# New Generation of Electrochemical Sensors Based on Multi-Walled Carbon Nanotubes
## 3 MWCNT Electrochemical Properties  
MWCNT-based electrochemical sensors have remarkable advantages for analytical applications, such as enhanced detection sensitivity, low charge-transfer resistance (\(R_{tc}\)), electrocatalytic effect, and reduced fouling. It is also speculated that traces of metallic catalysts can add remarkable electroanalytical characteristics (positive synergistic effect) to the nanotubes, playing a supporting role for the development of the devices [6; 7; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. There are still controversies about the electron transfer mechanism along the structural network: Some researchers believe that the sidewalls of the CNT are inert (i.e., the edge-plane-graphite-like open ends and defect sites control the phenomenon) [27; 26; 28], but some studies conducted with two-dimensional SWCNT have shown evidence of charge-transfer on the basal-plane [7; 23; 25]. Although the uncertainties about the fundamental contributions of MWCNT in electrochemical sensors, the number of MWCNT successful applications in highly complex systems does not stop growing [2; 9; 17; 26]. Understanding the size-dependent features is an indispensable requirement to explore the fascinating properties of nanotubes in analytical tools and to move the electroanalytical researches forward. In this sense, it must be assumed that the electronic band structure of CNT as a hexagonal lattice, where each carbon atom is covalently bonded to three others via sp\({}^{2}\) molecular orbitals (the fourth valence electron in a \(p_{2}\)-orbital hybridizes with all others to form a delocalized \(\pi\)-band) [27]. Like this, an even number of electrons are contained in the basic nanotube structure, whose conduction character will depend on its behavior as metal or semiconductor. This property can be assessed through the tight-binding electronic structure calculations, considering the translational vector (\(C_{h}\)) that connects two equivalent crystallographic sites (unitary vectors \(a_{1}\) and \(a_{2}\)) and its coefficients (\(n_{1}\) and \(n_{2}\)), according to the following Equation (1):  
\[C_{h}=n_{1}a_{1}+n_{2}a_{2} \tag{1}\]  
so that a metallic behavior is observed when the result is an integer multiple of three, while a non-metallic/semiconductor profile is observed for other cases; a direct consequence of the different conduction band gaps [6]. The electron-transfer along the MWCNT are also influenced by the tube diameter (\(d_{t}\)), chiral angle (\(\theta\)), and basic translation vector (\(T\)), as shown in Figure 3. Thus, the manner in which a graphene sheet is rolled determines the main electronic properties of the resulting nanotube [6; 15; 23; 27]. Observing the referred parameters on the tubulene lattice, \(d_{t}\) values can be determined by Equations (2) and (3), which relate this variable with the circumferential length of the nanotube (\(L\)) and a hexagonal network constant (\(a\)):  
\[d_{t}=\frac{L}{\pi}=\frac{C_{h}}{\pi}=\frac{a\left(n_{1}^{2}+n_{1}n_{2}+n_{2}^{ 2}\right)^{\frac{1}{2}}}{\pi} \tag{2}\]  
considering \(a_{\circ}=1.42\) A for carbon nanotubes [6]:  
\[a=\sqrt{3}a_{\circ}=2.49\text{\AA} \tag{3}\]  
Regarding the \(\theta\), the expression that describes its magnitude can be written as follows:  
\[\cos\theta=\frac{2n_{1}+n_{2}}{2\sqrt{n_{1}^{2}+n_{1}n_{2}+n_{2}^{2}}} \tag{4}\]  
The vector \(T\) is used to specify the orthogonal orientation to the \(C_{h}\) axis of the nanotube, and its value can be calculated by:  
Figure 3: Honeycomb lattice structure of graphene and the main parameters that control the chirality and electronic properties of the resultant nanotube (adapted by permission from References [8; 23]).