Self-similar Carpets Research Articles

Quasisymmetrically rigid self-similar carpets

Abstract We define balanced self-similar quasi-round carpets and compare the carpet moduli of some path families relating to such a carpet. Then, using some known results on quasiconformal geometry of carpets, we prove that the group of quasisymmetric self-homeomorphisms of every balanced self-similar quasi-round carpet is finite. Furthermore, we prove that some balanced self-similar carpets in the unit square with strong geometric symmetry are quasisymmetrically rigid by using the quasisymmetry of weak tangents of carpets.

Mechanical reinforcement by bridging chains in polymer nanocomposites

The adsorption-induced bridging of polymer chains between adjacent nanoparticles was studied in poly(vinyl acetate)/silica nanocomposites with different polymer molecular weights, interaction strengths, nanoparticle sizes, and silica loading fractions. The polymer bridging contribution is successfully decoupled from the apparent rubbery plateau modulus by subtracting the hydrodynamic and trapped chain entanglement contributions. The bridging modulus depends on the nearest interparticle surface distance through a power-law relation with a universal exponent −3. Polymer molecular weight can enhance the bridging modulus only when nanoparticles have a high density of adsorption sites due to the emergence of direct bridging of multiple particles by one chain in addition to the regular bridging through self-similar carpets. Our results illustrate that bridging chains can dominate modulus enhancement at nanoparticle's loading at about 10 vol% in polymer nanocomposites with long polymer chains and high surface silanol density.

Polymer physics view of peripheral chromatin: de Gennes' self-similar carpet.

Using scaling arguments to model peripheral chromatin localized near the inner surface of the nuclear envelope (NE) as a flexible polymer chain, we discuss the structural properties of the peripheral chromatin composed of alternating lamin-associated domains (LADs) and inter-LADs. Modeling the attraction of LADs to NE by de Gennes' self-similar carpet, which treats the chromatin layer as a polymer fractal, explains two major experimental observations. (i) The high density of chromatin close to the nuclear periphery decays to a constant density as the distance to the periphery increases. (ii) Due to the decreasing mesh size towards the nuclear periphery, the chromatin carpet inside NE excludes molecules (via nonspecific interactions) above a threshold size that depends on the distance from the nuclear periphery.

Temperature-dependent storage modulus of polymer nanocomposites, blends and blend-based nanocomposites based on percolation and De Gennes’s self-similar carpet theories

Temperature-dependent storage modulus of polymer nanocomposites, blends and blend-based nanocomposites based on percolation and De Gennes’s self-similar carpet theories

Manhattan property of geodesic paths on self-affine carpets

For any Bedford-McMullen self-affine carpet, the geodesic path on the carpet between points $$(x_{1},y_{1})$$ and $$(x_{2},y_{2})$$ has length greater than or equal to $$|x_{1}-x_{2}|+|y_{1}-y_{2}|.$$ This property fails for self-similar carpets.

Strong Selective Adsorption of Polymers.

A scaling theory is developed for selective adsorption of polymers induced by the strong binding between specific monomers and complementary surface adsorption sites. By "selective" we mean specific attraction between a subset of all monomers, called "sticky", and a subset of surface sites, called "adsorption sites". We demonstrate that, in addition to the expected dependence on the polymer volume fraction ϕbulk in the bulk solution, selective adsorption strongly depends on the ratio between two characteristic length scales, the root-mean-square distance l between neighboring sticky monomers along the polymer, and the average distance d between neighboring surface adsorption sites. The role of the ratio l/d arises from the fact that a polymer needs to deform to enable the spatial commensurability between its sticky monomers and the surface adsorption sites for selective adsorption. We study strong selective adsorption of both telechelic polymers with two end monomers being sticky and multisticker polymers with many sticky monomers between sticky ends. For telechelic polymers, we identify four adsorption regimes at l/d < 1 that are characterized by the fraction of occupied adsorption sites and whether the dominant conformation of adsorbed chains is a single-end-adsorbed "mushroom" or double-end-adsorbed loop. For l/d > 1, we expect that the adsorption layer at exponentially low ϕbulk consists of separated unstretched loops, while as ϕbulk increases the layer crosses over to a brush of extended loops with a second layer of weakly overlapping tails. For multisticker chains, in the limit of exponentially low ϕbulk, adsorbed polymers are well separated from each other. As l/d increases, the conformation of an individual polymer changes from a single-end-adsorbed "mushroom" to a random walk of loops. For high ϕbulk, adsorbed polymers at small l/d are mushrooms that cover all the adsorption sites. At sufficiently large l/d, adsorbed multisticker polymers strongly overlap. We anticipate the formation of a self-similar carpet and with increasing l/d a two-layer structure with a brush of loops covered by a self-similar carpet. As l/d exceeds the threshold determined by the adsorption energy, the brush of loops under the carpet reaches a saturated state, resulting in a l/d-independent brush-under-carpet structure, which can also be applied to describe adsorbed multisticker polymers in nonselective adsorption where a sticker can strongly bind to any place on the adsorption surface. We examine the adsorbed amount Γ of multisticker polymers in different regimes for selective adsorption. If the adsorbed multisticker polymers are nonoverlapping mushrooms, the adsorbed amount Γ increases linearly with the surface density of adsorption sites Σ ≈ 1/d2. In the self-similar carpet regime, Γ increases sublinearly as Σ0.15 in a good solvent, while only logarithmically in a theta solvent. Formation of a brush layer under the carpet contributes an additional adsorbed amount. This additional amount increases linearly with Σ and eventually dominates the overall adsorbed amount Γ before saturating at a plateau value controlled by the adsorption energy.

Strong Selective Adsorption of Polymers

A scaling theory is developed for of polymers induced by the strong binding between specific monomers and complementary surface By selective we mean specific attraction between a subset of all monomers, called sticky, and a subset of surface sites, called adsorption sites. We demonstrate that, in addition to the expected dependence on the polymer volume fraction ϕbulk in the bulk solution, strongly depends on the ratio between two characteristic length scales, the root-mean-square distance l between neighboring sticky monomers along the polymer, and the average distance d between neighboring surface The role of the ratio l/d arises from the fact that a polymer needs to deform to enable the spatial commensurability between its sticky monomers and the surface sites for adsorption. We study strong of both telechelic polymers with two end monomers being sticky and multisticker polymers with many sticky monomers between sticky ends. For telechelic polymers, we identify four regimes at l/d 1, we expect that the layer at exponentially low ϕbulk consists of separated unstretched loops, while as ϕbulk increases the layer crosses over to a brush of extended loops with a second layer of weakly overlapping tails. For multisticker chains, in the limit of exponentially low ϕbulk, adsorbed polymers are well separated from each other. As l/d increases, the conformation of an individual polymer changes from a single-end-adsorbed mushroom to a random walk of loops. For high ϕbulk, adsorbed polymers at small l/d are mushrooms that cover all the At sufficiently large l/d, adsorbed multisticker polymers strongly overlap. We anticipate the formation of a self-similar carpet and with increasing l/d a two-layer structure with a brush of loops covered by a self-similar carpet. As l/d exceeds the threshold determined by the energy, the brush of loops under the carpet reaches a saturated state, resulting in a l/d-independent brush-under-carpet structure, which can also be applied to describe adsorbed multisticker polymers in nonselective where a sticker can strongly bind to any place on the surface. We examine the adsorbed amount Γ of multisticker polymers in different regimes for adsorption. If the adsorbed multisticker polymers are nonoverlapping mushrooms, the adsorbed amount Γ increases linearly with the surface density of sites Σ ≈ 1/d2. In the self-similar carpet regime, Γ increases sublinearly as Σ0.15 in a good solvent, while only logarithmically in a theta solvent. Formation of a brush layer under the carpet contributes an additional adsorbed amount. This additional amount increases linearly with Σ and eventually dominates the overall adsorbed amount Γ before saturating at a plateau value controlled by the energy.

Self-similar carpets over finite fields

In [Dann E. Passoja, Akhlesh Lakhtakia, Carpets and rugs: An exercise in numbers, Leonardo 25 (1) (1992) 69–71] an informal algorithm ‘to display interesting numeric patterns’ is described without any proof. We generalize this algorithm over arbitrary finite fields F q of characteristic p and we prove that it always generates some self-similar sets. For the prime fields F p the generalized algorithm produces p − 1 different self-similar sets. These sets are classified according to their arithmetic and their groups of symmetry.

The critical exponent v of directed TSAWs and TSALFs on Sierpinski carpets: an analytic approach

It is shown rigorously that the critical exponent vperpendicular to for a directed true self-avoiding walk on an arbitrary self-similar carpet is 1. Hence the simulation values by Yao et al (1990) represent an artefact due to finite-size effects. The corresponding exponent for a Levy flight with parameter u is max(1, 1/u).