Adsorption of a flexible self-avoiding polymer chain: Exact results on fractal lattices

The large-scale behavior of surface-interacting self-avoiding polymer chains placed on finitely ramified fractal lattices is studied using exact recursion relations. It is shown how to obtain surface susceptibility critical indices and how to modify a scaling relation for these indices in the case of fractal lattices. We present the exact results for critical exponents at the point of adsorption transition for polymer chains situated on a class of Sierpinski gasket-type fractals. We provide numerical evidence for a critical behavior of the type found recently in the case of bulk self-avoiding random walks at the fractal to Euclidean crossover.     

Thresholds and Universality of the Site Percolation on the Sierpinski Carpets

Following the methods proposed by Yonezawa, Sakamoto and Hori, we have calculated the percolation thresholds P_c, their error bars ΔP_c, and the correlation length exponents v of a family of the Sierpinski carpets for the site percolation problems by making use of MonteCarlo simulations and finite size scaling. We have found the dependence of P_c and v on the fractal dimensionality D_f and the lacunarity. We ascertain that the site percolation problems on a family of Sierpinski carpets with central cutouts and different D belong to different universal classes, and those on Sierpinski carpets with same D_f but of different lacunarities belong to different universal classes.     

Ising thermodynamics on the Sierpinski gasket

We report a calculation of the thermodynamic properties of an Ising system on a fractal lattice, the Sierpinski gasket (SG). The scale-invariant geometry of SG leads to a wider critical region compared to that in translationally invariant systems. We calculate exactly the near-neighbor correlation function and specific heat and discuss their critical behaviour.     

Mean Field Theory of Ising Model on Sierpinski

We study Ising model on Sierpinski carpets by using mean field theory. We find a phase transition at T_c > 0 which is dependent on the geometrical factors. The critical exponents are calculated and found to be the same as the values for translationally invariant lattices.     

Absorbing phase transition in contact process on fractal lattices

We investigate the critical behavior of nonequilibrium phase transition from an active phase to an absorbing state on two selected fractal lattices, i.e., on a checkerboard fractal and on a Sierpinski carpet. The checkerboard fractal is finitely ramified with many dead ends, while the Sierpinski carpet is infinitely ramified. We measure various critical exponents of the contact process with a diffusion-reaction scheme A→AA and A→0, characterized by a spreading with a rate λ and an annihilation with a rate μ, and the results are confirmed by a crossover scaling and a finite-size scaling. The exponents, compared with the ?-expansion results assuming , being the fractal dimension of the underlying fractal lattice, exhibit significant deviations from the analytical results for both the checkerboard fractal and the Sierpinski carpet. On the other hand, the exponents on a checkerboard fractal agree well with the interpolated results of the regular lattice for the fractional dimensionality, while those on a Sierpinski carpet show marked deviations.     

Adsorption of Ideal Polymers on an Infinitely Ramified Fractal

We study ideal polymer chains interacting attractively with the borders of the lacunas of an infinitely ramified fractal, the Sierpinski carpet. Ideal chains are simulated on finite stages of construction of this fractal at various temperatures. The mean-square displacement and the mean number of adsorbed monomers of N-step chains are estimated in these lattices, and extrapolations to the fractal limit (infinite lattice) consider the exact forms of finite-size corrections as previously predicted by the series expansion method. In the noninteracting case, a finite fraction of the monomers is adsorbed, and this fraction increases as the temperature decreases. However, there is evidence that the critical exponent v which governs the growth of the chains varies with the temperature in a nonmonotonic way. At high temperatures v increases with decreasing temperature, and thus the chains are more stretched than in the noninteracting case. At an intermediate temperature, v starts to decrease and is still positive at very low temperatures, when the chains grow along the borders of several lacunas, occasionally crossing the bulk between them.     

Force-induced desorption of self-avoiding walks on Sierpinski gasket fractals

In this work we investigate force-induced desorption of linear polymers in good solvents in non-homogeneous environment, by applying the model of self-avoiding walk on two- and three-dimensional fractal lattices, obtained as generalization of the Sierpinski gasket fractal. For each of these lattices one of its boundaries represents an adsorbing wall, whereas along one of the fractal edges, not lying in the adsorbing wall, an external force acts on the self-avoiding walk. The hierarchical nature of the lattices under study enables an exact real-space renormalization group treatment, which yields the phase diagram of polymer critical behavior. We show that for this model there is no low-temperature reentrance in the cases of two-dimensional lattices, whereas in all studied three-dimensional cases the force-temperature dependance is reentrant. We also find that in all cases the force-induced desorption transition is of first order.     

Sound diffraction at Sierpinski carpet

The sound field scattered by a fractal surface in the form of a Sierpinski carpet is calculated in the framework of the perturbation method. The Sierpinski carpet has an alternating acoustic admittance preset at its squares, which sequentially scale down. It is demonstrated that such a Sierpinski carpet scatters sound almost uniformly in all directions.     

Plasmonic fractals: ultrabroadband light trapping in thin film solar cells by a Sierpinski nanocarpet

Plasmonic Sierpinski nanocarpet as back structure for a thin film Si solar cell is investigated. We demonstrate that ultra-broadband light trapping can be obtained by placing square metallic nanoridges with Sierpinski pattern on the back contact of the thin film solar cell. The multiple-scale plasmonic fractal structure allows excitation of localized surface plasmons and surface plasmon polaritons in multiple wavelengths leading to obvious absorption enhancements in a wide frequency range. Full wave simulations show that 109 % increase of the short-circuit current density for a 200 nm thick solar cell, is achievable by the proposed fractal back structure. The amount of light absorbed in the active region of this cell is more than that of a flat cell with semiconductor thickness of 1,000 nm.     

Analysis of fractal dimensions in the express diagnostics of bacterial colonies

The features of the formation of speckle structures under irradiation of a model fractal (Sierpinski carpet) have been investigated. The relationship between the fractal properties of the diffraction pattern and the scattering structure parameters (model fractal geometrical sizes, fractal depth) has been analyzed for the irradiation by a focused light beam, whose size is comparable with that of the irradiated object. The results of the computer simulation of the Gaussian beam scattering in bacterial colonies are compared with the experimental data.     

The Critical Behavior of Partially Directed SAW on Sierpinski Carpets

Using the fractal-cell generation method we perform a numerical simulation study for partially directed self-avoiding walks (PDSAW) on Sierpinski carpets. The obtained critical exponents v_H is found to be independent of the fractal dimension of Sierpinski carpet d_f, but v_⊥ is dependent on d_f . This result indicates that PDSAW on different Sierpinski carpets belong to different universality classes. Compared with the fully directed self-avoiding walks (FDSAW) on the same carpets, the obtained results indicate that PDSAW and FDSAW belong to the same universality class.     

Voter model on Sierpinski fractals

We investigate the ordering of voter model on fractal lattices: Sierpinski Carpets and Sierpinski Gasket. We obtain a power-law ordering in all cases, but the dynamics is found to differ significantly for finite and infinite ramification order of investigated fractals.     

纹理高阶分形特征在海面舰船目标检测中的应用

针对复杂海面环境下的舰船目标检测，分析了高阶分形特征缝隙在纹理分类中的应用，提出了一种基于分形维与缝隙的目标检测新方法，并利用该方法对海面舰船目标进行了检测。实验结果表明利用纹理分形维与缝隙特征进行海面舰船目标检测，可以取得较单一分形维检测更高的准确率。     

Calculation of Spectral Dimension on Inhomogeneous Fractal Lattices

We propose a scheme to derive the spectral dimension of inhomogeneous fractal lattice via renormalization procedure, in which the distribution of masses on the sites of fractals is introduced. The spectral dimension of diamond-type hierarchical lattice and Sierpinski gasket with b = 3 are re-investigated in this way. Moreover, the variants of Sierpinski gaskettype fractals are studied, the results show that the spectral dimension is independent of the details of internal structure of fractal, and thus implies the existence of universality. The source of the universality is also analyzed.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Tunneling in quantum superlattices with variable lacunarity

Fractal superlattices are composite, aperiodic structures comprised of alternating layers of two semiconductors following the rules of a fractal set. The scattering properties of polyadic Cantor fractal superlattices with variable lacunarity are determined. The reflection coefficient as a function of the particle energy and the lacunarity parameter present tunneling curves, which may be classified as vertical, arc, and striation nulls. Approximate analytical formulae for such curves are derived using the transfer matrix method. Comparison with numerical results shows good accuracy. The new results may be useful in the development of band-pass energy filters for electrons, semiconductor solar cells, and solid-state radiation sources up to THz frequencies.     

基于分形特征和双层光子带隙结构的宽阻带低通滤波器

提出一种新颖的基于分形特征和双层光子带隙（PBG）结构的宽阻带低通滤波器. 该滤波器在接地板上刻蚀一阶Sierpinski carpet PBG结构，在顶层微带线与接地板之间增加一层具 有三阶Sierpinski gasket PBG结构的金属层，该金属层经过打通孔与接地板连通. 这种双 层PBG结构的低通滤波器，具有良好的宽阻带特性，且电路尺寸小、结构紧凑. 对比了单层 普通方孔PBG结构的低通滤波器、单层一阶Sierpinski carpet PBG结构的低通滤波器和双层 分形PBG结构低通滤波器的传 关键词： 低通滤波器 双层PBG结构 分形 宽阻带特性     

Aperiodic extended surface perturbations in the Ising model

We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power of the distance l from the free surface with an oscillating amplitude where follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent , the wandering exponent which governs the fluctuation of the sequence and the bulk correlation length exponent . Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the -plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed. Received 1 December 1998     

Fractal properties of aggregates of metal nanoclusters on solid surface

AFM images are used to determine and analyze fractal characteristics (cluster fraction dimension and lacunarity) of aggregates of Au and Ag nanoclusters on metal films of the same metal produced with the aid of thermal vacuum deposition on mica surface. A fractal dimension of 1.6 that corresponds to typical samples with relatively uniform distribution of nanoclusters on the film surface is in agreement with the mean value calculated from experimental data of Belko et al., who studied the fractal dimension of Au nanoclusters on a different dielectric (quartz) surface. When a compact single aggregate of Au nanoclusters is formed on a certain active center or defect, the fractal cluster dimension decreases to 1.4. The experimental data are compared with the results of existing theoretical models of association of nanoclusters in 2D systems.     

Percolation on infinitely ramified fractals

We present a family of exact fractals with a wide range of fractal and fracton dimensionalities. This includes the case of the fracton dimensionality of 2, which is critical for diffusion. This is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal. These fractals include the cases of finite and infinite ramification characterized by a ramification exponentp. The infinite ramification makes the problem of percolation on these lattices a nontrivial one. We give numerical evidence for a percolation transition on these fractals. This transition is tudied by a real-space renormalization group technique on lattices with fractal dimensionality ¯d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.