Ordering and association of patchy particles in quasi-one-dimensional channel

TL;DR

This work addresses how confinement and anisotropic patchy interactions control self-assembly and orientational ordering of particles in quasi-one-dimensional channels. It develops an exact theoretical framework by generalizing Wertheim's TPT1 to q1D sticky, anisotropic particles and proves its equivalence to the transfer matrix method (TMM) in the thermodynamic limit, with pressure as a natural driving variable. The authors derive a generalized free-energy density, mass-action laws for unbonded-site fractions, orientational distributions, and the equation of state, demonstrating exactness in the sticky limit for additive hard-core interactions and extending the approach to continuous orientational freedom. The results provide a rigorous basis for predicting self-assembly and ordering in confined patchy systems and offer a pathway to exact DFT formulations and SAFT-like theories for anisotropic particles in narrow geometries.

Abstract

We show that the formalism of Wertheim's first order thermodynamic perturbation theory can be generalised for the fluid of rotating sticky particles with anisotropic hard core confined to a quasi-one-dimensional channel. Using the transfer matrix method, we prove that the theory is exact if the hard body interaction is additive, only the first neighbors interact and the particles can stick together only along the channel. We show that the most convenient treatment of association in narrow channels is to work in NPT ensemble, where all structural and thermodynamic quantities can be expressed as a function of pressure and fraction of sites unbonded.

Figures (3)

• Figure 1: An example of q1D systems of anisotropic sticky hard particles. The particle has 6 possible orientations and bonding sites. The orientational unit vector is denoted by thick brown arrow, which is in a 2D plane in this example but can be arbitrary in 3D in general. The bonding sites are denoted by blue circles. A trimer (3 particles are in bond) is also shown with filled circles of binding pair of sites, numbered with (4,2) and (5,3), respectively. The numbering of orientations and sites is harmonised such a way that in orientation $i$, sites ${\text{i}}$ (on the right) and ${\bar{\mathrm{\text{\i}}}}$ (on the left) are allowed to form bond. All the relevant information about the shape of the particle bounded by the red curve is contained in the $\sigma_i$ radii (shown by the blue skeleton), because they determine the contact distance. The intersection of the blue lines is considered as the position of the particle, which is constrained to the $x$ axis.
• Figure 2: Boltzmann factor of associating hard particles as a function of $x'-x$, where $x$ ($x'$) is the position of the particle with orientation $i$ ($j$). The interaction is hard repulsive in the overlapping region, while the interaction between bonding sites is square-well attraction in the regions with $\delta$ length. The area of the right (left) shaded regions is forced to be $\gamma^{{\text{i}}{\bar{\mathrm{\text{\j}}}}}$ ($\gamma^{{\text{j}}{\bar{\mathrm{\text{\i}}}}}$) in the sticky limit: $\delta\rightarrow 0$, $\varepsilon^{{\text{i}}{\bar{\mathrm{\text{\j}}}}}\rightarrow\infty$, and $\varepsilon^{{\text{j}}{\bar{\mathrm{\text{\i}}}}}\rightarrow\infty$.
• Figure 3: Some orientationally discrete (cube and rectangular board), partially discrete (cylinder) and continuous (2D disk and sphere) hard bodies with some patches whose q1D systems can be studied exactly using TMM or generalised TPT1.