Nonclassical transport in fractal media with a diffusion barrier

We investigate the impurity transport in a randomly heterogeneous fractal medium with a diffusion barrier. The barrier is due to low permeable medium surrounding the source. The transport regimes and asymptotic (large-distance) concentration distributions are found. The presence of the diffusion barrier results in the retardation of the transport regimes at short times. As regards the asymptotic concentration distribution, the barrier influence persists for long times as well.     

Impurity transport in percolation media

An equation describing the impurity transport in a percolation medium is obtained and the inferences drawn from this equation are analyzed based on the scale invariance concept. A determining part in this analysis is allowance for the sinks inherent in such media. At distances shorter than the correlation length, the particles are transferred in the regime of subdiffusion; at large distances, the concentration asymptotics exhibits a characteristic “tail” shape. In the medium occurring in the state above the percolation threshold, the impurity transport over time periods longer than the characteristic time related to the correlation length is well described by the classical equation with a renormalized diffusion coefficient. In this case, the concentration tail has a Gaussian shape at moderate distances and tends to subdiffusion asymptotics at very long distances. A relation is established between the factor determining renormalization of the diffusion coefficient and the factor determining a decrease in the number of active impurity particles at large times.     

Surface tracer diffusion in the presence of a solid barrier: a Monte Carlo study

We investigate, by way of Monte Carlo simulations, the diffusion of adsorbates on a square lattice. The lattice contains inaccessible sites forming linear barriers. The adsorbate atoms do not interact among themselves, except for the hard core. The tracer diffusion coefficients, reduced by their values in the absence of the barrier, are plotted versus the length of the barrier. We find that the diffusion along the barrier is not affected by its presence, whereas that perpendicular to it decreases linearly, for a large interval of the barrier, with its length. The reduced diffusion coefficient does not depend on the concentration of the diffusing particles, but its value increases with the perpendicular distance between the barriers.     

Impurity diffusion in a gas flow in a channel

The impurity concentration N in a gas flow in a channel is approximately calculated. In a number of cases, the wall losses of impurities in the channel can be taken into account by introducing an effective lifetime τ that describes the diffusion drift of the impurity from the flow toward the walls.     

Physical mechanisms of power fractal asymptotic forms of dispersion transport in disordered media

Particle drift in systems with anomalous diffusion is investigated. Physical mechanisms of power fractal asymptotic forms in dispersion transport are established and the physical meaning of the characteristic changeover time for asymptotic forms is clarified. It is shown that long-term power fractal asymptotic forms for particle mobility in subdiffusion problems corresponding to the behavior of transition currents in disordered systems (i.e., having different asymptotic forms for short and long time intervals) are associated with capture in traps (ribs in the comb structure).     

Impurity diffusion in a harmonic potential and nonhomogeneous temperature

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.     

Impurity transport in a dual-porosity medium with sorption

A nonstationary model of impurity transport in a dual-porosity (fractured porous) medium with sorption is proposed. It is shown that the equilibrium between dissolved and adsorbed phases is absent for a prolonged time, which leads to the development of nonclassical impurity transport regimes. Parameter values are found for which the behavior of the system cannot be described using the conventional model of equilibrium sorption even at large times for which the equilibrium between dissolved and adsorbed components should be established.     

Impurity diffusion in NiO grain boundaries

The diffusion of ¹³⁹Ce and ⁵¹Cr in polycrystalline NiO has been studied in the temperature range 600–1100°C in oxygen at a pressure of 1 atm. These impurities were chosen because of their different effective charges and segregation behaviour and because of their relevance to the oxidation of metals at elevated temperature. The solubility of Ce in the NiO lattice is negligible, but Ce is soluble at NiO grain boundaries and dislocations. Consequently Ce tracer only diffused along these pathways and the resulting profiles were analysed straightforwardly to give grain boundary and dislocation diffusion coefficients. The lattice solubility of Cr, on the other hand, is not negligible and Cr also segregates strongly to NiO grain boundaries. A procedure for analysing penetration profiles of such an impurity has been developed in which parameters describing segregation and grain boundary diffusion are deduced self-consistently. When combined with previous measurements of Ni and Co diffusion it is found that grain boundary coefficients decrease in the order Co, Ni, Cr, Ce as is also found for lattice diffusion (except for Ce, which has negligible lattice solubility).The implication of the results for the distribution of Cr and Ce in NiO films formed by metal oxidation is also discussed.     

Transfer processes in fractal media

An irreversible process in fractal media involves coupling relation between the space and the time. The present note displays how the fractional derivation has to be introduced to describe this effect. As a result the law of the chemical diffusion to a fractal is given.     

Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media

We consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media. It is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions (in addition to the fractal dimensiond _f ): (1)d _f ^BB , the fractal dimension of the backbone, arises in connection with electric current flow, (2)d _red, the fractal dimension of the singly connected bonds in the backbone, arises in connection with its equivalence to the thermal scaling power, (3)d _E, the fractal dimension of the of the elastic backbone, (4)d _u, the fractal dimension of the unscreened perimeter, arises in connection with the viscosity singularity at the gelation threshold, (5)d _min the fractal dimension of the minimum path (or chemical distance) between two sites, arises in co-nnection with the Aharony-Stauffer conjecture, (6)dw, the fractal dimension of a random walk, (7)d _G, the fractal dimension of growth sites that arise as a random walk creates a cluster. Relations among these fractal dimensions are discussed, some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases.Supported in part at the Center for Polymer Studies by grants from ONR and NSF.     

Spectral properties and transport mechanisms of partially chaotic bounded flows in the presence of diffusion

The spectral properties of the Poincaré operator associated with the advection-diffusion equation for partially chaotic periodic flows defined in bounded domains are analyzed in this Letter. For vanishingly small diffusivities (i.e., for the Peclet number tending to infinity) the dominant eigenvalue Lambda exhibits the scaling Lambda approximately Pe-alpha, where the exponent alpha in (0,1) depends on the global property of the flow (shape, geometry, and symmetry of quasiperiodic islands). The value of the exponent alpha is an indicator of qualitatively different transport mechanisms and depends on the localization properties of the corresponding eigenfunctions.     

Acoustic-elastodynamic interaction in isotropic fractal media

This research explores the acoustic-elastodynamic interaction in isotropic fractal media. The analysis discusses the direct coupling of two constitutive models under dynamic loading: a continuous solid and an isotropic fractal medium. We consider two situations where in the first, the fractal medium is enclosed within a thin spherical shell (interior problem), while in the second, the fractal medium extends infinitely outside the shell (exterior problem). The two problems are simulated analytically, and the exact solution for the shell displacement is expressed in closed form in the Laplace domain. The formulation of the radiation condition for infinite fractal media is essential to derive the exterior problem’s solution. This study represents a meaningful idealization of real-application problems involving the interaction of multi-constitutive media, e.g. the human brain, whereby fractal features affect the response of this body under various excitations.     

Transmission of quantum particles through a one-dimensional fractal potential barrier

We consider scattering of a nonrelativistic quantum particle by a one-dimensional fractal potential barrier carried by a generalized Cantor set. We obtain recurrence relations for the reflection coefficient and examine the scaling properties as functions of the wave number.V. D. Kuznetsov Siberian Physicotechnical Institute, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 120–127, July, 1993.