Polymer Model

 In our model the polymer is represented by an N-unit self-avoiding walk on a simple cubic lattice. There is an attractive energy, $\epsilon$, between non-bonded polymer units on adjacent lattice sites and an energetic penalty, $\epsilon_g$, for kinks in the chain. The total energy is given by  

$$E=-n_{pp}\epsilon+n_g\epsilon_{g}$$ (1)

where n_pp is the number of polymer-polymer contacts and n_g is the number of kinks or `gauche bonds' in the chain. $\epsilon$ can be considered to be an effective interaction representing the combined effects of polymer-polymer, polymer-solvent and solvent-solvent interactions, and so our model is a simplified representation of a semi-flexible polymer in solution. The behaviour of the polymer is controlled by the ratio $kT/\epsilon$; large values can be considered as either high temperature or good solvent conditions, and low values as low temperature or bad solvent conditions. The parameter $\epsilon_g$ defines the stiffness of the chain. The polymer chain is flexible at $\epsilon_g$=0 and becomes stiffer as $\epsilon_g$ increases. In this study we only consider $\epsilon_g\geq 0$.

When $\epsilon_g=0$, this model corresponds to one originated by Orr[17] and has been much used to study homopolymer collapse.[3,14,18,19,20] The system with positive $\epsilon_g$ has been recently studied theoretically by Doniach et al.[15] and using simulation by Bastolla and Grassberger.[16] In our work we pay special attention to the structural changes of a polymer of a specific size. In this sense our work is complementary to that of ref. 16 which focussed on the accurate mapping of the phase diagram.

Our model was chosen because we wished to have the simplest model in which we could understand the effects of stiffness, and because it gives low energy states that have chain folds resembling those found in lamellar homopolymer crystals.[21] Similar structures have previously been seen in a diamond-lattice model of semi-flexible polymers,[11,12,13] and in molecular dynamics simulations of isolated polyethylene chains.[22,23,24] Particular mention should be made of the studies by Kolinski et al. on the effect of chain stiffness on collapse[11,12,13] since they found evidence for many of the phenomena that we explore systematically here.

The global minimum at a particular (positive) $\epsilon_g$ is determined by a balance between maximizing n_pp and minimizing n_g. If the polymer is able to form a structure that is a cuboid with dimensions $a\times b \times c$ (N=abc), where $a\le b \le c$, then

n_pp=2N-ab-ac-bc+1

and

 

n_g^min=2ab-2 (3)

The structures that correspond to n_g=n_g^min have the polymer chain folded back and forth along the longest dimension of the cuboid. By minimizing the resulting expression for the energy one finds that the lowest energy polymer configuration should have  

$$a=b \quad\hbox{and}\quad {c\over a}=1+{2\epsilon_{g}\over\epsilon}.$$ (4)

Therefore, at $\epsilon_g=0$ the ideal shape is a cube and for positive $\epsilon_g$ a cuboid extended in one direction, the aspect ratio of which increases as the chain becomes stiffer. As the ideal aspect ratio of the crystallite is independent of N, its squared radius of gyration, R_g², will scale as N^2/3; this scaling is the same as for the disordered collapsed globule.

Substitution of the optimal dimensions of the cuboid [Equation (4)] into Equations (1)-(3) gives a lower bound to the energy of the global minimum,  

$$E_{opt}/\epsilon=-2N+3N^{2/3}(1+2\epsilon_g/\epsilon)^{1/3}-1-2\epsilon_g/\epsilon.$$ (5)

However, at most sizes and values of $\epsilon_g$ it is not possible to form a cuboid with the optimal dimensions, and so the energy of the global minimum will be higher than given by the above expression. Nevertheless, it is easy to find the global minimum just by considering the structures which most closely approximate this ideal shape.