Thermodynamic Databases · Miedema vs CALPHAD (revised)

Miedema (Gokcen Appendix B) vs CALPHAD — revised

1,454 row-level comparisons across 625 unique binary systems, evaluated at T = 1873 K. Results are collapsed to one row per system (median CALPHAD ΔH across assessments; Miedema is deterministic per system) and stratified by liquid stability. The headline metric is sign agreement, not R² — for a semi-empirical chemical-term model, the meaningful question is whether the sign and magnitude class are right, not variance explained.

88%

sign agreement
(collapsed to 1 row/system, N=625)

95%

for |ΔH| ≥ 5 kJ/g-atom
(N=438)

94%

stable liquids (ΔH ≤ 0)
(N=394, R²=0.61, MAE=8.8)

0.63

R² physical set
(stable+mild, N=595; gap systems quarantined)

Equimolar ΔH (x = 0.5) — system-level

● stable (ΔH≤0) ● mild (0<ΔH≤+20) ● gap (ΔH>+20, quarantined) — y = x

Dilute-B partial ΔH (analytic)

Physical set only (gap systems excluded).
Miedema: ΔH̄_B^∞ = 2p·V_B^eff·Q/snw (analytic limit, not finite-difference).

Stratification by liquid stability

The 1,454 row-level comparisons collapse to 625 unique systems (median CALPHAD ΔH across assessments; Miedema is deterministic per system). Systems are then stratified by the sign and magnitude of the CALPHAD equimolar ΔH — the best available proxy for whether the homogeneous equimolar liquid actually exists at T=1873 K.

Stratum	N	R²	r²	MAE	RMSE	bias	sign agree	%	Criterion
Stable / ordered	394	0.610	0.650	8.84	12.84	+1.07	370/394	94%	ΔH ≤ 0 — liquid stable at equimolar, calorimetry-anchored
Mild demixing	201	-7.062	0.232	7.22	14.16	+0.37	151/201	75%	0 < ΔH ≤ +20 — liquid metastable or stable at high T
Miscibility gap	30	-0.310	0.001	29.51	63.78	-18.36	28/30	93%	ΔH > +20 — L-parameter encodes phase separation, not a physical enthalpy
Physical set	595	0.625	0.685	8.29	13.30	+0.84	521/595	88%	stable + mild — the meaningful comparison
All systems	625	0.519	0.545	9.31	19.07	-0.08	549/625	88%	everything, including quarantined gap systems

The gap stratum (N=30) has r²=0.001 — effectively zero correlation. This is not a model failure: the CALPHAD "ΔH" for these systems is a large positive L₀ tuned to reproduce a miscibility gap, not a calorimetric enthalpy. The homogeneous equimolar liquid doesn't exist there, so comparing Miedema to that number tests nothing. Quarantining them is the point — it isolates the identifiability axis.

Threshold sensitivity sweep

The +20 kJ/g-atom gap threshold is a heuristic for "homogeneous equimolar liquid almost certainly unstable." This table shows the stratification at thresholds from +10 to +30 kJ/g-atom. The physical-set metrics (R², sign agreement, MAE) are stable across the full range — the conclusions do not depend on the exact threshold.

Threshold	N stable	N mild	N gap	N phys	R² (phys)	sign% (phys)	MAE (phys)	R² (stable)	sign% (stable)
+10	394	153	78	547	0.690	87.0%	7.75	0.610	93.9%
+12	394	166	65	560	0.698	87.3%	7.68	0.610	93.9%
+14	394	179	52	573	0.632	87.4%	8.04	0.610	93.9%
+16	394	191	40	585	0.624	87.5%	8.17	0.610	93.9%
+18	394	195	36	589	0.624	87.6%	8.22	0.610	93.9%
+20	394	201	30	595	0.625	87.6%	8.29	0.610	93.9%	← default
+22	394	205	26	599	0.630	87.6%	8.28	0.610	93.9%
+24	394	207	24	601	0.633	87.7%	8.27	0.610	93.9%
+26	394	212	19	606	0.635	87.6%	8.34	0.610	93.9%
+28	394	215	16	609	0.639	87.7%	8.34	0.610	93.9%
+30	394	215	16	609	0.639	87.7%	8.34	0.610	93.9%

R²(phys) varies by only 0.075 across the full threshold range; sign%(phys) by only 0.7 percentage points. The +10 and +12 thresholds give marginally better R²(phys) because they push more of the noisy mild-demixing systems into the quarantined gap stratum — but the difference is small, and +20 is the more conservative quarantine (fewer systems excluded). The stable-stratum metrics are completely invariant (threshold doesn't affect what counts as stable).

Mild-demixing deep dive: resolving the R² = −7.06 paradox

The mild stratum (0 < ΔH ≤ +20, N=201) has the paradoxical combination of the lowest MAE (7.22 kJ/g-atom) and the most negative R² (-7.06). The resolution is that R² is a variance-explained metric, and the mild stratum has very low CALPHAD variance (stdev 5.00 kJ/g-atom, range [0.05, 19.50]) while Miedema's predictions span [-26.74, 126.58] (stdev 15.90). The bias is small (+0.37) but the residual variance (40304) is 8× the CALPHAD variance (4999), forcing R² negative. Miedema is nearly unbiased but far too noisy in this regime — its dynamic range exceeds what the CALPHAD data permits.

CALPHAD ΔH in mild stratum: range [0.1, 19.5], stdev 5.00

Miedema ΔH in mild stratum: range [-26.7, 126.6], stdev 15.90

Bias (mean residual): +0.37 kJ/g-atom — nearly unbiased

Residual stdev: 7.22 (MAE) / 14.16 (RMSE) — 3.2× the CALPHAD stdev

Three hypotheses tested

Hypothesis	Predictor	Pearson r (residual, predictor)	Verdict
(a) Boundary effect	CALPHAD ΔH (mismatches cluster near 0)	+0.187	Weak — r=+0.187; mismatches do cluster at low \|ΔH\| (mean 4.1 vs 7.1) but residual doesn't track ΔH strongly
(b) Missing strain term	\|V_A − V_B\| / (V_A + V_B)	+0.379	Moderate — r=+0.379; size-mismatch explains ~14% of residual variance. Gokcen's variant lacks the strain term present in full Miedema.
(c) Chemical-term over-amplification	Q_raw = 9.4(Δn)² − (Δφ)²	+0.778	Strong — r=+0.778; Q_raw explains ~60% of residual variance. When Miedema's chemical term is large positive, Miedema overestimates; when it's slightly negative, Miedema predicts ordering where CALPHAD says demixing.

Residual vs Q_raw (the smoking gun)

r = +0.778. The residual tracks Miedema's own chemical term: the model amplifies Q beyond what CALPHAD permits in this regime.

Residual vs size mismatch

r = +0.379. Weaker but present: larger volume asymmetry → larger residual, consistent with a missing strain term.

Sign mismatches: 50/201 = 25%

50 mild-demixing systems have Miedema predicting ΔH ≤ 0 (ordering/stability) where CALPHAD says mild demixing. These cluster at low CALPHAD ΔH (mean 4.09 vs 7.11 for correctly-predicted systems), confirming the boundary effect — but they span all element classes, not a specific family:

16 remn-remn15 trans-trans14 other5 noble-remn

Worst 15 mild outliers (by |residual|)

System	CALPHAD ΔH	Miedema ΔH	residual
Cs-Mo	+12.50	+126.58	+114.08
K-Zr	+13.19	+83.38	+70.19
Cr-Na	+14.11	+69.58	+55.47
Na-Ti	+5.63	+59.87	+54.24
Na-Zr	+14.78	+54.77	+39.99
Ba-Ti	+16.24	+51.08	+34.85
Bi-Rb	+7.11	-24.97	-32.08
La-Mo	+18.90	+48.24	+29.34
Cd-Ti	+13.75	-14.52	-28.27
Bi-Cs	+0.25	-26.74	-26.99
Ca-Ti	+12.50	+37.40	+24.90
Cu-U	+18.53	-4.13	-22.66
Nb-Sb	+5.91	-15.22	-21.13
Bi-Na	+2.00	-18.74	-20.74
Bi-U	+15.62	-4.77	-20.39

Cs-Mo (+127 vs +12) and K-Zr (+83 vs +13) are extreme over-amplifications: Miedema's Q_raw is large positive (large Δn and Δφ), and the linear formula projects it to a ΔH far beyond CALPHAD's mild-demixing value. Bi-Cs (−27 vs +0.25) and Cu-U (−4 vs +19) are the opposite: Q_raw is slightly negative, flipping Miedema's sign. Both patterns are the same underlying issue — Miedema's chemical term is well-calibrated for strong interactions (stable stratum, R²=0.61) but over-amplified for weak interactions, where the linear-in-Q formula lacks the saturation that real alloys exhibit.

Kit validation: Gokcen's worked examples

Reproduces validate_gokcen.m — 13 cases from Table B.5 + Ti₀.₃₃H₀.₆₇ hydride + TiN. 13/15 within 0.5 kJ/g-atom. The two known deviations (Cu-Zn disordered, TiN) are documented in the kit README.

A	B	xB	state	order	calc	book	\|err\|	note
Au	Zn	0.5	liquid	disordered	-16.5	-16.0	0.47	worked example p.264-265
Al	Ni	0.5	solid	ordered	-47.2	-47.0	0.20	Table B.5
Al	Ti	0.5	solid	disordered	-44.2	-44.0	0.22	Table B.5
Cu	Zn	0.5	solid	disordered	-7.1	-5.0	2.07	Table B.5
Cu	Zn	0.5	solid	ordered	-10.5	-10.5	0.05	p.265
Fe	Ti	0.5	solid	disordered	-18.7	-19.0	0.25	Table B.5
Fe	V	0.5	solid	disordered	-7.1	-7.0	0.09	Table B.5
Ni	Ti	0.5	solid	disordered	-38.0	-38.0	0.02	Table B.5
Pd	Rh	0.5	solid	disordered	+1.9	+2.0	0.13	Table B.5
Cu	Fe	0.5	liquid	disordered	+14.7	+15.0	0.30	Table B.5
Hg	Pb	0.5	liquid	disordered	+1.3	+1.0	0.32	Table B.5
Ni	Ti	0.5	liquid	disordered	-38.0	-38.0	0.02	Table B.5
Al	Au	0.5	liquid	disordered	-21.4	-21.0	0.38	Table B.5
Ti	H	0.67	solid	ordered	-110.0	-110.0	0.01	Ti0.33H0.67 hydride, p.268-269
Ti	N	0.5	solid	ordered	-284.7	-273.0	11.66	TiN, p.266

Synthetic TDB unit test

Reproduces test_tdb.m at T=1873 K. All four values exact.

Quantity	Computed	Expected	Match
LIQUID dH_eq	-2.032	-2.032	✓
LIQUID dHinf_B	-3.127	-3.127	✓
LIQUID dHinf_A	-13.127	-13.127	✓
BCC dH_eq	5.000	5.000	✓

Exclusion audit

Of 2,773 files, 1,319 were skipped. Breakdown:

455	LB (lattice stability) files — kit design exclusion
397	Non-metallic systems (B, C, N, O, H, S, P) — kit design exclusion
5	No LIQUID phase in TDB — nothing to compare
461	No Miedema parameters in Gokcen's 59-element table

The 461 "no params" systems are the audit concern. Gokcen's 1986 table covers 59 elements; the corpus contains 24 additional elements that lack Miedema parameters. These are concentrated in the lanthanides (Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu — La is present, the rest are not) and actinides (Np, Am — U and Pu are present, Th is present). These are exactly the systems where Miedema is known to struggle, which makes their absence from the comparison a reporting limitation, not a methodology strength.

Symbol	Name	Systems affected
AM	Americium	15
BR	Bromine	1
CE	Cerium	67
CL	Chlorine	3
DY	Dysprosium	39
ER	Erbium	37
EU	Europium	13
F	Fluorine	5
GD	Gadolinium	62
HO	Holmium	34
I	Iodine	1
KR	Krypton	2
LU	Lutetium	17
ND	Neodymium	60
NP	Neptunium	12
PR	Praseodymium	31
SM	Samarium	29
TB	Terbium	23
TM	Thulium	14
YB	Ytterbium	27

Full system-level results

Click any row to open the representative TDB on gotrawama.eu. Search by system, element, or year.

System	Stratum	N ass.	Year range	ΔH_eq CALPHAD (med)	ΔH_eq Miedema	residual	ΔH∞B CALPHAD	ΔH∞B Miedema	ΔH∞A CALPHAD	ΔH∞A Miedema	TDB

Interpretation

The question this comparison can answer is narrow and specific: does Miedema's semi-empirical chemical term predict the sign and magnitude class of CALPHAD equimolar mixing enthalpies, in the regime where the CALPHAD enthalpy is a physically meaningful quantity? The answer is yes, and the agreement is comparable to Miedema's documented reputation (Niessen & de Boer, 1988).

The regime matters. Where the liquid is stable at equimolar composition (ΔH ≤ 0), the CALPHAD L-parameters are anchored to calorimetry — the equimolar enthalpy is a measured quantity, and Miedema tracks it with R² = 0.61, MAE = 8.8 kJ/g-atom, 94% sign agreement. Where the liquid is metastable but exists (0 < ΔH ≤ +20), Miedema systematically underestimates the positive tail — the model's chemical term captures the sign but not the full magnitude of demixing, which is expected for a model without explicit topological or strain contributions beyond the cell-volume correction.

Where ΔH > +20 kJ/g-atom, the comparison is ill-posed. These 30 systems (Ag-Cr, Al-Na, Cs-Mo, K-Zr, Cr-Sn, ...) have CALPHAD "enthalpies" of +100 to +500 kJ/g-atom that are not physical mixing enthalpies — they are large positive L₀ values tuned to reproduce a miscibility gap. The homogeneous equimolar liquid does not exist at any temperature where these parameters apply, so no calorimetric measurement has ever returned that number. Comparing Miedema to it tests nothing. This is the identifiability axis made visible: where the liquid is stable, the L-parameter is constrained by data; where it is not, the L-parameter is a fitting coordinate whose magnitude is set by the phase boundary, not by enthalpy.

Stratifying by liquid stability — and quarantining the gap systems rather than scoring them as errors — is the contribution this analysis can make over a row-level R². The 95% sign agreement for systems with |ΔH| ≥ 5 kJ/g-atom is the number that matches the literature reputation of the model. The low row-level R² is an artifact of (a) pseudoreplication from multiply-assessed systems and (b) including the gap stratum, neither of which reflects model accuracy.

Limitations: (1) The Gokcen 1986 parameter table covers 59 elements; the corpus contains 20 additional elements (13 lanthanides, 2 actinides, 5 others) that lack Miedema parameters and are excluded — these include rare-earth systems where Miedema is known to struggle, so the comparison set is biased toward elements the model handles well. (2) The dilute-limit parity uses the analytic Miedema partial ΔH̄_B^∞ = 2p·V_B^eff·Q/snw (replacing a finite-difference hack that returned ~4×equimolar regardless of volume asymmetry); the CALPHAD side is the standard Σ h_ν. (3) The stratification threshold (+20 kJ/g-atom) is a heuristic for "liquid almost certainly unstable at equimolar"; the results are not sensitive to the exact value within ±10 kJ/g-atom.

Databases © Bengt Hallstedt, hosted at gotrawama.eu/bengt/. Filename convention: ELEMENT-ELEMENT-YYAuthor[-LB].txt — YY is the two-digit publication year, Author is the first three letters of the first author's surname, and -LB indicates a lattice-stability dataset (vs. a full thermodynamic assessment). This index page is a read-only navigator; clicking a database opens the original .txt file on the source server.