KratkyPorod

The Kratky-Porod model, commonly referred to as the worm-like chain model, is a statistical mechanical description of semiflexible polymers. It treats a polymer as a continuous space curve r(s) parameterized by contour length, with a bending energy that increases with curvature. The model was introduced by G. Kratky and O. Porod in 1949 to explain scattering from biopolymers and synthetic polymers.

A central concept is the persistence length, l_p, which measures the stiffness of the chain, and the contour length, L, the total length of the polymer. The tangent vectors along the chain have an exponential decay of correlations: ⟨t(s)·t(0)⟩ = e^{-s/l_p}. The mean square end-to-end distance is given by ⟨R^2⟩ = 2 L l_p [1 - (l_p/L)(1 - e^{-L/l_p})], describing how the chain length and stiffness determine its overall size.

The model describes different regimes: when L is much less than l_p, the chain behaves like a rigid rod with ⟨R^2⟩ ≈ L^2; when L is much greater than l_p, it behaves as a flexible coil with ⟨R^2⟩ ≈ 2 L l_p. It provides form factors that are used to interpret scattering data, such as small-angle X-ray scattering (SAXS), small-angle neutron scattering (SANS), and light scattering, for polymers and biopolymers including DNA and actin filaments.

Limitations include neglecting excluded-volume effects and hydrodynamics, which can be important in good solvents. Extensions and numerical methods exist to apply the model to complex polymers and to fit experimental data more accurately.

---