Dissociation kinetics of doublets of aerosol particles

Dissociation Kinetics of Doublets of Aerosol Particles G A N E S A N N A R S I M H A N 1 AND ELI R U C K E N S T E I N 2 Department of Chemical Engineering, State University of New York, Buffalo, New York 14260

Received April 14, 1986; accepted July 1, 1986 If the dispersion forces, responsible for coagulation, are not too strong (i.e., if the aerosol particles are sufficiently small), then the coagulated aerosol doublets can derive enough energy from the collisions with the molecules of the suspending medium to overcome the interaction potential well, thus leading to their dissociation. Because of the great disparity between the time scales of oscillation in the potential well and Brownian motion of the coagulated particle pairs, the Fokker-Planck equation can be averaged with respect to the relative position of the constituents of the doublet. One thus obtains a one-dimensional Fokker-Planck equation in terms of the energy of the relative motion of the two constituents. As a result, the average lifetime of a doublet can be calculated as a first passage time in the energy of the relative motion of its constituents. The average dissociation time of the doublets, in air at 1 atm and 298 K, for a Hamaker constant of 10 -12 erg, has been thus calculated for different radii of the constituent particles. The averagedissociation time is found to increase dramatically, from 10 -7 t o 10 - l S,as the radius changes from 15 to 50 A. This is a result of the rapid increase in the depth of the interaction potential well with increasing radii. The doublets consisting of particles whose radii are greater than 50 A are found to be extremely stable. © 1987 Academic Press, Inc. INTRODUCTION The coagulation o f aerosols is mainly a result o f the short-range dispersion forces between particles. Since the dispersion forces between large particles are strong, it is customary to assume that every collision between two such particles results in irreversible coagulation. This assumption is reasonable because the interaction potential well (due to the van der Waals attraction and Born repulsion) is, in this case, very deep. F o r sufficiently small particles, however, not every collision leads to coagulation, since the dispersion forces between t h e m are not strong enough (i.e., the interaction potential well is not sufficiently deep). The effectiveness o f particle collisions, as characterized by the sticking probability, has been calculated in terms o f the interaction potential between t h e m (1, 2). The coagulated particles are subjected, in addition, to collisions with the molecules o f the suspending m e d i u m . If the dispersion 1G. Narsimhan is now associated with the Department of Agricultural Engineering, Purdue University, West Lafayette, Indiana. 2 To whom correspondence should be addressed. 0021-9797/87 $3.00 CopYright © 1987 by Academic Press, Inc. All rightsof reproduction in any form reserved.

forces, responsible for coagulation, are not sufficiently strong, i.e., if the particles are sufficiently small, the molecules o f the suspending m e d i u m can impart sufficient energy to the coagulated particles to o v e r c o m e the interaction potential well leading to their dissociation. In other words, the coagulation o f sufficiently small particles is reversible. The aim o f the present paper is to develop a theory capable o f predicting the average dissociation time o f a doublet. I n f o r m a t i o n regarding the rate o f dissociation o f coagulated particle pairs is useful for the calculation o f the evolution o f particle size distribution. The effect o f doublet dissociation rates o n aerosol growth due to coagulation is discussed in a subsequent paper. The average lifetime o f a coagulated pair (doublet) o f small particles can be calculated by examining the relative Brownian m o t i o n o f the particle pair in the interaction potential well. If the energy imparted by the collisions o f the molecules o f the m e d i u m is smaller than the depth o f the interaction potential well, the particle pair will exhibit an oscillatory m o t i o n within the potential well. It will be shown later that, for sufficiently small particles, the time scale o f these oscillations is m u c h smaller than

278 Journal of Colloid and Interface Science, Vol. 116, No. 1, March 1987

279

KINETICS OF AEROSOL PARTICLES

the time scale of the Brownian motion. This disparity in the time scales enables one to describe the relative Brownian motion of the particle pair within the potential well only in terms of a stochastic differential equation for the energy of their relative motion. The particle pair will dissociate as soon as the energy of their relative motion equals the depth of the interaction potential well. The calculation of the average lifetime of a doublet of equalsize particles can, therefore, be formulated as a first passage time (the time at which the energy of the relative motion becomes equal to the depth of the potential well) in the particle energy space. The first passage time concept has been widely used in the dynamic description of a variety of physical problems (3). The derivation of the dynamic equations describing the relative Brownian motion of the particle pair is presented in the next section. The subsequent section discusses the first passage time approach for the evaluation of the mean dissociation time of a doublet. The rates of dissociation of doublets of equal-size particles are calculated in the last section.

calculated through the integration of the intermolecular potential, modeled as the Lennard Jones 6-1 2 potential, under the assumption of pairwise additivity (1). The two particles of the doublet experience Brownian motion in the interaction potential well. Their individual Brownian motions can, therefore, be described by the Langevin equations (4)

avl

mp-~7= -mp~'vl + F , +

[1]

and

dv~

m p - ~ = -mp~v2 + F2 +

V2mmp~kTW2(t'),[2]

where ¢ is the velocity, ~"is the friction coefficient, t' is the time, T is the absolute temperature of the suspending medium, k is the Boltzmann constant, F is the interparticle force, ~(t') is a white noise vector, I is the identity matrix, and the subscripts 1 and 2 refer to the particles 1 and 2, respectively. The interaction forces F1 and F2 act on the particles 1 and 2, respectively, along their line of centers and are related via

DYNAMIC EQUATIONS FOR THE RELATIVE BROWNIAN MOTION OF A PAIR OF EQUAL-SIZE PARTICLES

Let us consider a doublet consisting of two equal-size aerosol particles of mass mp and radius R. The formation of the doublet is a result of the interparticle van der Waals attractive forces between its two constituents. These constituents are, however, subjected to random collisions with the molecules of the suspending medium. For large particles, the interparticle dispersion forces are strong. Therefore, the molecular collisions cannot impart sufficient energy to dissociate the doublet. On the other hand, for a doublet consisting of sufficiently small particles, the molecular collisions can impart sufficient energy to its constituents to overcome the interaction potential well, thus leading to dissociation. The lifetime of the doublet is determined by the time spent by its constituents in the potential well before their eventual escape. The overall interaction potential between the two constituents was

2f~mp~kTlli'l(t')

F1 = -F2

-

F(r'l -r'2) {r'l- ¢21 (ldl

- - rt2)'

[3]

where r'~ and r'2 are the position vectors of partides 1 and 2, respectively, Ir[ is the magnitude of the vector r, and F is the magnitude of the interaction force. Taking the center of mass of the doublet as the origin of the coordinate system and introducing the relative coordinates r' = lal - r'2, ¢ = ¢ 1 - ¢ 2 , the Langevin equation for the relative motion of the two particles of the doublet can be written as de

mrS7 =

¢ d~ , ~ l ~ ' ( t ' ) , --mr~'Vl-- 7 ~rr-r

[41

where mr(=mp/2) is the reduced mass and q~ is the interaction potential (~'(t))=0

and

(~(t')~(t'o))=lr(t'-t'o).

Here 6 is the Dirac delta function and ( . ) denotes the ensemble average. Journal of Colloid and Interface Science, Vol. 116, No. 1, March 1987

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NARSIMHAN AND RUCKENSTEIN

The relative motion of the two particles of the doublet can, therefore, be described by the Brownian motion of a fictitious particle of reduced mass rn~ in the interaction potential well with the same friction coefficient as that of the particles. The motion of the fictitious particle of reduced mass (referred to hereafter as the particle) around the sphere of influence of radius Rs(=2R) is divided into two regions, which are shown in Fig. 1. The thickness of region I is equal to the decay length (expressed here as the shortest distance between the surfaces of the two particles) of the interaction potential. As long as the particle moves in this region, it remains a doublet. In region II, the interaction potential is negligible, i.e., the particle experiences negligible interaction force and, therefore, the particle will dissociate. This occurs if the particle gains sufficient energy from the colliding molecules of the suspending medium to overcome the potential well, i.e., to escape from region I and enter region II (Fig. 1). If the particle, on the other hand, does not gain sufficient energy, it will experience an oscillatory motion within the potential well. The relative magnitudes of the time scales of oscillation and of its Brownian motion can be evaluated by comparing the decay length ha of the interaction potential and the correlation length X, (the distance over which the Brownian motion of the particle is correlated). The

REGION I

Xd

/

t ! I

.~'~,~

'

/ \

\

Rs

I..-S" /

/

correlation length ~'r is related to the time scale ~--l of Brownian motion via ~kr ,~, ~--1

y

Journal of Colloid and Interface Science, Vol. 116, No. 1, March 1987

[5]

=

For sufficiently small particles, the decay length ?,d is much smaller than the correlation length Xr. Indeed, for particles of 100 A radius and unit density, in air at 1 arm and 298 K, and for a Hamaker constant of 10-12 erg, Xa is of the order of 50 A, whereas ~r is of the order of 300 A. Therefore, the time scale of oscillation of a captured particle is much smaller than the time scale of its Brownian motion. Consequently, a captured particle will undergo very few collisions with the medium during one period of oscillation. As a result, the total energy of the captured particle (potential and kinetic) will nearly be conserved during one period of oscillation. Moreover, the depth of the overall interaction potential well, even for doublets of sufficiently small particles, can be of the order of or larger than k T . For instance, the depth of the potential well for two particles of 50 A radius and of Hamaker constant 10-12 erg is of the order of 17 k T . Therefore, the particle in the potential well must undergo a very large number of collisions with the medium before acquiring sufficient energy to escape from the potential well. In other words, the time scale of dissociation of a doublet is expected to be much larger than the time scale of the Brownian motion and, therefore, even much larger than the time scale of oscillations. Because of this great disparity between the time scales of dissociation and oscillation, the lifetime of a doublet can be estimated without a detailed description of the particle motion during a period of oscillation. In other words, the lifetime of a doublet can be estimated by averaging the equation over the oscillatory motion. Let us define the following dimensionless variables: r = r'/},d,

FIG. 1. Differentregionsfor the motion of the fictitious particle of reduced mass.

k\/_~,.

t = t'lro,

v = v ' / ~ ,

rO -- l / - k T i m ~ @ = dp/kT,

[6]

281

KINETICS OF AEROSOL PARTICLES

The associated equation for the evolution of the position r of the particle is

and A - - - - TO __ ~kd

~--1

)'r'

where zo is the time scale of oscillations. The Langevin equation (Eq. [4]) for the motion of the particle can, therefore, be written as d2r dr r d~ l dt 2 FA~+r-dT=(2A)/21~(t),

[7]

where the white noise vector ~(t) satisfies the following conditions:

(~(t)) = 0,

(~(t)~(to)) = 16(t- to).

[8]

Taking the scalar product of Eq. [7] with the velocity vector i, one obtains d(1..) +d~=(2A)l/Z~.l,(t). ~ r - r +A(~.~) dt

[9]

Since the right-hand side ofthe above equation is a linear combination of the three uncorrelated components of the white noise vector ~(t), it represents a scalar uncorrelated white noise (5), i.e., k- I~(t) = :C~l(t) + y~2(t) + z~3(/), where ~?, 3), and ~ are the three velocity components and ~l(t), ~2(t), and ~3(t) are the three components of the white noise vector ~(t). Therefore, i . I~(t) = (i-/r)l/z~(t),

d 1.. ~ ( ~ r . r + ~ ) +A(~-~) [10]

Since the dimensionless velocity v and energy E of the particle are given by ~.~=v 2

[11]

and E=lI32q-

~,

Eq. [10] can be recast in the form dE - - = - 2 A ( E - ~) + V4A(E- ~)~(t). dt

[ 13]

where o~is the angle between the velocity vector v and the position vector r (Fig. 1). Equations [12] and [13] are the stochastic differential equations which describe the motion of the particle in the potential well. Since the ratio of the time scales of oscillation and Brownian motion A < 1, Eqs. [12] and [13] show that the time scale of change of the position r is much shorter than the time scale of change of the energy E. Therefore, the particle experiences several oscillations during the time scale in which its energy changes. As already pointed out, the details of the particle motion over the time scale of oscillation do not appreciably influence the evolution of its energy. Consequently, in order to calculate the evolution of the energy E, one can approximate Eq. [ 13] by replacing cos o~ by the average value V(cos2o)). This average value is calculated by assuming that the velocity vector can be oriented in any direction with equal probability. Therefore, Eq. [ 13] can be rewritten as ~ - - - ~)V(cos20) _ )

[ 13a]

and, further, since (cos2o:) = 3, as

where (~(t)) = 0, (~(t)~(t - to)) = ~(t - to). Equation [9] can now be rewritten as

= (2A)1/2(1"- r)l/2~(t).

dr r - - = v- - -- v cos ~o, dt r

[12]

dr ~

{2(E_ ~)}m.

[141

FIRST PASSAGE TIME FORMULATION FOR THE CALCULATION OF THE MEAN DISSOCIATION TIME OF A DOUBLET

Consider a particle whose initial dimensionless position and energy are r0 and E0, respectively. Since this particle experiences Brownian motion in the potential well, both its position and its energy will be random at any time. Therefore, the evolution of its position and energy should be described by the conditional joint probability density w(r, E; t[ro, E0; 0), defined as w( r, E; t]to, Eo ; O)drdE, Journal of Colloid and Interface Science, Vol. 116,No. 1, March 1987

282

NARSIMHAN

AND

which is the probability that at time t the position of the particle is between r and r + dr and its energy between E and E + dE, when its initial position and energy are ro and Eo, respectively. Employing the stochastic differential equations [ 12] and [ 14], the Fokker-Planck equation for the evolution of the conditional joint probability density w(r, E; tlro, Eo; 0) has the

RUCKENSTEIN

q(rlE) f

C(E)

= ~V2(~-- ~)/3 ' [0,

for

~(r) < E

for

~(r) ~>E,

t171

where C(E) is a normalizing constant which is given by 1

form (6)

C(E) = f~-'(E)

L

= - O[{ 2 ( E , ~)}1/2 Ow(r,E;t[ro,Eo,O)

dr VZ(E- ~)/3

The conditional joint probability density can, therefore, be written as

× w(r, E; tlro, E0, 0)] + 2A O [ ( E - ~I,)

p(E; tlEo;0) w(r,E;tIro,Eo;O)= b,(E)~ ,

[181

× w(r,E; tlro,Eo; 0)] + 2A~ [(E- ~) ×w(r,E;tlro,Eo;O)]. [15] As already noted, the time scale of oscillation of the particle is much smaller than its time scale of Brownian motion. Therefore, the particle undergoes very few collisions and its energy is nearly conserved during many periods of oscillations. Consequently, the conditional joint probability density can be decomposed as

w(r,E;tlro,Eo;O) =q(dE)p(E;t[ro,Eo;O),

where /,~-~(E)

'(E) = jol

1

~dr.

Combining Eqs. [15] and [18], one obtains 1

0

E

ff'(E)VZ(E- ~) ~ p( ; tiE0;0)

A 0 [V2(E-~)p(E;t[Eo;O)]

=

Journal of Colloid and Interface Science, Vol. 116, No. 1, March 1987

J I

A O2 fV2(E-~)P(E;tlE°;O)I [ j

[16]

where q(rlE)dr is the conditional probability that the position of the particle (which is affected by oscillations) is between r and r + dr when its energy is E, and p(E; tlro, go; O)dE is the conditional probability that the energy of the particle at time t is between E and E + dE when its initial position and energy are ro and Eo, respectively. The evolution of the energy of the particle is weakly dependent on its initial position as the energy is almost conserved during many periods of oscillations, i.e., p(E; tlro, g0; 0) = p(E; tiE0; 0). The time spent by the particle at any position r is inversely proportional to its velocity (6) and is given by Eq. [14]. Therefore,

[ 191

[20]

The Fokker-Planck equation for the conditional probability density p(E; t[Eo; 0) can be obtained by averaging Eq. [20] with respect to the particle position to yield 0 ~ 11 -1 ~--p(g;tl ~/ (E) \ V2(E- ~)[ Ot

E o; 0 )

A 0 [(~)p(E;tlEo;O)] =

-O-EL

,~(E)

A 02 [ ( ~ ) p ( E ; t l E o ; O ) } ,

+ oE~[

~'(E)

J [21]

KINETICS

OF AEROSOL

where ( - ) denotes the average with respect to the particle position r,

o

283

PARTICLES

,_^,_

,

,¢(Eo)

g PtL;UILo;--r): a ~

1

0

O-Eo

× p(E;OlEo;-z) - A ~P(Eo) 02 ~V(Eo) OE~

and

×p(E; 01E0; - r ) , r, ek-t(E)

=1

~dr:~b(E).

[22]

Equation [21] can be recast in the form

[24a]

where r = t - t'. Further, p(E; 0[Eo; - r ) = p(E; rlEo; 0) and Eq. [24a] becomes 0

ff~p(E; rlEo;O)

-- - A ~b(E0) 0

~'(Eo) OEop(E; rIE°;O)

~tP(E;tlEo;O ) = A ~-~l~,---~ptz~,'--'tE'0,0)} 0 fff(E) + A ~P(Eo) 02 0 z fff(E)

_

+ A-ff-ESl~-7~p(L;tlEo;O)

t

[251

[231

The particle leaves the potential well as soon as its energy becomes equal to the depth of the potential well. In other words, the dimensionless energy E becomes equal to the dimensionless depth of the potential well ~0 at the time of dissociation. In order to calculate the average dissociation time, the probability that the dissociation time is r must be calculated as a function of the initial conditions. Hence, the final energy and time of a dissociating doublet are fixed. It is, therefore, more convenient to deal with the backward evolution of the conditional probability density p(E; tiE0; t') (i.e. evolution with respect to t' for fixed E and t). The corresponding backward FokkerPlanck equation (Kolmogorov equation) for the conditional density p(E; t[Eo; t') is given by (5)

Op(g; tlgo;t') = 3. @(go) 0 p(g; tlgo;t') '(Eo) ~,~o

~(Eo) 02 . A.L,-57-~w~~2P(E;t[Eo;t').

E r

~,---~o)O--ff~p( ; IEo;O).

Let r be the dimensionless time at which the particle with initial dimensionless energy Eo leaves the potential well (region I). Since the particle will remain in the potential well as long as its total energy is less than the depth of the potential well Prob (r >i tJEo;O) = G(Eo, t) =

kL'0) uz~,0

Since the particle motion is time homogeneous (i.e., the autocorrelation function for the energy of the particle is independent of the initial time), the above equation can be rearranged in the form

[26]

where ~0 is the dimensionless depth of the potential well. Integrating Eq. [25] with respect to E, yields

o

=

_A•(Eo)

0 ~_,=

A ~(Eo) 0 2

+ ~O-E~G(E°'r)"

[271

Since p(E; 0lEo; 0) = 6(E - Eo), it is clear that {;

[24]

I °°p(E;tlEo;0)dE,

G(Eo, 0) =

f°r 0~
[28]

The particle whose initial energy is equal to the depth of the potential well leaves the potential well immediately. Therefore, G(~o, r) = 0

for all z.

[29]

Journal ofColloidandlnterface Science, Vol. 116, No. 1, March 1987

284

NARSIMHAN

AND RUCKENSTEIN

The flux of the particles that leave the potential well with zero initial energy is zero, i.e.,

-~EG(E, T) I = 0 for all T E = 0.

[30]

One may note that the boundary conditions [29] and [30] represent absorbing and reflecting boundary conditions in the particle energy space at E0 = ~0 and E0 -- 0, respectively. Since G(Eo, r) is the probability that the first passage time of the particle (with initial dimensionless energy E0) is >r, the mean first passage time T*(Eo) is given by

constituent particles. The initial energy of the relative motion of the particles of a doublet (or, the initial energy of the fictitious particle of reduced mass) is, however, random being a result of the collisions with the molecules of the medium. The initial energy, therefore, is described by the probability density f(Eo) defined as

f(Eo)dEo = Probability that the initial dimensionless energy is between E0 and E0 + dEo.

Let us consider an ensemble containing a large number of isolated doublets exposed to a heat bath of temperature T at time t = 0. If T*(Eo) = r G(Eo,r)d'r the time scale for the doublet to reach thermal dO (?IV equilibrium with the heat bath is much smaller than the time scale of dissociation of the = G(Eo,r)dr. [311 doublet, the initial energy distribution can be assumed to be Maxwellian. Since the depth of Integrating Eq. [27] with respect to r and the potential well is of the order of or greater taking into account that G(Eo, m) = 0, one than kT, even for doublets of sufficiently small obtains particles, the time scale to reach thermal equilibrium (which is of the order of the relaxation - l = - A ~(Eo) d T*(Eo) time for Brownian motion) is much smaller than the time scale of dissociation. Consequently, it is reasonable to approximate the A ff(Eo) d 2 ~ * ' E " + ~ - ~ - o 2 j r o), [321 initial energy distribution as Maxwellian, i.e., with the associated boundary conditions T*(~o) = 0

f(Eo) = ~ E~/Ee-e°.

and

d T* (Eo)[ = 0. eo=o

dEo

[34]

The dimensionless average dissociation time (T*) of a doublet is therefore given by _

Solving Eq. [32] with the boundary conditions [33]-[34], yields

t 1f~ X

T*(Eo) =

[36]

[331

(T*)

A f~zJo E1/2e e°JEo

eXJoe-y

,) × ~ (Y dydxdEo,

[37]

o e ~ j : e-y -~,(y) ~ dYdx , for

0~
[351

which, rewritten for the average dissociation time ( T ) of a doublet becomes 97.-1 r~0

0

for

Eo> ~o.

The above equation provides the average dissociation time of a doublet for a specified initial energy of relative motion of the two Journal of Colloid andlnterface Science, Vol. 116, No. 1, March 1987

30 × f%ex f X e - Y ~ d y d x d E o . J Eo ,Io ~( y )

[38]

KINETICS OF AEROSOL PARTICLES THE MEAN LIFETIMES OF DOUBLETS CONSISTING OF EQUAL-SIZE AEROSOL PARTICLES The average dissociation time ( T ) for doublets of equal-size particles, in air at 1 atm and 298 K, has been calculated using Eq. [38]. The overall interaction potential between the constituent particles has been obtained by the integration of the Lennard-Jones 6-12 intermolecular potential, under the assumption of pairwise additivity. The expression for this overall interaction potential in terms of the Hamaker constant can be found elsewhere (1). All the calculations have been performed for a Hamaker constant of 10 -12 erg. The time scale of Brownian motion ~--1 of the constituent particles was calculated from ~--1 _

kT mp D '

[39]

where D is the diffusion coefficient o f the particle of mass mp. Accounting for the Philips (7) slip correction factor, the diffusion coefficient D can be expressed in terms of the Knudsen number (Kn = ~g/R, k s being the mean free path of the suspending medium) and viscosity of the medium n as

D

k T /5 + 4Kn + 6Kn2 + 18Kn3~ 5 - K n + ( 8 + T r ) K n 2 ]"

= ~

[40]

The values of the time scale o f Brownian motion ~--1 for doublets are given in Table I, as a function of the radius R of the constituent particles. The calculated values of the average dissociation time of the doublet for different values of the radius R are plotted in Fig. 2. One may note that the average dissociation time in-

285

creases exponentially with increasing particle size. It changes by six orders of magnitude, i.e., from 10 -7 to l0 -l s, as the particle radius increases from 15 to 50 A. Such a dramatic increase in the average dissociation time with only a moderate increase in the particle size is a result of the rapid increase in the depth of the overall interaction potential well. As the potential well becomes deeper, the particles must acquire more energy (equal to the depth of the potential well) from the collisions with the medium in order to dissociate and, hence, they need more time to dissociate. The calculated values o f the average dissociation times for panicles of radius greater than 50 A are found to be extremely large and, hence, are not reported. In other words, the calculations indicate that the doublets composed of particles greater than 50 ~ in radius are extremely stable. The dimensionless average dissociation time (expressed in terms of the time scale o f the Brownian motion) as well as the dimensionless depth of the overall interaction potential well are plotted as functions of the radius of the constituent particles in Fig. 3. The dimensionless depth of the potential well is found to increase from 2 to a value as large as 15 as the radius was varied from 15 to 50 A. Except for the doublets containing particles of 15/~ radius, the average dissociation time is much greater than the time scale of Brownian motion. The values of the parameter A (the ratio of the time scales of oscillation and Brownian motion) are plotted against R in Fig. 4. Since the first passage time analysis involves the assumption that the time scale of oscillation is much smaller than the time scale of Brownian motion, the values of the parameter k must be much smaller than unity. The cal-

TABLE I Radius (A)

~--1(s)

~0

A

(T) (s)

15 20 25 30 50

2.01 × 10-9 2.69 × 10 -9 3.37 X 10 -9 4.06 X 10 -9 6.84 X t 0 -9

2.2368 3.7674 5.4679 7.286 15.2447

5.67 × 10-2 7.51 × 10-z 9.30 × 10-2 0.1096 0.181

6.05 × 10-8 6.22 × 10 -7 4.95 X 10 -6 3.79 × 10-5 0.1729

Journal of Colloid and Interface Science, Vol. 116, No. 1, March 1987

286

NARSIMHAN AND RUCKENSTEIN t

_~ 4o-2

t0_4

I 40- (

"q t0-6

8, o o) 40-e I 20

I 40

0

Portiele

I 30

I 40

I 50

Radius (~)

FIG. 2. The variation of the average dissociation time with the radius of the constituents of a doublet consisting of equal-size particles of unit density, in air at 1 atm and 298 K, for a Hamaker constant of 10-L2erg. culated values o f the parameter A satisfy this condition, since they vary f r o m 0.057 to 0.181 as the particle radius increases f r o m 15 to 50 A (Table I). The dimensionless dissociation time (expressed as the n u m b e r o f time scales o f Brownian motion) is plotted as a function o f the initial energy for different particle sizes in Fig. 5. The dissociation time is f o u n d to rem a i n almost the same for initial energies smaller than 1kT, and to decrease sharply for 20 400 46 40s t2

v

:8

40z

4

I t0

I 20

I 30

I 40

I 50

0

Perticle Radius (~,)

FIG. 3. The variation of the dimensionless average dissociation time as well as the dimensionless depth of the interaction potential well with the radius of the constituents of a doublet consisting of equal-size particles of unit density, in air at 1 atm and 298 K, for a Hamaker constant o f 1 0 -12 erg. Journal of Colloid and Interface Science, Vol. 116, No. 1, March 1987

40"

[ t0

I 20

Particle

I 50

1 40

I 50

Radius (~)

FIG. 4. The variation of the ratio of the time scales of oscillation and Brownian motion A with the radius of the constituents of a doublet consisting of equal-size particles of unit density, in air at 1 atm and 298 K, for a Hamaker constant of 10-12 erg. larger values, b e c o m i n g zero for an initial energy equal to the depth o f the potential well. The range o f the initial energies for which the dissociation time is constant increases with increasing particle sizes. Except for doublets consisting o f particles o f 15 A radius, the dissociation time is independent o f the initial energy when the latter is as large or even larger than ~kT. In fact, for doublets c o m p o s e d o f particles o f 50 A radius, the dissociation time is f o u n d to remain constant for initial energies as large as 10 kT. If the time scale that is necessary to attain thermal equilibrium is indeed m u c h smaller than the time scale o f dissociation, the particles with small initial energies will attain rapidly the average kinetic energy o f the m e d i u m (3kT) t h r o u g h collisions with the molecules. As a result, the dissociation time is expected to be fairly independent o f the initial energy, when the latter is less than 3kT. Therefore, the insensitivity o f the dissociation time to initial energies as large as the m e a n kinetic energy o f the suspending med i u m is consistent with the assumption that the time scale necessary to attain thermal equilibrium is m u c h shorter than the time

KINETICS OF AEROSOL PARTICLES Curve LL?

287

Particle Radius (~}

t0

~5 2O 25 30 50

2 3 4 5 LIJ 3

4

o=

~0-t c:s l=

t0-2 40

I t02 Dimensionless

I t0 s

// 104

I t0 r

t0 a

Dissociation Time < T >

FIG. 5. The variation of the dimensionless dissociation time with the initial dimensionless energy for different values of the radius of the constituents of a doublet consistingof equal-sizeparticles of unit density, in air at 1 atm and 298 K, for a Hamaker constant of 10-12 erg.

scale of dissociation. Consequently, the assumption that the initial energy distribution is Maxwellian is reasonable. CONCLUSIONS The relative Brownian motion between the constituents of doublets consisting of sufficiently small equal-size aerosol particles is described by a one-dimensional Fokker-Planck equation in the particle energy space. A first passage time approach is employed for the calculation of the average lifetime of the doublets. This calculation is based on the assumption that the initial distribution of the energy of the relative motion of the constituent particles is Maxwellian. The average dissociation time of doublets, in air at 1 a t m and 298 K, for a H a m a k e r constant of 10 -12 erg has been calculated for different sizes of the constituent particles. The calculations are found to be consistent with the assumption that the

initial energy distribution of the particle pair is Maxwellian. The dissociation rate is found to decrease dramatically by several orders of magnitude with increasing particle size, as a result of the rapid increase in the depth of the interaction potential well. The doublets consisting of particles whose radii are greater than 50 A are found to be extremely stable. REFERENCES 1. Narsimhan, G., and Ruckenstein, E, J. Colloid Inte(face Sci. 104, 344 (1985). 2. Narsimhan, G., and Ruckenstein, E., J. Colloid Interface Sci. 107, 174 (1985). 3. Weiss,G. H., Adv. Chem. Phys. 13, 1 (1967). 4. Chandrasekhar, S., Rev. Mad. Phys. 15, 1 (1943). 5. Gardiner, C. W., "Handbook of Stochastic Methods," Springer-Veda.g, New York/Berlin, 1983. 6. Stratonovich,R. L., "Topicsin the Theoryof Random Noise," Vol. I, Gordon and Breach, New York, 1963. 7. Philips, W. F., Phys. Fluids 18, 1089 (1975).

Journal of Colloid and Interface Science, Vol. 116, No, 1, March 1987