Limits of the numerical estimation of the adsorption energy distribution from adsorption isotherm data using the expectation-maximization method

The limits of the use of the expectation-maximization (EM) method for the study of the heterogeneity of adsorbent surfaces were tested by calculating the adsorption energy distribution of systems having known degrees of heterogeneity. Connecting on-line two different columns allows the simulation of a heterogeneous system. The two columns used were endcapped, C(18)-bonded silica used as stationary phases and having different degrees of C(18) chain coverages (0.42 and 2.03 micromol/m(2)). The adsorption constants of phenol measured by frontal analysis (FA) are significantly different on these two columns. On each column, the adsorption behavior was best accounted for by a bi-Langmuir isotherm model, corresponding to a heterogeneous surface with a bimodal energy distribution. The difference between the adsorption energies on the weak adsorption sites of the two columns is 1.5 kJ/mol. The energy difference of their high energy sites is 2.2 kJ/mol. The EM method can readily distinguish between adsorption sites having energies that differ by more than 5 kJ/mol after more than 10 million iterations, but it cannot distinguish between adsorption sites for which this energy difference is less than 2 kJ/mol, even after 100 million iterations. For highly heterogeneous systems, (e.g., those with more than three different types of adsorption sites), the EM program does not converge necessarily towards the actual energy distribution function but toward a simpler one, having fewer adsorption sites that are almost equally spaced in the energy space. This failure of the EM program is related to the fact that, despite the excellent precision of the FA measurements (<1%), any series of adsorption data can be represented by several distinct AEDs. Thus, the degree of heterogeneity of RPLC adsorbents determined with the EM method might often be minimized, resulting in erroneous values of the isotherm parameters.