Dynamic Scaling in the Kinetics of Clustering

Abstract

In irreversible aggregation processes without a gelation transition the cluster size distribution approaches a scaling form, c"(t) -s 2@(k/s). Usking Smoluchowski's coagulation equation we determine the exponents in the mean cluster size s(t) -t' (t~) and in the small-and large-x behavior of the scaling function @(x). Depending on certain characteristics of the coagulation coefficients, @(x) -x ' (x 0) or $(x) -exp( -x") (x 0) with p, some negative constant. In aggregation processes with gelation a similar scaling form is obtained as t approaches the gel point.PACS numbers: 64.60. -i, 05.50. +q, 64.75. +g, 82.35.+t To study the kinetics of irreversible aggregation and clustering phenomena, in particular the time evolution of the cluster size distribution ck(t), Smoluchowski s coagulation equation is one of the few available, and also one of the most widely used, theoretical tools in many fields of physics, astronomy, polymer physics, colloid chemistry, atmospheric physics, biology, and technology. ' 5 It reads Ck= 2 g K(l,J)ctcJ Ck XK(k,J) Jc, i+ j=k j=l where the coagulation kernel K (i j ) represents the rate coefficient for a specific clustering mechanism between clusters of sizes i and j. We distinguish gelling and nongelling mechanisms.In the former the mean cluster size s(t) diverges as t approaches the gel point t, ; in the latter s (t) keeps increasing with time.It is known from exact solutions, ' coagulation experiments, and computer simulations that the size distribution approaches a scaling form, c"(t) -s '@(k/s), as soon as s(t) becomes large compared to the characteristic size at the initial time. The important point is that the k and t dependence of ck (t) is given through a universal function of a single variable, k/s(t), that does not depend on the initial distribution. For a limited number of coagulation mechanisms, all belonging to class III (see below), Friedlander's theory of selfpreserving spectra (SPS theory) gave a satisfactory explanation of the experimental observations on Brownian coagulation in the hydrodynamic and molecular regime, although the experimental data at large k and t are rather poor.By generalizing the SPS theory we can give a unifying description of the scaling behavior occurring in gelling and nongelling systems, described by Smoluchowski's equation.Our generalization covers all coagulation kernels K (i j) that are homogeneous functions of i and j, and includes large classes of models, for which the original SPS theory is not valid, e.g. , K(ij) =i +j.Since Smoluchowski's equation with a homogeneous kernel is invariant under a group of similarity transformations, it admits exact similarity or scaling solutions, 2 3 that can be solved from a nonlinear integral equation and whose properties are analyzed in this Letter. The basic assumption of our method is that the solutions for general initial distributions indeed approach the special similarity solution. With the help of the integral equation we can determine scaling functions and related exponents, analytically or nu...

Models of chemical reactions with participation of polymers

“…(33) can be solved explicitly for arbitrary kernels fl, which decrease with an increase of I. It was found [94] that the MWD P(N,t) has the multinomial form This is exactly the result obtained by means of scaling arguments in [95,96]. Let us also examine the form of the MWD in the limit of small molecular weights N (or large times).…”

“…(33) have been analysed using scaling arguments in [95,96] for algebraically decreasing kernels in Eqn. (34).…”

Numerical evaluation of self-preserving spectra in smoluchowski's coagulation theory

“…In the latter case, the integral equation for the scaling function, resulting from Friedlander's theory, is ill-defined and must be modified (6). This point has not been noticed in the older literature.…”

“…These findings can be explained from Friedlander's theory of self-preserving spectra (1), which is essentially based on the assumption that, after some initial transient period, the solution of Smoluchowski's equation approaches a similarity or scaling form depending on time and cluster size only through a single argument. Smoluchowski's equation indeed admits similarity solutions, provided the coagulation rates obey certain homogeneity properties (1,6).…”

“…However in the kinetic collision rates for (fractal) soot agglomerates (6), one must account for the fractal geometry of the cluster with a Hausdorff dimension D -~ 1.9. One can show (15) that the appropriate collision rate for D ~< 2 be-…”