Here, we use molecular dynamics simulations as a tool to investigate vacancy clustering in pure aluminum single crystals. A 1% superconcentration of single vacancies are randomly introduced into an otherwise perfect lattice, and the system is allowed to evolve for 500 ns at an elevated temperature of 728 K. Under these conditions, the individual vacancies rapidly agglomerate into larger clusters to reduce their overall energy. The systems are then subject to mechanical deformation to failure. The results of a total of 35 molecular dynamics simulations are reported. The mechanical behavior of these systems is found to be highly sensitive to the vacancy cluster microstructure, with the largest cluster size being most closely correlated with the cavitation strength. Since the largest cluster size evolves, an interesting time–structure–property coupling governs the behavior of these supersaturated metals. Despite the idealizations of the microstructure and loading conditions, we find a remarkably favorable agreement with laser-driven spall experiments.

## I. INTRODUCTION AND BACKGROUND

The equilibrium vacancy concentration in metals at ambient temperature and pressure is extremely low, e.g., $O(10\u221210)$ for lead (low melting temperature) and $O(10\u221222)$ for copper (high melting temperature). At such low concentrations, their direct effects on the mechanical behavior are fairly negligible. However, supersaturated vacancy concentrations $O(10\u22124)\u2212O(10\u22122)$ can be metastable under extreme conditions, e.g., high temperature (Berger *et al.*, 1979; Adams and Wolfer, 1993), high tensile pressure (Gavini, 2009; Reina *et al.*, 2011), radiation environments (Song *et al.*, 2011; Zinkle and Busby, 2009; Zinkle and Was, 2013; Zhang *et al.*, 2018; Zinkle and Farrell, 1989), or severe plastic deformation (Gray and Huang, 1991; Rose and Berger, 1968; Nancheva and Saarinen, 1986; Kanel, 1998; Nancheva *et al.*, 1986). Such high concentrations of point defects can have significant deleterious effects on the mechanical behavior of metals, e.g., accelerated creep deformation rates, embrittlement, loss of ductility, and some positive effects like improved hardness. Furthermore, under extreme conditions, the vacancies may diffuse and rearrange themselves into lower energy states through the formation of vacancy clusters (Ho *et al.*, 2007; Gavini, 2009). Given that size effects are ubiquitous in the strength of materials, it is reasonable to expect that the mechanical behavior of metals with supersaturated concentrations of vacancies will be somehow correlated with the evolving size and/or spacing distribution of these vacancy clusters.

The vacancy concentration in aluminum subject to a tensile pressure of $4.5$ GPa at room temperature is $C\u223c0.06%$ and increases even further at higher tensile pressures (Reina *et al.*, 2011). This is approximately the same vacancy concentration achieved in aluminum near its melting temperature at ambient pressures. Moshe *et al.* (2000) subjected aluminum foils to short pulses of tensile pressures as large as $8$ GPa via dynamic loading, which will generate extremely large equilibrium vacancy concentrations in excess of 5%, even at room temperature. One of our primary interests here include studying the maximum tensile pressure a material can sustain before unstable failure as a function of vacancy microstructure. This property is termed the cavitation strength (or the spall strength) and is an important property in a range of applications from ballistic performance (cf. Antoun *et al.*, 2003) to tree transpiration (cf. Cochard, 2006; Wang *et al.*, 2017). Contrary to conventional wisdom, Wilkerson and Ramesh (2016a) and earlier Reina *et al.* (2011) analyzed the experiments of Moshe *et al.* (2000) and concluded that vacancy-driven processes are likely to play an important role in failure at such high tensile pressures, even on these extremely short timescales. This claim is further supported by observations of spall fracture surfaces of high-purity aluminum reported in Dalton *et al.* (2011) and Pedrazas *et al.* (2012) and reprinted here in Fig. 1. For samples subjected to tensile pressures in excess of $\u223c3.5$ GPa, Pedrazas *et al.* (2012) found that the majority of fracture sites have no obvious initiation sites and instead fracture seems to spontaneously appear. This is contrary to fracture samples subjected to lower tensile pressures, e.g., $\u22723$ GPa, in which all fracture sites have an obvious associated second-phase particle as their fracture initiation site. This transition was captured in idealized models, assuming pre-exisiting vacancy clusters (Wilkerson, 2014, 2017). Similar observations of seemingly spontaneous void nucleation have been observed ahead of blunt crack tips in high-purity FCC metals (Lyles and Wilsdorf, 1975; Wilsdorf, 1982; Wang and Anderson, 1991) and austenitic stainless steel (Bauer and Wilsdorf, 1973), even at room temperature.

Size effects are ubiquitous in structure–property relationships. Orowan strengthening, (Orowan, 1948; Embury *et al.*, 1989; Nie and Muddle, 1998; Dixit *et al.*, 2008) e.g., in over-aged 2000-, 6000-, and 7000-series aluminum alloys, is governed by the mean spacing of precipitates, which serve as obstacles to dislocation motion. The Taylor hardening law assumes that strain hardening is dictated by the evolving dislocation density (i.e., the material hardens as the mean free-path between forest dislocations decreases) (Taylor, 1938; Bishop and Hill, 1951; Kocks and Mecking, 2003; Ma and Roters, 2004). The Hall–Petch relationship established an empirical relationship between grain size and yield strength (cf. Hall, 1951; Petch, 1953). A similar relationship between grain size and cavitation strength has been recently derived (Wilkerson and Ramesh, 2016b). Solid solution strengthening of the yield strength, e.g., in 3000-series Al-Mn alloys and 5000-series Al–Mg alloys, is typically found to scale with roughly the concentration of point defects raised to the $n$th power, where $n\u223c0.75$ for binary Al–Mg alloys (Cahn and Haasen, 1996) and $n\u223c1$ for commercial 3000-series and 5000-series aluminum alloys (Ryen *et al.*, 2006). Yield strength of nanoporous materials generally scales with both ligament size (Dou and Derby, 2009; El-Awady, 2015; Wilkerson, 2019) and relative density (Ashby and Gibson, 1997). In fracture mechanics, the fracture strength of brittle materials scales roughly with the inverse square root of the largest flaw size (Griffith, 1921). In all these examples, the smaller size of volume defects and/or smaller spacing between defects generally induces a higher strength.

Here, we report a systematic study to demonstrate that the mechanical behavior of aluminum is primarily sensitive to the size of these vacancy clusters. This diffusion-driven vacancy clustering process is time-dependent, and for aluminum with $C=1%$ at $T=728$ K is shown to be fast enough for significant degradation of the cavitation strength over nanoseconds of clustering time. This paper is organized as follows. Section II provides an overview of the modeling approach, which makes use of the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) molecular dynamics (MD) package. Section III discusses results and analysis of the molecular dynamics simulations. Finally, Sec. IV provides a brief summary and conclusions.

## II. PARTICULARS OF MOLECULAR DYNAMICS SIMULATIONS

In the present work, molecular dynamics (MD) simulations are performed using the open source package LAMMPS (Plimpton, 1995). The model system under consideration is a face centered cubic (FCC) aluminum (Al) single crystal containing a $C=1%$ concentration of monovacancies. Our MD system is held fixed at a constant temperature of $T=728$ K chosen to be consistent with related lattice kinetic Monte Carlo (LKMC) simulations on a similar system (Reina *et al.*, 2011). The system is subjected to either a prescribed pressure $(P)$ or volume $(V)$. One particularly powerful advantage of studying these systems via MD rather than LKMC is that mechanical behavior may be analyzed with ease in MD, whereas LKMC is limited to defect evolution. That said, LKMC has significant advantages over MD in the study of large systems over long timescales. In order to study the mechanical behavior of these materials as a function of evolved vacancy cluster structure, we deform the system to failure following a desired clustering time ($\Delta tc$). The effect of clustering time on the mechanical behavior and cavitation strength is reported. A total of 35 MD simulations are carried out. To our knowledge, this is the first study of its kind to systematically study nonequilibrium structure–property relationships in metals with nonequilibrium concentrations of vacancies.

### A. Interatomic potential and ensembles utilized

The atomic interactions are calculated using the Finnis/Sinclair embedded atom method (EAM) (Mendelev *et al.*, 2008). It is a well developed potential for studying the behavior of Al and accurately captures experimentally measured elastic constants and vacancy formation energy of Al. The initial material defect structure is prepared according to the schematic shown in Fig. 2. First, a perfect Al system consisting of 108 000 atoms in a cubic periodic cell (with a side length of 12.3 nm) is equilibrated at the desired temperature with the pressure held constant at ambient conditions ($P=0$ GPa) by a Nosé–Hoover isothermal-isobaric (NPT) barostat. Once the system comes to equilibrium, a prescribed number of atoms are randomly removed from the system to produce a structure with the desired supersaturated concentration of vacancies. After removing the atoms, the ensemble is switched to an isothermal-isochroic (NVT) canonical ensemble in order to maintain a constant volume of the periodic cell. A Laplace (tensile) pressure is generated by the creation of vacancies. To study the kinetics of the system in driving toward its lowest energy state, we allow the system to evolve under NVT conditions for a desired clustering time. Following clustering for a prescribed clustering time, triaxial extension deformation is carried out at a constant strain rate of $108$ s$\u22121$ via an NVT ensemble. This particular strain rate is motivated by a suite of laser-driven spall experiments on aluminum carried out at similar strain rates (Moshe *et al.*, 2000). These spall experiments identified an abrupt transition in strain rate-sensitivity at a strain rate of $\u223c108$ s$\u22121$, cf. Fig. 3, which could be indicative of a transition in the underlying failure mechanism.

### B. Identification and visualization of vacancy clusters

The random distribution of vacancies is a nonequilibrium state that is far from the lowest energy state for the system. A lower system energy is achievable through the formation of vacancy clusters. Here, a vacancy cluster is defined as a single volume defect composed of a closed set of $\u2113$ vacancies with nearest neighbor connectivity. We denote a vacancy cluster of size $\u2113$ as a cluster containing $\u2113$ vacancies, and we utilize $\u2113^$ to denote the largest cluster size at a given time. Note that this particular definition of clusters does not take into account the geometry of the clusters.

To identify vacancies in simulation cells, the geometric method of Wigner–Seitz cells (Nordlund *et al.*, 1998) is used. In this method, the atom positions in a simulation cell are analyzed with respect to a lattice defined by undisturbed simulation cell regions. A lattice site which includes an empty Wigner–Seitz cell is labeled a vacancy. Likewise, a Wigner–Seitz cell with multiple atoms is labeled an interstitial. After identifying specific vacancies, the system can be decomposed into disconnected sets of vacancies (clusters of vacancies) based on a distance criterion. The cut-off distance for vacancy clustering here is taken to be 2.86 Å, which is the first neighbor distance of pure aluminum in the simulations. The samples are compared with a reference configuration, which is taken here to be the state of the system after NPT ensemble at the particular temperature and pressure being analyzed. To calculate the vacancy evolution, the state of the system is taken to be characterized by the spatial distribution of vacancies on a frozen lattice. The visualization of MD simulation results is performed with OVITO (Stukowski, 2010).

## III. RESULTS AND THEORETICAL ANALYSIS

The present section provides results of the microstructure evolution of aluminum with a supersaturated concentration of vacancies as predicted by molecular dynamics calculations. Since this particular material system is in a nonequilibrium state, the kinetics of microstructure, e.g., vacancy cluster evolution is of particular interest and is the focus of Subsection III A. Additionally, the effect of these evolving nonequilibrium microstructures on the mechanical response of the system is investigated in Subsection III B. Finally, Subsection III C provides some implications of this diffusion-failure coupled process for the structure–property modeling of the mechanical behavior and cavitation strength under triaxial loading.

### A. Microstructure evolution via vacancy aggregation

Once the material system is seeded with the prescribed number of vacancies, the system begins to evolve toward a more energetically favorable state. In a real material system with a vast hierarchy of points, lines, and surface defects, the material may lower its overall energy by (i) reducing the overall vacancy concentration via recombination with interstitials and vacancy annihilation at vacancy sinks, e.g., dislocations and grain boundaries; (ii) aggregating vacancies to form vacancy clusters via diffusion; or (iii) a combination of both mechanisms. However, our particular simulations contain no vacancy sinks or interstitials; as such, mechanism (i) is inoperative in our molecular dynamics simulations, and the system is forced to evolve toward lower energy states via mechanism (ii) exclusively. Again, although this is not necessarily the physical situation for most applications, it is very useful for understanding the influence of clustering on the system in the absence of other competing mechanisms.

Figure 4 shows an example of the kinetic evolution for an aluminum system with $1%$ vacancy concentration subject to ambient pressure and a constant temperature of 728 K. The evolution of the microstructure is remarkably fast. In this particular simulation, relatively large vacancy clusters composed of as many as 12 aggregated single vacancies are formed within about 5 ns of clustering time. The clusters continue to grow with time. For example, the largest clusters reach a size of 58 in 100 ns and 109 in 500 ns. Even after 500 ns, the clusters have not yet reach an equilibrium state. This is not particularly surprising given that the LKMC simulations of Reina *et al.* (2011) revealed that similar systems continued to evolve even after 100 $\mu $s. The clustering kinetics are driven by the energetics, and in this particular simulation shown, the overall energy of the system reduces by 500 eV in evolving from the randomly seeded structure of monovacancies ($\Delta tc=0$) to the defect structure shown in Fig. 4 at $\Delta tc=500$ ns. This energy relaxation is primarily associated with a reduction in the elastic strain energy (or bond energies), which reduces by approximately 0.14% in the first 500 ns after vacancy generation.

In studying Fig. 4, some interesting questions arise pertaining to the spatial correlation between the initial conditions and the evolving structure. One might assume that the spatial location of the randomly generated trivacancies (the largest clusters at $t=0$) will be the likely location of the largest clusters at later times. However, this does not necessarily seem to be the case. Perhaps, because the trivacancies themselves are somewhat mobile. In addition, it may be seen that even rather large clusters $(>20)$ are absent from spatial locations that they previously occupied. In this case, it is less likely that such large clusters are mobile and more likely that these clusters are being cannibalized by even larger ones.

Further insight may be gleaned from statistically quantifying the kinetic evolution of each cluster size as shown by the histogram in Fig. 5. The histogram shows the fraction of vacancies belonging to each cluster size as a function of time. Initially, nearly $100%$ of the vacancies are single vacancies; however, this fraction very rapidly drops down to only about $15%$ of the vacancies with the remaining $85%$ of vacancies belonging to clusters. Another interesting observation from this analysis is a relative dearth of clusters in the size range of $\u2113=$3$\u2212$10 for all time after about 10 ns. After this time, very few $(<5%)$ of vacancies belong to this intermediate size range, with about $80%$ of the vacancies belonging to clusters larger than 10. In addition, after about 500 ns, most of the clusters in the size of $\u2113=$3$\u2212$40 are cannibalized by the larger clusters. This seems to indicate that the intermediate size clusters are the least energetically favorable. Perhaps, this phenomenon is associated with the large Laplace pressure (and associated strain energy penalty) generated by such intermediate clusters. Due to the constant volume constraint of the NVT ensemble, a tensile Laplace pressure is induced by the creation of the vacancies. The induced Laplace (tensile) pressure is $\u223c0.25$ GPa in magnitude.

Closer inspection of Fig. 5 shows that the percentage of monovacancies rapidly drops in the first 10 ns of clustering time, from around 100% of total vacancies to around 50% and this decrease continues rapidly to about 15% in 100 ns of clustering time. This is also reflected in the evolution of the number of discrete clusters (including monovacancies) as shown in Fig. 6. In Fig. 5, the percentage of monovacancies continues to slowly decrease to around 8% after 580 ns of clustering time. It is interesting to note that although a high percentage of monovacancies (around 500) cluster together in the first 10 ns of clustering time, it takes about 400 ns for a cluster as large as 100 to be grown. A possible explanation for this might be that the mobility of vacancy clusters likely decreases with their size.

As a practical matter, for vacancy clustering to play any role in real applications involving a supersaturated vacancy concentration, the timescales for vacancy clustering must be faster than the time required for vacancies to diffuse to vacancy sinks, e.g., dislocation cores and grain boundaries, or to recombine with interstitials. Otherwise, the supersaturated vacancy concentration will drop back down toward the equilibrium concentration before any significant clustering takes place. Moreover, some applications in which vacancy clustering is speculated to play a role, e.g., shock and spall failure (Reina *et al.*, 2011; Wilkerson and Ramesh, 2016a), involve very short timescales $O$(1 ns–1 $\mu $s). As such, the kinetics of vacancy clustering must be relatively fast for clustering to play any meaningful role in such applications. At least for this case of extremely high vacancy concentrations and fairly high temperatures, the results shown in Figs. 4 and 5 seem to suggest that vacancy clustering is indeed fast enough to play a role in shock loading applications.

As noted, these particular simulation results (shown in Figs. 4 and 5) represent a fairly high temperature relative to the melting temperature of perfect aluminum (933 K). Additionally, the vacancy concentration is likely one- to two-orders of magnitude higher than what is expected to be achievable in any of the motivating applications discussed earlier in Sec. I. For comparison, the equilibrium vacancy concentration of aluminum at 728 K is $\u223c10\u22124$. Nevertheless, we argue that the insights that may be gleaned from such molecular dynamics calculations are quite valuable in the development of continuum-level constitutive models, as will be discussed further in Sec. III C.

### B. Mechanical behavior and structure–property relationships

Thus far, this paper has focused on the kinetics of vacancy clustering in otherwise defect-free pure Al with a supersaturated concentration of vacancies. We have reported that the clustering kinetics are quite fast, even on nanosecond timescales. This section will discuss how vacancy clusters of various sizes and spacing distributions affect the mechanical behavior of these systems. In particular, it will be shown that the rearrangement of randomly seeded monovacancies (and divacancies, trivacancies, etc.) into lower energy states composed of large vacancy clusters weakens the cavitation strength of these systems. Thus, since this vacancy clustering process is time-dependent, it must follow that the strength is also time-dependent, provided the microstructure and strength are correlated. To study this time evolving microstructure–strength relationship, after a prescribed amount of clustering time to allow a particular degree of microstructural evolution, the MD samples are then deformed under triaxial (volumetric) tensile loading at a constant engineering strain rate of $\epsilon \u02d9=108$ s$\u22121$.

Figure 7 shows the true mean stress ($\Sigma m\u225c\u2212P$) vs longitudinal engineering strain ($\epsilon $) with response of the FCC Al samples for an initial vacancy concentration of $C=1%$ at a constant temperature of $T=728$ K as a function of clustering time $\Delta tc$. In all cases, the stress starts at a value higher than zero (an initial tensile stress), then increases nonlinearly with deformation to a peak tensile (cavitation) strength, followed by a complex softening response. The nonlinear response is essentially pure elastic until right before the peak stress is achieved. This general nonlinear elastic response may be captured by an appropriate equation of state that accounts for the nonlinear relationship between pressure and volume at high elastic strains, e.g., the equation of state (Mie, 1903; Grüneisen, 1912). The stiffness (nonlinear bulk modulus) is independent of the clustering time, i.e., independent of the size distribution of the vacancy clusters at a particular vacancy concentration.

On the other hand, the peak tensile (cavitation) strength, denoted here as $\Sigma m\u2217$, is quite sensitive to the clustering time, as shown in the inset of Fig. 7. For example, for $\Delta tc=0$ ns (immediately after vacancy generation and without any clustering time), the tensile cavitation strength is about 4.2 GPa at an engineering strain of about 2.8%. After $\Delta tc=500$ ns of clustering time, the tensile cavitation strength is about 3.5 GPa at 2.4% engineering strain. It is noteworthy that this cavitation strength agrees remarkably well with the laser-driven spall strength experiments carried out by Moshe *et al.* (2000) on aluminum at a similar strain rate. It is quite impressive that a diffusion-driven $\u223c17%$ (700 MPa) weakening of the tensile cavitation strength is achieved in less than a microsecond for aluminum with $C=1%$ at $T=728$ K. It should be noted that this $\u223c17%$ (700 MPa) weakening of the tensile cavitation strength cannot be explained by the initial (zero strain) tensile (Laplace) pressure alone, i.e., $\Sigma m(\epsilon =0)$, which only reduces by $150$ MPa from $\Delta tc=0$ ns to $\Delta tc=100$ ns. In other words, even if the initial tensile stresses in Fig. 7 are zeroed out, there will still be a 550 MPa weakening of the cavitation strength.

This observed drop in the strength of samples by increasing the clustering time is obviously associated with the change in microstructure, i.e., creation of large vacancy clusters at the expense of monovacancy populations. In examining the dislocation nucleation process (via OVITO), we find that dislocations tend to most favorably nucleate at the largest vacancy clusters, i.e., the largest flaw theory as gleaned by Griffith (1921) nearly 100 years ago. More recently, Lubarda *et al.* (2004) derived a theory for dislocations nucleating from a nanovoid surface, which was later expanded upon by Wilkerson and Ramesh (2016a). Their models predict that the macroscopic stress required for heterogeneous dislocation nucleation at the surface of a nanovoid is a function of the size of said nanovoid. Generally speaking, the nucleation stress reduces nonlinearly with increasing size of the nanovoid and is generally easier than homogeneously nucleating dislocations within the perfect lattice. This size effect is primarily driven by size effects associated with surface tension, dislocation image stresses, length scales associated with stress field perturbations around the nanovoid, and the hop distance of dislocation nuclei under thermal excitation, cf. Lubarda *et al.* (2004) and Wilkerson and Ramesh (2016a) for details. Once dislocations are nucleated, the nanovoids rapidly grow and drive a sudden (nearly brittle) relaxation of the macroscopic stress. From our MD simulations here, it seems that this process is (to first order) essentially the same for a material with a complex distribution of vacancy clusters that have agglomerated via diffusion processes. Section III C will quantify these relations in greater detail.

Following the above discussion, it is now apparent that the clustering time–strength relationship is essentially a size effect that would be analogous to the Hall–Petch size effects for a microstructure experiencing dynamic grain growth. As such, the correlation of interest is between the shape of the largest cluster size evolution vs clustering time (which can be seen in Fig. 5 by tracing out the extreme value of the histogram for each time, which is shown in the inset of Fig. 5) with the general shape of the cavitation strength vs clustering time shown in the inset of Fig. 7. In comparing these two curves, it is interesting to note that most of the diffusion-driven weakening is achieved in the first 100 ns of clustering, with a fairly small change over the next 400 ns of clustering. That said, Fig. 5 demonstrates that there is still substantial evolution of vacancy cluster size distribution during this interval from $\Delta tc=100$ ns to $\Delta tc=500$ ns. If we assume, as argued above, that this evolving structure–strength relationship is fundamentally governed by a vacancy cluster size effect, then we can conclude that the size effects are strongest for very small cluster sizes, e.g., $\u2113\u223c1\u2212$30, and the size effect is not as strong as the clusters continue to grow, which comports with the models of Lubarda *et al.* (2004) and Wilkerson and Ramesh (2016a).

It should be noted that our simulations of clustering kinetics are not truly subject to ambient pressure conditions due to the tensile Laplace pressures generated by the creation of vacancies following NPT equilibration. As the vacancies cluster into larger sizes, this tensile stress relaxes. For example, the Laplace pressure prior to any vacancy clustering ($\Delta tc=0$) is about 0.25 GPa. After $\Delta tc=500$ ns of clustering time, the Laplace pressure has relaxed to about 0.1 GPa. This is to be expected as the Laplace pressure scales with the inverse of the void radius, i.e., $\gamma /a,$ with $\gamma $ denoting the surface energy. Recall from Fig. 5 that the largest vacancy cluster (denoted here as $\u2113^$) grows from an initial size of $\u2113^=3$ to $\u2113^\u223c100$ after $\Delta tc=500$ ns of clustering time for $C=1%$ at $T=728$ K. Hence, the initial Laplace pressure is fairly sensitive to the size distribution of vacancy clusters, as expected. Since vacancy mobility increases with tensile pressure (Ho *et al.*, 2007), it is anticipated that this Laplace pressure plays some role in slightly accelerating the vacancy clustering process from what it would otherwise be under truly ambient conditions. That said, the Laplace pressure is a physical phenomena that would occur in real systems, so it should not be neglected.

Our overall assessment thus far is that the rearrangement of monovacancies into larger vacancy clusters generally weakens the cavitation strength (and associated failure strain), relaxes the Laplace pressure, but does not affect the general elastic response (as expected). Any trends in the post-peak stress softening response are fairly difficult to discern in Fig. 7.

### C. Implications for reduced-order structure–property modeling

Having discussed how the evolving vacancy microstructure affects mechanical behavior, the final technical piece of this paper addresses implications of our findings for reduced-order continuum models. The fundamental questions we aim to address here:

Does the largest cluster size dominate the cavitation strength of these supersaturated FCC metals?

Is it possible to adequately predict the cavitation strength of these systems by tracking the largest cluster size?

Or is it necessary to track the full statistics of the vacancy cluster size distributions?

To help answer these questions, we study an idealized set of microstructures with a single (periodic) spherical void as shown in Fig. 8 (on the right). These idealized microstructures are constructed such that the volume of the single void is identical to the volume of the largest vacancy cluster, i.e., $\u2113^\Delta \Omega v$, in the random clustering simulation under comparison, shown in Fig. 8 (on the left). Next, the simulation box size of the idealized microstructure is chosen such that the void volume fraction is identical to the vacancy concentration in the random clustering simulation, i.e., $C=1%=\u2113^\Delta \Omega v/w\u20323$. We may deduce that $\u2113^$ is the most important parameter governing the cavitation strength, if the cavitation strength of the random clustering microstructure agrees reasonably well with that of the idealized microstructure. In other words, that the details of the vacancy cluster size distribution are secondary in comparison to $\u2113^$, since the idealized microstructure has a Dirac delta size distribution, and the random vacancy cluster has a complex (evolving) size distribution as shown in Fig. 5. To compare the cavitation strength of the random clustering microstructure and the idealized microstructure, we generated several idealized microstructures and subjected them to the same loading conditions, i.e., triaxial expansion loading at a constant strain rate of $108$ s$\u22121$. Representative stress–strain curves for the idealized microstructure are shown in Fig. 9.

In Fig. 10, the cavitation strength is plotted as a function of $\u2113^$, equivalent to the volume of the largest vacancy cluster. The crosses correspond to the cavitation strength of the random vacancy clustering microstructures (the peak stresses in Fig. 7). The closed circles correspond to the cavitation strength of the idealized (single periodic void) microstructures (the peak tensile stresses in Fig. 9). Since the largest vacancy clusters may grow during the mechanical loading phase, the cluster sizes are extracted at a tensile longitudinal strain of $\epsilon =0.02.$

By comparing the crosses and closed circles in regions of overlap ($20\u2272\u2113^\u2272110$) in Fig. 10, we see that the idealized microstructures are only slightly weaker than the random vacancy clustering microstructure. Overall, the trends are in strong agreement with a similar size dependence in both microstructures. Therefore, we conclude that this is strong evidence that $\u2113^$ is adequate to capture the dominate trends in the cavitation strength. Thus, we can safely state that the details of the vacancy cluster size distribution are secondary when it comes to the cavitation strength. This finding has important implications for developing atomistically informed constitutive models of the mechanical behavior of these supersaturated materials, given that tracking full size distribution statistics is typically fairly computationally expensive.

## IV. SUMMARY AND CONCLUSIONS

Here, we have utilized molecular dynamics simulations to study vacancy clustering processes in pure aluminum single crystals. A superconcentration of single vacancies are randomly introduced into an otherwise perfect lattice, and the system is allowed to evolve for 500 ns under (nearly) adiabatic pressure. The individual vacancies will rapidly agglomerate into larger clusters to reduce the system’s overall energy.

Furthermore, the mechanical behavior of these systems is found to be strongly dependent on the microstructure, e.g., the manner in which the randomly seeded monovacancies rearrange themselves into lower energy states composed of distributions of vacancy clusters. Thus, since this vacancy clustering process is time-dependent, it must follow that the mechanical behavior is also time-dependent. To study these nonequilibrium microstructure–property relationships, after a prescribed amount of clustering time to allow a particular degree of microstructural evolution, the MD samples were then deformed under triaxial (volumetric) tensile loading at a constant engineering strain rate of $\epsilon \u02d9=108$ s$\u22121$. Some of the key findings are summarized below:

The timescales of clustering are sufficiently fast to be applicable even under shock loading conditions.

An initial Laplace tensile pressure is generated by the creation of vacancies, which relaxes as clusters grow.

The nonlinear elastic response is fairly insensitive to the arrangement of vacancies into clusters.

The cavitation strength is found to be strongly sensitive to the arrangement of vacancies into clusters.

The relationship between microstructure and the post-peak stress softening response is indiscernible.

Cavitation strength is most sensitive to the largest cluster size, with the details of the size distribution playing a secondary role.

Our MD predicted cavitation strength after $\Delta tc\u2273100$ ns of clustering time agrees remarkably well with the laser-driven spall strength measurements reported in Moshe

*et al.*(2000).

## ACKNOWLEDGMENTS

This material is based upon work supported by the Army Research Laboratory under the MEDE Collaborative Research Alliance through Cooperative Agreement No. W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. This work was supported by the TACC Computational Resource Center through the use of its high performance computing facilities. Portions of this research were conducted with high performance research computing resources provided by the Texas A*&*M University (https://hprc.tamu.edu).

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