Diffusion-limited Regime Research Articles

Timescales determining the degree of kinetic isotope fractionation by evaporation and condensation

A first order characteristic of the relative abundance of the elements in solar system materials ranging in size from inclusions in primitive meteorites to planetary sized objects such as the Earth and the Moon is that they are very much like that of the Sun for the more refractory elements but systematically depleted to varying degrees in the more volatile elements. This is taken as evidence that evaporation and and/or condensation were important processes in determining the distinctive chemical properties of solar system materials. In some instances there is also isotopic evidence suggesting evaporation in that certain materials are found enriched in the heavy isotopes of their more volatile elements. Here model calculations are used to explore how the relative rates of various key processes determine the relationship between elemental and isotopic fractionation during partial evaporation and partial condensation. The natural measure of time for the systems considered here is the evaporation or condensation timescale defined as the time it would take under the prevailing conditions for evaporation or condensation to completely transfer the element of interest between the two phases of the system. The other timescales considered involve the rate of change of temperature, the rate at which gas is removed from further interaction with the condensed phase, and the rates of diffusion in the condensed and gas phases. The results show that a key determinant of whether or not elemental fractionations have associated isotopic effects is the ratio of the partial pressure of a volatile element ( P i ) to its saturation vapor pressure ( P i,sat ) over the condensed phase. Systems in which the rate of temperature change or of gas removal are slow compared to the evaporation or condensation timescale will be in the limit P i ∼ P i,sat and thus will have little or no isotopic fractionation because at the high temperatures considered here there is negligible equilibrium fractionation of isotopes. If on the other hand the temperature changes are relatively fast, then P i ≠ P i,sat and there will be both elemental and isotopic fractionation during partial evaporation or partial condensation. Rapid removal of evolved gas results in P i ≪ P i,sat which will produce isotopically heavy evaporation residues. Diffusion-limited regimes, where transports within a phase are not sufficiently fast to maintain chemical and or isotopic homogeneity, will typically produce less isotopic fractionation than had the phases remained well mixed. The model results are used to suggest a likely explanation for the heavy silicon and magnesium isotopic composition of Type B CAIs (as due to rapid partial melting and subsequent cooling at rates of a few °C per hour), for the uniformity of the potassium isotopic composition of chondrules despite large differences in potassium depletions (as due to volatilization of potassium by reheating in regions of large but variable chondrules per unit volume), and that the remarkable uniformity of the potassium isotopic composition of solar system materials is not a measure of the relative importance of evaporation and condensation but rather due to the solar nebula having evolved sufficiently slowly that materials did not significantly depart from chemical equilibrium.

Analysis of radial segregation in directionally solidified Hg 0.89Mn 0.11Te

Bridgman growth experiments were performed on Hg 0.89Mn 0.11Te (MMT) to determine the extent of radial Manganese segregation during directional solidification. MMT crystals were directionally solidified at rates of 0.09 and 0.18 μm/s and in axial thermal gradients of 83 and 68 °C/cm. Wavelength dispersive spectroscopy and Fourier Transform infra-red analytical techniques were used to determine the radial homogeneity in all boules and the deflection of the solid–liquid interface (SLI) in two boules that were rapidly quenched after 5–6 cm of directional solidification. For all growth runs, the measured radial compositional variations were on the order of 0.01 molar percent MnTe in the steady-state region of growth. Comparison of the measured radial compositional results of the crystals to predicted values in the diffusion-limited regime indicate a strong influence of convection near the SLI. This conclusion is supported by the weak influence of the translation rates and axial thermal gradients utilized in this study upon radial compositional homogeneity.

Self-organized formation of regular nanostripes on vicinal surfaces

We explore the mechanism of self-organized formation of regular arrays of nanostripes on vicinal surfaces by using a phase-field model. Epitaxial growth during deposition usually results in both nanostripes and islands on terraces of a vicinal substrate. Postdeposition annealing at elevated temperatures induces growth of the nanostripes but makes the islands shrink. It is a ripening process of the mixed system of the nanostripes and the islands, being dependent upon the temperature and strain. It is accompanied by a transition from the diffusion-limited regime to the detachment-limited regime induced by the strain at high temperatures. This ripening makes the islands diminish and on the other hand makes the nanostripes smoother. As a result, the islands disappear completely and the regular arrays of nanostripes are formed on the vicinal substrate. This theory can explain the self-organized formation of nanostripes and nanowires on vicinal surfaces, such as the intriguing regular arrays of Fe nanostripes on the vicinal W surfaces.

Influence of natural organic matter and ionic composition on the kinetics and structure of hematite colloid aggregation: implications to iron depletion in estuaries.

The stability and aggregation behavior of iron oxide colloids in natural waters play an important role in controlling the fate, transport, and bioavailability of trace metals. Time-resolved dynamic light scattering experiments were carried out in a study of the aggregation kinetics and aggregate structure of natural organic matter (NOM) coated hematite colloids and bare hematite colloids. The aggregation behavior was examined over a range of solution chemistries, by adjusting the concentration of the supporting electrolyte-NaCl, CaCl2, or simulated seawater. With the solution pH adjusted so that NOM-coated and bare hematite colloids were at the same zeta potential, we observed a significant difference in colloid stability which results from the stability imparted to the colloids by the adsorbed NOM macromolecules. This enhanced stability of NOM-coated hematite colloids was not observed with CaCl2. Aggregate form expressed as fractal dimension was determined for both NOM-coated and bare hematite aggregates in both NaCl and CaCl2. The fractal dimensions of aggregates formed in the diffusion-limited regime indicate slightly more loosely packed aggregates for bare hematite than theory predicts. For NOM-coated hematite, a small decrease in fractal dimension was observed when the solution composition changed from NaCl to CaCl2. For systems in the reaction-limited regime, the measured fractal dimensions agreed with those in the literature. Colloid aggregation was also studied in synthetic seawater, a mixed cation system to simulate estuarine mixing. Those results describe the important phenomena of iron oxide aggregation and sedimentation in estuaries. When compared to field data from the Mullica Estuary, U.S.A., it is shown that collision efficiency is a good predictor of the iron removal in this natural system.

Diffusing capacity reexamined: relative roles of diffusion and chemical reaction in red cell uptake of O2, CO, CO2, and NO.

This paper presents an analytical expression for the diffusing capacity (Theta(t)) of the red blood cell (RBC) for any reactive gas in terms of size and shape of the RBC, thickness of the unstirred plasma layer surrounding the RBC, diffusivities and solubilities of the gas in RBC and boundary layer, hematocrit, and the slope of the dissociation curve. The expression for Theta(t) has been derived by spatial averaging of the fundamental convection-diffusion-reaction equation for O(2) in the RBC and has been generalized to all cell shapes and for other reactive gases such as CO, NO, and CO(2). The effects of size and shape of the RBC, thickness of the unstirred plasma layer, hemoglobin concentration, and hematocrit on Theta(t) have been analyzed, and the analytically obtained expression for Theta(t) has been validated by comparison with different sets of existing experimental data for O(2) and CO(2). Our results indicate that the discoidal shape of the human RBC with average dimensions of 1.6-mum thickness and 8-mum diameter is close to optimal design for O(2) uptake and that the true reaction velocity in the RBC is suppressed significantly by the mass transfer resistance in the surrounding unstirred layer. In vitro measurements using rapid-mixing technique, which measures Theta(t) in the presence of artificially created large boundary layers, substantially underpredicts the in vivo diffusing capacity of the RBC in the diffusion-controlled regime. Depending on the conditions in the RBC, uptake of less reactive gases (such as CO) undergoes transition from reaction-limited to diffusion-limited regime. For a constant set of morphological parameters, the theoretical expression for Theta(t) predicts that Theta(t,NO) > Theta(t,)(CO(2)) > Theta(t,)(O(2)) > Theta(t,CO).

Reaction–diffusion with a time-dependent reaction rate: the single-species diffusion–annihilation process

We study the single-species diffusion-annihilation process with a time-dependent reaction rate, lambda(t)=lambda_0 t^-omega. Scaling arguments show that there is a critical value of the decay exponent omega_c(d) separating a reaction-limited regime for omega > omega_c from a diffusion-limited regime for omega < omega_c. The particle density displays a mean-field, omega-dependent, decay when the process is reaction limited whereas it behaves as for a constant reaction rate when the process is diffusion limited. These results are confirmed by Monte Carlo simulations. They allow us to discuss the scaling behaviour of coupled diffusion-annihilation processes in terms of effective time-dependent reaction rates.

Amphiphilic derivatives of dextran: adsorption at air/water and oil/water interfaces.

Ionic amphiphilic dextran derivatives were synthesized by the attachment of sodium sulfopropyl and phenoxy groups on the native polysaccharide. A family of dextran derivatives was thus obtained with varying hydrophobic content and charge density in the polymer chains. The surface-active properties of polymers were studied at the air-water and dodecane-water interfaces using dynamic surface/interfacial tension measurements. The adsorption was shown to begin in a diffusion-limited regime at low polymer concentrations, that is to say, with the diffusion of macromolecules in the bulk solution. In contrast, at long times the interfacial adsorption is limited by interfacial phenomena: adsorption kinetics or transfer into the adsorbed layer. A semiempirical equation developed by Filippov was shown to correctly fit the experimental curves over the whole time range. The presence of ionic groups in the chains strongly lowers the adsorption kinetics. This effect can be interpreted by electrostatic interactions between the free molecules and the already adsorbed ones. The adsorption kinetics at air-water and oil-water interfaces are compared.

Diffusion-limited deposition of dipolar particles.

Deposits of dipolar particles are investigated by means of extensive Monte Carlo simulations. We found that the effect of the interactions is described by an initial, nonuniversal, scaling regime characterized by orientationally ordered deposits. In the dipolar regime, the order and geometry of the clusters depend on the strength of the interactions and the magnetic properties are tunable by controlling the growth conditions. At later stages, the growth is dominated by thermal effects and the diffusion-limited universal regime obtains, at finite temperatures. At low temperatures the crossover size increases exponentially as T decreases and at T=0 only the dipolar regime is observed.

Precursor Aggregate Distributions for Particulate Gels

We analyze the aggregation process leading to gel formation in colloidal dispersions under quiescent conditions in order to identify possible precursor aggregate distributions for particulate gels. The aggregation process is described with an experimentally validated population balance model, which allows us to calculate how the aggregate mass distribution evolves in time. When the cumulative occupied volume fraction of fractal aggregates reaches a certain limit, the random cluster−cluster aggregation regime crosses over to an interconnection regime, and a gel network forms. The aggregate distribution at the end of the aggregation regime is the precursor for the subsequently formed particulate gel. We demonstrate that, when aggregation proceeds fully in the reaction-limited regime, the precursor aggregate distribution depends only on the solid volume fraction while being insensitive to both the primary particle size and the stability ratio. However, the stability ratio is an important operating parameter controlling the overall aggregation and gelation time. The situation can be significantly changed by entering the diffusion-limited aggregation regime. The interplay among these various factors is illustrated through an example of the optimal design of the precursor aggregate distribution. Primary particle size, solid volume fraction, and stability ratio are used as design parameters, and the objective function is formulated in terms of specific requirements on the precursor cluster mass distribution. These requirements were determined on the basis of qualitative arguments about the relation between the precursor structure and the gel mechanical properties.

Effect of cross relaxation on the 1470 and 1800 nm emissions inTm3+:TeO2–CdCl2 glass

We have used spectroscopic methods to investigate the effect of cross relaxation on the 1470 and 1800 nmemissions in Tm3+:TeO2–CdCl2 glass for Tm3+ ion concentrations of 0.05, 0.1, 0.25, 0.5 and 1.0 mol%. The emission spectra obtained under800 nm excitation reveal the existence of energy transfer via cross relaxation among theTm3+ ions. As a result, as the thulium concentration is increased, the strength of the 1470 nm emission isfound to decrease in relation to that of the 1800 nm emission. The analysis of the time evolution ofthe 3F4 emission due to the transition further shows that the electronic mechanism responsible for the ion–ioninteraction can be identified as an electric dipole–dipole energy transfer process and occursin the diffusion-limited relaxation regime. The average critical distance parameter whichprovides a measure for the strength of cross relaxation was further determined to be 17.9 Å.

Modelling of aggregation kinetics of colloidal systems and its validation by light scattering measurements

Complex behavior of aggregating colloidal systems far from equilibrium is due to the interplay between the particle–particle interactions, the cluster population, the fractal structure of clusters and the resulting aggregation mechanism and kinetics. In this work, a modelling approach has been proposed to simulate the aggregation kinetics of colloidal systems under quiescent conditions, based on the combinations of the cluster mass distribution (CMD) computed using a mathematical model based on population balance equations (PBE) with the structure properties of individual clusters determined by Monte-Carlo (MC) simulations. The effects of the particle–particle interactions, the mobility of the fractal clusters and the fractal dimension on the aggregation rate, have been considered in the aggregation kernel used in the PBE model. It is found that in both the diffusion-limited and reaction-limited aggregation regimes, the average sizes and structure properties predicted using the proposed model are in good agreement with those determined experimentally by light scattering measurements. The results indicate that the proposed PBE modelling approach has a good chance to predict correctly the CMD of the aggregating system, and the structure properties computed from the MC simulations can well represent those of real clusters. This approach has also been extended to analyze the gelation behavior of colloidal systems at relatively large particle volume fractions.

Hydrogen Gas Driven Permeation through Asymmetric Membranes in Diffusion Limited and Surface Limited Regimes: Interplay between Analytical and Numerical Calculations

Validity of analytical solutions for the gas driven permeation of H in the Diffusion Limited Regime (DLR) and Surface Limited Regime (SLR) is analyzed by comparison with numerical calculations. Margins for analytical formulas have been established in terms of the permeation factors W = KLSp1/2/D on the inlet (W1) and outlet (W2) sides of the membrane. The DLR analytical formula gives perfect result (error less than 0.5%) if both W2 ≥ 104 and W1 ≥ 102 conditions are satisfied simultaneously. Decrease of both margins by two orders of magnitude leads to 10% error. The SLR analytical formula gives a very good result (error less than 0.5%) if both W1 ≤ 10-2 and W2W1 ≤ 10-3 conditions are satisfied simultaneously. Increase of both margins by two orders of magnitude leads to 10% error. It has been shown that the inlet side and the outlet side conditions are different in their importance for validity of the analytical formulas. In DLR the condition is softer on the inlet side and more rigid on the outlet side, while in SLR the condition is softer on the outlet side and more rigid on the inlet side.

Interactions of Hydrating Tricalcium and Dicalcium Silicate using Time-Resolved Quasielastic Neutron Scattering

ABSTRACTHydrating tricalcium and dicalcium silicate mixtures were studied using time-resolved quasielastic neutron scattering. Application of a hydration model allowed the extraction of several parameters describing the hydration mechanics. The hydration rate during the nucleation and growth period, diffusivity during the diffusion limited regime, and the maximum amount of product able to be formed were not related linearly to the mixture combinations. Addition of dicalcium silicate at approximately 20 wt. % caused a sharp alteration to these parameters, including reduced hydration rate and reaction layer permeability. These two factors lead to the prediction that more reaction products would be ultimately manufactured during the reactions at this composition.

Fluorescence Quenching Studies of Curcumin by Hydrogen Peroxide in Acetonitrile Solution

Steady state quenching studies of curcumin, 1,7-bis(4-hydroxy-3-methoxyphenyl)-1,6-heptadiene-3,5-dione, fluorescence by hydrogen peroxide were conducted in acetonitrile solution. A quenching rate constant, k q, of 1.05 × 1010 M −1 · s−1 was obtained with a short fluorescence lifetime of 347 ps. The reaction rate constant, which is within the diffusion-limited regime, is activation-controlled. The rate constant of deactivation of the thermally excited curcumin was 1.2 orders of magnitude more nonradiative (2.67 × 109 s−1) than radiative (2.16 × 108 s−1). The reaction was exothermic with a ΔG° of −1.97 eV and solvent reorganization energy of 1.37 eV. These values indicate that the electron transfer reaction is solvent-mediated with electron transfer rate constant, k ET, of 2.16 × 1010 s−1.

Kinetic study of copper precipitates under VVER-type reactor conditions

The copper-rich cluster evolution in the neutron-irradiated VVER steels is investigated beginning at the nucleation stage. For this, typical VVER-type reactor conditions are considered. The cluster dynamics approach is used for calculation of the density distribution of copper precipitates related to the number of Cu-atoms or radius, mean radius, volume content, number density of precipitates and the concentration of free Cu-atoms in dependence on the irradiation time. The results for time of one year are compared with the results of small angle neutron scattering experiments which were carried out on specimens irradiated at the surveillance positions of VVER reactors. It has revealed the intermediate type of the evolution kinetics between diffusion and interfacial kinetics limited regimes. The duration of the nucleation and deterministic stages is estimated. The coarsening stage does not occur.

Diffusion–reaction modelling of DNA hybridization kinetics on biochips

An analytical expression for the rate of DNA hybridization on the surface of DNA biochips is established for the case of a finite 1-D diffusion space. The expression allows to account for the diffusion-limited supply of unreacted probe DNA and was obtained by solving the continuous 1-D reaction–diffusion mass balance using a Finite Fourier Transform technique. The extrapolation of the presently considered 1-D case to the full 3-D case is outlined as well. By bringing the obtained result into concurrence with the results of a stochastic random walk study, the kinetic constant k of the continuous reaction–diffusion model could be expressed as a function of the basic physico-chemical parameters (persistence length and jump frequency of the Brownian motion, collision frequency, binding probability) of the individual molecules. With the availability of an analytical expression describing the full time course of the 1-D hybridization process, and by using the Damköhler number Da= kh/ D mol, the different reaction- and diffusion-limited regimes occurring during the course of a hybridization process can now be described in general, dimensionless terms, allowing to establish some very simple rules for the design of DNA biochips and flow-through biosensors.

Chemical brecciation processes in the Sue unconformity-type uranium deposits, Eastern Athabasca Basin (Canada)

Chemical brecciation in sandstone is common in many unconformity-type uranium deposits of the Athabasca Basin, and is expressed in some of them as ball zone breccia. Ball zones are composed of rounded argillized sandstone fragments, varying in size from several centimeters to 1 m, wrapped in a clay matrix. The Sue C open pit provided a unique opportunity to map and to study such ball zones. Here, they were up to 5 m wide with a 20–30 m vertical extension. They were mainly observed along a reverse fault controlling the Sue A and B uranium deposits, and were well developed at intersections with dextral NE-trending structures. Their maturity, characterized by the matrix percentage, increased toward the unconformity and at fault intersections. They are characterized by massive quartz dissolution, hematite leaching, (Ca,Sr,LREE)Al-phosphates crystallization and replacement of dickite by illite. Illite composition indicates formation temperatures of 240–280 °C, close to peak diagenesis conditions in the basin. Mass balance calculations show that V, K, Rb, B, LREE, Mg, Cr, Sr, U and Y were added and Si and Fe leached out with up to 85% volume loss. Ball zones were initiated by tectonic fracturing in sandstone during reverse faulting. Consecutive permeability increase induced basement fluid circulation in the sandstone with quartz dissolution along fractures. With a silica saturation of the fluid of 90%, a minimum fluid/rock ratio of 38,000 is obtained. The rounded morphologies of the breccia fragments are attributed to a diffusion-limited regime of dissolution. The resulting increase of clay content led to self-sealing of the hydrothermal system. Seismic reactivation may have been periodically rejuvenated the permeability. These processes seem to be coeval with the formation of structurally controlled high-grade unconformity-type uranium mineralization. Formation of the ball zones required probably more than 1 million years.

Hydrogen Permeation Through Liquid LiPb

Permeation through a sandwich structure “martensitic steel-liquid LiPb alloy” has been modeled by numerical calculations. Diffusion-limited, absorption-limited, and recombination-limited regimes have been observed in various conditions. Features of permeation dynamics, steady state permeation rate, concentration profile, and the net accumulation have been analyzed in these regimes. A comparison of permeation from different sides of the sandwich has been made.

Hydrogen permeation through metal membrane with protective coating in contact with atomic or ionized hydrogen

The use of protective coatings is one of the most common methods to reduce the hydrogen permeation through structural materials. Considering the hydrogen permeation through such multilayer membrane it is usually suggested that the ratio of hydrogen concentrations at the interface is equal to that of the hydrogen equilibrium solubilities in these materials. However, during diffusion permeation the hydrogen concentrations at the interface are not equilibrium values but they are determined by diffusion input D 1/ d 1 and output D 2/ d 2 rates and by passage rate through interface which is proportional to the ratio of equilibrium solubilities S 1 and S 2 ( d i , D i – thickness of and hydrogen diffusivity in i-layer, respectively). Therefore the ratio of hydrogen concentrations at the interface can be adopted by the ratio ( S 1 D 1/ d 1)/( S 2 D 2/ d 2). By this approximation the analysis of hydrogen permeation through the duplex membrane in a diffusion-limited regime showed that the coating of the same materials can both decrease and increase the hydrogen permeability of the system depending on the ratio of these parameters. In this case it is possible to explain the contradictory results of experiments on hydrogen permeation through duplex membranes.

Approach to quantum Kramers’ equation and barrier crossing dynamics

We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.