Compact aerosol aggregate model (CA2M): A fast tool to estimate the aerosol properties of fractal-like aggregates

Abstract The structure of fractal-like aggregates is of increasing interest to a variety of fields extending from combustion soot formation to synthesis of engineered nanoparticles. Nanoparticle aggregates and agglomerates formed of solid primary particles have been extensively studied, along with their subsequent processing, using increasingly sophisticated modeling techniques over the last decade. However, there is still a need for the community to have a simple and versatile tool, able to run on a single processor, which can estimate aerosol properties and compare them to experimental results. Many aerosol studies have focused on measuring the properties related to particle mass and momentum transfer and exchange with their surroundings. Often, simplified theories are available to support the experimental results (e.g., charging or transport) but show discrepancies due to the omission of complex phenomena related to particle morphology. Thus, there is a tradeoff when one wants to use computer simulations to generate and process nanoaggregates through realistic scenarios in a limited amount of time. The tool presented here provides an efficient and realistic means to estimate mass, volume, projected-area, surface-area, mobility diameter, aerodynamic diameter, and effective density. Critical features treated here include the overlap and polydispersity of primary particles composing the aggregate. Computer-generated micrographs are also available for Scanning Electron Microscopy and Transmission Electron Microscopy data comparison. The code – available on request – originally runs on MATLAB and is built such that users can adapt it and extend it for their own studies. Graphical Abstract


Introduction
The behavior of aerosol nanoparticles is commonly characterized using intrinsic properties, including mass and geometric size, as well as non-intrinsic properties such as the mean free path of the molecules of the surrounding gas (Sorensen 2011). While determining the intrinsic parameters for numerically simulated spherical particles is straightforward, even state-of-the-art techniques are limited for non-spherical particles. As a consequence, it is common to use equivalent diameters which compare the physical behavior of the irregular particle to that of a sphere with some defined properties. These lengthscales depend on both intrinsic and non-intrinsic properties and are strictly dependent on the flow regime.
Aerosol classification techniques often employ external transverse forces to perturb and deflect the laminar trajectory of the particles. Common coupled forces for aerosol classification include drag, electrostatic, and centrifugal used in different combinations. In a differential mobility analyzer (Knutson and Whitby 1975), the particles are exposed to an electric force perpendicular to the flow direction which balances the orthogonal drag force. For a given electric field intensity and particle charge, only one specific diameterthe mobility diameterallows the particle to follow the correct trajectory. Although the relationship between electrical mobility and mobility diameter requires knowledge of the particle charge state, which is not directly measurable for arbitrarily-shaped nanoparticles, it has been a useful standard in aerosol metrology for decades. An alternative metric is the aerodynamic diameter, defined as the diameter of a unit density sphere having the same terminal velocity as the given non-spherical particle. The aerodynamic diameter is related to the relaxation time, which describes how quickly a particle can accommodate to a new set of flow conditionseffectively comparing the relative magnitude of the external drag and inertial effects (see 2.3). Common techniques include time-of-flight (Griffiths, Patrick, and Rood 1984) and centrifugal (Tavakoli and Olfert 2013) for aerodynamic diameter characterization. The combination of centripetal and electrical force allows the determination of the intrinsic particle mass provided the charge state is known. Generally, the relationship between extrinsic measured properties and intrinsic properties or the relationship between various extrinsic particle properties are challenging to determine without the aid of numerical models of the aggregate structure, particularly when relating fluid-particle interactions (Gopalakrishnan, Thajudeen, and Hogan Christopher 2011).
The flow regime impacts the force balance on the particles, with the kinetic theory of gases applying for particles much smaller than the mean free path. In the free molecular regime, the friction coefficient is dependent on the orientationally averaged projected area (Vincenti and Kruger 1965). In the continuum regime, the drag forces acting on a non-spherical particle can be expressed in terms of the equivalent radius of a sphere having the same diffusion rate as the true particle. Previous works have shown that the latter equivalent parameter, the socalled hydrodynamic radius, can be seen as an analog of the mass transfer equivalent radius, i.e., the Smoluchowski radius (Douglas, Zhou, and Hubbard 1994;Given, Hubbard, and Douglas 1997). Collision kernels between vapor molecules and non-spherical solid bodies have been modeled early on via the "adjusted sphere" model (Dahneke 1973), and then for chains of monomers using Langevin dynamics (Isella and Drossinos 2011) and for any arbitrary shape particles using Brownian dynamics (Chan Kim and Torquato 1991;Gopalakrishnan, Thajudeen, and Hogan Christopher 2011). By combining both the Smoluchowski radius and the orientationally averaged projected area, the latter recent approach has been used to calculate the mobility of a broad range of aggregate morphologies in the transition regime of Knudsen number. In this work, the need to calculate the mobility of nanoaggregates with complex morphologies in the transition regime, where no direct universal relation exists, motivates the randomized simulation approach for the calculation of the orientationally averaged projected area and Smoluchowski radius.
Solid aggregates consisting of tens to hundreds of primary particles are usually generated using particlecluster (PC) and cluster-cluster (CC) aggregation mechanisms driven by diffusion or assuming a ballistic behavior (Meakin et al. 1989;Ball and Witten 1984;Sorensen 2001;Schaefer and Hurd 1990;. These approaches, which describe different kinetics of agglomeration, depend on the Knudsen regime and lead to drastically different fractal properties, e.g., fractal exponent D f and fractal prefactor k f : To widen the applicability of the tool presented here, a sequential tunable algorithm is preferred over the selection of a specific agglomeration mechanism. The fractal properties are thus defined by the user during the calculation initialization. A number of tunable algorithms have already been established to generate particles via PC (Mitchell and Frenklach 1998;Chakrabarty et al. 2011b;Singh and Tsotsas 2021) or via CC aggregation (Filippov, Zurita, and Rosner 2000;Yon et al. 2019;Tomchuk, Avdeev, and Bulavin 2020). While the former implementation is more straightforward, it becomes time consuming when large aggregates are generated, e.g., one hour for aggregates with 25 primary particles using a single processor in Chakrabarty et al. (2011a). In addition, the complexity of polydisperse aggregates leads to convergence difficulties with this approach. In this work, a two-step approach, i.e., PC followed by CC, is used as proposed by Filippov, Zurita, and Rosner (2000) and others (Yon et al. 2019;Tomchuk, Avdeev, and Bulavin 2020). To reduce computational time, the cluster-cluster aggregation trial procedure neglects any rotation, and is only governed by translations in the 3D domain while satisfying the quasi-fractal relation, expressed for a monodisperse aggregate by Sorensen (2001) where N is the number of primary particles composing the aggregate, R g is the gyration radius, and R pp is the constant primary particle radius. This Generation Module allows users to obtain monodisperse and polydisperse aggregates generated by the collision of up to several hundreds of particles with D f ranging from 1.0 (lacy aggregate) to 3.0 (spherical cluster) and a broad range of geometric standard deviations for the size distribution of the primary particles. For diffusionlimited cluster-cluster aggregation (DLCA) aggregates, increasing fractal pre-factor k f leads to an overall decrease in particle shape anisotropy (Heinson, Sorensen, and Chakrabarti 2010;Liu et al. 2013). While D f and k f are parameters describing the true fractal nature of particles from a geometrical viewpoint, it is crucial to link these quantities to those used in experiments. The mass-mobility relation, whose expression can be written m ¼ k a d D a m , includes a pre-factor k a and an exponent D a and is determined through the measure of electrical mobility and mass simultaneously, e.g., using a combination of differential mobility analyzer and particle mass analyzer, both upstream of a particle counter. The fundamental differences in pairs fD f , k f g and fD a , k a g have been addressed in Liu et al. (2013), where D a is shown to be sensitive of changes in D f while k a is more affected by variations of k f : A mass-mobility exponent reaching a value of three means that the particle is spherical, while more irregular particles have an exponent below three (Park et al. 2003;Sorensen 2011;Leung et al. 2017).
Various studies have shown that, besides polydispersity, fractal nanoparticles exhibit overlap between primary particles Yon, Bescond, and Liu 2015;). This phenomenon is normally a consequence of the formation process, e.g., particle sintering or reheating. A variable low-to-moderate degree of particle overlap can be implemented within the Processing Module by shifting the center coordinates of each primary particle in succession, while taking the solid material redistribution into account.
Finally, the Properties Module visualizes the final aggregate and calculates and displays the relevant aerosol properties to the user prompt. The average projected area is calculated first by averaging its value over multiple ($100) orientations. The projected area results from the projection on a two-dimensional plane of all visible areas of the aggregate at a given orientation. Therefore, this parameter influences the particle drag, albeit neglecting any screening effects. For each of these orientations, an option allows the user to display the corresponding micrograph.
The calculated value for the average projected area is then used in the calculation of the mobility as proposed in Gopalakrishnan, Thajudeen, and Hogan Christopher (2011) via the intermediate calculation of a hydrodynamic radius (capacity) and an equivalent Knudsen number, also valid in the transition regime.
The relation used to link the mobility to the mobility diameter is taken from Rogak, Flagan, and Nguyen (1993) and was validated on laboratory flame-generated soot in the original paper. Based on the volume and the mass of the aggregate, assuming an a priori known bulk material density, and the previously computed mobility, the relaxation time of the aggregate transported in a laminar flow is obtained, hence its aerodynamic diameter is determined (Tavakoli and Olfert 2013;Johnson et al. 2018). Finally, the derived mass-mobility relation allows the effective density to be calculated.
The key contribution of this work is to provide a fast, direct, and computationally cheap combined algorithm to generate mono/polydisperse aerosol particles and to calculate their properties, allowing near real-time analysis. This simple platform might be useful in particular for researchers working with aerodynamic and mobility diameters, particle mass, and microscopy. Users are encouraged to build upon this code to develop more sophisticated algorithms for their own studies.

Model definition
2.1. Two-step particle aggregation Fractal aggregates are usually not represented by uniform size distributions but rather lognormal distributions which consider the non-idealities of the aggregation processes (Boies et al. 2015;. To initiate the tunable aggregation, radii values are sampled from the primary particle size distribution whose probability density function is where d pp is the primary particle diameter, d gm and r are the logarithm of the geometric mean primary particle diameter and of the geometric standard deviation, respectively. Once the primary particles are selected, the first step is to aggregate small clusters of particles and the second step is the aggregation of these sub-clusters together such as done by Filippov, Zurita, and Rosner (2000). This two-step process is depicted in Figure 1.
The calculation of the gyration radii for the primary particles R g, pp and aggregates R g, agg are The validity of this approach is based on the respect of the quasi-fractal relation which is expressed here as the distance d cm between the center of mass of each aggregate via where m t is the total mass, m 1 and m 2 are the masses of aggregates 1 and 2, respectively, and R g, f is the gyration radius of the resulting aggregate. The distance d cm can be represented as the radius of a sphere whose center is located at the center of mass of the Aggregate 1 and whose envelope contains the center of mass of Aggregate 2 (see Figure 1). This geometric constraint allows the candidate aggregate to translate in the domain within a fixed d cm , validating the criterion of quasi-fractal aggregation.
The second criterion is the contact between the primary particles without overlap, i.e., point-touching, meaning that the surfaces of at least one pair of primary particles from different aggregates should be at a distance 0 < < 10 À3 nm from each other.

Overlap-induced material redistribution
In practice, primary particles are bonded together through chemical reactions which produce a contact surface instead of an ideal point contact (Kruis et al. 1993;Schmid et al. 2004). A low overlap value can be specifiedalbeit violating the quasi-fractal relation due to the effective reduction of the radius of gyration by defining the new distance between the primary particles centers as where c ov is the constant overlap coefficient. Unlike the majority of existing models (Brasil et al. 2001;Al Zaitone, Schmid, and Peukert 2009;Eggersdorfer et al. 2011), here we account for a realistic redistribution of the solid material for the volume to be conserved as a result of the overlap. This scenario corresponds to sintering without mass transfer to the surroundings, which is seen for instance when operating below the material melting point, and which only involves internal transfer via surface and volume diffusion. This phenomenon has been observed experimentally to create necking between primary particles . For monodisperse aggregates with known primary particle radius and overlap degree, the excess material is redistributed through a catenoidal surface model, which is a case of minimal surface. The overlapping phenomenon is depicted in Figure 2.
Considering an axial section S A of a dimer with an orthogonal contact surface S C between the two primary particles, the point P 0 2 ðS A \ S C Þ creates an angle w 0 with the major axis of the dimer A catenary curve is inscribed in a triangle with base 2a 1 and height a 2 defined as where w g is a guess angle based on which the equation Gðw g Þ is minimized Figure 1. 2D schematic of Cluster-Cluster aggregation trial procedure which occurs in 3D. The trial aggregate 2 is separated from the reference aggregate 1 by the distance d cm : The parameters R agg , R g, agg , R g, pp , and d pp define the aggregate and gyration radii, the primary particle gyration radius, and the primary particle diameter of aggregate 1, respectively.
In Equation (10), the left-hand side defines the area G to be minimized. The parameters h and c constrain the aspect ratio of the catenary curve and take the values c ¼ 10 c ov = cos w g and h ¼ c Â 10 À3 throughout this work, to ensure realistic results across the range of overlap values considered. An iterative algorithm allows finding the unique value of w g that minimizes Gðw g Þ and outputs a unique catenary profile that is subsequently propagated by axisymmetric rotation. The associated change in total surface area is expressed with k 1 ¼ R sin ðw g ÞÀmaxðccoshðx=cÞÞ and k 2 ¼ Rðcos ðw 0 Þ À cos ðw g ÞÞ: This approach is limited to the following physical scenarios: (i) contact between only two primary particles at a time, and (ii) low values of c ov for which there is no radial growth of the particles in addition to necking.

Equivalent diameters of interest
The quasi-fractal aggregates considered in this paper are in a quiescent air environment at temperature T ¼ 300 K and pressure p ¼ 1 atm, therefore they are likely to fall in the transition regime where the Knudsen number is expressed by Gopalakrishnan, Thajudeen, and Hogan Christopher (2011) where k is the gas mean free path (% 66.5 nm), R s is the Smoluchowski radius of the particle which is assumed to be equivalent to the mean hydrodynamic radius (see below), and PA is the aggregate orientationally averaged projected area. Based on the projected area, the platform provides an option to extract a micrograph for any orientation. The mobility B is a fundamental parameter representing the inverse of the drag coefficient, thus reflecting interactions between the aggregate and the surrounding gas. The mobility is calculated as (Rogak, Flagan, and Nguyen 1993;Thajudeen, Gopalakrishnan, and Hogan 2012) with the Cunningham slip correction factor expressed as in which l is the dynamic viscosity (% 1.82 Â10 À5 Pa.s). The coefficients to correct for slip are taken from the original work by Davies (1945). The electric mobility Z p is derived from the mechanical mobility incorporating the charge state of the aggregate (Hinds 1999) where p is the number of charges and e is the elementary charge. From the calculation of Z p , the mobility diameter d m is inferred as (Decarlo et al. 2004) The Knudsen numberwhich is function of R s and PAis therefore key to calculating the mobility. To obtain PA, the aggregate is fixed at the center of the domain and is rotated around (x, y, z) for N o orientations during which the projected binary image is processed and the projected area is calculated.
The Smoluchowski radius is calculated via Brownian simulations as proposed early on by Chan Kim and Torquato (1991). As described in Figure 3, the aggregate is at the center of a spherical domain whose radius is R sph ¼ 1.5 R g, agg : A massless Brownian walker is then randomly placed on the sphere envelope. One simulation step is represented as the random path that the walker is undertaking. The minimum distance between the walker and the aggregate is computed and defines the travel distance. The direction is computed via randomly choosing the angle in (x, y, z) and the walker is moved the travel distance in the direction of the chosen angle. If the walker stays inside the sphere, the simulation is repeated; otherwise, a value a is randomly picked in a uniform distribution and the walker is eventually lost from the simulation and counted as a non-colliding walker if where R w represents the distance between the walker and the domain origin. When the walker reaches a distance equal or less than d pp =4 to a primary particle, it is counted as a collider. The Smoluchowski radius is finally obtained as R s ¼ X col R sph , with X col being the ratio of the collision events to the total events, i.e., to the sum of the collision and non-collision events. Following the calculation of the mechanical mobility, the aerodynamic diameter d ae is determined. This property is commonly measured using time-of-flight and impaction instruments (e.g., ELPI, Dekati Ltd.) or with the Aerodynamic Aerosol Classifier (AAC, Cambustion Ltd.). Numerous studies have been undertaken using the AAC as the classification depends solely on the aerodynamic properties (shape, density) of the particles, without an a priori knowledge of the particle charging state (Tavakoli and Olfert 2013). The aerodynamic diameter can be inferred from the definition of the particle relaxation time s, which is (Johnson et al. 2018;Johnson et al. 2021) where m t is the particle total mass, q p is the material density of the primary particles, d ve is the volume equivalent diameter, and v is the shape factor. The effective density q eff has been defined in Decarlo et al. (2004) and supported by experimental measurements (Olfert, Symonds, and Collings 2007;Rissler et al. 2013;Olfert et al. 2017;Kazemimanesh et al. 2019). The total mass is readily obtained where V m is the total particle volume.

Initialization and validation
To validate the generation and mobility calculation, a set of aggregates with N ¼ 50 and N ¼ 80 (D f ¼ 2.00 and 1.95, respectively) is compared with the work of Gopalakrishnan et al. (2013) which is also based on the generation approach of Filippov, Zurita, and Rosner (2000). The fractal prefactor is taken constant k f ¼ 1.3. For all calculations in this paper, the mobility is calculated assuming 10 3 collision events between the Brownian walkers and a charged particle (þ1). Orientationally averaged projected areas were calculated from 10 3 independent projections. Furthermore, the scale of the structures is defined by taking the gyration radius equal to the major axis of an ellipse having the same second central moment as the region of interest. For each primary particle diameter, the corresponding mobility diameter is calculated according to the fractal properties. Final results represent the average mobility diameter assuming a convergence criterion of 1 nm. The results of the primary particle validation are shown in Figure 4. Both models agree within 20% on the entire range of mobility diameters. The primary particle diameter for the set (D f ¼ 2.0, N ¼ 50) is always larger than for the second set to obtain comparable mobility diameters. For both fractal pairs, the present approach overestimates the mobility from d m ¼ 20 to $100 nm and underestimates beyond d m ¼ 100 nm. The maximum deviation occurs at d m ¼ 500 nm for which the model underestimates by $20 %:

Properties of fractal aggregates
3.2.1. Fractal morphology of aggregates Fractal aggregates are defined by their fractal dimension D f and prefactor k f which govern their compactness, from lacy structures for D f ! 1 to compact structures for D f ! 3, and the particle anisotropy, respectively. The gyration radius is largely dependent on the primary particle radius d pp and the total number of particles N composing the aggregate. The material density of a typical soot core (q p ¼ 1800 kg/m 3 ) is used throughout this work (Rissler et al. 2013). Figure 5 shows the dependence of the morphology on fractal parameters for aggregates with N ¼ 100 and d pp ¼ 10 nm. Nine aggregates were generated with different pairs of fractal parameters: D f ¼ 1:3 ! 2:1 and k f ¼ 1:2 ! 1:7: The particles are more compact when D f and k f are increased. Fewer branchesreflecting less open structuresare observed for k f ¼ 1.7. The interaction of the primary particles with the surrounding gas molecules generates drag, which reduces with aggregate compactness due to screening effects. Conversely, when D f tends to one and k f is decreased, the three dimensional effects are reduced and the drag maximized. The fractal nature ultimately governs the transport of particles, as well as their deposition.
Figure 6 (left) shows the evolution of particle mobility and aerodynamic diameters when the aggregate size is increased. Three morphologies are studied, i.e., a lacy (D f ¼ 1:3, k f ¼ 2:0), DLCA-like (D f ¼ 1:78, k f ¼ 1:4), and compact (D f ¼ 2:5, k f ¼ 0:7) structures. Independently of the morphology, the mobility and aerodynamic diameters increase monotonically with aggregate size. In this regime, the mobility is continuously higher than the aerodynamic diameter and their ratio d m =d ae increases with N. The effect of the fractal parameters is dependent on the aggregate size. For aggregates composed of 200 or fewer primary particles, the lacy structure shows a larger aerodynamic diameter than the other aggregates, while the opposite is true for the mobility diameter. For N ¼ 16, d ae and d m vary by 15:6% and 9:8% respectively based on the structure. Conversely, for aggregates with more than 200 primary particles, the compact structure has the largest aerodynamic diameter and smallest mobility diameter. For the largest aggregate size, N ¼ 512, d ae is 18:7% lower and d m is 11:6% higher for the lacy structure compared to the compact one. Figure 6 (right) shows the effect of the fractal parameters on the mass-mobility relationship, studied using the three pairs of fractal parameters (D f , k f ) presented above. Aggregates of different sizes are achieved either by varying N from 16 to 512 while keeping d pp ¼ 10 nm, or by varying d pp at constant N. The range 5 -50 nm is chosen for d pp : The upper limit of d pp used here is representative of experimental observations by Bond and Bergstrom (2006). A linear dependence of the mass on the mobility is found in log-log space for both enlargement scenarios; however, a steeper curve is observed for the radial growth scenario. Lacy aggregates have a larger mobility diameter than compact aggregates for m > 0:1 fg (primary particle addition) or m > 1 fg (radial growth). The model developed by Lall and Friedlander (2006) only agrees for low mobility diameters, whereas significant discrepancies are seen from d m % 30 nm. When d pp ¼ 10 nm, reasonable agreement is found between the present model and the semiempirical relation proposed by Sorensen (2011). Curve fitting based on this relation leads to the well-known mass-mobility exponent and prefactor, both of which can be obtained experimentally after measurement of mass and electrical mobility, e.g., using a tandem DMA-CPMA setup, and further data inversion.
The correspondence between the fractal (D f ) and mass-mobility (D a ) exponents of the three pairs of studied fractal parameters shows a quasi-linear (R > 0.99) relationship, i.e., D a ¼ a 1 D f þ a 2 with a 1 ¼ 0.165 and a 2 ¼ 2.069. For compact structures D a approaches D f , whereas the mass-mobility prefactor k a is found to be k a > 2 k f for all pairs of fractal parameters.  . Left: evolution of the mobility and aerodynamic diameters, d m and d ae respectively, with aggregate size (N) for three pairs of fractal parameters. Right: mass-mobility relationship for three pairs of fractal parameters -fD f , k f g ¼ f1:3, 2:0g, f1:78, 1:4g, and f2:5, 0:7gfor particle enlargement via radial growth (N ¼ 60) or primary particle addition (d pp ¼ 10 nm).

Influence of the polydispersity
Freshly emitted aggregates such as soot are likely to exhibit a significant degree of polydispersity. Numerical studies have investigated the influence of polydispersity on the fractal dimension and on the mobility diameter of aggregates for different collision mechanisms and Knudsen numbers . Dastanpour and Rogak (2014) observed geometric standard deviation values ranging from 1.2 to 1.7 for single and ensemble of soot aggregates sampled from gasoline direct injection (GDI) and high pressure direct injection engines, aviation turbines, and inverted burners.
In order to study the influence of primary particle polydispersity, one particular aggregate is chosen as a reference with D f ¼ 1.78, k f ¼ 1.4, and the geometric mean diameter (GMD) of the primary particles is fixed at 10 nm. Based on this reference aggregate, the polydispersity (represented by r) is changed, leading to 30 simulation cases, each averaging properties over one hundred aggregates. Figure 7 (left) describes the changes in mobility and aerodynamic diameters when the number of primary particles is increased for different degrees of polydispersity. Quasi-linear relationships are found in the semi-logarithmic space. Highly polydisperse aggregates, i.e., with r ¼ 1.8, show a significant increase of both d m and d ae as compared with monodisperse aggregates. For these particular fractal parameters and material density, the resulting mobility diameter is always higher than its aerodynamic counterpart, with the difference between the two increasing with N as far as $63 % for r ¼ 1.0 and $56 % for r ¼ 1.8 at N ¼ 256. The corresponding reduced diameters with respect to N ¼ 16 are d m, r ¼ 3.23 (r ¼ 1.0) and 3.35 (r ¼ 1.8), and d ae, r ¼ 1.81 (r ¼ 1.0) and 2.03 (r ¼ 1.8). Equations (22)  (right) represents the evolution of both mass m / Nd 3 pp and surface area SA / Nd 2 pp with aggregate enlargement. Any degree of polydispersity causes the mass and surface area to increase compared with the monodisperse case. This change is increasingly pronounced for highly polydisperse aggregates, here for r ¼ 1.8. Figure 8 (left) shows the mass-mobility relationship which is linear the log-log space, allowing a simple expression of m ¼ f ðd m Þ: This relation is shifted toward higher mass values for polydisperse aggregates in the entire mobility diameter range. A slight increase in mass-mobility exponent D a ($1.3 %) and prefactor k a ($8.2 %) is obtained for high polydispersity. This observation is consistent with the results of  who also found a small enhancement of mass-mobility parameters for r ¼ 1.6 in both the free molecular and continuum regimes. As a result, the mass-mobility exponent D a ¼ 2.46 is 38 % larger than D f and k a ¼ 2.64 is 88 % larger than k f : It should be noted that all curve fits present an Rsquared value > 0.99. Figure 8 (right) describes the evolution of the effective density with the mobility diameter as a function of the polydispersity. The presence of external voids alone is sufficient for the effective density to be smaller than the material density in this setting. A well-known decreasing curve is obtained which depends on the distribution of the primary particle radius. Monodisperse aggregates have the lowest effective density with values comparable to experimental measurements using an inverted burner as shown by Ghazi et al. (2013) and Olfert and Rogak (2019). Highly polydisperse aggregates have higher effective densities, similar to measurement from a Santoro flame (Pagels et al. 2009). The wide range of effective densities obtained from a GDI engine by Graves, Koch, and Olfert (2017) covers the entire span of polydispersity presented in this study, highlighting the complex morphology of fresh nanoparticles released by such engines.

Influence of the overlap degree
Upon aggregation, agglomerates originally in point contact coalesce to create chemically-bonded (sintered) aggregates, resulting in a change of particle transport, radiative, and electron transfer properties . The aggregates morphology evolves following the extent of the sintering process, from the formation of necks between primary particles to a spherically-shaped particle. By analogy with the excess free energy responsible for the gas-liquid phase transition in the vicinity of negatively curved surfaces building capillary bridges (Orr, Scriven, and Rivas 1975;Crouzet and Marlow 1995), a gradient of stress (solid phase) results in material diffusion toward the necks.
The influence of the overlap degree on the aerosol properties is studied for a lacy aggregate (D f ¼ 1.3, k f ¼ 2.0) with variable sizes N ¼ 16, 32, and 64.
The mass-mobility relation is plotted for five values of c ov (0.0 to 0.3) in Figure 9 (left). The evolution of the shapes of the necks is shown in insets of Figure 9 (left), where the maximum c ov ¼ 0:3 corresponds to a neck filling the gap completely. The overlap process impacts more the small particles (N ¼ 16) with a 22 % difference in mobility diameter between c ov ¼ 0:3 and c ov ¼ 0, while there is only an 8 % difference for larger particles (N ¼ 64). The overlap is very sensitive to the particle size as even an overlap of 0.01 has a major impact on the mobility for the smallest aggregate considered here. Similar conclusions were obtained with a more compact aggregate with D f ¼ 1.78 and k f ¼ 1:4: As the overlap increases, the surface area of the resulting aggregate reduces. This reduction is expressed as a function of the overlap degree and presented in Figure 9 (right) for an aggregate with (D f ¼ 1.78, k f ¼ 1.4, N ¼ 32) for different primary particle  . Left: mass-mobility relationship during initial stages of sintering (c ov ) for a lacy aggregate (D f ¼ 1.3, k f ¼ 2.0) with N ¼ 16 À 64: Right: total surface area change (%) as a function of the overlap degree c ov for four primary particle radii, d pp ¼ 10, 20, 30, and 40 nm. diameters (d pp ¼ 10, 20, 30, and 40 nm). Assuming a constant aspect ratio parameter c (see definition in Section 2.2), the surface area change with maximum overlap is $32.8 À 36.4 % for the particle diameter range studied. The influence of the primary particle size is more pronounced for light overlap scenarios for which smaller particles lose more surface area. In most casesexcept for d pp ¼ 10 nmthe surface area change becomes independent of primary particle size as overlap increases.
The overlap scenario corresponds to the initial stages of sintering where the bonding is formed through necking before any radial growth of the primary particles begins. The insets in Figure 9 (right) show Molecular Dynamics (MD) simulations within the NVT ensemble of the sintering of a silver dimer using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) (Thompson et al. 2022) with d pp ¼ 10 nm at a temperature of T ¼ 1100 K. At this high temperature, Ag atoms diffuse radially from the contact point toward the external layer to maximize the contact surface between the two monomers.

Conclusions
This study focused on the development of a compact algorithm to generate and measure nanoscopic aggregates that can include some non-idealities. The generation algorithm is based on the two-step particle-cluster and cluster-cluster aggregation. This method allows generating mono or polydisperse aggregates on a wide size range (up to 512 primary particles in this study) at reasonable computational cost, i.e., for most cases less than an hour.
As these aggregates are not observed as monodisperse with point contact in nature, the polydispersity and the overlapping can be adjusted independently. After the particles are numerically synthesized, three dimensional interactive figures and two dimensional micrograph representations are available. Eventually the fundamental properties of the particles, such as their mass, mobility and aerodynamic diameters, effective density, and surface area are calculated in line with semi-empirical relations from validated literature results.
The study of bare aggregates revealed that the prefactor and the fractal exponent independently play a role in determining the final morphology. Increasing the value of these parameters led to more compact particles overall, however the prefactor itself had a role in driving the structure opening with more or less branches. The calculation of the mobility and aerodynamic diameters for these aggregates showed an increase with particle size, with the electrical mobility always greater than the aerodynamic diameter. Similarly, the mass-mobility relationship allowed to find a quasi-linear correspondence between the mass-mobility exponent and the fractal exponent and both approached each others for compact structures. When the polydispersity is accounted for, higher values of themobility and aerodynamic diameters are found with 4 % and 12 % increases, respectively. In addition, mass, surface area, effective density, and mass-mobility exponent of polydisperse aggregates are larger than their monodisperse counterparts. Finally, when primary particles are overlapping, a minimization strategy is used to shape a minimal surface and conserve the total volume of the aggregates. An important observation from the overlapping is the decrease in aggregate total surface area which can reach more than 30 % even for lightly overlapped particles.

Disclosure statement
No potential conflict of interest was reported by the authors.