Fractal Kinetics Research Articles

A critical discussion of the analytical approach to the normal grain growth of materials in aD-dimensional space with some possible extensions to other growth phenomena

Abstract The normal grain growth of materials such as metals, ceramics or even biomembranes is reconsidered, based on the formalism obtained by Louat and improved by Mulheran and Harding. A critical discussion of the model is presented and some interesting extensions, leading to the description of the growth process in a time-dependent regime (i.e. when the so-called fractal kinetics are introduced) and/or by utilizing the fractal dimension concept, are provided. As an example of the methods applied, a general power and logarithmic law of the average radius of the domain against time has been proposed.

Influence of fractal kinetics on molecular recognition.

Molecular recognition is a central issue for nearly every biological mechanism. The analysis of molecular recognition to date has been conducted within the framework of classical chemical kinetics, in which the kinetic orders of a reaction have positive integer values. However, recent theoretical and experimental advances have shown that the assumptions inherent in this classical framework are invalid under a variety of conditions in which the reaction environment may be considered nonideal. A good example is provided by reactions that are spatially constrained and diffusion limited. Bimolecular reactions confined within two-dimensional membranes, one-dimensional channels or fractal surfaces in general exhibit kinetic orders that are noninteger. An appropriate framework for the study of these nonideal phenomena is provided by the Power-Law formalism, which includes as special cases the Mass-Action formalism of chemical kinetics and the Michaelis-Menten formalism of enzyme kinetics. The Power-Law formalism is an appropriate representation not only for fractal kinetics per se, but also for other nonideal kinetic phenomena, provided the range of variation in concentration is not too large. After defining some elementary concepts of molecular recognition, and showing how these are manifested in classical kinetic terms, this paper contrasts the implications of classical and fractal kinetics in a few simple cases. The principal distinction lies in the ability of fractal kinetics to nonlinearly transform, rather than proportionally transmit, the input S/N ratio. As a consequence, fractal kinetics create a threshold for the input signal below which no recognition occurs and above which amplified recognition takes place.(ABSTRACT TRUNCATED AT 250 WORDS)

Nonlinear kinetics of ferritin adsorption

The adsorption of ferritin at a methylized quartz surface was measured with off-null ellipsometry and transmission electron microscopy. An initial lag-phase was seen, followed by an accelerating adsorption leading to mass transport limitation of the reaction. The rate of adsorption then decreased at a surface concentration far below monolayer coverage, and a continuously decreasing rate of binding was seen. The slope of the binding rate was linear with the logarithm of time (fractal kinetics). The adsorbed ferritin molecules were distributed in clusters as seen by transmission electron microscopy. Clusters grown during the mass transport limited adsorption had crystalline structure at short range and low fractal dimensions (df = 0.89) over long range. Clusters grown during adsorption with fractal kinetics showed random structure at short range and a high fractal dimension df = 1.86 over all ranges. These findings indicate some new important mechanisms responsible for the complex kinetics of macromolecular reactions at solid-liquid interfaces. The results are discussed in relation to recently developed theories of self-organized criticality.

Hydrogen dimerization process: A probe for investigation of the α-SiO2structure

We discuss a possible mechanism leading to the dimerization of hydrogen, produced by irradiation, in OH-containing \ensuremath{\alpha}-${\mathrm{SiO}}_{2}$. We describe the hydrogen-annealing behavior by combining both fractal and classical kinetics to account for the ${\mathit{H}}_{0}$+${\mathit{H}}_{0}$\ensuremath{\rightarrow}${\mathit{H}}_{2}$ process. The proposal for the combined kinetics is based on a possible picture of the \ensuremath{\alpha}-${\mathrm{SiO}}_{2}$ atomic network where the lattice is seen not as a continuous but instead as a ``granular'' random lattice as indicated also by Raman-scattering experiments. Many experimental data, previously reported by Tsai et al., concerning electron-spin-resonance measurements of the radiolytic-hydrogen-annealing behavior in \ensuremath{\alpha}-${\mathrm{SiO}}_{2}$ are well reproduced by the model. The physical meaning of the parameter values utilized in the fitting procedure is discussed. Finally it is suggested that the analysis of the anomalous diffusion may help in understanding the atomic structure in amorphous solids.

A study of fluorescence quenching in homomicelles and mixed micelles analyzed by diffusion-controlled and fractal reaction model kinetics

Fluorescence data-emission intensities and decay profiles of benzyl 1-pyrenoate in the presence of dimethylaniline in anionic homomicelles and in mixed micelles have been analyzed with two different models: a kinetic model of diffusion-controlled reactions in micelles with a reflecting boundary surface and a fractal reaction kinetic model. A constant value obtained for the spectral fractal dimension, d s = 1.6, implies that the extent of the local heterogeneity is small. The dependence of the rate constant on time shows that this interaction is very fast in the first nanoseconds but at longer times reaches diffusion-controlled rate constants of (7–9) × 10 8 M −1 s −1 in this medium. A correlation between kinetic parameters derived from either model was found to be possible in the systems studied.

Adsorption Kinetics of Dye Molecules on Polymer Membrane Filters

e kinetics of the adsorption of dye molecules on membrane filters made of cellulose nitrate have been studied. In order to obtain thermodynamical data the temperature has been varied from 283 K to 333 K. Because kinetic laws of 1. or 2. order cannot describe the experimental results two different theories were applied. In addition to a classical concept a new equation based on the fractal theory has been checked and has proved to be suitable for adsorption data. We have shown that obtained fractal parameters, fractal dimension Ds and fractal dimension Df, are in excellent agreement with other fractal kinetics experiments.

Fractal Kinetics of Mechanical Alloying

Fractal Kinetics of Mechanical Alloying

Fractal kinetics in surface reactions

The kinetics of the surface reaction A + B n / n → s u r f a c e c a t a l y t i c A B , n = 1 , 2 has been investigated by means of Monte Carlo simulations. This kind of three color reactive percolation kinetics should be tentatively described in terms of parameters (fractal dimension, perimeters,…) of the specific clusters generated by the model reactions for different values of the external parameters. The islands of adsorbed molecules are found to be fractal with dimension D ≈ 5/3, for the different kinetic schemes considered.

Medium-range structural order and fractal annealing kinetics of radiolytic atomic hydrogen in high-purity silica.

La recuisson thermique de l'hydrogene atomique induit par radiation dans des silices de haute purete contenant OH, est etudiee par resonance magnetique electronique, dans l'intervalle 75-185 K. Les donnees sont interpretees en termes de cinetique bi-moleculaire, en supposant une dimerisation du H o radiolytique et/ou sa reaction avec d'autres especes paramagnetiques dans le verre

Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies

Ion channels in the cell membranes of the corneal endothelium, hippocampal neurons, and fibroblasts, and gramicidin channels in lipid bilayers have open and closed times that can be fit, in whole or part, by power law distributions. That is, the gating is self-similar when viewed at different time scales. Hence, kinetic processes at slow and fast time scales are not independent but rather are interrelated. To study how such a relationship can arise we analyze a closed ⇌ open channel with the fractal dimension for leaving the closed state D CO ≈2 and the fractal dimension for leaving the open state D OC ≈1. This special case can be analyzed because it can be represented by equivalent Markov processes. We show that it is equivalent to Markov chains with forward and backward kinetic rate constants approximately equal at each stage, and forming an approximate geometric progression along the different stages. These kinetic rates determine the energy levels and activation energy barriers separating those levels. We find that there are many conformational states (substates) separated by high activation energy barriers. This is similar to the energy structure found for globular proteins such as myoglobin. However, the novel feature reported here is that the activation energy barriers are not independent but are interrelated and form an arithmetic progression. Because of this relationship the fast processes across the low activation energy barriers are linked to slow processes across the high activation energy barriers.

Diffusion to fractal surfaces—II. Verification of theory

Decay of the diffusion controlled current of particles diffusing from an initially homogeneous medium to a completely absorbing fractal boundary was previously shown to exhibit t −α time-dependence instead of the conventional t − 1 2 one with the exponent α being determined by the fractal dimension, D f, of the interface as α =( D f−1)/2. In electrochemical terms this corresponds to a generalized Cottrell equation (or Warburg impedance) and can be used to describe the frequency dispersion caused by surface roughness effects. We verify the predicted behaviour for fractal surfaces with D f>2 (rough interface), and D f<2 (partially blocked surface or active islands on inactive support). In addition, the fractal decay kinetics is shown to be valid for both contiguous and non-contiguous surfaces. Computer simulation, a mathematical model, and direct experiments on well defined fractal electrodes are the tools for verifying the fractal decay law for the different surfaces. The predicted power law behaviour is observed, and the predicted α( D f) relationship was seen to prevail in each case.

Fractal and Markov behavior in ion channel kinetics

Kinetic analysis of ion channel recordings attempts to distinguish the number and lifetimes of channel molecular states. Most kinetic analysis assumes that the lifetime of each state is independent of previous channel history, so that open and closed durations are Markov processes whose probability densities are sums of exponential decays. An alternative approach assumes that channel molecules have many configurtions with widely varying lifetimes. Rates of opening and closing then vary with the time scale of observation, leading to fractal kinetics. We have examined kinetic behavior in two types of channels from human and avian fibroblasts, using a maximum likehood method to test the dependence of rates on observational time scale. For both channels, openings showed mixed fractal and Markov behavior, while closings gave mainly fractal kinetics.

Simulations of one-dimensional and fractal luminescence kinetics

Exciton and electron-hole recombination reactions were simulated on one-dimensional and fractal networks. Particle distributions depend on dimensionality and electric fields. Geminate vs. non-geminate and pulsed vs. steady-state generation give very different reaction orders and population distributions.

Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes

Ion channels have been modeled as consisting of a small number of discrete conformational states such as closed ⇌ closed ⇌ open, and the transitions between the states treated as a Markov process. We derive an alternative model based on a fractal scaling of the kinetic rate constants k, namely that k = At 1- D , where t is time, A the kinetic setpoint, and D the fractal dimension. By measuring the effective kinetic rate constants at different time scales, we show how to determine if single channel records are best represented by models with discrete Markov states or continuum fractal kinetics. The fractal model suggests that the multiple closed states postulated by Markov models to fit the histograms of closed times may not exist but are an artifact of trying to fit the sum of exponentials to histograms that are not the sum of exponentials. The fractal model is also more consistent with the conformational dynamics of proteins. Analysis of patch clamp recordings from an epithelium, the corneal endothelium, shows that its channels have fractal rather than Markov kinetics.

Fractal kinetics in the luminescence decay of solids

The power-law time dependence of luminescence decay in solids is interpreted in terms of kinetic equations in which the rate constant is governed by the efficiency with which the reactants explore a discrete lattice. It is shown that the bimolecular reaction gives the best description for a substantial number of cases, but the power-law exponent does not have a universal value.

Matrix interactions with organic triplet states

Phosphorescence kinetics of ketones in ambient temperature polymer solid solutions are reviewed. The aromatic ketones benzophenone, anthraquinone, phenanthraquinone and benzil, under flashed laser excitation, demonstrate non-exponential kinetics. For these compounds, fractal kinetics are found to fit some of the systems examined. In contrast, the aliphatic diketones diacetyl and camphorquinone demonstrate exponential kinetics. Such kinetics, when perturbed in the presence of oxygen as a quencher, have been the subject of preliminary study.

Fractal chemical kinetics: Binary steady-state reaction on a percolating cluster.

Monte Carlo simulations confirm the conjecture that the order X of the elementary reaction A+A\ensuremath{\rightarrow}products in a fractal medium with random-walk exponent f is given by X=1+${f}^{\mathrm{\ensuremath{-}}1}$. Specifically for a percolating cluster (f=(2/3), X=(5/2)) we get X=2.5\ifmmode\pm\else\textpm\fi{}0.1.

Fractal chemical kinetics: Simulations and experiments

We relate the effective order X of the elementary binary reaction, A+A→ products, on a fractal, to the spectral dimension. For ds&amp;lt;2: X=1+2/ds. We regain the calsssical value (X=2) for the cubic lattice but get X=3 for the linear lattice. Supercomputer simulations (transient and steady state) and exciton fusion experiments are in quantitative agreement with the above. For percolating clusters X=2.5 and for the Sierpinski gasket X=2.45. Relevance to heterogeneous kinetics is discussed.

Luminescent fractal reactions: Exciton fusion on percolation clusters

Experimental measurements of exciton annihilation in isotopic mixed crystals of naphthalene-h 8 in naphthalene-d 8 exhibit fractal reaction kinetics on samples well below and slightly above the percolation threshold. The effective spectral dimensions are in very good agreement with the superuniversality conjecture and 2-dim. random walk simulations. Crystals well above the percolation threshold exhibit the expected classical behavior.

Fractal chemical kinetics: Reacting random walkers

Computer simulations on binary reactions of random walkers (A + A → A) on fractal spaces bear out a recent conjecture: (ρ −1 − ρ0 −1) ∞ tf, whereρ is the instantaneous walker density andρ 0 the initial one, andf=d s /2, whered s is the spectral dimension. For the Sierpinski gaskets:d=2, 2f=1.38 (d s =1.365);d=3, 2f=1.56 (d s =1.547); biased initial random distributions are compared to unbiased ones. For site percolation:d= 2,p=0.60, 2f= 1.35 (d s =1.35); d=3,p=0.32, 2f=1.37 (d s =1.4); fractal-to-Euclidean crossovers are also observed. For energetically disordered lattices, the effective 2f (from reacting walkers) andd s (from single walkers) are in good agreement, in both two and three dimensions.