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Division of ferrofluid drops induced by a magnetic field Carlo Rigoni, Stefano Bertoldo, Matteo Pierno, Delphine Talbot, Ali Abou-Hassan, and GIAMPAOLO MISTURA Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02399 • Publication Date (Web): 30 Jul 2018 Downloaded from http://pubs.acs.org on August 5, 2018
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Division of Ferrofluid Drops Induced by a Magnetic Field Carlo Rigonia, Stefano Bertoldoa, Matteo Piernoa, Delphine Talbotb, Ali Abou-Hassanb† and Giampaolo Misturaa* a
Dipartimento di Fisica e Astronomia “G.Galilei”, Università di Padova, via Marzolo 8, 35131 Padova, Italy
b
Sorbonne Université, CNRS, Laboratoire PHysico-chimie des Electrolytes et Nanosystèmes, PHENIX, F-75005 Paris,
France.
ABSTRACT We report a comprehensive study of the division of ferrofluid drops caused by their interaction with a permanent magnet. As the magnet gradually approaches the sessile drop, the drop deforms into a spiked cone and then divides into two daughter droplets. This process is the result of a complex interplay between the polarizing effect caused by the magnetic field and the magnetic attraction due to the field gradient. As a first attempt to describe it, during each scan we identify two characteristic distances between the magnet and the drop: zmax, corresponding to the drop reaching its maximum height, and zsaddle, corresponding to the formation of a saddle point on the drop peak identifying the beginning of the drop breakup. We have investigated the location of these two points using sessile drops of ferrofluid water solutions at various concentrations and volumes, deposited on four surfaces of different wettability. An empirical scaling law based on dimensionless variables is found to accurately describe these experimental observations. We have also measured the maximum diameter of the drops right before the division and found that it is very close to a critical size which depends on the magnetic attraction. ACS Paragon Plus Environment
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INTRODUCTION Ferrofluids are colloidal suspensions of sub-micrometer-sized magnetic particles dispersed in a solvent, which respond to a magnetic field by orienting along the field. As a result, the field can exert strong forces on the fluid, radically changing its shape as the fluid moves to maximally fill its volume with magnetic field.1, 2 These effects are both exotic and useful for a wide range of applications, ranging from computer disk drives to rotary vacuum seals and speaker-damping technology.3 More recently, ferrofluid droplets are starting to be used in microfluidics4,
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where external magnetic fields are
exploited to handle drops,6-8 carry out continuous flow cell separation,9, 10 perform chemical ferrofluid synthesis11 and study the drop breakup dynamics.12 They are also used to control the wetting properties of ferrofluid sessile drops and to drag them along a surface.
13-16
For instance, the shape and contact
angles of ferrofluid drops can be continuously varied by moving a permanent magnet.16 More recently, drops of commercial ferrofluid placed below a magnet shows a temporal evolution of their shape on a time scale of several minutes due to the migration and accumulation of the magnetic nanoparticles at the apex of the drop that eventually leads to its detachment.17 The interplay between magnetic fields and slippery surfaces yields a quite rich dynamics of gravity driven ferrofluid drops. If ferrofluid drops move on superhydrophobic surfaces in an external magnetic field, their interaction force with the substrate is measured to be very low18 and is defined by the processes occurring in the vicinity of the triple line only.19 The speed of gravity driven ferrofluid drops on oil impregnated surfaces and in the presence of a patterned magnetic field can be accurately tuned by the magnetic interaction. More interestingly, drops are found to undergo a stick-slip motion whose contrast and phase can be easily tuned by changing either the strength of the magnetic field or the ferrofluid concentration.20 Ferrofluid drops on superhydrophobic surfaces represent also a model system to study and bridge the gap from static to dynamic self-assembly.21 The droplets self-assemble under a static external magnetic field into simple patterns that can be switched to complicated dynamic ACS Paragon Plus Environment
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dissipative structures by applying a time-varying magnetic field. In a typical experiment, one drop of aqueous ferrofluid is placed on a superhydrophobic surface and subjected to a confining field of a cylindrical permanent magnet below the substrate. Gradually increasing the field strength B and the vertical gradient dB/dz acting on the drop (by decreasing the gap between the magnet and the surface) eventually leads to the breakup of the drop in two or more smaller droplets. The division takes a few tens of milliseconds, after which the daughter droplets briefly oscillate before settling at their equilibrium separation. The drop break-up is related to the normal-field instability of ferrofluids (socalled Rosensweig instability), but differs from it in two ways. 21 The characteristic Rosensweig pattern is observed by applying a perpendicular uniform magnetic field to a pool of ferrofluid which produces the spontaneous generation of an ordered pattern of surface peaks having a critical wavelength:1, 22 = 2
(1)
and being the density and the surface tension of the ferrofluid, when the magnetization exceeds the
critical magnetization:1, 22
= 1 +
where
() is the vacuum (ferrofluid) magnetic permeability.1,
(2) 22
When the magnetization is
increased from zero by rising the applied magnetic field, the fluid interface is perfectly flat over a range of field intensities up to the point when the transition suddenly occurs. Conversely, no increase in the applied field, no matter how large, can cause the interface to be unstable if the saturation magnetization of the fluid is less than .
In contrast to this scenario, the gravitational force in the droplet experiment21 is negligible compared with the magnetic force due to the vertical field gradient; that is, the effective gravitational acceleration16, 23 ∗ = +
∇
, where M(B) is the ferrofluid magnetization corresponding to
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B and ∇(B) is the gradient of the magnetic field, is up to two orders of magnitude larger than . Thus, the revised formula of the critical distance ! = 2 ∗
(3)
does not determine the periodicity of the pattern but rather provides the criterion for the division: a droplet divides when the droplet diameter becomes larger than ! .21 In further contrast, the Rosensweig pattern is reversible (the pattern decays when the field is removed), but the daughter droplets do not coalesce back to a single drop.21 Numerical simulations of a thin ferrofluid film subjected to an applied uniform magnetic field show that the subtle competition between the applied field and the van der Waals induced dewetting determines the appearance of satellite droplets.24 Apart from these studies, the division of ferrofluid drops has not been further analyzed and, more generally, there is very little in the literature on interfacial instabilities of ferrofluid drops.25 To shed more light on this phenomenon, we have thus decided to systematically study the division of a ferrofluid drop in two daughter droplets induced by a magnetic field exploring its dependence on the major quantities involved, i.e. the nature of the substrate, the magnetic interaction and the drop volume.
METHODS Ferrofluid drops of known volume Ω (on the order of tens of µL) are deposited on a horizontal substrate by using a micropipette and are subject to an external magnetic field. The magnetic field B is generated by a cylindrical neodymium magnet (diameter 45 mm, thickness 30 mm), of grade N40 and residual magnetic field 1.25 T. The diameter of the magnet is always much larger than the maximum size of the ferrofluid drops. This guarantees that the magnetic field can be considered as homogeneous in the horizontal plane and only dependent on the vertical distance z between the magnet and the bottom of the drop.16 The left graph of Figure 1 shows the values of B measured with a Hall probe
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(LakeShore 460 3-Channel Gaussmeter with a 3-Axis Probe) along the axis of the magnet. The continuous line represents the spatial dependence of the axial component of the magnetic fields calculated using the quoted magnet data. The experimental results agree very well with the theoretical prediction:16, 26 " =
#
$
%&'
) ( & %&' ) )
−
'
) ( &' ) )
+
(4)
where ", is the residual magnetic field of the magnet (in our case 1.25 T), - is its thickness and ! is its diameter. Accordingly, the gradient of the magnetic field is calculated from the spatial derivative of the analytical formula and its variation along the axis of the magnet is plotted in the inset. Figure 1 (center) shows a schematic representation of the set-up layout comprising substrate, ferrofluid drop and magnet. An actuator varies the distance of the permanent magnet from the bottom of the glass slide in a controlled way. The drops are illuminated by two backlight-collimated led sources and their profile is viewed using two CCD video cameras (Manta G-146, Allied Vision Technologies) mounting a telecentric lens. The two cameras are oriented orthogonally to each other. The acquired images and videos are analyzed off-line using a custom made program that calculates the apparent contact angle, the height and the lateral extension of the contact line of the ferrofluid drops.27 Four different substrates are chosen to span an ample wettability range and their behavior is characterized by measuring the apparent contact angle of water drops ϴ and the contact angle hysteresis CAH defined as the difference between the advancing and receding contact angles, which determines the wetting dynamic behavior. 28, 29
The four surfaces are: hydrophilic glass slides (ϴ~60°, CAH~5°), hydrophobic Teflon tapes
stretched on glass slides (ϴ~115°, CAH~20°), natural Lotus leaves (ϴ~155°, CAH~4°) and slippery liquid impregnated surfaces (LIS, ϴ~100°, CAH=4°, viscosity of the silicone oil 100 cSt) produced following the recipe described in 30. Since the oil surface is intrinsically smooth and free from chemical and morphological defects typical of solid surfaces, LIS are recognized to hardly pin sessile drops.13, 31, ACS Paragon Plus Environment
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32
The ferrofluids used in this work are stable aqueous suspensions of maghemite (γ-Fe2O3) nanoparticles in aqueous acidic solutions at pH ∼2 with NO3‾ counterions synthetized according to the Massart
process,33 which guarantees stable suspensions over long time intervals and in the presence of strong magnetic fields (see Supplemental Material of Ref.20 ). The detailed synthesis procedure is described in our previous work.16 The resulting nanoparticles have a rock like shape with a crystal diameter of about 8.5 nm. The iron concentrations, the magnetic susceptibilities (χ) and the magnetizations at saturation (Msat) of the ferrofluid solutions used in this study are: 7.5 mol/L (χm = 1.56, Msat = 38.2 kA/m), 6.0 mol/L (χm = 1.28, Msat = 34.2 kA/m), 5.0 mol/L (χm = 1.11, Msat = 29.6 kA/m) and 4.0 mol/L (χm = 0.76, Msat = 24.3 kA/m). We point out that our highest Msat is less than half that used in the original work.21
Figure 1. (Left) Experimental (open squares) and theoretical (continuous line) values derived from Equation 4 of the axial magnetic field as a function of the vertical distance z from the top surface of the magnet. Inset: analytical calculation of the gradient of the magnetic field as a function of z. (Centre) Schematic representation of the disposition of the substrate, ferrofluid drop and magnet. There are indicated the generic distance z between the magnet and the base of the drop and the two characteristic distances zmax and zsaddle. Colors highlight the intensity of the B in arbitrary units. (Right) Snapshots of water drops on the four different surfaces used in this study. Volume of the drops is 4 µL. For each surface, the apparent contact angle is reported.
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RESULTS AND DISCUSSION In a typical experiment, one drop of aqueous ferrofluid is deposited on the substrate and its contour is recorded as the distance z with the permanent magnet is gradually decreased (see Figure 1). The consequent increase of the magnetic interaction with the drop leads to a deformation of the drop into a spiked cone and a division into smaller drops at specific z values. With our magnet, if the volume of the mother drop is less than about 40 µL, it divides into two daughter droplets. If 40
453∗
(5)
This equation suggests converting the raw data of Figure 2 to the dimensionless numbers 453∗ and S
and fit the data according to the scaling law =A = B453∗ , where B ≡ 8 < +
>2 >
is taken as a free
parameter. The corresponding results are shown in the log-log graph of Figure 3. The data are found to nicely collapse on parallel straight lines having slope of ½ as expected. Interestingly, the resulting curves seem to fit rather well both the saddle points (graphically enclosed in the white region) and the maximum points (enclosed in the dashed region). Similarly, the data on different surfaces are found to collapse on the same curves. Instead, there is a marked dependence on the ferrofluid concentration and thus, more generally, on the magnetic interaction: as the concentration increases, the straight line moves upward. The inset shows that the coefficient B is linearly dependent on the ferrofluid
magnetization. We have compared these values with those derived from B = 8 < +
>2 >
using the
measured ferrofluids susceptibilities. The numerical agreement is fair and, more important, the formula predicts a decrease of α with the ferrofluid magnetization. This clearly suggests that the formula of the parameter α valid for a ferrofluid pool must be suitably corrected for a ferrofluid drop.
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Figure 3. Saddle points (graphically enclosed in the white region) and maximum points (enclosed in the dashed region) plotted in terms of the dimensionless parameters 453∗ and S. Lines are power-law fits according to the formula =A = B453∗ , where B is a free parameter. Its value is reported in the inset as a linear function of the magnetization at saturation of the ferrofluid solutions.
We have also analyzed the connection between the drop size at breakup and the critical size defined in Eq. (2), which can be rewritten in terms of the magnetic capillary length16,
23
D∗ E = 63∗ E as :
FA = 8GD∗. In the absence of a magnetic field, D∗ reduces to the standard capillary length representing the typical size below which capillarity dominates gravity. The snapshots in figure 4 show the characteristic profiles assumed by ferrofluid drops of volume Ω=10 µL deposited on the four surfaces when the saddle point is observed. The green horizontal lines indicate the maximum lateral diameter D ACS Paragon Plus Environment
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exhibited by the drops. Experimentally, it is obtained by taking the maximum value between the two diameters measured from the two orthogonal cameras just before the saddle point is observed. The direction along which the maximum diameter is measured is always orthogonal to the division
direction. The graph in Figure 4 displays D as a function of the critical length FA deduced from the distance zsaddle for drops of different volume and concentration, deposited on the four surfaces. Within the intrinsic scatter of the measurements, there do not seem to be systematic variations attributable to volume, concentration or substrate. The data lie on a straight line of equation H =
