Tensile Strength of Liquids: Equivalence of Temporal and Spatial

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Tensile Strength of Liquids: Equivalence of Temporal and Spatial Scales in Cavitation Yang Cai, Junyu Huang, Heng-An Wu, William A. Goddard III, and Sheng-Nian Luo J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02798 • Publication Date (Web): 17 Feb 2016 Downloaded from http://pubs.acs.org on February 18, 2016

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The Journal of Physical Chemistry Letters

Tensile Strength of Liquids: Equivalence of Temporal and Spatial Scales in Cavitation Y. Cai,† J. Y. Huang,† H. A. Wu,∗,† M. H. Zhu,‡ W. A. Goddard III,§ and S. N. Luo∗,¶ CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230027, China , Key Laboratory of Advanced Technologies of Materials, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan 610031, P. R. China, The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610031, P. R. China, and Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, USA E-mail: [email protected]; [email protected];[email protected]

∗

To whom correspondence should be addressed CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230027, China ‡ Key Laboratory of Advanced Technologies of Materials, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan 610031, P. R. China ¶ The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610031, P. R. China § Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, USA †

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Abstract It is well known that strain rate and size effects are both important in material failure, but the relationships between them are poorly understood. To establish this connection, we carry out molecular dynamics (MD) simulations of cavitation in LennardJones and Cu liquids over a very broad range of size and strain rate. These studies confirm that temporal and spatial scales play equivalent roles in the tensile strengths of these two liquids. Predictions based on smallest-scale MD simulations of Cu for larger temporal and spatial scales are consistent with independent simulations, and comparable to experiments on liquid metals. We analyze these results in terms of classical nucleation theory, and show that the equivalence arises from the role of both size and strain rate in the nucleation of a daughter phase. Such equivalence is expected to hold for a wide range of materials and processes, and useful as a predictive bridging tool in multiscale studies. Table of Contents Image

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Keywords : molecular dynamics, size effects, strain rate effects, classical nucleation theory, multiscale science

In such physical processes as cavitation in liquids and ductile fracture in solids, system size and loading rate play equally important roles in when and how a material fails. 1–4 Various experimental or simulation tools are utilized to probe certain segments in the breadth of temporal or spatial spectra, and there exist wide scale gaps to cross. For instance, con2


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ventional molecular dynamics (MD) simulations involve 100 nm and sub-ns scales, while experimental length scales exceed µm with time scales over ns. Such development as accelerated molecular dynamics now allows extending the MD time scales into the µs regime of experiments. 5,6 Even so, it is inappropriate to compare directly tensile strengths from accurate MD simulations against high strain-rate experiments, without considering the drastic differences in spatial scales. Extensive investigations of size effects 7–12 or loading rate effects 13–16 have been reported for various materials and processes, including deformation (deformation twinning, dislocation, indentation) of alloys 7 and metals, 4,12–15 melting 9,11 and crystallization 8 of metals. In particular, both size and rate are involved simultaneously in the nucleation of a daughter phase, but their effects have been investigated separately so that the links between them have not been well addressed. It is essential to develop schemes that enable predictions from one scale to another, in both time and/or space. Cavitation or the formation of bubbles in metastable liquids is the key mechanism for failure of liquids, 17–22 and can be considered as a simple “phase transition” amenable to the precise description by classical nucleation theory (CNT). 23–26 Tensile strengths of liquids under high strain rate loading (spallation) have been examined with experimental, simulation and theoretical approaches, including water, 27,28 glycerol, 29 ethanol, 30,31 metallic melts, 26,32–39 and other liquids. 40–42 However, strain rates and sample sizes can not be set at arbitrary values due to experimental limitations. While MD simulations are capable of revealing cavitation processes on atomistic scale, 26,39–42 only strain rate effect is considered when investigating spall strength. Models such as nucleation and growth 43 are developed to calculate spall strength at different strain rates, and are in agreement with experimental data in some cases. 40 Such a comparison is somehow inconclusive without considering size effects. Here, we choose the Lennard-Jones (LJ) liquid as a general case, and Cu liquid as a specific example representative of metallic liquids, to investigate size and rate effects on tensile strength of liquids in cavitation. Based on CNT, we show the equivalence of temporal

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and spatial scales for tensile strength, which we confirm by MD simulations over a wide range of system sizes and strain rates. Such equivalence allows predictions from the smallest MD scales to be applied to much longer temporal and larger spatial scales, in agreement with independent simulations and experiments. In CNT, 24 the nucleation rate, J, is expressed as ∆G J = J0 exp − , kB T

(1)

where J0 is a prefactor, kB is the Boltzmann constant, and T is temperature. J and J0 refer to per atom per second in our discussion. ∆G is the Gibbs free energy difference driving cavitation, and 4 ∆G = 4πr2 γ + πr3 P. 3

(2)

Here P is pressure (negative for tension), γ is surface tension, and r is the radius of a cavity nucleus. Under isothermal expansion at a constant strain rate ε, ˙

P = P (ε, T ) = P (εt)| ˙ T,

(3)

where ε is volumetric strain and t is time. For constant-rate tension, the probability for a given amount (N atoms) of liquid phase R containing no cavities, is exp −N Jdt . 44 Since dt = dε/ε, ˙ we obtain the probability of cavitation as

Z ⋆ N ε p = 1 − exp − J(ε)dε , ε˙ 0

(4)

where ε⋆ denotes the strain at the onset of failure. The corresponding pressure defines tensile strength, P ⋆ ≡ −P (ε⋆ ). The above equation states that p, ε⋆ and P ⋆ depend on both system size (spatial scale) and strain rate (temporal scale). Since system size and strain rate have

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opposite effects on the probability of cavitation, we introduce a size−rate parameter,

ζ≡

ε˙ . N

(5)

p decreases (increases) with increasing (decreasing) ζ. We consider two different cases, ζ1 and ζ2 . Given p1 = p2 , it follows that 1 ζ1

Z

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1 J(ε)dε = ζ2

Z

ε⋆2

J(ε)dε.

(6)

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If ζ1 = ζ2 , we have ε⋆1 = ε⋆2 , and P1⋆ = P2⋆ . In other words, regardless of the exact values of strain rate and size, the tensile strength of a liquid remains constant if their ratio ζ is fixed. In this regard, spatial scale (in terms of N ) and temporal scale (in terms of ε) ˙ are reciprocal; increasing system size is equivalent to decreasing strain rate proportionally, or vice versa. This equivalence of temporal and spatial scales allows one to predict P ⋆ for other sizes and rates via numerically solving Eq. (6), from that obtained at a specific system size and strain rate. Although its derivation is straightforward, Eq. (6) is rarely exploited as a bridging tool in multiscale studies. To verify the equivalence of spatial and temporal scales in cavitation, and to predict tensile strengths at experimental scales from MD simulations, we choose LJ and Cu liquids as representative examples and perform MD simulations using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). 46 The LJ system is described with characteristic length and energy scales, σ and ε, and the cutoff distance is 3σ. An accurate, thoroughly tested, embedded-atom method potential 45 is used to describe the atomic interactions in Cu. We apply the constant pressure-temperature ensemble to construct the initial liquid configurations, and constant volume-temperature ensemble for cavitation simulations. All simulations are conducted under three-dimensional periodic boundary conditions. The time step for integrating the equation of motion is 4τ and 1 fs in the LJ and Cu simulations, respectively. The initial configurations are equilibrated at zero pressure and a temperature 5



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Figure 1: Pressure evolution during cavitation in liquid Cu via isotropic tension at 1400 K. Insets: snapshots showing void nucleation and growth. Only atoms with coordination number ≤ 5 are plotted. N ≈ 106 atoms, and ε˙ = 1.5 × 108 s−1 . of ε/kB (for LJ simulations; kB is the Boltzmann constant) or 1400 K (for Cu simulations), and then subjected to isotropic tension at a constant strain rate. For each size and strain rate, we carried out 20 independent runs in order to provide sufficient statistics. For each run, we use an independent random number seed in the initial velocity assignment to atoms. During cavitation, void nucleation and growth processes are largely similar for the LJ and Cu systems. Figure 1 shows the pressure evolution during tension for liquid Cu with N ≈ 106 atoms and ε˙ = 1.5 × 108 s−1 , corresponding to ζ = 1.5 × 102 s−1 , along with the snapshots of void nucleation and growth at representative moments of cavitation (insets). The horizontal axis also represents increasing strain as time evolves (ε = εt). ˙ Initially, only randomly distributed subcritical nuclei are expected for homogeneous nucleation, so the critical nucleation events ensue via thermal fluctuations. 26 The formation and rapid growth of supercritical nuclei lead to pressure pullback at maximum tension (530 ps, ε = 0.0795), indicating the onset of liquid failure. The pressure at the maximum tension is taken as the tensile strength of the liquid for this particular system size and strain rate, i.e., P ⋆ ≈ 4.3 GPa. At the pullback, a single void dominates many other supercritical nuclei. However,

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Figure 2: (a) Tensile strength of the LJ liquid for different system sizes (105 − 107 atoms) at different strain rates (4×10−8 −4×10−5 τ −1 ), in the LJ units. (b) Tensile strength of liquid Cu for different system sizes (104 − 108 atoms) at different strain rates (1.5 × 104 − 1.5 × 108 s−1 ). Filled and unfilled symbols refer to simulated and predicted values, respectively. Predictions are made from ζ = 1.5 × 104 s−1 via Eq. (6). The dashed lines in (a) and (b) denote the averages over the MD runs with the same-ζ values. The error bars are calculated based on 20 independent runs. catastrophic growth follows via void coalescence, with the pressure stabilizing approximately at zero after several oscillations showing the complete failure of the liquid. The largest time and length scales achieved in our simulations are 108 τ and 250σ for the

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LJ liquid, and 1 µs and 0.1 µm for liquid Cu. For the strain rates and system sizes explored, ζ varies from 4 × 10−12 to 4 × 10−9 τ −1 for the LJ system, and from 1.5 to 1.5 × 104 s−1 for liquid Cu. The tensile strength for a given system size and strain rate is obtained as exemplified in Fig. 1. The simulation results of P ⋆ as a function of N and ε, ˙ and thus ζ, for the LJ and Cu liquids, are summarized in Fig. 2(a) and 2(b), respectively. The error bars for P ⋆ are calculated from 20 independent statistical runs; the relative errors are small (ranging from 2–5%) as a result of thermal fluctuations. As expected from CNT [Eqs. (4) and (6)], P ⋆ increases with increasing strain rate for the same system size, and it decreases with increasing system size for the same strain rate. And P ⋆ increases with increasing ζ, while it stays approximately unchanged for the same ζ (dashed lines). Therefore, our MD simulations supply direct support to the equivalence of spatial and temporal scales in cavitation of both LJ and Cu liquids. For liquid Cu simulations, equation (6) provides a quantitative way to predict P ⋆ for different ζ(N, ε), ˙ from a known P ⋆ value at a specific N and ε, ˙ using parameters including surface tension from a previous study. 26 Starting from the case of N = 104 atoms and ε˙ = 1.5 × 108 s−1 , i.e., ε⋆1 = 0.096, P1⋆ = 5.11 GPa, and ζ1 = 1.5 × 104 s−1 , we predict unknown P2⋆ at a different system size or a different strain rate (ζ2 ) via integration along the isotherm [Eq. (6)], P (ε, 1400 K), up to ε⋆2 . We first predict P ⋆ at the conditions simulated in Fig. 2(b), and compare the predictions against direct simulations. For fourteen (N, ε) ˙ cases, the predictions (unfilled symbols) are in excellent agreement with direct simulations, which corroborates the predictive capability of Eq. (6) across multiple temporal and spatial scales. Strain rates range from 1 s−1 in low strain rate experiments to 109 s−1 in laser-driven shock experiments, while system sizes vary from 1019 to 1022 atoms in “bulk-scale” experiments. For our MD simulations of Cu, the strain rates vary in the range of 104 − 109 s−1 , while the system sizes, over the range of 104 − 108 atoms. The largest dimensions in our simulations are approximately 110 × 110 × 110 nm3 . The ζ-ranges are 1 − 104 s−1 and 10−20 − 10−10 s−1 in our MD simulations and experiments, respectively. In a typical dynamic cavitation or liquid

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Figure 3: N − ε˙ − P ⋆ colormap of liquid Cu. The dotted lines denote typical iso-ζ lines for experimental (e.g., N = 1022 atoms and ε˙ = 106 s−1 ) and MD scales (e.g., N = 106 atoms and ε˙ = 107 s−1 ). spallation experiment, ε˙ = 105 s−1 and N = 1022 atoms (e.g., 10 × 10 × 1 mm3 ), and thus, ζ = 10−17 s−1 . In a typical MD simulation, ε˙ = 108 s−1 , N = 106 atoms, so ζ = 102 s−1 . The temporal scales in MD simulations and dynamic experiments normally differ only by three orders of magnitude, while the difference in the spatial scales is much more pronounced (by 16 orders of magnitude), and ζ differs by 19 orders of magnitude. It is extremely useful to predict P ⋆ on experimental scales from MD simulations, as well as on the scales in between, as a way of bridging gaps in spatial and temporal scales. We also use Eq. (6) to predict P ⋆ at much larger temporal and spatial scales (smaller ζ) for liquid Cu, again based on the MD simulations at ζ = 1.5 × 104 s−1 . The results are shown in Figs. 3 and 4, where ζ varies by more than 20 orders of magnitude. In the N − ε˙ − P ⋆ colormap (Fig. 3), a linear iso-ζ curve represents an iso-P ⋆ curve (e.g., the dotted lines). This shows the equivalence of an increase in temporal scale (decrease in strain rate) and a proportional increase in spatial scale, or vice versa, i.e., P ⋆ (N, ε) ˙ = P ⋆ (ζ). With decreasing ζ, P ⋆ decreases from over 5 GPa at MD scales (ζ ∼ 10 s−1 ) to less than

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Figure 4: Tensile strength of liquid Cu as a function of ζ (a), of N for different strain rates (numbers, b), and of ε˙ for different system sizes (numbers, c). 3 GPa at experimental scales (ζ ∼ 10−16 s−1 ), as shown in Fig. 4(a). At a fixed strain rate (108 s−1 ), the tensile strength decreases with increasing system size, e.g., by ∼50% when N increases from 105 atoms to 1030 atoms [Fig. 4(b)]. At a fixed system size (N = 1022 atoms), P ⋆ shows an increase from about 2.5 GPa at ε˙ = 1 s−1 to 3.0 GPa at ε˙ = 109 s−1 [Fig. 4(c); experimental scales]. Figure 4(c) also compares the ε-dependences ˙ of tensile strength for different system sizes: the rate of increase in P ⋆ with N is more pronounced for small system sizes. Thus, despite a likely match in strain rates, it is not appropriate to compare directly MD simulations with experiments, without correcting for the size difference between them (16 orders of magnitude). MD results should be lowered for such a comparison. For instance, a tensile strength of 4.3 GPa for a system of N = 106 atoms at 106 s−1 is corrected to 2.6 GPa for N = 1022 atoms at the same strain rate. Small system size in MD simulations leads to an overestimate of Ps when compared with experiments, and this discrepancy is usually only attributed to strain rate differences. Since the difference in size (tens orders of magnitude) is much larger than that in strain rate (several orders of magnitude), size effects outweigh rate effects. Experimental data on tensile strength of liquids are scarce, mostly because of the technical difficulties in experiments. The experiments by Carlson reported P ⋆ = 1.9 GPa for Hg at ε˙ ≈ 106 s−1 . 32 Based on experiments and computations, Agranat et al. 33 found P ⋆ = 2.9 GPa for Al melts at ε˙ ≈ 109 s−1 . Our prediction for bulk Cu liquid at ε˙ ≈ 109 s−1 and

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1400 K is P ⋆ ∼ 3 GPa. While experimental data on Cu liquids are not available for a direct comparison, our predictions appear reasonable by comparison to similar bulk-scale experiments on other metallic liquids. Although only two cases, i.e., cavitation of LJ and Cu liquids, are considered here, the equivalence between temporal and spatial scales is likely common for any physical processes that involve nucleation of a new “phase” and hence should apply to a variety of processes for many materials, including melting and crystallization. 47 Equation (6) is quite general and can be used to make upward or downward extrapolations from one scale to another, either in time or in space. In conclusion, we demonstrate the equivalence between temporal and spatial scales inherent in nucleation events and show the relationship with classical nucleation theory. Based on large-scale MD simulations of cavitation, this equivalence is confirmed for tensile strengths of LJ and Cu liquids, leading to predictions for longer temporal and larger spatial scales that are consistent with independent simulations and comparable to experiments. Such equivalence is expected to hold for a wide range of materials and processes, providing a useful predictive bridging tool in multiscale studies.

Acknowledgments This work is partially supported by the 973 Project (No. 2014CB845904), and NSF (No. 11472253) of China.

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