Learning atomic interactions for the thermodynamics of liquid mixtures

Marco Hoffmann; Sebastian Schmitt; Hans Hasse; Fabian Jirasek

Figure 1: Overview of the COGNAC model architecture and its application. a , Molecular graphs constructed from SMILES [ 30 ] using RDKit [ 31 ] are encoded by a message-passing neural network (MPNN). Heavy atoms and hydrogens bound to non-carbon atoms form explicit nodes, allowing hydrogen-bond donors and acceptors to be resolved. Atoms with identical molecular neighborhoods within the message-passing radius obtain identical embeddings. b , Two prediction heads map the context-aware atom embeddings

\mathbf{h}_{a}

to volume and surface-area parameters

R_{a}

and

Q_{a}

, and pairs of embeddings to the symmetric interaction coefficients

a_{ab}

,

b_{ab}

, and

c_{ab}

of the exported parameter table (Methods). Summing the respective atom-level geometry parameters gives the molecular quantities

r_{i}

and

q_{i}

. c , The learned parameters and the state point (composition

\mathbf{x}

and temperature

T

) are processed in a local-composition framework. The combinatorial contribution accounts for size and shape effects, while atomic surface areas and pair interactions determine the residual contribution. Together, the two contributions yield the activity coefficient of component

i

. d , For a specified component set, parameters need to be generated only once and can be reused across compositions and temperatures without further neural network evaluations. The obtained atom-level parameter vectors

\mathbf{R}

and

\mathbf{Q}

and interaction matrices

\mathbf{a}

,

\mathbf{b}

, and

\mathbf{c}

enable, e.g., the prediction of vapor–liquid (VLE), liquid–liquid (LLE), and solid–liquid (SLE) equilibria, azeotropes, and residue curve maps for binary and multi-component mixtures, and can easily be implemented in process simulations.

Figure 2: Results of COGNAC and comparison to mod. UNIFAC and HANNA. a , System-wise mean absolute errors

\mathrm{MAE}_{\mathrm{sys}}

for the binary data, resolved by data type. Results of COGNAC and HANNA are obtained by system-wise ten-fold cross-validation, ensuring unseen test systems. Data types: activity coefficients from vapor–liquid equilibria with (TPXY) and total pressures without (TPX) reported vapor composition, activity coefficients at infinite dilution (ACT), liquid–liquid equilibria (LLE), excess enthalpies (HE), and activity coefficients from solid–liquid equilibria (SLE). The scores are calculated from the absolute deviations in the logarithmic activity coefficient (TPXY, ACT, SLE), logarithmic pressure (TPX), liquid mole fractions (LLE), and excess enthalpies in kJ mol -1 (HE). Circles denote the median, diamonds the mean, and the shaded bars the interquartile range of the distribution of

\mathrm{MAE}_{\mathrm{sys}}

over all systems;

n

is the number of systems in the respective panel, restricted to those systems that all three models can describe. The percentage above each panel shows the relative change in the median

\mathrm{MAE}_{\mathrm{sys}}

of COGNAC relative to HANNA; negative values indicate that COGNAC is more accurate. Mod. UNIFAC describes 50% of the binary systems of the full data set (Extended Data Fig. 1 ). COGNAC and HANNA were trained and evaluated on the same data, mod. UNIFAC was used as published. b , Same comparison for the ternary data set, predicted with the full-data models of COGNAC and HANNA. Mod. UNIFAC describes 52% of the ternary systems of the full data set. c – e , Exemplary COGNAC predictions for different binary data types against experimental data from unseen systems: solid–liquid equilibrium with eutectic point ( c ; the dashed line is the ideal-solution liquidus), heteroazeotrope with three-phase equilibrium at atmospheric pressure ( d ; the horizontal line marks the temperature at which vapor and two liquid phases coexist; below it, the green lines are the compositions of the two coexisting liquid phases), and excess enthalpies at different temperatures ( e ).

Figure 3: COGNAC predictions of azeotropes and residue curve maps. a,b , Classification matrices for azeotropes in binary ( a ) and ternary ( b ) mixtures: predicted class against experimental class. The diagonal marks correct prediction of azeotrope existence and class with COGNAC; the corresponding matrices for HANNA and mod. UNIFAC are shown in Extended Data Fig. 7 . Azeotrope types are minimum-boiling (

p_{\mathrm{max}}

), maximum-boiling (

p_{\mathrm{min}}

), double azeotrope (double), saddle-point (saddle), and zeotropic (none).

n

denotes the number of data points. c , Predicted compositions at the azeotropic point, compared with experimental values, for all components in the mixture. Only azeotropes for which existence and type were predicted correctly are shown;

n

is the number of azeotropes, each contributing one point per component. d,e , Isothermal residue curve maps predicted by COGNAC for acetone + chloroform + hexane at 318.15 K ( d ) and for 2-propanol + nitromethane + 1,1,2,2-tetrachloro-1,2-difluoroethane (R112) at 347.65 K ( e ), a mixture outside the scope of mod. UNIFAC (no group decomposition for R112). Orange lines are residue curves, black lines are the distillation boundaries. The pressures at the vertices are the pure-component vapor pressures at the given temperature; the pressure next to a ternary azeotrope is its predicted value. Predicted singular points are marked by symbols encoding their type (unstable node, stable node, saddle). Experimental azeotropes are shown in red: the ternary azeotrope with its reported type, binary azeotropes measured within 5 K with the symbol of the nearest predicted singular point on that edge.

Figure 4: Interpretability of COGNAC. a , Molecular graph of 4-methylhexanoic acid. The surface area of the orange circles surrounding each atom is scaled with the predicted atom surface area from COGNAC; carbon-bound hydrogens are folded into their carbon atom, and the hollow gray circle has an area of 20 Å 2 . b , Matrix of averaged exchange energies

\overline{\Delta u_{ab}}

at 298.15 K between selected atom types, obtained from all atom pairs of all binary mixtures in the data set, including pairs of atoms of the same component. Pairs of the same atom type are not shown; their

\overline{\Delta u_{ab}}

would not vanish either, since each type pools atoms with different molecular neighborhoods and

\Delta u_{ab}=0

holds only for two atoms with identical embeddings. The color scale is centered at zero and scaled separately to the most negative and the most positive value. c , Exchange energy

\Delta u_{ab}

at 298.15 K between each carbon atom of 1-hexanol and the oxygen atom of water, over the number of bonds between the carbon atom and the hydroxyl oxygen. d , Exchange energy between the CH 2 carbon atom bonded to the substituent X of 1-substituted hexanes (C 5 H 11 –CH 2 –X) and the oxygen atom of water. The dashed lines in c and d mark the central CH 2 groups of hexane (C3 and C4, equivalent by symmetry); the colors follow the color scale of b .

Figure 5: Dynamic simulation of a batch distillation using a Julia-based process simulator and the process model from Werner et al. [ 45 ] . a,b , Liquid mole fractions

x_{i}^{\mathrm{res}}

in the reboiler, i.e., in the liquid residue, over the course of the batch distillation of a ternary mixture (1-butanol + 2-propanol + water, head pressure 70 kPa; a ) and a five-component mixture (methanol + ethanol + 2-propanol + 1-butanol + acetone, head pressure 50 kPa; b ), simulated with COGNAC (lines). Symbols denote experimental data from a laboratory-scale plant [ 44 ] . c , Wall-clock time that the integrator spends per time step for five thermodynamic models and the two mixtures. Median of up to 10 repeated solves; the compilation time of the generated residual functions is measured separately and excluded.

Extended Data Fig. 1: Benchmark of COGNAC against HANNA. a , System-wise mean absolute errors

\mathrm{MAE}_{\mathrm{sys}}

for the binary data, resolved by data type; each system is predicted by the COGNAC and HANNA models on the ten-fold cross-validation set it did not see during training. Circles denote the median, diamonds the mean, and shaded bars the interquartile range of the distribution over systems.

n

is the number of systems in each panel. For LLE, a system is included if both models predict a phase split for at least one of its data points. The percentage above each panel shows the relative change in the median

\mathrm{MAE}_{\mathrm{sys}}

of COGNAC relative to HANNA; negative values indicate that COGNAC is more accurate. Both models were trained on the same binary data set. The systems outside the coverage of mod. UNIFAC, included here but not in Fig. 2 , contain components that mod. UNIFAC cannot decompose or for which interaction parameters are missing; many of them are ionic liquids. b Same comparison for the ternary data set, predicted with the full-data models of COGNAC and HANNA.

Extended Data Fig. 2: Benchmark of COGNAC against mod. UNIFAC and HANNA on the quaternary data set, analogous to Fig. 2 . The data set comprises 21,221 data points for 1,032 systems: 3,750 TPXY data points (134 systems), 1,113 TPX data points (38 systems), and 16,358 LLE data points (888 systems, including 234 three-liquid data points). COGNAC and HANNA were trained only on binary data, so every quaternary prediction extrapolates the number of components; mod. UNIFAC was used as published and is the only model whose parameters were also fitted to multi-component data. LLE predictions use the same phase-equilibrium procedure as for the ternary mixtures, with the four-component composition space discretized into 6,545 points. System-wise mean absolute errors

\mathrm{MAE}_{\mathrm{sys}}

of the logarithmic activity coefficients (TPXY), the logarithmic pressure (TPX), and the liquid mole fractions (LLE, all phases and components). a , Shared subset: systems that all three models can describe; for LLE, only systems for which all three models predict a phase split are included. Relative to mod. UNIFAC, COGNAC reduces the median

\mathrm{MAE}_{\mathrm{sys}}

by 9% (TPXY), 11% (TPX), and 18% (LLE). b , Full data set: all systems that COGNAC and HANNA both describe, without restricting the comparison to the coverage of mod. UNIFAC. Circles denote the median, diamonds the mean, and shaded bars the interquartile range of the distribution over systems;

n

is the number of systems. The percentage above each panel shows the relative change in the median

\mathrm{MAE}_{\mathrm{sys}}

of COGNAC relative to HANNA; negative values indicate that COGNAC is more accurate. The number of TPX systems is small, so these values should be read with caution.

Extended Data Fig. 3: Benchmark of HANNA and COGNAC for aqueous and non-aqueous mixtures. System-wise mean absolute errors

\mathrm{MAE}_{\mathrm{sys}}

for binary ( a , b ) and ternary ( c , d ) mixtures that contain water ( a , c ) or not ( b , d ), on all systems that both models describe. Binary: out-of-fold predictions of the ten-fold cross-validation; ternary: full-data models. The data points are those of Extended Data Fig. 1 , split by whether a component is water. Circles denote the median, diamonds the mean, and shaded bars the interquartile range across systems;

n

is the number of systems. The percentage above each panel shows the relative change in the median

\mathrm{MAE}_{\mathrm{sys}}

of COGNAC relative to HANNA; negative values indicate that COGNAC is more accurate. For every data type and both mixture classes, COGNAC gives the lower median

\mathrm{MAE}_{\mathrm{sys}}

. For the binary mixtures, the relative reduction is similar with and without water except for TPX; for the ternary mixtures, it is considerably larger for aqueous mixtures in TPXY and ACT, larger in TPX, and smaller in LLE and HE.

Extended Data Fig. 4: Effect of the hydrogen representation on COGNAC accuracy, resolved by mixture and data type. Bars show the median

\mathrm{MAE}_{\mathrm{sys}}

of the production model, in which polar hydrogen atoms are explicit graph nodes, and of an ablated model in which all hydrogen atoms are treated implicitly, so that water is a single node. Architecture, training data, loss weights, hyperparameters, and random seed are identical; the only further difference is that the sign regularization of the donor–acceptor interactions, which is defined on the explicit polar hydrogen atoms, has no counterpart in the ablated model, so the comparison measures the combined effect of the explicit hydrogen nodes and this regularizer. We trained the ablated model in every fold of the ten-fold cross-validation and evaluated it like the production model: binary mixtures on out-of-fold predictions, and ternary mixtures with the full-data model, on all systems in these data sets that both models describe;

n

is the number of systems. For aqueous mixtures, the implicit treatment increases the error for every data type, most strongly for binary LLE (

+101.8\%

) and TPX (

+62.2\%

); without water, the two representations differ by less than

5\%

except for ternary ACT and HE. For reference, the bars also show the median

\mathrm{MAE}_{\mathrm{sys}}

of HANNA, trained on the same folds. HANNA is scored on all systems for which it yields a prediction; for LLE, it predicts no phase split for some systems, which are excluded (binary: 1,840 of 1,998 non-aqueous and 906 of 917 aqueous systems; ternary: 2,414 of 2,482 and 2,897 of 2,933). Even the ablated model is more accurate than HANNA in 18 of the 22 cells; HANNA is ahead only for binary TPXY, LLE, HE, and SLE of aqueous mixtures.

Extended Data Fig. 5: Activity coefficients at infinite dilution of alkanes (top) and alcohols (bottom) in water, predicted with mod. UNIFAC, mod. UNIFAC 2.0, HANNA, COGNAC, and COGNAC-IMP, against the experimental values: binary ACT data with water as the solvent, with the solutes classified by their DDB component class. COGNAC, COGNAC-IMP, and HANNA: out-of-fold predictions of the ten-fold cross-validation. Mod. UNIFAC 2.0 [ 20 ] is a version of mod. UNIFAC in which the entire interaction-parameter matrix was fitted simultaneously via matrix completion rather than sequentially; both mod. UNIFAC versions are used as published and were fitted to DDB data that likely include these data points, which favors them. Only data points for which all models yield a prediction are shown (367 data points of 41 alkane–water and 784 data points of 67 alcohol–water systems). MAE is the mean absolute deviation in

\ln\gamma_{i}^{\infty}

. Mod. UNIFAC underestimates

\ln\gamma_{i}^{\infty}

of every alkane in water; mod. UNIFAC 2.0 reduces the error but still underestimates

97\%

of the alkane data points, so the deficiency is not an artifact of the sequential fitting but of the static group decomposition itself. COGNAC, which evaluates the same equations with atom-level parameters, and HANNA, which does not build on a local-composition model, describe both families well.

Extended Data Fig. 6: Activity coefficients of the three benzenediol isomers at infinite dilution in diisopropyl ether ( a ) and in 4-methyl-2-pentanone ( b ) at

298.15\,\mathrm{K}

, predicted with mod. UNIFAC and COGNAC. In mod. UNIFAC, the three isomers have identical group counts, so it predicts one value for all benzenediols. Experimental values are shown as filled circles, and COGNAC predictions are shown as open diamonds connected by a thin line to the corresponding measurement. The substitution pattern of each component is given in parentheses. All six COGNAC activity coefficients are out-of-fold predictions; i.e., each was predicted by a cross-validation model that did not see the respective mixture during training.

Extended Data Fig. 7: Azeotrope predictions of HANNA ( a – c ) and mod. UNIFAC ( d – f ), analogous to Fig. 3 a–c. a,b,d,e , Classification matrices for binary ( a,d ) and ternary ( b,e ) mixtures, predicted class against experimental class; the diagonal marks correct prediction of azeotrope existence and class. c,f , Predicted against experimental compositions at the azeotropic point, over all components, for the azeotropes whose existence and type were predicted correctly, excluding double azeotropes. Each panel shows the model on all data points for which it yields a prediction, so

n

is smaller for mod. UNIFAC; the classification shares compared in Supplementary Note are evaluated on the data points that all three models describe.

Extended Data Fig. 8: COGNAC cavity geometries compared with COSMO references from the CHAOS database [ 35 ] , predicted by the full-data COGNAC model. a , molecular volume

V_{i}

; b , molecular surface area

A_{i}

; and c , atomic surface area

A_{a}

. Molecular values are the sums of the atom-level geometry-head outputs,

V_{i}^{\mathrm{pred}}=\sum_{a\in i}V_{a}

and

A_{i}^{\mathrm{pred}}=\sum_{a\in i}A_{a}

. The comparison is restricted to the 45,488 CHAOS molecules that occur in no mixture of the thermodynamic data and consist only of elements in the data set; since the COSMO regularizer used in training covers only molecules in the thermodynamic data, none of the compared values entered training. Atom-level areas are available for 45,427 of these molecules (675,822 atoms); carbon-bonded hydrogen areas are folded onto their parent atoms, and polar hydrogens are explicit, consistent with the model’s molecular graphs. Superscripts

\mathrm{pred}

and

\mathrm{COSMO}

distinguish predictions and reference values; dashed lines indicate parity. The axis limits run from the first percentile of the smaller coordinate in each pair to the 99th percentile of the larger one, retaining about

98\%

of the points; the reported counts and statistics include points outside the displayed limits, excluding 701 zero-area references from the logarithmic comparison.

R^{2}_{\log}

denotes the coefficient of determination against parity in logarithmic space; MAE is calculated in the physical units shown and amounts to 11.02 Å 3 (molecular volume), 14.13 Å 2 (molecular surface area), and 1.69 Å 2 (atomic surface area).

Temperature dependence	Closure	TPXY	TPX	ACT	LLE	SLE	HE
UNIFAC	explicit	$0.083$	$0.090$	$0.213$	$0.066$	$0.134$	$0.147$
UNIFAC	implicit	$0.085$	$0.092$	$0.201$	$0.067$	$0.129$	$0.159$
free form	explicit	$0.078$	$0.088$	$0.198$	$0.062$	$0.128$	$0.147$
free form	implicit	$\mathbf{0.077}$	$\mathbf{0.085}$	$0.188$	$0.060$	$0.124$	$0.139$
mod. UNIFAC	explicit	$0.080$	$0.086$	$0.194$	$\mathbf{0.059}$	$0.124$	$0.145$
mod. UNIFAC	implicit	$0.078$	$\mathbf{0.085}$	$\mathbf{0.185}$	$0.060$	$\mathbf{0.121}$	$\mathbf{0.138}$

Extended Data Table 1: Ablation study comparing six COGNAC model variants. The models differ in the implemented temperature dependence (UNIFAC, free form, mod. UNIFAC) and the closure of the residual activity coefficients (explicit, implicit). Each variant was trained on the ten folds of the cross-validation (Methods) with identical hyperparameters and the same single random seed per fold, so that the variants differ only in the stated switches (the UNIFAC variants also in the combinatorial term, see Supplementary Note ); differences of a few percent are within the variation between independently trained models. Scores are the mean

\mathrm{MAE}_{\mathrm{sys}}

over the out-of-fold predictions of all binary systems; as for the comparison of the closures in Supplementary Note , the mean over the systems is used rather than the median of the benchmarks, so that the tail of the error distribution is included. The lowest value per data type among the variants is printed in bold. HANNA denotes the HANNA results of the same ten-fold cross-validation; for LLE, they yield a prediction for 2,653 of the 2,784 systems and are scored on these. COGNAC is the mod. UNIFAC variant with explicit closure; COGNAC-IMP is the one with implicit closure (Supplementary Note ).

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