G Orlandi

P Kuzhir
email: kuzhir@unice.fr

Y Izmaylov

J Alves Marins

H Ezzaier

L Robert

F Doutre

X Noblin

C Lomenech

G Bossis

A Meunier

G Sandoz

A Zubarev

Microfluidic separation of magnetic nanoparticles on an ordered array of magnetized micro-pillars

Microfluidic separation of magnetic particles is based on their capture by magnetized micro-collectors while the suspending fluid flows past micro-collectors inside a micro-channel. Separation of nanoparticles is often challenging because of strong Brownian motion. Low capture efficiency of nanoparticles limits their applications in bio-analysis. However, at some conditions, magnetic nanoparticles may undergo field-induced aggregation that amplifies the magnetic attractive force proportionally to the aggregate volume and considerably increases nanoparticle capture efficiency. In this paper, we have demonstrated the role of such aggregation on an efficient capture of magnetic nanoparticles (about 80 nm in diameter) in a microfluidic channel equipped with a nickel micro-pillar array. This array was magnetized by an external uniform magnetic field, of intensity as low as 6-10 kA/m, and experiments were carried out at flow rates ranging between 0.3-30 µL/min. Nanoparticle capture is shown to be mostly governed by Mason number Ma, while the dipolar coupling parameter α does not exhibit a clear effect in the studied range, 1.4 < α < 4.5. The capture efficiency Λ shows a strongly decreasing Mason number behavior, Λ∝Ma -1.78 within the range, 32 ≤ Ma ≤ 3250. We have proposed a simple theoretical model, which considers destructible nanoparticles chains and gives the scaling behavior, Λ∝Ma -1.7 , close to the experimental findings.

I. Introduction

Magnetic fluids composed of nanoscale (ferrofluids) or micron-sized magnetizable particles (magnetorheological suspensions) attract considerable interest of researchers and engineers due to a rich set of physical properties that are valuable for many industrial and bio-medical technologies.

Development of the modern and perspective applications of these systems requires detailed study of internal mechanisms of macroscopic phenomena and behavior of magnetic fluids. Magnetic separation is one of the fields for the practical use of magnetic fluids. Since large-scale applications in ore beneficiation and power industry [START_REF] Svoboda | Magnetic Techniques for the Treatment of Materials[END_REF], magnetic separation is nowadays gaining a new interest in view of emerging applications to bioanalysis [START_REF] Zborowski | Magnetic Cell Separation[END_REF]. The basic idea consists of adsorption of biological cells or molecules onto a surface of magnetic bead functionalized with specific chemical groups. The beads bearing the target cells or molecules are then separated from the solvent containing other cells/molecules by means of a magnetic field gradient. Microfluidic technology allows integration of the whole bioanalysis process onto a single microchip along with considerable reduction of the reagent consumption. Since last decade, an important number of microfluidic devices have been developed and applied to cell manipulation, nucleic acid assays and various immunoassays [START_REF] Gijs | [END_REF]. Among different technological solutions, microfluidic channels with an ordered array of nickel or iron columns (micropillars) provide an effective retention of micron-sized magnetic beads. Micro-pillars are magnetized by external electromagnets or on-chip integrated micro-magnets. Analyzed aqueous sample is pushed through the micro-pillar array. The magnetic beads are attracted to micro-pillars and separated from the solvent. Such arrays are usually created by electroplating coupled with soft lithography [4][5][6] and have been successfully applied for the controllable capture and release of cancer cells [7][8][9], isolation of lymphocytes [10], in situ generation of cancer cell patterns [11], and magnetically actuated adhesion process [12].

Efficiency of the magnetic separation depends mostly on the magnetic bead size (radius r n ) or rather on the ratio of the magnetic-to-thermal energy called dipolar coupling parameter [START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Magnet | [END_REF]:

2 2 3 0 0 2 3 n n B H r k T πβ μ α = , (1)
where μ 0 =4π•10 -7 H/m -the magnetic permeability of vacuum; H 0 -the magnetic field intensity Micron-sized beads possess large values of α and are easily separated. Nanoparticles have a smaller α and their capture often requires a high-gradient magnetic separation, which is not easily integrated into small-scale devices. Magnetic nanoparticles with a diameter of about 250 nm can still be effectively captured by a micro-pillar array at moderate magnetic fields (20kA/m) [5,7,9]. However, microfluidic capture of nanoparticles as small as 50 nm in diameter requires much higher fields (200 kA/m) [15,16] and, to the best of our knowledge, has never been reported for micro-pillar systems.

Colloidal stability and high specific surface area of nanoparticles is their unconditional advantage in comparison to microbeads. Thus, substantial improvement of microfluidic separation of nanoparticles is a key feature for their successful applications in bioanalysis.

Despite the physical limitation related to small values of the α-parameter, we have recently shown that the capture of magnetic nanoparticles (as small as 50-60 nm in size) is still possible if they are able to undergo field-induced phase separation caused by dipole-dipole interactions [17]. In this case, nanoparticles gather into bulk drop-like aggregates having 1 α . These aggregates are separated from the solvent much easier than isolated nanoparticles. Studies of the nanoparticle capture by a single collector or by a packed bed of magnetized micro-spheres of radius r m have shown that the capture efficiency is still efficient at magnetic fields as low as 10 kA/m [START_REF] Magnet | [END_REF]18] and is mostly governed by the ratio of the magnetic-to-hydrodynamic forces -Mason number, defined as follows [18,[START_REF] Bossis | Magnetorheology: Fluids, Structures and Rheology[END_REF]:

0 2 2 0 0 3 m n n ur Ma H r η β μ = , (2)
where η 0 is the solvent viscosity, and u is the filtration speed.

In this paper, we aim to demonstrate the feasibility of the low-field separation of magnetic nanoparticles on an ordered array of nickel micro-pillars integrated into a microfluidic channel. To this purpose, we develop a theoretical model predicting nanoparticle capture efficiency as function of Mason number. The role of the field-induced aggregation of nanoparticles on the capture efficiency is analyzed.

To validate the theoretical model, the capture efficiency is quantified experimentally and compared to the calculated one.

II. Theory

Consider a micro-filter composed of a microfluidic channel containing an ordered array of cylindrical micro-pillars oriented perpendicularly to the channel walls and spanning all the channel width. Assume a laminar flow of the nanoparticle suspension (dilute ferrofluid) through the micro-filter along its symmetry axis, z, with a filtration speed u=Q/A, as shown in Fig. 1, where Q and A are respectively the flow rate and the cross-section area of the filter. Micro-pillars are magnetized by an external magnetic field of an intensity, H 0 , applied parallel to the flow such they can capture a certain amount of nanoparticles situated in their vicinity. Depending on its initial position at the filter inlet, as well as on u and H 0 , a given nanoparticle is either captured and retained in the filter or passes through the filter. The capture efficiency of the micro-filter is defined through the ratio of the number of nanoparticles entering the filter to the number of particles leaving the filter. In this section, we seek to find the capture efficiency of the filter as function of Mason number. This dependency will be found using a two-scale approach frequently used in filtration theory [START_REF] Tien | Granular Filtration of Aerosols and Hydrosols[END_REF]. First, temporal evolution of the amount of the captured nanoparticles inside the micro-filter will be predicted on a macroscopic scale [Sec. II-A]. On this scale, microscopic mechanism of the capture (magnetic attraction with micro-pillars in our case) will be discarded and the capture efficiency will be characterized by some unknown coefficient Λ. Then, on a microscopic scale, nanoparticle flux to each micro-pillar will be evaluated at different Mason numbers, Ma, and the coefficient Λ will be calculated as function of Ma. This calculation will be done under assumptions of isolated nanoparticles [Sec. II-B] and chained nanoparticles [Sec. II-C], the latter case allowing estimation of the effect of the fieldinduced aggregation on the capture efficiency.

A. Macroscopic scale: amount of captured nanoparticles

Filtration process implies some variation in particle concentration along the filter, which evolves with the time t and is described by the concentration profiles φ(t, z) and σ(t, z) of non-captured and captured particles, respectively. These quantities are obtained from the set of phenomenological filtration equations [START_REF] Tien | Granular Filtration of Aerosols and Hydrosols[END_REF][START_REF] Herzig | [END_REF]:

( ) F z σ λ σ σ ∂ = - ∂ ; ( 3a
) 0 0 0 ( ) d u F dq σ λ σ φ = ; (3b) 0 0 ( , ) ( , ) ( ) q z q z q φ σ φ σ = (3c)
where φ 0 and σ 0 are concentrations of non-captured and captured nanoparticles at the filter inlet;

( , )

/ q t z t c z u = -is the corrected time; c is a fraction of filter volume occupied by micro-pillars;

hereinafter we put ( , ) q t z t ≈ for the present case / 1 ut L ; λ is the phenomenological filter coefficient;

( ) 1 / m F σ σ σ = -
is the correction factor taking into account a decrease of the capture efficiency with a progressive particle deposition; σ m is the maximum value of σ when the filter is saturated.

Equations (3a), (3b) were solved with respect to the desired quantity, σ, under initial and boundary conditions as follows: 0 ( 0) 0 q σ = = ; 0 ( , 0) ( ) q z q σ σ = = . The resulting expression is obtained in the dimensionless form: In the limit 1 Λ , the volume fraction σ of captured particles becomes nearly-constant along the filter, and the equation (4) reduces to:

0 0 1 / / ( , ) 1 1 m m Z m e Z e φ τ σ φ τ σ σ τ σ - Λ -Λ -Λ ⎡ ⎤ = + ⎢ ⎥ - ⎣ ⎦ (4)
( ) 0 / ( ) 1 m m e φ τ σ σ τ σ -Λ ≈ - (5)
Comparison of the last two equations shows that the equation ( 5) works reasonably well beyond the limit 1 Λ

, until 1 Λ ≈ . The majority of our experimental data fits into the range 0 1 ≤ Λ ≤ , thus, equation (5) will be employed hereinafter for the data processing in Sec. IV-A.

B. Microscopic scale: single particle capture

The Mason number dependency of the capture efficiency Λ can be estimated by considering the nanoparticle capture process at the micro-pillar scale. First, consider the simplest case of capture of isolated nanoparticles, which are not agglomerated under dipole-dipole interactions.

Since the micro-pillars are spaced relatively far apart from each other, a cell representation of the micro-filter should be a reasonable approximation taking into account the effect of neighboring micropillars on particle capture by a given micro-pillar. According to this model, the space around a given micro-pillar is represented by a homogeneous effective medium except for an annular layer surrounding the micro-pillar and whose inner-to-outer diameter ratio is equal to the fraction c of filter volume occupied by micro-pillars [Fig. 1]. Using this geometry, the relationship ( ) f Ma Λ = is obtained from the following considerations. The capture rate / t σ ∂ ∂ in the beginning of the capture process (τ=0) is related to the filter coefficient λ and to the flux J of nanoparticles captured by a single micro-pillar by the following expression [START_REF] Tien | Granular Filtration of Aerosols and Hydrosols[END_REF]:

A c o l l u n J t σ λφ ∂ = = ∂ (6) where 2

/( )

coll m n c r h π =
is the number density of the micro-pillars of the radius r m and height h; φ A is the volume fraction of nanoparticles arriving to the cell. Neglecting diffusion flux, the flux J of captured particles can be calculated by integration of the magnetophoretic flux density over a fraction Ω of the micro-pillar surface at which the flux is inward:

r J Fd S φξ Ω = -∫ (7)
where 0 1/(6 ) n r ξ πη = is the nanoparticle mobility, η 0 is the viscosity of the suspending liquid; φ is the nanoparticle concentration on a micro-pillar surface and F r is the radial component of the magnetic force acting on nanoparticles (possessing an induced dipole moment) at the micro-pillar surface and is given by the following expression [START_REF] Magnet | [END_REF]:

2 3 0 2 m r nn r r H F r r πμ β = ⎛ ⎞ ∂ = ⎜ ⎟ ∂ ⎝ ⎠ (8)
The distribution of the magnetic field intensity H inside the concentric cell around a magnetized micro-pillar is given by a solution of the magnetostatic problem in the concentric cylinder geometry from the mean field Maxwell-Garnett theory [START_REF] Berthier | Optique des Milieux Composites[END_REF] applied to the cylindrical geometry. The magnetic field distribution takes the following form in polar coordinates (r,θ), with the origin set at the micropillar center and the polar angle θ counted from the direction of the applied external field H 0 :

2 0 1 c o s 1 m r m m H r H c r β θ β ⎛ ⎞ ⎛ ⎞ = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ ; 2 0 1 s i n 1 m m m H r H c r θ β θ β ⎛ ⎞ ⎛ ⎞ = - - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ ; 2 2 2 r H H H θ = + (9)
The integration of Eq. ( 7) is considerably simplified in the limit of high Mason numbers, , the magnetophoretic flux [Eq. (7)] and the capture efficiency can be found without the need of knowing the velocity distribution of the suspending liquid around micro-pillars. Combining together Eqs. ( 6) -( 9) we arrive, after some algebra, at the final expression for the capture efficiency of isolated nanoparticles:

( )

2 1 2 8 1 ( a c o s ) (1 ) m m m m m m c L L M a c r β λ β βπ β π β - Λ = = - + - - , (10)
valid for 1

Ma

. Note that, in the limit of small micro-pillar concentrations, 1 c

, our model predicts the capture efficiency two times larger than that calculated by Gerber under assumption of potential flow within the framework of the theory of high gradient magnetic separation [START_REF] Gerber | Applied Magnetism[END_REF]. This discrepancy comes from the fact Gerber has considered the nanoparticle capture only by the front side of the micro-pillars, while in reality the nanoparticles are captured by both the front and the rear sides at 1 Ma , whatever the flow type -potential or creeping. Finally, in the present case of micro-pillars with high magnetic permeability, 1 m μ and 1 m β ≈ , the last equation is reduced to:

1 2 8 (1 ) m cL Ma c r - Λ ≈ - , (11)
This last equation ( 11) will be used for comparison with experiments in Sec. IV-B.

C. Microscopic scale: capture of chain-like aggregates

The theoretical correlation [Eq. (11)] is obtained for the case of isolated nanoparticles. Field-induced clustering of nanoparticles is probably responsible for higher capture efficiency at low Mason numbers, while the clusters are destroyed by the flow at higher values of Ma. Nanoparticle chaining likely occurs at slow flows in the considered experimental range of dipolar coupling parameters and particle volume fractions (1.4<α<4.5, φ=3.2•10 -3 ), according to the phase behavior predicted for dipolar particles [START_REF] Yu | [END_REF]25]. The magnetic force exerted by micro-pillars to N -particle chains is roughly N times the force experienced by isolated nanoparticles. Particle chaining is therefore expected to increase significantly the capture efficiency, Λ, while the flow destroys the chains leading to a strong decrease of Λ.

To check this hypothesis, let us assume that the nanoparticles are gathered to single straight chains containing N particles having a variable length depending on the intensity of the flow and of the magnetic field. The advective flux of the chains,

r v dS φ Ω ∫
, at the creeping flow in the vicinity of the micro-pillar is much smaller than the magnetophoretic flux [Eq. (7)] because the suspending liquid velocity (as well as its radial component v r ) is close to zero. Behavior of the chains situated near the micro-pillar surface is therefore supposed to be governed mostly by the magnetic field. The chains are assumed to be oriented along the magnetic field lines, and their demagnetizing field is associated to the longitudinal component, n , of the demagnetization factor tensor. Since, in high permeability limit, 1 m μ , the magnetic field lines are almost perpendicular to the micro-pillar surface, the chains are almost perpendicular to the flow streamlines in the vicinity of the micro-pillar. Thus, their hydrodynamic drag will be described by a transverse component ξ ⊥ of the mobility tensor. The magnetophoretic flux of the chains captured by the micro-pillar is still given by Eq. ( 7), in which the single-particle mobility ξ should be replaced by the chain mobility, ξ ⊥ . The quantities ξ ⊥ and F r intervening into Eq. ( 7) are given by the following expressions, derived under commonly used approximation where the particle chain is represented by an ellipsoid of revolution with diameter, length and aspect ratio equal to 2r n , 2r n N and N, respectively [26,[START_REF] Landau | Electrodynamics of Continuous Media[END_REF]:

( )

0 2 16 2 (1 )/( 1) n r N N πη ξ γ γ ⊥ = + - - ( 12a
) 2 3 0 2 m r n r r H F N r r πμ β = ⎛ ⎞ ∂ = ⎜ ⎟ ∂ ⎝ ⎠ (12b) 2 1 /2 acosh ( 1) N N N γ = - , ( 12c
) 1 1 3 1 ( 1) n n n μ β μ - = + - (12d) 2 1 /2 2 1 /2 2 3 /2 ( 1) ( 1) atanh ( 1)
N N N n N N N ⎡ ⎤ - - = - ⎢ ⎥ - ⎣ ⎦ (12e)
Combining together Eqs. ( 6), ( 7), ( 9), ( 12), we show that the capture efficiency for chained nanoparticles, Λ ch , can be found from Eq. ( 11) by introducing a correction factor, K(N)≥1, increasing with the chain aspect ratio N (or number of particles per chain):

1 2 ( ) 8 ( ) ( 1
)
ch m cL K N K N Ma c r - Λ = Λ ≈ - (13a) 2 1 ( ) 2 8 1 ( 1) 1 n n N K N n N μ γ γ μ + - ⎛ ⎞ = + ⎜ ⎟ + - - ⎝ ⎠ (13b)
with γ and n given respectively by Eqs. (12c) and (12e). In the present case of the magnetic nanoparticles with relatively high magnetic permeability, 30

1 n μ ≈
, the correction factor [Eq. (13b)] can be fitted by a simple power law trend valid at a maximal error of 25% within the range of the chain aspect ratio 1 8 N ≤ ≤ :

7/5 ( ) K N N ≈ (14)
Both exact [Eq. (13b)] and approximate [Eq. ( 14)] expressions for the correction factor are compared in Fig. 2 where they are plotted against the chain aspect ratio N. Notice that both expressions have the same first order term of the series expansion around the point N=1 in the limit The tensile hydrodynamic forces tend to destroy the chains while the compressive hydrodynamic force and magnetic force between particles inside the chains tend to consolidate the chains. At this moment, we are unable to predict the chain length at any point of the micro-filter. Instead, we will provide a scaling behavior of the chain length with Mason number. To this purpose, we suppose that an average (over some effective layer adjacent to the micro-pillar surface) chain length is defined by a balance of the tensile hydrodynamic force and the magnetic attractive force. These both forces scale as 2 0 ( )

1 n μ , 2 ( ) 1 (7/5)( 1) (( 1
h n F Nr η ε& and 2 2 0 0 m n F H r μ [28]
, where / m u r ε& is the order of magnitude of the extension rate along the chain axis. Equalizing these forces, we arrive at the following scaling of the chain aspect ratio:

1/ 2 2 1/ 2 0 0 0 m H r N M a u μ η - ⎛ ⎞ ∝ ⎜ ⎟ ⎝ ⎠ (15)
Such an approach is often used in magnetorheology for estimation of the length of magnetic particle chains under magnetic and shear fields [START_REF] Martin | [END_REF]. Finally, substituting Eq. ( 15) into Eq. ( 14) and then into Eq. (13a), we get the following scaling behavior of the capture efficiency in frames of the chain model, approximately valid for the chain aspect ratios 1 8 N ≤ ≤ :

Λ = Λ = = ⋅ (16)
This equation ( 16) will then be fitted to experimental data [Sec. IV-B] and the numerical multiplier b at Ma -1.7 will be used as a single adjustable parameter.

III. Materials and methods

To carry out filtration experiments, we fabricated a microfluidic channel with a nickel micropillar array created on a glass substrate by electroplating and soft photolithography, according to a general procedure described by Deng et al. [4]. First, the glass substrate was covered with a seed layer to permit the electrodeposition of nickel. Then, a 50 µm thickness of a positive photosensitive resist AZ 9260 (MicroChemicals GmbH, Germany) was applied onto the metalized glass substrate by spray coating. A chromium glass mask (with a square array of circles) was used for the lithography step to expose the resist with ultraviolet radiation of an energy density of 1.3 J/cm 2 . The exposed part of the resist was developed with the aqueous alkaline solution AZ 400K (MicroChemicals GmbH, Germany).

Then, oxygen plasma was used for the removal of organic residues at the bottom of the patterns. At the end of this stage, ordered arrays of cylindrical holes were formed in the resist layer. After that, the nickel micro-pillars were formed in the pre-designed holes by electroplating using a nickel sulfamate (Sigma Aldrich, France) bath. At the end, the photosensitive resist was stripped with acetone, and the seed layer was removed by ionic etching in Argon plasma. Finally, a polydimethylsiloxane (PDMS, Sigma Aldrich, France) cover was fabricated by a photolithography with a photosensitive resist SU8-2075 (Microchem, United States) and glued to the glass substrate (bearing the nickel micro-pillars) using oxygen plasma. Aluminum needles were introduced to the PDMS cover at both its extremities allowing the channel inlet and outlet. The resulting microfluidic channel is schematically presented in Fig. 3a along with an optical microscopy image of the micro-pillar array [ Water-dispersible iron oxide nanoparticles, or rather aqueous ferrofluids, were synthesized by a co-precipitation method, as described in details in [17]. At the issue of the synthesis, instead of isolated nanoparticles, we obtained permanent isotropic nanoclusters of a mean diameter of 2r n ≈84 nm and an initial magnetic permeability μ n ≈30, as inferred from magnetization measurements. The prepared ferrofluid was diluted by distilled water (viscosity η 0 =10 -3 Pa.s) to the particle volume fraction of φ 0 =0.32%. The experimental procedure can be summarized as follows. After careful filling of the system, the diluted ferrofluid was pushed through the channel at a desired flow rate for 5 min in the absence of the magnetic field. Then, the magnetic field, of a desired intensity, was applied. The applied field magnetized the micro-pillars and they became capturing the ferrofluid nanoparticles. As a result, a progressive formation of nanoparticle deposits was observed in the vicinity of the magnetic poles of

( ) 0 /( ) ( ) 1 m s m s s e φ τ τ -Λ Φ ≈ - (17)
valid for 1 Λ ≤ . The capture efficiency, Λ, was used as an adjustable parameter of this fit, and was compared to the theoretical one [Eq. (11) or Eq. ( 16)] in function of Mason number.

IV. Results and discussion

A. Kinetics of nanoparticle capture

Progressive capture of nanoparticles is shown in Fig. 4 where a sequence of snapshots of the micro-pillar array is presented at different elapsed times since the application of the magnetic field (each row corresponds to t=0, 15, 30, and 45 min), at different filtration speeds u and at the same external magnetic field H 0 =6.05 kA/m oriented along the flow, horizontally with respect to the figure. In the absence of field (t=0, the 1 st row of the snapshots), a homogeneous medium filling the space between pillars is observed. In the presence of field and flow (t>0), nanoparticles form deposits extended along the applied field around each pillar. The size of these deposits increases with time and decreases with the filtration speed because of increasing hydrodynamic forces eroding the particles from the interface between the deposit and the ambient fluid. At higher speeds (u>1.88 mm/s), the deposits achieve a steady shape and size corresponding to a balance between nanoparticle fluxes arriving to and leaving the deposits. At steady state, our micro-filter is saturated and no more captures the nanoparticles. At sufficiently long elapsed times, stripe patterns of percolated deposits are observed at lower speeds, u≤1.88 mm/s. They are replaced by separate patterns at higher speeds, u≥3.41 mm/s, whose oval shape is qualitatively similar to the one predicted and observed for a single spherical collector [START_REF] Magnet | [END_REF]17].

Unfortunately, we were unable to reliably measure the nanoparticle concentration downstream the micro-filter. However, the quantity of the captured particles is quantified in terms of the relative deposit area s [cf. Sec. III], plotted as function of the dimensionless elapsed time τ in Fig. 5 Figure 5 shows that experimental dependencies s(τ) are fitted reasonably well with Eq. ( 17) (solid lines) in all cases except for the lowest Mason number, Ma=323. This signifies that the classical filtration equations (3) can be safely used for the macroscopic description of the microfluidic filtration of magnetic nanoparticles. In the case of Ma=323, corresponding to Λ>1, the flow was stopped (because of time limitation) long before achieving the steady state, and the data were fitted with Eq. ( 4), in which 17) and ( 4), the maximum deposit relative area s m and the capture efficiency Λ quantify the filtration process and are studied in details in the next Section.

B. Retention capacity and capture efficiency

The maximum deposit relative area s m describes the maximum amount of nanoparticles that the micro-filter is able to retain at the given magnetic field intensity H 0 and filtration speed u, thus at a given value of Mason number Ma. The values of s m are plotted against Mason number in Fig. 6 for two values of the applied magnetic field, H 0 =6.05 kA/m and 10.9 kA/m. . Such behavior agrees qualitatively with the one of the size of nanoparticle deposits formed around a single spherical collector [START_REF] Magnet | [END_REF]. Quantitatively, at Mason numbers as high as Ma=2×10 3 (at u≈2 and 7 mm/s for H 0 =6.05 and 10.9 kA/m, respectively), the deposit size is about 10% of the micro-pillar size and the particle capture is still guaranteed. The experimental value 0 0.22 s ≈ of the relative deposit area at Ma→0 is consistent with the value The theoretical correlations, obtained in the frame of isolated particle model [Eq.( 11)] and chain model [Eq. ( 16)], correspond to the dashed black and dotted blue lines in Fig. 7. We remark a quantitative difference between the single particle model ( 1Ma - Λ ∝

) and the experimental behavior ( 1.78

Ma - Λ ∝

). Such discrepancy cannot be explained only by the interplay between magnetic and hydrodynamic forces acting on isolated nanoparticles, even including Brownian motion into consideration. As, stated in Sec. II-C, field-induced clustering of nanoparticles is responsible for higher capture efficiency at low Mason numbers, while the clusters are destroyed by the flow at higher values of Ma. The chain model takes into account these effects and gives the scaling law, [Eq. (11)] while, in the present work, the Mason numbers are lower and the chains resist to the flow.

V. Conclusions

In summary, we attempted to demonstrate efficient capture of magnetic nanoparticles (about 80 nm in diameter) in a microfluidic channel equipped with a micro-pillar array at small applied magnetic fields. At filtration speed u=0.034 mm/s and applied field, H 0 ≈10.9 kA/m (Mason number Ma=32), the capture efficiency was Λ≈5.5, indicating that more than 99.5% of particles were retained by the microfilter during a characteristic time 0 /( ) 5 m t s L u φ Φ Λ ≈ min, allowing analysis of a few-µ-liter samples.

Alternatively, similar retention capacity can be achieved at lower magnetic fields and higher speeds using a closed-loop configuration, i.e. forcing the sample to pass several times through the micro-filter [18].

From the physical point of view, nanoparticle capture is mostly governed by Mason number and the capture efficiency exhibits a strongly decreasing Mason number behavior, Despite of such a strong loss of the capture efficiency with increasing filtration speed, we believe that nanoparticle clustering is a key feature that will allow in future an efficient microfluidic separation of magnetic nanoparticles. An appropriate magneto-fluidic design could promote nanoparticle aggregation before they arrive to the micro-pillar array. Thus, a separate fundamental study is required for the better understanding how kinetics of nanoparticle aggregation affects the capture process. On the other hand, the use of optimized geometry of the micro-pillar array (hexagonal lattice, transverse magnetic field) is expected to significantly improve the nanoparticle capture efficiency as compared to that reported in the present paper.

  factor, with n μ -initial relative magnetic permeability of beads; k B =1.38•10 -23 J/K -Boltzmann constant and T -absolute temperature.

Fig. 1 (

 1 Fig. 1 (Color online). Schematic description of the capture of magnetic nanoparticles on macroscopic (left) and microscopic (right) scales

  the dimensionless capture efficiency defined as a logarithm of the ratio of the inlet to outlet particle concentration at the beginning (τ=0) of the capture process: Λ depends on the micro-filter geometry and on the forces driving nanoparticle capture, therefore, on Mason number, as shown below.

[Fig. 1 ]

 1 with the relative magnetic permeabilities of three different domains taking the following values:(i) μ m for the inner cylinder (nickel micro-pillar); (ii) μ=1 for the annular fluid layer; and (iii) magnetic contrast factor of the micro-pillar. The expression for μ e comes

1

 1 Ma, encountered in our experiments. Analysis shows that in this case: (1) the nanoparticle concentration is nearly homogeneous in the whole cell and A φ φ ≈ ; (2) the nanoparticles are captured by almost the whole surface of the micro-pillar[18], except for the regions where the magnetic interaction between nanoparticles and micro-pillar is repulsive; these regions correspond to the range of polar angles [ ] with the critical angle 0 (1/ 2)( acos )

  the exact function K(N) has the same slope at N=1 in a loglog scale, equal to 7/5, as the slope of the approximate power law dependency

Fig. 2 (

 2 Fig.2 (Color online). Dependency of the correction coefficient of the capture efficiency on the chain aspect ratio Since the micro-pillars perturb the flow streamlines, the chains travelling through the filter are

  Fig.3b]. In experiments, we used a square 10x10 array with micro-pillar diameter of 2r m =50µm and expected height of h=50µm and a distance between micro-pillar axes of Δ=100µm. The fraction of the filter volume occupied by the of the PDMS channel was equal to 3 mm, the height was fitted to the micro-pillar height (50µm) and the distance between inlet and outlet was 12 mm.

Fig. 3 ( 5 [

 35 Fig.3 (Color online). Sketch of the microfluidic channel with a micro-pillar array (a). An optical microscopy image of a fragment of the micro-pillar array is shown on figure (b), the deposits of nanoparticles captured by the micro-pillars are schematically presented in figure (c); the deposit area is S is hatched. The diluted ferrofluid was pushed through the microfluidic channel by a syringe pump Harvard

1 τ

 1 each micro-pillar, as presented schematically in Fig. 3c. The snapshots of the micro-pillar array were taken each minute during 1h of observation. The snapshots were processed with the ImageJ software allowing calculation of the surface S occupied by nanoparticle deposits [Fig. 3c], as well of the relative deposit area s=S/S tot as function of dimensionless time / ut L τ = , flow rate and field intensity, where L=1 mm is the length of the micro-pillar array and S tot is the total area of the filter including the micropillars and the voids between them. For comparison with the theoretical model, the relative deposit area s was related to the volume fraction of captured particles σ [Eq. (5)] through the deposit internal volume fraction Φ (ratio of the total volume of captured particles to the deposit volume). Namely, we get s σ = Φ ⋅ for any elapsed time τ, and m m m s σ = Φ at steady state (saturated filter, τ → ∞ ), with Φ mand s m being the values of Φ and s at steady state. In the considered limit the maximum packing fraction of the body-centered tetragonal (bct) lattice, as inferred from the theoretical analysis[START_REF] Magnet | [END_REF]. This allows fitting the experimental dependencies ( ) s τ by the following equation:

Fig. 4 (

 4 Fig. 4 (Color online). Snapshots of the micro-pillar array with deposits of the captured nanoparticles at the applied magnetic field, H 0 =6.05 kA/m, oriented along the main flow, horizontally with respect to the snapshots. Each row of snapshots corresponds to the fixed elapsed time equal to t=0, 15, 30, and 45 min from the upper to the bottom row. The first row at t=0 corresponds to micro-pillars clear from nanoparticle deposits. Each column corresponds to different filtration speeds, u, whose value is given on the top of the figure. The scale bar of 100µm is presented on the left upper snapshot

Fig. 5 (

 5 Fig.5 (Color online). Experimental time dependencies of the relative deposit area for the applied magnetic field H 0 =6.05kA/m and at different filtration speeds u corresponding to different values of Mason number. The inset shows two upper curves in an extended time scale. Symbols stand for experimental data; solid lines correspond to the fit of the data by Eq.(17). The upper curve at Ma=323 was fitted by Eq. (4).

σ

  and σ m were replaced by Φ•s and Φ•s m , respectively, with 0

Fig. 6 (

 6 Fig. 6 (Color online). Experimental Mason number dependency of the relative deposit area As is seen from this figure 6, experimental dependencies s m (Ma), are gathered along a single

  -pillar axes) estimated under assumption of straight stripe patterns observed in steady state at slow flows [Fig.4].The second fitting parameter, Λ, describes capture efficiency at the beginning of the separation process and is plotted in function of Mason number in Fig.7for two values of the applied magnetic field, H 0 =6.05 kA/m and 10.9 kA/m. The data for both values of the magnetic field (corresponding to α=1.4 and 4.5) collapse onto a straight line in a log-log scale. Such a collapse shows that the dipolar coupling parameter α [Eq. (1)] does not influence the capture process in the considered range, 1.4<α<4.5; this justifies the neglect of Brownian motion in our model. The master curve shows a power law decay of Λ with Mason number:

Fig. 7 (

 7 Fig.7 (Color online). Experimental and theoretical Mason number dependencies of the nanoparticle capture efficiency

3 ( 3 .

 33 ] fitted to the experimental data with the free parameter 00 0.22) 10 b = ± × . This scaling law (blue dotted line in Fig.7) appears to be very close to the initial power-law fit ). This qualitatively supports our hypothesis on a strong decrease of the capture efficiency with Mason number because of destruction of particle aggregates (chains) with increasing filtration speeds. A more precise determination of the chain length taking into account a local force balance and kinetics of their growth and destruction is required for quantitative predictions of the capture efficiency. Notice that the previous study of the nanoparticle separation in a three-dimensional (3D) filter composed of a packed bed of micron-sized magnetizable beads showed a less strong decrease of the capture efficiency with Mason number[18]: probably because, in the range of such high Mason numbers, all the nanoparticle chains are destroyed by the flow and the capture behavior is well reproduced by the single particle

  model assuming destructible nanoparticle chains allowed us to recover this experimental trend. The predicted scaling 1.7 Ma - Λ ∝ is very close to the experimental one and is explained by a product of the two following effects: (a) a decrease of the capture efficiency with increasing filtration speed washing the nanoparticles away from the micro-pillars and giving a contribution to the capture efficiency proportional to 1 Ma -; (b) a decrease of the capture efficiency with a decreasing chain length providing a term scaling with 0.7Ma -.

  for the fixed value of the applied magnetic field, H 0 =6.05 kA/m, and for six different filtration speeds u (or Mason numbers Ma). The relative deposit area increases progressively with time and tends to saturation at high Mason numbers, Ma≥1790. Apparently, one hour experiments were too short for achieving steady state plateau at relatively low speeds corresponding to Ma≤818. As expected, at any fixed time, the relative deposit area increases with a decrease in flow rate and consequently with a decrease in

	Mason number.

Acknowledgements

The authors are very grateful to the LPMC director Prof. F. Mortessagne for financial support. This work was partly supported by the Project PICS 6102 CNRS/Ural Federal University as well as by the French RENATECH network and its FEMTO-ST technological facility which has fabricated the micropillar arrays. One of the authors (J. AM) acknowledges CNPq through its post-doctoral fellowship (ref.

203100/2014-0), another author (H.E.) acknowledges FSB through its PhD collaboration fellowship and the author A.Z. acknowledges the Program of Ministry of Science and Education of the Russian Federation (3.12.2014/К) as well as the Russian Fund of Fundamental Investigations, project 14-08-00283.


P Kuzhir
email: kuzhir@unice.fr

C Magnet

H Ezzaier

A Zubarev

G Bossis

Magnetic filtration of phase separating ferrofluids: from basic concepts to microfluidic device

Keywords: phase separation, ferrofluid, magnetic colloids, magnetic separation

Magnetic filtration of phase separating ferrofluids: from basic concepts to microfluidic device

P Kuzhir, C Magnet, H Ezzaier, A Zubarev, Georges

Introduction

Magnetic separation is a process in which magnetically susceptible particles are extracted from a mixture using a magnetic force. It has found numerous applications in ore beneficiation industry [START_REF] Svoboda | Magnetic Techniques for the Treatment of Materials[END_REF], in bio-analysis for separation or detection of biological cells or molecules [START_REF] Zborowski | Magnetic cell separation[END_REF] as well as in water purification from organic or inorganic micro-pollutants [START_REF] Ambashta | Water purification using magnetic assistance: A review[END_REF].

The last two applications are based on adsorption of biological or pollutant molecules onto a surface of magnetic particles bearing specific chemical groups, followed by particle separation from the suspending fluid by a magnetic force created by an applied non-uniform magnetic field. For biomedical applications, magnetic separation is often realized in microfluidic scale allowing analysis of small volume bio-analytes and considerable reduction of high-cost chemicals [START_REF] Gijs | Microfluidic Applications of Magnetic Particles for Biological Analysis and Catalysis[END_REF]. In both biomedical and environmental applications, the use of nanoparticles instead microbeads is beneficial because the nanoparticles has a much larger specific area and can capture more molecules per unit volume of the suspension that will result in either a better sensitivity of biological assays or a lower consumption of nanoparticles for water purification. However until now, the broad use of nanoparticles has been limited by their strong Brownian motion making inefficient their magnetic separation.

In this work, we will review our recent investigations showing that the magnetic separation of nanoparticles of the size of about 50 nm can still be efficient at low-to-moderate applied magnetic fields (5-10 kA/m). In fact, the applied field could promote their phase separation manifesting through appearance of drop-like or chain-like aggregates; these aggregates have a large enough volume to become non-Brownian and they are separated from the suspending fluid much easier than individual nanoparticles. In practice, we deal with a separation of the solid phase of a ferrofluid (colloidal suspension of magnetic nanoparticles) from its liquid phase. We will start with a brief description of the field induced phase transition in ferrofluids extending existing theories [START_REF] Tsebers | Magnetostatic instabilities in-plane layers of magnetizable liquids[END_REF][START_REF] Buyevich | Equilibrium properties of ferrocolloids[END_REF][START_REF] Liu | Field-Induced Structures in Ferrofluid Emulsions[END_REF][START_REF] Zubarev | Theory of structural transformations in ferrofluids: Chains and -gas-liquid phase transitions[END_REF] to the present case of medium sized nanoparticles with multipolar magnetic interactions [Sec. 2]. Then, separation of nanoparticles in the simplest geometry -capture by a spherical magnetizable microbead -will be considered in Sec. 3. Sections 4 and 5 will be devoted to filtration of nanoparticles on a packed bed of magnetized microbeads and to the microfluidic filtration on an ordered array of magnetized micro-pillars, respectively. The last geometry approaches the systems currently used for realization of magneto-microfluidic immunoassays with magnetic microbeads [START_REF] Gijs | Microfluidic Applications of Magnetic Particles for Biological Analysis and Catalysis[END_REF].

However, replacement of microbeads by much smaller nanoparticles changes the physics of the magnetic separation process, especially if nanoparticles undergo a field-induced phase transition. The conclusions and perspectives will be outlined in Sec. 6.

Field-induced phase transition

Magnetic nanoparticles of ferrofluids exhibit dipole-dipole magnetic interactions. If the particle concentration and/or the applied magnetic field are strong enough, dipole-dipole interactions can lead to a phase separation of the ferrofluid, i.e. appearance of dense particle aggregates (concentrated phase) separated by a suspending liquid containing a small amount of nanoparticles (dilute phase). The two key parameters governing the phase separation (or phase transition) are the nanoparticle volume fraction φ and the dipolar coupling parameter defined as the ratio of the energy of dipole-dipole interactions between nanoparticles to the thermal energy k B T [START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Orlandi | Microfluidic separation of magnetic nanoparticles on an ordered array of magnetized micro-pillars[END_REF]:

α ∼ 2 2 3 2
0 0 3 0 induce ddipoles only permanent and induced dipoles [START_REF] Zborowski | Magnetic cell separation[END_REF] 3

n n B n B H r m k Tr k T πβ µ µ = , ( 1
)
where m is the absolute value of the particle dipole moment; µ 0 =4π•10 -7 H/m is the magnetic permeability of vacuum, r n is the nanoparticle radius;

( 1)/( 2)

n n n β µ µ = - + and µ n are,
respectively, the magnetic contract factor and relative magnetic permeability of nanoparticles with induced dipole moments; H 0 is the intensity of the applied field.

In the general framework of the phase transition theory, the equilibrium between the two considered phases is found by the equilibrium of nanoparticle chemical potentials, ζ, and osmotic pressures, p, in each phase [11]:

( , ) ( , ) c d ζ φ α ζ φ α = ; ( , ) ( , )
c d p p φ α φ α = , (2)
where the subscripts "c" and "d" stand for the concentrated and dilute phases, respectively.

The thermodynamic functions ζ and p depend on the interaction potential between nanoparticles. The existing theoretical models propose different ways to estimate magnetic and short-range non-magnetic interactions between nanoparticles that results in different expressions for ζ and p and leads to different shapes of the phase diagram [START_REF] Tsebers | Magnetostatic instabilities in-plane layers of magnetizable liquids[END_REF][START_REF] Buyevich | Equilibrium properties of ferrocolloids[END_REF][START_REF] Liu | Field-Induced Structures in Ferrofluid Emulsions[END_REF][START_REF] Zubarev | Theory of structural transformations in ferrofluids: Chains and -gas-liquid phase transitions[END_REF]. In particular case of our experiments, we deal with permanent near spherical clusters of a medium size ranging between 50 and 80 nm and composed of iron oxide nanoparticles of a medium size of 10 nm. These nanoclusters can be seen as multi-domain particles without permanent dipole moment but having a relatively strong initial magnetic permeability, µ n =10-30 [START_REF] Magnet | Haloing in bimodal magnetic colloids: the role of field induced phase separation[END_REF][START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized microsphere under magnetic and flow fields[END_REF]. This implies multi-polar interactions between nanoclusters at short distances between them characteristic for the concentrated phase [START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized microsphere under magnetic and flow fields[END_REF]. Both from hydrodynamic and magnetic point of views, the nanoclusters behave as individual spherical particles possessing their proper hydrodynamic mobility and induced magnetic moment. Thus, their dipolar coupling parameter α is defined by the right-hand side of Eq. (1). For the sake of clarity the nanoclusters will be hereinafter called magnetic particles.

Implementing multi-polar and hard-sphere interactions into the model and assuming that the dilute ferrofluid phase corresponds to a disordered fluid and the concentrated phase to a face centered cubic (fcc) solid, we get appropriate expressions for the functions ζ and p in both phases. Substituting these functions to Eq. ( 2), we get a system of two algebraic equations that is solved numerically with respect to the particle concentrations φ c and φ d at different dipolar coupling parameters α. So obtained functions, φ c (α) and φ d (α) correspond to the binodal curves of the α-φ diagram plotted in Fig. 1. These two curves separate the phase diagram into the three regions, as follows: the disordered fluid situating below the left binodal curve; the fcc-solid situating below the right binodal curve and the fluid-solid mixture occupying the space between the two binodals [START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized microsphere under magnetic and flow fields[END_REF]. This mixture corresponds to drop-like aggregates dissolved in a dilute suspension of isolated magnetic particles. For our application, we are more interested by the left binodal curve, showing the lower limit of particle aggregation. At magnetic fields H 0 >10 kA/m (α >2), the particles get aggregated at low concentrations, φ<8•10 -3 (<0.8%vol.) This is beneficial for the magnetic separation since the aggregates are easier separated from the suspending fluid than individual particles.

Separation on a single microbead

To understand the basic mechanisms of the magnetic filtration of a phase separating ferrofluid, we start with the simplest geometry where the ferrofluid flows past a single microbead magnetized by an external magnetic field. Depending on the balance between hydrodynamic forces and magnetic forces between ferrofluid particles and micro-bead, the particles are either captured by the micro-bead or carried away from the micro-bead by the flow. During time, the captured particles build dense deposits around the micro-bead. These deposits grow with time until some steady state size when the quantity of arriving particles equilibrates the quantity of the particles eroding from the deposit surface by hydrodynamic forces. In this Section, we describe the effect of the flow and the magnetic field on the steady- 1b]. At the highest concentration, the ferrofluid undergoes the phase separation: long string-like particle aggregates were observed in the ferrofluid bulk; these aggregates progressively migrated to the microbead and formed dense and long deposits extended along the applied magnetic field [Fig. 1c]. The content of the deposits can be seen as a concentrated ferrofluid phase, while the surrounding transparent medium corresponds to the dilute phase. These two snapshots clearly show that the phase separation enhances significantly the particle capture efficiency. Under flow, the size and shape of the particle deposits evolve. The snapshots 1d and 1e show that their size decreases with the filtration speed because the hydrodynamic forces, eroding the particles from the deposit surface, become stronger with increasing speed. Asymmetry of the front (facing the flow) and back deposits likely comes from the asymmetry of the velocity profile resulting in difference in hydrodynamic forces acting on the front and on the back deposits. Finally, the spikes on the deposit surface (both in the presence and in the absence of field) can be explained by a surface instability appearing because of negative surface energy in the vicinity of the deposit extremities where the magnetic field is nearly orthogonal to the deposit surface [START_REF] Lobkovsky | Surface energy anisotropy for dipolar lattices[END_REF][START_REF] Promislow | Low-energy suspension structure of a magnetorheological fluid[END_REF].

Let us now estimate the deposit's size and shape and compare them to the ones observed in experiments. Analysis shows that in the considered range of experimental parameters, the steady-state deposit shape is not significantly influenced by the flow. It can be determined from the continuity of the normal component of the stress tensor on the surface excluding the viscous term. This gives us the following differential equation for the deposit surface described by the geometric locus [R(θ), θ] in the polar coordinate system [START_REF] Magnet | Haloing in bimodal magnetic colloids: the role of field induced phase separation[END_REF]: In the present model, the anchoring angle defines the deposit volume. In the absence of flow, we measure the angle θ 0 ≈69 deg, while in the presence of flow the balance of particle fluxes on the deposit surface allows us to find approximate relationships (not given here for brevity) between the anchoring angle (and, consequently, the deposit size) and the Mason number describing the ratio of hydrodynamic-to-magnetic forces exerted on magnetic particles [START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized microsphere under magnetic and flow fields[END_REF]:

2 2 2 0 2 ( / ) 1 ( / ) r h h R R h h R R θ χ ′ - -= ′ + , (3) where
2 0 0 m n n u Ma H r η µ β β = , ( 4
)
where η is the viscosity of the suspending liquid of the ferrofluid (water in our case).

The calculated deposit shape is compared to the experimentally observed one on Fig. 3 for

Separation on a packed bed of microbeads

In a real magnetic separator, magnetic particles are separated from the suspending liquid not by a single micro-bead but by numerous more or less closely spaced collectors magnetized by an external magnetic field. The particle capture on a given collector is affected by neighboring collectors because they modify the velocity and the magnetic field around a given collector. A packed bed of magnetizable micro-beds is one of possible configurations of the magnetic separation system. In this Section, we study magnetic filtration through such a system and seek for the Mason number effect on the capture efficiency.

In experiments, strongly diluted ferrofluid (solid phase concentration φ 0 =1.6•10 -5 (0.0016%vol.) is subjected to a recirculation flow in a closed loop circuit containing a magnetic filter whose ends are connected with each other by flexible tubes, as depicted in Fig. 5a. A peristaltic pump is included into the circuit and ensures the flow with a constant flow rate Q corresponding to the filtration speeds, / u Q A = , inside the filter, ranging from 0.03 to 0.12m/s. The filter itself is a cylindrical column, filled with nickel microbeads of a mean diameter 2r m =45±5 µm at a volume fraction of about c≈0.3. A solenoid is placed around the filter and generates a magnetic field parallel to the flow direction and having an intensity up to H 0 =32 kA/m. Once the field is applied and the flow is started, the filter becomes to capture the magnetic particles such that their concentration φ outside the filter (in the tubes) starts to decrease with the time. The temporal evolution of the concentration φ(t) is measured by a turbidimeter as function of the filtration speed u and of the applied field H 0 . The measured φ(t) curves are then fitted to the phenomenological filtration law derived from the solution of the filtration equation [START_REF] Tien | Granular Filtration of Aerosols and Hydrosols[END_REF] taking the following form for the closed loop geometry at low initial particle concentrations1 0 φ <<1 and at low-to-moderate capture efficiencies 1 Λ < [START_REF] Magnet | Closed-loop magnetic separation of nanoparticles on a packed bed of spheres[END_REF]:

0 exp( / ) V V φ φ τ Σ ≈ -Λ , (5)
where / ut L τ = is the dimensionless time; L and V are the filter length and volume, respectively; V Σ is the total volume of fluid phase in the closed-loop circuit. The single adjustable parameter of this fit is the capture efficiency Λ -a phenomenological coefficient defined through the ratio of particle concentrations at the filter inlet ( in

The effect of the neighbourhood of the microbeads of the packed bed on the capture efficiency of magnetic particles is described by the term α ≈ , at which the phase separation is expected to occur for the considered particle concentration φ 0 =1.6•10 -5 , while the data for Ma>4 correspond to 5 α ≈ , for which the particles do not undergo phase separation. The effect of a possible aggregation of magnetic particles on their capture efficiency will be analyzed in the next Section 5 for a simpler and more important (for biomedical applications) microfluidic magnetic separation system.

Microfluidic separation on an ordered array of micro-pillars

For applications to bioassays, it is often beneficial to realize ordered and more or less sparse arrays of magnetic collectors and reduce the size of the separator to microfluidic scale.

In this Section we will describe magnetic filtration using one of such systems and will characterize the capture efficiency paying special attention to field-induced particle chaining.

To this end, we fabricated a nickel micro-pillar array on a glass substrate by electroplating and soft photolithography, according to a general procedure described by Deng et al. [START_REF] Deng | Fabrication of magnetic microfiltration systems using soft lithography[END_REF]. A polydimethylsiloxane (PDMS) cover was glued to the glass substrate bearing the nickel micro-pillars using oxygen plasma. Aluminum needles were introduced to the PDMS cover at both its extremities allowing the channel inlet and outlet. The resulting microfluidic channel is schematically presented in Fig. 5b

Fig. 1 .

 1 Fig.1. Phase diagram of the ferrofluid composed of magnetic particles having an induced dipole moment and a constant magnetic permeability equal to µ n =30 (reprinted from Magnet et al., Phys. Rev. E 89, 032310 (2014), with kind permission from American Physical Society).

  Fig.2. S nickel unifor partic illust presence filtration hours afte

  are respectively, the radial and the polar components of the magnetic field intensity, both normalized by the applied field H 0 value of the normalized magnetic field at a given point (R, θ) of the deposit surface and value of h at the point (R=1 and θ=θ 0 ) where the deposit surface joins the microbead; the angle θ 0 [cf. Fig.3] is called the anchoring angle. The ordinary differential equation (3) is solved numerically at the initial condition R(θ 0 )=1.

  the longitudinal magnetic field and two different values of the Mason numbers equal to 0.08 and 0.35. Our model reproduces qualitatively the elongated shape of the deposits. It correctly predicts an asymmetry of the front and rear deposits in the longitudinal field, as observed in experiments. Because of the simplifying input hypotheses, the model is unable to reproduce the surface instability manifested through appearance of conical spikes on the deposit surface.

Fig. 3 .

 3 Fig. 3. Comparison between the calculated (red solid line) and the experimental (black wavy line) shapes of the particle deposits attached to the microbead in the presence of an external magnetic field and flow both oriented horizontally with respect to the figure. The Mason number Ma is set to 0.08 (a) and 0.35 (b) (reprinted from Magnet et al., Phys. Rev. E 89, 032310 (2014), with kind permission from American Physical Society).

Fig. 4 .

 4 Fig.4. Theoretical and experimental Mason number dependencies of the deposit length normalized by the microbead radius. The upper solid line stands for the theoretical prediction for the front deposit in the longitudinal field. The lower solid line stands for the prediction for the back deposit in the longitudinal field (reprinted from Magnet et al., Phys. Rev. E 89, 032310 (2014), with kind permission from American Physical Society).

Fig. 5 .

 5 Fig. 5. Sketch of the experimental setups used for the filtration through a packed bed of microbeads (a) and through a microfluidic channel with a micro-pillar array (b). Optical microscopy images (c) -(e) show the evolution of the particle deposits around micro-pillars with time at the magnetic field intensity H 0 =6.05 kA/m, filtration speed u=1.88•10 -3 m/s and initial particle concentration in the ferrofluid φ 0 =0.32%vol. The snapshot (c) at t=0 corresponds to the micro-pillars free of magnetic particles.

.

  This coefficient is finally analyzed as a function of Mason number. From the theoretical point of view, the capture efficiency can be related to the Mason number by calculating magnetic particle flux arriving to one of the microbeads situating inside the packed bed. Analytical solution of this problem is still possible in the frame of the concentric-sphere model of the packed bed, in the limit Ma / m n r r >>1 and under assumption of negligible Brownian motion. Furthermore, we consider magnetic beads with a high magnetic permeability ( 1 m β ≈ ) and assume that all the particle aggregates are destroyed in the considered range of Mason numbers. This gives us the following expression for Λ,

2 /

 2 the efficiency with the volume fraction c of the microbeads. The theoretical and experimental (obtained by fitting experimental φ(t)-curves to Eq. (5)) Mason number dependencies are shown on Fig. 6. Both the theory (dashed curve) and experiments (open triangles) show a decreasing Mason number dependency of the particle capture efficiency. Such behaviour is easily explained by increasing hydrodynamic forces transporting the particles through the filter and hindering their capture by the microbeads. Quantitatively, our model [Eq. (6), dashed curve] allows a correct prediction of the capture efficiency only at Mason numbers Ma>4, while it strongly underestimates the value of Λ at lower Mason numbers. The discrepancy rises likely because of the field-induced phase separation occurring at Ma≤1 and ignored in the present model. In fact, the experimental point at Ma≈1 (encircled point on Fig.6) corresponds to the value of the dipolar coupling parameter, 20

  and an optical microscopy image of the micropillar array is shown in Fig.5c. In experiments, we used a square 10x10 array with micropillar diameter of 2r m =50µm and expected height of h=50µm and a distance between micropillar axes of ∆=100µm. The fraction of the filter volume occupied by the pillars was 0.2 c ≈ .The diluted ferrofluid (composed of nanoclusters of a mean diameter 2r n ≈84 nm dispersed in water at volume fraction of φ 0 =3.2⋅10 -3 ) was pushed through the microfluidic channel by a syringe pump at the flow rates Q corresponding to the range of filtration speeds u=Q/A=3.4×10 -5 -3.4×10 -3 m/s. Accumulation of deposits of magnetic particles around the micro-pillars in the presence of flow and of an external magnetic field oriented along the flow was visualized and recorded using the equipment described in Section 3. A few snapshots showing progressive growth of the deposits are shown in Figs.5c-e. At certain conditions, oval shape of the deposits at short times from the beginning of the capture [Fig.5d] evolves to a continuous wavy pattern when the deposits join together all the micro-pillars along the direction of the applied field [Fig.5e]. Image processing allowed us to follow the evolution of the surface S occupied by nanoparticle deposits with time. This surface was then normalized by the total area of the filter S tot . The time dependency of the relative deposit area s=S/S tot was then fitted to the s(t)-dependency obtained by integration of the phenomenological filtration equations[START_REF] Tien | Granular Filtration of Aerosols and Hydrosols[END_REF] under assumption of relatively low capture efficiency 1 Λ <[START_REF] Orlandi | Microfluidic separation of magnetic nanoparticles on an ordered array of magnetized micro-pillars[END_REF]:

Approximate expression (5) is obtained by setting γ

to unity in Eq. (3) of the original paper[START_REF] Magnet | Closed-loop magnetic separation of nanoparticles on a packed bed of spheres[END_REF]

Acknowledgements

We are grateful to master students Y. Bourras, S. Nave, M. Acouala and G. Orlandi for experimental measurements. This work has been partially supported by the French RENATECH network and its FEMTO-ST technological facility, by the PEPS-INSIS project of CNRS and by the Russian Scientific Foundation (project n° 14-19-00989). One of the authors (H.E.) acknowledges FSB through its PhD collaboration fellowship.

where m s is the maximal relative deposit area when the filter is saturated with captured particles, 0.74 Φ ≈ is estimated internal volume fraction of deposits; the dimensionless time τ and the capture efficiency Λ are introduced in the same way as for filtration through a packed bed [see Sec . 4]. The capture efficiency, Λ, was used as an adjustable parameter of this fit, and its value was analyzed as a function of Mason number.

The theoretical dependency Λ(Ma) was obtained in the limit Ma / m n r r >>1 in a similar way as in Sec. 4, i.e. by integration of the particle flux on the surface of the micro-pillar but the model was extended to the case of field-induced aggregation of magnetic particles that is expected for larger particle concentrations used for the micro-pillar geometry (φ 0 =3.2⋅10 -3 vs 1.6•10 -5 for filtration through a packed bed). We supposed that magnetic particles assembled in straight chain-like aggregates with the number of particles per aggregate denoted by N. The aggregate length (or, equivalently, N) was defined by the balance of an average hydrodynamic tensile force rupturing the chain and an average attractive magnetic force between particles inside the chain. Such reasoning was borrowed from the chain model of electrorheology [START_REF] Martin | Chain model of electrorheology[END_REF] and allowed the following scaling for the particle number per chain:

Incorporating this issue to the capture efficiency, we obtain the following Mason number law, approximately valid for relatively short chains ( 8 N ≤ ) and for the Mason numbers corresponding to the limit Ma / m n r r >>1 [START_REF] Orlandi | Microfluidic separation of magnetic nanoparticles on an ordered array of magnetized micro-pillars[END_REF]:

The experimental Mason number dependencies of the capture efficiency (obtained by fitting the s(t) curves to Eq. ( 7)) were fitted to the theoretical law [START_REF] Zubarev | Theory of structural transformations in ferrofluids: Chains and -gas-liquid phase transitions[END_REF] and are shown in Fig. 6 by solid triangles and open squares for two values of the applied magnetic field, H 0 =6.05 and 10.9 kA/m. The data for both fields (corresponding to the dipolar coupling parameter α=1.4 Ma - Λ ∝

) related to the fact that there is no phase separation or particle aggregation in the first case (at very small particle concentration, φ 0 =1.6•10 -5 , except for the encircled point), while particle aggregation is expected in the second case. Finally, (Ma) Λ -curve for the packed bed (dashed curve on Fig. 6) lies above the (Ma) Λ -curve for the micro-pillar geometry (solid line). This could be explained by geometrical differences between both configurations, such as higher collector volume fraction c and higher length L of the packed bed filter. [Eq. ( 6)] at Ma>3. Notice that a slightly different definition of Ma (using another length-scale) was used in the original papers [START_REF] Orlandi | Microfluidic separation of magnetic nanoparticles on an ordered array of magnetized micro-pillars[END_REF][START_REF] Magnet | Closed-loop magnetic separation of nanoparticles on a packed bed of spheres[END_REF] leading to expressions for Λ different from Eqs. ( 6) and ( 8) of the present paper.

Conclusions and perspectives

In this work, we have briefly reviewed magnetic separation of ferrofluids composed of large magnetic particles (spherical nanoclusters of a size of the order of 60 nm) possessing an induced dipole moment. In the presence of an applied uniform magnetic field, these ferrofluids undergo a fluid-solid phase transition fully governed by the particle volume fraction φ and the dipolar coupling parameter α. At moderate magnetic fields, H 0 ∼10 kA/m (α∼2), the phase transition occurs at relatively low particle volume fraction (φ∼0.8%vol.) and is accompanied by appearance of particle drop-like or chain-like aggregates. Such fieldinduce aggregation strongly enhances the capture efficiency of magnetic particles both by single and multi-collector systems. At favorable conditions, the aggregates become large enough to be non-Brownian and their capture by magnetized collectors depends only on the ratio of hydrodynamic-to-magnetic forces -the Mason number. The capture efficiency on a single microbead is characterized by the steady-state size of magnetic particle deposits around the microbead while it is defined through the ratio of the particle concentration at the inlet and outlet for filtration through multi-collector systems (packed bed or micro-pillar array). In all considered geometries, the capture efficiency decreases with an increasing Mason number because of hydrodynamic forces washing away magnetic particles from the collectors. This decrease may become stronger for aggregated magnetic particles ( 1.7

Ma - Λ ∝

) than for individual ones ( 1Ma - Λ ∝

) if the shear fields are strong enough to provoke aggregate rupture, as is expected for the microfluidic filtration through the micro-pillar array. However, the filter geometry could be optimized to promote particle aggregation well before they arrive to the collectors and to avoid the aggregate rupture in the vicinity of the collectors. This imposes combined engineering and fundamental studies of the kinetics of particle aggregation and of aggregate behaviors in the microfiltration devices. These future studies should allow development of new magneto-microfluidic immunoassays based on magnetic nanoparticles offering a much better sensitivity as compared to presently used magnetic microbeads.


X Garbet

P Donnel

C Ehrlacher

E Caschera

R Dumont

M Faganello

V Grandgirard

P Ghendrih

M Idouakass

M Lesur

Y Sarazin

D Zarzoso

On the relationship between residual zonal flows and bump-on tail saturated instabilities

A connection is established between two classical problems: the non linear saturation of a bump-on tail instability in collisionless regime, and the decay of a zonal flow towards a finite amplitude residual. Reasons for this connection are given and commented.

Introduction

 and references therein). It turns out that these two results, zonal flow residual and saturated level of bump on tail instability, are connected. The reason for this relationship, which is not entirely fortuitous, is detailed and commented in this paper.

The paper is organized as follows. Section 2 summarizes the O'Neil calculation of Landau damping in non linear regime and its extension to the bump on tail instability. The Rosenbluth-Hinton calculation of the time evolution of zonal flow is reminded in section 3. Finally the relationship between residual zonal flows and bump-on tail saturated instabilities is discussed in section 4. A conclusion follows.

Landau damping in non linear regime -extension to the bump on tail instability

The seminal O'Neil paper [3] aimed at studying Landau damping in collisionless regime in the non linear regime where particle trapping matters. The time evolution of the distribution

function and electric field were calculated. O'Neil realized that the asymptotic state can be captured by calculating the energy transfer from the wave to particles. This calculation was a source of inspiration for many subsequent works (e.g. [7,8]). The O'Neil work was soon extended to the bump-on tail instability. The saturated state can be calculated by stating that the kinetic energy of resonant particles is fully converted to wave energy. The key point is then to calculate the distribution function. O'Neil prescription consists in using a phase mixing ansatz. The later states that the relaxed distribution function is the average of the unperturbed one along trajectories, due to phase mixing. Hence it is flat within the island, and joins smoothly the unperturbed distribution function outside the separatrix. This prescription is consistent with the detailed calculation of the mode evolution done by O'Neil in the same work. Berk's prescription rather corresponds to a discontinuous distribution function that is flat in the island, and joins the unperturbed distribution right at the separatrix [5]. A third option is to choose a flat distribution function within an annulus, whose width is the island width [START_REF] Galeev | Nonlinear Plasma Theory[END_REF]. Of course other choices are possible as to the contribution of particles near the island separatrix.

Summary of the O'Neil calculation

Let us start from an arbitrary integrable Hamiltonian system that can be described by a set of angle/action variables (Θ, J) such that

dΘ dt = ∂H eq ∂J = Ω eq (J) dJ dt = - ∂H eq ∂Θ = 0
where H eq (J) is the unperturbed Hamiltonian. This system is perturbed by an oscillation, i.e.

H(Θ, J, t) = H eq (J) -h cos (n • Θ -ωt) (1)
We look for an asymptotic steady state , i.e. the amplitude h of the perturbed Hamiltonian is a constant (in the original O'Neil paper, the evolution of h(t) was computed). We also introduce the resonance frequency Ω(J) = n • Ω eq (J) -ω. The resonant surface is defined by the condition Ω(J) = 0 in the action space. Near the resonant surface, the actions are of the form J = J R +nI, where J R spans the resonant surface, while the action I measures the distance to the resonant surface. The resonant frequency reads Ω = CI, where C is related to the Hessian matrix of the Hamiltonian calculated at the resonant surface

C = n i n k ∂ 2 H ∂J i ∂J k Ω=0 (2)
We assume that C is positive without any loss of generality. The trajectory equations read

dξ dt = Ω dΩ dt = -ω 2 b sin ξ
where ξ = n • Θ -ωt is an angle, and ω b = √ Ch is the bounce frequency of deeply trapped particles. The variable Ω is quite convenient since it scales as a pulsation. Nevertheless the variable I presents also some advantages as it is an action conjugate to ξ. It is reminded here that the Jacobian is equal to 1 when moving from one to another set of conjugate variables. The new Hamiltonian H is defined as

H = 1 2 Ω 2 -ω 2 b cos ξ (3)
H is a motion invariant, so that the system is still integrable. Its contour lines exhibit the shape of an island (Fig. 1). Eqs. [START_REF] Neil | [END_REF] show that (ξ, Ω) are conjugate variables with respect to the Hamiltonian H. The set (ξ, Ω) can be completed by a set of conjugate variables (Θ R , J R ), which span the resonant surface. The phase space volume element reads

d 3 Θd 3 J = d 2 J R d 2 Θ R dIdξ (4)
To simplify the notations, the explicit dependence on (Θ R , J R ) will be omitted in the following.

In other words, the problem becomes 2D, and decoupled from the dynamics along the resonant surface. In the original O'Neil calculation, which addresses a 1D electrostatic wave, the following correspondences hold

h = eφ ; Ω = kv -ω ; ξ = kx -ωt ; C = k 2 m ; ω b = k eφ m 1/2 ( 5
)
where k is the wavenumber, x the position, v th velocity, and φ the electric potential. We introduce an average ("bounce" average) over time for any function f (H, ξ, Ω ) where Ω is the sign of Ω, which can be written as

f = 1 τ π -π dξ 2π 1 |Ω| f (H, ξ, Ω ) (6)
for passing particles (ω

2 b ≤ H ≤ ∞), while f = 1 2τ ξ 0 -ξ 0 dξ 2π 1 |Ω| (f (H, ξ, 1) + f (H, ξ, -1)) (7)
for trapped particles (-

ω 2 b ≤ H ≤ ω 2 b ).
Here Ω is to be understood as a function of (H, ξ, Ω )

Ω = Ω √ 2 H + ω 2 b cos ξ 1/2 (8)
The bounce angle ξ 0 is the positive root of ω 2 b cos ξ 0 = H. The normalizing time τ is defined as

τ = ξ 0 -ξ 0 dξ 2π 1 |Ω| (9)
with ξ 0 = π for passing particles. Note that τ is a continuous function of κ, and differs from the bounce period T b . More precisely τ = T b 2π for passing particles and τ = T b 4π for trapped particles.

Saturation level

A subtle point is that the kinetic energy should be calculated in the laboratory frame, while the calculation of the distribution function is done in the wave frame. The Hamiltonian in the laboratory frame H lab is related to the one in the wave frame H via the relation H = H lab -ωI, or equivalently H lab = H+Ω ω C . The Hamiltonian H does not participate in the resonant transfer of energy due to parity reason. Indeed f = F -F eq , where F eq is the unperturbed distribution function, is an odd function of Ω. Therefore the resonant transfer of kinetic energy is Moreover F eq which can be expanded near the resonant surface Ω = 0, i.e. F eq = F 0 + F 0 Ω. The phase mixing prescription states that the relaxed distribution function is the average of the initial one over trajectories, i.e.

∆E = ω C +∞ -∞ dI +π -π dξ 2π f Ω (10)
f = F -F eq = F 0 [ Ω -Ω] (11)
Note that Ω is zero for trapped particles, since Ω changes sign when the particle moves back and forth due to trapping. A schematic view of the distribution function at the O point of the island is shown in Fig. [START_REF] Galeev | Nonlinear Plasma Theory[END_REF]. A change of variables yields the following result

∆E = 2 ω C 2 F 0 +∞ -ω 2 b dHτ Ω 2 -Ω 2 (12)
It appears that most of the energy given by particles to the mode comes from trapped particles, and to a lesser extent from barely passing particles. When balancing the energy transfer with the mode energy density Λ ω h 2 , one gets a prediction for the field amplitude ∆E + Λ ω h 2 = 0. The final result can be formulated as

ω b = K BT γ L (13)
where

γ L = π 4 ω F 0
Λω is the linear growth rate. The constant K BT is given by the following relation

K BT = 32 π ∞ 0 dκ κ 3 τ ω b Ω 2 -Ω 2 (14)
where

Ω = 2ω b κ Ω 1 -κ 2 sin 2 ξ 2 1/2 ( 15
)
and κ is the the trapping parameter defined as

κ 2 = 2ω 2 b H+ω 2 b . Trapping corresponds to 1 ≤ κ 2 ≤ ∞
while passing particles are characterized by 0 ≤ κ 2 ≤ 1.

Zonal flow residual 3.1. Formal expression

In 1998 Rosenbluth and Hinton calculated the time evolution of a zonal flow and predicted the asymptotic value, called zonal flow residual [1]. The calculation consists in initializing the electric potential with a perturbation of the form φ(x, t = 0) = φ 0 e ikψ , where k is the radial wavenumber, and ψ is minus the poloidal magnetic flux normalized to 2π, also equal to -A ϕ R, where A ϕ is the toroidal component of the vector potential, and R the major radius. It is a function of the radial coordinate, labeled r. The evolution of φ(x, t) is calculated by solving the gyrokinetic Vlasov equation coupled to the electroneutrality equation. The ratio of the potential residual to its initial value is

φ ∞ φ 0 = 1 -d 3 pF eq J 2 0 1 -d 3 pF eq J 2 0 J 2 b ( 16
)
where J 0 and J b are related to averages over the cyclotron and drift motions, F eq is the unperturbed distribution function. These are scalars for Fourier modes.

Field shielding

The gyroaverage function is the usual one, and is approximated by the following form in the large scale limit

J 2 0 1 -1 2 k 2 ⊥ ρ 2 c
, where ρ c is the kinetic ion Larmor radius. The guiding center radial coordinate reads ψ = ψ + ψ, where ψ is the time average position and ψ describes the shift with respect to the reference magnetic surface ψ = ψ. For a periodic potential, the transit average operator is defined as

J b = e ik ψ 1 -k 2 ψ2 ( 17
)
where the bracket is an average over the drift motion. At first order in k 2 ρ 2 c , the shielding term reads 1 -

d 3 pF eq J 2 0 J 2 b k 2 1 2 ρ 2 c + d 3 pF eq ψ2 = k 2 ρ 2 i + δ 2 (18)
where ρ i = mT eB is the thermal ion Larmor radius, and dψ dr

2 δ 2 = d 3 pF eq ψ2 (19)
is the radial size of the drift orbit squared. The expression of the radial displacement ψ is obtained by using the conservation of the canonical toroidal kinetic momentum

P ϕ = m I B v -eψ, namely ψ = m e I v B - v B b (20)
where the index 'b' indicates a bounce average for particles trapped in the magnetic field due to the mirror force, while it labels a transit time average for passing particles. Here I = B T R is the product of the toroidal field by the major radius, B is the modulus of the magnetic field, v is the particle velocity along the magnetic field, m and e are the ion mass and charge. This gives an expression of the width δ dψ dr

2 δ 2 = 1 2 mI e 2 d 3 pF eq v B 2 b - v B 2 b (21)
This is the main result derived by Rosenbluth and Hinton in arbitrary geometry. We now restrict the analysis to circular concentric magnetic surfaces with large aspect ratio, so that r is now the minor radius of a magnetic surface, and θ is the poloidal angle. The hamiltonian reads

H = 1 2 mv 2 + µB 0 (1 -cos θ) (22)
where = r R 0 is the inverse aspect ratio. This Hamiltonian is obviously identical to the one addressed in the previous section, with the following correspondences

h = µB 0 ; Ω = v qR 0 ; ξ = θ ; C = 1 mq 2 R 2 0 ; ω b = 1 qR 0 µB 0 m 1/2 (23)
and H = C(H -µB 0 ). Hence the problem is related again to particle trapping. However the reader should be warned that trapping here is due to a magnetic mirror force, not to the mode itself. The amplitude of the latter is very small, so that the problem can be treated linearly with respect to φ. Moreover, it can be easily verified that for a Maxwellian distribution function

F eq d 3 p = (2 ) 1/2 2 √ π dEE 1/2 e -E 2dκ [(1 -)κ 2 + 2 ] 3/2 ω b τ ( 24
)
where E is the energy normalized to the temperature. In the limit of large aspect ratio → 0 and precluding that the largest contribution comes from the separatrix neighborhood κ 1, a compact expression of the zonal flow residual Eq.( 16) is found

φ ∞ φ 0 = 1 1 + C RH q 2 √ ( 25
)
where

C RH = 3 √ 2 ∞ 0 dκ κ 3 τ ω b Ω 2 -Ω 2 (26
I = 2 1 0 dκ κ 4 2 π E κ 2 - π 2 1 K (κ 2 ) + 2 +∞ 1 dκ κ 3 1 κ 2 -1 2 π K 1 κ 2 + 2 π E 1 κ 2 (28)
where K and E are complete elliptical functions of first and second kinds. A numerical calculation yields I 0.38 [12], which implies C RH 1.61 and K BT 3.84. The value C RH 1.6 has been recovered in a number of gyrokinetic codes, and is now considered as a "must do" verification test in gyrokinetic computation. However less success was met for the bump-on tail instability since numerical simulations rather find K BT 3.2 [START_REF] Fried | National Technical Information Service Document[END_REF]11]. An alternative recipe consists in postulating that trapped particles only contribute to the mode saturation. It turns out that I is analytic in that specific case, namely I = 8 9π , and K BT = 256 9π 2 2.9 [5]. Another calculation based on O'Neil prescription yields K BT 3.5 [START_REF] Levin | [END_REF]. Finally a radical option is to choose a flat distribution function within an annulus, whose width is the island width. This procedure is fully analytic and leads to

K BT = 32 √ 2 3π
4.8 [START_REF] Galeev | Nonlinear Plasma Theory[END_REF]. All these results differ from the numerical findings. A value that is too low is found when minimizing the role of passing particles, while an overweight of passing particles leads to an estimate that is too large. So it appears that only a fraction of particles near the separatrix participate to the energy exchange. In particular it looks like barely passing particles do not contribute as expected. This is not surprising since the bounce time is very large near the separatrix (and infinite right on it), so that a mixing argument is questionable for particles in this region of the phase space. This is consistent with the analysis of the contributions of particles to the Rosenbluth-Hinton constant C RH (and therefore of K BT ). It appears that barely trapped particles provide the largest contribution (see Fig. 3). This structure in the velocity space was verified numerically by gyrokinetic codes [13]. Another important test is the dynamics of the mode evolution, which is also analytical for both problems, and quite different. Time evolution has been verified numerically with success in both cases. As a final note, we stress that a steady solution is not unique by far for bump-on tail instabilities, in particular when dissipation is accounted for. Quite often the system evolves to unsteady solutions, which can be periodic, or chaotic, thus leading to frequency chirping [14,15,11,6]. These important developments are beyond the scope of the present paper.

Pure coincidence or some common physics?

The previous sections show that a relationship connects the bump-on tail constant with the Rosenbluth-Hinton constant, i.e. K BT = 16 √ 2 3π C RH . This is rather unexpected as the two results correspond to different physics. On the one hand, the bump-on tail saturation comes from a flattening of the distribution function near the resonance surface, leading to a weakening of the drive until saturation. On the other hand, the zonal flow residual comes from a shielding effect of the initial perturbation due to the particle response to the field. So the fast conclusion is that this relationship is fortuitous. There are nevertheless common features : particle trapping, absence of collisions, and more importantly momentum conservation. The relation Eq. [START_REF] Levin | [END_REF] shows that the energy transfer in the bump-on-tail instability per unit time can be related to an exchange of momentum between the wave and particles. The power is the drag force F = dP dt times the wave phase velocity ω k . Here P is the momentum exchange between the wave and resonant particles

P = k +∞ -∞ dI +π -π dξ 2π If = m 2 λ 0 dx λ +∞ -∞ dv v - ω k f (29)
where λ = 2π k is the wavelength. The zonal flow residual is also related to momentum conservation, though in a different way. As mentioned above, the shielding effect comes from the particle displacement due to drift motion. The poloidal variation of the magnetic field in a tokamak can be seen as a static wave, that exchanges momentum with particles. The change of parallel velocity induces a radial displacement, which results from the conservation of toroidal kinetic momentum. This establishes a link between the particle radial shift and the parallel velocity Eq.( 20). The displacement is directly proportional to the variance of velocity, hence the I integral. In the case of the bump-on tail instability, it comes from the structure of the distribution function. This is probably the reason why the analytic calculation works better for the Rosenbluth-Hinton shielding effect than for the bump-on tail mode saturation: the distribution function results from a complex dynamics, and the assumption of phase mixing is easily broken, in particular near the separatrix.

Conclusion

In summary it is found that the saturation of a bump-on tail instability in collisionless regime, and the zonal-flow residual are connected. More precisely, the two constants that appear in these classical results, namely the ratio of the trapping frequency to the linear growth rate, and the zonal flow residual normalized to its initial value, are connected via a simple formula. From the numerical point of view, the prediction for the zonal flow residual works better than the one for the bump-on tail instability. This indicates that for zonal flow decay, all particles play their expected role, while for the bump-on tail problem, particles near the separatrix do not behave as foreseen. This means that the distribution function is flat within the island, as predicted, but exhibits a transition near the separatrix sharper than expected on the basis of phase mixing argument. The relationship between the two problems seems at first sight fortuitous, since the physics that is involved is different. However particle trapping and momentum conservation appear as common denominators, thus explaining partially the link between these expressions.

Figure 1 .

 1 Figure 1. Island shape (ω b = 1).

Figure 2 .

 2 Figure 2. Schematic view of the distribution function at the island O point

) 4 . 4 . 1 . 3 √

 4413 Relationship between residual zonal flows and bump-on tail saturated instabilities Numerical values and metrologyThe two constants C RH and K BT are related to the quantity I the relations K BT = 32 π I and C RH = 2I. A straightforward calculation shows that

Figure 3 .

 3 Figure 3. Contour lines of the integrand of I in the v ,v ⊥ space at θ = 0.

Acknowledgements

Drs H. Berk and A. Smolyakov are warmly acknowledged for fruitful discussions. This work has also taken benefit from discussions within the NLED Eurofusion Enabling Research project.