Percolation on the square lattice in aL×M geometry

A study of the site percolation model on the square lattice in aL×M geometry at critically is presented. ForL?M one observes the growth of numerous percolation colation clusters in theL-direction in contrast to the absence of percolation in theM-direction. Consequently, relevant properties of these clusters such us for example the average number of clusters (N _CL ), the cluster length distribution (P(l,L), withl=cluster length in theM direction) and average cluster length (l _CL ), are studied by means of the Monte Carlo technique and analyzed on the basis of finite-size scaling arguments. The following behavior is found:N _CL ?(3/8) (L/M)^?δ, with δ=1; andl _CL ?2.0L. Also the distributionP(l, L) is of the exponential-exponential type and their characteristic exponents are evaluated.     

Percolation in a network with long-range connections: Implications for cytoskeletal structure and function

Cell shape, signaling, and integrity depend on cytoskeletal organization. In this study we describe the cytoskeleton as a simple network of filamentary proteins (links) anchored by complex protein structures (nodes). The structure of this network is regulated by a distance-dependent probability of link formation as P=p/d^s, where p regulates the network density and s controls how fast the probability for link formation decays with node distance (d). It was previously shown that the regulation of the link lengths is crucial for the mechanical behavior of the cells. Here we examined the ability of the two-dimensional network to percolate (i.e. to have end-to-end connectivity), and found that the percolation threshold depends strongly on s. The system undergoes a transition around s=2. The percolation threshold of networks with s<2 decreases with increasing system size L, while the percolation threshold for networks with s>2 converges to a finite value. We speculate that s<2 may represent a condition in which cells can accommodate deformation while still preserving their mechanical integrity. Additionally, we measured the length distribution of F-actin filaments from publicly available images of a variety of cell types. In agreement with model predictions, cells originating from more deformable tissues show longer F-actin cytoskeletal filaments.     

Formation of an ensemble of nanoclusters under rapid deposition of atoms on a surface

The results of an experimental study of the formation of nanometer-size Au clusters on NaCl(100) and HOPG(0001) surfaces under pulsed laser deposition are presented. No clusters of small sizes (d ≤ 1 nm) have been found in the cluster size distribution. The distribution itself at d < 5 nm has the form of a percolation distribution. It has been established that the perimeter of clusters with sizes d < 5 nm has a fractal structure. The fractal dimension of clusters is different for NaCl(100) and HOPG(0001) surfaces with different symmetries; it decreases with increasing cluster size from D _f ≈ 1.2–1.4 at d ≈ 1.5 nm to D _f ≈ 1 at d ≈ 5 nm. A physical mechanism of nanocluster formation is suggested. Under pulsed laser deposition, the attainable densities of adatoms are close to the percolation threshold in the region of thermodynamically unstable states and many-particle correlation regions are formed in a spatially inhomogeneous adsorbate. Clusters are formed on the surface from many-particle correlation regions in several diffusion jumps. The suggested mechanism allows the fractal dimension of the clusters forming on surfaces with different symmetries, its dependence on cluster size, and the cluster size distribution functions to be calculated.     

2D-to-3D percolation crossover of metal-insulator composites

Percolation properties and d.c. conductivity were determined for an L²×h-random resistor network model of metal-insulator composite films. The effects of the thickness h on the percolation threshold and conductivity were studied numerically in the limit of an infinite size of the L²-plane parallel to the film. For thicknesses ranging from h/L=0.01 to h/L=0.24, a crossover between a finite-size regime and a saturation regime was observed at h/L≈0.1. In the finite-size regime (h/L?0.01), the percolation threshold scales as pc(h)−pc3∝h−1/x, the exponent x being compatible with that of the critical exponent of the 3D correlation length, ν₃. The conductivity exponent t appeared to depend linearly on the ratio h/L with a slope νD compatible with 2+ν₂, where ν₂ is the 2D correlation length exponent. In the saturation regime, a scaling correction for the percolation threshold was found with an exponent 1+1/ν₃. In this regime we also observed a logarithmic dependence of the conductivity exponent on h/L.     

Height-height correlations for surface growth on percolation networks

The height-height correlations of the surface growth for equilibrium and nonequilibrium restricted solid-on-solid (RSOS) model were investigated on randomly diluted lattices, i.e., on infinite percolation networks. It was found that the correlation function calculated over the chemical distances reflected the dynamics better than that calculated over the geometrical distances. For the equilibrium growth on a critical percolation network, the correlation function for the evolution time t?1 yielded a power-law behavior with the power ζ^′, associated with the roughness exponent ζ via the relation ζ=ζ^′d_f/d_l, with d_f and d_l being, respectively, the fractal dimension and the chemical dimension of the substrate. For the nonequilibrium growth, on the other hand, the correlation functions did not yield power-law behaviors for the concentration of diluted sites x less than or equal to the critical concentration x_c.     

Current distribution on a three-dimensional,bond-diluted,random-resistor network at the percolation threshold

Current and logarithm-current distributions on a three-dimensional random-bond percolation cubic network were studied at the percolation threshold by computer simulations. Predictions of a hierarchical model that combine fractal structure and randomness agree with our numerical simulations. In the thermodynamic limit the logarithm-current distribution exhibits ann(ln(i))i ^1/3 dependence below some characteristic currenti _c. This distribution may scale with lni/lnL, but the data are insufficient to make this a definite conclusion. Due to the small range of lnL considered, a study of the moments does not reveal this behavior and a study of the distribution itself is required.     

Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (t_e). t_e is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is d_f=2.0 and the size distribution has a power law decay with exponent τ=1.5. At larger values of t_e transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension d_f=2.7 and the power law exponent of the size distribution is τ=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.     

Quark trapping for lattice U(1) gauge fields

We show for lattice U(1) gauge fields in d = 3 dimensions, that 〈exp(i∮_CAdx)〉 ? exp (? const.T lnL), where C is a rectangle of dimension T × L, T ? L. This indicates quark trapping, by a potential at least as strong as Coulomb.     

Configurational statistics of densely and fully packed loops in the negative-weight percolation model

By means of numerical simulations we investigate the configurational properties of densely and fully packed configurations of loops in the negative-weight percolation (NWP) model. In the presented study we consider 2d square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where edge weights are drawn from a Gaussian distribution. For a given realization of the disorder we then compute a configuration of loops, such that the configurational energy, given by the sum of all individual loop weights, is minimized. For this purpose, we employ a mapping of the NWP model to the “minimum-weight perfect matching problem” that can be solved exactly by using sophisticated polynomial-time matching algorithms. We characterize the loops via observables similar to those used in percolation studies and perform finite-size scaling analyses, up to side length L = 256 in 2d, L = 48 in 3d and L = 20 in 4d (for which we study only some observables), in order to estimate geometric exponents that characterize the configurations of densely and fully packed loops. One major result is that the loops behave like uncorrelated random walks from dimension d = 3 on, in contrast to the previously studied behavior at the percolation threshold, where random-walk behavior is obtained for d ≥ 6.     

A two dimensional theoretical model for describing gain coefficient in N2-lasers and the model validity for CVL and excimer lasers

Based on our previously reported measurements on the gain-value in a N₂- laser and numerical calculations, we introduce a method to obtain an analytical expression for the small signal gain, g₀, where the dependency of g₀ on the laser geometrical configuration, including electrodes length and gap separation, is demonstrated. For this study one- and two-dimensional approaches for the photon density have been applied independently to determine gain-parameter, where for explaining the observed dependency of the gain-parameter on the laser electrodes separation, d_AMP, which was found experimentally and explained by an empirical expression of the type g₀ = r + q/d_AMP, with r and q some constants, realization of introducing an extra dimension along the gap separation was unavoidable. For the electrodes length, l_AMP, we have already shown that an empirical equation of the type g₀ = m + n/l_AMP, with m and n some constants, is consistent with the measurements corresponding to N₂-lasers. With this realization, it is proved that the gain-parameter in N₂-lasers can be written as g₀^{above threshold} = m″ + g_0z(γ_L^z, z) + g_0y(γ_L^y, y), where it consists of two independent gain-values along the electrodes length (z) and gap separation (y) with the corresponding power losses given by γ_L^z and γ_L^y. m″ is a very small quantity showing that laser is operating slightly above the threshold. The results of this calculation are consistent with our recent measurements and also other reported N₂-laser gain values measured under moderate current density conditions. To check the validity of the model for other types of lasers, the reported gain-values for copper vapor lasers of different laser tube radii, R_AMP, and tube lengths, l_AMP, have been examined using the one-dimensional model of either g₀(R_AMP) or g₀(l_AMP) and the consistency with the observed measurements was found to be quite satisfactory. The model was also found to be valid for the excimer lasers of different types, different gas mixtures and pressures at a constant input operational voltage. Due to limited numbers of the reported experimental measurements, for the graphs preparation of g₀(l_AMP) in excimer lasers we used the observed data at V₀ = 30 kV and also some variations of the input voltages in the range of ΔV ≅ 20 kV have been adopted. The results for both cases were found to be consistent with the proposed one-dimensional model.     

Glass formation in amorphous SiO<Subscript>2</Subscript> as a percolation phase transition in a system of network defects

Thermodynamic parameters of defects (presumably, defective SiO molecules) in the network of amorphous SiO₂ are obtained by analyzing the viscosity of the melt with the use of the Doremus model. The best agreement between the experimental data on viscosity and the calculations is achieved when the enthalpy and entropy of the defect formation in the amorphous SiO₂ network are H _d =220 kJ/mol and S _d =16.13R, respectively. The analysis of the network defect concentration shows that, above the glass-transition temperature (T _g ), the defects form dynamic percolation clusters. This result agrees well with the results of molecular dynamics modeling, which means that the glass transition in amorphous SiO₂ can be considered as a percolation phase transition. Below T _g , the geometry of the distribution of network defects is Euclidean and has a dimension d=3. Above the glass-transition temperature, the geometry of the network defect distribution is non-Euclidean and has a fractal dimension of d _f =2.5. The temperature T _g can be calculated from the condition that percolation arises in the defect system. This approach leads to a simple analytic formula for the glass-transition temperature: T _g =H _d /((S _d +1.735R). The calculated value of the glass-transition temperature (1482 K) agrees well with that obtained from the recent measurements of T _g for amorphous SiO₂ (1475 K).     

Angular dependence of coercivity and remanence of Ni nanowire arrays and its relevance to magnetoviscosity

Ni nanowire arrays with varying wire dimensions (diameter d, length l) and center-to-center distances d_CC were synthesized by pulsed electrodeposition of Ni in porous Al templates. The magnetization-reversal behavior of the arrays was investigated by means of magnetometry for different angles θ between the wire axes and the applied magnetic field. The functional dependences of the characteristic parameters coercivity H_C(θ) and reduced remanence m_R/m_S(θ) exhibit a strong dependence on the wire dimensions and the center-to-center distance. For instance, for nanowire arrays with d=40 nm, d_CC=100 nm, and for θ=0°, the coercivity takes on a rather large value of μ₀H_C=85 mT and m_R/m_S≅94%; reducing d_CC to 30 nm and d to 17 nm results in μ₀H_C=49 mT and m_R/m_S≅57%, an observation which suggests an increasing magnetostatic interwire interaction at increased (d/d_CC)-ratio. The potential application of nanowires as the constituents of ferrofluids is discussed.     

Topological destruction of the phase transition to the spin-glass state in amorphous alloys with an asymmetric distribution of exchange interactions

It is shown that an increase in the asymmetry in the distribution of exchange interactions reduces the real magnetic dimension of bulk samples of amorphous spin glasses FeNi and FeMn, which attains its lowest critical value D _L =2.51±0.12 near the percolation threshold of infinite ferromagnetic clusters. The experimentally obtained result is in good agreement with calculations made using computer experiments.     

Estimation of Critical Point in Branching Reactions: Further Examination of Gel Point Formula

The theory of gel point in real polymer solutions is examined with the empirical correlation between the reciprocal of the percolation threshold and the coordination number given by the percolation theory. Applying a larger value of the relative frequency of cyclization, an excellent agreement is obtained between the present theory and the percolation result. This suggest that while the ring distribution on lattices is similar to that in real systems, ring production is more frequent in the lattice model than in real systems. To confirm this conjecture, we derive the ring distribution function of the lattice model as a limiting case of d→∞, and show that the solution is in fact identical to the asymptotic formula of C→∞ in real systems except for the coefficient C, which has a maximum at d = 5, in support of the above conjecture. To examine the validity of the asymptotic solution for the lattice model, we apply it to the critical point problem of the percolation theory, showing that the solution works well in high dimensions greater than six.     

The Bose-Einstein condensation in a finite one-dimensional homogeneous system of noninteracting bosons

Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l (L ? l) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N(l/L)⁴ ? l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t=T/T ₁, where T ₁ is a certain characteristic temperature depending only on the longitudinal size L. Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ～ 1 and acquires the shape of the Bose-Einstein distribution for t ? 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N(l/L)⁴ ? 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.     

Perpendicular giant magnetoresistance of Co91Fe9/Cu exchange-biased spin-valves: further evidence for a unified picture

Measurements are reported of the increase in specific resistance, AΔR, with increasing Co₉₁Fe₉ layer thickness in current-perpendicular to the plane (CPP) exchange-biased spin-valves (EBSVs) of Co₉₁Fe₉ and Cu. Analysis of these measurements yields a spin anisotropy parameter for Co₉₁Fe₉ of β=0.65±0.05, and a spin diffusion length of l^CoFe_sf=12±1 nm. The value of β agrees reasonably well with those obtained experimentally and theoretically for dilute CoFe alloys, thus providing additional support for a unified picture of spin-polarized transport in CPP-MR and bulk alloys. This value of l^CoFe_sf, and the previously determined l^Py_sf for Permalloy, scale approximately inversely with the residual resistivities of the two alloys.     

Conductance distribution near the Anderson transition

Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L ^d–1 × L _z , characterizing by parameters L/ξ and L _z /L (ξ is the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d = 2 + ? obtained in the framework of the σ-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g = 1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d = 3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/ξ.     

Computational analysis of factors influencing thermal conductivity of nanofluids

Numerical investigations are conducted to study the effect of factors such as particle clustering and interfacial layer thickness on thermal conductivity of nanofluids. Based on this, parameters including Kapitza radius and fractal and chemical dimension which have received little attention by previous research are rigorously investigated. The degree of thermal enhancement is analyzed for increasing aggregate size, particle concentration, interfacial thermal resistance, and fractal and chemical dimensions. This analysis is conducted for water-based nanofluids of Alumina (Al₂O₃), CuO, and Titania (TiO₂) nanoparticles where the particle concentrations are varied up to 4 vol%. Results from the numerical work are validated using available experimental data. For the case of aggregate size, particle concentration, and interfacial thermal resistance, the aspect ratio (ratio of radius of gyration of aggregate to radius of primary particle, R _g/a) is varied from 2 to 60. It was found that the enhancement decreases with interfacial layer thickness. Also the rate of decrease is more significant after a given aggregate size. For a given interfacial resistance, the enhancement is mostly sensitive to R _g/a < 20 indicated by the steep gradients of data plots. Predicted and experimental data for thermal conductivity enhancement are in good agreement. On the influence of fractal and chemical dimensions (d _l and d _f) of Alumina–water nanofluid, the R _g/a was varied from 2 to 8, d _l from 1.2 to 1.8, and d _f from 1.75 to 2.5. For a given concentration, the enhancement increased with the reduction of d _l or d _f. It appears a distinctive sensitivity of the enhancement to d _f, in particular, in the range 2–2.25, for all values of R _g/a. However, the sensitivity of d _l was largely depended on the value of R _g/a. The information gathered from this study on the sensitivity of thermal conductivity enhancement to aggregate size, particle concentration, interfacial resistance, and fractal and chemical dimensions will be useful in manufacturing highly thermally conductive nanofluids. Further research on the refine cluster evolution dynamics as a function of particle-scale properties is underway.