Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening

doi:10.21203/rs.3.rs-5003860/v1

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Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening

https://doi.org/10.21203/rs.3.rs-5003860/v1

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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 Ni₅₀Pd₅₀ 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

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 1a 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 Ni₅₀Pd₅₀, Ni_63.2V_36.8, Cr₃₃Co₃₃Ni₃₃, Cr₃₀Co₃₀Ni₃₀Pd₁₀, and Cr₃₀Co₃₀Ni₃₀V₁₀, chosen for significant lattice distortion and/or electronegativity differences (Fig. 1b). Despite Ni_63.2V_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.

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 Ni₅₀Pd₅₀, Ni_63.2V_36.8, Cr₃₃Co₃₃Ni₃₃, Cr₃₀Co₃₀Ni₃₀Pd₁₀, and Cr₃₀Co₃₀Ni₃₀V₁₀, 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_63.2V_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^− 3s^− 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.

3.1 MICROSTRUCTURAL CHARACTERIZATION

The thermodynamic calculations depicted in Fig. 2a-e illustrate the FCC single-phase field for the Ni₅₀Pd₅₀ Ni_63.2V_36.8, Cr₃₃Co₃₃Ni₃₃, Cr₃₀Co₃₀Ni₃₀Pd₁₀, Cr₃₀Co₃₀Ni₃₀V₁₀ 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. 3a-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. 3d-e with the addition of Pd and V to the CrCoNi system (Fig. 3c). 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 Cr₃₃Co₃₃Ni₃₃, 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:

$$\:0.74=\frac{QV}{{a}^{3}}$$

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:

$$\:{A}_{av}=\frac{\text{0,74}{a}^{3}}{4}$$

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]:

$$\:\delta\:=\:\sqrt{\sum\:_{i=1\:\:\:}^{n}{c}_{i}\:{\left(1-{r}_{i}/\stackrel{-}{r}\right)}^{2}}$$

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, Ni₅₀Pd₅₀, and Ni_63.2V_36.8 alloys.
Material	a (Å)	Δa* (%)	A_av (Å³)	ΔA_av**(%)	δx100
Ni	3.5246	-	8.1003	-	-
Ni₅₀Pd₅₀	3.7396	6.10	9.6747	19.44	4.97
Ni_63,2V_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 Cr₃₃Co₃₃Ni₃₃, Cr₃₀Co₃₀Ni₃₀Pd₁₀, and Cr₃₀Co₃₀Ni₃₀V₁₀ alloys.
Material	a (Å)	Δa* (%)	A_av (Å³)	ΔA_av**(%)	δx100
Cr₃₃Co₃₃Ni₃₃	3.5621	-	8.3616	3.23	1.40
Cr₃₀Co₃₀Ni₃₀Pd₁₀	3.6091	1.32	8.6971	4.01	3.13
Cr₃₀Co₃₀Ni₃₀V₁₀	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:

$$\:\varDelta\:{\chi\:}=\sqrt{\sum\:_{x}{c}_{x}{\left({{\chi\:}}_{x}-{⟨{\chi\:}⟩}_{element}\right)}^{2}}$$

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	-
Ni₅₀Pd₅₀	0.150
Ni_63,2V_36,8	0.175

Table 4

Electronegativity difference caused by the addition of Pd and V to CrCoNi system
Material	δχ
Cr₃₃Co₃₃Ni₃₃	0.100
Cr₃₀Co₃₀Ni₃₀Pd₁₀	0.118
Cr₃₀Co₃₀Ni₃₀V₁₀	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 Ni₅₀Pd₅₀ 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 (Ni₅₀Pd₅₀ and Cr₃₀Co₃₀Ni₃₀Pd₁₀) should exhibit the highest contribution of SSS. Conversely, the Oh model posits alloys with vanadium addition (Ni_63.2V_36.8 and Cr₃₀Co₃₀Ni₃₀V₁₀) 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:

$$\:{{\tau\:}}_{0}=0.051{{\alpha\:}}^{-\raisebox{1ex}{$1$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}\text{G}{\left(\frac{1+{\nu\:}}{1-{\nu\:}}\right)}^{\raisebox{1ex}{$4$}\!\left/\:\!\raisebox{-1ex}{$3$}\right.}{\text{f}}_{1}\left({\text{W}}_{\text{c}}\right){\left(\sum\:_{\text{n}}\frac{{\text{x}}_{\text{n}}{\varDelta\:\stackrel{-}{V}}_{n}^{2}}{{\text{b}}^{6}}\right)}^{\raisebox{1ex}{$2$}\!\left/\:\!\raisebox{-1ex}{$2$}\right.}$$

$\:{\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.

$$\:{\tau\:}_{y}\left(T,\dot{ϵ}\right)={\tau\:}_{0}exp\:\left(-\frac{1}{0.51}\frac{kT}{\varDelta\:{E}_{b}}ln\frac{\dot{{ϵ}_{0}}}{\dot{ϵ}}\right)$$

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:

$$\:{\sigma\:}_{ss\:}\propto\:\varDelta\:{\chi\:}\:$$

More specifically, Oh model suggests that the SSS could be predicted by the following equation:

$$\:{\sigma\:}_{ss}=\left(4293\pm\:448\right)\varDelta\:\chi\:+\left(84\pm\:37\right)\:MPa\:$$

Therefore, taking into consideration Eq. 7, Fig. 4a 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. 4b illustrates the SSS contribution values as per Oh model.

Figure 4a shows that adding Pd to pure Ni results in higher SSS, compared to the addition of V, with Ni₅₀Pd₅₀ demonstrating the highest SSS. Similarly, in the CrCoNi system, Cr₃₀Co₃₀Ni₃₀Pd₁₀ 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. 4b shows that adding V to pure Ni results in higher SSS than adding Pd, with Ni_63.2Pd3_6.8 demonstrating the highest SSS. Similarly, in the CrCoNi system, Cr₃₀Co₃₀Ni₃₀V₁₀ 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. 4b) 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_63.2V_36.8 alloy, in contrast to the equiatomic CrCoNi and CrCoNiMnFe alloys (all with similar grain sizes). Notably, the Ni_63.2V_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.

$$\:\sigma\:={\sigma\:}_{0}\:+\:{K}_{hp}{d}^{-\text{0,5}}$$

In this context, σ_o represents the friction lattice stress, K_hp is the Hall-Petch coefficient, and d denotes the grain size. Theoretically, σ₀ 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 σ₀. Additionally, Eq. 10 can analyze the influence of grain size on hardness (H) by replacing σ_o with H₀.

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 5a-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. 5a-b, it is evident that both H_o and σ₀ values for the binary alloys are higher than those for pure Ni (H_o = 68.6 Hv and σ₀ = 14 MPa ) [35, 36]. Additionally, the binary alloy with Pd shows higher H_o and σ₀ 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 σ₀ values compared to the Cr₃₃Co₃₃Ni₃₃ 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 H₀ and σ₀ 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_63.2V_36.8 has a significantly higher K_hp than Ni₅₀Pd₅₀. Similar trends are observed in Cr₃₀Co₃₀Ni₃₀Pd₁₀ and Cr₃₀Co₃₀Ni₃₀V₁₀ 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_63.2V_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 6a-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_63.2V_36.8 alloy has higher hardness and σ_y values at this grain size compared to other alloys. Despite Ni₅₀P₅₀ exhibiting the lowest K_hp for σ_y (Fig. 5b), its σ_y value (Fig. 6b) exceeds those of the ternary and quaternary alloys due to its highest σ₀ 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. 5a-b, except for the Ni₅₀Pd₅₀ 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 Ni₅₀Pd₅₀ 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 Ni₅₀Pd₅₀ alloy requires more in-depth analysis.

Figure 7 presents a comparison between the experimental and calculated σ₀ values using the Varvenne and Oh models, in Fig. 7a and 7b, 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 Ni₅₀Pd₅₀ 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 Ni₅₀Pd₅₀ alloy promising material for future fundamental metallurgical studies due to its minimal mechanical property sensitivity to grain growth.

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 Ni₅₀P₅₀ alloy are warranted because its hardness and yield strength exhibit minimal variation with grain growth, making it promising from an engineering perspective.

Data availability

The raw/processed data required to reproduce these findings cannot be fully shared at this time due to technical or time limitations.

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Exploring the Relative Influence of Atomic Parameters on Solid Solution Strengthening

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Abstract

Figures

1. INTRODUCTION

2. METHODS

3. RESULTS AND DISCUSSION

3.1 MICROSTRUCTURAL CHARACTERIZATION

3.2 INFLUENCE OF ATOMIC VOLUME

3.3 INFLUENCE OF ELECTRONEGATIVITY

3.4 PREDICTION OF SSS SOLUTION USING THE EVALUATED MODELS

3.5 EXCLUDING THE EFFECT OF GRAIN BOUNDARY STRENGTHENING

4 CONCLUSIONS

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Data availability

References

Additional Declarations

Supplementary Files

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