The self-avoiding walk (SAW) model discussed in Chapter 9 is much more realistic than an ideal chain, as it takes into account the excluded volume (EV) interaction. However, due to complexity, none of its parameters (e.g., the partition function) has a strict analytical solution. For SAWs, the exponent ν (which describes the expansion of the end-to-end distance, R, R ~ N ν) depends on the dimension d and the effect of attractive interactions (other critical exponents have a similar dependency). Thus, for SAWs with both EV and attractive interactions, the following three regimes of the chain global shape are characterized by ν: (1) the good solvent region (T > θ), where the EV repulsions are dominant and the chain is thus open (ν ~ 0.588). As the temperature is lowered to the Flory θ-temperature, the attractions increase to a point that they cancel the repulsions, and in d = 3, a long chain behaves to a large extent as a random walk (ν = 1/2). For T < θ, the attractions are dominant and the chain collapses (ν = 1/3) – corresponding to a folded protein. The cross-over behavior around θ is described in detail. Various exponents for SAWs in different dimensions are discussed including their relation to the n-vector model. These three phases are typical for realistic macromolecules.