Twisted fibers enable drop flow control and enhance fog capture

Significance

Transport of drops on fibers is fundamental to the design of atmospheric fog capture systems, which are currently used to relieve water scarcity in arid regions of the world. Here, we find how twisted dual fiber systems can stabilize asymmetric drops, consequently enhancing drop transport speeds, and create a rich array of new drop sliding flow patterns. Through the newfound understanding of drop flow on twisted fibers, we demonstrate its use in making anticlogging meshes and show that the twisted fiber system significantly enhances fog capture as compared to straight fibers.

Abstract

Drop–fiber interactions are fundamental to the operation of technologies such as atmospheric fog capture, oil filtration, refrigeration, and dehumidification. We demonstrate that by twisting together two fibers, a sliding drop’s flow path can be controlled by tuning the ratio between its size and the twist wavelength. We find both experimentally and numerically that twisted fiber systems are able to asymmetrically stabilize drops, both enhancing drop transport speeds and creating a rich array of new flow patterns. We show that the passive flow control generated by twisting fibers allows for woven nets that can be “programmed” with junctions that predetermine drop interactions and can be anticlogging. Furthermore, it is shown that twisted fiber structures are significantly more effective at capturing atmospheric fog compared to straight fibers.

Drop on fiber systems are prevalent in nature (1) and used to stave off water scarcity via direct atmospheric fog capture in arid regions (2–4). Common materials for fog capture devices are extruded, uniform cross-section steel wires (5), Raschel meshes of plastic fibers and strips (2), and woven poly-yarn (4). Nature, however, has evolved slender, grooved geometries (6) and bumpy structures on plants and animals such as cactus spines (7, 8), nepenthes plants (9, 10), Syntrichia caninervis (11), spider spindle silk (1) and Namib desert beetle shells (12, 13) to enhance fog capture and promote fast drop transport (6, 14, 15) and drainage (16). Though despite relevant applications and natural prevalence, drop interactions with nonuniform radius fibers remain understudied (17–20).

A drop deposited on a cylindrical fiber can either assume a symmetric “barrel” or an asymmetric “clamshell” shape, cf. Fig. 1. Much prior work regarding the transport of drops on fibers considers perfectly wetting barrel-shaped drops on cylindrical fibers (21–24). Weyer et al. (22) found that the manipulation of fiber diameters at fiber junctions could be used to deterministically control a drop’s path as it flows down a fiber mesh, and Gilet et al. (23) characterized a drop crossing a fiber junction, enabling the production of compound drops. However, in many situations drops preferentially form clamshell shapes, especially when the liquid considered has a finite liquid–solid contact angle $\alpha$, e.g., water. Adam (25) was first to notice how increasing $\alpha$ caused a drop to transition from the barrel to the clamshell shape, which Carroll (26) solved analytically. Since then, numerical simulations and experiments have uncovered a bistability regime between the shapes, both with (27) and without (28) gravitational effects.

Fig. 1.

Dual fibers stabilize asymmetric drop shapes. Side-view images of asymmetric clamshell and symmetric barrel drops of silicone oil ($\mathrm{\Omega }=0.75{\text{mm}}^3$, $\alpha =0^{°}$, $\mu =20$ cSt) on either (A) one single or (B) two parallel nylon fibers ($d=200\mathrm{\mu }$m, acting as the scale bar). Gravity $g$ acts in the negative $z$-direction. Unlike on single fibers, perfectly wetting drops on two parallel fibers can exist in the clamshell configuration despite gravity. (A) Gravity-free Surface Evolver simulations show the drop’s scaled total surface energy $\widehat{E}=E/(\sigma {(4\pi )}^{1/3}{(3\mathrm{\Omega })}^{2/3})$ against its scaled center of mass $\widehat{z}/d$. As a drop spreads around a single fiber, the energy pathway smoothly descends until the drop engulfs the fiber at $\beta =0^{°}$ and $\widehat{z}/d\approx 1.6$. The drop then quickly transitions to the barrel shape $\widehat{z}/d\approx 0$ with associated energy $\widehat{E}=0.81$, releasing energy $\mathrm{\Delta }\widehat{E}\approx 0.03$. Here, the maximal extant of the contact line $\beta$ is varied and the drop’s center of mass is left free. (B) For drops on two parallel fibers the energy pathway is discontinuous during fiber engulfment, allowing for deposited drops to adopt clamshell shapes in spite of the drop’s perfect wettability. Here, $\beta =0^{°}$ at $\widehat{z}/d\approx 1.8$.

These previously reported drop shape transitions, however, hinge on either changing the liquid’s physical properties or the surface it wets. In this work, we report how twisting a fiber pair with a wavelength $\lambda$ provides a facile way to control the clamshell to barrel transition. We characterize a drop’s dynamics as it slides along the twisted fibers and show how twisted fibers can preprogram the drop’s behavior at fiber junctions, as well as showcase how they can promote atmospheric fog capture.

Static Shapes

For a perfectly wetting oil drop on a single fiber ($\alpha \approx 0^{°}$), the only stable shape is the barrel shape (28). The clamshell shape is unstable for all reduced drop volumes $\mathrm{\Omega }/d^3$, where $\mathrm{\Omega }$ is the drop’s volume and $d$ the fiber’s diameter (28). This stability can be seen by numerically minimizing a static clamshell drop’s total free surface energy as the maximum position of its apparent contact line is fixed about a fiber, sketched in Fig. 1A. By using the open source finite element solver, Surface Evolver (29, 30) (SI Appendix), we computed these clamshell drop surface energies and compared them against the position of the drop’s apparent contact line on the fiber $\beta$, shown in SI Appendix. Here, the drop’s energy landscape appears continuous from $\beta =[\pi -0]$, i.e., going from a small wetted area to the barrel shape. A plot of this energy landscape against the scaled position of the drop’s center of mass $\widehat{z}=z/d$ reveals the same smooth, continuous behavior, cf. Fig. 1A.

For drops spreading around fibers with nonuniform radius, Carroll postulated that they would encounter a series of nonequilibrium “jumps” based on the fiber’s geometry, each requiring an activation energy to overcome (31, 32). Such activation energies are analogous to those which are posited to cause hysteresis during smooth surface contact line spreading (32, 33), wetting/dewetting transitions on heterogeneous surfaces (34–36), corner pinning (37), the Haines jump in porous media flow (38, 39), as well as other phenomena (40, 41).

In Fig. 1B, the system’s energy is presented for a drop on two parallel fibers. A discontinuity in the energy is found as the drop’s shape sweeps the clamshell shapes, $\beta =[1.2\pi -0]$. To overcome this discontinuity in energy, the position of the drop’s center of mass appears to lift off the fibers before its contact line can move to decrease the drop’s energy and reach the barrel shape at $\widehat{z}=0$.

Presuming a silicone oil drop’s contact line spreads around the fibers, we can also expect this drop to spread until reaching the numerically predicted energy barrier. Fig. 1B shows a drop of silicone oil ($\mathrm{\Omega }=0.75{\text{mm}}^3$) deposited on two parallel nylon fibers ($d=200\mathrm{\mu }$m, $\lambda =\infty$). Despite the presence of gravity (negative $z$-direction), the drop maintains the clamshell shape. Instead, a clamshell-to-barrel transition was found only when the system was significantly disturbed, e.g., by rapid fiber oscillation. These drop shapes are shown as insets in Fig. 1B, corresponding to the preoscillation drop sitting in the clamshell local energy minimum and the postoscillation drop that has transitioned to the barrel shape, shown in Movies S1–S7.

Sliding Dynamics

By playing with the idea of an asymmetrically stabilized drop, as in Fig. 1B, we can consider the deposition of a drop on two twisted fibers oriented in the direction of gravity, as shown in Fig. 2. Here, a single drop of silicone oil ($\mathrm{\Omega }=2\mathrm{to}6{\text{mm}}^3$, $\alpha =0^{°}$, $\mu =10\mathrm{to}101$ mPa-s) is deposited on one side of a pair of vertical nylon fibers ($d=200\mathrm{\mu }$m) twisted with a wavelength $\lambda$. Analogous experiments for partially wetting glycerol/water drops ($\alpha ={65}^{°}$) are presented in SI Appendix. Post deposition the drop assumes either the clamshell or barrel shape and slides with a velocity $u_{||}$ in the direction of gravity $g$. The deposited drop’s resultant flow, i.e., its trajectory about the fiber pair, can be classified into three regimes, Fig. 2: groove flow, skipping flow and barrel flow. These flows are defined by both the drop’s shape (clamshell or barrel) as well as the coupling between $\lambda$ and the drop’s rotational velocity $u_{\perp }=\omega r$, where $\omega$ is the drop’s angular velocity and $r$ the drop’s radial center of mass (perpendicular with respect to gravity). For both groove and skipping flow ($r\ne 0$), drops adopt a clamshell shape, whereas in barrel flow drops adopt a barrel shape ($r=0$ and $u_{\perp }=0$). For groove flow, the drop’s flow path remains coupled to the fibers’ twisted wavelength $\lambda$, following the path traced by the groove formed between the two fibers $\omega =\pi u_{||}/\lambda$. Whereas for skipping flow, the drop freely slides down the fiber with its trajectory uncoupled from $\lambda$, $0<\omega <\pi u_{||}/\lambda$.

Fig. 2.

Twisting dual fibers control drop flow regimes. Complex drop sliding dynamics emerge upon fiber reorientation, here visualized by shadowgraphs of silicone oil drops ($\mathrm{\Omega }=5{\text{mm}}^3$) sliding due to gravity with velocity $u_{||}$ in the negative $z$-direction. Varying the twisting wavelength $\lambda$ controls the transition between three distinct flow regimes: 1) groove flow. The drop slides down the fiber with an angular velocity coupled to $\lambda$ as $\omega =\pi u_{||}/\lambda$; 2) skipping flow. The drop slides with a finite $\omega$ uncoupled to $\lambda$; 3) barrel flow. The drop fully engulfs the fiber and $\omega =0s^{-1}$.

Our interest lies in understanding these flow regimes based on the twisting geometry of the fibers as well as the liquid’s physical properties, i.e., viscosity $\mu$, volume $\mathrm{\Omega }$, surface tension coefficient $\sigma$, contact angle $\alpha$, and density $\rho$. The simplest flow in our system is barrel flow, as it is qualitatively the most similar to a barrel-shaped drop sliding down a single fiber (24). By using Newton’s 2nd law (24, 42), the drop’s $z$-component of motion can be written as $\text{d}{u}_{||}/\text{d}t=\rho \mathrm{\Omega }g-F_{\mu }-F_{\sigma }$, where $g$ is gravity and $F_{\mu }\sim \mu u_{||}d$ is the viscous sliding friction. $F_{\mu }$ is proportional to $\mu$, the drop’s sliding velocity $u_{||}$ and the fiber’s diameter $d$ as postulated and shown in prior studies on single fibers (23, 24, 33, 42). We assume the surface tension force $F_{\sigma }$ to be negligible as these drops perfectly wet the fibers, meaning at steady flow $F_{\mu }$ balances the drop’s weight $\rho \mathrm{\Omega }g$, giving $u_{||}\sim \rho \mathrm{\Omega }g/\mu d$. This means that for barrel drops, barring changes to the drop’s wettability, only variations in $\mu$, $\mathrm{\Omega }$, $\rho$ and $d$ modify $u_{||}$. A constant of proportionality $\mathcal{C}$ for $u_{||}$ is then determined to be $\mathcal{C}\approx 0.0034$ (SI Appendix), giving a characteristic sliding velocity $V_o=\mathcal{C}(\rho \mathrm{\Omega }g)/(\mu d)$ for a barrel-shaped oil drop during barrel flow.

Surprisingly, despite the different flows’ complexities, a drop’s downward transport velocity $u_{||}$ seems unaffected by $\lambda$. Instead, only the shape of the drop, i.e., barrel shape for barrel flow or clamshell shape for groove and skipping flow, sets $u_{||}$. The scaled downward transport velocities $u_{||}/V_o$ can be found in SI Appendix. It can be seen that the downward barrel velocities gather about $u_{||}/V_o=1$, whereas the drops exhibiting an asymmetric flow (groove, skipping) transport at a constant, yet faster, $u_{||}/V_o>1$ velocity than their symmetric barrel counterparts. For drops with similar physical properties ($\mathrm{\Omega },\rho ,\sigma ,\mu$) these variations in transport velocities can be understood through the value of $\mathcal{C}$, which is a function of the drop’s geometry and wettability. As clamshell shapes have less wetted area than barrel drops for the same $\alpha$, $\mathrm{\Omega }$, $d$, and $\lambda$, it follows that drops in groove and skipping flow would generate less viscous friction as compared to those in barrel flow, leading to an increase in $\mathcal{C}$ and a subsequent increase in their transport velocity (43). It is important to note that the observed invariance of $u_{||}/V_o$ to $\lambda$ is only true for drop/surface combinations with minimal sliding contact angle hysteresis, as presented here for silicon oil on nylon. The influence of hysteresis on $u_{||}$ can be seen affecting the glycerol/water data in SI Appendix.

For a drop to transition away from groove flow into either skipping flow or barrel flow, it is necessary for the drop to overcome the energy barrier presented in Fig. 1B. For drops following the groove, it is clear from geometry that the rotational velocity $u_{\perp }$ couples to the downward sliding velocity $u_{||}$ as $u_{\perp }=r\omega =\pi r{u}_{||}/\lambda \sim {\mathrm{\Omega }}^{1/3}{u}_{||}/\lambda$. Thus, an upper limit becomes $u_{\perp }/u_{||}\sim 1$, or in other terms ${\mathrm{\Omega }}^{1/3}/\lambda \sim 1$, to maintain groove flow. If ${\mathrm{\Omega }}^{1/3}/\lambda >1$, the flow is expected to transition from groove flow to either skipping or barrel flow, as seen in Fig. 3A when scaling $u_{\perp }$ by $V_o$. The rotational velocity then either decreases to $0$ indicating barrel flow, or remains $>0$ and no longer increases as $\lambda$ decreases. The Inset to Fig. 3A provides a closer look at a silicon oil drop in skipping flow as it skips over and is rotated by the twisted fibers’ texture.

Fig. 3.

Drop dynamics. (A) Scaled rotational velocities $u_{\perp }/V_0$ for silicone oil drops ($\mathrm{\Omega }=2\mathrm{to}6{\text{mm}}^3$, $\mu =10\mathrm{to}101$ mPa-s). The transition from groove flow to either skipping or barrel flow is delineated by the dimensionless length ${\mathrm{\Omega }}^{1/3}/\lambda$. (B) Center of mass trajectory plots in the $y$-$z$ plane for a silicone oil drop in either groove or skipping flow. For groove flow, the center of mass oscillates with a single wavelength $2\lambda$. For skipping flow, the drop’s center of mass oscillates into and away from the fiber as it skips over the fiber’s twisted structure, causing a short wavelength oscillation set by $\lambda$ superimposed onto a longer wavelength oscillation set by $u_{\perp }$ and $u_{||}$. (C) The drop’s center of mass position against time $t$. During skipping flow, the drop may spontaneously transition into barrel flow. Its downward velocity $u_{||}$ decreases due to the additional wetted area and its center-of-mass moves toward the fiber $\widehat{x}=\widehat{y}=0$. Only transitions away from skipping flow are observed.

For glycerol/water drops, $\alpha ={65}^{°}$, a similar transition from groove flow to skipping flow was observed; see SI Appendix. However, the transition from groove flow to barrel flow was not observed. This absence of barrel flow can most likely be attributed to a global instability of the barrel shape for $\alpha$ near neutral wetting, as is the case for a drop on a single fiber (28). For hydrophobic fibers a similar absence of the barrel shape is likely for most liquid/fiber combinations; however, we note that both groove and skipping flow were still observed. This result is striking as it implies wicking into the grooved structure is not the sole factor inciting the groove and skipping flow patterns. Here, wicking is only expected as dictated by the Concus–Finn condition (44) for capillary wicking in an interior corner, $\alpha <\pi /2-\zeta$, where $\zeta$ represents the corner’s interior half angle and $\zeta =0$ for two touching cylinders (45). Thus for $\alpha >\pi /2$, wicking is not expected.

Fig. 3B shows the position of the drop’s center of mass in time as it slides down the fiber in both skipping flow and groove flow. Note how for groove flow the drop’s center of mass motion follows a smooth curve and is coupled to $\lambda$. For skipping flow, however, the rotational motion oscillates with two frequencies. The first frequency appears coupled to $\lambda$ and represents the drop’s center of mass periodically moving toward and away from the fibers. This behavior is reminiscent of the behavior predicted in Fig. 1B that suggests that for the drop to spread around the fibers its center of mass must lift off the fibers. The second frequency is lower and represents the drop’s macroscopic rotation $\omega$.

For ${\mathrm{\Omega }}^{1/3}/\lambda \le 0.4$ it was not observed that a drop which reached steady state in groove flow ever transitioned to either skipping or barrel flow, illustrating the robustness of the flow pattern. For ${\mathrm{\Omega }}^{1/3}/\lambda >0.4$, however, it was observed that drops which began in the skipping flow regime sometimes decayed into barrel flow after a certain amount of time, Fig. 3C. This decay illustrates the tenuousness of skipping flow in comparison with the robust stability of barrel flow and is reminiscent of a drop which begins by asymmetrically sliding on a single fiber before transitioning to the barrel shape (24).

Applications

Programmable Junction Crossings.

Twisted fibers have potentially broad benefits for drop-on-fiber technologies such as the burgeoning field of drop-on-fiber digital microfluidics (23). However, hurdles in their implementation include controlling the sliding drop’s dynamics as it crosses the fiber junctions as well as controlling the volume of liquid residue it leaves (21–23). One potential benefit of twisted fiber systems lies in their ability to stabilize asymmetric flows, allowing a user to weave together a fiber mesh that encodes which junctions a drop can pass across and at which junctions the drop leaves or does not leave a liquid residue.

We demonstrate this concept in Fig. 4 where two fibers are woven between an array of horizontal single fibers. A drop of 0.01M (1-Tetradecyl)trimethyl-ammonium bromide (TTAB) in glycerol/water of viscosity $\mu =50$ mPa-s and volume $\mathrm{\Omega }=5.0{\text{mm}}^3$ is deposited on the two fibers and slides due to gravity. The surfactant TTAB is used to reduce the drop’s contact angle and avoid pinning effects at the fiber junctions, which could alternatively be achieved by surface functionalization. Since ${\mathrm{\Omega }}^{1/3}/\lambda$ is controlled such that the drop remains in groove flow, we know from the phase space of flow behaviors in Fig. 3 that the drop will follow the groove. Upon encountering a junction, if the crossing fiber passes through the drop’s flow path, liquid is deposited. However, at junctions where the crossing fiber is on the opposite side of the drop’s flow path, liquid is not deposited. By controlling which junctions receive deposited drops, a weaving process such as this creates a facile way of fabricating a fiber mesh that through careful design can selectively deposit liquid at junctions, opening avenues for the creation of fiber meshes that can make multiple different compound drops using the same mesh (21) or encode a series of different reactions in parallel, similar to traditional microfluidic systems (46). This deposition process is further illustrated by Movies S1–S7.

Fig. 4.

Programmable junction crossings. Time sequenced images recorded at 250 fps of a 0.01M TTAB water/glycerol drop ($\mathrm{\Omega }=5.0{\text{mm}}^3$, $\mu =50$ mPa-s) deposited on two twisted vertical nylon fibers (A) ($d/\lambda =0$) or (B) ($d/\lambda =0.05$) woven between an array of single horizontal nylon fibers ($d=200\mathrm{\mu }\text{m}$). $d/\lambda$ is set to control the drop in groove flow. (A) The asymmetric nature of the drop’s flow causes it to selectively interact with the horizontal fibers, allowing a user to “program” at which fiber junctions fluid is deposited. (B) Twisting the two fibers creates a downward path for the drop sans horizontal fiber interactions.

Fog Harvesting.

Twisted fibers also show promise in applications relying on droplet transport and capture such as fog harvesting from the atmosphere. To explore a twisted fiber’s fog capture potential, an array of twisted fibers was placed in a wind tunnel with fog, as illustrated in Fig. 5.

Fig. 5.

Enhancing fog collection. Measured twisted fiber collection efficiencies ${\eta }_{\lambda }$ over the predicted collection efficiencies for single straight fibers of similar projected areas ${\eta }_c=[0.19-0.34]$ against the twisting ratio $d/\lambda$. Bulk averaged wind velocities $U=1.5\pm 0.2$ m/s and fog drop diameters $d_d=2.5\mathrm{\mu }$m. The Reynolds number was $\text{Re}={\rho }_{a}U{d}_f/{\mu }_a=33\pm 4$, where ${\rho }_a$ and ${\mu }_a$ are air’s density and viscosity at ${20}^{°}\text{C}$ and $d_f$ the fibers’ diameter. The test section’s cross-sectional area was $0.24{\text{m}}^3$, and fog generation averaged $9\text{L}/\text{h}$. The fibers’ fog capture efficiency clearly increases as the fibers are increasingly twisted. As fog collects on single fibers, individual drops grow before coalescing and shedding. However, on twisted fibers liquid transport happens within the grooved structure, facilitating drainage, and avoiding drops to reside on the fibers leading to enhanced efficiency. See Movies S1–S7 for illustration.

The twisted fiber’s individual capture efficiency is defined as ${\eta }_{\lambda }=Q_c/Q_f$, where $Q_c$ is the measured amount of water captured by the fog net over time and $Q_f$ the fog flux across the net. $Q_f$ is defined as $Q_f=qNL{d}_f/A$, where $q$ is the total fog generation over time, $N$ the number of fibers in the net, $d_f$ the diameter of a cylinder and $L$ the cylinder length that would have the same projected area as a pair of twisted fibers, and $A$ the cross-sectional area of the tank in which the fog net was placed.

In order to normalize ${\eta }_{\lambda }$ against the capture efficiency of a single fiber, we compare our results against the classical Stokes’ correlation model (14, 47–49). In our experiments the Stokes’ numbers are between $St=\rho d_d^{2}U/(9\mu d_f)=[0.54-0.89]$, where $d_d$ is the diameter of the fog drops and $U=1.5\pm 0.2\text{m/s}$ the wind velocity. Details on the measurement of $d_d$ and $U$ are found in SI Appendix. Consequently, the single fiber’s capture efficiency can be described by the equation ${\eta }_c=0.466{({log}_{10}(8St))}^2$ for $0.125<St<1.1$ (47).

It has previously been shown and can be seen in Fig. 5 that fog collection on a single fiber occurs as a series of beaded drops on the fiber’s surface (49). One interpretation of the influence on capture efficiency is that these beaded drops are thought to increase the fiber’s effective diameter, subsequently decreasing an individual fiber’s capture efficiency as compared to the dry fiber case. However, for twisted fibers in this range of Stokes numbers, as can be seen in the inset to Fig. 5, a film is stabilized within the groove consequently suppressing the formation of larger drops on the fiber’s surface. Because of this observed drop suppression in the twisted fiber case, the dry fiber diameter was used as a length scale in the Stokes’ correlation for the single-fiber case to calculate ${\eta }_c=[0.19-0.34]$. If using hydrophobic twisted fibers there will be no liquid transport facilitated through the grooves (44) and larger drops may be generated, which would effectively lead to a larger $d_f$ and, thus, an expected reduced efficiency (49).

Fig. 5 shows ${\eta }_{\lambda }/{\eta }_c$ against the fiber’s geometry $d/\lambda$ (14, 47, 48) (SI Appendix). For two parallel fibers, $d/\lambda =0$, the individual capture efficiency of the parallel fiber pair is increased as compared to the theoretical prediction for a single, smooth, straight fiber. When the fibers are slightly twisted together, the capture efficiency initially decreases which could be caused by the reduction in cross-sectional surface area for long twist wavelengths $\lambda$. However, by increasingly twisting the fibers, i.e., reducing $\lambda$, we find that the total amount of captured water clearly increases with twisting ratio $d/\lambda$. Intuitively one might expect that the twisted fiber’s increased total surface area per unit length would cause this efficiency increase; however, this assumption cannot account for the increase as the added surface area is no more than 6%. Instead, it is the geometry of the twisted fibers that plays a crucial role in facilitating the capture and drainage of incoming fog droplets in its wetted grooves, suppressing the formation of drops with increasing fiber twist (Fig. 5 and Movies S1–S7).

A fog net’s efficiency is thought to be modeled as $\eta ={\eta }_a{\eta }_c{\eta }_d$, where ${\eta }_a$ represents a net’s aerodynamic efficiency, ${\eta }_c$ its capture efficiency, and ${\eta }_d$ its drainage and re-entrainment efficiency (13, 48, 50–52). As our system is designed as isolated fibers spaced at a relatively large pitch and since no drops formed on the fibers are observed to be re-entrained, it can be assumed, as have previous authors, that the efficiencies ${\eta }_a\approx {\eta }_d\approx 1$ (48, 50). These assumptions would then leave us with the individual fiber’s droplet capture efficiency to primarily be representing the total efficiency of the net. It can be seen from Fig. 5 that the capture efficiency ${\eta }_{\lambda }$ of the parallel and twisted fiber systems exceeds the theoretical limit ${\eta }_c$ for a single-fiber system. ${\eta }_{\lambda }/{\eta }_c$ should then in principle not exceed unity, which here is likely linked to dependencies in the experimental setup that was not designed for the intent to provide a precise measure of these efficiencies and may stem from fluctuations in the flow or flow disturbances from the frame inserted in the wind tunnel. Nevertheless, the key observation is the clear trend that the water harvesting capacity of the twisted fibers increases with increasing twist, i.e., increasing $d/\lambda$. As the theoretical ${\eta }_c$ is used for the sake of normalizing the measured ${\eta }_{\lambda }$, the data in Fig. 5 should rather be interpreted as a system approaching an optimal droplet impaction efficiency/capture thus approaching an upper theoretical limit. Essential to this process is the capillary drainage in the grooves, leading to a reduction in the number of drops on the fibers thus increasing fog collection efficiency.

Concluding Remarks

A drop’s flow along two fibers can be passively controlled by twisting the fibers. We delineate three distinct flow regimes: groove, skipping, and barrel, which are formed by varying the fibers’ twisting wavelength $\lambda$, without changing the fibers’ material or the liquid’s properties ($\sigma$, $\rho$, $\mu$, $\mathrm{\Omega }$). We build this passive control based on our found understanding that perfectly wetting asymmetric clamshell shaped drops can be stabilized in groove flow, and that the ratio of the drop radius and twist wavelength sets the flow transition.

Single-fiber-based microfluidic systems have been suggested as a passive, low electrolysis risk alternative to current 2D implementations (23). Now the ability to stabilize asymmetric drops on fibers has potential for these systems by allowing for the creation of a single “programmable” mesh that could perform an array of different reactions, create a multitude of unique compound drops, or provide effective sorting of variably sized drops. Additionally, selective deposition at fiber junctions in mesh networks could enable the selective deposition of UV-curable cross-linker, enabling the fabrication of meshes with designer properties.

For fog capture, twisted fiber systems show even more promise as they provide a facile method to boost current fog capture rates and can be easily implemented using existing technological solutions (53, 54). These twisted fiber systems utilize no harsh chemicals or surface modifications which may degrade over time and potentially pollute captured drinking water. They would also simultaneously decrease the amount of raw material, e.g., steel, used in construction both saving resources and lowering shipping and manufacturing costs. Additionally, the grooves formed by twisting together fibers have the potential to suppress large drop growth and improve drop drainage, allowing for the creation of self-draining, nonclogging meshes which maintain their fiber capture efficiency in a variety of fog conditions.

Materials and Methods

Materials.

1000 St silicone oil VIS-RT100K (European Article# 99515.271), 20 cSt silicone oil (CAS# 63148-62-9), and glycerol (CAS# 56-81-5) were purchased from VWR International (Oslo, Norway). Acetone (CAS# 67-64-1) was purchased from Merck (Darmstadt, Germany). 10 cSt Silicone oil (CAS# 63148-62-9, Article# 378321), 50 cSt Silicone oil (CAS# 63148-62-9, Article# 378356) and 100 mPa-s Silicone oil AP (CAS# 63148-58-3, Article# 10838) were purchased from Sigma Aldrich/ Merck (St. Louis, Missouri). Abulon top $200\mathrm{\mu }$m and $400\mathrm{\mu }$m fishing line produced by Abu Garcia was purchased from a local sporting goods store in Oslo, Norway. (1-Tetradecyl)trimethyl-ammonium bromide (TTAB) (CAS# 1119-97-7) was purchased from Alfa Aesar (Kandel, Germany).

Sliding Dynamics Experimental Methods.

Experiments were recorded at up to 2,500 fps as needed. Prior to each experiment, the fibers were rinsed with acetone and deionized water, isopropanol was avoided as it was found to damage the nylon fiber’s surface. Care was taken to ensure the fibers had no built-up static electricity, as residual surface static could increase sliding velocities (55). The experimental fluids used were either bought silicon oils or created by mixing glycerol and deionized water or 1000 St silicone oil with 20 cSt silicone oil. Glycerol/water and oil/oil mixture viscosities were measured using an Anton Paar 702 rheometer and all mixtures were confirmed to behave as Newtonian fluids. Static contact angles $\alpha$ were measured by first cleaning a nylon fiber with acetone and DI water and then cutting it into 2 to 5 mm fragments that were layered on top of a glass slide cleaned first with deionized water and then with isopropanol. The nylon-covered slide was then placed in an oven at just above the nylon’s melting point for a few minutes. The oven was then cooled and the nylon removed from the glass slide. The surface of glass slide is found to create a flat, smooth surface of nylon. 2 ${\text{mm}}^3$ drops of water were then deposited on top of the flat nylon surface and the contact angle $\alpha$ of the resulting sessile drop was measured using the secant method in Matlab. Silicone oil surface tensions were assumed to be $0.035\text{N}/\text{m}$ and glycerol/water surface tensions and densities were taken from literature (56, 57).

The downward velocity $u_{||}$ was obtained by tracking the maximum extension of the drop in the radial direction in time and was extracted using subpixel edge detection techniques in Matlab (58, 59). The angular speed $\omega$ was measured by tracking the average of the drop’s minimum and maximum $y$-projection in time $t$, c.f. Fig. 2B of the main text, and relating it to a helix as $y\sim \text{sin}(\omega t+\delta )$, where $\delta$ represents the time shift of the first measurement. The radial center of mass $r$ for the clamshell shapes is taken to be the radius of a sphere of equal volume for simplicity $r={(3\mathrm{\Omega }/(4\pi ))}^{1/3}$. The angular velocity is then $u_{\perp }=r\omega$.

Note that near the boundary of the transition away from groove flow that the system becomes sensitive to how the drop is initially deposited on the fiber; however, for gentle depositions, the probability of deposition into barrel flow where groove flow would be expected was slim to none.

Fog Harvesting Experimental Methods.

Water from a reservoir is gravity fed into a pool containing 12 ultrasonic humidifiers that produce fog at 9 L/h. Downstream outlet fans pull the fog first through a 3D printed hexagonal mesh to remove some vorticity then through a fog collector composed of 19 pairs of vertical $200\mathrm{\mu }$m diameter, $23$ cm long twisted fibers spaced at a pitch of $1$cm. The bulk velocity of the fog flow $U$ was between $1.2$ and $2.0$ m/s for each experiment and estimated using particle tracking velocimetry just upstream of the fog collection device. The collected fog is funneled to a graduated cylinder and a reading from the cylinder is extracted every 5 min. Each experiment was run for 3 h, 1 h for the system to reach equilibrium, and 2 h to collect the data from which the capture rate is extracted. See Movies S1–S7 for illustration. As the nylon fibers have some inherent elasticity, they are first prestretched and held taught for a few days in order for their structure to plasticize. This minimizes their elastic vibrations in the fog flow, as vibrations have been shown to modify the transport of drops on fibers (60, 61).

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Transition from clamshell shape to barrel shape drop when exposed to oscillations.

Movie S2.

Droplet sliding on a mesh with designed deposition at the junctions.

Movie S3.

Demonstrating how twisted fibers can be used to avoid deposition and pinning at junctions in a mesh.

Movie S4.

Programmed drop interactions on a mesh with multiple drops.

Movie S5.

Drop dynamics of two parallel fibers in fog flow.

Movie S6.

Drop dynamics on a single fiber in fog flow.

Movie S7.

Drop dynamics on twisted fibers in fog flow.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information and/or research data depository (62).