Coagulation–Agglomeration of Fractal-like Particles: Structure and

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Coagulation−Agglomeration of Fractal-like Particles: Structure and Self-Preserving Size Distribution Eirini Goudeli, Maximilian L. Eggersdorfer,† and Sotiris E. Pratsinis* Particle Technology Laboratory, Institute of Process Engineering, Department of Mechanical and Process Engineering, Eidgenössische Technische Hochschule Zürich, Sonneggstrasse 3, CH-8092 Zürich, Switzerland S Supporting Information *

ABSTRACT: Agglomeration occurs in environmental and industrial processes, especially at low temperatures where particle sintering or coalescence is rather slow. Here, the growth and structure of particles undergoing agglomeration (coagulation in the absence of coalescence, condensation, or surface growth) are investigated from the free molecular to the continuum regime by discrete element modeling (DEM). Particles coagulating in the free molecular regime follow ballistic trajectories described by an event-driven method, whereas in the near-continuum (gas-slip) and continuum regimes, Langevin dynamics describe their diffusive motion. Agglomerates containing about 10−30 primary particles, on the average, attain their asymptotic fractal dimension, Df, of 1.91 or 1.78 by ballistic or diffusion-limited cluster−cluster agglomeration, corresponding to coagulation in the free molecular or continuum regimes, respectively. A correlation is proposed for the asymptotic evolution of agglomerate Df as a function of the average number of constituent primary particles, np̅ . Agglomerates exhibit considerably broader self-preserving size distribution (SPSD) by coagulation than spherical particles: the number-based geometric standard deviations of the SPSD agglomerate radius of gyration in the free molecular and continuum regimes are 2.27 and 1.95, respectively, compared to ∼1.45 for spheres. In the transition regime, agglomerates exhibit a quasi-SPSD whose geometric standard deviation passes through a minimum at Knudsen number Kn ≈ 0.2. In contrast, the asymptotic Df shifts linearly from 1.91 in the free molecular regime to 1.78 in the continuum regime. Population balance models using the radius of gyration as collision radius underestimate (up to about 80%) the small tail of the SPSD and slightly overpredict the overall agglomerate coagulation rate, as they do not account for cluster interpenetration during coagulation. In the continuum regime, when a recently developed agglomeration rate is used in population balance equations, the resulting SPSD is in excellent agreement with that obtained by DEM.

1. INTRODUCTION Agglomeration is the formation of clusters of primary particles by coagulation without such particle deformation (e.g., fusion or sintering or coalescence). It occurs in industrial (e.g., gasphase particle synthesis, flocculation, and fluidization) and natural1 (smog formation, volcanic ash transport) processes, especially at low temperatures, resulting in filamentary structures. Such loose structures are attractive in nanocomposites and particle suspensions in paints or microelectronic polishing slurries.2 Agglomeration also takes place during flocculation and clearing of liquid suspensions3 and with atmospheric aerosols having fractal-like structure, like diesel soot4 and fly ash.5 Such aerosols have a direct effect in air pollution by absorbing and scattering light6 or an indirect effect by changing cloud properties7 and lifetimes.8 The spread of the agglomerate size distribution strongly affects aerosol transport9 and behavior,4 including optical properties10,11 related to aerosols’ radiative characteristics. For example, scattering of solar radiation limits the visibility in the troposphere and increases the Earth’s albedo.12 Understanding agglomerate dynamics can facilitate optimal process design for aerosol synthesis of materials, flocculation of suspensions, coagulation of colloids, and monitoring of combustion © XXXX American Chemical Society

emissions and atmospheric aerosols. Also, agglomerate morphology and structure affect handling and processing and eventually agglomerate performance,13 while mobility determines agglomerates’ transport and dispersal properties.5 Even though Brownian coagulation is reasonably well understood for fully coalescing particles at both dilute14 and high concentrations,15,16 little is known about the dynamics of agglomerates during coagulation, especially in the transition regime where the gas mean free path is comparable to the agglomerate radius. Vemury and Pratsinis17 showed the attainment of self-preserving size distributions (SPSDs) by coagulation of agglomerates with constant Df in the free molecular and continuum regimes. Dekkers and Friedlander18 showed by Monte Carlo (MC) simulations that such agglomerates attain a narrower quasi-SPSD in the transition regime. Kostoglou and Konstandopoulos19 showed by bivariate MC simulations that agglomerates of different Df approach the above SPSDs by Brownian coagulation at a rate that increases with decreasing number of initial primary particles per Received: October 31, 2014 Revised: December 19, 2014

A

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np > 100.5,13 Large DLCA-made agglomerates have34 kfm = 1.11 ± 0.01 and Dfm = 2.15 ± 0.003. In the free molecular regime, Dfm becomes asymptotically5 2.17 ± 0.10. 2.2. Numerical Implementation. Agglomerate formation by coagulation is investigated with the assumption that every particle is in thermal equilibrium with the surrounding fluid.35 Thus the particle velocity has a Maxwellian distribution, and particle trajectories are calculated36 neglecting van der Waals, electric,37 and hydrodynamic interactions. Two thousand spherical particles are randomly distributed in a cubic cell at 300 K with periodic boundary conditions.15 Once two particles collide (with a sticking probability of unity), they form a cluster of primary particles by agglomeration or a sphere by full coalescence. These collisions reduce continuously the number of particles in the simulation cell. When that number is halved, the cell volume is doubled15 consecutively in the x-, y-, and zdirections so that the number of particles always lies between the initial number, N0, and 0.5N0. The velocity of each particle in the newly generated volume is reversed to conserve momentum. The time between successive collisions is calculated by an eventdriven (ED) method38 in the free molecular and transition regimes and by Langevin dynamics (LD) in the near-continuum (gas-slip) and continuum regimes. The trajectory of the newly formed clusters or spheres is calculated by use of the Stokes friction coefficient with the Cunningham correction factor describing the drag force from the free molecular to the continuum regime. A particle changes direction once it collides with another particle or has traveled its persistence length (stopping distance). Here, particle coagulation is investigated at full coalescence (for validation) and pure agglomeration (no sintering, fusion, or coalescence). The overall collision frequency is numerically obtained from the free molecular to the continuum regime by DEM as15

agglomerate. Mitchell and Frenklach20 showed by MC simulations that agglomerates attain a power law when they contain 10−100 primary particles. Recently, Thajudeen et al.21 have proposed a rate for agglomeration from the free molecular to continuum regime by MC simulations. Here the detailed asymptotic structure and size distribution of fractal-like particles undergoing Brownian coagulation (monodisperse primary particles in point contact) in the absence of coalescence, sintering, or surface growth is investigated by discrete element modeling (DEM) and highly accurate population balances in the entire particle size range. The DEM simulations are validated by comparison to the coagulation rate14 and SPSD17 of spherical particles as well as to the attainment of the well-known fractal particle structures, Df, by ballistic22 and diffusion-limited23 cluster−cluster agglomeration. The purpose of this study is to quantify the evolution of particle structure from spherical to fractal-like agglomerates by coagulation as well as the transition of the asymptotic fractallike structure and the corresponding quasi-SPSD from the free molecular to the continuum limit. That way agglomerate particle dynamics can be described by single-moment particle models (e.g., monodisperse24) in detailed fluid−particle simulations upon completion of new particle formation without the need of population balances. Then the detailed size distributions can be recovered from the average size and the corresponding SPSD geometric standard deviations for aerosol reactor design,25 comparison to experimental data,26 and assessment of the significance of various particle growth pathways. Thus the average minimum number of primary particles per agglomerate, n̅p, for attainment of fractal structure is investigated, and a simple expression relating Df to np̅ is proposed. The transition of the asymptotic Df from the free molecular to the continuum regime is given by a linear relationship in terms of the agglomerate diffusive Knudsen number. Attainment of the agglomerate SPSD is investigated and its geometric standard deviation is given in terms of agglomerate mobility, gyration, and volume-equivalent radii.

βsim = 2

βFuchs,i , j

1 N1

t 2 − t1

(3)

−1 ⎡ dc,i + dc,j 8(Di + Dj) ⎤ ⎢ ⎥ = βco + cr̅ (dc,i + dc,j) ⎥⎦ ⎢⎣ dc,i + dc,j + 2δr

(4)

where βco is the collision frequency function in the continuum regime with

⎛ dc,i dc,j ⎞ βco = 4π ⎜ + ⎟(Di + Dj) 2 ⎠ ⎝ 2

2.1. Agglomerate Structure. Particles that stick upon collision without any coalescence form agglomerates.27 Depending on collision mechanism and duration, these agglomerates form fractals that exhibit a power law between their radius of gyration, rg, and constituent number of primary particles, np:28,29

(5a)

and δ12 + δ2 2

δr = (1)

δi =

where kn is a proportionality constant (lacunarity) related to the agglomerate shape30 and rp is the primary particle radius. In ballistic cluster−cluster agglomeration (BCCA) that corresponds to coagulation in the free molecular regime, kn = 1.4 and Df = 1.91 ± 0.03,22 while in diffusion-limited cluster−cluster agglomeration (DLCA) that corresponds to coagulation in the continuum regime, kn = 1.3 and Df = 1.78 ± 0.05.23 Typically, rg is the collision radius31 defined as the rootmean-square radius of the fractal-like agglomerate.32 The agglomerate mobility radius, rm, scales with np by the massmobility exponent, Dfm:33

⎛ r ⎞ Dfm np = k fm⎜⎜ m ⎟⎟ ⎝ rp ⎠

−

where N1 and N2 are the particle number concentrations at times t1 and t2, respectively, and compared to that by Fuchs:14

2. THEORY

⎛ rg ⎞ Df np = k n⎜⎜ ⎟⎟ ⎝ rp ⎠

1 N2

Di =

kBTCs,i 3πμdm

(5b)

1 ⎣⎡(dc,i + λα,i)3 − (dc,i 2 + λα,i 2)3/2 ⎦⎤ − dc,i 3dc,i λα,i

λα,i =

8Di π ci̅

cr̅ =

c1̅ 2 + c 2̅ 2

(5c)

(5d)

ci̅ =

8kBT πmi

(5e)

Particles are initially monodisperse, having solid volume fraction ϕs = 10−4 at 1 atm and density of 2.2 g/cm3 corresponding to silica (fly ash or fumed silica). The DEM simulations are compared to those of a highly accurate population balance model39 for coagulation, using the radius of gyration as collision radius in eqs 4 and 5a−5e with vi+1/vi = 21/q and q = 4, where vi is the average volume of the ith section. For full coalescence, dc = dm in eqs 4 and 5a−5e.

(2)

where kfm and Dfm are the mass-mobility prefactor and exponent, respectively, that should not be equated to kn and Df since rg ≥ rm for B

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3. RESULTS AND DISCUSSION 3.1. Validation. Figure 1 shows the evolution of the DEMobtained collision frequency function (eq 3; black lines, three

Figure 1. Collision frequency function, β, of fully coalescing spheres as a function of Knudsen number (Kn = λair/rp). The DEM-derived β values (black lines) closely follow the Fuchs14 (red line) interpolation formula that bridges the free molecular and continuum regimes. The DEM-derived β (eq 3) exhibits an enhancement from the free molecular to the continuum regime due to polydispersity of the attained self-preserving size distribution by coagulation. Figure 2. (a) Self-preserving size distributions (SPSDs) by coagulation−coalescence of spherical particles (Df = 3) in the free molecular and continuum regimes by discrete element modeling (DEM, symbols) are in agreement with population balance equation (PBE) simulations (lines).17 (b) Evolution of DEM-derived numberbased (σg,n) and volume-based (σg,v) geometric standard deviations (top and bottom clustered black solid lines, respectively) for coagulation of spheres as a function of gas Knudsen number (Kn = λair/rp). All particles reach the corresponding self-preserving σg,n and σg,v in the free molecular43 and continuum regimes,42,44 whereas in the transition regime σg,n follows Otto et al.45 (orange line).

separate simulations) of fully coalescing spherical particles upon collision as a function of Knudsen number Kn = λair/rp. The present βsim closely follows the Fuchs14 interpolation formula for coagulation of monodisperse particles (eq 4; red solid line). This DEM-derived βsim exhibits an enhancement of ∼18% in the free molecular regime (Kn ≈ 10−100) and about 7% in the continuum regime (Kn < 0.01) over the Fuchs interpolation, due to polydispersity of the rapidly attained SPSD of 17.9%40 in the free molecular regime and 6.5−7.34%41,40 in the continuum regime, at dilute conditions. The variability of the results is practically unaffected by the initial position or number (500− 3000) of particles in the simulation cell. Figure 2a shows the SPSD of coagulating particles undergoing full coalescence by DEM in the free molecular (triangles) and continuum regimes (circles) to be in good agreement with the SPSD of coagulating spherical particles (Df = 3) by Vemury et al.42 in the continuum (solid line) and free molecular (broken line) regimes. The results are shown in terms of the dimensionless concentration, Ψ = νn(ν, t)/N∞, and particle ̅ volume, η = ν/ν,̅ where N∞ = ∫ ∞ n(ν, t) dν is the total number 0 of particles per unit volume. There is some scatter at the smallest particle sizes (η < 0.01) due to low particle numbers there by DEM. Figure 2b shows the evolution of the geometric standard deviation of number-based (σg,n) and volume-based (σg,v) SPSDs of fully coalescing spherical particles (Df = 3, kn = 1) by DEM (top and bottom thin black lines, respectively) as a function of Knudsen number. At both regime limits, the DEMderived σg converges nicely to the corresponding SPSD in the free molecular43 and continuum regimes.42,44 In the transition regime, σg,n and σg,v exhibit minima of 1.35 and 1.27, respectively, at Kn ≈ 1 and nicely match the minimum of the evolving quasi-self-preserving geometric standard deviation there45 (orange line). The present DEM simulations are validated also by their attainment of the well-known fractal dimension by BCCA and DLCA. For both collision mechanisms the DEM-calculated Df are 1.89 ± 0.025 and 1.765 ± 0.019, in excellent agreement with Ball and Jullien22 and Jullien et al.,23 respectively, for

agglomerates consisting of monodisperse, noncoalescing primary particles. 3.2. Agglomerate Dynamics. Figure 3 shows the evolution of the agglomerate geometric mean radius of gyration rg (□), mobility radius rm (▽), and volume-equivalent radius rv (○) as a function of np̅ in the (a) free molecular and (b) continuum regimes, respectively. When agglomerates consist of only a few primary particles (n̅p less than 30 and 10 in the free molecular and continuum regimes, respectively) all radii do not differ significantly, consistent with Sorensen.5 Above these np̅ , there is little difference between the rg and rm that are significantly larger than rv as the effective density of agglomerates is decreasing with increasing np̅ . For n̅p > 100, however, rg becomes progressively larger than rm, consistent with the results obtained by Sorensen5 and Eggersdorfer and Pratsinis.13 For example, for fine fumed silica (A300 by Evonik)46 of np = 85 and dp = 9 nm, rg is about 44 nm, rm = 34 nm, and rv = 22 nm (as obtained for agglomerates with rp,0 = 5 nm; not shown here). In contrast, for coarser fumed silica (OX50, Evonik)46 of np = 5−6 and dp ≈ 55 nm, all agglomerate radii are equal (within 12% as obtained from DEM simulations for rp,0 = 30 nm; not shown here). So caution is needed when rg or rm is used as rv in materials process design or to relate emissions to ambient aerosol mobility measurements, especially for agglomerates composed of many primary particles (e.g., aged atmospheric aerosols). 3.3. Evolution of Fractal Dimension. Even though the asymptotic values of Df are well-known for a given collision C

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The transition of Df from 3 to that of DLCA, Df = 1.78 ± 0.05, is rather an average of various agglomerate structures, as these agglomerates are too small to exhibit a true power law scaling (e.g., n̅p = 2.4 at Df = 2.5 in Figure 4). Figure 5 shows the evolution of the above DEM-derived agglomerate (a) Df and (b) kn as a function of np̅ for various

Figure 3. Evolution of geometric mean radius of gyration rg (□), mobility radius rm (▽), and volume-equivalent radius rv (○) as a function of the number of primary particles per agglomerate n̅p, for primary particle radius rp,0 of (a) 1 nm and (b) 1 μm, corresponding to the free molecular and continuum regimes, respectively. For small agglomerates (n̅p < 10) there is hardly a difference between agglomerate radii. Error bars show the spread of the agglomerate self-preserving rg distribution.

Figure 5. Evolution of DEM-derived (a) effective fractal dimension Df and corresponding (b) power-law exponent kn of coagulating agglomerates as a function of the average number of primary particles per agglomerate n̅p. Df evolution to fractal-like particles for 2 < n̅p < 400 is approximated by eq 6 (black line). For np̅ > 400, Df = 1.8 with kn = 1.4 ± 0.2.

mechanism,22,23 little is known about the evolution of Df from spherical to fractal-like particles. Figure 4 shows snapshots of

monodisperse primary particles of rp,0 = 1, 2, 3, 7, 15, 30, 100, and 500 nm (average of five simulations for each primary particle size). Initially Df = 3 and kn = 1, as all particles are spheres. When agglomeration starts, n̅p increases and Df decreases, attaining its asymptotic value of ∼1.8 at np̅ = 10− 30 regardless of rp,0. This is in agreement with Mitchell and Frenklach,20 who reported attainment of a power law by 10− 100 agglomerating monodisperse particles. kn increases for np̅ up to about 20 for all rp,0 without reaching true asymptotic value as Df, but it can be bracketed by 1.4 ± 0.2 for np̅ > 10. There is small deviation of Df and kn for any set of agglomerates for a given primary particle radius. For example, the variation for rp,0 = 1 nm at n̅p = 2700 is Df = 1.799 ± 0.022 and kn = 1.523 ± 0.109. The above Df evolution from spherical particles to fractal-like agglomerates can be summarized quantitatively as a function of n̅p from Figure 5a (black line) as 1.9 ln 2 n p̅ − 2.6 ln n p̅ + 3 Df = ln 2 n p̅ − ln n p̅ + 1 (6)

Figure 4. Snapshots of coagulating agglomerate distributions in terms of their number of primary particles (with radius rp,0 = 15 nm) per agglomerate, np, and the corresponding radius of gyration rg at various times t (symbols). The slope corresponds to a distinct fractal dimension Df, pre-exponent factor kn (eq 1), and n̅p at each t. Power law dependence is attained in the range np = 10−10 000, when Df and kn change rather little.

for 2 < n̅p < 400. For n̅p ≥ 400, Df attains a constant asymptotic value of 1.8 for all rp,0 employed here. Equation 6 can be used to accurately model detailed particle dynamics where the evolving particle structure is typically neglected and constant Df = 1.8 is assumed.24 Figure 6 shows the evolution of the DEM-obtained asymptotic Df (black lines, five simulations) from the free molecular to the continuum regime as a function of diffusive Knudsen number KnD,m = λα/rm (ratio of agglomerate stopping distance to mobility radius). KnD,m is used to characterize the

the evolution of agglomerate distribution in the np−rg space having rp,0 = 15 nm. At any given time t, Df and kn are calculated from eq 1 with Df being the slope and kn the intercept in a log np−log rg plot. So initially when all particles are still spheres (squares, t = 10−5 s), Df = 3 and kn = 1 (black line). Df steadily decreases as t and n̅p increase (inset). At large enough t and n̅p (e.g., t > 10−3 s and np̅ > 19), Df and kn change rather little as they approach the well-known fractal-like structure by DLCA.23 D

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Figure 6. Evolution of mass fractal dimension Df as a function of the mobility radius-based diffusive Knudsen number (KnD,m = λα/rm) as obtained from DEM (black lines) from free molecular to continuum regime. In the free molecular limit, agglomerates attain Df of 1.91 ± 0.03 (green line) corresponding to the ballistic cluster−cluster agglomerate structures.22 This decreases linearly to 1.78 ± 0.05 (blue dotted line) in the continuum limit, corresponding to the structure of agglomerates formed by diffusion-limited cluster−cluster agglomeration.23 Blue and green highlighted regions represent the standard deviation in the two regimes. Evolution of Df in the transition regime (0.16 ≤ KnD,m ≤ 3) is fitted with eq 7 (red line).

Figure 7. Self-preserving size distributions (SPSDs) of coagulating agglomerates by DEM in the free molecular (△) and continuum regimes (○) are consistent with SPSDs from PBE models for Df = 1.91 and 1.78, using as coagulation rate either the Fuchs interpolation14 with rg as the collision radius51 (solid lines) or a recently developed one21 (dotted lines) in both regimes. The DEMobtained SPSDs are higher than PBE ones for smaller particles (η < 0.1) in the free molecular regime because the collision kernels neglect penetration of small into large agglomerates during coagulation. In contrast, in the continuum regime, the collision kernel of Thajudeen et al.21 accounts for the effect of particle interpenetration.

free molecular, continuum, and transition regimes for nonspherical particles47 and serves as a gauge of agglomerate size. Blue and green shaded regions in Figure 6 represent the standard Df variation in continuum and free molecular regimes, respectively, as discussed under Theory.22,23 Figure 6 shows only the asymptotic Df in each regime, as the power law of eq 1 is strictly valid for larger agglomerates48,49 (np ≥ 100).50 The DEM simulations (thin black lines) show that in the transition regime the Df rather linearly shifts from ∼1.91 to ∼1.78 from the free molecular to the continuum regime and can be approximated by eq 7 (Figure 6, red line):

Gotor and Rosner54 observed a reduced collision rate for Df = 1.8 and kn = 1.3 and attributed it also to interpenetration between colliding agglomerates of significantly different sizes. The SPSDs from PBE using a recently developed21 coagulation kernel (dotted lines), however, are in excellent agreement with DEM (○) in the continuum regime, pointing out its capacity to account for cluster interpenetration during agglomeration. In contrast, in the free molecular regime, PBEs using that kernel21 predict similar SPSDs to those using the Fuchs interpolation (eq 4) with rg as the collision radius, as these two kernels hardly differ (Figure S1, Supporting Information). In the continuum regime, the kernel by Thajudeen et al.21 is substantially larger (up to a factor of 3) than that by the Fuchs interpolation using rg as collision radius51 (Figure S1, Supporting Information). From the SPSDs of Figure 7, one can obtain SPSDs based on rg and rm in the free molecular and continuum regimes (Tables S1 and S2, Supporting Information) by use of eqs 1 and 2. The solid volume-based SPSD of agglomerates in the free molecular regime is broader than that of spherical particles (Figure 2a, broken line) while in the continuum regime they have about the same geometric standard deviation, consistent with the results obtained by Vemury and Pratsinis17 and Tandon and Rosner.55 Figure 8 shows the number-based geometric standard deviation (five simulations at each KnD,m range) of the DEMobtained SPSD based on the radius of gyration σg,g, mobility radius σg,m, and volume-equivalent radius σg,v, from free molecular to continuum regime as a function of KnD,m. σg,g is 10−15% larger than σg,m throughout the size spectrum. The self-preserving values of σg,g, σg,m, and σg,v in the free molecular regime are 2.27, 2.03, and 1.41, respectively. The last one is in good agreement with σg,v = 1.42 by Vemury and Pratsinis17 for fractal-like agglomerates of Df = 2, as the present one was calculated with22 Df = 1.91. σg,g and σg,m in the free molecular limit are consistent with σg,g and the geometric

Df (KnD,m) = 0.1 log(KnD,m) + 1.86

(7)

for 0.16 ≤ KnD,m ≤ 3, reaching the asymptotic values of Df (KnD,m = 0.16) = 1.78 and Df (KnD,m = 3) = 1.91. 3.4. Self-Preserving Size Distribution by Coagulation−Agglomeration. Figure 7 shows the SPSD of agglomerating fractal-like particles obtained by DEM (Table 1) in the free molecular (△) and continuum regimes (○) by BCCA and DLCA, respectively. The SPSD is broader in the free molecular than in the continuum regime, consistent with literature.17 These SPSDs are also in agreement with population balance equation (PBE)39 simulations using as β either the Fuchs interpolation with rg as the collision radius51 (eq 4, solid lines) or that by Thajudeen et al.21 (dotted line) for Df = 1.91 and kn = 1.4 in the free molecular regime and Df = 1.78 and kn = 1.3 in the continuum regime. For η < 0.1 (small particles), however, the DEM-obtained SPSDs (symbols) are up to 80% higher than those predicted by PBE using the radius of gyration as collision radius in β (solid line). This difference stems from the fact that the PBE model neglects the penetration of small through large agglomerates when rg is used as collision radius.52 For Df < 2, such penetration reduces the number of actual particle collisions53 given by a collision frequency function based on rg. ZuritaE

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Langmuir Table 1. Self-Preserving Size Distribution of υ-Based Concentration by Agglomeration from DEM Simulations in Free Molecular and Continuum Regimesa free molecular regimeb

continuum regimec

solid particle volume η

concn Ψ based on υ

solid particle volume η

concn Ψ based on υ

0.0027 0.0035 0.0047 0.0062 0.0082 0.0109 0.0144 0.0191 0.0254 0.0336 0.0446 0.0591 0.0783 0.1038 0.1376 0.1825 0.2419 0.3207 0.4251 0.5636 0.7472 0.9906 1.3132 1.7409 2.3080 3.0597 4.0562 5.3774 7.1289 9.4509

0.2559 0.3257 0.6824 1.2697 2.0294 2.6164 3.0989 2.8152 2.7521 2.7624 2.3355 2.2235 1.8568 1.5046 1.3480 1.1283 0.8736 0.7117 0.5330 0.4133 0.3006 0.2189 0.1550 0.1060 0.0717 0.0425 0.0246 0.0119 0.0055 0.0020

0.0106 0.0141 0.0187 0.0247 0.0328 0.0435 0.0577 0.0764 0.1013 0.1343 0.1781 0.2361 0.3130 0.4149 0.5501 0.7292 0.9668 1.2817 1.6991 2.2525 2.9862 3.9588 5.2483 6.9577 9.2239

0.0773 0.1634 0.1987 0.3314 0.4451 0.6121 0.7976 0.7960 0.8205 0.8502 0.9140 0.9032 0.8385 0.7140 0.6254 0.4831 0.3768 0.2742 0.1819 0.1056 0.0499 0.0179 0.0041 0.0006 0.0001

Figure 8. SPSD number-based geometric standard deviation of the radius of gyration σg,g, mobility radius σg,m, and volume-based radius σg,v as a function of the mobility radius-based diffusive Knudsen number (KnD,m = λα/rm) from DEM (top to bottom clustered black lines). σg,g is larger than σg,m in the entire size regime. The SPSD σg,g, σg,m, and σg,v values in the free molecular regime are 2.27, 2.03, and 1.41, respectively, with the last one in good agreement with the literature17 (green solid line) for fractal-like agglomerates of Df = 2. In the continuum regime, values of 1.95 and 1.31 are attained for σg,g and σg,m, respectively. The evolution of σg,g in the entire particle size range is described by eq 8 (red solid line), exhibiting a minimum similar to that of spherical particles (Figure 2b).

for 0.001 < KnD,m < 100. σg,g,fm and σg,g,co are the self-preserving σg,g of 2.27 and 1.95 in the free molecular and continuum regimes, respectively. Equation 8 bridges the free molecular with the continuum regime, similar to Otto et al.45 (Figure 2b, orange line) for fully coalescing spheres.

4. CONCLUSIONS Brownian coagulation (agglomeration) in the absence of any coalescence (sintering) is investigated at standard dilute conditions (ϕ ≪ 0.01) by discrete element modeling (DEM) in the entire particle size range. Evolution of the resulting agglomerate radius of gyration rg, mobility radius rm, and volume-equivalent radius rv is traced along with that of the average number of primary particles per agglomerate np̅ . These characteristic radii differ significantly only for large (n̅p ≫ 10− 30) fractal-like agglomerates. New routes for agglomerate characterization are enabled with the quantification of the difference among rg, rm, and rv during agglomerate growth, as measuring two of these radii helps to obtain important agglomerate properties, like the specific surface area and primary particle diameter.34 Ballistic cluster−cluster agglomeration (BCCA) and diffusion-limited cluster−cluster agglomeration (DLCA) result in particles with asymptotic Df ≈ 1.91 and 1.78, respectively, in agreement with the literature, whereas in the transition regime Df shifts linearly from 1.91 to 1.78. The Df evolution from spherical to fractal-like agglomerates occurs through a smooth but rapid transition from 3 to an asymptotic value of about 1.8 for n̅p that ranges from 10 to about 30 depending on the size regime. A simple relationship is proposed for Df evolution from spheres to fractal-like particles, which is a function only of n̅p. Coagulating fractal-like agglomerates in the free molecular regime obtain a self-preserving size distribution (SPSD) with number-based geometric standard deviation of rg-based σg,g = 2.27, rm-based σg,m = 2.03, and rv-based σg,v = 1.41. In the transition regime, agglomerates attain a quasi-SPSD that goes through a minimum at mobility-based diffusive Knudsen number KnD,m ≈ 0.2 with rg-, rm-, and rv-based σg values of 1.65, 1.50, and 1.26, respectively. The corresponding σg,g and

In terms of solid particle volume, η = Nυ/V (Figure7. bFree molecular regime: rp,0 = 1 nm, Df ≈ 1.91, and kn = 1.4. cContinuum regime: rp,0 = 1000 nm, Df ≈ 1.78, and kn = 1.3. a

standard deviation of the projected area-equivalent radius σg,pa, as obtained by transmission electron microscopy of flame-made zirconia aggregates at 10 cm above the burner.26 Both σg,g and σg,pa reach their asymptotic self-preserving values for KnD,m > 10 with σg,g > σg,pa as here. The corresponding PBE-obtained selfpreserving σg,g and σg,m values in the free molecular regime are 2.17 and 1.97, respectively, smaller than the DEM-obtained ones when the Fuchs collision kernel is used, due to penetration of small agglomerates through larger ones as discussed in Figure 7. The SPSDs based on rg and rm are significantly broader than those of equivalent spheres. In the continuum regime, agglomerates attain σg,g = 1.95 and σg,v = 1.31, versus 1.44 and 1.307 for spheres, respectively. σg,v = 1.31 is in good agreement with the PBE-obtained value σg,v = 1.30 for agglomerates of Df = 1.8.17 However, similar to spherical particles (Figure 2b), σg,g, σg,m, and σg,v exhibit minima of 1.65, 1.50, and 1.26, respectively, in the transition regime (KnD,m ≈ 0.2). This evolution of σg,g as a function of KnD,m can be approximated by eq 8 (Figure 8, red solid line): σg,g,co σg,g,fm σg,g = + 1 + 2KnD,m 1 + KnD,m−4/3 (8) F

DOI: 10.1021/la504296z Langmuir XXXX, XXX, XXX−XXX

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Langmuir σg,v values in the continuum regime are 1.95 and 1.31. Population balance equations (PBEs) with coagulation rates based on the Fuchs interpolation collision kernel using rg as the collision radius underpredict σg,g and σg,m by 2−5%, as these PBEs neglect penetration of small agglomerates into larger ones. However, when a recently-developed collision rate21 for agglomerates in the continuum regime is used, such PBEs result in a SPSD identical to that obtained by DEM. Evolution of asymptotic Df and geometric standard deviation of SPSD from the free molecular to the continuum regime are quantified by simple relationships in terms of the mobilitybased diffusive Knudsen number. These relationships can be readily used in detailed particle dynamics simulations coupled to fluid dynamics for industrial process design, air pollution, meteorology, and climate dynamics.

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n np np̅ q rc rg rm rp rv T t v

Greek Letters

β δi δr η λα λ μ ρp σg,g

ASSOCIATED CONTENT

S Supporting Information *

One figure, showing collision frequency function of coagulating agglomerates as a function of Kn, and two tables, listing selfpreserving size distributions of agglomerates based on radius of gyration and mobility radius. This material is available free of charge via the Internet at http://pubs.acs.org.

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number primary particle number per agglomerate average primary particle number per agglomerate spacing exponent collision radius, m radius of gyration, m mobility radius, m primary particle radius, m volume-equivalent radius, m temperature, K time, s particle/agglomerate volume, m3

σg,pa

AUTHOR INFORMATION

Corresponding Author

σg,m

Present Address

σg,v ϕ Ψ

*E-mail [email protected]; phone +41 (0) 44 632 31 80; fax +41 (0) 44 632 15 95. †

School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138.

collision frequency, m3·s−1 mean distance, m average mean distance, m dimensionless volume agglomerate stopping distance ⟨λα = (8D/πci̅ )⟩ gas mean free path, m fluid viscosity, kg·s−1·m−1 particle density, kg·m−3 number-based geometric standard deviation of radius of gyration distribution number-based geometric standard deviation of projected area-equivalent radius distribution number-based geometric standard deviation of mobility distribution volume-based geometric standard deviation volume fraction dimensionless concentration

Subscripts

Notes

0 ∞ air c co fm g i, j m p s sim

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has received funding from the Swiss National Science Foundation (Grants 200021_149144 and 148643) and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013, ERC Grant Agreement 247283). For this article, E. Goudeli received the best oral presentation award by a PhD student at the 2014 Symposium on Flame and High Temperature Synthesis of Functional Nanomaterials during the Fall Meeting of the Materials Research Society, Dec. 1−4, Boston, USA.

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initial total air properties collision continuum regime free molecular regime geometric averaging particle indices mobility primary particle solid simulation

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NOMENCLATURE Cs Cunningham slip correction, unitless ci̅ mean velocity of particle i, m/s cr̅ average mean particle velocity, m/s d particle diameter, m D diffusion coefficient, m2/s Df mass fractal dimension Dfm mass-mobility exponent kB Boltzmann constant, m2·kg·s−2·K−1 kn mass fractal prefactor mass-mobility prefactor kfm Kn gas Knudsen number KnD gyration-based particle diffusive Knudsen number KnD,m mobility-based particle diffusive Knudsen number m mass, kg N number concentration, m−3 G

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DOI: 10.1021/la504296z Langmuir XXXX, XXX, XXX−XXX