Fast Diffusion Limit Research Articles

Analysis of Island Dynamics in Epitaxial Growth of Thin Films

This work is concerned with analysis and refinement for a class of island dynamics models for epitaxial growth of crystalline thin films. An island dynamics model consists of evolution equations for step edges (or island boundaries), coupled with a diffusion equation for the adatom density, on an epitaxial surface. The island dynamics model with irreversible aggregation is confirmed to be mathematically ill-posed, with a growth rate that is approximately linear for large wavenumbers. By including a kinetic model for the structure and evolution of step edges, the island dynamics model is made mathematically well-posed. In the limit of small edge Péclet number, the edge kinetics model reduces to a set of boundary conditions, involving line tension and one-dimensional surface diffusion, for the adatom density. Finally, in the infinitely fast terrace diffusion limit, a simplified model ofone-dimensional surface diffusion and kink convection is derived and found to be linearly stable.

Influence of Diffusion on the Kinetics of Donor−Acceptor Electron Transfer Monitored by the Quenching of Donor Fluorescence

The problem of photoinduced donor−acceptor electron transfer in liquid solution is analyzed to obtain an understanding of the relationship between approximate treatments of the role of diffusion in electron transfer, that is, the Collins−Kimball approach, and a detailed analysis of the problem. It is shown why previous analyses of experimental data have yielded distance dependences of electron transfer that are much too long range. From an appropriate fitting of the nonstationary kinetics of donor fluorescence quenching by diffusion-assisted electron transfer, the effective radii and the steady-state constants associated with electron transfer are found for a donor−acceptor system studied experimentally in seven solvents with different viscosities. The dependence of diffusion agrees with the one predicted theoretically for electron transfer having a distance-dependent transfer rate initially taken to be exponential with distance. In the fast-diffusion limit, the dependence on the rate of diffusion is well approximated by the Collins−Kimball relationship, which permits the kinetic rate constant and the effective radius associated with diffusion-induced quenching to be extracted from the experimental data. The effective radius is then related to the electron transfer rate with arbitrary distance dependence. From this relationship, the tunnelling length for both exponential and Marcus-type rates is obtained from the data analysis, and it is demonstrated that the latter is almost twice as long as the former. For the Marcus transfer rate, it is found that the Marcus parameter β = 1.2 Å-1 (β = 2/tunnelling length), which is in accord with previous measurements on a variety of systems. The theoretical analysis presented here resolves the apparent discrepancies between early measurements of very long tunnelling lengths in liquid systems and physically reasonable values of β ≈ 1 Å-1.

Modeling base voltammetry and CO electrooxidation at the Pt(111)-electrolyte interface: Monte Carlo simulations including anion adsorption

We perform dynamic Monte Carlo simulations to understand the effect of anion adsorption on CO oxidation at the Pt(111)–electrolyte (sulfuric acid) interface. Our simulations are performed on a model for CO electrooxidation, where oxygen-containing species (adsorbed OH) formed on the Pt surface reacts with adsorbed CO by a Langmuir–Hinshelwood mechanism to form CO2. In our site-blocking model, discharged anions adsorb on fcc and hcp sites, while CO and OH occupy the atop sites. Our simulations of blank voltammograms show a disorder to order phase transition for HSO4 adsorption on Pt(111) surfaces near 0.4 V, RHE. This transition is observed when the difference in binding energy of HSO4 on hcp and fcc sites, Δe ≫ kBT. Here T is the temperature and kB is the Boltzmann constant. We attribute this transition to the formation of the sulfate ‘butterfly’ observed in experimental base voltammograms. The ordered state () is composed of antiphase anion islands separated by domain walls. OH adsorption is observed at a potential near 0.75 V along the atop sites near the domain walls. Our simulations indicate the quenching of this phase transition for relatively high rates of OH adsorption. This phenomena is also observed in experiments for small concentrations of sulfuric acid solutions. For CO oxidation stripping voltammetry in the limit of slow CO diffusion, our simulations show a prewave (slow CO oxidation) followed by a sharp peak (rapid CO oxidation). The observed features are the result of strongly correlated kinetic events, and is explained using a nucleation–growth model. In the limit of fast diffusion, CO moving rapidly on the electrode surface washes out these effects resulting in a voltammogram with no prewave.

One‐gas models with height‐dependent mean molecular weight: Effects on gravity wave propagation

Many models of the thermosphere employ the one‐gas approximation where the governing equations apply only to the total gas and the physical properties of the gas that depend on composition (mean molecular weight and specific heats) are height‐dependent. It is further assumed that the physical properties of the gas are locally constant; thus motion‐induced perturbations are nil. However, motion in a diffusively separated atmosphere perturbs local values of mean molecular weight and specific heats. These motion‐induced changes are opposed by mutual diffusion of the constituent gases, which attempts to restore diffusive equilibrium. Assuming that composition is locally constant is equivalent to assuming that diffusion instantaneously damps the changes that winds attempt to produce. This is the limit of fast diffusion. In the limit of slow diffusion, gas properties are constant (conserved) following the motion but are perturbed locally by advection. An analysis of the static stability shows that composition effects significantly change the static stability, with greater changes for the slow‐diffusion limit than for the fast‐diffusion limit. We have used a one‐gas full‐wave model to examine the effects of wave‐perturbed composition on gravity waves propagating through the lower thermosphere. We have augmented the conventional system (fixed gas properties) with predictive equations for composition‐dependent gas properties. These equations include vertical advection and mutual diffusion. The latter is included in parameterized form as second‐order scale‐dependent diffusion. We have found that the fast diffusion implied by locally fixed properties has a significant effect on the dynamics. Predicted temperatures are larger for locally fixed composition than for conserved composition. The simulations with parameterized mutual diffusion gave results that are much closer to the results for conserved gas properties than for fixed properties. We found that the divergence between the fast and slow limits was greatest for fast waves and for colder thermospheres. This is because the propagation characteristics of fast waves are sensitive to changes in the static stability and because compositional gradients are stronger for colder thermospheres. We conclude that future models that use the one‐gas approximation for fast waves in the lower thermosphere should include, at minimum, the simplification of conserved rather than fixed properties, especially for colder thermospheres.

Effect of diffusional relaxation in random sequential adsorption of polymer chains

We study by Monte Carlo computer simulations random sequential adsorption with diffusional relaxation of polymer chains of size N onto a square lattice. The coverage θ( t) is found to grow to full saturation with a power-law behaviour i.e., θ( t) ∼ 1 − t − y , where the exponent y ≈ 1/( N − 1) in the fast diffusion limit. This result agrees with recent mean-field results for a corresponding chemical reaction.

Adsorbate order-disorder effects on recombinative thermal desorption: Equivalence between dynamic Monte Carlo simulations and self-consistent cluster approximations

The thermally activated desorption of dissociated diatomic species from a metallic surface is described as a lattice-gas problem on a square lattice with nearest- and next-nearest neighbor interactions between the adsorbates and investigated within dynamic Monte Carlo simulations. In the limit of fast diffusion with respect to desorption, it can be shown that the desorption rate depends directly on the local order induced by the interactions within the adsorbate layer. Therefore, by employing an appropriate quasi-equilibrium cluster approximation for the local order (beyond the quasi-chemical approximation), a differential equation can be derived that depends on self-consistently calculated structure forms, reproducing quantitatively the temperature-programmed desorption spectra simulated with the Monte Carlo procedure. In this way it can be shown that the time evolution obtained from the dynamic Monte Carlo algorithm is indeed “correct,’’ and on the other hand, that it can be successfully substituted by a “cheaper’’ cluster approximation.

Modeling the spectra of superconducting tunnel junction x-ray detectors

SIS tunnel junctions (STJD) are predicted to have exceptionally good energy resolution as x-ray detectors. This paper spectra illustrative of the current state of the art. They were obtained when /sup 55/Fe and /sup 109/Cd x-ray sources bombarded an epitaxial base layer Nb/Ta/Al/AlO/sub x//Al/Nb junction having a diamond shape and sides 15 microns long. Our model of energy loss via the escape of photoelectrons early in the energy cascade is demonstrated to lie qualitatively correct. Earlier results based on fine-grained junctions suggested that such small area devices will have a large characteristic foot (asymmetric broadening of low energy side) on the peaks in the spectra that correspond to complete capture of the incident energy in the device. Our device is small, yet its full energy peaks lack significant feet. We conclude that this device is in the fast diffusion limit. This result contradicts the suggestion that Andreev reflection should severely restrict lateral diffusion in clean superconducting bilayers. >

Nuclear magnetic resonance properties of rocks at elevated temperatures

Although the NMR properties of fluid-saturated rocks have been studied for decades, the temperature dependence of these properties has never been systematically investigated. A hot-air-flow loop has been constructed that can elevate the temperature of a core sample. An algorithm to estimate the distribution of NMR relaxation times from Fast Inversion Recovery/CPMG data has been developed and used to analyze the effects of temperature. Proton magnetic resonance measurements have been made on an assortment of water-saturated sandstones and carbonates and an oil-saturated sandstone at temperatures ranging from 25 to 175°C. The relaxation times of all the water-saturated sandstones and most of the carbonates depend only weakly on temperature. This indicates that the water-saturated rocks studied are in the fast-diffusion limit and the intrinsic surface relaxation strength is independent of temperature. In all the rocks studied the magnitude of the NMR signal, which is proportional to porosity, decreases with increasing temperature in accordance with the Curie Law. Data from oil-saturated sandstone clarify the roles of molecular diffusion, material-specific surface interactions, and bulk fluid relaxation.

The slow diffusion limit for the survival probability in reactive diffusion equations

We consider the Smoluchowski equation with a coordinate-dependent reactivity and obtain asymptotic expansions for the survival probability and the mean lifetime as power series in the diffusion constant. The coefficients in this expansion may depend on both temporal and spatial variables. Using the present results together with the previously derived expansion for the fast diffusion limit, we are able to obtain useful Padé approximations for the dependence of the mean lifetime on the diffusion coefficient.

1H spin‐lattice relaxation and observation of interfacial material in solid polyethylenes

AbstractThe 1H NMR laboratory‐ and rotating‐frame (LF,RF) spin‐lattice relaxation (SLR) behaviour in solid polyethylenes is reviewed in light of the two region spin diffusion coupled model. Highly crystalline high‐density materials show single T1 values indicative of the fast diffusion limit whilst RFSLR is close to the slow diffusion limit, allowing direct determination of the dimensions of the crystalline and disordered regions. The LFSLR and RFSLR of the HDPE's suggest the existence of an interfacial material which the SLR behaviour includes as part of the crystalline region in contrast to XRD and DSC. 13C CPMAS measurements of 1H RFSLR in a LLDPE provide an additional insight into this problem. The 1H RFSLR is best represented by a superposition of two exponentials, measured by either 1H or 13C NMR. The 1H RFSLR, measured at different chemical shifts in the 13C NMR spectrum, provides direct evidence of the interfacial material. The possible use of spin‐locking at angles other than 90° in the rotating frame to scale the spin diffusion rates is reviewed and its possible utility for LLDPE materials explored.

Viscosity expansions in reactive diffusion processes

Properties of chemical reactions in systems undergoing diffusional motion depend on the ratio of chemical to diffusional rates. The present work deals with perturbation expansions in this quantity. For bounded diffusion, the Laplace transformed survival probability, mean lifetime, eigenvalues, and eigenfunctions are expanded in this ratio. The theory is developed mainly in the fast diffusion limit. In this limit, the survival probability for an initial equilibrium state is shown to be exponential up to linear order. For unbounded diffusion, expansions are derived for the steady-state concentration profile and rate coefficient. By inverting the series one obtains Padé-like approximations for rate coefficients with much improved convergence. Several examples are worked out in detail. These include the ‘‘radiation’’ boundary condition, barrierless isomerization, steady-state binding, and Förster quenching.

Energy transfer study between (Yb, Pr), (Yb, Sm) and (Yb, Dy) in fluorohafnate glasses

The energy transfer mechanism among Yb and Pr, Sm or Dy in fluorohafnate glasses has been examined. The Yb 3+ ions are the donor ions and Pr 3+, Sm 3+ or Dy 3+ are the acceptor ones. Donor luminescence decay measurements are anakyzed using the kinetic models developed for direct energy transfer and excitation diffusion. For low Yb 3+ concentration (0.1 mol%) the excitation is transferred directly to the acceptors and migration of energy between the Yb 3+ ions is negligible. The results are compatible with a dipole-dipole interaction mechanism among donors and acceptors. By increasing the donor concentration, it has been possible to change the energy diffusion parameter D by over two orders of magnitude, but the limit of fast diffusion was not reached. Finally the donor-acceptor and donor-donor transfer parameters obtained from luminescence decay measurements are given and they are compared with those obtained from spectroscopic data for the donor and acceptor ions.

Vibrational energy transfer in solutions: From diffusive to impulsive binary collisions

The effect of diffusion on energy transfer from excited donor to acceptor molecules in liquid solutions is studied with particular attention focused on vibrational energy transfer between solute molecules in dilute solutions. Such processes are often discussed in the independent binary collision (IBC) framework and diffusion effects are assumed to be negligible. We introduce the concept of diffusive collisions (encounters between acceptor and donor molecule within an effective energy transfer range) and investigate the conditions under which the cross section for the energy transfer process may be affected by the cross section for the diffusive collision as opposed to the more common fast diffusion limit where the energy transfer is dominated by direct binary collisions. We conclude that while in most common situations vibrational energy transfer is indeed dominated by binary collision events, pronounced diffusion effects should exist at moderately high pressures. Explicit estimates are provided for the HCl/Xe system.

1H nuclear magnetic resonance and spin–lattice relaxation in solid, high-density polyethylene

1 H spin–lattice relaxation, in both laboratory and on-resonance rotating frames, are reported for five samples of a high-density, linear polyethylene prepared so as to produce a range of lamellar crystalline widths. The samples are also characterised by measurements of density, Raman longitudinal acoustic modes (LAM) crystallinity (DSC) and small-angle X-ray scattering (SAXS). The laboratory-frame spin–lattice relaxation (LFSLR) is well represented by a single process, and the relaxation rates are linearly related to the crystalline fraction (DSC), which is consistent with the prediction of the fast-diffusion limit of the two-region, spin-diffusion-coupled model of relaxation in these systems. The rotating-frame relaxation (RFSLR) is, as found in previous studies, a multi-exponential process. It is shown that a fit to four components is optimum in terms of quality of fit and signal-to-noise ratio and that this leads to accurate values for the amplitude and time constant of the dominant, longest relaxation-time component. It is also shown that the use of a solid-echo refocussing pulse, following the spin-locking substantially increases the total signal observed but has a much smaller effect on the amplitudes and time constants. The values of the relaxation times and amplitudes for the two longest relaxation time components for each sample are compared with the predictions of the slow-diffusion limit of the spin-diffusion model. The longest relaxation-time component is found to be proportional to the square of the lamellar thickness determined from Raman LAM, and this is used to deduce a value for the spin-diffusion coefficient in the crystalline region, D2≈ 2 × 1016 m2 s–1. Assuming the slow-diffusion limit, the amplitudes of the long-time components in the RFSLR are used to derive values for the size of the disordered region, l1, using both the Raman and NMR derived values of l2, the size of the crystalline region. The l1 and l2 values obtained from the RFSLR are used to derive densities for regions 1 and 2 from the measured sample densities. The values of (l1+l2) from NMR are compared with those obtained from SAXS, and good agreement is found. The validity of the spin-diffusion model is further supported by the data reported here. The relaxation processes (both LF and RF) seem to imply a higher degree of ‘crystallinity’ than revealed by DSC or X-ray diffraction. The LFSLR and RFSLR are internally self-consistent, and this suggests that for SLR the crystalline component includes a fraction of the material characterised as non-crystalline by other methods (XRD, DSC).

Reactive dynamics in energy transfer problems

An approach to energy transfer due to microscopic interactions between moving acceptor and donor molecules is described. The theory is based on a set of coupled evolution equations, assuming that the displacements of the reacting particles obey a diffusion equation. A general expression for the transfer rate holding for any microscopic interaction form is derived and its asymptotic behavior at long and moderately long times is examined. Special emphasis is given to the influence of the molecular motion. In the fast diffusion limit, the transfer rate is independent of the mobility of the particles and given by the integral of the microscopic interaction form over space. In a finite diffusion limit, a characteristic transfer length is introduced; it may be interpreted as the radius of a sphere within which trapping is complete in spite of outward diffusion. Its exact expression is derived for multipolar and exponential interactions.

Energy Transfer between the Low-Lying Energy Levels ofPr3+andNd3+in LaCl3

The energy-transfer mechanisms acting between the low-lying energy levels of ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Nd}}^{3+}$ in La${\mathrm{Cl}}_{3}$ have been investigated. In all cases studied the ${\mathrm{Pr}}^{3+}$ ions acted as donors and the ${\mathrm{Nd}}^{3+}$ ions acted as acceptors. Fluorescence-intensity measurements have been made on the ${\mathrm{Pr}}^{3+}^{3}F_{3}$ and $^{3}H_{6}$ donor systems. The shape of the time decay of the ${\mathrm{Pr}}^{3+}^{3}F_{3}$ level has been investigated. For low Pr concentrations (2%) the excitation is transferred directly to the ${\mathrm{Nd}}^{3+}$ ions without significant migration between the ${\mathrm{Pr}}^{3+}$ ions. The experimental results are consistent with a dipole-dipole interaction mechanism with donor-acceptor energy-transfer constants of $C=6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}38}$ ${\mathrm{cm}}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for the ${\mathrm{Pr}}^{3+}^{3}F_{3}$ donor system and $C=2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}38}$ ${\mathrm{cm}}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for the ${\mathrm{Pr}}^{3+}^{3}H_{6}$ donor system. For Pr concentrations \ensuremath{\ge} 2 at.% the effects of donor-donor transfer between ${\mathrm{Pr}}^{3+}$ ions are evident from both the fluorescence-intensity measurements and the shape of the decay curves; diffusion constants were obtained from these decays. The diffusion rates indicate a donor-donor transfer constant of $C\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}38}$ ${\mathrm{cm}}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for ${\mathrm{Pr}}^{3+}$ ions excited to the $^{3}F_{3}$ manifold. Diffusion-limited decay was observed but the limit of fast diffusion was not reached.