# What pins edge dislocations in random alloys? Comparing size and elastic heterogeneities

Bassem Sboui, Pierre-Antoine Geslin, David Rodney

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Bassem Sboui, Pierre-Antoine Geslin, David Rodney. What pins edge dislocations in random alloys? Comparing size and elastic heterogeneities. Scripta Materialia, 2025, 268, pp.116879. 10.1016/j.scriptamat.2025.116879. hal-05183396

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# What pins edge dislocations in random alloys? Comparing size and elastic heterogeneities

Bassem Sboui<sup>a,b, ID</sup>, Pierre-Antoine Geslin<sup>a,c, ID, *</sup>, David Rodney<sup>b, ID, *, 1</sup>

## ARTICLE INFO

### Keywords:

- Solid solution strengthening

- Complex concentrated alloys

- Atomistic modeling

- Edge dislocations

## ABSTRACT

This work compares the impact of size mismatch and elastic heterogeneities on solute pinning of edge dislocations in random body-centered cubic alloys – solid solutions in which different atomic species occupy lattice sites at random. Using atomic-scale modeling of both parametric alloys and more realistic binary systems (W–Ta and W–Nb) across the full composition range, we show that the mismatch of elastic constants between the alloy constituents can have an effect comparable to, or even greater than, the size mismatch. This contrasts with some solid solution strengthening theories, which assume a homogeneous elastic medium. Our results further demonstrate that the combined effects of size and modulus mismatches are well captured by a quadratic form, consistent with historical phenomenological approaches.

Predicting solid solution strengthening is a classical topic in metallurgy [1] that has gained renewed interest since the 2010s with the rise of high-entropy alloys (HEAs) and compositionally complex alloys (CCAs) [2,3]. Classical theories, such as Friedel-Fleischer [4,5], Mott-Nabarro-Labusch [6], and Suzuki [7], provide a foundation for understanding solute strengthening in binary and more complex alloys, and the emergence of HEAs and CCAs has prompted a reassessment of these theories, resulting in frameworks that replace empirical inputs with atomistic calculations [8,9].

A widespread assumption in both classical and modern theories is the homogeneity of elastic properties among alloy constituents. This simplification arises partly because analytical expressions can only be rigorously derived for elastically uniform media [10]. However, many alloys exhibit substantial elastic heterogeneity—particularly refractory BCC alloys, where elastic modulus mismatches between constituents can exceed 100% [11–13].

An early attempt to incorporate elastic mismatch was made by Fleischer [4,14] and Labusch [6], who proposed a phenomenological formula relating the yield stress increase in binary alloys to a quadratic combination of size and modulus mismatches. Although this model has had a significant impact [15,16], it lacks clear physical justification and has remained largely untested at the atomic scale.

Here, we explore the respective roles of atomic size and elastic modulus mismatch in solute pinning of edge dislocations in binary BCC random alloys. We use a parametric Embedded Atom Method (EAM) potential that allows independent variation of size and modulus contrasts. This potential was used in a recent study [17] to investigate lattice distortions.

Here, we explore the respective roles of atomic size and elastic modulus mismatch in solute pinning of edge dislocations in binary BCC random alloys. To this end, we use a model binary alloy composed of tungsten (W) and a fictitious element $\bar{W}$. While $\bar{W}$ does not correspond to a real element, it is defined through a parametric Embedded Atom Method (EAM) potential based on a realistic potential for W. This construction allows independent variation of size and modulus contrasts, while maintaining all other features of the W potential unchanged. Such a strategy provides a controlled and physically grounded framework to investigate fundamental mechanisms of dislocation pinning. The potential was used in a recent study [17] to investigate lattice distortions.

Solute pinning is typically quantified by the critical resolved shear stress (CRSS), the minimum shear stress needed to initiate plastic flow. In atomistic simulations, CRSS refers to the smallest resolved shear stress (RSS) enabling a dislocation to traverse the simulation domain. However, this quantity is strictly defined only in the limit of long dislocations

<footnote><sup>a</sup> Univ. Lyon, CNRS, INSA Lyon, Université Claude Bernard Lyon 1, MATEIS, UMR5510, 69100 Villeurbanne, France</footnote>

<footnote><sup>b</sup> Univ. Lyon, CNRS, Université Claude Bernard Lyon 1, Institut Lumière Matière, UMR5306, 69622 Villeurbanne, France</footnote>

<footnote><sup>c</sup> ELYTMAX IRL3757, CNRS, Univ. Lyon, INSA Lyon, Centrale Lyon, Université Claude Bernard Lyon 1, Tohoku University, Sendai, Japan</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: pierre-antoine.geslin@insa-lyon.fr (P.-A. Geslin), david.rodney@univ-lyon1.fr (D. Rodney).</footnote>

<footnote n="1">¹ David Rodney was an Editor of the journal during the review period of the article. To avoid a conflict of interest, David Rodney was blinded to the record and another editor processed this manuscript.</footnote>
<caption>**Table 1**</caption>

Lattice parameter (in Å) and shear modulus (in GPa) of the pure elements in W-Ta [25] and W-Nb [23,24] alloys, with the corresponding $\alpha_{size}$ and $\alpha_{modulus}$ parameters.

<table><thead><tr><th>Alloy</th><th>a<sub>W</sub></th><th>a<sub>Ta/Nb</sub></th><th>μ<sub>W</sub></th><th>μ<sub>Ta/Nb</sub></th><th>α<sub>size</sub></th><th>α<sub>modulus</sub></th></tr></thead><tbody><tr><td>W-Ta</td><td>3.16</td><td>3.3</td><td>160</td><td>68.88</td><td>1.04</td><td>0.43</td></tr><tr><td>W-Nb</td><td>3.16</td><td>3.3</td><td>160</td><td>49.45</td><td>1.04</td><td>0.31</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Illustration of the simulation cell forming a periodic array of edge dislocations in a random BCC alloy.</figcaption></figure>

and large glide distances [18,19], a regime difficult to access with atomistic simulations due to computational constraints [20]. To address this, we focus on the Local Critical Resolved Shear Stress (LCRSS), the stress required to unpin a dislocation from a local energy minimum. Our simulations use short, mostly straight edge dislocations with lengths below the Larkin length, $L_c$. In this regime, the dislocation remains essentially straight and the LCRSS scales with dislocation length $L$ as $1/\sqrt{L}$ [19], making it a size-dependent rather than universal quantity. Nonetheless, we expect the observed trends, particularly with respect to size and modulus mismatch and concentration, to be relevant to the true CRSS. Finally, we restrict our study to edge dislocations to avoid the complex interplay between stress fluctuations and the intrinsic lattice resistance (Peierls stress), which is especially high for screw dislocations in BCC metals [21].

Employing the tunable EAM potential introduced in Ref. [17], we simulated a hypothetical BCC binary alloy, W–$\tilde{W}$, where interactions among W atoms follow the potential developed by Marinica et al. [22]. The second species, $\tilde{W}$, is a hypothetical BCC element with adjustable lattice parameter and elastic constants. Its lattice parameter is scaled by a factor $\alpha_{size}$ relative to W, i.e., $a_{\tilde{W}} = \alpha_{size} a_{W}$, and its elastic constants by $\alpha_{modulus}$, such that $\mu_{\tilde{W}} = \alpha_{modulus} \mu_{W}$. The user-defined parameters $\alpha_{size}$ and $\alpha_{modulus}$ can be freely tuned to explore various strengthening scenarios. In what follows, we use $\alpha_{size}$ values from 1 to 1.05 and $\alpha_{modulus}$ from 1 to 1.50.

We also studied two real BCC alloys, W-Nb and W-Ta, modeled using EAM potentials from Refs. [23,24] and [25], respectively. For each pure element, we computed the lattice parameter and shear modulus to extract the corresponding size and modulus mismatches, shown in Table 1. The size contrasts fall within the range of the parametric study, while modulus contrasts are notably larger, enabling us to probe the effect of extreme modulus mismatch on solute pinning.

The simulation cell consists of a random W-$\tilde{W}$ BCC lattice oriented along the crystallographic axes $X = \frac{1}{\sqrt{3}}[111]$, $Y = \frac{1}{\sqrt{2}}[101]$, and $Z = \frac{1}{\sqrt{6}}[12\bar{1}]$. As shown in Fig. 1, after introducing an edge dislocation, free surfaces were introduced along $Y$ (normal to the glide plane), while periodic boundary conditions were applied within the glide plane to

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Histogram of $\tau_{LCRSS}$ distribution for an edge dislocation calculated for a binary $W_{0.5}\tilde{W}_{0.5}$ alloy, with size mismatch taken as 5% and homogeneous elasticity ($\alpha_{modulus} = 1$). The distribution was constructed from 500 configurations. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

generate a periodic array of dislocations [26,27]. We considered two dislocation lengths, $L_z$: a “short” cell with a single repeat unit, $L_z = 2\sqrt{2b}$, and a “long” cell with 14 units, $L_z = 28\sqrt{2b}$. Both share the same height, $L_y = 50\sqrt{2/3b}$, and glide length, $L_x = 60b$.

We performed calculations at 0 K and triggered dislocation motion using quasi-static loading. To this end, a resolved shear stress was applied by adding forces to atomic slabs near the upper and lower free surfaces. The stress was incrementally increased, each step followed by energy minimization, until the dislocation moved from its initial pinned state. Calculations were repeated with at least 300 random alloy realizations to obtain statistically meaningful LCRSS values. Simulations were carried out using LAMMPS [28].

The position of the dislocation center of mass was tracked using the common neighbor analysis (CNA) [29,30]. The dislocation was considered to have escaped its initial pinning position when its center of mass moved by more than one Burgers vector, with a 5% tolerance to account for structural fluctuations due to lattice distortions. We used an initial stress increment of 100 MPa, but once a jump was detected, we reverted to the last equilibrium configuration before the event and repeated the procedure with a finer increment of 10 MPa to refine our measurement of the LCRSS.

In a random alloy, the LCRSS is an inherently random variable. To illustrate its statistical nature, we conducted 500 independent simulations in a $W_{0.5}\tilde{W}_{0.5}$ alloy with $\alpha_{modulus} = 1$ and $\alpha_{size} = 1.05$. The resulting LCRSS distribution is presented in Fig. 2. The histogram reveals a highly skewed distribution that extends down to near-zero stress. A significant fraction of dislocation configurations are therefore only weakly pinned in shallow energy minima. Conversely, the long tail of the distribution highlights the presence of rare but strongly pinned configurations, which correspond to deep wells in the energy landscape. We observed similarly broad and skewed distributions for all size and modulus contrasts and alloy compositions.

In the following, we characterize the magnitude of solute pinning using the average LCRSS, $\langle\tau_{LCRSS}\rangle$, shown as a vertical red dashed line in Fig. 2. We evaluated the statistical uncertainty associated with this measure using the classical statistical dispersion formula:

$$ \text{uncertainty} = \frac{\Delta\tau_{LCRSS}}{\sqrt{N_{\text{sim}}}}, \qquad (1) $$

where $N_{sim}$ is the number of simulations (i.e., the number of independent LCRSS values sampled), and $\Delta\tau_{LCRSS}$ is the standard deviation of the LCRSS distribution. For the case presented in Fig. 2, we have $\langle\tau_{LCRSS}\rangle = 1141 \pm 49$ MPa.
<figure><img src="image_5.png" /><figcaption>Fig. 3. Mean LCRSS as a function of $\tilde{W}$ concentration for different contrasts in size and elastic modulus listed in the legend. All curves result from averages over 300 configurations with the shaded regions indicating statistical uncertainties (Eq. (1)).</figcaption></figure>

Fig. 3 summarizes our calculations of $\langle \tau_{LCRSS} \rangle$ in W-$\tilde{W}$ alloys for varying combinations of $\alpha_{modulus}$ and $\alpha_{size}$. Bell-shaped curves are obtained, starting and ending near zero stress in pure W and $\tilde{W}$, as expected since edge dislocations have a negligible lattice friction in pure BCC metals.

For elastically homogeneous alloys (green and blue curves in Fig. 3 with $\alpha_{modulus} = 1$ and $\alpha_{size} = 1.01$ and 1.05), varying the size mismatch yields strong pinning, peaking around 1200 MPa near $W_{0.5}\tilde{W}_{0.5}$ for $\alpha_{size} = 1.05$ in the short cell and around 300 MPa in the long cell. As noted in the introduction, this size effect is expected and consistent with the theoretical $1/\sqrt{L}$ scaling, since $300 \times \sqrt{14} \sim 1100$.

To study the effect of heterogeneous elastic constants, we considered two cases. First, alloys with no size misfit ($\alpha_{size} = 1$) and elastic heterogeneities $\alpha_{modulus} = 1.5$ (purple curve in Fig. 3) show significant strengthening. Second, combining size and modulus mismatches ($\alpha_{size} = 1.01$ and 1.05 with $\alpha_{modulus} = 1.5$, pink and red-shaded curves) yields significant strengthening, reaching 1750 MPa in the short simulation cell. Similar trends are seen in the long cell, albeit with lower mean LCRSSs due to the size effect.

When there is an elastic modulus contrast, the mean LCRSS curves become asymmetric, with stronger strengthening in the $\tilde{W}$-rich region. This arises from the gradual increase of the alloy average shear modulus, as $\tilde{W}$ atoms are stiffer than W atoms.

When only a size contrast is present, the dependence of the mean LCRSS on dislocation length and $\tilde{W}$ concentration can be captured by a classical elastic model. Considering the elastic interaction energy between the pressure field of a straight edge dislocation and misfitting size inclusions, we show in section 1 of the Supplementary Materials [31] that

$$ \langle \tau_{LCRSS} \rangle \propto \mu |1 - \alpha_{size}| \sqrt{c_{\tilde{W}}(1 - c_{\tilde{W}})} \sqrt{\frac{b}{L_z}} \quad (2) $$

where $\mu$ is the alloy shear modulus. The rescaled LCRSS, $\langle \tau_{LCRSS} \rangle / (\mu |1 - \alpha_{size}| \sqrt{b/L_z})$, should thus follow a master $\sqrt{c_{\tilde{W}}(1 - c_{\tilde{W}})}$ curve, independent of dislocation length and size mismatch. Assuming that the superposition principle holds, the model can be extended to include interactions with elastic heterogeneities [32,33]. In contrast with the size interactions, this di-elastic interaction is short ranged and strongest in the direct vicinity of the dislocation core (see [31]). As such, it is often labeled as chemical interactions between the dislocation and solute atoms [34,35].

In sections 2 and 3 of the Supplementary Materials [31], we employ several approximations to show that the mean LCRSS scales as:

<figure><img src="image_6.png" /><figcaption>Fig. 4. Mean LCRSS scaled with a quadratic combination of size and modulus effects. Same color and line conventions as in Fig. 3.</figcaption></figure>

$$ \langle \tau_{LCRSS} \rangle \propto \bar{\mu}(c_{\tilde{W}}) \sqrt{4 \left( \frac{1 - \alpha_{\text{modulus}}}{1 + \alpha_{\text{modulus}}} \right)^2 + \eta^2 (1 - \alpha_{\text{size}})^2} \times \sqrt{c_{\tilde{W}}(1 - c_{\tilde{W}})} \sqrt{\frac{b}{L_z}} \quad (3) $$

where $\bar{\mu}$ is the alloy average shear modulus that now depends on the alloy composition. This expression holds in the limit $|\alpha_{size} - 1| \ll 1$ and $|\alpha_{modulus} - 1| \ll 1$, where higher-order terms can be neglected [31]. Eq. (3) reduces to Eq. (2) if there is only a size mismatch and introduces a nonlinear dependence on $\alpha_{modulus}$ that reflects the classical quadratic combination of size and modulus misfits proposed by Fleischer and Labusch [4,6,14].

A key result of our study is to justify this quadratic form and the absence of cross-terms between size and modulus effects, which is due to the distinct symmetries of dislocation interactions with dilatation inclusions and elastic heterogeneities. As this result derives from general symmetry arguments, it is applicable to systems other than BCC binary alloys. As in classical models, we introduce a dimensionless parameter $\eta$ to quantify their relative influence. While the Supplementary Materials [31] outline a theoretical method to estimate $\eta$, we determine it here by fitting to atomistic data.

In Fig. 4, we compiled all computed mean LCRSS values and rescaled them using $\bar{\mu} \sqrt{4 \left( \frac{1 - \alpha_{\text{modulus}}}{1 + \alpha_{\text{modulus}}} \right)^2 + \eta^2 (1 - \alpha_{\text{size}})^2 \sqrt{b/L_z}}$ from Eq. (3). We then fitted the parameter $\eta$ to collapse all data onto a single master curve proportional to $\sqrt{c_{\tilde{W}}(1 - c_{\tilde{W}})}$, shown as a dashed black line. The
<figure><img src="image_7.png" /><figcaption>Fig. 5. Mean LCRSS as a function of Nb and Ta content in random (a) W-Nb and (b) W-Ta alloys, respectively. Dashed curves show predictions of elasticity considering only the size misfit or only the modulus misfit or both effects combined. A 'short' simulation cell was used for these calculations.</figcaption></figure>

fit yields $\eta = 14.8$, close to Labusch's historical estimate of 16 for edge dislocations [6]. The quality of the collapse confirms that Eq. (3) successfully captures solute strengthening due to both size and modulus contrasts, as well as the effect of the asymmetric average shear modulus $\bar{\mu}$.

To assess the transferability of the scaling (3), we repeated our analysis for W-Ta and W-Nb alloys. A key difference is their nonlinear dependence of the lattice parameter on composition. As shown in the Supplementary Materials [31], this requires replacing the $(1 - \alpha_{size})$ term in Eq. (3) with the normalized lattice parameter derivative, $a'(c_B)/a(c_B)$, where $a(c_B)$ is the average lattice parameter at solute concentration $c_B$ (B = Ta or Nb), and $a'(c_B)$ its derivative.

Fig. 5 shows the computed mean LCRSS for W-Nb and W-Ta. Using Eq. (3) with the same prefactor and $\eta$ value as for W-$\bar{W}$, we predict the mean LCRSS for three scenarios: only a size misfit (blue dashed), only a modulus misfit (green dashed), or both effects combined (red dashed). For W-Nb (Fig. 5(a)), the full prediction agrees well with atomistic results, without parameter adjustment. Notably, the modulus mismatch dominates the strengthening across the composition range. The agreement is less satisfactory for W-Ta (Fig. 5(b)), although the model still captures the magnitude of strengthening. Here, the modulus effect dominates at low Ta content, consistent with the weak composition-dependence of the lattice parameter (see Fig. 3 of the Supplementary Materials [31]), while size misfit dominates at higher concentrations, yielding a nearly symmetric prediction, unlike the asymmetric atomic data. The discrepancy at low Ta content suggests that the modulus effect is overestimated in the model, likely due to additional nonlinear or atomic-scale effects not captured by the elastic model. However, we note that these nonlinearities may be artifacts of the interatomic potentials, as recently highlighted in the context of screw dislocation glide in BCC metals [36].

In this work, we examined edge dislocations in concentrated BCC alloys, focusing on the roles of atomic size misfit and elastic heterogeneities in solute strengthening. Through both idealized parametric alloys and realistic binary systems, we showed that modulus mismatch, due to large elastic contrasts typical in BCC alloys, can be as important as, or even dominate, the size effect contribution. This contrasts with many strengthening models that assume a homogeneous elastic medium. Notably, such modulus contrasts are less significant in FCC alloys, where this assumption is more reasonable. All calculations were performed at 0 K. At finite temperatures, thermal activation lowers the LCRSS by helping dislocations overcome energy barriers, though the barrier heights, set by elastic mismatches, should remain largely unchanged.

We found that the mean LCRSS scales with a quadratic combination of size and modulus mismatches. Interestingly, in Ref. [17], lattice distortions in different model W-$\bar{W}$ alloys were found to follow a mul-

tiplicative scaling. The contrast between both scalings challenges the common assumption that the CRSS scales directly with lattice distortion [37].

Due to computational limitations, our study was restricted to short, straight edge dislocations—below the Larkin length. Beyond this length scale, dislocations roughen and the CRSS becomes length-independent if the glide distance increases simultaneously [19,38]. With size misfit alone, the CRSS above $L_c$ is expected to scale as $\tau_c \sim \mu|1 - \alpha_{size}|^{4/3}[c(1-c)]^{2/3}$. Compared to the scaling derived here, this introduces an exponent of 4/3. By analogy, we propose the following scaling for extended dislocations with both size and modulus mismatches:

$$ \tau_{\text{CRSS}} \propto \bar{\mu} \left[ 4 \left( \frac{1 - \alpha_{\text{modulus}}}{1 + \alpha_{\text{modulus}}} \right)^2 + \eta^2 (1 - \alpha_{\text{size}})^2 \right]^{2/3} [c(1-c)]^{2/3}, \quad (4) $$

but further modeling and simulations are needed to validate this relation. Lastly, while we focused on edge dislocations, screw dislocations may behave differently due to their core structure, high lattice friction in BCC metals, and distinct interactions with solutes. Extending this analysis to screw dislocations is an important direction for future work.

## CRediT authorship contribution statement

Bassem Sboui: Writing – original draft, Visualization, Investigation, Formal analysis. Pierre-Antoine Geslin: Writing – review & editing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. David Rodney: Writing – review & editing, Supervision, Methodology, Formal analysis, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

David Rodney is an Editorial Board Member/Editor-in-Chief/Associate Editor/Guest Editor for this journal and was not involved in the editorial review or the decision to publish this article.

## Acknowledgement

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-20-CE08-0019 (project INSPIRA). We gratefully acknowledge support from the CNRS/IN2P3 Computing Center (Lyon - France) for providing computing resources needed for this work.
# Appendix. Supplementary material

Supplementary material related to this article can be found online at
https://doi.org/10.1016/j.scriptamat.2025.116879.

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