Smoluchowski Rate Equations Research Articles

Modified Smoluchowski Rate Equations for Aggregation and Fragmentation in Finite Systems.

Protein self-assembly into supramolecular structures is important for cell biology. Theoretical methods employed to investigate protein aggregation and analogous processes include molecular dynamics simulations, stochastic models, and deterministic rate equations based on the mass-action law. In molecular dynamics simulations, the computation cost limits the system size, simulation length, and number of simulation repeats. Therefore, it is of practical interest to develop new methods for the kinetic analysis of simulations. In this work we consider the Smoluchowski rate equations modified to account for reversible aggregation in finite systems. We present several examples and argue that the modified Smoluchowski equations combined with Monte Carlo simulations of the corresponding master equation provide an effective tool for developing kinetic models of peptide aggregation in molecular dynamics simulations.

Combined Experimental and Theoretical Investigation on Formation of Size-Controlled Silver Nanoclusters under Gas Phase

Metallic nanoclusters (NCs) have been predicted to achieve the best Surface-Enhanced Raman Scattering (SERS) due to the controllable amount of atoms and structures in NCs. The Local Surface Plasmon Resonance (LSPR) effect on silver metal NCs (Ag) enables it to be a promising candidate for manipulating the LSPR peak by controlling the size of NCs, which in turn demands a full understanding of the formation mechanism of Ag. Here, we apply an extended Smoluchowski rate equation coupled with a fragmentation scheme to investigate the growth of size-selected silver NCs generated via a modulated pulsed power magnetron sputtering (MPP-MSP). A temperature-dependent fragmentation coefficient D is proposed and integrated into the rate equations. The consistency between the computational and experimental results shows that in relative low peak power ( W), the recombination of cation and anion species are the dominant mechanism for NC growth. However, in the higher region (≥800 W), the fragmentation mechanism becomes more impactful, leading to the formation of smaller NCs. The scanning electron microscopy observation shows the Ag is successfully soft-landed and immobilized on a strontium titanate crystal, which facilitates the application of the Ag/STO to the SERS research.

Role of energy in ballistic agglomeration.

We study a ballistic agglomeration process in the reaction-controlled limit. Cluster densities obey an infinite set of Smoluchowski rate equations, with rates dependent on the average particle energy. The latter is the same for all cluster species in the reaction-controlled limit and obeys an equation depending on densities. We express the average energy through the total cluster density that allows us to reduce the governing equations to the standard Smoluchowski equations. We derive basic asymptotic behaviors and verify them numerically. We also apply our formalism to the agglomeration of dark matter.

Application of Extended Smoluchowski Equations to Formation of Silver Nanoclusters Generated by Direct Current Magnetron Sputtering Source

We apply the extended Smoluchowski rate equations with neutralization process between nanocluster (NC) anion and cation to study the formation of silver NCs generated by direct current magnetron sputtering (DC-MSP). By comparing the experimental and simulation results, it was found that the origin of size tuning by the cell length was prolonged aggregation time. Size tuning by DC power (P) was attributed to the enhanced initial number density and ionization fraction of the sputtered materials. It was revealed that parts of NC ions were “lost” after aggregation and the amount of lost ions increased with power (20 W < P < 100 W), which was caused by the accelerated neutralization process. The high consistency between experiment and simulation suggests that the extended Smoluchowski rate equations are capable to describe the NC formation in gas phase.

Dynamic aggregation evolution of competitive societies of cooperative and noncooperative agents

We propose an evolution model of cooperative agent and noncooperative agent aggregates to investigate the dynamic evolution behaviors of the system and the effects of the competing microscopic reactions on the dynamic evolution. In this model, each cooperative agent and noncooperative agent are endowed with integer values of cooperative spirits and noncooperative spirits, respectively. The cooperative spirits of a cooperative agent aggregate and the noncooperative spirits of a noncooperative agent aggregate change via four competing microscopic reaction schemes: the win-win reaction between two cooperative agents, the lose-lose reaction between two noncooperative agents, the win-lose reaction between a cooperative agent and a noncooperative agent (equivalent to the migration of spirits from cooperative agents to noncooperative agents), and the cooperative agent catalyzed decline of noncooperative spirits. Based on the generalized Smoluchowski's rate equation approach, we investigate the dynamic evolution behaviors such as the total cooperative spirits of all cooperative agents and the total noncooperative spirits of all noncooperative agents. The effects of the three main groups of competition on the dynamic evolution are revealed. These include: (i) the competition between the lose-lose reaction and the win-lose reaction, which gives rise to respectively the decrease and increase in the noncooperative agent spirits; (ii) the competition between the win-win reaction and the win-lose reaction, which gives rise to respectively the increase and decrease in the cooperative agent spirits; (iii) the competition between the win-lose reaction and the catalyzed-decline reaction, which gives rise to respectively the increase and decrease in the noncooperative agent spirits.

Kinetic evolutionary behavior of catalysis-select migration

We propose a catalysis-select migration driven evolution model of two-species (A- and B-species) aggregates, where one unit of species A migrates to species B under the catalysts of species C, while under the catalysts of species D the reaction will become one unit of species B migrating to species A. Meanwhile the catalyst aggregates of species C perform self-coagulation, as do the species D aggregates. We study this catalysis-select migration driven kinetic aggregation phenomena using the generalized Smoluchowski rate equation approach with C species catalysis-select migration rate kernel K(k;i,j) = Kkij and D species catalysis-select migration rate kernel J(k;i,j)= Jkij. The kinetic evolution behaviour is found to be dominated by the competition between the catalysis-select immigration and emigration, in which the competition is between JD0 and KC0 (D0 and C0 are the initial numbers of the monomers of species D and C, respectively). When JD0 −KC0 > 0, the aggregate size distribution of species A satisfies the conventional scaling form and that of species B satisfies a modified scaling form. And in the case of JD0−KC0 < 0, species A and B exchange their aggregate size distributions as in the above JD0−KC0 > 0 case.

Growth kinetics of Al clusters in the gas phase produced by a magnetron-sputtering source

A magnetron-sputtering (MagS) cluster source was used to produce metal clusters of different size distributions by varying individual source parameters. Selectivity of the size of the Al clusters was observed and the mass distribution presents wide-range controllability by the MagS-source, which enables experimental determination of the growth process of Al clusters. By implementing the extended Smoluchowski rate equation, here we propose an interpretation of the cluster-growth kinetics in the gas phase environment. A collision-dependent growing time domain is demonstrated.

Competing role of catalysis-coagulation and catalysis-fragmentation in kinetic aggregation behaviours

We propose a kinetic aggregation model where species A aggregates evolve by the catalysis-coagulation and the catalysis-fragmentation, while the catalyst aggregates of the same species B or C perform self-coagulation processes. By means of the generalized Smoluchowski rate equation based on the mean-field assumption, we study the kinetic behaviours of the system with the catalysis-coagulation rate kernel K(i,j;l) ∝ lν and the catalysis-fragmentation rate kernel F(i,j;l) ∝ lμ, where l is the size of the catalyst aggregate, and ν and μ are two parameters reflecting the dependence of the catalysis reaction on the size of the catalyst aggregate. The relation between the values of parameters ν and μ reflects the competing roles between the two catalysis processes in the kinetic evolution of species A. It is found that the competing roles of the catalysis-coagulation and catalysis-fragmentation in the kinetic aggregation behaviours are not determined simply by the relation between the two parameters ν and μ, but also depend on the values of these two parameters. When ν > μ and ν ≥ 0, the kinetic evolution of species A is dominated by the catalysis-coagulation and its aggregate size distribution ak(t) obeys the conventional or generalized scaling law; when ν < μ and ν ≥ 0 or ν < 0 but μ ≥ 0, the catalysis-fragmentation process may play a dominating role and ak(t) approaches the scale-free form; and in other cases, a balance is established between the two competing processes at large times and ak(t) obeys a modified scaling law.

A parameter-free description of the kinetics of formation of loop-less branched structures and gels

We study, via Brownian dynamics simulation, the kinetics of formation of branched loop-less structures for a mixture of particles with functionalities of two and three, the three-functional ones providing the branching points in the resulting network. We show that for this system, by combining the appropriate Smoluchowski rate equations, including condensation and fragmentation terms, with the thermodynamic perturbation theory of Wertheim, it is possible to provide a parameter-free description of the assembly process, even in the limit of irreversible aggregation (low T). Our work provides evidence of a connection between physical and chemical gelation in low-valence particle systems, properly relating aging (or curing) time with temperature.

Scaling dynamics for solvable coagulation equations with dust and gel

AbstractWe study limiting behavior of rescaled size distributions that evolve by Smoluchowski's rate equations for coagulation, with rate kernel K=2, x+y or xy. We find that the dynamics naturally extend to probability distributions on the half‐line with zero and infinity appended, representing populations of clusters of zero and infinite size. The “scaling attractor” (set of subsequential limits) is compact and has a Levy‐Khintchine‐type representation that linearizes the dynamics and allows one to establish several signatures of chaos. In particular, for any given solution trajectory, there is a dense family of initial distributions (with the same initial tail) that yield scaling trajectories that shadow the given one for all time. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Dynamics of vesicle self-assembly and dissolution

The dynamics of membranes is studied on the basis of a particle-based meshless surface model, which was introduced earlier [Phys. Rev. E 73, 021903 (2006)]. The model describes fluid membranes with bending energy and-in the case of membranes with boundaries-line tension. The effects of hydrodynamic interactions are investigated by comparing Brownian dynamics with a particle-based mesoscale solvent simulation (multiparticle collision dynamics). Particles self-assemble into vesicles via disk-shaped membrane patches. The time evolution of assembly is found to consist of three steps: particle assembly into discoidal clusters, aggregation of clusters into larger membrane patches, and finally vesicle formation. The time dependence of the cluster distribution and the mean cluster size is evaluated and compared with the predictions of Smoluchowski rate equations. On the other hand, when the line tension is suddenly decreased (or the temperature is increased), vesicles dissolve via pore formation in the membrane. Hydrodynamic interactions are found to speed up the dynamics in both cases. Furthermore, hydrodynamics makes vesicle more spherical in the membrane-closure process.

Dynamics of a rate equation describing cluster-size evolution

In this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three.

Application of the Smoluchowski equation to the formation kinetics of cluster ions

Cluster ions with various sizes and compositions have been generated from direct laser vaporization. The Smoluchowski rate equation is extended to describe the formation kinetics of the cluster ions, which have been assumed to be produced from ion–molecule reactions with same-rate constants. By solving the kinetic equation analytically, we obtained the distribution function of the cluster ions. Analysis of the distribution function showed that the function can characterize the statistical size distribution of the cluster ions and can be reduced to the asymptotic solution suggested by Jullien [1].

Smoluchowski ripening of Ag islands on Ag(100)

Using scanning tunneling microscopy, we study the post-deposition coarsening of distributions of large, two-dimensional Ag islands on a perfect Ag(100) surface at 295 K. The coarsening process is dominated by diffusion, and subsequent collision and coalescence of these islands. To obtain a comprehensive characterization of the coarsening kinetics, we perform tailored families of experiments, systematically varying the initial value of the average island size by adjusting the amount of Ag deposited (up to 0.25 ML). Results unambiguously indicate a strong decrease in island diffusivity with increasing island size. An estimate of the size scaling exponent follows from a mean-field Smoluchowski rate equation analysis of experimental data. These rate equations also predict a rapid depletion in the initial population of smaller islands. This leads to narrowing of the size distribution scaling function from its initial form, which is determined by the process of island nucleation and growth during deposition. However, for later times, a steady increase in the width of this scaling function is predicted, consistent with observed behavior. Finally, we examine the evolution of Ag adlayers on a strained Ag(100) surface, and find significantly enhanced rates for island diffusion and coarsening.

Carbon clustering kinetics in detonation wave propagation

Much of the nonideality of insensitive carbon rich explosives such as TATB is thought to be caused by a late-time slow release of energy associated with diffusion limited growth of carbon clusters. We have adapted Shaw and Johnson’s approximate analytic solution of the Smoluchowski rate equations for the evolution of the cluster size distribution function for use in hydrodynamic calculations and in Wood and Kirkwood approximate theory for the detonation velocity-curvature relationship. Solutions for the effect of carbon cluster growth on the structure of the reaction zone are obtained, and the results are compared with available experimental data.

Influence of island diffusion on submonolayer epitaxial growth

We investigate the kinetics of submonolayer epitaxial growth which is driven by a fixed flux of monomers onto a substrate. Adatoms diffuse on the surface, leading to irreversible aggregation of islands. We also account for the effective diffusion of islands, which originates from hopping processes of their constituent adatoms, on the kinetics. When the diffusivity of an island of mass k scales as ${k}^{\ensuremath{-}\ensuremath{\mu}},$ the (mean-field) Smoluchowski rate equations predicts steady behavior for $0&lt;~\ensuremath{\mu}&lt;1,$ with the concentration ${c}_{k}$ of islands of mass k varying as ${k}^{\ensuremath{-}(3\ensuremath{-}\ensuremath{\mu})/2}.$ For $\ensuremath{\mu}&gt;~1,$ a quasistatic approximation of the rate equations predicts a slow continuous evolution, in which the island density increases as $(\mathrm{ln}{t)}^{\ensuremath{\mu}/2}.$ A more refined matched asymptotic expansion reveals unusual multiple-scale mass dependence for the island size distribution. Our theory also describes basic features of epitaxial growth in a more faithful model of growing circular islands. For epitaxial growth in an initial population of monomers and no external flux, a scaling approach predicts power-law island growth and a mass distribution with a behavior distinct from that of the nonzero flux system. Finally, we extend our results to one- and two-dimensional substrates. The physically relevant latter case exhibits only logarithmic corrections compared to the mean-field predictions.

Logarithmic islanding in submonolayer epitaxial growth

We investigate submonolayer epitaxial growth with a fixed monomer flux and irreversible aggregation of adatom islands due to their effective diffusion. When the diffusivity D_k of an island of mass k is proportional to k^{-\mu}, a Smoluchowski rate equation approach predicts steady behavior for 0<\mu<1, with the concentration c_k of islands of mass k varying as k^{-(3-\mu)/2}. For \mu>1, continuous evolution occurs in which c_k(t)~(\ln t)^{-(2k-1)\mu/2}, while the total island density increases as N(t)~(\ln t)^{\mu/2}. Monte Carlo simulations support these predictions.

Measurements of cluster-size distributions arising in salt-induced aggregation of polystyrene microspheres

We have measured the temporal evolution of cluster-size distributions c n(t) during salt-induced aggregation of polystyrene microspheres, 258 nm in radius. Two regimes of irreversible aggregation were studied: fast aggregation ( W = 1.9) and slow aggregation ( W = 420). Distributions were measured using a single-particle light-scattering technique. At long times, we find that c n(t) for both fast and slow aggregation exhibits a dynamic scaling form given by c n ( t) ∼ s −2 φ( n/ s). For fast aggregation, s(t) ∼ n̄ n(t) the number-average mean cluster size, φ( x) is bell-shaped with a peak occuring at x ≈ 0.1, and s( t) ∼ t. For fast slow aggregation, s(t) ∼ n n 2(t) ∼ t 2, φ(x) decrease monotonically with x, and for x < 1, φ( x) ∼ x −1.5. We interpret our results using the Smoluchowski rate equation, which enables us to quantify the measured distributions in terms of two key parameters.

Nonscaling and source-induced scaling behaviour in aggregation model of movable monomers and immovable clusters

General discussion of the aggregation kinetics for the wide class of aggregation models in which cluster growth occurs by bonding reactions between movable monomers and immovable clusters is presented. The study is carried out in terms of Smoluchowski's rate equations. The authors treat a general homogeneous case where the monomer-cluster reaction rates vary as kgamma , with the cluster size k. For models with 0< gamma <1 without a source they find that systems evolve to a final 'frozen' state. Evolution behaviour of the system appears to be nonscaling, but the deviation from a frozen state has a self-similar form. For the systems with a source, they have found that the solution has a scaling form in the most important party of the cluster-size distribution except for an asymptotically ignorable tail. They also carried out the analysis of the structure of the tail and of the thin boundary layer separating the scaling and nonscaling tail regions. The qualitative explanation of nonscaling and source-induced scaling behaviour may be made in terms of 'internal' time inherent for the models of these types.

Numerical solutions to the smoluchowski aggregation—fragmentation equation

We numerically solved the Smoluchowski rate equation for different combinations of fragmentation and aggregation kernels. The average cluster size, number of clusters, and cluster size distribution were calculated as functions of time and found to reach equilibrium values. Scaling and functional form of the cluster size distribution have been obtained. Detailed comparison with previously reported analytical predictions was made and satisfactory agreement was found.