Conclusion

In this paper we have provided an example of an order-disorder transition that can be observed for an isolated homopolymer. Its basis in the interactions of the model is clear. The global minimum is orientationally ordered in order to minimize the energetic penalty for gauche bonds. At some finite temperature this order must be lost in a transition to the higher entropy globule or coil. As the stiffness of the polymer increases the energy gap between the ordered and disordered states increases contributing to an increase in the temperature at which this `melting' transition occurs. This effect, coupled with the weak dependence of the $\theta$-point on the stiffness parameter, leads to the disappearance of the disordered globule for sufficiently stiff polymers; there is no longer a collapse transition from the coil to globule. This behaviour is analogous to the disappearance of the liquid phase of simple atomic systems as the range of the potential is decreased. The phase diagram we obtain is in good agreement with recent theoretical[15] and simulation[16] studies of this polymer model.

One of the intriguing aspects of our model is the folded structures that form at low temperature. Firstly, these states are not artifacts of our lattice model. In simulations of isolated polyethylene chains, for the same energetic reasons as in our model--the extra energy from polymer-polymer contacts outweighs the energetic cost of forming a fold--relaxation to folded structures was observed.[22,23] Furthermore, as here, it was also found that the aspect ratio of the crystallites increased with increasing stiffness.[23] However, we know of no experiments on a homopolymer system where evidence has been found for a transition from a disordered globule to a crystalline structure with chain folds. The most likely signature for such a transition would be an increase in the radius of gyration with decreasing temperature in very dilute polymer solutions.

It is also natural to ask whether this study can provide any insights into polymer crystallization, since polymer crystals have a lamellar morphology in which the polymer is folded back and forth in a similar manner to the folded structures we observe. However, the formation of the folded structures for the isolated polymer is a purely thermodynamic effect--the structures are the global minima--whereas in the bulk case it is a kinetic effect[45,46]--it is generally accepted that the global minimum is a crystal where all the chains are in an extended conformation. The study, though, may have a more indirect relevance to the crystallization of polymers from solution. Although little consideration has been given to the structure of the polymer arriving at the surface in theories of polymer crystallization, it is not implausible that the existence of significant ordering in the adsorbing polymer would have a considerable effect on the crystallization process.

Although the present study only examines the homopolymer thermodynamics, some comments concerning the dynamics can be made. Klimov and Thirumalai made the interesting observation that model proteins are most likely to be good folders when $T_f/T_\theta$ is large, where T_f is the temperature at which the transition to the native state of the protein occurs.[47] For our system $T_m/T_\theta$ increases with $\epsilon_g$[Figure 8]. Therefore, if Klimov and Thirumalai's relation also holds for our homopolymers, one would expect crystallization of the polymer to become more rapid as the stiffness increases--crystallization is easier direct from the coil than via the disordered dense globule. However, whereas at T_f there is a transition to a single state, the globule minimum, at our T_m a transition to an ensemble of crystalline structures occurs. Therefore, it does not necessarily follow that the homopolymers would be able to reach the global minimum more rapidly as the stiffness increased--indeed it might be that trapping in low-lying crystalline structures is more pronounced for stiff polymers.