the inset to Fig. 2, the rate of density increase is more pronounced for rapidlycooled glasses, and the difference in the average glass densities after 2 × τ is less than0 . − ε/k B τ , 10 − ε/k B τ ,10 − ε/k B τ , and 10 − ε/k B τ , the glasses were subjected to repeated cycles of heating andcooling with respect to the reference temperature T LJ = 0 . ε/k B . The examples of theselected temperature profiles measured during the first five cycles are presented in Fig. 3 forthe glass that was initially annealed with the cooling rate of 10 − ε/k B τ . In what follows,we denote the maximum deviation from the reference temperature T LJ = 0 . ε/k B , or thethermal amplitude, by ∆ T LJ . In the present study, we considered a wide range of thermalamplitudes but ensured that temperature remained above zero and below the glass transi-tion temperature. Thus, the maximum value of the thermal amplitude is ∆ T LJ = 0 . ε/k B .We also remind that the simulations were carried out at constant pressure ( P = 0), thusallowing significant variation in volume during each cycle. Here, we emphasize the key dif-ferences in the choice of parameters from the previous MD study on thermally cycled binaryglasses [29]. Specifically, the thermal treatment was performed with a smaller oscillationperiod, T = 2000 τ , higher reference temperature, T LJ = 0 . ε/k B , and larger number ofthermal amplitudes to resolve more accurately the neighborhood around the minimum ofthe potential energy after 100 cycles (discussed below).We next present the variation of the potential energy at the beginning and the end ofthermal treatment in Figs. 4, 5, 6, and 7 for glasses initially annealed with the cooling5ates 10 − ε/k B τ , 10 − ε/k B τ , 10 − ε/k B τ , and 10 − ε/k B τ , respectively. For reference, theblack curves in each figure denote the data at the constant temperature T LJ = 0 . ε/k B (the same data as in Fig. 2). It can be seen that the amplitude of the potential energyoscillations increases at larger thermal amplitudes. From Figs. 4-6, it is apparent that forrelatively quickly annealed glasses, the minima of the potential energy become progressivelydeeper over consecutive cycles for all thermal amplitudes. Moreover, a small difference inthe potential energy between aged and thermally cycled glasses is developed after each cycle.This discrepancy becomes especially evident in the enlarged view of the data during the lastcycle shown in the insets to Figs. 4-6. By contrast, in the case of slowly annealed glass,shown in Fig. 7, it is difficult to visually detect any changes in the minima of the potentialenergy from cycle to cycle from the main panels. However, a more detailed view of the datain the inset to Fig. 7 reveals that there is a noticeable deviation in the potential energy afterthe last cycle for the thermal amplitudes ∆ T LJ = 0 . ε/k B and 0 . ε/k B . This implies anonmonotonic dependence of U (100 T ) as a function of ∆ T LJ .We next summarize the data for the potential energy after 100 thermal cycles in Fig. 8 asa function of the thermal amplitude for the cooling rates 10 − ε/k B τ , 10 − ε/k B τ , 10 − ε/k B τ ,and 10 − ε/k B τ . The data at ∆ T LJ = 0 correspond to the potential energy of glasses agedduring the time interval 2 × τ = 100 T at T LJ = 0 . ε/k B . The data points in Fig. 8 wereobtained by linearly extrapolating the potential energy values in the vicinity of 100 T . Itis evident that with increasing thermal amplitude, the potential energy first decreases andthen acquires a local minimum at about ∆ T LJ = 0 . ε/k B . From the data presented inFig. 8, it is difficult to conclude with certainty whether the minimum in U (100 T ) versus∆ T LJ depends on the cooling rate. Notice also that the dependence of U (100 T ) is nearlythe same for glasses initially annealed with cooling rates 10 − ε/k B τ and 10 − ε/k B τ .The increase of the potential energy U (100 T ) at ∆ T LJ = 0 . ε/k B and 0 . ε/k B in Fig. 8can be ascribed to relative proximity of the system temperature after a quarter of a cycle,0 . ε/k B + ∆ T LJ , to the glass transition temperature. In the latter case, the temperatureapproaches T g from below, the role of thermal fluctuations temporarily increases, and thesystems is then annealed with the effective cooling rate 4∆ T LJ /T = 0 . ε/k B τ duringeach cycle. Note that the potential energy of the glass annealed with the cooling rate10 − ε/k B τ and aged at T LJ = 0 . ε/k B is nearly the same as U (100 T ) at ∆ T LJ = 0 . ε/k B .6 qualitatively similar trend in the dependence of U (100 T ) on ∆ T LJ was reported in therecent study, although the decrease in potential energy due to aging was not so pronounced,since the reference temperature was much lower, i.e. , T LJ = 0 . ε/k B [29]. Overall, itcan be concluded from Fig. 8 that the sequence of 100 thermal cycles with the referencetemperature T LJ = 0 . ε/k B did not grant access to rejuvenated states and resulted only inrelaxed states for all cooling rates and thermal amplitudes considered in the present study.The analysis of atomic displacements during consecutive cycles revealed that the relax-ation process proceeds via irreversible rearrangements of groups of atoms. We first examinethe distribution of displacements during selected cycles for thermal cycling with the am-plitude ∆ T LJ = 0 . ε/k B . The results are presented in Fig. 9 for binary glasses annealedwith cooling rates 10 − ε/k B τ and 10 − ε/k B τ . It can be observed from Fig. 9 (a) that in therapidly cooled glass, the distribution of displacements is relatively broad during the first fewcycles, implying that a large number of atoms with displacements much greater than thecage size, r c ≈ . σ , undergo irreversible rearrangements, or cage jumps, after one cycle.However, after 100 thermal cycles, the distribution function becomes more narrow, indicat-ing progressively more reversible particle dynamics. By contrast, as evident from Fig. 9 (b),the shape of the distribution of displacements for slowly annealed glass is rather insensitiveto the cycle number, and only a relatively small number of atoms might change their cagesduring one cycle.We next discuss the distribution of atomic displacements at the beginning of the thermaltreatment, i.e., during the second cycle, but consider instead the effect of the thermal ampli-tude. The probability distributions are plotted in Fig. 10 for selected values of the thermalamplitude and cooling rates 10 − ε/k B τ and 10 − ε/k B τ . As shown in Fig. 10 (a), the distri-bution of displacements is much broader at the thermal amplitude ∆ T LJ = 0 . ε/k B , and itnarrows upon decreasing ∆ T LJ towards the limiting case of aging at constant temperature.A similar trend can be observed for the slowly cooled glass in Fig. 10 (b) but the effect is lesspronounced. In general, these results are consistent with the analysis of nonaffine displace-ments during thermal cycling with respect to a very low reference temperature reported inthe previous study [29]. In particular, it was shown that atoms with large nonaffine dis-placements after one cycle are organized into transient clusters, whose size is reduced withincreasing cycle number or decreasing cooling rate and thermal amplitude [29].7fter the thermal treatment, the samples were strained along the ˆ x direction with con-stant strain rate ˙ ε xx = 10 − τ − at T LJ = 0 . ε/k B and zero pressure. The resulting stress-strain response for different cooling rates is presented in Fig. 11. For reference, the datafor tensile stress as a function of strain for glasses aged at T LJ = 0 . ε/k B during 2 × τ are also included in Fig. 11 and indicated by black curves. All samples were strained up to ε xx = 0 .
25 until the tensile stress is saturated to a constant value independent of the pro-cessing history. Following the elastic regime of deformation, the stress-strain curves exhibita pronounced yielding peak at about ε xx = 0 .
05. It can be observed in each panel in Fig. 11that the magnitude of the peak becomes larger with increasing thermal amplitude up to∆ T LJ ≈ . ε/k B . Notice, however, that the stress overshoot is generally reduced fromthe maximum value when ∆ T LJ = 0 . ε/k B for all cooling rates. These results indicate anonmonotonic variation of the yield stress on the thermal amplitude.The stress-strain curves reported in Fig. 11 were used to extract the values of the yieldingpeak, σ Y , and the elastic modulus, E , which are plotted in Fig. 12 as a function of the thermalamplitude. It can be seen that σ Y increases with ∆ T LJ and it has a maximum at ∆ T LJ ≈ . ε/k B for all cooling rates. Note that the data are somewhat scattered as simulationswere performed for only one sample in each case due to computational limitations. We alsocomment that the yield stress only weakly depends on the cooling rate except for the case10 − ε/k B τ . Similar trends can be observed for the dependence of the elastic modulus onthe thermal amplitude and cooling rate, as illustrated in the inset to Fig. 12. Generally,we find an inverse correlation between the dependencies of the yielding peak in Fig. 12 and U (100 T ) in Fig. 8 as functions of the thermal amplitude ∆ T LJ . In other words, the lowerthe energy state, the higher the values of σ Y and E . Finally, in comparison with the resultsof the previous study, where thermal cycling was performed with respect to a much lowerreference temperature of 0 . ε/k B [29], the relative increase in σ Y and, correspondingly,decrease in U (100 T ) for rapidly cooled glasses is less pronounced for cycling with respectto the reference temperature of 0 . ε/k B , considered in the present study. IV. CONCLUSIONS
In summary, the response of amorphous alloys during the sequence of quenching, pe-riodic thermal treatment, and tensile loading was investigated using molecular dynamics8imulations. The amorphous material was represented via a binary mixture of atoms withhighly non-additive cross interactions that prevents crystallization upon cooling. The ther-mal quenching was performed at constant pressure by cooling the binary mixture from theliquid state into the glassy region with different rates. The reference temperature was cho-sen to be approximately half the glass transition temperature, and the simulations wereperformed at zero pressure.After the thermal quench to different energy states, the glasses were either set to age atthe reference temperature or subjected to one hundred thermal cycles of spatially uniformheating and cooling at constant pressure. It was found that thermal cycling leads to relaxedstates with the potential energy levels lower than those in the aged samples for a given valueof the cooling rate. The potential energy after one hundred cycles acquired a minimum atthe thermal amplitude just below the reference temperature. The results of uniaxial tensileloading demonstrated than the stress overshoot and the elastic modulus only weakly dependon the cooling rate except for the lowest rate. Overall, the inverse correlation between thepotential energy levels and mechanical properties for different thermal amplitudes agreeswell with the results of the previous study [29], although the magnitude of the effects areslightly reduced due to the higher reference temperature considered in the present study.
Acknowledgments
Financial support from the National Science Foundation (CNS-1531923) is gratefullyacknowledged. The article was prepared within the framework of the Basic Research Pro-gram at the National Research University Higher School of Economics (HSE) and supportedwithin the framework of a subsidy by the Russian Academic Excellence Project ‘5-100’. Themolecular dynamics simulations were performed using the LAMMPS numerical code devel-oped at Sandia National Laboratories [34]. Computational work in support of this researchwas performed at Wright State University’s Computing Facility, the Ohio SupercomputerCenter, and the HPC cluster at Skoltech. [1] T. Egami, T. Iwashita, and W. Dmowski, Mechanical properties of metallic glasses, Metals ,77 (2013).
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Thelarger atoms of type A are denoted by red spheres and smaller atoms of type B are indicated byblue spheres. t / τ -8.02-8-7.98-7.96-7.94-7.92-7.9 U / ε
10 100 1000 10000 1e+05 t / τ ρ σ −2 −3 −4 −5 FIG. 2: (Color online) The variation of the potential energy per atom for binary glasses preparedwith cooling rates 10 − ε/k B τ (black), 10 − ε/k B τ (red), 10 − ε/k B τ (green), and 10 − ε/k B τ (blue).The simulations are performed at constant temperature T LJ = 0 . ε/k B and zero pressure. Theinset shows the glass density as a function of time for the same samples. t / T T L J FIG. 3: (Color online) The variation of temperature T LJ (in units of ε/k B ) during 5 periods, T = 2000 τ , for the thermal amplitudes ∆ T LJ = 0 . ε/k B (red), 0 . ε/k B (green), 0 . ε/k B (blue), and 0 . ε/k B (orange). The black line denotes the data at the constant temperature T LJ = 0 . ε/k B . The data are taken in the binary glass initially annealed with the cooling rate of10 − ε/k B τ to the temperature T LJ = 0 . ε/k B . t / T -8.4-8.2-8-7.8-7.6 U / ε
92 94 96 98 100 t / T t / T -8.04-8.02-8 U / ε FIG. 4: (Color online) The potential energy series during the first and last ten cycles with thethermal amplitudes ∆ T LJ = 0 (black), 0 . ε/k B (red), 0 . ε/k B (green), 0 . ε/k B (blue), and0 . ε/k B (orange). The sample was initially annealed with the cooling rate of 10 − ε/k B τ . Theenlarged view of the same data at the end of the last cycle is displayed in the inset. t / T -8.4-8.2-8-7.8-7.6 U / ε
92 94 96 98 100 t / T t / T -8.04-8.02-8 U / ε FIG. 5: (Color online) The variation of the potential energy for the glass annealed with thecooling rate of 10 − ε/k B τ and subjected to thermal cycling with the amplitudes ∆ T LJ = 0 (black),0 . ε/k B (red), 0 . ε/k B (green), 0 . ε/k B (blue), and 0 . ε/k B (orange). The inset shows U at t ≈ T . t / T -8.4-8.2-8-7.8-7.6 U / ε
92 94 96 98 100 t / T t / T -8.04-8.03-8.02-8.01 U / ε FIG. 6: (Color online) The potential energy per atom,
U/ε , for glasses cycled with the amplitudes∆ T LJ = 0 (black), 0 . ε/k B (red), 0 . ε/k B (green), 0 . ε/k B (blue), and 0 . ε/k B (orange). Thecooling rate is 10 − ε/k B τ . The same data are resolved near t ≈ T and shown in the inset. t / T -8.4-8.2-8-7.8-7.6 U / ε
92 94 96 98 100 t / T t / T -8.04-8.03-8.02 U / ε FIG. 7: (Color online) The time dependence of the potential energy during thermal treatment withamplitudes ∆ T LJ = 0 (black), 0 . ε/k B (red), 0 . ε/k B (green), 0 . ε/k B (blue), and 0 . ε/k B (orange). The sample was initially cooled with the rate 10 − ε/k B τ . The inset shows the same datain the vicinity of t ≈ T . ∆ T LJ -8.03-8.02-8.01-8-7.99 U −2 −5 FIG. 8: (Color online) The dependence of the potential energy after 100 cycles, U /ε , as afunction of the thermal amplitude ∆ T LJ (in units of ε/k B ) for glasses initially annealed with thecooling rates of 10 − ε/k B τ (black), 10 − ε/k B τ (red), 10 − ε/k B τ (green), and 10 − ε/k B τ (blue). N ∆ r / σ N −2 (a) 10 −5 (b) FIG. 9: (Color online) The distribution of atomic displacements during one cycle for the thermalamplitude ∆ T LJ = 0 . ε/k B . The cycle numbers for computing the displacements are tabulatedin the legend. The data are reported for glasses initially annealed with cooling rates (a) 10 − ε/k B τ and (b) 10 − ε/k B τ . The period of thermal oscillations is T = 2000 τ . N ε / k B ε / k B ε / k B ε / k B ε / k B ε / k B ε / k B ε / k B ∆ r / σ N ε / k B ε / k B ε / k B ε / k B ε / k B ε / k B ε / k B ε / k B (a)(b) 10 −2 −5 FIG. 10: (Color online) The probability distribution of atomic displacements during the secondcycle for the indicated values of the thermal amplitude ∆ T LJ . The samples were initially preparedwith cooling rates (a) 10 − ε/k B τ and (b) 10 − ε/k B τ . ε xx σ xx σ xx ε xx −2 −3 −4 −5 (a) (b)(d)(c) FIG. 11: (Color online) The variation of tensile stress, σ xx (in units of εσ − ), as a function of strain, ε xx , for thermally treated glasses that were initially annealed with cooling rates (a) 10 − ε/k B τ ,(b) 10 − ε/k B τ , (c) 10 − ε/k B τ , and (d) 10 − ε/k B τ . The strain rate is ˙ ε xx = 10 − τ − . The tensiletests were performed after the thermal treatment with amplitudes ∆ T LJ = 0 (black), 0 . ε/k B (red), 0 . ε/k B (green), 0 . ε/k B (blue), 0 . ε/k B (orange), 0 . ε/k B (brown), and 0 . ε/k B (dashed violet). ∆ T LJ σ Y ∆ T LJ E FIG. 12: (Color online) The dependence of the stress overshoot σ Y (in units of εσ − ) as a functionof the thermal amplitude for glasses annealed with cooling rates 10 − ε/k B τ (black), 10 − ε/k B τ (red), 10 − ε/k B τ (green), and 10 − ε/k B τ (blue). The variation of the elastic modulus E (in unitsof εσ − ) versus thermal amplitude is shown in the inset for the same samples.) versus thermal amplitude is shown in the inset for the same samples.