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Drops and jets of complex fluids
Javadi, A.
Publication date
2013
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Final published version
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Javadi, A. (2013). Drops and jets of complex fluids.
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Drops and jets of complex fluids
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2013
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam
op gezag van de Rector Magnificus prof. dr. D.C. van den Boom
ten overstaan van een door het college voor promoties ingestelde commissie,
in het openbaar te verdedigen in de Aula der Universiteit op vrijdag 5 juli 2013, te 15:00 uur
door
Arman Javadi
Promotor: Prof. dr. D. Bonn
Overige leden: Prof. dr. G.H. Wegdam, emeritus Dr. R. Sprik, UHD
Prof. dr. M. Bonn Prof. dr. N. Ribe Dr. S. Moulinet Dr. M. Habibi
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Cover Page: by Arman Javadi c
copyright 2013 by Arman Javadi. All rights reserved. The author can be reached at “arman.javadi@gmail.com”
The research reported in this thesis was carried out at the Institute of Physics, University of Amsterdam. The work was financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
1 General Introduction 1
1.1 Drops . . . 1
1.1.1 Surface tension . . . 2
1.1.2 Wetting . . . 3
1.1.3 Dynamic of partial wetting . . . 4
1.2 Jets . . . 5
1.3 Contents . . . 6
2 Effect of Wetting Properties on Capillary Pumping in Microchan-nels 11 2.1 Introduction . . . 11
2.1.1 Microfluidics and its applications . . . 12
2.1.2 Micropumps . . . 12
2.2 A surface tension driven micropump . . . 13
2.3 3-phase flow . . . 14
2.3.1 2-drops system numerical analysis . . . 16
2.4 Backflow . . . 18
2.4.1 Backgrounds . . . 18
2.4.2 Experiments and results . . . 20
2.5 Electrowetting : bi-directional active pumping . . . 22
2.5.1 Theoretical backgrounds . . . 25
2.5.2 Experiments and setup . . . 28
3 Dynamics of Ferrofluid drops 33
3.1 Introduction . . . 33
3.2 Ferrofluids . . . 34
3.2.1 Stability requirements . . . 34
3.2.2 Some applications of ferrofluids . . . 36
3.3 Drop Characteristics . . . 38
3.4 Dynamics: Sliding Shaped drops on an inclined plane . . . 40
3.4.1 Discussion on the acting forces . . . 42
3.4.2 Extracting the perimeter . . . 45
3.4.3 Analyzing the data . . . 46
3.5 Conclusion and perspectives . . . 48
4 Delayed capillary breakup of falling viscous jets 51 4.1 Introduction . . . 51
4.2 History and backgrounds . . . 52
4.3 Rayleigh Instability . . . 54
4.4 Long wave-length approximation . . . 56
4.5 Experiments of the viscous falling jet . . . 58
4.6 Theoretical investigation . . . 59
4.6.1 Dimensional analysis . . . 59
4.6.2 WKB analysis . . . 65
4.6.3 High-viscosity limit . . . 67
4.7 conclusion . . . 70
5 The Non-Newtonian Hydraulic Jump 71 5.1 Introduction . . . 71
5.2 Newtonian hydraulic jump . . . 72
5.2.1 Inviscid jump . . . 72
5.2.2 Viscous Theory . . . 74
5.2.3 Surface tension . . . 76
5.3 Non-Newtonian hydraulic jumps . . . 77
5.4 Rheological properties . . . 80
5.5 Non-Newtonian experiments . . . 81
6 Coiling of Yield Stress Fluids 87
6.1 Introduction . . . 87
6.1.1 Coiling of Newtonian fluids . . . 89
6.2 Coiling of elastic ropes . . . 93
6.2.1 Regimes of coiling . . . 96
6.3 Yield stress fluids . . . 98
6.4 Coiling of yield stress fluids . . . 98
6.4.1 Experimental procedure and observations . . . 99
6.4.2 Results and discussion . . . 101
6.5 Conclusion . . . 106 Bibliography 107 Summary 115 Samenvatting 117 List of publications 119 Acknowledgements 120
General Introduction
Drops and jets are perhaps the most common shapes a liquid can take when it leaves its reservoir or pipe line. Generally, drops are round-shaped portions of liquid floating in another fluid, like air, or adhering to a solid surface and jets are columnar stream of liquid. Small drops (of size . 1 mm), usually take the shape of a sphere in air (or another fluid), and the shape of a spherical cap on a solid surface, while jets usually take the shape of a straight or bent cylinder. Despite their simple shapes, the dynamics of drops and jets in different situations are quite complex. These complexities have been the focus of a good portion of the literature in soft matter science, especially in recent times. As an example, wetting and spreading of drops is of major interest in both science and technology. Many practical processes require the wetting or spreading of a drop on a solid. The liquid drop maybe a paint, a lubricant, an ink or a dye (1). The solid may either show a simple surface or be finely divided (suspensions, porous media, fibers). Buckling, coiling and folding of viscous jets when they impact a solid surface are other examples of complex situations. In this thesis, we will explore some of these complexities of drops and jets of both Newtonian (water and silicon oil) and non-Newtonian fluids (ferrofluid, polymer solutions and foam).
1.1
Drops
This thesis (like most of the literature) will consider small drops of size ∼ 1 mm and thin jets of diameter . 1 cm. In these cases, the surface to volume ratio is relatively large. Therefore, the surface properties of the fluid, and in particular the surface tension, would be an important parameter of the problem. Surface tension is in fact responsible for the shape of liquid droplets and the breakup of liquid jets into drops.
Liquid
Figure 1.1: The forces acting on a liquid molecule inside the bulk and on the surface.
1.1.1
Surface tension
Surface tension (γ) is measured as the energy required to increase the surface area of a liquid by a unit of area. The surface tension of a liquid results from an imbalance of intermolecular attractive forces, such as the Van der Waals forces, between molecules. A molecule in the bulk liquid experiences cohesive forces with other molecules in all directions, while a molecule at the surface of a liquid experiences only net inward cohesive forces (Fig. 1.1).
The unbalanced attraction of molecules at the surface of a liquid tends to pull the molecules back into the bulk of the liquid, leaving the minimum number of molecules on the surface. Energy is required to increase the surface area of a liquid because a larger surface area contains more molecules in the unbalanced situation. This is the reason why a small drop has a sphere shape (in equilibrium conditions). A sphere has the minimum free surface (compared to other geometrical shapes) for a given volume of liquid. It is also the reason why a cylindrical jet of a liquid tends to break into spherical drops: surface minimization.
Gravity can play a role in the shape of a static drop. A bond number is defined as the ratio of gravity to surface tension forces. For a drop of size R it can be written as
Bo = ρgR
2
γ , (1.1)
where ρ is density, g the gravity acceleration and γ the surface tension. Since Bo ∝ R2, for small sizes the gravity is negligible and surface tension determines
the shape of the drop. For instance, a water drop of size R = 1 mm, has a Bond Number Bo ∼ 0.1.
Figure 1.2: A small droplet in equilibrium over a horizontal surface. (a) and (b) correspond to partial wetting, the trend towards wetting being stronger in (a) than in (b). (c) corresponds to complete wetting (θe= 0) (1).
1.1.2
Wetting
A liquid (L) when deposited on a flat, impermeable, solid surface (S), may show two types of equilibrium behaviour: partial wetting (Fig. 1.2(a) and (b)) or total wetting (Fig. 1.2(c)). The choice is dictated by interfacial energies γSL, γSV and
γLV, where V stands for the vapor phase.
When the combination:
S = γSV − (γSV + γLV), (1.2)
is positive, the energy of the solid/vapor interface is lowered by interaction of a flat liquid film: this corresponds to complete wetting. But when S is negative, a liquid drop does not spread on the solid: it terminates in the form of a wedge, with a well-defined contact angle θe (Fig. 1.2). We call this partial wetting.
Balancing the tensions γ (projected along the solid surface, which defines the al-lowed direction of motion) Young found the relation (2)
γSV − γSL= γLVcosθe. (1.3)
Equation (2.12) is best derived by considering a reversible change in contact line position, using global energetic arguments (1). Thus the nature of the contact line
U
Figure 1.3: A wedge of liquid moving with velocity U.
region, over which intermolecular forces are acting, is not considered. Accordingly, θe is understood to be measured macroscopically, on a scale above that of
long-ranged intermolecular forces.
1.1.3
Dynamic of partial wetting
Equation (2.12) holds at equilibrium. What happens if we move out of equilibrium, for instance by forcing a droplet on a surface as in chapter 3 of this thesis ? Let us discuss this for the case of partial wetting.
If the contact line of Fig. 1.3 moves at a velocity U , we expect a dissipation per unit length
T ˙S = F U, (1.4)
where F is the non-compensated Young force:
F = γSV − γSL− γLVcosθd= γLV(cosθe− cosθd), (1.5)
θd being the dynamic contact angle. If we can find the dissipation mechanism, we
end up with a relation between the driving force and the velocity.
The dissipation may have different origins: either molecular processes very near the contact line, or viscous processes in the whole moving fluid. The first may be sensitive to the chemical details of the molecules making liquid and the solid. The second is more universal. There is one limit, where viscous flows must be dominant: namely when the dynamic contact angle is small (θd 1). We can understand
this by the following argument.
Inside the moving wedge of Fig. 1.3, the velocities u range from u ∼ U at the free surface and u ∼ 0 at the lower surface. Therefore the viscous dissipation is of order
T ˙S = Z dxη U y 2 y, (1.6)
where y = θdx is the local thickness. Equation (1.6) gives a logarithmic integral
l = ln(xmax/xmin). Inserting the correct coefficients:
T ˙S = 3lηU
2
θd
U
Figure 1.4: Dynamic contact angle θd versus velocity U.
The logarithmic factor l is of order 12; it has worried the experts in fluid mechanics for many years. However, it is not the dominant feature of equation (1.7). The really important feature is the presence of θd in the denominator. At small wedge
angles, the viscous dissipation becomes very large, and dominates over all molecular processes.
Combining equations (1.4) and (1.7), we end up with a basic dynamic formula for partial wetting (3):
F ≡ γ(cosθe− cosθd) =
3lη θd
U, (1.8)
valid for θd 1 (γ ≡ γLV). A vast number of experiments can be understood in
these terms (4).
Fig. 1.4 shows the relation between U and θd in partial wetting. Of course, U
vanishes at the equilibrium angle U (θe) = 0, but it also vanishes at small θd, where
the dissipation is large.
1.2
Jets
Jets are present in our everyday environment in kitchens (Fig. 1.5(a)), showers, pharmaceutical sprays and cosmetics. The study of jets is also motivated by many
practical applications such as improving and optimizing liquid jet propulsion, diesel engine technology, agricultural sewage and irrigation (Fig. 1.5(b)), manufacturing (Fig. 1.5(c)), powder technology, ink-jet printing (Fig. 1.5 (d)), medical diagnos-tics and DNA sampling. On the other hand, jet dynamics probes a wide range of physical properties, such as liquid surface tension, viscosity or non-Newtonian rheology and density contrast with its environment.
Here, our focus will be on the dynamics of falling jets. When a vertical jet em-anates from a nozzle and falls under gravity, it exhibits a number of complex and interesting phenomena downstream. Depending on the boundary conditions and parameters of the problem, such as viscosity, flow rate and falling height, different phenomena are observed:
Break-up: If the height of the fall is large enough, the inevitable destiny of the liquid jet is breakup. The jet breaks into larger main drops and some smaller satellite drops downstream (Fig. 1.6). The main driving force here is the surface tension. Inertia and viscosity oppose the breakup; jets with larger viscosity and flow rate have larger breakup lengths. Supposedly, gravity helps the breakup by thinning the profile of the jet. But since it irons out the perturbations which lead to the breakup of the jet at the same time, it can actually make the breakup length larger (see Chapter 4).
Hydraulic jump: When a falling cylindrical jet impacts a solid surface before its breakup, different behaviours are observed. Typically, hydraulic jump occurs for high flow rates and low viscosities (Fig. 1.5(a)): right after the impact the jet spreads symmetrically in a thin layer, then there is an abrupt increase in the fluid depth at a well-defined Radius from the impact point Rj. This abrupt rise of the
fluid surface, flowing from a shallower and higher velocity to a deeper and lower velocity zone is called hydraulic jump.
Buckling: At the other extreme, at low flow rates and very high viscosities, the thin jet buckles as it impinges the plate; an interesting solid-like behaviour. As the flow continues a rotating helical coiling or folding of the thin filament is observed. Here, the jet exhibits very rich dynamics and different regimes of motion.
1.3
Contents
In chapter 2 of this thesis we show how a simple feature of water droplets on a surface, i.e. Laplace pressure, can be exploited to build a micropump. We also use electrowetting on water droplets to make a bi-directional micropump. The dynamics and velocity of the contact line of the drops does not play an important role in this chapter. Instead, we show that wetting properties are the key to controlling the flow. In the next chapter, sliding drops on an inclined plane are
(a)
(b)
(c)
(d)
Figure 1.5: a) A jet of tap water falling into a sink. The jet is too thick and its falling time too short for breakup to occur, yet it has become rough. The continuous jet hits the sink floor, where it expands radially in the form of a thin sheet boarded by a hydraulic jump. b) Sprays produced by jets are widely used in agricultural irrigation. c) Higher speed water jets are also used to cut tissues, meat, and even metal plates. d) Drops emerging from a bank of ink-jet nozzles. The drop heads are 50 µm across and the tails are less than 10 µm wide (5).
Figure 1.6: Plate I of Rayleigh’s ‘some applications of photography’ (6) showing: a) the destabilization of a jet of air into water and b) of a water jet in air. Rayleigh notes that the air jet destabilizes faster than the water jet. c), d) The breakup of a falling jet. Larger main and smaller satellite drops can be seen in these pictures.
studied to investigate the dynamics of wetting. Ferrofluid and magnets are used to make drops which do not follow the steepest decent.
The subsequent chapters make a study of 3 scenarios which can occur for a falling jet; the breakup of viscous threads of silicone oil, the spread of a non-Newtonian liquid jet (dilute polymer solutions) over a horizontal plate and the coiling of yield stress fluids (foam and gel). A more detailed abstract of what is done in each chapter is as follows:
Chapter 2: We investigate capillary pumping in microchannels both experimen-tally and numerically. Putting two droplets of different sizes at the in/outlet of a microchannel, will generally produce a flow from the smaller droplet to the larger one due to the Laplace pressure difference. We show that an unusual flow from a larger droplet into a smaller one is possible by manipulating the wetting properties, notably the contact line pinning. In addition, we propose a way to actively control the flow by electrowetting.
Chapter 3: ‘Shaped drops’ are made by adding ferrofluid to small magnets (∼ 1 cm). The important feature of these drops is that the magnet preserves the shape of the drop and the contact line. Therefore, we can impose the shape of the contact line and observe the consequences on the dynamics of sliding drops on an inclined plane. In this chapter, we observe how a liquid object, submitted to its weight and viscous forces, can adopt a different direction than the steepest descent.
Chapter 4: Thin jets of viscous fluid like honey falling from capillary nozzles can attain lengths exceeding 10 m before breaking up into droplets via the Rayleigh-Plateau (surface tension) instability. Using a combination of laboratory exper-iments and WKB analysis of the growth of shape perturbations on a jet being stretched by gravity, we determine how the jet’s intact length lb depends on the
flow rate Q, the viscosity η, and the surface tension coefficient γ. In the asymp-totic limit of a high-viscosity jet, lb∼ (gQ2η4/γ4)1/3, where g is the gravitational
acceleration. The agreement between theory and experiment is good except for very long jets.
Chapter 5: This chapter contains a brief review of hydraulic jump for viscous and inviscid flow. Expressions for the radius of the jump Rj(from the impact point
of the jet) are derived. Correction to the viscous expressions due to the surface tension is also presented. The same approach is pursued for the case of non-Newtonian power law fluids. Some data for the hydraulic jump of dilute polymer solutions are also shown, however, these experiments are still incomplete.
Chapter 6: We present an experimental investigation of the coiling of a filament of a yield stress fluid falling on a solid surface. We use two kinds of yield stress fluids: shaving foam and hair gel, and show that the coiling of the foam is similar
to the coiling of an elastic rope. Two regimes of coiling (elastic and gravitational) are observed for the foam. Hair gel coiling, on the other hand, is more like the coiling of a liquid system; here we observe viscous and gravitational regimes. No inertial regime is observed for either system because of instabilities occurring at high flow rates or the break up of the filament in large heights.
Effect of Wetting Properties
on Capillary Pumping in
Microchannels
2.1
Introduction
Microfluidics deals with the behavior, precise control and manipulation of fluids that are geometrically constrained to a small, typically sub-millimeter, scale. Typ-ically, micro means one of the following features: small volumes (nl, pl, fl), small size, low energy consumption, effects of the micro domain.
The behavior of fluids at the microscale can differ from ‘macrofluidic’ behavior in that factors such as surface tension, energy dissipation, and fluidic resistance start to dominate the system. Microfluidics studies how these behaviors change, and how they can be worked around, or exploited for new uses (7; 8; 9). At small scales (channel diameters of around 100 nanometers to several hundred micrometers) some interesting and sometimes unintuitive properties appear. In particular, the Reynolds number (which compares the effect of momentum of a fluid to the effect of viscosity) can become very low, making laminar flow a typical property of the micro domain. A key consequence of this is that fluids, when side-by-side, do not necessarily mix in the traditional sense; molecular transport between them must often be through diffusion (10). High specificity of chemical and physical properties (concentration, pH, temperature, shear force, etc.) can also be ensured resulting in more uniform reaction conditions and higher grade products in single and multi-step reactions (11; 12).
2.1.1
Microfluidics and its applications
As mentioned, microfluidics allows to manipulate and control flow of small quan-tities of liquids (10; 13). It is a multidisciplinary field intersecting engineering, physics, chemistry, microtechnology and biotechnology, with practical applications to the design of systems in which such small volumes of fluids will be used. Mi-crofluidics emerged in the beginning of the 1980s and is used in the development of inkjet printheads, DNA chips, lab-on-a-chip technology, microelectromechanical systems (MEMS). As an example, Fig. 2.1 shows two commercialized devices based on the microfluidicd technology. The field is evolving at a rapid pace and has the potential to influence diverse subject areas from chemical synthesis and biological analysis to optics and information technology. However microfluidic techniques are still not used on a large scale (14), which is at least in part due to the complex equipments these systems rely on.
a
b
Figure 2.1: a) Biosite chip. A droplet of blood is placed in an opening on the chip. This drop is pulled towards the microfilter and the microseparator (in the direction of the arrow) by capillarity. The results of the analysis are given after data is analysed on a microcomputer. In just 15 min, this diagnostic can determine whether or not a heart attack has taken place. b) This chip, commercialized by Agilent Technologies, is a few centimeters long, and permits the identification of specific genetic sequences in a 1-µl sample of roughly purified DNA. This process takes place in just 10 minutes (10).
2.1.2
Micropumps
For a growing number of applications there is a large demand for flow control and pumping methods in microchannels (15). As a consequence, micropumping
has become one of the main developing areas in microfluidics and lab-on-a-chip technology. There have been many pumping methods explored for moving the fluid inside microchannels (16; 17; 18; 19), but most of these still need rather complex external equipments.
In general there are two types of pumps: active and passive. Active pumps need external energy consumption to operate. An important example of these are the piezoelectric pumps, which are used in some inkjet printers (Fig. 2.2). In passive pumps, the fluid is transferred through the channels by taking advantage of its intrinsic properties such as surface tension, molecular diffusion or osmotic pressure, which are the simplest examples of micropumps.
Figure 2.2: Classification of piezo inkjet (PIJ) printhead technologies by the defor-mation mode used to generate the drops (20).
2.2
A surface tension driven micropump
Putting two droplets of different sizes at the ends of a capillary tube or a micro channel, there will in general be a flow from the smaller droplet to the larger one due to the Laplace pressure difference (21).
Figure 2.3: Passive pumping chip containing a small input drop and a large output drop. (A) Before the placement of a drop. (B) and (C) During the flow (21).
Beebe (2007) proposed a simple pump which consists of a microchannel with two outlets connected to water droplets of different sizes. Due to its smaller radius, the smaller of the two droplets generates a larger Laplace pressure γ/R, with R the radius of curvature and γ the liquid-vapor surface tension. This should therefore result in a flow from the smaller drop into the larger one (21; 19), and the flow will stop once the smaller drop has been pumped entirely into the larger one (Fig. 2.3). For simplicity, they took the size of the larger droplet to be much larger than the other one so that they could neglect the Laplace pressure inside the larger drop. Hence, they did the calculations of the volume change just for the smaller drop. They observed that the shape of the drop changes in 2 different phases, as depicted in Fig. 2.4. At the beginning of the process the droplet is shrinking by only changing its contact angle while preserving its contact area radius. In the second phase the contact angle is fixed while the radius contact area is decreasing.
2.3
3-phase flow
The first part of our experiments were performed on polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning) blocks with a channel, made by pouring the uncured PDMS over a strained wire as a mold and baking it in an oven at 80◦ for 5 hours.
Figure 2.4: Recession of a drop during passive pumping modeled in two phases, each pictured in a different shade (21).
After pulling out the wire through the PDMS, we close the two ends (where the wire was taken out) with small drops of instant glue. We punch two holes perpendicular to the channel through the PDMS for the inlet and outlet. By this method we manufactured cylindrical channels of length L = 25.5 mm and diameter D = 110 µm (Fig. 2.7a). After the channel was filled completely with water, using a syringe, we put two droplets with different sizes (each on the order of few µl) at the outlets of the channel. As the smaller droplet is emptying into the larger one, the height of the droplets and their contact radii were measured by analyzing the movies made with a CCD camera at 20 frames/s.
In contrast to the Beebe et al. (2007) experiments we took the size of the two drops to be initially of the same order. In this case we need to take care of the shape of both droplets to achieve the dynamics of the system.
Due to the pinning of contact line the shrinking or growing of a droplet may happen either at constant contact area or at constant contact angle. Combining the possibilities for each droplet there are 4 possible situations. In our experiments we observe three that happen consecutively in time. First Phase: In the beginning of the process, the contact areas remain constant for both drops, but the contact angles change. Second Phase: The smaller droplet has fixed contact area but the contact angle of the larger drop has reached the advancing contact angle θadv =
110◦; therefore the contact area starts to increase at constant contact angle (Fig.
2.6). Third Phase: The smaller drop has reached the receding contact angle θrec=
46◦, and its contact area starts to decrease at fixed contact angle. The important observation here is that this sequence happens because the receding contact angle is significantly smaller than the advancing one (Fig. 2.6). This is probably due to surface roughness and/or chemical heterogeneity.
2
H
2 R2 Rw 1 Rw1 R1H
1Figure 2.5: The parameters of the 2-drops experiments.
substrate. Therefore, starting from this angle, the larger droplet will reach its advancing contact angle much sooner than the smaller drop reaches its receding contact angle (Fig. 2.7(d) and (e)). This is also the reason why ‘phase 4’ is not observed in our experiments.
The experimental data for the three different phases compare favorably to the numerical solutions of the Stokes equation (see next subsection) for the geometry considered here (Fig. 2.7(b)-(g)). The favorable comparison shows that solely the wetting properties of the drops determine the flow speed and hence, inversely, controlling the wetting properties allows to control the flow.
2.3.1
2-drops system numerical analysis
Experimentally, the shapes of the droplets are spherical caps of height H; we need not consider gravity because for all cases considered here the Bond number, Bo = ρgHγ 2is significantly smaller than unity; here g is the gravitational acceleration and ρ the fluid density. For the numerics, we consider two droplets at the outlets of a channel of length L, with the Laplace pressures, P1 and P2 for drops of radii
R1 and R2, we have: ∆P = P1− P2= 2γ R1 −2γ R2 = 2γ Rw1 sinθ1− 2γ Rw2 sinθ2, (2.1)
where Rw1, Rw2are the contact radii and θ1, θ2 are the contact angles of drops 1
and 2, respectively (Fig. 2.5). For Poiseuille flow, the flow rate is then Q = Kch∆P ,
where for a cylindrical channel Kch= πD
4
128ηL, and for a rectangular channel Kch=
wb3
12ηL, with a channel height b and width w, where w >> b. Generally Kch depends
on the geometry of the channel and the fluid viscosity. The larger the value of Kch
the higher the average flow rate in the channel for a given drop pressure. Therefore this parameter can be used to tune the flux in the microchannel.
To quantify the dynamics, we solve this problem numerically. The flow rate is equal to the rate of volume change for each drop, i.e. Q = −dV1/dt = dV2/dt, where
generally V1 is the volume of the shrinking droplet (usually the smaller drop),
H
2R
2R
w2θ
2θ
adv H1 R1 Rw1 θ1 θ recFigure 2.6: The parameters of the two connected drops system are sketched. θadv
and θrec are also shown. The drop on the right is expanding while the left one is
shrinking. The initial contact angles are the same for both drops.
equations we arrive at: −dV2 dt = dV1 dt = Kch 2γ R1 − 2γ R2 . (2.2)
Using trigonometric relations for spherical caps, one can find expressions for the volume of a droplet in terms of two independent parameters such as the height H, the contact radius Rwor contact angle θ of the drop. As an example we have these
three formulas V (H, Rw) = π H3 6 + R2wH 2 , (2.3) V (Rw, θ) = πR3 w sin3θ 1 3(2 + cos 3θ) − cosθ , (2.4) V (H, θ) = πH 3cot3(θ/2) sin3θ 1 3(2 + cos 3θ) − cosθ . (2.5)
Therefore, using equation (2.2) for each phase, we will end up with 2 coupled differential equations for the height of the droplet for each phase. Hence, the equations of the first phase are
dH1 dt = 8γKch π H 2 (H2 2+R2w2)(H12+R2w1)− H1 (H2 1+R2w1)2 , dH2 dt = − 8γKch π H2 (H2 2+R2w2)2 − H1 (H2 1+R2w1)(H22+Rw22 ) , (2.6)
where Rw1 and Rw2 are constants. The second phase equations would be
dH1 dt = 4γKch π 1 − cosθ 2 H2(H12+ R2w1) − 2H1 (H2 1+ Rw12 )2 , dH2 dt = − 2γKch π 1 − cosθ2 2 + cosθ2 1 − cosθ2 H3 2 − 2H1 (H2 1+ R2w1)2) , (2.7)
with constant Rw1 and θ2. Finally, for the third phase we have dH1 dt = − 2γKch π 1 − cosθ1 2 + cosθ1 1 − cosθ2 H2H12 −1 − cosθ1 H3 1 , dH2 dt = − 2γKch π 1 − cosθ2 2 + cosθ2 1 − cosθ2 H3 2 −1 − cosθ1 H1H22 , (2.8)
where θ1and θ2are constants. With the initial conditions given by the experiments,
we can solve the equations numerically using Runge-Kutta methods. For each set of the equations (2.6), (2.7) and (2.8) we need two initial values H1(t = 0) and
H2(0), and three constants, Kch and e.g. Rw1, Rw2 for ‘phase 1’, which are given
by the experiment. We took the last point in the numerical solution of the previous phase as the starting point of the next phase in order to maintain the continuity of our solution (Fig. 2.7).
The nonlinear equations (2.6), (2.7) and (2.8) are very sensitive to the choice of initial values and parameters. A small error for example in the initial value of the contact radius can propagate though the phases and make larger deviations from the data, as can be seen in Fig. 2.7(f) and (g).
Using scaling arguments, we generalize the above equations. Scaling the length parameters with lν= η
2
ργ, and the time with τν =
η3 ργ2, we arrive at dHi0 dt0 = Afi(H 0 1, H 0 2, R 0 w1, R 0 w2), i = 1, 2 (2.9)
where H1,20 = H1,2/lν are the dimensionless heights, R0w1,w2 = R0w1,w2/lν the
di-mensionless contact area radii and t0 = t/τν is the scaled time and A is a
dimen-sionless parameter defined as γKchτν/l4ν. f1,2 are functions of H10, H20, R0w1 and
R0w2 that can be defined using equations (2.6), (2.7) and (2.8) for each phase.
2.4
Backflow
In the previous section we showed that controlling the wetting properties allows to control the flow. In this section we show how to create an unusual flow from a larger into a smaller droplet (Fig. 2.10).
2.4.1
Backgrounds
Ju et al. (2008) (22) claimed that they have seen a relatively small backward flow, (flow from the larger to the smaller drop) in the 2 drops capillary pump near the end of the process (Fig. 2.8). They first proposed the inertia in the channel is responsible for this backward flow, but they showed that it can not be the case. They also stated that the inertia of rotational flow in a droplet might be the reason of backflow. Using goniometer they just showed that this rotational flow entering the larger droplet exists, while there is no such flow inside the smaller droplet, Fig.
(c)
(b)
Phase 2 Phase 3 Phase 1 Phase 2 Phase 3(a)
0 2 4 6 8 10 12 0.2 0.4 0.6 0.8 1 1.2 Time(s) Experimental data Numerics 0 2 4 6 8 10 12 1.75 1.8 1.85 1.9 Time(s) (m m ) Experimental data Numerics 0 2 4 6 8 10 12 14 40 60 80 100 120 Time(s) ( Theoritical data Experimental data 0 2 4 6 8 10 12 14 108 110 112 Time(s) (deg) Theoritical dataExperimental data
(e)
0 2 4 6 8 10 12 14 0.65 0.7 0.75 0.8 Time(s) Experimental data Theoritical data 0 2 4 6 8 10 12 14 1.25 1.3 1.35 1.4 Time(s) Experimental data Theoritical data(f)
(g)
Phase 1 Phase 2 Phase 3 Phase 3 Phase 2 Phase 1 Phase 1 Phase 2 Phase 3 Phase 1 Phase 2 Phase 3 (m m ) (m m ) (m m ) (de g) Phase 1 Phase 2 Phase 3(b)
(c)
Phase 3 Phase 2 Phase 1(d)
Figure 2.7: a) Picture of the cylindrical micro-channel in PDMS. b), c) The heights, d), e) the contact angles f), g) the contact radii of the smaller and the larger droplets respectively. The first phase change happens at t = 2.5 s and the second at t = 9 s. The initial values for the numerics are H1(t = 0) = 1.11 mm, H2(0) = 1.76 mm,
2.9. But they failed to show how this inertia of the rotational flow is responsible for the backflow and if this effect is large enough enough to produce backward flow or not. They just made a qualitative description that: ‘The fluid entering the output droplet during the flow creates a rotational motion which can store inertia and cause an increase of pressure according to the NavierStokes equation.’ However,
Figure 2.8: Schematic view of backflow as seen in the experiments of (22) the Reynolds number in their experiment being Re = ρU R/η ' 10−2, where U is the velocity, casts some doubt on this interpretation of their experiments. Furthermore, in their experiments the ratio of inertial to surface tension forces is so small and of the order of ρRU2/γ ∼ 10−5.
2.4.2
Experiments and results
We show that an unusual flow from a larger droplet into a smaller one is possible by manipulating the wetting properties, notably the contact line pinning. If the drops are spherical caps, the pressure can be written as R2γ
wsinθ, where Rw and θ are the radius of contact line area and contact angle respectively, two experimentally controllable parameters. The dependence on the contact angle is interesting since the sine function and hence the Laplace pressure is maximum at θ = 90◦. Therefore, flow from a larger drop, i.e. the one with larger volume, into a smaller one is in principle possible by manipulating the contact angles. Notably, if the contact angle of the larger drop is close to 90◦, and the small drop is of similar size but has a significantly smaller contact angle, a flow from the large into the small droplet should be expected. Although the basic idea that the flow is controlled by the radius of curvature and not the drop size is of course well established (23; 21), we show here how it can be used for passive and active pumping.
To achieve this, we make a drop of volume V1and another one with a smaller volume
V2, but put the larger droplet on a hydrophobic Teflon surface, so that θ1 ∼ 90◦.
The smaller drop is on a hydrophilic glass surface; we thus have Rw1 . Rw2 but
sinθ2 < sinθ1. Under these initial conditions, the pressure in the larger droplet
will be larger (P1 > P2), and backflow will occur, as is indeed observed in the
experiments (Fig. 2.10).
The backflow experiments were performed on channels made by sandwiching a Parafilm, with the channel carved out in it, between two microscope slides. We
Figure 2.9: (a) Observation of the rotational flow in the outlet drop using a go-niometer. Each picture was taken at 1 s intervals, approximately. The scale bar is 500 µm. (b) There is rather stable flow in the inlet droplet and rotating flow in the outlet droplet. (22)
used a sand jet perforating technique to make in/outlets through one of the glass slides. To change the wetting properties we attached a layer of Teflon (PTFE) tape around one of the outlets, leaving the other side to be just glass.
(a)
(b)
(c)
Figure 2.10: Illustration of the flow from the larger drop to the smaller one, left drop is on glass, right one on PTFE a) t = 0.0 s b) 6.6 s c) 9.24 s.
In these experiments the initially larger drop has a constant contact radius (Rw1),
i.e. the contact line is pinned, and the initially smaller one has a constant contact angle (θ2). The results of analyzing our movies made by a fast CCD camera
operating at 125 frames/s plotted along with the numerical solutions are shown in Fig. 2.11.
It follows that the larger the difference in wetting properties of the two surfaces beneath each drop, the more likely it is to observe backflow. A few examples of the numerics are shown in Fig. 2.12, where we consider the second phase (θ2,
Rw1 = const), with V1 > V2 initially, and plot the pressure difference ∆P , as a
function of the ratio of the volumes, V1/V2. Backflow occurs when ∆P = P1− P2
is positive and V1
V2 > 1; therefore the plots in Fig. 2.12 show that for smaller values of the contact angle of the initially smaller drop, backflow is more readily observed. This shows that the wetting properties can be used as a method for active pumping.
2.5
Electrowetting : bi-directional active
pump-ing
To be able to actively control the flow and its direction, we use electrowetting tech-niques to manipulate the wetting properties of a surface instantly and temporarily (24; 25; 26; 27). We use it here for building a bi-directional active pump. Directional control using this micropump with such a simple design can have many applica-tions. For instance, due to its biological compatibility, it is suitable for practical applications in microfluidic systems including micro total analysis systems (mTAS) and lab-on-achip systems (28; 29; 30).
0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x 10 −8 t (s) Volume (m 3) V 2 V 1 V 2 V 1 0 5 10 15 20 25 30 35 40 45 t (s) 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 9x 10 −10 t (s) Q (m 3/s) 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 −9 t (s) Volume (m 3) V2 V1 V2 V 1 0 2 4 6 8 10 12 14 −1 0 1 2 3 4 5x 10 −10 t (s) Q (m 3/s)
Figure 2.11: lhs: Rectangular channel with w = 4.2 mm, b = 0.1 mm and L = 24.75 mm. Viscosity is η = 1 mPa.s. We are in “phase 2”, with the constants Rw2 = 1.96 mm for the smaller and advancing angle on glass for the larger drop
θ1= θadv= 56◦. Evolution of the volumes of the two droplets (upper lhs), the flow
rate in time (lower lhs). rhs: Rectangular channel with w = 0.76 mm, b = 0.1 mm and L = 8.75 mm. Viscosity is η = 1 mPa.s. We are in “phase 2”, with the constants Rw2 = .955 mm for the smaller and advancing angle on glass for the
larger drop θ1= θadv= 56◦. Evolution of the volumes of the two droplets (upper
Figure 2.12: ∆P = P1− P2 as a function of the ratio of the volumes. The total
volume is V1+V2= 25 µl and Rw1= 1.6 mm for all the curves, we are in “phase 2”.
For smaller contact angles of the initially smaller droplet (θ2), ∆P is positive in a
wider range of V1/V2> 1, so the backflow can be seen with a larger probability. The
grey area corresponds to the range of parameters for which backflow is expected to occur.
Figure 2.13: Generic electrowetting set-up. Partially wetting liquid droplet at zero voltage (dashed) and at high voltage (solid). See the text for details (27).
2.5.1
Theoretical backgrounds
The spreading of a drop is determined by the equilibrium contact angle, which is set by intermolecular forces. However, the drop can be made to spread by the application of an electric field (31). This effect is called electrowetting.
Electrocapillarity, the basis of modern electrowetting, was probably first explained by Gabriel Lippmann in 1875 (32). He found that the capillary depression of mercury in contact with electrolyte solutions could be varied by applying a voltage between the mercury and electrolyte. The term electrowetting was first introduced in 1981 to describe an effect proposed for designing a new type of display device (33).
Electrowetting using an insulating layer on top of the bare electrodes was studied by Berge in 1993 (34) in order to eliminate the problem of electrolysis. Electrowetting on this dielectric-coated surface is called electrowetting-on-dielectric (EWOD) to distinguish it from the conventional electrowetting on the bare electrode.
In electrowetting, one is generically dealing with droplets of partially wetting liquids on planar solid substrates (Fig. 2.13). In most applications of interest, the droplets are aqueous salt solutions with a typical size of the order of 1 mm. The ambient medium can be either air or another immiscible liquid, frequently an oil.
Lippmann’s classical derivation of the electrowetting or electrocapillarity equation is based on general Gibbsian interfacial thermodynamics (35). Unlike in the re-cent applications of electrowetting where the liquid is separated from the electrode by an insulating layer, Lippmann’s original experiments dealt with direct metal (in particular mercury)-electrolyte interfaces (32). For mercury, several tenths of a volt can be applied between the metal and the electrolyte without any current flowing. Upon applying a voltage dE, an electric double layer builds up sponta-neously at the solid-liquid interface consisting of charges on the metal surface on the one hand and of a cloud of oppositely charged counter-ions on the liquid side of the interface. Since the accumulation is a spontaneous process, for instance the
adsorption of surfactant molecules at an air-water interface, it leads to a reduction of the (effective) interfacial tension γslef f:
dγslef f = −ρsldE, (2.10)
where ρsl(E) is the surface charge density of the counter-ions. The voltage
depen-dence of γslef f is calculated by integrating equation (2.10). We make the simplifying assumption that the counter-ions are all located at a fixed distance dH(of the order
of a few nanometres) from the surface (Helmholtz model). In this case, the double layer has a fixed capacitance per unit area, cH = ε0εl/dH, where εlis the dielectric
constant of the liquid. We obtain
γslef f = γsl− Z E Epzc ρsld ˜E = γsl− Z E Epzc cHEd ˜˜ E = γsl− ε0εl 2dH (E2− E2 pzc). (2.11)
Here, Epzc is the electric potential (difference) of zero charge. Note that
mer-cury surfaces, like those of most other materials, acquire a spontaneous charge when immersed into electrolyte solutions at zero voltage. The voltage required to compensate for this spontaneous charging is Epzc (27). The chemical
contri-bution γsl to the interfacial energy, is assumed to be independent of the applied
voltage. The interfacial energies (per unit area) γsv (solid-vapor), γsl (solid-liquid)
and γlv (liquid-vapor) are related to the Young’s equilibrium contact angle θY by
the following equation
cosθY =
γsv− γsl
γlv
. (2.12)
To obtain the response of the contact angle, equation (2.11) is inserted into Youngs equation (2.12). For an electrolyte droplet placed directly on an electrode surface we find cosθ = cosθY + ε0εl 2dHγlv (E2− E2 pzc). (2.13)
For typical values of dH (2 nm), εl(81), and γlv (0.072 mJ m−2) we find that the
ratio on the rhs of equation (2.13) is on the order of 1 V−2. The contact angle thus
decreases rapidly upon the application of a voltage.
It should be noted, however, that equation (2.13) is only applicable within a volt-age range below the onset of electrolytic processes, i.e. typically up to a few hundred millivolts. As mentioned already, modern applications of electrowetting usually circumvent this problem by introducing a thin dielectric film, which in-sulates the droplet from the electrode. In this EWOD configuration, the electric double layer builds up at the insulator-droplet interface. Since the insulator thick-ness d is usually much larger than dH, the total capacitance of the system is reduced
tremendously. The system may be described as two capacitors in series, namely the double at the solid-insulator interface (capacitance cH) and the dielectric layer
with cd = ε0εd/d (εd is the dielectric constant of the insulator). Since cd cH,
Figure 2.14: The apparent contact angle as a function of the root mean squared amplitude of the applied voltage (ac field) for a drop of oil. The filled squares are for increasing field (advancing contact line), the open circles for decreasing field (receding contact line). The solid line is the theoretical curve (27).
the finite penetration of the electric field into the liquid, i.e. we treat the latter as a perfect conductor. As a result, we find that the voltage drop occurs within the dielectric layer, and equation (2.11) is replaced by
γslef f(E) = γsl−
ε0εd
2d E
2. (2.14)
Here and in the following, we assume that the surface of the insulating layer does not give rise to spontaneous adsorption of charge in the absence of an applied voltage, i.e. we set Epzc= 0. In this equation the entire dielectric layer is considered part of
one effective solid-liquid interface with a thickness of the order of d, i.e. in practice typically O(1µm). In that sense, the interfacial energy in equation (2.14) is clearly an ‘effective’ quantity. Combining equation (2.14) with equation (2.12), we obtain the basic equation for EWOD:
cosθ = cosθY +
ε0εd
2dγlv
E2. (2.15)
Fig. 2.14 shows a typical experimental example. As in many other experiments, equation (2.15) is found to hold as long as the voltage is not too high. Beyond a certain system dependent threshold voltage, however, the contact angle has always been found to become independent of the applied voltage (36). This is called the contact angle saturation phenomenon.
2.5.2
Experiments and setup
In the previous section we observed flow from a larger droplet into a smaller one based on modifying the wetting properties of the surface at one side. However we did not observe a flow in both directions during a single experiment using only passive pumping. Even if in principle one should be able to create a change in flow direction during a single experiment, the flow rate would probably be so small (22) that the bi-directional pumping is very inefficient.
Teflon
V
Al
Figure 2.15: Schematic of the bi-directional pumping setup using electrowetting techniques.
To have a well-performing bi-directional pump in microchannels, we propose to control the wetting properties dynamically using electrowetting techniques (24; 25; 26; 27).
Putting an electrode inside a droplet of water and another one underneath an insulating surface, one can apply a voltage difference between the electrodes, which leads to an instantaneous decrease of the contact angle and hence Laplace pressure of the drop. This happens because the charges makes it more attractive for the fluid to wet the surface, due to the polarizability of water molecules (31). Therefore, turning the voltage on and off induces an abrupt change of the contact angle and can change the flow direction inside our microchannels.
If we have two droplets but the smaller one (left hand side of Fig. 2.15) has a larger curvature, the flow goes from the smaller droplet to the larger one. Turning on the voltage, due to the electrowetting effect the droplet will spread abruptly, have a larger contact radius and smaller contact angle, and hence a smaller Laplace pressure. If we make the curvature small enough (smaller than that of the larger drop on the right hand side), a flow in the reverse direction is indeed observed.
E = 0 V
E = 900 V
a
b
c
d
E = 0 V
E = 900 V
flow
Figure 2.16: a) The two drops before turning on the electric potential t = 0.1 s, b) t = 11.7 s. c) A potential difference of E = 900 V is applied, the contact angle and the direction of flow have changed, t = 14.8 s, d) t = 41.0 s.
Fig. 2.16 show that at the beginning of the experiment, the drop on the left hand side is evidently smaller and has a smaller curvature than the drop on the right, and a flow from left to right is observed. Turning on the voltage (900 V), the smaller drop spreads; the pressure difference ∆P changes sign followed by a change in flow direction.
The electrowetting setup is sketched in Fig. 2.15. We attach a layer of Teflon tape of thickness 0.25 mm to the upper glass surface, with an aluminized film under the Teflon tape around one of the outlets, that functions as one of the two electrodes. The results for the dynamics compared to the numerical solutions are shown in Fig. 2.17, for an experiment done in “phase 1” for which Rw2= const, and there
0 5 10 15 20 25 30 35 40 45 0.5 1 1.5 2 2.5x 10 −3 t (s) Height (m) H 1(OFF) H 2(OFF) H 1(ON) H 2(ON) H 2 H 1 0 5 10 15 20 25 30 35 40 45 2 4 6 8 10 12 14 16x 10 −9 t (s) Volume (m 3) V2(OFF) V1(OFF) V2(ON) V1(ON) V 2 V 1
Figure 2.17: a) H1 and H2 are the heights and b) V1, V2 the volumes of the
initially smaller and larger drops respectively. The dimensions of the rectangular channel are width w = 2.5 mm, height b = 0.12 mm, length L = 16.3 mm. The electrical potential E = 900 V is turned on at t = 15 s. Rw1(OF F ) = 1.59 mm,
2.6
Conclusion
In summary, we have shown experimentally that a system of two droplets of dif-ferent sizes at the ends of a microchannel can behave in three difdif-ferent ways. Con-sidering these different possibilities, we developed a numerical scheme needed for dealing with the system and the importance of the different phases for observing a flow from the larger droplet into the smaller one was shown. In addition, by changing the properties of the surface beneath each droplet we can generate situ-ations in which we get a backflow by choosing the appropriate initial conditions. This provides a different possible interpretation of the experiments by Berthier, Beebe and Ju et al. (22), who observed backflow in situations similar to the ones described here. They argue that inertia, originating from a rotating flow in the larger droplet, is responsible for the backflow. However, the Reynolds number in their experiment being Re = ρU R/η ' 10−2, where U is the velocity, ρ the density and η the viscosity, casts some doubt on their interpretation of the experiments. In addition, we have shown by taking advantage of dynamic manipulation of the effective surface properties by electrowetting, that a bidirectional micropump can be made. The possibility of changing the flow direction back and forth by simply turning the voltage off and on, can make the capillary pumping method a powerful tool in microfluidic technologies.
Dynamics of Ferrofluid drops
3.1
Introduction
In the previous chapter, we focused on the Laplace pressure inside a liquid drop deposited on a solid surface. We showed how the wetting properties of the solid surface and the pinning of contact line can be used to manipulate the Laplace pressure inside drops and hence the flow rate of our capillary micropump. In doing this, we did not take into account the effect of contact line dynamics of the drops, because the contact line speed was very small.In this chapter, we focus on the contact line dynamics of a drop.
Understanding the dynamics of contact lines has become of a major interest as it plays an important role in problems related with industrial applications (31). A large research effort has gone into the control of the interaction between solid substrate and liquids in order to help or prevent the spreading of liquids. Coating and drying solid surfaces are important problems as well. Often, these phenomena are controlled by what happens at the frontier between the wetted and dry part of the substrate: i.e. the contact line (C.L).
One of the key questions is how a contact line changes in response to the dynamics of the spreading/withdrawal of the liquid. Several features emerge: the wriggling dynamics of a C.L. on heterogeneous substrate, or the instability of a receding con-tact line leading to the formation of a film, known as the Landau-Levitch transition (37).
The study reported in the present chapter deals with the interaction between the shape and the dynamics of the C.L. But, instead of imposing the motion between liquid and solid, as is usually done, we impose the shape of the contact line and ob-serve the consequences on its dynamics. To achieve this goal, we have built objects we have called “shaped drops”. These are obtained by pouring some ferrofluid on a small neodynium magnet. The ferrofluid wraps around the magnet, and forms a film between the bottom of the magnet and the supporting substrate. Thus,
the magnet levitates in the core of a drop that keeps the geometry imposed by the magnet. Therefore, contact lines of unusual shapes can be made by changing the geometry of the magnet as desired. As a result, one can study the motion of contact lines of arbitrary shapes (Fig. 3.8).
3.2
Ferrofluids
The study of various fields and interactions with fluids may be divided into three main categories (38):
1. electrohydrodynamics (EHD), the branch of fluid mechanics concerned with electric force effects;
2. magnetohydrodynamics (MHD), the study of interaction between magnetic fields and fluid conductors of electricity;
3. ferrohydrodynamics (FHD), the study of fluid motion influenced by strong forces of magnetic polarization.
It is important here to emphasize the difference between ferrohydrodynamics and the relatively better known discipline of megnetohydrodynamics. In MHD the body force acting on the fluid is the Lorentz force that arises when electric current flows at an angle to the direction of an imposed magnetic field. However, in FHD there need not be an electric current flowing in the fluid and usually there is none. The body force in FHD is due to polarization force, which in turn requires material magnetization in the presence of magnetic field gradients or discontinuities. Ferro-hydrodynamics began to be developed in the early to mid-1960s, motivated initially by the objective of converting heat to work by no mechanical parts (39).
Several types of magnetic fluids can be used for FHD; the principal type is col-loidal ferrofluid. A colloid is a suspension of finely divided particles in a continuous medium, including suspensions that settle out slowly. However, a true ferrofluid does not settle out, even though a slight concentration gradient can become es-tablished after long exposure to a force field (gravitational or magnetic). Such ferrofluids are composed of small particles (3-15 nm) particles of solid, magnetic, single-domain particles coated with a molecular layer of dispersant and suspended in a liquid carrier (see Fig. 3.1). Thermal agitation keeps the particles suspended because of Brownian motion, and the coating prevent the particles from sticking to each other.
The colloidal ferrofluid must be synthesized, since it is not found in nature. A typical ferrofluid contains 1023 particles per cubic meter and is opaque to visible
light.
3.2.1
Stability requirements
Dimensional reasoning may be used to arrive at criteria for physicochemical sta-bility. To begin it is useful to write expressions for various energy terms. These energies per particle are (38):
Figure 3.1: Schematic representation of coated, subdomain, magnetic particles in a colloidal ferrofluid. Collisions of suitably coated particles are elastic (38).
thermal energy = kT magnetic energy = µ0M HV
gravitational energy = ∆ρV gL
where k is Boltzman’s constant and equals 1.38×10−23 N · M · K−1, T is abso-lute temperature in Kelvin, µ0 is the permeability of free space an has the value
4π × 10−7 H · m, volume V = πd3/6 for a spherical particle of diameter d, and
L is the elevation in the gravitational field. Ratios of one term to another yield dimensionless quantities that inform about the stability of the ferrofluids.
Stability in a magnetic field gradient
Consider the stability against settling of particles in a field gradient due to an ex-ternal magnetic source. Particles are attracted to the higher intensity regions of a magnetic field, while thermal motion counteracts the the field force and provides statistical motions that allow the particles to sample all portions of the fluid vol-ume. The magnetic energy µ0M HV represents the reversible work in removing a
magnetized particle from a point in the fluid, where the field is H, to a point in the fluid that is outside the field:
W = − Z
(µ0M
dH
Provided some part of ferrofluid volume is located in a field-free region, then sta-bility against segregation is favored by a high ratio of the thermal energy to the magnetic energy: thermal energy magnetic energy = kT µ0M HV ≥ 1. (3.2)
Rearranging and substituting for the volume of a sphere gives an expression for the maximum particle size:
d ≤ (6kT /πµ0M H)1/3. (3.3)
Consider the conditions existing in a beaker of magnetic fluid containing mag-netite (F e3O4) particles subject to magnetic gradient field of a typical hand-held
permanent magnet:
H = 8 × 104A · m−1, (3.4)
M = 4.46 × 105A · m−1, (3.5)
T = 298 K, (3.6)
the particle size then follows as d ≤ 8.1 × 10−9 m or 8.1 nm. The actual particle
size of stable colloids range up to about 10 nm. Stability against settling in a gravitational field
The relative influence of gravity to magnetism is described by the ratio gravitatinal energy
magnetic energy = ∆ρgL µ0M H
. (3.7)
Again for a beaker of fluid, typical values of parameters are L = 0.05 m and ∆ρ = ρsolid− ρf luid = 4300 kg m−3; with g = 9.8 ms−2, the ratio is 0.047. Thus
gravity is less of a threat to the segregation of these magnetic fluid than is magnetic field.
3.2.2
Some applications of ferrofluids
Applications of ferrofluids span a very wide range. Commercial use presently in-cludes novel zero leakage rotary shaft seals used in computer disk drives (Bailey 1983) (40), vacuum feed-throughs for semiconductor manufacturing and related uses (Moskowitz 1975) (41), and more. Also in use are liquid cooled loud-speakers that employ mere drops of ferrofluid to conduct heat away from the speaker coils (Hathaway 1979) (42). This innovation increases the amplifier power the coils can accommodate and hence the sound level the speaker produces. A magnetic field
Figure 3.2: Shape of the drops at rest, when ferrofluid is added to a magnet with height l=2 mm and diameter d=10 mm, on a horizontal glass surface. a) The magnet. b) Added fluid volume = 400 µl, c)1000 µl, d) 2800 µl.
5 10 15 20 25 30 35 40 5 10 15 20 25 30 X(mm) Y(mm)
Figure 3.3: Experimentally obtained surface shapes for the magnet l = 2 mm, d = 10 mm on a solid plastic surface with volumes of fluid: 0, 600, 1200, 1800, 2400, 3000, 3600, 4400 µl.
can also pilot the path of a drop of ferrofluid in the body, bringing drugs to a tar-get site (Morimoto, Akimoto and Yotsumoto 1982) (43) , and ferrofluid serves as a tracer of blood flow in noninvasive circulatory measurements (Newbower 1972) (44).
An especially promising application under study is the use of magnetic fluid ink for high speed, inexpensive, silent printers (Maruno, Yubakami and Soga 1983) (45) . In one type of these printers, as many as 104drops per second issue from a
tiny orifice and are guided magnetically to form printed characters on a substrate (Kuhn and Myers 1979) (46).
Figure 3.4: Ferrofluid added to the magnet l = 2 mm, d = 10 mm suspended in the air with its symmetry axis parallel to the gravity direction with different volumes of the fluid a) v = 400 µl b) 1200 µl c) 2600 µl d) 4000 µl.
3.3
Drop Characteristics
In our experiments, we have used a commercial ferrofluid (EFH1 by Ferrotec) on a flat NdFeB magnet with a thickness of 1 mm and typical size of 1 cm (Fig. 3.2 and 3.3). The overall shape of the liquid surface of the drop is strongly dominated by the magnetic forces. We have checked that gravity has only a small influence by obtaining a similar drop shape on a magnet suspended by strings (Fig. 3.4 and 3.5). The capillary forces appear to play a role only in the vicinity of a substrate to which the liquid surface connects under a small contact angle. The interface is there deformed over a typical distance of 0.1 mm, that we can call a magneto-capillary distance lmc= γ/µ0M H, where γ is the surface tension, H the magnetic field and
M is the magnetic polarisation of the ferrofluid. The measured lmc is consistent
with γ = 0.025 N.m−1, H = 2 × 103A.m−1and M = 105A.m−1. We measured the elevation of the magnet inside the drop by attaching a flat index under the magnet poking out of the ‘shaped drop’ surface (Fig. 3.7).
It would be an interesting challenge to find the equilibrium shape of the ferrofluid around magnets of different geometries. In principal this can be done by minimizing the magnetic bulk energy along with the free surface energy of the ferrofluid. We studied these shapes, for different magnet geometries both when the magnet was on a solid surface and when it was suspended in air.
Throughout this section and the next, l will stand for the height of the cylindrical magnet, d for its diameter and v for the volume of the ferrofluid added to the magnet.
We checked that the viscosity of our ferrofluid is not affected by the magnet. Ac-cording to Shliomis (47), the change in effective viscosity due to the magnetic field |H| = H and the magnetic moment of the particles M , if the Langevin parameter ξ = M H/kT 1, is
∆η ∼=3 2ηφξ
5 10 15 20 25 −2 0 2 4 6 8 10 12 14 16 18 20 X(mm) Y(mm)
Figure 3.5: Disk magnet of l = 2 mm, d = 10 mm suspended in the air in horizontal position (with its symmetry axis parallel to the gravity direction), the volumes are : 200, 600, 1200, 1800, 2400, 3000, 3600, 4000 µl, respectively. 0 200 400 600 800 1000 1200 1400 1600 1800 0.0 0.5 1.0 1.5 2.0 2.5 C m = A + B*v A -0.31036 ±0.14343 B 0.00146 ±0.00014 Experiments Linear fit C m ( m m ) v ( L
Figure 3.6: Vertical component of the center of mass of the ferrofluid on the magnet l = 2 mm, d = 10 mm as a function of the fluid volume.
Figure 3.7: Measurement of elevation height of the magnet, the wire is attached to the bottom of the magnet l=1, d=10 mm, and it is sticking out.
where φ is the particle volume concentration and δ is the angle between the vectors H and Ω = ∇ × v/2. Since in our experiments φ ∼ 10−1, ξ ∼ 10−1 and sin2δ ∼ 1 at most, hence ∆η ∼ 10−3η. This makes the influence of magnetization on viscosity
negligible in our experiments. Our test fluid is a Newtonian fluid with η = 7.8 mPa.s.
3.4
Dynamics:
Sliding Shaped drops on an
in-clined plane
Since the magnet is not touching the surface, the overall drop has a very small static friction coefficient, so that it will slide down the plate even with the slightest inclinations. In our experiment, we let the drop slide down a flat plate with an inclination with respect to the horizontal θ ranging between 2.2◦ and 18◦. The motion of the drop is observed on a typical distance of 70 cm. After a short distance, of the order of 10 cm, the drop has a constant velocity. The most striking observation is the direction taken by the drops.
As a drop made with a disc-shaped magnet slides down the plate along the steepest descent, drops that do not have a plane of symmetry will take other directions. As an example, a drop built from a Tetris-like assembly of four square magnets in a shape of a ‘S’ deviates to the left of the the steepest descent under a deviation angle ϕ. The symmetric shape ‘Z’ deviates to the right, taking a path symmetric to the one of the ‘S’ drop with respect to the steepest decent (Fig. 3.8). A more surprising behaviour is observed with drops made from a half-disc magnet (obtained by breaking a disc magnet in half). When put in motion, its planar symmetry is broken, each drop deviates from the steepest slope with a 50-50 distribution to the left and the right side.
Due to wetting, the magnet leaves a thin film of fluid behind as it slides down the surface. We have weighted the mass of fluid left behind, on a distance of 50 cm, a drop loses around 3% of its total mass which can be assumed constant.
d
c
b
a
Figure 3.8: Top: picture of the magnet used to make shaped drops. Bottom: Pictures of the moving shaped drops, the vertical side of the picture is parallel to the steepest descent. a) A drop made from a disc slide down the steepest decent. A drop made from four square magnets (sticking together by dipolar interaction) in a shape of a ‘S’ Tetris piece (b) deviates to left, while a ‘Z’ piece (c) deviates in the symmetric direction. Made from a half-disc magnet (d) deviates from the steepest descent with a constant angle, but with a random side.
Figure 3.9: The schematic of the setup for the ferrofluid drop moving on the tilted plate
Figure 3.10: The movie of the fluid ball (magnet geometry: l =2, d =8 mm), sliding down the tilted plate. With tilt angle: 0.149 rad. Rate of the movie capture is 125 frames/s. The camera is rotated to make the horizontal axis of movie frames parallel to the shaped drop’s velocity.
The motion was captured with a CCD at 125 frames/s, placed vertical to the path of the ball, to measure its speed. The setup and a sequence of our movies with the drop moving on the tilted plate are shown in Fig . 3.9 and 3.10, respectively. To measure the location of the drop in each frame we used a technique to achieve sub-pixel accuracy (Fig. 3.11). This method helps for getting smoother time derivatives of position. We used the function ‘find-edge’ in ‘Labview’, which ba-sically assigns higher values to the edges in the image, by taking a derivative of the image matrix values (Fig. 3.11c). Apparently, The edges of the drop has a thickness (of a few pixels), and there is a maximum value somewhere in the mid-dle, the coordinate of this maximum is the exact location of the edge of the drop (Fig. 3.11d). By fitting a gaussian curve to the points nearby each peak, we find the position of the edge (and hence the drop) with sub-pixel accuracy (Fig. 3.11). When performed at different slopes θ the velocity is found to be proportional to sin θ (Fig. 3.12). For a given drop (same magnet and amount of ferrofluid), the deviation angle ϕ is independent of the slope.
3.4.1
Discussion on the acting forces
To understand this surprising behaviour, we have to list the forces acting on the drop. As it reaches a constant velocity, the net force acting on the shaped drop vanishes. The component of its weight parallel to the plate is mg sin θ. One may
Figure 3.11: a) The l =2, d = 8 mm on the tilted plate while moving down. Ferrofluid mass is 1.005 g. b) The intensity values on the line defined in (a). c) The edges of the image (a). d) The intensity values on the line defined in (c), this diagram can be considered as the derivative of the diagram (b). The precise position of the borders of the droplet is the location of the peaks in this diagram. Sub-pixel accuracy is achieved by fitting a gaussian curve to the points nearby each maximum peak, and taking the curve’s peak x-coordinate.
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 sin V ( m / s ) l=1 mm, d=10 mm, m=1.216 g l=2 mm, d=8 mm, m=1.00 g l=1 mm, d= 6 mm, m=0.475 g V ( m / s ) l=1 mm, d= 10 mm, m=0.51
Figure 3.12: Steady velocity of the drop as a function of Sine of the tilt angle of the plate. Top: different disk magnets. The total mass of the shaped drop is m. Bottom: a half disk magnet.
