NUMERICAL ESTIMATION OF ADSORPTION ENERGY-DISTRIBUTIONS FROM ADSORPTION-ISOTHERM DATA WITH THE EXPECTATION-MAXIMIZATION METHOD

The expectation-maximization (EM) method of parameter estimation is used to calculate adsorption energy distributions of molecular probes from their adsorption isotherms. EM does not require prior knowledge of the distribution function, or the isotherm, requires no smoothing of the isotherm data, and converges with high stability toward the maximum-likelihood estimate. The method is therefore robust and accurate at high iteration numbers. The EM algorithm is tested with simulated energy distributions corresponding to unimodal Gaussian, bimodal Gaussian, Poisson distributions, and the distributions resulting from Misra isotherms. Theoretical isotherms are generated from these distributions using the Langmuir model, and then chromatographic band profiles are computed using the ideal model of chromatography. Noise is then introduced in the theoretical band profiles comparable to those observed experimentally. The isotherm is then calculated using the elution-by-characteristic points method. The energy distribution given by the EM method is compared to the original one. The results are contrasted to those obtained with the House and Jaycock algorithm HILDA and shown to be superior in terms of both robustness, accuracy, and information theory. The effect of undersampling of the high-pressure/low-energy region of the adsorption is reported and discussed for the EM algorithm, as well as the effect of signal-to-noise ratio on the degree of heterogeneity that may be estimated experimentally.