**4. Discussion**

## *4.1. Thermal Activation Analysis*

Here we further analyze the lattice-trapped delamination rate data and demonstrate that lattice-trapped delamination is thermally activated. For a system containing a crack of length *a* loaded by hydrostatic stress *σH*, the theory of thermally activated crack growth says that the growth rate is [43]

$$\dot{a}\left(\sigma^{H},a\right) = \dot{a}\_{0}\exp\left(\frac{-E\_{a} + A^{\*}K\left(\sigma^{H},a\right)^{2}/E}{k\_{B}T}\right) \tag{1}$$

where *Ea* is the activation energy for bond breaking, *K* is the stress intensity factor, *A*∗ is the activation area, *<sup>E</sup>* is the modulus of elasticity, . *a*<sup>0</sup> is the exponential prefactor related to the attempt frequency, *T* is the absolute temperature, and *kB* is Boltzmann's constant. It is important to note that Equation (1) is only valid when *K* > *Kc*, the critical stress intensity factor. If *K* < *Kc*, then the free energy of the system increases when the crack grows,

violating the second law of thermodynamics [44]. The stress intensity factor can always be written in the form

$$\mathcal{K} = \mathcal{Y}\sigma^H \sqrt{\pi a} \tag{2}$$

where *Y* is a geometric factor dictated by the geometry of the problem. To relate the crack volume to the crack length, we approximate the crack as an ellipsoid with two axes of radius *a* and the other of radius *a*/2 (consistent with cracks observed in our simulations). The volumetric crack growth rate is then . *<sup>V</sup>* <sup>=</sup> <sup>4</sup>*πa*<sup>2</sup> . *a*. In our simulations where latticetrapped delamination occurs, the crack length does not increase significantly (cracks remain relatively small during lattice-trapped delamination, see Figure 2). Hence, for simplicity, we neglect changes in *a* and assume that *K* = *K σH*, *a* and . *<sup>V</sup>* <sup>=</sup> <sup>4</sup>*πa*<sup>2</sup> . *a*, where *a* is the average crack length during the simulation. With this assumption and using Equation (1), we obtain that the activation enthalpy (numerator in the exponential) is

$$
\Delta H\_{\rm d} = E\_{\rm d} - Ck\_{\rm B} \left( \sigma^{\rm H} \right)^{2} \tag{3}
$$

where *C* = *A*∗*Y*2*πa*/(*EkB*) and simple algebraic manipulation further shows that

$$
\ln \dot{V} = \ln \dot{V}\_0 - \frac{E\_d}{k\_B T} + \mathcal{C} \frac{\left(\sigma^H\right)^2}{T} \tag{4}
$$

where . *<sup>V</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>*πa*<sup>2</sup> . *a*0. Hence, if we plot ln . *V* from our simulations as a function of *σ<sup>H</sup>* <sup>2</sup> /*T*, the dataset for each temperature should form a straight line with slope *C* if growth is thermally activated. In Figure 10 we plot the data in this way and see a consistent linear trend across all datasets. Specifically, we find that the same slope fits all datasets, indicating that *C* = 258 K/GPa2. Next, we extract the *y*-intercept from each of these linear fits, and Equation (4) indicates that these intercepts should scale with 1/*T*. Figure 11 plots the *y*intercepts from Figure 10 as a function of 1/*T* and once again a linear behavior is recovered as expected. The slope of Figure <sup>10</sup> is <sup>−</sup>*Ea*/*kB* and the *<sup>y</sup>*-intercept is ln . *V*0; we obtain values of *Ea* <sup>=</sup> 1.37 eV and . *<sup>V</sup>*<sup>0</sup> = 1.23 × <sup>10</sup><sup>9</sup> Å3/ps. The fact that our data so strongly reproduces the behaviors predicted by Equation (4) indicates that the lattice-trapped delamination observed in our MD simulations is indeed thermally activated, and that our neglect of the crack length dependence of *K* does not introduce any significant errors into our analysis.

**Figure 10.** Lattice-trapped delamination rate data from Figure 8 plotted based on the theory of thermally activated crack growth. Dashed lines are linear fits with identical slopes.

**Figure 11.** Arrhenius plot of ordinate intercepts from linear curve fits in Figure 10, showing Arrhenius behavior consistent with the theory of thermally activated crack growth.

The thermally activated delamination mode that we observe is likely governed by the so-called lattice trapping phenomenon [42]. Under lattice trapping, a crack which is loaded supercritically (i.e., with *K* > *Kc*) grows in a step-wise manner as atomic bonds at the crack tip are sequentially broken, often by a kink-pair mechanism [43–46]. As the crack grows, it experiences an oscillating potential energy landscape due to the periodicity of the lattice, and the height of these oscillations dictates the activation energy for growth. To our knowledge this crack growth mode has only been observed in brittle materials like Si [45] and glass [47], but not in ductile materials like Al considered here. This is nonetheless reasonable, since in effect the Al system is acting in a brittle manner in the absence of dislocation activity.

To further analyze our extracted parameters, we need to estimate the stress intensity factor. Tan and Gao [48] numerically determined the stress intensity factor for an axisymmetric interfacial crack on a sphere which forms an angle *φ* from the radial axis, as shown in the inset in Figure 12a, with various modulus ratios *Ep*/*Em* where *m* stands for matrix and *p* for precipitate. We can express their results in the form

$$K\_0 = \mathcal{Y}\left(\phi, E\_p/E\_m\right) \sigma^H \sqrt{\pi R} \tag{5}$$

where *K*<sup>0</sup> = *K*2 *<sup>I</sup>* + *<sup>K</sup>*<sup>2</sup> *I I* is the "overall" stress intensity factor for the mixed-mode loading (the crack is generally mixed mode) and *Y φ*, *Ep*/*Em* was determined by Tan and Gao via numerical boundary integral methods. Using Equation (5) enables us to obtain the activation area as

$$A^\* = \mathbb{C} \frac{k\_B E}{\pi R Y \left(\Phi, E\_p / E\_m\right)^2} \tag{6}$$

where *E* = 2*EmEp*/ *Em* + *Ep* is the bimaterial modulus for interfacial fracture [49]. For approximation purposes, we employ the typical Young's modulus of untextured polycrystals instead of the anisotropic single crystal elastic constants. Estimating *Em* = 70 GPa (experimental value for pure Al) and *Ep* ≈ 120 GPa [50] gives *E* ≈ 88 GPa and *Ep*/*Em* ≈ 1.7. Tan and Gao found that for cracks varying from *φ* = 22.5 to 67.5 degrees with a modulus ratio of *Ep*/*Em* = 2, *Y φ*, *Ep*/*Em* varied from 1.31 to 1.79. Unfortunately, cracks observed in our simulations typically had angles *φ* < 22.5◦, so it is difficult to apply Tan and Gao's solution. Note that according to Equation (5), *Y* → 0 as *φ* → 0 since *K*<sup>0</sup> must go to zero when the crack length is zero (i.e., *φ* = 0). Hence, we expect the *Y* values in our simulations to be less than 1.31. Nonetheless, to gain insight into orders of magnitude for the thermal

activation parameters we assume *Y* ≈ 1.31 in the analysis below. Using these parameter values with *<sup>R</sup>* = 50 Å in Equation (6) leads to *<sup>A</sup>*<sup>∗</sup> ≈ 1.17 *<sup>A</sup>*2. According to Schoeck [43]

$$A^\* = (1 + \beta)\Delta A\tag{7}$$

where Δ*A* is the atomistic area of crack advance between the equilibrium and saddle position of the crack front and *β* is a factor in the range between 0 and 2. Schoeck argued that for crack advance by breaking of individual bonds, <sup>Δ</sup>*<sup>A</sup>* ≈ <sup>1</sup> *<sup>A</sup>*2; hence, our *<sup>A</sup>*<sup>∗</sup> value gives the correct order of magnitude, further bolstering our conclusion that lattice-trapped delamination is thermally activated.

**Figure 12.** Predictions of lattice-trapped delamination for a crack with *φ* = 22.5◦. (**a**) Stress range over which latticetrapped delamination may operate. (**b**) Lattice-trapped delamination rates for particles of radius *R* = 1 μm and 10 μm at three temperatures.

Finally, we note that an important aspect of thermally activated crack growth is the athermal stress intensity factor, *K*ath, at which the activation enthalpy goes to zero. At and above this load, the crack growth rate is no longer governed by thermally activated bond breaking. According to Equation (1) and accounting for the modulus mismatch, the athermal stress intensity factor is

$$K\_{\rm ath} = \sqrt{\frac{E\_a \overline{E}}{A^\*}}.\tag{8}$$

Using our estimates for parameters above, we obtain that *<sup>K</sup>*ath ≈ 1.29 MPa·m1/2. According to Equation (5), *K*ath is reached when

$$
\sigma\_{\text{ath}}^H = \frac{K\_{\text{ath}}}{\Upsilon \sqrt{\pi R}} \tag{9}
$$

and using the estimates above we obtain *σ<sup>H</sup>* ath ≈ 7.8 GPa. This value is consistent with our data since we did not observe lattice-trapped delamination above 7.2 GPa (although we did not attempt to obtain a maximum stress where lattice-trapped delamination rates could be obtained).
