Quantitative visualization of flow inside an evaporating droplet using

INSTITUTE OF PHYSICS PUBLISHING

MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 15 (2004) 1104–1112

PII: S0957-0233(04)71729-4

Quantitative visualization of flow inside an evaporating droplet using the ray tracing method Kwan Hyoung Kang1, Sang Joon Lee1, Choung Mook Lee1 and In Seok Kang2 1

Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea 2 Department of Chemical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790-784, Korea E-mail: [email protected]

Received 11 November 2003, in final form 25 February 2004 Published 13 May 2004 Online at stacks.iop.org/MST/15/1104 DOI: 10.1088/0957-0233/15/6/009

Abstract Liquid droplets possess many practically important applications and academically interesting issues. Accurate flow data are necessary to correlate the hydrodynamic characteristics with the physicochemical processes occurring inside a droplet. However, the refraction of light at the droplet surface makes it difficult to measure the flow field inside the droplet accurately. To resolve this problem, two correction methods based on the ray tracing technique are employed. One is the image mapping method and the other is the velocity mapping method. For this, a mapping function between the image plane and the object plane is derived. The two correction methods are applied to the flow inside evaporating droplets of different ethanol concentrations for measuring their velocity fields, using a PIV method. The results obtained with the two methods are nearly identical. The major differences between the original results and the corrected results are found in the locations of the vortex centres and the magnitude of velocity vectors. Between the two correction methods, the velocity mapping method is recommended, because it is more convenient and recovers a greater number of velocity vectors, compared with the image restoration method. Keywords: fluid flow velocity, PIV, ray tracing, droplet, evaporation

1. Introduction Droplets have many interesting applications associated with microfluidic problems, e.g., DNA molecule imaging [1, 2], micro-pumps and ink-jet printing. The details of dropletrelated phenomena in micro- and nanoscales such as evaporation process, Marangoni effects, contact angles with solid substrates and electrowetting are not well known. There is a consensus of opinion that the fluid flow inside a droplet may play an important role in the overall transport phenomena. Some of the interesting issues concerning the fluid flow inside a droplet are described below. 0957-0233/04/061104+09$30.00

Uno et al [3] investigated the adsorption characteristics of colloidal suspensions during the evaporation of a droplet on hydrophobic and hydrophilic surfaces. As they pointed out, the particle accumulation phenomena will be closely related to the flow pattern inside a droplet. Together with this, the mechanism of DNA stretching in an evaporating droplet [1, 2] is not completely known. As will be shown later in this paper, a regular flow pattern is formed due to evaporation, which is believed to be a key factor for the DNA stretching. Another concern is about the contact angle of evaporating sessile droplets placed on a solid substrate. According to Erbil

© 2004 IOP Publishing Ltd Printed in the UK

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Quantitative visualization of flow inside a droplet

(a)

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Figure 1. Typical particle image and velocity vectors of the flow inside an evaporating droplet. 3

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et al [4, 5], the behaviour of an evaporating sessile droplet is significantly dependent on the initial contact angle. That is, if the initial contact angle is smaller than 90◦ , the contact area is almost constant during the overall evaporation stage. On the other hand, if the initial contact angle is greater than 90◦ , the contact area shrinks, while the contact angle remains almost invariant. As suggested by Rowan et al [6] and McHale et al [7], this intriguing phenomenon can be closely related to the Marangoni force and Marangoni convection (generated by surface-tension gradient), near the three-phase contact line of air, liquid and solid. One effective method for characterizing the electrochemical characteristics of redox active solid is to deposit a small droplet, and then obtain the voltammetric curve for measuring the ionic transport. It is reported that convective motion exists inside droplets, which may be generated by the Marangoni effect, the EHD effect or the evaporation of species [8]. The redox process at the interface is significantly dependent on the convective transport of ionic species [9]. For an accurate assessment of the electrochemical characteristics of surfaces, the flow characteristics and the correlation with the ionic transport should be revealed. Considering its importance to fluid flows, only a few systematic investigations have been performed in the past towards the understanding of overall transport phenomena inside a droplet. Among the few who have investigated the flow structure inside droplets, Chung and Trinh [10] visualized the flow behaviour inside an ultrasonic–electrostatic hybrid levitation system, and Zhang and Yang [11] visualized the unstable flow due to Marangoni instability of an evaporating droplet of fluid mixture. Savino and Monti [12] investigated the flow inside sessile and pendant droplets numerically. We provide detailed information on the flow field inside a droplet in various situations, by using the PIV (particle image velocimetry) technique. Many problems are encountered, and one of the most significant problems is the image distortion due to the refraction of light at the droplet surface. It requires a proper procedure to eliminate the distortion effect to obtain accurate data. Without this correction, an accurate quantitative analysis of the internal flow may be nearly impossible. We developed a velocity correction method on the basis of the ray tracing method. For this, a mapping function between the points on the image plane and the object plane is derived for an axisymmetric droplet. The velocity correction method is further divided into two methods. One is to use the mapped particle images for obtaining the velocity vectors by the PIV process (image mapping method). The other is what we call the velocity mapping method. In the latter method, the velocity vectors obtained from the original particle images by PIV are directly mapped onto the object plane, without the image restoration procedure. The developed correction methods are applied to visualize the natural convection inside an axisymmetric droplet. So far as we know, we are the first ones to demonstrate the existence of such a flow and measure the flow field quite accurately. The flow is conjectured to be driven by uneven evaporation of species in a multi-component mixture. This kind of flow is especially beneficial for the validation of the present method because it is easier to generate and ensure a high degree of reproducibility without a sophisticated control mechanism.

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Figure 2. Tracing of light rays emanating from a point in a hemispherical droplet.

2. Mapping functions for an axisymmetric droplet Figure 1 shows a typical image of seeded particles in a cross-sectional plane of a (nearly axisymmetric) evaporating droplet and the velocity vectors obtained with the PIV method. Figure 1(b) shows a strong upward flow in the centre region and a relatively weak downward flow at the boundary region of the droplet. With the velocity vectors alone, the continuity requirement appears to be violated. This results from the distortion of particle images due to the lens effect of the droplet itself. Thus, it becomes obvious why the flow field should be corrected. As the wavelength of the light beam becomes much smaller than the size of the droplet and the lens, the effect of diffraction becomes less significant [13]. Then, the geometrical optics can be used to trace the propagation of the light beam without solving the wave equation. Figure 2 shows the computed result of the propagation of rays emanating from a point in the cross-sectional plane of a hemispherical water droplet by using the ray tracing method. For computation, the refractive indices of the droplet and air are chosen as 1.33 and 1, respectively, and the refractive index of the lens is arbitrarily chosen as 2. The radius of curvature of the spherical lens is 6R, where R is the radius of the hemispherical droplet. Among the many rays emanated from each point, only a small portion of the ray 1105

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y Pi ' Po

(x, y, z) Cartesian coordinate system and the(r, θ, φ) spherical coordinate system which have the origin at the centre of the base circle of the droplet (see figure 4). The unit vectors in the x, y and z directions are denoted by i, j and k, and those in the r, θ and φ directions are denoted by er , eθ and eφ , respectively. The shape of an axisymmetric droplet can be represented by the function F (r, θ) defined as

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F (r, θ) = r − ζ (θ ) = 0.

Figure 3. Tracing of a ray normally incident to the object plane. (a)

The shape function ζ (θ ) is represented by a sum of the cosine series as shown below

y Object plane

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The droplet surface is usually well represented by ζ (θ ) = R + b cos θ, and all the higher order terms of the cosine series are neglected for simplicity of analysis. The unknown constants R and b are determined using a digitized image of the droplet. The least-square method is employed for curve fitting of the droplet boundary to equation (3). We will employ two methods for the velocity correction. The first method is to obtain the velocity vector field from the mapped particle images (image mapping method). The other method is to map the velocity vectors in the original image plane onto the object plane (velocity mapping method). 2.1. Image mapping method

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Since most of the rays passing through the aperture will converge to the focal point of the lens, it is enough to trace the representative ray. We trace the normal incident ray to the lens. First, we draw a straight line (parallel to the z-axis) from a point on the image plane Pi (xi , yi , zi ) to the point on the droplet surface Ps (xs , ys , zs ), at which xs = xi and ys = yi . −−→ Then, the direction vector of Ps Po can be obtained by using Snell’s law, and the coordinates of Po (xo , yo , zo ) are related to those of Pi (xi , yi , zi ) by the following equations (see the appendix, for detailed procedures):

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Figure 4. Coordinate system and the vector relationship at the surface of the droplet.

bundles is collected by the aperture (i.e., the lens), depending on the aperture size and the distance from the object. The rays are refracted as they pass through the droplet surface. The incident and the transmitting angles of a ray passing through the droplet surface are related by the following Snell’s law of refraction nd sin ψd = na sin ψa ,

(1)

where na and nd are the refractive indices of air and the fluid inside the droplet, respectively, and ψa and ψd are the respective incident and transmitting angles of the rays (see figure 3). The geometrical relationship between the point Pi on the image plane and the point Po on the object plane (see figure 4) for an axisymmetric droplet is derived. We introduce the 1106

N
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(2)

xo = xi + zs Bx ,

(4a)

yo = yi + zs By ,

(4b)

where Bx = −f tan(ψa − ψd )(rs − b cos θs )sin θs sin φs , By = −f tan(ψa − ψd )(rs cos θs + b sin2 θs ),  f = cos2 φs (rs cos θs + b sin2 θs )2  −1/2 + sin2 φs rs2 + b2 sin2 θs ,  zs = rs2 − xs2 − ys2 ,   rs = 12 R + R 2 + 4bys . All the points on the image plane are transformed to the object plane using the above equations. The velocity vectors can be obtained by using the transformed particle images.

Quantitative visualization of flow inside a droplet

2.2. Velocity mapping method

Micro pipette

Alternatively, the velocity vectors can also be obtained by a direct mapping of the velocity vectors obtained in the image plane to the object plane. That is, if we take the time derivative of equation (4), we get ∂(zs Bx ) ∂(zs Bx ) dxo d(zs Bx ) = ui + = ui + u i + vi , uo = dt dt ∂xi ∂yi (5a) d(zs By ) ∂(zs By ) ∂(zs By ) dyo = vi + = vi + ui vo = + vi , dt dt ∂xi ∂yi (5b) where (ui , vi ) and (uo , vo ) denote the velocity vectors in the (x, y) directions in the image plane and the object plane, respectively. On the other hand, we will check the validity of the image restoration method for the case of a hemispherical lens. For a hemispherical droplet having a radius of R, b becomes zero in the shape function. Then, equation (4) becomes xs zs f tan(ψa − ψd ), (6a) xo = xi −  xs2 + ys2 ys zs yo = yi −  f tan(ψa − ψd ). (6b) xs2 + ys2 Let us introduce ηi and ηo to represent the distance to the points from the origin in the image plane and the object plane, respectively, that is  ηi = xi2 + yi2 ,  ηo = xo2 + yo2 . Then, the relation between ηi and ηo can be obtained as follows: ηo = ηi − zi tan(ψa − ψd ), where zs =

(7)

 R 2 − ηi2 .

3. Experimental apparatus and method A droplet of 3 µl (the diameter of which is about 2 mm) is placed with a micro pipette on a glass substrate having a thickness of 2 mm (see figure 5). The liquid tested in this investigation is ethanol mixtures with deionized water. The ethanol concentration is varied to have a volume ratio of 1%, 5% and 20%. The substrate is coated, by the dip coating method, to make the surface hydrophobic, that is, to increase the contact angle of a droplet. An amorphous fluoropolymer R called Teflon AF1600 of Du Pont is used as the coating material. The thickness of the amorphous fluoropolymer is about 50 µm. The static contact angle of water droplets on the coated surface is about 114◦ under standard atmospheric conditions. To prevent external disturbances to the flow inside a droplet and for the control of humidity, a base substrate is placed inside a test chamber whose dimension is 200 W × 300 L × 200 H (mm3). The temperature and the relative humidity inside the chamber are kept in the range of 25 ◦ C ± 0.5 ◦ C and 45% ± 5%, respectively.

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Droplet Test chamber

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Figure 5. Experimental setup to measure the flow field inside a droplet.

The two-frame cross-correlation PIV method is employed to obtain the velocity vectors [14]. The interrogation window has a size of 48 × 48 pixels and overlapped 50%. When we used the scattered light to obtain the flow images, the strong reflection of the laser light on the droplet surface deteriorates the quality of the image, especially near the bottom of the droplet. This problem is resolved by capturing the fluorescence light emitted from fluorescent seeding particles by filtering out the scattered light using an optical filter. A dual-head Nd:YAG laser (Minilase, New Wave) is used for illumination. The Nd:YAG laser (λ = 532 nm) has a maximum output energy (per pulse) of 25 mJ. The Nd:YAG laser is synchronized with a CCD camera and a pulse-delay generator. The time delay between the two consecutive images is changed to 1 s, 0.5 s and 0.01 s for the ethanol concentrations of 1%, 5% and 20%, respectively. The beam is expanded to make a laser light sheet of about 0.2 mm thickness by passing through cylindrical lenses. A cooled CCD camera (SensiCam, PCO) having 1280 × 1024 pixels array is used to capture the particle images. We attached a long distance zoom lens (QM100, Questar) in front of the CCD camera. According to the specification of the zoom lens, the depth of focus of the present imaging system is about 1 mm. As tracer (seeding) particles, fluorescent (polystyrene) particles having a mean diameter of 3 µm are used. Their density is 1.05 g cm−3 and the refractive index is 1.59 at λ = 589 nm. The aqueous suspensions are packaged as 1% solids in a multi-component dispersing system which prevents clumping. The number density of seeding particles is 6.7 × 108 ml−1. In the experiment, the concentration of the particle suspension is about 2% in volume ratio, so that the number density of the particles in the solution is about 1.3 × 104 µl−1. For the restoration of distorted images, the ray tracing method is used. For image restoration, the droplet shape should be known in advance. The droplet shape can be approximated as part of a sphere only when the capillary length √ ρg/γ is much smaller than the dimension of the droplet. Here, γ is the surface tension at the air–water interface, ρ is the liquid density and g is the constant of gravitational acceleration. For water, the capillary length is about 2 mm. In the present experiment, the diameter of the droplet is typically in the range of 2 mm. Hence, the flattening of the droplet due to the gravity should be considered. McHale et al [15] approximated the shape of droplets as an elliptical cap. In the 1107

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present investigation, the droplet shape is approximated by a sum of cosine series, as shown in equation (3).

4. Results and discussion

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Figure 7. A hemispherical lens and its image distortion. (a) Hemispherical Plexiglas lens; (b) original image; (c) restored image. 1

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Figure 6 shows the boundary of a droplet surface for a typical case, together with the curve obtained using the least-square curve fitting. The shape approximated by a part of a sphere is also included. In the figure, the open circles denote the edge point of the raw image, the solid line represents the case of N = 1 in equation (3), while the dash-dotted line is for the case of a spherical surface (N = 0). It is shown that the boundary surface of the droplet can be well represented by the cosine series. The standard deviation between the experimentally obtained profile of the droplet shape and the curve-fitting profile is less than 0.01% which is smaller than the image resolution. The other curve-fitting methods such as those of Erbil et al [4] and Li et al [16] do not account for the deformation due to other forces such as electromagnetic force. The present method does not have such a limitation, and will be especially suitable for assessing the electromagnetic influence on the contact angle. The usefulness of the present image restoration methods which incorporate the ray tracing method is checked by a hemispherical Plexiglas lens shown in figure 7(a). Figure 7(b) shows the distorted image of 30 × 15 meshes due to the presence of the hemispherical Plexiglas lens. In the figure, the distorted mesh region corresponds to the vertical crosssectional plane of a droplet. The centre region is magnified about 1.5 times with a moderate distortion, while the edge region is significantly distorted. The image restored using the ray tracing method is shown in figure 7(c). The centre region is well restored, while the accuracy of image restoration is not so good in the outer region of ηo /R > 0.75, in which ηo denotes the radial distance from the centre to a point in the bottom plane of a hemispherical lens, and R is the radius of the hemispherical lens. This indicates that it is difficult to restore the distorted meshes in the edge region accurately. The limited resolution of the digitized image also contributes to the restriction in the image restoration in the edge region. Despite this shortcoming at the edge region, the ray tracing method provides accurate flow images in the inner region of the droplet, i.e., ηo /R < 0.75.

0.4 Air (n=1.00) Water (n=1.33) Plexiglas (n=1.55) Diamond (n=2.42) Experiment (Plexiglas)

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Figure 8. Distortion effect of a hemispherical lens.

Figure 8 shows the distortion effect of the hemispherical lens for four different materials: water (n = 1.33), Plexiglas (n = 1.55), air (n = 1) and diamond (n = 2.42). The results obtained from figure 7(b) are also shown by rectangular symbols. In the figure, ηi is the radial distance from the centre to a point in the photographed image. The predicted results by using the ray tracing method agree well with the results obtained from figure 7(b). This reconfirms that the ray tracing method is useful in assessing the image distortion effect of the known shape of geometry. For the case of Plexiglas, the outer region 0.75 < ηo /R < 1 in the object plane is shrunken to 0.95 < ηi /R < 1 in the image plane. The greater the degree of distortion the greater the refractive index. However, the refractive index of water is 1.33 which is smaller than that of Plexiglas of 1.55. Therefore, it is anticipated that the

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Figure 9. Comparison of the instantaneous velocity field inside an ethanol droplet of 1% concentration: (a) raw image method; (b) velocity mapping method; (c) image mapping method.

Figure 10. Comparison of the instantaneous velocity field inside an ethanol droplet of 5% concentration: (a) raw image method; (b) velocity mapping method; (c) image mapping method.

uncertainty in image restoration will be reduced to a certain degree. Figures 9, 10, 11 and 12 show the comparisons of instantaneous velocity fields obtained by using the three methods for ethanol concentrations of 1%, 5%, 20% and 20%, respectively. The case of 20% is shown for two different instances to show the unsteady nature of the flow in that concentration. In each figure, (a) corresponds to the velocity field obtained from the original image plane, (b) represents those obtained by velocity mapping method and (c) is for those obtained by using the image mapping method. In (a), the velocity vectors at a point (xj , yj ) close to the droplet boundary are discarded. This condition is imposed by

of 0.8 is chosen instead of 0.95 in the above equation, based on the reason that the points close to the droplet boundary tend to move towards the centre region. There is no manipulation of the raw PIV results except for discarding a few apparent error vectors. A pair of vortices is clearly shown in figures 9 and 10 for the cases of 1% and 5% mixture. The flow moves upward in the centre region for these cases. It is worthwhile to note that the magnitude of the upward velocity component in the centre region of the droplet is reduced after the velocity correction. Additionally, as shown for the concentrations of 1% and 5%, the centre of vorticity in the modified velocity field is moved towards the origin. The corrected velocity fields of (b) and (c) look similar to each other. In principle, if the image restoration is perfectly done, the two results obtained by the velocity mapping and image mapping methods should be identical. However, the

ηj /R > 0.95[1 + (b/R) cos θj ] √ in which ηj = xj2 + yj2 and cos θj = yj /ηj . In (c), the value

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Figure 11. Comparison of the instantaneous velocity field inside an ethanol droplet of 20% concentration: (a) raw image method; (b) velocity mapping method; (c) image mapping method.

Figure 12. Comparison of the instantaneous velocity field inside an ethanol droplet of 20% concentration: (a) raw image method; (b) velocity mapping method; (c) image mapping method.

velocity mapping method does not require the tedious image restoration procedure. In addition, the number of total velocity vectors obtained from the same flow image is about 40% larger for the case of the velocity mapping method. The image restoration brings about squeezing of the domain (image), and consequently, there is a loss of useful pixels for the PIV measurement. In these aspects, the velocity mapping method is recommended for resolving the image distortion problems encountered in the droplet-related researches. For the case of the 5% mixture, the flow is initially unsteady at the periphery of the droplet. After about 1 min, however, the stable flow pattern shown in figure 10 is established. As the ethanol concentration is increased to 20%, the irregular flow pattern becomes more dramatic, as shown in figures 11 and 12. In all the cases considered here, regular flow patterns such as those shown in figures 9 and 10 for 1%

and 5%, respectively, are formed after a moderate time. If we further increase the ethanol concentration, more complex flow patterns appear and persist for a longer time. When KCl (sodium chloride) mixed with deionized water is used, a similar flow pattern to figure 9 can be observed too. Such a flow inside a multi-component droplet can be generated by the Marangoni convection and the buoyancyinduced convection. The Marangoni convection is usually caused by temperature and concentration gradients along the fluid interface. For the pure water case, however, only negligible fluid motion is observed, although there is no remarkable difference in the evaporation rate, compared to the case with additives. This result implies that the flow does not originate from the thermal Marangoni effect. The faster evaporation of ethanol makes the fluid density at the droplet surface greater than at the core

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region. This statically unstable density distribution can induce the buoyancy-induced convection when the speed of diffusional transport, which would homogenize the density distribution, is much slower than that of the convection. The existence of such convection in a sessile droplet has already been suggested by Savino and Monti [12] through a numerical analysis. More firm evidence that the flow is strongly influenced by gravity was obtained from the hanging-droplet experiment. In this case, the flow direction was again upward at the axis of symmetry of a droplet. However, the concentration gradient along the droplet surface will also be involved in the fluid motion as explained by Pratt and Kihm [17]. A detailed numerical investigation to find out the role of buoyancy and the Marangoni effect under the present experimental conditions is being done by the present authors. On the other hand, this kind of flow can also be observed in the thin layer multi-component film [18, 19]. The investigations to explore the inter-relationship between the solute concentration, evaporation rate, flow pattern and contact angle of a droplet will be valuable.

5. Concluding remarks Two different correction methods are developed on the basis of the ray tracing method, for a quantitative measurement of the velocity fields inside a small axisymmetric droplet. One is based on the restoration of distorted images. The other is the velocity mapping method, utilizing the mapping relation of coordinates and velocity vectors without image restoration. A relatively rapid and regular fluid motion is generated spontaneously inside an evaporating droplet containing a small amount of additive. Two stable counter-rotating vortices are observed at moderate concentrations. With the velocity correction based on the ray tracing method, the internal flow field up to 80% (in radius) is obtained quite reasonably. The applicability and effectiveness of the correction methods are verified by comparing the PIV velocity fields. The location of vortex centres and the magnitude of flow velocity are different from those obtained without velocity correction. The two correction methods produce nearly identical results. However, the velocity mapping method is more convenient, because the image mapping method requires a tedious image restoration procedure. Moreover, the number of total velocity vectors correctly obtained is about 40% larger for the case of the velocity mapping method. It is recommended, therefore, to use the velocity mapping method for resolving the image distortion encountered during the flow analysis of the droplet-related problems.

Acknowledgments The present investigation was supported by the Brain Korea 21 Project in 2003 and by the POSCO technology development fund in 2002 (contract no 1UD02013). ISK was also supported by a grant from the Korea Science and Engineering Foundation (KOSEF) (contract no R01-2001-00410). The authors deeply appreciate Dr Hee Chang Lim for his contribution during the initial stage of the present investigation. The authors acknowledge the help of Mr Seok Kim and Mr Young Gil Jang during the PIV experiment and analysis of data.

Appendix. Derivation of mapping function in equation (4) A point Pi is defined as the projected point of Pi onto the −−→ −−→ −−→ object plane. The vectors OPi , OPs , OPo are written as −−→ OPi = xi i + yi j, −−→ OPs = xs i + ys j + zs k, −−→ OPo = xo i + yo j. Since xs = xi and ys = yi , and Ps is on the droplet surface, we can obtain unknown zs for given xi and yi by using equations (2) and (3) as rs = R + b cos θs = R +

bys , rs

(A1)

 1/2 where rs = xs2 + ys2 + zs2 . From the above equation, it becomes  R + R 2 + 4bys , (A2) rs = 2 and  zs = rs2 − xs2 − ys2 . (A3) −−→ −−→ Let A ≡ Pi Ps and B ≡ Ps Po , then the vector A is represented, in both the Cartesian and the spherical coordinate systems, as A = −k = −sin θs cos φs er − cos θs cos φs eθ + sin φs eφ , where ys , rs  xs2 + zs2 sin θs = , rs zs cos φs =  , xs2 + zs2 xs . sin φs =  2 xs + zs2 cos θs =

(A4a) (A4b) (A4c) (A4d)

The vectors A, B and N exist on the same plane S, in which N denotes the outward unit normal vector on the droplet surface. From the shape function for the droplet shown in equation (3), N becomes N=

∇F b sin θs = er − eθ . |∇F | rs

(A5)

A unit vector normal to the plane S becomes D = N×A. If we define a vector C which is parallel to the plane S and normal to the vector A, it becomes (see figure 4) B = A − tan(ψa − ψd )ec A × (N × A) , = A − tan(ψa − ψd ) |A × (N × A)|

(A6)

where ec is the unit vector in the direction of C. From the geometrical relation, it becomes A · N = −cos ψa in which A·N = −

sin θs cos φs (rs − b cos θs )  . rs2 + b2 sin θs2 1111

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Therefore, ψa = cos−1





sin θs cos φs (rs − b cos θs )  . rs2 + b2 sin2 θs

From Snell’s law, it becomes 

−1 na 2 1 − cos ψa . ψd = sin nd

(A7)

(A8)

After the manipulation of formulae, we can show that ec becomes ec = f1 {(rs − b cos θs ) sin θs sin φs i + (rs cos θs + b sin2 θs )j}, (A9) where 1 f ≡  . cos2 φs (rs cos θs + b sin2 θs )2 + sin2 φs rs2 + b2 sin2 θs (A10) In equation (A9), since C is normal to A, the vectors ec and C have no component in the z direction. Then, from equation (A6), the vector B becomes B = −f tan(ψa − ψd ){(rs − b cos θs )sin θs sin φs i + (rs cos θs + b sin2 θs )j} − k. The line passing through the points Ps and Po having the direction vector B is written as x − xs y − ys z − zs = = , Bx By Bz where Bx , By and Bz are the vector components of B in the Cartesian coordinate system. The coordinate of the intersecting point between the line and the plane z = 0 (i.e., Po ) becomes zs Bx xo = xs − Bz = xi − zs f tan(ψa − ψd )(rs − b cos θs )sin θs sin φs , (A11a) zs By yo = ys − Bz = yi − zs f tan(ψa − ψd )(rs cos θs + b sin2 θs ).

(A11b)

References [1] Jing J et al 1998 Automated high resolution optical mapping using arrayed, fluid-fixed DNA molecules Proc. Natl Acad. Sci. USA 95 8046–51

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