Diffusive model to assess the release of chemicals from a material under intermittent release conditions

www.nature.com/scientificreports OPEN Diffusive model to assess the release of chemicals from a material under intermittent release conditions Diego Frezzato1*, Gianluca Stocco2, Enrico Boscaro2, Marco Ferraro3 & Andrea Tapparo1* We consider the archetype situation of a chemical species that diffuses in a material and irreversibly escapes through the interface. In our setup, the interface switches between two states corresponding to ‘release phase’ (absorbing boundary) during which the species is released to the exterior, and ‘pause phase’ (reflecting boundary) during which the species is not released and its concentration profile inside the material partially relaxes back to uniformity. By combining numerical solution of the diffusion equation and statistical analysis of the outcomes, we derive upper and lower bounds and an empirical approximation for the amount of species released up to a certain time, in which the only information about the release-pause alternation schedule is the number of release phases and the average duration of a release phase. The methodology is developed thinking especially to dermal exposure assessment in the case of a slab-like homogeneous material irreversibly releasing chemicals during a number of contacts. However, upon proper extensions, this approach might be useful for inspecting other situations that are encountered, for instance, when dealing with leakage of chemicals in environmental contexts and regulatory toxicology. Characterizing the release of chemicals from a material through the interface is a crucial problem in disparate situations. Just to mention a few, think to environmental ambits in which the species is released from the material into a different phase (e.g., volatile species released in the air from floors1 or other building materials2,3, or released from buildings into receiving water under wet weather c onditions4), in the risk assessment of near-field exposure to consumer p roducts5 (e.g., release of chemicals from consumer a rticles6 and cotton w ipes7 into derm, from plastics into drinking w ater8, from packaging films into f ood9), and in the context of controlled drug release from pharmaceutical d evices10,11,12. Although the leakage is typically continuous, in some cases it might be intermittent with alternation of release-pause phases. For instance, chemicals in consumer objects are intermittently released to the derm through repeated contacts of limited duration; just think to rings or earrings releasing Nickel while they are worn, to a baby that puts a toy in the mouth from time to time, to phthalates released from plastic handles when the tools are used, etc. Yet, containers that release chemicals only when they are filled with liquids, or species that might pass into the environment from building materials or abandoned waste only under wet weather conditions. In all these examples, the release is intermittent and the (partial) redistribution of the species inside the material between two consecutive phases might affect, to some extent, the subsequent release rate. An accurate assessment of the amount of the released chemicals should make use of appropriate models, and the outcomes could be relevant for the risk assessment in occupational medicine, in REACH Regulation, or in studies of exposure to environmental pollutants. Despite the potential relevance of the issue, to our knowledge a systematic quantitative analysis is still lacking. Such analysis would be clearly case-dependent, hence to start facing this topic we must select a specific relevant situation. Among the variety of contexts in which the intermittent release from a material into a hosting phase is important, here we focus on simple but relevant situations like that of a consumer material, or a tool, which comes into contact with the skin. We shall move the first steps in the release assessment by treating the simplest but realistic scenario that allows us to make basic modeling and numerical explorations. Specifically, we will assume that the release is irreversible and controlled by the diffusion in the material. A crucial target is to evaluate the total 1 Department of Chemical Sciences, University of Padua, Via Marzolo 1, 35131 Padua, Italy. 2Normachem S.r.l., Via Roma 14, 35014 Fontaniva, Padua, Italy. 3Department of Pharmacy, University of Naples Federico II, Via D. Montesano 49, 80131 Naples, Italy. *email: ; Scientific Reports | (2022) 12:3466 | https://doi.org/10.1038/s41598-022-07144-0 1 Vol.:(0123456789) www.nature.com/scientificreports/ Figure 1.  Scheme of the release-pause phases. quantity mext of species released through the interface up a certain time ttot , given that the system is initially at equilibrium. How is mext related with the diffusivity of the species in the bulk material, with the geometrical features of material and interface, and especially with the history of the contacts between material and exterior? To tackle the problem on theoretical grounds, it proves convenient to consider the simplest archetype setup of a slab-like homogeneous material with given diffusion coefficient of the chemical species in it, and with uniform volumetric concentration at the initial equilibrium. The slab geometry is the simplest and natural setup if we refer to macroscopic objects whose exposed surface, from the viewpoint of the diffusion processes, looks locally planar. Assuming a constant diffusion coefficient is the most natural choice if the material is meant to remain uncontaminated by the external medium during the contact phases and if its homogeneity is preserved (a typical example is that of leakage of additives from plastics in the rubbery state, to which these assumptions are normally applied). The homogeneous initial load is also the most common situation thinking to such a kind of material after a preceding long resting phase. In addition, it is assumed that the chemical species passes irreversibly through the interface treated as a perfect absorbing boundary. This means to refer to the situation in which the external medium is quickly renewed at the interface so that the external concentration of the species can be assumed to be vanishingly small. Based on these assumptions, as stated above our purpose is to work out bounds on mext and even a likely approximation getting rid of the details of the release-pause schedule. Facing this problem goes much beyond the mere solution of the diffusive model at given schedule. The critical (and novel) aspect, in fact, consists in devising a global statistical synthesis of the schedule-dependent outcomes. This is precisely the target of the present theoretical-methodological work. We proceed as follows. In “Theoretical model and approach” Section we present the theoretical model and our approach. “Results” Section presents the results; the methodological details and the analysis that support the findings are provided in “Methodological aspects and analysis” Sectio (...truncated)