The Coil-Globule Transition

Let s explore the dependence a(x), given by (9.7), in more detail in the range y j/cr- When x 0 (so that the solvent is good), this dependence provides the correct qualitative description of how a polymer coil swells due to excluded volume interactions (see Section 8.6). In this case we can neglect the second terms on both sides of Equation (9.7) mathematically, this neglect is obviously justified at a 1, but one can check that it works OK everywhere at x 0. Then we shall have  

what if X 0 Let s look at the region x Xcr first, which corresponds to higher temperatures than that where the jump in polymer size occurs. We can deduce from (9.7) that a. is close to one in this case. This means that the molecule takes the shape of a nearly Gaussian coil, and is hardly at all disturbed by volume interactions. What happens if X Xcr (on the bottom branch of the loop ) In this case, normally, the equilibrium swelling coefficient is very low (a 1 ). So the molecule looks terribly squashed by attraction between the monomers, when compared with an ideal coil. The terms on the left-hand side of Equation (9.7) will be much smaller than those on the right. Where did the terms on the left come from You can easily trace that they have to do with the entropy, Equation (9.5). 

From here we conclude that the entropy contribution Uef[ a) to the free energy is not significant for x x. Thus, the equilibrium size of the molecule is only controlled by the free energy of the monomer interactions U a). If we neglect the terms a and a in (9.7), we shall have  

Then the concentration (or density) n of monomers inside the globule is estimated as  

So far, we have only considered the coil-globule transition for y ycv This is when it is accompanied by a jmnp in the molecule s size. Can the 

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In contrast to PNIPAM, direct observation of the coil-globule transition of a single PVME chain has not been reported. However, a number of reports... 

Partial vitrification may affect kinetic processes during the coil-globule transition. Thus, at very high dilution, macroscopic phase separation well above the LCST might be stopped by partial vitrification of the polymer-rich phase. At this point we can only speculate whether vitrification interferes with the coil-globule transition or not. This problem is open for discussion and needs experimental confirmation. 

Random distribution of a significant number of hydrophilic NVIAz units along the polymer chain could result in uniform hydrophilization. This, in turn, could lead to a loss of ability for the coil-globule transition, which is caused by the hydrophobic interactions. As a result, such copolymers should be water-soluble over a wide temperature range. 

Note that the paper [ 1 ] by I.M. Lifshitz forestalled the corresponding experiments by at least 10 or rather 20 years. Experimental observation of such transitions is even now far from a routine procedure. Here we shall limit the discussion to mentioning several studies, which are in our opinion the most important achievements in this field [31-34]. We shall also refer to very informative reports on computer simulation of the coil-globule transition, namely, recent paper [35], and a very good reference list therein. 

As we mentioned earlier, I.M. Lifshitz was the first to realize that the coil-globule transition is not just a decrease of the chain size, but a phase transition to a condensed phase. Considering the multiplicity of known and possible condensed states, such a bro ad view on the coil-globule transition opened a perspective for unified understanding of a great number of physical phenomena in a variety of polymer systems. This gave rise to the concept of coil-globule-type transitions. Below we discuss several examples of such transitions. 

If network chains carry charged units, the coil-globule transition becomes sharper the amplitude of the jump increases, the critical value of pcr decreases down to unity pcr 1 (see Fig. 10). 

Before the volume phase transition was experimentally demonstrated in synthesized gels, its existence was theoretically predicted by Dusek and Patterson [4]. They suggested that the volume phase transition of gels is similar to the coil-globule transition of polymer chains and could be regarded as a first-order phase transition. 

Yoshikawa, K., Kidoaki, S., Takahashi, M., Vasilevskaya, V.V. and Khokhlov, A.R. (1996b) Marked discretness on the coil-globule transition of single duplex DNA. Ber. Bunsen-Ges. Phys. Chem., 100, 876-880. 

In a series of papers [216,217], Nakata and Nakagawa have studied the coil-globule transition by static light scattering measurements on poly(methyl methacrylate) in a selective solvent. They have found that the chain expansion factor, a2 = R2/R20, plotted against the reduced temperature, r = 1 - 0/T, first decreases with decreasing r, as it should be, but then begins to increase (see, e.g., Fig. 2 presented in [217]) In the authors opinion, the increase of... 

The above analysis implies that the coil-globule transition is essentially a gas-liquid transition within a single chain. Unlike usual molecular gases, the translational entropy is absent due to the chain connectivity, and instead, the conformational entropy shows up. The collapsed state is a spherical droplet, that is, globule, to minimize the surface area, the size of which is self-adjusted to satisfy the mechanical balance between the inside and the outside of the globule. 

It is known that the coil-globule transition in flexible polymers is well explained by the theory of the type discussed [22]. Note that the chain length and the solvent quality come into the theory in the following combined form x = BN1/2/l3, which is the only dimensionless parameter governing the transition. The presence of the master curve (see Fig. 3.5 below) implies that the phase behavior of the thermodynamic limit with N —> oo is readily discussed from the measurement of shorter chains via finite-size scaling. 

Thus, at T > T3, the long semiflexible macromolecule is in the coil state while at T < T3 it is in the globular state. Consequently, the temperature T3, which is determined by Eq. (3.7), is the temperature of the coil-globule transition for the long freely jointed macromolecule (see Fig. 6). It is clear that this transition is the first order phase transition with a considerable bound of the coil dimensions, it leads simultaneously to the transformation of the coil into the globule and to the formation of the liquid-crystalline ordering in the globule. 

In particular, it is well known that, if the macromolecule is supercooled below the 0 temperature, the phase transition isotropic coil-isotropic globule occurs. We emphasize that for the semiflexible macromolecule this is the peculiar phase transition between two metastable states. It should be recalled that the theory of the transition isotropic coil-isotropic globule for the model of beads is formulated in terms of the second and third virial coefficients of the interactions of beads , B and C24). This transition takes place slightly below the 0 point and its type depends on the value of the ratio C1/2/a3 if Cw/a3 I, the coil-globule transition is the first order phase transition with the bound of the macromolecular dimensions, and if C1/2/a3 1, it is a smooth second order phase transition (see24, 25)). 

The parameters of the effective model of beads for the polymer chain models under consideration (Fig. 7b-d) can easily be found using the methods of Refs.25, 33). We omit here the corresponding trivial analysis and present only the final results for the value of C1/2/a3, which determines the type of the transition isotropic coil-isotropic globule. For the models of Figs. 7 b, d it turns out that at ( > d, always C1/2/a3 coil-globule transition is always of the first order. At the same time, for the model of Fig. 7 c, the type of the transition depends on the ratio da at da 1, this is the first order phase transition and at da S l the transition is of the second order. 

This result can be interpreted as follows. The coil-globule transition takes place when the number of rods in the fringe becomes N (the dimensions of the fringe are (a2 + z)1/2). 

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