Article pubs.acs.org/Langmuir
Predictive Model of Supercooled Water Droplet Pinning/Repulsion Impacting a Superhydrophobic Surface: The Role of the Gas−Liquid Interface Temperature Morteza Mohammadi,*,†,‡ Moussa Tembely,‡ and Ali Dolatabadi‡ †
Islamic Azad UniversityNour Branch, Nour, Mazandaran, Iran Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, Canada
‡
S Supporting Information *
ABSTRACT: Dynamical analysis of an impacting liquid drop on superhydrophobic surfaces is mostly carried out by evaluating the droplet contact time and maximum spreading diameter. In this study, we present a general transient model of the droplet spreading diameter developed from the previously defined mass−spring model for bouncing drops. The effect of viscosity was also considered in the model by definition of a dash-pot term extracted from experiments on various viscous liquid droplets on a superhydrophobic surface. Furthermore, the resultant shear force of the stagnation air flow was also considered with the help of the classical Homann flow approach. It was clearly shown that the proposed model predicts the maximum spreading diameter and droplet contact time very well. On the other hand, where stagnation air flow is present in contradiction to the theoretical model, the droplet contact time was reduced as a function of both droplet Weber numbers and incoming air velocities. Indeed, the reduction in the droplet contact time (e.g., 35% at a droplet Weber number of up to 140) was justified by the presence of a formed thin air layer underneath the impacting drop on the superhydrophobic surface (i.e., full slip condition). Finally, the droplet wetting model was also further developed to account for low temperature through the incorporation of classical nucleation theory. Homogeneous ice nucleation was integrated into the model through the concept of the reduction of the supercooled water drop surface tension as a function of the gas−liquid interface temperature, which was directly correlated with the Nusselt number of incoming air flow. It was shown that the experimental results was qualitatively predicted by the proposed model under all supercooling conditions (i.e., from −10 to −30 °C).
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INTRODUCTION Understanding droplet wetting behavior on a low-wettable surface is a crucial issue for various industrial applications ranging from electrical transmission power lines to the aerospace industry. Under cold atmospheric conditions, the impacts of microsized supercooled water drops in the troposphere might result in the formation of an ice layer on target solid substrates, which is similar to what happens on the leading edge of aircraft wings.1,2 Various studies have shown that using superhydrophobic surfaces is a key prerequisite for the mitigation of ice accretion.3,4 Therefore, the dynamical analysis of an impacting drop on a superhydrophobic surface is taken into account by the evaluation of two important parameters, namely, the droplet maximum spreading diameter and the contact time. The only predictive model of the maximum spreading diameter for bouncing drops was presented by Clanet et al.5 and was based on the scaling law analysis. In the scaling law analysis, it was claimed that the Ps factor (i.e., We/Re0.8) determines the dynamics of wetting, in which We and Re are Weber and Reynolds numbers. They are defined as a ratio of the inertia force to the surface tension (i.e., We) and viscous force (i.e., Re), respectively. For an impacting droplet with Ps < 1, the maximum wetting diameter is © XXXX American Chemical Society
influenced only by the capillarity; it is independent of the viscosity and can be approximated by Dmax/D = 0.9We0.25. On the other hand, for Ps > 1, the droplet dynamics is followed by the viscous force and represented by Dmax/D = 0.9Re0.2.5 With respect to the droplet contact time, the previous studies were limited to the experimentation of bouncing drops where the droplet contact time was measured experimentally.6,7 However, the concept of the theoretical value of the droplet contact time was proposed only in the study of Okumura et al.,8 where it was asserted that the droplet contact time can be obtained through the harmonic solution of the mass−spring model. Although Okumura et al.8 proposed the concept of the mass−spring model, it was properly formulated in the study of Mishchenko et al.,9 which was further explained in the study of Bahadur et al.10 later on. Indeed, inertia and capillary forces were interestingly incorporated into the mass and spring terms, respectively.9,10 A novel technique was also used to relate the droplet mass−spring model to classical nucleation theory at low temperature where the liquid droplet viscosity is increased as a Received: December 7, 2016 Revised: January 19, 2017 Published: February 8, 2017 A
DOI: 10.1021/acs.langmuir.6b04394 Langmuir XXXX, XXX, XXX−XXX
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Figure 1. (a) Schematic of the experimental test setup. (b) Different parts of the droplet accelerator.
result of the supercooling condition.10 It was asserted that even the Ps factor (i.e., We/Re0.8) becomes lower than 1 and viscosity effect cannot be ignored.11,12 A recent study of a highly viscous water droplet impacting superhydrophobic surfaces showed that by increasing the water droplet viscosity by up to 4-fold, which shows Ps < 0.6, the maximum spreading diameter was reduced by up to 25% for high-impact velocity (i.e., 3 m/s), and the droplet contact time was also increased by up to 2-fold.11 However, the maximum spreading diameter was reduced by up to about 8% for the lower inertia water droplet impact condition (i.e., 1.3 m/s).11 Therefore, it is highly appreciated to define a model in which the effect of viscosity variation is taken into account. In this study, a dashpot term is added to the mass−spring model to account for the viscosity of a supercooled drop. Consequently, it is solved by defining the appropriate initial conditions. The proposed mass−spring− dashpot model is further modified where stagnation air flow is presented through the classical Homann flow approach.13 In the real scenario of supercooled droplet impingement on solid surfaces, stagnation air flow is present (i.e., the leading edge of an aircraft), which may promote another thermodynamically preferred ice nucleation mechanism.14 In fact, under real atmospheric cold conditions, induced homogeneous ice nucleation can occur even below the critical temperature of homogeneous ice nucleation (i.e., −37 °C).15,16 It is related to the presence of air flow that systematically increases the rate of evaporative cooling under low- and even high-humidity environmental conditions.14 Both induced homogeneous ice nucleation14,17 (i.e., due to evaporative cooling) and heterogeneous ice nucleation (i.e., due to a solid substrate) are related to the gas−liquid and solid−liquid interface temperatures,9 respectively. Therefore, the gas−liquid interface temperature becomes as important as the solid−liquid interface temperature where the evaluations of both ice nucleation mechanisms are taken into account.14 The proposed mass−spring−dashpot model is incorporated with ice nucleation mechanisms (i.e., both homogeneous and heterogeneous ice nucleation) and is validated against the experimental results of supercooled water drops at various air temperatures and velocities.
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used in the proposed mass−spring−dashpot model under the supercooling condition, are extracted. The first parameter is the average or root-mean-square roughness of the substrate. The second parameter is related to the surface area ratio of the location with the highest peak to the lowest valley, which is called the ϕ ratio (Supporting Information). This information is become necessary where knowing the interface temperature of substrate−water is important to the proper prediction of the phase change (i.e., solidification). This topic is described in detail in the Theoretical Basis section. Experiments were conducted first under room temperature and atmospheric pressure and humidity conditions (i.e., relative humidity was 25%) and at low temperatures ranging from −10 to −30 °C with an almost negligible relative humidity ratio of
