ARTICLE pubs.acs.org/JPCC
Polymer Transports Inside Evaporating Water Droplets at Various Substrate Temperatures Jung-Hoon Kim,† Sang-Byung Park,† Jae Hyun Kim,‡ and Wang-Cheol Zin*,† † ‡
Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang 790-784, Korea Manufacturing Technology Team, Infra Technology Service Center, Device Solution Business, Samsung Electronics Co., Ltd., San no. 16 Banwol-Ri, Taean-Eup, Hwasung-City, Gyeonggi-Do, Korea, 445-701
bS Supporting Information ABSTRACT: The flow dynamics inside evaporating water droplets containing a polymer solute were investigated at various substrate temperatures. A new experimental method (laser absorbance monitoring system) was established to estimate the flow dynamics by measuring the central concentration of the solute in the evaporating droplet. The system showed that the polymer transports inside the evaporating droplets near the end of evaporation changed direction depending on the substrate temperature. The resulting deposit patterns after the evaporation were also changed from center-concentrated to edgeconcentrated deposit patterns by the flow dynamics. The experimental results are discussed with the calculation results obtained from a simple heat transfer model. This study clearly shows that controlling substrate temperature is an effective way to manipulate the flow dynamics inside evaporating water droplets.
’ INTRODUCTION Over the past several decades, the dynamics of water droplets evaporating on surfaces has been extensively studied because of its scientific interest110 and importance to develop the inkjet printing technologies for manufacturing processes in many applications such as electronic devices,11,12 biomolecules arrays,13,14 and templates for microlenses.15 Picknett and Bexon suggested two evaporation modes.16 One is the “constant contact area mode” in which the contact area stays constant while the contact angle decreases, and the other is the “constant contact angle mode” in which the contact angle remains fixed while the contact area decreases. Yu et al. observed that the evaporation process of water droplets on self-assembled monolayers occurs in two distinct stages: the constant contact area mode and then the constant contact angle mode.17 They also reported that the transition between the two evaporation modes originates from the contact angle hysteresis, which is dominated by the surface roughness.18 Deegan et al. first reported a mechanism for the solute transport in evaporating droplets on surfaces yielding a ring-like stain, which is known as the “coffee stain phenomenon”.19,20 The ring-like stain is produced because the contact line is pinned, so solvent lost by evaporation at the droplet’s edge must be replaced by solvent drawn from the center of the droplet. The outward flow from the droplet’s center to its edge causes the accumulation of solute at the contact line. We also reported three distinct stages in evaporation process of water droplets on two different r 2011 American Chemical Society
polymer surfaces.21 The three distinct stages are constant contact area mode, the constant contact angle mode, and a “mixed mode” in which both contact angle and contact area decrease as the last stage in the evaporation process. The mixed mode had an outward direction of flow, resulting in ring-like stains on polymer surfaces after evaporation. The transition between the constant contact angle mode and the mixed mode is caused by the Marangoni instability, which arises from the emergence of a local concentration gradient of solute along the air/water interface. It has also been shown that the resulting stain morphology can be controlled by adjusting the initial concentration of surfactant in a droplet.22 Therefore, the Marangoni flow in evaporating droplets should be regarded as an important factor to determine the final resulting patterns of solute. Hu and Larson numerically analyzed the effects of Marangoni stress caused by the temperature gradients along the surface on the flow in an evaporating water droplet.23 They also showed that Marangoni flow in an evaporating octane droplet can reverse the coffee stain phenomenon and produce a solute deposition at the droplet center rather than the edge.24 In recent years, many researchers have focused on the effect of substrate on the temperature distribution of an evaporating droplet to predict the direction of Marangoni flow. Ristenpart et al. derived a Received: March 15, 2011 Revised: June 26, 2011 Published: July 10, 2011 15375
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The Journal of Physical Chemistry C quantitative criterion, which is the ratio of thermal conductivities of the liquid and substrate, for the direction and magnitude of Marangoni flow.25 Xu et al. claimed that Marangoni flow in an evaporating droplet depends sensitively not only on the ratio of thermal conductivities but also on the ratio of the substrate thickness to the contact radius of the droplet.26 The Marangonidriven convection inside the evaporating droplet on a heating substrate was described quantitatively by Girard et al.27 They numerically investigated the temperature distribution along the droplet surface for different values of substrate temperature and showed that the temperature gradients in an evaporating droplet create a Marangoni stress, resulting in the convective phenomena inside the droplet. In addition, they successfully observed the temperature profiles on the droplet surface using infrared thermography and reported that the apex of a droplet on a heating substrate is colder than elsewhere at all investigated substrate temperature (3060 °C), which is consistent with their numerical analysis.28 Despite the great deal of recent progress, the flow dynamics inside evaporating droplets on temperature-controlled substrates is not fully understood because of the lack of experimental data for the flow dynamics. Recently, Kajiya et al. successfully visualized the concentration field in evaporating droplets of fluorescent polystyrene anisole solution by combining a fluorescence measurement and a lateral profile measurement.29 They observed that a strong concentration region is created in the vicinity of the contact line in an early stage, whereas the polymer concentration in the central region remains almost constant until the late stage of evaporation. This means that the fluid in the central region was removed mainly by outward flow. They found only the ring-like deposit of polymer after the evaporation occurred on an open surface or in a box containing solvent bath. Here we report not only a new experimental method to estimate the flow dynamics by measuring the central concentration of a solute in a droplet but also the effect of substrate temperature on the flow dynamics inside the evaporating droplet. The final patterns after the evaporation of droplets change from centerconcentrated deposit patterns to edge-concentrated deposit patterns depending on the substrate temperature. The experimental results are also discussed with the calculation results obtained from a simple heat transfer model. This study clearly shows that controlling substrate temperature is an effective way to manipulate the flow dynamics inside evaporating water droplets.
’ EXPERIMENTAL SECTION Substrates Preparation and Controlling the Substrate Temperature. The experiment using the laser absorbance
monitoring system employed a surface-modified glass substrate to measure the intensity of the laser traveling through the center of droplet on the substrate. Square glasses (no. 1, Corning) with 0.13 to 0.16 mm thickness having 25 mm 25 mm dimensions were cleaned by immersion in H2SO4/H2O2 (7:3) solution for 20 min at ∼90 °C and then rinsed with deionized water. The cleaned glasses were treated with the layer-by-layer assembly method using silica nanoparticles (11 nm diameter) and poly(allylamine hydrochloride) (Mw ≈ 70 000 g/mol). After onebilayer-coated glasses were dried, the glasses were sintered at 500 °C for 1 h to remove the organic materials and then hydrophobically treated by immersing them in octadecyltrichlorosilane toluene solution with a concentration of 10 mM for 3 h at
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Figure 1. Atomic force microscopy image of the surface-modified glass.
room temperature in a nitrogen environment. The glasses were removed from the solution and rinsed repeatedly with toluene and ethanol and then baked in an oven at 120 °C for 20 min. After baking, the glasses were again rinsed thoroughly with toluene and ethanol and finally dried. The static contact angle of a water droplet on the initial surface modified glass was 129 ( 2°. The advancing and receding contact angles of a water droplet on the surface were determined with the needle-syringe method to be ∼155 and ∼82°, respectively (contact angle hysteresis = 73°). By applying a UV-ozone treatment, the static contact angle was reduced to ∼96 ( 5°, and the advancing and receding contact angles were also reduced to ∼114 and ∼19°, respectively (contact angle hysteresis = 95°). The surface-modified glass was found to have a 20 nm rootmean-square roughness in area of 3 μm 3 μm by atomic force microscopy (Dimension 3100, Veeco), as shown in Figure 1. The substrate temperature was controlled using an aluminum stage connected to a bath circulator (RW-2025G, Lab. Companion) that could maintain the stage temperature between 22 and 42 °C. A 5 mm diameter hole in the stage allowed the incident laser to pass through the droplet to the detector. A thermocouple was used to measure the temperature of the stage. The glass was placed on the stage after the temperature of the stage was stabilized and left at least 10 min before the experiment to ensure the thermal equilibrium with the stage at each temperature. Laser Absorbance Monitoring System. The flow dynamics inside the evaporating droplet were investigated using the laser absorbance monitoring system shown in Figure 2. The absorbance is linear with the concentration according to A = εlc (Beer’s law), where A is absorbance, ε is absorptivity of the absorber, l is path length, and c is the concentration of absorbing species in the solution.30 Therefore, if the absorptivity is known and both the path length and the absorbance are measured, then the concentration of absorber can be deduced. Poly(3,4-ethylenedioxythiophene)poly(styrenesulfonate) (PEDOT-PSS, total molecular weight = 560 000 g/mol)31 water solution was employed to monitor the concentrations of the absorbing species inside the solutions. The PEDOT-PSS water solution with concentration of 5 104 weight fraction was prepared by diluting BAYTRON P, which has a concentration of ∼1 102 weight fraction and viscosity of ∼60 mPa 3 s. The absorption spectrum of the solution was obtained with a Shimadzu UV-2550 spectrometer. The absorbance was measured using a polarized HeNe laser having a wavelength of 633 nm with output power of 5 mW 15376
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Figure 2. Schematic diagram of the laser absorbance monitoring system to investigate the changes in flow direction inside the evaporating PEDOT-PSS water droplets. Both the laser absorbance (A) and path length (l) at the center of PEDOT-PSS water droplets on the substrates were simultaneously measured every minute.
(Melles Griot) and a silicon photodetector (818-SL, Newport) as a light source and a detector, respectively. The diameter of the incident laser beam and the output power were adjusted by using a slit and polarization filter, respectively. The incident laser beam was focused onto the substrate using a plano convex lens. The laser beam was aligned perpendicularly by the mirror so that it passed through the two iris diaphragms with diameters of ∼0.5 mm. The surface-modified glass was placed over the hole in the temperature-controllable stage, which was positioned between the iris diaphragms. After the droplet was put onto the glass substrate, the glass was aligned so that the laser traveled through the center of the droplet. After passing through the droplet, the laser was focused to the detector using a plano convex lens. The detector was connected to an optical power meter (model 835, Newport), and the intensity was recorded using the LabVIEW program (National Instruments). The experiments were carried out in a darkened chamber to prevent the intensity fluctuation of laser and the air convection. The absorbance was calculated by the formula A = log(I/I0), where I0 and I are the intensity of the laser after passing through the glass and the droplet, respectively. The path length of the laser through the evaporating droplet was also monitored by capturing the side view of the evaporating droplet. A blue LED backlight was employed to avoid any interference with the laser absorbance. Droplets of 3032 μL were deposited as a sessile drop from a micropipet onto the substrate at each temperature; then, the laser absorbance and the shape of the droplets were simultaneously recorded every minute. After all of the images had been acquired, precise digital image analysis was performed as described in our previous papers.21,22 The room temperature and the relative humidity during the experiment were in the range of 2629 °C and 4866%, respectively. The morphology and the height profiles of the resulting PEDOT-PSS deposits on the substrates after evaporation were observed by optical microscopy, surface profiler (Alpha-Step 500, KLA-Tencor), and 3-D profiler (Wyko NT1100, Veeco Touson).
’ RESULTS AND DISCUSSION To check the validity of Beer’s law in the laser absorbance monitoring system and to evaluate the absorptivity of PEDOTPSS, we measured normalized absorbances (A/l) by varying the
Figure 3. (a) Method to measure the normalized absorbance by varying the concentration of PEDOT-PSS water solution for the laser absorbance monitoring system. (b) Absorbance variation of PEDOT-PSS water solution with a concentration of 2 104 weight fraction in the range of 300800 nm wavelength. (c) Changes in normalized absorbance of PEDOT-PSS water solutions at 633 nm wavelength as a function of concentration. The solid line in part c is obtained by a linear fitting of the normalized absorbances in the range of 5 105 to 1.5 103 weight fraction. The absorptivity of PEDOT-PSS at 633 nm wavelength (395 ( 2 mm1 3 weight fraction1) is obtained from the slope of the solid line.
concentration of PEDOT-PSS water solution. To minimize the movement of fluid in PEDOT-PSS water solution, we controlled the meniscus of the solution to be flat, as shown in Figure 3a, by adjusting the wettability of the side wall in the glass well as in the study of Chen et al.32 Although the PEDOT-PSS water solution does not have a local maximum in the absorbance around 633 nm wavelength, as shown in Figure 3b, the normalized absorbance shows very good linearity against the concentration of PEDOTPSS, as shown in Figure 3c. The linearity of the normalized absorbance in the solution inside the well is applied to the concentration of ∼1.5 103 weight fraction. Therefore, we concluded that Beer’s law is valid for our system with dilute 15377
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and the solute is uniformly distributed in the evaporating droplet. ω¼
Figure 4. Surface profiles of an evaporating water droplet with initial volume of ∼31 μL on surface-modified glass at 26 °C in substrate temperature. The solid lines running through the data are fits obtained from the ellipsoidal cap model (0 min, 16 min elapsed time) and the spherical cap model (from 32 to 80 min elapsed time).
concentrations of PEDOT-PSS water solutions, and the absorptivity of the solution at 633 nm wavelength is 395 ( 2 mm1 3 weight fraction1 from the slope of the linear fitting of data. The typical surface profiles of an evaporating droplet are shown in Figure 4. The initial contact angles are ∼96 ( 5°. The (5° variations in the contact angle value on the UV-ozonetreated surface with larger than (2° variations in that on the initial surface-modified glass are acceptable, because larger contact angle hysteresis introduced by applying UV-ozone treatment may result in larger variations in the contact angle value. After the droplet is deposited on the glass, the contact area of the droplet is pinned for quite a long time; then, the contact area shrinks just before the completion of the evaporation. The deponing of contact line occurs when the contact angle has reached the receding angle as demonstrated in our previous study. Two different polymer substrates were employed in the study to test for the relatively high receding angle of ∼72 and ∼57°.21 In this study, however, we conclude that both the relatively low receding angle of ∼19° and the presence of PEDOT-PSS in the droplet are responsible for the pinning of contact area until the last stage of evaporation.20 The surface profiles in Figure 4 were calculated from the ellipsoidal model at 0 and 16 min and from the spherical cap model for all higher times.5 All profiles are in good agreement with the observed ones. This indicates that gravity affects the initial droplet shape and the effect of gravity becomes negligible as the evaporation proceeds. Although the large volume of droplet may result in the distortion of the surface profile from spherical cap model, we employed 3032 μL of droplets for a higher resolution of the results obtained from laser absorbance monitoring system. The constant contact area mode in the evaporating droplet on the surface-modified glass is significantly prolonged compared with the evaporation process on the polymer surfaces used in our previous studies.21 This may be caused by the fact that the surface roughness and the chemical heterogeneity in the surface-modified glass are quite larger than those of the polymer surfaces. The path length in the evaporating droplet with much lower contact angle (including when the contact area is shrinking) was not measured because of the insufficient clarity of the image. Therefore, the flow dynamics studied using the laser absorbance monitoring system corresponds to only the constant contact area mode. The flow field in the evaporating droplet can be discussed by comparing the concentration profile near the center of the droplet obtained from the experiment and the concentration profile calculated with ex 1, which assumes that no flow occurs
ωi ωi þ ð1 ωi Þð1 T:=tf Þ
ð1Þ
Here ω is the weight fraction of PEDOT-PSS, ωi is the initial weight fraction of PEDOT-PSS, T. is the evaporation time, and the tf is the elapsed time to complete the evaporation. Equation 1 was derived from the assumption that the weight fraction of solvent decreases linearly with time during the evaporation. This assumption is valid for our study because the weights of droplets decreased almost linearly with time until the last stage of evaporation. This method is quite similar to the fluorescence microscopy developed by Kajiya et al.29 They observed that the concentration at the center of the evaporating anisole droplet did not increase until ca. 80% of total evaporation time, meaning a strong outward flow transferred the polymer from the central region to the edge of the droplet until the last stage of drying. The laser absorbance results for the center of evaporating droplets are shown as a function of evaporation time for various substrate temperatures in Figure 5. The variation in the absorbance was observed during the evaporation of the droplet at each substrate temperature, and the absorbance did not significantly vary after the completion of evaporation. The final evaporation time (tf) was determined by the finishing time of the absorption variation. The final evaporation time decreases as the substrate temperature increases because the evaporation rate increases with increasing temperature. The evaporation rate should also be affected by the relative humidity, but we found that it was mainly governed by the variation of the substrate temperature rather than that of the relative humidity in our experimental range. The path length of the incident laser is decreased linearly until the end of evaporation at each substrate temperature. The normalized absorbance was found to gradually increase as a function of the evaporation time. The concentrations in weight fraction are obtained by dividing the normalized absorbance in Figure 5 by the absorptivity values of 395 ( 2 mm1 3 weight fraction1. The evaporation times are also normalized by the final evaporation time. Figure 6 shows the variation with the normalized evaporation time of experimental concentrations of PEDOT-PSS in the evaporating droplets and the concentration calculated using ex 1 at each substrate temperature. The initial experimental concentrations of the droplets are quite similar to the intended initial value of 5 104 weight fraction. Therefore, we concluded that the concentrations were successfully determined by the laser absorbance monitoring system. Although Beer’s law is not obeyed at the concentration above ∼1.5 103 weight fraction of the solution inside the well, it would be still applicable for the droplets of higher concentration above 1.5 103 weight fraction because the absolute number of absorber in the path length of incident laser beam would not be significantly increased because of the continuous decrease in the path length. Our results show that regardless of the variation in substrate temperatures the concentration profiles in all evaporating droplets obtained from experiments accorded well with the concentration profiles calculated by using ex 1 in the early stage of evaporation. Moreover, the concentration profiles in the droplets at 26 and 30 °C accorded well with the calculated concentration profiles until the end of evaporation. However, the concentration profiles in the evaporating droplets at other substrate temperatures started to deviate gradually from the calculated concentration profiles as 15378
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Figure 6. Concentration changes at the center of the evaporating PEDOT-PSS water droplets as a function of the normalized evaporation time (T./tf) at various substrate temperatures. The solid lines are obtained using ex 1. Each concentration series is offset for clarity, but the initial concentrations are effectively equal. The y axis increment is 0.001 weight fraction. For the substrate temperature below ambient temperature (22 and 24 °C), the experimental concentrations at the center rise above the calculated concentrations, whereas for the substrate temperature above ambient temperature (34, 38, and 42 °C), the experimental concentrations fall below the calculated concentrations near the end of evaporation.
Figure 5. Experimental results at the center of the evaporating PEDOTPSS water droplets with concentration of 5 104 weight fraction at various substrate temperatures obtained from the laser absorbance monitoring system: (a) changes in the absorbance (A), (b) changes in the path length (l), and (c) changes in the normalized absorbance (A/l) as a function of evaporation time (T.). The arrows in part a indicate the final evaporation time (tf). Each absorbance series in part a is offset for clarity. The y axis increment is 0.25 absorbance units.
the evaporation went on. Near the end of evaporation, the concentration profiles in the droplets at 22 and 24 °C were slightly higher than the calculated concentration profiles, and those at 34, 38, and 42 °C in substrate temperatures were slightly lower than the calculated concentration profiles. It should be noted that the substrate temperatures of 22 and 24 °C are below the ambient temperature (2629 °C), whereas the substrate temperature above 30 °C is clearly above the ambient temperature; these phenomena will be discussed later in detail. Also, we confirmed that small variations in the initial contact angle value and in relative humidity have minor effects on the above results, and the results are highly reproducible in repeated experiments. The good agreement between the experimental and the calculation results is different from Kajiya et al.’s results, where the concentration at the center in the evaporating anisole droplet
Figure 7. Concentration changes at the center of the evaporating poly(3-dodecylthiophene-2,5-diyl) anisole droplet and PEDOT-PSS water droplet as a function of the normalized evaporation time (T./tf) with a initial concentration of ∼5 104 weight fraction at room temperature. The solid lines are obtained from ex 1. Each concentration series is offset for clarity, but the initial concentrations are effectively equal. The y axis increment is 0.001 weight fraction.
does not increase until ∼80% of the total evaporation time. The laser absorbance monitoring system was also used to test poly(3dodecylthiophene-2,5-diyl) (Mw ≈ 16 200 g/mol) anisole droplets with room-temperature substrate to compare with Kajiya et al.’s result (Supporting Information). As shown in Figure 7, the concentration profile in the evaporating anisole droplet does not significantly increase during the evaporation and was quite lower than the calculated concentration profiles near the end of evaporation. The final solute deposit shows a clear ring-like pattern after the evaporation. These results are consistent with Kajiya et al.’s result. Therefore, we believe that the different experimental results were caused by different flow dynamics in the water and anisole droplets. The flow inside the evaporating droplet can be interpreted in terms of an evaporation-induced 15379
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Figure 8. (a) Optical microscope and 3-D images of resulting deposits of PEDOT-PSS on the substrates. The scale bar inserted in each optical microscope image represents 2 mm. 3-D images are the magnifications around the perimeters of the deposits. (b) Height profiles of the deposits at various substrate temperatures obtained from surface profiler.
outward flow and a recirculated flow (i.e., Marangoni flow), which brings the solute back to the apex of droplet from the edge. The outward flow would be generated in both anisole and water droplets because the contact lines are pinned during the evaporation. A difference in Marangoni flow between the anisole and water droplets would be responsible for the different flow dynamics. From our previous study, it is found that the solutes generating the surface tension gradients should be positioned at the airwater interface, and the initial contact angle of droplets should be decreased as increasing the concentration of the solutes.22 Because the initial contact angles of PEDOT-PSS water droplets is not significantly decreased as increasing the concentration of PEDOT-PSS from 0 to 1 103 weight fraction, we concluded that the PEDOT-PSS in water droplets would not be responsible for Marangoni flow. We suggest that the low temperature coefficient of surface tension (0.1657 dyn cm1 °C1)23 and high contact angle (96 ( 5°) in the water droplet in this study would produce the Marangoni flow, which can compensate for the outward flow by carrying the solute from the edge to the central region of the water droplet. Whereas, both the temperature coefficient of surface tension
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(0.1163 dyn cm1 °C1)33 and contact angle (66°) in the anisole droplet would be insufficient to generate the Marangoni flow to carry the solute from the edge to the central region of droplet, so only the outward flow would transport the solute from the center to the edge of the anisole droplet. The difference of temperature coefficient between the two solvents would be caused from the difference of intermolecular interaction between hydrogen bonding and dipoledipole interaction.34 Any gradient in a vertical direction at the center of a droplet cannot be accounted for, but the concentration change in radial direction can be effectively discussed by the results obtained from the laser absorbance monitoring system. As described above, the good agreement between the concentration profiles obtained from experiment and calculation indicates the Marangoni flow, which brings the solute back to the center of droplet from the edge. The outward flow would draw the solute from droplet’s center to its edge, and the Marangoni flow would compensate for the outward flow by transporting the solute from droplet’s edge to its top along the surface of droplet. Thus, the outward flow in the central region of droplets at the substrate temperatures near ambient temperature is speculated to be roughly as strong as the inward Marangoni flow until near the end of evaporation. The lower concentration profiles near the end of evaporation of droplets at the substrate temperatures, which above ambient temperature indicate that the outward flow dominates over the inward Marangoni flow. As the contact angle decreases, the Marangoni flow would be reduced by the decreasing temperature difference between the droplet’s apex and its edge, but the velocity of the outward flow would be enhanced by the increasing evaporation flux. Therefore, the outward flow should be more intense than the Marangoni flow near the end of evaporation. In contrast, the higher concentration profiles near the end of evaporation of droplets at the substrate temperatures, which below ambient temperature mean the solute is drawn by inward flow from the edge to the central region of droplets. These results indicate that the flow dynamics in the evaporating water droplets are significantly affected by the substrate temperatures. The different flow dynamics in evaporating droplets should result in different solute deposits. Figure 8 shows both the images and the height profiles of the solute deposits on the substrate after the evaporation of the droplets in Figure 6. Clearly, the circular solute distribution is gradually changed from centerconcentrated deposit patterns to edge-concentrated deposit patterns as the substrate temperature increased, as expected from the results obtained from the laser absorbance monitoring system. Not only are the characteristic solute deposits determined by the flow dynamics at a given substrate temperature but also the deposit heights near the center are also affected. The rise near the center of patterns is probably caused by the solute deposit after the shrinking of contact area just before the completion of the evaporation because this rise in deposit height was not observed in poly(3-dodecylthiophene-2,5-diyl) anisole droplets, which did not shrink in terms of contact area during evaporation. The above experimental studies clearly show that the flow dynamics in the evaporating water droplet can be manipulated by controlling the substrate temperature. The Marangoni flow caused by the temperature distribution along the droplet surface is a significant factor to determine the convective cell in water droplets on substrates. To predict the direction of Marangoni flow in the droplets at various substrate temperatures, we estimated the temperature difference between the edge and the 15380
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droplet, respectively, and hS and h(r) are the substrate thickness and height along a radial component of the droplet, respectively. The height is determined from the spherical cap model, which is given by h(r) = (R2/sin2 θ r2)1/2 R/tan θ, where θ is the contact angle and R is the contact radius. The latent heat flux of the vaporization of a droplet surface can be expressed as Qevap(r) = ΔHvapJ0(θ)(1 r2/R2)Λ(θ), where ΔHvap is the vapor latent heat, J0(θ) is the evaporation flux at the center of a droplet, and Λ(θ) is a function of the contact angle, which can be expressed as Λ(θ) = 0.2239(θ π/4)2 + 0.3619.36 The evaporation flux at the center of the droplet may be written as J0(θ) = D(1 H)cv(0.27θ2 + 1.30)(1 Λ(θ))/R, where D is the vapor diffusivity, H is the relative humidity of the ambient air, and cv is the saturated vapor concentration. Considering that Qcond(r) and Qevap(r) are balanced at the vaporliquid interface, as shown in Figure 9b, the surface temperature of a droplet on a room temperature or heating substrate can be expressed as Ti ðrÞ ¼ TSL ðrÞ
hðrÞΔHvap J0 ðθÞð1 r 2 =R 2 ÞΛðθÞ kL
ð2Þ
In an evaporating droplet on a cooling substrate, the energy balance at the droplet surface should be different. As shown in Figure 9c, the direction of conductive heat flux would be from the top of droplet to the bottom of substrate, and the conductive heat flux would be suppressed by the latent heat flux of the vaporization at the droplet surface. Considering that the conductive heat fluxes are balanced at the solidliquid interface, the conductive heat flux in a droplet on a cooling substrate may be expressed as Qcond(r) = kS[TSL(r) T0]/hS = {kL[Ti(r) TSL(r)]/h(r)} ΔHvapJ0(θ)(1 r2/R2)Λ(θ). Then, the surface temperature of a droplet on a cooling substrate may be given by Figure 9. (a) Droplet with the shape of a spherical cap on a flat substrate. The contact angle is θ, the local height is h(r), the substrate thickness is hS, the thermal conductivities of the substrate and the liquid are kS and kL, respectively. Schematic diagrams for heat transfer in the immediate vicinity of the symmetry axis of the evaporating droplets (b) on room temperature and heating substrates and (c) on a cooling substrate. The temperatures in the stage, at the solidliquid interface, and at the vaporliquid interface of the droplet are T0, TSL, and Ti, respectively.
apex of a droplet with the shape of a spherical cap on a flat substrate, as shown in Figure 9a by employing a rough assumption. The temperature was calculated according to the 1D heat transfer model suggested by Xu et al.26,35 The assumption used in the model was that the conductive heat flux vector was oriented perpendicular to the substrate surface, as shown in Figure 9b,c. This assumption is valid for the central region in the droplets but not so suitable for the region close to the contact line. However, under the assumption that the surface temperature changes monotonously, which was verified by Hu and Larson,23 a criterion for the direction of temperature gradient along the whole droplet surface and consequently for the direction of Marangoni flow inside the droplet is then acquired. As shown in Figure 9b, the conductive heat flux on a room temperature or heating substrate could be expressed as Qcond(r) = kS[T0 TSL(r)]/hS = kL[TSL Ti(r)]/h(r), where kS and kL are the thermal conductivities of the substrate and liquid, respectively, T0, TSL(r), and Ti(r) are the temperatures of the stage, at the solidliquid interface, and at the vaporliquid interface of the
Ti ðrÞ ¼ TSL ðrÞ þ þ
kS hðrÞ ½TSL ðrÞ T0 kL hS
hðrÞΔHvap J0 ðθÞð1 r 2 =R 2 ÞΛðθÞ kL
ð3Þ
The temperature at the solidliquid interface of a droplet is calculated using eqs 2 and 3 under the assumption that the temperature at the solidliquid interface is equal to the imposed temperature (i.e., TSL(r) = T0), for simplicity. This assumption is reasonable because the thermal energy would be readily supplied to the solidliquid interface through the very thin glass substrate. The temperatures are obtained as a function of the contact angle of a droplet using the following parameters: droplet radius R = 1 mm, substrate thickness hS = 0.15 mm, thermal conductivity of water kL = 1.4536 103 cal cm1 s1 K1, latent heat of vaporization ΔHvap = 541 cal g1, vapor diffusivity D = 26.1 mm2 s1, relative humidity H = 0.6, and saturated vapor concentration cv = 2.32 108 g mm3.23,36 The radial components r = 0 mm and r = 0.6, 0.9, and 0.99 mm were used to describe the apex and the interface around the edge of a droplet, respectively. Figure 10 shows the calculation results of the temperature difference between the apex and the edge as a function of the contact angle. The positive and negative values in the y axis indicate higher and lower temperatures, respectively, around the edge relative to the temperature at the apex of a droplet. On the room temperature and heating substrates, the temperature difference produced positive values, regardless of the magnitude of the contact angle. The magnitudes of the temperature difference 15381
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Figure 11. Schematic diagram of the Marangoni convective cells in water droplets on substrates: (a) evaporating droplet on room temperature and heating substrates and (b) evaporating droplet on a cooling substrate.
Figure 10. Calculation results of the temperature differences between the apex (r = 0 mm) and the interface around the edge (r = 0.99, 0.9, and 0.6 mm) in a droplet as a function of the contact angle. The solid lines were obtained from droplets on room temperature and heating substrates, and the dashed lines were obtained from droplets on cooling substrates. The upper and lower insets represent the direction of Marangoni flow in the droplet on the room temperature and heating substrates and on cooling substrates, respectively.
between the surfaces at r = 0.99, 0.9, and 0.6 mm and at r = 0 mm slowly decayed to zero as the contact angle decreased. In contrast with the droplet properties on the room temperature or heating substrates, the temperature difference in a droplet on a cooling substrate displayed the reverse behavior. Above ∼70° in contact angle, unrealistic temperature differences are obtained because the assumption as shown in Figure 9 would be greatly inappropriate for the region close to the contact line of droplet with high contact angle. In addition, the temperature difference transition from positive to negative values at a 14° contact angle on the room temperature substrate, which was described in a previous study,23 was not observed because of the rough assumptions used in this calculation. These problems indicate that there are limitations to a quantitative discussion of our calculated results, shown in Figure 10. However, qualitative conclusions may be drawn. The cooling substrates reversed the temperature distribution in the droplet relative to the distributions in droplets on the room temperature and heating substrates. Figure 10 indicates a higher temperature around the edges relative to the temperature at the apex of a droplet on the room temperature or heating substrates, which was qualitatively consistent with previous results.27,28 The temperature difference leads to a surface tension gradient and will always induce Marangoni flow from the edge to the top along the surface of a droplet, as described in the upper inset of Figure 10. In contrast, the reversed temperature distribution in a droplet on a cooling substrate will always induce Marangoni flow from the top to the edge along the surface of a droplet, as described in the lower inset of Figure 10. The velocity of the Marangoni flow would be proportional to the temperature difference. Therefore, the velocity of Marangoni flow should decrease with decreasing contact angle. The temperature differences in Figure 10 did not change with varying substrate temperature. However, if the temperature dependences of the latent heat of vaporization and vapor diffusivity are considered in the calculation, then the temperature differences should differ slightly from those described in Figure 10 with respect to varying substrate temperature. The convective cell in an evaporating droplet can be determined by combining the Marangoni flow and the evaporation-induced
outward flow. By considering the temperature difference along the droplet surface, the Marangoni convective cell inside an evaporating water droplet on room temperature and heating substrates can be estimated, as shown in Figure 11a. Petsi and Burganos claimed that the velocity of the outward flow is proportional to the evaporation flux.37,38 Therefore, with heating of the substrate, the velocity of the outward flow is expected to increase, although the velocity of Marangoni flow would not be expected to change significantly. The velocity of the outward flow would decrease, but the velocity of Marangoni flow, the direction of which is reversed relative to the direction of flow in a droplet on a heating substrate, would not significantly change upon cooling the substrate. Then, a reversed Marangoni convective cell inside an evaporating water droplet on a cooling substrate would result, as shown in Figure 11b. In addition to the above discussion, the substrate temperature dependence of the degree of nonuniformity in an evaporation flux profile along the surface of a droplet would increase the differences between Marangoni convective cells on heating or cooling substrates. Here the nonuniformity of the evaporation flux indicates a dramatically enhanced evaporation flux at the edge compared with that at the center of a droplet.19,20,36 Heating substrates generate higher temperatures at the edge, and more evaporation at the edge would enhance the nonuniformity of the evaporation flux profile; thus the outward flow would increase on a heating substrate. In contrast, cooling substrates would reduce the nonuniformity in the evaporation flux profile by lowering the temperature at the edge; thus the outward flow would decrease accordingly on a cooling substrate. Therefore, the Marangoni convective cell in an evaporating droplet on a cooling substrate should be determined mainly by the Marangoni flow because of the significant reduction of the outward flow, as described in Figure 11b.
’ CONCLUSIONS We have demonstrated that the flow dynamics inside evaporating water droplets containing a polymer solute can be manipulated by controlling the substrate temperature. The laser absorbance monitoring system clearly showed the polymer transports inside the droplets near the end of evaporation changed direction depending on the substrate temperature. The resulting deposit patterns after the evaporation were also changed, as expected for the flow dynamics from the laser absorbance results. On a room-temperature substrate, the Marangoni convective cell in the evaporating water droplets produces that the outward flow is roughly as strong as the inward Marangoni flow at the central region of the droplets during the evaporation. Because the edge-concentrated deposit patterns are formed after the evaporation, the outward flow can be expected to be more intense than the Marangoni flow just before the end of evaporation of droplets. On a heating substrate, the outward flow 15382
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The Journal of Physical Chemistry C is more intense than the Marangoni flow near the end of evaporation of droplets. This flow dynamics is responsible for the edge-concentrated deposit patterns after the evaporation of water droplets containing a polymer solute. The inward flow is more intense than the outward flow near the end of evaporation of droplets on cooling substrate. Because the solute is drawn from the edge to the central region of droplets by the inward flow, center-concentrated deposit patterns are formed after the evaporation of water droplets containing a polymer solute. Although the exact flow dynamics in the evaporating water droplet on the cooling substrate is not fully examined in this study, it is expected that the inward flow just above the water/substrate interface is more intense than the flow along the surface near the end of evaporation. This study shows that a substrate temperature lower than the droplet temperature can clearly make a Marangoni convective cell with the opposite direction to those in droplets on room temperature and heating substrates. The remarkably low temperature coefficient of surface tension and high contact angle in a droplet would be necessary to manipulate the flow dynamics by controlling the substrate temperature. The results of our study may be applicable to inkjet printing technologies.
’ ASSOCIATED CONTENT
bS
Supporting Information. Experimental results of the evaporating poly(3-dodecylthiophene-2,5-diyl) anisole droplets obtained from laser absorbance monitoring system, optical microscope images, and a 3-D image of resulting solute deposits on the substrates. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Tel: +82-54-279-2136. Fax: +82-54-279-2399. E-mail: wczin@ postech.ac.kr.
’ ACKNOWLEDGMENT This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0016361).
ARTICLE
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