D A R I O V I N C E N Z I

Orientation of non-spherical particles in an axisymmetric random flow

The dynamics of non-spherical rigid particles immersed in an axisymmetric random flow is studied analytically. The motion of the particles is described by Jeffery's equation; the random flow is Gaussian and has short correlation time. The stationary probability density function of orientations is calculated exactly. Four regimes are identified depending on the statistical anisotropy of the flow and on the geometrical shape of the particle. If λ is the axis of symmetry of the flow, the four regimes are: rotation about λ, tumbling motion between λ and -λ, combination of rotation and tumbling, and preferential alignment with a direction oblique to λ.

Introduction

Non-spherical solid particles suspended in a moving fluid rotate and orient themselves under the action of the velocity gradient. Even at low concentrations, the orientational dynamics of non-spherical particles can influence the rheological properties of a suspension, namely the intrinsic viscosity and the normal stress coefficients (Bird et al. 1977;[START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF]. This phenomenon has diverse practical applications. In the turbulent regime, for instance, the injection of rodlike polymers in a Newtonian fluid can produce a considerable reduction of the turbulent drag, with this effect being routinely exploited to reduce energy losses in pipelines [START_REF] Gyr | Drag Reduction of Turbulent Flows by Additives[END_REF]. The study of the orientation of particles immersed in a fluid also has numerous applications in the natural sciences. Amongst them it is worth mentioning the swimming motion of certain biological micro-organisms [START_REF] Saintillan | Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations[END_REF][START_REF] Koch | Collective hydrodynamics of swimming microorganisms: Living fluids[END_REF] and the formation of ice crystals in clouds [START_REF] Chen | The theoretical basis for the parametrization of ice crystal habits: growth by vapor deposition[END_REF]. This latter phenomenon plays a crucial role in processes such as rain initiation and radiative transfer.

The starting point for understanding the properties of a dilute suspension is the motion of an isolated particle in a given flow field. Analytical results on the dynamics of a non-spherical particle have been obtained for various laminar flows, both steady and unsteady. [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] derived the equations of motion for an inertialess ellipsoid in a steady uniform shear flow at low Reynolds number. For a spheroid (i.e. an ellipsoid of revolution), Jeffery showed that the axis of symmetry of the particle performs a periodic motion on a closed orbit. [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] subsequently extended Jeffery's analysis to particles of a more general shape: he demonstrated that, except for certain very long particles, the dynamics of any body of revolution transported by a low-Reynolds-number shear is equivalent to that of a spheroid with an effective aspect ratio. Amongst bodies of revolution, rigid dumbbells and rods have received a systematic investigation [START_REF] Bird | Kinetic theory and rheology of dumbbell suspensions with Brownian motion[END_REF][START_REF] Doi | The Theory of Polymer Dynamics[END_REF].

In [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF] and [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] derivations, not only fluid and particle inertia are disregarded, but also Brownian fluctuations due to the collisions of the particle with the molecules of the fluid. Nevertheless, if a particle is sufficiently small, molecular diffusion does influence its orientational dynamics, as was shown by [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF] and [START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF]. The review article by [START_REF] Brenner | Rheology of a dilute suspension of axisymmetric Brownian particles[END_REF] collects analytical results on the motion of rigid neutrally buoyant bodies of revolution subject to a uniform velocity gradient and to Brownian fluctuations. A more recent review on this problem can be found in [START_REF] Petrie | The rheology of fibre suspensions[END_REF]. Finally, even in simple laminar flows, the orientation of particles can form complex patterns; this behaviour was studied by [START_REF] Szeri | Pattern formation in recirculating flows of suspensions of orientable particles[END_REF] in the context of the theory of dynamical systems.

In turbulent flows and in some chaotic flows, the velocity gradient exhibits small-scale fluctuations. Thus, depending on the size of the particles, Jeffery's assumption of a spatially uniform velocity gradient may not be applicable. To measure the probability of orientations, experiments have generally used relatively large particles, and therefore the velocity gradient could vary appreciably over the size of a particle (e.g. [START_REF] Krushkal | On the orientation distribution function of nonspherical aerosol particles in a general shear flow-II. The turbulent case[END_REF][START_REF] Bernstein | Direct determination of the orientation distribution function of cylindrical particles immersed in laminar and turbulent shear flows[END_REF][START_REF] Newsom | Orientational properties of fibrous aerosols in atmospheric turbulence[END_REF][START_REF] Parsheh | On the orientation of stiff fibres suspended in turbulent flow in a planar contraction[END_REF][START_REF] Parsa | Rotation and alignment of rods in two-dimensional chaotic flow[END_REF]Zimmermann et al. 2011a,b). Accordingly, theoretical studies were mainly concerned with the derivation of model equations for the probability of orientations, in which turbulent fluctuations were treated as an effective isotropic diffusion term [START_REF] Olson | The motion of fibres in turbulent flow[END_REF][START_REF] Shin | Rotational and translational dispersion of fibres in isotropic turbulent flows[END_REF]). Jeffery's approach, however, remains applicable to chaotic or turbulent flows provided that the particles are sufficiently small. The orientation dynamics of tiny elongated particles was studied numerically in channel flows [START_REF] Zhang | Ellipsoidal particles transport and deposition in turbulent channel flows[END_REF]Mortensen et al. 2008a,b), in isotropic turbulence [START_REF] Shin | Rotational and translational dispersion of fibres in isotropic turbulent flows[END_REF][START_REF] Pumir | Orientation statistics of small particles in turbulence[END_REF][START_REF] Parsa | Rotation rate of rods in turbulent fluid flow[END_REF], and in chaotic velocity fields [START_REF] Wilkinson | Fingerprints of random flows[END_REF]. In contrast with the case of laminar flows, few analytical results seem to exist for the probability distribution of orientations of small particles transported by a turbulent or chaotic flow. [START_REF] Turitsyn | Polymer dynamics in chaotic flows with a strong shear component[END_REF] examined the tumbling motion of rodlike polymers in a random flow resulting from the superposition of a mean shear and of white-in-time isotropic fluctuations. [START_REF] Wilkinson | A model for alignment between microscopic rods and vorticity[END_REF] recently studied the alignment of rods with vorticity in a turbulent isotropic flow.

Here, the probability density function (p.d.f.) of orientations is derived exactly for small particles transported by a random flow with axisymmetric statistics. The particles are general bodies of revolution possessing fore-aft symmetry. The axisymmetry of the flow means that the velocity field is statistically invariant under arbitrary rotations of the axes about a given direction as well as under reflections in planes containing that direction or normal to it [START_REF] Batchelor | The theory of axisymmetric turbulence[END_REF][START_REF] Chandrasekhar | The theory of axisymmetric turbulence[END_REF]. Axisymmetry is the simplest form of statistical anisotropy [START_REF] Biferale | Anisotropy in turbulent flows and in turbulent transport[END_REF][START_REF] Chang | Experimental study of the influence of anisotropy on the inertial scales of turbulence[END_REF], and is found in rotating, stratified, or wind-tunnel turbulence [START_REF] Lindborg | Kinematics of homogeneous axisymmetric tubulence[END_REF]. Furthermore, the random flow is assumed to be Gaussian and to have zero correlation time. The assumption of temporal decorrelation is adequate when the correlation time of the flow is short compared to the characteristic time scale of material-line-element stretching. This assumption, albeit restrictive, allows a fully analytical solution of the problem.

The evolution equation for the orientation vector of a non-spherical rigid particle is introduced in § 2. Section 3 is devoted to the derivation of the Fokker-Planck equation for the p.d.f. of the orientation angle. Its stationary solution is studied in § 4. Some conclusions are drawn in § 5.

Orientation dynamics

The particles considered here are rigid bodies of revolution possessing fore-aft symmetry (although in the literature such particles are commonly referred to as "axisymmetric", this terminology will be avoided here not to generate confusion; the term "axisymmetry" will be reserved to the statistical invariance of the velocity field). The particles are of uniform composition and are suspended in a Newtonian fluid of the same density. Furthermore, the inertia of the particles as well as hydrodynamic particle-particle interactions are disregarded, and no externally imposed force or couple influences the dynamics. In particular, it is appropriate to disregard hydrodynamic interactions when the suspension is sufficiently dilute.

The undisturbed motion of the fluid is described by the velocity field v(x, t). The size of the particles is assumed to be small compared to the typical length over which the velocity gradient ∇v = (∂ j v i ) 1 i,j 3 changes (∂ j ≡ ∂/∂x j ). For turbulent flows, this assumption is satisfied if the particles are smaller than the Kolmogorov scale, where viscosity smooths out the velocity fluctuations; the Reynolds number of the flow at the scale of a particle is thus less than 1. Given their small size, the particles also experience Brownian collisions with the molecules of the fluid.

In a sufficiently dilute suspension, attention can be restricted to the dynamics of a single isolated particle. The configuration of a body of revolution is determined by the position of its centre of mass, r c (t), and by the orientation of its axis of revolution, which is specified by a unit vector N (t) parallel to the axis itself. As the particles are subject to Brownian fluctuations, their dynamics is random even in a laminar flow. Consider first a deterministic velocity field or a given realisation of a random velocity field. On the above assumptions, the centre of mass moves according to the following equation (e.g. [START_REF] Doi | The Theory of Polymer Dynamics[END_REF]:

ṙc (t) = v(r c (t), t) + D T ζ(t), (2.1)
where D T > 0 is the translational diffusion coefficient and ζ(t) is three-dimensional white noise, i.e. a Gaussian stochastic process with

ζ(t) = 0 and ζ i (t + τ )ζ j (t) = δ ij δ(τ ) (2.2)
for all t, τ > 0 and i, j = 1, 2, 3. The orientation vector satisfies the following stochastic differential equation (summation over repeated indexes is implied):

Ṅ i = κ ij (t)N j -κ pq (t) N p N q |N | 2 N i + D R Σ ij (N ) • ξ j (t), |N (0)| = 1, (2.3)
where

κ(t) = Ω(t) + γE (t) (2.4)
with

Ω(t) = G (t) -G T (t) 2 , E (t) = G (t) + G T (t) 2 , (2.5)
and G (t) = ∇v(r c (t), t). Thus, Ω(t) and E (t) are the vorticity tensor and the rate-ofstrain tensor evaluated at r c (t) and κ(t) is an effective Lagrangian velocity gradient.

The scalar constant γ depends on the geometrical shape of the particle. For |γ| < 1, the evolution equation can be mapped into that of a spheroid with aspect ratio equal to (1 + γ)/(1γ) [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF]. Prolate spheroids are obtained for 0 < γ < 1, oblate spheroids for -1 < γ < 0. Special cases are: spheres (γ = 0), rigid dumbbells (γ = 1), rods (γ = 1), and disks (γ = -1). Furthermore, it was shown by [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] that in principle there exist very long particles for which |γ| > 1. In (2.3), the random vector ξ(t) is three-dimensional white noise and hence has the same properties as ζ(t), but is statistically independent of it. The matrix Σ (n) has the following form:

Σ (n) = I -nn/|n| 2 (2.6)
and D R > 0 is the rotary diffusion coefficient. By using the Cauchy-Schwarz inequality, it is easy to check that Σ (n) is positive semi-definite. The symbol • indicates that the stochastic term in (2.3) is understood in the Stratonovich sense. Equation (2.3) is Jeffery's equation for the orientation vector of a body of revolution with the addition of a stochastic term modelling Brownian fluctuations. The stochastic term is chosen in such a way as to produce isotropic diffusion of N (t) on the unit sphere, so that |N (t)| is preserved in time (see appendix A for more details). It is worth remarking that a Brownian term of the same form has been used to model the turbulent fluctuations of the velocity gradient [START_REF] Krushkal | On the orientation distribution function of nonspherical aerosol particles in a general shear flow-II. The turbulent case[END_REF][START_REF] Olson | The motion of fibres in turbulent flow[END_REF][START_REF] Shin | Rotational and translational dispersion of fibres in isotropic turbulent flows[END_REF] or to describe particle-particle interactions both in semi-dilute suspensions [START_REF] Doi | The Theory of Polymer Dynamics[END_REF] and in concentrated suspensions [START_REF] Doi | Dynamics of rod-like macromolecules in concentrated solution[END_REF][START_REF] Kuzuu | Nonlinear viscoelasticity of concentrated solution of rod-like polymers[END_REF].

Equation ( 2.3) can be generalised to the case of a homogeneous axisymmetric random flow. The velocity field transporting the particle is Gaussian and has zero mean and correlation:

v i (x + r, t + τ )v j (x, t) = Q ij (r)δ(τ ), i, j = 1, 2, 3. (2.7)
The form of the correlation guarantees that v(x, t) is statistically homogeneous in space.

Additionally, the velocity field is assumed to be incompressible (∇ • v = 0) and statistically axisymmetric with respect to the direction specified by the unit vector λ. The tensor Q(r) must then take the form:

Q ij (r) = Ar i r j + Bδ ij + Cλ i λ j + D(r i λ j + λ i r j ), (2.8)
where A, B, C, D are smooth functions of |r| 2 and (r •λ); A, B, and C are even in (r •λ), while D is odd in (r • λ) [START_REF] Batchelor | The theory of axisymmetric turbulence[END_REF][START_REF] Chandrasekhar | The theory of axisymmetric turbulence[END_REF]. Furthermore, the functions A, B, C, D are not independent and satisfy certain differential relations [START_REF] Batchelor | The theory of axisymmetric turbulence[END_REF][START_REF] Chandrasekhar | The theory of axisymmetric turbulence[END_REF]. The velocity field defined above is an axisymmetric generalisation of the isotropic random flow introduced by [START_REF] Kraichnan | Small-scale structure of a scalar field convected by turbulence[END_REF] in the context of passive turbulent transport. The same axisymmetric velocity field was used by [START_REF] Shaqfeh | Polymer stretch in dilute fixed beds of fibres or spheres[END_REF] to study polymer stretching in flows through random beds of fibres.

The velocity gradient is also Gaussian and zero-mean; the single-point two-time correlation can be derived by using the statistical homogeneity of the velocity field:

∂ j v i (x, t + τ )∂ q v p (x, t) = Γ ijpq δ(τ ) i, j, p, q = 1, 2, 3 (2.9) with Γ ijpq = - ∂ 2 Q ip ∂r j ∂r q r=0 .
(2.10) Substituting (2.8) in (2.10) yields (see equation (5.12) in [START_REF] Batchelor | The theory of axisymmetric turbulence[END_REF]):

Γ ijpq = (d + 4a)δ jq δ ip -a(δ pq δ ij + δ iq δ pj ) + (b + c + 5d)δ jq λ i λ p -bδ ip λ j λ q -d[(δ pq λ i + δ iq λ p )λ j + (δ pj λ i + δ ij λ p )λ q ] -cλ i λ p λ j λ q , (2.11)
where a, b, c, d are real constants. For the sake of simplicity, λ is taken in the direction of the third axis, i.e. λ = (0, 0, 1). Then, the coefficients in Γ ijpq are written:

2a = Γ 1212 -Γ 1111 d = Γ 3333 -Γ 1111 b = Γ 1212 -Γ 1313 2a + b + c + 4d = Γ 3131 -Γ 3333 .
(2.12)

The two-time correlation of the components of the vorticity ω = ∇ × v can be expressed in terms of a, c, and d as follows (Batchelor 1946, p. 490):

ω 1 (x, t + τ )ω 1 (x, t) = ω 2 (x, t + τ )ω 2 (x, t) = (10a + c + 9d)δ(τ ) (2.13)
and

ω 3 (x, t + τ )ω 3 (x, t) = (10a + 2d)δ(τ ).
(2.14)

The coefficients a, b, c, d are not free; they are constrained by the following inequalities (see appendix B):

a + d > 0 if a 0, 5a + d > 0 if a < 0, (2.15a) 4a -b + d > 0, 4a + b + c + 6d > 0, (2.15b) 15a 2 -b 2 -(b -d)(c + 5d) + 2a(2c + 13d) > 0. (2.15c)
If a > 0 and b = c = d = 0, then Γ ijpq gives the single-point correlation of the gradient of an isotropic velocity field [START_REF] Robertson | The invariant theory of isotropic turbulence[END_REF]). Thus, a determines the intensity of the isotropic part of the gradient and b, c, d control the statistical anisotropy of the flow.

The orientation dynamics of the particle depends on the velocity gradient evaluated at r c (t), which was denoted as G (t) in § 2. In virtue of the δ-correlation in time and the statistical homogeneity of the flow, G (t) has the same temporal statistics as ∇v(x, t) for any given x (Falkovich, Gawȩdzki & Vergassola 2001) (the presence of white noise in (2.1) does not modify the statistics of G (t)). The components of κ(t), defined in (2.4), are a linear combination of the components of G (t), and consequently κ(t) is a Gaussian process with zero mean and correlation:

κ ij (t + τ )κ pq (t) = K ijpq δ(τ ) (2.16) with K ijpq = 1 4 [Γ ijpq -Γ ijqp -Γ jipq +Γ jiqp +2γ(Γ ijpq -Γ jiqp )+γ 2 (Γ ijpq +Γ ijqp +Γ jipq +Γ jiqp )].
(2.17) The form of K ijpq can be derived by substituting (2.4) and (2.5) into the left-hand side of (2.16) and by using (2.9). The tensor K ijpq is positive semi-definite, for it is the covariance of a second-order Gaussian tensor. Thus, in the evolution equation for the orientation vector, κ(t) plays the role of a multiplicative tensorial white noise. As κ(t) can be thought of as an approximation of a real noise process in the limit of zero correlation time, the corresponding terms in (2.3) must be interpreted in the Stratonovich sense (e.g. Kloeden & Platen 1992, p. 227). The stochastic differential equation for the orientation vector of a non-spherical particle can then be rewritten as follows:

Ṅ i = M ipq (N ) • κ pq (t) + D R Σ ij (N ) • ξ j (t), (2.18)
where

M ipq (n) = (δ ip δ jk -δ ik δ jp )n j n k n q /|n| 2 and the initial condition N (0) is such that |N (0)| = 1.

Fokker-Planck equation for the probability density function of the orientation angle

As v(x, t) is statistically invariant under spatial translations, the p.d.f. of N (t) taking the value n = (n 1 , n 2 , n 3 ) at time t is independent of r c and is thus denoted by f (n; t). The Itô equation equivalent to (2.18) is with

Ṅ i = β i (N ) + M ipq (N )κ pq (t) + D R Σ ij (N )ξ j (t) (3.1)
β i (n) = 1 2 K mnpq M jpq (n) ∂ ∂n j M imn (n) + D R 2 Σ jk (n) ∂ ∂n j Σ ik (n). (3.2)
Consequently, f (n; t) satisfies the Fokker-Planck equation:

∂f ∂t = - ∂ ∂n i [β i (n)f ] + 1 2 ∂ 2 ∂n i ∂n j [α ij (n)f ] (3.3) with α ij (n) = K mnpq M imn (n)M jpq (n) + D R Σ ik (n)Σ jk (n) = K mnpq M imn (n)M jpq (n) + D R Σ ij (n), (3.4)
where the last equality follows from (A 3). Equations (3.1) and (3.3) can be derived from (2.18) by using the formal rules κ ij (t)dt = O( √ dt) and κ ij (t)dt κ pq (t)dt = K ijpq dt and by proceeding as in the case of a vectorial white noise (see [START_REF] Gardiner | Handbook of stochastic methods[END_REF] and the appendix in [START_REF] Falkovich | Particles and fields in fluid turbulence[END_REF]. The diffusion tensor α is positive semi-definite as a consequence of the positive semi-definiteness of K and Σ .

To study the orientation dynamics of a non-spherical particle, it is convenient to move from Cartesian coordinates (n 1 , n 2 , n 3 ) to spherical coordinates (n, ϑ, ϕ) according to the usual transformations:

n = n 2 1 + n 2 2 + n 2 3 , ϑ = arctan n 2 1 + n 2 2 /n 3 , ϕ = arctan(n 2 /n 1 ) (3.5)
with 0 n, 0 ϑ π, and 0 ϕ < 2π (figure 1). On account of the fixed length of the orientation vector, the probability density function of orientations must take the form f (n, ϑ, ϕ; t) = ψ(ϑ, ϕ; t)δ(n-L) with L = 1. Thus, ψ(ϑ, ϕ; t) sin ϑ dϑ dϕ is the probability of the particle being oriented at time t within an elementary solid angle sin ϑ dϑ dϕ of (ϑ, ϕ). In addition, the following normalisation holds:

π 0 2π 0 ψ(ϑ, ϕ; t) sin ϑ dϑ dϕ = 1 ∀ t 0. (3.6)
The function Ψ (ϑ, ϕ; t) = ψ(ϑ, ϕ; t) sin ϑ satisfies a new Fokker-Planck equation, which can be derived from (3.3) by using the transformation formulae for the drift and diffusion coefficients under a change of variables (Risken 1989, p. 88), by writing f (n, ϑ, ϕ; t) = ψ(ϑ, ϕ; t)δ(n -L) with L = 1, and by integrating the resulting equation with respect to n. The final result is:

∂Ψ ∂t = - ∂ ∂ϑ [B ϑ (ϑ)Ψ ] + 1 2 ∂ 2 ∂ϑ 2 [A ϑϑ (ϑ)Ψ ] + 1 2 A ϕϕ (ϑ) ∂ 2 Ψ ∂ϕ 2 , (3.7)
where

B ϑ (ϑ) = β i (n) ∂ϑ ∂n i + α ij (n) 2 ∂ 2 ϑ ∂n i ∂n j = µ 1 + µ 2 sin 2 (ϑ) + µ 3 sin 4 (ϑ) cot ϑ,(3.8a) A ϑϑ (ϑ) = α ij (n) ∂ϑ ∂n i ∂ϑ ∂n j = 2µ 1 + µ 4 sin 2 (ϑ) + µ 3 sin 4 (ϑ), (3.8b) A ϕϕ (ϑ) = α ij (n) ∂ϕ ∂n i ∂ϕ ∂n j = µ 5 + 2µ 1 csc 2 (ϑ) (3.8c) with µ 1 = 1 8 2(5 + 3γ 2 )a -4γb + (1 -γ) 2 c + (9 -10γ + 5γ 2 )d + 4D ′ R , (3.9a
)

µ 2 = 3 4 γ [2b + (1 -γ)c + (5 -γ)d] ,
(3.9b)

µ 3 = cγ 2 , (3.9c) µ 4 = γ[2b + (1 -γ)c + 5d],
(3.9d)

µ 5 = 1 4 4bγ -(1 -γ) 2 c -(1 -γ)(7 -3γ)d , (3.9e)
and D ′ R = D R /L 2 with L = 1 (the numerical values of D ′ R and D R coincide, but their physical dimensions are different). Equations (3.8) involve the Jacobian and the Hessian of the transformation from Cartesian to spherical coordinates, which can be calculated from (3.5). Inequalities (2.15) guarantee that A ϑϑ (ϑ) and A ϕϕ (ϑ) are strictly positive (appendix B).

The contribution to (3.7) due to the isotropic part of Γ is a Laplace-Beltrami term with diffusion coefficient proportional to a; this contribution is of the same form as that coming from ξ(t). The isotropic component of the flow and the Brownian fluctuations therefore have the same effect on the orientation statistics of the particle.

Since ϑ and ϕ are angular variables, the boundary conditions for Ψ (ϑ, ϕ; t) are periodic:

Ψ (ϑ, ϕ; t) = Ψ (ϑ + 2π, ϕ; t) and Ψ (ϑ, ϕ; t) = Ψ (ϑ, ϕ + 2π; t) (3.10)
for all ϑ, ϕ, and t. The long-time properties of Ψ (ϑ, ϕ; t) can be deduced from (3.7). If the partial derivative with respect to time is dropped, then (3.7) is invariant under the transformations ϑ ↔ 2πϑ (reflections with respect to planes containing λ) and ϑ ↔ πϑ (reflections with respect to planes orthogonal to λ). Moreover, the coefficients B ϑ , A ϑϑ , and A ϕϕ do not depend on ϕ (invariance under rotations about λ). These properties of (3.7) are a natural consequence of the statistical axisymmetry of the velocity field and translate into analogous properties of the stationary p.d.f. of orientations. The invariance of (3.7) under rotations about λ can be used to derive a one-dimensional Fokker-Planck equation for the marginal p.d.f.: Ψ (ϑ; t) = ψ(ϑ; t) sin ϑ with ψ(ϑ; t) = 2π 0 ψ(ϑ, ϕ; t)dϕ. Integrating (3.7) with respect to ϕ from 0 to 2π and making use of (3.10) yield:

∂ Ψ ∂t = - ∂ ∂ϑ B ϑ (ϑ) Ψ + 1 2 ∂ 2 ∂ϑ 2 A ϑϑ (ϑ) Ψ . (3.11)
The solution of the above equation must be normalised and periodic: Ψ (ϑ; t) = Ψ (ϑ+2π; t) for all ϑ, t. A direct consequence of (3.11) is that, along the trajectory of the particle, the time evolution of ϑ(t) is decoupled from that of ϕ(t) and is described by the stochastic ordinary differential equation: (3.12) where η(t) is white noise.

θ(t) = B ϑ (ϑ(t)) + A ϑϑ (ϑ(t)) η(t), 0 ϑ(t) π,

Stationary statistics of the orientation angle

It was argued in § 3 that the stationary solution of (3.7), Ψ st , does not depend on ϕ. Therefore, Ψ st solves the equation:

1 2 d 2 dϑ 2 [A ϑϑ (ϑ)Ψ st ] - d dϑ [B ϑ (ϑ)Ψ st ] = 0. (4.1)
Thanks to the periodic boundary conditions (3.10), Ψ st (ϑ) takes the following simple form (see appendix C):

Ψ st (ϑ) = N A ϑϑ (ϑ) exp 2 ϑ ϑ0 B ϑ (z) A ϑϑ (z) dz , (4.2)
where 0 ϑ 0 π and N is a normalisation constant such that

2π π 0 Ψ st (ϑ)dϑ = 1. (4.3)
In (4.2), the choice of ϑ 0 is in fact unimportant, since it only modifies the normalisation constant.

Only the case µ 3 = 0 is considered here; the case µ 3 = 0 is examined in appendix D, even though no new physical regimes emerge in this latter case. For µ 3 = 0, the integral in (4.2) can be calculated by using the change of variable y = sin 2 (ω) and formulae 2.172, 2.175(1), and 2.177(1) of [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]. The final result is:

ψ st (ϑ) = N χ(ϑ) [A ϑϑ (ϑ)] 3/4 , (4.4)
where b, c, d, andγ. The stationary p.d.f. of orientations satisfies ψ st (2πϑ) = ψ st (ϑ) and ψ st (πϑ) = ψ st (ϑ). These properties are a consequence of the statistical symmetries of the carrier flow, as was noted after (3.10).

χ(ϑ) =                        exp 2 √ ∆ µ 2 - 3 4 µ 4 arctan µ 4 + 2µ 3 sin 2 (ϑ) √ ∆ (∆ > 0) exp 3 4 µ 4 -µ 2 2 µ 4 + 2µ 3 sin 2 (ϑ) (∆ = 0) √ -∆ + µ 4 + 2µ 3 sin 2 (ϑ) √ -∆ -µ 4 -2µ 3 sin 2 (ϑ) 1 √ -∆ ( 3 4 µ4-µ2) (∆ < 0) (4.5) with ∆ = 8µ 1 µ 3 -µ 2 4 . It is shown in appendix B that ψ st (ϑ) is bounded for all values of a,
Since ∆ can be written as ∆ = a 2 F (γ, b/a, c/a, d/a, D ′ R /a), ψ st (ϑ) only depends on the ratios b/a, c/a, d/a, D ′ R /a (and on γ). The same conclusion could have been reached by rescaling t by a -1 in (3.7).

For a spherical particle (i.e. γ = 0), µ 2 and µ 4 vanish. Consequently, χ(ϑ) = 1, A ϑϑ (ϑ) = const., and hence ψ st (ϑ) = (4π) -1 , in accordance with the fact that all orientations are equally probable for a sphere. Similarly, if a or D ′ R are much greater than b, c, d, i.e. if the isotropic component of the velocity field or the Brownian fluctuations prevail on the anisotropic component of the flow, then µ 1 µ i , i = 2, . . . , 5 and ψ st (ϑ) weakly depends on ϑ regardless of the shape of the particle. In the following, therefore, γ is assumed to be nonzero and b, c, d are of the same order of magnitude as a and D ′ R or greater.

The behaviour of ψ st (ϑ) can be deduced from that of its first derivative. By using A ϑϑ (ϑ) > 0, it can be shown that, for all values of ∆,

d dϑ ψ st (ϑ) = γ sin(2ϑ)[cγ cos(2ϑ) -σ]h(ϑ), (4.6)
where

σ ≡ 2b + c + (5 + γ)d (4.7)
and h(ϑ) is a strictly positive function for all 0 ϑ π. Four regimes are then identified depending on the properties of the axisymmetric flow and on the geometrical shape of the particle: (i) Rotation about the axis of symmetry of the flow. For the following values of the parameters:

|σ| > |cγ| and γσ < 0, (4.8)
the function ψ st only has three extrema in ϑ = 0, π/2, π (indeed the equation cγ cos(2ϑ) = σ has no solution). More precisely, ψ st (ϑ) has a maximum in π/2 and two minima in 0 and π (figure 2). Thus, in this regime the particle rotates about the direction λ; the level of alignment with the plane perpendicular to λ decreases as the degree of anisotropy of the flow vanishes or the shape of the particle approaches the spherical one (figure 2).

(ii) Tumbling motion. In the following regime:

|σ| > |cγ| and γσ > 0, (4.9)

ψ st has three extrema: a minimum in π/2 and two maxima in 0 and in π (figure 2). The particle tumbles between the direction parallel to λ and that antiparallel to λ. The probability of the orientation angle ϑ being in the neighbourhood of 0 or π depends on the anisotropy degree of the flow and on the shape of the particle (figure 2).

(iii) Preferential alignment with a direction oblique to the axis of symmetry of the flow. If |σ| < |cγ| and c > 0, (4.10) then ψ st has three minima in 0, π/2, π and two maxima in ϑ ⋆ and πϑ ⋆ , where 0 < ϑ ⋆ < π/2 is such that

sin ϑ ⋆ = 1 2 1 - σ cγ .
(4.11)

The particle therefore spends most of the time at an angle ϑ ⋆ (or πϑ ⋆ ) with respect to λ (figure 3).

(iv) Combination of rotation and tumbling. For |σ| < |cγ| and c < 0, (4.12) the function ψ st has three maxima in 0, π/2, π and two minima in ϑ ⋆ and πϑ ⋆ with ϑ ⋆ defined in (4.11). In this regime, the particle preferentially lies either in the plane perpendicular to λ, in the direction parallel to λ, or in the direction antiparallel to it. Naturally, the coefficient a controlling the intensity of the isotropic component of the flow does not play any role in the above classification; the dynamical regime is selected by b, c, d, and by the shape coefficient γ. Whereas sufficiently elongated or flattened spheroids can be strongly aligned in regimes (i) and (ii), the ability of the flow to orient particles is weaker in regimes (iii) and (iv). In these regimes, stronger alignment can be obtained for |γ| > 1 (figure 4). Nevertheless, [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF] observed that particles with |γ| > 1 may be unrealistic, albeit conceivable from a purely geometrical point of view.

The form of ψ st (ϑ) simplifies considerably if τ )∂ 3 v 3 (x, t) , i.e. if d = 0. In this case, µ 2 = 3µ 4 /4 and hence ψ st (ϑ) = N [A ϑϑ (ϑ)] -3/4 ; furthermore, σ does not depend on γ. It then follows from (4.8) and (4.9) that if the particles with shape coefficient γ rotate in the plane orthogonal to λ (resp. tumble), the particles with shape coefficientγ tumble (resp. rotate in the plane orthogonal to λ). By contrast, regimes (iii) and (iv) are independent on the sign of γ, i.e. on whether the particle is elongated or flattened, although ϑ ⋆ changes when the sign of γ changes. Moreover, if the particles with shape coefficient γ rotate about λ (resp. tumbles), then all particles with |γ| < | γ| and sgn(γ) = sgn( γ) rotate about λ (resp. tumble). Similarly, if the particles with shape coefficient γ are in regime (iii) (resp. in regime (iv)), then all particles with |γ| > | γ| are in regime (iii) (resp. in regime (iv)). These properties, however, do not generally hold true if d is nonzero. For instance, if b/a = -4.4, c/a = 0.5, d/a = 1.5, then particles tumble for 1 γ > 0.8, they have a preferential orientation for 0.8 > γ > 0.4, they rotate for 0.4 > γ > 0, and they tumble again for 0 > γ -1.

∂ 1 v 1 (x, t + τ )∂ 1 v 1 (x, t) = ∂ 3 v 3 (x,

Conclusions

Axisymmetric turbulence arises as one of the simplest frameworks in which to study the orientation dynamics of non-spherical particles. On the assumptions of Gaussianity and short correlation in time, it was shown analytically that the dynamics of a non-spherical particle immersed in a random axisymmetric flow exhibits four regimes: rotation around the axis of symmetry of the flow, tumbling, combination of rotation and tumbling, and preferential alignment with a direction oblique to the axis of symmetry of the flow. The regime is selected by the form of the anisotropic component of the flow and by the geometrical shape of the particle. If the flow is weakly anisotropic or if the particle is almost spherical, the mathematical description of ψ st (ϑ) in terms of minima and maxima remains formally valid, but the above physical classification loses its meaning, since ψ st (ϑ) does not differ appreciably from the uniform distribution.

The tumbling motion of a non-spherical particle in the axisymmetric random flow differs from that of a rod immersed in the flow resulting from the superposition of a uniform shear and of a short-correlated isotropic random component (Puliafito & Turitsyn 2005;[START_REF] Turitsyn | Polymer dynamics in chaotic flows with a strong shear component[END_REF]). In the presence of a strong mean shear, the tumbling dynamics of a rod consists of aperiodic transitions between two unstable states: the one aligned with the direction of the shear and that anti-aligned with it. When a fluctuation takes the rod away from the aligned or anti-aligned state and moves it into the unstable region of the flow the mean shear makes the rod flip. By contrast, in the axisymmetric case, large deviations of the orientation of the particle from the axis of symmetry of the flow do not necessarily result into sudden flips of the particle (figure 5). Simply, the orientation vector of the particle fluctuates randomly, but the orientations aligned and anti-aligned with the axis of symmetry of the flow are much more probable than the other orientations. Thus, there can be excursions of ϑ(t) from ϑ ≈ 0 to ϑ ≈ π/2 followed by a return to ϑ ≈ 0 (figure 5). This behaviour would not be possible in the presence of a strong mean shear. The axisymmetric random flow and the laminar uniform shear differ in the dependence of the orientation dynamics on the geometrical shape of particles. In a uniform shear flow, the motion of a rod or of a disc represents a degenerate case of the dynamics of non-spherical particles [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF]. Moreover, the dynamics is qualitatively different for |γ| < 1 and |γ| > 1 [START_REF] Bretherton | The motion of rigid particles in a shear flow at low Reynolds number[END_REF]). In the axisymmetric random flow, the dynamics of particles changes smoothly as a function of the shape coefficient.

Finally, it is worth remarking that the function χ defined in (4.5) also determines the probability distribution of Jeffery's orbits in the presence of weak Brownian fluctuations [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF]. There does not seem to be, however, a simple relation between the orientation dynamics in the uniform shear and that in the axisymmetric random flow.

The author is grateful to Jérémie Bec, Antonio Celani, François Delarue, Prasad Perlekar, Raymond Shaw, and Michael Wilkinson for useful suggestions. This work was supported in part by the French National Research Agency (ANR) under grant BLAN07-1 192604 "Dynamique et statistique des particules en turbulence" and by the EU COST Action MP0806 "Particles in turbulence". the tensor are non-negative (Gantmacher 1959, p. 307). When this condition is applied to Γ ℓ(i,j)ℓ(p,q) it yields the following inequalities: In this paper, the above inequalities are assumed to hold strictly; (2.15) follows from this assumption. Some useful inequalities can be derived from (2.15). Here, γ = 0 (the case of a spherical particle is indeed trivial), D ′ R = 0 (it is easily seen that a positive D ′ R does not alter the inequalities below), and µ 3 = 0 (the case µ 3 = 0 is treated separately in appendix D).

Firstly, A ϑϑ (ϑ) and A ϕϕ (ϑ) are positive for all values of a, b, c, d, and γ satisfying (2.15). Indeed, the quantities 2µ 1 and 2µ 1 + µ 3 + µ 4 are quadratic polynomials in γ, their discriminant is equal to -̟ 3 < 0, and for γ = 1 they are equal to ̟ 1 > 0 and to ̟ 2 > 0, respectively. Moreover, 2µ 1 + µ 5 is a quadratic polynomial in γ with discriminant equal to -(3a + d)(5a + d) < 0 and takes the value 4a + d > 0 for γ = 1. Hence A ϑϑ (ϑ) µ 4 sin 2 (ϑ) + (2µ 1 + µ 3 ) sin 4 (ϑ) (2µ 1 + µ 3 + µ 4 ) sin 4 (ϑ) > 0 (B 9) A ϕϕ (ϑ) 2µ 1 + µ 5 > 0 (B 10) for all 0 < ϑ < π.

Secondly, ψ st (ϑ) is bounded for all a, b, c, d, and γ and for all 0 ϑ π. For ∆ > 0, this property is obvious. For ∆ < 0, note that ( √ -∆ + µ 4 )( √ -∆µ 4 ) = -µ 3 P 1 (γ), (B 11) and

( √ -∆ + µ 4 + 2µ 3 )( √ -∆ -µ 4 -2µ 3 ) = -µ 3 P 2 (γ), ( B 12)
where P 1 (γ) and P 2 (γ) are quadratic polynomials in γ such that P 1 (1) = 4̟ 1 > 0 and P 2 (1) = 4̟ 2 > 0. Furthermore, the discriminants of P 1 (γ) and P 2 (γ) are equal to -16̟ 3 < 0. Therefore P 1 (γ) and P 2 (γ) are positive for all γ, and the products on the left-hand-sides of (B 11) and (B 12) have the same sign as -µ 3 . Also observe that

( √ -∆ + µ 4 + 2µ 3 )( √ -∆ -µ 4 ) = -µ 3 P 3 (γ) -2 √ -∆ , ( B 13)
where P 3 (γ) is a quadratic polynomial satisfying: P 2 3 (γ) + 4∆ = P 1 (γ)P 2 (γ) > 0. Hence P 3 (γ) -2 √ -∆ = 0 for all γ. Moreover, for γ = 0, P 3 (0) -2 √ -∆ = P 3 (0) = ̟ 1 + ̟ 2 + 2(a + d) > 0. As a result P 3 (γ) -2 √ -∆ > 0 for all γ. The left-hand-side of (B 13) therefore has the same sign as -µ 3 .

Figure 1 .

 1 Figure 1. Orientation of a non-spherical particle.

Figure 2 .

 2 Figure 2. Stationary p.d.f. of the orientation angle ϑ. (a) Rotation about λ: b/a = 4.8, c/a = 55, d/a = 0.85, D ′ R /a = 10 -2 , and γ = -1 (solid line), γ = -0.75 (dashed line), γ = -0.5 (dot-dashed line), γ = -0.25 (dotted line). (b) Tumbling motion between λ and -λ: b/a = 4.75, c/a = 10, d/a = 0.9, D ′ R /a = 10 -2 , and γ = 1 (solid line), γ = 0.75 (dashed line), γ = 0.5 (dot-dashed line), γ = 0.25 (dotted line). The normalisation coefficient N has been computed numerically according to (4.3).

Figure 3 .

 3 Figure 3. Stationary p.d.f. of the orientation angle ϑ. (a) Preferential alignment with a direction oblique to λ: γ = 1, D ′ R /a = 10 -2 , b/a = -6, d/a = -0.9, and c/a = 55 (dashed line), c/a = 25 (solid line), c/a = 11 (dot-dashed line). (b) Combination of rotation and tumbling: γ = 1, D ′ R /a = 10 -2 , c/a = -10, d/a = 1, and b/a = 1.1 (dot-dashed line), b/a = 2 (solid line), b/a = 3 (dashed line). The p.d.f. has been normalised numerically according to (4.3).

Figure 4 .

 4 Figure 4. Stationary p.d.f. of the orientation angle ϑ for |γ| > 1. (a) Preferential alignment with a direction oblique to λ: γ = 5, D ′ R /a = 10 -2 , b/a = -6, c/a = 25, d/a = -0.9. (b) Combination of rotation and tumbling: γ = 5, D ′ R /a = 10 -2 , b/a = 2, c/a = -10, d/a = 1.

Figure 5 .

 5 Figure 5. A typical time evolution of ϑ(t) in the tumbling regime (γ = 1, b/a = 4.75, c/a = 10, d/a = 0.9, D ′R /a = 10 -2 ); ϑ(t) has been computed by numerically integrating (3.12).

  4ab + d 0, ̟ 2 ≡ 4a + b + c + 6d 0, (B 4)̟ 3 ≡ 15a 2b 2 -(bd)(c + 5d) + 2a(2c + 13d) 0. (B5)

µ 1

 1 > 0, 2µ 1 + µ 5 > 0, 2µ 1 + µ 3 + µ 4 > 0 (B 6)for all a, b, c, d, and γ, and consequentlyA ϑϑ (0) = A ϑϑ (π) = 2µ 1 > 0

i,j,p,q 3U ij Γ ijpq U pq 0 (B 1)for all second-order tensors U. Inequality (B 1) can be reinterpreted within the theory of positive semi-definite second-order tensors (e.g. Moakher

2008). Consider an invertible map ℓ which assigns to each pair of indices (i, j), 1 i, j 3, a single index ℓ(i, j) ranging from 1 to 9. By means of the map ℓ, U can be regarded as a 9-dimensional vector; likewise, Γ can be regarded as a symmetric 9 × 9 second-order tensor, whose symmetry follows from Γ ijpq = Γ pqij . Accordingly, (B 1) can be rewritten in the following form:1 ℓ(i,j),ℓ(p,q) 9 U ℓ(i,j) Γ ℓ(i,j)ℓ(p,q) U ℓ(p,q) 0.(B 2) Inequality (B 2) is the definition of positive semi-definiteness for second-order tensors.The theory of such tensors says that a necessary and sufficient condition for an Hermitian second-order tensor to be positive semi-definite is that all the principal minors of

Appendix A. Rotary diffusion

Rotary diffusion is introduced in the dynamics of N (t) by assuming that its effect is to produce isotropic Brownian fluctuations of the direction of N (t), but not of its length. Mathematically, this is obtained by adding a Laplacian term to the equation for f (n; t) which acts only on the orientation of n. For a deterministic flow, f (n; t) thus satisfies (e.g. [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF][START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF][START_REF] Brenner | Rheology of a dilute suspension of axisymmetric Brownian particles[END_REF]Bird et al. 1977;[START_REF] Doi | The Theory of Polymer Dynamics[END_REF]:

where

The differential operator ∇ is the angular part of the gradient or, equivalently, the restriction of the gradient to the sphere of radius |n|.

The matrix Σ satisfies:

By using properties (A 3) and w(n, t) • n = 0, it is possible to rewrite (A 1) as follows:

3) is the Stratonovich stochastic differential equation associated with (A 4).

It is worth noting that given that the orientation vector has unit length, f (n; t) must take the form f (n; t) = F (n; t)δ(|n| -1) with ∂F/∂n = 0. It would therefore be more natural to consider the evolution equation for F (n; t) instead of that for f (n; t); this is indeed the usual approach in the literature [START_REF] Leal | The effect of weak Brownian rotations on particles in shear flow[END_REF][START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF][START_REF] Brenner | Rheology of a dilute suspension of axisymmetric Brownian particles[END_REF]Bird et al. 1977;[START_REF] Doi | The Theory of Polymer Dynamics[END_REF]). In the present context, however, it is easier to formulate the problem in R 3 and to move to angular variables afterwards.

Appendix B. Positive semi-definiteness of the covariance tensor

The fourth-order tensor Γ is the covariance of a Gaussian second-order tensor, and must therefore be positive semi-definite, i.e.

Three cases should now be distinguished:

(c) µ 3 > 0 and µ 4 < 0: for these values of the parameters, (B 12) and (B 13) yield the following relations:

Inequalities (B 14), (B 15), and (B 16) guarantee that if ∆ < 0, the function χ(ϑ) is bounded for all a, b, c, d, and γ.

For ∆ = 8µ 1 µ 3µ 2 4 = 0, µ 4 cannot be zero since µ 1 > 0 and µ 3 = 0. Moreover, µ 3 and hence c must be positive, in that µ 1 > 0. Therefore, for µ 4 > 0 the function χ(ϑ) is bounded. The case µ 4 < 0 requires a more detailed analysis. ∆ can be rewritten thus:

(Note that, for γ = 0, µ 1 = (10a + c + 9d)/8 and hence 10a + c + 9d > 0 for all a, c, d; this is consistent with the positivity of the variance of the components of the vorticitysee (2.13).) Provided that ρ > 0, ∆ vanishes for γ = ±γ ⋆ with γ ⋆ = √ ρ (the case γ = 0 is not considered here). Now note that both µ 4 and 2µ 3 + µ 4 are quadratic polynomials in γ. If γ ⋆⋆ ≡ 1 + (2b + 5d)/c is positive, then

where P 4 (c) is a quadratic polynomial in c whose coefficients depend on a, b, and d and such that P 4 (0) = (6a + 5d)(2b + 5d) 2 > 0 and lim c→∞ P 4 (c)/c 2 = 4̟ 1 > 0. Moreover, P ′ 4 (c) = 0 if and only if c = c ≡ -(2b + 5d)(12a -2b + 5d)/(8̟ 1 ), and for c = c:

Therefore, P 4 (c) > 0 for all c > 0 and γ 2 ⋆⋆ > γ 2 ⋆ for all c > 0. As a conclusion, if ∆ = 0, then (B 20) holds, and for µ 4 < 0:

This result proves that χ(ϑ) is bounded also for ∆ = 0 and µ 4 < 0.

Appendix C. Stationary probability density function of the orientation angle

Consider the functions:

and

The stationary solution of (3.7) is written (see Risken 1989, p. 98):

where N and S are constants. As A ϑϑ (ϑ) and B ϑ (ϑ) are periodic, also e -Φ(ϑ) is periodic. By contrast, g(ϑ) cannot be periodic given that A ϑϑ (ϑ) > 0 and hence g ′ (ϑ) > 0. Therefore, Ψ st (ϑ) satisfies (3.10) if and only if S = 0. Equation (4.2) then follows from (C 3) with S = 0.

Appendix D. The case µ 3 = 0

If µ 3 = 0 and γ = 0 (i.e. c = 0), three cases should be distinguished.

For µ 3 = 0, µ 4 = 0, and µ 2 = 3µ 4 /2, the integral in (4.2) can be easily calculated by means of the transformation y = sin 2 (ω) to yield:

The stationary p.d.f. is bounded as a consequence of (B 6). By examining the first derivative of the above function, it can be shown that, depending on the value of the parameters, ψ st has either three or five extrema in the interval 0 ϑ π. Therefore, the four regimes identified for µ 3 = 0 also describe the dynamics of the particle for µ 3 = 0. For µ 3 = µ 4 = 0 (i.e. c = 2b + 5d = 0), the stationary solution is:

For µ 2 /µ 1 > 0, ψ st has two minima in 0 and π and one maximum in π/2, and hence the particle rotates in the plane orthogonal to λ. For µ 2 /µ 1 < 0, ψ st has two maxima in 0 and π and one minimum in π/2; therefore the particle tumbles between the direction parallel to λ and that antiparallel to λ. Finally, for µ 3 = 0 and µ 2 = 3µ 4 /2 (i.e. c = 2b + (5 + γ)d = 0), (D 1) implies that the stationary statistics of orientations is isotropic: ψ st (ϑ) = (4π) -1 .


Sylvain Grangeon

Christophe Tournassat

Catherine Lerouge

Eric Giffaut

Thorsten Schäfer

Prediction of nickel mobility in a clayey formation requires accurate knowledge of the mineralogical assemblage

Introduction

In the perspective of nuclear waste storage, 63 Ni is one of the radio-elements of primary concern. Modeling of its retention in clay formations is usually performed using modeling approaches which consider that only phyllosilicates participate to retention. However, isotopic exchange with stable nickel contained in phases (i.e. minerals and organic matter) from the clay formation may play a role. Thus, a sound understanding and prediction of Ni retention also requires detailed knowledge of Ni distribution amongst the different phases. Importance of such knowledge is reinforced by the fact that the distribution of stable Ni between solution and some reactive phases could be considered as representative of 63 Ni distribution after long-term interaction. The present study aimed at elucidating the relative contributions of fast reversible (e.g. sorption on phyllosilicates) and slowly reversible/irreversible (e.g. isotopic exchange) processes to Ni retention capacities of the Callovian-Oxfordian clay formation from Bure. To achieve this goal, we combined (i) short and long-term (up to 180 days) 63 Ni batch sorption experiments to determine the contribution of irreversible to total sorption and (ii) chemical and physical methods to determine Ni-bearing phases that may contribute to irreversible sorption through isotopic exchange.

Results and discussion

Short-term (24 hours) batch experiments led to 63Ni Kd of ~100 L kg -1 , that can be successfully predicted using a blind modeling approach by taking into account competition with stable Ni and Zn naturally present in the solution. However, 63Ni Kd increased with time up to ~600 L kg -1 after 6 months reaction time, with a fraction of irreversible sorption also increasing with time (Fig. 1). This result cannot be explained by reversible sorption on phyllosilicates only: an additional slowly reversible or irreversible retention mechanism must be invoked. This mechanism could be linked to Ni incorporation in a preexisting phase or to a (co-)precipitation process. Ni is naturally present in the rock (~30 ppm) and its solubility in the experiment synthetic pore-water is far greater than the added 63 Ni concentration (~0.6 µmol/L vs. 1.5 nmol/L). This range of concentration is also in agreement with the value determined in pore-water extracted on site (~0.5 µmol/L). It is thus necessary to understand natural Ni chemical behavior to interpret 63Ni Kd experiment on the complex COx phase assemblage. Gaucher, E., Robelin, C., Matray, J.M., Négrel, G., Gros, Y., Heitz, J.F., Vinsot, A., Rebours, 665 H., Cassagnabère, A., Bouchet, A. (2004a)

Figure 1 :

 1 Figure 1 : Evolution of 63Ni

  ANDRA underground research laboratory: interpretation of the Oxfordian 667 formation by Lakshtanov, L.Z., Stipp, S.L.S. (2007) Experimental study of nickel(II) interaction with 694 calcite: Tessier, A., Campbell, P.G.C., Bisson, M. (1979) Sequential extraction procedure for the 716


S Betelu
email: s.betelu@brgm.fr

I Ignatiadis

Calibration of antimony-based electrode for pH monitoring into underground components of nuclear repositories

Nuclear waste repositories are being installed in deep excavated rock formations in some places in Europe to isolate and store radioactive waste. In France, Callovo-Oxfordian formation (COx) is potential candidate for nuclear waste repository. It is thus necessary to measure in situ the state of a structure's health during its entire life. The monitoring of the near-field rock and the knowledge of the geochemical transformations can be carried out by a set of sensors for a sustainable management of long-term safety, reversibility and retrievability. Among the chemical parameters, the most significant are pH, conductivity and redox potential.

Based upon the reversible interfacial redox processes involving H + , metal-metal oxides electrodes should be regarded among the promising technologies to be devoted to the observation and monitoring of pH into the underground components of nuclear repositories due to their physical and chemical stability, with regards to temperatures, pressures and aggressive environments [START_REF] Pourbaix | Atlas d'équilibres[END_REF]. Metal-metal oxides electrodes present furthermore the advantage of being easily miniaturised.

Among the metal-metal oxide group, antimony-antimony oxide system, for which improved properties were obtained using mono-crystalline antimony, has been the first and then the most investigated and disputed for pH sensing; the fact remains that it has been the most frequently used in practical pH measurements. Nevertheless, numerous conflicting data exist concerning the disturbances of their potential by various physical and chemical parameters, which require calibrating the electrode under conditions similar to those in which it is to be applied (Caflisch et al. 1978;[START_REF] Głab | Effects of some complex antimony pH-sensors[END_REF].

This work aimed to calibrate mono-crystalline Sb electrode (99.999 %, m = 500 mg, d = 6.7) for pH measurements into the underground components of nuclear repositories. The electrode presented the advantage of being strong in the conception: it presented an important reserve of material. Monocrystalline Sb was used without any (pre)treatment over the experiments (16 month).The high binding energy led to a low corrosion rate and therefore the surface was only slowly changed and occluding oxide was almost completely avoided. The uniform binding energy promoted a uniform (or generalized) corrosion on the surface.

Performances, reliability and robustness were examined by potentiometric measurements at 25°C. Investigation has been limited in pH, ranging from 5.5 to 13.5, close to those encountered in the environment of the nuclear repositories. The feasibility of measuring pH with Sb-based electrodes was first tested in NH 4 Cl/NH 3 buffer solutions, leading to electrode calibration over the widest range of pH, from around neutral to basic pH. The influence of the presence P(V), Cl(-I), N(V), on the analytical signal was studied from solutions plug NaH 2 PO 4 /NaHPO 4 (Ionic Strength (IS): 0.1), NaCl (0.1 mol.L -1 ), NaNO 3 (1 to 6 10 -3 mol.L -1 ). Experiments were also conducted in NaHCO 3 /Na 2 CO 3 buffer samples, similar to conditions prevailing in the COx formation. Ionic strength was investigated from 0.05 to 0.2. When necessary, NaOH was used to extend the curve to the region of high pH value. In addition to measurements done under atmospheric oxygen saturation (with PO 2 about 0.2 atm) while the influence of the O 2 /H 2 O redox couple was being studied, several measurements were also done in a glove box (GB or BAG) in an oxygen-free atmosphere (100 % nitrogen). These were done to the anticipated decrease in the redox potential in the COx pore water caused by the decrease in the O 2 concentration. The same measurements were also done using a 10-mm disk platinum electrode with a surface of 78.54 mm². Figure 1 only presents the results obtained for the linear range part. The general convergence of the stabilization potential revealed the occurrence of an equilibrium state under the experimental conditions. Sb-based electrodes rapidly responded to changes in pH (≤ 30 s) with regard to the relative standard deviation (RSD) based on repeatability (RSD≤3%). The equilibrium state remains unchanged with regard to pH variations, considering the reproducibility (RSD≤10 %). Whatever the ligand, no complex has competed with both the antimony hydrolysis and the oxygen depletion pH. Electrodes exhibit almost ideal Nernstian response (sensitivity sufficient oxidation at the metal surface to ensure the equilibration conditions. somewhat affected by hysteresis. oxygen saturation is fixed by the O PtO/Pt couple via PtO + 2H + + 2e pH and under aerobic conditions, a difference of about 700 mV (pH 0) is observed between E Sb . This difference is reduced to about 330 mV under anoxic conditions where O predominant redox couple with a very low quantity of oxygen. Moreover, the slope of the remains the same regardless of the medium, which is very important, while the intercept giving the potential at pH = 0 differs by about 50 mV. This shows the dependence, although weak, based electrodes on the O 2 /H 2 concerning the simultaneous but different behavior of two different electrodes under identical conditions and the same reference electrode. Knowledge of to draw potential-pH diagrams when observing and monitoring pH in the underground components of radioactive waste repositories where referenc electrode appears to be appropriate progress to demonstrate the robustness of the

Figure 1 :

 1 Figure 1: Sb-based calibration curve obtained at 25°C conditions carried out in a glove box (BAG NaH 2 PO 4 /NaHPO 4 , NaNO 3 , NaHCO 3 /Na

Acknowledgement

The research leading to these results has received funding from ANDRA under the BRGM partnership (CAPTANDRA project 2009

based calibration curve obtained at 25°C under atmospheric oxygen saturation carried out in a glove box (BAG). Data include measurements conducted in NH /Na 2 CO 3 (+ (NaCl 0.1 mol L -1 )) buffer solutions.

he equilibrium state remains unchanged with regard to pH variations, considering the Whatever the ligand, no complex has competed with both the antimony depletion. The Sb electrode potential (E Sb ) varied linearly exhibit almost ideal Nernstian response (sensitivity -(48 ± 1)mV/pH) sufficient oxidation at the metal surface to ensure the equilibration conditions. omewhat affected by hysteresis. The potential of the platinum electrode (E Pt oxygen saturation is fixed by the O 2 /H 2 O redox couple. Under these conditions, E + 2e -⇔ Pt + H 2 O with E PtO/Pt (mV/NHE) = 60.5 pH + 902. At the same pH and under aerobic conditions, a difference of about 700 mV (pH 0) is observed between . This difference is reduced to about 330 mV under anoxic conditions where O h a very low quantity of oxygen. Moreover, the slope of the remains the same regardless of the medium, which is very important, while the intercept giving the potential at pH = 0 differs by about 50 mV. This shows the dependence, although weak, O redox couple in the solution. This work provides information concerning the simultaneous but different behavior of two different electrodes under identical conditions and the same reference electrode. Knowledge of their electrochemical curves will allow us pH diagrams when observing and monitoring pH in the underground components of radioactive waste repositories where reference electrodes will not be used.

appropriate for accurate measurements of pH in these conditions demonstrate the robustness of the Sb-electrode into COx over a long period.

The research leading to these results has received funding from ANDRA under the BRGM (CAPTANDRA project 2009-2011, Dr. Johan Bertrand).

Caflisch, C. R., L. R. Pucacco, et al. (1978). "Manufacture and utilization of antimony pH electrodes." Kidney International 14: 126-141. [START_REF] Głab | Effects of some complex antimony pH-sensors[END_REF]. "Effects of some complex-forming ligands on the potential of sensors." Talanta 28: 301-311. [START_REF] Pourbaix | Atlas d'équilibres[END_REF]. Atlas d'équilibres électrochimiques. Paris, Gauthier-Villars. he equilibrium state remains unchanged with regard to pH variations, considering the RSD based on Whatever the ligand, no complex has competed with both the antimony linearly as a function of (48 ± 1)mV/pH). There was sufficient oxidation at the metal surface to ensure the equilibration conditions. Moreover, E Sb was Pt ) under atmospheric O redox couple. Under these conditions, E Pt is governed by the ) = 60.5 pH + 902. At the same pH and under aerobic conditions, a difference of about 700 mV (pH 0) is observed between E Pt and . This difference is reduced to about 330 mV under anoxic conditions where O 2 /H 2 O is the h a very low quantity of oxygen. Moreover, the slope of the E Sb -pH line remains the same regardless of the medium, which is very important, while the intercept giving the potential at pH = 0 differs by about 50 mV. This shows the dependence, although weak, of the Sbprovides information concerning the simultaneous but different behavior of two different electrodes under identical their electrochemical curves will allow us pH diagrams when observing and monitoring pH in the underground components of e electrodes will not be used. Mono-crystalline Sb these conditions. Work is in electrode into COx over a long period.

The research leading to these results has received funding from ANDRA under the BRGM-ANDRA utilization of antimony pH electrodes." forming ligands on the potential of


S Betelu

C Lerouge

Mechanistic and kinetic interaction at 25

After the closure of the underground nuclear waste repository, aqueous corrosion of the steel canister and, to a lesser extent, radiolysis of water would produce significant amounts of H with materials from the repository and with the surrounding clay host formation. contains pyrite (FeS 2 ), which has been demonstrated to react with H 2010) at temperature ranging from 90°C to 180°C

This work aims at understanding these interactions at 25°C for the two systems S-H per liter (about pH 2 S=1bar), FeS transform into Pyrrhotite (FeS 1-x and at pH higher than 12.5 FeS 2 at pH higher than 9, in agreement with the

After pyrite electrodes had been assembled applied to this material (and to platinum for comparison) while it was submerged in a partially reconstituted solution of COx pore water (LPTR, figure 1B), in the absence and in the presence µm) and H 2 (PH 2 = 0 or 1 bar) (Ignatiadis situ by water electrolysis by using an external generator and two platinated titane electrodes (anode & cathode) (figure 1B). In addition to the electrochemical behaviour of the platinum and the pyrite, the pH, temperature and pressure of the liquid medium were monitored. After the closure of the underground nuclear waste repository, aqueous corrosion of the steel canister extent, radiolysis of water would produce significant amounts of H with materials from the repository and with the surrounding clay host formation.

), which has been demonstrated to react with Hydrogen gas (H at temperature ranging from 90°C to 180°C.

This work aims at understanding these interactions at 25°C. With regards to E-pH equilibrium diagrams H 2 O and Fe-S-H 2 O with a total dissolved S concentration of 0.1 mole S S=1bar), FeS 2 must be an oxidant for H 2 at pH higher than 9, and FeS x ), according to: FeS 2 + (1-x) H 2 = FeS 1+x + (1-x) H should transform into Mackinawite. Investigation , in agreement with the alkaline perturbation in the clay-rock pore After pyrite electrodes had been assembled (figure 1A), various electrochemical distur applied to this material (and to platinum for comparison) while it was submerged in a partially reconstituted solution of COx pore water (pH 9.5), enclosed in a Low Pressure in the absence and in the presence of pyrite grains (particle size between 40 and 63 (Ignatiadis et al., 2012). The H 2 present in the LPTR by using an external generator and two platinated titane electrodes (anode & In addition to the electrochemical behaviour of the platinum and the pyrite, the and pressure of the liquid medium were monitored.

Pyrite electrodes B)

The Low Pressure Thermo-Reactor and control unit. linear sweep polarization (LSP) (figure 2) clearly shows the metastable behaviour of its surfacic S°. When pyrite is at the corrosion potential (E Py ) , surfacic S° simultaneously oxidizes to thiosulfates Potentiometric measurements demonstrated that both E to reach a stable redox potential. After the closure of the underground nuclear waste repository, aqueous corrosion of the steel canister extent, radiolysis of water would produce significant amounts of H 2 . This H 2 can interact with materials from the repository and with the surrounding clay host formation. The COx formation ydrogen gas (H 2 ) (Truche et al.

pH equilibrium diagrams oncentration of 0.1 mole S at pH higher than 9, and FeS 2 should

x) H 2 S (with 0<x<0.125) transform into Mackinawite. Investigations were thus conducted rock pore-water.

, various electrochemical disturbances were applied to this material (and to platinum for comparison) while it was submerged in a partially ressure Thermo-Reactor of pyrite grains (particle size between 40 and 63 present in the LPTR was produced in by using an external generator and two platinated titane electrodes (anode & In addition to the electrochemical behaviour of the platinum and the pyrite, the Reactor and control unit.

(figure 2) clearly shows the metastable behaviour of its surfacic surfacic S° simultaneously oxidizes to thiosulfates E Pt and E Py decrease in Experiments are in progress in order to prospects for this work are, therefore, the exploitation of these data and their extrapolation to sto conditions.

Figure 1 :

 1 Figure 1: A) Pyrite electrodes B) T

  Claude Guillemin, BP 36009, 45060 Orléans Cedex, Division, Environmental Survey and Disposal Monitoring Department (DS/OS), 1-7, rue

Figure 2 :

 2 Figure 2: Significant FeS 2 electrode reactionsIn comparison, E Py remains higher alkaline dissolution/reduction of pyrite by H pyrrhotite (results provided by MEB/ agreement with results provided by couple which equilibrium at 25°C 0.0074×Log[c]; c represents the total concentration of sulphur. Spectrometry demonstrated that t increase with the imposed cathodic potentials. The initial reduction reactivity of pyrite surface, hence the rapid increase in the HS to hinder the progress of the reaction up to a rapidly reached HS dissolution and pyrrhotite precipitation are coupled reactions. The fluid composition remains fairly steady when pyrite dissolution balances pyrrhotite precipitation, the solution.

and kinetic study of pyrite (FeS 2 )-hydrogen interaction at 25°C using electrochemical techniques , C. Lerouge 1 , G. Berger 2 , E. Giffaut 3 and I. Ignatiadis

  Pyrite linear sweep polarization S°. When pyrite is at the corrosion potential (E and reduces to sulphide. Potentiometric measurements the presence of H 2 to reach a stable redox potenti

	es Division, 3 Avenue Claude Guillemin, BP 36009, 45060 Orléans
	i.ignatiadis@brgm.fr)
	Pyrénées 14 av. E. Belin, 31400 Toulouse, France
	Division, Environmental Survey and Disposal Monitoring Department
	Malabry, France

Acknowledgement

Research has received funding from i) the European Union's European Atomic Energy Community

 's FP7/2007's FP7/ -2011under grant agreement n° 212287 (RECOSY project) and ii) the ANDRA under the BRGM-ANDRA partnership (TRANSFERT References Betelu S., Lerouge C., Berger G. Ignatiadis I. (2012) hydrogen in NaCl 0.1 M at 90 °C using electrochemical techniques. Scientific and Technical contribution in the 4AWS of RECOSY in Karlsruhe Ignatiadis I., Betelu S., Lerouge C., Berger G. (2012) by hydrogen gas produced in situ by water electrolysis in NaCl 0. /S 2- og[c]; c represents the total concentration of sulphur. Electrochemical Impedance demonstrated that the corrosion currents are strong on pyrite in the presence of H increase with the imposed cathodic potentials. The initial reduction-reaction rate is rapid due to the high reactivity of pyrite surface, hence the rapid increase in the HS -content in the bulk solution, which seems er the progress of the reaction up to a rapidly reached HS -concentration plateau. dissolution and pyrrhotite precipitation are coupled reactions. The fluid composition remains fairly steady when pyrite dissolution balances pyrrhotite precipitation, the iron content remaining very low in Experiments are in progress in order to determine the kinetics of FeS 2 reduction by H prospects for this work are, therefore, the exploitation of these data and their extrapolation to sto esearch has received funding from i) the European Union's European Atomic Energy Community's 2011 under grant agreement n° 212287 (RECOSY project) and ii) the ANDRA under the partnership (TRANSFERT project).


est destinée au dépôt

Pyrite-electrode potential measurements in reference solutions versus measured and computed redox potential: at the roots of an innovative generic pH electrode for nuclear waste repositories

Stéphanie Betelu, Christophe Tournassat, J. Bertrand, Ioannis Ignatiadis

To cite this version:

Stéphanie Betelu, Christophe Tournassat, J. Bertrand, Ioannis Ignatiadis. Pyrite-electrode potential measurements in reference solutions versus measured and computed redox potential: at the roots of an innovative generic pH electrode for nuclear waste repositories. International meeting "Clays in Natural and Engineered Barriers for Radioactive Waste Confinement", Oct 2012, Montpellier, France. hal-00703579

Pyrite-electrode potential measurements in reference solutions versus measured and computed redox potential: at the roots of an innovative generic pH electrode for nuclear waste repositories Within the framework of the geochemical observation and monitoring into the underground components of nuclear repositories, the purpose of the study was to investigate, in a semi-continuous mode of monitoring, the redox potential of reference and natural solutions via a bunch of electrodes.

Because, each electrode response depends upon its material nature as well as its surface properties, gold (Au) and glassy carbon (GC) electrodes were investigated as inert indicator electrodes (IIE) in addition to platinum (Pt). Indeed, platinum differs from gold as it has the higher exchange current density (10 mA/cm² for Pt, against 0.3 mA/cm² for Au). However, gold is known as a more inert material: Pt is susceptible to catalyze reactions, form oxides and adsorb H2; in contrast, gold does not.

Gold has moreover a greater potential range (-0.8 to 1.8 V/NHE) towards the positive potentials (≈ 400 mV) than Pt. In comparison, glassy carbon possesses a larger potential range (-1.0 to 1.0 V/NHE) towards the negative potentials (≈-200 mV) than Au and Pt. In addition to inert indicator electrodes pure pyrite (FeS 2 ) was selected as alterable or alternative electrode. Its interest for potentiometric measurements in bio-leaching has already been demonstrated by [START_REF] Cabral | Mechanistic study of the pyrite-solution interface during the oxidative bacterial dissolution of pyrite (FeS2) by using electrochemical techniques[END_REF]. It was even more interesting because pyrite occurs naturally in the Callovo-Oxfordian (COx) formation. It was thus of a great interest to integrate it to the bunch of electrodes and to identify the relevance of different redox reactions at the vicinity of the pyrite electrode surface.

The test offered an advantage of great importance, because it provided information on the simultaneous behaviour of different electrodes under identical conditions. Electrode performances and robustness were examined versus time "long term measurements". This work also aimed to:

identify the redox reactions controlling the measured potential, both into solution and at the electrode/solution interface compare the obtained results with speciation measurements and geochemical modelling. Because Fe 3+ /Fe 2+ couple has thus a decisive influence on the behaviour of pyrite electrode, the robustness of the complementary approaches to obtain the voltage measurement it-self and its interpretation was investigated in low organic matter content samples where iron redox-active species dominated redox reactions. [Fe 3+ ]/[Fe 2+ ] ratios were analysed in different supporting electrolytes (NaCl, CaCl 2 and MgCl 2 ) at different pH (ranging from 1 to 12) and/or ionic strengths (supporting electrolytes concentration ranged from 0.1 to 5 mol.L -1 ). Investigations were also conducted in not balanced systems: i) Fe° powder immersed in pH-buffered and unbuffered solutions, ii) mackinawite suspension. PHREEQC® geochemical code was used to determine Eh and aqueous speciation by thermodynamic methods with the appropriate associated thermodynamic database (THERMODEM® thermodynamic database generated by BRGM).

Potentiometric measurements demonstrated that the potential acquired by all the immersed IIE generally converges to a value of the same order of magnitude, independently of the sample. Acquired potentials were in good agreement with modelling. This fact demonstrates that, under the same experimental conditions, the redox couples fixe identically the potential of inert electrodes (i.e. outcome of the same equilibrium state whatever the inert electrode). In the case when acquired potential did not converge, interface phenomena occurred (adsorption, oxidation, reduction, precipitation, accumulation of ions...).

Concerning FeS 2 electrode, different behaviours have been observed as a function of pH:

-Pyrite behaviour in acidic media (pH ≤ 2): pyrite electrode acquired a potential which was increasingly lower than that of the platinum, which measures the redox potential of the solution. The lower value of pyrite at this pH can be explained by the interfacial pyrite oxidation, leading to the increase in Fe 2+ at the vicinity of pyrite electrode. As the pyrite electrode has Nernstian behaviour [START_REF] Cabral | Mechanistic study of the pyrite-solution interface during the oxidative bacterial dissolution of pyrite (FeS2) by using electrochemical techniques[END_REF], the equilibrium value is decreased because the Fe 3+ /Fe 2+ ratio at the electrode interface is lower than that of the solution.

-Pyrite behaviour at circum-neutral pH (4.7 ≤pH ≤7.0): it is important to note that, at low ionic strength, the potential of both pyrite and platinum electrodes were practically identical. The similar voltage measured by the two electrodes implies that the processes at the vicinity of the electrodes were the same. This result is in agreement with those already obtained by [START_REF] Cabral | Mechanistic study of the pyrite-solution interface during the oxidative bacterial dissolution of pyrite (FeS2) by using electrochemical techniques[END_REF], demonstrating that, in particular conditions, pyrite can act as redox electrode. These conditions can, for example, be encountered in the initial COx formation. In comparison with acidic media, at low ionic strength, pyrite oxidation/dissolution is lowered. The increase in pH diminishes the oxidizing capacity of Fe 3+ . The redox potential is thus fixed, for the two electrodes (Pt and Py), via the Fe(OH) 3 /Fe 2+ couple, which is dependent of Fe(OH) 3 solubility as a function of pH. High ionic strength, leads to an increase in the oxidizing capacity of Fe 3+ versus pyrite. It thus leads to an accumulation of Fe 2+ at the vicinity of pyrite electrode, consequently generating progressive decrease in the Fe(III)/Fe(II) ratio at the pyrite/solution interface and thus in the electrode potential.

-Pyrite behaviour under alkaline conditions (pH ≈ 9 or higher): a higher potential value of pyrite at these pH in comparison with the potential acquired by platinum can be explained by an anoblishment process on the pyrite surface. Indeed, at higher pH than 7 iron hydroxide coagulates and stabilizes chemically to form an iron hydroxide coating at the pyrite electrode surface. The combination of equation 1 and 5 leads to Fe 3+ depletion in solution and in the pyrite/solution interface. Fe(OH) 3 coating onto the pyrite electrode responsible for the lowering of pyrite oxidation/dissolution. According to the following reaction Fe 3+ + 3H 2 O ↔ Fe(OH) 3 + 3H + , pyrite resulting anoblisment lead to the decrease in pH at the electrode. When the solution redox potential (Pt) is fixed by Fe(OH) 3(s) /FeOH + couple (with E eq = -1.14 -0.06 log[FeOH + ] -0.03log[OH -]), the pyrite electrode potential is fixed by Fe(OH) 3(s)/ Fe 2+ couple (with E eq = -1.46 -0.06 log[Fe 2+ ] -0.01 log[OH -]).

These results confirmed, with absolute clarity, that the behaviour of pyrite is a function of the Fe(III)/Fe(II) ratio at the interface. These results are even more interesting because they lead to highlight the double interest of pyrite electrode as both redox and pH indicator when integrated in the bunch of electrodes. In fact, in natural conditions, i.e. those of initial COx formation, pyrite can operate as redox electrode (its potential is similar to that of an IIE). An increase of its potential versus the potential of an IIE can be correlated to an alkaline environment: pyrite electrode can thus behave as a pH indicator. This can be applied in the case of COx formation during its envisaged evolution as nuclear waste repository.

Acknowledgement

The research leading to these results has received funding from i) the European Union's European Atomic Energy Community's (Euratom) FP7/2007-2011 under grant agreement n° 212287 (RECOSY project) and ii) the ANDRA under the BRGM-ANDRA partnership (CAPTANDRA project 2009(CAPTANDRA project -2011). ).


A Boussadi

Stéphanie Betelu

F Silva

Ioannis Ignatiadis

H-terminated polycrystalline boron doped diamond electrode for geochemical sensing into underground components of nuclear repositories

H-terminated polycrystalline boron doped diamond electrode for geochemical sensing into underground components of nuclear repositories

A. Boussadi 1,2 , S. Betelu **1 , F. Silva 2 and I. Ignatiadis 1

1 BRGM, Environment and Processes Division, 3 Avenue Claude Guillemin, BP 36009, 45060 Orléans Cedex, France (**corresponding author: s.betelu@brgm.fr) 2 Laboratory of Materials Science and Process (LSPM -CNRS UPR 3407) -University of Paris XIII, Villetaneuse, France (*amine.boussadi@lspm.cnrs.fr)

Nuclear waste repositories are being installed in deep excavated rock formations in some places in Europe to isolate and store radioactive waste. In France, Callovo-Oxfordian formation (COx) is potential candidate for nuclear waste repository. It is thus necessary to measure in situ the state of a structure's health during its entire life. The monitoring of the near-field rock and the knowledge of the geochemical transformations can be carried out by a set of sensors for a sustainable management of long-term safety, reversibility and retrievability. Among the chemical parameters, the most significant are pH, conductivity and redox potential.

Wide band gap semiconductors are favored materials for chemical sensing because of their high stability to many chemical agents. Among the wide band gap materials, Chemical Vapor Deposition (CVD) boron doped diamond (BDD) benefits from a large band gap (5.45 eV), which gives rise to a wide electrochemical potential window (~3 V/Saturated Calomel Electrode(SCE)) [START_REF] Ignatiadis | Conducting Diamond Electrodes: Applications in Electrochemistry[END_REF].

It is moreover described as a radiation, corrosion and bio-corrosion resistant. These remarkable properties, in addition to a low double layer capacity and a low residual current, make BDD a promising material for geochemical sensor elaboration. This work aimed to investigate BDD-based electrodes coated with p-type polycrystalline BDDhydrogen-terminated surfaces (1 cm 2 ) for pH and/or redox measurements into the underground components of nuclear repositories. The boron-doped p-type channel was grown in a microwave plasma reactor (BJS 150) (Silva et al. 2009). The boron-doped channel was hydrogen terminated by a hydrogen plasma treatment in the CVD reactor, resulting in full saturation of the surface carbon bonds with hydrogen atoms.

Figure 1 shows the Scanning Electron Microscopy (SEM) of the polycrystalline BDD coating with a Bore/Carbon ratio of 500 ppm and its Raman spectrum. SEM micrograph illustrates the typical columnar growth of the polycrystalline CVD diamond. A homogeneous surface was observed concerning the crystallite size which average was 1.5 microns. On the Raman spectrum of a single crystal diamond intrinsic film (undoped), the diamond peak is usually observed at 1332 cm -1 . In Figure 1, the intense peak at 1327 cm -1 corresponding to diamond is shifted due to the "Fano" effect according to doping, which is observed through a broad peak at 910 cm -1 . Its intensity shows that the investigated sample was highly doped. [START_REF] Gheeraert | Effect of boron incorporation on the "quality" of MPCVD diamond films[END_REF] suggested that the peaks at 500 and 1230 cm -1 appears when the boron concentration reaches the critical value of 3×10 20 at.cm -3 corresponding to a metallic conductivity. The lack of peak around 1350 cm -1 and 1570 cm -1 , which corresponds respectively to D and G graphite peak of impurity phases of non-diamond carbon (sp2), attests to the crystalline quality of the deposit.

The slight width at half maximum of the characteristic peak of diamond compared to that of natural diamond reflects the degree of organization and structural perfection of this phase indicating that the coating was of high quality. Electrodes made in this way have been used for 8 month without any surface treatments or conditioning.

The electrochemical behavior of Hydrogen-terminated BDD was studied by cyclic voltammetry.

Electrodes showed a wide potential range of about 2 V/SCE. They also showed and a rapid reversible charge transfer in the presence of redox probes such FeCN 6 3-/ 4-and Ru(NH) 6 3+ / 2+ .

Performances, reliability and robustness for pH or redox monitoring were examined by potentiometric measurements at 25°C under anaerobic conditions (oxygen-free atmosphere, 100 % nitrogen) in a glove box. Investigation has been limited in pH, ranging from 5.5 to 13.5, close to those encountered in the environment of the nuclear repositories. The feasibility of measuring pH with BDD electrodes was first tested in NH 4 Cl/NH 3 -NaCl (0.1mol L -1 ) buffer solutions, leading to electrode calibration over the widest range of pH, from around neutral to basic pH. Experiments were also conducted in NaHCO 3 /Na 2 CO 3 buffer samples, similar to conditions prevailing in the COx formation. For redox measurements, [Fe 3+ ]/[Fe 2+ ] ratios were analysed at different pH and/or ionic strengths (supporting electrolytes concentration ranged from 0.05 to 1 mol.L -1 ). The same measurements were also done using a 10-mm disk platinum electrode with a surface of 78.54 mm².

No pH sensitivity was observed, thus the energy level of the state was not moved. However, for redox measurements the potential acquired by Hydrogen-terminated BDD and Platinum electrode converged to a value of the same order of magnitude, independently of the sample. This fact demonstrates that, under the same experimental conditions, the redox couples fixe identically the potential of the electrodes. Investigations with reference to ionic strength in thermodynamically equilibrated Fe(III)/Fe(II) samples were highly interesting. Independently of the electrode, the voltage measurement was not or little affected, whereas both the solution conductivity as well as the speciation were affected, due to the increase in salinity. This means that the term [Fe 3+ ]/[Fe 2+ ] is practically unaffected. This implies that assuming the ratio of the activity coefficients, γFe3+/ γFe2+ as equal to 1 has a minor effect on the measured redox potential. H-terminated BDD electrode appears well suited for redox monitoring. Work is in progress to demonstrate the robustness of the H-terminated BDD electrode for redox monitoring into COx over a long period.

Figure 1 :

 1 Figure 1: SEM micrograph of the B/C=500 ppm coating (left part) and its RAMAN spectrum (right part).Excitation wavelength: 514nm.

Acknowledgement

The research leading to these results has received funding from ANDRA under the BRGM-ANDRA partnership (CAPTANDRA project 2009-2011, Dr. Johan Bertrand). A special thanks to Riadh Issaoui who made the boron doped diamond films within Laboratory of Materials Science and Process (LSPM -CNRS UPR 3407) -University of Paris XIII.