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Langmuir 2008, 24, 9102-9109
Internal Flow in Polymer Solution Droplets Deposited on a Lyophobic Surface during a Receding Process Masayuki Kaneda, Kentarou Hyakuta, Yuu Takao, Hirotaka Ishizuka, and Jun Fukai* Department of Chemical Engineering, Kyushu UniVersity, 744 Motooka, Nishi-ku, Fukuoka 819-0395 Japan ReceiVed October 16, 2007. ReVised Manuscript ReceiVed May 27, 2008 When a polymer solution droplet is deposited on a lyophobic surface, the contact line is moved back to some degree and subsequently pinned. An experimental setup is constructed to investigate not only the receding process but also an internal flow of polystyrene-acetophenone and -anisole solutions. As a result, the time variation of the evaporation rate per unit area during receding does not strongly depend on the initial solute concentration. The average solute concentration at the pinning of the contact line increases as the initial solute concentration increases. A convective circulation flow that is upward at the axis of symmetry is observed. This flow pattern is different from those of pure liquids such as water, acetone, benzene, and so forth, which have been previously reported. Furthermore, the observed flow is enhanced as the initial solute concentration increases, contrary to an increase in the fluid viscosity. To resolve these discrepancies, the mechanism of the flow is numerically investigated using a hemispherical droplet model considering the density and surface tension distributions. The numerical results demonstrate that the circulation flow that is experimentally observed is actually caused. It is also found that the solutal Rayleigh effect initially induces the internal flow, and subsequently the solutal Marangoni effect dominates the flow. Both effects are enhanced as the initial concentration increases because of the evaporative mass balance at the free surface.
Introduction In recent years, the application of inkjet printing has been studied for the manufacturing of electronic devices such as organic light emitting devices, polymer electroluminescent devices, and so forth. Inkjet printing can be used to print electronic devices by ejecting small polymer solution droplets directly onto a target substrate to form continuous tracks, thereby reducing the number of manufacturing processes and reducing waste. A key requisite of manufacturing these devices by the inkjet printing method is the ability to achieve polymer-film thickness uniformity. Thus, the control of the film formation is a key factor for the aforementioned applications. The film formation is governed by the solute transport inside the droplet during evaporation. In the case of the pinned periphery, Deegan et al.1,2 reported a mechanism for the solute transport toward the periphery of the droplet yielding a ring-like stain. This flow is numerically discussed by Hu and Larson3,4 with and without the consideration of Marangoni effect. The buckling instability during evaporation also affects the film shape reported by Pauchard et al.5 and Kajiya et al.6 Furthermore, the evaporation rate affects the film formation. De Gans and Shubert7 and Kaneda et al.8 reported the dependence of the evaporation rate on the film shape. Both reports concluded that the slow evaporation rate yields a dot-like film rather than a ring-like one. Kaneda et al.8 also suggested a model where the evaporation rate affects the solute transport in the droplet on which the film shape depends. * Corresponding author. E-mail:
[email protected]; telephone number: +81-92-802-2744; fax number: +81-92-802-2794. (1) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827–829. (2) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. F.; Witten, T. A. Phys. ReV. E 2000, 62, 756–765. (3) Hu, H.; Larson, R. G. Langmuir 2005, 21, 3963–3971. (4) Hu, H.; Larson, R. G. Langmuir 2005, 21, 3972–3980. (5) Pauchard, L.; Parisse, F.; Allain, C. Phys. ReV. E 1999, 59, 3737–3740. (6) Kajiya, T.; Nishitani, E.; Yamaue, T.; Doi, M. Phys. ReV. E 2006, 73, 011601. (7) de Gans, B. J.; Schubert, U. S. Langmuir 2004, 20, 7789–7793. (8) Kaneda, M.; Ishizuka, H.; Sakai, Y.; Fukai, J.; Yasutake, S.; Takahara, A. AIChE J. 2007, 53, 1100–1108.
The solute transport inside the droplet can be investigated by the experimental flow observation inside the evaporating droplet. Zhang and Yang,9 Rymkiewicz and Zapalowicz,10 and Chao et al.11 observed a three-dimensional structured flow for the pure liquid droplet. They reported that the thermal capillary force governs the flow. However, the droplet size in their study was in millimeter order and is too large for the inkjet droplet. Furthermore, the dissolved polymer is predicted to affect the flow such as the solutal convection, which has not been fully discussed for the droplet on the substrate. This study focuses on the internal flow of a submicron sessile droplet of a polymer solution deposited on a lyophobic surface. The suspension method is used to observe the flow pattern at the cross section and aid characterization of the droplet evaporation. The effects of the initial solute concentration and the evaporation rate on the flow pattern and evaporation process are discussed. Then, the key factor to induce the flow inside the droplet is numerically discussed through these results.
Experimental Procedure Polystyrene (PS) is provided as a solute, while acetophenone (Ap) and anisole (Ani) are provided as solvents. The average molecular weight of PS (Acros Organics) is 250 000 g/mol. Initial PS concentrations are in the range between 0 and 0.20 kg-solute/ kg-solution. Physical properties of Ap and Ani are shown in Table 1. It should be noted that the saturation vapor pressure of Ani is higher than that of Ap. Thus, the evaporation rate of Ani is higher than that of Ap. Figure 1 shows the solute-concentration dependence on physical properties of the polymer solutions. The density, the surface tension, and the viscosity are measured using a densimeter (Anton Paar, DMA38), a surface tensiometer (Kyowa Interface Science, CBVP-Z), and a viscometer (Brookfield, DV-II), respec(9) Zhang, N.; Yang, W. J. J. Heat Transfer 1982, 104, 656–662. (10) Rymkiewicz, J.; Zapalowicz, Z. Int. Commun. Heat Mass Transfer 1993, 20, 687–697. (11) Chao, D. F.; Sankovic, J. M.; Zhang, Z. J. Thermophys. Heat Transfer 2006, 20, 620–624.
10.1021/la801176y CCC: $40.75 2008 American Chemical Society Published on Web 07/10/2008
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Table 1. Representative Physical Properties of Acetophenone and Anisole at 298 K and 101.3 kPa9 density surface tension viscosity vapor pressure (mN/m) (mPa · s) (Pa) (kg/m3) acetophenone anisole
1023.4 989.4
38.8 34.6
1.66 0.984
49.0 472
tively. The properties increase with increasing concentration. The concentration dependency of the viscosity is especially large. To visualize the flow pattern in the droplets, nylon particles (Kanomax, model 0456, average diameter ) 4.1 µm, F ) 1.02 × 103 kg/m3) are mixed into the solutions to act as a tracer. The mass fraction of the tracer is 0.001 kg/kg-solution. The estimated terminal velocities of the nylon particles in the solutions are 1.92 × 10-2 to 2.44 × 10-4 µm/s (PS-Ap) and 0.977-1.42 × 10-3 µm/s (PS-Ani) in the concentration range between 0 and 0.20. These values are low enough to recognize the particle velocity as the flow velocity. Figure 2 shows an experimental schematic. The substrate used is a silicon wafer whose surface is chemically modified with a fluoroalkylsilane (C6F13(CH2)2Si(OCH3)3) monolayer.12 A plastic vessel (φ ) 45 mm × 20 mm) is placed on the substrate as shown in Figure 2b to prevent a weak air flow in the surroundings from disturbing the internal flow in the droplet. This vessel has slits in its vertical side to aid visualization of the droplet and at the top so as not to intercept the laser sheet light. The substrate is placed horizontally on a heat exchanger where water flows from/to an isothermal bath. A droplet formed at the top of a capillary tube (i.d. 0.13 mm × o.d. 0.47 mm) attached to a microliter syringe (Hamilton, 701N), is gently dropped onto the horizontal substrate. The initial volume of the droplet is below 0.606 µL (d0 ) 1.05 mm). A YAG laser sheet light (Kanomax, CW532-600M), whose minimum thickness is 54 µm, illuminates the cross section of the droplet from above. The droplet frame and the visualized flow pattern from the side view are photographed using a digital CCD camera 1 (Victor, KY-F550, 30 fps) with a magnifying lens (×400), and then recorded. A droplet even on homogeneous surfaces sometimes moves while keeping a spherical cap during evaporation.8 However, the displacement velocity of our droplet is low enough to manually adjust the degree of laser sheet light reaching the droplet center using a precision positioning stage and CCD camera 2 (Sony, Iris, 30 fps; as shown in Figure 2b). This camera is placed horizontally with respect to the substrate, where the axis of the lens is perpendicular to that of CCD camera 1. The local surroundings and isothermal bath are kept at 297 K.
Results Characterization of the Droplet Evaporation. Time variations of the wetting diameter dc and droplet height h are measured from the recorded images. The contact angle Ψc is calculated from dc and h assuming a spherical cap. Each experimental case was carried out two to four times to identify any reproducibility in time variations of the experimental contact angle Ψc and the wetting diameter dc. The reproducibility of the flow pattern was also identified. The results shown below are typical. Shown in Figure 3 are data to investigate the effect of the flow visualization method on the evaporation process.The data of open symbols are taken from samples without the tracer particle and laser irradiation. The wetting diameter is normalized with the initial droplet diameter d0. The figure demonstrates that the evaporation process is not influenced by the tracer and laser irradiation. This figure also shows that the evaporation time of the anisole solutions is roughly 10 times shorter than that for the acetophenone solutions. (12) Morita, M.; Koga, T.; Otsuka, H.; Takahara, A. Langmuir 2005, 21, 911–918.
Figure 4 shows the effect of the initial solute concentration c0. In this figure, the dimensionless time is defined as µBt/(FBd02) by choosing the droplet initial diameter d0 as the characteristic length.13,14 The evaporation process obviously depends on c0 for both droplets. For the acetophenone solution in Figure 4a, the contact line recedes with an almost constant contact angle until the contact line is pinned. After that, Ψc rapidly decreases. The pinning occurs earlier as c0 increases. At c0 ) 0.005 and 0.03, dot-like films were formed. At c0 ) 0.08-0.20, a buckling5,6 was observed resulting in the formation of ring-like films. For the anisole solutions in Figure 4b, the pinning time also becomes earlier as c0 increases. However, dc and Ψc simultaneously decrease until the contact line is pinned. When c0 e 0.03, ringlike films form as a result of the high evaporation rate.8 At c0 ) 0.08, dot-like films are formed. At c0 ) 0.15-0.20, buckling is observed resulting in the formation of ring-like films. The difference in the receding behavior between two solutions is clarified by observing the contact line velocity. Figure 5 shows the receding velocity of the contact line uc, which is calculated using the numerical two points-deviation method from the data shown in Figure 4. For each experiment, uc rapidly increases at the initial stage: uc for the acetophenone solutions is independent of c0, while those for the anisole solution tend to decrease as c0 increases. An increase in uc with time results from the corresponding increase of the evaporation rate as described later. It is difficult to explain the reasonable mechanism of the above behavior from physical properties of solutions alone. Indeed, contact angles for PS-xylene solutions were almost constant during receding even though the viscosity and surface tension of xylene are 0.58 × 10-3 Pa · s and 28.1 × 10-3 N/m,15 respectively, all values which are less than those of anisole. If the viscosity near the contact line is associated with the change in the contact angle during receding, then the change in the solute concentration distribution must be a key factor to explain this observation. The concentration distribution in the droplet is contributed by the diffusion and convection, which are generally characterized by the temperature and concentration dependencies of the diffusion coefficient, the density, the viscosity, the surface tension, and so forth. Advances in research from the viewpoint of transport phenomena are necessary for solving this problem. Figure 6 shows time variations in the volumetric evaporation rate per unit area m ˙ that is calculated from the difference between the instantaneous droplet volumes for the specific timeframes. The open and filled symbols in the figure represent the processes before and after the pinning, respectively. It is observed that m ˙ increases with time before the pinning and then rapidly decreases after the pinning. The obvious dependence of m on c0 is not recognized before the pinning. An increase in m ˙ results from a decrease in the droplet height, which in turn elevates the free surface temperature due to the heat transfer from the substrate. The decrease in m ˙ after the pinning results from an increase in the solute content on the free surface. In other words, in the dry process field it is most likely that a constant-rate drying period appears before the pinning and a falling-rate drying period appears subsequently after. It is found that the transition between these periods agrees with the pinning time. Kaneda et al.8 experimentally found that the average solute concentration cav at the pinning increases as the evaporation rate decreases. Kaneda et al. surmised from this result that not only (13) Fukai, J.; Ishizuka, H.; Sakai, Y.; Kaneda, M.; Morita, M.; Takahara, A. Int. J. Heat Mass Transfer 2006, 49, 3561–3567. (14) Fukai, J.; Ishizuka, H.; Sakai, Y.; Kaneda, M.; Morita, M.; Takahara, A. Exp. Heat Transfer 2007, 20, 137–146. (15) Marcus, Y. The Properties of SolVents; Wiley Series in Solutions Chemistry; Wiley: New York, 1998.
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Kaneda et al.
Figure 1. Solute concentration dependences of measured density, surface tension, and viscosity at 25 °C.
Figure 2. Schematics of the experiment.
Figure 4. Dimensionless time variations of contact angle Ψc and dimensionless wetting diameter dc/d0 for the PS-Ap and PS-Ani droplets.
Figure 3. Time variations of contact angle Ψc and dimensionless wetting diameter dc/d0 for the PS-Ap and PS-Ani droplets. Open and filled symbols represent with and without tracer and laser irradiation.
the diffusion, but also the convection flow, contributes to the transfer of the solute in the droplet. However, the effect of the initial solute concentration was not clear in their experiments. Accordingly, the calculated cav for the various c0 in this study is shown in Figure 7. cav increases as the evaporation proceeds, as indicated by the end plot at the pinning. Comparing panels a and b of Figure 7 for the same c0, cav at the pinning for the acetophenone droplet has a large value. Considering the
evaporation rate, this result agrees well with the result found by Kaneda et al.,8 irrespective of the high viscosity of the Ap droplet in Figure 1. Furthermore, cav at the pinning increases for high c0 droplets both in Figure 7a,b. Therefore, not only the evaporation rate, but also the initial solute concentration contributes to the mass transfer in the droplet. Flow Visualization. For the flow visualization in the droplets, xylene was also provided as a solvent. However, xylene droplets have a lower visualization quality due to the formation of small contact angles, which quickly decrease with evaporation (see Appendix). Figure 8 shows the flow patterns at the initial stage. These images are compiled from frames obtained during periods of 640 and 63 s, respectively. The period of time for the visualization is represented in each image. The stream line near the bottom of the figure corresponds to the reflected image on the substrate surface (see Appendix). At c0 ) 0 for both solutions, complicated and unstable flows are observed during the whole evaporation process. These flows are not attributed to the sedimentation of the tracer particles. At c0 g 0.005, the flow patterns become
Internal Flow of PS-Ap/Ani on Lyophobic Surfaces
Figure 5. Dimensionless time variations of the contact line receding velocity uc until the pinning.
symmetrical and stable. The convective flow gradually develops after the droplet is placed on the substrate. After a while, the flow turns to slow down, and then the contact line is pinned. After the pinning, however, the flow pattern is not clearly observed because of the image distortion at the free surface and the aggregation of the tracer particles. For the acetophenone solutions under the present experimental condition (c0 e 0.20), the flow direction is upward at the center and downward near the free surface. This flow pattern does not agree with the flow of an experimental pure liquid droplet during receding9–11 and an experimental and analytical suspension droplet after pinning.1–4 Circulation flows in the same direction as those in acetophenone solution are observed for the anisole solution at c0 e 0.03. However, inverse flows are observed for the anisole solution at c0 g 0.08. The flow in the droplet is characterized by the average velocity near the center: First, a series of frames taken during periods of 300 (PS-Ap) and 30 (PS-Ani) seconds are chosen for analysis. Second, five particles that are sparkling in the whole series of the frames and located in a region of 175 µm wide × 200 µm high at the center, as indicated by the white frame in c0 ) 0.08 in Figure 8a, are chosen. The locations of the particles, which are distorted by the refraction at the liquid-gas interface, are
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Figure 6. Dimensionless time variations of the volumetric evaporation rate per unit surface area. Filled symbols represent during the pinning.
theoretically corrected.16 As shown in the Appendix, particle positions in the recorded image shift to the bottom in practice. The travel distances of the identical particles during the travel time are calculated. Each particle velocity is estimated from the travel distance and time. Finally, the velocities of five particles are averaged. The particle velocity can be approximated to the flow velocity due to the low terminal velocity as mentioned earlier. Figure 9 shows the time variation of the average velocity. The positive and negative values indicate the ascending and descending flows, respectively. The flow pattern at c0 ) 0 cannot be measured because it is unstable. For the acetophenone solution, the velocity increases to reach a maximum and then subsequently decreases. Here, the curves are inflected just prior to the point where the velocity becomes zero. However, a close inspection of the compiled images reveals that, although there is a slight recovery of the velocity, no significant change in the flow contraction is observed. It is found that the higher c0 is, the stronger the circulation flow. The behavior of the up-t curves for the anisole solutions at c0 ) 0.005 and 0.03 is similar to that of the acetophenone solutions, (16) Kang, K. H.; Lee, S. J.; Lee, C. M.; Kang, I. S. Meas. Sci. Technol. 2004, 15, 1104–1112.
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Figure 7. Dimensionless time variations of the average solute concentrations cav.
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Figure 9. Dimensionless time variations of the averaged tracer velocity ascending at the center up. Table 2. Computed Cases for PS-Ap Droplet c0
Sc0
Le0
0.005 0.03
2.94 × 1.83 × 104 103
1.31 × 3.58 × 102 102
Rac,0
Mac,0
1.63 28.6
3.69 349
maximum velocities are 7 to 27 times higher than those of the acetophenone solutions at the corresponding c0.
Numerical Simulation
Figure 8. Visualized flow pattern in PS-Ap and PS-Ani droplets. Arrows represent the direction of the flow.
although the maximum velocities of the former is about 2.5 times as large as those of the latter. However, the flow at c0 ) 0.08-0.2 is notably reduced at the center and then inversed near the end of the evaporation period. In this case, the absolute
As mentioned above, the observed flows inside the polymer solution droplet disagree with those in the prior works, which reported a downward flow at the center and upward along the surface. Furthermore, it is enhanced as c0 increases, in spite of increasing the fluid viscosity as indicated in Figure 1. These discrepancies are numerically discussed in this section. A constant contact angle during evaporation is assumed for simplicity. An axisymmetric undeformable hemispherical droplet is presumed and numerically solved in the spherical coordinate (R, φ). The governing equations consist of continuity, momentum, mass diffusion, and energy equations. Obvious circulation flow would have been observed at c0 ) 0 if it was strongly contributed by the thermal effect. Therefore the buoyancy and surface tension forces caused by the temperature distribution are ignored. Instead,
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those caused by the concentration distribution are considered. The dimensionless governing equations are given by
Le0 )
()
DU µ r0 )-∇P+ Dτ µ0 r
2
∇·U)0 (1) Sc0Rac,0 cos φ Sc0 2 ∇ U+ C -sin φ Mac,0 Ma 2
(
c,0
)
(2) DAB 1 DC ∇2C ) Dτ DAB,0 Mac,0
(3)
Le0 2 DT ∇T ) Dτ Mac,0
(4)
at
0 < R < 1,
φ ) 0 (5)
At the bottom of the droplet, nonslip, isothermal, and insulated concentration conditions are presumed.
U)V)
∂C )T)0 ∂φ
0 < R < 1,
at
˙ ∂T m ) - - BiT ∂R ˙ m0
at
R ) 1,
0 < φ < π/2
(7)
where m ˙ ) hmFvap,sat(θ). hm and Fvap,sat represent the mass transfer coefficient into the air and the saturated vapor density. The Biot number in the air is given at 0.02 because of the heat transfer. Simultaneously, the evaporation of the solvent increases the solute concentration at the surface. This effect is considered to conserve the solute mass balance depending on the local evaporation rate, local diffusion coefficient, and local concentration.
˙ DAB,0 c ∂C m ) ∂R m ˙ 0 DAB c0
at
R ) 1,
0 < φ < π/2
(8)
The effect of the solute concentration on the surface tension can be reduced as the following equation. The concentration dependence on the surface tension gradient is considered.
∂ V (∂σ/∂c) 1 ∂C ) ∂R R (∂σ/∂c)0 R2 ∂φ
()
at
R ) 1,
0 < φ < π/2 (9)
The representative dimensionless parameters are as follows:
Sc0 )
ν0 DAB,0
(10)
g(∂F ⁄ ∂c)∆c0r03 Rac,0 ) DAB,0µ0
(11)
∆c0r0(∂σ/∂c)0 DAB,0µ0
(12)
Mac,0 )
µ ) e34.41c-6.419
at
(17) Ruiz, O. E.; Black, W. Z. J. Heat Transfer 2002, 124, 854–863.
0 e c e 0.10 -5
∂σ/∂c ) 5.60 × 10 c + 4.00 × 10
at
(14)
0 e c e 0.20 (15)
Although the DAB values of the present solutions were unfortunately not found, DAB is in inverse proportion to µ according to Wilke and Chang:18
DAB ) 1.173 × 10-16
(ΦMB)0.5θ µVA0.6
∝ µ-1
(16)
Φ is the association parameter of the solvent. This value is assumed to be 1 because it is unknown for acetophenone. MB and VA are 120.15 g/mol and 0.1404 m2/(kg · mol) for acetophenone, respectively. The characteristic concentration difference ∆c0 in the direction normal to the free surface is derived from the diffusion equation. That is,
φ ) π ⁄ 2 (6)
At the surface of the droplet, the evaporation of the solvent decreases the temperature by the latent heat. The heat flux at the surface is presumed depending on the local evaporation rate and the convective heat transfer from the ambient air.17
(13)
where g is the gravity acceleration, r0 is the initial droplet radius, DAB,0 is the initial diffusion coefficient, and µ0 is the initial solution viscosity. The viscosity µ and the surface tension gradient ∂σ/∂c are approximated according to the results in Figure 1. -2
The definitions of the parameters are listed in the Nomenclature. µ and DAB imply the local values depending on the concentration. For the boundary conditions, an axisymmetric condition is employed at the axis of symmetry.
∂U ∂T ∂C )V) ) )0 ∂φ ∂φ ∂φ
R DAB,0
∆c0 )
˙ c0r0 m DABF0
(17)
The initial values of the nondimensional parameters are listed in Table 2, which corresponds to those of the PS-Ap droplets at c0 ) 0.005 and 0.03. The governing equations are approximated by the finite difference equations with the mesh number of (R, φ) ) (24, 24). The mesh width in the radial direction is nonuniform, and becomes finer near the surface. The third-order upwind scheme is applied to the inertial term, and the pressure term is solved with the HSMAC method.19 The velocity vectors and isoconcentration contours are shown in Figure 10. Streak lines are added in some results to reveal the flow pattern. Because the present purpose is to investigate the effect of the solute concentration such as the increase in viscosity and surface tension, the results at the initial stages are focused on and shown in the figure. The computed results show a circulating flow with a downward trend along the free surface. As the evaporation proceeds, this flow becomes strong. A comparison between panels a and b of Figure 10 shows that a high initial solute concentration enhances the convective flow. The strong flow induces the concentration distribution in Figure 10b. The flow pattern as well as its tendency toward the initial solute concentration qualitatively agrees with that in the experimental results. Next, the contributing factors on this flow are investigated. To compare the effect of the density and the surface tension, time variations of the maximum velocity with and without the surface tension effect at c0 ) 0.03 is shown in Figure 11. When no surface tension is considered, velocity boundary condition at the surface is given by the following equation instead of eq 9.
∂ V )0 ∂R R
()
at
R ) 1,
0 < φ < π/2
(18)
At the beginning, the velocities are almost equivalent. This means that the flow is initially induced by the density difference in the (18) Geankoplis, C. J. Transport Process and Separation Process Principles; Prentice Hall Professional Technical Reference: Upper Saddle River, NJ, 2003. (19) Hirt, C. W.; Nichols, B. D.; Romero, N. C. Technical Report Los Alamos Scientific Laboratory, LA-5852, 1975.
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Figure 10. Velocity vectors and isoconcentration contours.
Kaneda et al.
(3) The flow in the solvent droplets (c0 ) 0) is unstable and much slower than that in the polymer solutions. This fact probably shows that the temperature dependences of the density and surface tension do not strongly contribute to the circulation flow in the polymer solution droplets. (4) A symmetric circulation flow is observed in the polymer solution droplets on the substrate. The fluid ascends at the axis of symmetry when the contact line continuously recedes, which is different from the prior works. An inverse flow is also observed, which probably occurs when the contact line hardly moves. An increase in the initial solute concentration enhances the circulation flow in the polymer solution droplets. (5) The mechanism of the flow in the polymer solution droplet is numerically investigated. In the case of the constant contact angle, the descent flow along the free surface is induced by the solutal Rayleigh effect first, and then the solutal Marangoni effect enhances the flow in the same direction. These effects are enhanced at high initial solute concentration, which is due to the rapid increase in the concentration on the free surface. Acknowledgment. This study is partially supported by the Japan Society for the Promotion of Science (JSPS), a Grantin-Aid for Scientific Research (B) No. 16360387, and by the Ministry of Education, Culture, Sports, Science and Technology, a Grant-in-Aid for Young Scientists (B) No. 19760528.
Appendix
Figure 11. Time variation of maximum velocity with and without the surface tension effect at c0 ) 0.03.
droplet. As the evaporation proceeds, the flow due to the surface tension effect becomes pronounced. The initial solute concentration increases the elevation rate of c on the free surface due to the mass balance. The solvent evaporation rate is independent of c0, as shown in Figure 6, resulting in an increase of c on the free surface as c0 increases. This enhances the density difference between the surface and the bulk, or Rayleigh effect. Simultaneously, the surface tension effect, or Marangoni effect, is emphasized because the dependency of surface tension on c becomes large as c increases, as shown in Figure 1. Therefore the Marangoni effect furthermore enhances the convective flow as experimentally observed. This mechanism must not be basically changed in the case where the contact angle is below 90°.
The positions of the visualized particles refracted on the free surface are numerically corrected according to the method reported by Kang et al.16 For the choice of the solvents, a PS-xylene solution of c0 ) 0.005 was first prepared for the visualization. The refractive index of the solvent (nB ) 1.497) is employed for the transform because of the lack of data for the solutions at various concentrations. The particle position from the captured image and the transformed position of the xylene droplet are shown in Figure A1a. In this experiment, the static contact angle of the xylene droplet on the substrate is approximately 60°. Therefore, compared to the acetophenone droplet, the transformed particle position shifts closer to the substrate, as shown in Figure A1b (nB ) 1.533). Furthermore, the reflected image from the substrate overlaps the particle images. Indeed in Figure A1a, some particles are transformed at the lower level of the substrate, and these correspond to the reflected image from the substrate. Thus, the particles in the reflected image should be eliminated from the particle capturing process, as indicated in Figure A1b. Additionally, in the case of a low contact angle, the reflected image is dominant in the droplet, which decreases the measurement accuracy. To the best of the authors’ knowledge, particle capturing when the contact angle is less than 45° becomes difficult for the above reasons. In this experiment, the contact angle of
Conclusion The significant findings in this study are summarized below: (1) The time variation of the evaporation rate per unit surface area is almost independent of the initial solute concentration until the contact line is pinned. The evaporation rate starts decreasing immediately after the pinning. (2) The average solute concentration at the pinning of the contact line increases with increasing initial solute concentration. This means that the solute is mixed in the droplet, and the mixing is enhanced at higher initial solute concentrations. This fact is also identified by visualization of the flow pattern.
Figure A1. The raw and transformed particle positions.
Internal Flow of PS-Ap/Ani on Lyophobic Surfaces
the xylene droplet having an almost constant wetting diameter decreased quickly during the evaporation. For the above reasons, the visualization quality of the xylene droplet was poor for the study herein. The evaporation process of the xylene droplet is different from the aforementioned studies.8,13,14 From optical observations of a xylene droplet, it can be seen that the tracer descends as time proceeds. The density of the tracer particle (1020 kg/m3) is larger than that of xylene (860.4 kg/m3), which loses the traceability of the flow pattern. This tracer sedimentation may pin the contact line and affect the evaporation process of the xylene droplet. Consequently, the acetophenone and anisole droplets (nB ) 1.518) are selected in terms of the negligible terminal velocity of the tracer and the reduced tracer interruption on the evaporation process. Furthermore, the particle capturing process is ranged in the upper center to avoid confusion with the reflected image.
Nomenclature
Bi C c d D g h k Le M Ma m ˙ P p R r Ra
Biot number ) hr/k dimensionless concentration ) (c - c0)/∆c solute concentration, kg-solute/kg-solution wetting diameter, m diffusivity, m2/s gravity acceleration, m/s2 transfer coefficient, m/s or W/(m2 · K) thermal conductivity, W/(m · K) Lewis number ) R/D molar weight, g/mol Marangoni number ) [∆cr(∂σ/∂c)]/Dµ evaporation rate per unit area, m3/(m2 · s) ) p/p0 pressure, N/m2 ) r/r0 radial coordinate, m Rayleigh number ) [g(∂σ/∂c)∆cr3]/Dµ
Langmuir, Vol. 24, No. 16, 2008 9109
Sc T t u U V
Schmidt number ) ν/D dimensionless temperature ) (θ - θ0)/∆θ time, s velocity vectors, m/s ) (U, V) ) u/u0 molar volume in eq 16, m3/mol
Φ φ µ ν θ F σ τ Ψ
Greek Letters thermal diffusivity, m2/s expansion coefficient due to temperature or solute concentration, K-1, (kg-solute/kg-solution)-1 characteristic concentration difference due to evaporation, kg-solute/kg-solution characteristic temperature difference due to evaporation, K association factor in eq 16 colatitude direction, rad viscosity, Pa · s kinematic viscosity, m2/s temperature, K fluid density, kg/m3 surface tension, N/m dimensionless time ) t/t0 contact angle, degree
0 A av B c m p s sat vap
Subscripts initial or reference value solute average solvent contact line or solute mass tracer particle substrate saturated vapor
R β ∆c ∆θ
LA801176Y
