Dynamics of Contact Line Depinning during Droplet Evaporation

Jan 16, 2015 - Comparison of experimentally measured and theoretically modeled receding contact angles indicated that the dynamics of contact line dep...
1 downloads 8 Views 2MB Size
Download PDF
Article pubs.acs.org/Langmuir

Dynamics of Contact Line Depinning during Droplet Evaporation Based on Thermodynamics Dong In Yu,†,∥ Ho Jae Kwak,† Seung Woo Doh,‡ Ho Seon Ahn,§ Hyun Sun Park,*,‡ Moriyama Kiyofumi,‡ and Moo Hwan Kim*,†,‡,⊥ †

Department of Mechanical Engineering, POSTECH, Pohang, Republic of Korea Division of Advanced Nuclear Engineering, POSTECH, Pohang, Republic of Korea § Division of Mechanical System Engineering, Incheon National University, Incheon, Republic of Korea ‡

S Supporting Information *

ABSTRACT: For several decades, evaporation phenomena have been intensively investigated for a broad range of applications. However, the dynamics of contact line depinning during droplet evaporation has only been inductively inferred on the basis of experimental data and remains unclear. This study focuses on the dynamics of contact line depinning during droplet evaporation based on thermodynamics. Considering the decrease in the Gibbs free energy of a system with different evaporation modes, a theoretical model was developed to estimate the receding contact angle during contact line depinning as a function of surface conditions. Comparison of experimentally measured and theoretically modeled receding contact angles indicated that the dynamics of contact line depinning during droplet evaporation was caused by the most favorable thermodynamic process encountered during constant contact radius (CCR mode) and constant contact angle (CCA mode) evaporation to rapidly reach an equilibrium state during droplet evaporation.

1. INTRODUCTION Droplet evaporation is one of the basic mass-transfer phenomena associated with phase change. Droplet evaporation is the dominant phenomenon in various research and engineering fields, such as inkjet printing,1−5 spray cooling,6,7 deoxyribonucleic acid/ribonucleic acid (DNA/RNA) microarrays,8,9 and microlenses.10−12 When a droplet is placed on a surface, it undergoes evaporation in a complex manner. Previous studies have shown that capillary flow is generated inside droplets on a surface.13−16 Without external flow and temperature deviations, the droplets evaporate diffusively because of the difference in vapor concentration between the liquid−vapor interface and the environment.17,18 Because of the singularity at the contact line of the droplet, the evaporation field close to a droplet on a surface can be conveniently described in toroidal coordinates. Therefore, the droplet evaporates nonuniformly at the liquid−vapor interface, and significant evaporation occurs near the contact line, depending on the contact angle.19−21 As the contact line is initially pinned, capillary flow in the droplet is generated to supply the evaporated volume near the contact line. In addition, a droplet on a surface has a different evaporation mode.8,22−26 During evaporation, the contact angle is initially reduced while the contact radius remains constant as the contact line is pinned [constant contact radius (CCR) mode]. When the decreased contact angle reaches the receding contact angle, the contact radius is reduced while maintaining a constant contact angle [constant contact angle (CCA) mode] as the pinned contact line is released. Interestingly, the transition from CCR to CCA modes is markedly influenced by the surface © 2015 American Chemical Society

conditions, such as the chemical composition and geometric morphology.27−32 These phenomena are related to pinning and depinning at the contact line. There have been a number of recent investigations of the dynamics of contact line depinning of droplets on textured surfaces during droplet evaporation.33,34 However, these studies analyzed the dynamics on the basis of the pinning force. The critical pinning force and the receding contact angle depinning the contact line were only inferred on the basis of experimental data. Other studies have analyzed the dynamics of contact line depinning using the energy barrier based on thermodynamics.35−39 However, the energy barrier required to release the pinning contact line was also experimentally inferred. Therefore, the dynamics of contact line depinning remains unclear. In this study, the dynamics of contact line depinning during droplet evaporation was investigated theoretically and experimentally based on thermodynamics. The dynamics of contact line depinning was analyzed considering the decrease in the Gibbs free energy of a system with different evaporation modes. A theoretical model was developed to estimate the receding contact angle at the mode transition as a function of the surface conditions. In addition, droplet evaporation was experimentally investigated on surfaces with various chemical and geometric conditions that were fabricated using the microelectromechanical system technique. The receding contact angle was automatically measured during evaporation Received: October 21, 2014 Revised: January 12, 2015 Published: January 16, 2015 1950

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir

Figure 1. History of an evaporated droplet on a surface. (a) States for the wetting and evaporation phenomena. (b) Mode transition during droplet evaporation.

on the prepared surfaces. Comparisons between the modeled and experimental results on textured surfaces were used to validate the thermodynamic mechanism of contact line depinning during droplet evaporation.

Gsystem = Gphase + Ginterface = G L + G V + GS + G LV + GSV + GSL =(U + PV − TS)L + (U + PV − TS)V + (U + PV − TS)S + (σA)LV + (σA)SV + (σA)SL

2. THERMODYNAMIC CONTACT LINE DEPINNING MODEL If the thermodynamic states from 1 to 3 are defined as a droplet in air, an equilibrium droplet on a surface, and a fully vaporized droplet, respectively, the history of the droplet can be simplified by the wetting and evaporation processes as shown in Figure 1a. Because a droplet on a surface reaches the equilibrium state rapidly enough to neglect the evaporation process, the wetting process from states 1 to 2 is dependent on isothermal and isochoric processes. Therefore, a system consisting of the droplet on the surface can be defined using the Helmholtz free energy. After the wetting process, the droplet on the surface undergoes evaporation. From state 2 to 3, if no mass transfer occurs because of external flow and temperature deviations, the droplet is diffusively evaporated by the difference in the vapor concentration and becomes fully vaporized. Because diffusive evaporation is a slow phenomenon, the temperature change due to evaporation is very small. In this case, we can assume that the evaporation process from states 2 to 3 is an isothermal and isobaric process, and that the system of the droplet on the surface can be defined by the Gibbs free energy.40 Considering such a system, the Gibbs free energy is composed of the Gibbs free energies of the phases and interfaces:

(1)

where G, U, P, V, T, and S are the Gibbs free energy, internal energy, pressure, volume, temperature, and entropy of each phase, respectively. Additionally, A and σ are the interface area and surface tension between phases, respectively. Subscripts L, V, and S denote the liquid, vapor, and solid phases, respectively, and subscripts LV, SV, and SL denote the liquid−vapor, solid−vapor, and solid−liquid interfaces, respectively. As shown in Figure 1a, the droplet undergoes mode transition from CCR to CCA modes during the evaporation process and becomes fully vaporized. As the system progresses thermodynamically to reach the equilibrium state rapidly, the droplet evaporates to decrease greatly the Gibbs free energy of the system and becomes fully vaporized (state 3). On the basis of the thermodynamic view described above, the dynamics of contact line depinning can be analyzed as shown in Figure 1b. A droplet with CCR evaporation initially shows a marked reduction in the Gibbs free energy compared to a droplet with CCA evaporation with an equivalent evaporated volume. The droplet undergoes mode transition from CCR to CCA and contact line depinning at the receding contact angle. Note that the decrease in the Gibbs free energy in the CCR mode is smaller than that in the CCA mode. On the basis of this thermodynamic analysis, a theoretical model was developed to clarify and validate the dynamics of 1951

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir

during evaporation is due to various complex phenomena and includes the energy loss at the moving contact line as shown in Figure 2a. Recently, it was demonstrated that the wetting is hindered by the roughness on the hydrophilic surfaces with microscopic corrugation. 41 In numerous research studies, it has been suggested that the moving contact line causes the additional energy due to the viscous dissipation.42−50 Kang and Jacobi proposed the work of adhesion w related with the energy loss at a moving contact line such as

contact line depinning. When we consider the decreased volume element of a droplet during the evaporation process from nth to nth + 1, the decrease in the Gibbs free energy in the CCR to CCA modes can be described, respectively, as ΔGn + 1|CCR = Gn(Vn) − Gn + 1(Vn + 1)|CCR , ΔGn + 1|CCA = Gn(Vn) − Gn + 1(Vn + 1)|CCA

(2)

During the evaporation process from nth to nth + 1, we consider an equivalent volume between CCR and CCA mode. Vn − Vn + 1 = ΔV = ΔVCCR = ΔVCCA

⎤ ⎡⎛ 2πr2 2 ⎞ ⎟ /(πr2 2)⎥ w = σlv cos θ0 + σlv ⎢⎜4πR12 − ⎥⎦ ⎢⎣⎝ 1 + cos θ ⎠

(3)

Thermodynamically, between CCR and CCA mode, a droplet evaporates in a greatly decreasing mode of the Gibbs free energy. The evaporation mode transits from CCR to CCA when the decrease in the Gibbs free energy in CCR mode is equivalent in that in CCA mode. At this transition, a droplet has the receding contact angle.

where θ0 and θ are the intrinsic and apparent (or initial) contact angles, respectively, and R1 and r2 are the radius of curvature at state 1 and the contact radius at state 2, respectively.51 If a droplet evaporates in the CCA mode, the contact line of the droplet moves and the energy lost at the contact line can be generated during evaporation. The energy loss can also be generated by capillary flow in the evaporated droplet as shown in Figure 2b. Nguyen et al.21 showed numerically that the evaporation flux at the liquid−vapor interface is dependent on the contact angle of the droplet. In their study, the evaporation flux at the liquid− vapor interface was nonuniform and significant evaporation occurred near the contact line of the droplet with an acute contact angle. However, the evaporation was uniform for a droplet with an obtuse contact angle. Therefore, if a droplet on a rough hydrophilic surface evaporates in CCR mode, a significant amount of liquid near the contact line is evaporated, and capillary flow is generated from the center to the contact line of the droplet to supply the deficiency of liquid volume.13−16 Because of the capillary flow in the droplet, a drag force is generated and the energy loss can be eventually generated. Unfortunately, the energy loss due to capillary flow has not been investigated in detail. The energy losses due to capillary flow in CCR mode and the motion of contact line in CCA mode are caused altogether by the viscous dissipation. When the equivalent evaporated volume in CCR and CCA mode is considered, it could be expected that the flow rates and flow direction near the contact line are comparable and opposite, respectively, in each mode. Therefore, it could be expected that the energy loss due to the capillary flow in CCR mode is comparable to the energy loss due to the motion of the contact line in CCA. In this study, we assumed that the energy loss due to capillary flow is quantitatively equal to the energy loss due to the moving contact line. (In previous research,16,19 it was reported that the capillary number is ∼10−8 for a small water droplet with contact line radii on a millimeter scale and slow flows around 1 μm/s.) In previous research, Shanahan et al. insisted that the mode transition from CCR to CCA is dominantly influenced by the energy barrier between “stick” and “slip” on the basis of the thermodynamics.35

θn ≈ θR , when ΔGCCR [θn − Δθ(ΔV ), rn] − ΔGCCA [θn , rn − Δr(ΔV )] = 0

(6)

(4)

where θn and θR are the contact angle of the nth state during droplet evaporation and the receding contact angle at mode transition, respectively. In a system with a diffusively evaporated droplet on the surface, the Gibbs free energy changes of the vapor and solid phases are smaller than the energy change of the liquid phase. Also, the internal energy change is negligible in an isothermal process. When the temperature and entropy terms in eq 1 are expressed in terms of the Gibbs free energy loss, the difference between the decreases in energy with each mode in eq 4 can be expressed as ΔGCCR |n + 1 − ΔGCCA |n + 1 = [Vn + 1(Pn + 1|CCA − Pn + 1|CCR )]L + [σ(A n + 1|CCA − A n + 1|CCR )]LV + σLV cos θ0(A n − A n + 1|CCA )SL − ΔG loss = ΔG bulk + ΔGinterface + ΔG loss ‐ CCR − ΔG loss ‐ CCA (5)

where the decreases in the Gibbs free energy terms of the bulk, interface, loss-CCR, and loss-CCA are the decreases due to the pressure difference between the different modes, the area difference between the different modes, the energy loss in the CCR mode, and the energy loss in the CCA mode, respectively. The following phenomena can be considered to describe the decrease in the Gibbs free energy due to the energy loss in the CCR and CCA modes as shown in Figure 2. The energy loss

ΔG = G(r0 + Δr ) − G(r0) ≈ σlvπ sin 2 θ0(2 + cos θ0)(Δr )2 ∼ U (energy barrier)

Figure 2. Energy loss of viscous dissipation due to (a) the moving contact line in CCA mode and (b) the capillary flow in CCR mode.

(7) 1952

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir θn = θR(a Wenzel droplet with acute contact angle),

⎞2 ⎛ r 2 n when σLVπ ⎜ ⎟ (1 − cos θn + 1)2 (2 + cos θn + 1) 3 ⎝ sin θn + 1 ⎠ ⎧ ⎡ (1 − cos θ )2 (2 + cos θ ) ⎤1/3⎫ ⎪ ⎪ n n ⎥ ⎬ × ⎨1 − ⎢ 2 ⎪ ⎪ ⎣ (1 − cos θn + 1) (2 + cos θn + 1) ⎦ ⎭ ⎩ ⎧ ⎪ sin 2 θn + 2σLVπrn 2⎨ 2 ⎪ ⎩ sin θn + 1 × (1 + cos θn) ⎫ ⎡ (1 − cos θ )2 (2 + cos θ ) ⎤2/3 ⎪ 1 n+1 n+1 ⎬ ⎥ ×⎢ − 2 ⎪ 1 + cos θn + 1 ⎭ ⎣ (1 − cos θn) (2 + cos θn) ⎦ ⎧ ⎛ sin θ ⎞2 ⎪ n + σLVπrn 2⎨1 − ⎜ ⎟ ⎪ θ sin ⎝ n+1 ⎠ ⎩ ⎡ (1 − cos θ )2 (2 + cos θ ) ⎤2/3⎫ ⎪ n+1 n+1 ⎥ ⎬fW cos θ0 ×⎢ 2 ⎪ ⎣ (1 − cos θn) (2 + cos θn) ⎦ ⎭ = 0(θn + 1 = θn − Δθ ) (9)

θn = θR(a Cassie−Baxter droplet with obtuse contact angle), ⎛ r ⎞2 2 n when σLVπ ⎜ ⎟ (1 − cos θn + 1)2 (2 + cos θn + 1) 3 ⎝ sin θn + 1 ⎠ ⎧ ⎡ (1 − cos θ )2 (2 + cos θ ) ⎤1/3⎫ ⎪ ⎪ n n ⎥ ⎬ × ⎨1 − ⎢ 2 ⎪ ⎪ ⎣ (1 − cos θn + 1) (2 + cos θn + 1) ⎦ ⎭ ⎩ ⎧ ⎪ sin 2 θn + 2σLVπrn 2⎨ 2 ⎪ ⎩ sin θn + 1 × (1 + cos θn) ⎫ ⎡ (1 − cos θ )2 (2 + cos θ ) ⎤2/3 ⎪ 1 n+1 n+1 ⎬ ⎥ ×⎢ − 2 ⎪ 1 + cos θn + 1 ⎭ ⎣ (1 − cos θn) (2 + cos θn) ⎦

Figure 3. (a) Flowchart of the theoretical model. (b) Estimated receding contact angle based on the theoretical model.

In this study, we insist that the mode transition is caused by the Gibbs free energy balance as in eq 5. This equation could be expressed as

⎧ ⎛ sin θ ⎞2 ⎪ n + σLVπrn 2⎨1 − ⎜ ⎟ ⎪ θ sin ⎝ 1⎠ n + ⎩ ⎡ (1 − cos θ )2 (2 + cos θ ) ⎤2/3⎫ ⎪ n+1 n+1 ⎥ ⎬ ×⎢ 2 ⎪ ⎣ (1 − cos θn) (2 + cos θn) ⎦ ⎭

ΔGCCA |n + 1 = G(θn , rn) − G(θn , rn − Δr )|n + 1 =ΔG bulk + ΔGinterface + ΔG loss ‐ CCR − ΔG loss ‐ CCA − ΔGCCR |n + 1 ∼ U (energy barrier)

(8)

⎡ ⎤ w ×⎢fC − B cos θ0 − (1 − fC − B ) − fC − B ⎥ σLV ⎣ ⎦ = 0(θn + 1 = θn − Δθ )

Comparing the energy barrier determined by Shanahan’s group and the equation given above, we show that the two opinions for the mode transition during droplet evaporation are similar. Additionally, it is expected that the energy barrier is intensively related to the energy loss due to capillary flow and the moving contact line. With the assumptions that a droplet on a surface is part of a sphere and the droplet is in the Wenzel and Cassie− Baxter state on hydrophilic and hydrophobic textured surfaces, respectively,52,53 the terms of eq 5 can be derived as shown in Table 1. The detailed derivation on these terms is described in the Supporting Information. In conclusion, the following equations for estimating the receding contact angle at the transition from CCR to CCA mode can be described as

(10)

3. COMPARISON WITH EXPERIMENTS The proposed theoretical model is a method for determining the most thermodynamically favored process in the CCR and CCA modes during evaporation. The model describes the life of a droplet during evaporation. Therefore, to estimate the receding contact angle, the initial contact angle must first be estimated. In this study, the initial contact angle is estimated as a function of the surface conditions using the model of Kang and Jacobi:51 1953

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir (Wenzel state) ⎡ (2 + cos θi)(1 − cos θi)2 ⎤2/3 1 − cos θi − 2⎢⎣ ⎦⎥ 4 sin 2 θi

= fW

⎡ (2 + cos θ0)(1 − cos θ0)2 ⎤2/3 1 − cos θ0 − 2⎢⎣ ⎥⎦ 4 sin 2 θ0

(Cassie−Baxter state) ⎡ (2 + cos θi)(1 − cos θi)2 ⎤2/3 1 − cos θi − 2⎢⎣ ⎥⎦ 4

1 (1 − fC − B ) 2 sin θi ⎡ (2 + cos θ0)(1 − cos θ0)2 ⎤2/3 1 − cos θ0 − 2⎢⎣ ⎥⎦ 4 2

= fC − B

+

Figure 5. Comparison between the experimental and modeled receding contact angles. In this figure, symbols are the experimental data and lines are estimated values based on a developed model.

sin 2 θ0 (6a)

where θi and θ0 are the initial and intrinsic contact angles, respectively, and f w and f C−B are the roughness ratios in the Wenzel and Cassie−Baxter states, respectively. The meaning of the roughness ratio in the Cassie−Baxter state is equivalent to that of the solid fraction in this study. On the basis of the estimated initial contact angle with the model of Kang and Jacobi, by the iterative numerical process, the receding contact angle depending on the surface conditions is estimated by the theoretical model in which the receding contact angle as a function of the surface conditions must satisfy the relation that the decrease in the Gibbs free energy in the CCR mode is equivalent to that in the CCA mode as shown in panels a and b of Figure 3. To validate the developed model, the receding contact angle during droplet evaporation was measured on hydrophilic and

hydrophobic textured surfaces. In this experiment, textured surfaces were fabricated with various chemical compositions and geometric morphologies to investigate the relation between the criteria of contact line depinning and the surface conditions. Surfaces with intrinsic contact angles of 34°, 58°, and 109° were prepared with a self-assembled monolayer coating, as shown in Figure 4a. The surfaces with micropillars were fabricated in the ranges of 1.0 < f W < 2.0 and 0.1 < f C−B < 1.0 using photolithography and conventional dry etching, as shown in Figure 4b. Considering three chemical compositions and nine geometric morphologies, the experiments were conducted for a total of 27 surface conditions. The initial and receding contact angles during the evaporation of 6.3 μL deionized water droplets were measured automatically with a goniometer. To check the wetting state of the droplet, the periphery of the

Figure 4. Textured surfaces and wetting state on the surfaces. (a) Three different intrinsic contact angles: mercapto-propyl-trimethoxy-silane (MPTS)-coated silicon surface/silicon surface/heptadecafluoro-1,1,2,2-tetrahydrodecyltrichlorosilane (HDFS)-coated silicon surface. (b) Textured surface with micropillars (scanning electron micrographs). (c) Wenzel and Cassie−Baxter states on the hydrophilic (θ0 = 58°; f W = 2.00) and hydrophobic (θ0 = 109°; f C−B = 0.13) textured surfaces (X-ray images). 1954

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir Table 1. Terms of the Decrease in the Gibbs Free Energy in CCR and CCA Modes CCR mode

CCA mode Evaporation from n to n + 1

θn + 1 = θn − Δθ , rn + 1 = rn

Vn + 1|CCR = Vn + 1|CCA , θn + 1 = θn

Vn + 1 = Vn − ΔV(Δθ)

R n+1 =

Vn + 1 =

3 π ⎛ rn + 1 ⎞ ⎜ ⎟ (2 + cos θn + 1)(1 − cos θn + 1)2 3 ⎝ sin θn + 1 ⎠

3

⎤ 3Vn + 1 ⎡ 4 ⎢ ⎥ 2 4π ⎣ (2 + cos θn + 1)(1 − cos θn + 1) ⎦

rn + 1 = R n + 1 sin θn + 1

Difference between the Decreases in the Gibbs Free Energy in the CCR and CCA Modes

ΔGCCR |n + 1 − ΔGCCA |n + 1 ≈ ΔG bulk + ΔGinterface + ΔG loss[ΔVCCR (Δθ) = ΔVCCA(Δr )]

ΔG bulk

Decrease in the Gibbs Free Energy Due to the Pressure Difference ⎛ ⎞ 1 1 = Vn + 1·2σLV ⎜ − ⎟ R n + 1|CCR ⎠ ⎝ R n + 1|CCA Decrease in the Gibbs Free Energy Due to the Interface Area Difference

hydrophilic, Wenzel state

ΔGinterface = σLV

2πrn + 12 1 + cos θn + 1

− CCA

2πrn + 12 1 + cos θn + 1

+ σLVπ(rn 2 − rn + 12|CCA )fW cos θ0 CCR

hydrophilic, Cassie−Baxter state

ΔGinterface = σLV

2πrn + 12 1 + cos θn + 1

− CCA

2πrn + 12 1 + cos θn + 1

+ σLVπ(rn 2 − rn + 12|CCA )[fC − B cos θ0 − (1 − fC − B )] CCR

*f w and f C−B are the roughness ratios in the Wenzel and Cassie−Baxter states, respectively. Decrease in the Gibbs Free Energy Due to the Energy Loss in the CCR and CCA Modes hydrophilic, acute contact angle

ΔG loss = ΔG loss ‐ CCR − ΔG loss ‐ CCA ≈ wfW π(rn 2 − rn + 12) − wfW π(rn 2 − rn + 12)|CCA = 0 hydrophobic, obtuse contact angle

ΔG loss = ΔG loss ‐ CCR − ΔG loss ‐ CCA = 0 − wfC − B π(rn 2 − rn + 12)|CCA = − wfC − B π(rn 2 − rn + 12)|CCA

droplet micropillars on the various surfaces was visualized with Synchrotron X-ray radiography at the 6D beam-line of the Pohang Light Source-II; the results indicated that the droplets were in the Wenzel and Cassie−Baxter states on hydrophilic and hydrophobic textured surfaces, respectively, as shown in Figure 4c. The receding contact angles on the experimental surfaces were also estimated using the proposed theoretical model as functions of the surface conditions. The initial contact angle was estimated using the wetting model of Kang and Jacobi.51 The details of the test section and estimation method are described in the Supporting Information. The experimental receding contact angle was estimated well by the developed model, as shown in Figure 5. When the chemical composition and geometric morphology were quantized by the intrinsic contact angle and roughness ratio, respectively, the receding contact angle decreased as the roughness ratio increased and the intrinsic contact angle decreased.

contact angle at the mode transition as a function of the surface conditions. The thermodynamic analysis of the contact line depinning was verified by comparison between the experimental and modeled receding contact angles.

■

ASSOCIATED CONTENT

S Supporting Information *

Difference between the decreases in the Gibbs free energy in the CCR and CCA modes (section S1) and specific test section information (section S2). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Present Addresses ∥

D.I.Y.: Division of Advanced Nuclear Engineering, POSTECH, Pohang, Republic of Korea. ⊥ M.H.K.: Korea Institute of Nuclear Safety, Deajeon, Republic of Korea.

4. CONCLUSION This study was performed to investigate the dynamics of contact line depinning during droplet evaporation based on thermodynamics. The depinning phenomena at the contact line were shown theoretically to be due to the most favorable thermodynamics process that caused the Gibbs free energy to rapidly reach an equilibrium state during droplet evaporation. The evaporation mode underwent a transition when the decrease in the Gibbs free energy was equivalent for the CCR and CCA modes. On the basis of the analysis described here, a theoretical model was developed to estimate the receding

Author Contributions

H.S.P. and M.H.K. contributed equally to this work. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by National Research Foundation of Korea (NRF) grants funded by the Korean government (MSIP) 1955

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir

(21) Nguyen, T. A. H.; Nguyen, A. V.; Hampton, M. A.; Xu, Z. P.; Huang, L.; Rudolph, V. Theoretical and experimental analysis of droplet evaporation on solid surfaces. Chem. Eng. Sci. 2012, 69, 522− 529. (22) Picknett, R. G.; Bexon, R. The evaporation of sessile or pendant drops in still air. J. Colloid Interface Sci. 1977, 61, 336−350. (23) Bourges-Monnier, C.; Shanahan, M. E. R. Influence of Evaporation on Contact Angle. Langmuir 1995, 11, 2820−2829. (24) McHale, G.; Rowan, S. M.; Newton, M. I.; Banerjee, M. K. Evaporation and the Wetting of a Low-Energy Solid Surface. J. Phys. Chem. B 1998, 102, 1964−1967. (25) Kim, J. H.; Ahn, S. I.; Kim, J. H.; Zin, W. C. Evaporation of Water Droplets on Polymer Surfaces. Langmuir 2007, 23, 6163−6169. (26) Kulinich, S. A.; Farazaneh, M. Effect of contact angle hysteresis on water droplet evaporation from super-hydrophobic surfaces. Appl. Surf. Sci. 2009, 255, 4056−4060. (27) Soolaman, D. M.; Yu, H. Z. Water Microdroplets on Molecularly Tailored Surfaces. J. Phys. Chem. B 2005, 109, 17967− 17973. (28) Shin, D. H.; Lee, S. H.; Jung, J. Y.; Yoo, J. Y. Evaporation characteristics of sessile droplet on hydrophobic and hydrophilic surfaces. Microelectron. Eng. 2009, 86, 1350−1353. (29) Shin, D. H.; Lee, S. H.; Choi, C. K.; Retterer, S. The Evaporation and Wetting Dynamics of Sessile Water Droplets on Submicron-Scale Patterned Silicon Hydrophobic Surfaces. J. Micromech. Microeng. 2010, 20, 055021. (30) Kim, H. M.; Kim, J. W. Evaporation characteristics of a hydrophilic surface with micro-scale and/or nano-scale structures fabricated by sandblasting and aluminum anodization. J. Micromech. Microeng. 2010, 20, 045008. (31) Lee, C. Y.; Zhang, B. J.; Park, J. Y.; Kim, K. J. Water droplet evaporation on Cu-based hydrophobic surfaces with nano- and microstructures. Int. J. Heat Mass Transfer 2012, 55, 2151−2159. (32) Xu, W.; Leeladhar, R.; Kang, Y. T.; Choi, C. H. Evaporation Kinetics of Sessile Water Droplets on Micropillared superhydrophobic Surfaces. Langmuir 2013, 29, 6032−6041. (33) Chen, X.; Ma, R.; Li, J.; Hao, C.; Luk, B. L.; Li, S. C.; Yao, S.; Wang, Z. Evaporation of Droplets on Superhydrophobic Surfaces: Surface Roughness and Small Droplet Size Effects. Phys. Rev. Lett. 2012, 109, 116101. (34) Xu, W.; Choi, C. H. From Sticky to Slippery Droplets: Dynamics of Contact Line Depinning on Superhydrophobic Surfaces. Phys. Rev. Lett. 2012, 109, 024504. (35) Shanahan, M. E. R. Simple Theory of “Stick-Slip” Wetting Hysteresis. Langmuir 1995, 11, 1041−1043. (36) Shanahan, M. E. R. Condensation Transport in Dynamic Wetting. Langmuir 2001, 17 (13), 3997−4002. (37) Shanahan, M. E. R. Spreading of Water: Condensation Effects. Langmuir 2001, 17 (26), 8229−8235. (38) Shanahan, M. E. R.; Sefiane, K. Wettability and Adhesion; Koninklijke Brill NV: Leiden, The Netherlands, 2009; Vol. 6, pp 19− 31. (39) Bormashenko, E.; Musin, A.; Zinigrad, M. Evaporation of droplets on strongly and weakly pinning surfaces and dynamics of the triple line. Colloids Surf., A 2011, 385, 235−240. (40) Caray, V. P. Liquid−Vapor Phase-Change Phenomena, 1st ed.; Taylor & Francis: Boca Raton, FL, 1992; p 4. (41) Malijevský, A. Does surface roughness amplify wetting? J. Chem. Phys. 2014, 141, 184703. (42) Huh, C.; Scriven, L. E. Hydrodynamic Model of Steady Movement of a Solid/Liquid/Fluid Contact Line. J. Colloid Interface Sci. 1971, 35 (1), 85−101. (43) Voinov, O. V. Hydrodynamics of wetting. Fluid Dyn. 1976, 11 (5), 714−721. (44) de Gennes, P. G. Wetting: statics and dynamics. Rev. Mod. Phys. 1985, 57 (3), 827−863. (45) Cox, R. G. The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 1986, 168, 169−194.

(2014M2B2A9031122). The experiments at the Pohang Light Source (PLS) were supported in part by MSIP and POSTECH. The English in this document has been checked by at least two professional editors, both native speakers of English.

■

REFERENCES

(1) Sirringhaus, H.; Kawase, T.; Friend, R. H.; Shimoda, T.; Inbasekaran, M.; Wu, W.; Woo, E. P. High-Resolution Inkjet Printing of All-Polymer Transistor Circuits. Science 2000, 290, 2123−2126. (2) Kawase, T.; Sirringhaus, H.; Friend, R. H.; Shimoda, T. Inkjet Printed Via-Hole Interconnections and Resistors for All-Polymer Transistor Circuits. Adv. Mater. 2001, 13, 1601−1605. (3) Cheng, K.; Yang, M.-H.; Chiu, W. W. W.; Huang, S.-Y.; Chang, J.; Ying, T.-F.; Yang, Y. Ink-Jet Printing, Self-Assembled Polyelectrolytes, and Electroless Plating: Low cost Fabrication of Circuits on a Flexible Substrate at Room Temperature. Macromol. Rapid Commun. 2005, 26, 247−264. (4) Ingrosso, C.; Kim, J. Y.; Binetti, E.; Fakhfouri, V.; Striccoli, M.; Agostiano, A.; Curri, M. L.; Brugger, J. Drop-on-demand of GaNbased light-emitting diode with a localized surface plasmon resonance. Microelectron. Eng. 2009, 86, 1124−1126. (5) Voigt, A.; Ostrzinski, U.; Pfeiffer, K.; Kim, J. Y.; Fakhfouri, V.; Brugger, J.; Gruetzner, G. New inks for the direct drop-on-demand fabrication of polymer lenses. Microelectron. Eng. 2011, 88, 2174− 2179. (6) Kim, J. H.; You, S. M.; Choi, S. U. S. Evaporative spray cooling and microporous coated surfaces. Int. J. Heat Mass Transfer 2004, 47, 3307−3315. (7) Bhattacharya, P.; Samanta, A. N.; Chakraborty, S. Spray evaporative cooling to achieve ultrafast cooling in runout table. Int. J. Therm. Sci. 2009, 48, 1741−1747. (8) Laurell, T.; Nilsson, J.; Marko-Varga, G. The Quest for HighSpeed and Low-Volume Bioanalysis. Anal. Chem. 2005, 77, 264A− 272A. (9) Dugas, V.; Broutin, J.; Souteyrand, E. Droplet Evaporation Study Applied to DNA Chip Manufacturing. Langmuir 2005, 21, 9130− 9136. (10) Bonaccurso, E.; Butt, H.-J.; Hankeln, B.; Niesenhaus, B.; Graf, K. Fabrication of microvessels and microlenses from polymers by solvent droplets. Appl. Phys. Lett. 2005, 86, 124101. (11) Li, G.; Houn, N.; Graf, K. Microtopologies in polymer surfaces by solvent drops in contact and noncontact mode. Appl. Phys. Lett. 2006, 89, 241920. (12) Pericet-Camera, R.; Bonaccurso, E.; Graf, K. Microstructuring of Polystyrene Surfaces with Nonsolvent Sessile Droplets. ChemPhysChem 2008, 9, 1738−1746. (13) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Two-dimensional Crystallization. Nature 1993, 361, 26. (14) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary flow as the cause of ring strains from dried liquid drops. Nature 1999, 329, 827−829. (15) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Contact line deposits in an evaporating drop. Phys. Rev. E 2000, 62, 756−765. (16) Hu, H.; Larson, R. G. Analysis of the Microfluid Flow in an Evaporating Sessile Droplet. Langmuir 2005, 21 (9), 3963−3971. (17) Langmuir, I. The Evaporation of Small Spheres. Phys. Rev. 1918, 12, 368−370. (18) Birdi, K. S.; Winter, A. A study of the evaporation rates of small water drops placed on a solid surface. J. Phys. Chem. 1989, 93, 3702− 3703. (19) Hu, H.; Larson, R. G. Evaporation of a Sessile Droplet on a Substrate. J. Phys. Chem. B 2002, 106 (6), 1334−1344. (20) Zheng, L.; Popov, Y. O.; Witten, T. A. Deposit growth in the wetting of an angular region with uniform evaporation. Phys. Rev. E 2005, 72, 046303. 1956

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957

Article

Langmuir (46) Mahadevan, L.; Pomeau, Y. Rolling Droplets. Phys. Fluids 1999, 11 (9), 2449−2453. (47) Ishino, C.; Okumura, K. Wetting transitions on textured hydrophilic surfaces. Eur. Phys. J. E: Soft Matter Biol. Phys. 2008, 25, 415−424. (48) Peters, A. M.; Pirat, C.; Sbragaglia, M.; Borkent, B. M.; Wessling, M.; Lohse, D.; Lammertink, R. G. H. Cassie-Baxter to Wenzel state wetting transition: Scaling of the front velocity. Eur. Phys. J. E: Soft Matter Biol. Phys. 2009, 29, 391−397. (49) Bormashenko, E.; Bormashenko, Y.; Oleg, G. On the Nature of the Friction between Nonstick Droplets and Solid Substrate. Langmuir 2010, 26 (15), 12479−12482. (50) Bormashenko, E.; Musin, A.; Whyman, G.; Zinigrad, M. Wetting Transitions and Depinning of the Triple Line. Langmuir 2012, 28 (7), 3460−3464. (51) Kang, H. C.; Jacobi, A. M. Equilibrium Contact Angles of Liquid Droplets on Ideal Rough Solids. Langmuir 2011, 27 (24), 14910− 14918. (52) Wenzel, R. N. Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 1936, 28, 988−994. (53) Cassie, A.; Baxter, S. Wettability of porous surfaces. Trans. Faraday Sci. 1944, 30, 546−551.

1957

DOI: 10.1021/la504971y Langmuir 2015, 31, 1950−1957