## 3 Results and Discussion
### Particle \(\zeta\)-Potential  
According to the results of the previous section, the NPs made of silver, gold, and platinum have essentially the same size and size distribution but differ considerably in the fraction of oxidized metal present on the particle's surface. It becomes, therefore, decisive to determine their electrophoretic mobility because, all else being equal, it depends only on the charges on the particles. The electrophoretic mobility was determined by means of LDE in the PALS mode. The average electrophoretic mobilities were: \(-2.33\pm 0.05\) um-cm/V-s for AgNP; \(-3.37\pm 0.02\) um-cm/V-s for AuNP and \(-3.02\pm 0.02\) um-cm/V-s for PtNP. Since the TEM has demonstrated the nanoparticles are spherical and very small (with respect to the Debye's length of 30 nm at \(10^{-4}\) M KCl) it is possible to evaluate the average \(\zeta\)-potential according to [PERSON]'s equation [27] and obtain the data listed in Table 4. However, an LDE-PALS experiment performed at low ionic strength also allows the evaluation not only of the average \(\zeta\)-potential, but also the \(\zeta\)-potential distribution values among different particles. Such distributions are shown in Figure 4 for all three systems. The electrokinetic behavior of all types of NPs is mutually close. In particular, the AuNP and the PtNP are essentially the same although they have very different percentages of oxidized metals on the particle's surface (less than 3% for AuNP and \(\sim\)22% for PtNP). The knowledge of the \(\zeta\)-potential value allows the estimation of the electrokinetic charge of the particle (i.e., the excess charge that is moving with the particle). The ionic atmosphere around the charged spherical particles forms the Gouy-Chapman diffuse double layer that extends over distances of the order of the so-called Debye's length, \(\lambda_{\mathrm{D}}\), defined as:  
\[\lambda_{\mathrm{D}}=\sqrt{\frac{\varepsilon\varepsilon_{0}k_{\mathrm{b}}T}{e^ {2}\sum_{i}\overline{z}^{2}c_{i}}} \tag{1}\]  
where \(k_{\mathrm{b}}\) is the Boltzmann's constant, \(e\) the elementary charge, \(\varepsilon\) is the relative dielectric constant of the medium, \(T\), the temperature, \(\varepsilon_{0}\), the vacuum permittivity, \(z_{i}\), and \(c_{i}\) are the valence and the concentration of the \(j\)-th ion, respectively. The sum in Equation (1) is twice the solution ionic strength. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Ion** & **E\({}^{*}\)/V** & **cc/uM** & **ccc/uM from Equation (3)** \\ \hline Ag\({}^{+}\) & 0.8 & \(3\pm 2\) & \(50\times 10^{3}\) \\ Cu\({}^{++}\) & 0.34 & \(6\pm 2\) & 800 \\ Pb\({}^{++}\) & \(-0.13\) & \(13\pm 5\) & 800 \\ Mg\({}^{+}\) & \(-2.36\) & \(150\pm 50\) & 800 \\ Na\({}^{+}\) & \(-2.71\) & \((35\pm 10)\times 10^{3}\) & \(50\times 10^{3}\) \\ Ca\({}^{++}\) & \(-2.87\) & \(150\pm 50\) & 800 \\ K\({}^{+}\) & \(-2.93\) & \((8\pm 2)\times 10^{3}\) & \(50\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Critical coagulation concentration (ccc) for copper, lead, magnesium, calcium, and sodium chlorides and for silver and potassium nitrate. The values are ordered according to the standard reduction potential (E\({}^{\circ}\)) of the cation. In the last column are listed the predictions of the DLVO theory (Equation (3)) for mono and divalent ions. Figure 3: Gold **(A)**, silver **(B)**, and platinum **(C)** nanoparticles prepared by laser ablation in liquid in presence of 100 \(\upmu\)M KCl. Au4f, Ag3d, and Pt4 fXP XP spectra (black) and the best fit (red). Also shown (in different colors) the different contributions to the fit; XP intensity is in arbitrary units. Note that the K2s signal in B falls in the same binding energy (BE) range. Imposing that the charge on the particle (\(Z_{\rm c}\)) balances the one on the solution, one obtains [27]:  
\[\zeta\ =z_{\rm c}\frac{I_{\rm B}}{R}\frac{k_{\rm b}T}{e}\frac{1}{1+\frac{R}{ \lambda_{\rm D}}} \tag{2}\]  
where \(I_{\rm B}\) is the Bjerrum length that in water is 0.7 nm and \(R\) denotes the particle radius.