**1. Introduction**

Void nucleation is the first step towards fracture in many different contexts, including quasi-static tearing, dynamic spall, creep rupture, irradiation creep, and wear debris generation. Voids are predominantly thought to nucleate at second phase particles, either when the particles crack or when the interface between the particle and matrix debonds [1]. Here, we focus on the earliest stages of void nucleation via particle delamination. Subsequent to nucleation, these voids grow until they induce fracture. While many studies have employed continuum models, such as finite element modeling [2–5], to evaluate the process of void nucleation, the atomistic mechanisms governing nucleation are less well studied. And yet, since void nucleation begins at the nanoscale, it is intrinsically an atomistic process [6]. Our goal in this work is to evaluate void nucleation in a model system with an incoherent, second-phase particle (*θ*-particle in Al) in an effort to reveal the underlying micromechanics and kinetically limiting processes.

Given its central role in fracture, continuum damage models commonly invoke void nucleation in their underlying formalisms. Perhaps the most popular approach is the porous plasticity model of Gurson, Tvergaard, and Needleman [7] which utilizes a yield

**Citation:** Zhao, Q.Q.; Boyce, B.L.; Sills, R.B. Micromechanics of Void Nucleation and Early Growth at Incoherent Precipitates: Lattice-Trapped and Dislocation-Mediated Delamination Modes. *Crystals* **2021**, *11*, 45. https://doi.org/10.3390/cryst11010045

Received: 21 November 2020 Accepted: 2 January 2021 Published: 7 January 2021

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criterion that is a function of the void volume fraction, *f* . In this model, the void volume fraction is governed by a differential equation of the form:

$$
\dot{f} = \dot{f}\_{\text{max}} + \dot{f}\_{\text{growth}}
$$

where . *<sup>f</sup>* nuc and . *f* growth account for the contributions from nucleation and growth events, respectively. Ideally, the term . *f* nuc would be derived from a fundamental, micromechanicsbased understanding of void nucleation. Given that this understanding is lacking, a more phenomenological approach is commonly utilized. For example, it is commonly assumed that void nucleation occurs at particles when a critical plastic strain, *ε<sup>P</sup> <sup>c</sup>* , is reached, and that *ε<sup>P</sup> <sup>c</sup>* varies from particle to particle according to a probability density function *F εP*/*ε<sup>P</sup> c* where *ε<sup>P</sup>* is the equivalent plastic strain [8,9]. The void volume fraction then increases in time as a result of void nucleation according to the expression:

$$\dot{f}\_{\text{nuc}} = F\left(\varepsilon^P / \varepsilon\_c^P\right) \dot{\varepsilon}^P$$

where . *ε <sup>P</sup>* is the equivalent plastic strain rate. Usually, *F εP*/*ε<sup>P</sup> c* is assumed to be a normal distribution with mean *ε<sup>P</sup> <sup>c</sup>* and standard deviation *sε<sup>c</sup>* [8], however there is no direct evidence which justifies this choice. Furthermore, *ε<sup>P</sup> <sup>c</sup>* and *sε<sup>c</sup>* are treated as empirical parameters that are fitted against experimental data (e.g., stress-strain curves). While this and other similar phenomenological approaches have been applied pervasively across the literature, the Sandia Fracture Challenges have recently shown that these models often fare poorly when making blind fracture predictions [10–12]. This motivates a deeper look at the micromechanics of void nucleation, so that strong assumptions about what governs fracture (a critical strain?) and how the propensity for fracture varies across the population of particles (normally distributed?) can be lifted.

Unfortunately, the critical strain *ε<sup>P</sup> <sup>c</sup>* is not easily studied using micromechanical simulations because plastic strain is really a homogenized, macroscale concept; at the microscale where discrete dislocations interact with particles, plastic strain is not a very relevant concept. On the other hand, some damage mechanics models employ a critical stress *σ<sup>c</sup>* at which nucleation occurs [9], which is more consistent with micromechanics modeling (the stress state can be specified in molecular dynamics, for example). Here, we argue, however, that rather than focusing on a "critical stress" at which void nucleation occurs, it makes more sense to consider how the *nucleation rate* varies with stress state, temperature, etc. In other words, void nucleation can occur over a range of stresses, with the nucleation rate increasing as the stress is increased. This view is more consistent with other works on crack nucleation, which focus on the nucleation rate [13,14]. Within this view, the critical stress is the stress at which the nucleation rate goes to infinity, meaning that nucleation occurs instantaneously. We note that the nucleation rate for a given state is likely only welldefined in an average sense, because nucleation is a stochastic phenomenon. This means that at each state, there is a distribution of nucleation rates (which could be interpreted in probabilistic terms). We argue that the possibility of "subcritical" nucleation, i.e., with *σ* < *σc*, and the statistical aspects of nucleation could be important to the overall nucleation process. For these reasons, our focus here is on the stress and temperature dependence of the void nucleation rate.

An important nuance to the study of void nucleation is deciding when exactly a void is said to "nucleate." As soon as a crack appears within the particle-matrix interface? Or after a significant fraction of the interface has delaminated? We may expect that a clear nucleation event occurs whereby a crack "suddenly" appears along the interface, allowing us to disentangle this terminological ambiguity, although the appearance of a crack is often limited by the spatial and temporal resolution of the techniques employed. In the present approach, with atomic-scale and picosecond resolution, the initial emergence of a crack is still difficult to define: we observed steady growth of an interfacial crack starting

from a vacancy-sized nucleus. We were unable to determine the precise mechanism by which the vacancy-sized nucleus appeared, however. Furthermore, the appearance of the vacancy-sized nucleus did not control the kinetics of void nucleation. Instead, we found that it was the subsequent growth of the crack that governed the overall delamination (e.g., void nucleation) process. Hence, we find that it is the delamination rate, controlled by the growth of a crack, which governs the void nucleation rate. For this reason, we refer to our simulations as studying void nucleation and "early growth."

Void nucleation has been studied in perfect crystals [15–22], at grain boundaries [23,24], ahead of crack tips [25], and at second-phase particles [26–29] using molecular dynamics. In most cases, void nucleation results from interactions between several crystallographic defects, such as grain boundaries and twins/dislocations [23,24], pairs of intersecting stacking faults [21], and particles and dislocations [26,27,30]. The previous work on particlemediated void nucleation is most relevant here. Coffman et al. [31] studied void nucleation in Si under uniaxial tension with a cubic nanograin "particle" that delaminated from the surrounding matrix. They first performed atomistic simulations to calibrate a continuum fracture model (a cohesive zone model), and then compared atomistic and continuum predictions of void nucleation. In general, they found poor agreement between the models, motivating the need for further studies of void nucleation with atomistic resolution. Pogorelko and Mayer [26–29] and Cui and Chen [30] studied void nucleation at spherical particles in a variety of material systems, considering the influence of strain rate, temperature, simulation box size, and particle volume fraction on the delamination behavior under a fixed uniaxial strain rate. In simulations with face-centered cubic (FCC) matrices [26,27,29,30], nucleation was observed to occur in two stages: first a crack nucleated at the top and bottom poles along the loading axis (similar to the behavior predicted by continuum models [2]), and then after some subsequent growth dislocations were emitted from the crack tips. On the other hand, nucleation with body-centered cubic and hexagonal close-packed (HCP) matrices seemed to initiate from defects in the matrix rather than at the particle-matrix interface [28]. While the tensile strength of these systems has been characterized extensively using these simulation results, the nucleation rate could not be estimated because of the fixed-strain-rate boundary conditions.

Our study here had two goals: (1) to assess the stress and temperature dependence of the void nucleation and early growth rate with MD and (2) identify the micromechanical processes which govern the kinetics. In contrast to previous work [26–30], we perform simulations here with a fixed stress state (and temperature), so that our results can be used to estimate the stress and temperature-dependent rates. Our findings indicate that void nucleation may be rate limited by the kinetics of crack growth processes rather than the kinetics of crack nucleation (e.g., the time it takes for a crack to appear). Furthermore, we show two distinct delamination modes with drastically different growth kinetics. Finally, we conclude that a brittle delamination mode which we term *lattice-trapped delamination* may be an important contributor to void nucleation. While the stress range employed here (5.7 to 7.2 GPa) is high relative to quasi-static loading, it is in the range where shock spallation is observed under high loading rates [32,33]. Furthermore, through a thermal activation analysis of our data we are able to extrapolate our results down to lower stress conditions. The remainder of the work is organized as follows. In Section 2 we discuss our simulation setup and analysis methods, in Section 3 we present our results, in Section 4 we discuss the implications of our findings and compare results with existing theories, and finally conclude the manuscript in Section 5.
