Sierpinski Gasket Family Research Articles

Self-interacting polymer chains terminally anchored to adsorbing surfaces of three-dimensional fractal lattices

We have studied the adsorption problem of self-attracting linear polymers, modeled by self-avoiding walks (SAWs), situated on three-dimensional fractal structures, exemplified by 3d Sierpinski gasket (SG) family of fractals as containers of a poor solvent. Members of SG family are enumerated by an integer b (b≥2), and it is assumed that one side of each SG fractal is an impenetrable adsorbing surface. We calculate the critical exponents γ1,γ11, and γs, which are related to the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing boundary, respectively. By applying the exact renormalization group (RG) method (for the first three members of the SG fractal family, b=2,3, and 4), we have obtained specific values of these exponents, for θ-chain and globular polymer phase. We discuss their mutual relations and relations with corresponding values pertinent to extended polymer chain phase.

Statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three-dimensional space

We study the adsorption problem of linear polymers, immersed in a good solvent, when the container of the polymer–solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals, embedded in the three-dimensional Euclidean space. Members of the SG family are enumerated by an integer b (2≤b<∞), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the surface critical exponents γ11,γ1, and γs which, within the self-avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with both, one, and no ends grafted to the adsorbing surface (adsorbing boundary), respectively. By applying the exact renormalization group method, for 2≤b≤4, we have obtained specific values for these exponents, for various types of polymer conformations. To extend the obtained sequences of exact values for surface critical exponents, we have applied the Monte Carlo renormalization group method for fractals with 2≤b≤40. The obtained results show that all studied exponents are monotonically increasing functions of the parameter b, for all possible polymer states. We discuss mutual relations between the studied critical exponents, and compare their values with those found for other types of lattices, in order to attain a unified picture of the attacked problem.

Comment on "Critical behavior of the chain-generating function of self-avoiding walks on the sierpinski gasket family: the euclidean limit"

We refute the claims made by Riera and Chalub [Phys. Rev. E 58, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.

Crossover exponent for piecewise directed walk adsorption on Sierpinski fractals

We study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the Sierpinski gasket family. By applying the exact real space renormalization group method, we calculate the crossover exponent $\phi$, associated with the number of adsorbed steps, for the complete fractal family. We demonstrate that our results are very close to the results obtained for ordinary self-avoiding walk, and discuss the asymptotic behaviour of $\phi$ at the fractal to Euclidean lattice crossover.

Critical behavior of the chain-generating function of self-avoiding walks on the Sierpinski gasket family: The Euclidean limit

We study self-avoiding walks (SAW's) on the generalized Sierpinski gasket family of fractals. Each fractal can be labeled by an integer b $(2l~bl~\ensuremath{\infty}),$ so that the fractal and spectral dimensions tend to the Euclidean value 2 when $\stackrel{\ensuremath{\rightarrow}}{b}\ensuremath{\infty}$. By using an exact enumeration technique to obtain the series expansion for the chain-generating function of SAW's on these lattices, we calculate the associated critical exponent ${\ensuremath{\gamma}}_{b}$ for $2l~bl~100$. The large-b behavior of ${\ensuremath{\gamma}}_{b}$ is the first numerical result consistent with the asymptotic convergence toward the Euclidean value ${\ensuremath{\gamma}}_{E}.$ We also give an analytic argument supporting the assumption that ${limop}_{\stackrel{\ensuremath{\rightarrow}}{b}\ensuremath{\infty}}{\ensuremath{\gamma}}_{b}\ensuremath{\rightarrow}{\ensuremath{\gamma}}_{E}.$

Connective constant of SAWs on the Sierpinski gasket family

Using a graph counting technique suitable for regular fractals, an exact evaluation of the total number of embeddings of self-avoiding walks on the generalized Sierpinski gasket is obtained. Numerical estimates for the connective constants are quoted for the first time, where b is the generation parameter of the gaskets. It is shown that the number of distinct n-step SAWs per site converges to the triangular lattice values when . Our analysis indicates that converges to the Euclidean value in the same limit and an asymptotic expression is given.

Critical exponents for numbers of differently anchored polymer chains on fractal lattices with adsorbing boundaries

We study the problem of polymer adsorption in a good solvent when the container of the polymer-solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integerb (2≤b≤∞), and it is assumed that one side of each SG fractal is an impenetrable adsorbing boundary. We calculate the critical exponents γ1, γ11, and γ s , which, within the self-avoiding walk model (SAW) of the polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method for 2≤b≤8 and the Monte Carlo renormalization group (MCRG) method for a sequence of fractals with 2≤b≤80, we obtain specific values for these exponents. The obtained results show that all three critical exponents γ1, γ11, and γ s , in both the bulk phase and crossover region are monotonically increasing functions withb. We discuss their mutual relations, their relations with other critical exponents pertinent to SAWs on the SG fractals, and their possible asymptotic behavior in the limitb→∞, when the fractal dimension of the SG fractals approaches the Euclidean value 2.

Critical exponent gamma for self-avoiding walks on the Sierpinski gasket family of fractals

The authors apply the Monte Carlo renormalization group (MCRG) analysis of self-avoiding walks (SAWs) on fractals to calculate the critical exponent gamma , associated with the total number of distinct SAWs. In the case of the Sierpinski gasket family of fractals (whose members are labelled by an integer b, 2<or=b< infinity ) they have calculated gamma for 2<or=b<or=80. Their MCRG results deviate at most 0.2% from the available exact results (2<or=b<or=8). The entire set of their results demonstrates that gamma , being always larger than the Euclidean value 43/32, monotonically increases with b.

Self-avoiding walks on fractals studied by the Monte Carlo renormalization group

The authors have applied the Monte Carlo renormalization group (MCRG) method to study the problem of self-avoiding walks on the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b, 2<or=b<or= infinity , and when b to infinity both the fractal df and spectral ds dimension approach their Euclidean value 2. They have calculated the critical exponent nu , associated with the mean square end-to-end distance, up to b=80. Their MCRG results deviate at most 0.03% from the available exact results (for 2<or=b<or=9). The obtained data show clearly that nu monotonically decreases with b and crosses the Euclidean value nu =3/4 at b approximately=27, that is, before entering the fractal to Euclidean crossover region that occurs in the limit b to infinity .

Piecewise directed random walk on the Sierpinski gasket family of fractals

We have introduced a new type of nontrivial directed random walk on the Sierpinski gasket family of fractals. We demonstrate that the behaviour of the critical exponent ν, associated with the mean squared end-to-end distance, at the fractal to Euclidean lattice crossover is governed by the fractal dimension alone.

Asymptotic form of the spectral dimension at the fractal to lattice crossover

The authors study the spectral dimension of the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b(2<or=b< infinity ), so that when b to infinity both the fractal and spectral dimension approach the Euclidean value 2. The asymptotic law for the spectral dimension was recently suggested to be dS approximately=2-B/(1nb)beta , with B and beta being some constants. Here the authors demonstrate that this form should be replaced by the asymptotic law ds approximately=2-1n(1nb)+ constant/1nb. Their analysis is based on the exact calculation of the electric resistances Rb for all members of the family up to b=650.