Confinement Gaussian chain

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

We now consider an ideal Gaussian chain confined between two Garge) flat plates with area A at a plate separation h, see Fig. 2.9. 

The model consists of a network of Gaussian chains connected in any arbitrary manner. The physical effect of the chains is assumed to be confined exclusively to the forces they exert on the junctions to which they are attached. 

Confinement of a Gaussian Chain We leam here how a Gaussian chain changes npon confinement by varions geometries such as a slit, a square tube, and a cylindrical tnbe. It is possible to obtain a formula for the partition coefficient in each of the three geometries. 

Figure 2.61. Density profile of the end of the Gaussian chain when the other end is at z = dj 2. The density is compared for the confined chain (solid line) in a slit of walls at z = 0 and z = d and the unconfined chain (dashed line). R, = d/A was assumed.

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. 

The total probability of a Gaussian chain, with the condition that all of its N segments can be anywhere inside a confining sphere of radius R (Figure 5.8a) without hitting the wall, is given by (e.g., Muthukumar 2003)... 

In this section we discuss the static properties of a Gaussian polymer chain without excluded volume interactions that is confined to a medium populated with quenched... 

K, when no large-angle dynamics are detected in bulk POE, evidence for chain motions in the nanotube-confined POE is observed. The line shapes representing this motion were modeled by a discrete 3-site jump using a Gaussian-type distribution of correlation times. 

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