We have explored the phase behaviour of a simple model for thermoresponsive supramolecular rods mixed with discotic particles. Possessing attractive tips the rod monomers reversibly associate into polymers that retain their basic slender rod shape and experience only a limited degree of backbone flexibility. The interaction between the species is assumed to be of steric origin such that basic shape differences between the constituents, more specifically the excluded-volume disparity, plays a key role in determining the prevailing liquid crystal symmetry. The principal ones are a polymer nematic ($N^{+}$) composed of nematic polymer interspersed with an anti-nematic organization of discs and a discotic nematic ($N^{-}$) in which the polymers are dispersed anti-nematically. Lowering temperature stimulates polymer growth which enlarges the stability window for the $N^{+}$ phase. The phase diagram develops a marked {\em azeotrope} upon increasing the mole fraction of added discs which indicated that the polymer nematic is stabilized by the addition of non-adsorbing rigid discs provided their mole fraction remains small.
The polymer-dominated nematic phase eventually becomes destabilized at larger mole fractions where mutual disc alignment disrupts the nematic order of the polymers in favour of the formation of a discotic nematic phase in which the polymers self-organize into an anti-nematic structure. The corresponding molecular weight distribution functions strongly deviates from the usual exponential form and becomes non-monotonic with a maximum probability associated with oligomeric aggregates. Enhancing the shape-asymmetry between the rod monomers and discs we observe a depletion-driven demixing of the isotropic fluid which opens up the possibility of a quadruple existence featuring two isotropic phase along with the fractionated polymer and discotic nematic phases. Such quadruple points occur in a wide range of mixed-shape nematics involving supramolecular rods templated by discs and highlight the possibility of multiple liquid symmetries (both isotropic and anisotropic) coexisting in mixtures of anisotropic colloids with reversible and thermoresponsive shape-asymmetry without cohesive interparticle forces. Future explorations should aim at a more careful assessment of biaxial nematic order, ignored in the present study, which could develop in near-equimolar rod-disc mixtures provided they are stable against global demixing (see Appendix B for tentative discussion). Polymerizing rods and discs with finite particle thickness and low shape asymmetry may favor the emergence of liquid crystals possessing lamellar, columnar or fully crystalline signatures \cite{peroukidis2010} which may be addressed using computer simulation models along the lines of Refs. \cite{kuriabova2010,nguyen2014,perouklapp2020}. Inspiration for such mixed-shape lamellar structures could be drawn from bio-inspired supramolecular liquid crystals \cite{safinya2013} such as, for example, the `sliding columnar phase' and similar stacked architectures observed in cationic lyposome-DNA complexes \cite{wong2000,ohern1998} which are essentially made up of mixed planar and rod-shaped architectures.
% TODO: transfer appendix A to general appendices
\section*{Appendix A: Renormalized ${\mathcal P}_{2}$ approximation for slightly flexible polymers}
We seek a simple perturbation theory for the one-body density \eq{lnrhor} of near-rigid polymers characterized by a finite persistence length $\ell_{p}$. Let us attempt the following generalization of the probability density distribution for the polymers:
https://github.com/mtorresl-works/TORRES_thesis_manuscript/blob/1d341cdeb3ab1b758f02ec20b5e65a6348ac53d6/chapters/chapter-2.tex#L453