Microscale liquid-based mechanical elements for multifunctional integration

Introduction

Multifunctional integration of energy harvesting and storage devices has been a subject of great interest over the past decade. A key challenge has been that most high-performance energy harvesting or storage devices incorporate brittle ceramic components as critical parts. Our previous attempt ¹ to directly integrate thin-film silicon solar cells onto fiber-reinforced polymer composites, for example, showed that even moderate cyclic loading can lead to the formation and propagation of microcracks in semiconductor and transparent ceramic electrode layers. Analogous observation was also made for thin-film batteries embedded into fiber-reinforced composites. ²

Previous studies of flexible electronics embedded thin semiconductor elements near the neutral plane of a supporting substrate or constructed elementary circuits from pre-deformed ultrathin semiconductor membranes or a sparse array of semiconductor islands interconnected via arc shaped bridges. ³ The first approach, however, cannot readily be adapted to solar cells because load-bearing composites are most often opaque. Furthermore, both the area and the thickness of solar cells and batteries cannot be arbitrarily scaled down without compromising their performance and/or capacity.

This motivates us to explore an alternative strategy where we exploit the unique mechanical characteristics of microscale liquid elements to enable more versatile multifunctional integration of brittle components into mechanical structures. In the present manuscript, we will present our recent progress in assessing the early feasibility of this strategy and in exploring the mechanical characteristics of liquid elements through a modeling study.

Alternative integration approach

Our basic idea is to mechanically decouple load-bearing structures from energy harvesting/storage or other functional devices by using liquid-based mechanical elements (Figure 1). Liquid elements offer many intriguing mechanical characteristics. They may break (rupture) but they can be ‘perfectly’ healed. They can withstand significant cyclic deformations without suffering from fatigue and may also be designed to change shapes and slip, merge or split on solid surfaces in response to external stimulations. OPEN IN VIEWER

The emergence of ionic liquids with virtually zero vapor pressure ⁴ and the dominance of surface tension over the inertia and gravitational forces at microscales allow us to effectively confine liquid elements without requiring physical seals. This helps effectively eliminate practical difficulties in confining liquids, which have traditionally been a major impediment to using liquids in mechanical systems.

Experimental setup and procedure

To characterize the mechanical characteristics of individual microscale liquid bridges, such as the force-gap relations and rupture distances, we constructed an experimental setup consisting of a dual-camera image capture station and an analytic balance (Figure 2). A bottom substrate was bonded to the weighing plate of the balance and a top substrate was attached to a rigid cantilever arm, which is in turn mounted on a z-stage. OPEN IN VIEWER

A non-volatile liquid droplet was first dispensed on the bottom substrate. The top substrate was then lowered and brought into contact with the droplet to form a liquid bridge. The distance between the two substrates was varied while monitoring the force, either attractive or repulsive, exerted by the liquid bridge.

Custom software was developed to acquire the time-synchronized top and side images of the liquid bridge and readings from the balance under quasi-static loadings. The acquired images were analyzed to extract the geometric parameters of the liquid bridge.

Multifunctional integrated structures, where 500 -µm-thick silicon (ceramic) chips are used to emulate brittle functional components, were also examined using the setup under either tension or bending. To demonstrate the mechanical behavior of integrated structures under extreme deformations (>10% strain), we use thin highly stretchable silicone (polydimethylsiloxane, PDMS) substrates. For testing under tension, an integrated structure was suspended between two precision translational stages. We then optically observe the mechanical response of the integrated structure while increasing the spacing between the two stages. For testing under bending, an integrated structure was placed around cylindrical rods of different diameters to achieve pre-scribed radii of curvature.

Sample preparation

Chemically patterning surfaces (e.g. hydrophilic islands surrounded by a hydrophobic region) is one convenient approach to form an array of discrete liquid droplets/bridges on pre-defined locations. To prepare a silicon chip that emulates a functional element, we first spin coat a thin layer of Teflon® on a silicon substrate and then lithographically patterned the layer to expose an array of circular hydrophilic islands. The microfabrication steps are illustrated in Figure 3. OPEN IN VIEWER

PDMS substrates are normally hydrophobic with a static contact angle >100°. Hydrophilic patterns matching those on the silicon elements are formed by exposing the PDMS substrates to oxygen plasma through a shadow mask.

Mechanical behavior of the integrated structures under tension and bending

Figure 4 shows representative optical images of the integrated structures under tension and bending. The images demonstrate that the liquid bridges can effectively accommodate both tension and bending while keeping the brittle silicon chips intact. OPEN IN VIEWER

Modeling of mechanical characteristics of liquid elements

Previous studies reported analytic and numerical approaches for modeling capillary phenomena involving liquid bridges.^5–9 We implement a modeling approach based on the surface energy minimization algorithm ¹⁰ to study the mechanical characteristics of liquid bridges under axial loading.

The simulation domain includes the chemically patterned top substrate, the chemically patterned bottom substrate and the liquid bridge (Figure 5). The top and bottom substrates are hydrophobic except within chemically patterned hydrophilic islands. The radius of hydrophilic island on the top substrate is denoted as r_s and that on the bottom substrate as r_b. We consider cases with both symmetric (r_s = r_b) and asymmetric (r_s < r_b) hydrophilic patterns. The liquid volume and the gap between top and bottom substrates are denoted as V and g, respectively. OPEN IN VIEWER

The equilibrium shape of a liquid bridge is determined by iteratively finding or evolving the shape of the liquid bridge that minimizes its total surface energy (summed contributions of the liquid–gas, liquid–solid and solid–gas interfaces). The capillary force F_c exerted by the liquid bridge is then calculated using the virtual work approach, that is, by taking the derivative of the total surface energy E with respect to the gap g: F_c = dE/dg. This procedure is repeated for different combinations of the liquid volume and gap. A complete input file for a publicly available software implantation of the surface energy minimization algorithm, Surface Evolver, ¹⁰ is included in the Appendix.

When the gap exceeds a certain critical value, the liquid bridge is no longer the configuration with the minimum surface energy. The liquid instead breaks into two separate droplets, one each on the top and bottom substrate. This critical gap is considered as the rupture distance. We find this distance by starting with an initial guess and applying the bi-section method, that is, by iteratively narrowing a possible range of the rupture distance until we reach the numerical tolerance limit.

In the present study, we limit ourselves to ‘micro’-scale liquid bridges where surface tension is the dominant force and the effect of the gravity on liquid bridges can be ignored. The validity of this assumption can be checked by evaluating the so-called Bond number, Bo = ρ g L²/γ, which is the ratio between the gravitational force and the surface tension force. For typical ionic liquids with surface tension γ of the order of 50 mN/m and size L < 1 mm, we estimate Bo < 0.1.

Modeling results and discussion

Figure 5(a) shows two representative images of liquid bridges experimentally obtained in the present study and the corresponding predictions from the surface energy minimization algorithm. The intrinsic surface contact angle θ on the hydrophilic pattern is set to be 10°, which is a value we estimated from independent experiments on our silicon substrates.

The intrinsic surface contact angle θ is determined by the physico-chemical characteristics of the surface itself. In contrast, the apparent angle α is an actual angle between the liquid–gas interface of a liquid bridge and the solid surface. The apparent angle is a function of the liquid volume and gap and can exceed the intrinsic surface contact angle if the liquid volume is sufficiently large.

Figure 5(b) shows the experimentally measured and predicted force F as a function of the gap, which agree well with each other. All the results presented in the present work are non-dimensionalized so that they can be readily applied to liquid bridges of different liquids and/or sizes: the non-dimensionalized gap between the two substrates G* = g /r_s, the non-dimensionalized force F* = F_c/(2πγr_s) and the non-dimensionalized liquid volume: V* = V/r_s ³ . As a numerical example, for an array of liquid bridges with radius r_s = 50 µm, the non-dimensionalized force F* = 2.5 corresponds to a force per unit area of approximately 5 kPa.

The capillary force has two components, the surface tension force and the Laplace pressure. When the gap is small and the liquid bridge is compressed between the top and bottom substrates, the (negative) repulsive component contributed by the Laplace pressure is dominant. The capillary force therefore increases rapidly with the decreasing gap (∼1/g). When, instead, the two substrates are pulled apart, stretching the liquid bridge, the (positive) attractive surface tension force becomes important.

Suppose we start at a small gap such that the repulsive Laplace pressure is the dominant component of the capillary force. As the gap is gradually increased and the liquid bridge is stretched, the apparent angle α between the liquid–gas interface and the solid surface is decreased. The contact lines on the two substrates remain pinned at the boundaries of the hydrophilic islands as long as the apparent angle α remains greater than the intrinsic surface contact angle θ within the hydrophilic island This is called the radius-controlled regime (Figure 6(a)). OPEN IN VIEWER

If the apparent angle α reaches θ before the liquid bridge ruptures, however, the liquid bridge will transition into the angle-controlled regime. In this regime, the contact lines are de-pinned and the contact radii decrease as the liquid bridge is further stretched while keeping the apparent angle constant (Figure 6(b)). For small liquid volumes (see case V* = 0.5 in Figure 7(a)), the liquid bridge may rupture while in this angle-controlled regime. OPEN IN VIEWER

For large liquid volumes (see cases V* = 1.6 and 2.6 in Figure 7(a)), the apparent angle may stay above θ until the liquid bridge ruptures. The liquid bridge will therefore remain in the radius-controlled regime during the entire stretching process.

For intermediate liquid volumes (case V* = 1 in Figure 7(a) and (d)), the liquid bridge first deforms in the radius-controlled regime but subsequently transitions into the angle-controlled regime where the contact radii decrease with increasing gap. When the gap is increased even further, however, the contact radii may increase rather than decrease with increasing gap. The contact line may eventually reach the perimeter of the hydrophilic pattern and a rupture occurs in the radius-controlled regime.

The increase in the contact radii with increasing gap, which is also observed for smaller liquid volumes (cases V* = 0.5 and 0.8 in Figure 7(d)), is a result of competition between the solid–liquid interface energy and the liquid–vapor interface energy. Using the Young–Laplace equation, one can write the total surface energy E_total as

E total = A LV γ LV - A SL γ LV cos θ = E 1 - E 2

When the gap exceeds a certain value and a liquid bridge develops a sufficiently elongated necking region, the increase in the liquid–vapor interface energy, E₁, with increasing contact radii is more than offset by the increase in the solid–liquid and solid–vapor interface energy, E₂ (that is, ΔE₂ > ΔE₁). The minimum total surface energy requirement therefore leads to increasing contact radii. A similar behavior is discussed for liquid bridges confined between circular disks of finite radii. ⁵

The preceding discussion is concerned with cases with symmetric patterns (r_s = r_b). If the pattern radii on the top and bottom substrates, r_s and r_b, are different, the contact line on one substrate may reach the angle-controlled regime before the contact line on the other substrate. As a result, the radius-controlled regime and the angle-controlled regime could co-exist (Figure 6(c)) in asymmetric pattern cases.

Figure 7(a) shows the predicted capillary force as a function of gap g for symmetric cases (r_s = r_b). The different curves correspond to different non-dimensionalized liquid volumes V*. The value of the gap where the liquid bridge transitions between the radius-controlled and angle-controlled regimes is a function of both the liquid volume and the intrinsic surface contact angles. Figure 7(d) show the corresponding contact diameter as a function of gap g with different non-dimensionalized liquid volumes V*.

Figure 7(b) and (c) shows the simulation results for asymmetric cases for different values of the pattern radius ratio η = r_b/r_s for a fixed liquid volume of V* = 1. Note that the transition from the radius-controlled to the angle-controlled regime occurs at smaller gaps for larger pattern radius ratios.

Figure 8 shows the force–gap relations for different intrinsic surface contact angles θ. At small enough gaps, the apparent angle α is greater than θ and is determined only by the liquid volume and geometry. As a result, for the symmetric cases examined in Figure 8(a), the force in the radius-controlled regime is independent of θ. For asymmetric cases examined in Figure 8(b), different intrinsic contact angles lead to different capillary forces when the liquid bridge deforms in the angle-controlled regime on one of the substrates. OPEN IN VIEWER

Controlled rupture

A key requirement for implementing liquid-based mechanical elements is the ability to tune the gap at which liquid bridges rupture or detach from one of the substrates. This allows liquid bridges to rupture at select locations first, so that multifunctional load-bearing structures can experience large local deformation while protecting brittle functional elements.

One potential approach to control the rupture distance is to systematically vary the size of hydrophilic patterns that define liquid bridges. Figure 9(a) shows the predicted rupture distance as a function of liquid volume for three different surface conditions: homogeneous substrates; hydrophobic substrates with symmetric chemically defined circular hydrophilic islands; hydrophobic substrates with symmetric physically etched circular hydrophilic islands. The liquid contact lines are assumed to be pinned at the perimeters of the physical patterns for the third case. The value of θ is fixed at 50° for all cases. OPEN IN VIEWER

When the liquid volume is sufficiently small (V* approximately below V₀ = 2.3 for the specific cases shown in Figure 10), the rupture occurs in the angle-controlled regime where α is determined by the intrinsic surface contact angle θ. In this case, there is no distinction between homogeneous and chemically patterned substrates. When the liquid volume is larger than V₀, however, the apparent angle of the liquid bridges on the chemically patterned substrates exceeds θ and the liquid bridges can be deformed further while in the radius-controlled regime. As a result, the rupture may occur at larger gaps for the chemically patterned substrates than the homogenous substrate when V* > V₀. OPEN IN VIEWER

When the liquid volume is smaller than V₀, the non-dimensionalized rupture distance is smaller for the physically patterned substrates than the other two cases because the liquid contact lines are pinned at the perimeters of the hydrophilic pattern and the bridge is forced to form an elongated necking region. Above this threshold volume, the physically and chemically patterned substrates behave identically because the rupture occurs in the radius-controlled regime for both cases.

We can also control the rupture distance by creating asymmetric patterns (r_s ≠ r_b) as illustrated in Figure 10. Figure 9(b) shows the rupture distances for chemically defined surfaces with different pattern radius ratios.

Summary

Liquid-based mechanical elements enable intriguing new design concepts for integrating functional devices with brittle components, such as energy harvesting and storage devices, into load-bearing mechanical structures. We experimentally demonstrate preliminary feasibility of this integration strategy using chemically patterned 500 -µm-thick silicon chips to emulate brittle functional components. A systematic study of the mechanical characteristics of microscale liquid bridges, including the force-gap relations under axial loading and the control of rupture distances, is discussed to help guide the systematic design of liquid-based mechanical elements for multifunctional integration. The work presented here motivates further systematic studies and practical implantation of these unique mechanical elements in various applications of multifunctional composites.