Polymer polyelectrolyte single chains,

Amphiphilic molecules contain a polar and an apolar part. As a result, such molecules have an ambiguous (amphi) affinity (philos) for water. The apolar parts behave hydrophobically the water molecules tend to escape from contact with these parts. The polar parts are hydrophilic. They interact favorably with water. The consequence of the amphiphilic character is that the molecules are preferably located at interfaces with water, where the polar parts are exposed to the aqueous phase and the apolar parts to the nonaqueous phase. Low-molecular-weight, amphiphilic compounds are often called surfactants. Well-known examples of surfactants are the classical soaps (single-chain fatty acids), phospholipids, cholesterol, bile acids, lung surfactant, and so on. In Figure 7.1, the chemical structures showing the polar and apolar parts of some of these surfactants are given. Monolayers may also be formed by polymers, polyelectrolytes, and proteins that contain polar and apolar parts. 

Let us consider a single chain of N monomers in volume V. Each monomer is monovalently charged, and i is the distance between two successive monomers along the polymer. Due to the electroneutrality condition, there are N monovalent counterions. Let M be the number of counterions adsorbed on the polyelectrolyte so that M/N is the degree of counterion adsorption and a = 1 — M/N) is the degree of ionization of the polyelectrolyte. In addition. 

In the presence of interactions between the connected segments of a single chain, aforementioned simple diffusion or random walks get affected and the walks are no more random. However, the intricate coupling of the different components such as monomers, solvent, or small ions in the case of polyelectrolytes via the interaction potentials complicates the theoretical analysis. In order to decouple different components, the conformations of the chain can be envisioned as the walks in the presence of fields, which arise solely due to the fact that there are interactions present in the system. This physical argument is the basis of the use ofcertain field theoretical transformations such as Hubbard-Stratonovich [60] transformation, which is well known in the field theory. So, the conformational characteristics of a polymer chain in the presence of different kinds of intrachain interactions can be described once the fields are known. In general, an exact computation of these fields is almost an impossible task. That is the reason theoretical developments resort to certain approximations for computing these fields, which work well for most of the practical purposes. Once these fields are known, the physical properties can be described in terms of these fields. It was shown by Edwards [50] that the similar analysis can be carried out for systems with many chains, where interchain interactions also affect the properties in addition to intrachain interactions. 

The next subsections describe the properties of the pearl-necklace structure and the elongation of the pearl-necklace polymer chain by an external force. We will then present numerical simulations of single polyelectrolyte chains in a poor solvent. 

---