Accessing a broader range of energy states in metallic glasses by variable-amplitude oscillatory shear
AAccessing a broader range of energy states in metallic glasses byvariable-amplitude oscillatory shear
Nikolai V. Priezjev , Department of Mechanical and Materials Engineering,Wright State University, Dayton, OH 45435 and National Research University Higher School of Economics, Moscow 101000, Russia (Dated: January 12, 2021)
Abstract
The influence of variable-amplitude loading on the potential energy and mechanical propertiesof amorphous materials is investigated using molecular dynamics simulations. We study a binarymixture that is either rapidly or slowly cooled across the glass transition temperature and thensubjected to a sequence of shear cycles with strain amplitudes above and below the yielding strain.It was found that well annealed glasses can be rejuvenated by small-amplitude loading if thestrain amplitude is occasionally increased above the critical value. By contrast, poorly annealedglasses are relocated to progressively lower energy states when subyield cycles are alternated withlarge-amplitude cycles that facilitate exploration of the potential energy landscape. The analysisof nonaffine displacements revealed that in both cases, the typical size of plastic rearrangementsvaries depending on the strain amplitude and number of cycles, but remains smaller than thesystem size, thus preserving structural integrity of amorphous samples.Keywords: metallic glasses, thermo-mechanical processing, yielding transition, oscillatory sheardeformation, molecular dynamics simulations a r X i v : . [ c ond - m a t . s o f t ] J a n . INTRODUCTION The fundamental understanding of the interrelationship between the amorphous struc-ture, mechanical and physical properties of bulk metallic glasses is important for numerousbiomedical and structural applications [1–3]. In contrast to crystalline materials, where ir-reversible deformation is controlled by topological line defects, it was realized that glassesdeform plastically via a sequence of collective rearrangements of groups of atoms, often re-ferred to as shear transformations [5, 6]. The advantageous properties of metallic glassesinclude relatively high strength, large elastic strain limit, high resistance to corrosion, andbiocompatibility, among others, but they typically suffer from catastrophic failure upon ex-ternal deformation, i.e. , they are brittle when in a well annealed state [1]. To remediate thelatter issue, a number of processing methods are employed in order to rejuvenate metallicglasses; for example, cold rolling, high pressure torsion, irradiation, elastostatic loading, andsurface treatments like shot peening [4]. More recently, it was found that a particularlyelegant and minimally invasive approach to enhance potential energy is to thermally cycleglasses between the room and cryogenic temperatures [7–17]. Despite considerable efforts,however, the development of efficient processing methods to access a broader range of energystates in metallic glasses and, at the same time, maintain their structural integrity remainsa challenging problem.In the last decades, atomistic simulations have played an important role in understand-ing relaxation, rejuvenation and yielding phenomena in disordered materials subjected toperiodic deformation [18–46]. Most notably, it was discovered that after a certain number ofcycles at zero temperature, disordered solids become locked into the so-called ‘limit cycles’where the trajectory of each atom is exactly reversible during one or more periods [22, 23].At finite temperatures, the structural relaxation, sometimes termed as mechanical anneal-ing , proceeds via collective irreversible rearrangements during hundreds of loading cycles,and the potential energy gradually approaches a constant value [32, 33, 35, 37]. Dependingon the initial energy state, it typically takes a number of transient cycles to yield whenthe strain amplitude is above the critical value, and the number of cycles is reduced uponperiodically alternating the loading direction [30, 33, 36, 40, 42, 43, 45]. What remains un-known, however, is how to apply cyclic loading and rejuvenate glasses without the formationof shear bands, and, on the other hand, how to access low-energy states in poorly annealed2lasses by mechanical agitation.In this paper, the effect of variable-amplitude oscillatory shear deformation of binaryglasses on their energy states and mechanical properties is investigated using moleculardynamics (MD) simulations. We consider a model glass former initially cooled well belowthe glass transition temperature and then periodically deformed with strain amplitudesalternating between values below and above the yielding amplitude. It will be shown that well annealed glasses become rejuvenated by cyclic loading when the strain amplitude is oncein a while increased above the critical amplitude, followed by a sequence of low-amplitudecycles. Remarkably, the same deformation protocol drives poorly annealed glasses to lowerenergy states as it allows for a more efficient exploration of the potential energy landscape.The rest of this paper is organized as follows. The molecular dynamics simulations,parameter values, and the deformation protocol are described in the next section. Theresults for the potential energy series under variable-amplitude cycle loading, mechanicalproperties, and the spatiotemporal analysis of irreversible displacements are presented insection III. The brief summary is given in the last section.
II. MOLECULAR DYNAMICS SIMULATIONS
In our study, we use the binary (80:20) Lennard-Jones (LJ) mixture model to represent anamorphous alloy in three dimensions. This popular model of a glass former was first intro-duced by Kob and Andersen (KA), who investigated its structural and dynamical propertiesnear the glass transition temperature [47]. The mixture consists of two types of atoms withstrongly non-additive interaction between different types, which suppresses crystallizationupon cooling across the glass transition. Specifically, any two atoms of types α, β = A, B interact via the truncated LJ potential, as follows: 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 parameters: ε AA = 1 . ε AB = 1 . ε BB = 0 . σ AA = 1 . σ AB = 0 . σ BB = 0 . m A = m B [47]. A similar parametrization of the pairwise interaction was used byWeber and Stillinger to study the amorphous metal-metalloid alloy Ni P [48]. In oursimulations, the cutoff radius of the LJ potential was set to r c, αβ = 2 . σ αβ . The systemconsists of N = 60 000 atoms. As usual, the simulation results are reported in terms of the3educed units of length, mass, and energy σ = σ AA , m = m A , and ε = ε AA . The equationsof motion were solved numerically using the velocity Verlet algorithm with the time step (cid:52) t MD = 0 . τ , where τ = σ (cid:112) m/ε is the LJ time [49, 50].The sample preparation protocol involved a thorough equilibration of the liquid phase ata constant density ρ = ρ A + ρ B = 1 . σ − in a periodic box of linear size L = 36 . σ . Thesystem temperature was regulated via the Nos´e-Hoover thermostat [49, 50]. For reference,the computer glass transition temperature at ρ = 1 . σ − is T c = 0 . ε/k B , where k B isthe Boltzmann constant [47]. Following the equilibration period, the binary mixture wascooled to the low temperature T LJ = 0 . ε/k B with a slow (10 − ε/k B τ ) and fast (10 − ε/k B τ )cooling rates in order to obtain well (low energy) and poorly (high energy) annealed samplesat ρ = 1 . σ − .After cooling to T LJ = 0 . ε/k B , both samples were subjected to periodic shear defor-mation along the xz plane at constant volume, as follows: γ xz ( t ) = γ sin(2 πt/T ) , (2)where the oscillation period is T = 5000 τ , and, correspondingly, oscillation frequency is ω = 2 π/T = 1 . × − τ − . Unless otherwise noted, the strain amplitude during n − γ = 0 .
06, while the amplitude during every n -th cycle is changed to γ = 0 . n = 10 is shown in Fig. 1. In the presentstudy, the following values of the periodicity n were considered n = 2, 5, 10, 20, 50, and100. The typical simulation run during 3600 cycles takes about 95 days using 40 processors.During production runs, the potential energy, atomic positions, and shear strain were storedfor post-processing. The simulations were performed only for one well annealed and one poorly annealed samples due to computational constraints. III. RESULTS
It has been long realized that the atomic structure and mechanical properties of amor-phous alloys strongly depend on the thermo-mechanical processing history [4]. In particular,upon rapid cooling from the liquid state, glasses freeze into highly unrelaxed configurationsand, when externally deformed, exhibit a gradual crossover from zero to a plateau level ofstress [1]. On the other hand, more slowly cooled glasses settle at lower energy states and4ecome more brittle. Furthermore, under periodic strain with an amplitude slightly above acritical value, slowly cooled glasses gradually form a narrow shear band during a number oftransient cycles and then undergo a yielding transition, whereas rapidly quenched glasses firstrelax during hundreds of cycles and then suddenly yield [29, 30, 33, 36, 40, 43]. By contrast,repeated cycling with an amplitude below the critical value typically leads to an explorationof progressively lower energy states via structural relaxation [21, 29, 32, 35, 37, 41]. Theseconclusions were obtained for binary glasses subjected to cyclic loading at a fixed strainamplitude.It can be hypothesized that periodic deformation with the strain amplitude that is oc-casionally alternated between values slightly above and below the critical amplitude, mightlead to either enhanced rejuvenation or accelerated relaxation, depending on the initial en-ergy state. The rationale for this hypothesis is as follows. In the case of well annealed glasses, one cycle with an amplitude slightly larger than the critical value will increase thepotential energy via cooperative irreversible rearrangements of atoms. Subsequently, one orseveral subyield cycles will relax the glass, thus avoiding the formation of system-spanningshear bands and material failure. Upon iteration, such deformation protocol might result insteady increase of the potential energy, and, possibly, a well-defined energy level after manycycles. On the contrary, poorly annealed glasses under periodic loading below the yieldingpoint tend to relax and ultimately get trapped in a local minimum of the potential energylandscape. Thus, one cycle with a large strain amplitude might relocate the system to anadjacent minimum with lower energy, leading to accelerated relaxation upon iteration of thesequence of alternating cycles.In our simulations, these scenarios were tested for the well-studied KA binary mixturemodel at the low temperature T LJ = 0 . ε/k B and fixed density ρ = 1 . σ − . Thus, it wasrecently found that the critical value of the strain amplitude for rapidly cooled KA glasses atthese T LJ and ρ is γ ≈ .
067 [43]. Furthermore, well annealed glasses under periodic shearwith the strain amplitude γ = 0 .
08 were shown to undergo a yielding transition after about20 transient cycles, while it might take many cycles to yield at γ = 0 .
07 [30]. Therefore, twovalues of the strain amplitude slightly below and above the critical point were considered;namely, γ = 0 .
06 and γ = 0 .
08. The deformation protocol consists of one cycle with thestrain amplitude γ = 0 .
08, followed by n − γ = 0 .
06. An example5f the shear strain variation for the sequence of alternating cycles, n = 10, is displayed inFig. 1. In this study, the simulations were carried out for the values of the integer n = 2,5, 10, 20, 50, and 100. For reference, the results for periodic deformation with fixed strainamplitudes γ = 0 .
06 and 0 .
08 are also reported.The potential energy minima after each cycle at zero strain are plotted in Fig. 2 for the wellannealed glass subjected to periodic deformation with periodicity n = 2, 5, 10, 20, 50, and100. For comparison, the two limiting cases with the fixed strain amplitudes γ = 0 .
06 and γ = 0 .
08 are also included. Note that the potential energy at γ = 0 .
06 remains essentiallyconstant during 3600 cycles, indicating that mechanical annealing becomes inefficient forwell annealed glasses even in the presence of thermal fluctuations. It was previously foundthat glasses under small-amplitude cyclic shear cannot access energy states below a certainthreshold [42, 45]. Moreover, it can be clearly observed in Fig. 2 that upon increasingperiodicity n , the yielding transition is delayed (up to about 1600 cycles for n = 20).After a sharp increase in potential energy due to the formation of a shear band across thesimulation domain, and two-phase system continues steady state deformation.By contrast, the energy curves for the well annealed glass appear to increase and levelout for cyclic loading with large periodicity, n = 50 and 100, as shown in Fig. 2. In thesecases, the energy series resemble a step-like pattern, where a sudden increase after a cycleat γ = 0 .
08 follows by relaxation during n − γ = 0 . n is sufficiently large. It can be seen in Fig. 2 thatthe increase in potential energy after 3600 cycles is about 0 . ε , which is comparable tothe energy change reported for the KA binary glass subjected to prolonged elastostaticloading [52, 53].In the case of the poorly annealed glass, the potential energy at the end of each cycleis displayed in Fig. 3 for the modulated deformation n = 20, 50, 100 and the fixed strainamplitude γ = 0 .
06. As is evident, all deformation protocols initially result in a rapid decayof the potential energy since many unstable clusters of atoms can be mechanically drivento lower energy configurations. When the strain amplitude γ = 0 .
08 is applied every 10-thcycle, or more frequently (not shown), the binary glass undergoes a yielding transition via6he formation of a shear band across the system. As shown in Fig. 3, the yielding transitionfor n = 10 occurs after about 800 shear cycles when the glass is mechanically annealed to U ≈ − . ε . These results are consistent with the critical behavior reported in the previousMD studies [33, 36, 40, 43]. Furthermore, the potential energy continue to gradually decayfor n = 20, 50, 100, and when the strain amplitude is always fixed at γ = 0 .
06 (shown by theblack curve in Fig. 3). It can be seen that upon including a cycle with the strain amplitude γ = 0 .
08 ( n = 20, 50, and 100), the energy levels are reduced on average by roughly 0 . ε with respect to the black curve. This trend can be explained by noticing relatively largespikes along the energy series due to large-scale plastic deformation at γ = 0 .
08, whichincrease the probability of relocating the system between local minima of the potentialenergy landscape.The changes in mechanical properties for the well annealed glass subjected to variable-amplitude deformation with n = 100 (the violet curve in Fig. 2) were evaluated by carryingout mechanical tests at selected time intervals. Thus, following a certain number of loadingcycles, the binary glass, initially at zero strain, was continuously strained at a fixed rate˙ γ = 10 − τ − along the xy , xz , and yz planes. In each case, the shear modulus G and the peakvalue of the stress overshoot σ Y were computed from the linear slope and the maximum of thestress-strain curve. The results for G and σ Y are shown in Fig. 4 during 3600 loading cycles.The data are somewhat noisy, since it were collected for only one realization of disorder, but,nevertheless, one can clearly see that the shear modulus reduces and acquires directionalanisotropy, and the yielding peak tends to decrease with the cycle number, which is consistentwith gradual rejuvenation reported in Fig. 2. We also comment that it was previously shownfor cyclically annealed binary glasses that anisotropy in mechanical properties is reducedwhen the loading direction is alternated in two or three spatial dimensions [37].We next perform a microscopic analysis of plastic deformation quantified via the so-called nonaffine displacements of atoms. In disordered solids, the decomposition of the totaldisplacement of an atom into affine and nonaffine components can be used to estimate itsrelative displacement with respect to neighboring atoms. More specifically, the nonaffinemeasure for an atom i can be computed using the matrix J i that linearly transforms a group7f atoms and minimizes the following expression: D ( t, ∆ t ) = 1 N i N i (cid:88) j =1 (cid:110) r j ( t + ∆ t ) − r i ( t + ∆ t ) − J i (cid:2) r j ( t ) − r i ( t ) (cid:3)(cid:111) , (3)where ∆ t is the time between two configurations, and the summation is carried over neigh-bors located within 1 . σ from the position of the i -th atom at r i ( t ). This definition was firstused by Falk and Langer in the analysis of localized shear transformations in sheared disor-dered solids [51]. Typically, if the nonaffine displacement of an atom becomes greater thanthe cage size, then the local rearrangement is irreversible, and it is often involves a groupof atoms. In recent years, the spatial and temporal correlations of nonaffine displacementswere extensively studied in amorphous materials under startup continuous [54–60] and timeperiodic [26, 32, 33, 37, 40, 41, 43, 45] deformation. In particular, it was demonstrated thatthe yielding transition is accompanied by the formation of a system-spanning shear bandthat can be clearly identified by plotting atoms with large nonaffine displacements.The atomic configurations of the well annealed glass subjected to variable-amplitudeoscillatory deformation are shown in Fig. 5 for periodicity n = 20 and in Fig. 6 for n = 100.The system snapshots are taken at selected cycle numbers, and, for clarity, only atoms withrelatively large nonaffine displacements during one full cycle are displayed. First, it can beobserved in Fig. 5 (a, b) that, upon continued loading, the typical size of clusters of atomswith large nonaffine displacements increases but remains smaller than the system size. Theappearance of compact clusters during 500 and 1500-th cycles, when the strain amplitude is γ = 0 .
08, correlates well with the gradual increase in potential energy shown by the browncurve in Fig. 2. By contrast, the sharp increase in potential energy (after about 1600 cycles)signals a yielding transition and formation of a shear band along the xy plane, which canbe clearly seen in Fig. 5 (c, d). Second, as illustrated in Fig. 6, the situation is qualitativelydifferent when the strain amplitude is changed to γ = 0 .
08 less frequently ( i.e. , n = 100;see also the violet curve in Fig. 2). In this case, the clusters remain finite but they becomelarger during cycles with γ = 0 .
08, shown in Fig. 6 (a, c), while the deformation is nearlyreversible during subsequent relaxation at γ = 0 .
06, see Fig. 6 (b, d). Hence, we concludethat the oscillatory deformation with alternating excitation and relaxation periods leads toa controllable rejuvenation without shear banding.Lastly, the sequences of stroboscopic snapshots of the poorly annealed glass under periodic8eformation are presented in Figures 7 and 8 for n = 10 and 100, respectively. The first twosnapshots in Fig. 7 (a, b) show interconnected networks during 100 and 800-th cycles whenthe strain amplitude is γ = 0 .
08. These large-scale irreversible rearrangements correspondto structural relaxation after rapid cooling, as indicated by the brown curve in Fig. 3. In thisloading protocol, the strain amplitude is changed to γ = 0 .
08 frequently enough ( n = 10) toinduce a yielding transition after about 900 alternating cycles. Thus, one can clearly observea single shear band (across periodic boundaries) in Fig. 7 (c, d) that is formed after yieldingand remains stable upon continued loading. In the case n = 100, the poorly annealed glasscontinues relaxation during every 99 cycles with γ = 0 .
06 [ e.g. , see Fig. 8 (b, d) ], which isinterrupted by one cycle with γ = 0 .
08 that induces large-scale plastic events, shown forexample, in Fig. 8 (a, c). The occasional perturbation with the strain amplitude γ = 0 . γ = 0 .
06 (see the red andblack curves in Fig. 3). Altogether, the results for well and poorly annealed glasses undercyclic deformation with periodicity n = 100, shown in Figs. 6 and 8, demonstrate that glassesbecome either highly rejuvenated or better annealed, thus extending the range of accessibleenergy states. IV. CONCLUSIONS
In summary, molecular dynamics simulations were carried out to study the effect ofvariable-amplitude periodic deformation on relaxation and rejuvenation of disordered solids.The model glass former in three dimensions was represented via the binary mixture, whichwas cooled from the liquid state deep into the glass phase with relatively slow and fastrates, producing well and poorly annealed samples. The binary glass was subjected to cyclicloading where the strain amplitude is fixed below the critical value but occasionally changedslightly above the critical strain. During such deformation protocol, one shear cycle withthe large strain amplitude typically induces collective plastic rearrangements but not shearbands if the frequency of large-amplitude cycles is sufficiently low.It was found that well annealed samples can be rejuvenated during a sequence of shearcycles provided that the strain amplitude is rarely changed above the critical value. Theincrease in potential energy is reflected in mechanical properties and leads to enhanced9uctility. Interestingly, the same deformation protocol drives poorly annealed glasses to pro-gressively lower energy states, since large-amplitude cycles occasionally perturb the systemand prevent trapping in local minima of the potential energy landscape. On the other hand,both well and poorly annealed glasses undergo a yielding transition after a number of tran-sient cycles when the strain amplitude is frequently changed above and below the criticalvalue. These conclusions were confirmed by visualizing spatial configurations of atoms withlarge nonaffine displacements that are organized either in compact clusters or planar shearbands.
Acknowledgments
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06, except that during every n -thcycle the amplitude is changed to γ = 0 .
08. In this study, the following values were considered n = 2, 5, 10, 20, 50, and 100.
800 1600 2400 3200 t / T -8.31-8.3-8.29-8.28-8.27-8.26 U / e FIG. 2: (Color online) The potential energy minima at zero strain as a function of the number ofshear cycles with the strain amplitude γ = 0 .
06 during n − γ = 0 .
08 during every n -th cycle. The values of periodicity n are listed in the legend.The oscillation period is T = 5000 τ . The black and orange curves indicate cyclic loading with thestrain amplitudes γ = 0 .
06 and 0 .
08, respectively. The well annealed glass was initially preparedvia cooling from the liquid state to T LJ = 0 . ε/k B with the rate 10 − ε/k B τ .
400 800 1200 1600 2000 t / T -8.28-8.27-8.26-8.25-8.24 U / e n = n = n = n = FIG. 3: (Color online) The time dependence of the potential energy (at zero strain) for the indicatedvalues of periodicity n . The black curve denotes cyclic loading with the strain amplitude γ = 0 . T = 5000 τ . The poorly annealed sample was initiallyprepared by cooling with the rate 10 − ε/k B τ to T LJ = 0 . ε/k B at ρ = 1 . σ − .
800 1600 2400 3200 t / T s Y G xyxzyz FIG. 4: (Color online) The shear modulus G (in units of εσ − ) and yielding peak σ Y (in units of εσ − ) versus cycle number for periodic deformation with n = 100. The startup continuous sheardeformation with the strain rate ˙ γ = 10 − τ − was imposed along the xy plane (blue circles), xz plane (red squares), and yz plane (green diamonds). The variation of the potential energy for thisloading protocol is indicated by the violet curve in Fig. 2. The glass was initially cooled with therate 10 − ε/k B τ to T LJ = 0 . ε/k B . IG. 5: (Color online) Snapshots of the well annealed glass subjected to oscillatory shear defor-mation with periodicity n = 20. The potential energy for the same protocol is denoted by thebrown curve in Fig. 2. The nonaffine measure, as defined by Eq. (3), is (a) D (500 T, T ) > . σ ,(b) D (1500 T, T ) > . σ , (c) D (1600 T, T ) > . σ , and (d) D (2500 T, T ) > . σ . Theoscillation period is T = 5000 τ . The color in the legend indicates the magnitude of D . IG. 6: (Color online) Plastic rearrangements in the well annealed glass cyclically driven withperiodicity n = 100. The corresponding energy series are indicated by the violet curve in Fig. 2.The nonaffine displacements are (a) D (1000 T, T ) > . σ , (b) D (1050 T, T ) > . σ , (c) D (3000 T, T ) > . σ , and (d) D (3050 T, T ) > . σ , where T = 5000 τ . The color shows themagnitude of D . IG. 7: (Color online) The sequence of snapshots of poorly annealed glass subjected to variable-amplitude periodic deformation with periodicity n = 10. The same loading protocol as in Fig. 3 (thebrown curve). The nonaffine measure is (a) D (100 T, T ) > . σ , (b) D (800 T, T ) > . σ ,(c) D (900 T, T ) > . σ , and (d) D (1400 T, T ) > . σ . The colorcode in the legend definesthe magnitude of D . IG. 8: (Color online) Spatial configurations of atoms with large nonaffine displacements after onecycle for the poorly annealed glass under periodic deformation with n = 100. See the red curve inFig. 3. The nonaffine quantity in Eq. (3) is (a) D (100 T, T ) > . σ , (b) D (150 T, T ) > . σ ,(c) D (1900 T, T ) > . σ , and (d) D (1950 T, T ) > . σ . The magnitude of D is defined inthe legend.is defined inthe legend.