## 4 Fundamentals of TF-SPME  
SPME is one of the most attractive extraction techniques commonly used today due to its high throughput, low-cost and solvent-less extraction. In recent years, TF-SPME has been developed to better meet the demands of onsite and ultra-trace level analysis, as its high surface area-to-volume ratio enables it to achieve enhanced sensitivity and greater extraction efficiency [12]. In common with other extraction technologies, sampling using TF-SPME devices is characterized by partition equilibria between the free form of the analyte in a sample and the extraction phase constituting the extraction device. Consequently, the properties of the extraction phase play a significant role in the efficiency of the extraction process. The total amount of extracted analyte in direct immersion SPME, where the analyte equilibrates between only two phases, is described by Equation (1) [12]:  
\[n^{eq}=\frac{K_{\mathrm{es}}V_{e}V_{s}}{K_{\mathrm{es}}V_{e}+V_{s}}C_{s}\, \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Application** & **TDU Split** & **Cryo-Trap Split** & **Rationale** \\ \hline Ultra-trace analysis & Splitless & Solvent-vent 1  & Maximum amount of analyte injected \\ High concentrations & Splitless & Solvent-vent & Sensitivity/less column load \\ Unknown analysis & Split & Split & Clean cryo-trap/column \\ Watery/fouled devices & Split & Solvent-vent 1  & High sensitivity/clean cryo-trap \\ \hline \hline \end{tabular}  
* Solvent-venting ends at the same time the cryo-trap begins to heat, allowing maximum desorption flow while still achieving a splitless injection.  
\end{table}
Table 1: Common parameters of two-stage thermal desorption and their applications.  
where \(n^{eq}\) is the total amount of extracted analyte at equilibrium, \(K_{es}\) is the distribution constant of the analyte between the matrix and extraction phase, \(V_{e}\) is the volume of extraction phase, \(V_{s}\) is the volume of the sample and \(C_{s}\) is the initial concentration of analyte in the sample. As can be seen in Equation (1), the greater the volume of the extraction phase (\(V_{e}\)), the larger amount of \(n^{eq}\) is extracted.  
In the following equation describing the equilibrium time for the analyte between the two phases (Equation (2)) [12],  
\[t_{e}=t_{95\%}=\frac{3\delta(b-a)}{D_{s}}, \tag{2}\]  
\(t_{e}\) is the required time for the analyte to reach equilibrium with the extraction phase, \(t_{95\%}\) is the time needed to extract 95% of the equilibrium amount of an analyte on the device, \(\delta\) is the thickness of the boundary layer, \((b-a)\) represents the thickness of the extraction phase and \(D_{s}\) is the diffusion constant of the analytes into the sample matrix. While an increase in \(V_{e}\) allows higher extraction efficiency and thus greater sensitivity according to Equation (1), Equation (2) demonstrates that the corresponding increase in coating thickness results in a longer equilibrium time. Hence, it is important to optimize these parameters to ensure the most efficient and practical mode of extraction [41].  
Considering Equation (3) [12],  
\[\frac{dn}{dt}=\Big{(}\frac{D_{s}A}{\delta_{s}}\Big{)}C_{s}, \tag{3}\]  
A thin film geometry can also enhance the sampling rate due to its high surface area-to-volume ratio, reducing the time it takes to reach equilibrium and enhancing the capacity of the extraction device. In this equation, n is the amount analyte extracted over the extraction time t, A is the area of the extraction phase, \(D_{s}\) is the diffusion constant of the analyte into the sample matrix and \(\delta_{s}\) is the thickness of the boundary layer [42]. In other words, employing TF-SPME devices allows rapid sampling with high extraction capacity, suitable for ultra-trace level analysis.