In liquid chromatography (LC), adsorption heterogeneity arises from the distribution of adsorption sites on stationary phases with varying interaction energies, affecting retention and separation performance. This heterogeneity can cause peak tailing, reduced resolution, and unpredictable retention times in analytical chromatography, as well as broad, asymmetric elution profiles in preparative systems. Adsorption heterogeneity depends on the combined effects of the stationary phase, the mobile phase composition, the analyte properties, and the chromatographic conditions. Traditional adsorption isotherms often fail to fully describe these complex interactions because they assume uniform adsorption energies. The Adsorption Energy Distribution (AED) framework offers a powerful alternative by modelling adsorption as a sum of independent homogeneous sites, each with a specific energy, offering a realistic representation of heterogeneous adsorption. This review introduces the theoretical foundations of AED, including its mathematical formulation and computational approaches, and discusses its application in interpreting retention mechanisms in LC. AED analysis is illustrated through its use in both chiral and achiral separations, as well as its ability to explain peak tailing and surface heterogeneity. Practical considerations, such as the range of concentration data in the adsorption isotherm, the selection of a suitable kernel function, and the number of iterations and grid points in AED analysis, are discussed. Special emphasis is given on how to visualize and interpret the AED. This review aims to provide chromatographers with a comprehensive understanding of AED, emphasizing its practical value in characterizing the chromatographic system and elucidating retention mechanisms in liquid chromatography.

This work presents global random walk approximations of solutions to one-dimensional Stefan-type moving-boundary problems. We are particularly interested in the case when the moving boundary is driven by an explicit representation of its speed. This situation is usually referred to in the literature as moving-boundary problem with kinetic condition. As a direct application, we propose a numerical scheme to forecast the penetration of small diffusants into a rubber-based material. To check the quality of our results, we compare the numerical results obtained by global random walks either using the analytical solution to selected benchmark cases or relying on finite element approximations with a priori known convergence rates. It turns out that the global random walk concept can be used to produce good quality approximations of the weak solutions to the target class of problems. 

This work introduces the Adsorption Energy Distribution (AED) calculation using competitive adsorption isotherm data, enabling investigation of the simultaneous AED of two components for the first time. The AED provides crucial insights by visualizing competitive adsorption processes, offering an alternative adsorption isotherm model without prior assuming adsorption heterogeneity, and assisting in model selection for more accurate retention mechanistic modeling. The competitive AED enhances our understanding of multicomponent interactions on stationary phases, which is crucial for understanding how analytes compete on the stationary phase surface and for selecting adsorption models for numerical optimization of preparative chromatography. Here, the two-component AED was tested on both synthetic and experimental data, and a very successful outcome was achieved.

In recent decades, there has been rapid development in digital technologies for automated assessment. Through enhanced possibilities in terms of algorithms, grading codes, adaptivity, and feedback, they are suitable for formative assessment. There is a need to develop computer-aided assessment (CAA) tasks that target higher-order mathematical skills to ensure a balanced assessment approach beyond basic procedural skills. To address this issue, research suggests the approach of asking students to generate examples. This study focuses on an example-generation task on polynomial function understanding, proposed to 205 first-year engineering students in Sweden and 111 first-year biotechnology students in Italy. Students were encouraged to collaborate in small groups, but individual elements within the tasks required each group member to provide individual answers. Students’ responses kept in the CAA system were qualitatively analyzed to understand the effectiveness of the task in extending the students’ example space in diverse educational contexts. The findings indicate a difference in students’ example spaces when performing the task between the two educational contexts. The results suggest key strengths and possible improvements to the task design. 

We present a fully discrete scheme for the numerical approximation of a moving-boundary problem describing diffusants penetration into rubber. Our scheme utilizes the Galerkin finite element method for the space discretization combined with the backward Euler method for the time discretization. Besides dealing with the existence and uniqueness of solution to the fully discrete problem, we assume sufficient regularity for the solution to the target moving boundary problem and derive a a priori error estimates for the mass concentration of the diffusants, and respectively, for the position of the moving boundary. Our numerical results illustrate the obtained theoretical order of convergence in physical parameter regimes.

For certain materials science scenarios arising in rubber technology, one-dimensional moving boundary problems with kinetic boundary conditions are capable of unveiling the large-time behavior of the diffusants penetration front, giving a direct estimate on the service life of the material. Driven by our interest in estimating how a finite number of diffusant molecules penetrate through a dense rubber, we propose a random walk algorithm to approximate numerically both the concentration profile and the location of the sharp penetration front. The proposed scheme decouples the target evolution system in two steps: (i) the ordinary differential equation corresponding to the evaluation of the speed of the moving boundary is solved via an explicit Euler method, and (ii) the associated diffusion problem is solved by a random walk method. To verify the correctness of our random walk algorithm we compare the resulting approximations to computational results based on a suitable finite element approach with a controlled convergence rate. Our numerical results recover well penetration depth measurements of a controlled experiment designed specifically for this setting.

This paper reports on a pilot study with the focus on (re)design of a digitized task environment utilizing two types of technology – a dynamic mathematics software and a computer-aided assessment system. The data consist of responses from 256 first year engineering students, taking their first Calculus course, on two different types of task. The results are discussed in relation to (re)design of tasks as well as possible feedback design options to enable a formative assessment approach.

This paper reports on the planning of a design-based research (DBR) study, where the main aim is to develop principles in designing technology-enhanced learning environment utilizing a combination of a dynamic mathematics software (DMS) and a computer-aided assessment (CAA) system. The focus is on the design of tasks and automated feedback of high quality so as to enhance first year engineering students’ engagement in and conceptual understanding of mathematical contents. The paper outlines the rationale for the project and highlights theoretical aspects that will be considered in the study. Moreover, some findings from a pilot study that will inform the first cycle of the DBR study are presented.

We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer.