Particle growth by coalescence and agglomeration

J. Aevoso~ sci., Vol. 21, Suppl. Printed in Great Britain.

i, pp. S73-S76,

1990.

0021-8502/90 $3.00 + 0.00 Pergamon Press plc

PARTICLE GROWTH BY COALESCENCE AND AGGLOMERATION W. Koch Fraunhofer-Institut fiir Toxikologie und Aerosolforschung D-3000 Hannover 61, FRG S. K. Friedlander Department of Chemical Engineering University of California at Los Angeles Los Angeles, CA 90024

KEYWORDS Agglomerates, size of primary particles, coalescence

INTRODUCTION The pathway of particle formation in high temperature processes can often be characterized by nucleation of condensable material and subsequent particle growth by coagulation. Depending on the temperature, particles formed in this way may have quite different morphological structures. Spherical particles are produced when the colliding particles coalesce. However, in the case when coalescence is quenched agglomerates are formed. The size of the primary particles composing the agglomerates depends on the temperature history of the particle formation process and on the temperature dependence of the material properties determining particle coalescence. These properties are, for example, the surface tension, a, and the viscosity, r/, or the solid state self-diffusivity, D, of the coalescing particles (Frenkel, 1945; Helble and Sarofim, 1989; Kingerey et al., 1978). The average size, alp, of the primary particles can be defined as 6V d, = ~ - ,

(1)

where V is the total aerosol volume per unit volume of gas (which stays constant as the system coagulates), and A is the aerosol surface area per unit volume of carrier gas. This quantity decreases with increasing coagulation time due to coalescence, hence leading to an increase of the average size, alp. For perfectly coalescing spheres, i.e. droplets which fuse together instantaneously upon collision, dp is equal to the average volume equivalent diameter d~ph = (6V/xN)U3, where N is the total number concentration. In the free molecule limit and for constant temperature, d~ph increases monotonically according to (Lai et al., 1972):

d;Ph= A(Vt)~/5,

where

A=

1.88 (6kT~ \ pp / 1/5

(2)

Here, T is the temperature and p~ is the density of the coagulating material. In many cases, particles are formed under non-isothermal conditions. Initially, the system is at high temperatures but subsequently it will cool down to lower temperatures. Under this conditions the particles may eventually coalesce in the high temperature regime whereas, in the low temperature section of the reactor, they

$73

$74

W.

KOCH

and

S.

K.

FRIEDLANDER

>.

:i ::i ~ • ;.

i:

~ i,

V,A

coalescence agglomeration

t • ~.~.~

r V,A~

V,A1

•

V,A.~,

Figure h Possible particle morphologies resulting from the growth process determined by agglomeration and coalescence. The initially formed stable nuclei are assumed to be of molecular size. Particle growth is by coagulation, only. may behave as colliding solid particles. The final agglomerates are then be composed of individual particles which may be much greater in size than the original nuclei formed by nucleation. (Fig. 1).

THE MODEL In this paper, we calculate the average size of the primary particles, dp, for a system which is cooling down according to a linear cooling relationship: T = To -

~t,

(3)

where t is the time, To is the initial temperature and s is the cooling rate. Coalescence reduces the surface area of a coagulating aerosol. The rate of change of the aerosol surface area is driven by the excess surface energy which is formed upon particle collision. Using arguments given in a previous paper (Koch and Friedlander, 1990) the following rate equation is obtained: da(t)

dt

-

1

r(d~)

( A ( t ) - Ao,a(t)).

(4)

Asph(t) is the surface area of the system with minimum surface energy undergoing complete and instantaneous coalescence, r(dp) determines the time of coalescence of two particles with diameter dp. For the viscous flow mechanism and the solid state diffusion this time scale is given by (Frenkel, 1945 , Kingerey, 1978)

r(d,)~-~d o " and r(~,)~ ~ kT d,. respectively. Here, a 3 is the ~tomic volume of the diffusing material.

(5)

Particle growth

! ~

.9

Figure 2: Solution of Eq.8 for a = 1 an for the case of viscous flow coalescence (solid line). The asymptotic value of dp is also indicated. The dashed line shows the relative increase (decrease) of the viscosity (diffusivity).

volume cIif¢uSlvlty; I/vi$cOSlty

.8 .7

'~

~',nil

size

OT t. hm p r , m a r y

$75

lOmr~.icles

.G .5 .d

.3 .2 .I J

0

*

~

i

i

2

]

dlmen~;

i

"2 .....

3 ~onles~)

T.....

T....

4 t. i ene

Compared to the other physical quantities the solid state diffusivity and the viscosity are extremely sensible to temperature changes. They show an exponential temperature behavior characterized by a dimensionless activation energy: ~ = E~ct/(kT). Therefore, these quantities will mainly determine coalescence. When the temperature decreases a substantial increase in viscosity or decrease in solid state diffusivity, respectively, occurs within a characteristic time period of tc = To/(e~). Within this period of time, particles with a characteristic diameter, de, defined by r(dc) = tc may fuse together. The explicit values of dc can be calculated from Eq. 5 for the viscous flow and the diffusion mechanism, respectively. The quantities, de and tc are determined by material properties, the temperature and the cooling rate, but they are independent of the volume fraction, V, of the condensed material. Introducing dimensionless variables, dp = dp/dc and t = t/tc and using Eqs. 1-5, the following dimensionless growth equations for the individual particles composing the agglomerates are obtained:

d--r =

-~-=

~,

for the viscous flow and volume diffusion coalescence (respectively). The parameter

dc a = ~(vt~)2/s

(T)

is the ratio of the size of the individual particles that are able to fuse together to the overall size that the agglomerates would grow to by collisions in a time period of to.

RESULTS As an example, a growth curve for the (dimensionless)_primary particle size, alp, is shown in Fig. 2 for a = 1. For long times, coalescence is quenched and dp approaches asymptotically a constant value, dp , which is a function of ~, only. This function is plotted in Fig. 3 for the two fusion mechanisms, considered here. From this figure and by inspection of Eq. 6 it follows that for a ~ 1 the final size of the individual constituents of the agglomerates is constant, independent of ~ and, hence, independent of the volume fraction, V, of the condensed material. It is easy to verify that ~ ~ 1 for the viscous flow mechanism, and d~°° ~ ~/3 for the solid state diffusion, respectively. In the large-a regime, large clusters are formed by cluster cluster collisions before the individual particles fuse together. This process is

$76

W. K O C H and S. K. F R I E D L A N D E R

Figure 3: The (dimensionless) final size of the primary particles as a function of the dimensionless parameter c~. Qualitatively, both sintering mechanisms considered here show similar patterns.

U • o~,d

It,le

vOlume

alfcuslon

I

12.

ppmv

190 .~

>, viSCOUS

40 E

flo~

.l~p ,I

v

Cpmv

CL O

o

s,l,ci

•

~agnes,um

Icoolln~

[

.~31

o~,~¢

rlte:

5~

K/IIC]

, 1

I

te

alpha

independent of the overall size of the particles, which is primarily determined by V, but depends only on material properties and the temperature pattern (i.e. the cooling rate). The size of the primary particles is governed by coalescence. In the regime a ~ 1 collision and coalescence control the final size of the primary particles. Here, the volume fraction, V, plays ~n important role. For a --- oo, the model approaches the theory of instantaneously coalescing spheres. The particle size is determined by particle collisions and the volume fraction, V, is the key parameter. In order to calculate absolute values of the primary particles of the agglomerates data on the material properties such as the solid state diffusivity and the viscosity are needed. These data are often not easily available. As an example, data reported by Helble and Sarofim (1989) have been used to estimate the size of the primary particles of silica and magnesium oxide agglomerates. The calculations were performed for two different volume fractions of 1 ppmv and 0.1 ppmv, respectively. It is assumed that gas to particle conversion occurs at 1800 K and that the system cools down at a rate of 500 K/sec.

CONCLUSIONS A simple model has been presented to estimate the size of the primary particles of agglomerates formed by coalescence and agglomeration in a process which is cooled down at a constant cooling rate. The model allows a presentation of the results in dimensionless form.

REFERENCES

Frenkel, J. (1945), J. Phys. 9:385A Helble, J.H. and Sarofim, A.F. (1989), J. Colloid Interface Sd. 128:348A Ki,gerey, W.D., Bowen, H.K., and Uhlmann, D.R. (1978), Introduction to Ceramics, Wiley, New York, Koch, W., Friedlander, S.K. (1990), J. Colloid Interface Sci., in press, Lai, F.S., Friedlander, S.K., Pich, J. mad Hidy, G.M. (1972), J. Colloid Interface Sci. 39:395A