ABSTRACT
“Ideal chain” (random walk) of N steps on a lattice can cross itself and go on itself, thus satisfying only the basic condition of a polymer – the chain connectivity. The global properties of an ideal chain can be solved analytically; for example, the growth of the end-to-end distance, R with N is ~N 1/2, where ν = 1/2 is a critical exponent, and the distribution of R, P Gauss(R), is approximately, Gaussian. This behavior characterizes any chain with short-range interactions, such as the freely jointed chain; calculation of the partition function of an ideal chain is trivial. These parameters (and others) constitute the basis for understanding more complex macromolecules. For a d = 1 chain, many properties can be calculated analytically using a slew of alternative methods. Thus, the Gaussian distribution, P Gauss(R), can be obtained from both “the most probable term method” and the central limit theorem. The entropic force required to hold the chain at R is calculated by the most probable term, in two different ways within the framework of the “Gibbs ensemble,” and from P Gauss(R) itself. This arsenal of methods demonstrates the versatility and flexibility in solving problems in statistical mechanics. In spite of its simplicity, an ideal chain describes correctly a polymer in a dense many-chain system and a real chain in the θ-point (see Chapter 9).
