Test of the bounds on the crossover exponent for polymer adsorption on fractals

Crossover exponent for piecewise directed walk adsorption on Sierpinski fractals

“…For one type (PDW1) of this model we were able to find φ exactly for the entire SG family, whereas in the other case (PDW2) we found exact values of φ for 2 ≤ b ≤ 1000. Comparing our results with the results obtained for the common SAW model on the same fractal family [20,21] for 2 ≤ b ≤ 100 we find that for each b the values of φ for both models are very close. This is not surprising since attractive wall at the fractal boundary brings anisotropy in SAW paths which is similar to the effects of the directedness imposed on random walks in PDW models.…”

“…(iii) Suppose that (A.1) is valid for b ≤ m and every possible i, as well as for b = m + 1 and i ≤ j < m. Then, we will prove that (A.1) is valid for b = m + 1 and i = j + 1. To this end we use the transformation rules that follows directly from (3.3): [20,21]) models on SG fractals. The full line is the theoretical estimate (3.23) obtained by the asymptotic analysis of the PDW1 data in the limit b → ∞.…”

“…Due to the extraordinary complexity of the underlying combinatorial problem, exact crossover exponents φ were found [20] only for 2 ≤ b ≤ 9. For larger values of b the Monte Carlo renormalization group (MCRG) method [21] was developed, and the limit b → ∞ was analyzed within the finite-size scaling hypothesis [22], with the conclusion that φ does not tend to its Euclidean value φ E = 1/2 [6].…”

Critical exponents of surface-interacting self-avoiding walks on a family of truncated n-simplex lattices

“…The most of theoretical efforts so far have been devoted to a study of suitable models of surface-interacting SAWs on standard homogeneous spaces. Recently, a considerable research activity have appeared in the study of SAWs placed on fractal spaces [6][7][8][9][10][11], as models of polymers in nonhomogeneous environment. Aside from being interesting in its own right, we believe that such studies may yield some insights into more difficult questions related to the behavior of polymers in disordered systems.…”

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

“…Early investigations of polymer behavior near attractive surfaces dealt with polymer chains immersed in homogeneous spaces with planar adsorbing boundaries (see [1] for a review). These studies have been subsequently extended to polymers immersed in porous (inhomogeneous) media, modeled by fractal lattices embedded in two-dimensional [2][3][4] and three-dimensional [5,6] space. In these studies, almost exclusively, only two critical exponents have been studied, that is, the end-toend distance critical exponent ν and the crossover exponent φ (that governs the number of contacts between the polymer and the surface).…”