Uncertainty Quantification in Masses of Alloy Components and Atomic Radii Modification in High-Entropy Alloys Design: Thermophysical Parameters Calculation Approach

Abstract

Given the vast universe of high-entropy alloys (HEAs), solid solution formation (SSF) prediction is increasingly relevant. The processing route leads to uncertainty in the mass of each alloy component, affecting SSF. Furthermore, investigations led to atomic radius modification under interaction with neighboring atoms, also influencing SSF. Therefore, this paper presents an uncertainty quantification framework implemented over the thermophysical parameters calculation (TPC) approach to verify the behavior of the SSF parameters as the mass of the alloy components vary and the atomic radii are modified. The AlCrFeMoNbTaTiVW alloy was subjected to this framework, being the tungsten mass the most influential, and tantalum the less influential overall. Moreover, the atomic radii modification does not work properly under TPC theory, implying in non-SSF prediction even when a solid solution is formed. Thenceforth, equimolar HEAs are now near-equimolar, and the SSF parameters may indicate that some samples of the same alloy batch may result in SSF, others not.

Keywords:
High-entropy alloys (HEAs); uncertainty quantification (UQ); thermophysical parameters calculation (TPC); atomic radii modification; multicomponent alloys

1. Introduction

New materials have been developed since ancient times by civilization¹. The development of materials has been one of the major drivers of evolution. New metals were discovered, and new metal alloys fabricated to solve several key issues in the most diverse areas. It is well known that the mechanical properties of pure metals can be enhanced by alloying each one with other metals or non-metallic elements¹. In view of this, metal alloys have been based on one principal element for a long time, e.g. iron in steel. Thenceforth, to improve properties, metal alloys were then proposed with two main elements².

Posteriorly, the multicomponent alloys, which contain three or more main elements, were then created. Despite their useful properties, in several situations, they may form intermetallic compounds (IM) with complex microstructures corresponding to mechanical properties of undesired magnitudes. Consequently, a novel alloy design paradigm was introduced about two decades ago³,⁴. The related concept involves the mixing of multiple elements in an equimolar or near-equimolar composition. Therefore, the so-called high-entropy alloys (HEAs)³ emerged, in which their design is based on at least five main elements (concentrations varying between 5 and 35% in molar terms). This denomination is due to their high configurational entropy in the solidification process, promoting random solid solutions, in contrast to complex phases. HEAs represent a new research field within multicomponent alloys and have great potential to combine properties uniquely to enable new engineering applications or enhance the possibilities in existing ones.

This promising research field is being increasingly investigated due to reports of high strength⁵, hardness⁶,⁷, ductility⁸, fracture and impact toughness⁹, corrosion resistance¹⁰, strength-ductility trade-off¹¹, strength-to-weight ratio¹², low cost of manufacturing¹³, among other properties. Variations in HEAs’ nomenclature emerged as the topic discussions were addressed, such as RHEAs (refractory)¹⁴, SPHEAs (single phase)¹⁵, LWHEAs (lightweight)¹², among others. Inside the universe of possible HEAs (which commonly has two or more different phases, some of them being IM), in a few cases, a stabilized microstructure consisting of either (a) random solid solution with a simple crystal structure (BCC, FCC, or HCP), or (b) duplex microstructure (two simple solid solutions), which provides more interesting properties to turn into materials for demanding applications. The stabilized solid microstructure normally addresses ductility concomitantly with solid solution hardening. Conversely, whereas adding more elements increases the configurational entropy of a solid solution, the probability of IM formation also augments¹⁶,¹⁷.

In this regard, the formation of random solid solutions in HEAs is due to the four core effects: (a) high entropy, (b) sluggish diffusion, (c) lattice distortion, and (d) cocktail¹⁸. The high-entropy effect occurs due to a high configurational entropy at elevated temperatures. In this condition, the Gibbs free energy of mixing is minimized (per the second law of thermodynamics), thus benefiting the dissolution of IMs¹⁹. The sluggish diffusion effect is due to lattice distortion, which provides a slow rate of atomic diffusivity²⁰. The lattice distortion effect is caused by introducing substitutional solute atoms into a solvent matrix, displacing the atoms from their theoretical lattice positions²¹. The cocktail effect refers to the interaction between the elements, causing a synergistic blend of properties, such as strength-to-ductility ratio, strength-toughness balance, fatigue resistance, and corrosion resistance, among others²².

Therefore, three possible states are the result of HEAs obtainment: (a) elemental phases, which is a solid solution based on only one element; (b) IM, a stoichiometric compound with specific superlattices; and (c) solid solution, in which a complete or an almost complete mixing of all the elements in the form of BCC (body-centered cubic), FCC (face-centered cubic), or HCP (hexagonal close-packed).

Since the first investigations in the field of HEAs, the prediction of phase formation and composition is one of the most reported branches in the specific literature²³-²⁵. In this regard, there are four main approaches commonly applied in this type of alloy design: (a) molecular dynamic (MD) simulations²⁶,²⁷; (b) calculation phase diagrams (CALPHAD)²⁸,²⁹; (c) machine learning³⁰; and (d) thermophysical parameters calculation (TPC)³¹,³².

The TPC approach (applied in this work) addresses the calculation of several parameters and the evaluation of the corresponding criteria that, theoretically, if attended, refers to random solid solutions. The selected set of parameters and criteria is applied in each alloy design to predict solid solutions and phases. In this work, 8 parameters and their corresponding criteria are calculated according to their relevance and availability related to the thermodynamic data of the elements.

2. Thermophysical Parameters Calculation (TPC) for HEAs’ Design

This section describes the classical TPC (without radii modification), the modified TPC (with radii modification), and the thermophysical parameters and their associated criteria.

2.1. Classical TPC (without radii modification)

The core concept intrinsic to the utilization of the parameters in TPC relies on minimizing the differences in physical and thermodynamic properties of the main elements that constitute the HEAs, conducting to the formation of random solid solutions, thus lowering the likelihood of precipitation of ordered IM or amorphous structures. Although the detailing degree of TPC is low due to not providing information about the phase diagrams, it is a relatively fast method if the objective is only to predict solid solution formation (SSF).

As far as this research could reach, there are at least 29 parameters available in the specific literature³³. Among them, there are means and standard deviations (e.g. mean melting temperature, standard deviation of density, etc.). Due to report frequency, simplification, and objectives of the analysis, in this paper, 8 parameters and their associated criteria are presented. These criteria provide statuses about the SSF. The application of TPC may predict if a specific combination of elements results in a random SSF or not (e.g. IM, metallic glass, etc.).

This approach does not demand experimental data in what refers to solid solution prediction, besides having a high predictive capacity. Conversely, as a large design space is often required, the referred approach requires a high computational cost. The classical TPC relies on the hard-sphere model atom, which describes the atoms and molecules as solid spheres with no internal structure and no change in radii of the elements involved³⁴.

2.2. Thermophysical parameters and their associated criteria

Eight parameters and their associated criteria are presented in this paper. The first 2 depend exclusively on the radii of the elements involved; the third depends simultaneously on the radii and masses of the alloy components. The remaining parameters are dependent only on the masses of the alloy components.

Atomic radii difference, ∆r(%), (radius dependent): it is defined by the difference between the largest (r_L) and smallest (r_S) atomic radii of the elements involved related to r_L (Equation 1). The first rule of Hume-Rothery addresses that SSF effectively occurs when this difference is lower than 15% (Equation 2)³⁴.

$\Delta r\left(\%\right)=\frac{r_L-r_S}{r_L}100.$ (1)

$SSF:\Delta r\left(\%\right)<15.$ (2)

Topological discrepancy, γ, (radius dependent): it quantifies the non-linear impact of the largest and smallest atoms in SSF. γ is defined as the result of the Fermi level being close to a van Hove singularity in the density of states (Equation 3), where r_M is the mean atomic radius³⁵. The criterion for SSF states that the parameter should be smaller than 1.175 (Equation 4):

$\gamma =\left(1-\sqrt{\frac{{\left(r_L-r_M\right)}^2-r_M^2}{{\left(r_L-r_M\right)}^2}}\right)/\left(1-\sqrt{\frac{{\left(r_S-r_M\right)}^2-r_M^2}{{\left(r_S-r_M\right)}^2}}\right).$ (3)

$SSF:\gamma <\mathrm{1.175.}$ (4)

Polydispersion of atomic radii or atomic size mismatch, δ(%), (radius and molar fraction dependent): it is the mean square deviation of the atomic radii involved (Equation 5) and directly influences the atomic packing, where f_i is the molar fraction of the i-th element involved in the solid solution, and n is the number of alloy components. The associated criterion to SSF is expressed in Equation 6 (with rare exceptions)³⁶. The consequence of bond distortion is a corresponding degree of strain³⁷. In other words, δ is a manner to quantify the deviation from the ideal lattice.

$\delta \left(\%\right)=\sqrt{{{\displaystyle \sum }}_{i=1}^n{f}_i{\left(\frac{1-r_i}{r_M}\right)}^2}100.$ (5)

$SSF:\delta \left(\%\right)<\mathrm{6.5.}$ (6)

Mean valence electron concentration or valence electronic configuration, V_M, (molar fraction dependent): it is related to the third rule of Hume-Rothery, that states that elements with the same type of crystalline structure may form extensive series of solid solution³⁸. V_M, representing the mean periodic table group number, is given by Equation 7, where V_i is the electron concentration value of the i-th alloy element. The criterion associated to V_M is that BCC structure is formed when V_M < 6.87, and FCC structure when V_M ≥ 8; otherwise, FCC predominantly occurs, however BCC may occur in some circumstances (Equation 8).

$V_M={{\displaystyle \sum }}_{i=1}^n{f}_i{V}_i.$ (7)

$SSF:\left\{\begin{array}{c}{V}_M\ge \mathrm{8,}FCC,\\ V_M<\mathrm{6.87,}BCC,\\ 6.87\ge V_M\le \mathrm{8,}FCCorBCC.\end{array}\right.$ (8)

Entropy of mixing or configurational entropy, ∆S_MIX, (molar fraction dependent): the entropy of mixing of the elements involved is expressed by Equation 9 from Boltzmann’s hypothesis³⁹, where 𝑅 = 8.314 J / mol K is the universal gas constant. The value of ∆S_MIX determines the degree of entropy of the alloy, i.e. low-entropy alloys (LEAs) correspond to ∆S_MIX < 0.69 R, medium-entropy alloys (MEAs) refer to 0.69 R ≤ ∆S_MIX ≤ 1.61 R, and HEAs to ∆S_MIX ≥ 1.61 R. Therefore, ∆S_MIX ≥ 1.61 R (13.38 J / mol / K) addresses an increasing likelihood to SSF⁴⁰, representing the associated criterion (Equation 10). However, concomitantly with a higher value of δ, the opposite tendency is observed, leading to IM formation⁴¹.

$\Delta S_{MIX}=-R{{\displaystyle \sum }}_{i=1}^n{f}_i\ln \left(f_i\right).$ (9)

$SSF:\left(11\le \Delta S_{MIX}\le 19.5\right)J/kmol.$ (10)

Electronegativity difference, ∆χ, (molar fraction dependent): it is the difference in electronegativities by Pauling, and it leads to a quantification of phase separation and new phase formation⁴². In this regard, this parameter represents the standard deviation of electronegativity, i.e. the fourth rule of Hume-Rothery (Equation 11), in which χ_i is the electronegativity of the i-th element, and χ_M is the mean of the electronegativity of the elements involved. Values of electronegativity difference lower than 6% correspond to SSF, as per Equation 12.

$\Delta \chi \left(\%\right)=\sqrt{{{\displaystyle \sum }}_{i=1}^n{f}_i{\left({\chi }_i-{\chi }_M\right)}^2}100.$ (11)

$SSF:\Delta \chi \left(\%\right)<6.$ (12)

Average mixing enthalpy, ∆H_MIX, (molar fraction dependent): governed by the regular solution model⁴³, i.e. Equation 13, where c_i and c_j are the normalized fractions of the i-th and j-th elements (Equation 14 and Equation 15, respectively), nC₂ is the number of atomic pairs, and ∆𝐻_{𝑐𝑖,𝑐𝑗} is the mixing enthalpy of binary liquid alloys formed by the i-th and j-th elements (stated in Equation 16), where 𝛺_k is the k-th factor of third order approximation polynomial (obtained by Miedema’s scheme, via thermodynamic tables for each atomic pair⁴³. The associated criterion for SSF is shown in Equation 17.

$\Delta H_{MIX}={{\displaystyle \sum }}_{i=1;i\ne j}^{n{C}_2}\Delta H_{c_i,c_j}.$ (13)

$c_i=\frac{f_i}{f_i+f_j}.$ (14)

$c_j=\frac{f_j}{f_i+f_j}.$ (15)

$\Delta H_{c_i,c_j}=4c_i{c}_j\left({{\displaystyle \sum }}_{k=0}^3{\Omega }_k{\left(c_i-c_j\right)}^k\right).$ (16)

$SSF:\left(-22\le \Delta H_{MIX}\le 7\right)kJ/mol.$ (17)

Entropy-enthalpy relation or interacting parameter, 𝛺, (molar fraction dependent): The relation between enthalpy and entropy is employed to predict the phase formation, especially for HEAs. The introduction of 𝛺 is to verify the contribution of entropy of mixing in the SSF. The parameter and its associated criterion are calculated by Equation. 18, and Equation 19, respectively³⁸, where T_M is the mean temperature.

$\Omega =\frac{T_M\Delta S_{MIX}}{\left|\Delta H_{MIX}\right|}.$ (18)

$SSF:\Omega >1.$ (19)

2.3. Modified TPC (with radii modification)

Regarding the modified TPC approach, 3 of the 8 parameters presented herein include the atomic radii of the elements involved in their formulation. The classical TPC approach in HEAs’ design assumes that the atomic radii of the elements involved are invariable, as when dealing with pure metals. However, in alloys, several investigations stated that a change in the local electronic environment occurs, suggesting a change in atomic radii⁴⁴. This type of alteration is based on the soft sphere model and is significant to the development of amorphous, crystalline, and IM because the radii modification impacts the lattice distortion⁴⁵, thus affecting the alloy properties. Therefore, the 3 topological parameters of HEAs and their associated criteria addressed herein are calculated by observing 2 distinct situations: (a) per classical TPC: atomic radii as those in pure elements (no modification in atomic radii); and (b) per modified TPC: atomic radii as those in alloys (with atomic radii modification). The impact of radii modification on HEAs is evaluated by comparing the results without and with radii modification. Both situations, which are subjected to uncertainty in the masses of the alloy components involved, are calculated via a specific code developed in Matlab^®. In this regard, the formulated code was based on one previously developed in Microsoft Visual Basic^®46.

As previously proposed⁴⁶, the atomic radius of the i-th element in the solution is modified according to Equation 20, in which the prime indication addresses its modified radius, E_i is the electronic shell number of the i-th element, and E_M is the mean electronic shell number. In the context of uncertainty, it is important to note that the uncertainty in mass also originates the uncertainty in the condition of modified radii of the elements due to involving the means of both electronic shell number and valence electron concentration.

$r_i^{\text{'}}=r_i\left[\left(\frac{E_M}{V_M}\right)/\left(\frac{E_i}{V_i}\right)\right].$ (20)

3. Uncertain Modeling of Aleatory-type Variable

This section describes the possible types and sources of uncertainty, besides the mathematical model of aleatory-type one. Moreover, the uncertainty quantification process is highlighted, with its input and response quantities.

3.1. Sources and types of uncertainty

Uncertainty is inherent to the most varied engineering applications. Materials, processes, assumptions, conditions, simplifications, etc. are some of the main focuses when investigating sources of uncertainty. The engineering decision-making processes often demand uncertainty quantification (UQ) analyses, in which full detail of the uncertainty is required. Uncertainty generally originates from the following sources: (a) model form (mathematical modeling, simplifications, assumptions, among others); (b) model input (input data, surroundings, restrictions, etc.); and (c) numerical approximation (domain discretization, number of algebraic operations, etc.)⁴⁷. In predictions based on mathematical models, the classification of uncertainties can be listed as the following: (a) epistemic: corresponds to a lack of knowledge about the behavior of the variable, which is commonly represented by an interval⁴⁷; (b) aleatory: variability inherent to the system behavior, which can be better represented and quantified, however not eliminated. The aleatory-type can be represented by a probability density function (PDF) and/or a cumulative density function (CDF)⁴⁸; and (c) mixed: a combination of both epistemic- and aleatory-type ones, which can be represented by a PDF and/or a CDF with an interval⁴⁹. This paper deals with the aleatory type only.

3.2. Aleatory-type uncertainty

The adoption of the aleatory-type uncertainty refers to the nature of the manufacturing route adopted for this HEA, as if it were on its specific production line. In view of this, the Gaussian model was selected in this paper. Regarding this type of uncertainty, the information generally comes from a candidate, selected, available, or known PDF and/or CDF. If n_k is the number of known parameters of the PDF of the i-th probabilistic uncertain input quantity (UIQ) X_i, k is the k-th known parameter, and the PDF of X_i is represented by f_Xi(x_i, x_i+1, …, k, …, n_k-1, n_k), then the realizations of X_i are obtained from the inverse function of the PDF, as per Equation 21, in which x_i is a possible outcome from a set χ_i containing all possible outcomes. Therefore, the CDF F_Xi can be attributed to every element x_i such that both conditions are concomitantly fulfilled (Equation 22 and Equation 23):

$x_i=f_{X_i}^{-1}\left(x_i,x_{i+1},\cdots ,k,\cdots ,n_{k-1},n_k\right).$ (21)

${{\displaystyle \sum }}_{x_i\in {\chi }_i}{F}_{X_i}\left(x_i,x_{i+1},\cdots ,k,\cdots ,n_{k-1},n_k\right)=1.$ (22)

$F_{X_i}\left(x_i,x_{i+1},\cdots ,k,\cdots ,n_{k-1},n_k\right)\in \left[0,1\right],\forall x_i\in {\chi }_i.$ (23)

3.3. Uncertainty quantification (UQ), uncertain input quantities (UIQs), and system response quantities (SRQs)

In the context of model-based predictions, between the real situation and the mathematical model selected to represent it there is a gap that partially originates the uncertainty. Initial and boundary conditions, analysis time, logical complexity, errors, rounding, multiplicative factors, among others, originate the uncertainty⁴⁷.

In view of safety and functionality issues, the impact of these sources of uncertainty should be investigated and predicted⁵⁰, starting by designating these sources as UIQs (with known parameters to define the corresponding PDF and/or CDF).

These UIQs should be propagated throughout the selected mathematical model to obtain the SRQs. Among the available uncertainty propagation methods, Monte Carlo sampling (MCS) is one of the most widely used⁵¹. Other relevant methods are also applied: quasi-Monte Carlo sampling (QMCS)⁵², subset simulation (SS)⁵³, polynomial chaos (PC)⁵⁴, fuzzy interval arithmetic (FIA)⁵⁵, analytical uncertainty propagation (AUP)⁵⁶, Latin hypercube sampling⁵⁷ (LHS, applied in this work), among others. The adoption of LHS is explained by the ratio of ease of implementation and computational cost, thus demanding reasonable sample sizes to converge.

Each UIQ or SRQ is represented by a column vector V_i in the form of Equation 24, where p is the p-th realization, the superscript between parentheses is the realization number, n_r corresponds to the number of realizations, and ℕ is the closed set of natural numbers.

$V_i={\left[v_i^{\left(1\right)},v_i^{\left(2\right)},\cdots ,v_i^{\left(p\right)},\cdots ,v_i^{\left(n_r-1\right)},v_i^{\left(n_r\right)}\right]}^T,p\in \mathbb{N}|1\le p\le n_r.$ (24)

Thenceforth, for each alloy, the mathematical model $\mathcal{M}$ that maps the n UIQs (X₁, X₂, …, X_n) addressed in this work (mass of each alloy component involved) into the p SRQs (Z₁, Z₂, …, Z_p) is presented in Equation 25. It is important to note that, given the uncertainty in the process, a previous equimolar HEA turns out now to have a near-equimolar composition.

$\mathcal{M}\left(X_1,X_2,\cdots ,X_n\right)\to Z_1,Z_2,\cdots ,Z_p.$ (25)

In addition, within the scope of this paper, a variable can be regarded as a deterministic quantity if: (a) its variation can be neglected; (b) the computational cost to include it in the uncertainty framework is prohibitive; (c) its value is known exactly; or (d) its variation is of low relevance to the objective of the analysis.

4. Example

The numerical example addressed in this paper refers to a 9-distinct-element alloy, AlCrFeMoNbTaTiVW, with a body-centered cubic (BCC) structure, named metallic diamond alloy 6, reported in⁷. Aiming at maximizing the alloy hardness, the manufacturing route of the alloy was the following: (a) powder mixing; (b) powder compaction; (c) arc plasma melting (casting); and (d) annealing⁷.

4.1. Problem description

Some of the input data of the elements involved in the alloy theoretical composition comes directly from the periodic table of elements, and the remaining data are from the literature already mentioned in the section concerning the thermophysical parameters and the associated criteria⁷,³⁴-⁴⁶,⁵⁸.

The uncertainty in mass proposed by this investigation refers to the following manufacturing conditions: (a) the mass of each alloy component is rounded in the stage of separation and weighing of the 9 distinct raw powder, which represents the minor loss of mass; (b) there is a loss of mass during the compaction process of the raw powder, which also represents a low degree of mass loss; and (c) the major loss occurs in the casting process, significantly explained by the elements with relatively low vapor pressure, and adhesion to the interacting apparatus.

4.2. Uncertainty quantification framework

In an uncertain context, the classical TPC and the modified TPC are both applied to compare the thermophysical parameters without and with radii modification, respectively, to check if they attend the associated criteria to SSF. Therefore, the mass of each alloy component involved can be represented by the mathematical structure of a UIQ. The entire UQ process applied throughout this work is presented in Figure 1, where a flowchart indicates the steps followed.

Figure 1
Flowchart of the uncertainty quantification process employed in this work.

According to Figure 1, the Gaussian mathematical model is attributed to the mass of each alloy component involved. An estimation of the uncertainty in the UIQs is made and propagated (via LHS) through both classical and modified TPC methods to predict SSF. The SRQs are quantified via the two distinct methods, and, finally, the information on SRQs obtained from both approaches are compared.

Table 1 presents the mean and standard deviation values of each UIQ in this paper. The deterministic mass value of each alloy component involved is assumed as the mean of the simulated probability distribution, whereas the standard deviation is 0.5% of the corresponding mean (partially justified by measurements comparing the outlet mass with the inlet mass). The sample deterministic mass of the alloy is defined as 20 g and 1 x 10⁵ realizations are performed. All the calculations and plots reported were performed in MATLAB^®. The LHS was implemented using a distinct seed number for each sample generated.

Table 1
Input data for the elements involved in the alloy AlCrFeMoNbTaTiVW related to the UQ process.

UIQ	Deterministic value and mean (g)	Standard deviation (% of the mean)
mAl	0.685	0.5
mCr	1.321	0.5
mFe	1.419	0.5
mMo	2.438	0.5
mNb	2.360	0.5
mTa	4.597	0.5
mTi	1.216	0.5
mV	1.294	0.5
mW	4.670	0.5

5. Results and Discussion

This section presents the calculation and discussion about the output for the problem of prediction of SSF of the alloy AlCrFeMoNbTaTiVW. This includes probabilistic distributions, graphical sensitivity analyses, relative errors, and Pearson correlation coefficients of the parameters dependent: (a) exclusively on the mass of each alloy component involved; (b) only on the atomic radius of each alloy component involved; and (c) on both mass and atomic radius.

5.1. Distributions for the UIQs and SRQs

The histogram, PDF, and CDF of the powder mass of the alloy are presented in Figure 2, Figure 3, and Figure 4, respectively, (in blue) aiming to illustrate its shape, dispersion, and convergence to the means of the UIQs to assure an adequate input to be propagated through the mathematical model. Regarding the legends of Figure 2, powder mass is represented by m_P, “DETERM.” is the deterministic value (and initial mean) of the parameter (represented by a vertical continuous red line), and “MEAN” is the simulated mean of the parameter after the simulation (vertical continuous green line). The graphical coincidence of both vertical lines indicates the convergence of the mean of m_P with a relative error of 3 x 10^-7%.

Figure 2
Histogram of AlCrFeMoNbTaTiVW powder mass.

Figure 3
PDF of AlCrFeMoNbTaTiVW powder mass.

Figure 4
CDF of AlCrFeMoNbTaTiVW powder mass.

Among the 8 parameters addressed in this paper, V_M is selected to represent those which depend exclusively on the mass of each alloy component involved. The PDF and CDF of this SRQ are shown in Figure 5 and Figure 6, respectively. γ is chosen to represent those parameters with only atomic radius dependence (Figure 7 and Figure 8), and δ is selected to represent the parameter concomitantly dependent on radius and mass (Figure 9 and Figure 10). In Figure 7 through Figure 10, although calculated, the deterministic values of γ and δ (without radius modification), respectively, are suppressed for means of clarification (they are far from the values covered by the distributions). In view of this, the “DETERM.” label represents the modified deterministic value of the referred parameter.

Figure 5
PDF of AlCrFeMoNbTaTiVW mean valence electron concentration (V_M), where “DETERM.” stands for the deterministic value of V_M, and “MEAN” for the mean of V_M.

Figure 6
CDF of AlCrFeMoNbTaTiVW mean valence electron concentration (V_M), where “DETERM.” stands for the deterministic value of V_M, and “MEAN” for the mean of V_M.

Figure 7
PDF of AlCrFeMoNbTaTiVW topological discrepancy (γ) with atomic radii modification, where “MEAN” stands for the mean value of γ.

Figure 8
CDF of AlCrFeMoNbTaTiVW topological discrepancy (γ) with atomic radii modification, where “MEAN” stands for the mean value of γ.

Figure 9
PDF of AlCrFeMoNbTaTiVW polydispersion of atomic radii (δ) with atomic radii modification, where “MEAN” stands for the mean value of δ.

Figure 10
CDF of AlCrFeMoNbTaTiVW polydispersion of atomic radii (δ) with atomic radii modification, where “MEAN” stands for the mean value of δ.

In view of the uncertainty in the masses of the alloy components (according to Table 2), the previously planned and designed equimolar AlCrFeMoNbTaTiVW alloy turns out to be the non-equimolar alloy Al_0.690Cr_1.316Fe_1.398Mo_2.441Nb_2.341Ta_4.563Ti_1.218V_1.299W_4.634, for example, and the also non-equimolar alloy Al_0.685Cr_1.325Fe_1.418Mo_2.448Nb_2.355Ta_4.588Ti_1.220V_1.289W_4.636, among 99998 other alloys obtained via simulation in this investigation.

Table 2
Input data for the alloy AlCrFeMoNbTaTiVW related to the UQ process, in which “LB” stands for lower bound, “std” standard deviation, and “UB” upper bound.

UIQ	Determ. (theoretical mean) (g)	Theoretical standard deviation (mg)	Simulated LB (g)	Simulated Mean (mg)	Simulated UB (g)	Simulated std (mg)	Relative error from mean (%)	Relative error from std (%)
mAl	0.685	3.425	0.670	685.444	0.700	3.427	0.065	0.058
mCr	1.321	6.605	1.292	1320.929	1.350	6.604	-0.005	-0.015
mFe	1.419	7.095	1.387	1418.735	1.451	7.070	-0.019	-0.354
mMo	2.438	12.190	2.387	2437.550	2.493	12.130	-0.018	-0.495
mNb	2.360	11.800	2.301	2360.155	2.412	11.787	0.007	-0.110
mTa	4.597	22.985	4.500	4596.791	4.700	23.034	-0.005	0.213
mTi	1.216	6.080	1.186	1215.991	1.242	6.089	-0.001	0.148
mV	1.294	6.470	1.267	1294.111	1.321	6.453	0.009	-0.263
mW	4.670	23.350	4.571	4670.328	4.767	23.349	0.007	-0.004

5.2. Statistical data of the problem

Table 2, Table 3, and Table 4 present several statistical data referring to the UIQs, SRQs without radii modification, and SRQs with modified atomic radii treated herein, respectively. Table 2 shows the masses of the alloy components involved in the alloy design. Observing Table 2, with respect to the mean, titanium mass presented the lowest relative error in absolute value, while aluminum mass showed the highest one. Related to the standard deviation, the tungsten mass presented the lowest relative error in absolute value, and molybdenum mass the highest one. The relative errors from the mean and from the standard deviation pointed out in Table 2 may serve as a quantification of the reliability of the input data. In the case of the reliability of the output data, the procedure adopted was to run the code 10 times with 1 x 10⁵ realizations each. This procedure yielded a relatively low maximum absolute error of mean and standard deviation related to 𝛺 of 0.004 and 3.210 x 10^-4, respectively, for example.

Table 3
Output data for the alloy AlCrFeMoNbTaTiVW related to the UQ process without radii modification, in which “LB” stands for lower bound, “std” standard deviation, “UB” upper bound, and CV coefficient of variation.

SRQ	Determ.	Simulated LB	Simulated Mean	Simulated UB	Simulated std (10-4)	Relative error from mean (%)	CV (10-4)
Δr (%)	14.286	14.285	14.285	14.285	~0	5.593 x 10-11	~0
γ	1.179	1.179	1.179	1.179	0.069	-1.631 x 10-6	585.241
δ (%)	5.216	5.199	5.215	5.233	41.428	328.614 x 10-4	7.944
VM	5.333	5.323	5.333	5.343	22.351	2.451 x 10-4	4.191
ΔSMIX (J/kmol)	18.268	18.267	18.268	18.268	0.467	-5.104 x 10-4	0.026
Δ𝛘	28.198	28.066	28.197	28.316	314.832	-1.407 x 10-3	11.165
ΔHMIX (kJ/mol)	-11.274	-11.354	-11.274	-11.183	202.307	2.388 x 10-4	17.944
Ω	3.903	3.870	3.903	3.941	86.960	-7.003 x 10-4	22.280
rM (pm)	134.614	134.570	134.615	134.659	116.420	2.433 x 10-5	0.864

Table 4
Output data affected by atomic radii modification for the alloy AlCrFeMoNbTaTiVW related to the UQ process, in which “LB” stands for lower bound, “std” standard deviation, “UB” upper bound, and CV coefficient of variation.

SRQ	Determ.	Simulated LB	Simulated Mean	Simulated UB	Simulated std	Relative error from mean (%)	CV (10-4)
Δr (%)	51.720	51.560	51.719	51.879	0.037	-0.002	7.154
γ	2.237	2.230	2.236	2.244	0.002	-0.045	8.944
δ (%)	22.356	22.220	22.355	22.490	0.032	-0.004	14.314
rM (pm)	134.614	135.601	135.924	136.275	0.074	0.972	5.445

Table 3 and Table 4 address the eight parameters from the TPC and the mean radius of the alloy, highlighting that Table 4 presents only the parameters that differ from those in Table 3, i.e. the parameters not shown in Table 4 are already presented in Table 3. Two distinct types of data are compared by means of the relative error: theoretical and simulated. The relative errors calculated refer to the quantification of the convergence of the mean and the standard deviation of each UIQ and SRQ. The low values of the relative errors indicate a relatively good agreement between the theoretical proposed data and the simulated ones. This can be partially explained by the sufficient number of realizations selected, although a lower number of realizations already ensures enough precision to deal with this type of problem using LHS. This corroborates with the symmetry observed in Figure 4, Figure 6, Figure 8, and Figure 10. The lower and upper bounds, respectively LB and UB, are collected as the lowest and highest values simulated within each variable. Moreover, the coefficient of variation (CV) is presented in the last column of Table 3 and Table 4. In the case with no radii modification (Table 3), Δr and γ presented the lowest and the highest dispersions, respectively. The modified radii case (Table 3 and Table 4) showed ∆S_MIX and 𝛺 as the lowest and highest dispersive parameters, respectively.

The reliability of the obtained results may be verified by observing Table 3 (without radii modification), in which the deterministic and probabilistic values of γ and Δ𝛘 do not attend their respective criteria to SSF, while the values of the other six parameters are within each range to SSF. In the case of radii modification (Table 4), only ΔS_MIX, ΔH_MIX, and Ω comply with SSF, while the other five parameters are out of their accepted ranges to SSF. Comparing the deterministic and simulated values between Table 3 and Table 4, it can be noted that the SRQs dependent on atomic radii (Δr, γ, δ, and r_M) did not attend the related criteria previously established. As a result of going from 6 criteria met (without radii modification) for the SSF to only 3 (with radii modification), in addition to not fulfilling all the radii-dependent SSF criteria, the atomic radii modification theory proved to be non-effective in the application in TPC approach (modified TPC) for the tested alloy AlCrFeMoNbTaTiVW. In other words, if the associated criteria do not change in its formulation when the atomic radii are modified, the SSF criteria will not be fulfilled in 5 out of 8 criteria for the alloy presented herein.

In what refers only to the parameter γ, for example, while the alloy Al_0.690Cr_1.316Fe_1.398Mo_2.441Nb_2.341Ta_4.563Ti_1.218V_1.299W_4.634 corresponds to solid solution formation, Al_0.685Cr_1.325Fe_1.418Mo_2.448Nb_2.355Ta_4.588Ti_1.220V_1.289W_4.636 does not form solid solution per classical TPC. This almost residual difference between the two multicomponent alloys which could happen depending on the adopted manufacturing route, determines whether SSF effectively occurs. In other words, in the same batch, under the proposed uncertainty framework context, some samples of this alloy can result in SSF, others not.

Figure 11 through Figure 16 present several possible graphical sensitivity analyses (SAs) aiming at pictorially analyzing the influence of some UIQs on some SRQs (Figures 11-16). Figure 11 Figure 12 through Figure 13 picture the influence of the mean atomic radius of the alloy on atomic radii difference (Δr), polydispersion of atomic radii (δ), and topological discrepancy (γ), respectively. Figure 14, Figure 15, and Figure 16 show the impact of the mass of aluminum (for example) on Δr, δ, and γ, respectively. In the case of process improvement initiatives, these mappings may aid in future dispersion reduction, if possible.

Figure 11
Scatter plot of r_M vs. Δr of AlCrFeMoNbTaTiVW.

Figure 12
Scatter plot of r_M vs. δ of AlCrFeMoNbTaTiVW.

Figure 13
Scatter plot of r_M vs. γ of AlCrFeMoNbTaTiVW.

Figure 14
Scatter plot of m_Al vs. Δr of AlCrFeMoNbTaTiVW.

Figure 15
Scatter plot of m_Al vs. δ of AlCrFeMoNbTaTiVW.

Figure 16
Scatter plot of m_Al vs. γ of AlCrFeMoNbTaTiVW.

In addition to the scatter plots already mentioned (qualitative analysis), a quantitative analysis is performed via Pearson correlation coefficients comparison, in which Table 5 presents the correlation coefficients between the UIQs (masses of the alloy components involved in the alloy design) and all the SRQs (the parameters and the mean radius of the elements involved) of the problem in the condition without atomic radii modification. By observing Table 5, it is possible to note that: (a) ∆r is almost not influenced by the masses of the alloy components; (b) although very low, the influence of chromium mass on γ is the highest; (c) in the case of δ, iron mass has the highest impact, such that to lower the value of δ, it is recommended to lower the iron mass; (d) V_M is also most influenced by titanium and tungsten masses, however the influence is low; (e) the highest impact on ΔS_MIX comes from vanadium mass; (f) in the case of Δ𝛘 , the most impactful UIQ is the tungsten mass; (g) aluminum mass influences ΔH_MIX the most; (h) Ω is mostly affected by aluminum mass; and (i) chromium mass is the most influencing UIQ on r_M , however in a low level.

Table 5
Pearson correlation coefficients between UIQs and SRQs addressed in AlCrFeMoNbTaTiVW design without atomic radii modification.

	Δr (%)	γ	δ (%)	VM	ΔSMIX (J/kmol)	Δ𝛘	ΔHMIX (kJ/mol)	Ω	rM (pm)
mAl (g)	9.387 x 10-13	-0.002	-0.235	0.001	0.001	-0.175	-0.755	-0.760	0.002
mCr (g)	0.477 x 10-13	-0.006	0.291	0.004	-0.002	-0.214	0.221	0.154	0.006
mFe (g)	-4.533 x 10-13	0.003	0.743	0.001	0.002	-0.235	-0.211	-0.233	-0.003
mMo (g)	2.638 x 10-13	-0.003	-0.355	-0.002	0.006	0.242	0.361	0.340	0.003
mNb (g)	-6.428 x 10-13	0.001	-0.011	0.002	0.001	-0.155	-0.055	-0.007	-0.001
mTa (g)	7.077 x 10-13	0.001	-0.006	-0.003	-0.002	-0.025	-0.044	0.058	-0.001
mTi (g)	-8.020 x 10-13	-0.005	0.105	-0.005	0.001	-0.092	-0.117	-0.143	0.005
mV (g)	0.603 x 10-13	0.004	-0.202	0.001	0.008	-0.188	0.261	0.186	-0.004
mW (g)	-3.212 x 10-13	0.001	-0.351	-0.005	-0.001	0.860	0.341	0.409	-0.001

Under another perspective, the following list describes which SRQ is the most influenced by each UIQ. From Table 5: (a) aluminum mass influences more Ω than any other SRQ; (b) chromium mass has its highest influence on δ; (c) iron mass affects δ the most; (d) molybdenum mass has its highest impact on ΔH_MIX; (e) niobium mass has its highest influence on Δ𝛘; (f) tantalum mass has its highest impact on Ω; (g) titanium mass has its highest influence on Ω; (h) vanadium has its highest impact on ΔH_MIX; and (i) tungsten mass affects Δ𝛘 the most.

In the case of atom radii modification and regarding only the SRQs with the radii in their formulation, as per Table 6: (a) ∆r is most influenced by the iron mass; (b) γ is also most impacted by iron mass; (c) δ is most affected by the iron mass (specifically, it is the highest correlation coefficient overall); and (d) although low, r_M is most influenced by titanium mass.

Table 6
Pearson correlation coefficients between UIQs and SRQs affected by atomic radii modification in AlCrFeMoNbTaTiVW design.

	Δr (%)	γ	δ (%)	rM (pm)
mAl (g)	0.314	-0.314	-0.132	0.001
mCr (g)	-0.095	0.095	-0.068	-0.001
mFe (g)	-0.362	0.362	0.918	0.001
mMo (g)	-0.087	0.087	-0.196	0.001
mNb (g)	0.053	-0.053	-0.155	-0.004
mTa (g)	0.050	-0.050	0.061	-0.001
mTi (g)	0.174	-0.174	-0.150	0.006
mV (g)	0.044	-0.044	-0.193	0.002
mW (g)	-0.093	0.093	-0.104	0.004

In the other way, yet per Table 6: (a) aluminum mass has its highest influence on both ∆r and γ; (b) chromium mass has its highest influence concomitantly on ∆r and γ; (c) iron mass affects δ the most; (d) molybdenum mass has its highest influence on δ; (e) niobium mass has its highest impact on δ; (f) tantalum mass has its highest influence on δ; (g) titanium has its highest impact concomitantly on ∆r and γ; (h) vanadium has its highest influence on δ; and (i) tungsten has its highest impact on δ.

An important result in the context of this research refers to the overall impact of the masses of the alloy components on the addressed parameters. This influence was quantified here by the impact factor, which refers to the summation of the absolute values of Pearson correlation coefficients. Table 7 shows each alloy component mass ranked by impact factor. Per classical TPC approach (without atomic radii modification), the tungsten mass is the most influential and tantalum mass is the less one, while per modified TPC approach (with radii modification) tantalum keeps on being the least influential and now the most influential is the aluminum mass. Table 7 also presents the percent variation of modified TPC related to classical TPC, in which in 7 out of 9 masses the trend is to increase the impact factor when the atomic radii is modified.

Table 7
Comparison of the impact factors with and without atomic radii modification of the masses of the alloy components in AlCrFeMoNbTaTiVW.

Mass of the alloy element	Impact factor (without atomic radii modification)	Rank (without atomic radii modification)	Impact factor (with atomic radii modification)	Rank (with atomic radii modification)	Percent variation (%)
mAl	1.931	2	2.453	1	27.033
mCr	0.898	5	0.854	7	-4.900
mFe	1.431	3	2.325	2	62.474
mMo	1.312	4	1.322	4	0.762
mNb	0.233	8	0.485	8	108.155
mTa	0.140	9	0.294	9	110.000
mTi	0.473	7	0.862	6	82.241
mV	0.854	6	0.927	5	8.548
mW	1.969	1	1.910	3	-2.996

Therefore, the Pearson correlation coefficients mapping, made in Table 5 and Table 6, may serve as a guide toward the SSF by means of changing the molar fraction of the most impactful alloy elements aiming to reach SSF related to all the parameters and associated criteria.

6. Conclusions

This paper presented an uncertainty quantification (UQ) analysis conducted over the thermophysical parameters calculation (TPC) approach to check the behavior of the parameters (system response quantities, SRQs) as the masses of the alloy elements involved vary and their atomic radii are modified (uncertain input quantities, UIQs). Among the 8 parameters and their associated criteria that were addressed to predict the solid solution formation (SSF), 2 are only radii dependent, 1 is concomitantly dependent on radii and masses, and the remaining ones are dependent only on masses. In the strategy proposed by the presented UQ framework, the UIQs are Gaussian aleatory-type and Latin hypercube sampling (LHS) was the selected method to propagate the uncertainty through the mathematical model presented. The AlCrFeMoNbTaTiVW alloy was subjected to the proposed framework, generating probabilistic distributions, graphical sensitivity analyses, relative errors, and Pearson correlation coefficients as the main results.

According to the research conducted herein, the main conclusions are the following: (a) the UQ framework conducted over TPC can generate a range of results related to SSF according to the input data; (b) in an uncertain paradigm, a previous equimolar HEA turns out now to be a near-equimolar HEA; (c) for each SRQ there is at least one UIQ as the most impactful. In case of computational burden, the most influencing UIQs should be designed toward the establishment of an SSF, if it is the design objective; (d) it is not recommended to apply the atomic radii modification theory in TPC due to the lack of support of the mathematical formulation in a reasonable manner; (e) if a HEA is designed near the boundaries of SSF acceptance in the criteria, some samples of the same batch of the HEA may result in SSF, and other samples may not.

In view of these conclusions, there is still an increasing demand toward the knowledge of prediction of SSF in HEAs’ design. The UQ framework and the parameters applied herein compose just a minor branch in the vast universe of HEAs’ design. In this scenario, uncertainty quantification methods may help in HEAs’ design, mainly regarding avoiding unnecessary experiments and unexpected properties of the alloys.

7. Acknowledgements

The authors would acknowledge the journal for the opportunity to share this work. The authors would also acknowledge the following research councils for the financial support: (a) CNPQ (proj. 408406/2021-6), (b) CatalisaICT SEBRAE (proj. 29083*128), and (c) FAPESP (proj. 2023/09818-8).

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Publication Dates

• Publication in this collection
  14 Mar 2025
• Date of issue
  2025

History

• Received
  15 Aug 2024
• Reviewed
  06 Dec 2024
• Accepted
  13 Jan 2025