Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening

Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening Pedro Oliveira, Caio Martins, Guilherme Stumpf, Julio Spadotto, and 4 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5003860/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Oct, 2025 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Abstract This study designed and produced FCC solid solution alloys with significant differences in atomic volume or electronegativity among the constituent elements, and subjected them to mechanical testing. The results demonstrate that atomic volume differences have a greater influence on solid solution strengthening (SSS) than electronegativity differences. Each solid solution system exhibits unique behavior, making a general model for predicting SSS challenging. Additionally, for a given solid solution system, there is a considerable difference in the critical grain size below which grain boundary strengthening dominates yield strength and hardness. Furthermore, both predicted lattice distortion values and the measured SSS components were greater for binary alloys, indicating that the presence of more elements in a solid solution does not always cause greater distortions in the crystal lattice. Finally, the study successfully engineered the novel Ni 50 Pd 50 alloy, which has not been previously studied and exhibits mechanical properties remarkably insensitive to variations in grain size, warranting further in-depth investigations. Physical sciences/Materials science/Structural materials/Metals and alloys Physical sciences/Materials science/Structural materials/Mechanical properties Solid solution strengthening Grain boundary strengthening High Entropy Alloys Atomic volume Electronegativity Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1. INTRODUCTION The lattice friction stress, quantified by the Peierls stress [ 1 – 4 ], describes the resistance a single dislocation faces while moving through an alloy's lattice, which is closely linked to atomic-scale lattice distortions. In metallic alloys with solid solution, numerous solute atoms with varying sizes and properties create distorted lattices, resulting in extensive lattice friction [ 5 – 7 ]. This distortion raises the energy barrier against dislocation mobility, thereby strengthening the material [ 8 – 13 ]. Important pioneering models to quantify solid solution strengthening (SSS) were originally introduced by Fleischer [ 14 ], latter followed by Labusch [ 15 , 16 ]. According to these models, solutes can be effectively represented as point defects that will distort the crystal lattice, giving rise to a solute-dislocation elastic interaction energy, which in turn is the primary factor governing the SSS mechanism. This interaction energy is the product of the dislocation pressure field and the misfit volume of the solute and solvent, which can be defined as the difference between the volume of the solute in the alloy and the average atomic volume. However, the models proposed by Labusch and Fleischer take into account only solutions with solvents up to 10 at% in a single component matrix [ 17 ]. In the context of High Entropy Alloys (HEAs) [ 18 – 21 ], distinguishing between solvent and solute atoms is challenging due to the random occupation of lattice sites by multiple atom species, leading some solid solution models to perform inadequately. Researchers have recently developed models to explain SSS in HEAs, which rely heavily on input parameters like atomic size and shear modulus mismatch [ 8 , 22 , 23 ], and more recently, electronegativity differences [ 24 , 25 ]. For example, the Varvenne model [ 8 ], based on first principles, posits that differences in atomic volume control the mechanical strength of alloys by calculating the interaction energy between dislocations and solute atoms. This energy is integrated into a conventional equation to account for thermal effects, yielding a comprehensive representation of critical resolved shear stress and thus, yield strength. The Varvenne model also takes into account the shear modulus and poisson’s ratio of each element in its final calculation to obtain the SSS component. This model's ability to accommodate strain rate and temperature variations enhances its predictive accuracy. Conversely, Oh’s model [ 24 ] attributes SSS in HEAs to differences in electronegativity among the constituent elements. Using a quantum-mechanical approach, it approximates atomic-level pressure to identify optimal element combinations for high SSS, focusing on electronegativity disparities rather than atomic volume differences. Although both the Varvenne [ 8 ] and Oh [ 24 ] models are applied to FCC systems, systems, they are fundamentally different. Hence, the present study aimed to solve the conflict between the two models, i.e which factor most enhances the SSS, by developing different alloys, engineered to have significant differences in electronegativity and/or atomic volume. Furthermore, given the growing interest in high-strength HEAs for engineering applications, this study aims to resolve the conflict between these models. To achieve this, an analysis of various 3d transition metal elements was conducted, focusing on those that form a solid solution in Ni and provide significant differences in electronegativity and/or atomic volume compared to pure Ni. V was chosen for its high electronegativity difference to Ni, while Pd was selected for its smaller electronegativity impact but greater atomic volume difference. Figure 1 a illustrates the selection process, highlighting the 3d elements forming an appreciable solid solution in Ni (above 30 at. % solute). To analyze the implications of adding Pd and V and their effects on the solid solution system, the equiatomic CrCoNi was also taken as a benchmark. The alloys used here were Ni 50 Pd 50 , Ni 63.2 V 36.8 , Cr 33 Co 33 Ni 33 , Cr 30 Co 30 Ni 30 Pd 10 , and Cr 30 Co 30 Ni 30 V 10 , chosen for significant lattice distortion and/or electronegativity differences (Fig. 1 b). Despite Ni 63.2 V 36.8 being unlikely to form a solid solution under many typical processing conditions [ 28 – 31 ], it was included for comparison with the model by Oh et al. [ 24 ]. These alloys are ideal for testing both models due to their variations in electronegativity and atomic volume. It is worth mentioning that the selection of Pd sets this study apart from previous research efforts concentrated on the inclusion of other 3d transition elements, which commonly share comparable atomic volumes. Finally, this study aims not only to resolve the scientific conflict between the Varvenne [ 8 ] and Oh [ 24 ] models but also to propose a more accurate strategy towards alloy design, focused on optimizing mechanical properties with greater precision. 2. METHODS To analyze the implications of the addition of Pd and V, and their effect on the SSS, it was crucial to take a pure element and a simple solid solution system as benchmark. In this work, as mentioned previously, Ni was used as the pure element and the CrCoNi was the solid solution system Computational thermodynamic calculations using the CALPHAD method identified the FCC single-phase regions for the alloys Ni 50 Pd 50 , Ni 63.2 V 36.8 , Cr 33 Co 33 Ni 33 , Cr 30 Co 30 Ni 30 Pd 10 , and Cr 30 Co 30 Ni 30 V 10 , utilizing the ThermoCalc® software and the TCHEA 5 and TCNI11 databases. These alloys, along with pure Ni, were prepared by arc melting high-purity elements under argon atmosphere and underwent multiple re-melting cycles for compositional homogeneity. To achieve a single-phase, fine-grained microstructure with low dislocation density, the ingots were homogenized, water quenched, cold rolled, annealed, and quenched again. Specific thermomechanical treatments are detailed indicated in Supplementary Table 1, with Ni 63.2 V 36.8 undergoing additional reduction to disrupt its eutectoid structure before the treatment in Supplementary Table 1. For each heat treatment, samples were encapsulated in vacuum-filled quartz ampoules to mitigate oxidation during high-temperature processing. Phase structure analysis of recrystallized alloys and pure Ni samples was performed using high-energy X-ray diffraction (HE-XRD) at Petra III, DESY in Germany. A monochromatic X-ray beam at 87 keV and a Perkin Elmer 2D detector captured the Debye-Scherrer rings. The setup included a 700 x 700 µm² beam and a 1.226 m sample-to-detector distance, calibrated with LaB6 powder. The 2D images were converted to 1D diffraction patterns via radial integration, and single-peak fitting was done using GSAS II software. Rietveld refinement was also applied for full spectrum fitting to determine lattice parameters. Scanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS) analyses were conducted on polished samples using a TESCAN MIRA FEG microscope, operating at 20 kV. Additionally, electron backscatter diffraction (EBSD) was performed on samples at 30 kV. EBSD data were processed using ATEX software©. Uniaxial tensile tests were conducted at room temperature using an INSTRON machine, following ASTM E8 standards with an optical extensometer and a nominal strain rate of 10 − 3 s − 1 . Flat dog bone-shaped tensile specimens (1.0 mm thick) were prepared by electrical discharge machining, aligned parallel to the rolling direction, and yield strength was determined using the 0.2% proof stress criterion. To evaluate solid solution behavior, Hall-Petch plots were created by measuring hardness and yield strength for different grain sizes. Each alloy underwent various annealing treatments to achieve different grain sizes, and Vickers Hardness measurements were taken on recrystallized samples with a 100 g load and a 15-second dwell time. 3. RESULTS AND DISCUSSION 3.1 MICROSTRUCTURAL CHARACTERIZATION The thermodynamic calculations depicted in Fig. 2 a-e illustrate the FCC single-phase field for the Ni 50 Pd 50 Ni 63.2 V 36.8 , Cr 33 Co 33 Ni 33 , Cr 30 Co 30 Ni 30 Pd 10 , Cr 30 Co 30 Ni 30 V 10 alloys. Subsequently, Fig. 3 displays the X-ray diffraction (XRD) patterns, for the recrystallized samples, confirming the exclusive presence of the FCC phase in all samples and the correspondent inverse pole figure (IPF-Z) maps from EBSD analysis showing the microstructure of the alloys. For comparison, XRD for pure Ni is shown in Supplementary Fig. 1. The SEM-EDS elemental distribution maps showcased in Supplementary Figs. 2–6 revealed the compositional homogeneity of the recrystallized microstructure at the micro scale on each alloy. 3.2 INFLUENCE OF ATOMIC VOLUME Analysis of the diffractograms in Fig. 3 a-b shows that adding Pd and V to pure Ni causes a noticeable shift of all reflections to lower 2θ values compared to pure Ni (Supplementary Fig. 1). A similar trend is seen in Fig. 3 d-e with the addition of Pd and V to the CrCoNi system (Fig. 3 c). According to Bragg's law [ 32 ], these shifts to lower 2θ values are related to an increase in atomic radius (atomic volume). Accurately determining atomic volume in solid solutions with multiple atoms is challenging due to variations in atomic radii depending on the environment of which a certain atom is inserted. For example, α-Fe (BCC) and γ-Fe (FCC) have different radii (1.239 Å and 1.287 Å, respectively), as do α and β Ti in different structures (1.475 Å and 1.432 Å, respectively) [ 26 ]. Pure Ni (FCC) has a radius of 1.243Å, but a radius of 1.257 Å in a binary FCC solution with Cr (30 at. %) [ 8 ]. To address this, we propose using the concept of apparent atomic volume (A av ), which relies on the principles that the unit cell represents the entire system and contains atoms in the same proportion as the solid solution. Considering a system with a composition of Cr 33 Co 33 Ni 33 , forming an FCC solid solution, the atomic volume of the unit cell is crucial because it must include the three elements in the same stoichiometry as the system. Thus, each element—Cr, Co, and Ni—occupies approximately one third of the atomic volume in the cell. For an FCC cell, and assuming the simplified spherical geometry for the atoms, the following relationship is established: $$\:0.74=\frac{QV}{{a}^{3}}$$ 1 Where \(\:Q\) is the number of atoms in a FCC unit cell, \(\:V\) is the volume of one atom, and \(\:{a}^{3}\) is the unit cell volume. Thus, taking into account that an FCC cell is occupied by four atoms, the \(\:{A}_{av}\) might be expressed as: $$\:{A}_{av}=\frac{\text{0,74}{a}^{3}}{4}$$ 2 Therefore, for solid solutions, \(\:{A}_{av}\) provides an effective and functional measure of the volume occupied by atoms in a unit cell, regardless of the quantity of atoms composing the solid solution. It should be noted that atoms are not spheres and the volume occupied by a single atom might be the entire 1/4th of the FCC unit cell if the atom is now viewed in this way. However, the simplification of taking atoms as spheres is convenient to directly apply the results to many equations in the literature that consider the atomic radii’ as the intrinsic measurement for the atomic volume and either option, atomic radii or volume, will lead to the same outcome using these models. Furthermore, a distorted lattice may result if a significant difference in atomic sizes is present. The distortion generated in the crystalline lattice ( \(\:\delta\:\) ) can be estimated by the model proposed by Zhang et al.[ 27 ]: $$\:\delta\:=\:\sqrt{\sum\:_{i=1\:\:\:}^{n}{c}_{i}\:{\left(1-{r}_{i}/\stackrel{-}{r}\right)}^{2}}$$ 3 Where n is the total number of elements in solid solution, \(\:{c}_{i}\) is the atomic fraction of the i th element, \(\:{r}_{i}\) is the atomic radius of element i and \(\:\stackrel{-}{r}\) ( \(\:\sum\:_{i=1}^{n}{c}_{i}{r}_{i})\) is the average atomic radius. The \(\:\delta\:\) parameter is very common in the HEA literature to describe the difference in atomic radii in solid solutions. Using the XRD results shown in Fig. 3 , the apparent atomic volume ( \(\:{A}_{av}\) ) was calculated for the investigated materials, detailed in Tables 1 and 2 . These tables also present the δ values for pure Ni and the CrCoNi system, including changes from adding Pd and V. For precise δ determination, we used ‘solution atomic radii’. These radii represent the size that an element would have in a face-centered cubic (FCC) system. These values were derived from binary FCC alloys of the element with Ni [ 8 , 33 ], as detailed in Supplementary Table 2. Table 1 – Lattice parameter (a), variation in the lattice parameter (Δa), apparent atomic volume (A av ), variation in the apparent atomic volume (ΔA av ), and crystalline lattice distortion (δ) for pure Ni, Ni 50 Pd 50 , and Ni 63.2 V 36.8 alloys. Material a (Å) Δa* (%) A av (Å 3 ) ΔA av **(%) δx100 Ni 3.5246 - 8.1003 - - Ni 50 Pd 50 3.7396 6.10 9.6747 19.44 4.97 Ni 63,2 V 36,8 3.6094 2.40 8.6988 7.39 4.53 Table 2 Lattice parameter (a), variation in the lattice parameter (Δa), apparent atomic volume (A av ), variation in the apparent atomic volume (ΔA av ), and crystalline lattice distortion (δ) for Cr 33 Co 33 Ni 33 , Cr 30 Co 30 Ni 30 Pd 10 , and Cr 30 Co 30 Ni 30 V 10 alloys. Material a (Å) Δa* (%) A av (Å 3 ) ΔA av **(%) δx100 Cr 33 Co 33 Ni 33 3.5621 - 8.3616 3.23 1.40 Cr 30 Co 30 Ni 30 Pd 10 3.6091 1.32 8.6971 4.01 3.13 Cr 30 Co 30 Ni 30 V 10 3.5841 0.62 8.5176 1.87 1.94 Table 1 reveals that adding Pd to pure Ni increases A av by 19.44%, much more than the 7.39% increase from adding V. The δ values are also higher with Pd additions. Similarly, Table 2 shows that Pd increases A av by 4.01% in the CrCoNi system, compared to 1.87% for V, with higher \(\:\delta\:\) values for Pd as well. This larger increase in A av and δ when adding Pd to both Ni and CrCoNi is due to Pd's larger atomic size compared to V. Consequently, considering only atomic volume differences, alloys with Pd are predicted to exhibit stronger SSS due to greater lattice distortion. Moreover, it is worth noting that these findings corroborate what has been shown before by other authors [ 34 , 35 ], indicating that the presence of a greater number of elements in solid solution does not invariably induce larger distortions in the crystal lattice. This is exemplified by the higher values of A av and δ observed for the binary alloys in comparison to their counterparts in the ternary and quaternary alloy systems. 3.3 INFLUENCE OF ELECTRONEGATIVITY Based on the assumption that it's possible to predict configurational fluctuations of charge transfer and atomic-level pressure, the model introduced by Oh et al. [ 24 ] depends on the average charge transfer of each element in a solid solution, approximated by the local difference in electronegativity. The electronegativity difference between the constituent elements ( \(\:\varDelta\:{\chi\:}\) ) can be calculated as follows: $$\:\varDelta\:{\chi\:}=\sqrt{\sum\:_{x}{c}_{x}{\left({{\chi\:}}_{x}-{⟨{\chi\:}⟩}_{element}\right)}^{2}}$$ 4 Where \(\:{{\chi\:}}_{x}\) represents the electronegativity of element X, and \(\:{⟨{\chi\:}⟩}_{element}\) stands for the weighted average electronegativity across the element. Using the model proposed by Oh et al. [ 24 ], the \(\:\varDelta\:{\chi\:}\) values were calculated for the alloys under investigation and are presented in Table 3 and Table 4 . In their model, the authors claim the Allen electronegativity scale should be the one to used. The values of electronegativity used for each element are listed in Supplementary Table 2. Table 3 Electronegativity difference caused by the addition of Pd and V to pure Ni Material δχ Ni - Ni 50 Pd 50 0.150 Ni 63,2 V 36,8 0.175 Table 4 Electronegativity difference caused by the addition of Pd and V to CrCoNi system Material δχ Cr 33 Co 33 Ni 33 0.100 Cr 30 Co 30 Ni 30 Pd 10 0.118 Cr 30 Co 30 Ni 30 V 10 0.126 Tables 3 and 4 show that the introduction of V into pure Ni results in slightly higher \(\:\varDelta\:{\chi\:}\) values compared to the addition of Pd. Similarly, V addition to the CrCoNi system leads to a higher \(\:\varDelta\:{\chi\:}\) values than the addition of Pd. According to the Oh model [ 24 ], alloys with V should exhibit stronger SSS due to the greater \(\:\varDelta\:{\chi\:}\) values. It is important to note that the values presented in Table 4 may vary depending on the electronegativity scale used. For example, based on the electronegativity according to the Pauling scale, the Ni 50 Pd 50 alloy would exhibit the greatest \(\:\varDelta\:{\chi\:}\) values (Supplementary Tables 2 and 3). In light of the exposition thus far, it becomes clear that the models proposed by Varvenne and Oh contain a fundamental contradiction. According to the Varvenne model, alloys containing palladium (Ni 50 Pd 50 and Cr 30 Co 30 Ni 30 Pd 10 ) should exhibit the highest contribution of SSS. Conversely, the Oh model posits alloys with vanadium addition (Ni 63.2 V 36.8 and Cr 30 Co 30 Ni 30 V 10 ) should have the greatest SSS contribution. Therefore, the following section delves into both models and attempts to clarify the fundamental difference between them. 3.4 PREDICTION OF SSS SOLUTION USING THE EVALUATED MODELS The Varvenne model evaluates the energy associated with the interaction between a dislocation and a solute atom. This energy calculation is subsequently included in a standard equation to adapt to thermally induced deformation. As a result, the determined interaction energy becomes part of an equation to handle thermally driven deformation. The resulting outcome is a model for the SSS component in the yield strength, which considers the impact of strain rate and temperature. This final formulation offers a direct method for calculating the activation energy needed for dislocation movement ( \(\:{\varDelta\:\text{E}}_{\text{b}}\) ) and the Peierls stress at absolute zero ( \(\:{{\tau\:}}_{0}\) ), as expressed in the following equations: $$\:{{\tau\:}}_{0}=0.051{{\alpha\:}}^{-\raisebox{1ex}{$1$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}\text{G}{\left(\frac{1+{\nu\:}}{1-{\nu\:}}\right)}^{\raisebox{1ex}{$4$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}{\text{f}}_{1}\left({\text{W}}_{\text{c}}\right){\left(\sum\:_{\text{n}}\frac{{\text{x}}_{\text{n}}{\varDelta\:\stackrel{-}{V}}_{n}^{2}}{{\text{b}}^{6}}\right)}^{\raisebox{1ex}{$2$}\!\left/\:\!\raisebox{-1ex}{$2$}\right.}$$ 5 \(\:{\varDelta\:\text{E}}_{\text{b}}=0.274{{\alpha\:}}^{\raisebox{1ex}{$1$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}\text{G}{\text{b}}^{3}{\left(\frac{1+{\nu\:}}{1-{\nu\:}}\right)}^{\raisebox{1ex}{$2$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}{\text{f}}_{2}\left({\text{W}}_{\text{c}}\right){\left(\sum\:_{\text{n}}\frac{{\text{x}}_{\text{n}}{\varDelta\:\stackrel{-}{V}}_{n}^{2}}{{\text{b}}^{6}}\right)}^{\raisebox{1ex}{$1$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}\) (6) Where \(\:{\nu\:}\) is the Poisson’s ratio, \(\:{\alpha\:}\) (0.123 in this work) represents a constant related to the value of the line tension of the dislocation (Γ = \(\:{\alpha\:}\) G𝑏²), G denotes the shear modulus, b represents the Burger’s vector and X n represents de fraction of n Th element in solid solution. The functions f \(\:\left({\text{W}}_{\text{c}}\right)\) , denoted as \(\:{\text{f}}_{1}\left({\text{W}}_{\text{c}}\right)\) and \(\:{\text{f}}_{2}\left({\text{W}}_{\text{c}}\right)\) , are termed to minimize dislocation core-related coefficients. They account for the curved nature of dislocations, which deviate to locate local energy minima; Varvenne employed values of 0.35 and 5.70 for \(\:{\text{f}}_{1}\left({\text{W}}_{\text{c}}\right)\) and \(\:{\text{f}}_{2}\left({\text{W}}_{\text{c}}\right)\) , respectively. In both equations, the pivotal term is \(\:\varDelta\:{\stackrel{-}{V}}_{n}\) , denoting the average volumetric misfit per atom. It is computed as the difference between the volume of the n th atom and the average atomic volume within the mixture. The atomic volume is derived from the FCC unit cell's volume divided by four. The resulting terms from equations 5 and 6 are then inserted into the subsequent equation, delineated as Eq. 7 , for the computation of yield strength. $$\:{\tau\:}_{y}\left(T,\dot{ϵ}\right)={\tau\:}_{0}exp\:\left(-\frac{1}{0.51}\frac{kT}{\varDelta\:{E}_{b}}ln\frac{\dot{{ϵ}_{0}}}{\dot{ϵ}}\right)$$ 7 Where k is the Boltzmann constant and \(\:\dot{{ϵ}_{0}}\) is a reference term for the strain rate, which was 10 − 3 s − 1 in this work. Hence, the Eq. 7 is a model for the SSS component to the critical resolved shear stress (converted into the respective contribution to the yield strength by the Taylor factor) that incorporates the dependence on strain rate and temperature. On other hand, Oh model suggests that the SSS component (σ ss ) in multicomponent alloys, comprising atoms from the 3d family, can be elucidated by the electronegativity difference between the constituent elements (Δχ), outlined as follows: $$\:{\sigma\:}_{ss\:}\propto\:\varDelta\:{\chi\:}\:$$ 8 More specifically, Oh model suggests that the SSS could be predicted by the following equation: $$\:{\sigma\:}_{ss}=\left(4293\pm\:448\right)\varDelta\:\chi\:+\left(84\pm\:37\right)\:MPa\:$$ 9 Therefore, taking into consideration Eq. 7 , Fig. 4 a presents the values of the SSS contribution alloys under study. The variation of A Av was used to express the difference in atomic volume in each alloy. Moreover, the σ ss value for pure Ni was considered to be 0. Moreover, based on Eq. 9 , Fig. 4 b illustrates the SSS contribution values as per Oh model. Figure 4 a shows that adding Pd to pure Ni results in higher SSS, compared to the addition of V, with Ni 50 Pd 50 demonstrating the highest SSS. Similarly, in the CrCoNi system, Cr 30 Co 30 Ni 30 Pd 10 exhibits higher SSS than with V addition. These results align with Varvenne model, which indicates that greater differences in A Av between constituent atoms lead to stronger SSS. On other hand, Fig. 4 b shows that adding V to pure Ni results in higher SSS than adding Pd, with Ni 63.2 Pd3 6.8 demonstrating the highest SSS. Similarly, in the CrCoNi system, Cr 30 Co 30 Ni 30 V 10 alloy shows higher SSS compared to Pd addition. These findings support Oh’s model, which suggests that greater Δχ differences result in stronger SSS. However, the model proposed in Eq. 9 should be applied with caution, once the intrinsic dispersion associated with the model (indicated in Fig. 4 b) could led to a widely distribution of σ ss values. Therefore, the inherent dispersion associated with the model and the choice of the electronegativity scale to be employed may lead to inconclusive results when applying Oh's model, potentially limiting its practicality. 3.5 EXCLUDING THE EFFECT OF GRAIN BOUNDARY STRENGTHENING In fact, Oh et al. [ 24 ] support their perspective by examining the σ y and σ uts values of the Ni 63.2 V 36.8 alloy, in contrast to the equiatomic CrCoNi and CrCoNiMnFe alloys (all with similar grain sizes). Notably, the Ni 63.2 V 36.8 alloy demonstrates higher yield strength and ultimate tensile strength values than the other alloys. However, in order to conduct an accurate analysis of the isolated contribution of the σ ss , it is essential to disregard the influence of grain refinement strengthening. This approach was adopted in the present study. According to the Hall-Petch model, the strengthening attributed to grain boundaries can be elucidated by Eq. 10 . $$\:\sigma\:={\sigma\:}_{0}\:+\:{K}_{hp}{d}^{-\text{0,5}}$$ 10 In this context, σ o represents the friction lattice stress, K hp is the Hall-Petch coefficient, and d denotes the grain size. Theoretically, σ 0 includes various contributions to material strength, but since only substitutional Solid Solution is considered here (all alloys are single-phase), we focus solely on the SSS contribution to σ 0 . Additionally, Eq. 10 can analyze the influence of grain size on hardness (H) by replacing σ o with H 0 . Therefore, examining Eq. 10 reveals that the same d can produce different hardness and σ y values across materials with different values for K hp . Consequently, assessing the σ ss contribution based on absolute σ y and σ uts for fine-grained materials may be misleading. Hence, to isolate the σ ss component, the grain boundary strengthening contribution K hp must be removed. This can be done using Hall-Petch plots, measuring hardness (or σ y ) for the same material at different grain sizes, as shown in Fig. 5 for the alloys assessed in this study. The σ y values were extracted from several stress-strain curves for each alloy, shown in Supplementary Fig. 7. Figures 5 a-b present hardness and σ y values, respectively, plotted against d − 0.5 for the alloys under study. Moreover, the dashed lines in both figures represent a linear regression (y = ax + b) based on the data for each material. Thus, by extrapolating the curve to an infinite grain size (d − 0.5 →0), it is possible to estimate the intrinsic hardness (or σ y ) of each alloy and, consequently, the SSS. Analyzing the data in Figs. 5 a-b, it is evident that both H o and σ 0 values for the binary alloys are higher than those for pure Ni (H o = 68.6 Hv and σ 0 = 14 MPa ) [ 35 , 36 ]. Additionally, the binary alloy with Pd shows higher H o and σ 0 values compared to those with V. This trend is also observed in the quaternary alloys, where those containing Pd and V exhibit higher H o and σ 0 values compared to the Cr 33 Co 33 Ni 33 alloy. These results suggest that adding V and Pd enhances the SSS of both pure Ni and the CrCoNi system. However, while V-based alloys show higher hardness and σ y , their H 0 and σ 0 values indicate that V contributes less to SSS compared to Pd. Examining the K hp values (extracted from hardness and σ y Hall-Petch plots) for the alloys it is revealed that Ni 63.2 V 36.8 has a significantly higher K hp than Ni 50 Pd 50 . Similar trends are observed in Cr 30 Co 30 Ni 30 Pd 10 and Cr 30 Co 30 Ni 30 V 10 alloys, with the V-based alloy showing a higher K hp than the Pd-based alloy. This indicates that grain boundary strengthening is much more pronounced in V-based alloys. Also, Ni 63.2 V 36.8 exhibited the highest grain boundary strengthening among the studied alloys. Thus, directly comparing absolute hardness and σ y values across different alloys is not feasible due to the varying contributions of grain boundary strengthening. Figures 6 a-b show the projected hardness and σ y of these alloys, assuming a uniform grain size of 10 µm, based on their respective K hp values. The Ni 63.2 V 36.8 alloy has higher hardness and σ y values at this grain size compared to other alloys. Despite Ni 50 P 50 exhibiting the lowest K hp for σ y (Fig. 5 b), its σ y value (Fig. 6 b) exceeds those of the ternary and quaternary alloys due to its highest σ 0 for a large range of grain sizes. This demonstrates that analyzing hardness or σ y values alone can lead to incorrect conclusions regarding the SSS effect in different metallic alloys. The K hp values should be approached with caution, as the same constant can exhibit different values depending on whether hardness or σ y is being analyzed. As illustrated in Figs. 5 a-b, except for the Ni 50 Pd 50 alloy, the K hp values associated with hardness are consistently higher than those for σ y . This discrepancy arises because hardness measurements are conducted in a localized region of the material, where the plastic deformation is restricted to only a few grains. Additionally, during hardness testing indentation, significant work-hardening effects occur, directly influencing the hardness values obtained. Specifically, the Ni 50 Pd 50 alloy exhibits an unusually low K hp for hardness (i.e., the hardness remains unchanged with increasing grain size), potentially indicating a low work-hardening coefficient for this alloy. Consequently, the mechanical behavior of the Ni 50 Pd 50 alloy requires more in-depth analysis. Figure 7 presents a comparison between the experimental and calculated σ 0 values using the Varvenne and Oh models, in Fig. 7 a and 7 b, respectively. The analysis indicates that, although both models fail to precisely predict the SSS, the theoretical σ ss values predicted by the Varvenne model exhibit a closer correlation to the experimental data. Hence, the analysis of the data presented so far shows not only that the Varvenne model is more suitable for predicting SSS, but also that the atomic volume difference between species in a solid solution is the more significant factor in SSS of FCC alloys. However, the Varvenne model has not yet been able to accurately predict the SSS component for all the alloys in study. This suggests that additional factors should be considered when developing models to predict SSS. For example, the dislocation line constants might change from case to case or maybe the non-linear variations in the atomic volume of each element. Interestingly, the results indicate that alloys exhibiting the highest δ values also demonstrated the highest SSS values, primarily observed in the binary alloy systems. (Supplementary Fig. 8). This indicates that, while solid solutions comprising a greater diversity of atoms may not always display the greatest lattice distortion, increased lattice distortions are likely responsible for the greater contribution to solid solution strengthening. It was also noted that, contrary to current literature on HEAs [ 5 , 11 ], SSS is not always the predominant strengthening mechanism, as shown in Fig. 8 . The SSS contribution values, obtained using supplementary Equations 1 and 2 , indicate that SSS becomes predominant (SSS > 50%) only above a critical grain size, while grain boundary strengthening (GBS) dominates below this size. This critical grain size varies for each alloy and is detailed in Supplementary Table 4. Notably, the Ni 50 Pd 50 alloy, due to its relatively low K hp value, has an extremely small critical grain size (Supplementary Fig. 9), making SSS nearly the sole hardening mechanism. These findings highlight that much work has yet to be done to fully understand SSS for different alloys, and suggest the Ni 50 Pd 50 alloy promising material for future fundamental metallurgical studies due to its minimal mechanical property sensitivity to grain growth. 4 CONCLUSIONS This study successfully designed and produced FCC solid solution alloys with significant differences in atomic volume or electronegativity among the constituent elements. The results indicate that the atomic volume difference between elements in a solid solution has a greater influence on the SSS than the electronegativity difference. Due to the inherent characteristics of each solid solution system, such as the fraction of each element and the accompanying atomic volume variation with changes in elemental fractions, each solid solution system exhibits unique behavior. Consequently, developing a general model capable of predicting SSS remains a challenging task. For a given solid solution system, there is a critical grain size below which grain boundary strengthening plays a major role in yield strength and hardness, compared to SSS. It was also concluded that both predicted lattice distortion and the measured SSS component were greater for binary alloys. This provides further evidence that increasing the number of elements in solid solution not always leads to greater lattice distortion, and consequently, higher hardness and yield strength. Finally, further in-depth studies on the Ni 50 P 50 alloy are warranted because its hardness and yield strength exhibit minimal variation with grain growth, making it promising from an engineering perspective. Declarations Data availability The raw/processed data required to reproduce these findings cannot be fully shared at this time due to technical or time limitations. References He QF, Wang JG, Chen HA, Ding ZY, Zhou ZQ, Xiong LH, Luan JH, Pelletier JM, Qiao JC, Wang Q, Fan LL, Ren Y, Zeng QS, Liu CT, Pao CW, Srolovitz DJ, Yang Y (2022) A highly distorted ultraelastic chemically complex Elinvar alloy. Nature 602:251–257 Utt D, Lee S, Xing Y, Jeong H, Stukowski A, Oh SH, Dehm G, Albe K (2022) The origin of jerky dislocation motion in high-entropy alloys. Nat Commun 13:4777 Li Y, Liu X, Zhang P, Han Y, Huang M, Wan C (2022) Theoretical insights into the Peierls plasticity in SrTiO3 ceramics via dislocation remodelling. Nat Commun 13:6925 Liu G, Cheng X, Wang J, Chen K, Shen Y (2017) Atomically informed nonlocal semi-discrete variational Peierls-Nabarro model for planar core dislocations. Sci Rep 7:43785 Tandoc C, Hu Y-J, Qi L, Liaw PK (2023) Mining of lattice distortion, strength, and intrinsic ductility of refractory high entropy alloys. npj Comput Mater 9:53 Zhang Z, Zeng Q, Wang N, Wang L, Wu Q, Li X, Tang J, Li R (2024) Influence of nano-BN inclusion and mechanism involved on aluminium-copper alloy. Sci Rep 14:6372 He X, Zhang Y, Gu X, Wang J, Qi J, Hao J, Wang L, Huang H, Wen M, Zhang K, Zheng W (2023) Pt-induced atomic-level tailoring towards paracrystalline high-entropy alloy. Nat Commun 14:775 Varvenne C, Luque A, Curtin WA (2016) Theory of strengthening in fcc high entropy alloys. Acta Mater 118:164–176 Oliveira PHF, Magalhães DCC, Izumi MT, Cintho OM, Kliauga AM, Sordi VL (2021) Evolution of dislocation and stacking-fault densities for a Cu-0.7Cr-0.07Zr alloy during cryogenic tensile test: An in-situ synchrotron X-ray diffraction analysis. Mater Sci Engineering: A 813:141154 Roy A, Sreeramagiri P, Babuska T, Krick B, Ray PK, Balasubramanian G (2021) Lattice distortion as an estimator of solid solution strengthening in high-entropy alloys. Mater Charact 172:110877 Zhang HL, Cai DD, Sun X, Huang H, Lu S, Wang YZ, Hu QM, Vitos L, Ding XD (2022) Solid solution strengthening of high-entropy alloys from first-principles study. J Mater Sci Technol 121:105–116 Winkens G, Kauffmann A, Herrmann J, Czerny AK, Obert S, Seils S, Boll T, Baruffi C, Rao Y, Curtin WA, Schwaiger R, Heilmaier M (2023) The influence of lattice misfit on screw and edge dislocation-controlled solid solution strengthening in Mo-Ti alloys. Commun Mater 4:26 Li H, Zong H, Li S, Jin S, Chen Y, Cabral MJ, Chen B, Huang Q, Chen Y, Ren Y, Yu K, Han S, Ding X, Sha G, Lian J, Liao X, Ma E, Sun J (2022) Uniting tensile ductility with ultrahigh strength via composition undulation. Nature 604:273–279 Fleischer RL (1963) Substitutional solution hardening. Acta Metall 11:203–209 Labusch R (1972) Statistische theorien der mischkristallhärtung. Acta Metall 20:917–927 Labusch R (1970) A Statistical Theory of Solid Solution Hardening, physica status solidi (b). 41:659–669 Freudenberger J, Thiel F, Utt D, Albe K, Kauffmann A, Seils S, Heilmaier M (2022) Solid solution strengthening in medium- to high-entropy alloys. Mater Sci Engineering: A 861:144271 Pei Z, Zhao S, Detrois M, Jablonski PD, Hawk JA, Alman DE, Asta M, Minor AM, Gao MC (2023) Theory-guided design of high-entropy alloys with enhanced strength-ductility synergy. Nat Commun 14:2519 Tsuru T, Han S, Matsuura S, Chen Z, Kishida K, Iobzenko I, Rao SI, Woodward C, George EP, Inui H (2024) Intrinsic factors responsible for brittle versus ductile nature of refractory high-entropy alloys. Nat Commun 15:1706 Huang C-W, Su P-Y, Yu C-H, Wang C-L, Lo Y-C, Jang JS-C, Hu H-T (2023) A micromechanical study on the effects of precipitation on the mechanical properties of CoCrFeMnNi high entropy alloys with various annealing temperatures. Sci Rep 13:3379 Hsu W-L, Tsai C-W, Yeh A-C, Yeh J-W (2024) Clarifying the four core effects of high-entropy materials. Nat Reviews Chem 8:471–485 Maresca F, Curtin WA (2020) Mechanistic origin of high strength in refractory BCC high entropy alloys up to 1900K. Acta Mater 182:235–249 Toda-Caraballo I (2015) Rivera-Díaz-del-Castillo, Modelling solid solution hardening in high entropy alloys. Acta Mater 85:14–23 Oh HS, Kim SJ, Odbadrakh K, Ryu WH, Yoon KN, Mu S, Körmann F, Ikeda Y, Tasan CC, Raabe D, Egami T, Park ES (2019) Engineering atomic-level complexity in high-entropy and complex concentrated alloys, Nature Communications, 10 2090 Wen C, Wang C, Zhang Y, Antonov S, Xue D, Lookman T, Su Y (2021) Modeling solid solution strengthening in high entropy alloys using machine learning. Acta Mater 212:116917 PEARSON WB (1986) Pearson’s Handbook of Crystallographic Data for Intermetallic Phases. Pergamon Zhang Y, Zhou YJ, Lin JP, Chen GL, Liaw PK (2008) Solid-Solution Phase Formation Rules for Multi-component Alloys. Adv Eng Mater 10:534–538 Tanner LE (1972) The ordering transformation in Ni2V. Acta Metall 20:1197–1227 Kras̈evec V (1974) On the relief of ordering strains by twinning in Ni1.05Mn0.95 alloy. Mater Res Bull 9:1357–1361 Tanner LEJD (1968) The Ordering of Ni3V Singh JB, Sundararaman M, Mukhopadhyay P (2004) Propagation of twins across Ni2V/Ni3V coherent interfaces in a two phase Ni–29at%V alloy. Scripta Mater 51:693–698 Cullity BD (2001) Elements of X-ray Diffraction, Third Edition, Prentice-Hall, New York Mishima Y, Ochiai S, Suzuki T (1985) Lattice parameters of Ni(γ), Ni3Al(γ') and Ni3Ga(γ') solid solutions with additions of transition and B-subgroup elements. Acta Metall 33:1161–1169 Owen LR, Jones NG (2018) Lattice distortions in high-entropy alloys. J Mater Res 33:2954–2969 Owen LR, Pickering EJ, Playford HY, Stone HJ, Tucker MG, Jones NG (2017) An assessment of the lattice strain in the CrMnFeCoNi high-entropy alloy. Acta Mater 122:11–18 Keller C, Hug E (2008) Hall–Petch behaviour of Ni polycrystals with a few grains per thickness. Mater Lett 62:1718–1720 Additional Declarations There is NO Competing Interest. Supplementary Files Supplementarymaterial.docx Supplementary Dataset 1 Cite Share Download PDF Status: Published Journal Publication published 06 Oct, 2025 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5003860","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":354491371,"identity":"8ea38216-82fd-4d6b-8b26-bea33406f69f","order_by":0,"name":"Pedro Oliveira","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYHACAxAhwwYkDjAw2DCwN4BFDxDUwgPVksbAc4BYLVDOYcJazNmbt334wGDDwyeRfPBwRc35xB7+ww8/VzDcycelxbLnWPHMGQxpPGwSaQkHzxy7ndgjkWYseYbhmWUDLlfdyDFm5mE4zMPGc8bgYAPb7cT9EjwMkg0Mhw1weuT+G5iW8x8ONvw7B3TYGeafeLXc4IFqYe9hONjYdiCxhyGHDa8tlj1pxYwzDIB+YW8zONjYl2wM9IuZZYPBM5xazNkPb2b4UGEjJ9/M/Phjwzc7WWCIPb7ZUHEHt8OQSExxPFpGwSgYBaNgFOADAH79VGVtuyhfAAAAAElFTkSuQmCC","orcid":"","institution":"Federal University of Sao Carlos","correspondingAuthor":true,"prefix":"","firstName":"Pedro","middleName":"","lastName":"Oliveira","suffix":""},{"id":354491372,"identity":"f041b591-fd41-4d88-9a79-c1a23ab2355c","order_by":1,"name":"Caio Martins","email":"","orcid":"","institution":"Federal University of Sao Carlos","correspondingAuthor":false,"prefix":"","firstName":"Caio","middleName":"","lastName":"Martins","suffix":""},{"id":354491373,"identity":"eee7fb9b-3f19-4270-8723-a28f7f1c5a69","order_by":2,"name":"Guilherme Stumpf","email":"","orcid":"https://orcid.org/0000-0002-9457-1968","institution":"Federal University of São Carlos","correspondingAuthor":false,"prefix":"","firstName":"Guilherme","middleName":"","lastName":"Stumpf","suffix":""},{"id":354491374,"identity":"8f665eb9-f7e7-4d90-ae9a-5a0a0953372f","order_by":3,"name":"Julio Spadotto","email":"","orcid":"","institution":"The University of Manchester","correspondingAuthor":false,"prefix":"","firstName":"Julio","middleName":"","lastName":"Spadotto","suffix":""},{"id":354491375,"identity":"93e0bcfe-80bc-4e53-b1a4-14521591540e","order_by":4,"name":"Ed Pickering","email":"","orcid":"https://orcid.org/0000-0002-7516-868X","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Ed","middleName":"","lastName":"Pickering","suffix":""},{"id":354491376,"identity":"d22437b5-450d-4803-a5de-60398dedee32","order_by":5,"name":"Walter Botta","email":"","orcid":"","institution":"Universidade Federal de São Carlos","correspondingAuthor":false,"prefix":"","firstName":"Walter","middleName":"","lastName":"Botta","suffix":""},{"id":354491377,"identity":"8085d2c2-3eee-47aa-ab60-43a7e41096b6","order_by":6,"name":"Claudemiro Bolfarini","email":"","orcid":"https://orcid.org/0000-0002-3099-3694","institution":"Federal University of Sao Carlos","correspondingAuthor":false,"prefix":"","firstName":"Claudemiro","middleName":"","lastName":"Bolfarini","suffix":""},{"id":354491378,"identity":"90731a92-c7f5-47a1-b409-886b2ea59bdd","order_by":7,"name":"Francisco Coury","email":"","orcid":"https://orcid.org/0000-0002-0457-2087","institution":"Federal Univeristy of São Carlos","correspondingAuthor":false,"prefix":"","firstName":"Francisco","middleName":"","lastName":"Coury","suffix":""}],"badges":[],"createdAt":"2024-08-30 12:20:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5003860/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5003860/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41467-025-63900-6","type":"published","date":"2025-10-06T04:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":64728406,"identity":"45ba187c-e010-452b-b580-ac0dea33825a","added_by":"auto","created_at":"2024-09-18 06:15:31","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":88971,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Variation in electronegativity (Allen scale) vs. atomic volume when adding elements to pure Ni, using Ni values as benchmarks. Atomic Volume were estimated based on atomic radius of pure elements [26], (b) Variation in crystalline lattice distortion (δ) vs. electronegativity difference (Δχ) for the chosen alloys, with estimated values for δ and Δχ according to [27] and [24], respectively.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/f9d28b3be575fe9168da7aee.png"},{"id":64728397,"identity":"84e1cb10-9273-4cc9-b828-ec0cab4b1c4b","added_by":"auto","created_at":"2024-09-18 06:15:29","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":934238,"visible":true,"origin":"","legend":"\u003cp\u003eCALHPHAD isopleths for the (a) Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, (b) Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, (c) Cr\u003csub\u003e33.3\u003c/sub\u003eCo\u003csub\u003e33.3\u003c/sub\u003eN\u003csub\u003e33.3\u003c/sub\u003e, (d) Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e and (e) Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloys.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/ae74abe6f3c3ebf57ddaef25.png"},{"id":64728403,"identity":"86954b02-bf25-4a94-8135-e6a649b7e244","added_by":"auto","created_at":"2024-09-18 06:15:31","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":1329062,"visible":true,"origin":"","legend":"\u003cp\u003eX-ray diffraction results confirming the single-phase FCC structure for the (a) Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, (b) Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, (c) Cr\u003csub\u003e33.3\u003c/sub\u003eCo\u003csub\u003e33.3\u003c/sub\u003eN\u003csub\u003e33.3\u003c/sub\u003e, (d) Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e, and (e) Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10 \u003c/sub\u003ealloys. The respective microstructures of each alloy, evaluated via EBSD, are depicted in the insets of each figure.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/fe1513f339b66c0546280091.png"},{"id":64728394,"identity":"d6fa8242-4ee9-4b0e-850d-97eaf3f42987","added_by":"auto","created_at":"2024-09-18 06:15:29","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":403627,"visible":true,"origin":"","legend":"\u003cp\u003e(a) σ\u003csub\u003ess\u003c/sub\u003e X A\u003csub\u003eAv\u003c/sub\u003e according to the Varvenne model [8] and (b) σ\u003csub\u003ess\u003c/sub\u003e X Δ\u003csub\u003eΧ\u003c/sub\u003e according to the Oh [24] model for the Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8,\u003c/sub\u003e Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10 \u003c/sub\u003ealloys\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/f3aafcbab3916058d8cb37e1.png"},{"id":64728417,"identity":"c0ee64c1-3da5-4860-bc5d-186dada15d69","added_by":"auto","created_at":"2024-09-18 06:15:32","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":645959,"visible":true,"origin":"","legend":"\u003cp\u003eHall Petch plots for Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloys. The Values of Kh and H0 were obtained through a linear regression of the relationship between hardness and d\u003csup\u003e-0,5 \u003c/sup\u003efor each alloy.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/8f56967f639f13ed9120a951.png"},{"id":64728416,"identity":"816a9f36-edfc-45aa-a8a9-50a23daae27a","added_by":"auto","created_at":"2024-09-18 06:15:31","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":311546,"visible":true,"origin":"","legend":"\u003cp\u003eHypothetical hardness, considering the K\u003csub\u003ehp\u003c/sub\u003e values (Figure 5) if the alloys Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e had the same grain size (d=10 um).\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/0a7e0211d777c9b637e52ab9.png"},{"id":64728398,"identity":"8137f1a2-d99a-4eaa-a0e9-e6dea527bae0","added_by":"auto","created_at":"2024-09-18 06:15:30","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":397477,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between the experimental and predicted yield strengths (σ\u003csub\u003e0\u003c/sub\u003e) for the five alloys studied in this work using (a) the Varvenne and (b) Oh models.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/81727f389632bb01738de6e2.png"},{"id":64728407,"identity":"84a3e4c9-b13b-451a-891d-9a98f726d8a1","added_by":"auto","created_at":"2024-09-18 06:15:31","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":105300,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of the SSS component as a function of grain size in the (a) Hardness and (b) σ\u003csub\u003ey\u003c/sub\u003e of the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloys.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/917564a4ce71c8690c8acdb6.png"},{"id":92921828,"identity":"3f9c4e2d-313e-4b62-9617-547e561d4b52","added_by":"auto","created_at":"2025-10-07 07:09:27","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5018054,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/f4d84313-1d1b-4b10-b626-391bf3c4a3d6.pdf"},{"id":64729168,"identity":"5db50f57-e91f-490a-8c00-ca226bf595b1","added_by":"auto","created_at":"2024-09-18 06:23:31","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":5158180,"visible":true,"origin":"","legend":"Supplementary Dataset 1","description":"","filename":"Supplementarymaterial.docx","url":"https://assets-eu.researchsquare.com/files/rs-5003860/v1/ee02d617bf722fde80ce3bcd.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening","fulltext":[{"header":"1. INTRODUCTION","content":"\u003cp\u003eThe lattice friction stress, quantified by the Peierls stress [\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], describes the resistance a single dislocation faces while moving through an alloy's lattice, which is closely linked to atomic-scale lattice distortions. In metallic alloys with solid solution, numerous solute atoms with varying sizes and properties create distorted lattices, resulting in extensive lattice friction [\u003cspan additionalcitationids=\"CR6\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. This distortion raises the energy barrier against dislocation mobility, thereby strengthening the material [\u003cspan additionalcitationids=\"CR9 CR10 CR11 CR12\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eImportant pioneering models to quantify solid solution strengthening (SSS) were originally introduced by Fleischer [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e], latter followed by Labusch [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. According to these models, solutes can be effectively represented as point defects that will distort the crystal lattice, giving rise to a solute-dislocation elastic interaction energy, which in turn is the primary factor governing the SSS mechanism. This interaction energy is the product of the dislocation pressure field and the misfit volume of the solute and solvent, which can be defined as the difference between the volume of the solute in the alloy and the average atomic volume. However, the models proposed by Labusch and Fleischer take into account only solutions with solvents up to 10 at% in a single component matrix [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn the context of High Entropy Alloys (HEAs) [\u003cspan additionalcitationids=\"CR19 CR20\" citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], distinguishing between solvent and solute atoms is challenging due to the random occupation of lattice sites by multiple atom species, leading some solid solution models to perform inadequately. Researchers have recently developed models to explain SSS in HEAs, which rely heavily on input parameters like atomic size and shear modulus mismatch [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e], and more recently, electronegativity differences [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFor example, the Varvenne model [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e], based on first principles, posits that differences in atomic volume control the mechanical strength of alloys by calculating the interaction energy between dislocations and solute atoms. This energy is integrated into a conventional equation to account for thermal effects, yielding a comprehensive representation of critical resolved shear stress and thus, yield strength. The Varvenne model also takes into account the shear modulus and poisson\u0026rsquo;s ratio of each element in its final calculation to obtain the SSS component. This model's ability to accommodate strain rate and temperature variations enhances its predictive accuracy. Conversely, Oh\u0026rsquo;s model [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] attributes SSS in HEAs to differences in electronegativity among the constituent elements. Using a quantum-mechanical approach, it approximates atomic-level pressure to identify optimal element combinations for high SSS, focusing on electronegativity disparities rather than atomic volume differences.\u003c/p\u003e \u003cp\u003eAlthough both the Varvenne [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and Oh [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] models are applied to FCC systems, systems, they are fundamentally different. Hence, the present study aimed to solve the conflict between the two models, i.e which factor most enhances the SSS, by developing different alloys, engineered to have significant differences in electronegativity and/or atomic volume. Furthermore, given the growing interest in high-strength HEAs for engineering applications, this study aims to resolve the conflict between these models. To achieve this, an analysis of various 3d transition metal elements was conducted, focusing on those that form a solid solution in Ni and provide significant differences in electronegativity and/or atomic volume compared to pure Ni.\u003c/p\u003e \u003cp\u003eV was chosen for its high electronegativity difference to Ni, while Pd was selected for its smaller electronegativity impact but greater atomic volume difference. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea illustrates the selection process, highlighting the 3d elements forming an appreciable solid solution in Ni (above 30 at. % solute). To analyze the implications of adding Pd and V and their effects on the solid solution system, the equiatomic CrCoNi was also taken as a benchmark.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe alloys used here were Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e, and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e, chosen for significant lattice distortion and/or electronegativity differences (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb). Despite Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e being unlikely to form a solid solution under many typical processing conditions [\u003cspan additionalcitationids=\"CR29 CR30\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], it was included for comparison with the model by Oh et al. [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. These alloys are ideal for testing both models due to their variations in electronegativity and atomic volume.\u003c/p\u003e \u003cp\u003eIt is worth mentioning that the selection of Pd sets this study apart from previous research efforts concentrated on the inclusion of other 3d transition elements, which commonly share comparable atomic volumes. Finally, this study aims not only to resolve the scientific conflict between the Varvenne [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] and Oh [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] models but also to propose a more accurate strategy towards alloy design, focused on optimizing mechanical properties with greater precision.\u003c/p\u003e"},{"header":"2. METHODS","content":"\u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eTo analyze the implications of the addition of Pd and V, and their effect on the SSS, it was crucial to take a pure element and a simple solid solution system as benchmark. In this work, as mentioned previously, Ni was used as the pure element and the CrCoNi was the solid solution system\u003c/p\u003e \u003cp\u003eComputational thermodynamic calculations using the CALPHAD method identified the FCC single-phase regions for the alloys Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e, and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e, utilizing the ThermoCalc\u0026reg; software and the TCHEA 5 and TCNI11 databases. These alloys, along with pure Ni, were prepared by arc melting high-purity elements under argon atmosphere and underwent multiple re-melting cycles for compositional homogeneity. To achieve a single-phase, fine-grained microstructure with low dislocation density, the ingots were homogenized, water quenched, cold rolled, annealed, and quenched again. Specific thermomechanical treatments are detailed indicated in Supplementary Table\u0026nbsp;1, with Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e undergoing additional reduction to disrupt its eutectoid structure before the treatment in Supplementary Table\u0026nbsp;1. For each heat treatment, samples were encapsulated in vacuum-filled quartz ampoules to mitigate oxidation during high-temperature processing.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e \u003cp\u003ePhase structure analysis of recrystallized alloys and pure Ni samples was performed using high-energy X-ray diffraction (HE-XRD) at Petra III, DESY in Germany. A monochromatic X-ray beam at 87 keV and a Perkin Elmer 2D detector captured the Debye-Scherrer rings. The setup included a 700 x 700 \u0026micro;m\u0026sup2; beam and a 1.226 m sample-to-detector distance, calibrated with LaB6 powder. The 2D images were converted to 1D diffraction patterns via radial integration, and single-peak fitting was done using GSAS II software. Rietveld refinement was also applied for full spectrum fitting to determine lattice parameters.\u003c/p\u003e \u003cp\u003eScanning electron microscopy (SEM) and energy-dispersive X-ray spectroscopy (EDS) analyses were conducted on polished samples using a TESCAN MIRA FEG microscope, operating at 20 kV. Additionally, electron backscatter diffraction (EBSD) was performed on samples at 30 kV. EBSD data were processed using ATEX software\u0026copy;.\u003c/p\u003e \u003cp\u003eUniaxial tensile tests were conducted at room temperature using an INSTRON machine, following ASTM E8 standards with an optical extensometer and a nominal strain rate of 10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003es\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e. Flat dog bone-shaped tensile specimens (1.0 mm thick) were prepared by electrical discharge machining, aligned parallel to the rolling direction, and yield strength was determined using the 0.2% proof stress criterion. To evaluate solid solution behavior, Hall-Petch plots were created by measuring hardness and yield strength for different grain sizes. Each alloy underwent various annealing treatments to achieve different grain sizes, and Vickers Hardness measurements were taken on recrystallized samples with a 100 g load and a 15-second dwell time.\u003c/p\u003e"},{"header":"3. RESULTS AND DISCUSSION","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 MICROSTRUCTURAL CHARACTERIZATION\u003c/h2\u003e \u003cp\u003eThe thermodynamic calculations depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea-e illustrate the FCC single-phase field for the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e, Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloys. Subsequently, Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e displays the X-ray diffraction (XRD) patterns, for the recrystallized samples, confirming the exclusive presence of the FCC phase in all samples and the correspondent inverse pole figure (IPF-Z) maps from EBSD analysis showing the microstructure of the alloys.\u003c/p\u003e \u003cp\u003eFor comparison, XRD for pure Ni is shown in Supplementary Fig.\u0026nbsp;1. The SEM-EDS elemental distribution maps showcased in Supplementary Figs.\u0026nbsp;2\u0026ndash;6 revealed the compositional homogeneity of the recrystallized microstructure at the micro scale on each alloy.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 INFLUENCE OF ATOMIC VOLUME\u003c/h2\u003e \u003cp\u003eAnalysis of the diffractograms in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea-b shows that adding Pd and V to pure Ni causes a noticeable shift of all reflections to lower 2θ values compared to pure Ni (Supplementary Fig.\u0026nbsp;1). A similar trend is seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ed-e with the addition of Pd and V to the CrCoNi system (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec). According to Bragg's law [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e], these shifts to lower 2θ values are related to an increase in atomic radius (atomic volume).\u003c/p\u003e \u003cp\u003eAccurately determining atomic volume in solid solutions with multiple atoms is challenging due to variations in atomic radii depending on the environment of which a certain atom is inserted. For example, α-Fe (BCC) and γ-Fe (FCC) have different radii (1.239 \u0026Aring; and 1.287 \u0026Aring;, respectively), as do α and β Ti in different structures (1.475 \u0026Aring; and 1.432 \u0026Aring;, respectively) [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. Pure Ni (FCC) has a radius of 1.243\u0026Aring;, but a radius of 1.257 \u0026Aring; in a binary FCC solution with Cr (30 at. %) [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. To address this, we propose using the concept of apparent atomic volume (A\u003csub\u003eav\u003c/sub\u003e), which relies on the principles that the unit cell represents the entire system and contains atoms in the same proportion as the solid solution.\u003c/p\u003e \u003cp\u003eConsidering a system with a composition of Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, forming an FCC solid solution, the atomic volume of the unit cell is crucial because it must include the three elements in the same stoichiometry as the system. Thus, each element\u0026mdash;Cr, Co, and Ni\u0026mdash;occupies approximately one third of the atomic volume in the cell. For an FCC cell, and assuming the simplified spherical geometry for the atoms, the following relationship is established:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:0.74=\\frac{QV}{{a}^{3}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:Q\\)\u003c/span\u003e\u003c/span\u003e is the number of atoms in a FCC unit cell, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:V\\)\u003c/span\u003e\u003c/span\u003e is the volume of one atom, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{a}^{3}\\)\u003c/span\u003e\u003c/span\u003e is the unit cell volume. Thus, taking into account that an FCC cell is occupied by four atoms, the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{A}_{av}\\)\u003c/span\u003e\u003c/span\u003e might be expressed as:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:{A}_{av}=\\frac{\\text{0,74}{a}^{3}}{4}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTherefore, for solid solutions, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{A}_{av}\\)\u003c/span\u003e\u003c/span\u003e provides an effective and functional measure of the volume occupied by atoms in a unit cell, regardless of the quantity of atoms composing the solid solution. It should be noted that atoms are not spheres and the volume occupied by a single atom might be the entire 1/4th of the FCC unit cell if the atom is now viewed in this way. However, the simplification of taking atoms as spheres is convenient to directly apply the results to many equations in the literature that consider the atomic radii\u0026rsquo; as the intrinsic measurement for the atomic volume and either option, atomic radii or volume, will lead to the same outcome using these models.\u003c/p\u003e \u003cp\u003eFurthermore, a distorted lattice may result if a significant difference in atomic sizes is present. The distortion generated in the crystalline lattice (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\delta\\:\\)\u003c/span\u003e\u003c/span\u003e) can be estimated by the model proposed by Zhang et al.[\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\delta\\:=\\:\\sqrt{\\sum\\:_{i=1\\:\\:\\:}^{n}{c}_{i}\\:{\\left(1-{r}_{i}/\\stackrel{-}{r}\\right)}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere n is the total number of elements in solid solution, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{c}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the atomic fraction of the i\u003csup\u003eth\u003c/sup\u003e element, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{r}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the atomic radius of element i and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\stackrel{-}{r}\\)\u003c/span\u003e\u003c/span\u003e (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\sum\\:_{i=1}^{n}{c}_{i}{r}_{i})\\)\u003c/span\u003e\u003c/span\u003e is the average atomic radius. The \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\delta\\:\\)\u003c/span\u003e\u003c/span\u003e parameter is very common in the HEA literature to describe the difference in atomic radii in solid solutions.\u003c/p\u003e \u003cp\u003eUsing the XRD results shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the apparent atomic volume (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{A}_{av}\\)\u003c/span\u003e\u003c/span\u003e) was calculated for the investigated materials, detailed in Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. These tables also present the δ values for pure Ni and the CrCoNi system, including changes from adding Pd and V. For precise δ determination, we used \u0026lsquo;solution atomic radii\u0026rsquo;. These radii represent the size that an element would have in a face-centered cubic (FCC) system. These values were derived from binary FCC alloys of the element with Ni [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e], as detailed in Supplementary Table\u0026nbsp;2.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u0026ndash; Lattice parameter (a), variation in the lattice parameter (Δa), apparent atomic volume (A\u003csub\u003eav\u003c/sub\u003e), variation in the apparent atomic volume (ΔA\u003csub\u003eav\u003c/sub\u003e), and crystalline lattice distortion (δ) for pure Ni, Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e, and Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e alloys.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaterial\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ea (\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eΔa* (%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eA\u003csub\u003eav\u003c/sub\u003e (\u0026Aring;\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eΔA\u003csub\u003eav\u003c/sub\u003e**(%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eδx100\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNi\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.5246\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.1003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNi\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7396\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e9.6747\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e19.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.97\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNi\u003csub\u003e63,2\u003c/sub\u003eV\u003csub\u003e36,8\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.6094\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.6988\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e7.39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e4.53\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLattice parameter (a), variation in the lattice parameter (Δa), apparent atomic volume (A\u003csub\u003eav\u003c/sub\u003e), variation in the apparent atomic volume (ΔA\u003csub\u003eav\u003c/sub\u003e), and crystalline lattice distortion (δ) for Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e, and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloys.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaterial\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ea (\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eΔa* (%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eA\u003csub\u003eav\u003c/sub\u003e (\u0026Aring;\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eΔA\u003csub\u003eav\u003c/sub\u003e**(%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eδx100\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.5621\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.3616\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.40\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.6091\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.6971\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.5841\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.5176\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.87\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.94\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e reveals that adding Pd to pure Ni increases A\u003csub\u003eav\u003c/sub\u003e by 19.44%, much more than the 7.39% increase from adding V. The δ values are also higher with Pd additions. Similarly, Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows that Pd increases A\u003csub\u003eav\u003c/sub\u003e by 4.01% in the CrCoNi system, compared to 1.87% for V, with higher \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\delta\\:\\)\u003c/span\u003e\u003c/span\u003e values for Pd as well. This larger increase in A\u003csub\u003eav\u003c/sub\u003e and δ when adding Pd to both Ni and CrCoNi is due to Pd's larger atomic size compared to V. Consequently, considering only atomic volume differences, alloys with Pd are predicted to exhibit stronger SSS due to greater lattice distortion.\u003c/p\u003e \u003cp\u003eMoreover, it is worth noting that these findings corroborate what has been shown before by other authors [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e], indicating that the presence of a greater number of elements in solid solution does not invariably induce larger distortions in the crystal lattice. This is exemplified by the higher values of A\u003csub\u003eav\u003c/sub\u003e and δ observed for the binary alloys in comparison to their counterparts in the ternary and quaternary alloy systems.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 INFLUENCE OF ELECTRONEGATIVITY\u003c/h2\u003e \u003cp\u003eBased on the assumption that it's possible to predict configurational fluctuations of charge transfer and atomic-level pressure, the model introduced by Oh et al. [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] depends on the average charge transfer of each element in a solid solution, approximated by the local difference in electronegativity. The electronegativity difference between the constituent elements (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\chi\\:}\\)\u003c/span\u003e\u003c/span\u003e) can be calculated as follows:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:\\varDelta\\:{\\chi\\:}=\\sqrt{\\sum\\:_{x}{c}_{x}{\\left({{\\chi\\:}}_{x}-{\u0026lang;{\\chi\\:}\u0026rang;}_{element}\\right)}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\chi\\:}}_{x}\\)\u003c/span\u003e\u003c/span\u003e represents the electronegativity of element X, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\u0026lang;{\\chi\\:}\u0026rang;}_{element}\\)\u003c/span\u003e\u003c/span\u003e stands for the weighted average electronegativity across the element. Using the model proposed by Oh et al. [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\chi\\:}\\)\u003c/span\u003e\u003c/span\u003e values were calculated for the alloys under investigation and are presented in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. In their model, the authors claim the Allen electronegativity scale should be the one to used. The values of electronegativity used for each element are listed in Supplementary Table\u0026nbsp;2.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eElectronegativity difference caused by the addition of Pd and V to pure Ni\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaterial\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eδχ\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNi\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNi\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.150\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNi\u003csub\u003e63,2\u003c/sub\u003eV\u003csub\u003e36,8\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.175\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eElectronegativity difference caused by the addition of Pd and V to CrCoNi system\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaterial\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eδχ\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.100\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.118\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.126\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTables\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e show that the introduction of V into pure Ni results in slightly higher \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\chi\\:}\\)\u003c/span\u003e\u003c/span\u003e values compared to the addition of Pd. Similarly, V addition to the CrCoNi system leads to a higher \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\chi\\:}\\)\u003c/span\u003e\u003c/span\u003e values than the addition of Pd. According to the Oh model [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], alloys with V should exhibit stronger SSS due to the greater \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\chi\\:}\\)\u003c/span\u003e\u003c/span\u003e values. It is important to note that the values presented in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e may vary depending on the electronegativity scale used. For example, based on the electronegativity according to the Pauling scale, the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy would exhibit the greatest \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\chi\\:}\\)\u003c/span\u003e\u003c/span\u003e values (Supplementary Tables\u0026nbsp;2 and 3).\u003c/p\u003e \u003cp\u003eIn light of the exposition thus far, it becomes clear that the models proposed by Varvenne and Oh contain a fundamental contradiction. According to the Varvenne model, alloys containing palladium (Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e) should exhibit the highest contribution of SSS. Conversely, the Oh model posits alloys with vanadium addition (Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e) should have the greatest SSS contribution. Therefore, the following section delves into both models and attempts to clarify the fundamental difference between them.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.4 PREDICTION OF SSS SOLUTION USING THE EVALUATED MODELS\u003c/h2\u003e \u003cp\u003eThe Varvenne model evaluates the energy associated with the interaction between a dislocation and a solute atom. This energy calculation is subsequently included in a standard equation to adapt to thermally induced deformation. As a result, the determined interaction energy becomes part of an equation to handle thermally driven deformation. The resulting outcome is a model for the SSS component in the yield strength, which considers the impact of strain rate and temperature. This final formulation offers a direct method for calculating the activation energy needed for dislocation movement (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varDelta\\:\\text{E}}_{\\text{b}}\\)\u003c/span\u003e\u003c/span\u003e) and the Peierls stress at absolute zero (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\tau\\:}}_{0}\\)\u003c/span\u003e\u003c/span\u003e), as expressed in the following equations:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:{{\\tau\\:}}_{0}=0.051{{\\alpha\\:}}^{-\\raisebox{1ex}{$1$}\\!\\left/\\:\\!\\raisebox{-1ex}{$3$}\\right.}\\text{G}{\\left(\\frac{1+{\\nu\\:}}{1-{\\nu\\:}}\\right)}^{\\raisebox{1ex}{$4$}\\!\\left/\\:\\!\\raisebox{-1ex}{$3$}\\right.}{\\text{f}}_{1}\\left({\\text{W}}_{\\text{c}}\\right){\\left(\\sum\\:_{\\text{n}}\\frac{{\\text{x}}_{\\text{n}}{\\varDelta\\:\\stackrel{-}{V}}_{n}^{2}}{{\\text{b}}^{6}}\\right)}^{\\raisebox{1ex}{$2$}\\!\\left/\\:\\!\\raisebox{-1ex}{$2$}\\right.}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varDelta\\:\\text{E}}_{\\text{b}}=0.274{{\\alpha\\:}}^{\\raisebox{1ex}{$1$}\\!\\left/\\:\\!\\raisebox{-1ex}{$3$}\\right.}\\text{G}{\\text{b}}^{3}{\\left(\\frac{1+{\\nu\\:}}{1-{\\nu\\:}}\\right)}^{\\raisebox{1ex}{$2$}\\!\\left/\\:\\!\\raisebox{-1ex}{$3$}\\right.}{\\text{f}}_{2}\\left({\\text{W}}_{\\text{c}}\\right){\\left(\\sum\\:_{\\text{n}}\\frac{{\\text{x}}_{\\text{n}}{\\varDelta\\:\\stackrel{-}{V}}_{n}^{2}}{{\\text{b}}^{6}}\\right)}^{\\raisebox{1ex}{$1$}\\!\\left/\\:\\!\\raisebox{-1ex}{$3$}\\right.}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(6)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nu\\:}\\)\u003c/span\u003e\u003c/span\u003e is the Poisson\u0026rsquo;s ratio, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}\\)\u003c/span\u003e\u003c/span\u003e (0.123 in this work) represents a constant related to the value of the line tension of the dislocation (Γ\u0026thinsp;=\u0026thinsp;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\alpha\\:}\\)\u003c/span\u003e\u003c/span\u003eG\u0026#119887;\u0026sup2;), G denotes the shear modulus, b represents the Burger\u0026rsquo;s vector and X\u003csub\u003en\u003c/sub\u003e represents de fraction of n\u003csup\u003eTh\u003c/sup\u003e element in solid solution. The functions f\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left({\\text{W}}_{\\text{c}}\\right)\\)\u003c/span\u003e\u003c/span\u003e, denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{f}}_{1}\\left({\\text{W}}_{\\text{c}}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{f}}_{2}\\left({\\text{W}}_{\\text{c}}\\right)\\)\u003c/span\u003e\u003c/span\u003e, are termed to minimize dislocation core-related coefficients. They account for the curved nature of dislocations, which deviate to locate local energy minima; Varvenne employed values of 0.35 and 5.70 for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{f}}_{1}\\left({\\text{W}}_{\\text{c}}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{f}}_{2}\\left({\\text{W}}_{\\text{c}}\\right)\\)\u003c/span\u003e\u003c/span\u003e, respectively.\u003c/p\u003e \u003cp\u003eIn both equations, the pivotal term is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varDelta\\:{\\stackrel{-}{V}}_{n}\\)\u003c/span\u003e\u003c/span\u003e, denoting the average volumetric misfit per atom. It is computed as the difference between the volume of the n\u003csup\u003eth\u003c/sup\u003e atom and the average atomic volume within the mixture. The atomic volume is derived from the FCC unit cell's volume divided by four.\u003c/p\u003e \u003cp\u003eThe resulting terms from equations \u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and 6 are then inserted into the subsequent equation, delineated as Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e7\u003c/span\u003e, for the computation of yield strength.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{\\tau\\:}_{y}\\left(T,\\dot{ϵ}\\right)={\\tau\\:}_{0}exp\\:\\left(-\\frac{1}{0.51}\\frac{kT}{\\varDelta\\:{E}_{b}}ln\\frac{\\dot{{ϵ}_{0}}}{\\dot{ϵ}}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere k is the Boltzmann constant and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\dot{{ϵ}_{0}}\\)\u003c/span\u003e\u003c/span\u003e is a reference term for the strain rate, which was 10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e in this work. Hence, the Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e7\u003c/span\u003e is a model for the SSS component to the critical resolved shear stress (converted into the respective contribution to the yield strength by the Taylor factor) that incorporates the dependence on strain rate and temperature.\u003c/p\u003e \u003cp\u003eOn other hand, Oh model suggests that the SSS component (σ\u003csub\u003ess\u003c/sub\u003e) in multicomponent alloys, comprising atoms from the 3d family, can be elucidated by the electronegativity difference between the constituent elements (Δχ), outlined as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:{\\sigma\\:}_{ss\\:}\\propto\\:\\varDelta\\:{\\chi\\:}\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMore specifically, Oh model suggests that the SSS could be predicted by the following equation:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:{\\sigma\\:}_{ss}=\\left(4293\\pm\\:448\\right)\\varDelta\\:\\chi\\:+\\left(84\\pm\\:37\\right)\\:MPa\\:$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTherefore, taking into consideration Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e7\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea presents the values of the SSS contribution alloys under study. The variation of A\u003csub\u003eAv\u003c/sub\u003e was used to express the difference in atomic volume in each alloy. Moreover, the σ\u003csub\u003ess\u003c/sub\u003e value for pure Ni was considered to be 0. Moreover, based on Eq.\u0026nbsp;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e9\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb illustrates the SSS contribution values as per Oh model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea shows that adding Pd to pure Ni results in higher SSS, compared to the addition of V, with Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e demonstrating the highest SSS. Similarly, in the CrCoNi system, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e exhibits higher SSS than with V addition. These results align with Varvenne model, which indicates that greater differences in A\u003csub\u003eAv\u003c/sub\u003e between constituent atoms lead to stronger SSS.\u003c/p\u003e \u003cp\u003eOn other hand, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb shows that adding V to pure Ni results in higher SSS than adding Pd, with Ni\u003csub\u003e63.2\u003c/sub\u003ePd3\u003csub\u003e6.8\u003c/sub\u003e demonstrating the highest SSS. Similarly, in the CrCoNi system, Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloy shows higher SSS compared to Pd addition. These findings support Oh\u0026rsquo;s model, which suggests that greater Δχ differences result in stronger SSS. However, the model proposed in Eq.\u0026nbsp;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e9\u003c/span\u003e should be applied with caution, once the intrinsic dispersion associated with the model (indicated in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb) could led to a widely distribution of σ\u003csub\u003ess\u003c/sub\u003e values. Therefore, the inherent dispersion associated with the model and the choice of the electronegativity scale to be employed may lead to inconclusive results when applying Oh's model, potentially limiting its practicality.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.5 EXCLUDING THE EFFECT OF GRAIN BOUNDARY STRENGTHENING\u003c/h2\u003e \u003cp\u003eIn fact, Oh et al. [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] support their perspective by examining the σ\u003csub\u003ey\u003c/sub\u003e and σ\u003csub\u003euts\u003c/sub\u003e values of the Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e alloy, in contrast to the equiatomic CrCoNi and CrCoNiMnFe alloys (all with similar grain sizes). Notably, the Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e alloy demonstrates higher yield strength and ultimate tensile strength values than the other alloys.\u003c/p\u003e \u003cp\u003eHowever, in order to conduct an accurate analysis of the isolated contribution of the σ\u003csub\u003ess\u003c/sub\u003e, it is essential to disregard the influence of grain refinement strengthening. This approach was adopted in the present study. According to the Hall-Petch model, the strengthening attributed to grain boundaries can be elucidated by Eq.\u0026nbsp;\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:\\sigma\\:={\\sigma\\:}_{0}\\:+\\:{K}_{hp}{d}^{-\\text{0,5}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn this context, σ\u003csub\u003eo\u003c/sub\u003e represents the friction lattice stress, K\u003csub\u003ehp\u003c/sub\u003e is the Hall-Petch coefficient, and d denotes the grain size. Theoretically, σ\u003csub\u003e0\u003c/sub\u003e includes various contributions to material strength, but since only substitutional Solid Solution is considered here (all alloys are single-phase), we focus solely on the SSS contribution to σ\u003csub\u003e0\u003c/sub\u003e. Additionally, Eq.\u0026nbsp;\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e10\u003c/span\u003e can analyze the influence of grain size on hardness (H) by replacing σ\u003csub\u003eo\u003c/sub\u003e with H\u003csub\u003e0\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eTherefore, examining Eq.\u0026nbsp;\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e10\u003c/span\u003e reveals that the same d can produce different hardness and σ\u003csub\u003ey\u003c/sub\u003e values across materials with different values for K\u003csub\u003ehp\u003c/sub\u003e. Consequently, assessing the σ\u003csub\u003ess\u003c/sub\u003e contribution based on absolute σ\u003csub\u003ey\u003c/sub\u003e and σ\u003csub\u003euts\u003c/sub\u003e for fine-grained materials may be misleading. Hence, to isolate the σ\u003csub\u003ess\u003c/sub\u003e component, the grain boundary strengthening contribution K\u003csub\u003ehp\u003c/sub\u003e must be removed. This can be done using Hall-Petch plots, measuring hardness (or σ\u003csub\u003ey\u003c/sub\u003e) for the same material at different grain sizes, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e for the alloys assessed in this study. The σ\u003csub\u003ey\u003c/sub\u003e values were extracted from several stress-strain curves for each alloy, shown in Supplementary Fig.\u0026nbsp;7.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigures \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea-b present hardness and σ\u003csub\u003ey\u003c/sub\u003e values, respectively, plotted against d\u003csup\u003e\u0026minus;\u0026thinsp;0.5\u003c/sup\u003e for the alloys under study. Moreover, the dashed lines in both figures represent a linear regression (y\u0026thinsp;=\u0026thinsp;ax\u0026thinsp;+\u0026thinsp;b) based on the data for each material. Thus, by extrapolating the curve to an infinite grain size (d\u003csup\u003e\u0026minus;\u0026thinsp;0.5\u003c/sup\u003e\u0026rarr;0), it is possible to estimate the intrinsic hardness (or σ\u003csub\u003ey\u003c/sub\u003e) of each alloy and, consequently, the SSS.\u003c/p\u003e \u003cp\u003eAnalyzing the data in Figs.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea-b, it is evident that both H\u003csub\u003eo\u003c/sub\u003e and σ\u003csub\u003e0\u003c/sub\u003e values for the binary alloys are higher than those for pure Ni (H\u003csub\u003eo\u003c/sub\u003e = 68.6 Hv and σ\u003csub\u003e0\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;14 MPa ) [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]. Additionally, the binary alloy with Pd shows higher H\u003csub\u003eo\u003c/sub\u003e and σ\u003csub\u003e0\u003c/sub\u003e values compared to those with V. This trend is also observed in the quaternary alloys, where those containing Pd and V exhibit higher H\u003csub\u003eo\u003c/sub\u003e and σ\u003csub\u003e0\u003c/sub\u003e values compared to the Cr\u003csub\u003e33\u003c/sub\u003eCo\u003csub\u003e33\u003c/sub\u003eNi\u003csub\u003e33\u003c/sub\u003e alloy. These results suggest that adding V and Pd enhances the SSS of both pure Ni and the CrCoNi system. However, while V-based alloys show higher hardness and σ\u003csub\u003ey\u003c/sub\u003e, their H\u003csub\u003e0\u003c/sub\u003e and σ\u003csub\u003e0\u003c/sub\u003e values indicate that V contributes less to SSS compared to Pd.\u003c/p\u003e \u003cp\u003eExamining the K\u003csub\u003ehp\u003c/sub\u003e values (extracted from hardness and σ\u003csub\u003ey\u003c/sub\u003e Hall-Petch plots) for the alloys it is revealed that Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e has a significantly higher K\u003csub\u003ehp\u003c/sub\u003e than Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e. Similar trends are observed in Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003ePd\u003csub\u003e10\u003c/sub\u003e and Cr\u003csub\u003e30\u003c/sub\u003eCo\u003csub\u003e30\u003c/sub\u003eNi\u003csub\u003e30\u003c/sub\u003eV\u003csub\u003e10\u003c/sub\u003e alloys, with the V-based alloy showing a higher K\u003csub\u003ehp\u003c/sub\u003e than the Pd-based alloy. This indicates that grain boundary strengthening is much more pronounced in V-based alloys. Also, Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e exhibited the highest grain boundary strengthening among the studied alloys. Thus, directly comparing absolute hardness and σ\u003csub\u003ey\u003c/sub\u003e values across different alloys is not feasible due to the varying contributions of grain boundary strengthening.\u003c/p\u003e \u003cp\u003eFigures \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003ea-b show the projected hardness and σ\u003csub\u003ey\u003c/sub\u003e of these alloys, assuming a uniform grain size of 10 \u0026micro;m, based on their respective K\u003csub\u003ehp\u003c/sub\u003e values. The Ni\u003csub\u003e63.2\u003c/sub\u003eV\u003csub\u003e36.8\u003c/sub\u003e alloy has higher hardness and σ\u003csub\u003ey\u003c/sub\u003e values at this grain size compared to other alloys. Despite Ni\u003csub\u003e50\u003c/sub\u003eP\u003csub\u003e50\u003c/sub\u003e exhibiting the lowest K\u003csub\u003ehp\u003c/sub\u003e for σ\u003csub\u003ey\u003c/sub\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb), its σ\u003csub\u003ey\u003c/sub\u003e value (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003eb) exceeds those of the ternary and quaternary alloys due to its highest σ\u003csub\u003e0\u003c/sub\u003e for a large range of grain sizes. This demonstrates that analyzing hardness or σ\u003csub\u003ey\u003c/sub\u003e values alone can lead to incorrect conclusions regarding the SSS effect in different metallic alloys.\u003c/p\u003e \u003cp\u003eThe K\u003csub\u003ehp\u003c/sub\u003e values should be approached with caution, as the same constant can exhibit different values depending on whether hardness or σ\u003csub\u003ey\u003c/sub\u003e is being analyzed. As illustrated in Figs.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea-b, except for the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy, the K\u003csub\u003ehp\u003c/sub\u003e values associated with hardness are consistently higher than those for σ\u003csub\u003ey\u003c/sub\u003e. This discrepancy arises because hardness measurements are conducted in a localized region of the material, where the plastic deformation is restricted to only a few grains. Additionally, during hardness testing indentation, significant work-hardening effects occur, directly influencing the hardness values obtained. Specifically, the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy exhibits an unusually low K\u003csub\u003ehp\u003c/sub\u003e for hardness (i.e., the hardness remains unchanged with increasing grain size), potentially indicating a low work-hardening coefficient for this alloy. Consequently, the mechanical behavior of the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy requires more in-depth analysis.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e presents a comparison between the experimental and calculated σ\u003csub\u003e0\u003c/sub\u003e values using the Varvenne and Oh models, in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003ea and \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003eb, respectively. The analysis indicates that, although both models fail to precisely predict the SSS, the theoretical σ\u003csub\u003ess\u003c/sub\u003e values predicted by the Varvenne model exhibit a closer correlation to the experimental data.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eHence, the analysis of the data presented so far shows not only that the Varvenne model is more suitable for predicting SSS, but also that the atomic volume difference between species in a solid solution is the more significant factor in SSS of FCC alloys. However, the Varvenne model has not yet been able to accurately predict the SSS component for all the alloys in study. This suggests that additional factors should be considered when developing models to predict SSS. For example, the dislocation line constants might change from case to case or maybe the non-linear variations in the atomic volume of each element.\u003c/p\u003e \u003cp\u003eInterestingly, the results indicate that alloys exhibiting the highest δ values also demonstrated the highest SSS values, primarily observed in the binary alloy systems. (Supplementary Fig.\u0026nbsp;8). This indicates that, while solid solutions comprising a greater diversity of atoms may not always display the greatest lattice distortion, increased lattice distortions are likely responsible for the greater contribution to solid solution strengthening.\u003c/p\u003e \u003cp\u003eIt was also noted that, contrary to current literature on HEAs [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], SSS is not always the predominant strengthening mechanism, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The SSS contribution values, obtained using supplementary Equations \u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, indicate that SSS becomes predominant (SSS\u0026thinsp;\u0026gt;\u0026thinsp;50%) only above a critical grain size, while grain boundary strengthening (GBS) dominates below this size. This critical grain size varies for each alloy and is detailed in Supplementary Table\u0026nbsp;4. Notably, the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy, due to its relatively low K\u003csub\u003ehp\u003c/sub\u003e value, has an extremely small critical grain size (Supplementary Fig.\u0026nbsp;9), making SSS nearly the sole hardening mechanism. These findings highlight that much work has yet to be done to fully understand SSS for different alloys, and suggest the Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy promising material for future fundamental metallurgical studies due to its minimal mechanical property sensitivity to grain growth.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4 CONCLUSIONS","content":"\u003cp\u003eThis study successfully designed and produced FCC solid solution alloys with significant differences in atomic volume or electronegativity among the constituent elements. The results indicate that the atomic volume difference between elements in a solid solution has a greater influence on the SSS than the electronegativity difference.\u003c/p\u003e \u003cp\u003eDue to the inherent characteristics of each solid solution system, such as the fraction of each element and the accompanying atomic volume variation with changes in elemental fractions, each solid solution system exhibits unique behavior. Consequently, developing a general model capable of predicting SSS remains a challenging task.\u003c/p\u003e \u003cp\u003eFor a given solid solution system, there is a critical grain size below which grain boundary strengthening plays a major role in yield strength and hardness, compared to SSS. It was also concluded that both predicted lattice distortion and the measured SSS component were greater for binary alloys. This provides further evidence that increasing the number of elements in solid solution not always leads to greater lattice distortion, and consequently, higher hardness and yield strength.\u003c/p\u003e \u003cp\u003eFinally, further in-depth studies on the Ni\u003csub\u003e50\u003c/sub\u003eP\u003csub\u003e50\u003c/sub\u003e alloy are warranted because its hardness and yield strength exhibit minimal variation with grain growth, making it promising from an engineering perspective.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eData availability\u003c/h2\u003e \u003cp\u003eThe raw/processed data required to reproduce these findings cannot be fully shared at this time due to technical or time limitations.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eHe QF, Wang JG, Chen HA, Ding ZY, Zhou ZQ, Xiong LH, Luan JH, Pelletier JM, Qiao JC, Wang Q, Fan LL, Ren Y, Zeng QS, Liu CT, Pao CW, Srolovitz DJ, Yang Y (2022) A highly distorted ultraelastic chemically complex Elinvar alloy. 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Scripta Mater 51:693\u0026ndash;698\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCullity BD (2001) Elements of X-ray Diffraction, Third Edition, Prentice-Hall, New York\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMishima Y, Ochiai S, Suzuki T (1985) Lattice parameters of Ni(γ), Ni3Al(γ') and Ni3Ga(γ') solid solutions with additions of transition and B-subgroup elements. Acta Metall 33:1161\u0026ndash;1169\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eOwen LR, Jones NG (2018) Lattice distortions in high-entropy alloys. J Mater Res 33:2954\u0026ndash;2969\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eOwen LR, Pickering EJ, Playford HY, Stone HJ, Tucker MG, Jones NG (2017) An assessment of the lattice strain in the CrMnFeCoNi high-entropy alloy. Acta Mater 122:11\u0026ndash;18\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKeller C, Hug E (2008) Hall\u0026ndash;Petch behaviour of Ni polycrystals with a few grains per thickness. Mater Lett 62:1718\u0026ndash;1720\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Solid solution strengthening, Grain boundary strengthening, High Entropy Alloys, Atomic volume, Electronegativity","lastPublishedDoi":"10.21203/rs.3.rs-5003860/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5003860/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study designed and produced FCC solid solution alloys with significant differences in atomic volume or electronegativity among the constituent elements, and subjected them to mechanical testing. The results demonstrate that atomic volume differences have a greater influence on solid solution strengthening (SSS) than electronegativity differences. Each solid solution system exhibits unique behavior, making a general model for predicting SSS challenging. Additionally, for a given solid solution system, there is a considerable difference in the critical grain size below which grain boundary strengthening dominates yield strength and hardness. Furthermore, both predicted lattice distortion values and the measured SSS components were greater for binary alloys, indicating that the presence of more elements in a solid solution does not always cause greater distortions in the crystal lattice. Finally, the study successfully engineered the novel Ni\u003csub\u003e50\u003c/sub\u003ePd\u003csub\u003e50\u003c/sub\u003e alloy, which has not been previously studied and exhibits mechanical properties remarkably insensitive to variations in grain size, warranting further in-depth investigations.\u003c/p\u003e","manuscriptTitle":"Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-18 06:15:21","doi":"10.21203/rs.3.rs-5003860/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"89ae9063-b51c-49af-aa58-90b236aa428f","owner":[],"postedDate":"September 18th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":37671530,"name":"Physical sciences/Materials science/Structural materials/Metals and alloys"},{"id":37671531,"name":"Physical sciences/Materials science/Structural materials/Mechanical properties"}],"tags":[],"updatedAt":"2025-10-07T07:09:12+00:00","versionOfRecord":{"articleIdentity":"rs-5003860","link":"https://doi.org/10.1038/s41467-025-63900-6","journal":{"identity":"nature-communications","isVorOnly":false,"title":"Nature Communications"},"publishedOn":"2025-10-06 04:00:00","publishedOnDateReadable":"October 6th, 2025"},"versionCreatedAt":"2024-09-18 06:15:21","video":"","vorDoi":"10.1038/s41467-025-63900-6","vorDoiUrl":"https://doi.org/10.1038/s41467-025-63900-6","workflowStages":[]},"version":"v1","identity":"rs-5003860","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5003860","identity":"rs-5003860","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}