Mathematical modelling of chemical detection in porous substrates using sampling coatings

• A modelling framework is developed for the Sampling Coating process. • An asymptotic early-time reduction is shown to be valid, and solved analytically. • The sampling efficiency is quantified, along with its time dependence. • Dependence on substrate and chemical properties are explored numerically. • Propose a method to quantify contamination from measurements in the field. Sampling coatings are a developing technology for detection and quantification of contaminants in porous materials (PM). The sampling coating is a substance of high contaminant affinity that is applied onto the PM and left to absorb contaminant. Upon removal, it is analysed to determine the amount of contaminant absorbed. Whilst sampling coatings have been shown to recover more contaminant from PM than traditional methods, the role of transport kinetics has not been recognized, hindering the quantitative interpretation of the measurements. We present a mathematical modeling framework for sampling coatings, incorporating coupled vaporisation and transport within the PM and absorption into the coating. We show that, on practical time scales (hours to days), only a fraction of the contaminant present in the PM enters the sampling coating and chemical equilibrium is not reached. The sampling efficiency, which we define to be the fraction of the contaminant in the coating, thus depends on the sampling time. We find an analytical solution of the model, valid for feasible sampling times in the physical parameter regime, along with an explicit expression for the sampling efficiency as a function of time. Dependence on the physical system parameters is investigated numerically: the height, diffusivity, and structure of the PM, and the type of chemical contaminant, are identified as key factors affecting sampling efficiency. We further propose a method to quantify contamination in the field using sampling-coating measurements. Our model and results enable practitioners to determine important properties when designing coatings and will inform future experimental work.