Exact solutions to the coagulation equation

Volume 92A, number 6

PHYSICS LETTERS

22 November 1982

EXACT SOLUTIONS TO THE COAGULATION EQUATION M.H. ERNST and E.M. HENDRIKS lnstituut voor Theoretische Fysica, Ri]ksuniversiteit Utrecht, The Netherlands 1 and R.M. Z I F F Department o f Mechanical Engineering, State University of New York, Stony Brook, N Y 11794, USA Received 31 August 1982

Explicit post-gelation solutions are presented for Smoluchowski's coagulation equation with factorizable transition kernels Ki] = sisj, when s k = kto (to > 1/2) and sk = exp[a(k - 1)] (c~ > 0). In such solutions the total mass of sol (finite clusters) is not conserved in time, as the sol is loosing mass to the gel (infinite cluster). For the kernels Ki] = i1~]~ +]t~i ~ (~ = 0, 1, v general) Smoluchowski's equation can be solved sequentially in terms of a transformed time variable.

In coagulation processes Smoluchowski's equation describes the time evolution of the size distribution function Ck(t ), denoting the concentration o f clusters of size k: c~

1E where the coagulation kernel K i / m o d e l s the coalescence mechanism of an i- and a j-cluster. Here we neglect fragmentation, source and sedimentation terms which are frequently added in studies o f aerosol coagulation. The exact solution for a given initial distribution Ck(O ) is only known for kernels of the general form Kij = A + B (i + ]) + Cij [ 1 - 4 ] . If C ¢ O, the solution c k ( t ) undergoes a phase transition, called gelation; viz. before the gelpoint t c the total mass of finite size clusters (sol), M 1 (t) = Y. k c k ( t ) , is constant; at t c an infinite cluster (gel) appears causing M 1(t) to decrease for t ~> t c, as the sol is loosing mass to the gel. The purpose o f the present letter is to show (i) that explicit post-gelation solutions, with M 1(t)=/= 0, can 1 Mailing address: Princetonplein 5, P.O. Box 80006, 3508 TA Utrecht, The Netherlands. 0 0 3 1 0 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland

be found for factorizable kernels Ki] = sis] under some further restrictions on Sk, and (ii) that for a few other kernels the coagulation equation can be solved sequentially. Details will be published elsewhere [5]. A special post-gelation solution of the coagulation equation is suggested by the exact solution for a monodisperse initial distribution in the cases s k = ( f - 2)k + 2 withf~> 3 [2], and s k = k [3]. Past the gel point the size distribution is simply given by: ck(t )=ck(tc)[1 +b(t-

tc)]-i

(t~>tc).

(2)

In the former case t c = 1 I f ( f - 2) and b = f 2 ( f _ 2)/ ( f - 1) and in the latter case t c = b = 1. Up to the gel point the mass is conserved (21/1 -- 1), and past the gel point M l ( t ) = [1 + b(t - tc) ] - 1 . Therefore we investigate whether a post-gelation solution o f the form (2) [satisfyingMl(te) = 1] is possible for the models Ki/ = sis/. On substituting (2) into (1) we find that the time part cancels, and we obtain a recursion relation for ck(tc) , where consistency for k = I requires b = ~ SkCk(tc) (a proper choice o f time unit gives s 1 = 1). By introducing n k = Ck(tc)/b it takes the simple form: (s k _ 1 ) n k = 1__ ~ sis]nin] , 2 i+]=k

(3)

which has to be solved subject to the condition: 267

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skn k = l

,

PHYSICS LETTERS

nk

or

2"

(4)

For a given n l , (3) determines all n2, n 3 . . . . . Choosing n 1 such that (4) is satisfied, one can then in principle find b f r o m M l ( t c ) = 1 through b -1 = 2; kn k. The solution will be obtained by introducing generating functions:

G(x) = ]~k nk ekx '

F(x) = ~k Sknk ekx "

(5)

If 1/s k can be written as a Laplace transform, i.e.

1/sk = f 0

dy o(y) e x p ( - k y ) ,

then F and G are related by

G(x) = ? 0

dy o(y)F(x - y ) ,

k-I nk=rtkl /~0 ( k + l + 1 ) ( k - - 1 ) ! k k - 3 l(@) l = ~ ( k ~ l + f ) f ( k ~ l - 1)! .

2,

orF= l -(1-

2G) 1/2,

(6a)

(9)

From G ' ( 0 ) = b -1 we obtain b = M2(tc) = x/cJ. The large-k dependence o f this solution is:

n k ~ (2nX/-J) -1/2 [k -7/2 + (3x,/3/8)k -9/2 + ...] . (10) Thus we have determined all parameters (except to) in tire post-gelation solution (2) for the model sk = k 2, and obtained what constitutes a new solution to the coagulation equation. In the general case s k = k c° , post-gelation solutions only exist for co > 1/2 [7,8], and the quantities b and n 1 have been determined numerically [5]. Analytically only a few limiting properties can be calculated: viz. (i) the small-x behavior o f F is F ( x ) ~ 1 - ( - 2 x / b ) l / 2 + .... implying n k ~ ( 2 n b ) - l / 2 k -~° -3/2 + ... at large k, and (ii) for large co

kWn k ~

and it follows from the recursion relation (3) that

F-G=~F

22 November 1982

1

( - ½ ) k / k ! +a2 w ( - 3 / 2 ) k / [ 3 ( k - 1 ) ! ]

b~ ( ~knk)

- t ,~,2 - 3 i v. -. w ..+

,

,

or

~l~nk,~ F(x) - ½F2(x) = ?

d y o(y)F(x - y ) .

(6b)

0 We discuss two classes of models: sk = k ~ , where o 0 ' ) = y t O - 1 / p ( c o ) [when co is a positive integer F and G are simply related by differentiation: F = (d/dx)~G], and s k = exp [ a ( k 1)], where o(y) = e a S ( y - a). In the case, s k = k 2, where F = G" = ~ k2nk ekx , (6a) reduces to a differential equation, which can be solved to yield:

F(x)

½ F 2 ( x ) = eC~F(x - c0 ,

(x/~ + 1)(V~ - r ) exp[,v/~(r _ 1)1 15U + 0

(7)

where r = ~ - 2 F . The n k follow by expanding F ( x ) in powers of ex, using Lagrange's expansion [6]. n 1 is obtained as: n1=

lim

e-XF(x) = 6(2

and the higher n k are:

268

Xfj)e,f5

3 ,

(8)

(12)

to be solved subject to F ( 0 ) = 1. The relevant quantities can be determined as follows: hm

=

(c~'~co),(ll)

where (a)k = P(a + k)/F(k). In the case s k = exp [a(k - 1)] post-gelation solutions only exist for a > 0. The integral equation (6b) reduces to a difference equation:

X---+ _

eX

1

~ +1/8(2 c ~ - 1)2 - w +...

e-XF(x)=nt , e~

and all n k are calculated from (3). Since F ( 0 ) = Y, skn k < ~ , all moments exist, and

b-1 = eaF'(__a);

~ klnk = e ~ F q ) ( ~ x ) ,

(13)

where F(l)(x) denotes the lth derivative. The solution is determined b y defining numbers Pk = eX~F(-kcO, which obey the recursion relation 1

Pk+l =pk( 1 -- ~ e - k a p k )

(P0 = 1 ) .

(14)

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PHYSICS LETTERS

The limiting value gives p = = n 1 . For a > 0 the Pk constitute a monotonically decreasing sequence of positive numbers, implying the existence o f p = , and the post-gelation solution (2) exists. (For a ~< 0 no post-gelation solutions are possible). If a is not too small, (14) has the solution:

p= =~(1 - ~ 1 e_C~ - ~ 1 e_2C~ - e - 3~/8 + ...) ,

(15)

and the n k can be determined as a power series in e-Or.

To obtain b we introduce another set of numbers qk = e k ~ F ' ( - a ) , where q ~ = p = = n 1 . A recursion relation for qk is obtained by differentiating (12):

qk+l = q k ( 1 -- e - k a p k )

(q** = n l ) ,

(16)

l'-I (1 - e - k a p k ) . k=l

(17)

so that b = ( q l ) -1 = ( n l ) - I

In general the recursion relations (14) and (16) must be solved numerically, yielding n 1 and b as a function of the parameter a [5]. The large& behavior is given by

n k ~ ( 2 n b ) - l / 2 k - 3 / 2 exp[(1 - k ) a ]

+ ....

The limiting case a >> 1 is completely analogous to the large-co case, and the corresponding results are obtained from (11) by replacing k~°nk on the first line by e ( k - 1 ) ~ n k , and 2 -c° everywhere by e - ~ . The question arises [5,8] whether the special solution (2) evolves from a monodisperse initial distribution, as is the case in the exactly solvable models. Mthough the solution to the initial value problem is not known, we can give a definite and negative answer to this question for the cases: s k = k ¢° with co ;~ 1.1 and s k = e x p [ a ( k - 1)] for ~ > log 2 by using inequalities. For the remaining range of a- and co-values (except co = 1) we expect also a negative answer, but the question has not been settled. In the second part of this letter we discuss the models Ki] = i]u +flu and Ki] = i x + i x , for which the coagulation equation can be solved sequentially. As far as gelation is concerned the following is known about these models. IfKi] ~ C(i +]) (here U ~< 0 and X ~< 1), White [9] has shown that the coagulation equation does not show a gelation transition. If t~ > 0 and 3, > 1 gelation occurs [10] ; possible pre-gelation solutions (t < tc) have the scaling form

22 November 1982

Ck(t ) ~ k-r~p(k(t c - t ) l / o ) as t ]" t c and k -+ ¢¢ with x = k (t c - t) 1/° fixed; and possible post-gelation solutions have the form ek(t ) k - r A ( t ) for large k. In the first model with p 2> 0 (where existence of global solutions follows from the work of Leyvraz and Tschudi [8] for/~ ~< 1) we found 1 T = 2 + g p and r + O = p + 2. The second model with X > 1 is only meaningful in the pre-gelation stage, where the exponents • and r in the scaling form are related as o + r = )t + 1. At a finite time t c the system undergoes a gelation transition. Post-gelation solutions do not exist, since the coagulation equation is no longer well-defined beyond the gel point, as it contains divergent quantities. The coagulation equation for the first model Ki] = i]u +flu reads:

Ck = ~ i]#¢iCj -- kctcM• - kPvkM1 , i+]=k

(18)

where M s =-- ~ kaCk . It has been solved by Lushnikov and Piskunov [ 11 ] for p < O, where M 1 (t) = 1 for all t/> O. Their method can be trivially extended to obtain pre-gelation solutions for the case p > 0 (where M 1 (t) = 1 for t ~< tc). With the help of the substitution

Ck(t ) = Vk(t ) exp [kMo(t)] and the moment equation/1~/0 = - M u (valid in the pregelation stage), the coagulation equation can be transformed into:

~k = ~ i / " u i v i - k " v k . i+]=k

(19)

The solution of (19) is given in ref. [11], where the unknown M o ( t ) can be determined from the transcendental equation:

1 = ~kc k = ~

k v k exp[kMo(t)] .

The above substitution does not lead to any simplifications in the post-gelation stage, neither does it enable us to locate the gel point exactly. However, in the range 0 < p <~ 1 we have found [5,10] the lower bound t b = [2/.ul4~(0)]-1 for t c. For p >/1 the lowerbound changes into an upperbound. The coagulation equation for the second model Ki i = i x + ] x , reading:

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Volume 92A, number 6

PHYSICS LETTERS

bk = ~ iXci c j - c k M x-kxckMO , i+j=k

(20)

can also be solved sequentially, as follows: we introduce new variables b y the t r a n s f o r m a t i o n (reminiscent of Lushnikov's t r a n s f o r m a t i o n [12] where Mo(t ) is replaced by c l (t)): t

~: f dt'Mo(t');

Vk(r)=ck(t)/Mo(t),

(21)

0 and with the help of the m o m e n t e q u a t i o n ~;/0 = - M o M x we transform (20) into:

duk(r)/dr= ~ ihuiuj--~kXuk, i+j:k

(22)

to be solved subject to the c o n d i t i o n , N uk = 1. Once all Uk(r) have been determined the f u n c t i o n Mo(t ) p 0 ( r ) can be found as a f u n c t i o n of T from N kuk = 1//J0(r ) = 1/Mo(t ) and the original t-variable can be recovered by differentiating r(t) in (21) and solving for t with the result T

t f d,' [.0(r')]-I =

.

0 For a monodisperse initial distribution where = ~ k l , the first few uk read explicitly:

vk(O)

"l(r) = e-T, u2(r ) = (2 ~° - 2) - 1 [exp(-2~-) - exp(--2C°T)] , u3(r ) = - - ( 2 w -- 2 ) - 1 ( 3 w -- 2 c°

1 ) - 1 ( 2 w + 1)

X ( e x p [ - ( 2 ~° + 1)r] -- exp(--3~°r)} + (2 ~° -- 2) - 1 X (3 ~° - 3 ) - 1 (2~o + 1) [exp(-3~-) - e x p ( - 3 ~or)] .

(23)

270

22 November 1982

For the case X ~ 1 the present m e t h o d can be used for all t. For the case X 2> 1 the kinetic equation is only well-defined below the gel point. We have not been able to determine t c exactly, b u t only obtained the bound t c ~< [2(X - 1 ) M ~ - I ( 0 ) I - 1 F u r t h e r m o r e , w i t h o u t an intervening gelation transition, Mo(t ) would become negative within a finite time to, obeying t c < t o ~< [M0(0)] ;t -1/(• 1), so that there would not exist a one-to-one relationship b e t w e e n t and 7. This work was supported in part by the Office of Basic Energy Sciences, US Department of Energy.

References [ 1 ] R.L. Drake, in: Topics in current aerosol research, Vol. 3, eds. G.M. Hidy and J.R. Brock (Pergamon, New York, 1972) Pt. 2. [2] R.M. Ziff and G. Stell, J. Chem. Phys. 73 (1980) 3492. [3] F. Leyvraz and H.R. Tschudi, J. Phys. A14 (1982) 3389. [4] R.J. Cohen and G.B. Benedek, J. Chem. Phys., to be published. [5] E.M. Hendriks, M.H. Ernst and R.M. Ziff, J. Stat. Phys., to be published. [6] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover Publications, New York). [7] R.M. Ziff, E.M. Hendriks and M.H. Ernst, Phys. Rev. Lett. 49 (1982) 593. [8] F. Leyvraz and H.R. Tschudi, J. Phys. A15 (1982) 1951. [9] W.W. White, Proc. Am. Math. Soc. 80 (1980) 273. [10] M.H. Ernst, E.M. Hendriks and R.M. Ziff, J. Phys. A, to be published. [ 11 ] A.A. Lushnikov and V.N. Piskunov, translated from Kolloidn. Zh. 37 (1975) 285. [12] A.A. Lushnikov, J. Colloid Interface Sci. 45 (1973) 549.