## 2 Theoretical Background  
In its classical operational mode (direct immersion extraction), FPSE mimics the extraction principle of solid phase microextraction (SPME) and similar microextraction techniques, including stir bar sorptive extraction (SBSE) and thin film microextraction (TFME). Extraction of the analyte(s) on the FPSE membrane is primarily governed by the difference in partition coefficient of an analyte between the sample matrix and the FPSE membrane, and the mass transfer of the analyte(s) from the bulk of the sample matrix towards the FPSE membrane continues until an equilibrium is established between the two phases. The mass of the analyte(s) extracted by the FPSE membrane (n), under equilibrium extraction conditions, is proportional to the partition coefficient between the FPSE membrane (which varies with different fabric substrates and the sorbent coatings on the substrate surface) and the sample matrix (K\({}_{\text{es}}\)), volume of the extracting phase (V\({}_{\text{e}}\)), volume of the sample (V\({}_{\text{s}}\)), and the initial concentration of the analyte (C\({}_{\text{o}}\)). The mass of the analyte extracted by the FPSE membrane at equilibrium (n) can be expressed as:  
\[\text{n}=\frac{\text{KesVeVsCo}}{\text{KesVe}+\text{Vs}} \tag{1}\]  
When the sample volume is too large compared to the volume of the extracting sorbent (V\({}_{\text{e}}<<\text{V}_{\text{s}}\)),  
Equation (1) can be simplified as:  
\[\text{n}=\text{K}_{\text{es}}\text{V}_{\text{e}}\text{C}_{\text{o}} \tag{2}\]  
As can be inferred from Equation (2), the mass of the analyte extracted by the FPSE membrane (n) is directly proportional to the volume of the extracting sorbent and is independent of the sample volume. As such, the value of n can be increased by increasing the volume of the extracting sorbent if the initial concentration of the analyte(s) (C\({}_{\text{o}}\)) is kept constant [7]. The extraction efficiency of an FPSE membrane depends on: (a) thermodynamic factors, and (b) kinetic factors [8]. The partition coefficient for an analyte between the FPSE membrane and the sample matrix is a thermodynamic criteria that depends on the material properties of FPSE substrate, coated sorbent, mass of the extracting sorbent, temperature at which the extraction is carried out, and chemical state of analyte in the sample matrix, as well as other factors. However, the most simplistic way to increase the mass of the extracted analyte by the FPSE membrane is to increase the volume of the extracting sorbent. The kinetic criteria determine the rate at which the equilibrium is reached and can be dramatically improved by maximizing the contact surface between the sample matrix and the extracting phase as well as by applying external energetics (e.g., stirring, sonication, orbital shaking) to diffuse the analyte(s) through the boundary layer between the bulk solution and the extracting phase. According to the kinetic theory of extraction, increasing the volume of the extracting phase by increasing the thickness of the coated sorbent may lead to unsustainably long extraction equilibrium time as demonstrated in Equation (3). As the extracting phase in a microextraction technique is generally immobilized on the substrate surface in the form of a thin film, the diffusion of the analyte(s) through the boundary layer regulates the rate of extraction (extraction kinetics). The time required to extract 95% of the equilibrium extraction amount of the analyte \(\mathfrak{t}_{\text{e},95\%}\) can be calculated as:  
\[\text{te},95\%=\frac{\text{B}\text{8b}\text{Kes}}{\text{Ds}} \tag{3}\]  
where, b = the thickness of the extracting sorbent; \(\delta\) = thickness of the boundary layer; \(\text{K}_{\text{es}}\) = distribution constant for the analyte between the extracting sorbent and the sample matrix; \(\text{Ds}\) = diffusion coefficient of the analyte in the sample matrix; \(\text{B}\) = geometric factors related to the geometry on which the extracting sorbent is immobilized.