ARTICLE pubs.acs.org/JPCC
Mechanism of Sessile Water Droplet Evaporation: Kapitza Resistance at the SolidLiquid Interface H. Ghasemi and C. A. Ward* Department of Mechanical and Industrial Engineering, Thermodynamics and Kinetics Laboratory, 5 King’s College Road, Toronto, Ontario M5S 3G8, Canada ABSTRACT: The cooling of a Au(111) substrate by an evaporating water droplet has been quantitatively studied. Each droplet was maintained in a thermal steady state and of constant size by heating the droplet and injecting water at its base while removing vapor from an enclosing chamber. The thermal energy transported to the solidliquid interface of the droplet was measured with thermocouples embedded in the substrate. Because of a large thermal (or Kapitza) resistance at the solid liquid interface, only a small portion of the thermal energy was transported perpendicular to the solidliquid interface to the bulk liquid. A much larger fraction—up to 87%—was transported parallel to the solidliquid interface to the three-phase line where thermocapillary convection distributed this energy along the liquidvapor interface to be consumed by the phase change process. For a given substrate, the Kapitza resistance is found to increase with the magnitude of the solidliquid interfacial temperature discontinuity and the pressure in the liquid phase at the solidliquid interface.
’ INTRODUCTION It is recognized that if electronic systems are to continue their advancement, improved cooling techniques are needed13 and cooling a hot, solid surface by spraying it with droplets that can evaporate is one of the techniques being considered.46 When a sessile droplet is heated by a solid and the liquid evaporates into its own vapor, the heat flux from the solid to the bulk liquid at the solidliquid interface is usually assumed continuous,7,8 but there is an adsorbed phase between the bulk solid and the liquid phases (see Figure 1A).911 Its presence gives rise to the possibility of a thermal or Kapitza resistance, RK, at the solidliquid interface and the possibility of a thermal flux both perpendicular and parallel to the solidliquid interface. That parallel flux would give a thermal flux to the three-phase line of the sessile droplet. The Kapitsa resistance could result from phonon scattering,12,13 but a theoretical description of the source of RK is not at present available, especially at room temperature.1416 However, if a temperature discontinuity exists at an interface, a Kapitza resistance may be defined whether the thermal flux is continuous or not. If the temperature discontinuity at the interface is denoted ΔT and the heat flux to the interface as q_ RK
ΔT q_
ð1Þ
The usual assumption is that the Kapitza resistance is negligible at room temperature because it decreases from its value at liquidHe temperatures as T3.12,13 If true, it would be negligible at room temperature. We measure RK at the Auwater interface and find that this resistance depends on both the interfacial r 2011 American Chemical Society
temperature discontinuity and the pressure in the liquid phase at the droplet base, PL(zb). Using an apparatus that allows water to be pumped into the base of a sessile droplet as the water evaporates steadily in the absence of buoyancy-driven convection,17 we measured the temperatures on the substrate side and on the water side of the Au(111)water interface. There is no reason to expect corrosion at this interface, but under steady-state conditions, we find the temperature is discontinuous—by as much as 2 °C and that the corresponding value of RK is of the same order as that measured at liquid-He temperatures, 104 m2 K/W.12,13
’ EXPERIMENTAL PROCEDURES AND RESULTS Water to be used in the experiments was distilled, deionized, and nanofiltered until it had a resistivity of 18.2 Ω-cm and a surface tension within 1% of the documented value. This water was used to rinse the components of the apparatus. The water to form the droplets was additionally degassed and its purity examined by measuring the vapor-phase pressure in the degassing vessel. It was within (1 Pa of the saturation pressure corresponding to its temperature. Afterward, the degassed water was transferred to the syringe of a pump without exposing the water to air. The syringe pump supplied the water to form the sessile droplet and maintain it evaporating steadily (Figure 1B). A substrate of Au(111) with a thickness of 4.4 mm and diameter of 18 mm was purchased from MaTeck GmbH. It was Received: August 4, 2011 Revised: September 20, 2011 Published: September 21, 2011 21311
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Figure 1. (A) Solidliquid interphase, (B) experimental chamber, and (C) thermocouple placement in the substrate. The radial position of the liquidvapor interface is indicated by y(ϕ) and its height above the substrate surface by z(ϕ) zb. The radial position along the solidliquid interface is shown by the x axis. The kinetic contact angle is denoted θk.
cleaned by soaking it in acetone (24 h) and detergent solution (24 h) and rinsing it with the processed water. The substrate was then enclosed in the chamber indicated schematically in Figure 1B. The chamber was connected to a vacuum system, and the pressure was decreased to ∼105 Pa. A (quadrapole) mass spectrometer was used to examine the gases inside the chamber. Only traces of impurities were found, and there was no indication of the presence of oil. The temperature profiles as a function of depth in the substrate were measured with 12 thermocouples implanted at the positions indicated in Figure 1B. These thermocouples allow the steady thermal flux from the substrate to the adsorbed layer to be determined at four positions. The temperature at the bottom of the substrate was maintained constant at 4 °C, Figure 1B, with a circulating bath.10,18 Using the apparatus shown schematically in Figure 1B, an axisymmetric degassed water droplet with radius of ∼9 mm was formed on the Au(111) substrate by injecting water through the central 0.8 mm diameter hole in the substrate with a syringe pump. The syringe-pumping rate and the vapor-phase pressure were adjusted to maintain the droplet evaporating steadily into its own vapor and the liquidvapor interface from moving by more than (10 μm during the course of an experiment. We emphasize that during our experiments there was no measurable air content within the chamber initially, and the chamber was evacuated continuously during an experiment. The gas content of the chamber was examined at the end of each experiment. Only water vapor was detected. The temperature at the droplet base was constant in each experiment and less than 4 °C. Since water has its maximum
density at this temperature and evaporation cooled the liquid vapor interface further, buoyancy-driven convection was surpressed within the liquid phase. The maximum Reynolds number inside the injection tube was 0.25, and in the vapor at the liquidvapor interface its maximum value was 2.43 102. In both cases, this flow would be classified as “creeping” and any effect of it would be negligible. The temperature in the water-injection stream was not measurably different than that of the water adsorbed around the small injection hole, and we neglect any thermal conduction from the entering water to the adsorbed layer, but when an energy balance for the droplet is taken into account, we include the energy convected in by the water supplied to the base of the droplet. With the water evaporating steadily, the temperatures in the liquid and vapor phases were measured with a microthermocouple (bead diameter ≈ 25 μm) mounted on a 4-axis-positioning micrometer. With the aid of a camera (pixel size 1.23 μm) the thermocouple bead could be positioned with an accuracy of (10 μm. Experimental Results at the SolidLiquid Interface. We consider the solidliquid interface, including the adsorbed layer, as being in the steady state with thermal energy fluxing in and out. The measurements that indicate an interfacial temperature discontinuity at this interface are shown for one experiment in Figure 2, and the solidliquid temperature discontinuities for all experiments are listed in Table 1. The measured temperature gradients in the substrate were used to calculate the thermal flux from the substrate to the solid liquid interface with its adsorbed layer, q_ SA, and those measured in the liquid phase were used to calculate that from the adsorbed layer to the bulk liquid, q_ AL. The values are listed in Table 2. Note 21312
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Figure 2. Measurements made during experiment EVA5. (a) At a particular horizontal position (8 mm from the center line), the measured, steady temperature profiles in the substrate, in the liquid phase, and the extrapolation of the temperature to the solidliquid interface. The adsorbed layer (of exaggerated thickness) is shown as the colored portion. Note the interfacial temperature discontinuity, TSA TAL. (b) Temperature discontinuity is present all along the solidliquid interface. In the substrate, no radial temperature gradient was measurable: at z = 0.9 mm, x = 2, 4, 6, and 8 mm, the measured temperatures were 1.981, 1.983, 1.980, and 1.981 °C respectively. The standard deviation of these measurements is 0.001 °C, but the thermocouples could only be calibrated to an accuracy of (0.03 °C.
Table 1. Summary of the Condition at the Au(111)Water Interface during Evaporation RK ( (2 105) exp.
Jin sp ( 0.005 (mg/s)
Q_ SA (W)
Q_ A3 (W)
PL(zb) (Pa)
EVA1
0.180
0.438
0.387
717
EVA2
EVA3
EVA4
EVA5
0.200
0.221
0.241
0.246
0.495
0.526
0.588
0.603
0.435
0.458
0.467
0.498
679
656
630
606
that at each horizontal position in each experiment q_ SA was 2 orders of magnitude greater than q_ AL. Thus, contrary to the usual assumption, the thermal fluxes are not continuous but there is no fundamental reason they should be because there can be energy flux parallel to the solidliquid interface toward the three-phase line. We investigate this possibility. At one depth in the substrate the temperatures were measured at four radial positions (see caption of Figure 2). The temperatures in the solid were not measurably different. Thus, 3TS 3 ir ≈ 0 and as seen in Figure 2, 3TS 3 iz ≈ constant. Thus, the total
x ( 0.01 (mm)
TAL L ( 0.03 (°C)
AL TSA S TL (°C)
(104 m2 K/W)
8.75
2.07
0.66
3.8
8.50
2.10
0.65
3.8
8.00
2.14
0.58
3.4
7.00
2.18
0.54
3.1
8.75
1.43
1.20
6.1
8.50
1.56
1.06
5.4
8.00
1.64
0.96
4.9
7.00 6.00
1.74 1.78
0.88 0.83
4.5 4.2
8.90
0.82
1.47
6.6
8.75
1.03
1.28
5.8
8.50
1.20
1.11
5.0
8.25
1.34
0.95
4.3
8.00
1.52
0.77
3.5
8.90
0.30
1.85
7.9
8.75 8.50
0.48 0.61
1.67 1.53
7.2 6.6
8.25
0.69
1.46
6.3
8.00
0.87
1.28
5.5
8.90
0.18
2.15
8.8
8.75
0.06
1.92
7.8
8.50
0.40
1.62
6.6
8.25
0.51
1.50
6.1
8.00
0.60
1.42
5.8
thermal flux from the substrate to the adsorbed phase, Q_ SA Q_ SA ¼ ð kS ∇T S 3 iz Þπðrs2 rh2 Þ
ð2Þ
where rs is the radius of the Au(111) substrate, 9 mm, and rh is the radius of the water injection hole, 0.4 mm. In the liquid phase, there was a measurable dependence of the temperature on the radial position. At the solidliquid interface, for each experiment, we take q_ AL = kL(rTL)AL 3 iz. The data obtained from the temperature listed in Table 2 was fit with a fourth-order 21313
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Table 2. Summary of Solid, Liquid, and Vapor Heat Fluxes exp.
interface
EVA1
SL
parameter q_ AL ( 55 (W/m2)
x = 0 mm
x = 2 mm
56
q_ SA ( 55 (W/m2)
EVA3
EVA4
x = 8 mm
59
66
83
191
1730
1650
1700
1780
x = 8.75 mm 513
0
0.063
0.157
0.310
0.616
0.870
qV(ϕ) ( 45 (W/m2)
65
66
66
69
81
103
PV(ϕ) (Pa)
686
685.51
684.03
687.98
689.35
689.06
vLV t (ϕ) (mm/s)
0
0.001
0.007
0.04
0.2
0.7
q_ tc(ϕ) (W/m2)
0
2.77
5.45
6.32
280.41
7253.82
SL
q_ AL ( 55 (W/m2) q_ SA ( 55 (W/m2)
66
89 1930
143 1900
226 2010
264 1980
598
LV
ϕ (rad)
0
0.055
0.126
0.266
0.609
0.909
qV(ϕ) ( 45 (W/m2)
74
74
77
76
78
81
PV(ϕ) (Pa)
651
652.39
652.85
656.12
661.54
660.83
vLV t (ϕ) (mm/s)
0
0.001
0.007
0.05
0.24
0.8
q_ tc(ϕ) (W/m2)
0
2.21
3.32
7.24
108.95
7905.87
SL
q_ AL ( 55 (W/m2)
74
85
108
134
260
896
LV
q_ SA ( 55 (W/m2) ϕ (rad)
0
2270 0.070
2210 0.169
2150 0.355
2190 0.699
1.01
qV(ϕ) ( 45 (W/m2)
78
79
81
82
84
84
PV(ϕ) (Pa)
621
621
621.66
622.35
625.60
632.81
vLV t (ϕ) (mm/s)
0
0.002
0.01
0.06
0.30
1.0
q_ tc(ϕ) (W/m2)
0
2.10
4.31
9.42
384.95
4466.97
q_ AL ( 55 (W/m2)
111
136
256
617
653
958
2460
2240
2210
2340
0.145 86
0.302 89
0.617 94
SL
q_ SA ( 55 (W/m2) LV
EVA5
x = 6 mm
ϕ (rad)
LV
EVA2
x = 4 mm
SL
ϕ (rad) qV(ϕ) ( 45 (W/m2)
0 80
0.065 85
PV(ϕ) (Pa)
601
599.25
599.23
601.39
603.48
608.69
vLV t (ϕ) (mm/s)
0
0.004
0.02
0.08
0.40
1.1
q_ tc(ϕ) (W/m2)
0
2.43
3.31
7.42
175.91
8841.96
q_ AL ( 55 (W/m2)
122
130
148
235
672
1072
2312
2380
2520
2560
q_ SA ( 55 (W/m2) LV
0.866 73
ϕ (rad)
0
0.064
0.142
0.300
0.638
0.947
qV(ϕ) ( 45 (W/m2) PV(ϕ) (Pa)
86 576
88 577.26
91 577.25
85 577.23
83 580.68
85 585.25
vLV t (ϕ) (mm/s)
0
0.005
0.02
0.1
0.45
1.2
q_ tc(ϕ) (W/m2)
0
3.12
5.24
8.22
1585.07
7785.04
polynomial in x. This q_ AL(x) function was integrated over the axisymmetric surface of the adsorbed layer to determine Q_ AL. As will be seen below (Figure 4), the temperature gradients in the bulk liquid were constant from the adsorbed layer to the liquid vapor interface Q_ LI . Thus Q_ LI ¼ Q_ AL Z rs 2πq_ AL ðxÞx dx Q_ AL ¼
ð3Þ
0
The total thermal energy transport from the adsorbed phase to the three-phase line, Q_ A3, may be determined from conservation of energy. Q_ A3 ¼ Q_ SA Q_ AL
ð4Þ
The values obtained for Q_ SA, Q_ AL, and Q_ A3 are compared in Figure 3, and the value of Q_ A3 for each experiment is listed in Table 1. Later in the manuscript, Figure 5, we compare Q_ A3 with the energy transported along the liquidvapor interface.
Figure 3. In each experiment, a portion of the energy transported from the solid to the adsorbed layer, Q_ SA, was transported to the bulk liquid, Q_ AL, but by far the larger portion was transferred to the three-phase line of the sessile droplet, Q_ A3.
Experimental Results at the LiquidVapor Interface. The conditions required for the liquidvapor phase to operate in the 21314
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Figure 4. For experiment EVA5, the position of the liquidvapor interface was measured at 12 points and are indicated in the upper figure by the solid dots. The calculated shape of the droplet is indicated by the solid line. The value of the kinetic contact angle, θk, is given. The interfacial temperatures in the liquid and vapor phases at the positions of the vertical lines in the upper figure are shown in the lower figures. The interfacial vapor temperature was greater than that of the liquid at each position, but the magnitude of the discontinuity varied along the interface with the largest discontinuity near the three-phase line and smallest at the droplet apex. For this experiment, the evaporation rate was the highest and the calculated ratio of PV(ϕ)/Ps(TVI ) was in the range of 0.991 e PV(ϕ)/Ps(TVI ) e 1.010.
boundary of this phase—there could not have been any buoyancy-driven convection in the vapor and the measured temperature profile in this phase was linear. Thus, we assume the energy transport was by thermal conduction. A similar situation existed in the liquid phase. The temperature in the liquid near the substrate was less than 4 °C, the temperature at which water has its maximum density. The temperature in the liquid at the liquid vapor interface was less than that near the substrate, and therefore, lighter than that near the substrate. Thus, there could not have been any buoyancy-driven convection in this phase either, and the measured temperature in the liquid phase was linear. Hence, we assume the energy transport in this phase was also by thermal conduction. Then the thermal energy conducted to an interface element, q_ I, is given by ð6Þ q_ I ¼ kL ð∇T L ÞI 3 iϕ þ kV ð∇T V ÞI 3 iϕ Figure 5. Comparison of the energy transport through the adsorbed layer with the energy transported along the liquidvapor interface by thermocapillary convection.
steady state are now examined. The expression for the net energy transport to an interface element by thermocapillary convection, q_ tc, may be expressed in terms of the turning angle, ϕ (Figure 1B) 0 ! !1 L L δT δT @ A ð5Þ sin ϕ q_ tc ¼ cσ vLV t ðϕÞ cos ϕ δy δz I
I
where vLV t (ϕ) is the thermocapillary speed along the liquid vapor interface and cσ is the surface-thermal capacity. Its value has been measured for water,19 30.6 kJ/K m2 for 10 e TLV I e 3.5 °C, and independently examined in other experiments.17,20,21 The temperatures measured in the liquid and vapor phases for one experiment are shown in Figure 4. The measurements in other experiments were similar. Since the lowest temperature in the vapor phase was at the liquidvapor interface—the lower
where kL and kV are the thermal conductivities of the liquid and vapor phases and iϕ is the unit vector normal to the liquidvapor interface. Thus, for the element to remain unchanging with time jev ðϕÞ½hV ðϕÞ hL ðϕÞ ¼ q_ I ðϕÞ þ q_ tc ðϕÞ
ð7Þ
This relation may be applied once methods for predicting jev and the interface shape are available. Statistical Rate Theory (SRT) Expression for the Evaporation Flux. If the entropy change per molecule resulting from the phase change is denoted Δslv and kb denotes the Boltzmann constant, the SRT expression for the evaporation flux can be written22,23 Δslv jev ¼ 2Ke sinh ð8Þ kb where Ke is the equilibrium exchange rate. It may be expressed in terms of the saturation-vapor pressure, Ps, and the pressure that would exist in the liquid phase if the liquid and vapor phases were allowed to evolve to equilibrium, PLe 21315
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Table 3. Summary of Steady-State Evaporation Experiments on Au(111) experiment TL0 (( 0.03 TV0 (( 0.03 1 CLV 0 (m )
EVA1
EVA2
EVA3
EVA4
EVA5
K)
274.77
274.05
273.38
272.93
272.35
K)
275.85 27.1
275.02 21.6
274.42 28.6
273.93 26.8
273.42 25.7
qI(0) ((5 W/m )
121.9
140.8
152.5
191.4
208.8
jev(0) ((0.1 mmol/m2s)
2.70
3.11
3.37
4.24
4.62
2
PV0 (( 1 Pa)
686
651
621
601
576
z0 zb (( 0.01 mm)
2.76
2.51
3.19
2.61
2.72
Cslv ((0.02 m1)
111.1
111.1
111.1
111.1
111.1
Shape of the LiquidVapor Interface. We now develop an iterative technique for predicting the droplet shape. The predictions will be compared with the measurements shown in Figure 4. Since the droplets are axisymmetric, at the apex the principal curvatures are equal LV LV CLV 1 ð0Þ ¼ C2 ð0Þ C0
vLV t (0)
and gives
1.11 61.0
1.35 60.1
1.13 72.3
1.09 56.9
1.17 61.5
θ ((0.3°)
58.5
52.8
68.4
55.0
57.5
Jsrt ev (mg/s) ( 0.001
0.185
0.200
0.222
0.241
0.248
Jsp (mg/s) ( 0.005
0.180
0.200
0.221
0.241
0.246
jev ð0Þ½hV ð0Þ hL ð0Þ ¼ kL ð∇T L 3 iϕ Þϕ ¼ 0 þ kV ð∇T V 3 iϕ Þϕ ¼ 0
ηPs ðTIL Þ Ke ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πmkb TIL
ð9Þ
jev ð0Þ ¼ f ½PIV ð0Þ, CLV 0 ð10Þ
and Δslv may be written
2 3 " # !4 ! V L Δslv T P ðT Þ qvib ðTIV Þ TIV s I 5 I 4 þ ln þ 4 1 L ¼ ln PIV qvib ðTIL Þ kb TIL TI 3 2 3n 6 6 7 1 1 pωl 7 6 pωl 7 6 L þ þ V 5 4 pω TI TI 2k l b l¼1 kb exp 1 kb TIV
dyðϕÞ cos ϕ ¼ LV dϕ C1 ðϕÞ dzðϕÞ sin ϕ ¼ LV dϕ C1 ðϕÞ sin ϕ CLV 2 ðϕÞ ¼ yðϕÞ
LV L þ γLV ðTIL Þ½CLV 1 þ C2 Ps ðTI Þ
ð13Þ The vibration frequencies of the water molecule have been previously reported: 24 3651, 3756, and 1590 cm 1 . The coupled system of equations, eqs 813, will be used to calculate jev(ϕ).
ð18Þ
and the Laplace equation gives LV PIL ðϕÞ PIV ðϕÞ ¼ γLV ðCLV 1 ðϕÞ þ C2 ðϕÞÞ
ð19Þ
where the subscript I on a property indicates it is to be evaluated at the liquidvapor interface. Since there is a temperature gradient in the liquid phase, we do not assume the pressure profile is hydrostatic but suppose the gradient of PL[z(ϕ)] is proportional to the product of the liquidphase density, the acceleration of gravity, g, and the depth in the liquid below the apex, z(0) z(ϕ). We define the proportionality factor, Kp, to be such that the pressure difference at the liquidvapor interface is given by PIL ðϕÞ PIV ðϕÞ ¼ PIL ð0Þ PIV ð0Þ þ Kp FL g½zð0Þ zðϕÞ
∑
vf ðTIL Þ V þ ½P kb TIL I
ð17Þ
The equations of differential geometry describing the interface shape may be written
ð11Þ
LV where CLV 1 (ϕ) and C2 (ϕ) are the steady-state principal curvatures of the liquidvapor interface. Note that they vary along the liquidvapor interface. If n is the number of atoms in the evaporating molecule and its internal molecular vibration frequencies are denoted ωl, the vibration partition function of the molecule may be expressed pωl exp 3nY 6 2k T b qvib ðTÞ ¼ ð12Þ pωl l ¼ 1 1 exp kb T
ð16Þ
Since the temperature gradients in each phase at the apex were measured, eq 16 can be used to determine jev(0) for each experiment. Their values are listed in Table 3. When the SRT expression for the evaporation flux is applied at this position, it contains two unknowns: PVI (0) and CLV 0 . Symbolically
and PLe must satisfy the relation LV L PeL ¼ γLV ðTÞ½CLV 1 ðϕÞ þ C2 ðϕÞ þ ηPs ðTI Þ
ð15Þ
The energy balance, eqs 57, at the apex reduces to
(yr,zr zb) (( 0.01 mm) (8,1.11) (8,1.05) (8,1.41) (8,1.01) (8,1.11)
where η is defined " # vf ðTIL Þ L L ðP Ps ðTI ÞÞ η exp kb TIL e
vanishes there. The Laplace equation at this position
PL ð0Þ PV ð0Þ ¼ 2γLV CLV 0
Kp θk ((0.3°)
ð14Þ
ð20Þ and determine the value of Kp as part of the general solution to the governing equations and the boundary conditions. From the Laplace equation applied at an arbitrary position on the interface one finds CLV 1 ðϕÞ ¼
PIL ðϕÞ PIV ðϕÞ CLV 2 ðϕÞ γLV
ð21Þ
We take CLV 0 and Kp to be the iterative parameters. The measured droplet height on the center line is denoted z(0), the curvature of 21316
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the three-phase line as Cslv, and the measured interface height at ϕr as z(ϕr) zb. The boundary conditions then may be stated ϕ ¼ 0 : yð0Þ ¼ 0; zð0Þ ¼ z0 zb ϕ ¼ ϕm : yðϕm ÞCslv ¼ 1 ϕ ¼ ϕr : yðϕr Þ ¼ 8 mm; zðϕr Þ ¼ zr
ð22Þ
where ϕm is the maximum value of ϕ and is equal to the kinetic contact angle, θk. The measured values of the parameters are listed in Table 3. We assume a value of Kp and determine the value of CLV 0 that gives a possible solution (z(ϕ;Kp);y(ϕ,Kp)) by solving eqs 1721 simultaneously. If the condition at ϕr is not satisfied by this solution, a new value of Kp is assumed and the process repeated until the measured height at ϕr agrees with that predicted. The values of C0LV and Kp determined by this procedure are listed in Table 3. For one experiment, the calculated droplet shape is compared with that measured in Figure 4. The agreement found for the other experiments were equally as good. The values of θk determined in each experiment are listed in Table 3. The corresponding equilibrium contact angle calculated from the measured droplet height at the apex and Cslv are also listed there. In each case the value of θk is larger, as has been seen before for other systems.17 Once the droplet shape, z(ϕ) and y(ϕ), is known, the interfacial temperatures in the liquid and vapor phases at 11 points along the interface were measured, PL(ϕ) may be expressed as a function of ϕ from eq 20, then the local evaporation flux, eqs 813, at each of the 11 points may be expressed in terms of the local vapor-phase pressure, PV(ϕ), jev[PV(ϕ)]. If this pressure, PV(ϕ), is assumed to be equal to PV(0), the local evaporation flux at each point may be calculated and when these values of jev[PV(0)] are fitted and integrated over the droplet surface one finds the predicted total evaporation flux is different than that measured with the syringe pump, Jsp; however, the discrepancy is not large if the values of PV(ϕ) are adjusted from PV(0) by less than 13 Pa and are listed in Table 2. The calculated value of the total evasrt , is within 3% of Jsp, Table 3. poration rate, Jev For each steady-state experiment, the pressures listed in the Table 2 were constant, but from one experiment to another, i.e., from one pumping rate to another, these pressures were changed, as indicated there. Ideally, the predicted pressure should be within the error bars of that measured, but in the case of the sessile droplet, it is not possible to measure the pressure with sufficient spatial resolution to determine the pressure variation along the surface of the droplet. Thermal Energy Transport by Thermocapillary Convection. Of the energy transported from the solid to the adsorbed layer, only a small portion was found to be transported to the bulk liquid (see Figure 3). The remainder, Q_ A3, was suggested to be transported parallel to the solidliquid interface, through the adsorbed phase to what in the Gibbs approximation is the threephase line. As a hypothesis, we suppose this energy, Q_ A3, is transported along the liquidvapor interface by thermocapillary convection, Q_ tc, where it is consumed by the latent heat of the phase change process, allowing the liquidvapor interface to be in the steady state. If this hypothesis is valid, then in each experiment Q_ A3 = Q_ tc. The values of Q_ A3 are listed in Table 1. We now calculate Q_ tc. If eqs 5 and 6 are used in eq 7 one obtains an equation with only one unknown, the speed along the interface: v LV t . Thus, from the
measured interfacial temperatures, temperature gradients, P LI [z(ϕ)] (eq 20) and P VI (ϕ), this speed may be determined by calculating the local evaporation flux, j ev , from eqs 813. The values of vLV t obtained at six positions along the liquid vapor interface in each experiment are listed in Table 2. Note that it is negative, indicating its direction is from the threephase line toward the apex. Its most negative value occurs near the three-phase line. _ tc at These values of vLV t may be used in eq 5 and the values of q each of these positions determined, Table 2. After fitting these values to a fourth-order polynomial in ϕ, the total energy transported by thermocapillary convection is given by Z 0 yðϕÞ dz 2πq_ tc Q_ tc ¼ ð23Þ sin ϕ ϕm The steady-state, thermal energy transport to the three-phase line, Q_ A3, is listed in Table 1. In Figure 5, Q_ A3 and Q_ tc found in each experiment are compared. They are seen not to be measurably different. This results support the hypothesis and indicate that because of the Kapitza resistance to the energy transport perpendicular to the solidliquid interface the transport from the solid to the adsorbed layer, Q_ SA, is, in large part ultimately transported along the liquidvapor interface by thermocapillary convection. Conservation of Mass and Energy for the Evaporating Sessile Droplet. We examine the energy transport further by considering the droplet as a control volume. If the mass flow rate into the droplet provided by the syringe pump is denoted Jin, then since the droplet is of constant size Jin must equal the total evaporation flux Z z0 yðϕÞ dz 2πjev Jin ¼ ð24Þ sin ϕ zb If the temperature of the flow entering is denoted Tth, the energy convected into the droplet by the flow from the syringe pump, Q_ p, may be written Q_ p ¼ hL ðTth ÞJin Z z0 yðϕÞ dz ¼ jev hL ðTth Þ2π sin ϕ zb
ð25Þ
The expression for the thermal conduction from the solid to the droplet, Q_ SA, is given in eq 2, and the thermal conduction from the bulk vapor to the liquidvapor interface Q_ VI may be expressed Z z0 yðϕÞ dz kV ∇T V 3 iϕ 2π Q_ VI ¼ ð26Þ sin ϕ zb The energy leaving the droplet is that carried by the evaporating vapor, Q_ ev Z z0 yðϕÞ dz jev hV ½TIV ðϕÞ2π Q_ ev ¼ ð27Þ sin ϕ zb If energy is conserved Q_ ev ¼ Q_ p þ Q_ SA þ Q_ VI
ð28Þ
Since the enthalpies are known only to within a constant, this equa21317
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Figure 6. Comparison of the left-hand and right-hand sides of eq 29. Since the two sides are equal, the conservation-of-energy principle for the droplet as a whole is satisfied.
tion may be rewritten to eliminate the constant Q_ ev Q_ p ¼ Q_ SA þ Q_ VI Z z0 yðϕÞ dz jev ½hV ½TIV ðϕÞ hL ðTth Þ2π sin ϕ zb Z ¼ ð kS ∇T S 3 iz Þπðrs2 rh2 Þ þ
z0
zb
k∇T V 3 iϕ 2π
then Q_ LV must supply the energy for the latent heat, Q_ fg Z z0 yðϕÞ _ dz jev ½hV ½T V ðϕÞ hL ½T L ðϕÞ2π Q fg ¼ ð31Þ sin ϕ zb yðϕÞ dz sin ϕ ð29Þ
In Figure 6, the left-hand-side of this equation is plotted as the ordinate and the right-hand-side as the abscissa. As seen there, these quantities had the same value, indicating the conservation-of-energy principle is satisfied when the droplet is considered as a control volume.
’ DISCUSSION Energy transport by thermocapillary convection during sessile droplet evaporation is generally neglected, but recent results indicate this assumption is invalid. For example, when an experimental procedure similar to that used in this study was used to maintain a water droplets on a Cu substrate as the water evaporated steadily, the dominate mechanism of energy transport to the liquidvapor interface was thermocapillary convection. Of the total energy transport to the liquidvapor interface, thermocapillary convection transported up to 98.4%.17 In the experiments of this study, the total thermal energy flux to the liquidvapor interface, Q_ LV, is the sum of that from the vapor, Q_ VI , that from the liquid, Q_ LI , and that transported by the thermocapillary convection, Q_ tc, but because of the constant temperature gradients in the liquid phase, Q_ LI was equal to Q_ AL . Thus Q_ LV ¼ Q_ tc þ Q_ VI þ Q_ AL
Figure 7. Comparison of the energy supplied to the liquidvapor interface, Q_ LV, with that required for the phase change, Q_ fg. The data points fall on the 45 line, indicating they are equal and that the conservation-of-energy principle is satisfied. Also shown are the values of Q_ VI + Q_ AL for each experiment. The difference between Q_ LV and Q_ VI + Q_ AL is the energy transported by the thermocapillary convection, Q_ tc. At the maximum evaporation rate, one finds Q_ tc is 87% of the total Q_ LV.
ð30Þ
If energy conservation at the liquidvapor interface is satisfied,
As seen in Figure 7, this energy-conservation condition (Q_ LV = Q_ fg) is met. Also shown in this figure is the value of Q_ VI + Q_ AL for each experiment. As seen in eq 30 Q_ tc ¼ Q_ LV ðQ_ VI þ Q_ AL Þ
ð32Þ
The highest evaporation rate in these experiments was that in EVA5 (Table 3). In it, thermocapillary convection supplied 87% of the energy to the liquidvapor interface required for steadystate evaporation. Thus, in these experiments, as with the waterCu and waterPDMS experiments, compared to thermal conduction, thermocapillary convection is the dominant mode of energy transport to the liquidvapor interface during sessile water droplet evaporation. This observation raises the question: why does the energy from the solid transfer through the adsorbed phase to the three-phase line rather than to the bulk liquid (see Figure 1). When the definition of the Kapitza resistance is applied at the solidliquid interface, it may be expressed RK ¼
TSSA TLAL q_ SA
ð33Þ
We suppose the direction of the energy transport in the solid liquid adsorbed phase is because of the Kapitza resistance acts to resist the energy transport across this interface. We examine the correlation of this resistance with the Gibbs adsorption at the solidliquid interface, nSL. In the Gibbs model of the solid liquid interphase the concentration of the liquid is assumed uniform up to the interface and the Gibbs adsorption, nSL, describes whether the actual concentration in the interphase is 21318
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change process at this interface. This consumption allows the sessile droplet to remain in a steady-state condition.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We are grateful for the support of the Natural Sciences and Engineering Research Council of Canada and the Canadian Space Agency. Figure 8. Measured Kapitza resistance for the waterAu(111) interface as a function of the temperature discontinuity at the solidliquid interface for different pressures at this interface, PL(zb).
more or less than that assumed. Thus, a negative value of nSL indicates the concentration in the interphase is less than that in the bulk liquid.25 We show that RK increases as the concentration interphase decreases, nSL becomes more negative. Kapitza Resistance at the Au(111)Water Interface. The data listed in Tables 1 and 2 were used with eq 33 to calculate the values of RK along the interface, and the results are summarized in Figure 8. The pressure in the liquid phase at the base of the droplet was calculated using eq 20 and measured droplet height, z0 zb, Table 1. Note the results indicate that for a given value of PL(zb), RK increases with the magnitude of the temperature AL discontinuity, TSA S TL , and for a given value of the temperature discontinuity, RK increases as PL(zb) is increased. Since in these experiments TSA S had approximately the same value in each experiment, 2 °C, the increase in the temperature discontinuity indicates a decrease TAL L . In other studies, a L SL decrease in TAL L and an increase P (zb) have been found to make n 9,10,25,26 more negative. Physically, this means the concentration of the fluid in the interphase is decreased by a decrease in TLAL and by an increase PL(zb). For a Cu substrate heating liquid He II at 1.6 K, Kapitza measured an interfacial thermal resistance of the same order as our measurements for the waterAu interface, but it was suggested that the Kapitza resistance decreased as T3.13 Our measurements are not in agreement with this extrapolation. If it were valid, the interfacial temperature discontinuities of this study would not be measurable.
’ CONCLUSION Because of a large Kapitza resistance at the interface of water with Au(111), the mechanism of sessile water droplet evaporation has been shown experimentally to be different than the conventional model that assumes thermal conduction as the mechanism transporting thermal energy from the hot surface to the liquidvapor interface. The magnitude of the Kapitza resistance is determined by the temperature discontinuity at this interface per unit heat flux (eq 33). This resistance impedes the energy transport perpendicular to the solidliquid interface and results in the larger fraction of the energy transport from the solid being transported parallel to the solidliquid interface to the three-phase line where thermocapillary convection distributes it along the liquidvapor interface, and it is consumed in the phase
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