Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from Triangular Lattice Models

Interfacial self-assembly of SiO2-PNIPAM core-shell particles with varied crosslinking density

Spherical particles confined to liquid interfaces generally self-assemble into hexagonal patterns. It was theoretically predicted by Jagla two decades ago that such particles interacting via a soft repulsive potential are able to form complex, anisotropic assembly phases. Depending on the shape and range of the potential, the predicted minimum energy configurations include chains, rhomboid and square phases. We recently demonstrated that deformable core-shell particles consisting of a hard silica core and a soft poly(N-isopropylacrylamide) shell adsorbed at an air/water interface can form chain phases if the crosslinker is primarily incorporated around the silica core. Here, we systematically investigate the interfacial self-assembly behavior of such SiO₂-PNIPAM core-shell particles as a function of crosslinker content and core size. We observe chain networks predominantly at low crosslinking densities and smaller core sizes, whereas higher crosslinking densities lead to the formation of rhomboid packing. We correlate these results with the interfacial morphologies of the different particle systems, where the ability to expand at the interface and form a thin corona at the periphery depends on the degree of crosslinking close to the core. We perform minimum energy calculations based on Jagla-type pair potentials with different shapes of the soft repulsive shoulder. We compare the theoretical phase diagram with experimental findings to infer to which extent the interfacial interactions of the experimental system may be captured by Jagla pair-wise interaction potentials.

Statistical Mechanics and Thermodynamics of Liquids and Crystals

Thermodynamic phases are the most prominent manifestation of emergent behavior [...].

Vector Arithmetic in the Triangular Grid

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [-1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.