J F Bonnans

THEORETICAL AND NUMERICAL STUDY OF THE PROBLEM OF ABORT LANDING IN THE PRESENCE OF WINDSHEAR IV ÁN CADENA GU ÁQUETA

We analyze both theoretically and numerically the problem of abort landing in the presence of windshear. A pertinent model of optimal control is constructed in order to allow numerical simulations through an open source software. We verify that the numerical results correspond to the mathematical theory and in particular we analyze as much as possible the junction conditions in presence of state constraints. The code written to analyze the problem will be available as a contribution in bocop.org.

Introduction

Windshear is a difference in speed or direction of wind in two points sufficiently close to the atmosphere. It is a very dangerous problem when the aircraft is close to the ground at a low speed, that is at take-off and landing. This phenomenon can lead to a sudden lose of lift, resulting in a crash of the plane with the ground if the pilot does not have the time to react or if his efforts were not successfull. Between 1964 and 1985, windshear was directly or undirectly related to al least 26 major civil air disasters, leaving 820 casualties. Among these accidents, 15 occured during take-off, 3 during flight and 8 during landing. Having this context in mind, we would like to address mathematically this situation in order to understand which flight strategy the aircraft should adopt whenever windshear is present.

We rely on the models proposed in articles [START_REF] Bulirsch | Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 1: Necessary Conditions[END_REF][START_REF] Miele | Optimal Abort Landing Trajectories in the presence of Windshear[END_REF] where optimal control techniques are used. In the following sections we present the mathematical model that describes a windshear situation while the aircraft is landing (Section 2), we state the necesarry optimality conditions for this specific problem (Section 3), we find with a numerical procedure a control that garantees a maximum and positive altitude of the plane with respect to the ground (Section 4) and we make an analysis betw.een the theory and the numerical results that have been obtained (Section 5).

Model

We consider a bidimensional model with coordinates (x, h). More precisely, we analyze the plane landing in the presence of windshear. We suppose that the decision taken by the aircraft has been to abort landing and regain altitude in order to retry landing.

2.1.

Wind structure. Our model for the windshear phenomenon is inspired from [START_REF] Bulirsch | Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 1: Necessary Conditions[END_REF], where nonsmooth are used. However, our numerical method only requires smooth functions. Thus, we consider the following representation of the wind, where the functions are smooth approximations of the ones used in reference [START_REF] Bulirsch | Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 1: Necessary Conditions[END_REF]:

W x (x) = -50 exp[-β 1 (x) 2 ], (1)
W h (x) = 0.02β 2 (x) -50 (2)
These functions are built through auxiliary fonctions β 1 and β 2 that stretch the gaussian W x at its lowest point in order to form a valley, and to make W h constant after 5000ft. These functions rely on the smooth approximation of the absolute value:

|x| ≈ |x| := x 2 + 2 .
This choice models a wind structure presenting windshear around 2000ft and 3000ft in the x coordinate. 2.2. Aerodynamic Forces. There are three forces that take part in the aerodynamic of the aircraft. They are the thrust T , the lift L and the drag D. Their equations, that depend on the speed of the plane v and the angle of attack α are the following:

T = A 0 + A 1 v + A 2 v 2 , (3)
L = (1/2)ρS(C 0 + C 1 α)v 2 , (4)
D = (1/2)ρS(B 0 + B 1 α + B 2 α 2 )v 2 (5)
The constants ρ are S correspond to the air density and the surface of the aircraft.

Dynamics.

Once the wind structure and the areodynamic forces have been set, we can describe the dynamics of the aircraft where the position (x, h), the speed v, the path inclination γ and the angle of attack α are the state variables and the variation of the angel of attack u is the control.

ẋ = v cos γ + W x , (6) ḣ = v sin γ + W h , (7)
v = T /m cos(α + δ) -D/m -g sin γ (8) -Ẇx cos γ -Ẇh sin γ, γ = T /(mv) sin(α + δ) + L/(mv) -g/v cos γ (9) +( Ẇx sin γ -Ẇh cos γ)/v, α = u. ( 10
)
We will require, as in the reference [START_REF] Bulirsch | Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 1: Necessary Conditions[END_REF] that the angle of attack and its variation are bounded in the following fashion:

α ≤ α M , (11) |u| ≤ u M . (12)
The termes for the thrust inclination δ, the gravity constant g, α M and u M are fixed values. Choosing α as the control, we consider the case of a commercial airplane where it is forbidden to change baldly the angle of attack of the aircraft.

2.4.

Cost Function and Problem. The of landing or taking off in the presence of windshear is to crash into the ground, and so in this situations the pilot has to maximize the altitude of the aircraft. In order to analyze this idea we introduce the following cost to be minimized:

(13) J(u) := -θ -ηh(τ ), η ≥ 0,
where time varies between 0 and τ and the value θ verifies:

(14) θ ≤ h(t) p.p t ∈ [0, τ ].
Here the idea of maximization of the altitude is captured by the value θ in the cost. However, if we do not add more terms the problem would have infinite solutions: after having reached the minimum altitude the aircraft does not have any criteria to set the rest of the trajectory. Adding the term h(τ ) corresponding to the final altitude, we restrict ourselves to a much specific class of trajectories. On the other hand, the term η allows the values θ and h(τ ) to have the same order of magnitude.

We fix the values of all the variables at t = 0 and the value of the path inclination and the final time, setting the initial conditions and a condition of stability at the end of the trajectory. This can be represented as follows:

x(0) = x 0 , h(0) = h 0 , v(0) = v 0 < 0, (15) γ(0) = γ 0 , α(0) = α 0 . γ(τ ) = γ τ . ( 16
)
In summary, the optimal control problem to be studied is:

(17) min J(u) subject to ( 6)-( 12), and ( 14)-( 16).

Necessary Optimality Conditions

This problem belongs to the class of optimal control with constraints on both the state and control. For an extensive treatment of optimal control theory reference [START_REF] Bonnans | Lecture Notes in Optimal Control[END_REF] can be consulted. First of all, we verify that the problem that we are analysing is qualified. To check this, we write the problem in a fashion that allows us to assure the constraint qualification.

We note the C ∞ application that shows the solution of the ordinary differential equation problem that corresponds to the dynamics ( 6)-( 12), and to the boundary conditions ( 14) -( 16) like this:

u -→ (. . . , h u , α u , . . .).

With this notation the optimal control problem (17) is equivalent to:

min {F(u, θ) | (u, θ) ∈ W, G(u, θ) ∈ K 1 × K 2 }
Where the sets are:

W = L ∞ [0, τ ] × R, K 1 = {u ∈ L ∞ [0, τ ]; |u| ≤ u M }, K 2 = {(w, v) ∈ C -[0, τ ] × C[0, τ ]; v ≤ α M },
and the functions,

F(u, θ) = -θ -ηh u (τ ), G(u, θ) = (u, θ -h u , α u ).
with this structure we assure that the problem is qualified thanks to Corollary 2.101 in [START_REF] Bonnans | Perturbation Analysis of Optimization Problems[END_REF].

We look then for a trajectory u whose associated state (x, h, v, γ, α) and costate (µ, p) verify:

Hamiltonian inequality:

(18) p α (t)u(t) = min {p α (t)v; |v| ≤ u M } p.p t ∈ [0, τ ]
• Once the aircraft has attained maximum and constant vertical wind force, the control has to be near zero for the trajectory to be stable. • In the region where the windshear loses magnitude, the aircraft should maximize once again the control in order to gain altitude taking advantage of the weak force of the wind. Then when the maximum allowed value of the angle of attack is reached, the control has to be zero. We find that the minimum altitude is 562ft, reached around second 30.65.

Analysis

We verify that the numerical results correspond with the underlying theory. For ease of reference we denote by t 1 < t 2 < t 3 the points where the control u is discontinuous. Concerning the measures of the constraints of the states, we find the following numerical results: 1) µ h = cδ t2 , c > 0, that is, it is a function with zero value except at time t 2 where the minimum altitude is reached, and 2) µ α is zero valued only before t 3 where a possible discontinuity might appear. Its graphic is shown below:

ζ(y) = θ -h, ξ(y) = α -α M .
The costate dynamics show that the possible jumps of the costate satisfy This allows us to conclude that the minimal altitude is just attained in one point, otherwise we would have seen jumps in other points in both p h and µ h . It is clear that µ α is continuous in every point except possibly at t 3 . The following analysis solves this issue.

Lemma 1. In the current set-up, we have [µ α (t 3 )] = 0.

Proof. By (18), and since our is affine in the control we have the following relation between the jumps of the switching function Ξ = p α and u, for t ∈ (0, τ ):

[p α (t)][u(t)] ≤ 0.

However, using equation (26) we get:

[µ α (t 3 )][u(t 3 )] ≥ 0, and since by inspection we have [u(t 3 )] < 0, it must be the case that [µ α (t 3 )] = 0.

In the case of a scalar state constraint and of a Hamiltonian affine with respect to the an m-dimensional control similar results were obtained in [START_REF] Bonnans | Commande Optimale. Article pour l'encyclopédie des techniques de l'ingénieur[END_REF], so the above lemma can be viewed as a partial extension of Lemme 8.4 in the former reference, and a special case of Lemma 4.1 in [START_REF] Maurer | On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly[END_REF]. This analysis has shown that both p α and µ α are continuous in (0, τ ). The observed structure of the previous function, negative-zero-negative-zero, corresponds to the obtained conditions of the control u in order for the Hamiltonian inequality to hold: whenever Ξ is negative, u is maximum-valued and whenever Ξ is zero, u has no restrictions. We can examine, as well, the singular arc studying the switching function. We know that differentiating an even number of times Ξ we can recover an expression for u. Indeed in the first derivative, where D α = ∂D/∂α,

Ξ = -p v ∂f v ∂α -p γ ∂f γ ∂α = -p v - T m sin(α + δ) - D α m -p γ T mv + L α mv
there is no dependence on u. To analyze the second derivative, since u = α, we can differentiate once again to obtain:

Ξ = ∂ Ξ ∂α α + Φ = - p v m T cos(α + δ) + ρSB 2 v 2 u + Φ,
where the terme Φ is independant of u. We can check numerically that the coefficient of the control u has nonzero values, as shown by the following two figures:

α

  Define the fonction of constraints ζ and ξ, where y is the state of the system as follows:

5

 5 Time t sec T cos(α + δ) + ρSB 2 v 2

  Hence, the continuity of costates p h and p α will depend on the continuity of the corresponding measures. This behaviour is indeed verified in p h where we find just a positive jump at time t 2 :

		-[p(t)] = [µ h (t)]∇ζ(y(t)) + [µ α (t)]∇ξ(y(t)).
	This expression yields both
	(26)	[p α (t)] = -[µ α (t)], and [p h (t)] = [µ h (t)].
		-1.4
		-1.6
	h	-1.8
	p
		-2
		-2.2
		-2.4
		0	10	20	30	40
				Time t sec

  The following figure shows the continuous structure of p α :

		•10 11
		0
		-2
	p α	-4
		-6
		-8
		0	10	20	30	40
				Time t

Within the framework of the Master of Applied Mathematics and Information Sciences at École Centrale Paris, from 7/10/2014 to 7/02/2015

CMAP, Centre de Mathématiques Appliquées -École Polytechnique

INRIA Saclay -Île de France COMMANDS

dynamics: Writing p = (p i ) and f = (f i ), where i makes reference to the states and f i is the dynamic of the state i, the following relations must hold:

Complementarity conditions: µ = (µ h , µ α ):

At last, we do not have to specify non-nullity conditions because we have the constraint qualification.

Numerical Solution

We take the problem (17) stated in the last section with the following numerical values:

Constants for the aerodynamic forces

Fixed values at the boundaries

All these values were taken from reference [START_REF] Bulirsch | Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 1: Necessary Conditions[END_REF]. We solve numerically the problem with the open-source software BOCOP [8]. As parameters for the numerical procedure we take 175 discretization points, with an implicit Euler method and an intial value for the control u = u M = 0.035. We find that the control follows the following structure:

• Within the first seconds of the trajectory the aircraft has to maximize the variation of the angle of attack, that is the control. This has to be done in order to compensate the upcoming force of the windshear.


Fabien Perineau

Céline Rosticher

Laurence Rozes

Corinne Chanéac
email: corinne.chaneac@upmc.fr

Clément Sanchez

Doru Constantin

Ivan Dozov

Patrick Davidson
email: patrick.davidson@u-psud.fr

Cyrille Rochas

Hybrid nanocomposites with tunable alignment of the magnetic nanorod filler

ABSTRACT.

For many important applications, the performance of (polymer -anisotropic particle) nanocomposite materials strongly depends on the orientation of the nanoparticles. Using the very peculiar magnetic properties of goethite (α-FeOOH) nanorods, we produced (goethite -PHEMA) nanocomposites in which the alignment direction and the level of orientation of the nanorods could easily be tuned by simply adjusting the intensity of a magnetic field applied during polymerization. Because the particle volume fraction was kept low (1-5.5 vol%), we used the orientational order induced by the field in the isotropic phase rather than the spontaneous orientational order of the nematic phase. At the strongest field values (up to 1.5 Tesla), the particles exhibit almost perfect anti-nematic alignment, as measured by optical birefringence and small-angle X-ray scattering. The results of these two techniques are in remarkably good agreement, validating the use of birefringence measurements for quantifying the degree of orientational order. We also demonstrate that the ordering induced by the field in the isotropic suspension is preserved in the final material after field removal. This work illustrates the interest, for such problems, of considering the field-induced alignment of anisotropic nanoparticles in the isotropic phase, an approach that is effective at low filler content, that avoids the need of controlling the nematic texture, and that allows tuning at will the orientation level of the particles simply by adjusting the field intensity. KEYWORDS: nanocomposites, nanorods, goethite, PHEMA, magnetic field, nematic

1.

Introduction.

Hybrid organic-inorganic functional materials are currently popular for innovative applications in various domains such as transportation, construction, sealant, and textile. 1 The attractiveness of hybrid materials arises from the possibility of combining the properties of each constituent and, in some cases, of even inducing a new property thanks to synergetic effects. In this study, we focus on one class of hybrid materials, namely hybrid Polymer NanoComposites (PNCs), obtained by dispersing inorganic nanoparticles in a polymer matrix. Due to the small particle size, such materials display a huge amount of interfaces provided that a good dispersion is achieved. 2 With nanoparticles, an improvement of the properties of the composite materials is expected for lower filler volume fraction (typically 1-10%) than using traditional micrometer-size filler (15-20%), which is very important since any significant weight reduction would be highly beneficial to the transportation industry for example. 3 Beyond the lower loadings and the quality of nanoparticle dispersion in the polymer matrix, the great challenge in the field of PNCs, in order to improve technological applications, is the ability to control the organization and the distribution of filler to obtain ordered nanocomposites. 4 Indeed, the preferential orientation of anisotropic filler particles is known to induce remarkable optical and mechanical properties. 5 Two main kinds of PNCs, based either on oxide or on clay nanoparticles, are currently being investigated for their specific properties (reinforcement, electric insulation, gas/liquid barrier, flame retardancy). 6 More specifically, in a previous article, we reported on the enhancement of the mechanical properties of a PHEMA (poly(hydroxyethyl methacrylate)) matrix reinforced with goethite (α-FeOOH) nanorods. 7 By combining different characterization techniques and methodologies, this mechanical reinforcement was attributed to strong interactions, at the iron oxide-PHEMA interface, between the polymer chains and the particle hydroxyl groups, combined with a quite homogeneous dispersion of the nanoparticles. 7 Moreover, nanocomposite spin-coated films, filled with goethite nanorods, displayed birefringence, which was explained by a flow-induced alignment, during the deposition process, of the nanorods within the organic glassy matrix. From an applied perspective, such PNCs are very interesting because the nanofillers are both cheap and not toxic. A remaining challenge in this domain is the precise control of the alignment degree of the nanorods in the composites. Two main approaches can be used to address this issue: 2 (i) the organization of the nanoparticles by applying an external field during matrix polymerization and material processing (called "external-in" approach); (ii) exploiting the spontaneous mesophase formation by anisotropic nanoparticles ("internal-out" approach). 7 The first approach is better suited to low filler contents and is very convenient for goethite-based PNCs because goethite nanorods are easily aligned by magnetic or electric fields, even at low concentrations. 8 Goethite nanorods, about 400 nm long and 30 nm wide, can easily be synthesized and suspended in water in order to produce stable colloids. 9 Moreover, depending on concentration, these suspensions spontaneously form various liquid-crystalline phases. 10 Indeed, beyond some concentration threshold that depends on particle dimensions, goethite nanorods self-organize in a nematic phase. 8,[START_REF] De Gennes | The Physics of liquid crystals[END_REF] Within a nematic single domain, all rods spontaneously tend to align in the same direction, in contrast with the more dilute, isotropic phase in which the rods point in random directions.

Furthermore, because of their antiferromagnetic structure, goethite nanorods have outstanding magnetic properties. On the one hand, they bear a remanent magnetization due to uncompensated surface spins but, on the other hand, their magnetic susceptibility is weaker along their main axis than in the perpendicular plane. 8 The remanent magnetization gives rise to a linear contribution in the field intensity to the magnetic energy of the nanorods and this contribution is minimum when the rods are parallel to the field. In contrast, the anisotropy of magnetic susceptibility gives rise to a quadratic contribution in the field intensity and that contribution is minimum when the rods are perpendicular to the field. Therefore, in suspension, goethite rods align parallel to a small magnetic field but they reorient perpendicularly at larger field intensities. The crossover between these two behaviors occurs around 0.35 T, independently of the nanorod concentration and organization. At this precise value of the field intensity, the samples are optically isotropic. Consequently, the nematic phase is easily aligned by a small magnetic field and its alignment axis can be switched from parallel to perpendicular by increasing the field intensity. [START_REF] Lemaire | [END_REF] The magnetic field has also a strong influence on the isotropic phase because of the intrinsic properties of the goethite particles. Indeed, the field induces orientational order in the isotropic phase that becomes birefringent, even at rather small goethite concentrations. 13 Considering the isotropic phase of goethite suspensions instead of the nematic phase allows using smaller filler contents, dispenses with the need of controlling the texture of the samples before applying the field, and provides lower visco-elasticity and response times. In this article, we report on the alignment of goethite nanorods induced within PNCs by a magnetic field, which allowed us to produce hybrid materials whose direction and level of anisotropy is directly controlled by the field intensity.

Experimental section.

Inorganic Nanoparticles. Goethite nanorods were obtained by a dissolution-crystallization process from 2-line ferrihydrite, a poorly-defined highly hydrated phase. 14 16.6g of Fe(NO 3 ) 3 , 9H 2 O were dissolved in 400 ml of distilled water, then the pH of this solution was adjusted to 11 with NaOH (1M) solution. Due to the hydrolysis of iron cations, a brown ferrihydrite precipitate appears. After 20 days of aging, the precipitate became ochre, the characteristic color of goethite nanocrystals. Nanoparticles were washed three times with distilled water, then with a HNO 3 (3M) solution to obtain positively-charged particles. Stable aqueous suspensions of non-aggregated goethite particles were obtained by repeated centrifugations of this mixture and dispersions in water. The final pH of the colloidal suspension was close to 3. At pH 11, corresponding to the synthesis conditions, the surface is poorly charged and the particles are flocculated. After centrifugation and peptization of particles in nitric acid, a stable colloidal suspension is obtained. The pH is then close to 3 and the surface charge density of goethite nanorods is +0.2 C/m 2 . 13 The particle volume fraction (i.e. the ratio of the volume of goethite nanorods to the total sample volume) was φ G = 9.4%. Goethite nanoparticles have a lath-like shape with aspect ratio L/D close to 10, with average length of 300 nm, width of 30 nm, and thickness of 10 nm. For these particle dimensions, the onset of the nematic order is around 7% so that all suspensions with nanorod volume fraction smaller than this value are isotropic.

Polymer matrix. Poly(2-hydroxyethyl methacrylate) (PHEMA) has been chosen as a glassy polymer matrix for its hydrophilic character, its transparency, and its absence of toxicity. In this study, PHEMA is obtained by UV-light polymerization of 2-hydroxyethyl methacrylate (HEMA, Aldrich, > 99%) monomer added to 2 wt% of a photo-initiator (2,2 Dimethoxy-2-phenylacetophenone (DMPA), Aldrich, 99%).

Hybrid nanocomposite precursor suspension. PHEMA-goethite nanocomposites have been prepared as previously described. 7 A goethite aqueous suspension was mixed with various amounts of HEMA (that already contains 2 wt. % DMPA) to obtain several series of samples with volume fraction, φ H , of goethite particles increasing from 0 to 5.58 %, in hybrid composites. Homogeneous stable hybrid suspensions were thus produced and could readily be polymerized under UV-light to form nanocomposites.

Preparation of nanocomposite samples. After vigorous shaking, the suspensions were introduced into flat, 0.05 mm thick, glass capillaries (VitroCom, NJ, USA) that were flame-sealed afterwards.

Then, all samples were polymerized under a UV lamp at 25°C for 30 minutes (λ = 254 nm, power: 30 W). A magnetic field, ranging from 0 T to 1.5 T, delivered by an electromagnet, was applied before and during polymerization. In these conditions, goethite nanorod orientation occurs before the crosslinking of the polymer matrix is completed. Series of samples at fixed goethite concentrations were prepared by applying different magnetic-field intensities (0; 0.18; 0.35; 0.7; 0.9; 1.5 T) during polymerization. The goethite volume fraction φ Η in the nanocomposite is used hereafter because it is a relevant physical parameter that affects orientational order in these materials. The values of goethite volume fractions for all samples are collected in Table 1.

Characterization methods.

The samples were examined with an Olympus BX51 polarizing microscope and their optical textures were photographed using an Olympus (Camedia C-3030) digital camera. Birefringence (∆n) measurements were performed under the microscope using a Berek (Olympus U-CBE) optical compensator. The sample birefringence, ∆n, is an important physical property because, for singledomain samples, it is proportional to the nematic order parameter S: ∆n = ∆n sat φ Η S where ∆n sat is the intrinsic birefringence of the nanorods (See the Supplementary Information for more details on the determination of S from birefringence data). S is defined as ( )

1 cos 3 2 2 1 - = ϕ S
where ϕ is the angle of a rod with the uniaxial symmetry axis n of the phase, defined by the magnetic field, and the brackets mean averaging over all rods. By definition, S takes values ranging between -1/2 and 1, with S = 0 corresponding to the disordered isotropic phase. When the nematic order parameter is positive, 0 < S < 1, the rods point on average along the axis n. Negative values of S are the sign of so-called "antinematic" order 15 where the rods lie on average in the plane perpendicular to n, with isotropic angular in-plane distribution.

Small-angle X-ray scattering (SAXS) experiments were performed at the BM02 experimental station of the European Synchrotron Radiation facility at Grenoble, France. The sample-to-detection distance was 1.62 m and the wavelength was λ = 0.116 nm. The beam size was 0.4×.4 mm 2 . The data were recorded using a CCD Peltier-cooled camera (SCX90-1300, Princeton Instruments) of 1340×1300 pixels. Data processing (dark current subtraction, flat-field correction, grid distortion, and normalization) was performed using homemade software. The accessible range of scattering vector modulus q (q = (4πsinθ)/λ, where 2θ is the scattering angle) was 0.056 < q < 1.13 nm -1 . The nematic order parameter S can also be calculated from the anisotropy of the SAXS patterns by using a classical procedure (See the Supplementary Information for more details on the determination of S from X-ray scattering data) based on a convolution of the rod form factor with the orientational distribution function. 16 Page 7 of 26

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Field Emission Gun Scanning Electron Microscopy (FEG-SEM, Hitachi SU-70, accelerating voltage 1 kV) was used to examine the organization of the nanoparticles in the composites. For this purpose, self-standing, ribbon-shaped, polymerized samples were carefully extracted from the flat glass capillaries. Images of the flat surface and the cross-section of these thin (50 µm thick) samples were recorded by setting either the sample surface or its section perpendicular to the electron beam.

Results and discussion

Homogeneous and transparent nanocomposites were obtained whatever the goethite volume fraction and the magnetic field applied during the polymerization. As previously described in detail, 7 the nanocomposites are comprised of nanoparticles embedded within a continuous polymer matrix. The presence of the iron oxide nanoparticles does not affect the conversion of HEMA in PHEMA (see Section 1 of the Supplementary Information) and the shape of the nanorods is not altered either.

Moreover, the existence of strong interactions between the goethite nanoparticles and the side-groups of the PHEMA macromolecules have been previously evidenced by combining different analytical techniques (FTIR, ellipsometry) with equilibrium swelling experiments. It was thus demonstrated that a fraction of ester functions in the PHEMA chains are hydrolyzed and form carboxylate groups that interact with the surface of the nanoparticles. Such interactions ensure the good dispersion of goethite nanoparticles in the polymer matrix, regardless of their concentration and the magnetic fields applied.

These specific interactions allow anchoring of the polymers to the fillers and avoiding macrosegregation and phase separation at high filler content, like in hybrid SiO 2 -PHEMA matrix 17 .

Consequently, a significant improvement of the mechanical properties of the nanocomposites has been observed [START_REF] De Gennes | The Physics of liquid crystals[END_REF] .

Four samples, with φ Η = 1.08 or 4.31%, without magnetic field or with high magnetic field (1.5T) applied during polymerization, have been observed in FEG-SEM in order to characterize the nanorod organization in the polymer matrix at a sub-micronic scale. Images of the flat surface of the samples In contrast, alignment of the rods is clearly observed in the bulk of the sample, as illustrated by the image of the ribbon cross section (Figure 1c). Therefore, close to the surface of the sample, a continuous variation of the particle orientation, over a depth of about 100 nm, shows the influence of surface effects (Figure 1d). The competition between the rod alignment induced by the surface and that induced by the field in the bulk gives rise to this distorted region, at the surface, the thickness of which should decrease with increasing field intensity. The effect of the magnetic field on the orientation of the particles in the bulk is shown in Figure 2 for lower filler content (φ Η = 1.08%). While the rods are completely disoriented for composites polymerized without field, rod alignment is observed in high field even at such low volume fractions.

In order to probe particle alignment at a more macroscopic scale, we examined the optical textures of a goethite/PHEMA hybrid composite sample (φ Η = 4.31%, B = 0.18 T) observed by polarized-light microscopy (Figure 3). The texture is very uniform and much darker when the sample axis is parallel to either the polarizer or analyzer directions (Fig. 3a) than when it makes an angle of 45° with them (Fig. 3b). This optical anisotropy is the sign of a strong alignment of the goethite nanorods throughout the whole sample, at a millimetric length scale, even though the rod volume fraction is well below that of the nematic phase. Such an observation can be interpreted as follows: even in the isotropic phase, goethite nanorods partially align under the influence of the magnetic field, 13 in agreement with the FEG-SEM observations. This alignment was held by the magnetic field during polymerization of HEMA under UV-light. Finally, when the magnetic field was removed, the alignment of the goethite nanorods remained due to the glassy behavior of the PHEMA matrix at room temperature (Tg=100°C for the neat polymer).

Detailed analysis of the birefringence of the samples revealed that, upon HEMA polymerization, the nanorods were trapped with their main axis aligned parallel to the magnetic field at low field intensities and perpendicular to the field at high field intensities, as expected. Almost no birefringence was detected in the hybrid materials polymerized in a field of ~ 0.35 T. This behavior is indeed similar to that of the isotropic aqueous suspensions of goethite nanorods submitted to a magnetic field. 13 The birefringence of all the samples was measured in order to investigate quantitatively the dependence of the nematic order parameter of the hybrid materials on the magnetic field intensity under which they were polymerized (Figure 4, left). We first observed that the samples polymerized in the absence of field nevertheless displayed a weak birefringence. This is most probably due to residual stresses arising from polymerization of the system in the confined environment of the flat capillary.

Otherwise, the alignment of the nanorods in the composites, removed from the field, closely follows the field-induced alignment in the isotropic aqueous suspensions of nanorods (Figure 4a, inset). In particular, small (≈ 0.1-0.2) positive values of S are measured at low field intensities, S vanishes around 0.35 T, and large (≈ -0.4) negative values of S are measured at high fields. The nematic order parameter, S ≈ -0.4 at high fields is actually very large since the maximum value is -0.5 for this type of "antinematic" orientation. 15 This means that most nanorods are very well aligned in the plane perpendicular to the magnetic field direction.

These conclusions are confirmed by our SAXS measurements that probe nanorod alignment at a local scale but averaged over the illuminated sample volume (~ 1mm 3 ). The SAXS patterns (Figure 5) display a diffuse scattering halo that is most often anisotropic. The scattered intensity regularly decreases with scattering vector modulus, which suggests that the inter-particle interferences are negligible so that the scattering is mostly governed by the nanorod form factor. The scattering is stronger in the direction perpendicular to the magnetic-field direction at low field and in the parallel direction at high field.

Again, this means that the nanorods are aligned parallel to the field below 0.35 T and perpendicular to the field beyond this value.

The nematic order parameter can classically be derived from the SAXS patterns through a fit of azimuthal circular scans (at constant scattering vector modulus) of the scattered intensity, assuming a Maier-Saupe orientational distribution of the nanorod axes. 16 The values of S of the hybrid materials are displayed in Figure 4 (right), as a function of the field intensity applied during polymerization. These values are quite consistent with those obtained from the birefringence measurements. The very good agreement between the SAXS and birefringence data proves that the latter method yields reliable values for the order parameter of the particles. This technique is much more easily implemented than SAXS or TEM and only requires knowledge of the optical properties of the particles (intrinsic birefringence) and of the matrix (refractive index). As such, it can be extended to a wide variety of composites based on anisotropic particles.

The persistence for months of the alignment of goethite nanorods at low volume fractions in the composite materials, after field removal, can be explained by the glassy behavior of the polymer matrix.

Otherwise, the nanorod alignment may have vanished over some time. Moreover, since no influence of temperature was observed, up to ~ 100°C, above the glass transition of PHEMA, the nanorods must strongly interact with the polymer network, as previously inferred. 7,17 This strongly suggests that the glass transition of PHEMA is higher around the goethite particles due to the surface specific interactions at the hybrid interface. These interactions lead to the formation of an interphase that improves the load transfer between the matrix and the fillers. 7 Therefore, the environment of the particles is too stiff to allow any motion.

Compared to more common epoxy/clay composites, goethite/PHEMA composites can be oriented with lower magnetic fields using simple and cheap commercial (NdFeB) permanent magnets that routinely provide about one Tesla. Thus, large areas may be aligned, which is more difficult to achieve with electromagnets. Moreover, epoxy/clay composites could also be aligned using electric fields 5b but this approach that involves direct contact with electrodes is much more invasive than the use of magnetic fields which do not require any contact with the samples. Furthermore, in contrast with clay/polymer nanocomposites aligned by shear flow, where the clay nanosheets are parallel to the flow, two different types of particle orientation, parallel or perpendicular to the field, could be achieved by exploiting the very peculiar magnetic properties of goethite nanorods.

Very importantly, in principle, the use of the isotropic phase rather than the nematic one has many advantages. There is no need to reach the concentration threshold beyond which the nematic phase spontaneously forms. In the case of nanoparticles of moderate aspect ratio such as the goethite nanorods considered here, the volume fractions required may be of the order of 10% whereas large alignment effects were already observed here at volume fractions of typically a few %, well below the nematic threshold. Moreover, the nematic phase always has a spontaneous nematic order parameter of S ~ 0.75-0.85 whereas, in principle, the order parameter induced by the field in the isotropic phase can be tuned at will, by increasing the field intensity, to any value 0 < S < 1 for parallel alignment and 0 > S > -0.5 for perpendicular alignment. In turn, all alignment-dependent properties (such as birefringence) can therefore be easily tuned. In addition, there is no need in this approach to address the recurrent problem of the control of the nematic texture, which requires expensive and delicate anchoring layers and surface treatments. Finally, spatial patterning of the samples can easily be achieved either by simply moving the sample in the field or by propagating a polymerization front while modulating the field intensity, as recently illustrated in the case of clay/polyacrylamide hydrogels submitted to a periodic electric field. 18 Finally, we showed that the degree of particle alignment can be robustly determined using optical birefringence measurement. This latter technique is much easier to implement than the usual small-angle scattering method that is classically employed to characterize orientational order in nanocomposites, provided that the optical indices of the nanoparticles are known and the system is transparent.

ACKNOWLEDGMENT.

These experiments were performed on the BM02 beamline (CRG D2AM) at the European

(

  Figure1a,b), in contact with the flat faces of the glass capillary, reveal a random orientation of the nanorods, without any alignment even when polymerization was performed under high magnetic field.

Figure 1 :

 1 Figure 1: SEM-FEG images of the flat surface of nanocomposite samples with φ H =4.31%, polymerized

Figure 2 :

 2 Figure 2: SEM-FEG images of the cross section of samples with φ Η = 1.08% samples polymerized (a) in

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Images in polarized-light microscopy of a sample with φ Η = 4.31% samples polymerized

Table 1 :

 1 Details of sample composition (contents of goethite and HEMA, corresponding goethite volume fraction φ Η ) (HEMA*: HEMA + 2 wt. % of DMPA)

	Synchrotron Radiation Facility (ESRF), Grenoble, France (beamtime request 2-01-799). This work was
	funded by the French ANR agency (Agence Nationale de la Recherche, Programme P2N, NASTAROD
	and Programme PNANO, MECHYBRID). Nicolas Chemin is kindly acknowledged by the authors for
	helpful discussions.

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