1. Introduction

Recently, the aperture total internal reflection (A-TIR) technique was proposed to characterize the microdroplet and fingerprint patterns, such as the coverage fraction of the droplets on the surfaces [1]. The proposed A-TIR technique can characterize microdroplets by placing different sizes of apertures in front of a light detector and selectively capturing dispersed beams from the curved liquid-air interface of the droplet. However, the reflectance from the curved liquid-air interface was simulated as 2-D modeling to cause the inaccuracy, and the parameter dependency of the aperture size and the working distance of the aperture from the droplet was not conducted. Hence, the proposed application was limited to only determining the coverage fraction of the droplets on the surface [1].

In this research, we employed 3-D ray tracing with Fresnel equation for the reflection from the curved liquid-air interface of the droplet and characterized the parameter dependency in the measured reflectance for the aperture size and the working distance of the aperture to show a good agreement between the experiment and the simulation. Based on this successfully developed modeling, the contact angles of the droplet were experimentally determined in the range of 1° to 15° for various sizes of droplets by obtaining the droplet thickness through the subtraction scheme between the measured reflectance from the two different-sized apertures and using the spherical droplet profile. The contact angles by A-TIR were compared with the contact angles measured by Fizeau interferometry for the microdroplets less than 5°, and the side-view imaging for the macro droplets more than 5°, respectively, to show a good agreement. In addition, the simulation shows that it can effectively measure the contact angle in the range of 0 ∼ 90° by adjusting the aperture size and the working distance of the detector.

The contact angle is one of the substantial physical properties determining the material’s surface energy in the classical Young’s equation [2], which plays a key role of wetting behavior in liquid-solid-vapor interactions; binding, coating, evaporation, condensation, super repellency for applications of micro/nano fluidics, ionic liquids separator and membrane, oil recovery, printing, etc. Most of physical and chemical phenomena in nature and man-made systems are involved with the contact angle and its corresponding surface energy interaction.

3-D ray tracing with Fresnel equation has been rarely used in droplet research [3,4]. Kang et al. resolved the image distortion issues in droplet dynamics using ray tracing [3], and Apriono et al. employed ray tracing for the droplet-like lens antenna simulation [4]. To the authors’ knowledge, 3-D ray tracing with Fresnel equation has never been applied to the droplet profile characterization.

The experimentally demonstrated droplet diameters are 70∼7000 micron with the droplet height of 1 ∼ 450 µm, covering various sizes regardless of the droplet diameter, which provides a significant advantage compared with the existing technique based on microscope [5–8]. Campbell et al. [5] showed low contact angle measurement using the lens effect, Cha et al. [6] measured various contact angles using focal plan imaging, Frankel et al. [7] observed the droplet profile using total internal fluorescence, and Sundberg et al. [8] measured contact angle using interference, all based on microscopy setup. The existing techniques have a limitation only for tiny size droplets as these are using a microscope setup. However, A-TIR can measure various-sized droplets based on internal reflection configuration, not requiring a microscope setup, which was experimentally demonstrated with 70∼7000 micron diameters in this study. In addition, A-TIR can be used for a single or an array of micro or nanodroplets less than 70 microns with the microscope objective by reducing the beam size to determine the droplet thickness and the contact angle.

The side view imaging using a goniometer and the top imaging technique [9] can detect contact angles with the various droplet sizes, but these techniques have a limitation in small contact angles. However, A-TIR can effectively measure small contact angles below 1°, verified by experiments, and even ultrasmall contact angles in submicron thickness, confirmed by 3-D modeling, which is only possible using the interference fringe method by counting the number of the fringes with the sophisticated microscope setup to the authors’ knowledge [10–12].

Furthermore, A-TIR can detect large contact angle up to 90° with a larger aperture and smaller working distance. Experimental measurement of the contact angles from 1° to 15° was demonstrated successfully using A-TIR in this study. This outcome can enable the measurement of the contact angle variation from a large contact angle close to 90° to a ultrasmall angle close to 0°, regardless of the droplet size, which would be very beneficial, for example, when a sessile droplet spreads on the surface to turn into a liquid film.

In this work, the A-TIR technique has been successfully verified through 3-D ray tracing with Fresnel equation modeling for the curved liquid-air interface and the characteristic parameter study considering the aperture size and the working distance of the aperture as well as the droplet surface coverage fraction. The subtraction scheme is employed between the measured reflectance at the two different-sized apertures to determine the droplet height or thickness and its corresponding contact angle using the spherical profile [8,13].

2. Aperture total internal reflection (A-TIR) fundamentals

The optical ray tracing of A-TIR is illustrated in Fig. 1 [1]. In the case of an incident angle larger than the critical angle of the liquid-glass interface, the propagated wave is total-internally reflected [14] at the interface (Rfl_1 and Rfl_2 in Fig. 1(a)). Meanwhile, if the incident angle is lower than the critical angle of the liquid-glass interface but higher than that of the glass-air interface (Fig. 1(b)), the wave propagation becomes more complicated. The wave striking the glass-air interface is totally reflected (Rfl_1), but the wave reaching the liquid-glass interface under a droplet will be divided into two components; one is reflected at the liquid-glass interface (Rfl_2), and the other is transmitted through the liquid. The transmitted though liquid will be reflected at the liquid-air interface, i.e., the upper curved boundary of the droplet or transmitted through the air (T__scatt). The waves reflected at the curved droplet surface (Rfl_3_total=Rfl_3 + Rfl__blocked) are not parallel each other as in Fig. 1(b) and reflected differently depending on the incident angle normal to the profile of the droplet surface. A portion of the waves striking the top of the droplet surface is internally reflected with a similar angle (Rfl_3) as the reflections at the interfaces of the glass-air (Rfl_1) and glass-liquid (Rfl_2). The other portion of reflection waves (Rfl__blocked) away from the center apex will have a large aberration angle and will be blocked from the aperture-mounted detector. The different waves striking the sides of the droplet surface will have a severe aberration in their propagation (T__scatt) and will not be captured in the aperture-mounted detector.

 

Fig. 1. The illustration of aperture total internal reflection (A-TIR) occurring with droplet on transparent substrate. Ray tracing with two different ranges of incident angle (a) θ*_liquid < θ_incident and (b) θ*_vapor < θ_incident < θ*_liquid.

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In Fig. 1(b), only a tiny portion of the waves around the droplet’s apex in the curved liquid-air interface can pass through the aperture with the radius of r_a and be captured at the detector. In contrast, other waves are blocked because of large aberrations. Different sizes of apertures can be mounted in front of a detector at a working distance of L from the droplet to capture the reflectance from the curved liquid-air interface (Rfl_3) selectively depending on the aperture sizes. The other factor to consider in the A-TIR technique is the beam path’s aberration due to Goos-Hänchen (G-H) effects occurring near the droplet edge [1]. The G-H effect is a wave phenomenon caused by the lateral shift of reflected waves in TIR mode and the axial shift of the penetrated waves into a medium [15]. The lateral shift corresponds to a displacement L_GH along with the interface of the incident plane. If the incident wave strikes close to the contact line of the droplet, the wave could come out of the liquid-air interface and be scattered into the air, not captured by the detector. The axial shift can be considered if the thickness is smaller than the general penetration depth of 300 nm [15]. Please refer to the former study [1] about the detail G-H effect as G-H effect doesn’t play a significant role in determining the droplet morphological features of thickness and contact angle in tis study.

The parallel-reflected waves of Rfl_1 and Rfl_2 are captured in the detector regardless of the sizes of the apertures. At the same time, Rfl_3 is reflected from the curved droplet profile with significant aberration. The total reflectance R measured at the aperture-mounted detector consists of three individual reflectances as a function of incident angles:

A-TIR measurement with different-sized apertures produces the unique reflectance curve by selectively capturing the reflecting rays (Rfl_3) from the curved droplet surfaces, distinguished from the classical TIR method. The morphological features of the droplet pattern can be obtained from the measured reflectance curves. Without an aperture, the reflectance curve would be just the same as the classical TIR measurements.

Each reflectance will be expressed with the Fresnel equation modeling with the corresponding parameters reflecting the geometric features of the droplet pattern. The parameters will be described in detail in the following subsections.

2.1 Experimental setup of A-TIR

The experimental setup is illustrated in Fig. 2 with a schematic illustration on the left and a photo of the setup on the right. An equilateral triangle prism (SF10, n = 1.732 at 633 nm) is mounted on a vertical post, and a slide glass (SF10) with samples on top of it is placed on the prism. A He-Ne laser (λ = 633 nm, Newport 10mW) is mounted on a rail that pivots around the prism and a digital protractor (resolution = 0.01°, accuracy = 0.05°) is attached to the rail. A collimated beam (d = 0.8 mm) from the laser is incident on the top surface of the slide glass through the prism. The reflected laser beam is measured using a power detector (Newport 918D-Sl OD 3) with an incident angle from 30° to 75°. Various sized apertures (r_a = 1, 2, 4, and 5 mm) are placed in front of the power detector to cut off the highly aberrated beams caused by the curved surface of the microdroplets. The reflectance is determined as the power measured behind the cut-off aperture to the original power measured before entering the prism. The triolein oil is used for the droplet liquid, and the refractive index of 1.477 was measured in the former study using the classical TIR technique [1]. The microcontact printing (µCP) is used to make the microdroplets with a droplet radius less than 250 µm on the slide [1,16], while the macro droplets are made by the needle dripping with the droplet radius of more than 3 mm. Please refer to the former study [1] for the detail experiment set-up.

 

Fig. 2. The schematic diagram (a) and the experimental setup photo (b).

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2.2 Reflectance at the glass-air interface (Rfl_1) and coverage fraction (β)

The ray reaches the glass-air interface only in areas that are not covered by liquid droplets on the glass surface. The ray is totally (if θ ≥ 35.3°) or partially (if θ < 35.3°) reflected depending on the incident angle (θ), which is indicated as “Rfl_1” in Fig. 1. The reflectance of this ray is based on Fresnel equations with some parameters as follows:

The subscripts ‘i’ and ‘t’ indicate the incident and transmitted media at the interface, respectively. ${|{{t_{12}}} |^2}$ and ${|{{t_{21}}} |^2}$ are the transmittances occurring at both side interfaces of the prism where the wave enters and exits, respectively. ${|{{r_{21}}} |^2}$ is the reflectance of the wave propagating from glass to air on the top of the prism, and it has a maximum of 1 at an incident angle larger than 35.3° normal to the glass-air interface.

The first parameter α is the transmission ratio through the substrate medium composed of a prism, a slide glass, and an index-matching fluid. In this experiment, α was empirically estimated as approximately 95% by comparing the intensities of the entering and exiting beams. Experimentally measured reflectance of Rfl_1 agrees well with the calculation by Eq. (2) with α = 95%. This 95% value of α is consistently applied to all equations in this study. The second parameter $\beta $ is the coverage fraction of droplets defined as the ratio of the measured contact area of the droplets (A_droplets) and total area of the illuminating area (A_ill):

The droplets contacting area, A_droplets is determined from the image processing of the droplet pattern using Matlab software (the functions of polyshape, subtract, and area), in the case of multiple microdroplets as in Fig. 3. The area of the laser illuminating area, A_ill is based on the plane-cylinder intersection shaping an ellipse with an enlarged illuminating length which varies with incident angles (Fig. 3(a)) and can be calculated as πbw where b is the laser illuminating length and w is the laser width. The coverage fraction $\beta $ slightly changes depending on the incident angles. The variation of coverage fraction $\beta $ with incident angles is less than 1% for the most droplet radii as in Fig. 3(b), which is small but not negligible, affecting the simulated reflectance curves. The oscillating pattern of the surface coverage fraction is caused by the different trends of the droplet contact area, A_droplets and the laser illuminating area, A_ill, which is shown in Fig. S1, Supplement 1. Equation (2) is formulated with the uncovered area portion, (1−β) since Rfl_1 is about the reflectance at the glass-air interface of ${|{{r_{21}}} |^2}$.

 

Fig. 3. The illustration of laser illuminating area on droplet pattern (a) and the coverage fraction of $\beta $ as a function of incident angles for an array of droplets with varying the droplet radius, a (b) [17,18].

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In a single droplet, the coverage fraction is directly determined from the geometry relation; β=a²/(bw) where a is the radius of the droplet. In a single macro droplet with the radius larger than 3 mm, the coverage fraction is just one as the droplet area is always same as the laser illuminating area.

2.3 Reflectance at the glass-liquid interface (Rfl_2)

The ray strikes a glass-liquid interface under a droplet, called “Rfl_2” in Fig. 1. It can be reflected totally (if θ ≥ 58.5°) or partially (if θ < 58.5°) depending on the incident angles. The reflectance of the second path is expressed as:

2.4 Reflectance at the liquid-air interface (Rfl_3_total)

The ray passes through the glass-liquid interface with an incident angle lower than the critical angle of 58.5°, which is shown as “Rfl_3_total” in Fig. 1. To be precise, this ray is caused by the transmitted ray at the glass-liquid interface at an incident angle from 35.3° to 58.5°. This transmitted ray is total-internally reflected at the upper boundary of the droplet, i.e., the liquid-air interface, and then turns back to the liquid-glass interface at the bottom of the droplet. A part of the light transmits through the liquid-glass interface, exits the prism, and finally reaches the aperture-mounted detector.

2.5 3-D ray tracing with Fresnel equation modeling

Three-dimensional (3-D) ray tracing is employed to calculate the Rfl_3_total exactly with the Fresnel equation modeling. For 3-D ray tracing, the laser intensity is experimentally measured as in Fig. 4(a) and curve-fitted with Gaussian functions in Fig. 4(b).

 

Fig. 4. The experimentally measured laser intensity distribution (a) and the centerline distribution along the center dash-dot line (b) with its curve fitted line, ${\kappa _{xz}} = 0.3521{e^{ - {{\left( {\frac{{x - 0.0001861}}{{135.4}}} \right)}^2}}} + 0.6131{e^{ - {{\left( {\frac{{x - 0.0004168}}{{316.2}}} \right)}^2}}}$ [17,18].

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With this measured intensity distribution, Rfl_3_total is calculated using the following expression;

Figure 5 illustrates the schematic of the 3-D ray-tracing calculation. The top view in the figure shows the lens effect resulting in the accumulation of the ray reflected on the top curved liquid-air interface in the off-axis locations. The laser intensity is divided with a 1µm interval for its 800 µm diameter. Ray tracing is conducted for each ray with Fresnel reflection and transmission calculation [Eqs. (3) and (4)]. 3-D ray tracing requires more intensive and complicated algorithms than the previously employed 2-D calculation [1]. The program is developed with Matlab software, and the calculated node is ∼ 6 million elements with the horizontal (x-axis), the depth (z-axis), and the vertical (y-axis) directions with the incidence angles from 30 to 75 degrees with around 1-degree interval. The zoom-in view of the overview shows the ray tracing on the screen with Rfl_1, Rfl_2, and Rfl_3 with blue, green, and red color, respectively, with Rfl_1 and Rfl_2 targeting around the center region as they are the reflections on the plane interfaces of the glass-air and glass-liquid, respectively. Rfl_3 presents the dispersed beam profile in vertical (mostly in the y-axis) and in-depth directions (in z-axis) as in the front and side views of Rfl_3. The spherical shape of the droplet causes the lens effect, and the reflected rays from the curved liquid-air interface diverge significantly as it propagates, especially in the vertical direction (in the y-axis) due to the prism effect as in the overview.

 

Fig. 5. The schematic of 3-D ray tracing modeling [17,18].

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Hence, the apertures with various sizes from 1 to 5 mm in the radius, placed in front of the detector with a diameter of 10.3 mm which is larger than the apertures, are expected to collect the signatures of the droplets profile such as the thickness and the contact angle. The aperture size, r_a, and the working distance of the detector, L are indicated in the overview and the side view. The reflectance at the curved liquid-air interface causes the lens effect with a dispersed profile in the side view, which is verified by comparing the simulation with the measured Rfl_3 beam profiles.

Figure 6 presents the flow chart to calculate the reflectance, R [Eqs. (1–7)], by following 3-D ray tracing modeling. Rfl_3 calculation is the central part as it is three-dimensional, requiring the simulation of the dispersed beam profile and its match with the experimentally photographed beam profiles (Fig. 7) as well as the agreement of the simulated Rfl_3 intensity with the A-TIR measurement with apertures (Fig. 8–11). The detail 3-D ray tracing algorithms could be published separately in the future.

 

Fig. 6. The flow chart using 3-D ray tracing with the Fresnel equation modeling.

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Fig. 7. The photos of macro droplet in top and side view (a), the beam profiles of Rfl_3_total at various angles from the experiment and the simulation (b), and the centerline beam profile (Rfl_3_total) (c). In Rfl_3_total beam profile, the experiment is from the captured image on the screen in Fig. 5 and the simulation is from the 3-D ray tracing modeling. The centerline profile is along the dash-dot line in (b) [17,18].

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Fig. 8. The different PDMS stamped microdroplets patterns and the surface coverage fraction depending on differently coated surfaces. Measurement and simulation are denoted in symbols and lines, respectively: the oleophilic coating (a), the plain glass (b), and the oleophobic coating (c). r_a = 1mm and L=300 mm [17,18].

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Fig. 9. The reflectance curve for the macro (a and c) and micro (b and d) droplets with various aperture sizes (r_a = 1, 2, and 4 mm) versus incident angles and the working distance (L) of 150 mm. The total reflectance, R, is plotted in a and b with solid diamond, triangle, and square symbols for the p-pol and hollow symbols for the s-pol experiment data. The simulation curves are in solid lines for p-pol and dashed lines for s-pol with the apertures of r_a=1,2, and 4 mm in blue, red, and green colors, respectively. Each reflectance of Rfl_1, Rfl_2, and Rfl_3 is shown in p-polarization in black, orange, and purple solid lines in c and d.

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Fig. 10. The effect of the working distance of detector (L) on total reflectance, R (a) and Rfl_3 (b) for the droplet with h = 10µm, a = 150µm, r_a=10 mm, and θc=7.6°. Reflectance is p-pol.

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Fig. 11. The effect of the working distance of detector (L) in macro (a) and micro droplet (b). Reflectance is in p-pol.

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A macro-size droplet (Fig. 7(a)) is used to effectively produce the dispersed beam profiles, which are big enough to be easily detected. The images at the top row in Fig. 7(b) show the experimentally measured Rfl_3_total beam profiles on the screen using a color camera at the representative incident angles; 54.2, 45.9, and 33.7 degrees, respectively. The images are recorded at the working distance of 150 mm away from the droplet on the prism, which gets more prominent with the decreasing incident angles as the optical path through the prism gets longer, resulting in more dispersion, especially in a vertical direction. The simulated images at the bottom row present a good agreement with the experiment in its beam dimension and its intensity. Figure 7(c) shows the centerline normalized intensity distribution for each angle in (b) along the center dash-dot line. The intensities are normalized against the maximum at the critical angle. The experiment data are in solid symbols. The simulation results are solid lines in red, green, and blue colors for 54.2, 45.9, and 33.7 degrees, respectively, which shows a good agreement, indicating our 3-D ray tracing modeling works well.

3. A-TIR characteristics by the parameter study

To investigate A-TIR characteristics, the parameter study is conducted: the coverage fraction ($\beta )$, the aperture size (r_a), and the working distance of the detector from droplets (L).

3.1 Coverage fraction (β)

Firstly, the coverage fraction effect was characterized using an array of microdroplets prepared by the PDMS-stamp printing method as in Fig. 8 which presents the measured reflectance on three different coating surfaces, i.e., oleophilic coating (Fig. 8(a)), plain glass (Figure 8(b)), and oleophobic coating (Fig. 8(c)). The presented data are the mean values of three individual measurements. The calculated reflectance is from Eqs. (1–7). The value of α is identically 0.95 for all substrates since it is not affected by the surface conditions. The surface coverage fraction, β, is obtained from Eq. (5) and shown in Fig. 3. Reflectance shows a decreasing trend in the middle angle range (35.3° < θ_incident < 58.5°) in the order of droplet (c), (b), and (a) with decreasing oleophobicity as the coverage fraction β increases. Reflectance curves don’t show very curved profiles around the critical angles because of the low G-H ratio γ (mostly less than 1%). The calculation by 3-D modeling agrees well with the measurement, meaning the proposed modeling works well. The effect of the coverage fraction for the single droplet is straightforward and not shown here.

3.2 Aperture size (r_a)

Secondly, the size of aperture is characterized to explore its effect on the A-TIR reflectance in p and s polarization for the macro and microdroplets as in Fig. 9(a) and Fig. 9(b). The inset photos in Fig. 9(a) show the top and side view images for the macro droplet with a diameter of 6.5 mm and a thickness of 343 µm. The measured contact angle from the side view is 12 degrees, which agrees with the calculation from the spherical relation, showing the macro droplet maintains a spherical profile. The diameter of the macro droplet is big enough so that the laser (diameter of 0.8 mm) is confined inside the liquid droplet with targeting at the apex of the droplet, reflecting the only Rfl_2 and Rfl_3 rays in Fig. 1 and 5. The reflectance measured at the apertures of 1, 2, and 4 mm in the radius are colored in blue, red, and green, respectively. The reflectance data in p and s polarization are shown as filled and hollow symbols for the experiment and solid and dash lines for the simulation. It shows an excellent agreement between the experiment and the simulation, indicating that the proposed A-TIR theory based on 3-D ray tracing modeling describes the optics phenomena in aperture total internal reflection (A-TIR) very well.

After verifying the proposed A-TIR theory for the macro droplet, the aperture size effect is also characterized for the microdroplets in Fig. 9(b), showing a good agreement between the experiment and the simulation for different sized apertures. The inset photo shows an inverted microscope image of the microdroplets from the microcontact printing (µCP) method with the diameter of 320 µm and the thickness of 2.4 µm where the thickness is measured by counting Fizeau interference fringes [19], while the side view image is not shown as the thickness is so thin and the thickness measurement is not possible from the side view imaging.

Figure 9(c) and d present the contribution of each reflectance, Rfl_1, Rfl_2, and Rfl_3_tot in black, orange, and purple colors, respectively, for the macro and microdroplets in Fig. 9(a) and b, in p-polarization. In the macro droplet (Fig. 9(c)), the reflectance of Rfl_2 and Rfl_3 are measured separately and denoted by symbols; Rfl_2 is by a circle, and Rfl_3 is by a diamond, a triangle, and a square for 1, 2, and 4 mm apertures, respectively. The corresponding simulation results are shown as solid lines in blue, red, and green, respectively. In Rfl_3, the aperture of 4 mm shows the maximum reflectance among the given apertures with the aperture of 1 mm the minimum, while Rfl_2 doesn’t change with the different apertures as Rfl_2 is from the plain glass-liquid interface, meaning the reflected rays will maintain the original laser beam size (φ=0.8mm). Rfl_1 doesn’t exist in the macro droplet as the laser beam is smaller than the droplet dimension. The combined reflectance (Rfl_2+Rfl_3) is Fig. 9(a). Figure 9(c) shows a good agreement between the experiment and the simulation.

Likewise, each reflectance of the microdroplets (Fig. 9(b)) is shown in Fig. 9(d). The laser beam is larger than the micro droplet dimension and the reflectance of Rfl_3 was not able to be measured separately from the total reflectance, R. However, the simulation verified by the macro droplet shows that the reflectance from the glass-air (Rfl_1) interface in the micro droplet is the dominant component above the critical angle of 35.3°, denoted by the black line (Fig. 9(d)).

3.3 Working distance of the detector (L)

Thirdly, the effect of the working distance of the detector (L) from droplets is characterized in Fig. 10 and 11. The simulation in Fig. 10(a) clearly shows that the shorter working distance of the detector (L) can capture more signal (total reflectance R), which is caused by the increasing collected intensity at the aperture-mounted detector with a shorter distance, shown as the increasing reflectance from the curved droplet surface (Rfl_3) in Fig. 10(b). Note that Rfl_1 and Rfl_2 are not affected by the working distance as they are reflections from the plain surfaces, glass-air and glass-liquid interface, respectively.

Figure 11 shows the experimental verification of the effect of the working distance (L) on the macro (Fig. 11(a)) and the microdroplets (Fig. 11(b)). With varying the working distance, L, only Rfl_3 from the curved liquid-air interface changes, while Rfl_1 and Rfl_2 don’t change as they are reflected from the plane interface. The detector is located at 150, 200, and 250 mm, respectively, with the experiments in the square, triangle, and diamond symbols and the simulation in the solid, dash, and dot lines. It shows a good agreement between the experiment and the simulation for macro and microdroplets. The macro droplet (Fig. 11(a)) shows clear distinction depending on the working distance as there is no reflectance from the plain glass-air interface (Rfl_1) with the droplet size larger than the laser beam size, while the effect of the reflectance from the curved liquid-air surface (Rfl_3) is dominant as in Fig. 9(c) and Fig. 10(b). In the case of the microdroplet, Rfl_1 is the main reflectance (Fig. 9(d)), and it mitigates the effect of the Rfl_3 depending on the varying working distance. The mitigating effect gets larger with the smaller droplet size as Rfl_1 increases. The figure also presents the results of different aperture sizes with a good agreement with the simulation as in section 3.2.

4. Reflectance subtraction, thickness (h) and contact angle (θ_c) determination

Once the A-TIR technique is verified through the parameter study and the simulation of the 3-D ray tracing modeling, the contact angle of the droplets in various sizes can be determined by obtaining the droplet thickness through the subtraction of the reflectance measured at the two different sized apertures and using the spherical profile relation.

4.1 Reflectance ratio, δ (=Rfl_3/Rfl_3_total)

The ratio of the reflectance from the curved droplet surface, δ, between the captured intensity through the aperture (Rfl_3) and the total intensity from the whole curved surface (Rfl_3_total) can be approximated as equivalent to the ratio between the intensity illuminating the effective zone ($\mathop \sum \nolimits_{eff} {I_{i,xz}}$ on A_eff) of the droplet and the intensity hitting the whole droplet area ($\mathop \sum \nolimits_{droplet} {I_{i,xz}}$ on A_drop);

 

Fig. 12. The reflectance ratio with different aperture sizes for the droplet with a = 150µm and h = 3µm.

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4.2 Effective area ratio (A_eff/A_drop) and intensity flux ratio ($I_{eff}^{\prime\prime} /I_{drop}^{\prime\prime} )$

The effective area ratio (A_eff/A_drop) can be approximated as the following geometric modeling relation as in Fig. 13;

 

Fig. 13. The geometric modeling of the effective area to obtain the reflectance ratio, δ. L is the working distance from the droplet to the aperture-mounted detector. r_a is the radius of the aperture.

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Figure 14(a) shows the area ratio at the same condition as Fig. 12, which increases as aperture size gets larger. In Eq. (8), the intensity flux ratio between the effective zone and the droplet area can be determined numerically as $\frac{{I_{eff}^{\prime\prime} }}{{I_{drop}^{\prime\prime} }} = \frac{{Rfl\_3}}{{Rfl\_{3_{total}}}}/\frac{{{A_{eff}}}}{{{A_{drop}}}}$ where A_eff/A_drop is from Eq. (10) and Rfl_3 and Rfl_3_total are calculated from Eq. (7) based on the 3-D ray tracing modeling. The intensity flux ratio is plotted in Fig. 14(b), increasing with the incidence angle as the reflectance ratio increases (Fig. 12). It also presents the increasing trend with the smaller aperture size as the area ratio decreases significantly (Fig. 14(a)) although the reflectance ratio decreases (Fig. 12).

 

Fig. 14. The area ratio (a) and the intensity flux ratio (b).

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4.3 Reflection subtraction and thickness (h) determination

The reflectance measured in the aperture-mounted detector is expressed by $Rf{l_{Exp,\; i}} = Rlf\_{1_{Exp,\; i}} + Rlf\_{2_{Exp,\; i}} + Rlf\_{3_{Exp,\; i}}$ where the subscript of i is for the different sizes of apertures (r_a = 1, 2, and 4 mm). Rfl_1 and Rfl_2 are reflectance from the plain interfaces of glass-air and glass-liquid, respectively, meaning they don’t change regardless of the different apertures, while only Rfl_3 changes with the different apertures. Using the reflectance measured at the two different sized apertures, the following subtraction relation can be obtained;

With Eqs. (10) and (12),

In Eq. (14), the droplet thickness (h) can be uniquely determined as L, r_a,i, and r_a,j are the known conditions, a is measured directly from the side view or top view imaging, $I_{eff,i}^{\prime\prime} $, $I_{eff,j}^{\prime\prime} $, and $I_{drop}^{\prime\prime} $ are calculated from the 3-D ray tracing, and $\frac{{Rfl\_{3_{Exp,i}}\; }}{{Rfl\_{3_{total}}}}\; \textrm{and}\; \frac{{Rfl\_{3_{Exp,j}}\; }}{{Rfl\_{3_{total}}}}$ are experimentally measured by two different sized apertures (r_a,i and r_a,j). With the measured reflectance from the curved droplet-air interface by two different sized apertures and the 3-D ray tracing simulation, the droplet thickness, h can be determined.

Figure 15 shows the determined thickness, h, distribution in the middle angle range (35.3° < θ_incidence< 58.5°) for the microdroplet in Fig. 9(b) using Eq. (14) based on the subtraction between two reflectances at three aperture sizes; r_a=1, 2, and 4 mm. The subtractions of ${\delta _{2mm}} - {\delta _{1mm}}$, ${\delta _{4mm}} - {\delta _{1mm}}$, and ${\delta _{4mm}} - {\delta _{2mm}}$ are obtained separately, and the quadratic equation [Eq. (14)] is solved for the thickness, which shows a good agreement with the reference thickness (2.4 µm) measured by Fizeau interferometry. In Supplement 1, Fig. S2 presents the exact match between the reference thickness and the thickness determination by Eq. (14) using the 3-D modeling results ($\frac{{Rfl\_{3_i}\; }}{{Rfl\_{3_{total}}}}\; \textrm{and}\; \frac{{Rfl\_{3_j}\; }}{{Rfl\_{3_{total}}}}$) to verify Eq. (14). Averaging is done for the angle between 40° and 50° and for the three subtractions to provide the determined thickness of 2.1 µm, slightly smaller than the reference thickness. A total of 17 microdroplets are tested. The determined thickness data are plotted in Fig. 16(a) to show a good agreement with Fizeau interferometry [19] with the thickness range of 1∼8 µm.

 

Fig. 15. The determined thickness from the subtraction method for the microdroplet in Fig. 9(b).

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Fig. 16. The determined thicknesses distributions of the micro (a) and macro (b) droplets.

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Furthermore, A-TIR measurement is applied to 7 macro droplets with droplet radii larger than 3 mm by the syringe needle dripping. Figure 16(b) shows a good agreement between the determined thickness by A-TIR and the thickness measured by the sideview imaging with the thickness range of 350∼450 µm. In general, Fizeau interferometry is more effective in the thin thickness less than 10 µm, and the side view microscopy is preferred for the thick thickness larger than 10 µm. Measurement was conducted at least 2 hours later since the droplet preparation to minimize the droplet spreading effect for all cases.

4.4 Contact angle (θ_c) determination

Once the droplet thickness (h) is determined by Eq. (14) with the known droplet radius (a) from side view or top view imaging, then the contact angle (θ_c) is determined using the following spherical profile relation;

The determined contact angles are compared with Fizeau interferometry for the microdroplets and the side view microscopy for the macro droplets, respectively, in Fig. 17(a), showing a good agreement with each other. The microdroplets by the microcontact printing method show the contact angle from 1° to 5° with the thickness less than 10 µm and the radius smaller than 250 µm. In comparison, the macro droplets by the needle dripping show 9° to 15° with the thickness more than 300 µm and the radius larger than 3 mm. The small contact angles for the microdroplets are caused by the characteristics of the microcontact printing method, and the large contact angles for the macro droplets are caused by the needle dripping method. The 95% confidence intervals are indicated for each data, showing reasonable confidence.

 

Fig. 17. The contact angle determination for the micro and macro droplets (a) and the detectable contact angle vs. incident angle for the thickness from 1 to 140µm with a = 150µm, L = 50mm, and r_a = 1, 2, 4, 5, 15, 25, and 35.mm (b).

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The modeling shows that the approach of the shorter working distance of detector (L) and the larger aperture size (r_a) is preferred for the large contact angles, while the approach of the longer working distance and smaller aperture size is preferred for the small contact angles, capturing more signal from the curved liquid-air interface, and increasing the subtracted reflectance at two different apertures, to enable the determination of the thickness and the contact angles. Figure 17(b) demonstrates the contact angle can be determined as 0.8° ∼ 86.1° as the thickness changes from 1 µm to 140 µm with the droplet radius (a) of 150 µm, the working distance of (L) 50 mm, and the aperture sizes of 1, 2, 4, 5, 15, 25, and 35 mm. The large apertures (5, 15, 25 and 35 mm) are more effective for the large contact angles as distinctive differences in the subtracted reflectance [Eq. (13)] can be obtained, while small apertures (1, 2, 4, and 5 mm) for the small contact angles. It also shows the decrease of the incidence angle range for the detectable contact angle as the droplet height increases due to the increasing internal reflection inside the prism, which is also shown in Fig. 18 that the incidence angle range for the detectable contact angle decreases from 23° to 18° as the droplet thickness increases from 40 µm to 80 µm.

 

Fig. 18. The reflectance curves for the droplet with h = 40µm (θ_c = 29.9°) (a) and 80µm (θ_c = 56.1°) (b) at the same droplet radius and working distance condition as Fig. 17(b).

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Figure 18 presents the reflectance curves with the four large different apertures (r_a=5, 15, 25, and 35 mm) at the droplet thickness (h) of 40 µm and 80 µm, respectively, at the same droplet radius and the working distance as in Fig. 17(b). Here the detector is assumed as larger than the largest aperture size (r_a=35mm). The droplet with the smaller thickness (Fig. 18(a)) shows a clear difference in the reflectance curves compared with the larger thickness (Fig. 18(b)). The droplet with the larger thickness and contact angle shows a minor difference in the reflectance curves. However, it is still distinguishable, enabling the determination of the droplet thickness and its corresponding contact angle.

In Supplement 1, Fig. S3 shows that A-TIR can characterize the 100 nm thickness of microdroplet (a=150 µm) at the contact angle of 0.076 degree, by successfully capturing the difference in the reflectance with the small aperture sizes (r_a=0.5, 1, and 2 mm) and the longer working distance (L=300 mm), which would successfully detect the submicron liquid film thickness under condensation and evaporation dynamics. Figure S3 clearly presents the difference in the measured reflectance depending on the different aperture sizes to determine the droplet height and contact angle by Eq. (14) and (15) as in Fig. 17(b) and Fig. 18. The smaller apertures are effective to obtain a large difference in the subtracted reflectance, which is because the intensity flux ratio gets larger with a smaller aperture (Fig. 14(b)) and there is more variation in the captured reflectance with the smaller apertures as the reflected beam (Rfl_3) from the curved liquid-air interface is more focused around the aperture center with less dispersion in the smaller contact angle droplets.

5. Conclusion

In this study, we verified the A-TIR technique through 3-D ray tracing with a Fresnel equation modeling. We investigated the characteristics of A-TIR for the parameters of the surface coverage fraction, the aperture size, and the detector working distance. The experiments agree well with the simulation from 3-D ray tracing modeling. Furthermore, an analytic quadratic solution for the droplet height was obtained based on the subtraction scheme between two different sized apertures. The contact angles were determined as 1 ∼ 15° for the micro and macro droplets using the spherical profile relation from the droplet thickness of 1 ∼ 450 µm, to show a good agreement compared with the Fizeau interferometry and the side view imaging, respectively. In addition, the simulation presents that A-TIR can determine the contact angle close to 90° by the larger aperture and the shorter working distance and ultrasmall angle close 0° by the smaller aperture and the longer walking distance. This outcome provides a new diagnostic tool to determine the contact angle of the droplet, especially effective for the ultrasmall contact angle and the submicron thickness of the liquid film dynamics.