Hypothesis of entropy-driven crystallization

Just like pattern-forming phenomena in dissipative systems,^59,60 the gelation was via the fluctuation instability, and the initial fluctuation predetermined the final morphology. We may wonder what “small effects” felt by polymer molecules induce such a self-organizing process.

Let us draw attention to the conformational change of the s-PS during the fluctuation stage, as shown in Figure 9 (a).

Figure 9 (a) Real-time trace of the conformational-sensitive IR band of the regular TTG⁺G⁺ sequence of the s-PS (572cm^-1) in the early stage nucleation. (b) Entropy-driven crystallization in the polymer solution. Left: The excluded volume effect leads to a crowded environment, and the maximum of the conformational entropy leads to an inaccessible region or a closed chain loop to confine the degree of translational freedom of the solvent molecules (the depleting spheres). Middle: The solvents act as “side chain” affixed to the hard-sphere chain to stabilize the self-organizing helix, thus favoring the global maximum of the system's entropy. Right: Again, the free-volume per depleting sphere is larger in the ordered (the crystal) than in the disordered (the self-organizing helix) phase.

The integrated IR intensity of the regular TTG⁺G⁺ conformation emerged immediately after quenching, increased linearly before any indication of forming observable fibrils (t t< _nucl ), and approached only asymptotically the gelation stage. Superficially, this observation seems to agree: whereas the polymer crystal is partially built up with the intramolecular periodic structure, when a polymer molecule is possible to self-organize into the regular sequence, the next "crystallization" through intermolecular packing of these sequences must proceed in a more continuous and collective fashion as the "spinodal"

process.⁶¹

However, two deep questions should be addressed. First, what does drive the self-organization within a polymer molecule? And secondly why does the solution show up as a long-lived metastable state, despite its fluctuation instability? The answers to both questions may lie in understanding how the s-PS chain behaves in the solution. Nevertheless, the

instability in the solution itself causes the experimental difficulties. To avoid the problem, the comparable atactic polystyrene (a-PS, M_w=2.116×10⁵, M_w/M_n=1.01, Sigma-Aldrich Inc.) was used as a substitute [the intrinsic viscosity [ ]

η

, which characterizes the hydrodynamic specific volume of a polymer coil, is 0.69 dL g^-1and the Huggins constant, which is related to the structure of polymer coil (for the self-avoiding chains, k<0.52), is 0.488]. Inasmuch as the two polymers are the stereoisomers, their behaviors should be essentially identical at least at t=0. Unexpectedly, in the present condition, the gelation occurred at [ ]η C =0.345 (much lower than the hydrodynamic overlapped concentration, [ ] ~ 1η C )! Obviously, the formation of the self-organized helix (i.e., the regular TTG⁺G⁺ sequence of the s-PS) accompanied the significant expansion of the s-PS coils and made gelation at the dilute situation. From Daniel et al.,³² the self-organizing helix can be stabilized by o-xylene molecules intercalated between the phenyl groups (the π—π interactions⁶²). It, however, seems rather doubtful that all was driven entirely by such "chemical" force. On the other hand, we concerned why the coil expansion is not an instantaneous conformational transition but again a continuous process.

From a different angle, we all know that the preferred polymer crystal is the chain-folded lamellar crystal.⁶³ The chain-folding not only is a most efficient way of packing polymer chains but also satisfies the homogeneous nucleus and the sharp interface as required for the classical nucleation theory. In contrast, the fibril, a so-called “fringed-micelle crystal,” could collapse due to the high cumulative strain energy⁶⁴ and the entropy effect of overhanging loose chains at the sterically nervous transition-interface-layers. Thus, in bulk crystallization, the fibril is forbidden on the kinetic grounds.^64,65 However, the fibril nucleation during the gelation is certainly a different physical process. Now that the fibril structure does not favor the minimization of the interface energy, the global entropy maximization may be the primary in its formation. Indeed, using the hard-sphere chain model, the Prigogine-Flory-Patterson theory⁶⁶ predicts the entropy-driven phase separation to occur when the loss in the mixing entropy is compensated by the translational entropy increment.^67,68 Snir and Kamien also show that a purely entropic force can be enough to bend the stiff polymer into a helix.⁶⁹ So we suggest a hypothesis of entropy-driven crystallization to understanding the fibril formation.

Figure 9 (b) illustrates how this may arise.

We first consider a hard-sphere chain (self-avoiding chain) surrounded by many depleting spheres (solvent molecules). The excluded volume of the hard-sphere chain reduce the free volume in which the depleting spheres may live, as shown in Figure 9 (b) left.

Clearly, the only way to heighten the spheres' free volume (translational entropy) at constant internal energy is to allow the significant expansion of the hard-sphere chain, while its conformational entropy is more or less lost. However, whether the translational entropy can have an appreciable effect on the expansion of the hard-sphere chain should depend on the polymer concentration. We predict that the maximum translational entropy could be obtained near the overlap threshold ([ ] ~1

η

C ), enough to outweigh the loss in the conformational entropy. Even so, the stability of the expanded hard-sphere chain still has to depend on its persistence time. The stabilization can be done in two ways: either by a higher activation energy barrier for the rotational isomerization, or by the special chemical forces such as the π-π interactions, as shown in Figure 9 (b) middle. Furthermore, as the self-organizing helices grow beyond certain critical length, the depletion attraction caused by them would be sufficiently large to induce the fluctuation instability, as shown in Figure 9 (b) right. We finally have known why the solution shows up as the long-lived metastable state.

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