On the Scaling Behavior of the Force/Extension Relation of a Chain

Applying an extending force F along the end-to-end vector r of a chain enlarges the initial size ρ_i ∼ |r_i| leading to a final state with ρ_f larger than ρ_i. Assuming a power law dependence of the size ρ ∼ N^ν of the chain on its length N, at the two different states with different exponents ν_i and ν_f, a scaling relationship is derived between the measure of the extending force F and the extension ρ of the chain. The exponent γ of the force/extension relation, ρ ∼ F^γ, depends on both exponents ν_i and ν_f of the initial and the final states. A relation between γ and the exponents ν_i and ν_f is derived which permits the explanation of previous results and predicts some more. The scaling behavior is checked with the exactly soluble model of a random walk under a force.