Chain translational entropy

Figure 16 shows the dependence of the osmotic pressure of salt-free polyelectrolyte solutions. Through almost the entire concentration range considered, the osmotic pressure is proportional to the polymer concentration, supporting that it is controlled by the osmotic pressure of counterions, both above and below the overlap concentration. There appears to be a weak chain length dependence of the osmotic pressure for short chains. However, this iV-dependence is consistent with jN correction to the osmotic pressure due to chain translational entropy. The deviation from linear dependence of the osmotic pressure n occurs around polymer concentration c 0.Qla, which is above the overlap concentration for all samples. At very high polymer concentrations, where electrostatic interactions are almost completely screened by counterions and by charges on the... 

To complete the expression of the osmotic pressure, the corrections due to chains translational entropy Tc-/N (i = A, B) must be added. In a... 

When the network chains contain ionic groups, there will be additional forces that affect their swelling properties. Translational entropy of counterions, Coulomb interactions, and ion pair multiplets are forces that lead to interesting phenomena in ion-containing gels. These phenomena were studied in detail by Khokhlov and collaborators [74-77]. The free energy of the networks used by this group is... 

Equation 2.16 contains contributions from the translational entropy of the mobile species, the conformational entropy of polymer chains, the free energy associated with the different chemical equilibria in the system, the polymer-polymer and polymer-surface van der Waals (vdW) interaction energies, the electrostatic interaction energies and the repulsive interactions between all the different molecular species. The expressions for each of these terms are shown in Table 2.2, while the definition of the symbols is given in Appendix. Note that in Table 2.2, the densities. 

At x > xcr, the network is in the very expanded state. The size of the chain between two junction points, R, is proportional to m R = aR0 ma/a /2 as it is for fully stretched chain. The reason for such an essential expansion is the osmotic pressure of counter ions which originates from their translational entropy. Trom the entropy consideration counter ions would like to leave the network, however, this is forbidden due to the condition of total electroneutrality. This effect was for the first time described in Ref. [7]. 

These results imply that homopolymer PS is not always miscible with the PS blocks of the copolymer, i.e. confinement of PS to an interface in a block copolymer can lead to immiscibility with homopolymer PS (Hashimoto et al. 1990). This has been interpreted in terms of the enthalpic and entropic contributions to the free energy (Hasegawa and Hashimoto 1996). For a < 1 uniform solubilization increases the translational entropy of the homopolymer, but chain stretching in the homopolymer and in the PS chain of the diblock leads to a decrease in conformational entropy. At the same time, the lateral swelling of microdomains leads... 

It increases with the charge of the polymers and as expected, decreases with the ionic strength. So far, we have ignored the translational entropy of the polymer chains that tends to zero in the limit where N tends to infinity. If N is smaller, the translational entropy of the polymer chains must be included in the pressure balance and stabilizes a homogeneous solution when it becomes of the order of the electrostatic attractive pressure. Complex formation thus only occurs if... 

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. 

First of all, in the solution of disconnected rods each rod has the freedom of an independent translational motion while in the case under consideration, only the chain as a whole (but not each segment) can move independently. Hence the contribution of the translational entropy to the free energy is equal to... 

For long chains (L > () this is much less than the corresponding contribution in the case of the solution of disconnected rods TN n c/e. Practically, for sufficiently long macromolecules the contribution ot the translational entropy to the free energy can be neglected (see also 15)). 

Thus, in the athermal limit the only difference between the equilibrium free energies of the solutions of separate rods and long chains of rods is due to the translational entropy term. Consequently, we can immediately conclude (analogously to Sect. 2) that the liquid-crystalline transition for the athermal solution of semiflexible chains takes place at 1/p. 

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