Confinement of a Real Chain

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 

Unlike a Gaussian chain, R inaeases linearly with N. Note that / g refers to the radius of gyration of unconfined chains. 

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. 

The linear dimension of the chain in the slit is different from the counterpart in the cylindrical pore. Because the confined chain follows the conformation of two-dimensional excluded-volume chain, 

Here we used the fact that, in two dimensions, the self-avoiding random walk has an exponent of 3/4 in the relationship between Rp and N (Problem 1.13). We can also derive the above relationship by applying Hory s method that we used to derive the chain dimension in three dimensions (Problem 2.34). 

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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 ... 

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