\section{Introduction}

\revis{The understanding of the crystallization behavior of single flexible and elastic polymer chains is of particular interest in natural and technological molecular building-block systems, where functionalization is based on the individual structural properties of small molecules forming larger-scale composites. Among these are, e.g., hybrid systems of solid substrates and "soft" molecules, where it is known that the formation of regular structures can be mediated by the interaction with the amorphous or crystalline surface of the substrate \cite{X1}. Tuning the self-assembly of individual polymers near substrates is relevant for molecular nanoelectronic applications and materials design which has become a huge field of applied research. As an example from life sciences, capsomer proteins of spherical viruses assemble to form a perfectly icosahedral capsid hull, which shelters the viral genome \cite{X2}. Thus, frozen structures of single polymer chains can serve as the basic elements of larger assemblies on nanoscopic scales. As we will show here, the crystallization behavior of elastic polymers exhibits strong similarities to the cluster formation of colloidal (or atomic) particles.}

The general behavior of a polymer in solvent has already been subject of numerous studies \cite{KremerBinder,BinderMilchev}. At high temperatures or in good solvent the interactions of different parts of the polymer are of little relevance and due to entropic effects the dominating structures are extended random coils. \revis{This changes at the tricritical $\Theta$-point where the polymer behaves in three dimensions effektively like a Gaussian chain, up to logarithmic corrections \cite{Hager}.} The effective repulsion caused by excluded-volume effects and the monomer-monomer attraction are in perfect balance. In the globular regime, the chain is collapsed and compact conformations dominate which, having almost no internal structure, resemble a liquid state. At lower temperature, energy optimization leads to a freezing towards crystalline structures. In the thermodynamic limit, this transition is expected to be of first order, whereas for the $\Theta$-collapse a second-order phase transition is predicted \cite{Lifshitz,Khokhlov}.

Under certain conditions, the two transitions also may coincide in the thermodynamic limit as it was shown in a recent investigation of a bond-fluctuation model \cite{Rampf}. However, this is not a generic feature of flexible \cite{Thomas} and elastic polymers \cite{letter}.

While the coil-globule transition is well analyzed and further progress in the analysis of logarithmic corrections to the scaling behavior would require the investigation of \revis{extremely} long chains, less is known for the freezing transition especially in the case of off-lattice polymers. Due to the intrinsic similarities of the applied model to colloidal clusters it is useful to interpret the crystallization of elastic polymers with reference to atomic Lennard-Jones clusters and to use the corresponding nomenclature. In fact, if one replaces `atoms' by `monomers' many statements from the cluster field remain also true for polymers. Besides the numerous studies that have been performed on the freezing transition and ground-state properties of Lennard-Jones clusters, there are only a few publications on Lennard-Jones homopolymers. The authors of Ref.\ \cite{calvo} were able to show principal relationships between both classes of systems and motivated the transfer of terms and concepts from the cluster case to the field of polymers. Recent studies \cite{Liang,Australien1,Australien2,Seaton,Seaton2} mainly focused on the general behavior of elastic polymers \revis{but} did not refer in detail to the polymer length dependence of the freezing transition, the ground-state conformations or solid-solid transitions.

The aim of this work is to close this gap and to compare the low-temperature properties of polymers and clusters. The rest of the paper is organized as follows. In the subsequent section we will introduce the model and mention some problems that arise in the simulations. We then give a short review of the thermodynamic properties of Lennard-Jones clusters in Sec.~III, followed by the introduction of a geometrical order parameter in Sec.~IV. After the description of the method and the explanation of a few technical details in Sec.~V, we will present and discuss our results in Sec.~VI. Finally, the main findings of the paper are briefly summarized in the concluding part.

\section{Model}

We employ a model for elastic flexible polymers, where all monomers interact pairwisely via a truncated and shifted Lennard-Jones (LJ) potential:
