Alloy Homogenization: Formula Alloys from the Pure-Metal Roster

Date: 2026-08-25 Purpose: What material::Alloy mixes and how, the validation on record, and — the part to read first — for which alloys the result can be trusted Module: src/physics/materials Implementation: cl_Material_Alloy.{hpp,cpp}, invoked by MaterialFactory::create_material for any label that parses as a composition

This document replaces the February 2026 design notes alloy_mechanical_mixing.md and alloy_transport_mixing.md; their proposals are now implemented, and this document describes the current behavior.

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1. What it is

An Alloy is built from a composition formula:

material Sn60Pb40 --rrr 10
material Fe71Cr19Ni10 --rrr 1.13

MaterialFactory tFactory ;
Material * tSolder = tFactory.create_material( "Sn60Pb40", 10.0 );   // label, RRR

The numbers are integer mass percent: to_pair in src/core/stringtools.hpp parses letters followed by digits. Every element must be one of the pure metals in the roster (Al, Cr, Fe, Ni, Cu, Ag, In, Sn, Pb). The factory instantiates one Metal object per component and hands them to the alloy, which owns them. The last component is the balance: its percentage is recomputed as 100 minus the others (Alloy::set_components, cl_Material_Alloy.cpp:149), so Fe70Cr19Ni10 silently becomes 70/19/11.

RRR is mandatory and must exceed 1; otherwise BELFEM_ERROR fires. It is the only alloying effect the model knows about, and it is fixed at construction: the inherited Material::set_RRR is not overridden and raises an error. The material executable defaults it to 10 for a formula and to 100 for a pure metal.

Three object details matter to callers. It is typed MaterialType::PureMetal, so FEM code treats it like a metal. Its components are constructed without lookup tables of their own. Unlike Metal, whose rho( T, B, beta ) can evaluate Kohler's rule on the fly, Alloy::rho( T, B, beta ) and lambda( T, B, beta ) read the tabulated database and assert its presence; construct the alloy with tables enabled if you will ask for field dependence.

The alloy is a SplineLookupTable: every property is tabulated once at construction on a 4 K grid from 0 K up to the first 4 K multiple at or above the lowest T_max of its components, and served from cubic splines afterwards. The reference density is the mass-weighted harmonic mean of the components' room-temperature densities; the molar mass is the molar-fraction average.

2. What is mixed and how

Mass fractions Y_k are fixed. At each temperature, the components' densities are used to recompute volume fractions v_k(T) = (Y_k / ρ_k(T)) / Σ_j (Y_j / ρ_j(T)) (Alloy::update_volume_fractions).

2.1 Elastic moduli: self-consistent scheme

Each component contributes isotropic stiffness from its own E_k(T), ν_k(T): K_k = E_k / (3(1 − 2ν_k)), G_k = E_k / (2(1 + ν_k)). The effective K and G are found by the self-consistent (Eshelby–Hill) scheme for spherical inclusions:

1. Voigt and Reuss bounds, K_V = Σ v_k K_k, 1/K_R = Σ v_k / K_k (likewise for G), give the starting point K = (K_V + K_R)/2 at the first temperature; every later temperature starts from the previous converged value.
2. With the current (K, G) as the reference medium, the Eshelby tensor for a sphere is S = γ δδ + δ' (sym), with γ = K / (3K + 4G) and δ' = 3(K + 2G) / (5(3K + 4G)) (Qu & Cherkaoui 2006, Eq. 4.3.40). The Hill polarization tensor is P = S : M, with M the compliance of the reference medium.
3. For every component, compute the strain-concentration tensor A_k = [I + P : (C_k − C)]⁻¹ and update C = Σ v_k C_k : A_k. K and G are read off the isotropic result (C₁₁₁₁ = K + 4G/3, C₂₁₂₁ = G), under-relaxed with ω = 0.99.
4. Iterate until the relative change in (K, G) falls below 1e-8 (Alloy::self_consistent_KG).

E and ν follow from the converged pair: E = 9KG / (3K + G), ν = (3K − 2G) / (6K + 2G). The arithmetic uses GPa for scaling; the result is stored in Pa.

Spherical inclusions describe an equiaxed microstructure — cast or sintered alloys. Directional or heavily worked microstructures would need an ellipsoidal Eshelby tensor and extra parameters, which the class does not have.

2.2 Thermal expansion: Levin (Schapery) with the self-consistent K

With α_V = Σ v_k α_k and α_R = K_R Σ v_k α_k / K_k, the effective coefficient is

α = α_V + (1/K − 1/K_V) / (1/K_R − 1/K_V) · (α_R − α_V),

using the self-consistent K (cl_Material_Alloy.cpp:372). The formula is exact for two phases and a tight estimate for more; when all K_k coincide the interpolation is undefined and the code takes the mean of the two bounds. At 0 K, α, c_p and λ are set to zero.

2.3 Specific heat and Sommerfeld coefficient: mass-fraction averages

c_p = Σ Y_k c_p,k(T) — exact for an extensive property. The electronic coefficient γ = Σ Y_k γ_k is stored as a constant and sets the slope of the c_p spline at 0 K.

2.4 Electrical resistivity: Matthiessen with one residual term

ρ(T) = ρ₀ + Σ v_k ρ_i,k(T)

where ρ_i,k is each metal's phonon (Bloch–Grüneisen) resistivity from Metal::rho_i_custom, and ρ₀ is a single residual resistivity for the whole alloy, fixed from the caller's RRR at room temperature:

ρ₀ = Σ v_k ρ_i,k(293.15 K) / (RRR − 1)

(Alloy::set_components). One ρ₀ for all phases is a deliberate simplification: it reproduces the macroscopic RRR by construction and needs no phase-diagram data; it loses the distinction between the residual scattering inside a Pb-rich and a Sn-rich lamella.

Linear (volume-fraction) mixing of the intrinsic parts was chosen over a Bruggeman effective medium after a comparison with the Hariharan et al. 1979 data for 60Sn–40Pb: linear mixing tracks the measured ρ(T) from 4 K to 300 K, Bruggeman's spherical-inclusion geometry suppresses the slope by 40–50 % on a lamellar eutectic. Nordheim's rule is not used — it applies to random solid solutions below the solubility limit, not to the two-phase solders the class was written for.

2.5 Thermal conductivity: linear mixing of thermal resistivities

λ(T) = 1 / (w₀ + Σ v_k w_i,k(T)), with w₀ = ρ₀ / (L₀ T) the residual (Wiedemann–Franz) term and w_i,k = hust(P_k, T) each metal's intrinsic phonon-limited thermal resistivity from its Hust coefficients (fn_hust.hpp). The pure-metal Hust form has a third, interaction term w_i0 whose empirical constant is unknown for a mixture; it is dropped, which slightly underestimates λ where w₀ and w_i are comparable and is immaterial when the alloy's large ρ₀ dominates.

2.6 Field dependence: per-phase Kohler, mixed on the alloy's own baseline

Alloy::populate_rho_database (cl_Material_Alloy.cpp:529) tabulates ρ(T, B, β) exactly as Metal does, but evaluates each component's Kohler curve at the similarity parameter it would see inside the alloy:

S_k = ρ_ref,k / (ρ₀ + ρ_i,k(T)), δ_k = kohler_k(B, S_k, β), ρ(T, B, β) = ρ(T) · (1 + Σ v_k δ_k).

Because ρ₀ of an alloy is far larger than any pure metal's, S_k is small and the magnetoresistance is weak — the expected behavior of a dirty conductor, though not one this path has been checked against field data. Every component must provide a Kohler curve; all nine roster metals do. λ(T, B, β) follows by Wiedemann–Franz scaling, λ(T) ρ(T) / ρ(T, B, β), the same closure Metal uses. The table is cached as <label>_RRR<n>.hdf5 in the run directory like a metal's, probed for the current format on load and rebuilt if it predates it.

2.7 Stored curves

Property	0 K start condition	End condition
E, ν	clamped, zero slope	parabolic
c_p	clamped, slope γ	parabolic
ρ	clamped, zero slope	parabolic
α, λ	parabolic	parabolic

3. What the homogenizer can and cannot do

The model is a two-phase composite rule plus one residual-resistivity knob. It is trustworthy exactly when the real alloy is one of those two things, and it fails when alloying creates something no constituent has: a new crystal structure, a magnetic state, an intermetallic. The dividing line in one sentence: the method holds when the alloy keeps its constituents' crystal structure and magnetic state; the residual resistivity is the only alloying effect it models.

Trustworthy

• Eutectic solders — Sn–Pb, Sn–Ag, In–Sn, Sn–Pb–In. Pb and Sn are nearly insoluble in each other, so a solder is a fine composite of two almost-pure phases; the mixture rule is the right physics, not an approximation. For Sn63Pb37: α ≈ 24.6×10⁻⁶ from the mix against ≈ 25 measured, E ≈ 30–35 GPa against ≈ 30, ρ ≈ 1.5×10⁻⁷ Ω·m against 1.45×10⁻⁷. This is the case the class was written and validated for.
• Cu–Ag. Negligible mutual solubility: a two-phase composite, same reasoning.
• Cu–Sn bronze, Al–Cu. Solid solutions in the majority element's own fcc lattice; the matrix keeps its phase, and the residual resistivity is exactly what RRR ≈ 1.1 (bronze) or ≈ 2–3 (2xxx aluminium) describes.
• Cu–Ni, Ni–Cr, and Ni-base superalloy matrices (e.g. Ni72Cr16Fe8). Fully miscible fcc solutions; Cr, or a copper majority, suppresses nickel's ferromagnetism, so the class's lack of a magnetic channel is correct here rather than accidental. Expect chromium's small α to pull the mixture ≈ 15 % low.

Right mechanically, wrong magnetically

• Ferritic Fe–Cr (e.g. Fe83Cr17). A bcc solution in bcc iron, so α, E, c_p mix well — but the alloy is ferromagnetic and the class returns μ₀. Alloy has no magnetic channel at all, whatever its components are.

Not to be trusted

• Austenitic stainless steels (Fe71Cr19Ni10 ≈ 304). The alloy is fcc; Fe and Cr are bcc. Whatever depends on the crystal structure rather than on the atoms is systematically off, see the worked example below.
• Invar (Fe64Ni36) and other magnetovolume alloys. Measured α ≈ 1×10⁻⁶ against a mixture value of ≈ 12×10⁻⁶ — a factor ten.
• 9 %-Ni cryogenic steels, intermetallics (Nb₃Sn). Martensitic or ordered phases; not mixtures of anything in the roster.

Worked example: Fe71Cr19Ni10 with RRR 1.13 against NIST 304

Reference values from the NIST cryogenic material fits (https://trc.nist.gov/cryogenics/materials/304Stainless/304Stainless_rev.htm):

T [K]	k [W/(m·K)]	c_p [J/(kg·K)]	E [GPa]	ΔL/L₂₉₃ [%]	α [10⁻⁶/K]
4	0.27	2.1	210	−0.297	—
77	7.9	205	214	−0.280	7.0
293	15.1	471	200	0	15.4

What the mixture gives, and why:

• E lands near 215–220 GPa: ≈ 10 % high, the Voigt–Reuss–Hill average of bcc Fe, bcc Cr and fcc Ni. Ballpark.
• c_p above ~50 K is fine (Dulong–Petit does not care about the phase). Below ~20 K it is low by a factor of 5–6: 304's electronic Sommerfeld coefficient is ≈ 0.5 J/(kg·K²), the mass-weighted γ of the components ≈ 0.08. The austenitic density of states is not a mixture property. For a quench study at 4 K this is the number that matters most.
• α comes out ≈ 10.7×10⁻⁶ at room temperature against 15.4, and the 293→4 K contraction ≈ 0.18 % against 0.30 %: 30–40 % low, the fcc-versus-bcc difference again.
• ρ and k are within a factor two: the RRR fixes ρ ≈ 7–8×10⁻⁷ Ω·m (304: 7.2×10⁻⁷), and Wiedemann–Franz then gives the electronic part of k; the missing remainder is the lattice conductivity that dominates in a dirty alloy.
• μ = μ₀ — correct for 304, by accident.

4. Validation on record

• 60Sn–40Pb: ρ(T) against Hariharan et al. 1979, 4–300 K (the linear-versus-Bruggeman decision above). No other alloy has been checked against measurement through this path; the 304 table above is an estimate from the pure-metal values, not a run.
• No test in check-fast exercises the alloy path. A test constructing Sn60Pb40 and Fe71Cr19Ni10 and asserting loose bounds against the values above would turn "ballpark" into an executable statement and would catch a regression in any component metal at once.

5. Known limitations

• One ρ₀ for all phases (§2.4); per-phase residual terms would need solubility limits and Nordheim coefficients.
• The Wiedemann–Franz closure assumes L = L₀; in alloys at low temperature L/L₀ ranges roughly 0.6–1.9. Inherited from the pure-metal model.
• The w_i0 interaction term of the Hust form is dropped (§2.5).
• No magnetic channel: every alloy is non-magnetic.
• No alloy-specific Debye temperature; the phonon resistivities are those of the components.
• Spherical inclusions in the elastic scheme; the transport mixing was chosen for lamellar eutectics on the strength of the one Sn–Pb comparison in §4, the elasticity remains a spherical-inclusion approximation for them.

References

• Eshelby, J. D. (1957), Proc. R. Soc. London A 241, 376–396
• Hill, R. (1965), J. Mech. Phys. Solids 13, 213–222
• Qu, J. & Cherkaoui, M. (2006), Fundamentals of Micromechanics of Solids, Wiley — Eshelby tensor for spheres, Eq. (4.3.40)
• Levin, V. M. (1967), Mech. Solids 2, 58–61; Schapery, R. A. (1968), J. Compos. Mater. 2, 380–404
• Voigt, W. (1928), Lehrbuch der Kristallphysik; Reuss, A. (1929), Z. Angew. Math. Mech. 9, 49–58
• Hust, J. G. & Lankford, A. B. (1984), NBS Special Publication 260-90 — thermal conductivity form
• Hariharan, Y. et al. (1979), Pramana 13, 117–125 — 60Sn–40Pb resistivity
• Bruggeman, D. A. G. (1935), Ann. Phys. 416, 636–664; Nordheim, L. (1931), Ann. Phys. 401, 607–640
• NIST Cryogenic Material Properties, 304 stainless steel — reference fits in §3