cs.LGFeb 26, 2026

Prediction of Diffusion Coefficients in Mixtures with Tensor Completion

Authors: Zeno Romero, Kerstin Münnemann, Hans Hasse, Fabian Jirasek

Organizations: Laboratory of Engineering Thermodynamics, RPTU Kaiserslautern, Erwin-Schrödinger-Str. 44, 67663 Kaiserslautern, Germany

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Abstract

Predicting diffusion coefficients in mixtures is crucial for many applications, as experimental data remain scarce, and machine learning (ML) offers promising alternatives to established semi-empirical models. Among ML models, matrix completion methods (MCMs) have proven effective in predicting thermophysical properties, including diffusion coefficients in binary mixtures. However, MCMs are restricted to single-temperature predictions, and their accuracy depends strongly on the availability of high-quality experimental data for each temperature of interest. In this work, we address this challenge by presenting a hybrid tensor completion method (TCM) for predicting temperature-dependent diffusion coefficients at infinite dilution in binary mixtures. The TCM employs a Tucker decomposition and is jointly trained on experimental data for diffusion coefficients at infinite dilution in binary systems at 298 K, 313 K, and 333 K. Predictions from the semi-empirical SEGWE model serve as prior knowledge within a Bayesian training framework. The TCM then extrapolates linearly to any temperature between 268 K and 378 K, achieving markedly improved prediction accuracy compared to established models across all studied temperatures. To further enhance predictive performance, the experimental database was expanded using active learning (AL) strategies for targeted acquisition of new diffusion data by pulsed-field gradient (PFG) NMR measurements. Diffusion coefficients at infinite dilution in 19 solute + solvent systems were measured at 298 K, 313 K, and 333 K. Incorporating these results yields a substantial improvement in the TCM's predictive accuracy. These findings highlight the potential of combining data-efficient ML methods with adaptive experimentation to advance predictive modeling of transport properties.

Figures & tables

Figure 1: Experimental data for liquid-phase diffusion coefficients Dij∞D_{ij}^{\infty} of solutes ii at infinite dilution in solvents jj at temperatures 298 K, 313 K, and 333 K in the database used as the starting point in the present work. Numbers identify solutes and solvents, cf. Tables S1 and S2 in the Supporting Information. Solutes are ordered with respect to their molar mass (bottom: low; top: high), and solvents are ordered with respect to their viscosity (left: low; right: high). The color code indicates the value of Dij∞,expD_{ij}^{\infty\text{,exp}} , and black cells denote missing data.
TT298 K313 K333 K
Number of data points2247556
Matrix occupation rate16.1 %16.1\>\%5.4 %5.4\>\%4.0 %4.0\>\%
Number of available solvents312418
Number of available solutes453533
Table 1: Information on the availability of Dij∞D_{ij}^{\infty} in our database for each studied temperature, cf. Figure 1 .
Figure 2: Experimental data for liquid-phase diffusion coefficients Dij∞,expD_{ij}^{\infty\text{,exp}} of solutes ii at infinite dilution in solvents jj at temperatures 298 K, 313 K, and 333 K in the database used as starting point for the AL study. Numbers identify solutes and solvents, cf. Tables S3 and S4 in the Supporting Information. Solutes are ordered with respect to their molar mass (bottom: low; top: high), and solvents are ordered with respect to their viscosity (left: low; right: high). The color code indicates the value of Dij∞D_{ij}^{\infty} , and black cells denote missing data.
Figure 3: Schematic representation of the hybrid TCM for predicting temperature-dependent Dij∞D_{ij}^{\infty} developed in this work. The TCM incorporates prior information from the SEGWE model Evans et al. (2018) and uses the Tucker decomposition for tensor factorization.
Figure 4: Active learning workflow for the targeted improvement of the TCM developed in this work.
Figure 5: Boxplot of the ARE ij of the predicted Dij∞D_{ij}^{\infty} with SEGWE Evans et al. (2018) , MCM Großmann et al. (2022) , and the developed TCM. MCM and TCM results were obtained using leave-one-out analysis, the SEGWE model was used as proposed by the original authors Evans et al. (2018) . Boxes represent interquartile ranges (IQR) and whiskers represent 1.5 IQR.
Figure 6: Temperature features w1(T)w_{1}(T) and w2(T)w_{2}(T) of a TCM trained on the full discrete data set as a function of TT and linear fits.
γ\gammaAγA_{\gamma}BγB_{\gamma}R2R^{2}MSE
111.195-6.179 ⋅10−3\cdot 10^{-3}0.99394.81 ⋅10−5\cdot 10^{-5}
228.271-2.648 ⋅10−2\cdot 10^{-2}0.99973.18 ⋅10−5\cdot 10^{-5}
Table 2: Regression statistics obtained from fitting equation ( 8 ) to the discrete wγw_{\gamma} , cf. Figure 6 .
Figure 7: Boxplot of the ARE ij of the predicted Dij∞D_{ij}^{\infty} with SEGWE Evans et al. (2018) and the TCM as a function of TT . The numbers above each box indicate the number of data points per bin, horizontal lines represent the median, boxes represent the IQR, and whiskers represent 1.5 IQR.
   Dij∞,exp / 10−9m2s−1D^{\infty\text{,exp}}_{ij}\>/\>10^{-9}\text{m}^{2}\text{s}^{-1}
No.Solute iiSolvent jj298 K313 K333 K
1Methyl isopropyl ketone1,2-Propanediol0.052 ±\pm 0.0020.115 ±\pm 0.0090.248 ±\pm 0.044
2Butyl acetate1,2-Propanediol0.059 ±\pm 0.0020.122 ±\pm 0.0030.267 ±\pm 0.006
3Benzaldehyde1,2-Propanediol0.059 ±\pm 0.0090.127 ±\pm 0.0010.296 ±\pm 0.022
4DimethoxymethaneAcetone4.064 ±\pm 0.0254.894 ±\pm 0.0416.161 ±\pm 0.102
5WaterDimethoxymethane5.737 ±\pm 0.1327.085 ±\pm 0.1319.270 ±\pm 0.177
Table 3: Liquid-phase diffusion coefficients at infinite dilution Dij∞,expD^{\infty\text{,exp}}_{ij} measured by PFG NMR spectroscopy in this work, including experimental uncertainty σijexp\sigma_{ij}^{\text{exp}} . The systems are sorted according to the order in which they were selected by the AL strategy, measured, and subsequently included in the TCM training.
Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1 : Availability of experimental data for liquid-phase diffusion coefficients Dij∞,expD_{ij}^{\infty\text{,exp}} of solutes ii in solvents jj at infinite dilution at temperatures 298, 313, and 333 K from the DDB [ 15 ] and our previous work Romero et al. (2025) ; Mross et al. (2024) ; Bellaire et al. (2022a) .
Nr.SoluteSMILES
1MethaneC
2WaterO
3MethanolCO
4Argon[Ar]
5AcetonitrileCC#N
6Carbon dioxideO=C=O
Appendix
Table S1 : Identifiers for solutes considered in the full literature database, cf. Figure , including their SMILES.
Nr.SolventSMILES
1HexaneCCCCCC
2AcetoneCC(C)=O
3DimethoxymethaneCOCOC
4AcetonitrileCC#N
5HeptaneCCCCCCC
6Butyl chlorideCCCCCl
Appendix
Table S2 : Identifiers for solvents considered in the full literature database, cf. Figure , including their SMILES.
Nr.SoluteSMILES
1WaterO
2MethanolCO
3AcetonitrileCC#N
4EthanolCCO
5AcetoneCC(C)=O
6Acetic acidCC(=O)O
Appendix
Table S3 : Identifiers for and solutes considered in the reduced database used for the active learning (AL) portion, cf. Figure including SMILES.
Nr.SolventSMILES
1HexaneCCCCCC
2AcetoneCC(C)=O
3DimethoxymethaneCOCOC
4AcetonitrileCC#N
5HeptaneCCCCCCC
6Butyl chlorideCCCCCl
Appendix
Table S4 : Identifiers for solvents considered in the reduced database used for the AL portion, cf. Figure , including SMILES.
ChemicalFormulaSupplierPurity
   %
1,2-PropanediolC 3 H 8 O 2Sigma-Aldrich≥\geq 99.50
2-Methyl-2,4-pentanediolC 6 H 14 O 2TCI>> 99.00
AcetoneC 3 H 6 OSigma-Aldrich≥\geq 99.90
AcetonitrileC 2 H 3 NSigma-Aldrich≥\geq 99.90
BenzaldehydeC 7 H 6 ORoth≥\geq 99.50
Appendix
Table S5 : Suppliers and purities of the chemicals used in this work. The purities were taken from the supplier’s data sheets.
Figure S2 : rMAE for predicting Dij∞D_{ij}^{\infty} (averaged over solutes, solvents, and temperatures) by TCM for different combinations of hyperparameters rur_{u} , rvr_{v} , and rwr_{w} evaluated by leave-one-out analysis based on the experimental data set in Figure .
Figure S3 : Boxplot of the ARE of the predicted Dij∞D_{ij}^{\infty} with SEGWE Evans et al. (2018) , MCM Großmann et al. (2022) , and the developed TCM obtained using leave-one-out analysis. Boxes represent interquartile ranges (IQR) and whiskers represent 1.5 IQR. Evaluated based only on data points whose components appeared in all training data sets.
Figure S4 : Histograms showing the number of systems NsystemsN_{\mathrm{systems}} for which Dij∞D_{ij}^{\infty} is predicted with a certain relative error with SEGWE, the temperature-specific MCMs, and the developed TCM for the three studied temperatures. The results of the MCMs and the TCM were obtained using a leave-one-out analysis.
Figure S5 : Performance of the TCM for predicting Dij∞D^{\infty}_{ij} in terms of relative mean absolute error (rMAE) and relative mean squared error (rMSE) as a function of the additional training data points selected using uncertainty sampling and measured in this work and comparison with SEGWE Evans et al. (2018) .
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