Surface diffusion of k-mers in one-dimensional systems

The surface diffusion of interacting k-mers is studied both through analytical and Monte Carlo simulation methods in one-dimensional systems. Adsorption isotherms, jump diffusion coefficients and collective diffusion coefficients are obtained for attractive and repulsive k-mers, showing a variety of behaviors as a function of the size of particles, k. The following main results are found: (a) diffusion coefficients increase with k for non-interacting particles; (b) for fixed k, diffusion coefficients increase as the interaction energy increases from negative (attractive) to positive (repulsive) values; (c) for attractive interactions diffusion coefficients increase with k in the whole range of coverage; (d) for repulsive interactions diffusion coefficients decrease with k up to moderately high coverage and increase with k at high coverage. Results are rationalized in terms of the behavior of the vacancy probability distribution.     

Influence of the degree of crystal perfection and deviation from the stoichiometric composition on the diffusion processes in samarium sulfide

The processes of diffusion of samarium and europium in nonstoichiometric samarium sulfide (SmS) at temperatures in the range 950–1600°C have been investigated by the radioactive isotope method and the method of weight loss upon evaporation of excess samarium. It has been found that there is a correlation between the diffusion coefficient D and the size of coherent X-ray scattering regions in SmS samples, as well as between the diffusion coefficient and the degree of deviation from the stoichiometric composition: the diffusion coefficient of impurities decreases as the size of coherent X-ray scattering regions increases and the stoichiometric composition is approached. The calculation of the diffusion coefficient of electrons in samarium sulfide at T = 77–300 K has demonstrated that the value of D increases with increasing temperature and increasing size of coherent scattering regions.     

Oxygen diffusion in SrZrO3

Oxygen diffusion coefficients in SrZrO₃ polycrystals were determined using the isotopic exchange method with ¹⁸O as oxygen tracer. Diffusion treatments were performed at different temperatures between 1173 K and 1473 K. Oxygen diffusion profiles were established by secondary ion mass spectroscopy (SIMS). Classical diffusion equations were used to fit experimental results and to determine bulk diffusion (D_vol) and surface exchange (k) coefficients of oxygen in SrZrO₃ polycrystalline materials. From these values, bulk diffusion and grain boundary diffusion coefficients as well as oxygen surface exchange coefficients were determined. The activation energy of oxygen diffusion in the bulk is 2.1 eV, while for the diffusion in the grain boundary, 1.8 eV was found. The surface exchange reaction has an activation enthalpy of 1.2 eV.     

On the analysis of coordinate-dependent diffusion

Coordinate-dependent diffusion in solids, in which case the probability of presence of a vacancy, φ, and the probability of migration, γ, depend on coordinate x is analyzed. The coordinate dependence of parameters φ and γ is shown to cause the drift component of the diffusion flux. In the presence of this component, the direction of atomic motion may coincide (enhanced diffusion) or not coincide (retarded diffusion and up-diffusion) with the direction of the diffusion component. The criteria of enhanced diffusion, retarded diffusion, and up-diffusion are established. These criteria are specified by the behavior of function F(x)=φ(x)/γ(x).     

In vivo lipid diffusion coefficient measurements in rat bone marrow

The diffusion coefficient of lipids, D_l, within bone marrow, fat deposits and metabolically active intracellular lipids in vivo will depend on several factors including the precise chemical composition of the lipid distribution (chain lengths, degree of unsaturation, etc.) as well as the temperature. As such, D_l may ultimately prove of value in assessing abnormal fatty acid distributions linked to diseases such as cystic fibrosis, diabetes and coronary heart disease. A sensitive temperature dependence of D_l may also prove of value for MR-guided thermal therapies for bone tumors or disease within other fatty tissues like the breast. Measuring diffusion coefficients of high molecular weight lipids in vivo is, however, technically difficult for a number of reasons. For instance, due to the much lower diffusion coefficients compared to water, much higher b factors than those used for central nervous system applications are needed. In addition, the pulse sequence design must incorporate, as much as possible, immunity to motion, susceptibility and chemical shift effects present whenever body imaging is performed. In this work, high b-factor line scan diffusion imaging sequences were designed, implemented and tested for D_l measurement using a 4.7-T horizontal bore animal scanner. The gradient set available allowed for b factors as high as 0.03 μs/nm² (30,000 s/mm²) at echo times as short as 42 ms. The methods were used to measure lipid diffusion coefficients within the marrow of rat paws in vivo, yielding lipid diffusion coefficients approximately two orders of magnitude smaller than typical tissue water diffusion coefficients. Phantom experiments that demonstrate the sensitivity of lipid diffusion coefficients to chain length and temperature were also performed.     

Grain Boundary Diffusion and Mechanisms of Creep of Nanostructured Metals

Comparative investigations of diffusion in coarse-grained (d = 20 μm), nanocrystalline (d = 0.04 μm) and nanostructured nickel (d = 0.3 μm) have been carried out in a temperature interval of 0.2–0.3 melting temperature. The reasons for difference of parameters of copper grain-boundary diffusion in the above materials are discussed. The effect exerted by grain boundary state and grain boundary diffusion fluxes of impurity on creep mechanisms of nanostructured nickel and copper in the temperature interval of 373–473 K have been studied. Significant change in the apparent creep activation energy under copper grain boundary diffusion fluxes is described as a consequence of different contribution of grain boundary sliding to overall deformation.     

A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes

In this paper, we embark on a new strategy for computing the steady state solution of the diffusion equation. The new strategy is to solve an equivalent first-order hyperbolic system instead of the second-order diffusion equation, introducing solution gradients as additional unknowns. We show that schemes developed for the first-order system allow O(h) time step instead of O(h²) and converge very rapidly toward the steady state. Moreover, this extremely fast convergence comes with the solution gradients (viscous stresses/heat fluxes for the Navier–Stokes equations) simultaneously computed with the same order of accuracy as the main variable. The proposed schemes are formulated as residual-distribution schemes (but can also be identified as finite-volume schemes), directly on unstructured grids. We present numerical results to demonstrate the tremendous gains offered by the new diffusion schemes, driving the rise of explicit schemes in the steady state computation for diffusion problems.     

Fick's law and Fokker-Planck equation in inhomogeneous environments

In inhomogeneous environments, the correct expression of the diffusive flux is not always given by the Fick's law Γ=−D∇n. The most general hydrodynamic equation modelling diffusion is indeed the Fokker-Planck equation (FPE). The microscopic dynamics of each specific system may affect the form of the FPE, either establishing connections between the diffusion and the convection term, as well as providing supplementary terms. In particular, the Fick's form for the diffusion equation may arise only in consequence of a specific kind of microscopic dynamics. It is also shown how, in the presence of sharp inhomogeneities, even the hydrodynamic FPE limit may becomes inaccurate and mask some features of the true solution, as computed from the Master equation.     

Diffusionless crystal growth in a eutectic system during rapid solidification

Experiments on nonequilibrium rapid eutectic growth are surveyed. The applicability limits of the modern theoretical models describing rapid solidification of binary systems are assessed. A problem of rapid eutectic growth when the local equilibrium is violated in the solute diffusion field (in the bulk liquid and at the solid-liquid interface) is formulated. An analytical solution to the problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that the diffusion-limited growth of a eutectic pattern ceases as soon as a chemically homogeneous crystalline phase begins to grow when the critical point V=V _D is achieved (V is the solid-liquid interface velocity and V _D is the solute diffusion speed in the bulk liquid). At V ≥ V _D, eutectic decomposition is suppressed and the nascent homogeneous crystalline phase has the initial (nominal) chemical composition of the binary system.     

Contribution a la theorie atomique de la diffusion chimique

Here we present an analysis and a development of the atomic theory of chemical diffusion as proposed by Manning for a binary system a/b.The general expression for the flux of a tracer in a concentration gradient is first established. This expression of the flux is identified with that deduced in the phenomenological theory. Thus a relationship between the partial correlation factors of vacancies with each of the a and b species is obtained.The effect of “vacancy flow” can be described in terms of these correlation factors. Thus the vacancy flow on species A leads to a correlation of the vacancy jumps with species B and vice versa.We shall see that the Nernst-Einstein equation can be extended to the case of chemical diffusion and that the ratio of the intrinsic diffusion coefficients is equal to the ratio of the mean jump frequencies W_A and W_B.Also, the activation energies of intrinsic diffusion coefficients are related very simply to the activation enthalpies of atomic jumps.In conclusion, we shall see that chemical diffusion in a binary system a/b can be completely described if either the thermodynamic factor and the coefficients of self diffusion, or the thermodynamic factor and the coefficients of intrinsic diffusion are known as functions of the concentration.     

A formalism for extracting diffusion coefficients from concentration profiles

A formalism is presented for extracting diffusion coefficients from concentration profiles measured by analytical techniques such as Auger electron spectroscopy, secondary ion mass spectrometry, low energy and high energy ion scattering. The formalism is based on the measurement of the concentration profile or gradient (dC/dx) at the interface before and after diffusion, and corrects for extraneous effects in the profile due to sputtering artifacts, initial diffusion, and interface and surface roughness. The formalism is shown to be quite insensitive to surface depletion caused by preferential sputtering. The process of correction involves a deconvolution which is valid for all compositions if the above effects can be represented by complementary error functions, as judged by the linearity on a “probability” plot of C versus x where C is concentration and x is distance in angstroms. The formalism is expressed in the form of a nomograph for the routine determination of diffusion coefficients from measured profiles. Examples of its use are given for profiles measured on the Ti-Pd, Ti-Au, Pd-Au, and Cu-Au thin film systems.     

Geometric analysis of the b-dependent effects of Rician signal noise on diffusion tensor imaging estimates and determining an optimal b value

The optimal diffusion weighting (DW) factor, b, for use in diffusion tensor imaging (DTI) studies remains uncertain. In this study, the geometric relations of DW quantities are examined, in particular, the effects of Rician noise in the measured magnetic resonance signal. This geometric analysis is used to make theoretical predictions for selecting a b value to reduce the influence of noise. It is shown that the optimal b value for DTI studies in healthy human parenchyma is approximately b=1200 s mm⁻², with a simple relation given as well for a given expected apparent diffusion coefficient. Monte-Carlo simulations on sets of realistic DTI measures are then performed, verifying the optimal DW for minimizing estimate errors. The effects of noise on various DTI parameters such as anisotropy indices (fractional anisotropy and scaled relative anisotropy), mean diffusivity, radial diffusivity, eigenvalues and the direction of the first eigenvector are investigated as well.     

Directional sensitivity of anomalous diffusion in human brain assessed by tensorial fractional motion model

Anisotropic diffusion in the nervous system is most commonly modeled by apparent diffusion tensor, which is based on regular diffusion theory. However, the departure of diffusion-induced signal attenuation from a mono-exponential form implies that there is anomalous diffusion. Recently, a novel diffusion NMR theory based on the fractional motion (FM) model, which is an anomalous diffusion model, has been proposed. While the FM model has been applied to both healthy subjects and tumor patients, its anisotropy in the nervous system remains elusive. In this study, this issue was addressed by measuring the FM-related parameters in 12 non-collinear directions. A metric to quantify the directional deviation was derived. Furthermore, the FM-related parameters were modeled as tensors and analyzed in analogy with the conventional diffusion tensor imaging (DTI). Experimental results, which were obtained for 15 healthy subjects at 3T, exhibited pronounced anisotropy of the FM-related parameters, although the effects were smaller than the apparent diffusion coefficient (ADC). The tensorial nature for α, which is the Noah exponent in the FM model, showed behavior similar to the ADC, especially the principal eigenvector for α aligned with the dominant white matter fiber directions. The Hurst exponent H in the FM model, however, showed no correlation with the major fiber directions. The anisotropy of the FM model may provide complementary information to DTI and may have potential for tractography and detecting brain abnormalities.     

Fractal diffusion in smooth dynamical systems with virtual invariant curves

Preliminary results of extensive numerical experiments with a family of simple models specified by the smooth canonical strongly chaotic 2D map with global virtual invariant curves are presented. We focus on the statistics of the diffusion rate D of individual trajectories for various fixed values of the model perturbation parameters K and d. Our previous conjecture on the fractal statistics determined by the critical structure of both the phase space and the motion is confirmed and studied in some detail. In particular, we find additional characteristics of what we earlier termed the virtual invariant curve diffusion suppression, which is related to a new very specific type of critical structure. A surprising example of ergodic motion with a “hidden” critical structure strongly affecting the diffusion rate was also encountered. At a weak perturbation (K ? 1), we discovered a very peculiar diffusion regime with the diffusion rate D=K ²/3 as in the opposite limit of a strong (K ? 1) uncorrelated perturbation, but in contrast to the latter, the new regime involves strong correlations and exists for a very short time only. We have no definite explanation of such a controversial behavior.     

Changes in skeletal muscle diffusion parameters owing to intramyocellular lipid

IntroductionSeveral studies investigated the changes in diffusion of water molecules in skeletal muscle cells of lifestyle-related-disease patients who performed a hybrid training (HYBT) for six months. They reported that the apparent diffusion coefficient (ADC) and all diffusion eigenvalues (λ₁, λ₂, and λ₃) increased after the HYBT, owing to the enlargement of the intramyocellular diffusion space (intracellular space) caused by the muscular hypertrophy.We assumed that the HYBT promoted metabolism of the whole skeletal muscle including lipids, which reduced the amount of intramyocellular lipid (IMCL), and led to a secondary enlargement of the diffusion space in the skeletal muscle cells. However, the IMCL has to be a diffusion limiting factor in order to verify this hypothesis. Until now, there is no report on whether IMCL is a diffusion limiting factor for water molecules.The objective of this study was to examine whether the IMCL is a diffusion limiting factor in skeletal muscle cells.Materials and methodsWe performed a three-dimensional quantification of the IMCL in triceps surae muscles of lifestyle-related-disease patients and healthy volunteers. In addition, we measured the ADC in the volume of interest (VOI), diffusion anisotropy (FA), and diffusion eigenvalues (λ₁, λ₂, and λ₃), and evaluated the correlations between these diffusion parameters and IMCL.ResultsThe results showed that the amount of IMCL was positively and negatively correlated with the FA and λ₃, respectively, in lifestyle-related-disease patients. In addition, there was a weak negative correlation between IMCL and ADC, λ₁, and λ₂. There was no correlation between the amount of IMCL and diffusion parameters of healthy volunteers.DiscussionAbove a certain amount, the IMCL correlates with the diffusion parameters. A higher amount of IMCL leads to smaller diffusion eigenvalues. This result suggested that IMCL possibility of influencing diffusion of water molecules in skeletal muscle cells. However, in order for the influence of IMCL to be reflected in the diffusion eigenvalues, it was needed large amount of IMCL existed, and we thought that the influence was smaller than the influence by the already reported cell membrane.     

Surface diffusion in an adsorbed system with competing order parameters

A spin-1 lattice gas model with stochastic dynamics is developed and applied to the surface diffusion in an adsorbed system possessing an internal degree of freedom. Both the chemical diffusion coefficient (D) and the relaxation rate (T) associated with the internal degree of freedom are examined. The coverage dependence of these quantities is studied within the conventional approximation, with the various static averages estimated by a Monte Carlo method. A mean field theory treatment is also given. Particular emphasis is given to the effects that the various phase transitions of the system have upon the behavior of D and T. The ordering of the internal degree of freedom is found to have an important influence on diffusion, leading to a highly reduced diffusion coefficient in the ordered regions. The connection to surface diffusion in adsorbed systems with solid orderings is pointed out. The importance of thermodynamic properties in determining the behavior of the diffusion coefficient in such systems is indicated.     

Reproducibility and variation of diffusion measures in the squirrel monkey brain,in vivo and ex vivo

PurposeAnimal models are needed to better understand the relationship between diffusion MRI (dMRI) and the underlying tissue microstructure. One promising model for validation studies is the common squirrel monkey, Saimiri sciureus. This study aims to determine (1) the reproducibility of in vivo diffusion measures both within and between subjects; (2) the agreement between in vivo and ex vivo data acquired from the same specimen and (3) normal diffusion values and their variation across brain regions.MethodsData were acquired from three healthy squirrel monkeys, each imaged twice in vivo and once ex vivo. Reproducibility of fractional anisotropy (FA), mean diffusivity (MD), and principal eigenvector (PEV) was assessed, and normal values were determined both in vivo and ex vivo.ResultsThe calculated coefficients of variation (CVs) for both intra-subject and inter-subject MD were below 10% (low variability) while FA had a wider range of CVs, 2–14% intra-subject (moderate variability), and 3–31% inter-subject (high variability). MD in ex vivo tissue was lower than in vivo (30%–50% decrease), while FA values increased in all regions (30–39% increase). The mode of angular differences between in vivo and ex vivo PEVs was 12 degrees.ConclusionThis study characterizes the diffusion properties of the squirrel monkey brain and serves as the groundwork for using the squirrel monkey, both in vivo and ex vivo, as a model for diffusion MRI studies.     

Thermal neutron scattering from a hydrogen-metal system in terms of a general multi-sublattice jump diffusion model—I: Theory

A Multi-Sublattice Jump Diffusion Model (MSJD) for hydrogen diffusion through interstitial-site lattices is presented. The MSJD approach may, in principle, be considered as an extension of the Rowe et al.[1] model. Jump diffusion to any neighbours with different jump times which may be asymmetric in space is discussed. On the basis of the model a new method of calculating the diffusion tensor is advanced. The quasielastic, double differential cross section for thermal neutron scattering is obtained in terms of the MSJD model. The model can be used for systems in which interstitial jump diffusion of impurity particles occurs. In Part II the theoretical results are compared with those for quasielastic neutron scattering from the αNbH_x system.     

Diffusion in thermotropic liquid crystals

Diffusion coefficients in mesophases of thermotropic liquid crystals have been measured to a considerable extent only during the last twelve years. Some theories on diffusion in mesophases have been developed too. Measurements in nematic, smectic A, smectic B, and smectic C mesophases have been carried out by mass transport techniques, detected by radio-tracers or optically, by NMR spin-echo techniques, and by quasi-elastic neutron scattering. The diffusion is anisotropic in most of the cases. In nematics self-diffusion parallel to the molecular director (D₆) is somewhat faster than perpendicular (D_⊥). Both diffusion coefficients show about the same activation energy. Impurity diffusion (small molecules dissolved in mesophases) demonstrates the same behaviour with less anisotropy. In smectics A and C the diffusion coefficients D₆ and D_⊥ can be nearly equal. The anisotropy is now shown in their respective activation energies with E₆>E_⊥. This effect is more pronounced in impurity than in self-diffusion. In addition impurity diffusion shows a strong anisotropy of the diffusion coefficients (D_⊥?D₆). In smectics B the activation energies E₆ and E_⊥ of self-diffusion seem again to be equal and the diffusion coefficients show a small anisotropy (D_⊥≥D₆).     

On the effects of a varied diffusion time in vivo: is the diffusion in white matter restricted?

The aim of this work was to study the diffusion-related signal attenuation curves (signal-vs.-b curves) measured perpendicular and parallel to the neuronal fibers of the corticospinal tract in vivo and to determine whether effects of restricted diffusion could be observed when varying the diffusion time (T_D). A biexponential model and a two-compartment model including exchange according to the Kärger formalism were employed to analyze the signal-vs.-b curves. To validate the two-compartment model, restricted diffusion with exchange was simulated for uniformly sized cylinders, using different diameters and exchange times. The model was shown to retrieve the simulated parameters well, also when the short gradient pulse approximation was not met. The in vivo measurements performed perpendicular to the tracts, using b values up to 28000 s/mm² and T_D values between 64 and 256 ms, did not show the effects of restricted diffusion as expected from previous ex vivo studies. The applied two-compartment model yielded an average axonal diameter of about 4 μm and an intracellular exchange time of about 300 ms, but did not fit statistically well to the data. In conclusion, this study indicates that if the diffusion is modeled as two compartments, of which one is restricted, exchange must be included in the model.