M. H. Ernst

Coagulation Equations with Gelation

Smoluchowskis equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernelKij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelationtc, and give an exact postgelation solution for the modelKij=(ijω) (ω>1/2) andKij=ai+j (a>1). For the modelKij=iω +jω (ω<1, without gelation) initial solutions are given. It is argued that the kernelKij∼ijω with ω≃1−1/d (d is dimensionality) effectively models the sol-gel transformation in polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (Ω=1). For allΩ the exponents,t=Ω+3/2 andσ=Ω−1/2,γ=(3/2−Ω)/(Ω − 1/2) andΒ=1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.

Scaling solutions of Smoluchowski's coagulation equation

We investigate the structure of scaling solutions of Smoluchowskis coagulation equation, of the formck(t)∼s(t)−τ′ ϕ(k/s(t)), whereck(t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, andϕ(x) is a scaling function. For the rate constantK(i, j) in Smoluchowskis equation, we make the very general assumption thatK(i, j) is a homogeneous function of the cluster sizesi andj:K(i,j)=a−λK(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j→0 asj→∞. We show that gelation occurs ifλ>1, and does not occur ifλ⩽1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation forϕ(x). We present a detailed analysis of the behavior ofϕ(x) at large and small values ofx. For all models, we find exponential large-x behavior: ϕ(x)∼Ax−λe−δx asx→∞ and, for different kernelsK(i, j), algebraic or exponential small-x behavior: ϕ(x)∼Bx−τ or ϕ(x)=exp(−Cx−|μ| + ...) asx↓0.

Coagulation processes with a phase transition

Abstract Smoluchowskis coagulation equation with a collection kernel K(x, y) ∼ (xy)ω with 1 2 describes a gelation transition (formation of an infinite cluster after a finite time tc (gel point)). For general ω and t > tc the size distribution is c(x, t) ∼ x−τ for x → ∞ with τ = ω + 3 2 . For ω = 1, we determine c(x, t) and the time dependent sol mass M(t) for arbitrary initial distribution in pre- and post-gel stage, where c(x, t) ∼ x − 5 2 exp (−x/x c ) for large x and t c(x, t) ∼ (− M ) 1 2 x − 5 2 for large xt and t > tc. Here xc is a critical cluster size diverging as (t - tc)−2 as t ↑ tc. For initial distributions such that c(x, 0) ∼ xp-2 as x → 0, we find M(t) ∼ t−p/(p+1) as t → ∞. New explicit post-gel solutions are obtained for initial gamma distributions, c(x, 0) ∼ xp-2e−px (p > 0) in the form of a power series (convergent for all t), and reducing for p = ∞ to the solution for monodisperse initial conditions. For p = 1, the solution is found in closed form.

Kinetics of reversible polymerization

This paper extends the kinetic theory of irreversible polymerization (Smoluchowskis equation) by including fragmentation effects in such a way, that the most probable (equilibrium) size distribution from the classical polymerization theories is contained in our theory as the stationary distribution. The time-dependent cluster size distributionck(a(t)) in Florys polymerization modelsRAf andAfRBg, expressed in terms of the extent of reactionα, has the same canonical form as in equilibrium, and the time dependence ofα(t) is determined from a macroscopic rate equation. We show that a gelation transition may or may not occur, depending on the value of the fragmentation strength, and, in case a phase transition takes place, we give Flory- and Stockmayer-type postgel distributions.

Kinetics of gelation and universality

The exact solution (size distribution ck(t) and moments M, (t)) of Smoluchowskis coagulation equation (S-model) and of a modified equation (F-model) with a coagulation rate K,, = ij for i- and j-clusters is obtained for arbitrary Ck(0) in the sol (t (,) phases, where tc is the gel point. The behaviour of ck(t) and M,(t) is given for k -+CO, t -+ CO and t + I,. The critical exponents, critical amplitudes and scaling function that characterise the singularities near the non-equilibrium phase transition are calculated. For short-range ck(0) (i.e. all M, <a) the F-model belongs to the universality class of classical gelation theories and of bond percolation on Cayley trees; the S-model does not.

Cluster size distribution in irreversible aggregation at large times

It is assumed that the size distribution ck(t) satisfies Smoluchowskis coagulation equation with rate coefficients K(i, j), behaving as K(i, j) approximately imu jnu (i 0) the general solution ck(t) approaches for t to infinity the exact solution Cbk/t(k=1, 2, . . .), where the bks are independent of the initial conditions ck(0), and can be determined from a recursion relation. In class II systems ( mu =0), ck(t)/c1(t) to bk (t to infinity , k=1, 2, . . .), but the bks depend on ck(0). Only in the scaling limit (k to infinity , s(t) to infinity with k/s(t)=finite; s(t) is the mean cluster size) does ck(t) approach a form independent of the initial distribution. Class III, where ck(t)/c1(t) to infinity (t to infinity , k=2, 3, . . .), has not been considered here.

Exactly soluble addition and condensation models in coagulation kinetics

Abstract The kinetics of clustering through addition ( a k + a 1 → a k +1 ) and condensation ( a j + a k → a j + k , j ≠ 1, k ≠ 1) for a model of cylindrically shaped monomeric units a 1 are studied, using Smoluchowskis coagulation equation, and analytic solutions for several limiting cases (flat disks and needles) with and without condensation reactions, were given. The condensation models of flat disks and needles include, respectively, the linear polymer model RA 2 and the branched polymer model A 2 RB ∞ (with a gelation transition). If condensation reactions are inhibited, we obtain exactly soluble addition models with a monomer-cluster rate constant independent of or proportional to the cluster size. The monomers are (i) supplied in a given amount at the initial time; (ii) generated by a steady source; or (iii) supplied by an infinite reservoir that keeps the concentration of monomers, c 1 ( t ), constant in time.

Exact solutions of the nonlinear Boltzmann equation

A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf(v, t); (ii) moment equations and polynomial expansions off(v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at υ→∞; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.

On the occurrence of a gelation transition in Smoluchowski's coagulation equation

It has been conjectured by Lushnikov and Ziff that Smoluchowskis coagulation equation describes a gelation transition, i.e., the mean cluster size diverges within a finite timetc (gelpoint) if the coagulation rate constantsK(i,j) have the propertyK(ai,aj)=aλK(i,j), with λ>1. The existing evidence was based on self-consistency arguments. Here we prove this conjecture for an appropriate class of physically acceptable rate constants by constructing a finite upper bound fortc and a nonvanishing lower bound. Apart from the exactly solved caseK(i,j)=ij this result provides the first solid proof of the occurrence of a gelation transition in a description based on Smoluchowskis coagulation equation.

Tail distribution of large clusters from the coagulation equation

For sizes much larger than the mean cluster size we determine the asymptotic solution of Smoluchowskis coagulation equation for coagulation kernels Kij, being homogeneous functions of the cluster sizes i and j. We find that the size distribution has the universal form ck(t) ∼ z(t)k−λexp(kz(t)) as k → ∞, where dependence on the initial distribution enters only through z(t). For certain classes of initial distributions transients appear that cross over to universal solutions within a finite time.