Investigating the origin of the far-field reflection interference fringe (RIF) of microdroplets

Wetting is ubiquitous in nature and technology. When a drop of liquid is in contact with a substrate, the drop spreads and is affected by the hydrodynamic bending^1,2 to form the outer region for the macroscopic flow, the intermediate region bridging the outer and the inner regions, and the inner region. The outer region follows the spherical (convex) profile for a small capillary length.³ The immediate region is where the profile turns from convex to concave including the inflection point, and the inner region has tens of nanometer thickness, such as a precursor film, around the contact line.^4–8 The geometrical profile of the individual droplet is primarily influenced by the surface energy difference between liquid and the substrate.

Microcontact printing (μCP) causes liquid drop wetting and dewetting.^9,10 During dewetting, the contact line recedes, and recently, it has been experimentally reported that the droplet makes the convex–concave profile.² The resultant droplet tends to form a dual profile (convex–concave) with an inflection point that partitions a droplet surface geometrically and contact line, including the precursor film.^4–7 The upper part of the droplet surface covered from its apex to the inflection line takes a nearly spherical cap shape like a convex lens. From the droplet's edge to the intermediate region or inflection point, conversely, the droplet profile is concave upward, which is formed with a foot region and a precursor film.

Identifying the droplet profile is of key importance in understanding the underlying mechanism of van der Waals, electrostatic, and hydrodynamics forces in wetting and spreading.^4,5,11 Furthermore, the measurement of the geometric shape of droplets is substantial in applications, such as the spreading of droplets,^4,5 the evaporation/condensation of sessile droplets,^12–14 microfluidics,¹⁵ inkjet printing,¹⁰ and microlens array.¹⁶ 

There have been various methods for droplet profile measurement, such as the droplet height or thickness, and the contact angle, based on goniometry,¹⁷ interferometry,^6,8,18–24 total internal reflection microscopy (TIR),^25,26 atomic force microscopy (AFM),^27–29 ellipsometry,³⁰ and optical microscopy.^31,32 The interferometry, AFM, and ellipsometry can measure the droplet profile on a nanometer to sub-micrometer scale, enabling the profile determination in the intermediate region of the droplet. However, these techniques are sophisticated ones that require complicated setups and time-consuming measurements. The goniometry, TIR, and optical techniques are lack of the resolution. The existing techniques of reflection and transmission interference^20,33–35 are for thick droplets and high contact angles, thus not being able to characterize microdroplet features such as the convex–concave profile, low contact angle, and thin height.

In this study, we introduce a simple optical method to determine the droplet features, such as convex–concave profile, thin height, low contact angle, and inflection point, of microdroplets using the far-field reflection interference fringe (RIF) configuration without complicated microscopy, producing interference fringes depending on the droplet profiles. The laser light is illuminated on the droplet on the prism from the bottom and reflected to the screen. The interference fringes are generated because of the convex–concave profile of the droplet. If the droplet profile is purely convex, the reflected laser diverges without any interference fringes, meaning that the formed microdroplets have convex–concave profiles, which is a remarkable finding as the convex–concave profile and sub-micrometer scale inflection point can be easily observed in real-time by simple configuration, not requiring a sophisticated one. Furthermore, we show RIF technique can determine the microdroplet features, such as the droplet thin height (thickness), the low contact angle, the inflection point, and the precursor film thickness.

To understand the generated interference fringe patterns on the far-field screen, the geometric formulation was conducted to show that the fringe is caused by the interferences from the convex at the center and the concave profiles at the contact line in the droplet. The relation between the droplet height or thickness (h) and the optical path difference (OPD) was obtained from the geometric formulation, which shows a similar form to Fizeau interferometry's relation.³⁶ The droplet height is determined by counting the fringe numbers in the OPD relation, showing a linear dependency on the fringe numbers and an excellent agreement with the experimental measurements and the ray tracing simulation. Furthermore, a raytracing simulation was performed based on the custom-designed MATLAB program^25,37 to provide the details of the droplet profiles. The interfering rays reflected near the inflection point in the intermediate region produce the outermost fringe of the concentric interference pattern on the screen, away from the sample droplet. The reflecting rays from the convex and the concave profiles of the droplet are projected on the far-field screen to form interference fringe patterns. The reflected rays are diverged at a larger area because of the droplet lens effect,^25,37 to facilitate counting the fringe number (FN) and radius.

A variety of droplets are prepared using the polydimethylsiloxane (PDMS) stamp printing method³⁷ and a syringe on different substrates with different oleophobicities and varying incident angles to show a good agreement between the experiments, the geometric formulation, and the ray tracing simulation in the interference fringe features, such as fringe number (FN), the fringe radius (FR), and the fringe location, to determine the droplet convex–concave profile, the droplet height, the contact angle, the inflection point, and the precursor film thickness. The employed tens of droplets have a droplet radius of less than 200 μm. Most droplets show the formation of interference fringes on the screen. The ray tracing simulation shows that most droplets have inflection points close to the edge within 5% droplet radius from the edge and 10% droplet height from the substrate. The precursor film thickness is estimated to be distributed from 2–∼100 nm. The RIF technique effectively measures the droplet height from 1.7–15 μm with a contact angle from 1.16° to 14.8°. The outcome of this research shows that it can effectively detect the thin height of microdroplets as tiny as ∼1 μm and the contact angle as low as ∼1°.

Additionally, an analytic expression is obtained between the droplet's apparent contact angle, which is determined at the inflection point, and the interference fringe radius from the geometric formulation to show a good agreement with the experiments and the ray tracing simulation. The outermost rays determine the fringe radius by the apparent contact angle from the reflection at the inflection point region.

The microdroplets are prepared by microcontact printing (μCP) and syringe methods on the representative oleophilic, plain, and oleophobic coated SF 10 glasses (substrates), which are provided by CEKO, Co. Ltd.^25,37 The interference fringe is formed on the far-field screen away from the droplet in internal reflection configuration with an equilateral triangle prism (SF10, n_p = 1.732 at 633 nm; Esco Optics) below the critical angle.^25,37 The index-matching liquid (Cargille Labs) is put between the glass substrate and the prism.^36,37 The study shows that the substrate material and the shape would not affect the interference fringe formation. The liquid of triolein oil (refractive index, n_d of 1.477 at 633 nm; viscosity of 38.2 cSt) is used. A droplet is placed on the upper side of the substrate. A He–Ne laser (λ = 633 nm; Newport) is used as the light source to produce the interferometric fringes of light on a screen away from the droplet. The screen is located away from the droplet by a working distance (WD) of around 200–300 mm. A coherent, parallel beam (d = 0.8 mm) of a laser is incident on the left side of the prism and reflects on the droplet–air interface. Internal reflection associated with a liquid droplet involves the optical aberration of the reflection beam due to the curved profile of the droplet surface at the droplet–air interface. The screen is mounted at a right angle against the reflected laser's propagation axis.

In internal reflection configuration, an unexpected fringe pattern is observed on the far-field screen, as in Figs. 1(a) and 1(b). Figure 1 shows the representative reflection interference fringe formation depending on the surface oleophobicities, the droplet height, and the apparent contact angle at the incidence angle (θ_inc) of 43.2°. The top row is the side-view images, and the bottom is the recorded RIF images. The side-view images show increasing contact angles with oleophobicities. The contact angle and the thickness measured by the side-view imaging are inserted in the images. FN and FR in the figure are the number of fringes and the fringe radius, respectively. The fringe radius is measured from the fringe center to the right outer edge of the fringe. We confirmed that the right-hand side of the fringe is caused by the left edge of the droplet from the geometry (Fig. 1). The observed interference images show the relation between the fringe number and the droplet thickness (h). The number of fringes increases as the droplet thickness thickens. With smaller droplet thickness, the number of fringes gets smaller. In addition, the fringe radius gets larger with increasing contact angle. The reason for the interference fringes [Figs. 1(a) and 1(b)] on the far-field screen in internal reflection configuration is conjected to be due to the existence of the dual convex–concave profile of the microdroplets, which will be explained by the geometric formulation (Sec. III) and the ray tracing (Sec. IV).

The oleophobic coating surface does not show the interference fringe [Fig. 1(c)], as it is estimated that the droplets on the oleophobic surface tend to form a pure convex profile because of the high contact angle and thick droplet height. Some droplets prepared by the syringe method do not show interference formation even on the non-coating surface with a low contact angle (∼10°), similar to Fig. 1(b), which is attributed to the fact that the droplet profile is pure convex, not dual convex–concave.

To explain the fringe formation in the internal reflection in Figs. 1(a) and 1(b), the dual convex–concave profile is introduced in Fig. 2(a), which shows the schematic of the far-field reflectance interference fringe (RIF) technique. The droplet in Fig. 2(a) has the dual convex–concave profile to form interference fringe on the screen away from the droplet as in Figs. 1(a) and 1(b), while Fig. 2(b) is the droplet with only a convex profile to generate an expanded and aberrated beam profile without interference fringe as in the authors' recent study²⁵ and Fig. 1(c). The laser is illuminated to cover the whole droplet as the projected laser size on the horizontal substrate is 2–3 mm, depending on the incident angles, while the diameter of most droplets is below 500 μm. In Fig. 2(a), the reflected light from the substrate–droplet interface is projected on the screen with the original size of 0.8 mm, much smaller than the fringe radius, which is determined by the contact angle at the inflection point. Hence, the generated interference fringe, much larger than the center laser spot size, with a fringe radius of more than 35 mm as in Figs. 1(a) and 1(b), are not affected by reflection from the substrate–droplet interface. Figure 2(b) shows that the convex or spherical profile of the droplet produces the expanded and aberrated beam as in Fig. 1(c), which cannot form the interference fringes on the far-field screen away from the droplet–air interface by 200–300 mm. The interference can be formed on the substrate–droplet interface between the incoming light and the refracted light from the reflection at the droplet–air interface, but this interference can be observed only using a microscope or magnifier, which the observation plane can be located at the substrate–droplet interface. Thus, the interferences occurring on the far-field screen, as in Figs. 1(a) and 1(b), are not caused by the pure convex profile of the droplet at the droplet–air interface, and it is reasonably conjectured that the droplet profiles would be dual convex–concave as in Fig. 2(a).

The physical origin of the far-field interference fringe in this internal reflection is different from the existing fringes observed in microscope configuration as it is formed far away from the sample caused by the interference between the reflected rays from the spherical and concave profiles at the liquid droplet–air interface, as in the inset figure in Fig. 2(a). The general microscope setup observes the interference fringe mostly on the substrate–droplet interface,^6,23 as in the inset figure in Fig. 2(b). This finding suggests that the far-field reflection interference fringe (RIF) in this study can be applied to determining the droplet's complex features, such as the dual convex–concave profile, the thin droplet height, the small contact angle, the inflection point, and the precursor film thickness, in a simple configuration.

The detailed geometrical formulation is conducted to analytically explore how the dual convex–concave profile works for the interference fringe formation, as in Fig. 3. The laser beam reflected on the droplet surface is no longer parallel while it passes out through the right side of the prism. Eventually, due to the convex–concave profile of the droplet surface, the laser beam arrives at different coordinates on a screen far away from the droplet according to the position of the droplet surface on which it reflects. The light reflected on the apex of the droplet on the upper convex profile (point C) and the horizon (or the precursor film) of the lower concave profile (point H) are targeted at the screen center like the light reflected on the substrate–droplet surface (Ray₁ and Ray₁’ of Fig. 3). On the other hand, the lights reflected apart from apex and precursor film (points Q and T) are directed to the surrounding area of the screen center (Ray₂ and Ray₂') to form an interference at location U. The off-focused illumination becomes severe as the reflection position is further away from the apex and precursor film of the droplet surface. The lights reflected close to the inflection point of the dual convex–concave profile (points R and S) are most widely diverged and illuminated at the outermost location of the diffraction pattern at location V (Ray₃ and Ray₃'), which will be discussed in detail later in this section. The exact inflection point is M (Fig. 5). Points R and S are slightly off the exact inflection point (M) to show the interference. The interference occurs when two optical rays reflected on the upper convex and the lower concave profiles meet at the far-field screen, far away from the droplet (WD: 200–300 mm), as also illustrated in Fig. 2(a). The intensity of interfered light depends on the phase difference between both rays, which changes periodically from the center to the outermost of light illumination. The consequential interfered light appears as the concentric fringe pattern in Figs. 1(a) and 1(b). In the figure, the left side of the droplet corresponds to the upper fringes, and the right side corresponds to the lower fringes against the center spot in the interference fringes formed on the screen. The lights reflected at the substrate–droplet interface are targeted around the screen center (point F) with the brightest intensity (first reflection inside the glass substrate) with the original laser spot diameter (ϕ ∼0.8 mm) size as the screen is mounted at a right angle against the laser propagation. In addition, please note that the interference occurs on the substrate–droplet interface, as shown in Fig. 2(b), but the observation plane must be located on the substrate–droplet interface. The current observation plane is situated far from the droplet by a working distance (200–300 mm). The experimentally observed interference fringes on this screen, as in Figs. 1(a) and 1(b), have to be non-convex profiles at the droplet–air interface. Hence, the dual convex–concave profile is introduced as in Figs. 2(a) and 3 in this study.

Figure 4 shows the relationship [Eq. (3)] between the droplet height and the interference fringe number by comparing the experimental observation, the geometric formulation, and the ray tracing results, which show an excellent agreement with each other out of tens of droplets. For calculation, the precursor film thickness t is assumed as 100 nm.^6,7 Later in this study [Fig. 10(a)], the measured fringe radius and droplet height will be used to determine the precursor film thickness, t. The symbols are the black solid circle for the experiment, the green hollow square for the geometric formulation, and the red hollow triangle for the ray tracing results. The droplet height (h) is measured by Fizeau interferometry and side-view imaging depending on the droplet height magnitude; Fizeau interferometry is preferred for thin height for less than ∼5 μm, and the side-view imaging is used above ∼5 μm.^25,36 The experiment uses the measured fringe numbers in internal reflection configuration, the geometric formulation uses Eq. (3) for the fringe number calculation, and the ray tracing uses the simulated result. The ray tracing simulation will be discussed in Sec. IV. The tested droplets' heights (h) are 1.7–15 μm, and the fringe numbers are 4.5–43. The incident angle (θ_inc) is 43.2°, and the working distance (WD) is 300 mm from the sample to the far-field screen. The linear fitted line based on the experiment is shown in the blue dotted with a slop of 2.795, which is very close to the theoretical value of 2.783 in Eq. (3) with only 0.4% deviation, showing an excellent agreement between the experiment and the theory of the geometric formulation. The figure also shows the experimental data’s 5% horizontal and vertical error bars. Some deviation might be caused by the inaccurate counting of the fringes, especially when the fringe is located near the center's bright laser spot, as it is challenging to identify the fringe. Figure 4 indicates that the droplet height (h) can be exactly determined by counting the fringe number in a simple internal reflection configuration. This finding is remarkable as the contact angle of the thin-height droplet with around 1 μm can be determined without difficulty, while the existing techniques require a sophisticated equipment, such as a microscope, to characterize the thin height. Furthermore, it suggests that the sub-micrometer droplet height can be measured in this RIF configuration.

In Eq. (6), once the fringe radius is measured with the known droplet thickness (h) and radius (a), the contact angle (θ_c) can be determined. For a droplet with a thin thickness of less than 5 μm, the droplet thickness (h) can be determined from Eq. (3) by counting the fringe number on the screen. For a droplet with a thick thickness, the side-view imaging can be used to determine the droplet thickness.

Figure 6 shows the relationship [Eq. (6)] between the droplet contact angle and the fringe radius. The figure shows the increasing contact angle with the fringe radius at the incident angle of 43.2°. The contact angles are measured from the microscope's side-view imaging or Fizeau interferometry.^25,36 The experiment is shown in the black solid circle symbol with the measured fringe radius from the internal reflection configuration. The geometric formulation is shown in the green hollow square symbol using Eq. (6) and the geometric equations in Fig. 5. With the measured contact angle, the fringe radius is determined using Eq. (6). On the other hand, Eq. (6) can be used to determine the contact angle from the measured fringe radius by a closed form and a trial-and-error method. Initially, the contact angle is assumed and the calculated fringe radius is compared with the measured fringe radius from the experiment. If they do not match, then the contact angle is adjusted slightly until they match with each other at 99% confidence interval. The ray tracing is shown in the red hollow triangle symbol. In ray tracing, the contact angle is given from the experiment, and the calculated fringe radius is compared with the experiment. If there is a deviation, then the contact angle is adjusted a bit until the fringe radii from the ray tracing and the experiment are matched within 95% confidence. The contact angle in the ray tracing is from the measurement. In low contact angles less than ∼3 degrees, an analytical relation at the inflection point [θ_c= tan⁻¹(2ah/(a²-h²)], based on RIF [Eq. (3)] in this study or Fizeau interferometry, can be used in comparison with the side-view measurement. The contact angle varies from 1.16°–14.8°, while the fringe radius varies from 17.2 to 140 mm. The figure shows some variation in the data, which can be attributed to the different working distance (WD) and rail angles (θ_RAIL) used under various experimental conditions. The data are combined with each other in the unified condition of WD = 300 mm and θ_RAIL. = 60°. The working distance has a directly proportional effect on the fringe radius, as shown in Eq. (6) and Fig. 5. Despite these variations, each dataset consisting of the experiment, the geometric formulation, and the ray tracing shows an excellent agreement with each other. The fitted line is shown as an exponential function in the blue dotted line. Figure 6 indicates that with the measured fringe radius, the contact angle of the droplet can be determined. Specifically, it would be of great help for ultra-small contact angles of less than 3°, which are challenging in side-view imaging or goniometer and require sophisticated measurement tools such as a microscope, the observation plane of which is located on the substrate–droplet interface.

The geometric formulation is conducted to show that interference fringe formation away from the droplet is caused by the droplet's non-convex profile, and a dual convex–concave profile is introduced. With the convex–concave profile, the experiment, the geometric formulation, and the ray tracing show that the interference fringe number and radius are directly related to the droplet height (thickness) and the contact angle.

Ray tracing was conducted to try to simulate the experimentally recorded interference fringe patterns in parallel with the geometric formulation, especially to verify the assumed dual convex–concave profiles. The calculation algorithm is shown in Fig. 7. The custom-designed MATLAB code (R2019b) is developed with the modified Fresnel equation.^25,36,37,39 For simplicity, a dual spherical-hyperbola profile is proposed for the droplet, as in Sec. III.

The fringe patterns, such as the fringe number, the fringe radius, and the fringe locations, are simulated by the ray tracing and compared with the experiment and the geometric formulation, as in Figs. 3 and 5, to show a good agreement. In the simulation, the features of the droplet and the interference fringe are from the measurement: the top imaging for the droplet radius (a), the side-view imaging for the droplet height (h) and the contact angle (θ_c), and the far-field RIF in internal reflection for the fringe number, the fringe radius, and the fringe locations. The refractive indexes of the substrate and the droplet (n_s and n_d) are from the known values in the literature.^40,41 In the case of a thin height of less than 5 μm, Fizeau interferometry is used to determine its height in brightfield mode in an upright microscope (Olympus GX 51),³⁶ which is also checked by counting the fringe numbers in RIF [Eq. (3)] as in Fig. 4. For small contact angles less than 3°, Fizeau interferometry is used using the analytic relation of the contact angle [2 tan⁻¹(h/a)] by assuming that the upper profile of microdroplets is spherical (capillary length, γ_c is ∼2 mm and most droplet diameter is below 0.5 mm).³ In addition, the fringe radius relation with the contact angle [Eq. (6)] is used as in Fig. 5.

Once the profiles are determined, the ray tracing is simulated by following the flow chart in Fig. 7. Hundreds of rays are drawn first on the upper spherical profile from the apex to the inflection point with equal intervals. Next, the same number of rays are drawn from the apex to the inflection point on the lower concave hyperbola profile by adjusting its interval to create the interference on the screen (Fig. 3). The criterion for interference formation between the two rays is set as the distance between them being less than ∼1 nm. Then, the optical path difference (OPD) between the upper profile and the lower profile in each ray is calculated, and the number of fringes is calculated using Eq. (3). The fringe radius is also calculated with the fringe location as in Figs. 8 and 9. Thus, the simulated fringe features, such as the fringe number, the fringe radius, and the fringe locations, are compared with the experiment. If there is a difference between the ray tracing and the experiment, the parameters of the droplet height and the contact angle are adjusted to reduce the difference. For example, if the simulated fringe number is larger than the measured fringe number in RIF, the droplet height is slightly reduced by considering the measurement error as the height is directly proportional to the fringe number as in Eq. (3). If the simulated fringe radius is larger than the measured fringe radius in RIF, then the contact angle is somewhat reduced.

Figure 8(a) shows the interference fringe formation from the experiment and its comparison with the ray tracing to show a good agreement, indicating that the assumed dual spherical-hyperbola profile is reasonable. The droplet is on the oleophilic coating, as in Fig. 1(a) at the rail angle of 60° (the incidence angle of 43.2°). The lower inlet figure in Fig. 8(a) is a centerline intensity along the blue dash arrow in the central figure with the green curve from the ray tracing simulation and the white dot line from the experiment by reading the image intensity along the centerline, which shows a reasonable agreement with each other. Figure 8(b) is the sketch of the ray tracing with the pink rays for the spherical profile and the blue rays for the hyperbola profile. The red ray passes through the droplet–substrate interface. Part of the rays are shown for display purposes. More studies need to be done to better understand the interference fringe formation with ray tracing.

Furthermore, the interference fringes are explored by varying the incidence angles for the droplets of Fig. 1, as in Fig. 9. In the case of an incidence angle of 60°, no interference fringe is formed as it is above the critical angle of 58.5°.²⁵ The interference fringes are formed below the critical angle. In Fig. 9, the three left columns are the experiment for the oleophilic, non-coating (plain glass), and oleophobic coating surfaces [Fig. 9(a)], and the two right columns [Fig. 9(b)] are the ray tracing simulation results for the oleophilic and the non-coating surfaces. The fringe number and radius are counted from the recorded photo and indicated on the bottom center of each interference fringe photo in Fig. 9(a). The calculated fringe number and radius are shown on the top of the simulated interference fringe profile in Fig. 9(b). The comparison of the fringe number and radius between the experiment and ray tracing shows a good agreement with each other in the oleophilic coating case with a thinner height [4.5 μm in Fig. 1(a)], as the thinner height causes fewer fringes. The non-coating substrate shows some discrepancy in the fringe number. The discrepancy is estimated to be because the thicker thickness [8 μm in Fig. 1(b)] causes many fringes, and some of the fringes in the center bright laser spot were not counted clearly. On the other hand, the fringe radius shows a good agreement as the fringe radius is dependent on the contact angle, and there is less uncertainty in the contact angle measurement if the contact angle is more than ∼3°. Please note that the fringe number is highly sensitive to the droplet height. The droplets on the oleophobic coating do not form any reflection interference fringe as it is estimated to have a pure convex profile with its high contact angle and thick height, not a dual convex–concave profile. Thus, the ray tracing was not conducted for the oleophobic coating case. In the current internal reflection configuration, the interference fringe radius increases with decreasing incident angles, which is also verified from Eq. (6). The experiment and the ray tracing show a good agreement with each other in terms of the fringe number, the fringe radius, and the fringe location. Tens of droplets were simulated using the ray tracing to show a good agreement between the ray tracing simulation and the experiments in the fringe number, the fringe radius, and the fringe location with the given contact angle and the droplet height (h), which is not shown here.

The ray tracing simulation can be applied more effectively in the droplet spreading and wetting with the convex–concave profile together with near-field detection technique of surface plasmon resonance (SPR) imaging.³⁶ More detailed ray tracing simulation will be reported separately.

Figure 10(a) presents the estimated precursor film thickness (t) from the geometric formulation and the ray tracing simulation. The precursor film thickness, t, in the lower droplet profile (Fig. 3) is determined from the geometric formulation [Eq. (3)] with the linearly fitted equation in Fig. 4, shown in the green solid circle symbol in Fig. 10(a). It uses the measured fringe number in internal reflection configuration and the droplet height (h) in Fizeau interferometry with upright microscopy or side-view imaging. It shows the precursor film thickness distribution of 70–150 nm for the microdroplets with a thickness of 1.7–15 μm, which agrees with the generally accepted value for the precursor film thickness of ∼100 nm in the contact line of the droplet.^6,7 The precursor film thickness distribution from the ray tracing simulation is shown in the red hollow square symbol. The simulation was conducted with the measured fringe number, fringe radius, contact angle, droplet radius, and droplet height to show that most droplets have a precursor film thickness of less than 100 nm. The reason for this precursor film distribution by ray tracing is estimated to be the assumed hyperbola characteristics in the contact line region in the dual convex–concave profile by trying to satisfy the fringe radius [related to the contact angle via Eq. (6)] and the fringe number (related to the formed interference under the given droplet height) conditions. The contact line region's profile is not verified in this study. However, the assumed hyperbola profile provides an inference that the precursor film thickness could be less than ∼100 nm, which corresponds with the existing report.^6,7

The ray tracing simulation also shows that the inflection point, M in Fig. 5, is located within the 5% droplet radius from the edge and 10% droplet height from the substrate, respectively, in most droplets, as in Fig. 10(b).

By the geometric formulation and the ray tracing, this research clearly shows that the experimentally observed reflection interference fringe (RIF) on a far-field screen is caused by the droplet's non-convex or dual convex–concave profile. With the dual convex–concave profile of the droplet, the droplet features, such as the droplet height and the contact angle, are compared and verified with good agreement through the experiment, the geometric formulation, and the ray tracing. The geometric formulation shows that the droplet height, the contact angle, and the precursor film thickness can be analytically determined by the measured fringe number and the fringe radius, respectively. The inflection point and the precursor film thickness are also determined by ray tracing. The developed RIF technique provides a new optical diagnostic tool to determine the microdroplet features, such as the dual profile, the droplet thickness, the contact angle, the inflection point, and the precursor film thickness, which is expected to give a new insight to understand the liquid thin film dynamics, such as microdroplet evaporation, microlayer in boiling, condensation, and lubrication.

Portions of this work were presented at the conferences of the 5th Thermal Fluids Engineering Conference in New Orleans, LA, USA, April 5–8, 2020 (TFEC-2020-31998),³⁸ the 2020 ASME IMECE at Portland, OR, USA, November 16–19, 2020 (IMECE2020-23901),⁴² and the Frontiers in Optics + Laser Science, WA, USA, October 9–12, 2023 (FiO-LS JTu7A.2).³⁹ The relevant research is patented as U.S. Application as of December 26, 2023 by TAMUS (P).⁴³ Sandia National Laboratories is a multi-mission laboratory managed and operated by the National Technology & Engineering Solutions of Sandia, LLC (NTESS), a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration (DOE/NNSA) under Contract No. DE-NA0003525. This written work is authored by an employee of NTESS. The employee, not NTESS, owns the right, title, and interest in and to the written work and is responsible for its contents. Any subjective views or opinions that might be expressed in the written work do not necessarily represent the views of the U.S. Government. The publisher acknowledges that the U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this written work or allow others to do so, for U.S. Government purposes. The DOE will provide public access to the results of federally sponsored research in accordance with the DOE Public Access Plan. This study was supported by research grants from CEKO Co., LTD, a Texas Comprehensive Research Fund (TCRF), TAMU-CC Division of Research & Innovation, Dunamis Science & Technology, LLC, NTESS, and NSF CBET TTP program (No. 2301973).

The authors have no conflicts to disclose.

Iltai Isaac Kim: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Yang Lie: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Methodology (supporting); Software (equal); Validation (supporting); Visualization (supporting). Jaesung Park: Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Software (supporting); Validation (supporting); Writing – original draft (supporting). Hyun-Joong Kim: Funding acquisition (supporting). Hong-Chul Kim: Funding acquisition (supporting). Hongkyu Yoon: Funding acquisition (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.