J.-C Loudet

A Choudhury

M Qiu

J J Feng

Particle trapped at the isotropic-nematic liquid crystal interface: Elastocapillary phenomena and drag forces

Keywords: Two-phase flows, particle/fluid flow, phase-field simulations, liquid crystals, elastic distortions, topological defects, interfacial deformations, drag force, elastocapillary phenomena

We present numerical simulations of a particle trapped at the isotropic-nematic liquid crystal (Iso-N) interface. We use our recent model, based on a phase-field approach (see Qiu et al., Phys. Rev. E, 103, 022706 (2021)), to couple the capillary forces acting on the interface with the elastic stresses in the nematic phase along with topological defects. A range of floating configurations are first investigated as a function of the contact angle and various anchoring conditions at the fluid interface. The results show that the response of the system is driven by the existence of an anchoring conflict at the contact line. Substantial particle displacements and/or interfacial deformations may occur in this case even for moderate anchoring strengths. These findings highlight the coupling between elastic and capillary forces.

In a second part, we compute drag forces exerted on a particle that moves along the Iso-N interface for several contact angles and a moderate Ericksen number. Because

Introduction

Colloidal dispersions in bulk liquid crystal (LC) phases, also known as 'LC colloids' [START_REF] Smalyukh | Liquid crystal colloids[END_REF], and colloidal particles attached to (isotropic) fluid interfaces [START_REF] Mcgorty | Colloidal self-assembly at an interface[END_REF] are both very rich and versatile systems on their own in soft matter science. They have been extensively investigated, but in a rather independent way so far. Yet, from a self-assembly perspective, both systems share quite a few similarities although the underlying driving physical mechanisms differ greatly. In both cases, the particles may undergo either attractive or repulsive interactions which originate from the deformations of the fluid matrix they are embedded in. In LC colloids, the so-called elastic interactions result from elastic distortions of the LC matrix [START_REF] Smalyukh | Liquid crystal colloids[END_REF][START_REF] Poulin | Novel colloidal interactions in anisotropic fluids[END_REF][START_REF] Lubensky | Topological defects and interactions in nematic emulsions[END_REF][START_REF] Loudet | Colloidal structures from bulk demixing in liquid crystals[END_REF][START_REF] Muševič | Two-dimensional nematic colloidal crystals self-assembled by topological defects[END_REF][START_REF] Ravnik | Entangled nematic colloidal dimers and wires[END_REF][START_REF] Tkalec | Reconfigurable knots and links in chiral nematic colloids[END_REF][START_REF] Blanc | Ordering nano-and microparticles assemblies with liquid crystals[END_REF][START_REF] Foffano | The dynamics of colloidal inclusions in liquid crystals: a simulation perspective[END_REF], whereas it is the overlap of interfacial deformations that is responsible for capillary interactions occurring between floating particles [START_REF] Mcgorty | Colloidal self-assembly at an interface[END_REF][START_REF] Kralchevsky | Capillary interactions between particles bound to interfaces, liquid films and biomembranes[END_REF][START_REF] Oettel | Capillary interactions at fluid interfaces[END_REF][START_REF] Danov | Capillary forces between particles at a liquid interface: General theoretical approach and interactions between capillary multipoles[END_REF][START_REF] Botto | Capillary interactions between anisotropic particles[END_REF][START_REF] Dasgupta | Nano-and microparticles at fluid and biological interfaces[END_REF][START_REF] Liu | Capillary assembly of colloids: Interactions on planar and curved interfaces[END_REF]. Both elastic and capillary interactions can be long-ranged, anisotropic in nature (e.g., of dipolar or quadrupolar symmetry), and feature energies that greatly exceed the thermal energy k B T . In both cases, a myriad of colloidal structures, either ordered or disordered, have been discovered depending on numerous factors such as the particle size and shape, surface chemistry (e.g., Janus particles), and confinement (see reviews e.g., [START_REF] Smalyukh | Liquid crystal colloids[END_REF][START_REF] Mcgorty | Colloidal self-assembly at an interface[END_REF][START_REF] Blanc | Ordering nano-and microparticles assemblies with liquid crystals[END_REF][START_REF] Dasgupta | Nano-and microparticles at fluid and biological interfaces[END_REF][START_REF] Liu | Capillary assembly of colloids: Interactions on planar and curved interfaces[END_REF]). However, so far only a handful of studies have attempted to bridge the gap between these two lines of research by studying the behavior of colloids trapped at liquid crystal interfaces. Such systems represent a new kind of soft material whose properties are expected to be governed mainly by the interplay of capillary and elastic phenomena.

On the experimental side, a few studies were dedicated to pattern formation, mostly in confined geometries with varying anchoring conditions. For instance, two-dimensional (2D) crystal-like structures and chain-like aggregates were observed at the air-nematic LC (NLC) [START_REF] Smalyukh | Ordered droplet structures at the liquid crystal surface and elastic-capillary interactions[END_REF][START_REF] Nych | Coexistence of two colloidal crystals at the nematic-liquid-crystal-air interface[END_REF][START_REF] Yamamoto | Self-assembled pseudo-hexagonal structures of colloidal particles at air-liquid crystal interface[END_REF][START_REF] Gharbi | Behavior of colloidal particles at a nematic liquid crystal interface[END_REF][START_REF] Wang | Controlled multistep self-assembling of colloidal droplets at a nematic liquid Crystal-air interface[END_REF] and water-NLC interfaces [START_REF] Lin | Ordering of Solid Microparticles at Liquid Crystal-Water Interfaces[END_REF][START_REF] Jr | Chemoresponsive assemblies of microparticles at liquid crystalline interfaces[END_REF]. Estimates of pair interaction potentials were derived for both spherical [START_REF] Smalyukh | Ordered droplet structures at the liquid crystal surface and elastic-capillary interactions[END_REF][START_REF] Gharbi | Behavior of colloidal particles at a nematic liquid crystal interface[END_REF] and nonspherical particles [START_REF] Liu | Elastocapillary interactions on nematic films[END_REF]. At the single particle level, Jeridi et al. [START_REF] Jeridi | Capillary-induced giant elastic dipoles in thin nematic films[END_REF][START_REF] Jeridi | Colloidal particles in thin nematic wetting films[END_REF] reported capillary-induced giant elastic dipoles in thin nematic wetting films, whereas the influence of interfacial curvature was addressed by Gharbi et al. [START_REF] Gharbi | Microparticles confined to a nematic liquid crystal shell[END_REF] by placing particles on NLC shells. More dynamical aspects such as the Brownian diffusion of micro-and nano-particles at the nematic-aqueous phase interface were first investigated by Abras et al. [START_REF] Abras | The mobilities of micro-and nano-particles at interfaces of nematic liquid crystals[END_REF], whereas the vibrational phonon modes of 2D crystalline packings of particles at the air-NLC interface were discussed by Wei et al. [START_REF] Wei | Dynamics of ordered colloidal particle monolayers at nematic liquid crystal interfaces[END_REF].

Despite the above discoveries, there is a lack of theoretical understanding of the observed phenomena. Only a few modelling studies have appeared on the subject. Tasinkevych & Andrienko performed the first numerical simulations dealing with particles trapped at an isotropic-nematic (Iso-N) interface in both 2D [START_REF] Andrienko | Effective pair interactions between colloidal particles at a nematic-isotropic interface[END_REF] and 3D [START_REF] Tasinkevych | Colloidal particles in liquid crystal films and at interfaces[END_REF]. However, they considered a peculiar situation where the interface separates the same mesogenic material in two phases coexisting under the same condition. No interfacial forces nor contact line were taken into account in their analysis. Nevertheless, pair interaction potentials exhibited a rather complex behavior due, in part, to the creation, annihilation and interaction of topological defects in the vicinity of the particles.

Oettel et al. [START_REF] Oettel | Effective interactions of colloids on nematic films[END_REF] considered a more complete physical picture and provided approximate analytical calculations for interacting colloids attached to a nematic free surface.

The primary goal was to rationalize the early observations of Smalyukh et al. [START_REF] Smalyukh | Ordered droplet structures at the liquid crystal surface and elastic-capillary interactions[END_REF], where hexagonal lattices of glycerol droplets were initially thought to result from a competition between an elastic repulsion and a capillary attraction. However, the theoretical analysis of Oettel et al. [START_REF] Oettel | Effective interactions of colloids on nematic films[END_REF] showed that such a competition is unlikely, at least within the model assumptions and the experimental conditions in [START_REF] Smalyukh | Ordered droplet structures at the liquid crystal surface and elastic-capillary interactions[END_REF]. Till today, this remains an open question, and we are not aware of any other detailed theoretical or numerical investigations dealing with these issues.

A possible reason for this is that the modelling of particles adsorbed at LC interfaces is a challenging task. The main difficulty arises from the presence of the interface which couples capillary and elastic phenomena. Indeed, elastic distortions, topological defects, the anchoring of LC molecules on a potentially moving and deformed fluid interface, the contact line dynamics and the particles' motion are all intertwined here. The combination of these phenomena makes these systems very rich but complicates enormously their theoretical treatment.

Recently, our group developed a model that may shed new light on the above problems.

The model, based on a phase-field (PF) method, was originally designed to describe elastocapillary flows of LCs [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]. In the present work, we use this model in numerical simulations to explore both the static and the dynamic behaviors of particles trapped at LC interfaces. In view of the dearth of theoretical and numerical studies in the literature, we will focus on 'basic' scientific questions at the single particle level. For instance, how do the LC elasticity and topological defects influence the partial wetting configuration of a particle adsorbed at a LC interface? How do capillarity and elasticity interact? How does interfacial deformation depend on the two factors? What is the drag force exerted on a particle straddling a LC interface? Answers to these questions will form the basis for studying more complex situations involving, e.g., a collection of particles (interactions), confined geometries (thin films), and curved interfaces.

The paper is organized as follows. The model is presented in Sec. 2. It is numerically solved with a finite element method in a 2D planar geometry. In Sec. 3, we first consider the equilibrium configuration of a partially wetting particle on the Iso-N interface, subject to different contact angles and anchoring conditions (Sec. 3.1). Then we study the drag forces for a range of particle positions across the interface (Sec. 3.2). Our results show that wetting configurations are primarily influenced by the existence of an anchoring conflict at the contact line (CL). Sizeable particle displacement and/or interfacial distortions may occur in this case even for moderate anchoring strengths. The drag force computations reveal linear force-velocity laws. The defect escapes from the particle and merges with the interface in certain circumstances. Furthermore, the drag coefficient at the Iso-N interface is almost always greater than its isotropic counterpart thanks to the coupling between the flow and the order parameter fields. We discuss our results within the framework of existing theories before concluding the paper in Sec. 4.

Theoretical model and numerical method

As aforementioned, we have adapted the recent model developed by Qiu et al. [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF][START_REF] Qiu | Computational studies on interfacial dynamics in complex fluids[END_REF] to describe the behavior of a colloidal particle trapped at the Iso-N fluid interface. The model of Qiu et al. [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF] uses a phase-field (PF) method to account for elastocapillary flows occurring at the Iso-N interface. It has two key features: (i) A tensor order parameter Q that provides a consistent description of the molecular and distortional elasticity of the N phase, including topological defects, and (ii) a PF formalism that accurately represents the Iso-N interfacial tension and the nematic anchoring stress by approximating a sharpinterface limit. Combining this with the equations of motion of the fluids, we obtain a model capable of describing the phenomena resulting from the coupling between capillarity and elasticity. All the details and validation examples of this model can be found in [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF][START_REF] Qiu | Computational studies on interfacial dynamics in complex fluids[END_REF]. In the following, we only give a brief account of the main ingredients. A notable difference between the present work and that of [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF][START_REF] Qiu | Computational studies on interfacial dynamics in complex fluids[END_REF] is the presence of a contact line where the three phases isotropic (Iso), nematic (N), and solid (S) meet. PF-based models can naturally handle the contact line dynamics thanks to intrinsic diffusive processes [START_REF] Seppecher | Moving contact lines in the Cahn-Hilliard theory[END_REF][START_REF] Jacqmin | Contact-line dynamics of a diffuse fluid interface[END_REF][START_REF] Qian | Molecular hydrodynamics of the moving contact line in two-phase immiscible flows[END_REF][START_REF] Yue | Sharp-interface limit of the Cahn-Hilliard model for moving contact lines[END_REF][START_REF] Wörner | Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications[END_REF]. Unless otherwise stated, parameter values for the model described in this section are listed in appendix A.

Free energies

The total free energy of the fluid system consists of the bulk elastic energy of the N phase, f b , the mixing energy of the Iso-N interface, f m , and the anchoring energy on the Iso-N interface, f a . Note that throughout this work, we assume an infinitely strong anchoring condition of the nematic mesogens on the solid particle and, consequently, the anchoring energy at the nematic-solid interface does not appear. f b is standard from the nematic order theory [START_REF] De Gennes | The Physics of liquid crystals[END_REF][START_REF] Kleman | Soft Matter Physics: An Introduction[END_REF] and can be described phenomenologically as [START_REF] Ravnik | Landau-de-Gennes modeling of nematic liquid crystal colloids[END_REF]

f b = A 2 Q ij Q ij + B 3 Q ij Q jk Q ki + C 4 (Q ij Q ij ) 2 + L 1 2 (∂ i Q jk )(∂ i Q jk ) . ( 1
)
Here, A, B, C are material property coefficients (A, B < 0, C > 0), and L 1 is the bulk elastic constant. As in [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF], it is useful to define a scalar order parameter:

q = 3/2∥Q∥ F = 3 2 Q : Q 1/2 , where ∥ • ∥ F is the Frobenius norm.
The mixing energy is standard from the PF model and may be written as [START_REF] Jacqmin | Calculation of two-phase Navier-Stokes flows using phase-field modeling[END_REF][START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF]

f m = λ 2 |∇ϕ| 2 + λ 4ϵ 2 ϕ 2 -1 2 , ( 2
)
where ϕ is the PF variable that marks the different fluids. ϕ = -1 denotes the nematic LC and ϕ = 1 denotes the isotropic phase. These two fluids mix in a thin diffuse interfacial region in which ϕ and all other variables transition smoothly. The Iso-N interface may be defined by the contour level ϕ = 0 . In Eq. ( 2), λ is the (constant) mixing energy density with the dimension of force, and ϵ is the (constant) capillary width governing the thickness of the diffuse interface. In the sharp-interface limit, the mixing energy gives rise to an isotropic surface tension given by σ = 2 √ 2λ/3ϵ [START_REF] Jacqmin | Calculation of two-phase Navier-Stokes flows using phase-field modeling[END_REF][START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF].

To describe the finite-strength anchoring of LC molecules at the Iso-N interface, we employ the following diffuse-interface variant of the Rapini-Papoular anchoring energy density [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]

f a = W 2 |∇ϕ| 4 Q -q e êê - 1 3 I 2 F , ( 3
)
where W is the (constant) anchoring strength, which can be viewed as an anisotropic surface tension [START_REF] De Gennes | The Physics of liquid crystals[END_REF], and ê is a unit vector along the easy direction on the Iso-N interface.

Both homeotropic and planar anchoring conditions are considered for ê with appropriate expressions of f a in each case (see [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF] for details). In Eq. ( 3), q e is the equilibrium scalar order parameter under the uniaxial assumption (q e = 0.81 with our parameters, see appendix A). Note that in Eq. ( 3), W has the dimension of energy×length. In the sharp interface limit, it can be related to the usual anchoring strength, W s (unit: J/m 2 ), via W = 35 8 √ 2 W s ϵ 3 as explained in Qiu et al. [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]. Hence, the total free energy of the fluid system in a domain Ω is given by

F = Ω f (ϕ, Q, ∇ϕ, ∇Q) dV = Ω f m + 1 -ϕ 2 f b + f a dV . ( 4
)
The factor (1 -ϕ)/2 gives the concentration of the N phase.

Governing equations

The governing equations include evolution equations for ϕ and Q , and equations of motion for the fluids and the solid particle.

Evolution equation for ϕ. The classical Cahn-Hilliard (CH) equation describes the evolution of ϕ [START_REF] Jacqmin | Calculation of two-phase Navier-Stokes flows using phase-field modeling[END_REF][START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF]

∂ϕ ∂t + v • ∇ϕ = ∇ • (γ∇µ) , ( 5
)
where v is the fluid velocity and γ the mobility constant. µ = δF m /δϕ is the chemical potential which, in our model, is defined only through the mixing energy, F m = Ω f m dV , i.e. without the contribution from the bulk elastic and anchoring energies. Consequently, the CH diffusion is decoupled from that of Q [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]. Such an approximation holds in the so-called sharp-interface limit and has been adopted in previous works [START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF][START_REF] Yue | Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing[END_REF].

Evolution equation for Q. We choose the Beris-Edwards (BE) formalism to describe the nematic hydrodynamics [START_REF] Beris | Thermodynamics of flowing systems with internal microstructure[END_REF]. Within our PF formulation, the evolution equation for Q in the LC bulk has the same general structure as in the original BE theory [START_REF] Denniston | Lattice Boltzmann simulations of liquid crystal hydrodynamics[END_REF] ∂Q ∂t

+ v • ∇Q = S + ΓH , ( 6
)
where Γ is the (constant) collective rotational diffusion coefficient of the N phase.

H(ϕ, Q, ∇ϕ, ∇Q) is the molecular field tensor, whereas S(∇v, Q) is the corotation tensor.

Both tensors have been modified in our PF method to take into account the anchoring conditions (either homeotropic or planar) at the Iso-N interface. The resulting lengthy expressions are omitted here but can be found in [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF].

Equations of motion for the fluids. Assuming incompressible fluids, we adopt the continuity and Navier-Stokes (NS) equations for the pressure (p) and velocity (v) fields in both phases:

∇ • v = 0 , ( 7
)
ρ(ϕ) ∂v ∂t + v • ∇v = -∇p + ∇ • T + µ∇ϕ , ( 8
)
where the last body force term in Eq. ( 8) (µ∇ϕ) is the diffuse-interface equivalent of the interfacial tension [START_REF] Jacqmin | Calculation of two-phase Navier-Stokes flows using phase-field modeling[END_REF][START_REF] Yue | Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing[END_REF]. ρ(ϕ) is the density of the two-phase system given by:

ρ(ϕ) = 1+ϕ 2 ρ i + 1-ϕ 2 ρ n
, where the subscript i (resp., n) refers to the isotropic (resp., nematic) phase. The total stress tensor T can be written in the following form:

T = -pI + (1 + ϕ)η i D + (1 -ϕ)η n D + T n , ( 9
)
where D = [(∇v) + ∇v] /2 is the rate of deformation tensor. The second term is the viscous stress from the isotropic phase while the third term is a viscous stress of the nematic phase with a constant effective viscosity η n , i.e. independent of the molecular orientation [START_REF] Beris | Thermodynamics of flowing systems with internal microstructure[END_REF][START_REF] Denniston | Lattice Boltzmann simulations of liquid crystal hydrodynamics[END_REF]. The last term (T n ) in Eq. ( 9) is the nematic stress tensor, which has been generalized from the bulk BE theory to take into account the anchoring constraints on the moving Iso-N interface [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]. Note that T n intrinsically contains effective anisotropic viscosities [START_REF] Denniston | Lattice Boltzmann simulations of liquid crystal hydrodynamics[END_REF].

Boundary conditions. For the continuity and NS equations, classical no-slip boundary conditions for v are imposed on all bounding walls and the particle, unless otherwise stated. On the outer boundaries, we need two BCs for the fourth-order CH equation. First, one often requires that there be no diffusive flux across all boundaries (∂Ω): that the interface is flat there but it is free to move up or down.

m • ∇µ| ∂Ω = 0 ,
Concerning the order parameter Q , we consider the following three types of BCs.

(i) On all bounding walls, we apply a homogeneous Neumann BC: m • ∇Q| ∂Ωw = 0 .

(ii) For simplicity, a Dirichlet BC is used on the particle surface: Q| ∂Ωp = Q D , where Q D corresponds to a uniaxial configuration perpendicular to the particle contour, i.e.

homeotropic anchoring, at equilibrium order (q e ). (iii) On the symmetry axis ∂Ω S , we impose the mixed BC [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]:

m • ∇ (Q ii )| ∂Ω S = 0, Q ij | ∂Ω S = 0 (i ̸ = j)
, where ii does not imply the Einstein summation. Finally recall that, at the Iso-N interface, the anchoring conditions are already accounted for in the expressions of H [Eq. ( 6)] and T [Eq. ( 9)].

Equations of motion of the particle. In some cases, the particle attached to the Iso-N interface is allowed to move in the y-direction (Fig. 1). Its translational velocity (U y ) is governed by Newton's equation of motion written here in the absence of gravity (see Sec. 2.3): M Uy = F y , with the initial condition U y | t=0 = 0 , the dot meaning differentiation with respect to time. M is the particle mass and F y is the total force exerted on the particle along the y-direction. F y consists of viscous, elastic, and capillary forces and may be written as

F y = ŷ • ∂Ωp (T + T c ) • m ds , ( 10
)
where ŷ is the unit normal vector along the y-axis and T is the stress tensor defined in Eq. ( 9). T c is the capillary stress tensor and can be derived using a variational procedure [START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF] T c = f mix I -λ∇ϕ∇ϕ , [START_REF] Kralchevsky | Capillary interactions between particles bound to interfaces, liquid films and biomembranes[END_REF] with f mix given by Eq. ( 2). Once U y has been determined, the particle's vertical position (y p ) can be updated by solving: ẏp = U y .

Geometry and parameters

In this section, we specify the parameters and the two geometries employed in our 2D simulations (Fig. 1). The first one, referred to as problem A hereafter, deals with the equilibrium configuration of a partially wetting particle trapped at the Iso-N interface, whereas the second one, referred to as problem B, is dedicated to the computation of drag forces exerted on the particle straddling the Iso-N interface. The small ellipsoids symbolize the LC molecules (not to scale). In all cases, a homeotropic (H) anchoring condition is prescribed at the particle surface, whereas either a homeotropic or planar (P) anchoring is enforced at the Iso-N interface. These two situations are de- 3)] will be considered on the Iso-N interface, while only an infinitely strong, or rigid, homeotropic anchoring is set on the particle surface (cf. "Boundary conditions" in Sec. 2.2). These two situations will be denoted hereafter as the H and P configurations, respectively. For the initial condition, in the former (resp., the latter) case, we require the far-field LC molecules to be oriented along the vertical (resp., horizontal) direction [Fig. 1(b)] with an equilibrium scalar order parameter q e (cf. appendix A). Two other configurations, namely a planar anchoring on the particle surface combined with either a planar or homeotropic anchoring at the interface, have been computed as well.

However, preliminary results indicate that no new qualitative insights emerge from such configurations, and therefore, we have decided not to include such setups in the present work.

The particle's center of mass is either fixed or allowed to move vertically. The contact angle on the particle surface [Fig. 1(a)] is equal to the equilibrium contact angle θ . The Iso-N interface is initially flat and kept horizontal at the left and right bounding walls of the domain in the course of simulations. However, due to the interplay of elastic distortions, surface tension and anchoring, the interface is likely to be deformed and may be displaced from its initial position as we will see below. Furthermore, since we are typically simulating the behavior of a micrometer-sized particle floating at the Iso-N interface (appendix A), the interfacial deformations arising from the particle's buoyant weight are negligible. Indeed, the Bond number Bo ≃ 10 -8 for typical parameter values (see e.g. [START_REF] Kralchevsky | Capillary interactions between particles bound to interfaces, liquid films and biomembranes[END_REF]).

Problem B concerns the drag on a circular particle straddling the Iso-N interface [Fig. 1(c)]. We use the same setup as in our previous work with isotropic fluid interfaces [START_REF] Loudet | Drag force on a particle straddling a fluid interface: Influence of interfacial deformations[END_REF], except that the bottom fluid is now replaced with the nematic LC. Instead of moving the particle horizontally parallel to the interface, we use a reference frame attached to the particle so that far from it, both liquids flow with a constant velocity v ∞ in the direction of the x-axis. In this case, the upper and lower bounding plane walls also move with the same velocity v ∞ from left to right in their own planes. Depending on the contact angle, the interface may be deformed near the particle, but it is kept horizontal at the inlet and outlet in all cases.

As aforementioned, our simulations are run with the base parameters listed in appendix A. When presenting the results, we use dimensionless variables marked by an asterisk. Of particular importance is the dimensionless anchoring energy defined by w * = W s /σ . For w * ≪ 1, the surface tension dominates anchoring effects and controls the shape of the interface by minimizing distortions. Conversely, interfacial deformations are expected for w * 1 , as reported with, e.g., nematic drops immersed in an isotropic fluid [START_REF] Van Bijnen | Texture and shape of two-dimensional domains of nematic liquid crystals[END_REF], or the reverse [START_REF] Lishchuk | A lattice Boltzmann scheme for a nematic-isotropic interface[END_REF]. Below, we will use w * as a measure of the anchoring strength.

Another important dimensionless quantity is the Ericksen number Er, which is relevant to Problem B. Er is defined as the ratio of viscous to elastic forces [START_REF] De Gennes | The Physics of liquid crystals[END_REF][START_REF] Kleman | Soft Matter Physics: An Introduction[END_REF]:

Er = γ 1 v ∞ R/K ,
where γ 1 = 1/Γ is the nematic rotational viscosity and K = q 2 e L 1 the elastic constant (appendix A). Er ≪ 1 (resp., Er ≫ 1) implies that the fluid flow has a negligible (resp., dominant) influence on the elastic distortions of the nematic texture.

Numerical method

The fluid equations, and their associated boundary conditions, together with the particle's equations of motion, are solved numerically until steady state with the finite element computational software COMSOL Multiphysics [START_REF]COMSOL Multiphysics v. 5.4 reference manual[END_REF]. Details of the numerical approximation can be found in [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF] and will not be repeated here. Whenever appropriate, we employ the built-in moving mesh module of COMSOL based on an Arbitrary Lagrangian-Eulerian (ALE) scheme to follow and resolve the particle's vertical motion. The amplitude of this motion is always moderate, i.e. typically 0.5R , and does not incur any remeshing event. As in Qiu et al. [START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF], we design nonuniform triangular meshes fitted with subdomains whose mesh size is adjusted to ensure a sufficient resolution of both the fluid interface [START_REF] Yue | Sharp-interface limit of the Cahn-Hilliard model for moving contact lines[END_REF][START_REF] Wörner | Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications[END_REF][START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF][START_REF] Yue | Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing[END_REF] and the topological defects as they move around. To appropriately resolve the defect core, we use a mesh size h 0.75 l nc , where l nc is the nematic coherence length. Outside of the subdomains, the mesh size is much coarser to save computational time.

Results and discussion

Problem A: floating particle

In problem A, we study how the interplay of surface tension, anchoring conditions and nematic elasticity affects the partial wetting configuration of a solid particle trapped at the Iso-N interface. We will first consider a simple situation with a fixed particle to explore the qualitative physical trends, mainly as a function of the anchoring type (H or P configuration) and strength. In a second step, we shall allow the particle to move vertically. Contact angles will range from 45 • to 135 • .

Fixed particle

We begin with an immobile particle whose center of mass lies in the middle of the box [(x p , y p ) = (L/2, H/2)] and the contact angle is set to 90 • on the particle surface. As explained in Sec. 2.3, the Iso-N interface is initially flat. We examine the influence of the anchoring strength at this interface for both the H and P configurations and we recall that a strong homeotropic anchoring is prescribed at the particle surface (Sec. 2.3). Thus, an anchoring conflict arises in the H configuration at the CL on the particle surface, whereas such a conflict is absent in the P situation [Fig. 1(b)].

Figure 2 depicts the interfacial deformation and the Q-field in the nematic phase for several w * values. In the H case, the Iso-N interface remains mostly flat for low values of w * (< 1) and, because of the rigid anchoring imposed on the particle, substantial Qfield distortions occur and extend away from the particle. In this case, the anchoring conflict at the CL is smoothed out and there are no topological defects. Equivalently, we may speak of a large anchoring extrapolation length. However, as we increase w * (> 1), the elastic distortions become more localized to the particle surface. As a result, the interfacial deformations grow, with the Iso-N interface bulging upwards in the vicinity of the CL, right where the anchoring conflict lies. For high anchoring strengths (w * ≫ 1), a topological defect appears at the particle surface very close to the CL [Fig. 2(b)].

In the P configuration [Fig. 2(c)], for which there is no anchoring conflict with θ = 90 • [Fig. 1(b)], the Iso-N interface remains nearly flat regardless of the anchoring strength.

A topological defect of winding number -1/2 forms beneath the particle, in accordance with the imposed boundary conditions. The defect position, which depends primarily on the anchoring strength prescribed on the particle surface, is not sensitive to w * for it is far from the Iso-N interface.

The interfacial deformations of Fig. 2(a) are quantified in Fig. 3 as a function of w * for the H configuration. The interfacial distortion ∆y I = y cl -y w is defined as the difference between the y-location of the interface at the CL (y cl ) and that at the bounding wall (y w ).

We see that ∆y I first grows steeply for w * 1 before levelling off at larger values with a crossover region around w * ≃ 1 . We can account for these two regimes via a scaling argument based on the competition between the bulk elastic energy of the nematic LC and the surface energies involving both surface tension and anchoring. The details of the derivation are presented in appendix B. We show that

∆y I = w * L σL L 1 (1 + w * ) , ( 12
)
where L is the box size [START_REF] Eq | implies that the interfacial deformations (∆y I ) diverge with the box size L, which is one of the characteristic length scales of the problem in the simple model of Fig. 11 (appendix B). But in more realistic situations, we should rather consider that L is about a few times the particle radius (R), say ∼ 10R, so that interfacial distortions[END_REF]. Thus, for w * ≪ 1, i.e. in the weak anchoring regime, we have ∆y I ≈ w * L σL/L 1 , indicating that ∆y I should be a linear function of w * . In the strong anchoring regime, w * ≫ 1 , and Eq. ( 12) yields ∆y ≈ L w * σL/L 1 , i.e. a square root profile is expected in this case. It follows that a crossover point at w * = 1 naturally appears between these two limits. As can be seen from Fig. 3, these predictions are very well confirmed by the numerical data. The inset of Fig. 3 provides a zoomed-in view of the crossover region, which indeed occurs around w * = 1 .

Thus, our analysis shows that, for w * 1 , the system resolves the anchoring conflict at the CL by deforming substantially the interface. In this regime, the energetic penalty incurred by the deformed interface is less than the cost associated with bulk elastic distortions and to deviations from the prescribed anchoring orientation. In doing so, the total

Free particle

In this section, we take into account the particle's equations of motion, which are solved with all the other governing equations, as outlined in Sec. 2.2. The particle is now free to move up or down depending on the forces acting on it. In turn, the shape of the Iso-N interface will be altered by the particle motion. We start with θ = 90 displacement alters the shape of the Iso-N interface which no longer bulges upwards as before [Fig. 2(a)], but now bends downwards into a concave meniscus. With a freely moving particle, the system manages the anchoring conflict at the CL by pushing the particle into the N phase, which is also likely to result in less pronounced interfacial distortions.

Thus, the particle position across the interface may be seen as a new degree of freedom that can be adjusted to minimize the overall energy. In contrast, the P-configuration of Fig. 4(c,d) hardly shows any interfacial deformation or particle displacement, even with a stronger anchoring (w * = 2). This is because of the lack of an anchoring conflict at the contact line. Therefore, the extent of particle motion and interfacial deformations appear to be directly correlated to the existence of an anchoring conflict at the CL. In the following, we will test further this conjecture by employing other contact angles and various anchoring strengths in both H and P setups. Contact angle θ ̸ = 90 • . We restrict our investigation to two distinct cases, one in which most of the particle body lies in the N phase (θ = 45 • ), and another one for which the particle is preferentially immersed in the Iso phase (θ = 135 • ). As previously, we will monitor the response of the system as we tune the anchoring strength at the Iso-N interface for both the H and P configurations.

We start with θ = 45 • and the H configuration. The series of snapshots exhibited in Fig. 5 reveals an interesting phenomenon. For w * = 0 [Fig. 5(a)], a topological defect of winding number -1/2 forms slightly below the equator of the particle as if it were repelled from the Iso-N interface. On the contrary, for a finite but weak anchoring strength [w * = 0.4, Fig. 5(b)], the defect gets attracted towards the interface but stabilizes itself at an equilibrium distance from it. This attraction has an elastic origin and probably results from the onset of an anchoring conflict at the CL, where the Q-field deformations differ greatly from the w * = 0 case. Further increasing w * leads to a stronger elastic attraction and eventually to the trapping of the defect within the interfacial area, very close to the CL [Fig. 5(c)]. The close-up view on Fig. 5(d) indeed shows that the defect has not disappeared. Also, notice the significant interfacial deformation together with the substantial sinking of the particle into the N phase.

Hence, for a high enough w * , it becomes energetically favorable for the system to absorb the defect at the interface rather than to sustain it in the bulk with costly director disturbances. This result is a further illustration of an otherwise well-known coupling effect between bulk elastic distortions and surface anchoring strength encountered in LC colloids (see e.g., [START_REF] Smalyukh | Liquid crystal colloids[END_REF][START_REF] Qiu | Phase-field model for elastocapillary flows of liquid crystals[END_REF]).

Other wetting configurations are presented in Fig. 6. All snapshots on the left side correspond to situations with w * = 0 , whereas those displayed on the right side are calculated for weak to intermediate anchoring strengths. For w * = 0 , there is no particle displacement, or only a minute one, and the Iso-N interface remains flat. This is in general not the case with w * ̸ = 0 . Note that an anchoring conflict now occurs in the P configuration since θ ̸ = 90 • . As aforementioned, the system resolves the anchoring conflict primarily by moving the particle across the Iso-N interface. In general, for a given The Iso-N interface also becomes more deformed due to a sizeable downward displacement of the particle. Grey scale: Q 2 22 . (d) Close-up view of (c) near the contact line. Grey scale: 1-ϕ 2 q . The dark blurry spot signals a topological defect lying within the interfacial area very close to the contact line. (e) Blown-up view of (d) showing a typical fine mesh in the vicinity of the interface (ϕ = 0) and the topological defect. As mentioned in Sec. 2.4, the mesh gradually coarsens away from the interfacial area. configuration (either H or P) and θ , the greater w * , the larger the particle displacement.

For example, in Fig. 6(b), the particle went up significantly (d * = 0.33) and a sizeable interfacial deformation remains at equilibrium. However, bulk elastic disturbances are very moderate as they are localized in the close proximity of the CL and there is no longer any defect nearby the particle, in contrast to Fig. 5. Next, in Fig. 6(d), the particle has sunk substantially (d * = -0.37) resulting in a slightly curved interface near the CL. This arises from an anchoring conflict at the CL, which is clearly signalled by additional elastic distortions of the Q-field [compare Fig. 4(c) and Fig. 6(d)]. Finally, a nearly flat interface results in Fig. 6(f) with an upward motion of the particle (d * = 0.18). Since only a small portion of the solid surface is submerged in the nematic, elastic distortion is limited to a small area, as compared with the case of θ = 90 • in Fig. 4(c). A defect of winding number -1/2 still lies beneath the particle, in agreement with the prescribed BCs.

To summarize, with finite anchoring on the Iso-N interface, potentially large interfacial deformations and particle displacements, on the order of 0.5R , may occur whenever there is an anchoring conflict at the CL. This finding has never been predicted before nor observed in experiments, and is the central result of Problem A. It is a direct consequence of the coupling between elastic, capillary (surface tension) and anchoring effects.

Problem B: drag force

Problem B concerns the drag force on a particle straddling an Iso-N interface. As mentioned in the introduction, a few experiments have touched upon this topic [START_REF] Liu | Elastocapillary interactions on nematic films[END_REF][START_REF] Abras | The mobilities of micro-and nano-particles at interfaces of nematic liquid crystals[END_REF][START_REF] Wei | Dynamics of ordered colloidal particle monolayers at nematic liquid crystal interfaces[END_REF] but no prior theoretical or computational work seems to have been reported so far. Fig. 1(c) specifies the setup, as already described in Sec. 2.3. A 2D planar geometry is employed in order to first capture some main qualitative physical trends. As in Problem A, a rigid homeotropic anchoring is prescribed at the particle surface. To simplify the analysis of the drag force, we make two further adjustments of the parameters. We increase the surface tension by a factor of 10 relative to that of Problem A to minimize interfacial deformation (appendix A). Besides, we adopt the P-configuration with soft planar anchoring (w * = 0.4) on the Iso-N interface to avoid complications from the an- choring conflict at the CL. In all cases, the motion of the particle is heavily overdamped since the Reynolds number Re = ρ n v ∞ R/η n is very small ( 10 -5 , cf. appendix A). Furthermore, a moderate coupling between the fluid velocity and the Q-field is considered with the Ericksen number Er in the range 0.28 -3 (cf. Sec. 2.3). Given the complexity of the problem and the large parameter space, we focus here on two key parameters, θ and Er . A more extensive exploration will be conducted in the future.

The drag force, F * D (LC) = F D (LC)/L 1 , consists of the x-component of elastic and viscous forces, which are evaluated by integrating the corresponding tractions along the particle contour, as in Eq. ( 10). As a reference, the drag force for an isotropic-isotropic (Iso-Iso) interface, F * D (iso) , was also calculated with the fluids having the same viscosities as those of the Iso-N system. This force corresponds to the limit of A, B, C, L 1 , W s → 0 (Sec. 2.1) and thus only contains a viscous component. In the next two sections, we investigate the influence of θ and Er on F * D (LC) . For convenience, we have used the ratio FD = F * D (LC)/F * D (iso) to assess the influence of the nematic ordering around the particle on the drag force. Note that the interface is maintained flat by shifting the center of the particle across the Iso-N interface. Gray scale plots of Q 2 11 show some illustrative steady [Fig. 7(b,c)] and transient [Fig. 7(d,e)] states obtained for a few θ values.

Influence of the contact angle θ

We first notice that the curve FD = f (θ) [Fig. 7(a)] is asymmetric with respect to θ = 90 • . This is of course expected since the Iso-N interface breaks the up-down symmetry.

We checked that for an Iso-Iso interface with matched viscosities between the two fluids, the curve FD = f (θ) has the shape of a symmetric concave parabola, in agreement with previous results [START_REF] Loudet | Drag force on a particle straddling a fluid interface: Influence of interfacial deformations[END_REF][START_REF] Danov | Viscous drag of a solid sphere straddling a spherical or flat interface[END_REF]. Second, for both Er values, we see that FD > 1 in most cases, and notably, FD is maximum for θ ≃ 50 • , i.e. when most of the particle body is immersed into the N phase [Fig. 7(c)]. In this case, the distorted nematic texture [see e.g., Fig. 7(b,c)] increases the drag force by about 15 -20 % compared to a purely In this case, we see that the defect, marked by the white arrow in (d), merges with the interface as a result of both flow advection and an attractive elastic interaction (see text for details). After the merging (cf. dashed circle in (e)), the system remains in this state till the end (t * end = 2 × 10 8 ). Gray scale: Q 2 11 . Bright (resp., dark) areas correspond to LC molecules aligned parallel (resp., perpendicular) to the horizontal axis (same meaning hereafter). The green arrow lines represent the flow streamlines. isotropic system. This drag enhancement may be explained by the fact that the fluid has to flow through distorted regions of the nematic field where the orientation of LC molecules changes. This is especially true near topological defects where large gradients of the order parameter usually occur. Reorientations of mesogens bring up an additional contribution to F * D (LC) via the rotational viscosity γ 1 (Sec. 2.3) [START_REF] Foffano | The dynamics of colloidal inclusions in liquid crystals: a simulation perspective[END_REF], which is specific to nematics. Heuristically, we may imagine that the particle and its companion defect [e.g., Fig. 7(b)] form a bigger effective particle in the N phase that has to be dragged along, thereby enhancing the friction.

Besides, we see that the main effect of the imposed external flow is to shift the topological defect downstream of the particle, to a steady state position if Er is not too large [Fig. 7(b,c)]. At this relatively low Er, the viscous forces from the fluid flow are strong enough to alter the orientation of LC molecules and yield a modified nematic texture. This is a prominent feature of the coupling between the flow and the LC orientation. Note that similar defect motions in bulk LC colloids were reported for Er ∼ 1 [START_REF] Araki | Surface-sensitive particle selection by driving particles in a nematic solvent[END_REF][START_REF] Khullar | Dynamic Evolution of Topological Defects around Drops and Bubbles Rising in a Nematic Liquid Crystal[END_REF][START_REF] Zhou | The rise of Newtonian drops in a nematic liquid crystal[END_REF][START_REF] Reyes-Arango | Defects around nanocolloids in nematic solvents simulated by Multi-particle Collision Dynamics[END_REF].

Next, as θ increases [Fig. 7(a)], FD decreases and tends towards unity, which goes in line with intuition since the particle is more exposed to the isotropic phase in this case.

However, an anomaly occurs for Er = 0.57 and θ = 135 • as FD jumps to a larger value (cf. dashed circle). This phenomenon may be ascribed to the merging of the defect with the Iso-N interface [Fig. 7(d,e)], which induces the formation of a pair of small vortices on either side of the interface downstream of the particle [see Fig. 7(e)]. These vortices originate from a Marangoni flow that develops along the interface because of the existence of an anchoring energy gradient near the contact line, as demonstrated in Fig. 8. Close to the contact line, the fluids flow along the interface from locations of low anchoring energy to those of high anchoring energy, in agreement with Rey's theory on Marangoni flow at LC interfaces [START_REF] Rey | Marangoni flow in liquid crystal interfaces[END_REF]. This anchoring energy gradient arises from the disturbed molecular orientation in this area as a consequence of the defect-interface merging. Note that the defect has disappeared from the interfacial area, giving rise to a nonsingular configuration.

The Marangoni eddies persist over long times and might be responsible for the enhanced drag force exerted on the particle in this case. The merging of the defect with the interface [Fig. 7(d,e)] may be caused by the combination of two effects: advection and elastic attraction. The merging only occurs for Er = 0.57 but not for Er = 0.28. In the former case, the flow is strong enough to sweep the defect sufficiently close to the interface so that it falls within the range of an attractive elastic interaction there. Indeed, from the dynamic motion of the defect [Figs. 7(d,e)],

it seems as if it is captured by the interface. This phenomenon appears consistent with what we reported previously in Fig. 5, where a defect trapping occurs, although in the absence of external flow.

Influence of the Ericksen number

In this section, we address in more detail the influence of the Ericksen number on the drag force [Fig. 1(b)]. In Sec. 3.2.1, only two different values of Er were considered for a broad range of θ (Fig. 7). Here, we carry out a complementary study and vary Er more systematically for only two different values of θ . Besides, the particle's center of mass is now kept fixed midway across the interface. Thus, the case θ = 90 • yields a flat interface, as before (Sec. 3.2.1), but a nonflat interface results if θ ̸ = 90 • . The latter setup is useful to examine the effect of interfacial deformations on the drag force. The numerical setup is exactly the same as before [Fig. 1(c)].

Contact angle θ = 90 • . Fig. 9(a) summarizes the results of our drag force computations for θ = 90 • as a function of the dimensionless far-field imposed velocity v * ∞ = τ v ∞ /l nc , where l nc is the nematic coherence length and τ a typical molecular time (see appendix A).

For the Iso-N interface, the maximum value of v * ∞ corresponds to Er = 1.71 . We could not investigate larger Er values here because, for Er 1.71, the defect 'escapes' from the particle and gets convected downstream continually by the flow, as shown in Fig. 9(b,c).

Even though no steady state can be reached in this case, the drag force acting on the particle remains about constant once the defect is sufficiently away from it. Previous 3D studies in bulk LC colloids reported similar defect detachments under the influence of flow [START_REF] Araki | Surface-sensitive particle selection by driving particles in a nematic solvent[END_REF][START_REF] Reyes-Arango | Defects around nanocolloids in nematic solvents simulated by Multi-particle Collision Dynamics[END_REF]. As in our study, it was found that defect separation typically occurs for Er 1, i.e. when the viscous forces become stronger than elastic forces. Otherwise stated, since the defect motion is governed by director rotation, the defect abandons the particle when the elastic torque can no longer prevent the director from rotating under the viscous torque due to the passing flow. In Fig. 9(c), notice the sizeable spatial extension of Q-distortions (dark areas) induced by the flow.

In addition, the graph on Fig. 9(a) clearly reveals a proportionality between F * D and v ∞ in both the Iso-Iso and Iso-N cases. While the linear behavior was expected in the isotropic situation (Reynolds number Re = ρ n v ∞ R/η n 10 -5 ), it was not guaranteed, a priori, that linearity would hold as well for the Iso-N interface in a regime where Er is not particularly small (Er ∼ 1). As far as we know, nonlinear drag forces were only reported by Stark & Ventzki [START_REF] Stark | Non-linear Stokes drag of spherical particles in a nematic solvent[END_REF] for a spherical particle exhibiting the dipolar configuration in a bulk nematic matrix.

Moreover, we see that the friction coefficient, F * D /v * ∞ , which is given by the slope of linear fits in Fig. 9(a), is slightly larger in the Iso-N case than in the Iso-Iso case. This result can be explained via the same arguments as those put forward previously (Sec. 3.2.1): as v ∞ increases, areas where the nematic alignment is distorted grow in size as a result of the coupling between the velocity and order parameter fields. Advection of the defect is also more pronounced, leading to significant molecular reorientations which contribute to the friction through the rotational viscosity.

Contact angle θ = 135 • , nonflat interface. Finally, we consider the case of a nonflat interface by setting θ = 135 • . Since the particle's center of mass is fixed, the Iso-N interface now deforms downwards so that most of the particle body is exposed to the isotropic phase. The graph in Fig. 10(a) compares the drag force data obtained in this configuration for the Iso-N and Iso-Iso interfaces, similarly to Fig. 9(a). First, in the Iso-Iso case, the friction coefficient (slope of the linear fit) is slightly larger than its counterpart determined in Fig. 9(a) for a flat interface (1.2 × 10 -4 vs. 1.1 × 10 -4 , respectively). This result agrees with our previous findings that a deformed fluid interface breaks the updown symmetry and leads to more bending of the flow streamlines, which enhances the friction and the drag coefficient [START_REF] Loudet | Drag force on a particle straddling a fluid interface: Influence of interfacial deformations[END_REF]. Second, in the Iso-N case, we observe again a linear relationship between F * D and v * ∞ . This linearity prevails up to Er = 2.86, corresponding to the maximum v * ∞ in the graph. As before, the friction coefficient for the Iso-N interface is slightly larger than that of the Iso-Iso interface, most likely for the same reasons as those already mentioned in the flat interface situation [Fig. 9(a)]. with the Iso-N interface for Er 0.57 , as illustrated in Fig. 10(b,c). Since the interface is now bent downwards, the flow can sweep the defect closer to it so that, at some point, an attractive elastic interaction between the defect and the interface, already encountered in Fig. 5 and Fig. 7(d,e), takes over and results in the trapping of the defect. As in Fig. 7(e), the defect disappears from the interfacial region once adsorbed, giving rise to a pair of Marangoni vortices located on either side of the interface [Fig. 10(c)]. The defect merging, as explained before, is expected to cause additional drag [Fig. 7(a)]. In fact, this seems to be the case in Fig. 10(a) as well. The merging does not occur for the first data point of Fig. 10(a) at v * ∞ = 2.7 × 10 -6 and Er = 0.28, but occurs for all the higher values of Er. The first data point falls a bit below the linear relationship manifested by the higher-Er data points by some 15% . This is less than the amount of drag enhancement (≃ 28%) observed in Fig. 7(a) due to defect merging but the two configurations are different.

Concluding remarks

We have performed numerical simulations on the behavior of solid colloidal particles attached to an Iso-N fluid interface. Building on our previous work, we have employed a model that combines a diffuse interface method (phase-field) to account for the properties of the Iso-N interface with a tensor order parameter description of the nematic phase, including topological defects. Such a model is well suited to describe elastocapillary phenomena, i.e. situations involving a coupling between elasticity and capillarity. We have first focused our attention on equilibrium floating configurations for both homeotropic and planar anchoring types at the Iso-N interface with various anchoring strengths and contact angles. A rigid homeotropic anchoring was assumed at the particle surface in all cases.

One of the key findings is that the anchoring conditions at the contact line drive the response of the system in terms of interfacial deformations and/or particle displacements.

We have discovered two regimes with two distinct scaling laws for the magnitude of interfacial distortions. A strong anchoring conflict generally yields a sizeable particle shift and interfacial deformation. Note that, because of their negligible buoyant weight, the particle we considered would not produce any interfacial distortion if adsorbed at isotropic planar interfaces. Thus, our results bring up the novelty of the Iso-N interface and highlight the interplay between elastic distortions, surface tension and the anchoring conditions.

In the second part, we have studied the dynamic situation of a particle being dragged along an interface. One of the primary objectives was to assess the influence of elastic distortions on the drag force exerted on the particle, and more generally, the effect of the coupling between the velocity and order parameter fields. We have computed drag forces as a function of the contact angle and the Ericksen number. Only a moderate planar anchoring has been considered at the Iso-N interface. Our investigation reveals that (i) drag forces at the Iso-N interface are always greater, albeit only slightly, than their isotropic counterparts and (ii) generally, greater drag enhancement tends to occur for smaller contact angles, when more of the particle body is immersed in the N phase. Defect sweeping, escaping, and merging with the interface, together with large distorted areas of the order parameter, contribute to the drag enhancement. For both flat and nonflat interfaces, the drag force is linearly proportional to the velocity in the entire range examined, up to Ericksen numbers of order unity.

The current 2D implementation of our model may be extended to address the case of a sphere trapped at the I-N interface in 3D, which can be more directly compared with experiments. Preliminary simulations of a floating sphere carried out in a 2D-axisymmetric geometry confirm qualitatively the main findings of the present 2D study: the presence of conflicting anchoring conditions at the contact line is responsible for the onset of interfacial deformations, and the stronger the anchoring, the greater the distortions. A detailed study will be reported in a forthcoming article. However, the computation of drag forces acting on a sphere straddling the I-N interface requires a full three-dimensional approach, which cannot be tackled yet by our current model implementation. Nevertheless, prior studies in bulk nematics [START_REF] Araki | Surface-sensitive particle selection by driving particles in a nematic solvent[END_REF][START_REF] Khullar | Dynamic Evolution of Topological Defects around Drops and Bubbles Rising in a Nematic Liquid Crystal[END_REF][START_REF] Zhou | The rise of Newtonian drops in a nematic liquid crystal[END_REF][START_REF] Reyes-Arango | Defects around nanocolloids in nematic solvents simulated by Multi-particle Collision Dynamics[END_REF] suggest that line defects may be distorted by the passing flow, detach from the particle, shrink or even transform into a point. As in the 2D case, defects merging with the interface could also be expected. These phenomena, if they occur, are likely to alter drag forces and perhaps make the force-velocity law deviate from a linear behavior.

Our two-dimensional results also form the basis for exploring more complex situations such as multiparticle interactions based on the elastocapillary coupling. Indeed, we have shown that sizeable interfacial deformations may take place due to the interplay of elastic and anchoring effects. Thus, in principle, we may expect the capillary and elastic interactions to contribute simultaneously to the self-assembly properties of the system.

As mentioned in the introduction, a number of experimental and theoretical studies shed some light on this topic over the past few years [START_REF] Smalyukh | Ordered droplet structures at the liquid crystal surface and elastic-capillary interactions[END_REF][START_REF] Gharbi | Behavior of colloidal particles at a nematic liquid crystal interface[END_REF][START_REF] Wang | Controlled multistep self-assembling of colloidal droplets at a nematic liquid Crystal-air interface[END_REF][START_REF] Oettel | Effective interactions of colloids on nematic films[END_REF], but there is still no clear demonstrations of the interaction mechanisms by accurate computations. We currently focus our simulation efforts in this direction.

Table 1: Definitions and base values of the parameters used in the simulations. The subscript "n" (resp., "i") stands for nematic (resp., isotropic). The superscript A (resp., B) refers to problem A (resp., B). "PS'" is the abbreviation for Phase-Field.

  where m is the outward unit normal vector to a given boundary. Next, we enforce the equilibrium value of the contact angle, θ, at the three-phase contact line on the particle via the geometric boundary condition: ∇ϕ • m = |∇ϕ| cos θ , on ∂Ω p [Fig. 1(a)]. We also impose a 90 • -contact angle on ∂Ω w (left side) [Fig. 1(a)], meaning

Fig. 1 (Figure 1 :

 11 Fig.1(a,b) specifies the geometry for Problem A. A solid particle of radius R is trapped at the interface between a Newtonian isotropic fluid and a nematic LC. Both fluids have

  noted H and P configurations, respectively. (c) Geometry for the computation of the two-phase flow drag force exerted on a particle confined at the Iso-N interface between two plane walls (Problem B). Box size: (H, L) = (20R, 30R). a matched density and the whole system is confined in a box of length L = 10R and height H = 8R. However, because of symmetry, we only need to simulate half of the domain. Fig. 1(b) depicts the prescribed anchoring conditions at the Iso-N interface and the particle surface. Both homeotropic (H) and planar (P) anchoring of variable strength W [see Eq. (

Figure 2 :

 2 Figure 2: Zoomed-in snapshots computed at steady state with the anchoring strength w * indicated in the white space inside the particle. The contact angle θ = 90 • for all cases.(a) H configuration. Contour plots of Q 2 22 . For w * = 16.2 , the dashed lines mark the cut-out area that is blown up in (b). (b) Zoomed-in view in the vicinity of the contact line for w * = 16.2 in (a). The grey scale shows the concentration-weighted scalar order parameter 1-ϕ2 q , with q e = 0.81 . The small blurry dark spot signals a topological defect with q 0.5 . (c) P configuration. Contour plots of Q2 11 .

Figure 3 :

 3 Figure 3: Interfacial deformation (∆y I /R) as a function of the anchoring strength (w * = W s /σ) for the H configuration. The shaded area marks the transition between a steep initial variation (weak anchoring regime) and a milder increasing trend (strong anchoring regime). The black dashed (resp., blue dash-dotted) line is a linear (resp., square root) fit to the data [Eq. (12)]. The inserted graph is a zoomed-in plot of the crossover region around w * = 1 . The red data points lie within the crossover area and were not taken into account in the fitting procedure. Contact angle: θ = 90 • .

  • before considering other contact angle values. Contact angle θ = 90 • . As before (Sec. 3.1.1), the particle's center of mass lies initially in the middle of the box and the fluid interface is flat, thereby realizing the condition θ = 90 • .Typical results obtained at steady state are displayed in Fig.4. In the H configuration [Fig.4(a,b)], we see that the particle has now sunk into the N phase, with an appreciable downward particle displacement d * = d/R. In this example, d * ≃ -0.3 for w * = 0.5 . This

Figure 4 :

 4 Figure 4: Order parameter snapshots and interfacial profiles with a moving particle. (a,b) H configuration. (c,d) P configuration. The bold black figures indicate the value of w * in each case. (a) (resp., (c)) Contour plot of Q 2 22 (resp., Q 2 11 ). (b),(d) Comparison of interfacial profiles (y * I = y I /R) computed in the fixed and moving particle cases (x * = x/R). Contact angle: θ = 90 • .

Figure 5 :

 5 Figure 5: (a-c) Steady state wetting configurations obtained with a moving particle for several anchoring strengths (w * , indicated by the bold black figures) in the H configuration with θ = 45 • . The particle displacement, d * = d/R, is also specified hereafter: (a) d * = 2 × 10 -3 , (b) d * = 0.05 , (c) d * = -0.5 . As w * increases, the defect is elastically attracted towards the Iso-N interface close to the CL and eventually gets trapped there.

Figure 6 :

 6 Figure 6: Additional steady state wetting configurations calculated with a moving particle. As in Fig. 5, the bold black figures (inside the particle) mark the values of w * and d * is also specified at the bottom right corner in each case. (a,b) H configuration, θ = 135 • . (c-f ) P configuration. (c,d) θ = 45 • . (e,f) θ = 135 • . Depending on θ and w * , the particle goes either up or down with relatively small interfacial deformations, except in (b). Color bar: Q 2 22 (a,b), Q 2 11 (c,f).

Fig. 7 (

 7 Fig. 7(a) shows the drag force as a function of the contact angle ([45 • -135 • ]) at two values of Er . For convenience, we have used the ratio FD = F * D (LC)/F * D (iso) to assess the influence of the nematic ordering around the particle on the drag force. Note that

Figure 7 :

 7 Figure 7: (Problem B) Results of drag simulations for various contact angles and a flat interface in the P configuration. (a) Normalized drag force F * D (LC)/F * D (iso) as a function of θ for Er = 0.28, 0.57 . (b)-(e) Flow and nematic fields for different contact angles and Er = 0.57 . (b) Steady state for θ = 90 • . (c) Steady state for θ = 45 • . (d,e) Transient states for θ = 135 • . t * = t/τ indicates the simulation time. In this case, we see that the

Figure 8 :

 8 Figure 8: (Problem B) Zoomed-in view of Fig. 7(e) near the contact line downstream of the particle. The gray scale shows the dimensionless anchoring energy density, f * a = f a /(L 1 l nc ) , which is defined across the entire diffuse interface [Eq. (3)]. As such, we see that it spreads both below and above the level set ϕ = 0 . The arrows indicate the velocity field. The anchoring energy gradient close to the contact line drives a Marangoni flow that is signaled by a pair of vortices located on either side of the interface (solid curve). These vortices may increase the drag force exerted on the particle. Parameters: Er = 0.57, θ = 135 • .

Figure 9 :

 9 Figure 9: (Problem B) P configuration, θ = 90 • . (a) Drag force (F * D ) as a function of the prescribed far-field velocity (v * ∞ ) for both the Iso-N and Iso-Iso interfaces. The solid and dashed lines are linear fits to the data. The numbers next to the fitting lines indicate the corresponding slopes. (b,c) Transient snapshots (gray scale: Q 2 11 ) of the Iso-N interface illustrating the defect escape from the particle for the last data point on (a) corresponding to Er = 1.71 . The defect, marked by the dashed yellow circle, is swept downstream by the flow, whose streamlines are represented by green arrow lines. The defect is still located close to the particle at time t * = 1.2 × 10 7 (b) but is being driven away from it at later times (c).

Figure 10 :

 10 Figure 10: (Problem B) P configuration, θ = 135 • . (a) Same as in Fig. 9(a). (b,c) Transient zoomed-in snapshots (Gray scale: Q 2 11 ) illustrating the merging of the defect with the Iso-N interface for Er = 0.57 . The yellow dashed circle marks the defect location just before (b) and right after (c) the merging event. The green arrow lines represent the streamlines and t * is the simulation time.

- 1 1 ) 2 )

 112 Box dimensions (height,length) (H, L) (8R, 10R)(A) , (20R, 30R)(B) µm conflict (cf. Fig.11). It follows that the change in the elastic energy density due to a nonzero anchoring strength can be written as δf e ∼ W 2 /L 1 . Furthermore, assuming that the area A ∼ L 2 of the nematic domain remains about constant (small variation), the change in the elastic energy reads Surface energies -In order to estimate the change in surface energies, we have to compute the excess length δL of a deformed interface caused by the existence of an anchoring conflict. The resulting excess surface energy is simply given by δF s = (σ+W )δL since both interfacial and anchoring energies will be affected by a change in L .From simple considerations of differential calculus, we havedL ′ = dL 1 + (dy/dx) 2 ,where dL ′ is a length element of the deformed interface and dL its projection along the x-axis. Assuming small interfacial deformations, and hence small slopes ( (dy/dx) ≪ 1), we may write dL ′ ≈ dL 1 + 1 2 (dy/dx) 2 resulting in δ(dL) = dL ′ -dL = 1 2 (dy/dx) 2 dx . In terms of orders of magnitude, (dy/dx) ∼ ∆y I /L , leading to δL ≈ ∆y I the (unimportant) prefactor 1/2 has been dropped. Consequently, we may write δF s ∼ (σ + W ) ∆y 2Balancing the two Eqs. (B.1) and (B.2) yields Eq. (12)∆y I = w * L σL L 1 (1 + w * ).

Acknowledgements

This work was financially supported by the EU Marie-Curie fellowship 'CoPEC' under grant No 794837-H2020-MSCA-IF-2017 and by the NSERC Discovery grant No 2019-04162. J.-C.L. is also indebted to the University of Bordeaux for further financial support thanks to the IdEx program entitled "Développement des carrières -Volet personnel de recherche". M.Q. acknowledges support from the UBC Four-year Doctoral Fellowship. A.C. was supported by an overseas visiting doctoral fellowship from SERB, Department of Science & Technology, Government of India. We acknowledge CMC Microsystems for software licensing. The IT staff of the Mathematics department of the University of British Columbia is also gratefully acknowledged for their valuable help and support.

Appendix A Simulation parameters

The parameters characterizing the nematic LC are close to those of the widely used compound 5CB [START_REF] Ravnik | Landau-de-Gennes modeling of nematic liquid crystal colloids[END_REF][START_REF] Kim | Interfacial Tension of a Nematic Liquid Crystal/Water Interface with Homeotropic Surface Alignment[END_REF][START_REF] Hung | Anisotropic nanoparticles immersed in a nematic liquid crystal: Defect structures and potentials of mean force[END_REF]. The numerical parameters for the CH dynamics (capillary width ϵ, mobility γ) are chosen according to the guidelines reported in [START_REF] Yue | Sharp-interface limit of the Cahn-Hilliard model for moving contact lines[END_REF][START_REF] Yue | A diffuse-interface method for simulating twophase flows of complex fluids[END_REF][START_REF] Yue | Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing[END_REF]. The dimensional base values of all simulation parameters are listed in the following table.

Appendix B Derivation of [Eq. (12)]

We derive Eq. ( 12) using simple scaling arguments based on the competition between the bulk elastic energy of the nematic LC and the surface energies (surface tension and anchoring). Fig. 11 specifies the geometry for our reasoning. To derive Eq. ( 12), there is no need to consider explicitly a particle attached to the interface (actually, the particle size is not a relevant length scale of the problem). The right wall of the box 'stands for' the particle surface and we assume that an anchoring conflict exists there resulting in a deformed interface (dashed curve in Fig. 11). The typical interfacial deformation is denoted ∆y I .

Figure 11: Geometry used for the scaling argument. The right boundary symbolizes the particle surface in the actual simulation. The small ellipsoids represent the LC molecules (not to scale). We assume they are oriented parallel to the y-axis everywhere except at the right wall. These conditions result in an anchoring conflict of the H type there [Fig. 1(b)], which makes the Iso-N interface bulge upwards.

Bulk elastic energy -As a first approximation, we know that the elastic free energy density, f e , scales as ∼ L 1 /ξ 2 , where L 1 = K/q 2 e (Sec. 2.3) is proportional to the elastic constant K of the nematic LC, and ξ is the characteristic length scale of the director deformations. In the presence of a finite anchoring strength W at the Iso-N interface, we may take ξ ∼ L W = L 1 /W , where L W is the anchoring extrapolation length. Indeed, this is the relevant characteristic length to describe director deformations due to an anchoring Author contribution statement J-CL and JJF designed the research. J-CL and AC performed the work; all authors analyzed the data, interpreted the results and collaborated to the manuscript written by J-CL.


Siyuan Niu

Aida Todri-Sanial

Exploring Multi-programming Applications in the NISQ Era

Key ideal: execute multiple circuits simultaneously to increase the hardware throughput and total runtime.

The multi-programming mechanism

Which factors should we consider to build a multi-programming technique?

• Results

• # parallel circuit executions: one to six • Hardware throughput: 7.7% to 46.2% • Significant fidelity loss when hardware throughput is over 38%.

Multiprogramming and VQE

• VQE limitation • Split the computation into O(N 4 ) sub-problems, introducing a large overhead of measurement circuits.

• Pauli grouping

• Grouping commuting Pauli terms and measure them simultaneously.

• Idea

• Apply multiprogramming to Pauli grouping (c) 12 optimizations with 24 measurements.

• Results

• Hardware throughput can be up to 73.8% with an error rate of less than 10%.

Multiprogramming and ZNE

• Zero noise extrapolation (ZNE) error mitigation.

• Noise-scaling • Extrapolation • Extra circuit overhead

• No ZNE (baseline) • ZNE+multiprogramming • ZNE • Results
• The error rate is reduced by 2x without any circuit overhead.

Introduction•

  Limited hardware connectivity • Unavoidable error rates • Only small circuits can obtain reliable results • Long waiting time • Hardware throughput: • Total pending jobs: Motivation (a) Hardware throughput = 40% (b) Hardware throughput = 80%

  •Crosstalk •Partition qubits to reliable regions •Qubit mapping (routing) •Task scheduling •Number of simultaneous circuits Hardware throughput vs Circuit fidelity (a) 4mod5-v1_22 result (b) alu-v0_27 result • Experiment setup • Execute different number of the same circuit on IBM Q 65 Manhattan. • Report the relation between hardware throughput and circuit fidelity.

  (a) 8 optimizations with 16 measurements. (b) 10 optimizations with 20 measurements.

Acknowledgment

This work is funded by the QuantUM Initiative of the Region Occitanie, University of Montpellier and IBM Montpellier.


Siyuan Niu
email: siyuan.niu@lirmm.fr

Aida Todri-Sanial

Exploring Multi-programming Applications in the NISQ Era

Key ideal: execute multiple circuits simultaneously to increase the hardware throughput and total runtime.

The multi-programming mechanism

Which factors should we consider to build a multi-programming technique? • Results

• # parallel circuit executions: one to six • Hardware throughput: 7.7% to 46.2% • Significant fidelity loss when hardware throughput is over 38%.

Multiprogramming and VQE

• VQE limitation • Split the computation into O(N 4 ) sub-problems, introducing a large overhead of measurement circuits.

• Pauli grouping

• Grouping commuting Pauli terms and measure them simultaneously.

• Idea

• Apply multiprogramming to Pauli grouping (c) 12 optimizations with 24 measurements.

• Results

• Hardware throughput can be up to 73.8% with an error rate of less than 10%.

Multiprogramming and ZNE

• Zero noise extrapolation (ZNE) error mitigation.

• Noise-scaling • Extrapolation • Extra circuit overhead

• No ZNE (baseline) • ZNE+multiprogramming • ZNE • Results
• The error rate is reduced by 2x without any circuit overhead.

Introduction•

  Limited hardware connectivity • Unavoidable error rates • Only small circuits can obtain reliable results • Long waiting time • Hardware throughput: • Total pending jobs: Motivation (a) Hardware throughput = 40% (b) Hardware throughput = 80%

•

  Crosstalk • Partition qubits to reliable regions • Qubit mapping (routing) • Task scheduling • Number of simultaneous circuits Hardware throughput vs Circuit fidelity (a) 4mod5-v1_22 result (b) alu-v0_27 result • Experiment setup • Execute different number of the same circuit on IBM Q 65 Manhattan. • Report the relation between hardware throughput and circuit fidelity.

  (a) 8 optimizations with 16 measurements. (b) 10 optimizations with 20 measurements.

This work is funded by the QuantUM Initiative of the Region Occitanie, University of Montpellier and IBM Montpellier.


Jonathan Bastard¹

Benjamin Taisne¹

Julie Figoni

Alexandra Mailles

Julien Durand¹

Myriam Fayad¹

Laurence Josset

Anna Maisa¹

Sylvie Van Der Werf

Isabelle Parent Du Châtelet¹

Sibylle Bernard-Stoecklin¹
email: sibylle.bernard-stoecklin@santepubliquefrance.fr

Sibylle Bernard

Jonathan Bastard

Taisne Benjamin

Figoni Julie

Mailles Alexandra

Durand Julien

Fayad Myriam

Rapid communication

Since the first reports in summer 2020, SARS-CoV-2 reinfections have raised concerns about the immunogenicity of the virus, which will affect SARS-CoV-2 epidemiology and possibly the burden of COVID-19 on our societies in the future. This study provides data on the frequency and characteristics of possible reinfections, using the French national COVID-19 testing database. The Omicron variant had a large impact on the frequency of possible reinfections in France, which represented 3.8% of all confirmed COVID-19 cases since December 2021.

Reinfections by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) have been reported early after its emergence and rapid spread worldwide. The aim of this study was to describe the frequency and characteristics of possible reinfections with SARS-CoV-2 detected in France since March 2021.

Temporal distribution of possible SARS-CoV-2 reinfections

We identified possible SARS-CoV-2 reinfections in the French national coronavirus disease (COVID-19) testing database. A case was defined as a person with at least two positive tests (by RT-qPCR, reverse transcription loop-mediated isothermal amplification (RT-LAMP) or antigen tests, excluding self-administered antigen tests, and irrespective of the variant involved) performed at least 60 days apart between 1 January 2021 and 20 February 2022. Because of changes in the patient anonymisation method in the database in early 2021, we could identify possible reinfections starting on 2 March 2021, but not those following a first episode or more in 2020.

From 2 March 2021 to 20 February 2022, 589,702 possible reinfections were identified, including 584,129 people with two episodes (99.1%), 5,485 (0.9%) with three, 66 (< 0.1%) with four, and 22 people (< 0.1%) with between five and seven distinct episodes. We excluded all possible cases of SARS-CoV-2 reinfection with more than two distinct episodes, because (i) they were more likely to reflect persistent viral shedding in possibly immunosuppressed people than actual reinfections, and (ii) they represented less than 1% of the possible cases of reinfection. Among the 584,129 analysed cases, 390,835 (66.9%) had a first episode of infection between 1 January and 20 June 2021 (i.e. before the epidemic wave in summer 2021) (Figure 1A), and 551,182 (94.4%) had a second episode from 6 December 2021 onwards, i.e. after the introduction and rapid spread of the Omicron (Phylogenetic Assignment of Named Global Outbreak (Pango) lineage designation B.1.1.529) variant in France (Figure 1B).

Possible cases of reinfection represented 3.1% of all confirmed COVID-19 cases detected between 2 March 2021 and 20 February 2022. This proportion varied over time, representing 0.8% in the period until week 48/2021 and 3.8% from the time of emergence of the Omicron variant (i.e. from week 49/2021 onwards, Figure 1C). In week 7/2022, possible cases of reinfection represented 4.5% of all confirmed cases.

Characteristics of possible cases of SARS-CoV-2 reinfection

The mean time interval between two episodes of infection was 244 days, and the median was 267 days (interquartile range: 166-314). In a majority of cases (69.1%), this time period was 180 days or more (see Supplementary Table S1 for the numbers of possible reinfections by time interval).

The proportion of women, healthcare professionals and people aged 18 to 40 years was significantly higher among possible cases of reinfection compared with all confirmed COVID-19 cases (p < 0.001). In contrast, people younger than 18 years and older than 40 years were less frequent among possible cases of reinfection (p < 0.001) (Table ).

The proportion of possible cases of reinfection reporting symptoms during their second episode was similar to that of all confirmed cases (p > 0.05) (Table ). However, the shorter the time between two episodes, the smaller this proportion (Figure 2A). Symptoms were also less frequent during the second episode in people younger than 18 years or older than 60 years (Figure 2B).

Impact of Omicron on possible reinfections with SARS-CoV-2

In France, an RT-PCR screening strategy for SARS-CoV S2). Possible reinfections with identical screening results for both episodes were rare, with a suspicion of Alpha, Delta and Omicron in both episodes in 613 (1.0%), 1,332 (2.2%) and 100 cases (0.2%), respectively.

Between week 46/2021 and week 3/2022, the proportion of infections suspected to be caused by the Omicron variant was systematically higher among possible reinfections than among all confirmed COVID-19 cases, even though the proportion of suspected Omicron infections rapidly increased in both categories (Figure 3C).

Discussion

This study provides original data on possible reinfections by SARS-CoV-2 (i.e. people with two positive tests for SARS-CoV-2 at least 60 days apart) in France, generated from a nationwide, exhaustive database of COVID-19 testing results. While possible reinfections were a rare event (< 1% of all confirmed COVID-19 cases) during most of the study period, their frequency increased dramatically after the emergence and spread of the Omicron variant (December 2021 to February 2022), representing more than 4% of total confirmed COVID-19 cases in mid-February. This phenomenon could be due to the waning of post-infection and post-vaccination immunity in people infected and/ or vaccinated earlier in the pandemic. Moreover, these reinfections most probably reflect the immune escape characteristics of Omicron. This is consistent with other reports published worldwide [START_REF] Altarawneh | Protection against the Omicron variant from previous SARS-CoV-2 infection[END_REF][START_REF] Eggink | Increased risk of infection with SARS-CoV-2 Omicron BA.1 compared with Delta in vaccinated and previously infected individuals, the Netherlands, 22 November 2021 to 19[END_REF][START_REF] Kuhlmann | Breakthrough infections with SARS-CoV-2 omicron despite mRNA vaccine booster dose[END_REF][START_REF]Weekly epidemiological update on COVID-19 -22[END_REF][START_REF] Planas | Considerable escape of SARS-CoV-2 Omicron to antibody neutralization[END_REF][START_REF] Pulliam | Increased risk of SARS-CoV-2 reinfection associated with emergence of the Omicron variant in South Africa[END_REF][START_REF] Cele | Omicron extensively but incompletely escapes Pfizer BNT162b2 neutralization[END_REF].

Interestingly, before the introduction and rapid diffusion of Omicron in France in early December 2021, the proportion of possible reinfections among confirmed COVID-19 cases fluctuated markedly, with high values observed during inter-epidemic periods, in June and September 2021. This could reflect the fact that during these periods of lower incidence, SARS-CoV-2 mostly circulated within populations with a higher exposure to the virus and who, possibly related to their profession, socioeconomic status or lower compliance to physical distancing, were more likely to have previously been infected and remained highly exposed during interepidemic periods.

The same hypothesis could explain the higher proportion of young adults and healthcare professionals among possible reinfections compared with all confirmed COVID-19 cases. On the contrary, the lower proportion of older adults could be due to a generally lower exposure to SARS-CoV-2 (e.g. better compliance to prevention measures such as physical distancing) and to a higher vaccination coverage (in Supplementary Table S3, we provide the vaccination coverage by age group during the study period), especially for those with a booster dose. This higher vaccination coverage could also explain why a lesser proportion of possible Error bars represent the 95% binomial proportion confidence interval. The dashed horizontal line represents the proportion of symptomatic infections in all confirmed COVID-19 cases detected throughout the study period (see Table ). Proportions are calculated among cases who provided information on the presence/absence of symptoms (values were missing in 8.3% of possible reinfections for the symptom reporting variable). reinfections were symptomatic in this group compared with younger adults.

The majority of possible reinfections occurred more than 6 months after the first infection episode. However, ca 6% of possible reinfections had an interval of 60-89 days between episodes, indicating that a 90-day cut-off in the case definition of a SARS-CoV-2 reinfection -a cut-off adopted by several health agencies [START_REF]Threat Assessment Brief: Reinfection with SARS-CoV: considerations for public health response[END_REF] -could lead to under-detection of reinfections.

On the other hand, a fraction of the possible cases of reinfection -especially those occurring within a 90-day period -may be false-positives caused by prolonged viral detection, particularly in immunosuppressed [START_REF] Corey | SARS-CoV-2 variants in patients with immunosuppression[END_REF][START_REF] Meiring | Prolonged shedding of SARS-CoV-2 at high viral loads amongst hospitalised immunocompromised persons living with HIV, South Africa[END_REF] or severely ill people [START_REF] Saud | Mechanically ventilated patients shed high titre live SARS-CoV2 for extended periods from both the upper and lower respiratory tract[END_REF][START_REF] Kim | Duration of culturable SARS-CoV-2 in hospitalized patients with Covid-19[END_REF]. This could explain the detection of several cases with five or more positive tests at least 60 days apart. However, the proportion of false-positive cases of reinfection is likely to be negligible with a 60-day threshold, based on available literature [START_REF] Cevik | SARS-CoV-2, SARS-CoV, and MERS-CoV viral load dynamics, duration of viral shedding, and infectiousness: a systematic review and meta-analysis[END_REF][START_REF] Kissler | Viral dynamics of acute SARS-CoV-2 infection and applications to diagnostic and public health strategies[END_REF].

The proportion of cases reporting symptoms during reinfection was similar to that of all confirmed COVID-19 cases. However, we observed a smaller proportion of symptomatic cases among possible reinfections with a shorter interval between infections. This suggests that the immune response may partly protect from a symptomatic reinfection shortly after a primary infection, but that such an effect tends to wane over time. Further work is needed to investigate this hypothesis.

This study has several limitations. Firstly, virological (e.g. viral load, sequencing data) and epidemiological (e.g. contact with a confirmed COVID-19 case before the second episode) information was not available, which could affect the specificity of our case definition for possible reinfection. Secondly, we did not have access to information on hospitalisation and vaccination status of possible cases of reinfection, which prevented a thorough characterisation of possible reinfections. In the future, matching the COVID-19 testing database with the national hospitalisation and vaccination databases would allow to further characterise the risk of SARS-CoV-2 reinfection. Finally, owing to technical specificities, we were not able to identify in this study possible reinfections with a first episode that occurred in 2020. However, the study period still allowed highlighting the impact of the Omicron variant on reinfections by comparing the periods before and after its emergence. Moreover, the available literature on the frequency of reinfections before the emergence of Omicron indicates that SARS-CoV-2 reinfections were rare events [START_REF]National flu and COVID-19 surveillance report: 3[END_REF][START_REF] Hansen | Assessment of protection against reinfection with SARS-CoV-2 among 4 million PCR-tested individuals in Denmark in 2020: a population-level observational study[END_REF][START_REF] Kojima | A systematic review of the protective effect of prior SARS-CoV-2 infection on repeat infection[END_REF][START_REF] Muzio | SARS-CoV-2 reinfection is a new challenge for the effectiveness of global vaccination campaign: a systematic review of cases reported in literature[END_REF][START_REF] Abu-Raddad | Assessment of the risk of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) reinfection in an intense reexposure setting[END_REF]. We therefore assume that the total number of undetected possible reinfections is small.

Conclusion

The data presented here on possible SARS-CoV-2 reinfections provide valuable information on their characteristics and frequency over time. As it appears very unlikely that SARS-CoV-2 will be eradicated, monitoring the frequency of reinfections will be particularly useful in the future, especially in case of a new emerging variant. These data will be useful for decision-making regarding prevention and control strategies.

License, supplementary material and copyright

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC BY 4.0) Licence. You may share and adapt the material, but must give appropriate credit to the source, provide a link to the licence and indicate if changes were made.

Any supplementary material referenced in the article can be found in the online version.

This article is copyright of the authors or their affiliated institutions, 2022.

Figure 1

 1 Figure 1Distribution of the first and second episode of infection for possible cases of SARS-CoV-2 reinfection, and proportion of possible reinfections among all confirmed COVID-19 cases, by week of sampling, France, January 2021-February 2022 (n = 584,129)

  -2 mutations has been in place since 23 January 2021, with 26% of all confirmed COVID-19 cases screened for mutations carried by different SARS-CoV-2 variants of concern from June 2021 to February 2022. The targeted mutations changed over time in order to adjust to the variants circulating in France and abroad. We defined five categories of screening results for this analysis: suspicion of Alpha (B.1.1.7), Beta/Gamma (B.1.351/P.1), Delta (B.1.617.2), Omicron and 'Other', the latter including any other interpretable mutation screening result(see the Supplement for a summary of the RT-PCR screening strategy in France). In this section, we analysed only the possible reinfections that occurred in mainland France (i.e. 565,149 possible cases of reinfection, or 97% of the total). An interpretable screening result was available for the first and/or the second episode in 53.5% (n = 302,604) of possible reinfection cases. First episodes were caused by several variants, while the large majority (86.1%) of the 117,601 possible cases of reinfection with an interpretable screening result on their second episode were most probably caused by Omicron. Moreover, 10.6% (n = 59,760) of the 565,149 possible reinfections had an interpretable screening result for both episodes (Figure3A and B). For 43.9% of them (n = 26,229), the results indicated a suspicion of infection with Alpha during the first episode and with Omicron during the second episode, and 31.5% (n = 18,842) had a suspicion of infection by Delta during the first episode and by Omicron during the second episode (see Figure3Aand and the combinations of mutation screening results for both episodes in Supplementary Table

Figure 2 COVID- 19 :

 219 Figure 2 Proportion of possible cases of SARS-CoV-2 reinfection reporting symptoms during the second episode of infection, stratified by time between episodes and by age, France, March 2021-February 2022 (n = 535,539)

Figure 3 COVID- 19 :

 319 Figure 3Suspected variants in possible cases of SARS-CoV-2 reinfection, by week of sampling, mainland France, January 2021-February 2022 (n = 302,604)

Table

  Characteristics of possible cases of SARS-CoV-2 reinfection, compared with all confirmed COVID-19 cases, France, 2 March 2021-20 February 2022 Proportion among cases with available information . The percentage of missing values was 1.0% for the sex variable, 6.8% for healthcare professional status, < 0.1% for the age variable and 7.8% for symptom reporting d Reporting symptoms at the time of sampling during the second episode of infection.

		Possible cases of SARS-CoV-2 reinfection	All confirmed cases of COVID-19
		(n = 584,129)		(n = 18,661,139)		p value a
		n	% b	n	% c
	Female	324,253	55.6	9,862,846	53.4	< 0.001
	Male	259,325	44.4	8,623,428	46.6	< 0.001
	Healthcare professional	33,298	6.2	615,074	3.5	< 0.001
	Symptoms reported	274,508 d	51.3	8,831,873	51.3	> 0.05
	Age group (years)
	< 18	137,915	23.6	5,052,111	27.1	< 0.001
	18-40	298,230	51.1	7,127,055	38.2	< 0.001
	41-60	116,799	20.0	4,495,255	24.1	< 0.001
	61-80	22,420	3.8	1,584,451	8.5	< 0.001
	> 80	8,265	1.4	397,037	2.1	< 0.001
	COVID-19: coronavirus disease; SARS-CoV-2: severe acute respiratory syndrome coronavirus 2.

a p value of a two-sample binomial proportion test. b Proportion among cases with available information. The percentage of missing values was 0.1% for the sex variable, 7.3% for healthcare professional status, 0.1% for the age variable and 8.3% for symptom reporting. c

Acknowledgements

We would like to thank Daniel Lévy-Bruhl, Didier Che, Bruno Coignard, Justine Schaeffer, Lucie Fournier, Patrick Rolland and Guillaume Spaccaferri (Santé publique France), Sylvie Behillil and Vincent Enouf (National Reference Center for Respiratory Viruses, Institut Pasteur, Paris) as well as Bruno Lina (National Reference Center for Respiratory Viruses, Hospices Civils de Lyon, Lyon), for their contribution to the surveillance of SARS-CoV-2 reinfections in France, as well as their input in the interpretation of reinfection indicators. We would also like to acknowledge all the people involved in COVID-19 surveillance activities at national and regional levels.

Ethical approval

An ethical approval for this study was not necessary, because data were not identifiable back to the patients from whom they came from.

Conflict of interest

None declared.

Authors' contributions

Jonathan Bastard, Julie Figoni, Alexandra Mailles, Laurence Josset, Sylvie van der Werf, Isabelle Parent du Châtelet and Sibylle Bernard-Stoecklin contributed to the implementation of the surveillance of SARS-CoV-2 reinfections in France. Benjamin Taisne, Julien Durand and Myriam Fayad provided the data used in this study and Jonathan Bastard performed the data analysis. Jonathan Bastard, Benjamin Taisne, Julie Figoni, Alexandra Mailles, Julien Durand, Myriam Fayad, Laurence Josset, Anna Maisa, Sylvie van der Werf, Isabelle Parent du Châtelet and Sibylle Bernard-Stoecklin designed this study and contributed to data interpretation. Jonathan Bastard and Sibylle Bernard-Stoecklin wrote the manuscript with input from all authors.