The Wertheim first-order thermodynamic perturbation theory (WTPT1) is generalized to obtain exact results for quasi-one-dimensional associating fluids, where the particles are allowed to move freely along a line and rotate in three dimensions. In this formalism, both the orientational distribution function and the fractions of sites not bonded are orientation-dependent functions determined by the minimization of the free energy. The theory is exact for a one-dimensional fluid of hard spheres decorated with n sticky sites, provided that each site is involved in a maximum of one bond. The reliability of the theory for non-sticky sites is examined with the help of the transfer operator method, by comparing the predictions for hard spheres decorated with one and two conical square-well sites. It is found that the generalized WTPT1 underestimates the orientational order and the fraction of bonded particles. Interestingly, the phase behavior of hard spheres decorated with sticky and non-sticky sites is very different, especially near close packing, as reflected in the pressure, the orientational order parameter, and the fraction of monomers.
Gurin et al. (Mon,) studied this question.