Single chain conformations 2 Ideal chains

Several experimental parameters have been used to describe the conformation of a polymer adsorbed at the solid-solution interface these include the thickness of the adsorbed layer (photon correlation spectroscopy(J ) (p.c.s.), small angle neutron scattering (2) (s.a.n.s.), ellipsometry (3) and force-distance measurements between adsorbed layers (A), and the surface bound fraction (e.s.r. (5), n.m.r. ( 6), calorimetry (7) and i.r. (8)). However, it is very difficult to describe the adsorbed layer with a single parameter and ideally the segment density profile of the adsorbed chain is required. Recently s.a.n.s. (9) has been used to obtain segment density profiles for polyethylene oxide (PEO) and partially hydrolysed polyvinyl alcohol adsorbed on polystyrene latex. For PEO, two types of system were examined one where the chains were terminally-anchored and the other where the polymer was physically adsorbed from solution. The profiles for these two... 

Polymers are random fractals, quite different from Koch curves and Sierpinski gaskets, which are examples of regular fractals. Consider, for example, a single conformation of an ideal chain, shown in Fig. 1.14. As will be discussed in detail in Chapter 2, the mean-square end-to-end distance of an ideal chain is proportional to its degree of polymerization. 

In Chapter 2, we studied the conformations of an ideal chain that ignore interactions between monomers separated by many bonds along the chain. In this chapter we study the effect of these interactions on polymer conformations. To understand why these interactions are often important, we need to estimate the number of monomer monomer contacts within a single coil. This number depends on the probability for a given monomer to encounter any other monomer that is separated from it by many bonds along the polymer. 

Both theories of single-chain adsorption, described above, ignore a very important effect—the loss of conformational entropy of a trand due to its proximity to the impenetrable surface. Each adsorption blob has jb contacts with the surface and each strand of the chain near these contacts loses conformational entropy due to the proximity effect. In order to overcome this entropic penalty, the chain must gain finite energy E er per contact between a monomer and the surface. This critical energy Ecr corresponds to the adsorption transition. For ideal chains Ecr A E. The small additional free energy gain per contact kT6 should be considered in excess of the critical value Ecr,... 

In an ideal model of single chains, the chain conformation can be treated in analogy to a trajectory of random walks in a lattice space. For random walks of certain steps, the total amount of possible paths is (f, where q is the coordination number of the lattice, and n the number of steps along the walk path. For a real single chain containing only the volume exclusion, the proper analogy becomes a self-avoiding walk. The mathematical treatment to SAW turns out to be a big... 

For the three-dimensional self-avoiding walks, the critical exponent of the polymer coil is 3/5, which is larger than the critical exponent of the ideal chain (1/2). This implies that the volume exclusion of the polymer chain leads to coil expansion. Such an expansion makes chain conformation deviate from its most probable state, causing a recovery force originated from the conformational entropy. Therefore, the single coil could not expand unlimitedly, and there exists a thermodynamic balance between the energy gain of volume exclusion and the entropy loss of chain conformation. 

The main asstrmptions used concern the Gaussian character of the chains and the absence of restrictions imposed by otber chains to tbe conformation of a given chain. They are based on the Flory theorem, which states that the statistical properties of polymer chains in a dense system are equivalent to those for single ideal chains. The reason is that in a imifoim, amorphous substance all the conformations of a certain chain are equally likely in a sense that they couespond to the same energy of interaction with other chains, because the surrotmdings of each emit are roughly the same. 

Calculations based on the RIS-model now exist for the majority of common polymers, thus providing a quantitative representation of the energetic and structural properties of single macromolecules. The prerequisite is a knowledge about the energies u((pi i,(pi) associated with the different pairs of conformational states. Information about these values has improved steadily with the number of carefully analyzed experiments. Clearly, the model does not account for the excluded volume interaction, but it provides a microscopic understanding for all situations with ideal chain behavior. 

Because the single macromolecules in the considered structures have a same periodical conformation, for a length of the chains which in the ideal case tends to infinity, the difference between the entropies of an ideally ordered structure and a disordered one is small. 

Self-avoiding random walks (SARW) statistics has been proposed [1] for single that is for non-interacting between themselves ideal polymeric chains (free-articulated Kuhn s chains [2]) into ideal solvents, in which the all-possible configurations of the polymeric chain are energetically equal. From this statistics follows, that under the absence of external forces the conformation of a polymeric chain takes the shape of the Flory ball, the most verisimilar radius Rf of which is described by known expression [3, 4]... 

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