## Abstract

Optical lenses with diameter in the millimeter range have found important commercial use in smartphone cameras. Although these lenses are typically made by molding, recent demonstration of fast-cured polymer droplets by inkjet printing has gained interest for cost-effective smartphone microscopy. In this technique, the surface of a fast-cured polydimethylsiloxane droplet obtains dynamic equilibrium via the interplay of surface tension, gravity, thermalization, and a steep viscosity hike. The nature of surface formation involves multiple physical and chemical domains, which represent significant challenges in modeling with the Young–Laplace theory, assuming constant surface tension and viscosity. To overcome these challenges, we introduce the concept of effective surface tension, which allows fast-cured polymer droplets to be modeled as normal liquid droplets with constant viscosity.

© 2018 Optical Society of America

## 1. INTRODUCTION

Optical lenses with up to 5 mm diameter and focal length are important in smartphone camera modules, which generally have a minimum working distance of 10–20 cm and a corresponding sampling resolution of 100 μm per pixel. A single additional external millimeter lens attached to the camera window can achieve 1 μm resolution, which effectively turns a smartphone camera into a microscope [1]. One simple idea among many others is to fabricate low-cost lenses by free-formed liquid surfaces. In fact, the development of free-formed curable-liquid lenses started in 1995 with moisture-cured polyurethane for surface doming [2]. In 2000, Komatsu *et al.* deposited high-surface-tension UV-cured resins on a surface to form micro droplets, and subsequently curing to form a microlens array. In 2014, Lee *et al.* developed a method of creating lenses by inverting the substrate with a deposited resin droplet and relying on gravity to produce significant curvature [3]. In these techniques, the liquid surface has already reached the endpoint of shape evolution and thus represents a steady-state curvature due to the long duration between dispensing and curing. Thus, the final surface curvature can be well predicted by the Young–Laplace (Y-L) model. In 2015, we demonstrated that heat-accelerated curing of polydimethylsiloxane (PDMS) can effectively stop droplet surface evolution, while it still possesses significant surface curvature. This single-step method has enabled high-throughput, high-quality, low-cost lens fabrication [4,5]. Since then, a variety of methods sharing the same or similar principles have been developed [6–9]. Most recently, DotLens smartphone nanocolorimetry has been demonstrated for lead sensing in drinking water [10]. In these techniques, it is crucial to understand the relationship between droplet surface geometry and fabrication parameters. To date, only empirical methods have been employed [11]. As mentioned earlier, the challenge of modeling surface curvature arises from the rapid viscosity hike during the heat-accelerated curing process, which violates the major assumption in the Y-L model. On the other hand, time-domain numerical modeling is difficult due to the multiple physical and chemical domain interactions of heat transfer, surface tension, viscosity and phase change, and of course, gravity [12]. Nevertheless, our observations suggest that polymer droplet lenses produced by fast curing does appear like normal liquid droplets as shown in Fig. 1. This insight has motivated us to introduce the concept of effective surface tension (EST) for a fast-cured droplet [13]. In other words, EST is the surface tension of the cured droplet as if it were still in the liquid form, and can be employed in the Y-L model.

## 2. MODELING LIQUID DROPLETS

To quantify the lensing effect of a droplet situated between an air–solid interface, we first consider the interfacial energy determined by the Young’s equation ${\gamma}_{\mathrm{sa}}-{\gamma}_{\mathrm{sl}}-{\gamma}_{\mathrm{la}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{c}=0$, where ${\gamma}_{\mathrm{sa}}$ is the solid surface free energy, ${\gamma}_{\mathrm{sl}}$ is the solid–liquid interfacial tension, ${\gamma}_{\mathrm{la}}$ is the liquid–air surface tension, and ${\theta}_{c}$ is the contact angle. Second, its curvature at equilibrium can be determined by solving the Y-L equation $\mathrm{\Delta}p=-{\gamma}_{\mathrm{la}}\overline{\nabla}\xb7\widehat{n}={\gamma}_{\mathrm{la}}({R}_{x}^{-1}+{R}_{y}^{-1})$ via surface energy minimization [14], where $\mathrm{\Delta}p$ is Laplace pressure, $\hat{n}$ is unit normal pointing out of surface, and ${R}_{x}$ and ${R}_{y}$ are radii of curvature in the axes parallel to surface. However, the Y-L equation has an analytical solution only in the simplest cases and does not consider other influential forces such as gravity; thus, the interface shape is usually calculated numerically. A widely used numerical modeling software is Surface Evolver [15], which simulates steady-state droplet shape by surface energy minimization subject to different forces and constraints in addition to surface tension, e.g., gravity [16–18]. Once the droplet surface is determined, lens parameters such as focal length ($f$) and numerical aperture (NA) can be calculated. Here we consider a droplet having the shape of a spherical cap resting on a flat substrate. The droplet shape is controlled by the Bond number $\mathrm{Bo}=(\rho g/{\gamma}_{\mathrm{la}})\times rh$, where $\rho $ is liquid density (${\rho}_{\mathrm{PDMS}}=960\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$), $g$ is gravitational constant, $r$ is radius, $h$ is height, and $\gamma $ is liquid surface tension. For small droplets (less than 5 μL), the Bond number is less than 1, and surface tension forces dominate, so droplets can be accurately regarded as a spherical cap described by $V=\frac{1}{6}\pi h(3{r}^{2}+{h}^{2})$, where $V$ is volume, $h$ is height, and $r$ is radius.

Here, water droplets of different volumes were deposited on glass surface with ${\theta}_{c}=90\xb0\pm 2.6\xb0$, as shown in Fig. 2(a) (inset, lower row). The generated model is shown for comparison (inset, upper row). The measured and simulated droplet height versus radius are shown in Fig. 2(a) (bottom line), and the corresponding volumes of the droplet show a good fit to the spherical cap equation for up to 10 μL before the gravitational effect starts to flatten the droplet [19]. For a different glass surface (Corning Gorilla Glass 3) that exhibits $\theta c=110\xb0\pm 3\xb0$, the measured and simulated droplet variables are shown in Fig. 2(a) (top line). For fixed surface tension, curvature is highly dependent on volume.

The curvature of the droplet was obtained by imaging and parabolic curve fitting from the droplet vertex. The numerical curvature was obtained as $1/R$ for a parabola in the form of $2y=\frac{1}{R}{x}^{2}$, and the results are shown in Fig. 2(b). The focal length can be obtained with the lensmaker’s equation, simplified as $f=R/(n-1)$ for a plano-convex lens. Smaller droplets possess larger curvatures that result in larger spherical aberrations; therefore, we defined the aperture as 1/3 the lens diameter. The curvature obtained this way typically deviates less than 5% from the curvature at the vertex.

In Fig. 2(a), as the droplet volume increases, its radius and height both increase as expected; however, the height eventually plateaus due to gravity, and additional volume creates only lateral expansion. The maximum puddle height that a liquid can attain on a surface is related to the energy minimization of surface tension and gravity, and its value can be calculated by Eq. (1):

## 3. MODELING CURABLE LIQUID POLYMER DROPLETS

Traditional methods of droplet modeling are applicable to slowly cured PDMS if the droplet is allowed to reach static equilibrium. For example, at 80°C, PDMS requires 1 min to cure, which ensures that static equilibrium is attained or nearly attained, as shown in Fig. 4(a). However, this model is oversimplified when a PDMS droplet undergoes fast curing on a heated substrate, while the surface evolution stops at a point of dynamic equilibrium, as shown in Fig. 4(b).

Despite this, observations indicate that fast-cured polymer droplets retain the appearance of a liquid droplet, and an empirical trend is observed between droplet geometries (radius, height, contact angle, and curvature) and its volume and substrate temperature, as shown in Fig. 4(c). For example, a 100 μL PDMS droplet shows decreasing radius and increasing height with increasing temperatures due to increased contact angle, similar to water droplets deposited on increasingly hydrophobic substrates. Droplets cured at 200°C show increasing radius and height (until plateau) with increasing volumes, also similar to water droplets with increasing volumes.

A series of cured PDMS lenses has been fabricated by a range of droplet volumes (12 values) and temperatures (13 values). In Fig. 5(a), the droplet radius was measured: for a fixed temperature, the radius increases with volume as expected due to the larger droplet. For a fixed volume, the radius decreases with temperature as the droplet cures before spreading out; this also indicates a height increase to offset for constant volume. In Fig. 5(b), the droplet height was measured: for a fixed temperature, the height increases with volume, but effectively plateaus in a fashion similar to the maximum puddle height equation. For example, at 300°C, the maximum height that can be achieved with very large volumes was 3.61 mm, while at 140°C the maximum height was 2.35 mm. Beyond these heights, the volume increase causes only lateral expansion. For a selected volume, the temperature increase causes height increase before it plateaus at high temperature.

In Fig. 5(c), the droplet contact angle was measured: for a fixed temperature, the contact angle is independent of the droplet volume, as surface energy is not related to volume in the Young’s equation. For a fixed volume, the contact angle displays itself as a function of substrate temperature, which is another mode of surface energy as defined by the Young’s equation. This is a unique phenomenon observed in fast-cured polymer droplets. The relationship between contact angle and temperature can be fitted by a sigmoidal function, shown in Eq. (2). The ability to tune the droplet contact angle by changing temperature makes it possible to obtain a wide range of usable PDMS droplet curvatures for lensing, overcoming the inherent problem of small contact angles in slowly cured PDMS droplets:

Two observations point to the similarities of fast-cured liquid PDMS droplets to normal liquid droplets. First, the droplets attain a maximum puddle height by which any volumetric increase results in strictly lateral expansion [Fig. 5(b)]. Second, the contact angle depends solely on temperature, i.e., surface energy [Fig. 5(c)], similar to the Young’s equation. Thus, we propose a droplet model that views fast-cured polymer droplets as normal liquid droplets with an EST, and circumvent the complex curing process. The EST is calculated by measuring the maximum puddle height attained by droplets at every temperature, and its contact angle at that temperature. Upon reorganization of Eq. (1), the EST is calculated by Eq. (3):## 4. CONCLUSION

Curable polymers such as PDMS possess low surface tension and exhibit small contact angles on the air–solid interface, leading to negligible surface curvature. However, heat-induced fast-curing causes accelerated solidification in a state of dynamic equilibrium results in significant surface curvature and lensing effect. Due to the dynamic nature of the fast-curing process, the Y-L model, which assumes constant surface tension, is not applicable. Here, we model fast-cured polymer droplets as normal liquid droplets that possess an EST, a static surface tension value that the cured droplets *appear* to exhibit. Using experimentally extracted EST, the final cured droplet surface can be modeled accurately using a standard surface evolution program.

## Funding

National Science Foundation (NSF) (1558240, 1643391).

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