On a conjecture about enumerating (2+2)(2+2)-free posets

Recently, Kitaev and Remmel posed a conjecture concerning the generating function for the number of unlabeled $(2+2)$-free posets with respect to number of elements and number of minimal elements. In this paper, we present a combinatorial proof of this conjecture

Anti-Swarming: Structure and Dynamics of Repulsive Chemically Active Particles

Chemically active Brownian particles with surface catalytic reactions may repel each other due to diffusiophoretic interactions in the reaction and product concentration fields. The system behavior can be described by a `chemical' coupling parameter $\Gamma_c$ that compares the strength of diffusiophoretic repulsion to Brownian motion, and by a mapping to the classical electrostatic One Component Plasma (OCP) system. When confined to a constant-volume domain, Body-Centered Cubic crystals spontaneously form from random initial configurations when the repulsion is strong enough to overcome Brownian motion. Face-Centered Cubic crystals may also be stable. The `melting point' of the `liquid-to-crystal transition' occurs at $\Gamma_c\approx140$ for both BCC and FCC lattices

Anisotropic swim stress in active matter with nematic order

Active Brownian Particles (ABPs) transmit a swim pressure $\Pi^{swim}=n\zeta D^{swim}$ to the container boundaries, where $\zeta$ is the drag coefficient, $D^{swim}$ is the swim diffusivity and $n$ is the uniform bulk number density far from the container walls. In this work we extend the notion of the isotropic swim pressure to the anisotropic tensorial swim stress $\mathbf{\sigma}^{swim} = - n \zeta \mathbf{D}^{swim}$, which is related to the anisotropic swim diffusivity $\mathbf{D}^{swim}$. We demonstrate this relationship with ABPs that achieve nematic orientational order via a bulk external field. The anisotropic swim stress is obtained analytically for dilute ABPs in both 2D and 3D systems, and the anisotropy is shown to grow exponentially with the strength of the external field. We verify that the normal component of the anisotropic swim stress applies a pressure $\Pi^{swim}=-(\mathbf{\sigma}^{swim}\cdot\mathbf{n})\cdot\mathbf{n}$ on a wall with normal vector $\mathbf{n}$, and, through Brownian dynamics simulations, this pressure is shown to be the force per unit area transmitted by the active particles. Since ABPs have no friction with a wall, the difference between the normal and tangential stress components -- the normal stress difference -- generates a net flow of ABPs along the wall, which is a generic property of active matter systems

The curved kinetic boundary layer of active matter

The finite reorient-time of swimmers leads to a finite run length $\ell$ and the kinetic accumulation boundary layer on the microscopic length scale $\delta$ on a non-penetrating wall. That boundary layer is the microscopic origin of the swim pressure, and is impacted by the geometry of the boundary [Yan \& Brady, \textit{J. Fluid. Mech.}, 2015, \textbf{785}, R1]. In this work we extend the analysis to analytically solve the boundary layer on an arbitrary-shaped body distorted by the local mean curvature. The solution gives the swim pressure distribution and the total force (torque) on an arbitrarily shaped body immersed in swimmers, with a general scaling of the curvature effect $\Pi^{swim}\sim\lambda\delta^2/L$

Growth mechanism of nanostructured superparamagnetic rods obtained by electrostatic co-assembly

We report on the growth of nanostructured rods fabricated by electrostatic co-assembly between iron oxide nanoparticles and polymers. The nanoparticles put under scrutiny, {\gamma}-Fe2O3 or maghemite, have diameter of 6.7 nm and 8.3 nm and narrow polydispersity. The co-assembly is driven by i) the electrostatic interactions between the polymers and the particles, and by ii) the presence of an externally applied magnetic field. The rods are characterized by large anisotropy factors, with diameter 200 nm and length comprised between 1 and 100 {\mu}m. In the present work, we provide for the first time the morphology diagram for the rods as a function of ionic strength and concentration. We show the existence of a critical nanoparticle concentration and of a critical ionic strength beyond which the rods do not form. In the intermediate regimes, only tortuous and branched aggregates are detected. At higher concentrations and lower ionic strengths, linear and stiff rods with superparamagnetic properties are produced. Based on these data, a mechanism for the rod formation is proposed. The mechanism proceeds in two steps : the formation and growth of spherical clusters of particles, and the alignment of the clusters induced by the magnetic dipolar interactions. As far as the kinetics of these processes is concerned, the clusters growth and their alignment occur concomitantly, leading to a continuous accretion of particles or small clusters, and a welding of the rodlike structure.Comment: 15 pages, 10 figures, one tabl