P. G. J. van Dongen

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.

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.

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.

On the possible occurrence of instantaneous gelation in Smoluchowski's coagulation equation

Considers possible solutions of Smoluchowskis coagulation equation if the rate constants K(i, j) behave as K(i, j) approximately imu jv as j to infinity , with an exponent nu satisfying nu >1. The author finds that, for such rate constants. Smoluchowskis equation predicts the instantaneous occurrence of a gelation transition. Thus the gel time tc=0 in such models. This result confirms recent, speculation in the literature. The author also studies the structure of post-gel solutions of Smoluchowskis equation, if they exist. For a given value of nu , the results depend on the value of the exponent mu . If mu >( nu -1), one finds that the cluster size distribution ck(t) approaches a universal form at large times (t to infinity ). No solutions exist if mu <or=( nu -1). Physically this means that the sol phase is depleted instantaneously.

Fluctuations in Coagulating Systems. II

Time-dependent fluctuations in a system of coagulating particles are studied, using the master equation for the probability distributionsP(m,t) for the occupation numbersm={mk} (k=1,2,...) of thek-cluster states. Van KampensΩ-expansion is used to determine the deterministic (orderΩ0) and fluctuating part (orderΩ−1/2) of the solution. We calculate the time-dependent behavior of the fluctuations in the cluster size distribution. The model under consideration is of special interest since it exhibits a phase transition (gelation). For monodisperse initial states we give explicit expressions for the probability distribution of the fluctuations and for the equal-time and two-time correlation functions also near the phase transition. For general initial conditions we study the fluctuations (1) for large cluster sizes, (2) in the scaling limit (near the critical point), and (3) for large times. Our results show that the deterministic approach to coagulation processes (Smoluchowski theory) is invalid very close to the gelpointtc and at large times (t≳tM), where the distance from the gelpoint and the timetM depend upon the size of the system.

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.

Spatial fluctuations in reaction-limited aggregation

A general method is used for describing reaction-diffusion systems, namely van Kampens “method of compounding moments,” to study the spatial fluctuations in reaction-limited aggregation processes. The general formalism used here and in subsequent publications is developed. Then a particular model is considered that is of special interest, since it describes the occurrence of a phase transition (gelation). The corresponding rate constants for the reaction between two clusters of sizei and sizej areKij=ij (i, j=1, 2,⋯). For thediffusion constants Dj of clusters of sizej the following class of models is considered:Dj=D if 1⩽J⩽s andDj=0 ifj>s. The casess=∞ ands<∞ are studied separately. For the models=∞ the equal-time and the two-time correlation functions are calculated; this modelbreaks down at the gel point. The breakdown is characterized by a divergence of the density fluctuations, and is caused by the large mobility of large clusters. For all models withs<∞ the density fluctuations remain finite attc, and the equal-time correlation functions in the pre- and in the post-gel stage are calculated. Many explicit and asymptotic results are given. From the exact solution the upper critical dimension in this gelling model isdc=2.

ABSENCE OF HVSTERESIS AT THE MOTT-HUBBARD METAL-INSULATOR TRANSITION IN INFINITE DIMENSIONS

The nature of the Mott-Hubbard metal-insulator transition in the infinite-dimensional Hubbard model is investigated by Quantum Monte Carlo simulations down to temperature T=W/140 (W=bandwidth). Calculating with significantly higher precision than in previous work, we show that the hysteresis below T_{IPT}\simeq 0.022W, reported in earlier studies, disappears. Hence the transition is found to be continuous rather than discontinuous down to at least T=0.325T_{IPT}. We also study the changes in the density of states across the transition, which illustrate that the Fermi liquid breaks down before the gap opens.

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.

Pre- and post-gel size distributions in (ir)reversible polymerisation

A class of irreversible coagulation processes can be modelled by Smoluchowskis coagulation equation with rate constants Kij=A+B(i+j)+Cij (non-negative A, B and C). For C not=0 a gelation transition occurs. The authors obtain explicit solutions for the size distribution ck(t) with ck(0)= delta k1. Next, they construct and solve the equations for reversible polymerisation by incorporating break-up processes in the kinetic equation with a unimolecular fragmentation rate Fij= lambda NiNjKij/Ni+j. The degeneracy factors Nk obey (k-1)Nk=1/2 Sigma KijNiNj with i+j=k and N1=1, and the strength parameter lambda =exp(g/kBT), where the binding energy g to - infinity for irreversible coagulation. Explicit results are only given for Florys polymerisation models RAf and BRAf-1. In the vicinity of the gel point the authors verify the scaling hypothesis and calculate critical exponents.