Firstorder diffusive effects in the coagulation of colloidal suspensionsStephen Baloga Citation: The Journal of Chemical Physics 70, 1129 (1979); doi: 10.1063/1.437612 View online: http://dx.doi.org/10.1063/1.437612 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/70/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in First-order virial expansion of short-time diffusion and sedimentation coefficients of permeable particlessuspensions Phys. Fluids 23, 083303 (2011); 10.1063/1.3626196 Diffusion into a nanoparticle with first-order surface reaction confined within a sphere J. Chem. Phys. 116, 5137 (2002); 10.1063/1.1453963 Alternative solution for diffusion to two spheres with first-order surface reaction J. Chem. Phys. 113, 10818 (2000); 10.1063/1.1323730 The firstorder concentration dependence of the mutual diffusion coefficient of Brownian particles J. Chem. Phys. 89, 6470 (1988); 10.1063/1.455366 The photoacoustic effect at firstorder phase transition J. Appl. Phys. 51, 6115 (1980); 10.1063/1.327641

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First-order diffusive effects in the coagulation of colloidal suspensions

Stephen Baloga

5 Adgate Drive, Ijamsville, Maryland 21754 (Received 28 August 1978)

Solutions to an infinite system of reaction-diffusion equations associated with colloidal coagulation are obtained by incorporating first-order diffusive effects into restricted diffusion-free initial value problem solutions. It is shown that this method of approximation yields explicit asymptotically exact solutions for sufficiently small values of time. In addition, the diffusion-corrected solutions are recast in closed form by assuming that the ditTusivity of the m-fold aggregates is given by Dm = Dm-a, where D and 8 are positive constants.

I. INTRODUCTION

To describe the simultaneous diffusion and chemical combination that occurs in the coagulation of colloidal suspensions, consider the system

(1.1)

i,j,m=1,2,3, .•• ,

where C", = C",(x, t) denotes the concentration of m-fold aggregates and D", and KlJ are positive constant parame­ters for all indicial values. The double summation ap­pearing in Eq. (1.1) represents the local formation of m-fold particles due to the coalescence of i- and j-fold aggregates (such that i + j = m), while the single summa­tion describes the decrease resulting from collisions between m-fold aggregates and all other species.

For many years, it has been known 1 that the broad features of colloidal coagulation follow from the diffu­sionfree speCialization of Eq. (1.1):

8C~ 1" 0 0 0 ~ 0 ar=-2 L.J KlJCiCJ-C",L.JK"'ICJ ,

h/=", 1=1 (1. 2)

and the mathematically simplifying assumption KiJ =K. Associated with this phenomenon are the initial condi­tions

o {a(x) , m = 1 C",(x, 0)=

o , m >1 (1. 3)

corresponding to an arbitrary initial distribution of one­fold particles in colloidal suspension, and the solution of Eq. (1. 2):

C~(x, t) = a"'(x) (2-1 Kt)"'-l [1 + 2-1 a(x) Kt]-m-l (1.4)

Provided the rate constants Kil and the initial condi­tions are such that the D", VZ C", terms are small com­pared to the other members of Eq. (1.1), diffusive ef­fects will manifestZ as perturbative corrections to the solutions of Eq. (1.2). The analysis presented in Sec. IT demonstrates that, for the special case Kil =K, first­order diffusive effects can be incorporated directly into the solutions of the initial value problem posed by Eqs. (1. 2) and (1. 3). It is established in Sec. IT! that this method of approximation yields explicit asymptotically exact solutions for suffiCiently small values of time. In addition, closed form diffusive perturbations are shown to be engendered by the assumption D", = Dm ~ , where D and 15 are positive constants. TYpical analytical closed form expressions are exhibited in the Appendix for the case 15 = 1 .

II. EVALUATION 'OF THE DIFFUSION MATRIX KERNEL

By setting Kil =K, space-time units can be chosen so that Eq. (1,1) takes the more convenient form

(2.1)

Recent theoretical investigations of diffusive effects in rate processesZ- 4 suggest the prescription

C'" := C~ [a(x), t] + [1 + a(x)t]-Z It t [1 + a(x) s]Z D"", [a(x), t, s] VZC~ [a(x) , s] ds , o k-l

(2.2)

where C~(a, t) satisfies the diffusionfree specialization of Eq. (2.1) and the elements of the so-called diffusion ma­trix kernel D"'k [a(x) , t, s] are to be determined. Substituting Eq. (2.2) into (2.1) and retaining terms linear in D"'k and D", produces the conditions

(2.3)

and

D"'k(a, t, t) =D",I5"'k , (2.4)

in which 15"", is the Kronecker delta. Equation (2.3) is recast in a more tractable form

J. Chem.Phys. 70(03), 1 Feb. 1979 0021-9606/79/031129-03$01.00 © 1979 American Institute of Physics 1129

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1130 Stephen Baloga: Coagulation of colloidal suspensions

8~t=-2DkC~+2 LC~Djk (2.5) hj=m

by summing over the index m and employing Eq. (2.4). To simplify the appearance of the integrals of Eq. (2.5),

let Hn(a, t, s)"'=' (t L 2j ft ft1 ... [1-1 C~ (a, t1) C~2(a, t2) ••. C~j(a, t j )dt1dt2 •·• dtj , n >0 , j=l (of •• • 1 (2.6) o , n"; 0 ,

where {q} denotes any set of j positive integers such that j

Lqr=n (2.7)

and ~(o) indicates summation over all distinct permutations of {q}. It is straightforward to show (by induction) that

(2.8)

satisfies the differential equation (2.5) as well as the boundary condition (2.4). By substituting Eq. (2.8) into (2.2), one obtains

Cm=C~(a,t)+(1+atr2Dm £' (1+as)2V2C~(a,s)ds o

t ~

+ (1 +at)-2 I L Dk [Hm..,,(a, t, s) - Hm(a, t, s)] (1 + as)2 V2 C2(a, s) ds o k=l

(2.9)

Since the diffusionfree solutions are given by Eq. (1.4), all quantities appearing in Eqs. (2.6) and (2.9) are known explicitly. Thus, analytic solutions of Eq. (2.1)' accurate to first order in the diffusivities, are obtained by perform­ing the integrals indicated in Eqs. (2.6) and (2.9).

III. ASYMPTOTICALLY EXACT AND CLOSED FORM SOLUTIONS

To clarify the validity of the linear diffusivity approxi­mation of the previous section, one can expand Eq. (2. 9) in ascending powers of t and compare the result with the expansion of the exact solution of Eq. (2.1). If terms to order tmare retained, Eq. (2.9) becomes

Cm(a, t) =am tm-1[1 - (m + l)at]

+am-1 tm f: DkAmk[V2a + (k _1)a-11 V a1 2] +0(tm+1) ,

k-1 (3.1)

in which

k>m

(3.2)

k<m

The set of positive integers {q}={q1' q2"'" qj} to be used in Eq. (3.2) must satisfy

j

Lqr=m-k, (3.3) r-1

but is otherwise arbitrary. As before, ~Iof denotes summation over all distinct permutations of the set {q}.

One can obtain the expansion of the exact solution in ascending powers of t by applying the Cauchy-Kovalevska theoremS to the differential equation (2.1) and the initial conditions stated in Eq. (1.3). It can then be verified that Eq. (3.1) is identical to the expansion of the exact

solution of Eq. (2.1) to order t m.

The diffusive perturbations exhibited in Eq. (2.9) can be recast in closed form by assuming that the diffusivity of the m-fold aggregates is given by

(3.4)

where D and 0 are positive constants. If Dm is inter­preted as an average over all diffusivities associated with the possible geometric configurations of the m particles, then Eq. (3.4) implies that the average radius of the m-fold aggregates is proportional to mO.

Suppose I V 2a I is continuous and uniformly bounded by a constant. It follows that the infinite series ~;=1 Dk V2C~(a, t) converges uniformly for all finite values of at and for any diffusivity prescription such that Dk > Dk+1 for all k. Under these conditions, the Laplacian can be factored out of the summation.

In view of Eq. (2.6) and the preceding paragraph, Eq. (2.9) will contain only a finite number of terms if a closed form can be found for the infinite series ~;=lDkC~, By evoking the prescription Dk=Dk-G and em­ploying well-known methods, 6 it is easy to demonstrate that the infinite series D L:;=1 k-G cg has the integral rep­resentation

~

L Dk C~(a, t) = Da(l + at)-l r-1(o) k=l

x I~ zO-l[e" +at(e.r -l)]-ldz (3.5) o

Substitution of Eq. (3.5) into (2.9) yields the desired closed form of the diffusion-corrected solutions

t t m-l

Cm=C~(a,t)+(1+atr2Dm-o r (1+as)2V2C~(a,s)ds+(1+att2D I Lk-OHm_k(a,t,s)(1+as)2V2C~(a,s)ds }o 0 k-1

J. Chern. Phys., Vol. 70, No.3, 1 February 1979

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Stephen Baloga: Coagulation of colloidal suspensions 1131

- (1 + at)"2 D f Hm(a, t, s) (1 +as)2 r-l(O) V2 {a(l + as)-l fa" zG-l [e6 + as(e" - 1 )]-1 dZ} ds+ O(n2)

I

(3.6)

Explicit analytic solutions for the corrected total parti­cle density and the corrected concentrations C1 and C2 can be found in the Appendix for the special case 0 = 1.

IV. CONCLUSIONS

This communication illustrates a method for incor­porating first-order diffusive effects into the initial-val­ue problem solutions of an infinite system of nonlinear rate equations. Owing to the severe restriction on the rate constants, the main results [Eqs. (2.9), (3.1), and (3.6)], represent only the broad qualitative features of the simultaneous diffusion and chemical reaction that occurs in the coagulation of colloidal suspensions. At­tempts are currently under way to extend the method to a more realistic aggregation model that would be suit­able for a detailed comparison with experimental re­sults. Moreover, it is likely that the mathematical technique established in Sec. II can be applied to other rate phenomena featuring comparatively small diffusive effects and an infinite system of governing equations.

and

APPENDIX

To illustrate the explicit structure of the diffusive ef­fects obtained by this method, let us set the parameter o equal to unity and evaluate the integrals appearing in Eq. (3.6).

With the infinite set of diffusivities so specified, Eq. (3.6) implies that the total particle density is given by .. .. L Cm(a,t)= L C~(a,t)+D(1+att2fo(a,t)+O(D2), (A1) mel m=l

where fo represents

fo =[2 at - (1 +at) In(l + at)] a-1 V2 a

+ [(5 + 2 at}ln(l + at) -ln2(1 + at) - 5 at] a-21 Va 12

Direct evaluation of Eq. (3.6) yields the following expres­sions for the diffusion-corrected dynamical behavior of the concentrations C 1 and C2 :

C2 =C~ + D(l +att2 [t gg - 2gt - f2 + 2 at(l +atfl (~+ 2f1) - 3a2 t2(1 +atf2 fo] + O(D2) ,

where

(A3)

11 =[2at- (2 +at)ln(l +at)+t In2(1 +at)]a-1V 2a +{[4+at - (1 +a/fl]ln(l +at)2 - 21n2(1 +at) - 6at+a2 t2(1 +atrl }a-21 val 2 , (A4)

12 = {[(1 + aWl - at - 3]ln(1 + at) + In2(1 + at) + 2 at}a-1 V2 a +{[ll + 2 at - 6(1 + aWl + (1 + a/t2]ln(1 + at)

and

{

k+m-l } g~ = (2m+k)ln(1 +at) - (k+m)at+ ~ [(k+m)(P + 1r1at _m]p-l al> tl>(l +at)-P a-1V2 a

+{-2(3m +k + l)ln(l + at) + 2(m+k + 1) at+ m (m -1) (m +kt1 (at/l + at)m+1I

m+1I + L [4m -2(m +k+ 1) (P + l)-l atJp-laJ> tl>(l +at)"l>} a-2 Iv al 2

1>=1

I

(A5)

IS. Chandrasekhar, Rev. Mod. Phys. 15, 59 (1943). 2G. Rosen, Phys. Lett. A 43, 450 (1973).

5p. Dennery and A. Krzywicki, Mathematics for Physicists

3G. Rosen, J. Chern. Phys. 63, 417 (1975). 4G. Rosen, SIAM J. Appl. Math. 29, 146 (1975).

(Harper and Row, New York, 1967), pp. 333-334. 6Z. Melzak, Companion to Concrete Mathematics (Wiley, New

York, 1973), pp. 109-110.

J. Chern. Phys., Vol. 70, No.3, 1 February 1979

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