It is proposed to revisit the inverse problem associated with Smoluchowski's coagulation equation. The objective is to reconstruct the functional form of the collision kernel from observations of the time evolution of the cluster size distribution. A regularised least squares method originally proposed by Wright and Ramkrishna (1992) based on the assumption of self-similarity is implemented and tested on numerical data generated for a range of different collision kernels. This method expands the collision kernel as a sum of orthogonal polynomials and works best when the kernel can be expressed exactly in terms of these polynomials. It is shown that plotting an "L-curve" can provide an a-priori understanding of the optimal value of the regularisation parameter and the reliability of the inversion procedure. For kernels which are not exactly expressible in terms of the orthogonal polynomials it is found empirically that the performance of the method can be enhanced by choosing a more complex regularisation function.
Temperature-dependent Smoluchowski equations describe the ballistic agglomeration. In contrast to the standard Smoluchowski equations for the evolution of cluster densities with constant rate coefficients, the temperature-dependent equations describe both -- the evolution of the densities as well as cluster temperatures, which determine the aggregation rates. To solve these equations, we develop a novel Monte Carlo technique based on the low-rank approximation for the aggregation kernel. Using this highly effective approach, we perform a comprehensive study of the phase diagram of the system a…
We revisit the scaling theory of the Smoluchowski equation with special emphasis on the dimensional analysis to derive the scaling ansatz and to give an insightful foundation to it. It has long been argued that the homogeneity exponent $λ$ of the aggregation kernel divides the aggregation process into two regimes (i) $λ\leq 1$ nongelling and (ii) $λ>1$ gelling. However, our findings contradict with this result. In particular, we find that the Smoluchowski equation is valid if and only if $λ<1$. We show that beyond this limit i.e. at $λ\geq 1$, it breaks down and hence it fails to describe a ge…
In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by comput…
We present a rigorous thermodynamic treatment of irreversible binary aggregation. We construct the Smoluchowski ensemble as the set of discrete finite distributions generated from the same initial state of all monomers upon fixed number merging events and define a probability measure on this ensemble such that the mean distribution in the mean-field approximation is governed by the Smoluchowski equation. In the scaling limit this ensemble gives rise to a set of relationships completely analogous to those of familiar statistical thermodynamics. The central element of the thermodynamic treatment…
The Smoluchowski equation for irreversible aggregation in suspensions of equally charged particles is studied. Accumulation of charges during the aggregation process leads to a crossover from power law to sub-logarithmic cluster growth at a characteristic time and cluster size. For larger times the suspension is usually called stable, although aggregation still proceeds slowly. In this regime the size distribution evolves towards a {\em universal} scaling form, independent of {\em any} parameter included in the theory. The relative width falls off to a universal value $σ_{\rm r}^{\infty}{\appr…
The paper compares the statistical description of physical-metallurgical processes and ceramic-polycrystalline evolutions, termed the normal grain growth (NGG), as adopted to soft- and chemically-reactive grains, with a Smoluchowski's population-constant kernel cluster-cluster aggregation (CCA) model, concerning irreversible chemical reaction kinetics. The former aiming at comprehending, in a semi-quantitative way, the volume-conservative (pressure-drifted) grain-growth process which we propose to adopt for hydrogel systems at quite low temperature (near a gel point). It has been noticed, that…
We consider the scaling solutions of Smoluchowski's equation of irreversible aggregation, for a non gelling collision kernel. The scaling mass distribution f(s) diverges as s^{-tau} when s->0. tau is non trivial and could, until now, only be computed by numerical simulations. We develop here new general methods to obtain exact bounds and good approximations of $τ$. For the specific kernel KdD(x,y)=(x^{1/D}+y^{1/D})^d, describing a mean-field model of particles moving in d dimensions and aggregating with conservation of ``mass'' s=R^D (R is the particle radius), perturbative and nonperturbative…
We study discontinuous percolation transitions (PT) in the diffusion-limited cluster aggregation model of the sol-gel transition as an example of real physical systems, in which the number of aggregation events is regarded as the number of bonds occupied in the system. When particles are Brownian, in which cluster velocity depends on cluster size as $v_s \sim s^η$ with $η=-0.5$, a larger cluster has less probability to collide with other clusters because of its smaller mobility. Thus, the cluster is effectively more suppressed in growth of its size. Then the giant cluster size increases drasti…
In a non-equilibrium system, a Constant Flux Relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean--field theory is applicable or not. We focus on cluster--cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a gen…
We present a detailed study of the statistics of a system of diffusing aggregating particles with a steady monomer source. We emphasise the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean-field description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterised by a constant flux of mass directed from small to large masses. It admits a phenomenological description based on the assumption of self-similarity and constant mas…
Kinetic Criticality in Linker-Mediated Colloidal Aggregation
Center for Functional Nanomaterials, Brookhaven National Laboratory（布鲁克海文国家实验室功能纳米材料中心）；Department of Chemical Engineering, Columbia University（哥伦比亚大学化学工程系）；Avram and Stella Goldstein-Goren Department of Biotechnology, Ben-Gurion University of the Negev（内盖夫本-古里安大学阿夫拉姆和斯特拉·戈德斯坦-戈尔恩生物技术系）；Department of Applied Physics and Applied Mathematics, Columbia University（哥伦比亚大学应用物理与应用数学系）；Center for Nanomedicine, Institute for Basic Science（基础科学研究所纳米医学中心）；Department of Nano Biomedical Engineering, Yonsei University（延世大学纳米生物工程系）
Impact of Initial Charge Distributions on the Kinetics of Charged Particle Coagulation
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins（奥希金斯大学工程科学研究所）；Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile（智利大学物理与数学科学学院物理系）
We revisit a simple lattice model of aggregation in which masses diffuse and coalesce upon contact with rate 1 and every nonzero mass chips off a single unit of mass to a randomly chosen neighbour with rate $w$. The dynamics conserves the average mass density $ρ$ and in the stationary state the system undergoes a nonequilibrium phase transition in the $(ρ-w)$ plane across a critical line $ρ_c(w)$. In this paper, we show analytically that in arbitrary spatial dimensions, $ρ_c(w) = \sqrt{w+1}-1$ exactly and hence, remarkably, independent of dimension. We also provide direct and indirect numerica…
This work outlines an exact combinatorial approach to finite coagulating systems through recursive equations and use of generating function method. In the classic approach the mean-field Smoluchowski coagulation is used. However, the assumptions of the mean-field theory are rarely met in real systems which limits the accuracy of the solution. In our approach, cluster sizes and time are discrete, and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters and applying monodisperse initial conditions, the exact expressi…
