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Aug 18, 2016 - Nanoparticles typically form irregular or fractal-like agglomer- ates of primary particles (PPs) by ballistic coagulation in volcanic p...
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Coagulation of Agglomerates Consisting of Polydisperse Primary Particles E. Goudeli, M. L. Eggersdorfer, and S. E. Pratsinis* Particle Technology Laboratory, Institute of Process Engineering, Department of Mechanical and Process Engineering, ETH Zurich, Sonneggstrasse 3, CH-8092 Zürich, Switzerland S Supporting Information *

ABSTRACT: The ballistic agglomeration of polydisperse particles is investigated by an event-driven (ED) method and compared to the coagulation of spherical particles and agglomerates consisting of monodisperse primary particles (PPs). It is shown for the first time to our knowledge that increasing the width or polydispersity of the PP size distribution initially accelerates the coagulation rate of their agglomerates but delays the attainment of their asymptotic fractal-like structure and self-preserving size distribution (SPSD) without altering them, provided that sufficiently large numbers of PPs are employed. For example, the standard asymptotic mass fractal dimension, Df, of 1.91 is attained when clusters are formed containing, on average, about 15 monodisperse PPs, consistent with fractal theory and the literature. In contrast, when polydisperse PPs with a geometric standard deviation of 3 are employed, about 500 PPs are needed to attain that Df. Even though the same asymptotic Df and mass-mobility exponent, Dfm, are attained regardless of PP polydispersity, the asymptotic prefactors or lacunarities of Df and Dfm increase with PP polydispersity. For monodisperse PPs, the average agglomerate radius of gyration, rg, becomes larger than the mobility radius, rm, when agglomerates consist of more than 15 PPs. Increasing PP polydispersity increases that number of PPs similarly to the above for the attainment of the asymptotic Df or Dfm. The agglomeration kinetics are quantified by the overall collision frequency function. When the SPSD is attained, the collision frequency is independent of PP polydispersity. Accounting for the SPSD polydispersity in the overall agglomerate collision frequency is in good agreement with that frequency from detailed ED simulations once the SPSD is reached. Most importantly, the coagulation of agglomerates is described well by a monodisperse model for agglomerate and PP sizes, whereas the detailed agglomerate size distribution can be obtained by scaling the average agglomerate size to the SPSD.

1. INTRODUCTION Nanoparticles typically form irregular or fractal-like agglomerates of primary particles (PPs) by ballistic coagulation in volcanic plumes and the manufacture of carbon black or fumed silica as well as an array of nanoparticle compositions at low pressure (e.g., inert gas condensation). Even though the dynamics of coagulating spherical particles (self-preserving size distribution (SPSD) and coagulation rate) are reasonably well understood, there is significant uncertainty for fractal-like agglomerates.1 For the latter, coagulation rates have been proposed,2−4 the agglomerate mobility5 and SPSDs6 have been determined, and even the time (or average number of PPs per agglomerate) needed to reach their asymptotic structure7 has been estimated.8 Nevertheless, all of these have been confined to agglomerates with monodisperse PPs. Realistic agglomerates, however, consist of polydisperse PPs (e.g., Scheckman et al.9). So little is known about the effect of constituent PP polydispersity on agglomerate structure10−12 and, most importantly, on coagulation dynamics. More specifically, Tence et al.10 showed experimentally and numerically no effect of PP polydispersity on the asymptotic Df in the © XXXX American Chemical Society

free molecular regime. This was confirmed for agglomerates consisting of three different PP sizes by Bushell and Amal,11 in the continuum regime whereas Eggersdorfer and Pratsinis12 showed that increasing PP polydispersity results in more open, ramified agglomerates for a finite number of constituent PPs. Such characteristics affect the environmental impact of agglomerates as in climate forcing and visibility impairment by soot13 and the performance of catalysts14 and gas sensors.15 Also, the agglomerate structure strongly affects the particle optical properties or settling rate,16 which often alter the macroscopic behavior of colloidal dispersions. For example, the fractal dimension, Df, affects the specific scattering of agglomerates (Dobbins et al.,17 eq 5) and the activation energy of heterogeneous reactions14 as well as the flowability and packing efficiency of granular materials.18 Furthermore, fractallike agglomerates exhibit a lower deposition efficiency in fibrous filters than do mass-equivalent spherical particles.19 Received: July 3, 2016 Revised: August 18, 2016

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ditions, and once the initial number of clusters N0 is halved, the system is mirrored consecutively in the x, y, and z directions so that after three duplications the domain is cubic again.32 The particle and fluid properties are summarized in Table 1.

Agglomerate properties are also affected by the PP polydispersity that changes the degree of particle anisotropy significantly.20 Charalompopoulos and Shu21 showed that the assumption of monodisperse PPs underestimates the real and imaginary parts of the refractive index. Compact agglomerates of narrow PP size distribution (PPSD) exhibit better dispersion at low flow rates, enhancing the efficiency of dry powder inhalers.22 Because environmental and industrial fractal-like agglomerates, however, typically consist of polydisperse PPs, there is significant interest in understanding their dynamics and structure and in particular their SPSD and agglomeration rate, which greatly affect the aerosol reactor design23 and climate forcing estimations.24 Furthermore, the overall collision frequency is used commonly to describe the evolution of particle number concentration N in monodisperse PBE models.25 In that way, the monitoring of the detailed particle size distribution can be reduced to one of its moments and readily interfaced with computational fluid dynamics for efficient process design for nanoparticle synthesis23 and the estimation of particulate air pollution in urban airsheds26 and atmospheric aerosol dynamics on large geographical scales (e.g., urban aerosol,27 climate modeling28). Here, the ballistic agglomeration of polydisperse nanoparticles in the absence of coalescence, sintering, or surface growth is investigated by an event-driven (ED) method. This is equivalent to coagulation in the free molecular regime and is most important during particle formation at high temperature or low pressure followed by rapid quenching to facilitate the collection of agglomerates and the retention of their ramified structure as in the manufacture of fumed SiO2, Ni, or TiO2. As a result, ballistic agglomeration dynamics determine the endproduct characteristics. The effect of PP polydispersity on agglomerate size (radius of gyration, mobility radius, and volume-equivalent radius), morphology (fractal dimension and mass mobility exponent and their prefactors), and the time needed for the attainment of the well-known asymptotic fractallike structure and self-preserving size distribution (SPSD) is investigated, and the dynamics of such agglomerates consisting of PPs of varying polydispersity and their evolving size and structure are studied for the first time. Furthermore, the effect of PP polydispersity on the agglomerate coagulation rate is elucidated quantitatively, and a simplified model for agglomerate growth is developed and benchmarked against the literature and detailed ED simulations.

Table 1. Simulation Parameters ρp ϕ na dp T p

u̅ =

density of SiO2 solid volume fraction number of agglomerates primary particle diameter temperature pressure

Agglomerates consisting of many PPs are often characterized by various equivalent radii of a sphere or a spherical shell that shares a physical property. For example, the radius of gyration, rg, is the radius of a spherical shell that has the same resistance against rotational motion as the agglomerate. The rg is welldefined numerically as the mean square displacement of mass from the agglomerate center of mass and can be measured experimentally by light-scattering experiments. The agglomerate structure is characterized by the mass fractal dimension Df and prefactor kn7 ⎛ rg ⎞ Df m υ = = kn⎜⎜ ⎟⎟ mp υp ⎝ rp ⎠

(2)

where rp is the PP radius and m, υ and mp, υp are the agglomerate and average PP mass and volume, respectively. By contrast, the mobility radius, rm, is the radius of a sphere with the same resistance against translational motion. rm is often measured in differential mobility analyzers for nanoparticles and corresponds in the free molecular regime to the rotationally projected area of an agglomerate. The agglomerate mobility radius rm also scales with the agglomerate mass (or volume) through the mass-mobility exponent Dfm and prefactor kfm, which quantify agglomerate structure and anisotropy similarly to Df and kn:33 ⎛ r ⎞ Dfm m υ = = k fm⎜⎜ m ⎟⎟ mp υp ⎝ rp ⎠

(3)

rm is calculated with the rotationally averaged projected agglomerate area in the free molecular regime.34 m/mp gives the actual number of PPs, np, for agglomerates of monodisperse PPs and an arithmetic average, n̅p, for agglomerates of polydisperse PPs with arithmetic average mass, m̅ p, diameter, d̅p. The overall collision probability, βn, is calculated from the ED simulations as (Heine and Pratsinis,32 eq 16)

2. THEORY Agglomerate formation in the absence of coalescence, sintering, or surface growth is investigated by ED.29 A brief overview of ED is given here, and details can be found in Goudeli et al.8 Spherical PPs having a number-based log-normal size distribution with a geometric standard deviation of σg,PP = 1− 3 are randomly distributed in a cubic simulation box.12 There, particles move in straight trajectories30 with a velocity according to the kinetic theory of gases31 8kBT πm

2200 kg/m3 10−7 1000−2000 1, 5, or 10 nm 300 − 1500 K 1.0133 × 105 Pa

(

2 βn =

1 N (t )

−

1 N (t − Δt )

Δt

) (4)

where N(t) is the agglomerate number concentration at time t. The collision probability equals the collision frequency by the kinetic theory for elastic collisions of homogeneously distributed agglomerates in space (which is the product of the collision cross section of the colliding agglomerates and their relative velocity) in the free molecular regime:2

(1)

where kB is the Boltzmann constant, T is the gas temperature, and m is the particle mass. The particles stick upon collision, forming agglomerates by ballistic cluster−cluster agglomeration (BCCA). The simulation box has periodic boundary conB

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Langmuir 8kBT ⎛ 1 1 ⎞⎟ ⎜⎜ + π (rg, i + rg, j)2 π ⎝ mi mj ⎟⎠

βij =

(5a)

For monodisperse coalescing spheres, it becomes 3kBT ⎛ 3υ ⎞1/6 ⎜ ⎟ ρp ⎝ 4π ⎠

βmono,s = 8

(5b)

and for monodisperse agglomerates, it is expressed as a function of their rg: βmono, r = 16 g

kBTπ 2 rg m

(6a)

Introducing eq 2 into eq 6a yields βmono = 8

3kBT ⎛ 3ϕ ⎞2/ Df − 1/2 2 − 6/ Df −2/ Df ⎜ ⎟ rp kn N (t )1/2 − 2/ Df ρp ⎝ 4π ⎠ (6b)

where ϕ is the agglomerate solid volume fraction, ϕ = υ(t)· N(t). For kn = 1, eq 6b reduces to that given by Matsoukas and Friedlander35 (eqs 3−6). Accounting for the SPSD, the coagulation rate of polydisperse agglomerates becomes (Friedlander,1 eqs 8.11, 8.18, and 8.19): α βpoly = βSPSD = β (7a) 4 2 mono

Figure 1. (a) Evolution of agglomerate structure by coagulation in dimensionless mass−radius of gyration, rg, space for agglomerates having a log-normal PPSD with a PP geometric standard deviation, σg,PP = 1.0 (a−c) and 3 (d−f). Agglomerates consisting of monodisperse PP (σg,PP = 1) attain fractal-like scaling with Df ≈ 1.91 (solid black line) faster than agglomerates of polydisperse PPs; for example, at t = 10−4 s, the former have already attained an overall Df of 1.83 (b), whereas the latter have a bimodal structure with an average effective Df = 2.41 (e). Once agglomerates have grown sufficiently large above their crossover size, asymptotic Df ≈ 1.91 is attained regardless of the PP polydispersity (Figure 1f). The dotted line corresponds to solid spheres (Df = 3, kn = 1).

where α is the dimensionless collision integral quantifying the contribution of the SPSD polydispersity to the overall free molecular βn1,36 ∞

α=

∫0 ∫0

∞

1/2 ⎛1 1⎞ (η1/ Df + η1/ ̃ Df )2 ⎜ + ⎟ Ψ(η)Ψ(η )̃ dη dη ̃ η̃ ⎠ ⎝η (7b)

where η and Ψ(η) are the dimensionless self-preserving particle size and concentration.1 Also, the collision frequency of agglomerates containing up to 50 monodisperse PPs by Thajudeen et al.4 is calculated based on the Smoluchowski radius (Thajudeen et al.,4 eq 21) instead of rg in eq 6a by using the ED-obtained normalized agglomerate volume (or np) and constant Df = 1.91 and kn = 1.41.37 The calculation of this frequency was validated8 by reproducing Figure 3a in Thajudeen et al.4

solid lines overlapping within the data scatter (e.g., at t = 10−3 s, Df = 1.88 ± 0.02 and kn = 1.45 ± 0.07), in excellent agreement with fractal theory,37 also validating the present ED simulations. For polydisperse PPs with σg,PP = 3, early on (t = 10−5 s) agglomerates follow a Df = 3 scaling as a few large PPs dominate the ensemble morphology by scavenging the smaller PPs (Figure 1d, inset). Later on (t = 10−4 s, Figure 1e), such agglomerates have an effective overall Df = 2.41 ± 0.05 and kn = 3.88 ± 0.45 (broken line). Looking, however, more closely at the agglomerates themselves (Figure 1e, miniaturized insets of agglomerates to accommodate their increasing size, rg/rp), there are relatively small but compact (triangle) and relatively large but open agglomerates (diamond) reflecting their broad spectrum of morphologies. The former are dominated by large PPs and thus exhibit a Df = 3 scaling (dotted line), whereas the latter follow a Df = 1.91 scaling (solid line). The agglomerates with Df = 3 scaling are dominantly spheres from the large tail of the initial PP size distribution decorated with smaller PPs (inset of Figure 1d). At the same time, agglomerates at the other end of the size spectrum exhibit an asymptotic BCCA fractal-like structure (diamond) having Df = 1.91. The change in Df from 3 to 1.91 represents the crossover from compact structures to open/fractal-like ones during agglomeration. This broad variation of particle structure is nearly absent for agglomerates made of monodisperse PPs (Figure 1b, inset) that fall rather quickly (t = 10−4 s) on a

3. RESULTS AND DISCUSSION 3.1. Agglomerate Structure and Dynamics. Figure 1 shows snapshots of coagulating agglomerates having a lognormal PP size distribution (PPSD) of number-based geometric standard deviation of PPs, σg,PP = 1 (monodisperse, a−c) and 3 (d−f), in their dimensionless mass (m/mp) vs radius of gyration (rg/rp) space. Both PPSDs have identical geometric mean PP radii rp. Each circle in Figure 1 corresponds to one agglomerate (for clarity, only 10% of them are shown). Initially (t = 0), all circles are at rg/rp = m/mp = 1 (not shown here), corresponding to spheres of Df = 3 (dotted line). For monodisperse PPs, early on (t = 10−5 s, average m/mp = n̅p < 10, Figure 1a), fractal-like agglomerates are formed with effective Df = 2.23 ± 0.09 and kn = 1.18 ± 0.02 (red broken line) as they approach their asymptotic Df = 1.91 ± 0.03 and kn = 1.41 (solid line) for BCCA.37 These agglomerates will eventually attain that asymptotic structure with the broken and C

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Langmuir straight line with Df = 1.83 ± 0.02 and kn = 1.41 ± 0.03 converging to their asymptotic BCCA values. A similar variation is noted for agglomerates of monodisperse PPs by the diffusion-limited cluster−cluster agglomeration of various PP sizes (Figure 5b of Goudeli et al.8). At t = 10−3 s, both agglomerates of monodisperse (Figure 1c) and polydisperse (Figure 1f, inset) PPs are approaching the asymptotic BCCA scaling with overall Df values of 1.88 ± 0.02 and 1.71 ± 0.02, respectively, but vastly different lacunarities, kn = 1.45 ± 0.07 and 27.8 ± 2.4, respectively. Figure 2 shows the evolution of effective (a) Df and (b) kn for agglomerates made by the coagulation of PPs having σg,PP = 1.0

faster with large ones than with equally sized ones.1 Such collisions, however, hardly affect the mass or radius of gyration of the large particle (inset in Figure 1d). When all PPs have the same size, even the formation of doublets has a strong effect on the particle mass. So the broader the PPSD, the longer it takes until large compact particles are affected by collisions with smaller ones to form fractal-like structures, thus delaying the attainment of asymptotic Df. For a given np̅ range (n̅p = 100− 1000), Df decreases with increasing PP polydispersity. This is also consistent with Figure 4a of Eggersdorfer and Pratsinis,12 who obtained the Df of agglomerates consisting of np = 16, 64, 256, 512 and 1024 PPs with σg,PP = 1−3 and np̅ ≈ 169. At such n̅p, however, agglomerates (especially those consisting of highly polydisperse PPs) have not yet reached their asymptotic Df. The minimum or crossover number of PPs in an agglomerate required to obtain fractal scaling, nc,s, increases with increasing polydispersity. This is consistent with Figure 3 of Hamsy et al.,38 who showed for reaction-limited cluster−cluster agglomeration that larger agglomerates are needed to attain the asymptotic Df as the constituent PP polydispersity increases. Figures 1f and 2a show that agglomerates with polydisperse PPSD (σg,PP > 1) reach the asymptotic Df = 1.91 of BCCA (horizontal line, Figure 2a) only when they become sufficiently large. Therefore, only agglomerates containing more PPs than nc,s have attained the asymptotic Df. The minimum Df (dDf/dn̅p = 0) is used here to estimate nc,s; for example, nc,s ≈ 35 PPs for σg,PP = 1.5 (broken line) and nc,s ≈ 150 for σg,PP = 2.5 (doubledotted broken line). Table 2 (third column) summarizes the nc,s Table 2. Crossover number of PPs per Agglomerate by Structure, nc,s, and Radius, nc,r, from Df = 3 to the Asymptotic Df ≈ 1.91 for Ballistic Cluster−cluster Agglomerates with Polydisperse Primary Particles (PP) Having a log-normal Distributiona

Figure 2. Evolution of the (a) mass fractal dimension, Df, and (b) prefactor, kn, as a function of the geometric mean number of PPs per agglomerate, np̅ , during the coagulation−agglomeration of polydisperse PPs having a log-normal size distribution with σg,PP = 1−3 at 300 K with dp = 1 nm. The broader the distribution, the longer it takes to reach the asymptotic Df ≈ 1.91 ± 0.05 (horizontal line). The change in Df is an effective fit to the ensemble of agglomerates and represents the crossover from compact to fractal-like agglomerates. The kn has not yet converged for σg,PP = 2−3.

PP geometric standard nc,m (minimum n̅p at which Df deviation, σg,PP = 1.91 is attained) 1.0 1.5 2.0 2.5 3.0

(solid line), 1.5 (broken line), 2.0 (dotted−broken line), 2.5 (double-dotted−broken line), and 3.0 (dotted line) as a function of the geometric mean number of PPs per agglomerate, n̅p. The time scale (top axis) is shown for one simulation of agglomerates with monodisperse PPs, and the relationship between time and np̅ (averaged over multiple ED simulations) is shown for σg,PP = 1−3 in Supporting Information Figure S2. The insets of Figure 2a show pictorially the evolution of agglomerates consisting of highly polydisperse PPs (σg,PP = 3). The gray shadow around the lines of Figure 2 shows an example of the variability of Df and kn from multiple ED simulations. The solid horizontal line shows the asymptotic Df by BCCA. Regardless of the PP polydispersity, the same asymptotic Df = 1.91 is obtained over a long time, consistent with Tence et al.10 who showed that PP polydispersity does not affect the fractal dimension of agglomerates. Narrower PPSDs result in agglomerates with the asymptotic Df ≈ 1.91 ± 0.0337 faster (or with lower n̅p) than those with broader PPSDs (as shown also at higher temperatures, Supporting Information Figure S1a,b). With increasing PP polydispersity (σg,PP), the difference in size (and mass) between PPs increases. The small particles collide

2.4 8 17 30 42

nc,s (dDf/ dnp̅ = 0) 15 35 140 150 315

nc,r (rg = rm) 15 50 160 300 500

a

The agglomerate size required to exhibit fractal scaling increases with increasing polydispersity.

obtained from the minimum Df (Figure 2a) for agglomerates of log-normal PPSD with σg,PP = 1−3 and dp = 1 nm. The minimum n̅p required to attain Df = 1.91 for the first time, nc,m, is also shown (Table 2, second column) for comparison. Even though nc,m increases with PP polydispersity, similar to nc,s, it is almost an order of magnitude smaller. Strictly speaking, the mass−radius relation does not follow a power law during the evolution from initial Df = 3 to final asymptotic 1.91.39−41 So in that period, the exponent Df (or Dfm) and prefactor kn (or kfm) are extracted as fitting (effective) parameters from the mass−radius relation (eq 2 or 3). The crossover size can be determined as a function of PP polydispersity by fitting power laws to the initial Df = 3 and asymptotic Df = 1.91 regions, as indicated in Figure 1 (black dotted and solid lines, respectively). The intersection of these lines defines the crossover size, rg,c (or rm,c), and number (mc/ mp). This crossover size is important in setting a lower size limit to fit asymptotic scalings for the experimental determination of D

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Langmuir Df (or Dfm) by electron microscopy images (or mass-mobility measurements). Even though Df converges to its asymptotic value for BCCA regardless of PP polydispersity (Figure 2a), the asymptotic kn depends on PPSD. The prefactor kn of the mass fractal scaling (eq 2) is related to the density and packing fraction of fractallike particles (Heinson et al.,20 eq 11). Figure 2b shows the evolution of kn using the geometric mean PP radius of the PPSD for rp and mp in eq 2. Increasing PP polydispersity results in larger kn for a given np̅ , especially for np̅ > 10. A higher asymptotic kn corresponds to a higher packing fraction as m ∝ kn and kn increases with increasing PP polydispersity. Thus, agglomerates of polydipserse PPs carry more mass within the same rg than those consisting of monodisperse PPs. However, the kn in the crossover region is a fitting parameter because a bimodal structure distribution exists, especially for polydisperse PPs (Figure 1e), so the power law scaling does not hold there. Only the asymptotic kn allows us to draw conclusions about the cluster density or packing that can be obtained by fitting eq 2 to clusters larger than the crossover size. At n̅p = 103, Df and kn of σg,PP = 1 (solid line) and 1.5 (broken line) have reached their asymptotic values for the whole cluster ensemble, whereas broader PP distributions have not yet fully converged; i.e., some clusters are smaller than the corresponding crossover size to that σg,PP. The minima in Df and maxima in kn are due to the presence of both compact structures that still evolve and fractallike clusters that have attained their asymptotic Df and are more pronounced for broader PP size distributions. In contrast, such minima/maxima are nearly absent for monodisperse PPs. Figure 3 shows the evolution of effective mass-mobility (a) exponent Dfm and prefactor, kfm, with time (top axis) and np̅ .

with growing agglomerates, similar to Figure 6a of Eggerrsdorfer et al.42 Collisions between large and small particles steadily increase the rotationally averaged projected area, monotonically decreasing the Dfm. For all PPSDs, Dfm decreases from 3 to the limit, 2.15 (red horizontal line5). In contrast, kfm (Figure 3b) increases steadily with growing agglomerates, e.g., PP number per agglomerate np̅ . Although agglomerates reach the same asymptotic Dfm regardless of PP polydispersity (Figure 3a), the prefactor kfm depends on σg,PP and for a fixed np̅ (or mass) might offer the possibility to obtain σg,PP from experimental mass-mobility data. It should be noted that kfm continuously increases with growing agglomerates, approaching a nearly asymptotic value (np̅ > 1000) somehow similar to kn that seems to approach an asymptotic value, at least, for σg,PP ≤ 2 (Figure 2b). 3.2. Comparison with Experiments. Figure 4 shows the Dfm measured by a differential mobility analyzer (DMA) and

Figure 4. ED-obtained Dfm for agglomerates consisting of monodisperse (solid line) and polydisperse PPs with σg,PP = 1.5 (broken line) and 3 (dotted line). The Dfm measured by a differential mobility analyzer (DMA) and aerosol particle mass (APM) analysis (symbols) increases with increasing relative particle density ρrel = ρeff/ρb. The experiments (symbols) show reasonable agreement with simulations for σg,PP = 1.5 that is traced to the self-preserving size distribution of agglomerates. The agglomerate consists of one spherical particle (Dfm = 3) once ρrel = 1, and Dfm approaches its asymptotic value when fractal-like agglomerates are formed (ρrel → 0).

aerosol particle mass (APM) analyzer (symbols) along with ED simulations for σg,PP = 1 (solid line), 1.5 (broken line), and 3 (dotted line). For fully evolved fractal-like agglomerates, the relative density is low (ρeff/ρb < 0.3) and the same asymptotic Dfm ≈ 2.15 is attained by ED, redardless of the PP polydispersity, as discussed in Figures 3a and 2a. At such low relative densities, the ED-derived Dfm is in excellent agreement with that of flame-made agglomerates (triangles43 and diamonds44) that typically have a PP polydispersity of about 1.45 as a result of the attainment of a self-preserving distribution for coagulating spheres.45,46 For example, flamemade ZrO2 nanoparticles generated by a 2 mL/min precursor flow rate atomized with 5 L/min O2 have a PP polydispersity from 1.33 (filled diamonds) to 1.53 (open diamonds), depending on the height above the burner as obtained by counting ∼3000 and ∼2600 PPs, respectively.44 The Dfm of these large fractal-like particles follows the evolution of the EDderived Dfm. At higher relative densities, Dfm increases as aggregates (sinter-bonded agglomerates) are formed, and once ρrel = 1, the aggregate consists of one spherical particle (Dfm = 3). Aggregates with a larger relative density (ρeff/ρb > 0.5) and a larger Dfm ≈ 2.6−3, such as Ag (squares47) and Cu (circles48), are typically produced at high temperatures. For these aggregates, the increase in Dfm is caused by the growing sinter necks among fewer PPs. It should be noted here that sampling

Figure 3. Evolution of the mass-mobility (a) exponent, Dfm, and (b) prefactor, kfm, as a function of the geometric mean number of PPs per agglomerate, n̅p. (a) The broader the distribution, the longer it takes to reach the asymptotic Dfm ≈ 2.15 ± 0.10 (horizontal line), and the corresponding prefactor kfm increases monotonically.

Dfm is obtained again by regressing eq 3 to the whole ensemble of agglomerates. Similar to Df, power laws exist for Dfm only for fractal-like structures much larger than the crossover size. In between, the so-called effective D fm is used because agglomerates on the order of the crossover size do not exhibit power law scaling. In contrast to Df, Dfm continuously decreases E

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Figure 6 shows the evolution of the number-based rg (a, c) and rm (b, d) distributions for σg,PP = 1 (a, b) and 3 (c, d) by

(symbols) takes place mostly in the transition and/or continuum regimes. However, the asymptotic Dfm hardly changes from the free molecular to the continuum regime, as shown for monodisperse PPs (Goudeli et al.,44 Figures 2a and 8). 3.3. Agglomerate Radii and Size Distributions. Figure 5 shows the evolution of the geometric mean rg (red lines), rm

Figure 6. Snapshots of rg- (a, c) and mobility radius rm-based (b, d) distributions of agglomerates during coagulation in the free molecular regime consisting of PPs with σg,PP = 1.0 (a, b) and 3.0 (c, d). The size distribution of agglomerates consisting of monodisperse PP continuously broadens until it attains the agglomerate self-preserving size distribution (SPSD) of (a) σg,g = 2.23 and (b) σg,m = 2.03 at t > 10−3 s. The agglomerate rg-based size distribution consisting of agglomerates with σg,PP = 3 broadens initially, and its rm-based SD narrows. Both distributions approach the agglomerate SPSD with σg,g and σg,m values identical to those attained with monodisperse PPs.

Figure 5. Evolution of agglomerate geometric mean radius of gyration, rg (red lines), mobility, rm (blue lines), and volume-equivalent radius, rv (black lines), as a function of n̅p for agglomerates having PPs of σg,PP = 1.5−3 (a−d). nc,r corresponds to the geometric mean number of PPs per agglomerate when rg = rm (yellow circles) and increases with increasing σg,PP. For agglomerates consisting of less than nc,r PPs, the three agglomerate radii do not differ significantly, but for larger n̅p, rg exceeds rm while both rg and rm become considerably larger than rv.

coagulation at t = 10−5 (circles), 10−4 (triangles), and 10−3 s (squares). For monodisperse PPs, the agglomerate σg,g and σg,m increase up to about 2.23 and 2.05, respectively, at t = 10−3 s, approaching their SPSD values of 2.27 and 2.03, respectively, in the free molecular regime, in agreement with Goudeli et al.8 For a PPSD with σg,PP = 3, the rg distribution at t = 10−4 s becomes even broader (σg,g = 3.28) than the PPSD (triangles, Figure 6c). This broadening is caused by the large agglomerates that have attained their fractal-like structure (Figure 1e), whereas the small and more compact ones have hardly changed their rg size. Later on (t = 10−3 s), the σg,g of agglomerates with polydisperse PPs decreases (σg,g = 2.50) and eventually approaches the self-preserving σg,g of monodisperse PP (Figure 6a). In contrast to σg,g, the rm distribution becomes narrower during agglomeration than the initial PPSD for σg,PP = 3 (at t = 10−5 s, σg,m = 2.11; Figure 6d), but at longer times (t = 10−3 s), it increases (σg,m = 1.99), also approaching the SPSD for BCCA of σg,m = 2.03 (Figure 6b). The difference in the evolution of the two distributions of the same agglomerate population highlights the significance of a consistent description of agglomerate dynamics in terms of their radii. Figure 7 summarizes the dynamics of Figure 6 by showing the coagulation dynamics of agglomerates consisting of PPs of σg,PP = 1−3 in terms of the geometric standard deviation of the agglomerate (a) gyration, (b) mobility and (c) volumeequivalent radii. The symbols indicate σg,g and σg,m of the distributions shown in Figure 6 at t = 10−5 (circles), 10−4 (triangles), and 10−3 s (squares) for σg,PP = 1 (filled symbols) and 3 (open symbols). Coagulating agglomerates eventually attain the same asymptotic SPSDs regardless of their PPSD. The agglomerates of narrower PPSDs reach their rg-based SPSD of σg,g = 2.27 faster (Figure 7a), but the PPSD polydispersity hardly affects the time needed to reach the rm-

(blue lines), and volume-equivalent radii rv (black lines) for agglomerates with identical geometric mean rp for σg,PP of (a) 1.5, (b) 2, (c) 2.5, and (d) 3 that should be compared to that for monodisperse PPs in Figure 3a of Goudeli et al.8 Initially, agglomerates consist of one PP, so rm and rv are practically the same and rg is slightly smaller (i.e., rg = 3/5 rm for perfect spheres) than the actual particle radius. 49 For highly polydisperse PPs, this early difference between rg and rm/rv becomes even more pronounced (e.g., Figure 5d at np̅ = 4) for σg,PP = 3. The rm as obtained from Sorensen5 (eq 24; Figure 5a, solid line) for monodisperse PPs in the free molecular regime is smaller than that of agglomerates consisting of polydisperse PPs. When agglomeration starts, rm exceeds rg as the coagulation between uneven particles affects the agglomerate cross-sectional area (and rm) more than it affects rg. This is the case until agglomerates attain a crossover size nc,r when rg = rm. This is similar to the crossover size nc,s. For larger agglomerates, there is little difference between rg and rm, though both become considerably larger than rv as a result of the decreasing relative particle density with increasing np̅ , consistent with Goudeli et al.8 It is worth noting, however, that nc,r increases with increasing PP polydispersity comparably to the crossover PP number, nc,s, determined by the minimum Df (Table 2, fourth and third columns, respectively), especially for σg,PP ≤ 2. These definitions of the critical np required to attain a fractal-like agglomerate are consistent with that of Sorensen,5 who suggested that large and small agglomerates can be separated by np = 100. F

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ment of the SPSD. The evolution of the volume-based σg,v is presented in Figure 7c. Although the agglomerates consisting of a broader PP, σg,PP, have more mass (Figure 1), their SPSD has the same shape, i.e., σg,v. As with the coagulation of spherical particles (Heine and Pratsinis,32 Figure 2b), the mass-based SPSD is attained faster than the number-based ones and even faster than the asymptotic structure (Figures 2a, 3a, and S2). 3.4. Simplified Agglomerate Dynamics. Figure 8 shows the effect of PP polydispersity (σg,PP = 1−3) on the overall

Figure 7. Coagulating agglomerates of PPs with various geometric standard deviations (σg,PP = 1−3) reach the same self-preserving size distribution (SPSD) that has number-based geometric standard deviations of the (a) radius of gyration σg,g = 2.27 and (b) mobility radius σg,m = 2.03, and (c) solid volume-based geometric standard deviation σg,v = 1.41. The mass- or volume-based distribution is attained much faster than number-based ones, even ahead of the attainment of asymptotic Df.

Figure 8. Evolution of the free molecular collision frequency function, βn, of agglomerates consisting of monodisperse PPs (Thajudeen et al.,4 solid green line) and of various polydispersities (σg,PP = 1−3) by the present ED (black lines). The primary particle size polydispersity has no effect on βn once the agglomerate self-preserving size distribution and fractal-like structure are attained (e.g., t = 4 × 10−4 s). Also, the evolution of a simplified βn is shown for quasi-monodisperse spherical particles (Fuchs,51 orange broken line) and agglomerates having the asymptotic structure (Df = 1.91 and kn = 1.4) and consisting of monodisperse PPs based on agglomerate N (eq 6b, blue dotted− broken line). Enhancing the latter by 35% (accounting for the SPSD polydispersity) brings its predictions into excellent agreement with those of detailed ED simulations once the SPSD has been attained (red dotted line).

based self-preserving σg,m = 2.03 (Figure 7b) of Goudeli et al.8 This indicates that the shape and width of the SPSD by agglomeration are determined solely by the collision mechanism: coagulating agglomerates always reach the same selfpreserving σg,g = 2.27, σg,m = 2.03, and σg,v = 1.41 (Figure 7), whereas fully coalescing aerosols attain σg,g = σg,m = 1.46 and σg,v = 1.33.45 The time required to reach the rg-based SPSD slightly increases, however, with increasing PP polydispersity, similar to fully coalescing aerosols with a broader-than-SPSD initial size distribution (Vemury et al.,46 Figure 4). As shown in Figure 6c, these rg-based agglomerate distributions become broader than their PPSD and exhibit a maximum above the asymptotic σg,g as the small tail takes more time to develop the self-preserving shape than the large tail consisting of fractal-like clusters (Figure 6), as with spherical particles.50 As discussed already, this overshoot is caused by the compact particles (Figure 1d, inset) even after several collisions with smaller particles and constitutes the small tail of the distribution (e.g., at t = 10−3 s in Figure 6c). Such compact particles have a small cross-sectional area and a low thermal velocity (eq 1) due to the large mass additionally reducing their collision probability. This trend is not observed for the mobility-based σg,m (Figure 7b). The mobility radius, rm, is based on the projected area (eq 3) and thus depends only on the projected cross-sectional surface area and to a lesser extent on the spatial distribution of the particles within an agglomerate compared to rg. Although the collision between a large and a small particle hardly affects rg, the rm (or projected area aa) is more sensitive, and each collision contributes to the develop-

collision frequency, βn (black lines), which is calculated by eq 4 from the ED simulations. As expected, broader PPSD results in an initially higher βn as collisions between larger and small particles are more likely than between equally sized ones.1 The evolution of βn for agglomerates with monodisperse PPs (solid black lines) is in agreement with that given by the detailed βn (solid green line) of Thajudeen et al.4 (eqs 21 and 22) in the early stages of agglomeration (t ≤ 10−4 s). The slight deviation of the latter from the ED-derived βn at longer times (t > 10−3 s) could be attributed to its employment of rather small agglomerates (np ≤ 50) because the average agglomerate size already contains 300 PPs at t = 10−3 s (Figure 2a). Monodisperse PPs (black solid line) form agglomerates that attain the asymptotic fractal-like structure earlier than polydisperse PPs (Figure 2a). This results in a larger collisional area for coagulation and a stronger increase in the overall collision frequency. So the βn from coagulating agglomerates of narrower σg,PP increases faster (at t ≈ 10−5 − 4 × 10−4 s) than that from broader σg,PP until all have nearly attained the SPSD (t > 4 × 10−4 s), e.g., Figure 7b, and converge to a single βn. From then on, agglomeration kinetics become independent of PPSD. The attainment of SPSD greatly facilitates the description of particle dynamics by neglecting the spread of the agglomerate and PP size distribution through monodisperse models.25 Such models can be readily incorporated into computational fluid dynamics for optimal reactor design.23 Having the above EDderived βn gives the opportunity to assess various monodisperse collision frequencies for such efficient simulations of agglomG

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Langmuir erate dynamics. Therefore, the evolution of ED-derived βn is compared to the collision frequency of monodisperse spherical particles (Df = 3, kn = 1; Fuchs,50 eq 5b, thick orange broken line) and agglomerates (Df = 1.91, kn = 1.41) based on their ED-obtained N (eq 6b: blue dotted−broken line) employed in monodisperse coagulation equations, assuming monodisperse PP and agglomerate size distribution.25 Initially (t < 10−5 s), all coagulation rates are in good agreement with ED simulations of agglomerates with monodisperse PPs (black solid line) because mostly spherical or compact particles are present (e.g., even for agglomerates, few have been formed). As expected, for coagulation under full coalescence (eq 5b), the βn is the lowest (orange broken line). The coagulation of agglomerates in the free molecular regime proceeds much faster than that of equivalent spheres as a result of the enhanced collisional area of agglomerates.6 This difference increases with time as the collision radius of agglomerates increases. The theoretical βmono (eq 6b) at t < 2 × 10−6 s, however, is slightly lower than that of ED simulations because it is calculated with constant Df = 1.91 and kn = 1.4. βmono (eq 6b) represents the coagulation of monodisperse agglomerates that consist of monodisperse PPs, and the EDobtained results for σg,PP = 1 represent the coagulation of polydisperse agglomerates consisting of monodisperse PPs. So once the SPSD is attained (t ≥ 4 × 10−4 s), the overall collision frequency of the SPSD by ED is 40% higher (t = 4 × 10−4 s) than the theoretical monodisperse kernel (eq 6b) of agglomerates with Df = 1.91 as a result of the agglomerate polydispersity and 30% higher than the detailed agglomeration kernel of Thajudeen et al.4 (green solid line). The present ED simulations can be used to develop a simplified but consistent collision frequency function for agglomeration. Here, the SPSD for coagulation−agglomeration with σg,v = 1.41 (Figure 7c) results in a 35% enhancement of the overall collision frequency (eq 7b: a ≈ 7.63 for Df = 1.91 vs 6.67 for spheres1), compared to the monodisperse βmono. Thus, eq 7a becomes

βSPSD = 1.35βmono

the PP polydispersity or simply on the corresponding Df that strongly varies with σg,PP as shown in Figure 2a. Therefore, accurate modeling of agglomerate dynamics requires knowing the evolution of Df (along with the βn) for a given PP polydispersity until the well-known asymptotic Df is attained.

4. CONCLUSIONS The effect of primary particle (PP) polydispersity on the structure, growth, and size distribution of coagulating agglomerates is investigated in the free molecular regime. Increasing the geometric standard deviation of the constituent PPs from σg,PP = 1 to 3 does not affect but only delays the attainment of the asymptotic mass fractal dimension, Df, and mass-mobility exponent, Dfm, of the resulting agglomerates. In contrast, the corresponding fractal, kn, and mass mobility, kfm, prefactors increase with increasing polydispersity. The attainment of asymptotic values by kn and kfm is delayed with increasing polydispersity but could be used to estimate the PP polydispersity from appropriate experimental data. The crossover agglomerate size and critical number of PPs per agglomerate that mark the attainment of Df = 1.91 increase with increasing PP polydispersity. Such crossover sizes are crucial for the analysis of experimental data of agglomerate structures because they determine at what aggomerate sizes standard power laws apply. Therefore, only agglomerates larger than the crossover size should be considered to ascertain their formation mechanism from their structure. Agglomerates smaller than the crossover size exhibit an exponent (Df or Dfm) that is typically larger than their asymptotic values (1.91 or 2.15) and might be either heavily sintered or consist of agglomerates that are bimodal in size that do not follow the asymptotic power law scaling, e.g., small, fully compact particles and ramified larger ones. Increasing the PP polydispersity slightly delays the attainment of the rg- but not the rm- or rv-based self-preserving size distributions (SPSD). Increasing the polydispersity of the PP size distribution accelerates the agglomeration rate. Once the SPSD is reached, agglomeration rates are identical because most agglomerates have attained the asymptotic (Df = 1.91 and Dfm = 2.15) structure. Then, increasing the coagulation rate of monodisperse agglomerates25 by 35% captures the asymptotic collision frequency function well in the free molecular regime, regardless of the PP size distribution. This 35% enhancement is traced to the enhancement of the monodisperse coagulation rate by the polydispersity of the agglomerate SPSD in that regime.

(8)

nicely describing the evolution of asymptotic βn in the free molecular regime (red dotted line), regardless of PP polydispersity, and can be used in monodisperse models. So enhancing the collision rate of monodisperse agglomerates by 35% brings its predictions into excellent agreement with those of detailed ED simulations once the SPSD has been attained (here t ≥ 4 × 10−4 s). Initially (t < 5 × 10−5 s), eq 8 closely follows the collision rate of Thajudeen et al.,4 and at t > 4 × 10−4 s, eq 8 converges to the asymptotic ED-derived βn. The enhancement calculated by eqs 7a and 7b is similar to that for the coagulation of spheres where the βn of polydisperse ones is 22% higher than that of monodisperse ones.52 Equation 8 holds for agglomerates coagulating in the free molecular regime regardless of mean PP size (e.g., for dp = 5 and 10 nm in Figure S3). Having the agglomerate N from dN/dt = −βSPSD × N(t)2/ 2 and a mass balance,25 one obtains the average size and can reconstruct the actual size distribution using the SPSD (Goudeli et al.,8 Table 1). However, applying simple monodisperse β models, such as eq 8, in population balance equations of realistic agglomerates with polydisperse PPs requires a knowledge of the critical np because only when np is attained does the agglomerate βn become independent of its PP polydispersity. For agglomerates consisting of fewer PPs than the critical np, the βn depends on

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b02455. Evolution of the mass fractal dimension and the prefactor as function of the arithmetic mean number of PPs per agglomerate, evolution of the geometric mean number of PPs per agglomerate with time, and ED-obtained evolution of βn of coagulating agglomerates with polydisperse PPs (PDF) H

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(3) Mulholland, G. W.; Samson, R. J.; Mountain, R. D.; Ernst, M. H. Cluster size distribution for free molecular agglomeration. Energy Fuels 1988, 2, 481−486. (4) Thajudeen, T.; Gopalakrishnan, R.; Hogan, C. J., Jr. The collision rate of non-spherical particles and aggregates for all diffusive Knudsen numbers. Aerosol Sci. Technol. 2012, 46, 1174−1186. (5) Sorensen, C. M. The mobility of fractal aggregates: a review. Aerosol Sci. Technol. 2011, 45, 765−779. (6) Vemury, S.; Pratsinis, S. E. Self-preserving size distributions of agglomerates. J. Aerosol Sci. 1995, 26, 175−185. (7) Mandelbrot, B. B. The Fractal Geometry of Nature; W.H. Freeman: San Francisco, 1982. (8) Goudeli, E.; Eggersdorfer, M. L.; Pratsinis, S. E. Coagulation − Agglomeration of Fractal-like Particles: Structure and Self-Preserving Size Distribution. Langmuir 2015, 31, 1320−1327. (9) Scheckman, J. H.; McMurry, P. H.; Pratsinis, S. E. Rapid characterization of Agglomerate Aerosols by In Situ Mass-Mobility Measurements. Langmuir 2009, 25, 8248−8254. (10) Tence, M.; Chevalier, J. P.; Jullien, R. On the measurement of the fractal dimension of aggregated particles by electron microscopy: experimental method, corrections and comparison with numerical models. J. Phys. (Paris) 1986, 47, 1989−1998. (11) Bushell, G.; Amal, R. Fractal aggregates of polydisperse particles. J. Colloid Interface Sci. 1998, 205, 459−469. (12) Eggersdorfer, M. L.; Pratsinis, S. E. The Structure of Agglomerates Consisting of Polydisperse Particles. Aerosol Sci. Technol. 2012, 46, 347−353. (13) Haywood, J. M.; Shine, K. P. Multi-spectral calculations of the direct radiative forcing of tropospheric sulphate and soot aerosols using a column model. Q. J. R. Meteorol. Soc. 1997, 123, 1907−1930. (14) Trypolskyi, A. I.; Gurnyk, T. M.; Strizhak, P. E. Fractal dimension of zirconia nanopowders and their activity in the CO oxidation. Catal. Commun. 2011, 12, 766−771. (15) Keskinen, H.; Tricoli, A.; Marjamäki, M.; Mäkelä, J. M.; Pratsinis, S. E. Size-selected agglomerates of SnO2 nanoparticles as gas sensors. J. Appl. Phys. 2009, 106, 084316. (16) Berry, M. V. Falling fractal flakes. Phys. D 1989, 38, 29−31. (17) Dobbins, R. A.; Mulholland, G. W.; Bryner, N. P. Comparison of Fractal Smoke Optics Model with Light Extinction Measurements. Atmos. Environ. 1994, 28, 889−897. (18) Aoki, R.; Suzuki, M. Effect of particle shape on the flow and packing properties of non-cohesive granular materials. Powder Technol. 1971, 4, 102−104. (19) Bałazy, A.; Podgórski, A. Deposition efficiency of fractal-like aggregates in fibrous filters calculated using Brownian dynamics method. J. Colloid Interface Sci. 2007, 311, 323−337. (20) Heinson, W. R.; Sorensen, C. M.; Chakrabarti, A. A three parameter description of the structure of diffusion limited cluster fractal aggregates. J. Colloid Interface Sci. 2012, 375, 65−69. (21) Charalampopoulos, T. T.; Shu, G. Effects of polydispersity of chainlike aggregates on light-scattering properties and data inversion. Appl. Opt. 2002, 41, 723−733. (22) Tong, Z. B.; Yang, R. Y.; Chu, K. W.; Yu, A. B.; Adi, S.; Chan, H. K. Numerical study of the effects of particle size and polydispersity on the agglomerate dispersion in a cyclonic flow. Chem. Eng. J. 2010, 164, 432−441. (23) Johannessen, T.; Pratsinis, S. E.; Livbjerg, H. Computational fluid-particle dynamics for the flame synthesis of alumina particles. Chem. Eng. Sci. 2000, 55, 177−191. (24) Zhang, R.; Khalizov, A. F.; Pagels, J.; Zhang, D.; Xue, H.; McMurry, P. H. Variability in morphology, hygroscopicity, and optical properties of soot aerosols during atmospheric processing. Proc. Natl. Acad. Sci. U. S. A. 2008, 105, 10291−10296. (25) Kruis, F. E.; Kusters, K. A.; Pratsinis, S. E.; Scarlet, B. A simplemodel for the evolution of the characteristics of aggregate particles undergoing coagulation and sintering. Aerosol Sci. Technol. 1993, 19, 514−526.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Ph. +41 (0) 44 632 31 80. Fax. +41 (0) 44 632 15 95. Author Contributions

E.G. and M.L.E. contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Swiss National Science Foundation (grant no. 200021_149144), ETH Research Grant (ETHIIRA) ETH-11 09-1, and the European Research Council under the European Union’s Seventh Framework Programme (FP7/ 2007-2013)/ERC grant agreement no. 247283 is gratefully acknowledged.

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NOMENCLATURE Df/fm mass-based scaling exponent for eqs 2 and 3 kB Boltzmann’s constant, m2 kg s−2 K−1 kn/m mass-based prefactor for eqs 2 and 3 m mass, kg N number concentration, number m−3 n number na agglomerate number np primary particle number r radius, m rp primary particle radius, m rg radius of gyration, m rm mobility radius, m T temperature, K t time, s u velocity, m s−1 v particle/agglomerate volume, m3 Greek letters

α β η ρ σg,g

σg,m σg,PP σg,v τ ϕ Ψ

dimensionless collision integral collision frequency, m3 s−1 dimensionless volume particle density, kg m−3 number-based geometric standard deviation of the radius of gyration distribution number-based geometric standard deviation of the mobility size distribution number-based geometric standard deviation of the primary particle size distribution volume-based geometric standard deviation time, s volume fraction dimensionless concentration

subscripts

0 c fm eff g

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initial crossover free-molecular regime effective geometric averaging

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J

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