A generalized treatment of the chemical diffusion coefficient in binary oxides as a function of the oxygen activity

The dependence of the chemical diffusion coefficient and of the self-diffusion coefficient of the metal on the oxide composition (a₀) in a binary oxide which is predominantly an electronic semiconductor and contains defects only in the cation sublattice has been examined theoretically. For the case where there is only one type dominant lattice defect following an ideal solution behavior the chemical diffusion coefficient is independent of a₀ When more than one type of defect is present instead, the chemical diffusion coefficient depends on a₀. Furthermore, it is shown that values of the chemical diffusion coefficient obtained by thermogravimetric analysis may differ from those obtained using changes in electrical conductivity. Results are compared with available experimental data for NiO and CoO at 1000°C and reasonable agreement is found.     

The effect of chain reactions on the diffusion of absorbates

A chain reaction mechanism is developed that can dramatically increase the predicted diffusion coefficient as a function of coverage for thermally activated submonolayer coverages of adsorbates weakly chemisorbed on surfaces. The key idea is that when a diffusing adatom encounters another adatom it will tend to forward scatter it (due to the channeling effect of the periodic potential of the substrate lattice) thus increasing the effective mean free path. The enhancement effect can be quite dramatic at large adsorbate coverages. The formalism is essentially that of a Markoffian random walk on a lattice which allows for “billiard ball” type collisions. A general formalism for calculating the chemical diffusion tensor for an unspecified lattice is developed. The diffusion tensor is then related to the effective diffusion coefficient measured by the field emission fluctuation method. The special case of the two-dimensional rhombic lattice with two adsorbate species is worked out and shown to agree qualitatively with theoretical predictions of the coverage dependence of the diffusion coefficient with the experimental data for hydrogen and deuterium on W(110)[1].     

Diffusion Lattice Boltzmann Scheme on a Orthorhombic Lattice

We present a diffusion lattice Boltzmann (DLB) scheme which is derived from first principles. As opposed to the traditional lattice BGK schemes the DLB is valid for orthorhombic lattices and it has two eigenvalues of the collision operator. It is shown that the diffusion coefficient depends only on one eigenvalue of the collision operator. Hence, the DLB scheme can be optimized with means of the additional eigenvalue of the collision operator and with different lattice spacing along the principal axes. The properties of the DLB scheme concerning consistency, stability, and accuracy are studied with eigenmode analysis. This analysis shows that the DLB scheme is consistent with diffusion for a wide range of diffusion coefficients, it has unconditional stability, and that it has third-order accuracy. Furthermore, it is shown that accuracy is improved by setting the additional eigenvalue to zero and by densifying the lattice spacing along the direction of the density gradient.     

Dynamical Arrest,Tracer Diffusion and Kinetically Constrained Lattice Gases

We analyze the tagged particle diffusion for kinetically constrained models for glassy systems. We present a method, focusing on the Kob–Andersen model as an example, which allows to prove lower and upper bounds for the self-diffusion coefficient D _S. This method leads to the exact density dependence of D _S, at high density, for models with finite defects and to prove diffusivity, D _S > 0, at any finite density for highly cooperative models. A more general outcome is that under very general assumptions one can exclude that a dynamical transition, like the one predicted by the Mode-Coupling-Theory of glasses, takes place at a finite temperature/chemical potential for systems of interacting particle on a lattice.     

Influence of a uniform driving force on tracer diffusion in a one-dimensional hard-core lattice gas

The influence of a uniform driving force on tracer diffusion is investigated for a one-dimensional lattice gas where particles jump stochastically to unoccupied neighboring sites. A new, simple calculation is presented for the diffusion coefficient of a tracer particle with respect to its average drift, obtained recently by rigorous methods by De Masi and Ferrari. A theoretical expression describing the tracer particle mean square displacement approximately for all times is derived and found to be in excellent agreement with the results of Monte Carlo simulations.     

Reflection of plane waves from an elastic layered medium: Resonance approach and numerical modeling

The resonance formalism developed earlier by H. Überall for a model of a liquid layer overlying a liquid halfspace is extended to a model of an elastic layer overlying an elastic halfspace. Using the Thomson—Haskell matrix technique, an exact analytical expression is obtained for the complex reflection coefficient. Characteristic equations are derived, and their roots, which determine the positions of the resonances of the reflection coefficient for longitudinal and transverse waves, are obtained analytically. In the resonance approach, the exact expression for the reflection coefficient is replaced by an approximate one that describes the behavior of the reflection coefficient near the resonances. The comparison of the exact and approximate values of the reflection coefficient shows good agreement between the results near the frequency and angular resonances.     

Quantum diffusion of atomic species in metals

The coefficient of quantum diffusion D(T) is calculated using exact states of the localized impurity in the deformed vibrating lattice. Without resorting to the Born-Oppenheimer approximation an explicit general expression for D(T), valid for impurities of arbitrary mass and holding in the whole range of temperature, is obtained in terms of the crystal properties. Interesting new features arise from retaining some terms of the Hamiltonian which were left aside in previous theories.     

Recent advances in the theory of chemical diffusion in defective solids

Recent advances in the theory of chemical diffusion in solids with high defect concentrations are reviewed and discussed with the object of establishing common ground among the various approaches. Emphasis has been placed on exact results gained from (1) phenomenological models of chemical diffusion including ambipolar diffusion, (2) atomistic models including lattice gas models.     

Lattice Models Solvable Through the Full Interval Method on Links

A two state model on a one dimensional lattice is considered, where the evolution of the state of each site is determined by the states of that site and its neighboring sites. Corresponding to this original lattice, a derived lattice is introduced the sites of which are the links of the original lattice. It is shown that there is only one reaction on the original lattice, which results in the derived lattice being solvable through the full interval method. And that reaction corresponds to the one dimensional stochastic non-consensus opinion model. A one dimensional non-consensus opinion model with deterministic evolution has already been introduced. Here this is extended to be a model which has a stochastic evolution. Discrete time evolution of such a model is investigated, including the two limiting cases of small probabilities for evolution (resulting to an effectively continuous-time evolution), and deterministic evolution. The formal solution to the evolution equation is obtained and the large time behavior of the system is investigated. Some special cases are studied in more detail.     

Computer simulation of the phenomenological transport coefficient in the lattice gas and Fick's First Law

A direct steady state computer simulation method for calculating the Onsager phenomenological transport coefficient from the gradient of the chemical potential in the one component lattice gas is presented. It is shown that the results are in good agreement with the Einsteinian method. A recent alternative formulation for Fick's First Law that had been proposed to replace the standard Fick's First Law formulation is also analysed using the same model. It is shown that the alternative formulation gives poor agreement with the simulation data whereas the standard Fick's First Law gives excellent agreement. Accordingly, the alternative formulation does not appear to have merit as a new definition of the diffusion coefficient. It is shown that the alternative formulation is a rough approximation for the dependence of the interstitial solute diffusion coefficient on solute concentration in an interstitial solid solution if information about the activity coefficient and solute diffusion coefficient at very dilute concentrations is available. However, in this role, this is not an entirely new idea.     

Diffusion in heterogeneous lattices

We present here the results of our investigations of particle diffusion over different heterogeneous lattices with deep and shallow adsorption sites. A general analytical expression for chemical diffusion coefficient has been derived for a number of inhomogeneous lattices of different dimensionality and symmetry. We have calculated coverage dependencies of diffusion coefficients. The analytical data have been compared with the numerical data obtained by the kinetic Monte Carlo simulations. Almost perfect agreement between the respective results has been found.     

Das Energiespektrum und die mittlere freie Bahn der deformierenden Excitonen in Molekularkristallen

A study is made of the energy spectrum of a deforming exciton and its temperature dependence in molecular crystals. The mean free path and diffusion coefficient of the deforming exciton with respect to multi-phonon collisions with crystal lattice vibrations are theoretically determined. The possibility of distinguishing the propagation of excitation in molecular crystals in the form of deforming excitons from that in the form of non-deforming excitons and from the propagation of excitation by random jumps from one molecule to another is discussed on the basis of the theoretically derived temperature dependence of the diffusion coefficient of the deforming exciton.     

Effect of lateral interactions on tunnel diffusion of adsorbed particles

A way of calculating the chemical diffusion coefficient for tunnel diffusion is discussed. It is shown that the famous “jump rate” formula may be used to describe the tunnel diffusion of interacting particles in the region of coverages and temperatures where phase transitions in the adsorbed overlayer are absent. The dependence of the tunnel diffusion coefficient on coverage is calculated for the square lattice taking into consideration the repulsive lateral interactions of nearest-neighbour particles.     

Determination of the chemical diffusion coefficient by Monte Carlo simulation of the center-of-mass propagation

The extendedZwerger method of determining thechemical diffusion coefficient, for lattice gasconventional diffusion in equilibrium andsuperdiffusion in stationary state, isverified by Monte Carlo simulation of thecentre-of-mass propogation. It has been found that the chemical diffusion coefficient for superdiffusion in one-dimensional, noninteracting lattice gas with the exclusion of double occupancy isconcentration independent like in conventional diffusion. Moreover, it has been proved that (chemical) superdiffusion ofuncorrelated particles isexactly described in this model by theBurgers nonlinear diffusion equation.     

Slip velocity during the flow of a liquid over a solid surface

An equation is derived for the slip velocity when a nonequilibrated drop is spreading over a solid surface. This velocity is obtained as the product of the force induced by the gradient of the chemical potential along the surface with Einstein’s expression for the mobility of the molecules. A similar approach was applied to the flow in a capillary. In that case, the surface diffusion coefficient evaluated by comparing the obtained theoretical expression with experiment is orders of magnitude larger than expected. For this reason, one concludes that the observed slip is apparent and is caused by the formation of a gap of gas separating the flowing liquid from the solid surface.     

On the diffusion of clusters of hard rods

It is shown that the diffusion coefficient of clusters of hard rods in one dimension is exactly independent of the number of rods comprising the cluster. An expression for the autocorrelation function of the velocity of the cluster centre of mass is obtained from a simple stochastic model, and asymptotically for large clusters from the exact autocorrelation function. Neither approximate nor exact velocity autocorrelation function decays exponentially for any size of cluster.     

Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

One-dimensional time-dependent distributions

A new derivation of two important one-dimensional time-dependent distributions for an infinite system of hard rods is presented. This derivation is simpler than previous derivations and it provides a direct physical interpretation of the individual terms in the final expressions. A new, more unusual distribution is also presented and discussed. Finally, an exact expression for the diffusion of a Brownian particle is obtained and compared with the exact expression for the self-diffusion coefficient.     

The relationship between chemical and tracer diffusion coefficients of aliovalent ions in an ionic lattice

The theoretical model developed by Lidiard was extended to describe the relationship between the chemical and tracer diffusion coefficients of aliovalent ions in an ionic lattice.It is shown that the relationship between the chemical diffusion coefficient, D, and the tracer diffusion coefficient, D¹, is D = 2D¹ if the migration of dimers is the principal mechanism of transport and for the migration of trimers D = 3D¹ if the concentration of impurity ion is relatively small. These relationships are valid regardless of the charge of the aliovalent or lattice ions.The chemical diffusion coefficients of Cr³⁺ in Cr-doped MgO were determined for three different temperatures, 1656, 1717 and 1768K, and for the concentration region 2.5×10^?2?2.8×10^?1 mole% Cr₂O₃. Using previously determined values for the tracer diffusion coefficient of ⁵¹Cr in Cr-doped MgO it was found that for the temperature and concentration region investigated D = (2.00±0.17)D¹ which indicates that diffusion proceeds primarily by the migration of dimers.