Correlation between Viscoelastic Moduli and Atomic Rearrangements

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Letter

Correlation between Viscoelastic Moduli and Atomic Rearrangements in Metallic Glasses Hai-Bin Yu, Ranko Richert, and Konrad Samwer J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b01738 • Publication Date (Web): 08 Sep 2016 Downloaded from http://pubs.acs.org on September 9, 2016

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The Journal of Physical Chemistry Letters

Correlation between Viscoelastic Moduli and Atomic Rearrangements in Metallic Glasses Hai-Bin Yu1*, Ranko Richert2 and Konrad Samwer3* 1

Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, WuHan 430074, China 2

School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, USA 3

I. Physikalisches Institut, Universität Göttingen, D-37077 Göttingen, Germany

AUTHOR INFORMATION Corresponding Author To whom correspondence should be addressed to H.-B.Y. Email: [email protected] or K.S. Email: [email protected]

ABSTRACT. Dynamical moduli, such as storage and loss moduli, characterize the viscoelasticity of materials (i.e., time-dependent elasticity) and convey important information about the relaxation processes of glasses and supercooled liquids. A fundamental question is what ultimately determines them in glassy materials. Here, for several model metallic glasses, we demonstrate that both the storage and loss moduli are uniquely determined by the most probable

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atomic non-affine displacements, regardless of temperature or frequency. Moreover, the fastmoving atoms (which contribute to dynamical heterogeneity) does not contribute explicitly to the moduli. Our findings provide a physical basis for the origin of viscoelasticity in metallic glasses.

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The elastic properties of many glassy materials, like polymers, metallic glasses (MGs)1-4 and colloids5, are time dependent over a wide range of temperatures (or volume fraction for colloids), which are in contrast to the well-known Hook elasticity and are called viscoelasticity6. To describe viscoelasticity, it is necessary to study the time-dependent dynamical moduli, such as storage and loss moduli as probed by dynamical mechanical experiments. Specifically, the storage modulus measures the stored energy, representing the elastic portion of the mechanical deformation; while the loss modulus measures the energy dissipated as heat, representing the viscous or the damping portion7,8. These global dynamical moduli are a major concern in many applications of glassy materials9 whose mechanical properties change with time. They also


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convey important information about the relaxation processes of glasses and supercooled liquids which are the central topic in glassy physics7,8,10-13. The infinite frequency shear moduli are the essential parameters in theoretical understanding the glass transition14 and the mechanisms of plastic deformation of glasses15. For crystalline materials, elasticity can be understood from the reversible atomic displacements on specific crystallographic planes (e.g., bond stretching). However, similar arguments cannot be safely applied to glassy materials. For example, in MGs, a recent study shows that even in the apparent elastic deformation regime, about 25% atoms undergo irreversible rearrangements16. The moduli of MGs are also found to be spatially inhomogeneous17,18. Even perplexed by the disordered atomic structure19-22 and the complex (and sometimes materials specific23,24) relaxation processes7,8,25, the question what ultimately determines the viscoelasticity and dynamical moduli is still not very clear in glassy materials26,27. In this work, we study the origin of viscoelasticity by investigating the global dynamical moduli of a model MG over a wide range of temperature and time scales, via an approach of molecular dynamics simulation of dynamical mechanical spectroscopy (MD-DMS) together with structural analysis based on the statistics of atomic displacements28-30. We find that the viscoelasticity is determined predominantly by the most probable atomic displacements. The influence of temperature and frequency on dynamical moduli can be understood in a unified way as they contribute to the atomic displacements. Interestingly, dynamical heterogeneity as induced by atoms whose displacements larger than the most probable one does not contribute explicitly to the global dynamical moduli. The details of our model system and the protocols of MD-DMS are presented in Methods. Briefly, at a temperature T, we apply a sinusoidal strain ε (t ) = ε A sin( 2πt / tω ) , with a period tω


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(related to frequency f = 1/ tω) and a fixed strain amplitude εA = 0.7 %, along the x direction of a model Ni80P20 MG and the resulting stress σ(t) is measured. To study the viscoelasticity and the dynamical moduli, we varied tω from 100 to 10,000 picoseconds (ps) and T from 20 to 780 K, covering both glass and supercooled liquid states. The storage- (E’) and loss (E”) moduli are calculated based on ε(t) and σ(t). a

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Figure 1 Dynamical modulus and statistics on atomic displacements. (a) Storage E’ and (b) loss modulus E” as a function of temperature T for different periods tω as indicated. (c) Probability density function p(u) of atomic displacements u for different T at tω = 100 ps, T ranges from 50 to 650K in a step of 50K (from left to right). The inset of (c) the peak position up of p(u) as a function of T for two typical tω as indicated. The storage modulus (d) and loss modulus (e) as a function of up.

Figure 1 (a) and (b) show the E’ and E” respectively as a function of T for five different tω. For low temperatures (e.g., T < 400 K) both E’ and E” are insensitive to tω and changes slowly with T, indicating that the system is deep in the glassy state. However, for higher T (e.g., T > 500 K), E’ and E” depends strongly on tω and changes rapidly with T, indicating the viscoelastic nature of the deformation. Specifically, at higher T, the values of E’ drop in a


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sigmoidal manner while those of E” develop asymmetrical peaks. Both of them signal the dynamics of the transition from a glassy state to a supercooled liquid state, i.e., the α relaxation. We note that all these features are qualitatively consistent with experimental dynamical mechanical measurements7,8, which confirm the validity and reliability of our MD-DMS in studying the viscoelasticity of MGs. To investigate the structural origin of the viscoelasticity and what determines the dynamical moduli, we conducted structural analysis based on the atomic displacements u,

r r defined as u (t ) = r (t ) − r (0) for each atom during a time interval of ∆t = tω for all the combinations of T and tω as reported in Fig. 1(a) and (b). This choice of ∆t is meant to avoid atomic displacements due to the overall deformations applied by the MD-DMS. Figure 1(c) shows a typical probability density function p(u) based on the statistics of u for different T (ranging from 50 to 650K), at a fixed tω =100 ps. Obviously, the peak position up of p(u) represents the most probable atomic displacements of all the atoms for the given tω. The inset of Fig.1(c) quantitatively summarizes how the up varies with T for two typical tω = 100 and 10,000 ps, respectively. We find that up has the T and tω dependence that qualitatively similar to those of E’ and E” as presented in Fig. 1(a) and (b). For example, the up changes slowly with T and insensitive to tω at low temperatures but it increases dramatically with T and becomes sensitive to tω at higher T. This observation implies that there might be connections between up and E’ and E”. To reveal such connections, Fig. 1(d) and (e) plot E’ and E” as a function of up for all the studied T and tω. Remarkably, we find that all the data of E’ and E” collapses onto two unique curves as shown in Fig. 1(d) and (e) respectively. This result establishes correlations between up and dynamical moduli (E’ and E”) in our model MG. The collapse of the data reminds the so-


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called “temperature-time superposition” where the spectra are shifted manually to construct a master curve31. Our result, however, does not involve any adjusting parameters. It suggests that the factors such as T and tω that affects E’ and E” can be understood in a unified manner as they contribute to up. This notion is validated by Fig.1 (d) and (e), which show that for any combinations of T and tω, if they have the same value of up, they would have uniquely the same dynamical moduli. Thus, the two major seemingly independent variables, T and tω, can be reduced into a single variable, up.

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Dynamical heterogeneity. (a) Probability density function p(u) for different pairs of

combinations of T and tω; from left to right (T = 560 K, tω = 100 ps), (T = 520 K, tω = 1,000 ps) and (T = 500 K, tω = 10,000 ps), respectively. They have the same value of up = 0.38±0.01 Å. (b) and (c) 2D slices of the atomic configurations correspond to (T = 560 K, tω = 100 ps) and (T = 500 K, tω = 10,000 ps), respectively.


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Given the above analysis, one immediate question is that how does atoms with different mobility (i.e., dynamical heterogeneity) contribute to E’ and E”? This question is particularly meaningful because the atomic motions in glasses and supercooled liquid are intrinsically heterogeneous32-38. To address this issue, Fig.2 (a) selectively plots three curves of p (u). They are the different pairs of combinations between T and tω [from left to right in Fig.2(a), these pairs are (T = 560 K, tω = 100 ps), (T = 520 K, tω = 1000 ps) and (T = 500 K, tω = 10000 ps), respectively] but they have the same value of up = 0.38±0.01 Å and therefore the same global values of E’ = 33±1 GPa and E” = 5.6±0.5 GPa, according to Fig.1(d) and (e). We find that they show appreciable difference at the tails of p(u) when u > 1.5 Å. Especially, the sub-peaks of p(u) around about u = 2.5 Å grow higher in the order of decreasing T (or increasing tω), which suggest the increases in dynamical heterogeneity and the fraction of atoms with atomic displacements larger than up. Figure 2(b) and (c) compare two atomic configurations for (T = 560 K, tω = 100 ps) and (T = 500 K, tω = 10000 ps), respectively. The color code represents the magnitudes of u. Heterogeneous dynamics are observed in both configurations. Compared to Fig. 2(b), Fig. 2(c) is found to be more heterogeneous over a wider range of length scales, and it involves more particles with larger u than up. Additionally, the regions with heterogeneous dynamics are observed to be string-like structures in Fig. 2(b) but compact structures in Fig.2(c). Overall, the above results suggest that for some specified combinations of T and tω, the model MGs might have the same values of up and E’ and E”, however, their dynamical heterogeneity and the portion of fast-moving atoms (those move farther than up ) are not necessarily the same. In other words, dynamical heterogeneity as induced by the faster atoms does not contribute explicitly to the dynamical moduli.


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Figure 3 Dynamical modulus and root mean square displacements. (a) Storage E’ and (b) loss E” moduli as a function of root mean square displacements 1/2 for different tω as indicated. The shadow region shows that E’ or E” do not collapse.

To provide more quantitative support for the above discussions, Fig. 3 reports another way to analyze the data. Instead of using up from statistical analysis, we calculated the root mean square atomic displacements 1/2 based on the u values, where < u 2 >= 1/ N ∑ ui2 is the mean square i

displacements and N = 32,000 is the number of atoms. Figure 3 (a) and (b) show E’ and E” as a function of 1/2, respectively. In contrast to Fig.1 (d) and (e), most of the data with different tω does not collapse onto unique curves [e.g., in the shadow region through Fig. 3(a) and (b)]. This suggests (i) that dynamical moduli could not be single variable functions of 1/2 and (ii) that only up fundamentally correlates with the viscoelasticity and the dynamical moduli. Since


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1/2 incorporates all the atomic displacements, including faster atoms with u > up, this result reinforces the above conclusion that faster atoms does not contribute to the viscoelasticity and dynamical moduli. By the way, a closer examine of the data in Fig.3 (a) and (b) reveals that the initial portion of the E’ and E” data indeed collapse when 1/2 is smaller than 0.5 Å. This is because at the conditions of small 1/2 (usually at low T or small tω), p(u) is narrowly distributed around up and therefore up ≈ 1/2. For other conditions up does not relate to 1/2 and lacks correlations between 1/2 and the dynamical moduli. We have also considered the mean atomic displacement < u >= 1 / N ∑ ui . It cannot collapse the data too, because of the same i

reason. In a colloidal glassy system39, Conrad et al identified that only the slow-moving particles contribute significantly to the dynamical shear moduli. This is consistent with the present findings, because in our model MG, atoms with u ~ up are indeed moving slowly, as can be seen in Fig.2. Therefore, the viscoelasticity of glassy materials and supercooled liquids could be considered as a consequence of the collective rearrangements of the atoms of the whole system. Possibly, the embedded fast-moving atoms are liquid like (with u >> up and E’ ~ E” ~ 0) and cannot restrain the stress and thus they do not contribute to the development of the viscoelastic modulus of a glass. We note that our findings are not limited to the Ni-P MG. In supporting online materials, Fig.S1-S3, we demonstrate similar conclusions hold for a Pd80Si20 MG (all the other conditions are same with the present Ni-P MG) as well. Additionally, our results are independent of the strain amplitudes (εA) conducted in the linear response regime (εA < 2%), Fig. S4. For the nonlinear response regime, plastic deformation is involved and the absolute values of E’ and E”


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are difficult to obtain with accuracy (i.e., one has to consider multiple Fourier components). Since the dynamical moduli in experiments are measured in the linear response regime, we do not discuss the nonlinear issue here. Finally, the system size of the simulation affect the results very little, as shown in Fig. S5. One interesting extension of our work could be to study model MGs with secondary relaxations (e.g. the β relaxations), and to see if the relations between E’, E’’ ~ up remain valid. Such a study could provide information about the mechanism of secondary relaxations8,9,23,40,41. In summary, we have shown that both of the storage and loss moduli in a model metallic glass and its supercooled liquid are uniquely determined by the most probable atomic non-affine displacements. The temperature and time that affect dynamical moduli can be understood in a similar way as they contribute to the atomic displacements. Our findings bridge the gap between the dynamical mechanical properties of metallic glass and the atomic motions. Hopefully, they could offer a new benchmark for constructing viscoelasticity theory of glassy materials in the future.

SIMULATION METHODS An open source LAMMPS package42 was used for all the MD simulations. The model system contains N = 32000 atoms with the composition Ni80P20 and the constituting atoms are interacting via an embedded atom method (EAM) potentials20. For the sample preparations, the system was melted and equilibrated at T = 2000 K for 100ps, then cooled down to T = 1 K with a cooling rate of 1010 K/s, during which the cell-sizes were adjusted to give zero pressure with an NPT ensemble. Periodic boundary conditions were applied for all the calculations.


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As in the case of experimental DMS, we apply a sinusoidal strain ε (t ) = ε A sin( 2πt / tω ) along the x direction of the model MG, where tω is the period and is selected as 100, 300, 1,000, 3,000 and 10,000 ps, while εA is fixed at about 0.7 % to ensure all the deformations are in the linear response regime. For each MD-DMS, 10 full cycles were used, i.e., t in the range [0, 10tω]. We fitted the resulting stress as: σ (t ) = σ 0 + σ A sin(2π t / tω + δ ) where σ0 is a constant term and usually small (σ0 < 0.1σA in the glassy state), δ the phase difference between stress and strain. Storage (E’) and loss (E”) moduli are calculated according to E ' = σ A / ε A sin(δ ) and E " = σ A / ε A cos(δ ) , respectively. The MD-DMS were carried out during the cooling processes

of the sample-preparations, and NVT ensemble was applied during the cyclic deformations. The probability density function p(u) is defined as p(u ) = [ P(u + ∆u) − P(u )] / ∆u , where P(u) is the cumulative distribution that quantifies the probability of finding X ≤ u. We employed ∆u = 0.01 Å for all the calculations. The probability density is normalized according ∞

to ∫ p (u )du = P(∞) = 1 . 0

ACKNOWLEDGMENT We thank Prof. Yue Wu for discussions. H.-B.Y. thanks the support from Huazhong University of Science and Technology of China. K.S. acknowledges the supports from the German Science Foundation within the FOR 1394, P1. The computational work was carried out at Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen (GWDG), Germany and the TianHe-1(A) of National Supercomputer Center in Tianjin, China.


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Supporting Information Available: We have considered other factors, such as the effects of different model MGs, the strain amplitude and the size of the system under simulation. They are shown in the supporting information.

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Correlation between Viscoelastic Moduli and Atomic Rearrangements

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