Stiffness dependence of critical exponents of semiflexible polymer chains situated on two-dimensional compact fractals

TL;DR

This study investigates how the stiffness of semiflexible polymer chains affects their critical exponents on two-dimensional fractals, revealing that these exponents depend on stiffness and fractal scale, with implications for understanding polymers on complex structures.

Contribution

The paper provides exact and Monte Carlo renormalization group calculations of critical exponents for semiflexible polymers on an infinite family of fractals, highlighting their dependence on stiffness and fractal scale.

Findings

Critical exponents depend on polymer stiffness parameter s.

Stiffer chains have larger nd smaller xponents.

Exponents vary monotonically with fractal scale parameter b.

Abstract

We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension $d_f$ is equal to 2 for all members of the fractal family enumerated by the odd integer $b$ ($3\le b< \infty$). For various values of stiffness parameter $s$ of the chain, on the PF fractals (for $3\le b\le 9$) we calculate exactly the critical exponents $\nu$ (associated with the mean squared end-to-end distances of polymer chain) and $\gamma$ (associated with the total number of different polymer chains). In addition, we calculate $\nu$ and $\gamma$ through the MCRG approach for $b$ up to 201. Our results show that, for each particular $b$, critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of $s$) display enlarged values of $\nu$, and diminished values of $\gamma$. On the other hand, for any specific $s$, the critical exponent $\nu$ monotonically decreases, whereas the critical exponent $\gamma$ monotonically increases, with the scaling parameter $b$. We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.