TAIL DISTRIBUTION FOR LARGE CLUSTERS IN IRREVERSIBLE AGGREGATION

FRA CTALS IN PHYSICS L. Pietronero, E. Tosatti (editors) © Elsevier Science Publishers Β. V.,

303

1986

TAIL DISTRIBUTION FOR LARGE CLUSTERS IN IRREVERSIBLE AGGREGATION

P.G.J. VAN DONGEN and M.H. ERNST Institute for Theoretical Physics, University of Utrecht, P.O. Box 80.006, 3508 TA The Netherlands

Utrecht,

The tail of the cluster size distribution c k( t ) at cluster sizes much larger than the mean cluster size, is determined from Smoluchowski's coagulation equation with rate constants K ( i , j ) . The latter are homogeneous functions of i and j. The tail is found to be exponential, c^(t) = A k(t)exp(kz(t)), with log A f (t) = o(k) as k «, and A ^ t ) may be calculated from a c recursion relation. Depending on the initial distribution, c k( 0) , and the rate constants, K(i,j), we distinguish universal solutions, coinciding with the scaling solutions as the average cluster size diverges, and non-universal solutions. Our general results have been verified in two exactly soluble models. 1. INTRODUCTION 1 In this paper1 we use Smoluchowski s coagu­ lation equation with rate constants K(i,j):

K(ai,aj) « a^K(i,j) = a*K(j,i) v K(i,j) = i ^ j (j>X, λ=μ+ν)

(2.a) (2.b)

00

I K(i,j)c c -c I K(k,j)c (1) J i+j=k j=l in order to determine the functional form of the cluster size distribution at large cluster sizes, for different types of initial distri­ butions, c k( o ) , and coagulation kernels, K(i,j). "Large" in this context means "much larger than the average cluster size s(t)". The asymptotic properties of the cluster size distribution have been studied exten­ sively in various limits: (i) short-time solutions for monodisperse initial 2 3 conditions , (ii) large-time behavior at fixed k, and (iii) scaling behavior in the limit k oo, s(t) », where k/s(t) remains finite^. To complete the available analytic information on the solutions of Smoluchowski*s equation (1), we have calculated the asymp­ totic size dependence of c k( t ) for (fixed) finite time. In gelling systems, where s(t) diverges at a finite time t c (gelpoint), our considerations are restricted to t < t c# The present asymptotic results coincide on the one hand with the short-time solutions (see (i)) and on the other hand with the scaling solutions (see (iii)), both taken at large cluster sizes (k > s ( t ) ) . This agreement relates and unifies the earlier results. The structure of the cluster size distri­ bution for large values of k, depends on the behavior of the rate constants K(i,j) at large cluster sizes i and j. As most 1 coagulation kernels used in the literature are homogene­ ous functions of i and j, at least for large cluster sizes, we restrict ourselves to such kernels, and we characterize K(i,j) by two exponents describing its i- and j- dependence for j » i: cK

The reactivity of large clusters should not increase faster than their size, hence ν < 1, λ < 2, but no restrictions are imposed on μ. We assume that K(x,l-x) is continuous and positive for all x G (0,1). The main result, to be found, is that the cluster size distribution show a universal exponential tail: c k( t ) = A k( t ) e

k

z ) (

t

(z(t) < 0)

(3.a)

with - 1

jLim k l o g A k( t ) = 0

(3.b)

where the large-k-behavior of the prefactor A k( t ) may be calculated from a recursion re­ lation. In solving this recursion relation for large k, we distinguish between kernels with ν < 1 and ν " 1· In either case, we further distinguish universal solutions, that reduce to the scaling solution in the scaling limit, and non-universal, or transient, solutions. Universal solutions have the form A k( t ) = A(t)k"^ (k + o o ) , except in certain models with ν • 1, where one finds the uni­ versal form A^(t) ~ e x p [ - A ( t ^ ] as k + «. The possibility and the shape of non-universal solutions are determined by the form of the initial distribution c k( o ) . 2. THE METHOD We consider in general solutions of the form C

c k( t ) - A k(t)exp(k z(t))

(4.a)

P.G.J, van Dongen, M.H. Ernst

304

with ζ > 0, z(t) < 0 and C

(4.b)

jLim k~ log A k( t ) = 0 If the initial condition is exponentially bounded for some constants C, ε > 0: c k( o ) < C e "

ek

(k-1,2,...)

(5)

then one can show^ that only the assumption ζ * 1, or (3.a,b), leads to consistent solu­ tions. By inserting (3.a,b) into Smolu­ chowski' s equation ( 1 ) , we find that z(t) and A^(t) are related through the following equa­ tion: oo

(6) zkA.+l =h I K(i,j)A.A.-A, J «k,j)c. K K J 1 J i+j=k j=l In solving equation (6) for large k, we dis­ tinguish between models with a v-exponent satisfying ν < 1, and models with ν == 1· 3. SOLUTIONS FOR ν < 1: Different behavior is found for Ζ Ψ 0 and for z • 0. If z Φ 0, then eq. (6) reduces for large k to the recursion relation:

Inspection of (10.a,b) for ν > 0 shows that this solution is possible only if A(t) < » as t ψ 0, i.e. if the initial condition satis­ v fies the requirement -k l o g A, (o) > constant as k », Similarly, tor ν • 0, one has the requirement £ k ^ A ^ o ) < » in order that Δ(ο) < For the v-values considered (0 », provided that J k ^ A ^ o ) < «. Consequently, the time development of A k( t ) (k fixed) cannot be explained on the basis of the large-k-behavior of the solutions only. Moreover, one cannot decide from the present analysis whether so­ lutions of the form (10) are either transient, or non-universal and long-lived. Next, we consider initial adistributions of the «) with α < l+μ, so form A ^ oy ) ^ A k ~ (k that £ k A«(o2 a = «, If we assume that \ ( t ) « A(£)k , we find from (6) as k «: 2

A/A =^k

zkA, *\

I K(i,j)A.A,3 (k co) (7) 1 i+j=k In the derivation of (7) we have used that \/\ =°M as k «, on account of (3.b), V

and I K(k, j)Cj=k M^ (k

«>), where

a c

M a( t ) - £ k k represents the a-th moment of c^(t). The solution of equation (7) satisfying (5.b) has the general form A k( t ) « A(t)k *

(k - »)

(8)

To determine A and & we substitute the asymp­ totic form (8) into (7), and equate leading orders in k. This gives, in combination with (3.a): X

c^t) » az(t)k" e

k z ) ( t

(k-«0

(9.a)

where z < 0, z > 0, and the constant a is determined by K

(9.b) J dx K(x,l-x)[x(l-x)] ο The integral in (9.b) converges provided v 0. Then eq. (6) allows for asymp­ totic solutions of the form: a

P

A k( t ) - A k( o ) exp[k A(t)]

(k-«>)

(10.a)

with β • ν and Δ(ο) • 0. The time dependence of A(t) is determined by i(t) = J

y

k Z

k A k( t ) ( l - e ° ) > 0

(10.b)

1 + Xa

a

~

/ dx K(x,l-x)[x(l-x)]" (k*«>) 0 (11) For a satisfying l+λ < α < l+μ one finds that A 0, or A k( t ) / A k( o ) > 1 as k > », and one arrives at the same conclusion as for α > l+μ. For α < l+λ, we conclude from (11) that A(o) - «, implying instantaneous cross­ over to the universal solution. In the special case α • l+λ one finds - 1transient asymptotic x behavior, c k( t ) = A ( t ) k " e x p ( k z 0) , with A(t) • l / ( t Q- t ) , crossing over to the uni­ versal solutions (9) at t = t Q. 4. RESULTS FOR ν = 1: The calculations for this special case are technically more complicated than for ν < 1. Here we give only the main results. Before doing so we note that the cases z • 0 and z Φ 0 need not be distinguished, because eq. (6) contains two terms of C ( k ) . For ν • 1 the asymptotic behavior of A k( t ) depends upon more details of K(i,j) than specified by the leading order (2.b). There­ fore, we introduce Q(i,j) which is defined by y

1 M

K(i,j) = (ij) (i+j) ' '[l+Q(i,j)]

(12.a)

and assume that the small-x-behavior of Q(x,l-x) is given by: Q(x,l-x) = q x

M

+

(x+0)

(12.b)

with ρ > ο and q non-vanishing and finite ( - c o < q < o o ) . The function Q(i,j) is homogene­ ous, with zero degree of homogeneity, as may be seen from (12.a).

Tail distribution for large clusters

Also for ν - 1 we distinguish universal and non-universal solutions. The criterion for universal behavior is that it reduces to scaling behavior (see below) as the average cluster size s(t) diverges, s(t) -* <». In either case one finds that the time-develop­ ment of z(t) is determined by the entire cluster size distribution, as follows: z(t)

k z

? k ^ A k( t ) ( l - e ) > 0 k=l

(13)

However, the asymptotic behavior of A k( t ) as k •* oo, is different for universal and nonuniversal solutions. The universal asymptotic behavior of A k( t ) for the various values of ρ and q has been listed in Table I. The large-k-behavior is either of algebraic form, A k = Ak~^, or of the stretched exponential form: A,(t) ~ exp(-A(t)kP)

(10.a).

>P 1

A. (t)=A(t)k

p=l,q>-2,J(24^)>l p0

l+>
B. p « l , q — 2 , . Ι ( 2 + μ ) < 0

A(t)-A(t)k #2+μ

C.

A.(t)«A(t)k"^ ιηΓ2)<τΧμ-ς

D.

p=l,q<-2

p
q -> - c o . Finally in class D one finds stretched exponential behavior with β = 1 - p . In the special case p=l, q=-2, there are some compli­ cations, that will not be discussed here. Next we consider transient solutions of the form A k( t ) - F(t) A,(o) (k-*-), with F(o) = 1 . They can be found if A k( o ) ~ k ° , with & > m(3) Ξ max{2+^, μ-q}, and ρ > 1 . The time dependence of F(t) may be expressed in terms of moments of the cluster size m distribution. 3 If k ^ A k( o ) + 0 ae k • ·, but k ( ) A k( o ) > constant, then the solution crosses over to the universal solution instantaneously. If, on the other hand, the initial distribution is such that A k(o)~exp[-D k^] as k + », with D Q > 0 and I > β > m ( 4 ^ = m a x { 0 , l - p j , then one finds transient behavior of the form P m ; A k( t ) ~ exp[-D(t)k - A ( t ) k ^ ] . Instantaneous cross-over to the universal solution occurs if β < m ( 4 ) .

(U)

(k-xx,)

with 0 < β < 1 and A(t) > 0 . Note that the exponent in ( 1 4 ) is negative, as opposed to

A.

P

5. RELATION WITH SHORT-TIME AND SCALING SOLUTIONS We discuss the way in which the universal large-k behavior of the cluster size distri­ bution c k( t ) is related to: ( 1 ) the short-time solution for monodisperse initial conditions, and (2) the scaling solution. Our arguments are concentrated on the class of models with ν < 1 , but the conclusions also hold for models with ν • 1 · 1 The short-time solutions of Smoluchowski s equation for a monodisperse initial distri­ bution, c k( o ) = 6 k-p has the form c k( t ) k 1 N kt " as t ψ 0 , where the coefficients N k satisfy the recursion relation:

A k(t)^exp[-k A(t)] β=1-ρ

(k-1)

Table I Algebraic behavior of A f (t) is found in the c classes A, Β and C, and stretched exponential behavior is found in class D. In class A, the exponent & is the solution of the transcendental equation:

and lying in the interval l+μ < & < m(l) = min { 2 + μ , 1 + μ + ρ } . Here J( #) is defined as \ 0

t>

00

t>

dx{K(x,l-x)[x(l-x)]" -x^ }-/ dx h

μ

χ "^ (15.b)

In class Β one finds a consistent algebraic solution with & = 2 + μ . For the models of class C one finds a solution with & in the interval m ( 2 ) < i 9 ^ - q , where ι η ( 2 ) = ι η 3 χ { 2 + μ , μ-q-l} · Since the parameter q may become arbitrarily large and negative in class C, we conclude that for any fixed value of μ, the exponent 00 $ may become arbitrarily large, i.e. δ -* as

N,

h

I K(i,j)N Ν i+j=k

J

1

(16)

with N, *l = 1 · The large-k-behavior of N k may be determined in a similar manner Xas that of k A k( t ) , with the result N k « ak" R"" , where a is given in (9.b) and R is left undetermined ( 0 < R < o o ) . This result for N k (k -> oo) shows that in the limit t ψ 0 , with k » 1 , c k( t ) has the form ( 3 ) , with

(15.a)

J(£)=J

305

A k( t ) - a k " V t

(17.a)

z(t) * log(t/R)

(17.b)

The same expressions ( 1 7 . a , b ) are obtained if we take the limit k -• oo first, i.e. if we start from eqs. (9.a,b), and make the identification z(t) ^ log(t/R) as t ψ 0 . We conclude that there exists a common region of validity where the two limiting solutions: (a) first t ψ 0 , next k •> oo, and: (b) first k o o , next t ψ 0 , coincide. Similarly one can show that there exist overlapping regions of validity of the largek-solution and the scaling solution. The

P.G.J, van Dongen, M.H. Ernst

306

latter applies in the scaling limit, where/ both the mean cluster size s(t), and the cluster size k are taken to infinity, while β the scaling argument χ k/s(t) remains finite. For gelling systems (1 < λ < 2 ) , the scaling solution has the form c k( t ) * (l/s(t))t „(k/e(t)) 2

s » ws "

(18.a)

T + x

(18.b)

β

(λ+3)/2 and w is a separation where τ constant. The mean cluster size diverges as t τ t c (gelpoint). The large-x-behavior of the scaling function φ(χ)1η (18.a) is given as: X

φ(χ) * w 6 a x " e "

6x

(x + »)

6 . EXACTLY SOLVABLE CASES We discuss the large-k-behavior in two exactly solvable models namely K(i,j) - 2, s which has ν 0, and K(i,j) = ij, which is a model with ν * 1. Detailed calculations are β given in reference 5 for K(i,j) 2. For K(i,j) - ij, details will be published elsewhere. Here we give only the results. The large-k-behavior in the model K(i,j) = 2 (the value 2 is chosen for convenience) may be expressed in terms of the generating function v(x) of the initial distribution:

I

c k( t ) = [ 2 * t V '

2

1

c, (t) » ( t v ' ( x n) x n) " x " * ο ο ο

k

(k-*») )

where the time dependence of x 0( t ) is deter­ mined by v(x ) - v(l) = 1/t ο

(21.b)

One easily verifies that (21) has the form (9.a,b). (ii) If, on the other hand, v(x ) < o o , or l Α, (ο) < «, one finds the following large-k-behavior: c k( t ) * [ l - t / t of

2

<^(ο)

where cross-over occurs at t Q

(k-~) β

,

[ν(χ £)-ν(1)] \

(23)

(24)

Furthermore, z(t) is determined by (13) with μ=1, which in this exactly solvable model may be integrated to yield^: z(t) - - ( s + t u(s ) - t) c c

(25)

f

In the case (ii), u ( - z Q) < then eqs. (23)(25) apply only if there exists a root s (t) of (24), i.e. if t > t Q = l / u ' ( - z 0) . For 0 < t < t Q, there exists transient behavior, depending on the shape of the initial distri­ bution. If we assume that the initial distri­ bution has the form c k(o)=A(o)k ° exp(kz 0) as k «, then the transient solution has the form: t?0

c k( t ) = A(t)k" exp(k z(t))

(k-*») (26.a)

with z(t) = z Q + [u(-z Q)-l]t

(26.b)

and (

A(t) = A(o) [ l - t / t 0] " ^ o - D

.

c)

Other initial conditions lead to different transient behavior. E.g. if c k( o ) ~ εχρ[-Δ(ο^β + k z Q] , one finds trans­ ( n a ients, behaving as: c k( t ) ~ exp[-A(t)kP + k z(t)]

(27.a)

with z(t) given by (26.b), and A(t) by A(t) = Δ(ο)[ΐ - t/t Q]P

(22)

k z

e (t)

u ( s c) = 1/t

k

c.
5

( s c) ] - V / 2

where the time dependence of s c( t ) is implic­ itly determined by

(19)

with a defined in (9.b). Comparison of (18.a,b), (19) with the large-k-solution (9.a,b) shows that both lead to the same ex­ pressions if we identify z(t) as z(t) « -6/s(t) as ζ ψ 0, or s co. The same arguments apply for non-gelling systems (λ<1) where τ in (18.a,b) should be replaced by 2, and s(t) diverges as t + ».

v(x) =

Clearly t Q is finite, on account of our assumption (5), or z(o) < 0. In the model K(i,j) = ij, the large-kbehavior of c k( t ) may be calculated from the exact solution". In this case the results are expressed in terms x of the generating function u(x) = £ k c k( o ) e ^ . f Again we distinguish the possibilities: (i) u ( - z ) - ·, and (ii) u'(-z ) < «, where Z Q = z(o). Case (i), u'(-z ) = c o , leads to universal asymptotic behavior of the form

(27.b)

Thus, the transients cross over to the uni­ versal , form (23)-(25) within a finite time t 0= l / u ( - z 0) , with t Q< t c, where t c-l/u'(o) is the gelpoint in this model. We conclude that the asymptotic ( k + c o ) behavior in the models K(i,j) • 2 and

( 2 6

Tail distribution for large clusters

K(i,j) = ij is in full agreement with the predictions for models with ν < 1 and ν • 1 respectively. 7. SUMMARY AND CONCLUSIONS We have shown for general homogeneous kernelsf with ν < 1 that the solution of Smolu­ chowski s equation has the form1 c k(t)=A k(t)exp(kz(t)), with k " l o g A k 0 as k •> °°, where the k-dependence of A k( t ) for k-values, much larger than the average cluster size, may be calculated from a recursion relation. The structure of the solutions of this recursion relation is different for ν < 1 and for ν = 1. Depending on the type of initial size distributions c,(o) we distin­ guish universal solutions, and (non-universal) transient solutions, the former reduce to scaling solutions as the average cluster size diverges, s(t) -* «>; the latter cross over to the universal solutions within a finite time. In some exceptional cases (v<0, v-1) we cannot exclude that non-universal solutions are long-lived. The results are as follows. For all kernels with vv 0 , or k A k( o ) + 0 if ν < 0. If, on the other hand, k ~ v i 0g A k( o ) > constant and 0 < ν < 1, or v=0 and £k^A k(o) < ®, one finds transients asymp­ v totic solutions c k( t ) ~ c k( o ) e x p [ k A ( t ) ] , that cross over to the universal solutions in finite time. For ν < 0, we find asymptotic solutions of the form A k( t ) / A k( o )1 _ •> 1 as k+oo, provided that A k( o ; = <9(k"" ^). Finally, the class of kernels with v=l shows fairly diverse universal and non-universal behavior, depending on the details of the coagulation kernel K ( i , j ) . Furthermore it was shown that for short times and monodisperse initial conditions, the universal large-k-solution coincides with the short-time solutions at large cluster sizes. Similarly the universal large-k-solution reduces to the scaling solution at large scaling arguments as the average cluster size diverges, s(t) -* <*>. Thus, the region of validity of the large-k-solution overlaps with those of the short-time solutions and the scaling solution, both taken at large cluster sizes. The predictions from the present theory for large cluster sizes have been verified in two exactly soluble models, viz. K(i,j) = 2 and K(i,j) = ij. We conclude that our present results, taken together with references 2-4 give a fairly complete analytic description of the asymptotic properties of the cluster size distribution if the coagulation kernel K(i,j)

307

is a homogeneous function of i and j, at least for large i and j. ACKNOWLEDGEMENT The work of one of us (P.G.J, v. D.) is part of a research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM),which is financially supported by the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (ZWO). REFERENCES 1. R.L. Drake, in: Topics in Current Aerosol Research, Vol. 3, eds. G.M. Hidy and J.R. Brock (Pergamon Press, New York, 1972) part 2. 2. A.A. Lushnikov, J. Coll. Interface Sci. 45 (1973) 549; M.H. Ernst, E.M. Hendriks and F. Leyvraz, J. Phys. A: Math. Gen. 17 (1984) 2137. 3. F. Leyvraz, Phys. Rev. A 29 (1984) 854; P.G.J, van Dongen and M.H. Ernst, J. Phys. A: Math. Gen., 18 (1985). 4. S.K. Friedlander and C S . Wang, J. Coll. Interface Sci. 22 (1966) 126; F. Leyvraz and H.R. Tschudi, J. Phys. A: Math. Gen. 15 (1982) 1951; P.G.J, van Dongen and M.H. Ernst, Phys. Rev. Lett. 54 (1985) 1396. 5. P.G.J, van Dongen and M.H. Ernst, J. Coll. Interface Sci., submitted. 6. R.M. Ziff, M.H. Ernst and E.M. Hendriks, J. Phys. A: Math. Gen. 16 (1983) 2293.