Alexis Poncet

Olivier Bénichou

Vincent Démery

Daiki Nishiguchi

Pair correlation of dilute Active Brownian Particles: from low activity dipolar correction to high activity algebraic depletion wings

We study the pair correlation of Active Brownian Particles at low density using numerical simulations and analytical calculations. We observe a winged pair correlation: while particles accumulate in front of an active particle as expected, the depletion wake consists of two depletion wings. In the limit of soft particles, we obtain a closed equation for the pair correlation, allowing us to characterize the depletion wings. In particular, we unveil two regimes at high activity where the wings adopt a self-similar profile and decay algebraically. We also perform experiments of self-propelled Janus particles and indeed observe the depletion wings.

I. INTRODUCTION

The pair correlation function has played a pivotal role in our understanding of the structure of equilibrium liquids [START_REF] Hansen | Theory of Simple Liquids[END_REF]; the same is to be expected for active liquids, such as bacterial colonies [START_REF] Zhang | Collective motion and density fluctuations in bacterial colonies[END_REF][START_REF] Nishiguchi | Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria[END_REF], flocks of starlings [START_REF] Cavagna | Scale-free correlations in starling flocks[END_REF][START_REF] Cavagna | The physics of flocking: Correlation as a compass from experiments to theory[END_REF] or colloidal rollers [START_REF] Bricard | Emergence of macroscopic directed motion in populations of motile colloids[END_REF][START_REF] Geyer | Sounds and hydrodynamics of polar active fluids[END_REF][START_REF] Geyer | Freezing a Flock: Motility-Induced Phase Separation in Polar Active Liquids[END_REF], or assemblies of Janus particles [START_REF] Palacci | Sedimentation and Effective Temperature of Active Colloidal Suspensions[END_REF][START_REF] Ginot | Nonequilibrium Equation of State in Suspensions of Active Colloids[END_REF][START_REF] Zhang | Active phase separation by turning toward the crowd[END_REF][START_REF] Iwasawa | Algebraic correlations and anomalous fluctuations in ordered flocks of Janus particles fueled by an AC electric field[END_REF]. It has indeed been used to quantify the order in bacterial colonies [START_REF] Zhang | Collective motion and density fluctuations in bacterial colonies[END_REF] and to infer the interactions in bird flocks [START_REF] Cavagna | Scale-free correlations in starling flocks[END_REF][START_REF] Cavagna | The physics of flocking: Correlation as a compass from experiments to theory[END_REF]. However, its broad utilization is precluded by the lack of analytical results on its general form, notably little is known about its regions of positive and negative sign, even in the homogeneous phase and in the absence of alignment interactions [START_REF] Bialké | Microscopic theory for the phase separation of self-propelled repulsive disks[END_REF][START_REF] Alexandre | Pressure and Phase Equilibria in Interacting Active Brownian Spheres[END_REF][START_REF] Marini | Towards a statistical mechanical theory of active fluids[END_REF].

The theoretical prediction of the pair correlation is indeed an important challenge, which has been undertaken for minimal models without alignment, such as Active Browian or Ornstein-Uhlenbeck particles (ABP or AOUP). So far, even for these minimal models, the study of the pair correlation focused mostly on the explanation of the phase separation that occurs at large activity [START_REF] Cates | Motility-Induced Phase Separation[END_REF][START_REF] Digregorio | Full Phase Diagram of Active Brownian Disks: From Melting to Motility-Induced Phase Separation[END_REF][START_REF] Klamser | Thermodynamic phases in two-dimensional active matter[END_REF]. Two main approaches have been followed. The first consists in computing the angular average of the pair correlation due to an activity represented by a persistent translational noise. An attractive term has been found, and interpreted as a higher tendency to phase separate [START_REF] Marini | Towards a statistical mechanical theory of active fluids[END_REF][START_REF] Farage | Effective interactions in active Brownian suspensions[END_REF][START_REF] Fodor | How Far from Equilibrium Is Active Matter?[END_REF]. However, this approach does not, by nature, retain the angular dependence of the correlations. The second approach consists of a quantitative prediction of the polar pair correlation using a closure of the many body Smoluchowski equation leading to the effective velocity, whose decrease with density may explain the phase separation [START_REF] Bialké | Microscopic theory for the phase separation of self-propelled repulsive disks[END_REF][START_REF] Alexandre | Pressure and Phase Equilibria in Interacting Active Brownian Spheres[END_REF][START_REF] Takatori | Swim Pressure: Stress Generation in Active Matter[END_REF][START_REF] Härtel | Three-body correlations and conditional forces in suspensions of active hard disks[END_REF][START_REF] Bickmann | Predictive local field theory for interacting active Brownian spheres in two spatial dimensions[END_REF]. This method requires a numerical solution of nonlinear equations, and does not provide explicit predictions. Finally, up to now * vincent.demery@espci.psl.eu there is no analytical characterization of the pair correlation.

In this article, we address the global shape of the pair correlation of ABP in the dilute and homogeneous regime [START_REF] Digregorio | Full Phase Diagram of Active Brownian Disks: From Melting to Motility-Induced Phase Separation[END_REF]. We first observe a winged pair correlation in numerical simulations of ABP: while particles accumulate in front of an active particle as expected, the depletion wake consists of two depletion wings (Fig. 1). We then resort to a linearized Dean equation to obtain a closed equation for the pair correlation. Solving this equation under different limits, we unveil three different regimes for the correlations, which we organize on a phase diagram. In two regimes, the wings adopt a self-similar profile and decay algebraically. Last, we measure the pair correlation in experiments of self-propelled Janus particles, and qualitatively observe the depletion wings.

II. MODEL AND NUMERICAL SIMULATIONS

First, we perform numerical simulations of ABP in dimension two interacting via a pairwise harmonic potential V (r) = 2 (1-r/a) 2 for r < a, where sets the interaction strength and a is the particle diameter (Fig. 1(a), see App. A 1 for details). The position r i (t) and orientation

θ i (t) of the particle i follow ṙi = -γ -1 ∇ i j =i V (r i -r j ) + U êθi + √ 2Dη i , (1) θi
= 2D r ν i , ( 2
)
where U is the propulsion velocity, D is the translational diffusion coefficient, D r is the rotational diffusion coefficient, êθ is the unit vector with polar angle θ and η i and ν i are normalized Gaussian white noises. The diameter a of the particles and the friction coefficient γ can be set to one through a rescaling; the parameters , U , D and D r are given in arbitrary units and only their relative values are important. We use a low area fraction φ 0.04 to fo- cus on two-body effects and remain in the homogeneous phase [START_REF] Cates | Motility-Induced Phase Separation[END_REF][START_REF] Digregorio | Full Phase Diagram of Active Brownian Disks: From Melting to Motility-Induced Phase Separation[END_REF][START_REF] Klamser | Thermodynamic phases in two-dimensional active matter[END_REF] (see discussion in App. A 3).

The polar pair correlation C(r, θ, θ ) is defined from the density field for the positions r i (t) and orientations

θ i (t), f (r, θ, t) = i δ(r i (t) -r)δ(θ i (t) -θ): C(r, θ, θ ) = f (0, θ)f (r, θ ) [ρ/(2π)] 2 - δ(r)δ(θ -θ ) ρ/(2π) -1, (3)
where ρ = 4φ/(πa 2 ) is the density; the correlation of a particle with itself is removed in the second term, and the r → ∞ limit is removed in the third. From rotational invariance, C(r, 0, θ ) contains all the information in C(r, θ, θ ). We focus on the density of particles in the reference frame of a given particle (blue axes in Fig. 1(a) [START_REF] Zhang | Collective motion and density fluctuations in bacterial colonies[END_REF]), which retains the polar character of the correlations:

B(r) = 1 2π 2π 0 C(r, 0, θ )dθ . (4)
Note that if translational and rotational degrees of freedom decouple, for instance in absence of propulsion, the isotropic pair correlation is recovered: C(r, θ, θ ) = B(r) = h(r) [START_REF] Hansen | Theory of Simple Liquids[END_REF].

U → 0 D → 0 D r → 0 x ∼ -1 3 p y 4 3
x ∼ . At smaller rotational diffusion, D r ≤ 1, as expected, the correlation is positive in front of the particle, indicating an accumulation of other particles. However, while a depletion wake is expected behind the active particle, as in active microrheology [START_REF] Meyer | Laser tweezer microrheology of a colloidal suspension[END_REF][START_REF] Démery | Generalized Langevin equations for a driven tracer in dense soft colloids: construction and applications[END_REF] or in driven binary mixtures [START_REF] Poncet | Universal long ranged correlations in driven binary mixtures[END_REF], the depletion concentrates in two "wings" on the sides of the particle. As D r decreases, the characteristic depletion wings appear, their length increases and their curvature decreases.

III. THEORY AND LIMITING REGIMES

To rationalize these different regimes, we consider the lengthscales and dimensionless parameters of the problem. In the dilute limit considered here, the structure of the pair correlation beyond the size of a particle is controlled by the parameters U , D and D r , which combine into three lengthscales

r = D U ; U = D D r ; p = U D r , (5)
whose relative values are set by the Péclet number

Pe = U √ DD r = U r = p U . (6)
The Péclet number varies between Pe = 10 and 316 in Fig. 1(b)-(e). For spherical particles the coefficients D and D r are related through D r ∼ D/a 2 ; here we regard them as independent parameters to disentangle the effects of translational and rotational diffusion. The form of the pair correlation thus depends on the Péclet number and the relative value of the observation length r with the three lengthscales, so that the numerical simulations can be placed on a parameter plane (blue arrows in Fig. 2).

To account for the observed correlations and characterize the shape of the depletion wings, we resort to a linearized Dean equation for the density field f (r, θ, t) [START_REF] Démery | Generalized Langevin equations for a driven tracer in dense soft colloids: construction and applications[END_REF][START_REF] Poncet | Universal long ranged correlations in driven binary mixtures[END_REF][START_REF] Dean | Langevin equation for the density of a system of interacting Langevin processes[END_REF][START_REF] Farrell | Pattern Formation in Self-Propelled Particles with Density-Dependent Motility[END_REF][START_REF] Dean | Relaxation of the thermal Casimir force between net neutral plates containing Brownian charges[END_REF][START_REF] Démery | The conductivity of strong electrolytes from stochastic density functional theory[END_REF], which is valid for weak interactions. At low density, we show in App. B 1 that the pair correlation satisfies:

2D∇ 2 + D r (∂ 2 θ + ∂ 2 θ ) + U (ê θ -êθ ) • ∇ C = - 2 γ ∇ 2 V.
(7) Solving this equation numerically (App. C), we obtain an excellent agreement with the numerical simulations (Fig. 1(b)-(e)). We now show that this equation captures the structure of the pair correlation by examining its limiting regimes U → 0, D r → 0, and D → 0.

For small propulsion velocity U , which corresponds to the left side of the phase diagram, Eq. ( 7) can be solved perturbatively as shown in App. B 2. To order one in U , we get a dipolar correction to the equilibrium radial correlation h eq (r): B(r, θ) = h eq (r) + U cos(θ)B 1 (r), where the first Fourier coefficient B 1 (r) decays exponentially over a length U . This prediction is compared to the numerical simulations in Fig. 3(a)-(c); a quantitative agreement with the prediction is obtained, without any adjustable parameter. Note that at order U 2 , an attractive correction to the equilibrium pair correlation h eq (r) is found, which is compatible with the results obtained when the activity is introduced in the form of a colored noise [START_REF] Marini | Towards a statistical mechanical theory of active fluids[END_REF][START_REF] Farage | Effective interactions in active Brownian suspensions[END_REF][START_REF] Fodor | How Far from Equilibrium Is Active Matter?[END_REF].

We turn to the large activity regime, and focus first on the limit D r = 0, where the wings are entirely deployed (Fig. 3(d)); this limit corresponds to the bottom right corner of the phase diagram (Fig. 2). Equation ( 7) can be solved in Fourier space (App. B 3), yielding

B(k) = - k 2 Ṽ (k) 2πU 2π 0 dθ 2 r k 2 + ik • (ê 0 -êθ ) . ( 8
)
The integrand has the form of the correlations in a driven binary mixture [START_REF] Poncet | Universal long ranged correlations in driven binary mixtures[END_REF], and the integration over θ mirrors that here there are particles moving in all directions. At distances below r , Eq. ( 8) yields a dipolar correlation, as in the small activity regime. Beyond r , the form of the correlations can be obtained by performing the integral and focusing on the singularity at the origin in Fourier space:

B(x, y) r r Ṽ (0) D 0 1 y 2 F x 1/3 r |y| 2/3 , (9)
where the explicit form of the scaling function F (u) is given in App. B 3. The scaling ( 9) can be tested by plot-ting rescaled cuts of the correlation at different values of y (Fig. 3(e)); as predicted, the cuts collapse to the scaling function F (u).

We now address the shape of the wings in presence of a finite rotational diffusion D r . We take the limit D = 0, which corresponds to the top right corner of the phase diagram (Fig. 2), where the only available lengthscale is the persistence length p (App. B 4). Beyond p , particles loose the memory of their orientation and their correlation vanish. Below p , the solution of Eq. ( 7) is dominated by angles close to (θ, θ ) (0, 0) and (π, π), and we obtain the following scaling form:

B(x, y) r p Ṽ (0) D r 4 p y 4 G 1/3 p x |y| 4/3 ; ( 10
)
the function G can be obtained numerically. This prediction is in very good agreement with the numerical simulations (Fig. 3(f),(g)). Last, the transition with the scaling form obtained in the limit D r = 0 can be obtained by matching the widths of the two profiles, x ∼ 1/3 r y 2/3 (Eq. ( 9)) and x ∼ -1/3 p y 4/3 (Eq. ( 10)), leading to y ∼ r p = U .

The structure of the correlations is now characterized (Fig. 2). For Pe < 1, the activity generates a dipolar correction to the equilibrium correlations for lengths r < U . For Pe > 1, the dipolar correction crosses-over to depletion wings at a scale r ; the shape of the wings is given by the scaling forms (9) for r < r < U and (10) for U < r < p ; the wings decay exponentially beyond p . The behavior of the correlations is summarized on the phase diagram, Fig. 2.

For weak interactions, we have obtained a quantitative agreement between our predictions derived from Eq. ( 7) and the numerical simulations. Moreover, the left hand side of Eq. ( 7), which controls the structure of the correlations, is also present in the two-body Smoluchowski equation [START_REF] Härtel | Three-body correlations and conditional forces in suspensions of active hard disks[END_REF], which is valid in the dilute limit for any interaction strength. We conclude that the structure of the correlations predicted here holds in the dilute limit for any interaction strength, which we check in App. A 2 with numerical simulations of hard particles.

We finally note that, in practice, D and D r are not taken as independent parameters. For spherical particles, where D ∼ a 2 D r , the Péclet number defined here takes the more common forms Pe = aU/D = U/(aD r ), and U ∼ a. At small Péclet, the dipolar correction, which decays over a length U , should thus be barely observable. At large Péclet, depletion wings with the shape [START_REF] Ginot | Nonequilibrium Equation of State in Suspensions of Active Colloids[END_REF] should be observed for a < r < p = aPe. Some theoretical studies assume D = 0 [START_REF] Solon | Active brownian particles and run-and-tumble particles: A comparative study[END_REF][START_REF] Basu | Active Brownian motion in two dimensions[END_REF][START_REF] Caprini | Spontaneous Velocity Alignment in Motility-Induced Phase Separation[END_REF] and define the Péclet number as Pe = U/(aD r ); here also depletion wings with the shape (10) should be observed below p .

IV. EXPERIMENTS

To confront our findings to real systems, we perform experiments with Janus particles, whose dynamics is similar to that of ABP [START_REF] Ginot | Nonequilibrium Equation of State in Suspensions of Active Colloids[END_REF][START_REF] Howse | Self-Motile Colloidal Particles: From Directed Propulsion to Random Walk[END_REF][START_REF] Mano | Optimal run-andtumble-based transportation of a Janus particle with active steering[END_REF]. Our Janus particles are propelled by a vertical AC electric field and swim in a two-dimensional horizontal plane [START_REF] Nishiguchi | Mesoscopic turbulence and local order in janus particles self-propelling under an ac electric field[END_REF][START_REF] Nishiguchi | Flagellar dynamics of chains of active Janus particles fueled by an AC electric field[END_REF] showing that they are a robust feature of pair correlations of self-propelled particles at high activity. At short range, qualitative differences with the numerical simulations can be noted, such as depletion behind the active particle (x < -a, y = 0) before correlations turn positive. These differences can be attributed to unavoidable hydrodynamic or electrostatic interactions [START_REF] Zhang | Active phase separation by turning toward the crowd[END_REF]. Depletion wings are present at larger distances, but their decay cannot be characterized due to insufficient statistics. Whether or not these wings follow the scaling laws predicted here or exhibit different exponents is an open question.

V. CONCLUSION

We have unveiled two regimes where active polar particles without alignment interactions have a pair correlation with a self-similar shape and an algebraic decay characterized by anomalous exponents. In presence of velocity-orientation coupling [START_REF] Iwasawa | Algebraic correlations and anomalous fluctuations in ordered flocks of Janus particles fueled by an AC electric field[END_REF][START_REF] Deseigne | Collective Motion of Vibrated Polar Disks[END_REF][START_REF] Dauchot | Dynamics of a self-propelled particle in a harmonic trap[END_REF] or alignment interactions [START_REF] Bricard | Emergence of macroscopic directed motion in populations of motile colloids[END_REF][START_REF] Zhang | Active phase separation by turning toward the crowd[END_REF][START_REF] Iwasawa | Algebraic correlations and anomalous fluctuations in ordered flocks of Janus particles fueled by an AC electric field[END_REF], we may expect distinct scaling laws to appear in the correlations. As we have shown, the cor-relations have a rich structure even without three-body interactions, it would therefore be instructive to measure them in a dilute configuration first, and then to see how they evolve as density increases. line on the phase diagram for ABP [START_REF] Digregorio | Full Phase Diagram of Active Brownian Disks: From Melting to Motility-Induced Phase Separation[END_REF]. As MIPS is a many-body effect, the density should be far below this line to remain in the dilute regime. For instance, for Pe = 100, ρ MIPS 0.18 (φ MIPS 0.14).

The second way is to look at measured pair correlations. By definition, in the dilute regime the pair correlation does not depend on the density. We plot the pair correlation for densities ρ = 0.01, 0.02 and 0.05 for D = 0 and Pe = 100 on Fig. 6. We observe an important difference between ρ = 0.02 and ρ = 0.05, while the difference between ρ = 0.01 and ρ = 0.02 is much smaller. We conclude that the limit of the dilute regime for this Péclet number sits around ρ 0.02 (φ 0.016).

= -∇ i j =i V (r i -r j ) + U êθi + √ 2Dη i (B1) θi = 2D r ν i (B2)
D is the translational diffusivity, D r is the rotational diffusivity, η i and ν i are Gaussian white noises of unit variance.

In this supplementary material, we set the particle diameter a = 1 and the friction coefficient γ = 1.

We define the density in position-orientation space f (r, θ, t) as

f (r, θ, t) = N i=1 ∞ m=-∞ f i (r, θ + 2mπ, t) (B3) f i (r, θ, t) = δ(r i (t) -r)δ(θ i (t) -θ). (B4)
We consider a smooth and fastly decaying test function ϕ(r, θ). By definition of f i ,

ϕ(r i (t), θ i (t)) = dr ∞ -∞ dθf i (r, θ, t)ϕ(r, θ). (B5)
Then, the time derivative of ϕ(r i (t), θ i (t)) can be written in two different ways:

d dt ϕ(r i (t), θ i (t)) = dr ∞ -∞ dθ ∂f i ∂t (r, θ, t)ϕ(r, θ) = dr ∞ -∞ dθf i (r, θ, t)(dt) -1 dϕ(r, θ), (B6)
with the differential dϕ given by the Itô formula [START_REF] Oksendal | Stochastic differential equations: an introduction with applications[END_REF],

dϕ = ∇ϕ • dr i + ∂ϕ ∂θ dθ i + 1 2 ∇ 2 ϕ(dr i ) 2 + 1 2 ∂ 2 ϕ ∂θ 2 dθ 2 i + ∂ ∂θ ∇ϕ • dr i dθ i (B7) = ∇ϕ •    -∇ i j V (r i -r j ) + U êθi    dt + Ddt∇ 2 ϕ + D r dt ∂ 2 ϕ ∂θ 2 . ( B8
)
The differentials are computed from Eqs. (B1) and (B2) (we assume ∇V (0) = 0). Performing integrations by part and recalling that ϕ is arbitrary, one obtains

∂f i ∂t = D∇ 2 f i + D r ∂ 2 f i ∂ 2 θ + ∇   f i N j=1 ∇V (r -r j (t))   -U e θ • ∇f i - √ 2D∇f i • η i -2D r ∂ ∂θ (f i ν i ). (B9)
Using Eq. (B3) and rearranging the noises like Dean [START_REF] Dean | Langevin equation for the density of a system of interacting Langevin processes[END_REF], we finally obtain the following Dean equation for f (r, θ, t),

∂ ∂t f (r, θ, t) = -∇J(r, θ, t) - ∂ ∂θ K(r, θ, t) (B10)
with the currents

J(r, θ, t) = -D∇f (r, θ, t) -f (r, θ, t) 2π 0 dθ(∇V * f )(r, θ, t) + f (r, θ, t)U êθ -f 1/2 (r, θ, t)η(r, θ, t) (B11) K(r, θ, t) = -D r ∂ ∂θ f (r, θ, t) -f 1/2 (r, θ, t)ν(r, θ, t). (B12)
The spatial convolution is defined by (f * g)(r) = dr f (r )g(r -r ). η and ν are Gaussian white noises with correlations

η α (r, θ, t)η β (r, θ, t) = 2Dδ αβ δ(r -r )δ(θ -θ )δ(t -t ), (B13) ν(r, θ, t)ν(r, θ, t) = 2D r δ(r -r )δ(θ -θ )δ(t -t ).
(B14)

b. Linearized Dean equation

The Dean equation for Active Brownian Particles is nonlinear with multiplicative noise. It is thus very difficult to tackle. Our approximation consists in linearizing it around an homogeneous density ρ which is the average density of particles. We write

f (r, θ, t) = ρ 2π + ρ 2π φ(r, θ, t). (B15)
The field φ is assumed to be of order 1 in ρ.

At the lowest order, the Dean equation (B10) becomes linear with additive noise.

∂φ ∂t = D∇ 2 + D r ∂ 2 ∂θ 2 -U êθ • ∇ φ + ρ 2π 2π 0 dθ (∇ 2 V * φ)(θ ) + ∇ • η + ∂ν ∂θ . (B16) c.

Equation for the correlations

We define the following correlations

C(r 1 , r 2 , θ 1 , θ 2 ) = φ(r 1 , θ 1 )φ(r 2 , θ 2 ) , (B17) C(r, θ, θ ) = 1 ρ {C(0, r, θ, θ ) -δ(r)δ(θ -θ )} . (B18)
One checks that the definition of C(r, θ, θ ) is consistent with Eq. ( 1) the article. We use Itô calculus to compute the time evolution of C.

C(r 1 , r 2 , θ 1 , θ 2 , t + δt) -C(r 1 , r 2 , θ 1 , θ 2 , t) = φ(r 1 , θ 1 , t)δφ(r 2 , θ 2 , t) + δφ(r 1 , θ 1 , t)φ(r 2 , θ 2 , t) + δφ(r 1 , θ 1 , t)δφ(r 2 , θ 2 , t) . (B19)
Computing the terms from the linearized Dean equation (B16), one shows that

∂ t C(r 1 , r 2 , θ 1 , θ 2 ) = D(∇ 2 1 + ∇ 2 2 ) + D r (∂ 2 θ1 + ∂ 2 θ2 ) -U (ê θ1 • ∇ 1 + êθ2 • ∇ 2 ) C(r 1 , r 2 , θ 1 , θ 2 ) + ρ 2π 2π 0 dθ ∇ 2 1 V * C(r 1 , r 2 , θ , θ 2 ) + ∇ 2 2 V * C(r 1 , r 2 , θ 1 , θ ) + [2D∇ 1 ∇ 2 + 2D r ∂ θ1 ∂ θ2 ] δ(r 1 -r 2 )δ(θ 1 -θ 2 ). (B20)
We use the invariance of the system by translation and write the equation in terms of C(r, θ, θ ),

∂ t C(r, θ, θ ) = 2D∇ 2 + D r (∂ 2 θ + ∂ 2 θ ) + U (ê θ -êθ ) • ∇ C(r, θ, θ ) + 2∇ 2 V (r) + ρ 2π 2π 0 dθ ∇ 2 V * [C(r, θ, θ ) + C(r, θ , θ )] . (B21)
The conventions are such that the pair correlation function C(r, θ, θ ) is normalized with respect to the density ρ. Focusing on the low density regime (ρ → 0), we can neglect the convolution of C with the potential V . The equation that we consider is

∂ t C(r, θ, θ ) = ρ→0 2D∇ 2 + D r (∂ 2 θ + ∂ 2 θ ) + U (ê θ -êθ ) • ∇ C(r, θ, θ ) + 2∇ 2 V (r). (B22)
Note that for a passive system, the solution of this equation is the direct correlation function (the one involved in the Ornstein-Zernike equation). In the following, we are only interested in the stationary correlations which satisfy the following linear partial differential equation.

2D∇ 2 + D r (∂ θ + ∂ θ ) + U (ê θ -êθ ) • ∇ C(r, θ, θ ) = -2∇ 2 V (r). ( B23
)
By rotational invariance, C(r, θ, θ ) = C(R -θ • r, 0, θ -θ). Like in the article, we define the "profile seen by a particle in its reference frame",

B(r) = 1 2π 2π 0 C(r, 0, θ )dθ . (B24)

Small activity

At small activity, U 1, one can expand C in power of U .

C(x, θ, θ ) = C (0) (x) + U C (1) (x, θ, θ ) + U 2 C (2) (x, θ, θ ) + . . . (B25)
Let us write Eq (B23) in Fourier space, using the convention C(k, θ, θ , t) = dre -ikr C(r, θ, θ , t).

-2Dk 2 + D r (∂ θ + ∂ θ ) C(k, θ, θ ) = 2k 2 Ṽ -iU k • (ê θ -êθ ) C(k, θ, θ ). ( B26
)
The passive correlation (U = 0) doesn't depend on the angles, the order 0 of the equation above leads to

C(0) (k) = -Ṽ (k) D . (B27)
This is the usual Random Phase Approximation solution for direct correlations. At order 1, the equation to solve and its solution are

-2Dk 2 + D r (∂ 2 θ + ∂ 2 θ ) C(1) (k, θ, θ ) = -ik • (ê θ -êθ ) C(0) (k), (B28) C(1) (k, θ, θ ) = ik • (ê θ -êθ ) D(1) (k), (B29) D(1) (k) = C(0) (k) 2Dk 2 + D r . (B30)
Next, we solve the second order,

-2Dk 2 + D r (∂ 2 θ + ∂ 2 θ ) C(2) (k, θ, θ ) = 2k 2 (1 -êθ • êθ ) D(1) (k), (B31) C(2) (k, θ, θ ) = -D(1) (k) D + k 2 D(1) (k) Dk 2 + D r (ê θ • êθ ). (B32)
At the end of the day, the expansions at order U 2 of C(k, θ, θ ) and B(k) are

C(k, θ, θ ) = Ṽ (k) D -1 + U 2 D 2 1 2k 2 + -2 U - U D ik • (ê θ -êθ ) 2k 2 + -2 U + U 2 D 2 (ê θ • êθ ) 1 2k 2 + -2 U - 1 k 2 + -2 U (B33) B(k) = Ṽ (k) D -1 + U 2 D 2 1 2k 2 + -2 U - U 2πD ik x 2k 2 + -2 U (B34)
with the typical length scale U = D r /D.

Our goal is now to look at large distances, that is to say, small wave number k. We assume that Ṽ (k) is regular at 0 (short-range potential) and we make the substitution Ṽ (k) → Ṽ (0). We define the function G(k) and its inverse Fourier transform G(r) as

G(k) = 1 2k 2 + -2 U ⇔ G(r) = 1 4π K 0 r √ 2 U , (B35)
with K 0 the modified Bessel function of the second kind of order 0. In real space, the expansion of B(r) becomes,

B(r) ∼ r→∞ - Ṽ (0) D + U 2 Ṽ (0) D 3 G(r) -U Ṽ (0) 2πD 2 ∂G(r) ∂x (B36) B(r, θ) ∼ r→∞ - Ṽ (0) D + U 2 Ṽ (0) 4πD 3 K 0 r √ 2 U + U Ṽ (0) √ D r 2π √ 2D 3/2 K 1 r √ 2 U cos θ (B37)
K 1 is the modified Bessel function of the second kind of order 1. We detail the meaning of the terms. The first one in the passive correlation. The second term, scaling as U 2 is a positive contribution to the isotropic part of the correlations. As stated in the article, this has been argued, in the literature, to account for the motility induced phase separation. The last term is the dipolar correlation, at order U , on which we focused in the article. Eq. (B37) corresponds to the prediction in Fig. 3b-c. At large distance, the Bessel functions decay exponentially,

K 0 (r) ∼ r→∞ K 1 (r) ∼ r→∞ π 2r e -r . (B38)
Both the dipolar contribution (cos θ) and the additional isotropic part thus decay exponentially over the length scale U .

No rotational diffusion

We now focus on the limit of no rotational diffusion D r = 0. We easily obtain the Fourier transform C(r, θ, θ ) from Eq. (B23),

2D∇ 2 + U (ê θ -êθ ) • ∇ C(r, θ, θ ) = -2∇ 2 V (r), (B39) C(k, θ, θ ) = -2k 2 Ṽ (k) 2Dk 2 -iU k • (ê θ -êθ ) . ( B40
)
This solution is very similar to the one for a binary mixture of particles forced respectively by U êθ and U êθ [START_REF] Poncet | Universal long ranged correlations in driven binary mixtures[END_REF]. We integrate over θ to obtain B, this gives,

B(k) = - k 2 Ṽ (k) πD 2π 0 dθ 2k 2 -i -1 r k • (ê x -êθ ) = -2k 2 Ṽ (k) D 2k 2 -i -1 r k x 2 + -2 r k 2 (B41)
with the characteristic length scale r = D/U . k x = k • êx , k y = k • êy where êx and êy are the unit vector along the horizontal and vertical axes of the plane. We consider the limit of large distance, that is to say k → 0. We assume that Ṽ (k) is regular so that we can replace it by Ṽ (0). Furthermore, an analysis of leading terms gives k 2 y ∼ k 3

x . We keep up these leading terms:

B(k) ∼ k→0 -2 Ṽ (0)k 2 x D -2 r k 2 y -4i -1 r k 3 x . ( B42
)
We can inverse Fourier transform with respect to k y :

B(k x , y) = - 2 Ṽ (0) r πD k 2 x K 0 |y| -4i r k 3 x . (B43)
We then Fourier transform with respect to k x : B(x, y) = (2π) -1 dk x e ikxx B(k x , y). We perform the changes of variables q = -( r y 2 ) 1/3 k x , w = xu 1/3 ( r y 2 ) -1/3 and obtain

B(x, y) ∼ Ṽ (0) D 1 y 2 F x 1/3 r |y| 2/3
, (B44)

F (w) = 1 π 2 ∞ -∞ e -iqw q 2 K 0 2 iq 3 dq. (B45)
This is the scaling form mentioned in the article, and used in Fig. 3e.

No translational diffusion

We set D = 0. Looking at distances large compared to the particle diameter, we replace Ṽ (k) by Ṽ (0). The equation we consider is

(∂ 2 θ + ∂ 2 θ ) + i p k • (ê θ -êθ ) C(k, θ, θ ) = 2k 2 Ṽ (0) D r . ( B46
)
with the persistence length p = U/D r . At small p k (large distance compared to p ), the following development can be obtained,

C(k, θ, θ ) = -Ṽ (0) D r 1 + i p k • (ê θ -êθ ) + 2 p k 2 êθ • êθ + O ( p k) 3 . (B47)
This is a hint that C(k, θ, θ ) is analytical around k = 0, meaning that C(r, θ, θ ) decays fastly (e.g. exponentially) at distances large compared to p .

We now consider distances below p but still large compared to the particle diameter. We define the angles γ and γ in the reference frame of k: k • êθ = k cos γ and k • êθ = k cos γ , where k is the norm of k. We obtain

(∂ 2 γ + ∂ 2 γ ) + i p k(cos γ -cos γ ) C(k, γ, γ ) = 2k 2 Ṽ (0) D r , (B48)
and we study it in the regime p k 1. A numerical resolution at constant k (see Fig. 7) shows that C(k, γ, γ ) concentrates around the two points (γ, γ ) = (0, 0) and (π, π). We focus on (0, 0) around which the equation reads

(∂ 2 γ + ∂ 2 γ ) - i 2 p k(γ 2 -γ 2 ) C(k, γ, γ ) = 2 Ṽ (0) D r k 2 . ( B49
)
We realize that we can inject the following scalings

C(k, γ, γ ) ∼ pk 1 2 Ṽ (0) D r ( p k) 3/2 H γ( p k) 1/4 , γ ( p k) 1/4 . ( B50
)
From Eq. (B49), the function H(u, v), for u and v unbounded, is independent of k and is the solution of the linear partial differential equation

∂ 2 u + ∂ 2 v - i 2 u 2 -v 2 H(u, v) = 1. (B51)
Around (γ, γ ) = (π, π), the scalings are

C(k, γ, γ ) ∼ p k 1 2 Ṽ (0) D r ( p k) 3/2 H * (γ -π)( p k) 1/4 , (γ -π)( p k) 1/4 . (B52)
with H * the complex conjugate of H.

The scaling for B around γ = 0 is

B(k, γ) = 1 2π 2π 0 dγ C(k, γ, γ ) ∼ p k 1 Ṽ (0) πD r ( p k) 5/4 HB γ( p k) 1/4 , (B53)
H B (u) = ∞ -∞ dv H(u, v). ( B54
)
And around γ = π, one checks that

B(k, π -γ) ∼ p k 1 Ṽ (0) πD r ( p k) 5/4 H * B (π -γ)( p k) 1/4 (B55)
with H * B the complex conjugate of H B . We now switch from polar coordinates (k, γ) to cartesian coordinates (k x , k y ). We approximate first with respect to k y , then with respect to k x . Using the appropriate changes of variables, we obtain a scaling form for B(x, y), a). This form corresponds to Eq. ( 10) of the article. The rescaled cuts are plotted on Fig. 3g with the prediction (gray line) computed from a rescaling of the numerical solution of Eq. ( 7).

k x = k cos γ +k if γ 0 -k if γ π , k y = k sin γ kγ if γ 0 k(π -γ) if γ π . ( B56
B(x, y) = Ṽ (0) πD r 4 p y 4 G 1/3 p x |y| 4/3 , (B59) G(w) = 1 2π ∞ 0 dze iwz z 2 H + B (z 3/4 ) + 1 2π 0 -∞ dze iwz z 2 H - B (|z| 3/4 ) (B60) H ± B is the inverse Fourier transform of H± B , one has H - B (a) = (H + B ) * (-
Appendix C: Numerical integration of the theoretical equation

Equation in terms of three parameters

We consider the time-dependent equation (B22) for the correlations C at low density,

∂ t C(r, θ, θ ) = 2D∇ 2 + D r (∂ 2 θ + ∂ 2 θ ) + U (ê θ -êθ ) • ∇ C(r, θ, θ ) + 2∇ 2 V (r). (C1)
In polar coordinates, we write r = re φ . The later equation depends on four coordinates (plus time): (r, φ, θ, θ ). By performing a rotation of angle θ, the symmetries allow to reduce the problem to three parameters (r, α, β),

α = φ -θ β = θ -θ. ( C2
)
We write x = re α = (x, y), C is then a function only of x and β. Its time evolution is given by

∂ t C(x, β) = 2D∇ 2 + D r L angles + U (1 -cos β) ∂ ∂x -sin β ∂ ∂y C(x, β) + 2∇ 2 V (x), (C3) ∇ 2 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 (C4) L angles = y 2 ∂ 2 ∂x 2 + x 2 ∂ 2 ∂y 2 -2xy ∂ 2 ∂x∂y -x ∂ ∂x -y ∂ ∂y + 2 -y ∂ ∂x + x ∂ ∂y ∂ ∂β + 2 ∂ 2 ∂β 2 (C5)
It is important to note that B is given by the integration over β, B(x) = pixels within it (the weights are the pixels' values) [START_REF] Iwasawa | Algebraic correlations and anomalous fluctuations in ordered flocks of Janus particles fueled by an AC electric field[END_REF]. The orientation of the particle is defined as the direction of the vector between the center of mass and the center of the circle. We will later give an estimate of the precision of this measure.

Experimental parameters

We now estimate the parameters of our experimental systems. a. Particle diameter. We captured the images with the resolution 0.12 µm px -1 . In the obtained images of the particle with the diameter a = 3.17 µm, the particle diameter appears a 26 px.

b. Density. There are on average 487 particles in a 3000 × 2400 pixels (∼ 360 µm × 290 µm) image. This gives the number density ρ 487 360 µm×290 µm 0.0047 µm -2 and the area fraction φ = ρπ(a/2) 2 0.04. c. Velocity. We use Trackpy [START_REF] Allan | soft-matter/trackpy: Trackpy v0[END_REF] to obtain the trajectories. The instantaneous velocities can be obtained by applying a Savitzky-Golay filter [START_REF] Virtanen | SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python[END_REF]. From the instantaneous velocities, we find an average velocity U = 56 ± 7 px s -1 (standard deviation given for different particles, see Fig. 8 (a)). Thus, U 6.7 µm s -1 , that is to say of the order of two particle diameters per second.

Alternatively, one can compute the mean square displacement as a function of time (Fig. 8 (b)). At the lowest order in time, ∆x 2 = (U ∆t) 2 . We find U 55 px s -1 6.6 µm s -1 . This is consistent with the previous result.

d. Rotational diffusion. To measure the rotational diffusion coefficient D r , we compute the mean square angular displacement (MSAD) as a function of time. At short time, we expect ∆θ 2 = 2θ 2 err + 2D r ∆t, (D1)

where θ err is the error made in the detection of the orientation of a given particle on a given frame. At times larger than the mean free time τ free ( 1 √ ρ -a)/U 1.7 s, hydrodynamic or electrostatic interactions may affect the orientation of the particles during collisions, resulting in a larger angular displacement. The MSAD is plotted on Fig. 8(c); fitting the MSAD with Eq. (D1) for 0 ≤ ∆t ≤ 1.5 s leads to D r = 0.123 ± 0.004 s -1 and θ err = 8.7 ± 0.3°.

e. Translational diffusion. This is the hardest quantity to evaluate. We give an estimate based on the Stokes-Einstein relation,

D = k B T 6πη(a/2)β , (D2)
η = 1.0 • 10 -3 Pa s -1 is the viscosity of water, T 300 K is the temperature, and β is a correction factor due to the proximity of the bottom electrode. Assuming that Faxen's law (Eq. (7-4.28) of Ref [START_REF] Happel | Wall effects on the motion of a single particle[END_REF]) is valid for h (distance to the wall) of the order of a, we obtain β 3 (one checks that the thermal fluctuations are negligible). At the end of the day,

Correlations

Once we have detected the positions and the orientations, we consider a given frame. For each particle far from the edges of the image, we consider every other particle and compute its position in the reference frame of the orientation of the first particle. We put the result in bins of size ∆x = ∆y = 0.1a. After processing all the frames and normalizing the bins, we obtain the correlation (Fig 4(b) of the main text).

Variation of electric field

  FIG. 1. (a) Snapshot of numerical simulations of ABP. The red arrows indicate the orientations of the particles. The blue axes represent the local frame used to compute the pair correlations. (b)-(e) Evolution of the correlations with decreasing Dr. φ = 0.04 and : D : U = 1 : 0.1 : 10, and Dr = 10, 1, 0.1, 0.01 (Pe = 10, 31.6, 100, 316). Top: simulations. Bottom: numerical integration of Eq. (7).

3 FIG. 2 . 3 .

 323 FIG. 2.Phase diagram showing the different regimes as a function of Péclet number and observation length. Blue arrows indicate the regimes explored in the simulations (Fig.1(b)-(e)) and in the experiments (Fig.4) and , the observation length ranges from the diameter a of the particles to 10a. Red arrows point to the limiting regimes explored in Fig.3.

FIG. 3 .

 3 FIG. 3. Limiting regimes. (a)-(c) Small propulsion (U → 0). (a) Numerical correlation (φ = 0.04 and : D : Dr : U = 1 : 0.1 : 0.5 : 0.05). (b) Cut of the previous correlation at radius r0 = 1.5 (dashed black line), and theoretical prediction (gray line). (c) Rescaled plot of the first Fourier coefficient B1(r) for φ = 0.04 and : D : U = 1 : 0.1 : 0.05 for Dr = 0.1, 0.2, 0.5, 1 (blue to red), the gray line is the theoretical prediction at large distance. (d)-(e) No rotational diffusion (Dr = 0). (d) Numerical correlation (φ = 0.04 and : D : Dr : U = 1 : 0.1 : 0 : 10). (e) Rescaled horizontal cuts of the correlation from (d), the gray line is the theoretical prediction (8). (f)-(g) No translational diffusion (D = 0). (f) Numerical correlation (φ = 0.04 and : D : Dr : U = 1 : 0 : 0.1 : 10). (g) Rescaled horizontal cuts of the correlation from (f) according to Eq. (10).

  (App. D, Fig. 4(a) and Movie 1 in the Supplemental Material). We estimate the experimental values of U , D and D r in App. D 3. This allows us to place the experiments on the phase diagram (dark blue arrow in Fig. 2), with a Péclet number Pe 90: depletion wings are expected. The measured pair correlation is shown in Fig. 4(b) (additional figures for other values of the electric field are provided in App. D 5). Depletion wings are present,

FIG. 4 .

 4 FIG. 4. Experiments (a) Experimental image, indicating the detected positions and orientations of the Janus particles (red dots and arrows). Scale bar: 10 µm. (b) Pair correlation B(r).

FIG. 5 . 1 BFIG. 6 .

 516 FIG. 5. Simulations in the limit cases Dr = 0 and D = 0 for dilute hard particles. (a) No rotational diffusion. Correlation B(r) for ρ = 0.05, : D : Dr : U = 50 : 0.1 : 0 : 10. (b) Rescaled cuts with the exponents predicted in the article. (c) No translational diffusion. Correlation B(r) for ρ = 0.02, : D : Dr : U = 50 : 0 : 0.1 : 10. (d) Rescaled cuts with the exponents predicted in the article.

Appendix B: Theoretical computations 1 .

 1 Equation for the correlations a. Dean equation for Active Brownian Particles The Langevin equations for Active Brownian Particles read, ṙi

)FIG. 7 .

 7 FIG. 7. Numerical resolution of Eq. (B48) with Ṽ (0) = U = Dr = 1, for k = 10 4 . (a) Real part of C. (b) Imaginary part of C. The solution concentrates around (γ, γ ) = (0, 0) and (π, π). The numerical resolution consists in discretizing γ and γ , and then solving the linear system corresponding to Eq. (B48).

FIG. 8 .

 8 FIG. 8. Determination of the experimental parameters. 670 particles are tracked for 35 frames on average (minimum: 20 frames). (a) Histogram of the average velocities of the particles, the vertical black line is the mean. (b) Mean square displacement. The estimated error bars are smaller than the symbol size. (c) Mean square angular displacement. Error bars represent standard errors estimated from the distribution of ∆θ 2 for each given ∆t, the solid line is a linear fit over 0 ≤ ∆t ≤ 1.5 s.

D 300 × 1 . 38 • 1 BFIG. 9 .

 13819 FIG. 9. Experimental pair correlations for different values of the voltage: 6 Vpp, 8 Vpp and 10 Vpp (left to right). The amplitudes of the electric field are 1.2 • 10 8 , 1.6 • 10 8 and 2 • 10 8 Vpp m -1 .

ACKNOWLEDGMENTS

The authors would like to thank K. A. Takeuchi for his hospitality and useful discussions, J. Iwasawa for his help with experimental and image processing protocols, and D. Bartolo, O. Dauchot and A. Solon for insightful discussions. D.N. was supported by JSPS KAKENHI Grant Numbers JP19K23422, JP19H05800 and JP20K14426.

We varied the strength of the electric field in the experiments setting the voltage to 6 V pp and 8 V pp in addition to the 10 V pp data that we report in the article. The velocity U of the particles is known to scale as the square of the strength of the electric field [36], thus the Péclet number ranges Pe = 32 (6 V pp ) to Pe = 90 (10 V pp ). The experimental correlations B(r) are show on Fig. 9. In all three cases the qualitative behavior is the same: the depletion is maximal in two wings behind the particle. Experiments with 10 V pp have been performed first, and when the system ages particles start to stick to the substrate, explaining the appearance of positive concentric circles in the correlations for 6 and 8 V pp .

We consider N particles (N = 5000) in a square periodic box of size L = N ρ with ρ the density (L ∼ 300 at ρ = 0.05). Initially the positions r i and the orientations θ i of the particles are assigned uniformly at random.

We use stochastic molecular dynamics and consider the following Langevin equations ṙi = -∇ i j =i

V (r i -r j ) + U êθi +

U is the velocity, D the transitional diffusion and D r the rotational diffusion. The mobility γ is set to 1. η i and ν i are Gaussian white noises with unit variance. During a time increment ∆t (∆t = 0.05 for the simulations presented in the article), the term √ 2D r ν i generates an increment υ √ 2D r ∆t with υ a random number generated from a standard normal distribution. Similarly for √ 2Dη i . We use the following soft-sphere potential, where the diameter a of a particle and the potential strength are set to unity,

We let the system evolve for a time t eq ∼ 500 before starting to record the correlations. B(x) is then recorded with a spatial resolution ∆x = 0.1, with one measure every ∆t c = 5, during an overall time period T ∼ 5 • 10 5 . The results are averaged over 100 realizations of the system.

Simulations of dilute hard particles

The theoretical approach that we use is a linearized Dean equation, a framework which is valid for weak interactions. Consequently, the simulations that we compare to our theoretical results have relative values of potential strength and velocity : U = 1 : 10. This means that the particles are able to interpenetrate. In this weak interaction regime, we are able to test quantitatively the scalings that we obtain.

That being said, it is interesting to test the scaling exponents that we obtain on simulations in the regime of strong interactions. We do it in Fig 5 and find that indeed the cuts of the correlation function obey the predicted scalings both in the limit of low rotational diffusion and in the limit of low translational diffusion. This leads us to state that the scaling exponents that we found are robust. Note that the limit curves are different from our prediction in the weak interaction regime.

Extension of the dilute regime

Here we discuss the range of density ρ where the dilute approximation that we use should be valid. We focus on hard particles, and expect range of density to be wider for soft particles. There are two ways to estimate where the dilute regime sits. The first is to look at the low density branch of the motility induced phase separation (MIPS)

Numerical integration

We consider the domain (x, y, β) ∈ [-x max , x max ]×[0, y max ]×[-π, π]. We discretize it with steps ∆x in x, ∆y = ∆x in y and ∆β in β.

We start from C(x, y, β, t = 0) = 0 and integrate Eq. (C3) in time using an explicit Euler scheme with time step ∆t. The differential operators are evaluted using finite differences valid at order (∆x) 2 and (∆β) 2 ,

and so on. The potential is

and its Laplacian is evaluated on the grid. The boundary conditions are as follow:

• Periodic boundary conditions for β;

• C(-x max , y, β) = C(+x max , y, β) = C(x, y max , β) = 0;

• We use the symmetry relation C(x, -y, -β) = C(x, y, β) to impose the additional points C(x, -∆y, β) = C(x, ∆y, -β).

The numerical integration in time converges to the stationary solution at sufficiently large time. When the increment on C over a time step ∆t is small enough, we output the stationary solution C eq (x, β) and its integral over β, B eq (x). This method is used on Fig 2a-d.

The experimental system is the one used in Ref. [START_REF] Nishiguchi | Flagellar dynamics of chains of active Janus particles fueled by an AC electric field[END_REF]. We used a suspension of Janus particles of diameter a = 3.17± 0.32 µm sandwiched between two ITO electrodes separated by a spacer of size H = 50 µm. The particles are suspended in a sodium chloride solution of concentration 10 -4 mol L -1 . The use of sodium chloride diminishes the temporal variation of the system, which enables long observation required for calculating pair correlations. Furthermore, because the addition of sodium chloride decreases the width of the screening double layer (Debye length) and consequently the strength of the induced hydrodynamic flow [START_REF] Todd | Breaking symmetries in induced-charge electro-osmosis and electrophoresis[END_REF], both the electrostatic and hydrodynamic interactions between the particles are diminished. This keeps the interactions indiscernible at the frequency range we used here [START_REF] Nishiguchi | Flagellar dynamics of chains of active Janus particles fueled by an AC electric field[END_REF]. For the correlations presented in the main text, we applied an electric field of frequency f = 5 kHz and amplitude 2 • 10 6 V pp m -1 (voltage 10 V pp ) in the vertical direction. In this range of frequency, the Janus particles move in the direction of the uncoated hemisphere due to the induced-charge electrophoresis [START_REF] Nishiguchi | Mesoscopic turbulence and local order in janus particles self-propelling under an ac electric field[END_REF] and exhibit neither of electrostatic attractive interactions nor strong hydrodynamic interactions [START_REF] Nishiguchi | Flagellar dynamics of chains of active Janus particles fueled by an AC electric field[END_REF]. Moreover no polar order is observed even at high density [START_REF] Nishiguchi | Mesoscopic turbulence and local order in janus particles self-propelling under an ac electric field[END_REF]; these experimental optimization justifies the theoretical modeling by Active Brownian Particles as a first approximation.

Videos of the system were captured by a CMOS camera (Baumer, LXG-80, 3000 × 2400 pixels, 8 bit grayscale) at the framerate 10 fps mounted on an inverted microscope (Olympus, IX70) equipped with a 40× objective lens (LUCPLFLN, NA=0.60). A green filter is inserted between the sample and the halogen lamp to increase the contrast of the hemispheres of the Janus particles. The acquisition length is 14 minutes (8400 frames).

We append an experimental movie at 10 V pp , captured at 10 fps. The field of view of this movie is cropped to 585 px × 409 px (70 µm × 49 µm) for visibility. The movie is played at the real speed.

Image analysis

Particles were detected using the Hough Circle Transform algorithm implemented in the OpenCV library [START_REF]The OpenCV Reference Manual[END_REF]. The positions of the particles are the centers of the circles. For each detected circle, we compute the center of mass of the


Dr Mireia Rodriguez-Zubiri

Rhodium

Dr Christine Baudequin

Aurélien Béthegnies

Dr Jean-Jacques Brunet

Intermolecular Rhodium-Catalysed Hydroamination of Non-Activated Olefins: Effect of Olefin, Amine, Phosphine and Phosphonium Salt

The right combination: RhCl 3 •3H 2 O/P(p-CH 3 C 6 H 4 ) 3 /nBu 4 PI/I 2 led to the highest catalytic activity for the intermolecular hydroamination of ethylene, 1-butene, and 1-hexene with aniline-type amines. Excellent activities were also found using Rh I Wilkinson's catalysts and the very efficient "in-situ generated" I <M-> from PPh 3 /I 2 , which can substitute pre-synthesized nBu 4 PI (see scheme; CE=catalytic efficiency). halides hydroamination non-activated alkenes phosphines rhodium

The catalytic system RhCl 3 •3H 2 O/2P(p-CH 3 C 6 H 4 ) 3 /65nBu 4 PI/2I 2 , which was discovered recently in our research group, allows the highest catalytic activity ever reported for the intermolecular hydroamination of ethylene, 1-butene, and 1-hexene with aniline-type amines (0.3^^mol^% catalytic precursor) to give the expected N-alkyl-(N-ethyl-, 1) and N,N-dialkylanilines (N,N-diethyl-, 2), along with 2-methyl quinolines (3) (in the case of ethylene). The effects of time and temperature, as well as the nature of the phosphonium salt, phosphine, and amine on the catalytic activity of this reaction have been studied. This system is particularly efficient for the hydroamination of ethylene with aniline in the presence of 2,2'bis(diphenylphosphino)-1,1'-binaphthyl (CE*=460) and tri(p-tolyl)phosphine (CE*=520).

Good to excellent activities were also found by combining Wilkinson's catalysts (Rh I complexes) with nBu 4 PI and I 2 . The simple association of PPh 3 and I 2 has been shown to be a very efficient "in-situ generated" source of I <M-> promoters. *CE (catalytic efficiency)=TON 1 +2TON 2 +2TON 3 .

Introduction

Amines are important chemicals used in industry and in everyday life. [1,^2] Chemists have developed many synthetic routes for the preparation of amines but most of them are not suitable for industrial purposes. [3] Amines are mostly synthesised from alcohols by the process discovered in 1909 by Paul Sabatier. [4,^5] As alcohols are mainly obtained from alkenes either by hydration or by the hydroformylation/hydrogenation sequence, [6,^7] the overall synthesis of amines requires two steps----the last step also produces water, which is an expensive and polluting side-product to be removed (Scheme^^1<schr1>).

In this context, the direct transformation of olefins into amines (the hydroamination reaction; Scheme^^2<schr2>) is obviously an interesting economic and ecologic alternative.

Nevertheless, this atom-economic reaction requires a catalytic process because such additions are characterised by a high activation barrier. [8] Great improvements have been reported on the design of catalytic systems for the intramolecular hydroamination of alkenes and for the intermolecular hydroamination of unsaturated substrates such as alkynes, vinylarenes, dienes, and allenes, which are more reactive than simple alkenes. [9] Indeed, the intermolecular hydroamination of non-activated alkenes is particularly challenging and remains the main challenge. [9,^10] A few years ago our research group reported the discovery of an original Pt II system, PtBr 2 /nBu 4 PBr, which involves the association of simple Pt II salts with ionic liquids (phosphonium halides). It proved to be the first platinum-based system that allows the catalytic hydroamination of ethylene with anilines [Eq.^^(1)]. Furthermore, the above association permits the catalytic hydroamination of higher alkenes, such as 1-hexene (still a remarkable result) with a surprisingly high Markovnikov regioselectivity (>95^%). [11] A breakthrough in this area was the discovery of such platinum-based systems that were equally effective in water using common salts such as NaBr instead of nBu 4 PBr as the source of bromide ions. [12] <ZS>(1) Aiming to improve the performance of this already efficient catalytic system, some mechanistic studies on the low-energy species of the catalytic cycle were carried out. The [Pt(C 2 H 4 )Br 3 ] <M-> complex was found to be the resting-state species. [13] In view of previous results on the platinum-based catalytic systems, we were prompted to examine the possible activating effect of nBu 4 PX derivatives on the rhodium(III)-catalysed hydroamination of ethylene with anilines. [14] Literature data on rhodium-catalysed hydroamination reactions of ethylene and/or higher non-activated olefins are scarce and exemplify the challenges in achieving good catalytic activity for this reaction.

In 1971, Coulson first reported that RhCl 3 •3H 2 O catalyses the hydroamination of ethylene under high-temperature (180--200^°C) and high-pressure (50--140^^bar) conditions. [15,^16] Nevertheless, this reaction was restricted to ethylene and secondary amines of high basicity and low steric bulk (e.g. piperidine, pK b =2.88). In contrast, Diamond et^^al. reported that RhCl 3 •3H 2 O, associated with PPh 3 (2^^equiv), catalyses the hydroamination of ethylene with aniline (pK b =9.37) at high temperatures. [17,^18] Although catalytic activity was quite low, the main interest of their study was to show that this reaction afforded 2methylquinoline as a secondary, highly elaborated product. Later on, Taube et^^al. studied the Rh III system reported by Coulson and concluded that it was limited because of decomposition of the active [RhCl(C 2 H 4 )pip 2 ] (pip=piperidine) species to yield metallic rhodium species. [19,^20] Taube and co-workers conceived a cationic rhodium(I) complex, [Rh(PPh 3 ) 2 (C 2 H 4 )(Me 2 CO)]PF 6 , which is active for the hydroamination of ethylene with piperidine at room temperature and at atmospheric pressure. Unfortunately, this system is completely deactivated within ten cycles. [20,^8b] These authors also reported relevant mechanistic investigations on a possible reaction pathway. [21] Taking into account the state-of-the-art mentioned above, our results in this area are particularly interesting. Indeed, the association of the usual " RhCl 3 •3H 2 O/2PPh 3 " catalytic precursor (0.3^^mol^%) with nBu 4 PI (65^^equiv with respect to Rh) and iodine (2^^equiv with respect to Rh) was found to be the most active catalytic system reported so far for the hydroamination of ethylene with aniline [Eq.^^(2)]. [14] <ZS>(2) This system proved particularly efficient for the hydroamination of ethylene with Nethylaniline [Eq.^^(3); turnover number (TON)>260 after 24^^h] and even more active, under comparable conditions, than the platinum systems (PtBr 2 /nBu 4 PX) previously developed in our group. [10--14] <ZS>(3)

In both cases, the halide ions (especially iodides for the rhodium catalyst) associated with the phosphonium salt promote a dramatic increase of the catalytic activity. Indeed, control experiments indicated that in the absence of nBu 4 PI and I 2 , the RhCl 3 •3H 2 O/2PPh 3 system allows only very low activity (TON<5) for the reaction of N-ethylaniline with ethylene, in agreement with observations reported by Diamond et^^al. [17,^18] Herein, we report a relevant study on the use of phosphorus ligands as co-catalysts for the rhodium-catalysed intermolecular hydroamination of ethylene and higher olefins such as 1-butene and 1-hexene with aniline (pK b =9.37). This simple one-pot catalytic system does not need to operate under inert atmosphere and does not need laborious experimental procedures.

The Rh III /phosphine/nBu 4 PI catalytic system provides great possibilities for an asymmetric version of this reaction, an almost unexplored area of research to date. [22,^23] Indeed, the only two literature reports on the asymmetric intermolecular hydroamination of non-activated alkenes are the unprecedented study reported by Widenhoefer and co-workers [22] using chiral [AuCl(phosphine)] pre-catalysts for ethylene and 1-alkene hydroamination with cyclic ureas (pK b =13.86--14.17), and later on the study by Hultzsch and co-workers [23] involved chiral catalysts based on rare-earth metals for the stereoselective addition of basic amines (benzylamine, pK b =4.66) to non-activated alkenes.

It is therefore important to note that our contribution concerns widely available amines, higher α-olefins, and low loading (0.3^^mol^%) of a stable "one-pot" ready-to-use metal--ligand catalyst precursor. Furthermore, we were able to generate an "in situ" source of I <M-> promoters and to show that this catalytic system is also efficient using Rh I complexes.

Results and Discussion

In this study, we were led to consider both the TON (moles of product formed by mole of catalyst used) and the catalytic efficiency (CE). The CEwas calculated on the basis of the number of cycles needed by the catalyst to yield each product (two cycles for the product of double hydroamination, 2, and for quinaldine, 3) [11a] [Eqs.^^(2) and ( 3)].

Hydroamination of ethylene by aniline: influence of the phosphine ligand

Our first experiments were aimed at investigating the effect of the relative amount of PPh 3 , an important parameter that was not extensively studied in our previous Communication [14] where we intended to test the experimental and analytical conditions (experimental details as in ref.^^<litrsn><litr11><litr_c></litrsn>) with regards to those reported earlier by Diamond et^^al. [17,^18] for the hydroamination of ethylene with aniline.

Using 2^^equivalents of PPh 3 , we supposed the active catalytic species to contain the {RhI(PPh 3 ) 2 } moiety necessary for either the coordination of the alkene followed by the external nucleophilic addition of the starting amine, or the N<C->H oxidative addition and subsequent insertion of the alkene moiety. [8b] Typically, catalytic runs conducted with variable amounts of PPh 3 (Table^^1, entries^^1--6) confirmed that, indeed, the best results are obtained with either a PPh 3 /Rh ratio of 2 or 3. Either lower or higher amounts of phosphine resulted in a drop of the catalytic activity in part because of a lower conversion of 1 into 2. These results suggest that the addition of PhNHEt to ethylene is favoured in the presence of I 2 and is more sensitive to the PPh 3 /Rh ratio than in the case of PhNH 2 .

The influence of the nature of several phosphorus ligands was examined. Different types of phosphorus ligands containing alkyl, aryl, or alkoxy substituents were used and the results are listed in Table^^2. In contrast with previous results on the platinum(II)-catalysed reactions of aniline with ethylene, [24] where phosphorus ligands (to the exception of trialkylphosphites) seemed to act as inhibitors, PPh 3 allows much better catalytic activity than trialkylphosphites (Table^^2, entries^^3--7). Aryloxy-substituted ligands (triaryl phosphites) and trialkylphosphines mediate the reaction but are inefficient co-catalysts (Table^^2, entries^^1, 2, and 8--11). In sharp contrast, the catalytic activity is partially recovered upon the introduction of phenyl groups as substituents of the phosphorus atoms as shown by the results obtained with mixed phosphorus ligands such as P(OMe)Ph 2 and dppe (1,2bis(diphenylphosphino)ethane; Table^^2, entries^^12 and 13) where the P atom contains two phenyl substituents. Interestingly, excellent activities were obtained in the presence of binap (2,2'-bis(diphenylphosphino)-1,1'-binaphthyl; Table^^2, entries^^14 and 15), probably owing to the presence of three phenyl substituents on each P atom to create a quite similar electronic and steric environment when compared to PPh 3 (1^^equiv of binap corresponds to 2^^equiv of P, which is analogous to the use of 2^^equiv of PPh 3 with respect to Rh). Thus, it was concluded that the presence of phenyl groups in the structure of the phosphorus ligands is a key factor for achieving good catalytic activity.

Next, the electronic effects of the substituents on the phenyl groups of the triarylphosphines (Table^^2, entries^^16--18) were considered. We found that the para^^substitution on the aryl ring with the strongly electron-withdrawing CF 3 group dramatically lowers the catalytic activity. In a minor way, the p-OMe group (mesomeric electron-donating and inductive electron-withdrawing) also causes a loss of activity. In sharp contrast, the para inductive electron-donating CH 3 substituent induces an important increase of the catalytic activity: this is in fact the best result ever reported for the intermolecular hydroamination of a non-activated olefin by aniline (CE=528; Table^^2, entry^^17).

In any case, formation of quinaldine 3 seems favoured when production of 1 is not, thus suggesting, once more, that 1 is not an intermediate in the pathway leading to 3. [14] These results are of special importance when considering the enantioselective version of the intermolecular hydroamination of non-activated olefins (see introduction). These related studies are in progress within our group.

Hydroamination of ethylene by aniline: influence of the amine

To potentially widen the scope of this catalytic process, the addition of other primary and secondary amines to ethylene was studied. Hence, the efficiency for the hydroamination of ethylene with more basic amines was considered (Table^^3). Using N-ethylaniline, the catalytic system afforded N,N-diethylaniline as the only reaction product [Eq.^^(3)] with higher TON than when the reaction was performed with aniline (Table^^3, entries^^1--3).

On the other hand, morpholine [Eq.^^(4)] and piperidine [Eq.^^(5)], two more-basic non-aromatic amines, afforded very low catalytic activities (Table^^3 entries^^5 and 6), possibly owing to strong coordination to the metal centre <ZS>(4) <ZS>( 5) Further experiments (Table 3, entry 4) showed an interesting result when performing the hydroamination of ethylene with benzylamine [Eq. ^^(6)]. Indeed, as may be seen in Table^^3, the catalytic efficiency of this reaction does not depend on the pK b ^^value of the amine because morpholine and piperidine afford very low catalytic activities when compared to benzylamine, an amine with an intermediate pK b ^^value. Analogously, the amine substitution does not seem to be a determining parameter by itself because aniline and benzylamine, two primary amines, allowed very different catalytic performance. The only structural factor that could somehow explain this result is perhaps, once more, related to the presence of an aryl group near to the nitrogen atom: for aniline and N-ethylaniline, the N atom is directly bonded to a Csp 2 ^^atom and for the benzylamine, only one Csp 3 ^^atom separates the N atom from the aryl Csp 2 ^^atom. Further studies (catalytic tests and theoretical calculations) are ongoing in our team to elucidate the parameters (nucleophicity, basicity, and/or steric hindrance) that influence the catalytic performance as a function of the nature of the starting amine.

<ZS>(6)

Hydroamination of ethylene by aniline: influence of the phosphonium salt

On the basis of the excellent catalytic results obtained with the association of nBu 4 PI with PPh 3 , we were prompted to evaluate the influence of the nature of the cation of the ionic salts, as we had done previously with the anion. [14] For this purpose, catalytic runs were conducted in the presence of Ph 4 PI, a phosphonium salt containing phenyl substituents on the P atom as in the case of the efficient PPh 3 ligand, and nBu 4 NI, an ammonium salts with the same nBu groups as our previous phosphonium candidate (Table^^4). Surprisingly, association of the RhCl 3 •3H 2 O catalyst with Ph 4 PI inhibited the efficiency of the reaction even in the presence of PPh 3 (Table^^4, entries^^1--4), showing that the PPh 4 + counter cation must also have a role on the reactivity Physical properties of this salt, such as its high melting point (338^°C), are not, a^^priori, responsible for its catalytic behaviour because Ph 4 PI is soluble in aniline even at room temperature and thus able to react under the catalytic conditions (150^°C).

In the same way, nBu 4 NI, strongly decreased the catalytic activity (Table^^4, entry^^5) in the presence of PPh 3 and I 2 , thus confirming our previously stated hypothesis on the relevance of the cation moiety of the promoter salt used.

Hydroamination of ethylene by aniline: "in-situ generated" source of I <M-> promoters

The rhodium-catalysed hydroamination reaction was tested in the presence of PPh 3 and I 2 (Table^^5). Our aim was to form "in situ" an ionic liquid similar to the one used previously (nBu 4 PI), and to avoid in this way its synthesis, purification, and conditioning (see the Experimental Section). This idea was based on literature data that have proved the formation of the corresponding phosphonium salts when reacting a phosphine with X 2 (Scheme^^3<schr3>). [25] Good results were obtained when using stoichiometric quantities of PPh 3 and I 2, with an optimum at 10^^equivalents of each (Table^^5, entries^^2--4 and 6). Moreover, only 5^^equivalents of each (Table^^5, entry^^2), PPh 3 and I 2 , afforded a better catalytic performance than the use of 20^^equivalents of nBu 4 PI (TON 1 =73, TON 2 =6, TON 3 =0, CE=85). [14] Hydroamination of ethylene by aniline: influence of temperature and reaction time

Studies on the rhodium(III)-based catalytic systems were initially conducted at 150^°C

because previous studies on the platinum(II)-catalysed addition of PhNH 2 to ethylene had shown that this temperature seemed a good compromise between activity and energetic input. [26] We wished then to explore the performance of this catalytic reaction under "milder" conditions to determine whether the catalytic efficiency could remain reasonably high in view of chiral processes. As may be seen in Table^^6, catalytic activity drops dramatically when the temperature of the reaction was decreased from 150^°C to 125^°C or 100^°C. As already stated, [11c] a possible explanation for this phenomena is as follows: because the reaction of aniline with ethylene coordinated to a transition-metal complex is disfavoured by the low nucleophilicity of aniline, the catalytic hydroamination is possible under high-temperature conditions to allow the zwitterionic addition complex (Scheme^^4<schr4>) to experience a temperature-promoted cleavage of the carbon--metal bond, and thus displacing all the steps of the reaction to the formation of N-ethylaniline. [11b] In the same way, it was observed that an increase of the ethylene pressure from 25^^bar to 45^^bar has a negative effect on the TON of the reaction. This outcome is in contrast with previous results on the optimisation of the platinum(II)-catalysed hydroamination of ethylene by aniline where catalytic activity was promoted in the presence of higher amounts of either of the substrates (aniline/Pt and/or ethylene/Pt). [26] Another important parameter to be studied was the evolution of the reaction with time.

Hence, various experiments were carried out under the usual conditions for 10^^hours and 93^^hours. The results are collected in Table^^7. As may be seen, after 10^^hours of reaction (Table^^7, entry^^1), the three hydroamination products were formed, although only small amounts of 2 and 3 were observed. The proportions are 19:2:1 for 1/2/3. On the other hand, after 93^^hours of reaction (Table^^7, entry^^2), the main product was 2, small amounts of 1

were detected compared to 2, and the amount of 3 did not increase after 10^^hours of reaction. This outcome is consistent with the idea that N,N-diethylaniline is the product of a second hydroamination cycle where N-ethylaniline is added to ethylene. Thus, along the course of the reaction, the amount of 1 increase and, at the same time, 1 is transformed into 2.

The proportions are 9:75:1 for 1/2/3 after 93^^hours of reaction.

As shown by the results stated in Table^^7, as the reaction evolves, amounts of 2 increase whereas 1 is consumed thus suggesting that hydroamination of ethylene by more basic anilines such as 1, is favoured. Amounts of the quinoline derivative 3 remained unchanged after more than 80^^hours, thereby confirming once more that 1 is probably not an intermediate in the pathway leading to 3.

Hydroamination of higher α-olefins by aniline

The next step in our study was to use the Rh III precursors to explore the intermolecular hydroamination of higher non-activated α-olefins such as 1-butene and 1-hexene. The products resulting from these reactions are of great relevance in industry and the reaction of higher α-olefins is itself an important scientific challenge. [15,^18,^27] Literature on the intermolecular hydroamination of 1-butene and higher α-olefins by simple amines is scarce, and only recently reported. Although catalytic activities were quite low when using organolanthanide complexes, Marks and co-workers [28,^9Br] successfully achieved the hydroamination of 1-pentene with n-propylamine, under anaerobic conditions, at 60^°C, using 20^mol% of the metal catalyst. These authors reported to deal with a challenging reaction, the intermolecular hydroamination of non-activated alkenes, in comparison to intramolecular hydroamination, 350x faster, and alkyne hydroamination, 1400x faster. In 2006, He and coworkers studied the addition of TsNH 2 to 1-octene with 5^^mol^% of [AuCl(PPh 3 )] /AgOTf in toluene at 85^°C. [29a] In 2007, Loh and co-workers studied the same reaction in the presence of InBr 3 (20^^mol^%) in toluene at 120^°C. [29b] Both research groups obtained the desired Markovnikov hydroamination product along with another Markovnikov isomer resulting from the hydroamination of an internal alkene formed by isomerisation of the initial terminal double bond. [29] Later on, Loh and Yin found that lanthanide salts in ionic solvents (2^^mol^% of Pr(OTf) 3 + nBu 4 PI + I 2 ) catalysed the hydroamination of 1-heptene and 1octene with aniline, at 160^°C, with up to 95^% of the Markovnikov addition products. [30] They reported the use of phosphonium salts as we had previously described in two unprecedented reports on the hydroamination of 1-hexene by aniline in the presence of a ligandless, inexpensive, and non-toxic PtBr 2 /nBu 4 PBr/H + catalyst, at 150^°C, leading to 95^% of the Markovnikov product without alkene isomerisation and with TON up to 200. [11c,^31] In 2010, Hultzsch and co-workers [23] succeeded in the use of sensitive rare-earth-metal asymmetric complexes (5^^mol^%) in toluene for the addition of simple amines to 1-alkenes at 150^°C, with good conversions and moderate ee^^values. Finally, Michon, Agbossou-Niedercorn, and co-workers succeeded in the combined use of a copper halide (5^^mol^%) and a silver salt (5--10^^mol^%) for the hydroamination of 1-hexene by TsNH 2 at 100^°C to yield the Markovnikov isomers (N-2-and N-3-hexylamines) in a ratio 4:6. [32] No studies have been reported using rhodium-based catalytic systems----a fact that led us to test our RhCl 3 •3H 2 O/nBu 4 PI/PPh 3 /I 2 system, which has turned out to be the best to date for the hydroamination of ethylene by simple amines. Thus, the addition of aniline to 1-butene (Scheme^^5<schr5>) first, and to 1-hexene next (Scheme^^6<schr6>) were studied.

Our results in this area are particularly promising. In fact, this system is also an efficient catalyst for the hydroamination of 1-butene with up to 150^^TON (Table^^8, entry^^6), and similar effects to the case of ethylene. Actually, the reaction of aniline with 1butene in the presence of the above rhodium(III)-based catalyst afforded the expected hydroamination butylanilines products, with a single Markovnikov isomer being highly predominant (99^%). Thus, the regioselectivity of this reaction appears even higher than for the PtBr 2 /nBu 4 PBr system, for which a 95:5 Markovnikov/anti-Markovnikov ratio (M/AM) was observed. [11b] This is a significant result when thinking about applying this catalytic system to the enantioselective version of this reaction.

As no dibutylaniline compounds were detected after the reaction, it seems that, under our experimental conditions, a second hydroamination reaction of 1-butene by N-secbutylaniline does not proceed, possibly owing to steric effects. In the same way, no quinoline product from an aromatic C<C->H activation was observed. An overview of the results stated in Table^^8 reveals that for the addition of aniline to 1-butene, the precursor RhCl 3 •3H 2 O is not active by itself (Table^^8, entry^^1), thus confirming the work done by Coulson on the hydroamination reactions catalysed by the same precursor, where the catalyst showed activity only for the reaction of ethylene with basic amines. [15,^16] In the presence of the phosphonium salts, moderate activities were observed (Table^^8, entries^^2 and 3), especially with the nBu 4 PI salt as was the case for the hydroamination of ethylene. [14] Additionally, PPh 3 and I 2 , in the absence of phosphonium salts, led to low activity (Table^^8, entry^^4), with a higher promoting effect than nBu 4 PBr, but lower than the effect of nBu 4 PI. An increase of the activity is once more observed when both promoters, nBu 4 PI and PPh 3 , are present (Table^^8, entry^^5), as was reported for the ethylene hydroamination with the same catalytic system. [14] Results are improved by the addition of molecular iodine (Table^^8, entries^^6 and 7), with an optimum at 65^^equivalents of nBu 4 PI.

The results in Table^^8 show that the RhCl 3 •3H 2 O/nBu 4 PI/PPh 3 /I 2 system appears to be a highly selective catalyst for the hydroamination of higher olefins such as 1-butene by anilines, thus affording approximately a quarter of the catalytic activity reached for the same process with ethylene.

Concerning the hydroamination of 1-hexene (Scheme^^6), preliminary results using the optimised conditions obtained for the reactions of ethylene and 1-butene in the presence of the RhCl 3 •3H 2 O/nBu 4 PI/PPh 3 /I 2 catalytic system, have shown (Table^^9) the catalyst to achieve only one catalytic cycle in the absence of phosphonium salt (Table^^9, entry^^3).

Nevertheless, the catalytic activity is recovered in the presence of nBu 4 PI (150^^equiv;

Table^^9, entry^^2). More importantly, the very high regioselectivity obtained, 100^% for the Markovnikov product, 8, with neither traces of the anti-Markovnikov isomer, 10, nor of quinoline derivatives. This is an important feature when compared to literature data where a great difference of regioselectivity between the (stoichiometric) hydroamination of propene (100^% Markovnikov) and 1-hexene (46^% Markovnikov, 54^% anti-Markovnikov) was reported using [cis-PtCl 2 (PR 3 )(olefin)] catalysts. [33] On the other hand, a significant amount of a secondary Markovnikov isomer (N-3hexylaniline, 9) resulting from the addition of aniline to 2-hexene (the isomerisation product of 1-hexene), was also obtained along with the expected Markovnikov isomer (N-2hexylaniline, 8). The ratio of the isomers 8 and 9 is approximately 50:50 (Scheme^^6).. This outcome is obviously a consequence of the recognised nature of many rhodium-based catalytic precursors as being good catalysts for olefin isomerisation. This was certainly not observed during the above-described hydroamination of 1-butene because the 2-butene isomer yields the same Markovnikov addition product as 1-butene.

Compared to our results with the PtBr 2 /nBu 4 PBr system (a non-olefin isomerisation catalyst), the rhodium(III)-based catalyst induces a higher M/AM regioselectivity with no other side products (quinoline) except the Markovnikov addition product; nevertheless it readily promotes olefin isomerisation and subsequent hydroamination at the 3-position.

Hydroamination of ethylene by aniline in the presence of Rh I precursors

Finally, we decided to explore the hydroamination of ethylene by aniline in the presence of Rh I precursors (Table^^10). This is an idea based on the hypothesis that the active species in the catalytic cycle of the rhodium-catalysed hydroamination are Rh I species. [15,^17a,^18] In view of the extremely important role played by phosphorus ligands in this reaction, especially aromatic phosphines such as PPh 3 , we were prompted to test the catalytic performance of both Wilkinson's catalysts: [RhCl(PPh 3 ) 3 ] and the corresponding dimer [{RhCl(PPh 3 ) 2 } 2 ], which were directly synthesised from RhCl 3 •3H 2 O and PPh 3 precursors used in this study. [34] Wilkinson and co-workers studied a series of relevant chemical processes involving the Rh I complexes, which could be partially applied to the hydroamination mechanism, or at least, to provide an insight on relative equilibria. [34] In fact, [RhCl(PPh 3 ) 2 ], which is formed from Rh I species reacts quickly with ethylene and slower with higher olefins. Furthermore, the formation of the olefin complex is favoured, as in our case, by the presence of I <M-> rather than Br <M-> or Cl <M-> as a consequence of the trans^^effect of X <M-> when in the position trans to ethylene. [34] As expected, Wilkinson's catalyst allows very high catalytic activities when performing the hydroamination of ethylene by aniline under optimised experimental conditions, using nBu 4 PI and I 2 promoters. Under similar conditions the RhCl 3 .3H 2 O/nBu 4 PI/PPh 3 /I 2 catalytic system afforded slightly higher TON (TON 1 =28, TON 2 =225, TON 3 =3, CE=484) mainly because it seems to better promote the formation of 2 from 1, and thus the hydroamination by more basic amines.

In the same way, Wilkinson's dimer appeared to be a good hydroamination catalyst although the selectivity of the reaction is different because less N,N-diethylaniline is formed either because the formation of 2 is less favoured than in the case of the corresponding monomer, or simply because the reaction is slower. It is important to note that no quinoline was observed when using either of the Wilkinson's complexes.

Our study on the performance of rhodium(I)-phosphine precursors, in the presence of phosphonium salts, could provide important access to a new and large family of asymmetric catalytic systems for the enantioselective intermolecular hydroamination of non-activated olefins by simple amines.

Conclusion

In summary, the association of RhCl 3 •3H 2 O/2PPh 3 /nBu 4 PI/I 2 has been shown to afford excellent catalytic activities for the hydroamination of ethylene and higher non-activated αolefins with aniline. Even more important is to note the very high regioselectivity of this rhodium-based catalytic system towards Markovnikov addition products (up to 100^% for the reaction of 1-hexene with aniline). It is also an excellent catalytic system for the addition of more basic anilines, such as N-ethylaniline, to non-activated olefins.

From a practical point of view, this system offers several advantages because it involves simple starting materials and tolerates the presence of common aromatic phosphines as potential chiral vectors for the enantioselective version of this reaction. Furthermore, good to excellent activities were achieved by "one-pot" addition of PPh 3 and I 2 , a source of an "insitu" generated halide promoter, in the absence of nBu 4 PI.

Research is now in progress to elucidate the nature and the energy of the intermediates of the catalytic cycle of this reaction and to take advantage of these promoting effects to explore the enantioselective intermolecular hydroamination of non-activated alkenes by simple amines in the presence of racemic and chiral Rh I -based N-heterocyclic carbene-phosphine precursors.

Experimental Section Instrumentation

The GC analyses were performed on a Hewlett--Packard HP^^4890 (FID) chromatograph (HP^^3395 integrator) equipped with a 30^^m HP1 capillary column. The GC-MS analyses were performed on a Hewlett--Packard HP^^6890 apparatus equipped with a HP^^5973 M ion detector.

The NMR analyses were performed on Bruker AM^^250 machine. Catalytic experiments were conducted in a 100^^mL stainless-steel thermoregulated (electric oven) autoclave with a glass liner and a magnetic stirring bar.

Methods and Materials

Tetra-n-butylphosphonium iodide was prepared from nBu 3 P and nBuI (see below), stored under argon, in a freezer, and protected from light. Tetra-n-butylphosphonium bromide (Aldrich, 98^%), tetraphenylphosphonium iodide (Alfa Aesar,>98^%), tetra-n-butylamonium iodide (Fluka,>99^%), 1-iodobutane (Alfa Aesar, 99^%), tri-n-butylphosphine (Acros Organics, 95^%), triphenylphosphine (Aldrich, 99^%), triphenylphosphite (Acros, 99^%), trimethylphosphite (Strem, 97^%), methyldiphenylphosphinite (Acros, 99^%), tri(4-methoxyphenyl)phosphine (Alfa Aesar, 98^%), tri((4-trifluoromethyl)phenyl)phosphine (Avocado, 98^%), tri(p-tolyl)phosphine (Fluka,>97^%), rac-2,2'-bis(diphenylphosphino)-1,1'-binaphthyl (Aldrich, 97^%), 1,2bis(diphenylphosphino)ethane (Strem, 99^%), and rhodium trichloride trihydrate (Johnson Matthey, 41.92^%) were used as received. Molecular iodine was purchased from Acros. Aniline (Acros Organics, 99^% for analysis ACS), N-ethylaniline (Acros Organics, 98^%), morpholine (Fluka, 99^%), piperidine (Acros Organics, 99^%), benzylamine (Aldrich,>99.5^%), N,N-di-n-butylaniline (Aldrich, 97^%), 4-n-octylaniline (TCI,>95^%), 1-hexene (Aldrich, 97^%) and triethylphosphite (Aldrich, 98^%) were distilled before use and stored under argon. Ethylene (N25) and 1-butene were purchased from Air Liquide. Tricyclohexylphosphine was recrystallised from toluene/methanol.

[RhCl(PPh 3 ) 3 ], [{RhCl(PPh 3 ) 2 } 2 ] were prepared according to established methods. [34] Preparation of tetra-n-butylphosphonium iodide

Typical hydroamination reactions

Hydroamination of ethylene: The autoclave was charged with RhCl 3 •3H 2 O (34.0^^mg, 0.13^^mmol), I 2 (2^^equiv with respect to Rh, 66.0^^mg, 0.26^^mmol), PPh 3 (2^^equiv with respect to Rh, 68.0^^mg, 0.26^^mmol), and nBu 4 PI (3.3^^g, 8.45^^mmol). The autoclave was closed and submitted to several argon-vacuum cycles. Distilled and degassed aniline (4.1^^mL, 45^^mmol) was then syringed into the autoclave. The ethylene pipe was connected to the autoclave, purged, and the ethylene pressure was adjusted to 25^^bar at RT (ca.^^100^^mmol). The temperature was then raised to 150^°C. After 96^^h, the autoclave was cooled to RT and slowly vented in a fume cupboard. The reaction mixture was poured into diethyl ether (120^^mL), stirred for 2^^h, and then filtered. The external standard (N,N-di-n-butylaniline, ca.^^0.15^^g) was added to the collected ethereal phases and the solution was analysed by GC and GCMS. -hexene (11.3^^mL, 91^^mmol) were then sequentially syringed into the autoclave. The temperature was then raised to 150^°C. After 96^^h, the autoclave was cooled to RT and slowly vented in a fume cupboard. The reaction mixture was then poured into diethyl ether (120^^mL), and stirred for 2^^h, then filtered. The external standard (N,N-di-n-butylaniline, ca.^^0.15^^g) was added to the collected ethereal phases and the solution was analysed by GC and GCMS.

<+>Taking into account the liquid nature of both substrates (aniline and 1-hexene), the hydroamination of 1-hexene by aniline could analogously be carried out using Schlenk glassware techniques.

Table^^1 <w=1>Effect of the amount of PPh 3 on the RhCl 3 •3H 2 O-catalysed hydroamination of ethylene by aniline. [a] Entry PPh 3 ^^[equiv] I 2 ^^[equiv] TON 1 [b] TON 2 [b] TON 3 [b] CE [c] 1 45.5^^mmol, 350^^equiv), C 2 H 4 (25^^bar, ca.^^100^^mmol, 770^^equiv), 150^°C, 96^^h.

[b]^^N-ethylaniline (1), N,N-diethylaniline (2), 2-methyl-quinoline (3).

[c]^^TON 1 + 2TON 2 + 2TON 3 .
Table^^2 <w=1>Effect of the nature of the phosphorus ligand on the RhCl 3 •3H 2 Ocatalysed hydroamination of ethylene by aniline in the presence of I 2 . [a] Entry Ligand^^[equiv] t^^[h] TON 1 [b] TON 2 [b] TON 3 [b] CE [c] 1 P(OPh) 3 ( 1 Table^^4 <w=1>Effect of the nature of the cation from the ionic liquid on the RhCl 3 •3H 2 O-catalysed hydroamination of ethylene by aniline. [a] Entry Ionic liquid^^[equiv] Ligand^^[equiv] I 2 ^^[equiv] TON 1 [b] TON 2 [b] TON 3 [b] CE [c] 1 Ph 4 PI ( 20 presence of an "in-situ" generated ionic liquid by one-pot addition of PPh 3 and I 2 . [a] Entry PPh 3 ^^[equiv] I 2 ^^[^equiv] TON 1 [b] TON 2 [b] TON 3 [b] CE [c] Entry T^^[°C] P^^[bar] TON 1 [b] TON 2 [b] TON 3 [b] CE [c] 1 Entry t^^[h] TON 1 [b] TON 2 [b] TON 3 [b] EC [c] 1 Table^^10 <w=1>Hydroamination of ethylene by aniline catalysed by Rh I complexes. [a] Entry Catalyst TON 1 [b] TON 2 [b] TON 3 [b] EC [c] 1

  150^°C. [b]^^N-ethylaniline (1), N,N-diethylaniline (2), 2-methyl-quinoline (3). [c]^^TON 1 + 2 TON 2 + 2 TON 3 .Table^^6<w=1>Effect of reaction temperature and ethylene pressure on the RhCl 3 •3H 2 O-catalysed hydroamination of ethylene by aniline.[a]

  Tri-n-butylphosphine (38^^mL, 0.15 mol) was slowly added to 1-iodobutane (40^^mL, 0.35 mol) under argon. The mixture was stirred for 1^^h at RT and then at 100^°C for 20^^h. The solution was heated with a heat gun for approximately 15^^min. After cooling, the resulting precipitate was washed with diethyl ether (4×50^^mL) until a white solid was obtained. Evaporation of residual solvent was carried out under vacuum overnight to afford pure nBu 4 PI as a white powder (96^%

	yield). m.p. 95--96^°C; 1 H^^NMR (250^^MHz, [D 6 ]acetone): δ=0.96 (t, 3^H, J=3^^Hz), 1.55 (m,
	2^H), 1.70 (m, 2^H), 2.54^^ppm (m, 2^H); 31 P^^NMR (250^^MHz, [D 6 ]acetone): δ=33.52^^ppm (s);
	elemental analysis calcd (%) for C 16 H 36 IP: C^^49.74, H^^9.39; found: C^^49.75, H^^9.57.

  RhCl 3 •3H 2O (34.2^^mg, 0.13^^mmol, 1^^equiv), PPh 3 , I 2 , aniline (4.1^^mL, 45.5^^mmol, 350^^equiv), C 2 H 4 (25^^bar, ca.^^100^^mmol, 770^^equiv), 96^^h,

	1	2<dp>	20<dp>	25<dp>	--<dp>	6<dp> 37<dp>
	2	5<dp>	5<dp>	94<dp>	6<dp>	--<dp>	106<dp>
	3	7<dp>.5	7<dp>.5	134<dp> 30<dp>	--<dp>	194<dp>
	4	10<dp>	10<dp>	21<dp>	189<dp> --<dp>	399<dp>
	5	12<dp>	10<dp>	65<dp>	96<dp>	--<dp>	257<dp>
	6	20<dp>	20<dp>	23<dp>	139<dp> --<dp>	301<dp>
	7	20<dp>	22<dp>	46<dp>	63<dp>	--<dp>	172<dp>
	[a]^^Reaction conditions:

  RhCl 3 •3H 2O (34.2^^mg, 0.13^^mmol), PPh 3 (68.2^^mg, 0.26^^mmol, 2^^equiv), nBu 4PI (3.3^^g, 8.45^^mmol, 65^^equiv), PhNH 2(4.1^^mL, 45.5^^mmol, 350^^equiv), C 2 H 4 , 96^^h. [b]^^N-ethylaniline (1), N,N-diethylaniline (2), 2methyl-quinoline (3). [c]^^TON 1 + 2TON 2 + 2TON 3 .Table^^7<w=1>Effect of reaction time on the RhCl 3 •3H 2 O-catalysed hydroamination of ethylene by aniline.[a]

		150	25	114<dp>	114<dp>	7<dp>	355<dp>
	2	100	25	56<dp>	8<dp>	--<dp>	72<dp>
	3	100	45	30<dp>	2<dp>	3<dp>	40<dp>
	4	125	45	46<dp>	4<dp>	4<dp>	62<dp>
	[a]^^Reaction conditions:

  RhCl 3 •3H 2 O (34.2^^mg, 0.13^^mmol), PPh 3 (68.2^^mg, 0.26^^mmol, 2^^equiv), I 2 (0.066^^g, 0.26^^mmol, 2^^equiv), nBu 4 PI (3.3^^g, 8.45^^mmol, 65^^equiv), PhNH 2 (4.1^^mL, 45.5^^mmol, 350^^equiv), C 2 H 4 (25^^bar, ca.^^100^^mmol, RhCl 3 •3H 2 O (34.2^^mg, 0.13^^mmol), PPh 3 (68.2^^mg, 0.26^^mmol, 2^^equiv), nBu 4 PBr or nBu 4 PI, PhNH 2 (4.1^^mL, 45.5^^mmol, 350^^equiv), I 2 , 1-butene (8.9^^mL, 100^^mmol, ca.^^770^^equiv), 150^°C, 96^^h. [b]^^Ratio of Markovnikov (M) and anti-Markovnikov (AM) isomers. n.d.=not determined.

	[a]^^Reaction conditions: Table^^9 <w=1>Hydroamination of 1-hexene by aniline catalysed by
	RhCl 3 •3H 2 O/PPh 3 . [a]
	Entry	10 Catalyst	56	5<dp> Salt^^[equiv] Ligand^^[equiv] t^^[h] TON 3 72<dp>	M/AM [b]
	2	93	28	225<dp>	3	484<dp>
	1	PtBr 2		nBu 4 PI (65)	--		10	6<dp> 95<dp>:5
	2 3 [a]^^Reaction conditions: RhCl 3 •3H 2 O (34.2^^mg, 0.13^^mmol), PPh 3 (68.2^^mg, RhCl 3 •3H 2 O nBu 4 PI (150) PPh 3 (2) 96 56<dp> 100<dp>:0 RhCl 3 •3H 2 O --PPh 3 (2) 96 1<dp> 100<dp>:0 0.26^^mmol, 2^^equiv), nBu 4 PI, PhNH 2 (4.1^^mL, 45.5^^mmol, 350^^equiv), 1-hexene [a]^^Reaction conditions: Table^^8 <w=1>Hydroamination of 1-butene by aniline catalysed by the (11.3^^mL, 91^^mmol, ca.^^770^^equiv), 150^°C. [b]^^Ratio of Markovnikov (M) and anti-
	RhCl 3 •3H 2 O/PPh 3 system. [a] Markovnikov (AM) isomers.
	Entry	Ligand	Salt^^[equiv] I 2 ^^[equiv]	TON	M/AM [b]
	1	--	--		--<dp>	1<dp>	n.d.
	2	--	nBu 4 PBr (65) --<dp>	18<dp>	99:1
	3	--	nBu 4 PI (65)	--<dp>	74<dp>	99:1
	4	PPh 3	--		2<dp>	23<dp>	99:1
	5	PPh 3	nBu 4 PI (65)	--<dp>	86<dp>	99:1
	6	PPh 3	nBu 4 PI (65)	2<dp>	153<dp>	99:1
	7	PPh 3	nBu 4 PI (150)	2<dp>	133<dp>	99:1

770^^equiv), 150^°C. [b]^^N-ethylaniline (1), N,N-diethylaniline (2), 2-methyl-quinoline (3). [c]^^TON 1 + 2TON 2 + 2TON 3 .

Acknowledgements

The <cgs>Centre National de la Recherche Scientifique (France)</cgs>, the <cgs>FSE (Fond Social European)</cgs>, and the <cgs>ANR</cgs> (<cgs>Agence National de la Recherche Scientifique</cgs>, <cgn>NT09_442499</cgn>) are acknowledged for financial support. We gratefully acknowledge Dr. J.-J. Bonnet and Dr. D. Neibecker for academic support and Dr. F. Malbosc (Solvionic) for helpful discussions.