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

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.