DOI: 10.1039/C7SM01729D
(Paper)
Soft Matter, 2018, Advance Article

Zhao Pan‡*^{a},
Floriane Weyer‡^{b},
William G. Pitt^{c},
Nicolas Vandewalle^{b} and
Tadd T. Truscott*^{a}
^{a}Splash Lab, Mechanical and Aerospace Engineering, Utah State University, Logan, UT 84322, USA. E-mail: panzhao0417@gmail.com; taddtruscott@gmail.com
^{b}GRASP, CESAM, Physics Department, University of Liège, Liège B-4000, Belgium
^{c}Department of Chemical Engineering, Brigham Young University, Provo, UT 84601, USA

Received
27th August 2017
, Accepted 6th March 2018

First published on 15th March 2018

Inspired by the huge droplets attached on cypress tree leaf tips after rain, we find that a bent fibre can hold significantly more water in the corner than a horizontally placed fibre (typically up to three times or more). The maximum volume of the liquid that can be trapped is remarkably affected by the bending angle of the fibre and surface tension of the liquid. We experimentally find the optimal included angle (∼36°) that holds the most water. Analytical and semi-empirical models are developed to explain these counter-intuitive experimental observations and predict the optimal angle. The data and models could be useful for designing microfluidic and fog harvesting devices.

Droplets attached to fibres are common in daily life and can be found in many situations (e.g., droplets hang on and/or between thin pine needles;

Within the broad range of these investigations, Lorenceau et al.^{14} reported the maximum volume of a droplet on a horizontal fibre, which focused on the fundamental question: how much liquid can be held by a horizontal fibre in a gravitational field? Likewise, we are inspired by the large water droplets attached to the “armpit-like” locations of tree leaves (Fig. 1(a) and (b)); and seek to answer the question how much liquid can be held by a bent fibre in a gravitational field? The volume of the droplets on these juniper leaves are about 60 μL, which is significantly larger than can be supported by a simple horizontal fibre. This observation implies that by changing the geometry of a fibre (e.g., bending a fibre to create a corner), the critical volume of water can be significantly increased. Further, the previous theoretical work does not account for a bent fibre which is debatably more common in nature.

Fig. 1 (a) Droplets attached on the leaves of a cypress tree after rain in Ditan Park, Beijing, China. (b) In the lab, a droplet is trapped on the interior fork of juniper leaf tips (collected from Cache Valley, UT) where the volume of the droplet is ∼60 μL. (c–e) Photographs of the morphology of droplets (SDS solution) on bent fibres (2b = 250 μm) with various half angles (θ) defined in Table 1, front views on the left and side views on the right: (c) θ = 4.3°; (d) θ = 58.2°; (e) θ = 21.5°. |

We hypothesize that the corner of a bent fibre can trap more liquid than a straight fibre, and the angle of the bent fibre affects the volume of the maximum liquid that the fibre can hold.

Two liquid solutions (a solution of 0.01 M sodium dodecyl sulfate (SDS) in water, a mixture of 25% glycerol in water) and pure water were used in these experiments. In all solutions, dye was added for better visualization. The surface tensions of the liquids were measured using the pendant drop method via a CAM 200 goniometer (KSV Instrument Ltd). The surface tensions were γ_{sw} = 36.34 mN m^{−1} for the SDS mixture, γ_{gw} = 67.14 mN m^{−1} for the glycerol mixture, and γ_{w} = 71.97 mN m^{−1} for water. Their densities and viscosities at 25 °C were 1.00 × 10^{3} kg m^{−3}, 1.19 × 10^{3} kg m^{−3} and 1.00 × 10^{3} kg m^{−3}, and 0.89 × 10^{−3} Pa s, 1.87 × 10^{−3} Pa s, and 1.00 × 10^{−3} Pa s respectively. On a flat nylon surface the contact angles are ∼40° (SDS–water solution), ∼30° (glycerol–water solution), and ∼59° (water).

Liquid was carefully measured by a micro-pipette and transported to the corner of the bent fibre via the thin vertical fibre (80 μm) which served as a drainage guide (ESI,† Fig. S1). The capillary effect from the vertical thin fibre is negligible when the droplet is large (ESI,† Fig. S2). The volume of the droplet was incrementally increased by 1 μL until the droplet detached from the fibre (ESI,† Fig. S3).

In addition, high-speed photography of the droplet falling off of the fibre, reveals two distinct droplet detachment processes. The first detachment occurs in regime II where a thin film develops at the apex of the bent fibre, which eventually ruptures, releasing the droplet as shown in the image sequence of Fig. 3(a) (ESI,† Video 1). The second type of detachment occurs for larger angles of regime III and releases the droplet slowly from one side of the fibre as shown in the sequence of images in Fig. 3(b) (ESI,† Video 2). It takes more than 20 times longer for this type of detachment to occur than the film rupture. During the detachment the film recedes from one side of the fibre (right hand side in Fig. 3(b)), but maintains relatively the same length on the opposite side (left hand side of Fig. 3(b)).

Fig. 3 High speed photographs of the instability at the critical droplet volume on a bent fibre (full videos found in ESI,† Videos 1 & 2). Time from the first droplet placement are labeled at the bottom of each image. The destabilized droplet detaches differently depending on the angle θ. (a) The thin film on the top of the droplet breaks and the droplet falls symmetrically between the bent fibre, releasing from both sides nearly simultaneously (θ = 3.9°). (b) The droplet slides down along the left side of the fibre for larger θ angles (θ = 53.9°). |

Regime I. The critical volume of a droplet that wets and is attached to a horizontal fibre (θ = 90°) can be modeled by either of two approaches. In the ESI† (Section 4) we consider the perturbation of free energy of the droplet-fibre system. The other method consists of balancing the drop weight and fibre adherence of an equivalent system.^{14} Both methods lead to

where β is the angle between horizontal and the line between the centre of the droplet and the 3-phase point where the fibre exits the droplet. Defining the capillary length of a fluid as , (1) is identical to the equation found by Lorenceau et al.,^{14} which we also validated experimentally. Eqn (1) implies that the maximum possible volume of a liquid held by a horizontal fibre will occur when sin β approaches unity. The maximum droplet size can now be estimated by normalizing (1) by a characteristic volume of a spherical droplet whose radius is the capillary length yielding:

which we label as model I. Note that the natural characteristic volume () will also be used to normalize the other two models. An important physical interpretation of eqn (2) is that the maximum volume of a liquid that can be held by a horizontal fibre is limited by fibre thickness. More physical insights from a free energy point of view can be found in the ESI† (Sections 4 and 7).

(1) |

(2) |

Regime II. When the fibre is bent at small angles (e.g., θ ≲ 18°), a droplet of critical size is characterized by a triangular thin film connected to the apex of the fibre (Fig. 1(c)). The same technique (see ESI† (Section 5) for more details) used for model I leads to an estimation of the critical volume of the droplet:

where L_{0} = L/λ is a length scale that characterizes the wetted length (L) compared to the capillary length (λ). A more detailed formulation of the model and an alternative derivation of model II made by balancing the weight of the droplet with the capillary force (similar to the method used in^{14}) can be found in the ESI† (Section 6).

Ω ≈ 4γLθ/ρg. | (3) |

Normalizing (3) by the characteristic volume (), we create model II that describes the critical volume for small angles:

(4) |

Eqn (4) has two implications for regime II: first, the maximum volume of water held by a bent fibre at small angles depends on the wetted length; and second, the maximum volume scales linearly with the angle between the fibre (Ω* ∼ θ). More generally, a larger angle (θ) increases the space between the fibre and a larger wetted length (L) increases the amount of liquid that can be held in the gap. Physical interpretation from a free energy point of view can be found in the ESI† (Sections 5 and 7). Note, we do not concern ourselves with the extreme case (θ = 0°, wetting of the parallel fibres) whose rich physics can be found in literature such as Protiere et al.^{15}

Regime III. We now turn our attention to empirical results before formulating the model for regime III. The non-axisymmetric geometry of the fibre, and more importantly, the complicated geometry of the droplet itself (Fig. 1(e)) make the problem analytically intractable; even approximate solutions such as (1) and (3) are not explicitly accessible. Therefore, we turn to experimental data to formulate a semi-analytical model.

indicating that the critical volume of liquid in regime III decreases when the angle increases (Ω* ∼ 1/sinθ).

We can illustrate that the characteristic length = Lsinθ is approximately the same for a wide range of angles (18° ≲ θ ≲ 90°) by superimposing photographs of droplets on a fibre as shown in Fig. 4(a). Physically, is a length scale that characterizes the critical size of the droplet, and geometrically measures the half width of the droplet trapped between a bent fibre. By comparing with the capillary length of the liquid we can formulate a normalized wetting length (L*) as

L* = /λ = Lsinθ/λ ≈ 1. | (5) |

Fig. 4 (a) Superimposition of photographs of SDS–water solution droplets at critical state on fibre bent to various angles (θ_{i} = 11.4°,36.8° & 67.8°, i = 1, 2, 3.). Non-superimposed photographs of more droplets can be found in the (Fig. S5, ESI†). (b) Normalized wetting length (L/λ) as a function of θ. Triangle, square, or round markers are experimental data, and the solid line represents the empirical model for Bo = 1. (c) L/λ as a function of sinθ in logarithmic scale. |

Experiments exhibit good agreement with the new parameter as shown in Fig. 4(b and c) (solid curve). The experimental data also reveal that the characteristic length of the drop-fibre system (Lsinθ) is comparable to the capillary length (λ) of the liquid. Not surprising, also yields a critical Bond number of unity (Bo = ρg^{2}/γ = 1). In this context, the Bo number can be considered a criteria of droplet stability. For example, Bo > 1 indicates the weight of a droplet dominates and tends to detach from the fibre. A Bo < 1 implies a stronger capillary force than gravitational force and thus a droplet stays on the fibre.

Experimental data also show that when θ is small (e.g., θ ≲ 10°), the normalized wetting length L/λ does not strongly depend on θ (Fig. 4(c)), which implies that L/λ ∝ θ^{0}. Data fitting (dashed line in Fig. 4(b and c)) reveals that L_{0} = L/λ ≈ 5.1, which provides a constant parameter for model II (4).

A natural way to formulate a model for the critical state for angles 18° ≲ θ ≲ 90° is to consider a scaling law where the weight of a droplet is balanced by the capillary force: ρgΩ ∼ 2Lγ, where 2L is the total wetted length of the droplet-fibre system. Recalling the definition of capillary length and substituting the empirical model of the wetting length (5) into the scaling law leads to Ω ∼ 2λ^{3}/sinθ. Normalizing the droplet volume by the characteristic volume () leads to a semi-analytical model (model III):

(6) |

This model is rather simplistic and over-predicts the critical droplet size for most experiments (Fig. 5), but it may be more useful for applications where an upper bound approximation is more practical (more analysis can be found in ESI†, Section 8). Physically, as the fibre is bent to smaller and smaller θ, the droplet is more firmly “confined” at the corner of a bent fibre. This is because any perturbation of the droplet mass is opposed by a larger portion of the force from the opposite side of the fibre as θ decreases. Thus, a droplet is less likely to move sideways, and will be more stable on the fibre (see ESI†, Section 9 for an alternative model based on this physical argument).

Fig. 5 The non-dimensional maximum volume, Ω*, held by a bent fibre as a function of the half-angle θ for several SDS solutions (blue markers), glycerol solutions (green markers) and water (black markers) from Fig. 2. The three proposed models are compared to experimental data. Region B (green) indicates the conditions necessary for a droplet to remain on a bent fibre. Region A (yellow) indicates values where a droplet will fall off a bent fibre with small fibre angles (Fig. 3(a)). Region C (pink) indicates values where a droplet stability first appears by sliding down one side of the fibre under small perturbations (Fig. 3(b)). Region D indicates the transient regime between A and C. Maximum volume for experiments occurs near θ ≈ 18° and for theory θ ≈ 21°. Note: model I is only applicable at θ = 90° and dependent on fibre diameter as shown in the lower right corner. |

However, the effect of this stabilization is maximized near θ ≈ 18°, where the thin film that develops at the apex of the fibre begins to play a more dominant role in the droplet attachment and detachment process as shown in Fig. 3(a) and governed by model II. Indeed we noticed that near θ ≈ 18° either droplet detachment mode could occur. As a final note, we expect a competition between the thinning of the film near the apex of model II and the liquid confinement of model III, wherein the optimal angle between the two allows a bent fibre to hold a maximum amount of liquid.

(7) |

2.4.1 Droplet detachment behavior. More interestingly, model II and III split the θ–Ω* space into four regions. In region B (green region in Fig. 5), droplets can be held steadily on a fibre. In regions A and C, droplets tend to detach from the fibre with two different modes. In region A (yellow area in Fig. 5), when a droplet exceeds the critical volume predicted by model II, the thin film on the top of a droplet breaks and the droplet falls off the fibre (t = 114 ms, Fig. 3(a) and Video 1, ESI†). This observation confirms one of the assumptions of model II that the main contribution of the droplet stability is the triangular film at the top of the droplet. In region C (red section in Fig. 5), when a droplet is larger than the critical volume predicted by model III, the droplet slides down along one of the two sides of the fibre (t = 11.49 s, Fig. 3(b) and Video 2, ESI†). Region D (orange area in Fig. 5) is the transition region where both modes could happen.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm01729d |

‡ These authors contributed equally to this work. |

This journal is © The Royal Society of Chemistry 2018 |