Confinement, free energy chains

Ro and are the end-to-end distances of unconfined ideal and real chains, respectively. These calculations can be generalized to confinement a polymer with fractal dimension lju from its original size bJST to a cylinder with diameter D. The confinement free energy in this case is (derived in Problem 3.16)... 

In order to gain this energy, the chain must pay the entropic confinement free energy Fconf, derived in the example above [Eqs (3.49) and (3.50)]. Therefore, the total free energy of a weakly adsorbing chain is... 

The first term on the right-hand side of Equation 5.10 is due to the fact that one end of the chain can be placed anywhere inside the volume of the cavity. The second term, called the confinement free energy, arises from all allowed conformations due to chain connectivity and confinement effects, provided that one end is already placed somewhere inside the cavity. For large values of N such that Rg > R, the leading term in Equation 5.11 dominates the sum. The... 

Basically, the confinement free energy of a polyelectrolyte chain inside a cavity is due to the translational entropy of counterions, electrolyte ions, and solvent molecules. Simple scaling formulas based on the radius of gyration of the polyelectrolyte, analogous to Equations 5.28 and 5.26, are not applicable for the confinement free energy of a polyelectrolyte in spherical cavities, unlike the case of uncharged polymers. 

Comparing this result with Equation 5.44, we conclude that the confinement free energy is hgT times the number of blobs in the confined chain. 

For an uncharged polymer in a good solution or a flexible polyelectrolyte chain in an aqueous solution with moderate amount of salt, the size exponent V is 3/5. For such cases, the radial extent of the confined chain in a channel geometry and the confinement free energy are given by... 

When the pore or channel is wide, the polymer undergoes non-single-file translocation. For such geometries, simple scaling formulas are derived for the confinement free energy and the spatial extent of the polymer, as functions of chain length, pore diameter or channel thickness, and the size exponent for the... 

Finally, an alchemical free energy simulation is needed to obtain the free energy difference between any one substate of system A and any one substate of system B, e.g., Ai- In practice, one chooses two substates that resemble each other as much as possible. In the alchemical simulation, it is necessary to restrain appropriate parts of the system to remain in the chosen substate. Thus, for the present hybrid Asp/Asn molecule, the Asp side chain should be confined to the Asp substate I and the Asn side chain confined to its substate I. Flat-bottomed dihedral restraints can achieve this very conveniently [38], in such a way that the most populated configurations (near the energy minimum) are hardly perturbed by the restraints. Note that if the substates AI and BI differ substantially, the transfomnation will be difficult to perform with a single-topology approach. 

AG is always negative, and the decrease in free energy can be due to adsorption effects (change in AH) or entropic interactions (change in AS). AS is always operating when the polymer chain cannot occupy all possible conformations in a pore (confined space) due to the limited size of the pore relative to the size of the macromolecule. In a real... 

The configurational free energy of the chain, G, is simply equal to -kT In n. The mean number of segments bound to the confining surfaces can be obtained by differentiating the free energy with respect to 0... 

Figure 8. The free energy of confinement of the chain, Ay/ki at constant composition, as a function of the width of the interfacial region, D, at 0 0 and 0.75.

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 is of interest to note that below (but in the vicinity of) the temperature, the minimum confinement distance for which the overlap reduces the free energy might exceed the maximum length of the chains, 2d , , > 2Na. Because for 2d > 2Na there are no interactions between brushes, since the chains cannot overlap, the interactions occur in this case for 2d < 2Na < 2dmm and hence are repulsive. Therefore, the brushes repel each other at all separations, as if they would have been immersed in a good solvent. This might explain the repulsion (and no attraction) between brushes observed recently in near- solvents.26... 

Most of the information on the photoreactions of the chromophores (cf. below) is for the a -trans form. The 13-d5 chromophores produced in the photoreaction are C=N anfi[85] and not as well accommodated by the binding pocket this results in the thermal relaxation to aW-trans and thus the cyclic reaction. Because the position of the 0-ionone ring is fixed, the displacement caused by the bond isomerization around the C13-C14 bond is confined to the chain near the Schiff base. As will be discussed below, it is the movements near the Schiff-base region of the retinal which store excess free energy to drive the reactions of the photocycle and the accompanying proton transport. 

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