Journal of Colloid and Interface Science233,117–123 (2001)doi:10.1006/jcis.2000.7222, available online at http://www.idealibrary.com on

Asymptotic Particle Size Distributions Attainedduring Coagulation Processes

S. H. Park and K. W. Lee1

Department of Environmental Science and Engineering, Kwangju Institute of Science and Technology, 1 Oryong-dong, Puk-gu, Kwangju 500-712, Korea

Received May 31, 2000; accepted September 11, 2000

The effects of the collision kernel on the self-preserving size dis-tribution and on the gelation phenomenon of aerosol coagulationwere investigated. An analytical solution for the asymptotic widthof log-normally preserving size distribution during coagulation wasobtained as a function of the degree of homogeneity using arbitraryshape of homogeneous collision kernels. From the solution obtained,it was shown that when the degree of homogeneity is larger than1, self-preserving size distribution does not exist, and gelation oc-curs. A very accurate numerical coagulation simulation method, thesectional method, was also used for calculating the self-preservingparticle size distribution for some specific classes of coagulationkernels and the results were compared with the analytical solutionobtained by the log-normal method. C© 2001 Academic Press

Key Words: coagulation; collision kernel; self-preserving; gela-tion.

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INTRODUCTION

An important mechanism that affects the particle sizetribution of aerosols is coagulation. Because many imporproperties such as light scattering, electrostatic charging, toity, and radioactivity of suspended particles depend on theirdistribution, the time evolution of size distribution due to tparticle coagulation is of fundamental interest in a wide raof applications in science, medicine, and engineering.

The time evolution of particle size distribution undergoicoagulation is governed by the following equation if partisize is taken to be continuous (1),

∂n(v, t)

∂t= 1

2

∫ v

0β(v − v, v)n(v − v, t)n(v, t) dv

− n(v, t)∫ ∞

0β(v, v)n(v, t) dv, [1]

wheren(v, t) is the particle size distribution function at time,t ,andβ(v, v) is the collision kernel for two particles of volumevandv. If the collision kernel satisfies the following relation, it

1 To whom correspondence should be addressed.

11

is-antxic-izeege

gle

s

called homogeneous,

β(av,av) = aγ β(v, v), [2]

whereγ is the degree of homogeneity.One of the interesting features of coagulation known to d

is that the shape of the size distribution of suspended partundergoing coagulation often does not change after a longand the distribution becomesself-preserving(2–4). The fact thataerosol particles attain self-preserving distribution is an eletheoretical result that has important applications in the conand design of particle synthesis processes because thesebutions indicate the narrowest possible width of the size dibution that can be obtained during particle production by coulation (5).

It has been pointed out that, if the collision kernelβ(v, v)is a homogeneous function with degree of homogeneitythan 1, an asymptotic solution to the coagulation equationexist (4, 6, 7), which has been called self-preserving distributFriedlander and Wang (4) further pointed out that the shapthe self-preserving spectrum is greatly influenced by the fof the collision kernel. Using a simple transformation methLushnikov (8) showed that the decrease of the total numconcentration of particles in the asymptotic stage reduces tfollowing functional form for larget ,

N

No∝ t−1/(1−γ ), [3]

whereN is the total number concentration of particles andNo isthe initial value ofN. Botet and Jullien (9) showed that there eists an asymptotic size distribution during coagulation of clusand it depends only on the degree of homogeneity. Assumthat the clusters attain the most probable distribution, theyderived Eq. [3]. Using the similarity solution method, Rogand Flagan (10) showed that the same result holds for thepreserving size distribution. Another parameter which affectsshape of the self-preserving distribution was introduced byDongen and Ernst (11). They characterized the homogencollision kernel by two exponents as

β(v, v) ∝ vµvγ−µ if v À v, [4]

7 0021-9797/01 $35.00Copyright C© 2001 by Academic Press

All rights of reproduction in any form reserved.

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118 PARK A

and showed that a bell-shaped self-preserving distributionoccur only forµ < 0 whereas a monotonically decreasing disbution occurs forµ ≥ 0. Gentry and Cheng (12) also examinthe effect of the collision kernel on the rate of coagulationspecial classes of kernels. Using a numerical method, theported that the asymptotic behavior during coagulation depeon the functional form of collision kernels.

Previous studies mentioned above have pointed out thadegree of homogeneity of the collision kernel indeed hasimportant role on the particle size distribution evolution duricoagulation. However, none of them could give a quantitaanalytical expression for the asymptotic width of size distribtion nor for the number concentration decay as functions ofdegree of homogeneity for arbitrary collision kernels.

Another problem of interest about coagulation is the pnomenon that is called gelation. Often the gelation phenomeis defined as the divergence of the second moment (13, 14)example, when the coagulation kernel isβ(v, v) = A(vv)γ /2,gelation occurs whenγ is greater than 1 (15, 16). When geltion occurs, particle size distribution does not become spreserving. Although gelation has been investigated by a nber of researchers, most theoretical studies have been devosome specific collision kernels. Therefore, further study usvarious forms of collision kernels is needed to generalizerelationship between the gelation phenomenon and the deof homogeneity.

In this study, we investigate the self-preserving size distrition and the gelation phenomenon of aerosol coagulation.first derive analytically the asymptotic width of log-normalpreserving size distribution during coagulation as a functionthe degree of homogeneity using an arbitrary shape of the hogeneous collision function. The derivation of number concention decay follows then. It is further shown that with a degreehomogeneity larger than 1, there is no self-preserving sizetribution, and gelation occurs att <∞. To overcome the criticaassumption of log-normal size distribution, a very accuratemerical coagulation simulation method, the sectional methis used for calculating the asymptotic particle size distributwith some specific kinds of coagulation kernels. A comparisof the results from two different methods is given.

DERIVATION OF ANALYTICAL SOLUTION

To represent an arbitrary homogeneous collision kernel wthe degree of homogeneityγ , we use the following equation,

β(v, v) =∑

x

Kxvx vγ−x, [5]

wherex is an arbitrary real constant andKx ’s are the collisioncoefficients. It should be noted that all the homogeneous csion kernels have functional forms of Eq. [5] or at least canexpressed as Eq. [5] by the Taylor series expansion method

In this study the initial size distribution is taken as being lonormal and in general this approach has been widely emplo

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for representing the size distribution of particles both theorically and experimentally. Further, subsequent time-dependsize distributions of the particles undergoing coagulation aresumed to remain log-normal or at least can be representedlog-normal function. Many experimental results and numericcalculations indicate that particle size distributions during coaulation fit approximately a log-normal function (2, 17, 18). Thsize distribution function for particles whose volume isv for alog-normal distribution is written as

n(v, t) = 1

3v

N(t)√2π ln σg(t)

exp

[− ln2{v/vg(t)}18 ln2 σg(t)

], [6]

whereN(t) is the total number concentration of particles,σg(t)is the geometric standard deviation, andvg(t) is the geometricnumber mean particle volume.

Thekth moment of the particle size distribution is written a

Mk =∫ ∞

0vkn(v, t) dv, [7]

wherek is an arbitrary real number. Among the moments,M0

represents the total number concentration of particles (=N) andM1 the total volume concentration of particles.

By multiplying Eq. [1] byvk and integrating from 0 to∞, wehave

d Mk

dt= 1

2

∫ ∞0

∫ ∞0{(v + v)k − vk − vk}

×β(v, v)n(v, t)n(v, t) dv dv. [8]

Substituting Eq. [5] into Eq. [8] and writing the results fork = 0and 2, we have

d M0

dt= −1

2

∑x

(Kx Mx Mγ−x), [9]

d M2

dt=∑

x

(Kx M1+x M1+γ−x). [10]

For k = 1, Eq. [8] becomes

M1 = const, [11]

which merely means that the total volume of particles is coserved during coagulation.

The log-normal function has the following properties (19):

Mk = M0vkg exp

(9

2k2 ln2 σg

), [12]

vg = M21(

M30 M2

)1/2 , [13]

( )

g-yed

ln2 σg = 1

9ln

M0M2

M21

. [14]

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ASYMPTOTIC PARTICL

After substituting Eq. [12] into Eqs. [9] and [10], we have

d M0

dt= −1

2

∑x

[Kx M2

0vγg exp

{9

2(γ 2− 2γ x + 2x2) ln2 σg

}],

[15]

d M2

dt=∑

x

[Kx M2

0v2+γg exp

{9

2(2+ 2γ + γ 2

− 2γ x + 2x2) ln2 σg

}]. [16]

After eliminatingdt from Eqs. [15] and [16], we have

d M2

Mγ

2

= −2M2(1−γ )1

d M0

M2−γ0

. [17]

Integrating Eq. [17], taking a limit condition,t →∞, and usingEq. [14], we find the asymptotic geometric standard deviafor γ < 1 as follows:

σg∞ = exp

[√1

9(1− γ )ln 2

]for γ < 1. [18]

Equation [18] shows that the self-preserving geometric standeviation depends not on the values ofx’s andKx ’s but on thedegree of homogeneity only. It coincides with the previous stby Botet and Jullien (9), but the detailed functional form forσg∞

is given in this work for the first time. It should be noted hethat Eq. [18] is the general solution that unifies all the previderivations ofσg∞ for various collision kernels (19–21). Oncan further notice thatσg∞ increases with increasing degreehomogeneity,γ . This also agrees with the widely known fact thgravitational or turbulent coagulation, which has a highly sidependent collision kernel, leads to a broader size distribucompared with Brownian coagulation whose collision kerneonly weakly size-dependent. Although Eq. [18] seems to bvery useful result, it should be noted at this point that Eq. [is an approximate solution based on the log-normal distribuassumption and that it can cause some errors when it isfor practical applications. In the Comparison and Discusssection, Eq. [18] will be compared with more accurate numerresults and its limitations will also be discussed.

For γ = 1, integration of Eq. [17] yields the following relation:

M2(t)

M2(0)={

M0(t)

M0(0)

}−2

for γ = 1. [19]

It is not possible to obtain explicit information onσg∞ in thiscase, but one should notice that Eq. [19] gives a very imporclue. Considering thatM0(t) goes toward 0 as time goes to ifinity during coagulation, the left-hand side of Eq. [19] wou

diverge after an infinitely long time. This means the gelatitime, tg, is infinity. Regarding this case as a limiting conditio

E SIZE DISTRIBUTIONS 119

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of Eq. [18], one can expect that the geometric standard dationσg will monotonically increase toward infinity excludingself-preserving distribution.

Forγ > 1, integration of Eq. [17] indicates thatM2(t) goes toinfinity (i.e., the gelation occurs) at some timetg <∞. There-fore, we can say at this point that whenγ > 1, the self-preservingsize distribution does not exist and the size distribution conuously becomes broader until the gelation time,tg, is elapsed.

In the self-preserving case,σg does not change, so Eq. [12for k = 1 combined with Eq. [11] yieldsvg = vgo(N/No)−1. Bysubstituting it into Eq. [15], we have

d N

dt= −1

2N2−γ (Novgo)

γ∑

x

Kx exp

{9

2(γ 2− 2γ x

+ 2x2) ln2 σg∞

}. [20]

Integrating Eq. [20], we obtain the total number concentratdecay of the self-preserving size distribution:

N

No=[1+ 1

2(1− γ )Nov

γgot∑

x

Kx exp

{9

2(γ 2− 2γ x

+ 2x2) ln2 σg∞

}]−1/(1−γ )

[21]

It is noted from Eq. [21] that for larget the total number con-centration of particles reduces to the functional form expresin Eq. [3] as was found by previous researchers. FurthermEq. [21] gives not only the functional form of the number cocentration decay but also its complete solution including allcoefficients needed for an arbitrary collision kernel.

NUMERICAL COMPUTATION USINGTHE SECTIONAL METHOD

To numerically investigate the asymptotic behavior of coaulating aerosols, we used the sectional code of LandgrebePratsinis (22) which did not assume any functional form foparticle size distribution. This code was used in our previostudy on Brownian and turbulent coagulation (23), and it wshown to be quite accurate by comparison with data availain the literature. In the sectional model, particles are groupesections whereas the section boundaries are defined as

vk = fsvk−1, [22]

wherevk−1 is the lower boundary of sectionk and fs is the sec-tion spacing factor. Herefs = 1.18. Two hundred sections werused and mass conservation was checked throughout the clation. It was checked that mass loss always remains within

onnduring the coagulation time of interest. Landgrebe and Pratsinis(22) showed that a sectional model that numerically conserves

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120 PARK A

the square of the particle volume (v2-based model) within eacsection predicts the evolution of particle size distribution maccurately than sectional models conserving the particle numor volume density (n-based orv-based models). Hence thisv2-based model is used for calculating the time evolution of thedistribution. In all numerical models, a time-efficient integratialgorithm subroutine DGEAR (24) was used. Furthermore,initial size distribution was assumed to be log-normally dtributed. In order to calculate the initial size distribution for tsectional model, we used the error function

n(k) = No1

2

[erf

{ln(vi+1/vgo)√

18 lnσgo

}− erf

{ln(vi /vgo)√

18 lnσgo

}], [23]

whereN, vg, andσg represent the total number concentratithe geometric mean volume, and the geometric standardation, respectively, while the subscript o indicates the inicondition. The rules for calculatingvg and σg are given byLandgrebe and Pratsinis (22).

The following form of kernels, which were called “idealizehomogeneous kernels” by Gentry and Cheng (12), were usthe collision kernel in this numerical simulation:

β(v, v) = C(va + va)(v−b + v−b). [24]

An example of Eq. [24] in wide use is the continuum Browncoagulation kernel, in whicha = b = 1/3 with the degree ohomogeneityγ = a− b = 0.

COMPARISON AND DISCUSSION

Conventionally, in the self-preserving formulation, the dimesionless particle volume is defined as

η = v/v, [25]

and the dimensionless size distribution density function as

9 = nv/N [26]

wherev [=M1/N] is the arithmetic mean particle volume. Sinthe plot of9 against a log scale ofη does not provide a correct shape of the size distribution, a plot of9η versusη wasused, so that a symmetric curve is obtained for a log-norsize distribution.

Figure 1 shows the self-preserving size distributions for difent values of the degree of homogeneity obtained by analyand numerical methods described in the previous sectionthis figure,a+ b was set to 0.6, which is close to the valuethe continuum Brownian coagulation, 2/3. In case of numericaresults, the self-preserving state was recorded when the geo

ric standard deviation became invariant with time. It is clearseen from both analytical and numerical results that increas

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FIG. 1. Comparison of self-preserving size distributions for different valuof degree of homogeneity obtained by analytical and numerical methodsa+b = 0.6).

the degree of homogeneity leads to a broader size distribuIn the numerical study for self-preserving size distribution dto free-molecular Brownian coagulation of fractal agglomates, Vemury and Pratsinis (5) showed that lower fractal dimsion leads to a broader self-preserving distribution. The colliskernel used by Vemury and Pratsinis (5) is as follows,

β(v, v) = C

(1

v+ 1

v

)1/2(v1/Df + v1/Df

)2, [27]

whereDf is the fractal dimension. It should be noticed thatEq. [27] a lower fractal dimension means a higher degreehomogeneity. Therefore, the result of Vemury and Pratsiniscan be interpreted as a higher degree of homogeneity leato a broader self-preserving size distribution, which agrees wFig. 1. Considering that Eq. [27] has a different form from E[24], we can expect this trend would hold also for other funtional forms of homogeneous collision kernels as was veriby analytical solution in this work. Figure 1 also shows ththe error of the analytical solution becomes more significanthe degree of homogeneity increases. This error results fthe symmetrical assumption of the log-normal size distributiAs is shown from Fig. 1, a higher degree of homogeneity leto a less symmetric distribution, causing a larger error ofanalytical solution.

Figure 2 shows the self-preserving size distributions for tfixed degrees of homogeneity,γ = −0.2 andγ = 0.2. The valueof a+ b varies from 0.4 to 1.0 in this figure. Compared wiFig. 1, the effect of varyinga+ b on self-preserving size distribution seems to be much less significant than that of chaing degree of homogeneity. At a higher degree of homogeity, γ = 0.2, the breakage of symmetry in distribution occurrmore significantly as was mentioned before, and it resultelarger differences between curves by numerical results asas a larger difference from the analytical solution.

lying

To show the effects ofγ anda+ b on the self-preserving sizedistribution more systematically, the values of self-preserving

ASYMPTOTIC PARTICLE SIZE DISTRIBUTIONS 121

f

-n

n in3%)

ving

sed

FIG. 2. Self-preserving size distributions at di

geometric standard deviation,σg∞ , obtained by the numerical method for differenta and b values are summarized iTable 1 compared with the analytical solution. It is shownthis table that the effect of varyinga+ b is not always negligi-ble although it is much less important than the effect of varyγ (=a− b). This is in contrast to the analytical result obtainby the log-normal method, which showed no dependence ofσg∞

on thea+ b value. In addition, the effect of varyinga+ b be-

comes more important asγ increases as was shown in Fig. 2.

)

by the collision kernel increasing with particle size, becauserate

For a fixed degree of homogeneity,γ , less than 0, σg∞ does not

TABLE 1Self-preserving Values of Geometric Standard Deviation, σg∞ , for the Collision Kernel of Eq. [24] with Different Degrees

of Homogeneity Obtained by the Sectional Method

a+ bAnalytical

γ (=a− b) 0 0.2 0.4 0.6 0.8 1.0 (log-normal

−0.6 1.276 1.276 1.275 1.273 1.271 1.268 1.245a = −0.3 a = −0.2 a = −0.1 a = 0 a = 0.1 a = 0.2b = 0.3 b = 0.4 b = 0.5 b = 0.6 b = 0.7 b = 0.8

−0.4 1.314 1.314 1.311 1.308 1.303 1.299 1.264a = −0.2 a = −0.1 a = 0 a = 0.1 a = 0.2 a = 0.3b = 0.2 b = 0.3 b = 0.4 b = 0.5 b = 0.6 b = 0.7

−0.2 1.379 1.376 1.370 1.361 1.351 1.341 1.288a = −0.1 a = 0 a = 0.1 a = 0.2 a = 0.3 a = 0.4b = 0.1 b = 0.2 b = 0.3 b = 0.4 b = 0.5 b = 0.6

0 1.530 1.514 1.484 1.454 1.427 1.405 1.320a = 0 a = 0.1 a = 0.2 a = 0.3 a = 0.4 a = 0.5b = 0 b = 0.1 b = 0.2 b = 0.3 b = 0.4 b = 0.5

0.2 NA NA 1.853 1.660 1.567 1.508 1.364a = 0.1 a = 0.2 a = 0.3 a = 0.4 a = 0.5 a = 0.6

b = −0.1 b = 0 b = 0.1 b = 0.2 b = 0.3 b = 0.4

0.4 NA NA NA NA 1.886 1.697 1.431a = 0.2 a = 0.3 a = 0.4 a = 0.5 a = 0.6 a = 0.7

b = −0.2 b = −0.1 b = 0 b = 0.1 b = 0.2 b = 0.3

0.6 NA NA NA NA NA NA 1.551a = 0.3 a = 0.4 a = 0.5 a = 0.6 a = 0.7 a = 0.8

rapid collisions among bigger particles continuously gene

b = −0.3 b = −0.2 b = −0.1

ferenta+ b values for (a)γ = −0.2 and (b)γ = 0.2.

in

inged

show a significant change with varyinga+ b value. Forγ = 0,however,σg∞ remarkably reduces with increasinga+ b value,approaching the analytical solution. Forγ > 0, the simulationcould not even give available data in some cases as showTable 1. This was because too much aerosol mass (more thandisappeared from the largest section before the self-preserstate was achieved due to a too high expectedσg∞ value. This isreferred to as a “runaway” mass. The runaway mass is cau

b = 0 b = 0.1 b = 0.2

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FIG. 3. Comparison of self-preserving geometric standard deviation vues obtained by analytical and numerical methods as a function of degrehomogeneity.

extremely large particles. This phenomenon is an inherent qity of the sectional method because it cannot deal with extrembig particles beyond the maximum size of the largest sectWith increasing degree of homogeneity, the runaway mass pnomenon occurs at a highera+ b value, and it is expectedthat for γ ≥ 1 this phenomenon will occur for alla+ b val-ues, which would mean the gelation. Figure 3 shows theof data in Table 1. As has been pointed out by previoussearchers (25, 26), the geometric standard deviation valuetained by the sectional method is always higher than thattained by the log-normal method, which, as mentioned beforesults from the symmetrical assumption of the log-normal sdistribution.

By comparing Eq. [24] with Eq. [4], one can seeµ = −bfor b > 0 and µ = 0 for b ≤ 0 if we assumea > −b forconvenience. Thus, the analysis of van Dongen and Ernstimplies that a bell-shaped self-preserving distribution occonly for b > 0. Further, it is also expected that the lowerb is,the more the self-preserving distribution deviates from a symetrical shape even forb > 0. Therefore, care must be takein applying the results of the present study to the case of hdegree of homogeneity with smallb in which the symmetricallog-normal assumption may not be valid, as demonstrateTable 1 and Fig. 3.

CONCLUSIONS

The effects of the collision kernel on the self-preserving sdistribution and on the gelation phenomenon of aerosol coagtion were investigated using analytical and numerical methoAn analytical solution for the self-preserving geometric standdeviation, Eq. [18], was obtained by assuming a time-dependlog-normal particle size distribution during coagulation forarbitrary shape of homogeneous collision kernels. This re

is important in the sense that Eq. [18] is the general solutithat unifies all the previous predictions forσg∞ derived for dif-

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ferent collision kernels. Numerical simulation was also carrout using the sectional method to obtain the self-preservingticle size distribution for some specific classes of coagulakernels. It was shown that increasing the degree of homogity always leads to a broader self-preserving size distribuand that when the degree of homogeneity is larger than 1,preserving size distribution does not exist, and gelation occWhile general agreement between analytical and numericasults was found, the main difference was that the self-presersize distribution was found by the numerical method to vfor different functional forms of collision kernels even with thsame degree of homogeneity. At higher degrees of homogity, the effect of the functional form of the collision kernel bcomes more important, causing a larger error of the analysolution due to the nonsymmetrical nature of the actual sdistribution.

NOMENCLATURE

a, b exponents used in Eq. [24] [—]Df fractal dimension [—]fs section spacing factor of sectional method [—]Kx collision coefficients used in Eq. [5] [m3(1−γ ) s−1]Mk kth moment of log-normal size distribution [m3(k−1)]N total number concentration of particles (=M0) [m−3]n particle size distribution density function [m−6]t time [s]tg gelation time [s]v, v particle volume [m3]v arithmetic mean particle volume [m3]vg geometric mean particle volume [m3]x arbitrary real constant used in Eq. [5] [—]β collision kernel [m3 s−1]γ degree of homogeneity [—]η dimensionless particle volume [—]µ exponent used in Eq. [4] [—]σg geometric standard deviation based on particle

radius [—]9 dimensionless size distribution density function [—]

Subscripts

o refers to initial condition∞ refers to condition att →∞

ACKNOWLEDGMENTS

This work was supported by the Brain Korea 21 program from the Minisof Education through the Graduate Program for Chemical and EnvironmeEngineering at Kwangju Institute of Science and Technology.

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E

al

ASYMPTOTIC PARTICL

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SIZE DISTRIBUTIONS 123

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