ARTICLE pubs.acs.org/ac
Self-Arraying of Charged Levitating Droplets Paul Kauffmann,†,‡ Jeremie Nussbaumer,† Alain Masse,† Christian Jeandey,§ Henri Grateau,§ Pascale Pham,§ Gilbert Reyne,† and Vincent Haguet*,‡ †
Grenoble Electrical Engineering Laboratory (G2Elab), UMR 5269 (Grenoble-INP, UJF, CNRS), BP 46, 38402 Saint Martin d’Heres Cedex, France ‡ Commissariat a l’Energie Atomique et aux Energies Alternatives (CEA), BGE (U1038 Inserm/CEA/UJF), 17 avenue des Martyrs, 38054 Grenoble Cedex 9, France § Commissariat a l’Energie Atomique et aux Energies Alternatives (CEA), LETI, 17 avenue des Martyrs, 38054 Grenoble Cedex 9, France
bS Supporting Information ABSTRACT: Diamagnetic levitation of water droplets in air is a promising phenomenon to achieve contactless manipulation of chemical or biochemical samples. This noncontact handling technique prevents contaminations of samples as well as provides measurements of interaction forces between levitating reactors. Under a nonuniform magnetic field, diamagnetic bodies such as water droplets experience a repulsive force which may lead to diamagnetic levitation of a single or few micro-objects. The levitation of several repulsively charged picoliter droplets was successfully performed in a ∼1 mm2 adjustable flat magnetic well provided by a centimeter-sized cylindrical permanent magnet structure. Each droplet position results from the balance between the centripetal diamagnetic force and the repulsive Coulombian forces. Levitating water droplets self-organize into satellite patterns or thin clouds, according to their charge and size. Small triangular lattices of identical droplets reproduce magneto-Wigner crystals. Repulsive forces and inner charges can be measured in the piconewton and the femtocoulomb ranges, respectively. Evolution of interaction forces is accurately followed up over time during droplet evaporation.
uring his Nobel lecture in 1989,1 Wolfgang Paul cited the German philosopher and physicist Georg Christoph Lichtenberg who wrote about 200 years ago: “I think it is a sad situation in all our chemistry that we are unable to suspend the constituents of matter free”. In Paul’s opinion, force measurement would be far more precise if the investigated particle could be isolated from the other particles and their environment. Levitation has emerged as a possible solution to manipulate single or very few objects by trapping them in a potential well. Levitating objects move within a controlled force field which prevents them to be in contact with solid surfaces, thereby protecting them from any contamination effects2 or sticking force. Levitation of droplets is of particular interest as it may lead to better understanding of the physics of airborne droplets which is involved in many natural or technical processes3 such as cloud formation,4 aerosols systems,5 protein crystallization systems,6,7 contactless digital microfluidics,710 or collectors of (bio)particles for air analysis.11 Many techniques of droplet levitation have been developed and rely on optical,12 acoustical,13 electrostatic,14,15 electrodynamic,16 or diamagnetic710,1719 force. For 15 years, diamagnetic levitation of droplets has been investigated using large superconducting1719 or Bitter20 coils. However, field gradients created by magnets follow a favorable reduction
D
r 2011 American Chemical Society
scale law.21 Consequently, diamagnetic levitation of microdroplets has been recently demonstrated above bulk permanent magnets79 and permanent micromagnets,10 thereby requiring no energy supply. In addition to single-particle manipulation, levitation has also been exploited for force and charge measurement. For instance, the determination of the charge of particles is of foremost importance for aerosol study.22 Many processes for measuring the charge of particles have been conceived to study aerosols since the mid-twentieth century. In a remarkable review, R. C. Brown splits charge measurement techniques into two categories.23 The first one gathers static measurements of the charge transfer, and the second one relates to dynamic measurements of the electrical mobility. The most famous dynamic technique remains the Millikan method where the particle charge is deduced from its deflection under an electric field.14 Methods for determining charges of aerosol particles has been increasingly improved so as to measure the absolute charge,22 to parallelize the measure by studying a population instead of one particle,24 or even to store the particles for further analysis.25 Received: February 1, 2011 Accepted: April 18, 2011 Published: April 18, 2011 4126
dx.doi.org/10.1021/ac2002774 | Anal. Chem. 2011, 83, 4126–4131
Analytical Chemistry
ARTICLE
Figure 1. (a) DPL device composed of 8 radially magnetized pieces assembled in a cylindrical magnet with a pierced iron piece placed above. (b) Diagram of the levitation plane representing three levitating droplets (numbered circles) in electrostatic repulsion. The sum of electrostatic forces Fij applied by the droplets i on the droplet j balances the centripetal diamagnetic spring force Fmag_j. (c and d) Comsol Multiphysics modeling of the total volume energy along the vertical axis uz (i.e., cylindrical axis of the DPL device) and the radial axis ur (i.e., the levitation plane) for air gap values between 1.2 and 2.0 mm. By increasing the air gap e, the depth ΔEz of the vertical magnetic well, the depth ΔEz of the vertical magnetic well decreases (c) whereas its horizontal component ΔEr increases (d). (e) Diamagnetic radial acceleration in the levitation plane vs distance between the droplet barycenter and the cylindrical axis of the DPL device, for an air gap e = 1.4 mm.
However, to the best of our knowledge, none of these techniques has ever allowed to produce, identify, and characterize any stable patterning of charged microparticles in suspension. In this article, self-assembled organizations of water droplets trapped in a magnetic well are performed and used to measure and characterize electrostatic repulsion forces between droplets. 40 and 70 μm droplets generated by piezoelectric microdispensers are accurately positioned in the potential well where the centripetal diamagnetic spring force stably balances the electrostatic repulsive forces between droplets. Various arrays of levitating droplets were obtained which reproduce bidimensional magneto-Wigner crystals.26 In addition, thanks to the calibration of the magnetic well, repulsive Coulombian forces and charges, respectively, in the piconewton and the femtocoulomb ranges, can be deduced from the droplet position. This technique accordingly provides new capabilities to handle and analyze without contact suspended droplets according to their charge, leading to promising applications, e.g. electric characterization of bioaerosols or contamination free protein crystallization.
’ THEORETICAL BACKGROUNDS Water is diamagnetic, its volume magnetic susceptibility χ is 9 106 SI. Under the point dipole approximation, the droplet of volume V, submitted to a nonuniform magnetic field B B, experiences in air the following magnetic force: χ ! 2 V rB B Fm ¼ 2μ0
ð1Þ
where μ0 = 4π 107 H m1 is vacuum permeability. As χ < 0, water droplets are repelled away from the high magnetic field regions. A centimeter-sized, hollow, and radially oriented NdFeB permanent magnet (Figure 1a), hereafter referred to as the droplet patterning levitation (DPL) device, traps water droplets. Levitation of droplets occurs at the equilibrium levitation height located in the magnetic well where the vertical magnetic force compensates for the droplets weight in a stable manner.8 The center of the magnetic well usually mixes up with the cylindrical axis of the DPL device (Figure 1b). The DPL device is designed to manage the equilibrium of both the vertical and the horizontal forces, independently. Consequently, two conditions must be fulfilled to obtain several droplets levitating above the cylindrical permanent magnet. First, to halt the droplets ejected by a piezoelectric dispenser, the vertical energetic magnetic barrier ΔEz (Figure 1c) has to be higher than their kinetic energy Ec. Second, the radial diamagnetic force (named hereafter “diamagnetic spring force” Fmag), which attracts the droplets toward the center of the magnetic well, must be sufficiently low to prevent coalescence of the repulsive droplets. A low diamagnetic spring force is therefore selected so that a quasi-flat magnetic well is achieved (red dashed line in Figure 1d). As a result, the sensitivity of the DPL device is enhanced and provides accurate measurements of repulsive forces between droplets.
’ EXPERIMENTAL SECTION The DPL device is a centimeter-sized cylindrical permanent magnet, composed of eight triangular shaped, radially oriented 4127
dx.doi.org/10.1021/ac2002774 |Anal. Chem. 2011, 83, 4126–4131
Analytical Chemistry NdFeB magnets (J ∼ 1.2 T) clamped in a brass cylinder, as depicted in Figure 1a. A piece of iron (polar piece), pierced with a hole 1.6 mm in diameter, is used to locally concentrate the magnetic flux. The polar piece is positioned above the cylindrical magnet at an adjustable height in order to fine-tune both the vertical and radial stiffnesses of the magnetic well. To adjust the thickness e of the air gap, an axisymmetric numerical model of the cylindrical magnet was computed by solving the magnetostatic equation using the finite element method (Comsol Multiphysics). As droplets (diameter < 100 μm) are much smaller than the DPL hole (∼ 1.6 mm), the magnetic volume energy was computed by assuming a point dipole approximation for the droplets. The resulting vertical and radial profiles of total volume energy are presented in Figure 1c,d for several air gap values. Water microdroplets of ∼40 μm and ∼70 μm in diameter were generated a few dozens of millimeters above the DPL device by annular piezoelectric dispensers MJ-AB-01-030 and MJ-ATP01-060 from MicroFab Technologies (Plano, Texas, USA) tilted at an angle of ∼20° from the vertical axis to allow visualization from above. Images and films were recorded using a CCD camera DP70 from Olympus (Tokyo, Japan) equipped with a zoom lens Optem 125 from Fairport (New York, USA) focused on the levitation area. The scale on the images was calibrated using an objective micrometer from Olympus. Deionized (DI) water with conductivity of σ1 = 3 μS cm1 and phosphate buffered saline (PBS) diluted 10 times with conductivity of σ2 = 2.8 mS cm1 were dispensed. The droplets are spontaneously charged during the droplet ejection process due to an electrokinetic separation of charges.31,32 Briefly, the high flow velocity required for ejecting the droplet is produced by successive compression and depression of the piezoelectric ceramic placed around the capillary tube. The burst generating the droplet flows out a portion of the counterions from the electrical double layer (EDL) at the vicinity of the dispenser aperture. The amount of charges in a droplet depends on the overlapping of the EDL and the Prandtl boundary layer (i.e., layer near the surface within which the velocity profile increases from zero to a uniform velocity). As the Debye length of the EDL is inversely proportional to the solvent conductivity, the amount of propelled counterions, and consequently the charge of droplets, is higher for nonconductive water than for conductive one. Thanks to the high reproducibility of the droplet formation process, the charge is the same for every droplet; so Coulombian forces between droplets are repulsive.
’ RESULTS AND DISCUSSION DPL Magnetic Well. The diamagnetic spring force in the levitation plane was modeled using Comsol Multiphysics for an air gap value varying between 1.2 and 2.0 mm (Figure 1d). With a negative slope of the corresponding curve, no droplet stability is observed for an air gap of 1.2 mm. For an air gap between 1.4 and 1.8 mm, a critical radius can be determined beyond which droplets are repelled outside the magnetic well. The amplitude and the radial position of this local maximum both increase with the air gap. The DPL device presented in this article is based on an air gap of about 1.4 mm so that the vertical energetic barrier ΔEz is high (Figure 1c) and the magnetic well is quasi-flat on a wide area (ΔEr f 0 in Figure 1d). The levitation height of a droplet within the magnetic well varies on less than 100 μm (Figure S-1 in the Supporting Information). This interval width is far smaller than the diameter of the magnetic well (∼1.4 mm in Figure 1d), thereby confirming the flatness of the magnetic well.
ARTICLE
The diamagnetic spring force was calibrated by progressively inclining the magnet from R = 0° to 4.6° from the horizontal plane. For a given angle R, the droplet moves away from the cylindrical axis of the DPL device to reach its corresponding new stable position where the diamagnetic spring force exactly compensates for the radial component of the particle weight due to the inclination.27 Considering a punctual approximation (i.e., the droplet diameter (
