We begin the discussion by formulating a qualitative picture that explains the trends observed above, obtaining a description of the depinning process that is very similar to that of (Patinet et al. 2011). First, we focus on the behavior prior to the depinning transition (*σ*<*σ*_{c}). Consider a periodic dislocation line of length *L* under applied shear stress *σ* and temperature *T*. At every position along the dislocation, the line may glide forward under the applied stress. Since the solid solution contains many pinning points (solute atoms), the line will tend to locally bow out between a set of “dominant” pinning points. We refer to the distance between these dominant pinning points as the *bowing length*, *l*_{b}, and the distance that the line bows out as the bowing amplitude. For dilute solid solutions (strong pinning), this interpretation is exact since solutes are encountered one-by-one. With a dense solid solution (weak pinning) as analyzed here, the story is more complicated given that the solutes interact with the dislocation line over the entire bowing length. None-the-less, we believe that the basic picture still holds in both cases.

At every point along the line, the roughness is controlled by the balance between the line tension which acts to straighten the line, the applied stress which acts to bow out the line, and the solute forces which act to induce a random (but correlated) line geometry. When the bowing length is small the dislocation line interacts with relatively few solutes, making it easier for the line to bow out coherently since the probability of encountering a “strong” interaction is lower. As a result, the line is able to bow out at small bowing lengths, which is the source of the roughness. This roughness (bowing amplitude) increases with bowing length, as predicted by Eq. (3), because the line is able to bow out more when the bowing length is longer. This gives rise to correlated roughness with 0.5<*H*_{1}<1. As the bowing length *l*_{b} increases, the probability of encountering a “strong” interaction increases, making it increasingly difficult for coherent bow outs of length *l*_{b} to form; the strong interactions “break up” long bowing lengths. Eventually, it becomes nearly impossible for such bow outs to form, causing the line roughness to become uncorrelated (or even anti-correlated) beyond a certain bowing length with 0<*H*_{2}<0.5. This is precisely what the correlation length *ξ* represents: 2*ξ* is the maximum length scale at which coherent bow outs can form (the factor of two accounts for the fact that roughness is measured in the middle of a bow out). A bow out with a bowing length above 2*ξ* is broken into smaller bowing lengths by the solute field. When *σ*=*σ*_{c}, the stress is high enough that the applied stress is able to overcome the solute pinning forces at all length scales, causing *ξ*→*∞*. Hence, bow outs at any length scale are able to form at the depinning transition.

At finite temperature the picture becomes complicated by the fact that the pinning induced by solutes is no longer deterministic, since the line may overcome a set of solute obstacles with the aid of random thermal fluctuations: motion of the dislocation line becomes thermally activated. None-the-less, the basic picture still holds, as our data show. The correlation lengths *ξ* and Hurst exponents *H*_{1} and *H*_{2} follow identical trends at *T*=5, 100, and 200 K, albeit with noisier data at higher temperatures. Interestingly, the variation of *ξ* with stress is shown to be temperature-dependent. For example, at *σ*=50 MPa we find that *ξ*=327, 363, and 439 Å when *T*=5, 100, and 200 K, respectively. This indicates that as temperature is increased, larger bow outs are able to form since thermal activation “reduces” the pinning strength of the solutes.

The influence of finite temperature also explains the trends we observe in *H*_{1} and *H*_{2}. Recently, the work of (Zhai and Zaiser 2019) showed how, prior to the depinning transition, the Hurst exponents behaves in two limits. In the absence of solutes at finite temperature, random thermal noise leads to Hurst exponents *H*_{1}=*H*_{2}=0.5. Geslin and Rodney obtained the same result by analyzing line geometries from MD simulations of pure Al (Geslin and Rodney 2018). At the other limit, a dislocation interacts with a solute field at *T*=0 K giving Hurst exponents of *H*_{1}=1 and *H*_{2}=0. Our results correspond to a mixture of these two limits, since we consider a solid solution system at finite temperature. Hence, prior to the depinning transition we should expect that 0.5<*H*_{1}<1 and 0<*H*_{2}<0.5, which is exactly what is observed in Fig. 10. At low stresses, where thermal noise strongly influences the roughness, *H*_{1}≈0.7. As the stress increases, so does *H*_{1} since the thermal noise is washed out by the sampling of the solute field; the dislocation line is able to find stronger and stronger pinning sites which overwhelm the random thermal forces. Finally at the depinning transition we find that *H*_{1}≈0.9.

After the depinning transition (*σ*>*σ*_{c}), the behavior simplifies a bit because the solutes simply introduce another random noise field which the dislocation samples (as is observed during domain wall migration in ferromagnetics (Bustingorry et al. 2010; Duemmer and Krauth 2005)). Accordingly, we observe that both *H*_{1} and *H*_{2} tend to 0.5 when *σ*>*σ*_{c}. Furthermore, we observe that *ξ*→0 above the depinning transition, which is also consistent with uncorrelated roughness; the line geometry converges to an uncorrelated fractal.

### Comparisons with solid solution strengthening theories

As mentioned in the Introduction, most theories for solid solution strengthening are formulated on the basis of length scales which govern the dislocation’s interaction with the solute field. Dating back to the variational analysis of Mott (Ardell; Mott 1952) and more recently formulated by Argon (2008), an ad hoc interaction range *ω* is commonly defined, beyond which the dislocation-solute interaction is assumed to go to zero. Using this approximation in conjunction with the statistics of random point distributions, estimates are obtained for the mean bowing length at zero applied stress, \(\bar {l}_{\text {b0}}\). For example, Mott obtained that (Mott 1952)

$$ \bar{l}_{\text{b0}} = \left(\frac{2\Gamma L_{\mathrm{s}}^{4}}{F_{\mathrm{s}} \omega}\right)^{1/3} $$

(7)

where *Γ* is the dislocation line tension, *F*_{s} is the strength of the solutes, and *L*_{s} is the mean solute spacing in the glide plane. Argon obtained a similar result (Argon 2008). The flow stress is then directly related to the mean bowing length \(\bar {l}_{\text {b0}}\) via a force balance or thermal activation analysis (Ardell; Argon 2008). Determination of the mean bowing length at zero-stress \(\bar {l}_{\text {b0}}\) is thus a critical feature of these classical models.

More recently, the model of Leyson et al. (2010) advanced theoretical understanding of solid solution strengthening by taking an atomistic view of the problem of determining \(\bar {l}_{\text {b0}}\). In their theory, the total energy of the dislocation line is estimated by accounting for the line energy and the interaction energy with the solute field. The mean bowing length and bowing amplitude is then obtained by minimizing the total energy (similar to Mott’s analysis) at zero applied stress. Again, once these length scales are determined strengthening due to solutes can be computed. A major benefit of Leyson et al.’s theory is that there is no need to assume an ad hoc interaction length scale *ω*; instead, solute interactions at all scales are rigorously incorporated.

The common feature among all of these theories is that strengthening is governed by the mean bowing length at zero-stress \(\bar {l}_{\text {b0}}\). As even the earliest theories acknowledged, however, the process of a dislocation line sampling the solute field is statistical leading to a distribution of bowing lengths. Our results bolster this notion, showing that many bowing length scales operate simultaneously. In fact, *all* bowing length scales below 2*ξ* are operative. This observation brings into question the geometric assumptions of previous solid solution strengthening theories. For example, in Leyson et al.’s theory it is assumed that the line adopts a quasi-sinusoidal profile characterized by a single bowing length. Our results indicate that this picture simply isn’t realistic: the line is characterized by many bowing lengths, none of which are clearly dominant, To help demonstrate this point, we plot the average (over all relevant DXA snapshots) power spectral density of the dislocation line at *T*=5 K and *σ*=0, 50, 75, and 97.5 MPa in Fig. 11. If a single bowing length *l*_{b} were dominant, we would expect to see a peak in the power spectral density at *q*=2*π*/*l*_{b} (*q* is the wavevector). Clearly no such peak exists. In a subsequent analysis, Leyson and Curtin (2016) considered the possibility of bowing at multiple length scales, concluding that at high temperatures a second bowing length scale becomes active. Again, this finding seems inconsistent with our results. None-the-less, it may be the case that the mean bowing length \(\bar {l}_{\text {b0}}\) does govern strengthening, so that these observations are inconsequential. It is unclear whether the specific form of the bowing length distribution affects solid solution strengthening or the mobility of dislocations in solid solutions. Future research should be focused on assessing the bowing length distribution and its influence in more detail, for example using the theoretical framework outlined below.

An additional point to raise is that since *ξ* increases with stress below the depinning transition, we expect that the mean bowing length will also increase with stress. Hence, it is not obvious that the mean bowing length at zero-stress—as is the focus of the theories mentioned above—is the relevant length scale which dictates a dislocation’s response under non-zero stresses. Somewhat troublingly, we note that at the depinning transition, i.e., the yield stress, *there is no mean bowing length* because *ξ*→*∞* so that all bowing length scales are operable. At this point it is unclear how consequential this fact is with respect to existing solid solution strengthening theories.

### A mobility model close to depinning

We can use the above interpretation of the dislocation interaction with the solute field to develop a mobility model. This mobility model is valid close to the depinning transition, where the dislocation interacts with the solute field in a thermally activated manner which leads to creep-style motion. Our goal in formulating this model is to clarify the fundamental physics at play and how they influence the overall mobility of the dislocation line. We also seek to better understand previous observations of length-dependent mobilities at low stresses and temperatures (Osetsky et al. 2019; Sills et al. 2020). We emphasize that while the mobility model which we develop below is physically motivated by our data, it is somewhat speculative in nature and is intended to demonstrate new concepts inspired by our MD data while motivating future work in this area. In contrast to previous mobility models where the mean bowing length is the focus and to statistical physics theories that are based on phenomenological arguments and seek for scaling laws between different exponents, we instead seek a more comprehensive picture where all relevant bowing lengths are considered.

We wish to answer the following question: for a given stress *σ*, temperature *T*, and local solute distribution \(\left \{\mathbf {x}_{i}^{s}\right \}\) (where \(\mathbf {x}_{i}^{s}\) is the coordinate of solute *i* with species *s* and brackets denote the set of all solute atoms), what is the mobility of the dislocation line at a point *x*? The analysis above indicates that line segments of length *l*_{b}<2*ξ* are able to bow out coherently. When *T*=0 a distribution of bow outs forms and then remains pinned if *σ*<*σ*_{c} in a final state of equilibrium; that is, below the critical stress the mobility is zero. Slightly above the threshold *σ*_{c} the motion of the dislocation line is intermittent and comprised of sudden jumps between consecutive metastable states, while being mostly quiescent between the jumps. When *T*>0, however, it becomes possible for a bow out to overcome its local solute environment via thermal activation and begin gliding in avalanche-style motion even for *σ*<*σ*_{c} (Patinet et al. 2011). Hence, determining the mobility of the dislocation line is akin to determining the release rate for a given bowing length *l*_{b} at stress *σ* and temperature *T*. This view is consistent with experimental observations of domain wall migration in ferromagnetic materials (Durin and Zapperi 2000; Grassi et al. 2018) and previous MD simulations of dislocation motion in a random alloy (Patinet et al. 2011).

We can put these statements into precise mathematical terms as follows. First, we define a *bowing density of states* for the dislocation line, *n*_{b}(*l*_{b}), such that *n*_{b}(*l*_{b})d*l*_{b} is the number of line segments with bowing length *l*_{b} per unit length of dislocation line. Next we invoke a stress, temperature, and bowing-length-dependent waiting duration \(\bar {t}_{\mathrm {w}}\left (l_{\mathrm {b}},\sigma,T\right)\), which is the average time that a segment of length *l*_{b} is pinned before being released by thermal fluctuations. Clearly, \(\bar {t}_{\mathrm {w}}\) is infinite at *T*=0 and for *σ*<*σ*_{c}, and finite otherwise. Finally, to combine these ingredients into a mobility model which describes the dislocation velocity, we must more precisely define what is meant by velocity. The “true” dislocation velocity is of course well-defined as the speed at which the line moves locally at every point along its length. In the picture of solute interactions here, the local velocity varies rapidly along the line and in time. Hence, it is not a tremendously useful quantity from the standpoint of understanding plasticity. Instead, we are often interested in an average velocity over some length scale. For example, in discrete dislocation dynamics velocities are computed for line segments of finite length (Sills et al. 2016). We must, then, clarify a coarse-grained velocity definition to specify how the averaging is accomplished. One option is to coarse-grain in terms of the area sweep rate of the dislocation line, \(\dot {A}\), since the plastic strain rate is proportional to \(\dot {A}\) (Argon 2008). Using this approach, we can employ the Orowan equation to define the average line velocity as

$$ \bar{v} = \frac{\dot{A}}{L} $$

(8)

where *L* is the length of the dislocation line. Hence, in this definition the average velocity is the area sweep rate per unit length of the dislocation line. The area sweep rate for the dislocation line described by density of states *n*_{b}(*l*_{b}) can be stated as

$$ \dot{A} = L\int_{l_{\text{min}}}^{2\xi} \left(\frac{\overline{\Delta A}(l_{\mathrm{b}},\sigma,T)}{\bar{t}_{\mathrm{w}}(l_{\mathrm{b}},\sigma,T)}\right) n_{\mathrm{b}}(l_{\mathrm{b}}){\text{dl}}_{\mathrm{b}} $$

(9)

where \(\overline {\Delta A}(l_{\mathrm {b}},\sigma,T)\) is the mean increment of swept area that results from release of a segment of bowing length *l*_{b} at stress *σ* and temperature *T*. \(\overline {\Delta A}(l_{\mathrm {b}},\sigma,T)\) can be thought of as the average avalanche magnitude associated with depinning of a bowing segment of length *l*_{b}. This integral is simply a sum over the swept area contributions from all bowing lengths. The upper limit of integration 2*ξ* is the length above which bow outs cannot form. The lower limit is a minimum segment length set by the discreteness of the crystal lattice (*l*_{min}∼*b*)^{Footnote 4}. Combining Eqs. (8-9) leads to the following mobility model:

$$ \bar{v} = \int_{l_{\text{min}}}^{2\xi} \left(\frac{\overline{\Delta A}(l_{\mathrm{b}},\sigma,T)}{\bar{t}_{\mathrm{w}}(l_{\mathrm{b}},\sigma,T)}\right) n_{\mathrm{b}}(l_{\mathrm{b}}){\text{dl}}_{\mathrm{b}}. $$

(10)

Equation (10) shows how the dislocation velocity is governed by three fundamental quantities which characterize the dislocation’s interaction with the solute field: the mean waiting time \(\bar {t}_{w}\), the mean swept area increment \(\overline {\Delta A}\), and the bowing density of states *n*_{b}(*l*_{b}). Our conclusion is that in order to establish a physics-based mobility model, all three of these quantities must be characterized. Further work is necessary, however, to demonstrate that our proposed constitutive quantities—\(\bar {t}_{\mathrm {w}}, \overline {\Delta A}\), and *n*_{b}(*l*_{b})—are well defined and computable atomistic or discrete dislocation modeling techniques.

In principle, Eq. (10) provides a route for constructing mobility laws derived from basic theoretical concepts. Our results above give some clues as to what the constitutive functions for \(\bar {t}_{\mathrm {w}}, \overline {\Delta A}\), and *n*_{b}(*l*_{b}) may look like. For example, one possibility is that \(\overline {\Delta A}\propto l_{\mathrm {b}}^{H_{1}+1}\), since the amount of swept area should scale with the product of the bowing length and the roughness (bowing amplitude) of the bow out (Patinet et al. 2011). Unfortunately, these constitutive functions are likely to depend on bowing length, stress, temperature, and the solute distribution in a complex manner. Patinet et al. (2011) demonstrated one possible technique for obtaining these parameters, considering the avalanche magnitude (swept area) and duration (waiting time) statistics associated with dislocation motion in a random alloy. Hence, in order to make the mobility model Eq. (10) useful, additional research is necessary on the fundamentals of dislocation avalanche statistics in solid solutions. None-the-less, the analysis above gives an explicit account of what governs dislocation mobility in the creep regime.

### Interpretation of length-dependent mobilities

Several recent works have shown using MD simulations that the dislocation mobility in solid solutions is length-dependent if the periodic line length is too small (Osetsky et al. 2019; Sills et al. 2020). In this length-dependent regime, the line is seen to occasionally become completely pinned during a simulation. In contrast, a longer dislocation line moves continuously in time. Sills et al. (2020) showed that at a given stress and temperature, a minimum line length could be identified above which the dislocation mobility was length-independent. They further showed that this minimum line length decreased with increasing stress and temperature. A conclusive justification for these observations has yet to be presented. Sills et al. showed a similar length-dependence in kinetic Monte Carlo simulations based on the solid solution strengthening model of Leyson et al. (2010), but a fundamental explanation is still lacking.

Using the mobility model presented above, we can identify a plausible physical explanation. According to Eq. (10), in order for the mobility to be length-independent, the system size must be large enough so that the mean values of \(\overline {\Delta A}(l_{\mathrm {b}})\) and \(\bar {t}_{\mathrm {w}}(l_{\mathrm {b}})\) in addition to the bowing density of states *n*_{b}(*l*_{b}) are converged. If the system size is too small, \(\overline {\Delta A}(l_{\mathrm {b}}), \bar {t}_{\mathrm {w}}(l_{\mathrm {b}})\), and *n*_{b}(*l*_{b}) may vary as the dislocation line glides through the simulation cell. For example, if the dislocation happens to move to a location where \(\bar {t}_{\mathrm {w}}(l_{\mathrm {b}})\) is rather large for all bowing lengths, then the entire dislocation line may become pinned. In other words, the concept of a length-dependent mobility may derive from a poor statistical sampling of the dislocation-solute interactions. We emphasize that this explanation is speculative and that additional research is necessary to see if it bears out. Specifically, the influence of line length on \(\overline {\Delta A}\left (l_{\mathrm {b}}\right), \bar {t}_{\mathrm {w}}(l_{\mathrm {b}})\), and *n*_{b}(*l*_{b}) must be quantified.