Partition-function-zero analysis of polymer adsorption for a continuum chain model

Polymer chains undergoing adsorption are expected to show universal critical behavior which may be investigated using partition function zeros. The focus of this work is the adsorption transition for a continuum chain, allowing for investigation of a continuous range of the attractive interaction and comparison with recent high-precision lattice model studies. The partition function (Fisher) zeros for a tangent-hard-sphere $N$-mer chain (monomer diameter $\ensuremath{\sigma}$) tethered to a flat wall with an attractive square-well potential (range $\ensuremath{\lambda}\ensuremath{\sigma}$, depth $\ensuremath{\epsilon}$) have been computed for chains up to $N=1280$ with $0.01\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}2.0$. In the complex-Boltzmann-factor plane these zeros are concentrated in an annular region, centered on the origin and open about the real axis. With increasing $N$, the leading zeros, ${w}_{1}(N)$, approach the positive real axis as described by the asymptotic scaling law ${w}_{1}(N)\ensuremath{-}{y}_{c}\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\phi}}$, where ${y}_{c}={e}^{\ensuremath{\epsilon}/{k}_{B}{T}_{c}}$ is the critical point and ${T}_{c}$ is the critical temperature. In this work, we study the polymer adsorption transition by analyzing the trajectory of the leading zeros as they approach ${y}_{c}$ in the complex plane. We use finite-size scaling (including corrections to scaling) to determine the critical point and the scaling exponent $\ensuremath{\phi}$ as well as the approach angle ${\ensuremath{\theta}}_{c}$, between the real axis and the leading-zero trajectory. Variation of the interaction range $\ensuremath{\lambda}$ moves the critical point, such that ${T}_{c}$ decreases with $\ensuremath{\lambda}$, while the results for $\ensuremath{\phi}$ and ${\ensuremath{\theta}}_{c}$ are approximately independent of $\ensuremath{\lambda}$. Our values of $\ensuremath{\phi}=0.479(9)$ and ${\ensuremath{\theta}}_{c}=56.8{(1.4)}^{\ensuremath{\circ}}$ are in agreement with the best lattice model results for polymer adsorption, further demonstrating the universality of these constants across both lattice and continuum models.