## 3 Materials and Methods
### Sampling Using Absorptive Coatings  
Inward (and outward) fluxes from (or backward to) air into an absorptive coating (\(Flux_{1}\) and \(Flux_{2}\), respectively) at the boundaries (marked by red lines in Figure 11) were simulated using the equation, previously proposed by [PERSON] and [PERSON] [32] for the water\(-\)air interface:  
\[Flux_{1}\ =\ k\times\left(C_{a}-\frac{C_{f}}{K_{d}}\right);\quad Flux_{2}\ =\ k\times\left(\frac{C_{f}}{K_{d}}-C_{a}\right) \tag{3}\]  
where: \(k\)--flux coefficient, m\(\cdot\)s\({}^{-1}\); \(C_{a}\) and \(C_{f}\)--concentrations of an analyte in air and coating at the boundary layer, respectively, mol\(\cdot\)m\({}^{-3}\); \(K_{d}\)--distribution constant for a VOC between SPME coating and air.  
The true value of the flux coefficient was unknown, but in this research, it was assumed to be sufficiently high for not affecting the flux, as was recently proposed by [PERSON] et al. [23]. Thus, the flux coefficient was set to 1000 m\(\cdot\)s\({}^{-1}\). A further increase of the flux coefficient did not affect the results of the modeling.  
### Sampling Using Absorptive Coatings  
For adsorptive coatings, the \"Adsorption\" mechanism was activated in the model. The isotropic diffusion coefficient (in the air inside pores) was the same as for air (set to 8.8, 8.7, 12.4 and 10.1 mm\({}^{2}\cdot\)s\({}^{-1}\) for benzene, toluene, acetone, and dichloromethane, respectively). The approach proposed by [PERSON] and [PERSON] [33] involving Knudsen diffusion in micro-pores was also tested. However, it was later rejected for model simplification because the diffusion of analytes inside coating is mainly driven  
Figure 11: The geometry of SPME device (retracted inside a protective needle for TWA sampling) used for modeling. Note: red lines indicate the boundaries between air and coating.  
by molecular diffusion inside macro-pores. The presence of PDMS binder was not considered in the model because: (1) it has much weaker affinity to analytes than Carboxen; and (2) the layer of PDMS in the coating is very thin and should not affect the diffusion of analytes [5]; (3) there is not enough published information about the exact structure of the coating.  
Adsorption was set to \"User defined\" with a distribution constant (\(K_{p}\), m\({}^{3}\cdot\)kg\({}^{-1}\)) calculated as a dimensionless distribution constant divided by a coating density (\(K_{d}/\rho\)). Coating porosities (\(\varepsilon\) = 0.685 for Car/PDMS and 0.775 for PDMS/DVB) were calculated using intra-particle porosities (0.37 for Car, and 0.55 for DVB [34]) and inter-particle porosity. The exact value of the latter is proprietary and not available in the open literature. Taking into account, the spherical shape of particles and available scanning electron microscope (SEM) photos, the inter-particle porosity of both coatings was set to the maximum possible value (0.50). A particle porosity (\(\varepsilon\)) was calculated as the total volume of pores (0.78 mL for Car, and 1.54 mL for DVB) divided by the total volume of one gram of material (2.13 mL for Car, and 2.78 mL for DVB). Densities of the coatings were calculated using free fall densities of the particles (470 kg\(\cdot\)m\({}^{-3}\) for Car, and 360 kg\(\cdot\)m\({}^{-3}\) for DVB) [34] and inter-particle porosity. Effective diffusion coefficients were calculated during the calculations by the COMSOL software using the Tortuosity model [33]:  
\[D_{e}\ =\ \frac{\varepsilon\times D_{p}}{\tau} \tag{4}\]  
where: \(\varepsilon\)--porosity; \(\tau\)--fortuosity calculated from the porosity [33]:  
\[\tau\ =\ \varepsilon+1.5\times(1-\varepsilon) \tag{5}\]  
For Car/PDMS and PDMS/DVB coatings, tortuosity was set to 1.16 and 1.1125, respectively.