Article pubs.acs.org/Langmuir
Thermal Patterns and Hydrothermal Waves (HTWs) in Volatile Drops Khellil Sefiane* School of Engineering, The University of Edinburgh, Kings Buildings, Edinburgh EH9 3JL, United Kingdom
Yuki Fukatani and Yasuyuki Takata Department of Mechanical Engineering and International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, Fukuoka, 819-0395 Japan
Jungho Kim Department of Mechanical Engineering, University of Maryland, College Park, Maryland 20742, United States ABSTRACT: Experimental measurements of temperature and heat flux at the liquid−wall interface during the evaporation of sessile FC-72 droplets have been reported for the first time using infrared (IR) thermography. Simultaneous high-speed imaging of the evaporating drop was carried out to monitor the drop profile. The study demonstrates that recently evidenced hydrothermal waves are actually bulk waves that extend across the entire droplet volume. More importantly, thermal patterns occurring in the bulk of the drop affect the temperature and heat-flux distributions on the solid substrate and ultimately influence the droplet evaporation rate. These effects were found to be increasingly pronounced as the substrate temperature was raised. The implications for heat-transfer mechanisms and energy transport are discussed.
1. INTRODUCTION Liquids are ubiquitous, and their heat-transfer and phasechange characteristics have direct consequences in nature, biology, and many industrial applications. The evaporation and interaction of liquids with solid materials often directly affect processes such as rain, sea mist, drying of printing ink, and sap in a tree trunk, to cite just a few cases. Two important types of liquid heat transfer can be highlighted. The first is evaporation, in which a phase change occurs, leading to motion of the vapor until it recondenses elsewhere. The second is dynamic wetting, in which the triple line, corresponding to the meeting of the three phases [liquid, solid (usually), and vapor/gas environment], moves over a solid. These two aspects of liquid transfer have been studied separately, but it is only fairly recently that the synergistic effects between the two phenomena have been studied. Wetting and evaporation of drops is now recognized as a fundamental phenomenon that plays a crucial role in a great variety of practical situations ranging from technological applications (e.g., printing, heat-transfer devices, and various coating processes) to a range of biological and geophysical situations (e.g., blood/serum drops, immunology, DNA stretching and sequencing).1−5 Other applications in which wetting and evaporation play important roles are found in the semiconductor industry (in which cleaning and drying of semiconductor wafers is a key issue), the oil industry (in which © 2013 American Chemical Society
wetting plays an important role in the tertiary recovery of oil), and heat transfer (in which the presence of fluid drops and/or films can have a dramatic effect on overall heat-transfer rates). In recent years, the apparently trivial phenomenon of evaporation of small liquid drops has proven to be quite difficult to comprehend satisfactorily. Overall considerations of heat and mass transfer are relevant, and interfacial phenomena and local hydrodynamics must also be taken into account. For instance, drying of a suspension such as coffee leaves stain rings of grounds near the pinned position of the wetting front.6 Advective flow to replace evaporated liquid transports solids that deposit in the triple-line region. The flow during the evaporation of such water drops has been characterized and found to be essentially an outward flow from the center to the contact-line region. The flow also accelerates as the drop evaporates and becomes thinner.7 The interfacial temperature of such evaporating water drops is uniform and does not show any variations. However, other organic liquids such as alcohols and refrigerants, when probed using IR spectroscopy, were found to exhibit thermal patterns and waves.8 One of the most fascinating aspects discovered recently is the existence of hydrothermal waves (HTWs) and regular patterns associated Received: April 11, 2013 Published: July 10, 2013 9750
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
Figure 1. (a) Sketch and (b) picture of multilayered substrate and mirror setup. Tb(t) = Tw is the temperature of the black coating (wall).
However, heat (energy) transfer into the drop takes place through the solid−liquid interface as well. The main objective of the present study was to experimentally investigate the extent of HTWs and their implication for energy transport in the underlying substrate.
with the evaporation of ethanol, methanol, and FC-72 (C6F14, perfluorohexane, 3M) drops as manifested by thermal patterns at the liquid−vapor interface.8 These patterns have since been confirmed by many researchers under normal gravity and microgravity conditions.9,10 The temperature and thermal properties of the substrate have been shown to affect the HTWs and patterns. In a first attempt to explain these observations, thermocapillary hydrothermal waves were proposed as a potential candidate mechanism for the occurrence of the observed thermal patterns. Conventional hydrothermal waves have so far been observed in shallow plane liquids and annuli whose interfaces were subjected to a lateral temperature gradient.11−14 The case of volatile drops, however, is unique as no lateral temperature difference is imposed; instead, the whole phenomenon is spontaneously self-driven by the phase-change process. Theoretical stability analysis has shown that this scenario is indeed plausible but might not be the only responsible mechanism.15 Despite the efforts of numerous researchers, many outstanding questions remain concerning the observed HTWs and patterns. The limited progress in gaining insight into this curious phenomenon is partly due to the fact that all of the experimental studies dedicated to this phenomenon, of which there are very few, have used IR imaging looking from the top down on evaporating drops.8,9 The recorded temperature field can thus reflect the temperature distribution on the liquid− vapor interface or a mean value over a depth adjacent to the interface within the liquid, depending on the optical properties of the liquid under investigation and its transparency to the IR wavelengths used. Many crucial open questions remain concerning these HTWs and patterns, for example, whether these are surface or bulk waves, whether they are thermal or physical in nature, what the underlying physical mechanisms are, and what the implications are regarding energy transport. The full range of mechanisms for heat and mass transfer is still poorly understood for such a fundamental phenomenon. The IR technique has thus far been used to reveal these waves by viewing the sessile drops from the top and reflects the heat and mass transfer occurring through the liquid−vapor interface.
2. EXPERIMENTAL TECHNIQUES AND PROCEDURES The substrate consisted of a silicon wafer with an insulating polyimide/ adhesive layer and black coating (Figure 1). The silicon substrate was heated by placing it on a hot plate with temperature control. The drops were deposited on the substrate using a syringe and allowed to evaporate freely into the surrounding air at 1 atm. The substrate was mounted between two gold mirrors inclined at 45° so both top and bottom surfaces of the drop could be viewed simultaneously using an IR camera (FLIR SC4000, 3.6−5.1 μm, recording at 30 fps at a resolution of 18 mK). A CCD camera (Eximer SImage mini 36) was used to monitor the drop profile as well. The IR data were synchronized with the profile measurements so an energy balance could be performed. The bottom-view IR data were used to obtain time- and spaceresolved temperature and heat-flux distributions at the liquid−solid interface. A multilayer consisted of a 500-μm-thick silicon substrate upon which was attached a thin thermal insulator (polyimide tape consisting of a 15-μm-thick polyimide layer with a 15-μm-thick silicone adhesive) was used as the substrate. The thickness of the tape (polyimide and adhesive) measured using a DektakXT Stylus surface profiling system was found to be about 30 μm. The top of the polyimide tape was coated with an opaque (τ = 0), high-emissivity (ε = 0.89), 6-μm-thick layer of black paint (Nazdar GV111 paint consisting of about 20% carbon black in a vinyl chloride/vinyl acetate copolymer after curing). The painted tape amplified the temperature variations and provided a strong thermal signal for the IR camera. The silicon was largely transparent to IR radiation, whereas the polyimide was partially transparent. The temperature of the bottom of the silicon wafer was measured by attaching the painted polyimide tape at selected locations. The temperature variations within the multilayer along with the heat flux at the solid−fluid interface were determined by solving a transient conduction−radiation problem. The energy measured by the IR camera consisted of the energy emitted by the black coating, polyimide tape, and silicon substrate, along with the energy reflected from the surroundings. Absorption within the polyimide and silicon layers and reflection at the various interfaces must be taken into account to obtain the temperature at the surface of the polyimide tape (the temperature of the black coating). Because the 9751
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
temperature profile within the substrate was initially unknown, an initial temperature profile was assumed and used to obtain an updated surface temperature at time t = 0. This updated temperature was used as the boundary condition for a transient conduction calculation, and an updated temperature profile was obtained at time t + Δt. This updated temperature profile was used to obtain an updated surface temperature. This procedure was repeated to obtain the instantaneous temperature profile within the substrate, from which the heat flux through the liquid−solid interface was obtained. The effects of the
initial temperature profile were found to effectively decay within 0.1 s, after which the true temperature profile could be determined. As a verification of the experimental technique, the temperature of the black coating as measured directly from the top view was verified to agree with the temperature determined from the bottom view before the drop was deposited. Details of the technique and the uncertainties in measurements are given in the Appendix. To ascertain the accuracy of the IR heat-flux measurements, the energy required to evaporate a drop, calculated as ρlVi[cp(Tw −Ti)+hfg] using the fluid properties (ρl is the liquid density) and the initial droplet volume (Vi), initial droplet temperature (Ti), and the wall temperature (Tw), was compared to the integrated IR heat flux over the droplet lifetime [∫ t0∫ A(t)q̇″ dA(t) dt, where A is the drop base surface area] for each test. The wall temperature was varied from 26 to 66 °C in 10 °C increments, and two tests per wall temperature were performed. The wall temperature (Tw) was the initial temperature of the wall before the drop was deposited. Because the silicon had a much larger thermal conductivity and the energy required to evaporate the drop was so small, the wall temperature did not change much due to droplet evaporation (the temperature change within the silicon wafer due to droplet evaporation was estimated to about 0.1 K). Ti was the initial temperature of the droplet, which was at room temperature. The agreement was within ∼23%, as shown in Figure 2, which is quite satisfactory considering the detailed data reduction procedure required for the IR measurements.
3. RESULTS AND ANALYSIS Small drops of FC-72 refrigerant deposited on the substrate were allowed to evaporate freely into the ambient atmosphere. The relevant physical properties of FC-72 are given in Table 1. FC-72 is highly volatile: At a substrate temperature of 26 °C, 2.5-μL drops took ∼5 s to evaporate completely. At a substrate temperature of 66 °C, the same drops took ∼1.5 s to evaporate. During the evaporation process, the three-phase contact line receded continuously because of the very smooth nature of the substrate and low surface tension of FC-72. The profile (height, angle, and base) of the drops as recorded by the CCD camera at 10 frames s−1 was used to deduce the droplet volume. The profile measurements show that the contact angle was almost constant for nearly the entire droplet lifetime whereas the base radius decreased monotonically (Figure 3). This behavior is typical of evaporation on smooth substrates. 3.1. Effect of Substrate Temperature on Heat-Flux Distribution. As discussed previously, the experimental setup allowed IR images of the evaporating drop to be captured, from
Figure 2. Comparison of energy required to evaporate droplets with IR measurements.
Table 1. Thermophysical Properties of FC-72a
a
property
value
saturation temperature (at 1 atm) heat capacity liquid density surface tension latent heat vapor pressure kinematic viscosity diffusion coefficient of FC-72 vapor in air saturation concentration (at 20 °C)
56 °C 1.05 kJ kg−1 K−1 1680 kg m−3 10 dyn cm−1 88 kJ kg−1 1.83 × 106 Pa 0.38 cSt 64 × 10−7 m2 s−1 3.285 kg m−3
3M data sheet.
Figure 3. Base and contact angle evolution of FC-72 drops at temperatures of (a) 26 and (b) 66 °C. Insets show comparisons of fits to experiments. Error bars on the experimental data are ∼10%. 9752
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
Figure 4. Evaporation of a 2.5-μL FC-72 droplet on a substrate at 26 °C as represented by a snapshot at t = 0.533 s of its lifetime. Upper panels show raw IR data from the top and bottom surfaces. Lower panels show the extracted temperature and heat-flux distributions on the bottom surface.
Figure 5. Evaporation of a 2.5-μL FC-72 droplet on a substrate at 66 °C as represented by a snapshot at t = 0.167 s of its lifetime. Upper panels show raw IR data from the top and bottom surfaces. Lower panels show the extracted temperature and heat-flux distributions on the bottom surface.
which the temperatures and heat fluxes distribution at the solid− liquid interface could be deduced. The effect of substrate temperature was investigated to ascertain the extent and effect of the observed hydrothermal waves and patterns on the substrate temperature and heat-flux distributions. Examples of the data before and after processing are shown in Figure 4 (26 °C) and Figure 5 (66 °C). It can be seen from Figure 4 that thermal patterns are clearly visible on the top view of the IR image even at the lowest investigated substrate temperature of 26 °C. The temperature and heat-flux distributions on the solid−liquid interface remained fairly uniform because of the small temperature difference between the substrate and ambient, but patterns were clearly present.
Figure 6. Temperature and heat-flux distributions on the bottom surface of a drop at t = 0.1 s at various substrate temperatures measured using the IR camera just before drop deposition: (a) 26, (b) 36, (c) 46, (d) 56, and (d) 66 °C. 9753
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
Figure 7. Evolution of the heat-flux distribution for a substrate temperature of Tw = 26 °C. Units for heat flux are W cm−2.
For the higher substrate temperature of 66 °C (Figure 5), many characteristic features can be noticed. First, the heatsink effect near the three-phase contact line is clearly visible. This corresponds to colder temperature and higher heat flux around the drop edge, which is consistent with the accepted wisdom that most of the evaporation occurs near the threephase contact line. In addition to the clearly identifiable thermal patterns on the top view, which are similar to those at lower temperature, these patterns are clearly visible in the bottom view. The distribution of the heat flux on the bottom surface clearly reflects the thermal patterns seen in the top view. The heat flux varied by as much as a factor of 2 between the lowest (∼1.5 W cm−2) and highest (∼3 W cm−2) values on the bottom surface. This observation implies that energy transport in the substrate was affected by the presence of HTWs triggered on the liquid−vapor interface where the phase change took place. The temperature at the three-phase contact line was close to the saturation temperature (57 °C), whereas the temperature within the bulk of the drop was as high as 60−61 °C. The temperature and heat-flux distributions on the bottom surface of the evaporating droplets at various substrate temperatures are shown in Figure 6. The differences in both temperature and heat flux became more pronounced as the substrate temperature increased. The measurements clearly reveal a high heat flux at the three-phase contact line, in agreement with the accepted understanding. However, an unexpectedly high heat transfer of the same order of magnitude as the contact-line heat transfer was also observed in the center of the droplet. This high heat flux is likely due to the convection cells within the droplet that circulate hot fluid to the liquid−vapor interface, where it cools by evaporation and then impinges onto the heated substrate at the center of the drop.
Videos illustrating the movement of the HTWs indicate movement of fluid from the center to the three-phase contact line to support this conclusion.16 The implications of the preceding experimental findings for understanding and properly predicting energy transport in such processes as evaporating drops is fundamental in that the effects of the HTWs on heat transfer extend into the underlying substrate. 3.2. Evolution of Heat-Flux Distribution. The heat-flux distributions at various time intervals and for the lowest and highest substrate temperatures are shown in Figures 7 and 8. The location of the three-phase contact line can be clearly identified. The drop base radius decreased and the contact line receded as evaporation proceeded. These results are similar to the data reported for top-down observations on heated water drops.2,3 One observation from these experiments is that the thermal patterns due to HTWs were more pronounced at the beginning of the experiments and tended to fade toward the end of the drop lifetime. This can be explained by the facts that drops become small as they evaporate and the available volume for the thermocapillary effects becomes more confined. Confinement is known to increase critical conditions and the threshold for the occurrence of such instabilities.15 The heat transfer as a function of radius is shown in Figure 9 at various times for a substrate temperature of 66 °C. It is clear that the heat transfer at the contact line remained relatively constant as the droplet evaporated. The peak in heat transfer at the center of the drop also remained relatively constant. A comparison of the radial heat-transfer distributions at the lowest and highest wall temperatures is shown in Figure 10. The heat-flux distributions both show peaks at the contact line and at the center of the droplet. 9754
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
Figure 8. Evolution of the heat-flux distribution for a substrate temperature of Tw = 66 °C. Units for heat flux are W cm−2.
3.3. Heat-Transfer Mechanisms and Analysis. The evaporation rate as predicted by quasi-steady-state vapor diffusion theory is given by17 −ṁ (t ) = πR bD(1 − H )Csat(0.27θ 2 + 1.30)
expressions that account for the contributions of solid and liquid show deviations from eq 2 for high evaporation rates and insulating substrates.18 The substrate used in the current experiment consisted of a thin polyimide layer on a highly conductive silicon wafer (kSi = 120 W m−1 K−1). Because the time constant of the polyimide tape was estimated to be on the order of 0.1 s, which is much shorter than the droplet evaporation times observed, the basic diffusion equation (eq 2) is expected to be valid. In the present experiments, the contact line receded as evaporation proceeded. Nonetheless, eq 2 can be used to estimate the order of magnitude of the evaporation rate based on diffusion alone. To perform the calculations, it was assumed that the far-field concentration of FC-72 vapor was zero. A comparison between this model and the heat flux obtained from the IR measurements is shown in Figure 11. The diffusion model (eq 2) greatly underpredicts the experimental data, implying that mechanisms in addition to steady-state diffusion (e.g., convection, unsteady mass diffusion) occurred in these experiments. The observed HTWs can indeed be responsible for enhancing the evaporation and energy transport within drops. The contribution of thermocapillary effects in energy transport has been demonstrated to be significant in some cases.19
(1)
where ṁ (t) is the mass evaporation rate, Rb is the drop base radius, D is the diffusion coefficient of FC-72 vapor in air, H is the relative humidity, Csat is the saturation concentration at the liquid−vapor interface, and θ is the liquid contact angle. Because the concentration of FC-72 in the ambient is zero, the evaporation heat rate can be written using the latent heat of evaporation as qdiff ̇ = hfg πDR b(t ) Csat(T ) f (θ )
(2) 2
where hfg is the latent heat and f(θ) = (0.27θ + 1.30). Equation 2 was developed for the steady-state diffusion of vapor into the ambient and does not account for contributions of the solid and liquid or for convection in the liquid or gas. It is valid for the evaporation of pinned sessile drops with no significant evaporative cooling (highly conductive substrates or slow evaporation) and has been validated against experiments of water droplets evaporating on glass and poly(methyl methacrylate) (PMMA). More recent 9755
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
10−6/(64 × 10−7) = 0.15 s, which is an order of magnitude smaller than the lifetime of the fastest evaporation case. (The drop lifetime for the fastest evaporation case at Tw = 66 °C was about 1.5 s.) This indicates that unsteady effects are unlikely, justifying the use of quasi-steady-state diffusion theory.
4. DISCUSSION IR thermography is an excellent tool for measuring droplet heat transfer at solid−liquid interfaces. The high spatial resolution along with the excellent thermal resolution enables detailed thermal information to be obtained at the previously inaccessible solid−liquid interface. IR thermography provides access to observations of droplets even when the droplets are very thin and when optical techniques are unable to resolve the profile. The data presented in this article show for the first time that the HTWs observed by previous researchers are not limited to a region close to the surface of the drop, but extend throughout the entire drop thickness to the wall. The findings from the present study indicate that, when HTWs are present, the evaporation rate of sessile droplets cannot be described by a diffusion mechanism of vapor alone. HTWs seem to contribute to energy transport and evaporation. Furthermore, they lead to a distribution of evaporation on the interface that is more uniform compared to cases where HTWs are not present. The direct results of this finding are that evaporation is not concentrated mainly near the contact line and overall evaporation is not linearly proportional to the base radius. The presence of HTWs and their extent as revealed in the present study have some significant implications. In heat-transfer applications, such as spray cooling, fire safety, and combustion, the evaporation of droplets resting on hot surfaces will be affected by the presence of these thermocapillary effects. Indeed, this study clearly shows that the observed effects extend to the liquid−solid interface. The heat-flux distribution is clearly nonuniform. Incorporating a nonuniform heat-flux distribution over the surface of contact of droplets evaporating on hot surfaces is undoubtedly paramount in understanding, predicting, and optimizing the numerous heat-transfer applications involving droplet evaporation. Existing theoretical models for droplets evaporating on a heated surface do not take into consideration the nonuniform distribution of the heat flux on the wall. The results obtained in the present study suggest that these models need to be improved to account for the presence of HTWs and a nonuniform heat-flux distribution on the contact surface between droplets and hot surfaces. Other areas that might also benefit from these finding are the numerous printing, patterning, and self-assembly techniques in which the evaporation of droplets is the main mechanism to form patterns and deposits on solids. The extent of HTWs as revealed in this investigation indicate that deposits from drying droplets can definitely be affected by the presence of HTWs. The heat-flux distribution on the solid−liquid interface is bound to influence the final pattern formed by volatile droplets. If a more uniform deposit is desirable, such as in printing, these HTWs could interfere with the uniform nature of the deposits. In this specific case, means to suppress HTWs should be sought. In other self-assembly applications where orderly mixing is desirable, HTWs can make a positive contribution to the generation of final patterns. Regarding the driving mechanism of the observed HTWs, it has previously been proposed that temperature gradients between the edge and the apex of the drop could be the driving force behind this thermocapillary phenomenon. The fact that HTWs were observed at ambient temperature
Figure 9. Evolution of the heat-flux profile for a substrate temperature of Tw = 66 °C.
Figure 10. Comparison of radial heat transfers for substrate temperatures of Tw = 66 and 26 °C at t = 0 s.
The area-integrated heat transfer as a function of the drop radius is shown in Figure 12. Previous work on volatile pinned sessile drops suggested that the overall evaporation rate is proportional to the radius because of the dominance of evaporation at the three-phase contact line (i.e., q̇″ = C1r, where C1 is a constant). The data shown in Figure 12 suggest that the overall evaporation rate is not linearly dependent on the radius, indicating significant evaporation over the entire liquid−vapor interface (i.e., q̇″ = C1r + C2r2, where C1 and C2 are constants). The discrepancy between the evaporation predicted by the diffusion model and that measured experimentally is more pronounced for higher substrate temperatures. The diffusion model employed makes the assumption of quasi-steady-state conditions. For this assumption to hold, the time scale for diffusion would need to be much faster than the change in profile of the drop. Some unsteady effects might be introduced as the contact line recedes. The vapor diffusion time can be estimated, taking a characteristic length (L) of 1 mm and using the diffusion coefficient of FC-72 vapor in air (D), as tdiffusion = L2/D ≈ 9756
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
Figure 11. Comparison of the evaporation heat rate obtained by multiplying the mass evaporation rate by the latent heat at any given time from the diffusion model and the experimental data obtained from the droplet profiles: Tw = (a) 26 and (b) 66 °C.
■
APPENDIX
A.1. Experimental Method for Measuring Temperature and Heat-Flux Distributions from IR Data
A multilayer wall consisting of a silicon substrate onto which polyimide tape (polyimide layer + acrylic adhesive) is attached is shown schematically on Figure A1a. An opaque black paint much thinner than the other layers was applied to the top of the polyimide tape. The droplets were deposited on the black surface. The polyimide tape was necessary to measure the heattransfer coefficient distributions of the expected magnitude because the high thermal conductivity of the silicon substrate would simply smear out any temperature variations through substrate conduction, reducing both the magnitude of the temperature differences and their spatial resolution. To obtain the heat-transfer coefficient at the fluid−wall interface, knowledge of the temperature gradient within the polyimide tape was required. If the time-varying temperatures of the black surface Ts1(t) and Ts2(t) could be found, the instantaneous temperature profile within the multilayer could be obtained through an unsteady heat-conduction simulation. Because the polyimide tape was thin compared with the spatial resolution of the camera and the temperature gradient was much larger in the x direction than in the y and z directions, a one-dimensional heat-conduction assumption was used. Assuming one-dimensional heat conduction, the governing equations within the layers are given by
Figure 12. Area-integrated heat transfer as a function of drop radius.
(i.e., with no imposed temperature constraint) raises the question of other possible mechanisms. Adsorption of humidity present in the ambient air might be an alternative mechanisms based on mass transfer. To explore these mechanisms further, experiments in a controlled environment must be undertaken. This is a future direction that could reveal more about the exact driving mechanism for the observed phenomenon.
5. CONCLUSIONS Two important conclusions can be drawn from this study: First, HTWs and thermal patterns observed in volatile drops are bulk waves that can extend throughout the drop volume. Second, these HTWs can affect the temperature of the solid surface as well as the solid heat-transfer distribution. This effect was found to be exacerbated at higher substrate temperatures. These new findings, in addition to showing new features of thermal patterns and HTWs in volatile drops, provide supplementary insight that can help in explaining the precise physical mechanism behind this phenomenon. Equally important is the conclusion that energy-transport concepts and theoretical models describing the evaporation of volatile drops must be revisited to account for these new observations.
ρSi c p,Si
∂T = k Si∇2 T + qSi̇ ∂t
(A1a)
ρA c p,A
∂T = kA ∇2 T ∂t
(A1b)
ρP c p,P
∂T = kP∇2 T ∂t
(A1c)
and the system was subject to the boundary conditions T = Ts1(y,z,t) at x = 0 and T = Ts2(y,z,t) at x = LSi + LA + LP. Consider now the calculation of the temperature of the black surface, Ts1(t). This temperature was not directly available because the energy measured by the IR camera consisted of emission from the black surface, emission from each of the layers (which depends on the temperature profile within them), and reflection from the surroundings. Because the optical 9757
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
Figure A1. Model descriptions.
properties of the polyimide and adhesive were similar, they were treated as a single layer in the radiation calculation, as indicated in Figure A1b. The total energy measured by the camera (Ec) was the sum of the energies emitted by each layer within the spectral bandwidth of the IR camera (from λ1 to λ2) Ec = ρ∞−c E∞ + εSi−cESi + εT−cE T + τs−cEs
τs−c =
(A2)
εSi−c =
εT−c =
(A3)
(1 − ρSi−∞)(ρapp,Si−T τSi) 1 − ρSi−∞ρapp,Si−T τSi 2
(A4)
(1 − ρSi−∞)(1 − ρSi−T )(1 + ρT−s τT)τSi (1 − ρSi−∞ρapp,Si−T τSi 2)(1 − ρSi−T ρT−s τT 2)
(A6)
(1 − ρSi−T )2 ρT−s τT 2 (1 − ρSi−T ρT−s τT 2)
(A7)
The derivation of these equations is given in ref 20. The optical properties of the various layers along with the temperature of the surroundings (T∞) were measured as described next. The temperature distributions within the silicon [TSi(x)] and the tape [TT(x)] and the temperature of the black surface (Ts,1) were not known initially, but could be obtained by solving the coupled conduction and radiation problems according to the following algorithm: (1) Assume an arbitrary temperature profile within the multilayer at t = 0. (2) Compute ESi and ET from the assumed temperature distribution and determine an updated Es and surface temperature Ts,1 from eq A2. (3) Solve the conduction equation using the updated Ts,1 to obtain a new temperature profile at t = Δt. (4) Repeat steps 2 and 3 for each successive time step. The effect of the assumed initial temperature profile within the multilayer decayed after a short time, after which the true temperature profile was obtained. For the conditions used for the experimental verification in this work, the initial transient was found to have completely decayed after 0.1 s. The heat flux from the wall to the fluid could be obtained from the derivative of the instantaneous temperature profile within the polyimide at the black surface according to q″ = −kP(∂T/∂x)|x=0.
(1 − ρSi−∞)2 ρapp,Si−T τSi 2 1 − ρSi−∞ρapp,Si−T τSi 2
(1 − ρSi−∞ρapp,Si−T τSi 2)(1 − ρSi−T ρT−s τT 2)
ρapp,Si−T = ρSi−T +
where E∞ = Fλ1−λ2σT∞4 is the blackbody radiation due to the surroundings, ESi = ∫ L0 SiαSiFλ1−λ2σ[TSi(x)]4 exp(−αSix) dx is the energy emitted by the silicon that reaches the Si−∞ interface, ET = ∫ L0 TαTFλ1−λ2σ[TT(x)]4 exp(−αTx) dx is the energy emitted by the tape that reaches the T−Si interface, and Es = Fλ1−λ2σTs,14 is the blackbody radiation of the black surface. Fλ1−λ2 is the total emitted energy from a blackbody contained with the wavelength interval λ1−λ2 and was obtained from tables or by integrating the Planck distribution. The coefficients ρ∞−c εSi−c, εT−c, and τs−c account for the attenuation and reflection within the multilayer wall and are given by ρ∞−c = ρSi−∞ +
(1 − ρSi−∞)(1 − ρSi−T )(1 − ρT−s )τSiτT
(A5) 9758
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
from which ρm−∞ and τm can be obtained. The absorption coefficient can then be obtained from the Beer−Lambert law, τm, and the layer thickness. The reflectivity of the tape was calculated using the indices of refraction of the polyimide (npolyimide = 1.7) and air to be
A.2. Radiant Property Determination
The model used in this work requires knowledge of the reflectivity and absorptivity of the materials used. These were determined experimentally using the two configurations shown in Figure A2. The blackbody consisted of a large cylindrical cavity at a controlled temperature with a small orifice; the emissivity of this blackbody was estimated to be >0.999. The apparent reflectivity was determined using the experimental setup shown in Figure A2a. The blackbody was
ρpolyimide−air
Table A1. Optical Properties of the Silicon and Polyimide Tape
positioned such that emission from it reflected from either the front or back surface of the layer to be tested. The total energy reaching the IR camera consisted of the blackbody energy reflected from the layer, the self-emission of the layer, and the transmission of energy from the surroundings through the layer Ec1 = ρapp, m−∞(Fλ1− λ2σTb 4) + εapp, m−∞(Fλ1− λ2σTm 4) + τapp, m−∞(Fλ1− λ2σT∞ )
(A8)
(A9)
Subtraction of eq A8 from eq A9 allowed the apparent reflectivity to be determined because the temperatures Tb and T∞ were known. The apparent transmissivity was determined using the setup shown in Figure A1b. The total amounts of energy reaching the camera with and without the blackbody are given by Ec3 = ρapp, m−∞(Fλ1− λ2σT∞4) + εapp, m−∞(Fλ1− λ2σTm 4) (A10)
Ec4 = ρapp, m−∞(Fλ1− λ2σT∞4) + εapp, m−∞(Fλ1− λ2σTm 4) 4
+ τapp, m−∞(Fλ1− λ2σT∞ )
(A11)
τapp, m−∞ =
0.34 0.07 0.14
(A15)
A.3. Uncertainty Analysis
The experimental uncertainty in the calculated heat flux was determined by perturbing each experimental parameter a small amount one at a time and running the algorithm to find the sensitivity to the variables at varying heat fluxes. Many of the assumed physical constants (e.g. thermal conductivity and thickness of silicon) introduced errors that were several orders of magnitude smaller than the important sources of error and were therefore omitted. The parameters that resulted in the highest errors along with their assumed values and uncertainties
(A12)
(1 − ρm−∞)2 τm 1 − ρm−∞2 τm 2
52.6 7110 reflectivity
from which the surface emissivity can be determined. The emissivity of the black coating was measured to be 0.90.
(1 − ρm−∞)2 ρm−∞τm 2 1 − ρm−∞2 τm 2
silicon polyimide tape interface
Ec = εFλ1− λ2σTs 4 + (1 − ε)Fλ1− λ2σT∞4
Subtracting one from the other allowed the apparent transmissivity to be determined. The apparent reflectivity and transmissivity for a single layer m sandwiched by two identical layers n are given by (see ref 18 for derivations) ρapp, m−∞ = ρm−∞ +
absorption coefficient (m−1)
The black coating (6 μm thick) was produced by silk screening onto the polyimide tape Nazdar GV111 paint consisting of about 20% carbon black in a vinyl chloride/ vinyl acetate copolymer after curing. The high carbon black content ensured that the coating was opaque, and its thinness and high thermal conductivity relative to the polyimide allowed its temperature to be assumed to be uniform. The emissivity of the black surface was determined experimentally. The polyimide tape with black coating was attached to a heated aluminum plate containing a thermocouple. The plate was heated to temperatures between 45 and 95 °C, and the energy emitted from the surface along with the energy reflected from the surroundings were measured using the IR camera. The total energy measured by the IR camera is given by
Ec2 = ρapp, m−∞(Fλ1− λ2σTb 4) + εapp, m−∞(Fλ1− λ2σTm 4)
+ τapp, m−∞(Fλ1− λ2σTb 4)
material
Si−air polyimide−air polyimide−Si
The measurement was then repeated without the blackbody
+ τapp, m−∞(Fλ1− λ2σT∞4)
(A14)
If the index of refraction of the acrylic adhesive was assumed to be that of acrylic (nadhesive = 1.5), the reflection would be ρadhesive−air = 0.04. The reflection between the polyimide and adhesive was very small, ρpolyimide−adhesive = 0.04. Because reflection within the tape was small and because the reflectivities of the polyimide and the acrylic adhesive were similar, the tape was considered to behave as a single optical layer with polyimide properties. The absorption coefficient of the tape was measured experimentally. A summary of the optical properties of the silicon and polyimide tape are given on Table A1. The absorption coefficient of the polyimide tape can be seen to be much larger than that of the silicon.
Figure A2. Schematics of the experimental setups used to measure the reflectivity and transmissivty of a single layer.
4
⎛ n polyimide − nair ⎞2 ⎜ =⎜ ⎟⎟ = 0.07 ⎝ n polyimide + nair ⎠
(A13) 9759
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
Langmuir
Article
(10) Brutin, D.; Shu, Z. Q.; Rahli, O.; Xie, J. C.; Liu, Q. S.; Tadrist, L. Evaporation of ethanol drops on a heated substrate under microgravity conditions. Microgravity Sci. Technol. 2010, 22 (3), 387−395. (11) Smith, M. K.; Davis, S. H. Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 1983, 132, 119−144. (12) Garnier, N.; Chiffaudel, A. Two dimensional hydrothermal waves in an extended cylindrical vessel. Eur. Phys. J. B 2001, 19, 87− 95. (13) Schwabe, D.; Möller, U.; Scheider, J.; Scharmann, J. A. Instabilities of shallow dynamic thermocapillary liquid layers. Phys. Fluids A 1992, 4, 2638−2381. (14) Riley, R. J.; Neitzel, G. P. Instability of thermocapillary− buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities. J. Fluid Mech. 1998, 359, 143−164. (15) Karapetsas, G.; Matar, O. K.; Valluri, P.; Sefiane, K. Convective Rolls and hydrothermal waves in evaporating sessile drops. Langmuir 2012, 28 (31), 11433−11439. (16) Videos can be seen at http://www.see.ed.ac.uk/∼ksefiane/ HTWs/HF(FC72_66C).avi. (17) Hu, H.; Larson, R. G. Evaporation of a sessile droplet on a substrate. J. Phys. Chem. 2002, 106, 1334−1342. (18) Sefiane, K.; Bennacer, R. An expression for droplet evaporation incorporating thermal effects. J. Fluid Mech. 2011, 12 (667), 260−271. (19) Ghasemi, H.; Ward, C. A. Energy transport by thermocapillary convection during sessile-water droplet evaporation. Phys. Rev. Lett. 2010, 105, 136102. (20) Kim, T. H.; Kommer, E.; Dessiatoun, S.; Kim, J. Measurement of two-phase flow and heat transfer parameters using infrared thermometry. Int. J. Multiphase Flow 2012, 40, 56−67.
are reported in Table A2. The maximum total error, calculated as the root mean square of each individual error, increased with heat flux and is summarized in Table A3 at various heat fluxes. Table A2. Parameters with Largest Effect on the Uncertainty Analysis parameter emissivity of the black coating absorptivity of the polyimide tape polyimide thickness adhesive thickness reflectivity of silicon-air interface reflectivity of silicon-polyimide interface thermal conductivity of polyimide thermal conductivity of adhesive
value
uncertainty
0.90 7110 m−1 15 μm 15 μm 0.34 0.14
0.01 500 m−1 2 μm 2 μm 0.017 0.007
0.12 W m−1 K−1 0.20 W m−1 K−1
0.01 W m−1 K−1 0.01 W m−1 K−1
Table A3. Summary of Uncertainty Analysis
■
heat flux (W cm−2)
max error (W cm−2)
2.0 4.0 6.0 8.0 10.0
0.35 0.47 0.62 0.80 0.98
AUTHOR INFORMATION
Corresponding Author
*E-mail: K.Sefi
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS K.S. acknowledges the support of the Japan Society for Promotion of Science (JSPS) to undertake a visit to Kyushu University in Japan and initiate this collaboration.
■
REFERENCES
(1) Sefiane, K.; Shanahan, M. E. R.; Antoni, M. Wetting and phase change: Opportunities and challenges. Curr. Opin. Colloid Interface Sci. 2011, 16 (4), 317−325. (2) Semenov, S.; Starov, V. M.; Rubio, R. G.; Agogo, H.; Velarde, M. G. Evaporation of sessile water droplets: Universal behaviour in presence of contact angle hysteresis. Colloids Surf. A 2011, 391 (1−3), 135−144. (3) Antoni, M.; Girard, F. Influence of substrate heating on the evaporation dynamics of pinned water droplets. Langmuir 2008, 24 (20), 11342−11345. (4) Girard, F.; Antoni, M.; Faure, S.; Steinchen, A. Evaporation and Marangoni driven convection in small heated water droplets. Langmuir 2006, 22 (26), 11085−11091. (5) Brutin, D.; Sobac, B.; Loquet, B.; Sampol, J. Pattern formation in drying drops of blood. J. Fluid Mech. 2011, 667, 85−95. (6) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827−829. (7) Marín, A. G.; Gelderblom, H.; Lohse, D.; Snoeijer, J. H. Rushhour in evaporating coffee drops. Phys. Fluids 2011, 23, 091111. (8) Sefiane, K.; Moffat, J. R.; Matar, O. K.; Craster, R. V. Self-excited hydrothermal waves in evaporating sessile drops. Appl. Phys. Lett. 2008, 93, 074103. (9) Carle, F.; Sobac, B.; Brutin, D. Hydrothermal waves in ethanol droplet evaporating under terrestrial and reduced gravity levels. J. Fluid Mech. 2012, 712, 614−623. 9760
dx.doi.org/10.1021/la402247n | Langmuir 2013, 29, 9750−9760
