Confined chains ideal

Since the ideal chains do not interact the partition function for N confined chains can be written as... 

Indeed the excess free energy in the globule state, P(R), involves two main contributions F = Fconf+Pmt- (The nearly ideal coil state is considered as the reference state with P= 0.) The conformational free energy F onf iS/ in fact, the minimal work required to confine the ideal chain in a sphere of radius R. To estimate it we represent the chain as a sequence of N/g ideal g-blobs such that each blob just fit in the sphere the blob size bgi/2 Obviously Pconf niust be proportional to the number of the blobs ... 

Secondly, I wish to counteract anticipated despondency which some of the complexities on the present theoretical scene may perhaps provoke. For this purpose, I wish to invoke the decisive simplicity and definiteness of some of the experimental effects observed within the confines of the above, near ideal systems. This, as I often pointed out elsewhere, is unmatched in the field of crystal growth of simple substances. Complicated as polymers may seem, and subtle as some of the currently relevant theoretical issues, this should not obscure the essential simplicity and reproducibility of the core material. To be specific, the appropriate chains seem to want to fold and know when and how, and it is hardly possible to deflect them from it. Clearly, such purposeful drive towards a predetermined end state should continue to give encouragement to theorists for finding out why Those who are resolved to persevere or those who are newly setting out should find the present review a most welcome source and companion. 

The pom-pom polymer reptation model was developed by McLeish and Larson (60) to represent long chain-branched LDPE chains, which exhibit pronounced strain hardening in elongational flows. This idealized pom-pom molecule has a single backbone confined in a reptation tube, with multiple arms and branches protruding from each tube end, as shown in Fig. 3.12(a). Mb is the molecular weight of the backbone and Ma, that of the arms. 

Figure 17. Simulations of confined polymer chains as ideal random walks between two hard impenetrable interfaces. Two populations exist free (nonimmobilized) chains, which contribute to the normal mode, and immobilized chains, which contribute to the confinement-induced mode via fluctuations of their terminal subchains.

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... 

As expected, the size of the ideal chain along the contour of the tube is not affected by the confinement. This is an important property of an ideal chain. Deformation of the ideal chain in one direction does not affect its properties in the other directions because each coordinate s random walk is independent. 

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. 

Earlier in this chapter, we have discussed the entropic cost due to confinement of an ideal chain into a cylindrical tube or in a slit between two parallel walls. A similar entropic penalty has to be paid if a chain is confined within a spherical cavity of size R < bN, Each compression blob corresponds to a random walk that fills the cavity. Thus, the number of monomers in each compression blob is determined by ideal chain statistics within the blob ... 

The Njg compression blobs of the ideal chain fully overlap for a chain confined in a spherical pore. The free energy cost of confinement within the spherical cavity is of the order of the thermal energy kT per compression blob ... 

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 ... 

Up to this point we have confined ourselves to ideally flexible chains. Thus, the theories developed on the models of such chains (for example, the spring-bead chain) should no longer be adequate for polymers whose chemical stmcture suggests considerable stiffness of the chain backbone. Many cheiin models may be used to formulate a theory of stiff or semi-flexible polymers in solution, but the most frequently adopted is the wormlike chain mentioned in Section 1.3 of Chapter 1 it is sometimes called the KP chain. This physical model was introduced long ago by Kratky and Porod [1] to represent cellulosic polymers. However, significant progress in the study of its dilute solution properties, static and dynamic, has occurred in the last two decades. 

In branching-chain reactions the activity of an inhibitor may be charaeterized by its eapability to shift the explosion limit. It should be noted that when stabilizing, for example, gas mixtures, in practice apart from the time of reaction retardation it is important for the inhibitor to ensure the reliable explosion safety of the mixture. Therefore, here the inhibitor activity characterizes to a certain degree its efficiency as an anti-igniting additive, as well. For this case let us confine ourselves to the ideal inhibition and consider the lower e qjlosion... 

Fig. 2.9 An ideal chain confined between two parallel flat plates...

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. 

We start from a melt of chains and confine it in a tube of diameter D. When D is large, we are dealing with a three-dimensional system, and we know from Chapter II that the chains are ideal, with a size Ro = N a. Let us now decrease D at fixed N, and reach the situation where D < Rq- Each chain is then confined to a linear dimension D for directions normal to the tube axis. Along the tube axis, the chain spans a certain length / y. We shall now define two essential parameters controlling the chain conformation. 

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