## 3 Results and Discussion
### Application as Anode Materials for Li, Na, K, Mg and Ca-Ions Batteries
The average voltage (\(\overline{V}\)) can be estimated in a coverage domain of \(x_{1}\leq x\leq x_{2}\) as follows [65]:  
\[\overline{V}\cong\frac{\left(E_{SM_{x_{1}}}-E_{SM_{x_{2}}}+(x_{2}-x_{1})E_{At} \right)}{(x_{2}-x_{1})|e|} \tag{5}\]  
where \(x_{1,2}\), \(E_{SM_{x_{1,2}}}\), \(E_{At}\), and \(|e|\) indicate the number of adsorbed metal ions, total energies of the systems, single adatom energy, and the number of valence electrons, respectively. Equation (5) calculates the theoretical average voltage of a system. The maximum change of Gibbs free energy by an electrochemical cell is equal to the cell potential multiplied by the total charge that can be transferred during the charging/discharging. The energy obtained by discharging between anode with x\({}_{1}\) and x\({}_{2}\) number of adsorbed ions is equal to the integral of the product of the cell potential(voltage) and the displaced charge. Therefore, the theoretical average voltage could be calculated by dividing the maximum amount of work by the maximum charge (q\({}_{\text{max}}=(x_{2}-x_{1})|e|\)) that can be transferred theoretically. Here \(|e|\) is the charge of an adatom and is equal to one for Li, Na, K ions, and for Ca is equal to two [34; 66; 67]. Figure 6 presents the open-circuit voltage profiles versus storage capacities. The voltage values drop immediately in the first coverage step (\(7\) ions adsorption) and remain under the value of \(-0.3\). It is a desirable value for increasing the output voltage of a cell. However, there is no experimental support for P-TpG to compare open-circuit voltage results to our best knowledge, graphdiyne exhibited theoretical specific voltage value between 0.12-0.25 V [66] after the first coverage step, and the fabricated device with graphdiyne as the host active material for perovskite solar cell applications exhibits an open-circuit voltage of 1.073 V [67]. Figure 6: Open-circuit voltage profiles versus storage capacity of Li, Na, K, and Ca ions over P-TpG. The P-TpG nanosheet exhibits ultra-high capacities of \(\sim\)2148, \(\sim\)1696, \(\sim\)1017, and \(\sim\)2035 mA\(\cdot\)h\(\cdot\)g\({}^{-1}\) for Li-, Na-, K-, and Ca- ions, respectively, while the same ion coverage for the N-TpG substrate revealed values of \(\sim\)2160, \(\sim\)1296, and \(\sim\)2592 mA\(\cdot\)h\(\cdot\)g\({}^{-1}\) for Na-, K-, and Ca-ions storage, respectively [53]. Besides, the P-TpG and N-TpG [53] nanosheets show a considerably higher storage capacity compared to the charge capacities of commercial graphite [68], and theoretical VS\({}_{2}\)/VSe\({}_{2}\)[21] 2D structures for the Li-ion storage reported as \(\sim\)372 mA\(\cdot\)h\(\cdot\)g\({}^{-1}\), and \(\sim\)466 mA\(\cdot\)h\(\cdot\)g\({}^{-1}\), respectively. Rechargeable batteries are continuously charged and discharged. The repeated increase/decrease in ion coverage may result in some structural changes in anode materials, which subsequently may reduce the battery performance. The structural stability of substrate materials during the ion adsorption plays a critical role in the application as a battery anode. Figure 7 depicts the stable structures of P-TpG for different ion coverages. Note that according to the ab-initio molecular dynamics calculations [37], the P-TpG nanosheets can stay intact at high temperatures of 2000 K, which is undoubtedly higher than the batteries' working temperatures. Metallic electronic characteristics are highly desirable for electrode materials in batteries to reach internal electronic resistances. Metal atoms generally have low ionization energy and low electronegativity. Hence, they tend to lose electrons. Therefore, we conducted total electronic density of states (DOS) calculations for all the configurations with three coverage levels of 1, 18, and 38 atoms over the P-TpG for each Li, Na, K, and Ca adatom. Figure 8 confirms that at zero-state energy (Fermi level), the DOS is not zero, verifying the composite structures' metallic electronic behaviour.