Packing density of rigid aggregates is independent of scale

Abstract

Large planetary seedlings, comets, microscale pharmaceuticals, and nanoscale soot particles are made from rigid, aggregated subunits that are compacted under low compression into larger structures spanning over 10 orders of magnitude in dimensional space. Here, we demonstrate that the packing density (θ f ) of compacted rigid aggregates is independent of spatial scale for systems under weak compaction. The θ f of rigid aggregated structures across six orders of magnitude were measured using nanoscale spherical soot aerosol composed of aggregates with ∼17-nm monomeric subunits and aggregates made from uniform monomeric 6-mm spherical subunits at the macroscale. We find θ f = 0.36 ± 0.02 at both dimensions. These values are remarkably similar to θ f observed for comet nuclei and measured values of other rigid aggregated systems across a wide variety of spatial and formative conditions. We present a packing model that incorporates the aggregate morphology and show that θ f is independent of both monomer and aggregate size. These observations suggest that the θ f of rigid aggregates subject to weak compaction forces is independent of spatial dimension across varied formative conditions. M any systems are comprised of elementary subunits packed within a defined volume. The simplest 3D system consists of uniform spheres, and despite its apparent simplicity, a rigorous mathematical description eluded researchers for nearly four centuries dating back to Kepler's conjecture in 1611. In 2005, Hales provided a definitive mathematical proof confirming the observed experimental maximum packing density of 74% (1). Packing of more complex structures is far more mathematically challenging and instead relies on empirical studies (2-9). One of the most ubiquitous packing systems in the universe are rigid aggregates composed of a collection of monomeric units joined together into a fractal structure and subsequently densified through omnidirectional applied force, as shown in Fig. 1 (10).The formation and compaction mechanism of disordered aggregates is presumed to be independent of dimension, composition, and spatial scale, and has been observed in a diverse range of materials and conditions, such as the accretion of material in interstellar space and the formation and compaction of aerosol in the Earth's atmosphere. Many interstellar formations comprise nano-or microscale dust particles that begin as disordered monomeric subunits, which electrostatically aggregate to form fractal (lacey) agglomerates that serve as foundries for comets and planetary seedlings (10-16). Soot, ubiquitous in the Earth's troposphere, is also comprised of nanometer sized carbonaceous monomers aggregated in a disordered lacey structure. Compaction into spheres occurs after trace gas and/or liquid adsorption and evaporation. In both cases, the resulting structure is constrained by aggregate rigidity (17).The systems described above are similarly constructed from single-unit building blocks assembled into larger disordered structures. The final structure ...

Particle Classification by the Tandem Differential Mobility Analyzer–Particle Mass Analyzer System

“…Figure 2b presents an example of the second case, which corresponds to a constant value of D m . As discussed in Section 2.1, the value of slope in the log(d m )-log(m p ) space is smaller than 3 for aggregated particles, such as soot (Park et al 2003;Cross et al 2010;Zangmeister et al 2014). In Figure 2b Park et al (2003) is shown as an example.…”

“…The log(d m )-log(m p ) relationship can also be represented using other metrics, such as the mass-mobility exponent (D m ), which is calculated by the following equation (Figure 1b) (DeCarlo et al 2004;Cross et al 2010;Sorensen 2011;Zangmeister et al 2014):…”

“…As indicated by the logarithmic form of Equation (3), D m corresponds to the value of the slope in the log(d m )-log(m p ) space. This metric is especially useful for characterizing the structure of aggregate particles, such as soot (Park et al 2003;DeCarlo et al 2004;Zangmeister et al 2014). r f is a pre-exponential factor for mass-mobility relationship.…”

“…In most of the DMA-APM operations, one operating parameter of either DMA or APM (e.g., V DMA , V APM , or v) is scanned to measure the particle population in the log(d m )-log(m p ) space. This is done to obtain the values or distributions of r eff and D m (McMurry et al 2002;Park et al 2003;Malloy et al 2009;Zangmeister et al 2014). An example of DMA voltage scanning is the APM-SMPS measurement (Malloy et al 2009).…”

“…Examples of these setups include the DMA-APM, DMA-CPMA, and APM-scanning mobility particle sizer (SMPS) systems (McMurry et al 2002;Malloy et al 2009;Cross et al 2010). In these setups, particles are classified by both electrical mobility diameter d m and m p , which allows for the quantification of important physical parameters, such as effective density r eff , dynamic shape factor, and mass-mobility exponent D m (Park et al 2003;Zangmeister et al 2014). The combination of these two techniques is useful in eliminating multiple charged particles because the classification regions for multiple charged particles of DMA and PMA do not overlap (Pagels et al 2009;Shiraiwa et al 2010).…”