Contact line stick-slip motion and meniscus evolution on micrometer-size wavy fibres
Graphical abstract
Introduction
Wetting forces and analysis of meniscus shape formed around fibres immersed perpendicularly into a liquid have been traditionally used to study the wettability of single fibres, which is essential in several technological applications, such as fibre reinforced composite design [1], [2], [3] and coating of textile fibres [4]. The measurement of the capillary force exerted by a liquid on a fibre, from which a contact angle can be calculated, is known as the Wilhelmy method (tensiometry) [5]. This pull or push force is produced by the weight of displaced liquid above or below the reference horizontal free surface [6]. An advancing meniscus is formed when the fibre is immersed into a liquid, while a receding meniscus is observed when the fibre is withdrawn from the liquid. In general, both situations correspond to different apparent contact angles and not to a single value, as theoretically predicted by Young’s equation for an ideal surface. This so-called hysteresis is related to the pinning of the contact line at physical or/and chemical heterogeneities [7], [8]. Moreover, the value of the experimentally measured contact angle is also influenced by the velocity of the advancing and receding fronts [3], [9], [10]. The Wilhelmy technique has high accuracy when measuring contact angles of fibres with regular shape [3], [11]. However, the analysis of the wetting behaviour of irregular fibres (i.e. wavy and rough natural fibres) using this technique is reported to be very challenging since small variations in the fibre perimeter can largely affect the capillary force and hence the calculated contact angles [12], [13].
Alternatively, the wetting behaviour of a single fibre can also be studied by goniometry, i.e. observing the shape of a steady meniscus formed when a vertical fibre is brought in contact with a liquid. This technique relies on the localization and modelling of the liquid/vapour interface when the fibre is partially immersed in a liquid volume. A simple modelling procedure could involve a proper fit of the meniscus profile by a polynomial or spline functions [14]. However, the contact angles obtained are then generally not accurate since the results are often highly influenced by the choice of a given mathematical function with no physical basis that cannot fit the drastic change in the interface curvature for a wide range of contact angles [2], [15], [16]. For instance, Bateni et al. [17] showed that contact angles calculated using the slope of a polynomial at the contact point vary according to the degree of the polynomial and the number of pixels used in the fitting procedure.
A more accurate approach is to use a mathematical model with a sound physical description of the shape of the meniscus [2], [18], [19]. In static and quasi-static conditions, the liquid/vapour interface is well described by the Young-Laplace equation [20], [21], [22]. If the gravitational effects are neglected (typically when the Bond number Bo ≪ , with the liquid density, the gravity acceleration, the fibre radius, and the liquid-air surface tension), an analytical solution to the Young-Laplace equation for a liquid wetting a cylindrical fibre can be used to describe the corresponding meniscus shape [20], [21]. Then, the contact angle can be simply extracted by measuring the meniscus height and fibre radius [21], [23]. For systems where gravity cannot be neglected, the Young-Laplace (non-linear second-order differential) equation has generally to be numerically solved by using, for instance, a first order Runge-Kutta algorithm [19]. The (quasi) static regime holds as long as the capillary number (with the liquid dynamic viscosity, and the contact line velocity) is small; otherwise, dynamical models should be used [24].
Almost all the models previously described for measuring the wettability of fibres have been developed for the characterization of the menisci on perfect circular cylinders, while fibres with more complex shapes have received less attention [25], [26], [27]. Irregular natural fibres [12] and the recent development of complex-shaped fibres for industrial applications, such as hollow glass [28] and expanded/ablated fibres [29], [30] for improving composite interfaces, call for new studies focusing on the particular features (e.g. diameter change along the fibre length) encountered during the motion of the contact line on fibres with non-circular and variable cross sections [25], [31], [32]. An analysis of the meniscus shape on such complex-shaped fibres needs the development of new experimental techniques and the validation or improvement of traditional theoretical models.
In this study, the dynamic wetting behaviour of an axisymmetric sinus-shaped fibre (wavy fibre) immersed vertically in a large liquid volume has been investigated. Fibres were computer-designed and 3D printed down to micrometre dimensions with Nanoscribe Photonics GT equipment, and the Wilhelmy method was used in parallel with meniscus shape analysis; these two independent techniques were being cross-validated by direct comparison of resulting contact angles. This methodology enabled monitoring the profile of the fibre and the liquid meniscus while the fibre was being immersed, and to correlate the contact angle variations with the motion of the contact line.
The immersion and withdrawal of the fibre resulted in stick-slip motion of the meniscus [27], [33], [34], [35], [36], which is predicted by a quasi-static model directly inspired by an analysis of the impact of chemical heterogeneities on stick-slip behaviour and hysteresis in a microchannel [37]. Static model solutions are here tracked upon varying the immersion depth in upward and downward directions, explaining why two different paths for the contact line position are observed, with pinning-depinning events occurring in both cases.
Section snippets
Materials
Sinus-shaped fibres were 3D printed with a Photonic Professional GT laser lithography system (Nanoscribe GmbH, Germany) by direct laser writing of an acrylic based negative tone resist (IP-G 780), with a 3D resolution of 600 nm. All fibres were printed following a sinusoidal wave with a diameter of 150 μm at peak amplitude and a period of 200 μm, while the diameter at the lowest amplitude was designed with 4 different values: 80, 100 (Fig. 1a), 120, and 140 μm. These fibres will be further
Wetting behaviour of an axisymmetric sinus-shaped fibre
Fig. 2a–e shows a 150/100 fibre being immersed in water (see movie clip MC1 in Supporting Material). When the fibre is put into contact with the liquid, the liquid surface deforms spontaneously and rises up on the fibre to form a meniscus, with a three-phase contact line located above the horizontal liquid surface of the reservoir (Fig. 2a). When the vessel containing the liquid moves up, the liquid surface induces a vertical displacement of the contact line; but the contact line does not move
Conclusions
Available experimental methodologies and models studying the meniscus shape and contact line movement during the wetting process of fibres deal almost exclusively with perfect circular cylinders [18], [19], [20], [21], [22]. While some work has been developed for shaped fibres [25], [31], [42], these have focused on fibres with complex cross section shapes but still constant along the fibre length.
In this paper, a novel experimental method combining tensiometry and goniometry was presented
Acknowledgments
We gratefully acknowledge the financial support from ESA: European Space Agency and BELSPO: The Belgian Federal Science Policy Office through the PRODEX (Evaporation and Heat Transfer) and MAP (Evaporation and MANBO) project.
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