Self-Avoiding Rotating Walks as Models of Crystals Made of Freely Rotating Polymers in Two Dimensions

The concept of random walk (RW) [1] and its variation in the form of the self-avoiding random walks (SAWs) [2] are important tools in studying complex processes in a wide variety of fields, including, among others, polymer physics. Recent molecular simulations on extremely confined, monolayer films made of athermal semi-flexible polymers have revealed a wealth of structural behavior as a function of surface coverage chain stiffness, in the form of equilibrium bending angle [3]. Inspired by this and to quantify the thermodynamic stability of the two-dimensional polymer crystals, we map them into self-avoiding rotating walks (SARWs), restricted to follow the geometric constraints of the corresponding lattices. For a reference crystal, and by selecting a compatible equilibrium bending angle, we enumerate all possible configurations of single-chain crystals and calculate the corresponding size distribution as a function of the number of steps. SARW enumeration has been performed on honeycomb, square, and triangular lattices, all being characterized by different coordination numbers and lattice connectivity. The scaling of the number of SARWs, as well as their average size, as a function of steps are fitted using exponential-power-law asymptotic expressions and critical amplitudes; the connective constant and the critical exponents are compared against the analogous ones for conventional SAWs, corresponding to freely jointed polymers, under the same conditions [4,5].

[1] R. Bhattacharya, and E. C. Waymire, Random Walk, Brownian Motion and Martingales, Springer-Verlag (2021).

[2] N. Madras, and G. Slade, The Self-Avoiding Walk; Birkhauser: Boston (1996).

[3] D. Martínez-Fernández et al., J. Chem. Phys. (2024, under review).

[4] N. Clisby, Phys. Rev. Lett. 104, 055702 (2010).

[5] O. Parreño et al., Polymers 12, 799 (2020).