Transient flow patterns in an evaporating sessile drop: A numerical study on the effect of volatility and contact angle

Abstract

In this paper, we numerically investigated the evolution of flow pattern inside an evaporating sessile drop using a transient numerical model based on the Arbitrary Lagrangian–Eulerian (ALE) frame that fully couples fluid flow and heat and mass transfer. Effects of both liquid volatility and initial contact angle on the transient flow pattern were explored. The results show that stronger volatile drops have more vortices whereas multi-vortices circulation lasts for shorter time. With the increasing of contact angle, the maximum number of vortices at first increases gently and then rapidly, while the duration of the multi-vortices circulation at first decreases rapidly and then increases a little bit. Marangoni flow is found to dominate the formation of vortices in most of the simulated cases, whereas natural convection plays a role in drops of large contact angles. These conclusions agree with the judgement criteria based on Marangoni number and Grashof number. Besides the classical symmetrical vortices, encompassing vortices are observed during the transient development of internal flow. These encompassing vortices normally form either through the merging of a big vortex with its adjacent small one or through the splitting of a big vortex into two smaller ones.

Introduction

Sessile drop evaporation has been wildly investigated over the past half century due to its scientific importance and practical applications in numerous fields, such as in DNA chip manufacturing [1], thin film coating [2], ink-jet printing [3], and electronic cooling [4]. Some utilizations [[1], [2], [3]] require precise control over the deposition patterns of particles contained in drops, which depends greatly on the internal flow of drying drops. Therefore, it's important to have a deep understanding of the mechanisms and affecting factors on internal flow inside evaporating sessile drops.

Deegan et al. [5] experimentally studied the well-known phenomenon of coffee-ring stain. Their research indicated when a drop evaporates in the constant contact radius (CCR) mode, a radially outward capillary flow is induced to replenish the mass loss at the contact line due to locally enhanced evaporation, leading to the accumulation of particles near contact line. Hu and Larson [6] observed a similar ring-like deposit after the complete evaporation of a water drop containing PMMA particles. However, the particles were found to preferentially deposit at the center area when an octane-based suspension drop was used. They explained that the difference in particle deposition pattern is originated from Marangoni flow driven by surface tension difference due to the interfacial temperature gradient. In a volatile drop like octane, Marangoni flow is dominant over capillary flow, thus the central deposition of particles is more easily formed. Further experimental studies [[7], [8], [9], [10]] revealed that the direction of Marangoni flow is changeable due to its inherent dependence on temperature field. For example, Ristenpart et al. [7] found that the direction of circulation current in drying drops can be controlled by varying the thermal conductivity ratio of liquid drop to solid substrate. By locally heating the center or the edge of drops, Kita et al. [8,11] observed a twin-vortices pattern of Marangoni flow, which travels azimuthally within the drop or remains pinned at the opposite side of the heating spot, respectively.

The internal flow inside an evaporative drop has also been widely studied using theoretical analysis and numerical simulations. The preliminary theoretical work was carried out by Hu and Larson [12]. Based on the lubrication theory of thin film flow, they deduced the flow field in a thin drying drop under the assumption of zero-stress at the liquid-vapor interface. The result agreed with Deegan's theory [5], i.e., the radially outward capillary flow along the substrate. Subsequently, they further considered the role of surface tension gradient by coupling it with the interfacial temperature distribution using a finite element algorithm and found that the induced Marangoni flow leads to a single-vortex circulation flow of anticlockwise direction in the right half of drops [13]. They also pointed out that for water drops there exists a critical contact angle of about 14°, below which the temperature gradient near contact line changes from a positive to negative value. Consequently, the circulation flow from the wetting edge to the drop apex will be weakened and even diminished. Since then, as the source of Marangoni flow, temperature distribution at drop surface has been extensively studied. Through an asymptotic analysis method for the vicinity of the triple contact line (TCL), Ristenpart et al. [7] pointed out that the critical contact angle exists only when the thermal conductivity ratio of solid substrate to drop liquid, k_r = k_s/k_l, lies between 1.45 and 2. For k_r > 2 (or k_r < 1.45), the circulation flow is always radially outward (or inward) along the substrate surface. Through an asymptotic analysis method for the center of drop, Xu et al. [14] found that the interfacial temperature gradient and the flow pattern also depend on the ratio of substrate thickness to contact radius of drops, h_r = h_s/r₀. However, the asymptotic analysis methods used in these two works [7,14] assumed a monotonous increasing or decreasing interfacial temperature distribution, therefore, could only predict one vortex flow in drop. In contrast, the numerical models of Zhang et al. [15] and Barash [16] obtained the whole profile of interfacial temperature distribution, and therefore captured the non-monotonous distributions of interfacial temperature. They found two-vortex or three-vortex flow patterns depending on the different combination of k_r, h_r and contact angle θ. Nevertheless, like the asymptotic analyses, these two numerical works still assumed a given evaporation flux profile at the interface [17] which is only valid for the slowly evaporation drop with contact angle small than 90^o. In summary, the above theoretical and numerical studies [7,[13], [14], [15], [16]] indicated that the non-uniform local evaporative cooling and heat conduction from substrate to drop surface are the two main factors to determine the temperature field in drying sessile drops. For a drop with small contact angle, the TCL region is coldest due to the strongest local evaporative cooling there, on the contrary, for a drop with relatively large contact angle, the drop apex is coldest due to the longest heat conduction path from substrate to drop surface. The competition between these two opposite effects under different conditions gives rise to different interfacial temperature distribution profiles and consequent flow patterns.

It should be noted that besides the assumption of a given evaporation flux profile at the interface that is only valid for slowly evaporating drop with contact angle smaller than 90°, the other drawback of aforementioned theoretical and numerical models [7,[13], [14], [15], [16]] is that they ignore the role of advection in heat transfer. In fact, fluid flow and heat transfer are highly coupled in the concerned issue, i.e., temperature gradient can generate Marangoni flow and natural convection, which in turn will enhance heat transfer in the form of heat advection and disturb the temperature distribution due to their intrinsic nonlinearities. Additionally, more complicated flow patterns such as Benard cell [18,19] and hydrothermal wave (HTW) [20] can be also induced when fluid flow and heat transfers are coupled. By coupling fluid flow and heat transfer, Yang et al. [21] numerically demonstrated that in the drop of the same properties as Hu's [12], the circulation flow consists of a pair of vortices with opposite rotation directions at the transitional state, rather than abruptly transits from an anticlockwise to a clockwise single-vortex flow near the critical contact angle value of 14°. Luo et al. [22] utilized the vorticity-stream function to solve the transient dimensionless momentum and energy equations for very thin drying drops and quantified the intensity of Marangoni flow and evaporative cooling effect by dimensionless numbers of Marangoni number (Ma) and Biot number (Bi), respectively. They reported that the increase of Ma and Bi strengthens the nonlinear effect of advection term in the momentum and heat transfer, leading to more vortices. Unfortunately, their model was established under the assumption of fixed drop shape. Barash et al. [23] evolved the drop shape by the quasi-stationary Laplace equation and implemented a quasi-steady simulation of the coupled heat-flow transport for an evaporating toluene drop. They identified a sequence of dynamic stages in the evolution of flow field characterized by different numbers of vortices, which are dependent on the heat advection induced non-monotonic interfacial temperature distribution.

The numerical models reviewed above are all based on the steady or quasi-steady drop evaporation, except the work by Yang et al. [21]. Additionally, the natural convection, originating from the inhomogeneous temperature field, is usually neglected in studying the internal flow of sessile drop evaporation, although as noticed by Volkov et al. [24] and Savino et al. [25,26], it will play an important role in the velocity field in drops if the temperature difference is high. Moreover, the transient stage has been clarified to occupy a large portion of drop life, especially for volatile drops on the solid substrates of poor thermal conductivity. For example, the transient stage accounts for about 33% of the evaporation lifetime for 3-methylpentane (3MP) drop without heating [26] and about 80% for ethanol drop with heating [27]. The effect of liquid volatility on the transient evolution of internal flow pattern during drop evaporation however is rarely explored.

In this study, we will adopt our previously developed transient numerical model that couples fluid flow and heat-mass transfer [26] to examine the effect of liquid volatility on vortex flow evolution of sessile drop evaporation. Four different drop volatilities will be employed, specifically, water, ethanol, methanol and 3MP. Besides, as another important factor closely related to evaporative cooling effect [28], contact angle will be also under investigation. The main content is organized as follows: in section 2, the mathematical model is introduced; in section 3, the numerical results of flow fields and temperature fields in the drops of different volatilities are discussed as the first part, followed by the results of different initial contact angles as the second part, and the detailed characteristics of different vortex types as the last part.