Article pubs.acs.org/Langmuir
Influence of Surface Wettability on Transport Mechanisms Governing Water Droplet Evaporation Zhenhai Pan, Justin A. Weibel, and Suresh V. Garimella* School of Mechanical Engineering and Birck Nanotechnology Center, Purdue University, 585 Purdue Mall, West Lafayette, Indiana 47907, United States S Supporting Information *
ABSTRACT: Prediction and manipulation of the evaporation of small droplets is a fundamental problem with importance in a variety of microfluidic, microfabrication, and biomedical applications. A vapor-diffusion-based model has been widely employed to predict the interfacial evaporation rate; however, its scope of applicability is limited due to incorporation of a number of simplifying assumptions of the physical behavior. Two key transport mechanisms besides vapor diffusion evaporative cooling and natural convection in the surrounding gasare investigated here as a function of the substrate wettability using an augmented droplet evaporation model. Three regimes are distinguished by the instantaneous contact angle (CA). In Regime I (CA ≲ 60°), the flat droplet shape results in a small thermal resistance between the liquid−vapor interface and substrate, which mitigates the effect of evaporative cooling; upward gas-phase natural convection enhances evaporation. In Regime II (60 ≲ CA ≲ 90°), evaporative cooling at the interface suppresses evaporation with increasing contact angle and counterbalances the gas-phase convection enhancement. Because effects of the evaporative cooling and gas-phase convection mechanisms largely neutralize each other, the vapor-diffusion-based model can predict the overall evaporation rates in this regime. In Regime III (CA ≳ 90°), evaporative cooling suppresses the evaporation rate significantly and reverses entirely the direction of natural convection induced by vapor concentration gradients in the gas phase. Delineation of these counteracting mechanisms reconciles previous debate (founded on single-surface experiments or models that consider only a subset of the governing transport mechanisms) regarding the applicability of the classic vapor-diffusion model. The vapor diffusion-based model cannot predict the local evaporation flux along the interface for high contact angle (CA ≥ 90°) when evaporative cooling is strong and the temperature gradient along the interface determines the peak local evaporation flux. where cosh α = (Rc sin θ/h) − cos θ, M is the droplet mass, J is the evaporation flux, Rc is the contact radius of the droplet, D is the diffusion coefficient, ρs is the saturated vapor density at the droplet interface, ρ∞ is the ambient vapor density, θ is the contact angle (CA), and r is the radial coordinate along the droplet interface. Although this model predicts the evaporation rate for some experiments conducted in an air ambient,3−6 the scope of its applicability is still debated, and it has been shown to either overpredict7−13 or underpredict7,10,14−20 the evaporation rate due to the simplifications imposed. Evaporative cooling is not considered in the vapor diffusionbased model and may introduce considerable discrepancies between predicted and measured evaporation rates. The droplet interface is cooled due to the absorption of latent heat, reducing the local vapor pressure and thus suppressing evaporation. This effect is typically significant on low thermal conductivity7−9 or heated substrates,10,11 where a large temperature drop can be maintained at the evaporating
E
vaporation of sessile droplets is widely encountered in various applications such as phase-change cooling systems, self-assembled surface coatings, inkjet printing, and microfluidic control. It is essential to understand the effects of the underlying transport mechanisms to accurately predict the global and local evaporation characteristics. A vapor-diffusionbased model1 is widely employed to predict the evaporation mass flux across the liquid−vapor interface of an evaporating droplet. For each quasi-steady instantaneous interface profile, the model assumes that the evaporation process is governed only by vapor diffusion through the surrounding gas phase. The overall evaporation rate and distribution of interface mass flux are calculated as2 dM sin θ = − πR cD(ρs − ρ∞)f (θ) where f (θ) = dt 1 + cos θ ∞ 1 + cosh 2θτ +4 tanh[(π − θ)τ ] dτ 0 sinh 2πτ
∫
J(r ) =
∫0
∞
D(ρs − ρ∞) ⎛ 1 ⎜ sin θ + ⎝2 Rc
(1)
2 (cosh α + cos θ )3/2 ×
⎞ cosh θτ tanh[(π − θ)τ ]P −1/2 + iτ(cosh α)τ dτ ⎟ ⎠ cosh πτ © 2014 American Chemical Society
Received: May 18, 2014 Revised: July 27, 2014 Published: July 31, 2014
(2) 9726
dx.doi.org/10.1021/la501931x | Langmuir 2014, 30, 9726−9730
Langmuir
Article
interface. Recent experiments12 have shown that even for water droplets on unheated high-conductivity silicon substrates, evaporative cooling may become important for certain droplet morphologies that are determined by surface wettability. The vapor-diffusion-based model was found to overestimate evaporation rates by up to ∼25% for droplets on superhydrophobic substrates (CA ∼160°). Further numerical simulations13 indicated that this deviation is attributed to evaporative cooling; the thermal resistance of the relatively tall liquid droplet sitting on a superhydrophobic substrate can sustain a large temperature reduction at the evaporating interface. Natural convection induced by buoyancy effects due to thermal and vapor concentration gradients in the gas domain is another important transport mechanism governing droplet evaporation. Recent experiments16,17 and numerical simulations18 have directly demonstrated that gas-phase flow influences vapor transport in the gas domain and thus the evaporation process. The vapor-diffusion-based model may underestimate the evaporation rate by up to ∼30% for a heated surface;10,19 even for unheated substrates, significant underestimation has been reported.7,14,16,17,20 In the present study, the influences of all of the above transport mechanisms are quantitatively determined for a 2 μL evaporating water droplet as a function of the surface wettability to establish the regimes for which the vapordiffusion-based model is applicable in an attempt to reconcile a lingering debate in the literature. A 2D axisymmetric, quasisteady model developed in the authors’ previous work13 is employed for this analysis. This numerical model agrees with experimental12 evaporation rates to within 3% on both hydrophobic and superhydrophobic substrates. The present model is also compared with droplet evaporation experiments21 on hydrophilic substrates with small contact angles (∼5°). The simulation results agree with the experimental data to within 3.7% (detailed comparison shown in the Supporting Information). In the model, evaporative cooling is included as an energy sink at the interface, and conjugate heat and mass transfer are solved throughout the substrate, droplet, and gas domains shown in Figure 1. The vapor concentration at the interface is assumed to be saturated, and transport is coupled to the local temperature and convection in the gas domain. Natural convection is considered in the fluid domains, and Stefan flow is considered in the gas domain. A detailed description of the numerical implementation is provided in ref 13. The ambient temperature is chosen as 21 °C with 29% humidity to match the conditions of previous experiments.12 The thickness of the solid substrate is 10 times the droplet contact radius and can be considered thermally semi-infinite; isothermal (at the ambient temperature) versus adiabatic boundary conditions at the bottom of the substrate do not influence the overall evaporation rate. (The deviation is
