## 4 Results and Discussion  
### Optimization of AuNP Size and Deposition Time to Yield Maximum EC Signal  
The ideal size for AuNPs was chosen based on two primary factors: the signal-to-mass ratio and the density of antibody conjugation. As the mean diameter of the particles decreases, the ratio of surface-exposed gold atoms to the given mass of gold increases. Theoretically, a higher proportion of surface-exposed gold for a specific mass should facilitate greater electron transfer, leading to an amplified normalized signal. As shown in Figure 2, 10 nm AuNPs produced the maximum normalized signal for all the dilutions under consideration. The 20 nm AuNPs produced the second-highest normalized signal for the three lower dilutions under consideration (D1, D2, D3). This result is expected given the relative increase in surface area to mass as the diameter of the particle decreases. However, the smallest particle diameter, 5 nm AuNPs, with the highest surface-to-volume ratio, did not yield the highest normalized signal as expected. This was previously observed by [PERSON] et al. and attributed to the Brownian effects governing smaller particles [32]. The rationale for selecting the 20 nm AuNPs and not the particles with the largest EC signal, the 10 nm AuNPs, was due to the antibody conjugation density and ease of conjugation. Our detection assay's effectiveness depends on the number of IgG antibodies that can conjugate to AuNPs. Previous experimental studies have shown that \(\sim\)3 antibodies conjugate to a single 20 nm diameter AuNP using our conjugation process [33]. We argue that the number of attached antibodies would be even lower for 10 nm AuNPs (given the jamming limit) [33]. This work also predicts that the conjugation of antibodies to 10 nm and 5 nm AuNPs will be 0.88 and 0.22, respectively (see Supporting Information), and insufficient for reliable labeling [33]. For these reasons, we selected the 20 AuNP as the most ideal for electrochemical labeling in our system. It was important to determine the optimal deposition time for the 20 nm AuNPs. Numerous factors can influence the optimal deposition time for an EC system. Through empirical testing, we observed that increased deposition times resulted in increased noise. As shown in Figure S2, the highest peak current was observed at 180 s. However, the blank signal also increased when the deposition time increased (observed to be highest at 300 s). This was possibly due to the supporting electrolyte (0.1 M HCl) affecting the carbon or reference electrodes over time [34]. The \(\Delta\)Signal illustrates the combined influence of the increased deposition time with the increase in noise. The \(\Delta\)Signal for the 120 and 180 s deposition times had similar values (0.60 and 0.62 uA, respectively). Hence, we chose the lower duration, viz--120 s, as the optimum deposition time. Figure 3 shows the absolute EC signal (DPV) from various dilutions (D0 to D3) of 20 nm AuNPs. The lowest signal that could be detected was at D2, which was 30% higher than the blank signal. However, ANOVA shows that the signal at D2 was not statistically significant vs. the blank (\(p\)-value = 0.558). The D1 signal was statistically significant compared with the blank (\(p\)-value = 0.0027). We also generated a calibration curve (Inset Figure 3). The curve was observed to be linear in the range of D3 = 1.4 \(\times\) 10\({}^{5}\) to D1 = 1.4 \(\times\) 10\({}^{7}\) (AuNPs in 2 uL of DI water) with the regression equation \(y~{}=~{}1.069\times 10^{-8}~{}x+0.3225\) (R\({}^{2}~{}=~{}0.99\), where \(y\) is the absolute DPV peak current in uA, and \(x\) is the AuNP dilution). The limit of detection (LOD) was equal to 9 \(\times\) 10\({}^{6}\) AuNPs (in a 2 uL volume). The LOD was calculated as 3.3*(\(\sigma\)/S), where \(\sigma\) is the standard deviation of the regression line, and S is the slope of the calibration curve. The LOD represents the lowest absolute number of 20 nm AuNPs we can detect (on a 3 mm dia. SPCE under given DPV parameters).