Confinement real chain

Problem 2.34 Hory s method, we learned in Section 1.4 to find the dimension of the real chain, can be extended to the confined real chain. Find the relationship between the dimension of the chain along the sUt wall or the tube wall, R, and the degree of polymerization N for the confinement by (1) a sUt of width d and (2) a tube of diameter d. 

Two simple examples comparing the properties of ideal and real chains are discussed in this section uniaxial and biaxial compression. A related of triaxial confinement shall be discussed in Section 3.3.2 for the... 

In the case of confinement of a real chain, the compression blobs repel each other and fill the pore in a sequential array. Therefore, the length of the tube 7 ]j occupied by a real chain is the size of one compression blob D times the number Njg of these blobs ... 

Note that in the case of a real chain confined to a tube, the occupied length of the tube R is linearly proportional to the number of monomers in the chain. The occupied length increases as the tube diameter D decreases. Ideal and real chains of the same length, confined in a cylinder of diameter D, are shown schematically in Fig. 3.10. There is no penalty to overlap the... 

Ideal and real chains of the same length, confined in a cylinder of diameter D. 

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)... 

The size of the real chain confined between plates is again much larger than that of an ideal chain (where R Ri bN ) because the compression blobs of the real chain repel each other. The maximum confinement cor-responds to thickness D of the order of the Kuhn monomer size b. In this case the chain becomes effectively two-dimensional with size... 

The adsorbed layer is thicker and bound less strongly for the real chain (since for weak adsorption 0 < < 1) because it pays a higher confinement penalty than the ideal chain. The excluded volume interaction of real chains make them more difficult to compress or adsorb than ideal chains. These scaling calculations can be generalized to adsorption of a polymer with general fractal dimension Ifir. 

The free energy of confining a real linear chain in a good solvent either into a slit of spacing Z) or to a cylindrical pore of diameter D is larger than for an ideal chain because the real chain has repulsive interactions ... 

Kuhn monomers with Kuhn length h. Recall that the unperturbed size of the real chain confined to the air-water interface in good solvent is hA [Eq. (3.54)],... 

In this case, the extremely confined polymer chain follows the scaling relationship of 2D SAWs, and exhibits the scaling law of the coil size for a 2D real chain. 

Rods and Hdices.—Most of the above is confined to flexible coil polymers, but there is considerable interest in stiffened chains e.g. rods, helices, not least because many polyelectrolytes and biological macromolecules are of this form (for a review see ref. 41). Yamakawa has published a series of papers, notably with Fujii and Shimada on the helical worm-like chain. His hypothesis is that on the bond length or somewhat longer scales, any real chain may be represented by the continuous helical worm-like chain, i.e. a hybrid of the three extreme forms of rod, coil, and helix . With this model, recent papers have covered the calculation of P( ) and comparison with scattering behaviour of poly(methyl metha-... 

We consider a real chain consisting of N monomers of size b and confined to a cylindrical pore of diameter d. When the chain dimension R in the free solution is smaller than the pore size, the chain does not feel much of the effect of the pore wall. As exceeds d, the chain must adopt a conformation extending along the pore because of the excluded volume effect. As R increases further, the confined chain will look like a train of spheres of diameter d (see Fig. 2.64). The excluded volume effect prohibits the spheres from overlapping with each other. Therefore, the spheres can be arranged only like a shish kebab. The partial chain within each sphere follows a conformation of a real chain in the absence of confinement. The number of monomers in the sphere is then given by... 

Figure 2.64. Real chain confined to a cylindrical pore of diameter d. The chain is regarded as a packed array of spheres of diameter d in one dimension. Within each sphere, the chain is three-dimensional.

The decrease in the entropy, -AS, grows linearly with N, i.e., a longer chain experiences a greater restriction on its conformation in the pore. It is interesting to see that the same power law, —AS N, also applies to the ideal chain if we replace 5/3 = 1/v by 2. The proportionality to N is common between the ideal chain and the real chain. This result is not a coincidence. If we follow the same discussion as above to calculate K for the ideal chain, the number of arrangement for the spheres in the pore is as opposed to 6 /" in the free solution. The ratio leads to -AS/kg = N/rii = (Rg/dy-. The confinement of the Gaussian chain gives the same relationship From K = and Eq. 2.136, we find —AS/k = (R /df. 

How about the confinement by the slit The spheres are arranged in the two-dimensional space. The nnmber of arrangements is now Then, - AS follows the same scaling relationship as Eq. 2.142 except the numerical coefficient. Figure 2.66 compares the partition coefficients of the Gaussian chain (solid Une) and the real chain (circles) with a radius of gyration in a slit of width d. The coefficients for the real chain were obtained in lattice computer simulations. ... 

As seen in the above examples, confinement lowers the number of dimensions available to a polymer chain. In the Gaussian chain, on the one hand, the confinement changes the confined components only. The root mean square end-to-end distance changes only by a numerical coefficient without changing the dependence of Rp on N. In the real chain, on the other hand, the decrease in the dimensionality changes qualitatively the relationship between N and R from that in the fiee solution. The confinement manifests the excluded volume effect more prominently. 

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... 

Consider a real linear chain confined to an air-water interface. The... 

The osmotic pressure is one of the colligative properties, which depends on the molar concentration of the particles present in the solution. For the polyelectrolytes, however, only the counterions contribute to the osmotic pressure because their number is much larger than the number of macroions. For the polyelectrolyte, the real osmotic pressure is much lower than that calculated by the van t Hoff law because a large portion of the counterions is confined around the polyelectrolyte chains [84, 85, 87-89], The ratio between the real osmotic pressure and the theoretical calculated one, the osmotic coefficient, q>, is a direct measure of the fraction of non-confined counterions. 

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