Permanent Magnet Generator Matlab

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Behavior of nanoparticle clouds around a magnetized microsphere under magnetic

And Flow Fields

C. Magnet1, P. Kuzhir1, G. Bossis1, A. Meunier1, S. Nave1, A. Zubarev2, C. Lomenech3 and

Russia

Stress), EA 4228, 28 avenue Valrose, 06108 Nice Cedex 2, France.

Abstract

When a micron-sized magnetizable particle is introduced into a suspension of nanosized magnetic particles, the nanoparticles accumulate around the microparticle and form thick anisotropic clouds extended in the direction of the applied magnetic field. This phenomenon promotes colloidal stabilization of bimodal magnetic suspensions and allows efficient magnetic separation of nanoparticles used in bioanalysis and water purification. In the present work, size and shape of nanoparticle clouds under the simultaneous action of an external uniform magnetic field and the flow have been studied in details. In experiments, dilute suspension of iron oxide nanoclusters (of a mean diameter of 60 nm) was pushed through a thin slit channel with the nickel microspheres (of a mean diameter of 50µm) attached to the channel wall. The behavior of nanocluster clouds was observed in the steady state using an optical microscope. In the presence of strong enough flow, the size of the clouds monotonically decreases with increasing flow speed in both longitudinal and transverse magnetic fields. This is qualitatively explained by enhancement of hydrodynamic forces washing the nanoclusters away from the clouds. In the longitudinal field, the flow induces asymmetry of the front and the back clouds. To explain the flow and the field effects on the clouds, we have developed a simple model based on the balance of the stresses and particle fluxes on the cloud surface. This model, applied to the case of the magnetic field parallel to the flow, captures reasonably well the flow effect on the size and shape of the cloud and reveals that the only dimensionless parameter governing the cloud size is the ratio of hydrodynamic–to–magnetic forces – the Mason number. At strong magnetic interactions considered in the present work (dipolar coupling parameter α ≥2), the Brownian motion seems not to affect the cloud behavior.

I. Introduction

Colloidal mixture of bimodal charged particles may exhibit a haloing phenomenon characterized by formation of thin clouds of small nanoparticles accumulated around bigger micron-sized particles. This phenomenon is attributed to the interplay between electrostatic

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and van der Waals interactions between the particles and ensures colloidal stability of the suspension . However, in such systems, the cloud thickness is only a few nanometers , that allows maintaining a good dispersion state of the suspension only within a narrow range of concentrations of both species.

Much thicker clouds appear in magnetic bimodal suspensions. An external magnetic field magnetizes large micron-sized particles, which attract small superparamagnetic nanoparticles, and the latter form thick anisotropic clouds extended at a distance of a few microparticle diameters in the direction of the applied field. At strong enough magnetic interactions, the ensemble of nanoparticles may undergo a gas-liquid or gas-solid phase transition and condense into highly concentrated domains (clouds) adhered to the microparticle surface . Such a phase transition has been proved to enhance significantly the capture efficiency of nanoparticles by magnetic microparticles. On the other hand, the nanoparticle clouds may completely screen dipole-dipole attraction between two micron-sized particles (with dipole moments oriented along the line connecting their centers) and even result in their effective repulsion. This effect has been explained by the interplay between local field modification due to the cloud formation around a pair of microparticles and the osmotic pressure induced by the nanoparticles .

Such a field-induced haloing accompanied with a condensation phase transition has at least two potential applications. First, it significantly improves colloidal stability of magnetorheological fluids based on bimodal magnetic particles . Second, in the domain of magnetic filtration, it is expected to broaden the size range of captured particles from micron- sized particles to nanoparticles. This could be an important breakthrough for biotechnology and magnetically assisted water purification . Both applications require detailed study of the behavior of nanoparticle clouds around a magnetized microsphere both under flow and in the presence of a magnetic field.

Up to now, theoretical investigations of the magnetic particle capture have been principally motivated by the development of magnetic separation technology. Usually accumulation of magnetic particles around a single magnetized wire or an ordered array of wires was considered. Capture cross-section along with the size and shape of magnetic particle deposits around a magnetized collector were determined. Two distinct approaches were used depending on the size of magnetic particles, or rather on the Péclet number (defined as a ratio of the hydrodynamic – to Brownian forces). For large enough non Brownian particles at large Péclet numbers, the mechanistic approach was employed on the basis of the balance of forces and torques acting on the particles. The capture cross-section was determined via the particle trajectory analysis while the size and the shape of the particle deposits were found from the mechanical equilibrium of the particles on the deposit surface, helpful reviews being given by Gerber and Birss , Svoboda . For smaller Brownian particles at low-to-intermediate Péclet numbers, statistical approach was used on the basis of either the convection-diffusion equation or the Langevin equation of particle motion. The former equation gave concentration profiles of the magnetic particles . The latter equation was integrated at fixed small time steps to obtain stochastic particle trajectories .

Both methods allowed calculation of the capture cross-section as function of the suspension

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speed and magnetic field strength. However, the steady-state size and shape of the nanoparticle clouds were only found in the limits of flow-dominated (infinite Péclet number) and diffusion-dominated (zero Péclet number) regimes [13, 19-21]. Recently, a quite rigorous approach has been proposed by Chen et al. who have considered the dynamic growth of the nanoparticle clouds as a moving boundary problem, with the field and the flow fields computed numerically. However, this model as well as most existing theories, did not take into account interactions between magnetic nanoparticles that might lead to underestimation of the capture efficiency and even to unphysical results like particle concentrations above the limit of the maximum packing fraction. A few attempts [19, 23, 24] to account for interparticle interactions in the problem of magnetic separation were restricted to non- Brownian particles and did not predict condensation phase transition, which is often observed in magnetic colloids .

Experimental investigations of the magnetic separation were mostly focused on visualization of the particle trajectories around a magnetized collector, see review by Gerber and Birss . On the other hand, the size and morphology of particle deposits (or clouds) were scarcely studied. Some results were briefly reported for the limits of diffusion- dominated and flow dominated regimes, for which condensation phase transitions were not observed [15, 30, 31]. Furthermore, the studies of particle deposits were restricted to some limited set of experimental parameters and general relationships in terms of dimensionless numbers were not established. Recently, Ivanov and Pshenichnikov have studied a rapid dynamics of accumulation of ferrofluid nanoparticles around a magnetized collector. The authors claim that the nanoparticles undergo the condensation phase transition around a collector and demonstrate a strong recirculation flows induced by the nanoparticle migration towards the collector. However these studies have been carried out in the absence of the external flow, so the flow effect on the behavior of the condensed magnetic phase is still unknown.

In view of the lack of information on this topic and its practical and fundamental interest, we have performed a detailed experimental study of the steady-state behavior of nanoparticle clouds accumulated on the single spherical microspheres in the presence of an external flow and an external magnetic field either aligned or transverse to the flow. To this purpose, we pushed a dilute suspension of magnetic nanoclusters through a microfluidic slit channel, and visualized nanocluster condensation and formation of dense solid-like clouds around a microsphere rigidly attached to one of the channel walls. Experiments have been done in a wide range of suspension velocities. The cloud size and shape have been analyzed as function of the Mason number defined as a ratio of hydrodynamic-to-magnetic forces. For a better understanding of the Mason number effect on the steady-state cloud behavior, we have developed a theoretical model based on the stress balance and particle flux balance on the cloud surface. In the limit of small filtration speeds, phase equilibrium between a solid- like particle cloud and a surrounding medium has been assumed and the nanoparticle concentration inside the clouds has been estimated from the condition of homogeneity of the chemical potential of nanoparticles.

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The present article is organized as follows. In the next section II, we present experimental techniques. An overview of visualization results is presented in section III. The size and shape of nanoparticle clouds is analyzed in Section IV in comparison with theoretical estimations. Finally the conclusions and perspectives are outlined in Section V.

Ii. Experimental

The experimental cell used for visualization of nanoparticle clouds around a magnetized microsphere is shown in Fig. 1. A dilute aqueous suspension of iron oxide nanoparticles (at volume fraction equal to 0.32%) was pushed through a slit channel by a syringe pump (KD Scientific KDS 100 Series) at imposed flow rates varying from Q=7·10-3 to 0.14 mL/min. This flow rate corresponds to the flow speeds,

, Varying In The

range 1.67·10-4≤v0≤1.79·10-3 m/s, with S being the cross-section area of the channel. These

Re  At The Microsphere Scale That Implies A

laminar flow in this scale. The flow channel was fabricated by squeezing of a silicon joint (GEB Silicone) between a flat Plexiglas substrate and a microscopic glass plate. Before manufacturing of the channel, spherical nickel microparticles (Alfa Aesar, 300 mesh, 99.8%, sieved to obtain the size ranging from 40 to 50 µm) were attached to the glass plate by heating at 700°C in an oven during two hours. Such a treatment did not cause a significant immersion of the microparticles into the glass plate but ensured a strong enough adhesion so that the particles did not move under suspension flux. The channel dimensions in the direction of the flow (length), fluid vorticity (width) and velocity gradient (height) were 60mm x 10mm x 70±5 µm, respectively. The channel height was measured by an optical microscope and its constancy along the channel walls was approximately adjusted by screws squeezing the glass plate to the Plexiglas substrate through the silicon joint. The flow channel was placed in the transmitted light microscope (Carl Zeiss Photomicroscope III) equipped with a camera PixelInk PL-B742U having a complementary metal oxide semiconductor (CMOS) color image sensor. A 20-fold objective (Olympus IC 20) was used for observations. A stationary magnetic field of an intensity H0=12 kA/m was applied by a pair of Helmholtz coils placed around a microscope and bearing iron yokes, as shown in Fig.1. Measurements showed that the magnetic field was homogeneous within a few percent tolerance in the location of the flow channel. The flow channel was put either along or perpendicularly to the coil axis, so, the magnetic field was either parallel to the flow (longitudinal field) or perpendicular to the flow and parallel to the fluid vorticity (transverse field).

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Fig.1. (Color online) Sketch of the experimental setup The following experimental protocol was adapted. Firstly, the suspension of magnetic nanoparticles was introduced into the syringe pump, which was then connected to the flow channel. The latter was filled with the suspension by pushing the syringe at a speed small enough to avoid entrapment of air bubbles. Once the channel was placed in the microscope, an external magnetic field of a chosen intensity was applied and the system was left at rest for ten minutes. During this time, some magnetic nanoparticles were attracted to nickel microparticles and formed clouds extended along the magnetic field direction. Then, the syringe pump was switched on and the suspension was pushed through the channel at a constant imposed flow rate during two hours. During this time, snapshots of the microparticle with nanoparticle clouds accumulated around were taken with a one minute interval, and three videos of a few-minute duration were recorded in the beginning, in the middle and at the end of the observation process. After the flow onset, we observed a rapid partial destruction of the clouds under hydrodynamic forces followed by their reconstruction on a time scale of about one hour. After this time, a quasi-steady state regime was achieved so that the cloud size, shape and morphology did not evolve significantly. At the end of the observation period, the field was switched off, the flow was stopped, the syringe pump was filled with a new portion of the magnetic suspension, and the experience was repeated at another flow rate. To check the reproducibility, all the measurements were conducted two times for the same set of experimental parameters. The steady-state shape of the clouds (at an elapsed time equal to two hours from the flow onset) was analyzed and quantified using ImageJ software. We also checked an eventual difference between the cases when the magnetic field was applied before and after the flow onset. The steady-state size and shape of the clouds were not affected by the sequence of field / flow switching on.

Aqueous solutions of iron oxide nanoparticles (ferrofluids) were synthesized by a coprecipitation of ferrous and ferric salts in an alkali medium and subsequently stabilized by an appropriate amount of oleic acid and sodium oleate using the method described in details by Wooding et al. and Bica et al. . Magnetic nanoparticles were characterized by the

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transmission electron microscopy (TEM), dynamic light scattering (DLS), ζ-potential measurements and vibrating sample magnetometry (VSM). The characterization results are described in details in . Briefly, TEM pictures and DLS measurements reveal that the iron oxide nanoparticles (of a volume mean diameter of 13 nm) were gathered into irregularly shaped nanoclusters of a mean sphericity close to the unity and of a volume mean diameter equal to 62 nm. Aggregation of nanoparticles occurred during the synthesis likely because of an uncontrolled kinetics of the second surfactant layer adsorption. The first surfactant layer (oleic acid deprotonated in alkali medium) was chemically adsorbed by its COO- group on the external surface of iron oxide nanoclusters, and the second layer (sodium oleate) was physically adsorbed onto the first one such that its polar COO- groups pointed outside the nanocluster towards the aqueous solvent. Such a steric double layer, bearing a quite strong negative charge (ζ-potential about -60 mV at a pH=8-9 and ionic strength ranging from 4 to 7 mM), ensured a rather good colloidal stability of synthesized ferrofluids: nanoclusters did not sediment during at least half a year. However, their relatively big size allowed a significant amplification of magnetic interactions and improved substantially their capture efficiency, as nanoclusters and was diluted by a distilled water (milli-Q, 18.2 MΩ·cm) in order to obtain Since the nanocluster behavior is principally governed by magnetic interactions, their magnetization properties are of particular importance. They are inspected in more details in Fig 2 where we plot the magnetization curve of the dry powder of iron oxide nanoclusters.

This curve has a shape reminiscent for Langevin magnetization law. Saturation magnetization and initial magnetic susceptibility (slope at the origin) are found to be equal to MS=290±10 kA/m and χi=9.0±0.5. The latter value allows us to estimate the initial magnetic permeability of the individual nanoclusters, µn≈30 – the value given by the model of multipole interactions between nanoclusters [see Section IV-A, Eq. (4)]. As inferred from the inset of Fig. 2, the magnetization curve of the iron oxide powder is nearly linear in the range of the magnetic field intensities H0=0-12 kA/m, used in our experiments. This allows us to suppose that, within the experimental field range, nanocluster magnetic permeability is independent of the applied magnetic field and equal to µn≈30.

Fig.2. Magnetization curve of the dry powder of iron oxide nanoclusters

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We have also checked by magnetization measurements that nickel microparticles preserved their strong magnetic properties after having been heated in the oven following the protocol similar to the one used for their adhesion on the glass plate. The initial magnetic permeability and saturation magnetization of thermally treated nickel particles is estimated to

Μ ∼

and MS,m≈450 kA/m, respectively.

Iii. Overview Of Observation Results

A sequence of pictures of the nanocluster clouds accumulated around a nickel microparticle in the presence of a magnetic field (of an intensity H0=12 kA/m) longitudinal to the flow is shown in Fig. 3 for the suspension of the solid phase volume fraction φ0=3.2·10-3 for different suspension flow rates Q, corresponding to the superficial velocities

Ranging From 0

to 1.79·10-3 m/s, with S being the flow channel cross-section. As a reference, a bare nickel microparticle in the absence of a magnetic field is shown in Fig. 3a. A picture of the microparticle bearing two nanocluster clouds in the presence of the external field but in the absence of flow is shown in Fig.3b. The applied magnetic field magnetizes the microparticle, and the latter attracts the iron oxide nanoclusters. We were unable to see single nanoclusters because of the optical resolution limit, but observed a change of the suspension optical contrast in the vicinity of the nickel microparticle because of the redistribution of nanocluster concentration. In more details, the nanoclusters accumulate near magnetic poles of the nickel microparticle and are repelled from the equatorial circumference of the microparticle. Such anisotropy of the nanocluster clouds in the absence of flow have been recently observed by Magnet et al. and explained by anisotropy of magnetic interactions favoring attraction within the region where the local magnetic field H is higher than the external field H0 and repulsion within the region where H

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Fig.3. (Color online) Visualization of nanocluster clouds around a nickel microparticle in the longitudinal magnetic field, H0=12 kA/m at the volume fraction of solids in the suspension equal to φ0=0.32%. Snapshot (a) shows a bare microparticle. Snapshot (b) illustrates nanocluster clouds formed in the absence of flow, but in the presence of an external horizontal magnetic field at the elapsed time equal to ten minutes after the field application. Snapshots (c)-(i) show the nanoclister clouds in the presence a magnetic field and in the presence of the flow oriented from the left to the right of the figure, parallel to the magnetic field direction. These snapshots were taken two hours after the moment of the flow onset. The superficial velocity, v0, of the flow is equal to 1.67·10-4 m/s (c), 2.38·10-4 m/s (d), 3.10·10-4 m/s (e), 4.05·10-4 m/s (f), 5.95·10-4 m/s (g), 1.19·10-3 m/s (h) and 1.79·10-3 m/s (i).

Figures 3 c-i show the cloud shape under flow, two hours after the flow onset, when the steady state regime was achieved. The flow is from the left to the right of the pictures in the same direction as the external magnetic field. We see that the flow induces an asymmetry of the clouds. The front cloud (facing toward the arriving suspension flux) appears to be somewhat larger than the back cloud (situating behind the nickel microparticle) and this difference depends on the suspension velocity. First, at low speeds, v0≤3.10·10-4 m/s, the cloud size seems to be almost constant, then it exhibits a step-wise increases at v0=4.05·10-4 m/s followed by a regular monotonic decrease at higher speeds. A relatively small cloud size at small speeds could be explained as follows. The external magnetic field induces a phase separation in the bulk of the magnetic suspension independently of the presence of nickel microparticles. This phase separation is manifested through the formation of the rod-like aggregates composed of magnetic nanoclusters. The aggregates grow rather quickly thanks to shear-induced collisions and quite strong magnetic interactions between nanoclusters. So, they become visible in optical microcope a few minutes after the field application. On the

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other hand, they are subjected to gravitational sedimentation because of the density difference with the aqueous solvent. At low suspension speeds their travel time from the channel inlet to the nickel microparticle appears to be larger than the time required for their settling across the channel thickness h≈70 µm. Because of friction with the channel bottom, the aggregates cannot move once they have been settled. Therefore, the clouds are principally built by the aggregates formed in the vicinity of the microparticle a few moments after the flow onset. At higher speeds, the settling time is longer than the travel time, and the aggregates continuously arrive to the microparticle and form relatively large clouds. Their size and shape achieve steady-state at much longer elapsed times (about one hour) and are defined by the interplay between magnetic and hydrodynamic interactions and, eventually, Brownian motion of nanoclusters, as will be shown in Section IV. With increasing velocity (Figs. 3f-i), hydrodynamic forces become more important, so that a stronger magnetic field is needed to maintain the nanoclusters on the cloud surface. Therefore the part of the cloud situating far from the microparticle is washed away and the cloud surface becomes closer to the microparticle where the magnetic field is high enough to maintain the nanoclusters.

At all suspension velocities, including zero, the nanoclouds are completely opaque. This does not allow us to estimate the nanocluster concentration inside the clouds by the measurements of the transmitted light intensity. Theoretical analysis (see also Section IVA) shows that this concentration is high enough, so that the nanoclusters likely undergo a condensation phase transition at the magnetic field, H0=12 kA/m, used in our experiments.

They form solid-like clouds around a microparticle and a dilute fluid-like phase around the clouds. A diffuse border of the clouds at zero and small speeds, v0<3.10·10-4 m/s [Figs. 3b-e], could be attributed to the polydispersity of the nanocluster suspension. Larger nanoclusters possess a higher magnetic energy and are accumulated in the vicinity of the microparticle forming a dense solid-like phase, while smaller nanoclusters form a diffuse layer around the solid region.

At higher speeds, v0>4.05·10-4 m/s, the diffuse layer seems to disappear and a smooth shape of the cloud is replaced by a sharp pattern with conical spikes on its surface [Figs. 3f-i]. Similar spikes have been observed in the vicinity of the magnetic poles of concentrated ferrofluid micro-droplets formed in the bulk ferrofluid because of the phase separation [35, 36]. Such a surface instability has been explained in terms of the surface energy anisotropy that favors some surface directions over others. Simulations, assuming arrangement of magnetic particles in body-centered-tetragonal (BCT) lattice, have revealed negative surface energies when the angle, δ, between the surface and the field direction becomes larger than 31deg . The flat surfaces with δ>31 deg are therefore absolutely unstable, while appearance of spikes with apex angles, δ<31 deg is energetically favorable. More recently, Cebers has carried out rigorous numerical simulations of the kinetics of the magnetic colloid phase transition and found a multi-spike shape of the droplets of the concentrated colloid phase attributing it to the surface tension anisotropy.

Formation of spikes is expected at any applied magnetic field strong enough to induce a solid-fluid phase separation. However, in our case, it is not observed in the absence of flow, neither at low speeds [Figs. 3b-e], provided that the flow should not affect significantly the

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balance of the surface stresses, according to the estimations presented in Section IV B. The absence of peaks is likely connected to the diffuse boundary layer, which probably destroys the source of the surface instability – the negative surface energy. On the contrary, disappearance of the diffuse layer at speeds v0≥4.05·10-4 m/s leads to a sharp interface of the clouds with an appropriate surface energy and may induce instabilities. Disappearance of this layer likely comes from hydrodynamic forces, which squeeze small nanoclusters to the solid- like phase of the clouds or wash them away from clouds. Finally note that the above considered surface instability appears in the case when the surface energy is principally governed by the magnetic interactions between the particles belonging to the surface layer – the case of the interface between two different phases of the same magnetic suspension. This instability should not be confused with the Rosensweig instability of the interface between two distinct magnetic suspensions (ferrofluids) subjected to an orthogonal magnetic field and whose surface energy is governed by molecular interactions and considered to be field-independent.

A video of the flow around the microparticle with attached nanocluster clouds corresponding to the picture shown in Fig. 3i (at H0=12 kA/m and v0=1.79·10-3 m/s) is presented in Supplemental material . As inferred from this video, the water flux arriving on the front cloud makes the spikes moving along the surface in the direction of the streamlines. The rear part of the back cloud has a tapered shape favorable to the flow. Such a shape likely induces only minor perturbations of the flow so that the spikes on the rear cloud seem to be quasi-immobile. We also observe some recirculation of nanocluster aggregates in the vicinity of the points where the cloud surface joins the microparticle. This recirculation is more pronounced on the lower side likely because of an imperfect alignment between the field and the flow.

Another point revealed in visualization experiments is appearance of bright white regions near the equatorial circumference of the microparticle. These regions correspond to aggregates of micelles of non-adsorbed oleic acid. As already stated, strongly magnetic nanoclusters are expelled from the equatorial region while nonmagnetic oleic acid aggregates are forced to move there because of the volume conservation of the whole suspension. The quantity of the captured oleic acid decreases with increasing flow speed, because hydrodynamic forces become more important as compared to the effective attraction.

We should also mention that the accumulation of nanocluster clouds around a magnetized microparticle is completely reversible process: once the magnetic field is switched off, the cloud is completely dissolved by Brownian motion and by the water flux streaming the microparticle. Destruction of the clouds after switching off of the magnetic field, of an intensity H0=12 kA/m, is demonstrated in the second video posted in Supplemental Material . The reversibility of the cloud formation / dissociation could be explained by the absence of remnant magnetization of the nanoclusters [Fig. 2] and by a presumably low solid friction between the nanoclusters covered by a surfactant double layer.

A sequence of pictures showing a steady-state shape and size of nanocluster clouds in the presence of a magnetic field perpendicular to the flow is shown in Fig. 4 for H0=12 kA/m,

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the suspension volume fraction, φ0=0.32%, and flow velocities ranging from v0=1.67·10-4 to 2.38·10-3 m/s. The shape of the clouds appears to be quite similar to the one in the longitudinal field. The clouds are extended along the applied magnetic field and conical spikes appear on their extremities because of the surface energy anisotropy. Both clouds attached to the microparticle have the same size and shape because of the symmetry of the streamlines with respect to the plain perpendicular to the applied magnetic field and passing through the microparticle center. The clouds seem however to be slightly asymmetric with respect to the microparticle axis aligned with the field. Such an asymmetry is caused by the hydrodynamic drag pushing the clouds in the direction of the flow. As inferred from Fig. 4, the cloud size decreases progressively with an increasing flow speed, v0, that is explained in terms of increasing hydrodynamic forces washing the nanoclusters away from the clouds.

Fig.4. (Color online) Visualization of the nanocluster clouds in the transverse magnetic field H0=12 kA/m at different flow speeds v0, equal to 1.67·10-4 m/s (b), 2.38·10-4 m/s (c), 5.95·10-4 m/s (d), 1.19·10-3 m/s (e) and 2.38·10-3 m/s. The snapshot (a) shows a bare nickel microparticle.

The mechanisms defining the cloud size and shape will be inspected in the next Section IV where experimental results will be compared with predictions of our model.

Iv. Theory And Discussion

The field-induced condensation phase transition is a distinguishing feature of our system having a strong impact on the nanocluster accumulation around a magnetized microparticle. Therefore, we begin with a thermodynamic description of this phase transition in the absence of flow (Sec. IV A). In the presence of flow, the cloud behavior, size and shape depend on Brownian motion, magnetic and hydrodynamic forces acting on nanoclusters. Simultaneous consideration of these three effects along with the condensation phase transition would substantially complicate the theoretical description. Fortunately, the thermodynamics governing the phase transition and the hydrodynamics defining the cloud size can be decoupled for relatively low suspension speeds, considered in our experiments. The validity of such decoupling is proved in Sec. IV B where we estimate the relative importance of hydrodynamic and magnetic forces, or rather their ratio, called Mason number. Based on this estimation, we calculate the cloud shape (Sec. IV C) and size (Sec. IV D) under the field and the flow in the steady-state regime. We consider the case of the field parallel to the flow.

Finally, we compare the calculated cloud size to the one observed in experiments (Sec. IV E).

A. Phase Transition

The appropriate parameter describing relative importance of magnetic interactions is the so- called dipolar coupling parameter. It is defined as the ratio of magnetic-to-thermal energy (kT)

(1)

where µ0=4π·10-7 H/m is the magnetic permeability of vacuum, Vn is the nanocluster volume,

+

is the magnetic contract factor of the nanocluster and µn is the

2 In The Last Equation Comes From The Energy

of dipole-dipole interaction between magnetic nanoclusters proportional to the square of their magnetic moment. The dipolar coupling parameter α is estimated to be of the order of 2 for the experimental value of the magnetic field intensity H0=12 kA/m. However, such relatively modest value of this parameter appears to be sufficient to induce a phase separation in the suspension of magnetic nanoclusters of the magnetic permeability as high as µn≈30. Since the magnetic field and the nanocluster concentration are not homogeneous around the microparticle, we should check the phase behavior of the ensemble of nanoclusters at different concentrations and applied magnetic fields. To this purpose, we shall construct a phase diagram, α - Φ, where different phases will be identified. A similar phase diagram has already been developed via Monte-Carlo simulations or analytical calculations for the magnetic particles exhibiting dipole-dipole interactions [42, 43]. In our case of magnetic nanoclusters with a strong magnetic permeability, µn≈30, we should take into account short- range multipolar interactions, which are especially important at moderate-to-high

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concentrations, since the dipolar interactions strongly underestimate the strength of the interactions between particles . To proceed with, we assume that all the nanoclusters are identical and only two phases of the nanocluster ensemble may exist: a disordered fluid and an ordered solid having a face- centered cubic (FCC) structure. Even though the body-centered tetragonal (BCT) lattice has the least energy in the presence of reasonably high magnetic fields , our choice for the FCC lattice is motivated by the desire to capture the disorder-order phase transition at zero field keeping in mind that that the energy of both structures differs insignificantly. Neglect of other possible ordered states should not cause substantial errors in determination of the nanocluster concentration profile around a magnetized microparticle. The equilibrium between the two considered phases is found by the equilibrium of nanocluster chemical potentials, ζ, and osmotic pressures, p, in each phase :

(2B)

where Φ is the nanocluster concentration in the suspension and the subscripts “s” and “f” stand for the solid and fluid phases, respectively. Taking into account the porous nature of the nanoclusters, their concentration is related to the “true” volume fraction of solids, φ, by the

Being The Internal Volume Fraction Of The

nanoclusters. The chemical potential ζ(Φ, α) and the osmotic pressure p(Φ, α), both contain the contributions coming from magnetic interactions (considered in details in ) and hard-sphere repulsion. The appropriate expressions for the later interaction have been developed by Zubarev and Iskakova on the basis of osmotic compressibility calculations carried out by Carnahan and Starling and Hall for semi-dilute and concentrated hard-sphere suspensions. Assuming that the steric interactions between our nanoclusters respect the Carnahan-Starling law in the fluid phase and the Hall law in the solid phase, we arrive to the following expressions for quantities ζ and p in both phases:

≈

is the maximum packing fraction of the FCC structure, A≈2.2 and C≈1.255 are the constants ensuring order-disorder phase transition at zero field in the concentration range 0.495<Φ<0.545 . The magnetic permeability of the nanocluster suspension, µ, intervening into the last equations is found assuming multipolar interactions between magnetic nanoclusters arranged in the FCC lattice. The interparticle distance in the three directions of the lattice is imposed by the suspension volume fraction. Performing numerical simulations using the numerical code developed by Clercx and Bossis , we obtain the following interpolating function for the magnetic permeability as function of the nanocluster concentration, Φ, and nanocluster magnetic permeability, µn:

−

Φ is the dilute-limit expression of the suspension magnetic permeability given by the Maxwell-Garnett mean field approach , c1≈0.408 and c2≈0.12 are numerical constants.

The magnetic permeability of the nanoclusters, µn, can be found from the experimental magnetization curve of the dry powder of nanoclusters [Fig.2]. The slope at the origin of this curve gives the initial magnetic susceptibility of the powder, χp≈9, which corresponds to the magnetic permeability µp=χp+1≈10. Assuming that the concentration dependency of the powder magnetic permeability is similar to that of the liquid suspension, i.e. defined by Eq.(4), we solve this equation with respect to µn having replaced µ(Φ, µn) by µp≈10 and Φ by Φm≈0.74. In this way, we obtain an estimation for the nanocluster magnetic permeability, µn≈30, at relatively low magnetic fields, H≤12 kA/m, considered in the present work.

Having defined all the terms intervening into Eqs. (2a), (2b), we obtain a system of two transcendental equations, which is solved with respect to Φs and Φf for given values of the dipolar coupling parameter α. So obtained functions, Φs(α) and Φf(α) correspond to the binodal curves of the α-Φ diagram plotted in Fig. 5. 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. As expected, at zero applied field (α=0) we recover the order-disorder transition in the well known concentration range, 0.495<Φ<0.545 . At the field parameter α >2, the solid phase exist only in a narrow range of concentrations, whose values are very close to the maximum packing fraction of the FCC lattice, Φm≈0.74, at least in thermodynamic equilibrium, i.e. in the absence of flow and at a long elapsed time after the moment of the field application.

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Fig.5. Phase diagram of the suspension of magnetic nanoclusters having a constant magnetic permeability equal to µn=30.

B. Mason Number

In what follows, we shall demonstrate that the cloud internal structure should not change drastically in the presence of flow at the flow speeds used in our experiments. To this purpose, we estimate the characteristic ratio σh/σm of hydrodynamic to magnetic stresses acting inside the solid-like cloud. These both stresses scale as

∼

. Here η0 and ηi are the viscosities of the suspending liquid (water) and of the particle suspension inside the cloud, v0 and vi are the suspension velocities far upstream from the microparticle and inside the cloud, respectively, rm is the microparticle radius. It is easy to

Η

of the suspension viscosity by its characteristic speed inside the clouds appears to be of the same order of magnitude as

If One Assumes A Recirculation

flow inside the clouds in the limit of high particle concentration (

(5)

Estimations show that this ratio is quite low in our experimental conditions:

<

<

. Small values of σh/σm (responsible for the internal flow and for the cloud surface behavior) allow us to conclude that the flow should have a minor effect on the cloud internal structure and on the shape of its surface. This will make possible to determine the internal volume fraction and the shape of the clouds in the limit of zero Mason number, in a similar way as in the absence of flow.

On the other hand, the cloud volume is governed by the flux of nanoclusters arriving on the clouds and leaving the clouds. The key parameter is the ratio of the convective to the magnetic migration flux, or rather the ratio of the hydrodynamic-to-magnetic forces acting on the nanoclusters in the vicinity of the cloud surface, so called Mason number:

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

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