The reversible adsorption of a polymer chain to an attractive surface is an important problem in materials science and biophysics. The location of this transition ($T_c$) is sensitive to both polymer flexibility ($l_p$) and the range of the attractive surface potential ($\lambda$) and, for long chains, simple scaling arguments predict $T_c\sim {\left(l_p\right)}^a{\left(\lambda \right)}^b$ with different power-law exponents for different regimes of $l_p$ and $\lambda$. Verification of these scaling laws for semiflexible polymers via computer simulation is challenging due to the long chain lengths ($N$) required to reach the asymptotic scaling regime. Here we propose a finite-size-scaling method using partition function zeros to obtain adsorption transition temperatures in the long-chain limit from simulations of chains of moderate length. By combining the real and imaginary parts of the leading partition function zeros it is possible to eliminate the size- and flexibility-dependent scaling function that describes the $N\to \infty$ approach of these leading zeros to the critical point in the complex inverse-temperature plane. Our model polymer is a flexible tangent-hard-sphere chain (sphere diameter $\sigma$) with a local bond angle restriction that sets a persistence length $l_p$. The chain is end-tethered to a flat surface that has a square-well attractive potential of range $\lambda \sigma$. We use a Wang-Landau simulation algorithm to obtain the density of states for chains up to length $N=2560$ with $1\le l_p/\sigma \le 13,100$ and $0.01\le \lambda \le 20$. We distinguish three distinct scaling regimes over this wide parameter space: (i) worm-like-chain behavior for $l_p/\sigma >max(10,10\lambda )$, (ii) expanded-coil behavior for $\lambda >1$ with $l_p/\sigma <\lambda$, and (iii) a single-bead-interaction region for $\lambda <1$ with $l_p/\sigma <10$, and we find scaling laws consistent with simple scaling expectations for each of these regions. In the rigid-rod limit ($l_p\to \infty$), an exact solution of the model shows that $T_c^{*}\to \infty$ for $N\to \infty$.

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• Adsorption
• Chain stiffness
• Polymer behavior
• Transition temperature
• Single polymer chains
• Entropic sampling methods
• Finite-size scaling
• Lee-Yang & Fisher zeroes
• Monte Carlo methods