#### Impact experiments

Based on the voltammetric stripping experiments, a potential of 0.3 V was chosen for the chronoamperometric impact experiments to assure oxidation of Ag NPs when contacting the working electrode (WE):9(1)

- (1)

Figure 2 b exemplarily shows one of the 115 obtained chronoamperograms. Ag NPs impacting the WE are oxidised, causing anodic spikes in the current. The charge (*Q*) assigned to each spike is the integral of the current (*I*) over time (*t*)(2)

- (2)

and the duration of a single spike is in the order of 1 ms to 20 ms.6, 7 This anodic charge is linked with the amount of oxidised silver by Faraday′s law:(3)

- (3)

where *n* is moles of silver, *z* is the number of exchanged electrons per oxidised silver atom (*z*=1; see [Eq. (1)]) and F is the Faraday constant (96 485 C mol^{−1}).

Because one electron is generated per oxidised silver atom, the charge (*Q*) assigned to each spike can directly be used to calculate the number of atoms (*N*) forming the impacting nanoparticle, taking the elementary charge *e*=1.602×10^{−19} C:(4)

- (4)

In the following, we assume that NP impact and APC experiments, like the majority of stochastic processes, obey the statistics of a normal distribution, that is, the detected charge, and hence the number of atoms, follows a Gaussian. The peak position is defined by the mean (expected value) for the number of atoms in a particle (*N*), and the full width at half maximum (FWHM) is determined by the standard deviation (*σ*).

To enable fitting of the experimental data, a histogram was drawn, containing the calculated number of atoms for all detected 1333 impacts (see Figure 3, ○). Finding a proper bin size is crucial for this. The smaller the bin size, the more precisely it describes the real distribution, but the corresponding decrease of counts per bin limits the stochastic description. The large number of analysed spikes allowed us to set the bin size to 0.05×10^{6} atoms, while still being able to fit the data appropriately. The bin centres and the accompanied number of impacts are displayed in Figure 3 (○). The fact that this distribution is not a Gaussian can be related to agglomeration of NP monomers, which yields a convolution of several individual Gaussians, as will be discussed in the following section.

Agglomeration and aggregation of particles describes the (reversible or permanent) sticking of at least two monomers. Consequently, the number of atoms of an agglomerate is an integer multiple of the number of atoms of a monomer (*N*_{M}). For example, the number of atoms of a dimer (*N*_{2M}) and a trimer (*N*_{3M}) is twice and three times that of a monomer, respectively.(5)

- (5)

with k denoting the number of monomers in a nanoparticle agglomerate.

For populations following a normal distribution, not only the mean values but also their standard deviations, that is, their FWHM, are directly linked. Consequently, the Gaussians for NP agglomerates (*N*_{kM}, *σ*_{kM}) can be derived from those of the NP monomer (mean=*N*_{M}, FWHM=standard deviation=*σ*_{M}):(6), (7)

- (6)

- (7)

Applying these correlations to the results of the impact experiments, we can deconvolute the data into the individual Gaussians. Thus, both information about NP monomers and also about the size and relative amount of agglomerates can be accessed. In Figure 3 the experimental data was fitted on this basis, showing the presence of NP monomers (*N*_{M}), dimers (*N*_{2M}), trimers (*N*_{3M}), tetramers (*N*_{4M}), pentamers (*N*_{5M}) and hexamers (*N*_{6M}). It should be noted that except for the intensity of each agglomerate, which describes its relative concentration (see below), no additional parameters were included in the fitting, since means and FWHM values for each agglomerate are defined by [Eq. (6) and (7)]. The good quality of the resulting fit indicates the validity of this new approach of APC analysis and demonstrates that more information about the nanoparticle agglomeration state in a specific solution can be extracted from electrochemical impact data than previously realised.9

The obtained mean for the number of atoms of a monomer (*N*_{M}) is 0.35⋅10^{6} atoms. Assuming spherically shaped NP monomers, this value is linked to the radius of a single nanoparticle (*r*_{M}), according to:(8)

- (8)

where *A*_{r} is the relative mass of silver atoms (*A*_{r}(Ag)=107.87 g mol^{−1}); N_{A} is the Avogadro constant (6.022×10^{23} mol^{−1}) and *ρ* is the density of silver (*ρ*(Ag)=10.49 g cm^{−3}).

Thus, the impact experiments yield a radius of *r*_{M}=11.3±0.6 nm for a single Ag NP, which is in excellent agreement with the results obtained by SEM and NTA in water/citrate (see Experimental Section).

#### NTA studies

To validate the APC results, NP agglomeration in the KCl/citrate electrolyte was also analysed by NTA using a NanoSight (LM 10, NanoSight Ltd.). This device tracks Brownian motion-driven particle movements in 2 D and assigns a diffusion coefficient (D) to each NP based on its individual displacement per time frame ().11(9)

- (9)

Afterwards, the Stokes–Einstein relation is employed to calculate a particle radius, taking the electrolyte viscosity (*η*=1.002×10^{−3} kg m^{−1} s^{−1} at 293 K12) and temperature (*T*=293 K) into account and assuming a spherical particle shape.11(10)

- (10)

where k_{B} is the Boltzmann constant (1.38×10^{23} kg m^{2} s^{−2} K^{−1}). This is done by the integrated NanoSight software for every NP that has been tracked for at least five times (this threshold value is automatically adjusted by the system depending on the size of the detected NPs and is meant to reduce measurement errors). Hence, a radius is assigned to each of these NPs, yielding the raw distribution shown in Figure 4 a. The internal NTA software fits this data assuming a spherical particle shape and introducing a sizedependent weighting factor. The resulting output data is plotted in Figure 4 b and, in the authors’ opinion, does not satisfactorily reproduce the raw data shown in Figure 4 a.

Consequently, we adapted the analysis of the NTA data according to our needs for agglomeration studies, as described below. Still assuming a spherical shape, the experimentally derived diffusion coefficient (*D*) can be used to determine the number of atoms (*N*) forming a NP:(11)

- (11)

Thus, the number of silver atoms per NP is proportional to and, as for the impact data this number of atoms is directly available from the NTA measurements, as long as the shape of the considered particle is close to a sphere. However, for agglomerates of few monomers, the assumption of a spherical shape is not fully appropriate, as shown schematically in Figure 5. While monomers likely represent almost perfect spheres (see Figure 7 in the Experimental Section), dimers more precisely have to be described as dumbbells. Trimers and tetramers also form nonspherical agglomerates.

Consequently, the diffusion coefficients, which actually are the physically meaningful experimental output of NTA measurements, need to be corrected to yield detailed information about particle agglomeration states. Hoffmann et al.13 determined the required correction factor for agglomerates consisting of up to four monomers both theoretically and experimentally and found excellent agreement for both approaches. Accordingly, we derived the diffusion coefficients of agglomerates (*D*_{kM}) from the value of the monomer (*D*_{M}) by applying the correction factors (*α*_{k})(12)

- (12)

For the agglomerates shown in Figure 5, these factors were reported by Hoffmann et al.13 for dimers (*α*_{2}=0.73), trimers (*α*_{3}=0.64) and tetramers (*α*_{4}=0.57), and we extrapolated these correction factors (see Figure 5 b) to correct for pentamers (*α*_{5}=0.52) and hexamers (*α*_{6}=0.50). In the following, these shape-corrected diffusion coefficients are used to fit the NanoSight data in light of agglomeration studies.

For better comparability with the impact data, we plotted the NTA data as counts against , since this is directly proportional to the number of atoms plotted at the x-scale in Figure 3, which is not true for the diffusion coefficient (*D*) itself. Please note that this means we consider the actual measured data obtained by the NTA, not introducing any restrictions or assumptions with respect to the shape or nature of the NPs.

since again they follow a normal distribution (like the number of atoms, to which is directly proportional).

The fitted value for is 0.102×10^{2} s^{3} cm^{−6}, that is, *D*_{M}=2.14×10^{−7} cm^{2} s^{−1}, which according to [Eq. (10)] corresponds to a monomer radius of 10 nm. Considering the precision of NanoSight data, which according to the producer is about 10 % of the NP size, this value is in excellent agreement with the value determined by the impact experiments (*r*_{M}=11 nm), proving that both techniques are suitable for characterisation of NP sizes and agglomeration states independently of each other. To gain precise information about the latter, the new approach suggested herein, which considers the stochastics of experimental data and the nonspherical shape of formed agglomerates, is necessary.

#### Agglomeration state of NPs in KCl/citrate

Besides analysing the size of the monomer and the various agglomerates present in the analysed electrolyte, it is also possible to quantify the relative concentration of monomers and agglomerates, taking the corresponding fitted peak intensities into account. For the impact experiments, these intensities directly display the number of detected impacts of NP of this specific kind. To derive the actual agglomeration state, that is, the ratio of the agglomerates, from this intensity ratio, the data has to be weighted by the respective diffusion coefficient (*D*_{kM}). This weighting is necessary since impacts occur due to Brownian motion of the NPs, and the smaller the NPs are, the faster they move and the more likely they are to impact at the electrode within the duration of an experiment. Consequently, smaller NPs will be counted more frequently than larger NPs, with the corresponding diffusion coefficient being the weighting factor. To correct for this, the experimentally detected intensities were divided by the shape-corrected diffusion coefficients (*D*_{kM}, see [Eq. (11)]). The resulting distribution of aggregation states in a KCl/citrate solution are summarised in Table 1. Thus, the impact data shows that only about 40 % of the NP in the solution are monomers, whereas one quarter of them are dimers and the remaining part are agglomerates of at least three monomers.

Table 1. Agglomeration state and relative concentrations of the NP monomers and agglomerates, as detected by impact and NTA experiments, with and without weighting by the corresponding diffusion coefficients (*D*_{kM}).No. monomers (*k*) in NP | Impact (fit) | Impact (*D*_{kM}) | NanoSight (fit) | NanoSight (*D*_{kM}) |
---|

1 | 49 % | 38 % | 55 % | 43 % |

2 | 23 % | 24 % | 20 % | 22 % |

3 | 10 % | 12 % | 9 % | 11 % |

4 | 7 % | 10 % | 8 % | 11 % |

5 | 7 % | 10 % | 5 % | 8 % |

6 | 4 % | 6 % | 3 % | 5 % |

Deriving the distribution of the various kinds of NPs from the NTA measurements also requires an additional weighting of the fitted intensity values. This weighting is automatically done by the implemented software for standard analysis. However, this does not take the nonspherical shape of agglomerates into account; thus, an alternative weighting needs to be applied. Using the track length, required for detected NPs to be considered in the statistics, was not appropriate for the present size distribution, since the automatically adjusted required minimum track length was 5 for all agglomerates. Taking into consideration that as for the impact experiments, diffusive motion of the NPs is crucial for them to be detected,2, 13 it seems reasonable to apply a similar weighting of the intensities fitted to the number of NP counted during the measurement, that is, dividing them by the shape-corrected value of the corresponding diffusion coefficient (*D*_{kM}). Doing so yields a distribution of the agglomeration state that is very similar to the one determined by the impact experiments. However, it has to be noted, that while this weighting was physically straight-forward for the impact experiments, it might introduce an unknown error to the NanoSight data.