Magnetic soap films and magnetic soap foams

Transcription

1 Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) Magnetic soap films and magnetic soap foams Florence Elias a,b,, Jean-Claude Bacri a,b, Cyrille Flament a,b, Eric Janiaud a,b, Delphine Talbot c, Wiebke Drenckhan d, Stefan Hutzler d, Denis Weaire d a Laboratoire des Milieux Désordonnés et Hétérogènes Université Paris 6, 140 rue de Lourmel, Paris, France b Matière et Systèmes Complexes UMR 7057, Université Paris 7, Paris, France c Laboratoire des Liquides Ioniques et des Interfaces Chargées Université Paris 6, 4 place Jussieu, Paris Cedex 05, France d Physics Department, Trinity College Dublin, Dublin 2, Ireland Received 27 October 2004; accepted 28 January 2005 Available online 2 March 2005 Abstract We investigate the physical properties of soap films and soap foams, the liquid phase of which contains a stable colloidal suspension of magnetic particles. The physical properties of such systems result from the equilibrium between the capillary forces, the gravity and the local magnetic forces. Therefore, they depend on the strength of an applied magnetic field or force, which makes it possible to act from outside on the system. We present the effect of the external field on the magnetic soap foams and films through two studies. First, we show that the dynamics of drainage in a freely suspended magnetic soap film is modified by an external magnetic field. Second, we present the control, via an external magnetic force, of the structure adopted by a monodisperse foam in a cylindrical tube. Those studies open possible applications of the ferrofluid foam, in the field of technological applications, but also as a model experimental system to study the drainage, the dynamical rheology, and the stability of soap foams Elsevier B.V. All rights reserved. Keywords: Ferrofluid; Magnetic interactions; Foam control; Soap film; Drainage 1. Introduction The physical properties of soap foams and soap films result from the interplay between gravity forces, local capillary forces and viscous dissipative forces [1]. Those forces depend on physical quantities, none of which is easy to vary in an experiment in order to study the dependence on the structure, rheology and dynamics of foams with respect to those quantities. For instance, capillary and viscous forces may be changed by using different soap solutions [2]. Gravity forces can be varied by using emulsions and adapting the relative density of the two liquids [3], or by modulating the local gravity field in parabolic flights [4]. However, it would be of great interest to have a tuneable control parameter which could modify the physical properties of foams in a single experiment at the scale of the laboratory. This Corresponding author. Tel.: ; fax: address: elias@ccr.jussieu.fr (F. Elias). may be achieved by replacing the continuous liquid phase of the foam by a magnetic liquid, or ferrofluid (FF). Therefore, the liquid phase of the foam acquires macroscopic magnetic properties, and the foam physical properties can be adapted by applying an external magnetic field, whose amplitude can be easily controlled. The properties (structure, dynamics) of such a magnetic cellular structure have already been studied in two dimensions (2D) as a function of an external applied magnetic field [5,6]. Such a 2D magnetic structure has been used to study the elastic properties of 2D foams, thanks to the possibility of manipulating the foam using small magnets [7]. In this article, we present a new experimental system: a threedimensional (3D) magnetic foam, made from a soap solution containing a FF. We show that the foam properties are modified by an external magnetic field via two kind of effects. First, a homogeneous applied field changes the dynamics of drainage in a freely suspended soap film. This effect is due to the magnetic dipole interaction within the FF, which depends on the relative orientation of the applied field with respect to /$ see front matter 2005 Elsevier B.V. All rights reserved. doi: /j.colsurfa

2 66 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) the film. Second, a gradient of an external magnetic field induces a volume magnetic force, which can compensate or amplify the gravity force. We investigate this effect in a magnetic soap foam confined in a tube. The use of local gradients of magnetic field then allows to control and manipulate the structure adopted by the foam. The article is organised as follows. Having presented the FF and the magnetic properties of the solution (Section 2), the effect of a homogeneous magnetic field on the dynamics of drainage of liquid in a magnetic soap film is presented in Section 3. Section 4 deals with the control and manipulation of the structure formed by a magnetic soap foam confined in a tube, using external magnetic fields. The implications of those studies are discussed in Section 5: in the domain of fundamental physics, magnetic soap foams may be used as a model experimental system for studying the physical properties of soap foams; in the technological domain, the possibility of manipulating the foam could have interesting applications for the realisation of microfluidics devices. 2. Ferrofluid and magnetic soap solution The continuous liquid phase of a magnetic soap foam is a solution made of a water-based FF and a surfactant. A FF is a stable colloidal dispersion of magnetic particles in a solvent. In our case it is a dispersion of maghemite ( -Fe 2 O 3 ) particles in water (average particle diameter: 10 nm), stabilised by the presence of a negative electrostatic charge at the surface of each particle [8]. Two solutions are used in the experiments, with two different concentrations of magnetic particle (respectively, 1.92% and 12.6% in volume). In order to achieve foamability, two percent (by weight) of sodium dodecyl sulfate (SDS) are added as surfactant. Since SDS is an ionic surfactant, negatively charged, the magnetic particles are repelled from the film interfaces, which avoids particle agglomeration at the surface. A sketch of the side view of a magnetic soap film is presented in Fig. 1. Each magnetic particle in the FF carries a permanent magnetic momentum, conferring macroscopic magnetic properties to the solution. In the presence of an external magnetic field, thermal agitation competes with the magnetic energy, which tends to align the particles in the same direction as the external field [9]. The resulting macroscopic magnetic behaviour of the solution is of paramagnetic type, the magnetisation M(H) of the FF having the same direction as the applied field H: M(H) increases linearly with the external field for low field, and saturates for high applied magnetic field, since all the magnetic dipoles are aligned. Magnetic forces can be induced in the FF, which will compete with the other forces in presence, and change the physical properties of the system in the presence of an applied magnetic field. Those magnetic forces are of two kinds. First, a volume magnetic force is generated by a gradient of the applied magnetic field (Kelvin force) [9]: F m = µ 0 V FF (M )H, where the notations are given in Appendix A. Second, Fig. 1. Sketch of the side view of a magnetic soap film. SDS molecules are present at the two interfaces of the film. Inside the film, the solution is made of a colloidal suspension of magnetic particles in water. The magnetic particles have an average diameter of 10 nm and carry a permanent magnetic moment. They are stabilised in the solvent by the means of a negative electrostatic surface charge. Positive counter-ions are present in the solution and maintain the electric neutrality of the solution. magnetic dipoles interact, once they are aligned by an external magnetic field. The magnetic dipole interaction is repulsive or attractive depending on the relative position of the magnetic dipoles with respect to their orientation: the particles repel if their relative position is perpendicular to the magnetic momentums, and attract if their relative position is parallel to the magnetic momentums. This magnetic dipole interaction is present even if the applied magnetic field is homogeneous, and generates macroscopic effects which depend on the geometry of the system. Finally, let us notice that no magneto-rheological effect is observed in the solutions used in this article. Magnetorheological effects are expected when the magnetic field is perpendicular to the vorticity, since the alignment of magnetic momentums along the field direction prevents the rotation of the particles in the shear [10]. The viscosity of the FF has been measured in a rheometer in a cone plane geometry, for different values of an external magnetic field applied in the direction perpendicular to the axis of the rheometer. Since the vorticity field is radial, the magnetic field is perpendicular to the vorticity in a plane perpendicular to the field and containing the axis of the rheometer, therefore, such a configuration is suitable for the study of magneto-rheological effects. Magneto-rheological effects increase with the concentration of magnetic particles and with the amplitude of the magnetic field. The measurement was performed on the concentrated magnetic solution, and the amplitude of the applied magnetic field was set between 0 and 450 G, corresponding to the typical field amplitude used in the experiments. No change in the viscosity has been observed as a function of the magnetic field. The linearity of the rheological response (strain rate versus applied stress) of the FF has been checked for

3 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) a shear rate ranging between 200 and 1300 s 1, i.e. 4 5 orders of magnitude larger that the typical shear rates involved in the flows studied in Section 3. Since non-linear rheological effects increase with the applied stress or shear rate, we can conclude that the magnetic solutions used in this article have a linear Newtonian rheological behaviour, without any magneto-rheological effect, in the range of magnetic fields and shear rate investigated. 3. Magnetic soap films: drainage under a magnetic field 3.1. Introduction In conventional soap films, drainage is due to the combined effects of gravity and capillary forces, and plays an important role in the stability of soap foams [11]. In this study, we show that if colloidal particles are added to the solution, the dynamics of drainage of a magnetic soap film is controlled by an external homogeneous magnetic field [12] Setup The solution used here is made with the concentrated FF (12.6% of magnetic particles in volume) and SDS. Glycerol (10% in volume) is added in the solution. The experimental setup is presented in Fig. 2. A freely suspended vertical soap film is created by pulling a rectangular frame out of the solution at a constant velocity. The bottom of the film is in contact with the solution. The box containing the film and the solution is sealed in order to prevent water from evaporating. An image of the soap film is presented on Fig. 3. The top of the film is occupied by the so-called black film. In this region, the film has reached its equilibrium thickness, of the order of 10 nm, and appears optically black. Below the black film, a light interference pattern is visible. In this region, the film Fig. 3. Vertical soap film freely suspended on a rectangular frame. The frame is visible at the top and the side edges of the film. The soap film is illuminated using a monochromatic light at normal incidence. thickness is of the order of a micrometer, and decreases while the liquid drains out. The soap film presented in Fig. 3 is illuminated using a monochromatic light at normal incidence. The interference pattern consists of fringes perpendicular to the thickness gradient and localised on the soap film. The dark fringes correspond to a film thickness e = pnλ, where p is an integer, n is the optical index of the solution and λ is the wavelength of light. In order to determine the order of interference p, short spots of white light are regularly used during the drainage until a silver-coloured fringe is observed, which is characteristic of the first interference order in white light illumination. At the bottom of the soap film, a meniscus links the film to the solution. The meniscus appears black on Fig. 3, which is the colour of the bulk solution, due to the absorption of light by the magnetic particles. The trough containing the soap film and the solution is placed between two magnetic coils, which create a homogeneous magnetic field. The amplitude of the magnetic field can be varied between 0 and 270 G. Two configurations for the external homogeneous magnetic field are used: the field is either applied vertically, parallel to the soap film H O z, or horizontally perpendicular to the film H O x. Drainage in the soap film is investigated through two studies: the dynamics of thinning of the region of the film of micrometric thickness, that we call the coloured film (Section 3.3), and the growth dynamics of the black film (Section 3.4) Drainage in the coloured film Fig. 2. Experimental setup and notations. The magnetic soap film is formed by pulling a copper frame (wire diameter mm; height 10 mm; width 16 mm) out of the solution at a constant velocity in a sealed glass box. The magnetic soap film is then submitted to an external homogeneous magnetic field, either parallel to the gravity (H O z ), or perpendicular to the film (H O x ). The dynamics of thinning of the coloured film is investigated by following in time the position of an interference fringe of constant thickness, which moves downwards when the film becomes thinner. Fig. 4 shows the time-evolution of the position of a fringe of constant thickness, for different amplitudes and orientations of the applied magnetic field. The velocity of the fringe depends on the external magnetic field: the drainage slows down under the effect of a parallel

4 68 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) Fig. 4. Time-evolution of the position of an interference fringe of order p =3, for different values of the intensity of the applied magnetic field and two orientations of the field: H O x (H perpendicular) and H O z.(h parallel). vertical magnetic field (H O z ), whereas it speeds up under the effect of a perpendicular applied field (H O x ). A simple model allows to reproduce the experimental data, taking into account the effect of the external magnetic field. The idea is the following. Since the applied magnetic field is homogeneous, no volume magnetic force is applied to the magnetic liquid: the dynamics of drainage depends on the external field via the magnetic dipole interaction. Macroscopically, the magnetic dipole interaction is repulsive when the magnetic momentums are perpendicular to an interface. In the case H O z, the magnetic dipole interaction energy is higher in the meniscus than in the film, therefore, the liquid tends to flow from the meniscus to the film, and the drainage slows down. On the contrary, in the case H O x, the magnetic dipole interaction energy is higher in the film than in the meniscus, thus the drainage speeds up. We present below the calculation based on this idea. Since the SDS surfactant is known to produce mobile interfaces [2], the flow of liquid is assumed to be a Poiseuille flow with sliding interfaces: the velocity profile is parabolic and goes to zero in the O x direction over a distance b (sliding length) larger than the film thickness e. In the case of low Reynolds number, quasi-stationary flow, in the lubrication approximation, and in the limit b e, the thinning rate of the film thickness writes [13]: e t = ( ) z P z 2η e2 b, (1) where z P is the vertical pressure gradient. z P contains three contributions: the hydrostatic, the capillary and the magnetic pressure gradient: z P = ρg + z P cap + z P mag. (2) The gradient of capillary pressure is due to the difference of curvature between the meniscus and the film. Far from the meniscus (z l, where l is the total height of the soap film), it can be written as: z P cap 2 Rl, (3) where R is the radius of curvature of the meniscus. The magnetic pressure gradient can be derived as follows. Macroscopically, the magnetic dipole interaction is taken into account by the demagnetising factor D, which depends strongly on the shape of the FF volume [14]. D varies between 0 and 1, and is an increasing function of the strength of the magnetic dipole repulsion. The magnetic dipole interaction energy can be written as: E (µ 0 /2) V FF DM 2. The pressure due to magnetic dipole interaction is the derivative of E with respect to V FF, assuming for simplicity than D does not depend on V FF : P mag (µ 0 /2) DM 2. Far from the meniscus (z l), the gradient of magnetic pressure is then: z P mag µ 0 DM 2, (4) 2l where D = D(z) D(l) is the difference of the demagnetising factor between the film and the meniscus. Approximating the film by an infinite plane, D(z) = 0 in the case H O z, and D(z) = 1 in the case H O x [14]. Approximating the meniscus by an infinite cylinder, D(l) = 0.5 for both orientations of the applied field. Therefore, D ε 2, where ε = { +1 if H Ox 1 if H O z, (5) Using Eqs. (1) (5) the position z(t) of a fringe of constant thickness e=pnλ can be computed. The velocity of the fringe of order p is then found to be constant, equal to: v p (M) = z t = pv 1(M), (6a) with v 1 (M) = v 0 [1 + εα(m)], v 0 = bnλ η ( ρg + 2γ ) Rl and α = µ 0 ( 4 ρgl + 2γ R )M 2. (6b) (6c) The fringe velocity depends on the applied magnetic field through the magnetisation M. The experimental data presented in Fig. 4 show that the velocity of the fringe of order 3 is roughly constant in time for small z. The experimental data is then analysed as follows. First, the velocity of all fringes is obtained by a linear fit of the data z(t) for z < 6 mm. v p (M) is then plotted versus p and fitted using Eq. (6a) for all values of M in order to obtain v 1 (M). The velocity v 0 and the parameter α are then deduced from Eq. (6b) for all values of M. One finds v 0 = (27 ± 2) ms 1. The radius of curvature of the meniscus is of the order of the capillary length: R γ ρg [13]. Using the values given in Appendix A for the parameters of the system, the

5 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) Fig. 7. Creation and upwards motion of a drop of black film inside the meniscus. Fig. 5. Parameter α, defined in Eq. (6b), as a function of the square of the FF magnetisation. The straight line represents a theoretical plot using Eq. (6c), with no fitting parameter. sliding length can be evaluated using Eq. (6c): b 40 m. This is the same order of magnitude as the value of the sliding length found in the literature [13] for a freely suspended soap film made of a water and SDS solution. The parameter α is then plotted versus M 2 in Fig. 5 for both configurations of the external magnetic field. For comparison, a line corresponding to Eq. (6c) is shown on the same graph, with no fitting parameter. The model reproduces very accurately the experimental data Growth dynamics of the black film We have investigated the growth dynamics of the black film at the top of the soap film, and the modification of this dynamics under the effect of an applied magnetic field. A plot of the time-evolution of the height of the black film is shown in Fig. 6. An important effect of the magnetic field is obtained when the field is applied in the direction perpendicular to the plane of the film (H O x ). Two regimes of growth are observed in this case. In the first tens of seconds, h(t) follows the same evolution than when no external field is applied. Then, a change of slope is observed: in the second regime, the growth velocity of the black film is higher under the effect of a perpendicular magnetic field. The growth velocity (i.e. the slope of the plot represented in Fig. 6) seems to be independent of the amplitude of the field. However the time at which the change of regime is observed depends on the field intensity: the higher the magnetic field, the sooner the change of regime. A direct observation shows that under the effect of a perpendicular magnetic field, large 2D drops of black film appear in the meniscus linking the film to the solution. Once created, these drops move up in the film towards the black film region, as shown in Fig. 7. The time of nucleation of the first drop of black film in the meniscus coincides with the time at which the change of regime is observed in Fig. 6: the increase of the growth rate of the black film under the effect of a perpendicular magnetic field is then due to the creation of large drops of black film in the meniscus. A possible explanation of this effect and its consequences is discussed in Section Magnetic soap foam in a tube: control of the structure using magnetic fields 4.1. Introduction Fig. 6. Time-evolution of the height of the black film, under the effect of a perpendicular magnetic field: H O x. Soap bubbles of equal size confined in a tube form ordered structures [1,15], as shown in Fig. 8. They consist of a tiling of gas bubbles, separated by liquid boundaries. The structures belong to a sequence of spiral arrangements, with, apart from the simplest structure, only hexagonal cells at the surface. They are indexed with the standard notation of phyllotaxis [16]. These structures offer promising applications in the transport through channels of small quantities of fluid enclosed in the foam cells. We have generated such foams confined in a cylindrical tube, using our magnetic liquid soap

6 70 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) Fig. 8. (Coming from Ref. [17].) Ordered structures of monodisperse FF foam. They consist of gas bubbles separated by liquid boundaries that appear black on the images. The structure adopted depends on the ratio Λ of the tube diameter to bubble diameter, but more than one structure may be stable for a given Λ. The tube diameter is 7.5 mm. (a) Bamboo or structure with Λ = 0.83; (b) structure, Λ = 0.83; (c) structure, Λ = 0.59; (d) structure; Λ = solution [17,18]. The foam can then be manipulated by external magnetic forces. We present here a review of the properties of those FF foams under the effect of external magnetic fields Experimental setup The experimental setup is represented in Fig. 9. The FF and SDS solution (1.92% of magnetic particles in volume) is injected into a vertical, cylindrical Plexiglas tube with an inner diameter of 7 mm. A monodisperse foam is produced by blowing air through a nozzle into the solution at a constant flow rate of 10 ml/min, using a micro pump. The foam is observed using video imaging. The bubble volume can be derived from the number of unit cells filling a chosen section of the tube. In the lower part of Fig. 9(a), the bubble size is controlled during the bubble formation using an external magnetic field (see Section 4.3). A vertical gradient of a horizontal magnetic field is created around the tip of the nozzle by placing it at the appropriate position between two magnetic coils fitted with soft iron cores of height 2L = 30 mm. The coils are supplied with a direct current I and generate a horizontal magnetic field. In the configuration shown in Fig. 9, where the tip of the nozzle in placed below the center of the coils (z 0 < L), the vertical gradient of the horizontal magnetic field points upwards at the tip of the nozzle. In the middle part of Fig. 9(b), the FF foam structure is manipulated using a local magnetic force (see Section 4.4). Two magnetic coils are fitted with soft iron pieces, and supplied with an electric current J. Contrary to the soft iron cores used in Fig. 9(a) of the setup, the soft iron pieces used here are sharp in order to create a strong but very local magnetic force. Alternatively, a permanent magnet can be used instead of the magnetic coils. Fig. 9. Experimental setup. (a) Bubble generation: bubbles are created by blowing air at a constant flow rate through a nozzle into the FF and SDS solution at the bottom of a vertical Plexiglas tube. The nozzle is placed in a horizontal magnetic field created by two coils fitted with soft iron cores. (b) Manipulation of the structure: two coils fitted with sharp soft iron cores create a local magnetic force in the FF foam, which can induce structural changes. (c) Structure detection: at the top of the tube the local electrical resistance of the generated foam structure is measured between two electrodes of different cross-section in contact with the foam. In the upper part of Fig. 9(c), the foam structure is detected independently of the video imaging by measuring the electrical conductivity of the foam (see Section 4.5) Bubble size control In this section, we show that magnetic fields can be used to very accurately regulate the volume of bubbles produced in a ferrofluid-surfactant solution over a range of at least 2 3 orders of magnitude.

7 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) Fig. 10. (Coming from Ref. [18].) Bubble diameter d as a function of current I through the coils at constant air flux. The two data sets correspond to measurements with different nozzle diameter δ at the same position in the region of positive vertical magnetic field gradient. Every data point is an average of four measurements. The lines represent the best fits of the data using Eq. (12). Circles: δ = 0.84 mm, Squares: δ = 2.5 mm. The nozzle is placed at a position z 0 in the magnetic field, and the variation of bubble volume with magnetic field strength is observed by varying the current through the coils. The effect of the strength of the current I on the bubble volume V is strongly dependent on where the bubbles are produced. For z 0 < L the bubble volume increases with I, for z 0 > +L it decreases with I, whereas it remains constant for L < z 0 <+L. This indicates the dominant role of the field gradient on bubble size control. Furthermore, above a critical current bubbles can even be observed to move downwards in the ferrofluid. The data presented in Fig. 10 was taken for different nozzle diameters δ within the region of positive field gradient (z 0 < L). For the purpose of fitting and plotting the data the bubble diameter d = (6/π(V)) 1/3 is used instead of the bubble volume V. As the current is varied between zero and approximately 1.7 A, the bubble diameter is increased by a factor of 2 3. This range could in theory be much larger, since the bubble diameter diverges as the current approaches a value of about 1.7 A. The measurements are found to be extremely reproducible. A simple model of force balance is developed below. The underlying idea is as follows. A magnetic field gradient generates a magnetic volume force on a bubble in a magnetic fluid. Depending on the direction of the field gradient, the magnetic force results in the reduction or magnification of the effective buoyancy of the bubble, which changes the critical bubble volume necessary for the detachment of the bubble from the nozzle. A bubble detaches from the nozzle when the interfacial force between nozzle and bubble, the buoyancy force and the magnetic force on the bubble equalize (Fig. 11). Fig. 11. Model and notations: the bubble (in white) detaches from the tip of the nozzle (in black), when the interfacial force, the buoyancy force and the magnetic force equilibrate. The interfacial force can be described by F S = πfγδe z, (7) where δ is the diameter of the nozzle, e z is a unit vector pointing upwards, and f is a correction factor close to unity [19]. This is a rather crude model, but seems sufficient for a very accurate description of the experiments conducted here. The buoyancy force is given by F G = ρgv e z, (8) where ρ is the density of the solution. The magnetic force on the bubble can be found by noting that a non-magnetic bubble inside a magnetic fluid of magnetisation M(H) is physically equivalent to a magnetic bubble of magnetisation M(H) inside a non-magnetic fluid [20], where H is the external magnetic field. Therefore, the magnetic force acting on the bubble is given by F M = µ 0 V(M(H) ) H [9]. For the coil system used in the conducted experiments, this can be written as H(z, I) F M = µ 0 VM(H) e z, (9) z where H(z,I) is the magnetic field strength as a function of position z and current I through the coils. Since H(z,I) is proportional to I it can be written as: H(z, I) = β(z)i (10) where β(z) depends on the position only and can be determined for the particular experimental setup using a Hallprobe. In Ref. [18], we have shown that the magnetisation of the solution M(H) can be accurately described using a simple empirical expression, which does not correspond to any theory but allows further analytical calculations: M(H) = χm SH M S + χh, (11)

8 72 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) where χ and M S are the magnetic susceptibility and the saturation magnetisation of the solution, respectively: χ = and M S =5230Am 1. Equating F S + F G + F M = 0 (Eqs. (7) (9)) and using Eqs. (10) and (11), the bubble diameter d(z 0, I) is determined as a function of nozzle position z 0 and the current I through the coils: 6δγf d(z 0,I) = ρg µ 0 M(z 0,I) β(z) z z0 1/3. (12) This formula is in accord with the experimental observation: the bubble diameter d increases with I when the gradient z β(z) z0 >0 [z 0 < L], whereas it decreases with I when z β(z) z0 <0 [z 0 > L], and remains constant when z β(z) z0 =0 [+L > z 0 > L]. If the applied field gradient is positive, the bubble diameter diverges when I approaches a critical value. For currents above this value the magnetic force on the bubble is strong enough to force it to move downwards in the ferrofluid. Eq. (12) is fitted to the data (Fig. 10) using two fitting parameters: the product γf and the gradient z β(z) z0. γf is of the same order of magnitude for all sets of data, on an average: γf = (41.3 ± 3.1) 10 3 Nm 1. This is a reasonable order of magnitude since f is of the order of unity [19]. The fitting results for z β(z) z0 are reasonably close to the measured values (see Ref. [18]), of the order of m Manipulation of the structure Once the ordered foam is formed, the imposition of a local magnetic field dramatically affects the moving structure. The field can, for example, induce transitions from one structure to another (Fig. 12), or twist them through 90 or 180 (Fig. 13). Fig. 13. (Coming from Ref. [17].) Magnetic fields may also be used to create localized twists (dispirations [24]) in a foam structure. The angle of twist appears to be 180 for the structure with two bubbles per unit cell (a) and 90 for the structure with four bubbles in the unit cell. The tip shown is a permanent magnet of strength 450 G/mm close to the edge. Once created, the twists do not unwind themselves even when the magnet is removed. The stability of these twists is yet to be understood [25]. Fig. 12. (Coming from Ref. [17].) Applying a magnetic field to an ordered FF foam can result in a change of the structure. The transitions are induced by a local oscillating magnetic field gradient (frequency 2 Hz, strength 450 G/mm). (a) Transition from to The bubbles (diameter 5.0 mm) move upwards in the tube at a rate of 30 bubbles per minute. (b) Transition from to bamboo. Here the bubbles (diameter 6.2 mm) move upwards at a rate of 20 bubbles per minute. Such structural transitions can also be induced using a static permanent magnet. The bubbles are very robust: we do not observe rupture of the films that separate them, even during rapid changes of the kind shown here. This use of a magnetic field provides precise local control over structural changes, which have in the past been promoted by forced drainage [21,22] or mechanical compression/dilation [23], applied to the whole system. The action of the magnetic field appears to be based on two effects, both arising from the attraction of ferrofluid to the region of maximum field. Firstly the foam develops a higher liquid fraction, which is enough in itself to provoke a morphological change between two metastable structures [21,22]. Using an oscillating magnetic field accelerates the transition (Fig. 12). Secondly, since most fluid resides in the Plateau borders (the channels which are formed between bubbles in a foam of low liquid fraction), the borders are pulled towards the region of high field, resulting in a net torque. This torque may be strong enough to twist the structure as shown in Fig. 13. Together

9 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) these two mechanisms offer many possibilities for propulsion and control of the bubble stream Structure detection For some technological applications (microscopic devices, opaque fluids or fluids with similar optical index), it may be useful to have a non-optical detection of the structure flowing in the tube, as well as a non-optical measurement of the bubble volume and of the number of bubbles passing through the tube. We show here that an electrical resistance measurement allows to perform those measurements, provided that only the bamboo structure (Fig. 8a) and the staircase structure are considered (Fig. 8b). As the ferrofluid is of ionic nature it is conductive and, therefore, ideal for electrical conductivity measurements. The local electrical conductivity σ F of the foam structure is measured between two electrodes at a fixed height (see top of Fig. 9). In order to achieve a sensitivity of the measurement to the foam structure, the electrodes are of different crosssection S 1 = 1.1 mm 2 and S 2 =12mm 2. They are made of conducting wire, their diameters being smaller than the spacing between the Plateau borders of the foam structure. σ F is measured indirectly by determining the voltage U across a reference resistance R =10k in series with R F for an applied voltage U 0 = 0.5 V. An alternating voltage (frequency: 10 khz) is applied to avoid electrolysis at the electrodes. A lock-in amplifier is used to measure the voltage U in phase with U 0. The liquid phase of the foam can be considered equivalent to a network of conducting wires between the electrodes [1]. Since the cross-sections of the electrodes are smaller than the separation of the Plateau borders, the detected signal U reflects the structure of the foam. As the structures are periodic and flow within the tube at a constant velocity, the signal U is periodic in time, being superimposed upon a constant signal due to a conducting wetting film of ferrofluid around the inside of the tube (Fig. 14). A peak in the electrical signal is recorded, whenever a Plateau border is in contact with one or both of the electrodes. In the case of the bamboo structure, both electrodes are in contact with the same Plateau border at the same time, giving a regularly spaced sequence of identical spikes in conductance (Fig. 14a). A more complicated signal characterises the staircase structure. When one electrode touches a Plateau border, the other one is only in contact with the wetting film. This leads to the occurrence of two alternating peak heights in the periodic signal (Fig. 14b). The small peak is recorded when the smaller electrode is in contact with a Plateau border, whereas the big peak corresponds to the case where the larger electrode is in contact with a Plateau border. Since in both structures each peak in the signal corresponds to one bubble the bubbles can then be counted from the electrical signal shown in Fig. 14. Knowing the flow rate of gas Q injected at the bottom of the tube, the bubble volume can also be obtained from the period τ of the Fig. 14. Signal U/U 0 as a function of time using two electrodes of different cross-section in contact with the foam flowing through the tube. (a) Bamboo structure; (b) staircase structure. electrical signal: V = cqτ, with c = 1 for the bamboo structure and c = 1/2 for the staircase structure. 5. Conclusion and outlooks 5.1. Drainage in a magnetic soap film The control of the drainage in a vertical freely suspended magnetic soap film using an external homogeneous magnetic field has been investigated through two studies: the dynamics of thinning of the coloured film, i.e. the part of the film, of micrometric thickness, in which light interference in observed, and the growth dynamics of the black film. In the coloured film, we have shown that a homogeneous parallel (respectively, perpendicular) applied magnetic field slows down (respectively, speeds up) the drainage of the liquid solution out of the film. We have developed a simple model in which the effect of the homogeneous applied field is taken into account via the magnetic dipole interaction energy, which depends in average on the relative orientation of the magnetic dipoles and the interface: a magnetic pressure drop is then induced between the film and the meniscus, which depends on the orientation and amplitude of the external field. This model reproduces very accurately the experimental data. The growth rate of the black film is also affected by an external magnetic field. Under the effect of a field applied perpendicularly to the soap film, millimetric drops of black film are created in the meniscus. The creation of black film, generally observed in classical non-magnetic soap films close to the frame, is known as marginal regeneration [11], and is still poorly understood. Here, the perpendicular magnetic field induces marginal regeneration in the meniscus. This effect, which could be due to the development of a magnetic instability in the meniscus [26], remains to be investigated. The

10 74 F. Elias et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 263 (2005) magnetic field could then be used as an external parameter to control the nucleation of black film in order to investigate the phenomenon of marginal regeneration in soap films, which plays an important role in the stability of soap films and foams. Is this study applicable to describe the internal structure and dynamics of a 3D FF foam? A recent investigation has shown that the dynamics of drainage of such a 3D FF foam created using the same solution is affected by a homogeneous applied magnetic field; this modification has been attributed to the anisotropy of the structure of the foam under the effect of the field [27]. However, in a 3D foam, the dynamics of drainage is dominated by the geometry of the multiconnected Plateau borders. Therefore, the study of the drainage in a single magnetic soap film is not sufficient for a complete understanding of the drainage in 3D FF foam at the scale of an individual cell. Finally, let us notice that the repartition of magnetic particles in the black film remains to be investigated. No study of the concentration of magnetic particles in the soap film has been performed so far. When the film thickness is large in comparison with the particle diameter, we have assumed that the concentration of magnetic particles in the film is the same as the concentration in the bulk. However, this assumption may not be valid when the film thickness reaches values comparable with the particle size. The particles, confined in the film, may adopt crystalline arrangements, which could modify the elastic properties of the film [28]. It would be very interesting to use magnetic soap films, in order to investigate the physical properties of soap films of nanometric thickness containing confined colloidal magnetic particles Magnetic soap foam in a tube Using a ferrofluid-surfactant solution as the liquid phase of a monodisperse foam confined in a tube, the structure of the foam can be controlled and manipulated using magnetic fields. First, the volume of the gas bubbles is set during the bubble generation by the strength of an electric current in a pair of magnetic coils surrounding the nozzle through which the gas is injected. This effect is due to the magnetic volume force applied to the gas bubble immersed into the magnetic solution, which acts against the buoyancy force. This study offers the possibility of controlling the bubble size, and thus the foam structure, independent of the gas flow rate at the tip of the nozzle. Second, the structure adopted by the foam can be manipulated. An external magnetic force attracts the ferrofluid locally, which may induce transitions between metastable structures. An external magnetic torque may twist the whole structure of an angle compatible with the symmetry of the structure. Finally, we have shown that if the foam flowing in the tube is either a bamboo or a staircase structure, the structure can be detected by an electrical conductivity measurement, as well as the frequency of the bubbles and the bubble volume, without visualising the foam. All those features (structure control, structure manipulation and structure detection) make of the FF foams confined in a tube good candidates for technological applications, such as the control of cell transport in capillaries. It should also be possible to scale down and apply those results for the conception of smart microfluidic devices. Acknowledgements Research was supported by CNRS (contract ATIP No. 0693) (FE, EJ, J.-C. B), an Enterprise Ireland Basic Research Grant (SC/2002/011) (DW,SH,WD), and the German National Merit Fundation (WD). The authors would like to thank Nicolas Rivier and Andrejs Cebers for stimulating discussions, and the referee of this article for enlightning comments. Appendix A. Notations g acceleration of gravity: g =9.81ms 1 µ 0 magnetic permittivity of void: µ 0 =4π 10 7 International Units H external applied magnetic field M magnetization of the solution χ magnetic susceptibility of the ferrofluid (in low fields) M S saturation magnetization of the ferrofluid D demagnetising factor γ interfacial tension of the solution: γ 30 mn m 1 ρ mass per unit volume of the solution: ρ = 1500 kg m 1 η viscosity of the solution: η 10 2 Pa s volume of ferrofluid V FF Notations of Section 3 l height of the soap film: l =10mm e thickness of the soap film b sliding length R radius of curvature of the meniscus λ wavelength of the monochromatic light source: λ = (589.3 ± 0.3) nm n optical index of the solution: n 1.33 p interference order V bubble volume Notations of Section 4 z 0 position of the nozzle δ internal nozzle diameter I current through the coils surrounding the nozzle f correction factor (close to unity) β(z) position-dependent coefficient which relates I to the local magnetic field strength

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