Due to their unique structural and mechanical properties, randomly cross-linked polymer networks play an important role in many different fields, ranging from cellular biology to industrial processes. In order to elucidate how these properties are controlled by the physical details of the network (e.g., chain-length and end-to-end distributions), we generate disordered phantom networks with different cross-linker concentrations C and initial densities ρ init and evaluate their elastic properties. We find that the shear modulus computed at the same strand concentration for networks with the same C, which determines the number of chains and the chain-length distribution, depends strongly on the preparation protocol of the network, here controlled by ρ init. We rationalize this dependence by employing a generic stress−strain relation for polymer networks that does not rely on the specific form of the polymer end-to-end distance distribution. We find that the shear modulus of the networks is a nonmonotonic function of the density of elastically active strands, and that this behavior has a purely entropic origin. Our results show that if short chains are abundant, as it is always the case for randomly cross-linked polymer networks, the knowledge of the exact chain conformation distribution is essential for correctly predicting the elastic properties. Finally, we apply our theoretical approach to literature experimental data, qualitatively confirming our interpretations.

Modelling of multi-wavelength metasurfaces relies on adjusting the phase of indi-vidual nanoresonators at several wavelengths.The traditional procedure neglects thenear-field coupling between the nanoresonators, which dramatically reduces the over-all diffraction efficiency, bandwidth, numerical aperture and device diameter.Anotheralternative design strategy is to combine a numerical optimization technique with full-wave simulations to mitigate this problem and optimize the entire metasurface at once.Here, we present a global multiobjective optimization technique that utilizes statisticallearning method to optimize RGB spherical metalenses at the visible wavelengths. Theoptimization procedure, coupled to a high-order full-wave solver, accounts for the nearfield coupling between the resonators. High numerical aperture RGB lenses(NA= 0.47and NA= 0.56) of 8μm and 10μm diameters are optimized with numerical average1focusing efficiencies of 55% and 45%, with an average focusing error smaller than 6%for the RGB colors. The fabricated and experimentally characterized devices present44.16% and 31.5% respective efficiencies. The reported performances represent thehighest focusing efficiencies for highNA >0.5 RGB metalenses obtained so far. Theintegration of multi-wavelength metasurfaces in portable and wearable electronic de-vices requires high performances to offer a variety of applications ranging from classicalimaging to virtual and augmented reality.

We clarify some analytical expressions existing in the literature, • Within the correct formulae we conclude that there is no need for an ad hoc improvement on the opposite to the title paper, • We highlight the symmetry properties of the function to be integrated in order to agree with the usual assumptions made to derive the Kramers-Kronig relations, • The analytical formula we provide may be used to increase the accuracy of the "Poor Man's Kramers-Kronig analysis" method and the "Kramers-Kronig constrained variational analysis" method.

We use ab initio molecular dynamics simulations to investigate the properties of the dry surface of pure silicaand sodium silicate glasses. The surface layers are defined based on the atomic distributions along the direction(z direction) perpendicular to the surfaces. We show that these surfaces have a higher concentration of danglingbonds as well as two-membered (2M) rings than the bulk samples. Increasing the concentration of Na 2 O reducesthe proportion of structural defects. From the vibrational density of states, one concludes that 2M rings have aunique vibrational signature at a frequency ≈850 cm −1 , compatible with experimental findings. We also findthat, due to the presence of surfaces, the atomic vibration in the z direction is softer than for the two otherdirections. The electronic density of states shows clearly the differences between the surface and interior and wecan attribute these to specific structural units. Finally, the analysis of the electron localization function allows toget insight on the influence of local structure and the presence of Na on the nature of chemical bonding in theglasses.

The ParABS system is composed of three components: a centromeric sequence, parS, and two proteins, the ParA (ATPase) and the ParB DNA binding proteins. Hundreds of ParB proteins assemble dynamically to form nucleoprotein parS-anchored complexes (ParBS) that serve as substrates for ParA molecules to catalyze positioning and segregation events. We address the theoretical modeling of ParBS with a Lattice Gas models and compare it to experiments [1]. We give evidences that (i) ParBS is formed via a Liquid-Liquid Phase Separation in the metastable region of the phase diagram and (ii) this nucleation is catalyzed with the parS sequence [1].[1] Guilhas, Walter,... & Nollmann (2020). ATP-driven separation of liquid phase condensates in bacteria. Mol. Cell, 79, 293-303.

Recently T. Bridgeland defined a complex hyperk\"ahler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants. This metric is encoded in a function $W(z,\theta)$ satisfying a heavenly equation, or a potential $F(z,\theta)$ satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both $W$ and $F$ in terms of solutions of that system. These expressions are recognized as conformal limits of the `instanton generating potential' and `contact potential' appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce's original construction of $F$ as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star-product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called $\tau$ function for arbitrary values of the fiber coordinates $\theta$, in terms of a suitable two-variable generalization of Barnes' $G$ function.