In PFA, particle number is applied as flow and stock metric rather than mass. Mass has been used to indicate magnitude of flows and stocks of chemical substances in substance flow analysis (Van der Voet 2002) as well as exposure and effect of chemicals in environmental and chemical risk assessments (Suter et al. 1993; Van Leeuwen and Vermeire 2007). There are strong indications, however, that mass may not be a relevant indicator of flow and stock magnitude, exposure, or toxic effects for the case of NPs (Oberdörster et al. 2005; Handy et al. 2008; Ju-Nam and Lead 2008; Van Hoecke et al. 2008; Arvidsson et al. 2011). When particle number is applied instead of mass as the flow and stock metric, relevant particle properties, such as size, can be accounted for. In addition, frameworks describing different types and properties of NPs, such as those by Hansen and colleagues (2007) and Jiang and colleagues (2009), can be utilized in the analysis. The framework for characterization of NPs by Hansen and colleagues (2007) includes particles that are surface bound, suspended in liquid, suspended in solids, or airborne. The categorization framework of NPs by Jiang and colleagues (2009) divides particles into primary particles, agglomerates (primary particles held together by weak, Van der Waals forces), and aggregates (primary particles held together by strong, covalent bonds). Processes that change particle number, such as agglomeration, melting of particles, dissociation of particles into ions, and grinding (which produces more particles), can be included by the addition of a source (or sink) factor to a substance flow model (see figure 2). Thus, the convenient law of mass conservation on which substance flow analysis is based is not sufficient to describe flows and stocks of particle numbers. Instead of the law of mass conservation, a similar equation can describe the particle flows and stocks of a compartment, with the source or sink term included:
where N denotes the particle number (particles) stock and n the particle number flow (particles per year). Note that the source or sink term (nS) can be either positive or negative. Due to lack of data, only the NP use phase (the same as the substance use phase in a substance life cycle) and related flows and stocks have been investigated in this study. The production phase relates to working environment, and NP emissions in that phase may be more dependent on companies’ management practices than on NP properties. The waste-handling process is not included due to poor knowledge of the fate of Ag NPs during that phase. The parameters estimated in this study are thus the use phase in-flow (nu), the use phase stock (Nu), and the use phase emissions (neu), as shown in figure 2. These parameters are estimated for all three Ag NP applications included. Data on Ag NP production is often reported on a mass basis, however. For the technologies selected in this article, no well-defined particle size distributions of Ag NPs have been found. Hence, the average particle diameter is used as proxy for particle size, and the following equation is used to convert mass to particle number:
where m is the mass flow; ρ is the density in kilograms per cubic meter (kg/m3), which is 10,500 kg/m3 for Ag NPs; and d is the average particle diameter in meters (m). Note, however, that although spherical silver NPs are used in the applications included in this article (see below), other shapes are possible (Pal et al. 2007). If nonspherical NPs were present, equation (2) would need to be modified.