1D Classical Density Functional Theory of a Tethered Polymer Layer

A simple application of classical density functional theory is derived and applied to a system of polymers grafted to a plane. The system is assumed to have symmetry in directions parallel to the grafting plane hence it being a '1-dimensional' problem. A quick introduction to using propagators in the theory of chains in the presence of external fields and a numerical algorithm for solving the modified diffusion (or Fokker-Planck) equation are presented. Density profiles are found for hard-sphere chains which agree with results in the literature (see Murat,.., Milner,.., Muhukumar and others) and align with physical expectations of brush formation due to excluded volume. The linear scaling $h \approx (\sigma)^{1/3} N$, of brush height with polymerisation $N$, is reproduced here.