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Local Porosity Theory for Electrical and Hydrodynamical Transport through Porous Media

R. Hilfer

Physica A 194, 406 (1993)
https://doi.org/10.1016/0378-4371(93)90372-B

submitted on
Sunday, August 2, 1992

The current status of local porosity theory for transport in porous media is briefly reviewed. Local porosity theory provides a simple and general method for the geometric characterization of stochastic geometries with correlated disorder. Combining this geometric characterization with effective medium theory allows for the first time to understand a large variety of electrical and hydrodynamical flow experiments on porous rocks from a single unified theoretical framework. Rather than reproducing or rephrasing the original results the present review attempts instead to place local porosity theory within the context of other current developments in theory and experiment.



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Scaling Theory and the Classification of Phase Transitions

R. Hilfer

Mod. Phys. Lett. B 6, 773 (1992)
https://doi.org/10.1142/S0217984992000855

submitted on
Monday, May 11, 1992

The recent classification theory for phase transitions (R. Hilfer, Physica Scripta 44, 321 (1991)) and its relation with the foundations of statistical physics is reviewed. First it is outlined how Ehrenfests classification scheme can be generalized into a general thermodynamic classification theory for phase transitions. The classification theory implies scaling and multiscaling thereby eliminating the need to postulate the scaling hypothesis as a fourth law of thermodynamics. The new classification has also led to the discovery and distinction of nonequilibrium transitions within equilibrium statistical physics. Nonequilibrium phase transitions are distinguished from equilibrium transitions by orders less than unity and by the fact that equilibrium thermodynamics and statistical mechanics become inapplicable at the critical point. The latter fact requires a change in the Gibbs assumption underlying the canonical and grandcanonical ensembles in order to recover the thermodynamic description in the critical limit.



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Classification Theory for Anequilibrium Phase Transitions

R. Hilfer

Physical Review E 48, 2466 (1993)
10.1103/PhysRevE.48.2466

submitted on
Monday, March 16, 1992

The paper introduces a classification of phase transitions in which each transition is characterized through its generalized order and a slowly varying function. This characterization is shown to be applicable in statistical mechanics as well as in thermodynamics albeit for different mathematical reasons. By introducing the block ensemble limit the statistical classification is based on the theory of stable laws from probability theory. The block ensemble limit combines scaling limit and thermodynamic limit. The thermodynamic classification on the other hand is based on generalizing Ehrenfest’s traditional classification scheme. Both schemes imply the validity of scaling at phase transitions without the need to invoke renormalizaton-group arguments. The statistical classification scheme allows derivation of a form of finite-size scaling for the distributions of statistical averages while the thermodynamic classification gives rise to multiscaling of thermodynamic potentials. The different nature of the two classification theories is also apparent from the fact that the generalized thermodynamic order is unbounded while the statistical order is restricted to values less than 2. This fact is found to be related to the breakdown of hyperscaling relations. Both classification theories predict the possible existence of phase transitions having orders less than unity. Such transitions are termed anequilibrium transitions. Systems near anequilibrium transitions cannot be described by conventional equilibrium thermodynamics or equilibrium statistical mechanics because of very strong fluctuations. Anequilibrium transitions are found to exist in statistical-mechanical model systems. The identification of the Lagrange parameter β in the canonical ensemble becomes invalid if a reservoir and a system of the same substance are in thermal contact and anequilibrium transitions are present. Based on the ergodic hypothesis and the theory of convolution semigroups it is shown that near anequilibrium transitions the equations of motion for macroscopic observables of infinite systems may involve modified time derivatives as generators of the macroscopic time evolution. The general solution to the modified equations of motion exhibits very slow dynamics as frequently observed in a nonequilibrium experiment.



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Microstructural Sensitivity of Local Porosity Distributions

F. Boger, J. Feder, R. Hilfer, T. Jøssang

Physica A 187, 55 (1992)
https://doi.org/10.1016/0378-4371(92)90408-I

submitted on
Wednesday, December 4, 1991

The recently introduced concept of local porosity distributions for the geometric characterization of arbitrary porous media is scrutinized using computer generated pore space images. The paper presents the first direct determination of local porosity distributions from digital images. Pore space images with identical two point correlation functions are employed to analyse the geometrical sensitivity of the local porosity concept. The main finding is that local distributions can be used to discriminate between images which are indistinguishable using standard correlation functions. We also discuss the question of length scales associated with the local porosity concept.



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Grain Size Distribution for Competitive Growth with Nucleation

R. Hilfer, P. Meakin

Zeitschrift für Physik B 88, 223 (1992)
https://doi.org/10.1007/BF01323576

submitted on
Monday, November 25, 1991

The paper introduces and discusses an idealized competitive growth model with nucleation for the microstructure formation during dense branching phase separation in thin Al/Ge films. Grain size and grain length distributions for the new model are obtained analytically and by simulation. These distributions exhibit a characteristic scaling form similar to cluster size distributions in many other growth models. The cutoff functions in these scaling forms and their influence on the determination of effective exponents are studied in detail. It is found that nucleation introduces a new length scale into the other-wise selfsimilar competitive growth model. This length scale appears only inside the cutoff function and diverges algebraically as the nucleation rate vanishes. We find both analytically and by simulation that the cutoff functions can exhibit stretched exponential behaviour ∼exp(−x α) for large arguments. Our analytical and simulation results for grain size and grain length distributions are in excellent quantitative agreement.



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Orientational Ordering in Lipid Monolayers: A Two-dimensional Model of Rigid Rods Grafted onto a Lattice

M. Scheringer, R. Hilfer, K. Binder

Journal of Chemical Physics 96, 2269 (1992)
https://doi.org/10.1063/1.462077

submitted on
Wednesday, October 23, 1991

A simple model for lipid monolayers on water surfaces at high spreading pressure is investigated in this work. In this model, the hydrophilic head group of the lipid molecules form a rigid regular triangular lattice, and the hydrophobic alkane chains (assumed to be in an all‐trans state) are represented by rigid rods with two angular degrees of freedom (θ, φ). The rods consist of ‘‘effective monomers,’’ and between the effective monomers on neighboring rods a Lennard‐Jones interaction is assumed. The model is studied by exact ground‐state calculations, mean‐field theory, and Monte Carlo simulations. Basic parameters are rod length a and lattice constant b. The ground‐state phase diagram shows the following phases: for small b, the rods are oriented perpendicularly to the monolayer plane (no‐tilt phase, 〈θ〉=0); for somewhat larger b, a sixfold degenerate uniform‐tilt state occurs with all rods tilted towards one of their next‐nearest neighbors. For still larger b, the rods are tilted nonuniformly and form a ‘‘striped’’ structure. These unexpected phases do not occur if we allow a rectangular distortion of the lattice. For T>0, the simplest mean‐field theory predicts a gradual disordering of the uniform‐tilt state via a second‐order phase transition. For the transition region, the Monte Carlo results disagree with this picture. Instead they show a strong asymmetric first‐order phase transition with pronounced hysteresis. The transition temperature increases with increasing rod length a, qualitatively similar to experiment.



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Geometry, Dielectric Response and Scaling in Porous Media

R. Hilfer

Physica Scripta T44, 51 (1992)
10.1088/0031-8949/1992/t44/007

submitted on
Sunday, September 15, 1991

Local porosity distributions and local percolation probabilities have been proposed as well defined and experimentally observable geometric characteristics of general porous media. Based on these concepts the dielectric response is analysed using the effective medium approximation and percolation scaling theory. The theoretical origin of static and dynamic scaling laws for the dielectric dispersion and enhancement including Archie’s law in the low porosity limit are elucidated. These well known experimental facts are unified within a theoretical framework based on a quantitative characterization of the pore space geometry. In the high porosity limit the zero frequency real dielectric constant is predicted to diverge as ε'(0) ∝ (1 − phgr)-m’ where phgr denotes porosity and m’ is analogous to the cementation exponent.



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Multiscaling and the Classification of Continuous Phase Transitions

R. Hilfer

Physical Review Letters 68, 190 (1992)
10.1103/PhysRevLett.68.190

submitted on
Monday, August 5, 1991

Multiscaling of the free energy is obtained by generalizing the classification of phase transitions proposed by Ehrenfest. The free energy is found to obey a new generalized scaling form which contains as special cases standard and multiscaling forms. The results are obtained by analytic continuation from the classification scheme of Ehrenfest.



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Critical phenomena Equilibrium Statistical Physics

Thermodynamic Scaling Derived via Analytic Continuation from the Classification of Ehrenfest

R. Hilfer

Physica Scripta 44, 321 (1991)
10.1088/0031-8949/44/4/002

submitted on
Tuesday, April 2, 1991

Scaling of the free energy is derived from thermodynamic arguments. The free energy is found to obey a new generalized scaling form which contains standard scaling as a special case. Contrary to standard scaling the new scaling form permits also nonuniversal exponents. The results are obtained by analytic continuation from the classification scheme of Ehrenfest.



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Disordered Systems fluid flow Porous Media Transport Processes

Local Porosity Theory for Flow in Porous Media

R. Hilfer

Physical Review B 45, 7115 (1992)
https://doi.org/10.1103/PhysRevB.45.7115

submitted on
Thursday, March 28, 1991

A recently introduced geometric characterization of porous media based on local-porosity distributions and local-percolation probabilities is used to calculate dc permeabilities for porous media. The disorder in porous media is found to be intimately related to the percolation concept. The geometric characterization is shown to open a possibility for understanding experimentally observed scaling relations between permeability, formation factor, specific internal surface, and porosity. In particular, Kozeny’s equation relating effective permeability and bulk porosity and the power lawrelation between permeability and formation factor are analyzed. A simple and general consolidation model is introduced. It is based on the reduction of local porosities and emphasizes the general applicability and flexibility of the local-porosity concept. The theoretical predictions are compared with the experimentally observed range for the exponents, and are found to be in excellent agreement.



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Modeling of Orientational Ordering in Lipid Monolayers

M.Kreer, M.Scheringer, K.Binder, K.Kremer, R.Hilfer

in: Computer Simulation Studies in Condensed Matter Physics IV
edited by: D.P. Landau and K.K. Mon and H.B. Schüttler
Springer Verlag, Berlin, 159 (1993)
https://doi.org/10.1007/978-3-642-84878-0_13
ISBN 978-3-642-84878-0

submitted on
Monday, February 18, 1991

Lipid monolayers at high densities are modelled as rigid rods grafted to an interface at the sites of a regular lattice. The transition between the state where the rods are uniformly tilted to a disordered state with no (average) tilt is studied by computer simulation methods. For the one-dimensional model, the molecular dynamics approach is found much less suitable to equilibrate the system rather than Monte Carlo methods. Both in d=2 discretized versions of Monte Carlo codes are much more efficient than continuum Monte Carlo methods, in spite of huge storage requirements. While in d=l the transition occurs at temperature T=0 via the spontaneous creation of solitons, at d=2 a finite temperature first order transition occurs.



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Lattice Models Statistical Physics

Analysis of Multilayer Adsorption Models without Screening

R. Hilfer, J.-S. Wang

Journal of Physics A: Mathematical and General 24, L389 (1991)
10.1088/0305-4470/24/7/013

submitted on
Tuesday, February 5, 1991

A class of recently introduced irreversible multilayer adsorption models without screening is analysed. The basic kinetic process of these models leads to power law behaviour for the decay of the jamming coverage as a function of height. We find the exact value for the power law exponent. An approximate analytical treatment of these models and previous Monte Carlo simulations are found to be in good agreement.



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dielectric relaxation Porous Media

Geometric and Dielectric Characterization of Porous Media

R. Hilfer

Physical Review B 44, 60 (1991)
https://doi.org/10.1103/PhysRevB.44.60

submitted on
Friday, October 12, 1990

This paper introduces local porosity distributions and local percolation probabilities as well-defined and experimentally observable geometric characteristics of general porous media. Based on these concepts the dielectric response is analyzed using the effective-medium approximation and percolation scaling theory. The theoretical origin of static and dynamic scaling laws for the dielectric response including Archie’s law in the low-porosity limit are elucidated. The zero-frequency real dielectric constant is found to diverge as as a power law in the high-porosity limit with an exponent analogous to the cementation exponent. Model calculations are presented for the interplay between geometric characteristics and the frequency-dependent dielectric response. Three purely geometric mechanisms are identified, each of which can give rise to a large dielectric enhancement.



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Frequency Dependent Response and Dynamic Disorder

R. Hilfer

Journal of Non-Crystalline Solids 131–133, 213 (1991)
https://doi.org/10.1016/0022-3093(91)90303-N

submitted on
Wednesday, June 20, 1990

This paper discusses selected aspects of the application of dynamic percolation models to ionic transport in mixed-ion superionic conductors. The discussion is based on an AB lattice gas model with hard-core repulsions and a ratio r between the transition rates of particles A and B. The frequency-dependent conductivity for a tracer particle is calculated within an effective-medium theory. The motion of the background B-particles is regarded as providing a fluctuating disordered environment for the tracer particles A. A crossover frequency separating high-frequency and low-frequency response is found which scales with the negative square root of r. The results for the dc-limit are compared with simulations and are found to be in very good agreement.



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Disordered Systems

Frequency Dependent Response and Dynamic Disorder

R. Hilfer

Journal of Non-Crystalline Solids 131–133, 213 (1991)
https://doi.org/10.1016/0022-3093(91)90303-N

submitted on
Wednesday, June 20, 1990

This paper discusses selected aspects of the application of dynamic percolation models to ionic transport in mixed-ion superionic conductors. The discussion is based on an AB lattice gas model with hard-core repulsions and a ratio r between the transition rates of particles A and B. The frequency-dependent conductivity for a tracer particle is calculated within an effective-medium theory. The motion of the background B-particles is regarded as providing a fluctuating disordered environment for the tracer particles A. A crossover frequency separating high-frequency and low-frequency response is found which scales with the negative square root of r. The results for the dc-limit are compared with simulations and are found to be in very good agreement.



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Disordered Systems

Random Walks with Short Memory in a Disordered Environment

R. Hilfer

Physical Review B 44, 638 (1991)
10.1103/PhysRevB.44.638

submitted on
Monday, May 21, 1990

The backward-jump model is investigated for the case of a bond-disordered lattice. The backward-jump model is a correlated nearest-neighbor random-walk model in which the walker has a different transition rate for jumps to its previously visited site than for jumps to all other nearest-neighbor sites. The standard formulation of the model must be modified if the disorder is introduced at the level of the usual master equation. The difficulties with the standard formulation are discussed in the paper. The first-order master equation for the disordered backward-jump model is established, and a symmetrized second-order equation that was suggested previously is derived from it.



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Disordered Systems Lattice Models Transport Processes

Correlation Effects on Hopping Transport in a Disordered Medium

R. Hilfer

in: Dynamical Processes in Condensed Molecular Systems
edited by: A. Blumen and J. Klafter and D. Haarer
World Scientific Publ.Co., Singapore, 302 (1990)
https://doi.org/10.1142/9789814540261
ISBN: 978-981-4540-26-1

submitted on
Monday, April 23, 1990

Correlated hopping transport through a disordered system is discussed in terms of a random walk model with memory correlations on a bond disordered lattice. Correlations will in general result in a difference between the transition rate to the previously occupied site and the rate for transitions to any other nearest neighbour site. Such a correlated process corresponds exactly to Fürth’s model for a random walk with a finite memory. This paper establishes a first order master equation for Fürth’s random walk on a bond disordered lattice. The equation is found to be equivalent to a symmetrized second order equation which was used previously as the starting point for an effective medium treatment.



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Disordered Systems Glasses

Theoretical Aspects of Realistic Spin Glass Models

R. Hilfer

in: New Trends in Magnetism
edited by: M.D. Coutinho-Filho and S.M. Rezende
World Scientific Publ.Co., Singapore, 32 (1989)

submitted on
Thursday, July 27, 1989

This note investigates the universality of spin glass models by calculating the distribution of instantaneous local magnetic fields, p(h). It is found that short range Ising models with Gaussian bond disorder fall into a different universality class than realistic models with RKKY-interactions and randomly positioned spins. The result is obtained from an analysis of p(h) at high temperatures where thelocal fields are sums of independent random variables. It is found that for realistic models these sums are in general not governed by the central limit theorem. In three dimensions a cutoff Cauchy distribution is obtained for p(h) instead of a Gaussian distribution. In general p(h) is a cutoff stable law whose characteristic exponent depends strongly on the dimension and the decay of the interactions. As a consequence a new short range model is proposed for dilute metallic spin glasses in three dimensions in which the bond disorder is taken to be a cutoff Cauchy distribution instead of a Gaussian. Preliminary considerations indicate a much smoother specific heat for models in this universality class and suggest the existence of strong precursor effects in qualitative agreement with experiment.



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Correlated Hopping in a Disordered Medium

R. Hilfer

Physical Review B 44, 628 (1991)
10.1103/PhysRevB.44.628

submitted on
Monday, March 6, 1989

This paper discusses random walks with memory on a percolating network as a model of correlated hopping transport through a disordered system. Correlations can arise from such sources as hard-core and Coulomb repulsions, correlated hops of groups of particles, or lattice-relaxation effects. In general these correlations will result in a difference between the hopping probability for return to the previously visited site and the probability to jump to another nearest neighbor of the currently occupied site. Thus the hopping process possesses a memory of its previous hop. Such a random walk is investigated in this paper for the case of bond percolation on a regular lattice. The frequency-dependent conductivity σ(ω) is calculated using a generalized effective-medium approximation. Results are presented for the linear chain and the hexagonal lattice. New features appear in both the real and the imaginary part of σ. These depend on the strength of the correlations and on the concentration of bonds. As an example, the possibility of a pronounced maximum in the real part of σ(ω) at finite frequencies is found, which is sometimes accompanied by a change of sign in the imaginary part. The results are found to agree qualitatively with experimental data on ionic transport in Na+ β-alumina, where both disorder and correlations are known to be important.



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Disordered Systems electrical conductivity Percolation Transport Processes

Correlated Hopping in a Disordered Medium

R. Hilfer

Physical Review B 44, 628 (1991)
10.1103/PhysRevB.44.628

submitted on
Monday, March 6, 1989

This paper discusses random walks with memory on a percolating network as a model of correlated hopping transport through a disordered system. Correlations can arise from such sources as hard-core and Coulomb repulsions, correlated hops of groups of particles, or lattice-relaxation effects. In general these correlations will result in a difference between the hopping probability for return to the previously visited site and the probability to jump to another nearest neighbor of the currently occupied site. Thus the hopping process possesses a memory of its previous hop. Such a random walk is investigated in this paper for the case of bond percolation on a regular lattice. The frequency-dependent conductivity σ(ω) is calculated using a generalized effective-medium approximation. Results are presented for the linear chain and the hexagonal lattice. New features appear in both the real and the imaginary part of σ. These depend on the strength of the correlations and on the concentration of bonds. As an example, the possibility of a pronounced maximum in the real part of σ(ω) at finite frequencies is found, which is sometimes accompanied by a change of sign in the imaginary part. The results are found to agree qualitatively with experimental data on ionic transport in Na+ β-alumina, where both disorder and correlations are known to be important.



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