Criterion for Reversal of Thermal Marangoni Flow in Drying Drops

Sep 18, 2009 - h(r), where TSL(r) and Ti(r) are the temperature at the so- lid-liquid interface and at the vapor-liquid interface respec- tively. Cons...
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Criterion for Reversal of Thermal Marangoni Flow in Drying Drops Xuefeng Xu, Jianbin Luo,* and Dan Guo State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China Received July 21, 2009. Revised Manuscript Received September 2, 2009 The thermal Marangoni flow induced by nonuniform surface temperature has been widely invoked to interpret the deposition pattern from drying drops. The surface temperature distribution of a drying droplet, although being crucial to the Marangoni flow, is still controversial. In this paper, the surface temperature in the drop central region is analyzed theoretically based on an asymptotic analysis on the heat transfer in such region, and a quantitative criterion is established for the direction of the surface temperature gradient and the direction of the induced Marangoni flow of drying drops. The asymptotic analysis indicates that these two directions will reverse at a critical contact angle, which depends not only on the relative thermal conductivities of the substrate and liquid, but also on the ratio of the substrate thickness to the contact-line radius of the droplet. The theory is corroborated experimentally and numerically, and may provide a potential means to control deposition patterns from drying droplets.

1. Introduction During droplet evaporation, the nonuniform temperature along the vapor-liquid interface usually generates a thermal Marangoni flow inside the drop.1-13 The Marangoni flow significantly affects the flow field and the deposit pattern of drying drops,5-8,14-16 and thus influences processes such as self-assembly17,18 and printing.19-22 Despite it being crucial to the Marangoni flow pattern, the surface temperature distribution of drying droplets is still controversial.4-7,14,23-25 Deegan et al.14 conjectured that the apex of the droplet ought to be coolest due to its longer conduction distance from the substrate, and then a radially inward surface flow should occur. Contrarily, Steinchen and Sefiane4 assumed *Corresponding author. E-mail: [email protected].

(1) Savino, R.; Fico, S. Phys. Fluids 2004, 16, 3738–3754. (2) Xu, X. F.; Luo, J. B. Appl. Phys. Lett. 2007, 91, 124102. (3) Girard, F.; Antoni, M.; Faure, S.; Steinchen, A. Langmuir 2006, 22, 11085– 11091. (4) Steinchen, A; Sefiane, K. J. Non-Equilib. Thermodyn. 2005, 30, 39–51. (5) Hu, H.; Larson, R. G. Langmuir 2005, 21, 3972–3980. (6) Hu, H.; Larson, R. G. J. Phys. Chem B 2006, 110, 7090–7094. (7) Ristenpart, W. D.; Kim, P. G.; Domingues, C.; Wan, J.; Stone, H. A. Phys. Rev. Lett. 2007, 99, 234502. (8) Barash, L. Yu.; Bigioni, T. P.; Vinokur, V. M.; Shchur, L. N. Phys. Rev. E 2009, 79, 046301. (9) Ward, C. A.; Duan, F. Phys. Rev. E 2004, 69, 056308. (10) Das, K. S.; Ward, C. A. Phys. Rev. E 2007, 75, 065303(R). (11) Duan, F.; Ward, C. A. Langmuir 2009, 25, 7424–7431. (12) Duan, F. J. Phys. D: Appl. Phys 2009, 42, 102004. (13) Murisic, N.; Kondic, L. Phys. Rev. E 2008, 78, 065301(R). (14) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Phys. Rev. E 2000, 62, 756–765. (15) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827–829. (16) Deegan, R. D. Phys. Rev. E 2000, 61, 475–485. (17) Narayanan, S.; Wang, J.; Lin, X. M. Phys. Rev. Lett. 2004, 93, 135503. (18) Schnall-Levin, M.; Lauga, E.; Brenner, M. P. Langmuir 2006, 22, 4547– 4551. (19) Kim, D.; Jeong, S.; Park, B. K.; Moon, J. Appl. Phys. Lett. 2006, 89, 264101. (20) Calvert, P. Chem. Mater. 2001, 13, 3299–3305. (21) Park, J.; Moon, J. Langmuir 2006, 22, 3506–3513. (22) Kawase, T.; Sirringhaus, H.; Friend, R H.; Shimoda, T. Adv. Mater. 2001, 13, 1601–1605. (23) Girard, F.; Antoni, M. Langmuir 2008, 24, 11342–11345. (24) David, S.; Sefiane, K.; Tadrist, L. Colloids Surf., A: Physicochem. Eng. Aspects 2007, 298, 108–114. (25) Dunn, G. J.; Wilson, S. K.; Duffy, B. R.; David, S.; Sefiane, K. Colloids Surf., A: Physicochem. Eng. Aspects 2008, 323, 50–55.

1918 DOI: 10.1021/la902666r

that the edge of the droplet is colder because of strong evaporation there. Numerical simulations by Hu and Larson5 have suggested that, for water droplet with contact line radius of 1 mm on glass substrate of thickness 0.15 mm, the radial surface temperature gradient reverses direction at a critical contact angle of approximate 14. Meanwhile, many researchers have studied the influence of the substrate on the surface temperature distribution of drying droplets. Girard and Antoni23 numerically investigated the effect of the size of the heating substrate, and showed that the contact line is warmer (colder) than the apex for L S/R > 1 (L S/R < 1), where LS is the radius of the substrate heated region and R is the radius of the droplet contact line. David et al.24 and Dunn et al.25 reported that thermal conductivities of the liquid and the substrate have a significant effect on the temperature field of drying droplets. An asymptotic analysis by Ristenpart et al. 7 further indicated that the critical contact angle depends on the relative thermal conductivities of the substrate and the liquid. However, the theory by Ristenpart et al., 7 which is devised only for substrates of infinite thickness, is not so appropriate for finite thickness substrates. For example, it can not explain why, for water droplets on glass substrates, recirculation occurs in droplets with a smaller contact angle of about 102 rather than a bigger one of about 14.15 To the best of our knowledge, the influence of the substrate thickness on the droplet temperature, especially on the direction of the surface temperature gradient of the droplet, has never been reported. In the paper, we establish a quantitative criterion for the direction of surface temperature gradient and the direction of thermal Marangoni flow of drying droplets on a finite thickness substrate. An asymptotic analysis shows that both directions will reverse at a critical contact angle, which in turn depends not only on the relative thermal conductivities of the substrate and liquid, but also on the ratio of the substrate thickness to the contact line radius of the droplet. The prediction is corroborated experimentally using water and isopropanol droplets on glass substrates. Our results demonstrate a simple way to predict and control the flow field and the deposition of drying droplets.

Published on Web 09/18/2009

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2. Theory 2.1. Nondimensional Analysis for Temperature Field in Evaporating Drops. We consider a small, pinned, and slowly evaporating liquid droplet with contact angle of θ and contact line radius of R resting on a flat substrate of thickness hS (Figure 1a). The thermal conductivities of the substrate and the liquid are kS and kL, respectively. In the axisymmetric configuration it is convenient to choose cylindrical coordinates (r, z). The droplet shape can be regarded as a spherical cap due to small Bond number and capillary number, and thus the height of the droplet is h(r)=[(R2/sin2 θ) - r2]1/2 - R/tan θ. The temperature at the lower boundary of the substrate is assumed to be a constant T0. For the slowly evaporating droplet, the evaporation flux along the droplet surface can be well approximated by the simple form J(r)=J0(1 r2/R2)-λ, where λ=1/2 - θ/π, and the prefactor J0 depends on the contact angle, the saturation pressure, the relative humidity, and the vapor diffusivity.14,15,26 The nonuniform evaporation rate and the nonuniform path lengths for heat conduction lead to a nonuniform temperature distribution along the vapor-liquid interface and hence a nonuniform surface tension, which drives a thermal Marangoni flow.5-8 For most liquids, the surface tension-temperature coefficient β  ∂γ/∂Τ, where γ is the surface tension and T is the temperature, is negative. During evaporation, heat is transferred from the substrate to the liquid surface to compensate for the heat loss due to evaporation. The heat transfer can be justified as quasi-steady processes, and the convective heat transfer can be neglected compared to the conductive heat transfer.5,7,27,28 This means that the temperature is thus governed by Laplace’s equation r2Τ=0. Assuming that the heat conduction and convection in the air can be neglected, on the drop surface, the heat flux is equal to the latent heat flux of the phase change at the vapor-liquid interface, i.e., -kLrΤ 3 n=HJ(r), where H is the latent heat of evaporation, and n is the unit normal. The Laplace’s equation can be rewritten in a nondimensional form as ~ 2 T~ ¼ 0 r ~ 2 ¼ D2 þ where r D ~r 2

1 D ~r D~r

þ

ð1Þ

D2 , D2 ~z

ðT -T0 ÞkL ~r ¼ r=R, ~z ¼ z=R, and T~ ¼ HJ0 R The nondimensional boundary conditions for eq 1 are as follows: ~ r Þ, 0 e ~r e 1 ð2Þ ~ T~ L n ¼ ð1 -~r 2 Þ -ð1=2 -θ=πÞ at ~z ¼ hð~ -r 3 ~ T~ S n ¼ 0 at ~z ¼ 0, ~r > 1 r 3

ð3Þ

~ T~ S n at ~z ¼ 0, 0 e ~r e 1 ð4Þ ~ T~ L n ¼ kR r T~ L ¼ T~ S , r 3 3 T~ S ¼ 0 at ~z ¼ -hR ; -hR < ~z < 0, ~r f ¥

ð5Þ

~ =er(∂/∂~r) þ ez(∂/∂~ z), with er being the radial unit vector where r and ez the axial, T~L is the nondimensional temperature in the ~ r)=h(r)/R=[(1/sin2 θ) liquid and T~S is that in the substrate, h(~ 2 1/2 r~ ] - 1/tan θ, kR =kS/kL, and hR =hS/R. (26) Hu, H.; Larson, R. G. J. Phys. Chem. B 2002, 106, 1334–1344. (27) Hu, H.; Larson, R. G. Langmuir 2005, 21, 3963–3971. (28) Girard, F.; Antoni, M.; Sefiane, K. Langmuir 2008, 24, 9207–9210.

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Figure 1. (a) A sessile spherical-cap droplet on a flat substrate in a cylindrical coordinate system with radial coordinate r and axial coordinate z. (b) The heat transfer in the immediate vicinity of the symmetry axis of the drying droplet. As described in the text, the heat transfer in this region can be considered as one-dimensional.

It can be seen that the above nondimensional equations are governed by three parameters: kR, hR, and θ. For a drop with a given geometry and given thermal properties, the temperature field inside the drop can be determined by both the relative thermal conductivities kR and the relative thickness hR. It implies that to give an appropriate criterion for the direction of surface temperature gradient of drying drops on finite thickness substrates, the influence of the substrate thickness must be taken into account. 2.2. Asymptotic Approach in the Drop Central Region. Since it is not easy to obtain an analytic solution for the above equations, asymptotic approaches are used to find a criterion for the surface temperature gradient direction of evaporating drops. Ristenpart et al.7 employ an asymptotic methodology to examine the heat transfer in the immediate vicinity of the contact line, and establish a quantitative criterion for the case of infinite substrate. However, the approach is not so suitable for a finite substrate because the temperature of the substrate lower surface is difficult to include in the model as a boundary condition. It is well-known that the drop surface is concave in the region very close to the contact line where the liquid merges gradually with a flat adsorbed film.29-32 The microscopic contact angle at the contact line would always be zero, and increases as the liquid film thickness increases and goes gradually to the macroscopic apparent contact angle. This microstructure of the drop in the region close to the contact line should be taken into account in the analysis on the heat transfer in such region,2 and will inevitably cause more difficulty. Contrarily, the microstructure in the droplet edge has negligible effect on the drop shape in the central region. Thus, examining the heat transfer in the central region is more convenient than doing so on the edge. Considering that the evaporation diverges at the contact line,14,15,26 the thermal gradients will be much larger in such region than in the drop central region. Therefore, a worry may arise as to whether an analysis focusing on the drop central region give an appropriate criterion for the direction of the temperature gradient along the whole drop surface. Fortunately, a numerical simulation by Hu and Larson5 showed that the temperature changes monotonously along the droplet surface, which means that the direction of the surface temperature gradient in the (29) Truong, J. G.; Wayner, P. C., Jr. J. Chem. Phys. 1987, 87, 4180–4188. (30) Sharma, A. Langmuir 1993, 9, 3580–3586. (31) Solomentsev, Y.; White, L. R. J. Colloid Interface Sci. 1999, 218, 122–136. (32) Gokhale, S. J.; Plawsky, J. L.; Wayner, P. C., Jr. Phys. Fluids 2004, 16, 1942–1955.

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Figure 2. The absolute ratio of the radial temperature gradient to the axial one in the region close to the droplet axis at contact angles of 10, 30, 50, 70, and 90.

central region is the same as that in the contact line region, and the same as along the whole drop surface. Although a stagnation point indicating a reversal of the surface temperature gradient may exist when considering the microstructure of the drop,2 the temperature profiles obtained by Hu and Larson5 will still hold on the droplet surface except in the reason very close to the contact line because the microstructure would have little effect on the heat transfer in the region far away from the contact line. Thus, when the contact line radius is much larger than the distance between the stagnation point and the contact line (≈8 μm),2 it is still reasonable to assume a monotonous temperature profile along almost the whole surface. In our analysis, the direction of the surface temperature gradient in the region close to the symmetry axis is first obtained by an asymptotic analysis on the heat transfer in the immediate vicinity of the axis. Under the assumption that the surface temperature changes monotonously, which was verified by Hu and Larson,5 a criterion for the direction of the temperature gradient along the whole drop surface and consequently for the Marangoni flow inside the drop is then acquired. 2.3. Expression for Surface Temperature in the Drop Central Region. The heat transfer in the region very close to the axis is sketched in Figure 1b. At the axis, the radial conductive heat flux must be zero because of the axisymmetry. Numerical result shows that, in the region where |r/R|,1, the radial heat flux can be neglected compared to the axial one (Figure 2). So, in such region, it is reasonable to assume that the conductive heat flux vector is perpendicular to the substrate surface, i.e., the heat transfer can be considered as one-dimensional (Figure 1b). Thus, the conductive heat flux in the region near the axis can be expressed as Qcond(r)=kS[T0 - TSL(r)]/hS =kL [TSL(r) - Ti(r)]/ h(r), where TSL(r) and Ti(r) are the temperature at the solid-liquid interface and at the vapor-liquid interface respectively. Considering that the conductive heat flux Qcond and the latent heat flux of the phase change at the vapor-liquid interface Qevap are balanced, i.e., Qcond(r)=Qevap(r)=HJ0(1 - r2/R2)-λ[1 þ (dh/dr)2]1/2, the surface temperature can be given by ! -λ ! -1=2 HJ0 R r2 r2 sin2 θ 1- 2 1Ti ðrÞ ¼ T0 kL R2 R 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 r 1 @ þ RN A - sin2 θ R2 tan θ 1920 DOI: 10.1021/la902666r

ð6Þ

Figure 3. The percent difference between the dimensionless surface temperature T~i calculated with the approximate form given in eq 6 and T~i,Calc from numerical calculations in the drop central region at contact angles of 10, 30, 50, 70, and 90. The parameters used here are as follows: kR =1.58 for water droplets on glass substrates, hR=0.15, and hence RN ≈ 0.095.

where RN  hR/kR  hSkL/RkS. We further define the dimensionless surface temperature as ðTi ðrÞ -T0 ÞkL ¼ -ð1 -~r 2 Þ -λ ð1 -~r 2 sin2 θÞ -1=2 T~i ð~r Þ ¼ HJ0 R ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 þ RN -~r ð7Þ tan θ sin2 θ The percent difference between the value of T~i given in eq 7 and that from numerical calculations in the drop central region is plotted in Figure 3. Considering that the temperature drop at the interface due to evaporation is often a tiny value, the figure indicates that the analytical results from eq 7 are reasonably consistent with the finite element result in a quite wide range in the drop central region, especially for the smaller contact angles. So, eq 6 and eq 7, despite their simple origin, give an approximate value for the surface temperature in the drop central region. As also can be seen below, the direction of the surface temperature gradient predicted by our theory is consistent well with the numerical results. The consistency of the theory with the numerical calculations corroborates the validity of the asymptotic method presented here. It should be noted that there are some limitations to the above derivation. To ensure that the evaporation flux along the droplet surface can be well approximated by the simple form mentioned above, the effect of the buoyant convection of the liquid vapor on the evaporation rate33,34 is neglected here. Furthermore, to reduced the evaporative cooling effect,25,33 an assumption is implicitly adopted that the surface temperature drop |Ti - T0| due to evaporation is a small value compared with T0. Consequently, the condition hS/kS , T0/HJ0 must be satisfied. This means that the theory given in this paper is only suitable for finite thickness substrates.

(33) Dunn, G. J.; Wilson, S. K.; Duffy, B. R.; David, S.; Sefiane, K. J. Fluid Mech. 2009, 623, 329–351. (34) Shahidzadeh-Bonn, N.; Rafai, S.; Azouni, A.; Bonn, D. J. Fluid Mech. 2006, 549, 307–313.

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3. Results and Discussions 3.1. A Criterion for Direction of Surface Temperature Gradient. From eq 6, we can also obtain dTi ðrÞ j ¼0 dr r ¼0

ð8Þ

d2 Ti ðrÞ HJ0 ð2λ þ sin2 θÞ½RðθÞ -RN  j ¼ dr2 r ¼0 kL R

ð9Þ

where R(θ)=[sin 2θ - 4λ tan(θ/2)]/(4λ þ 2 sin2 θ). Given that 0 < θ < π/2, the inequality (2λ þ sin2 θ)HJ0/kLR > 0 is satisfied. So, (1) if RN < R(θ), d2Ti(r)/dr2|r=0 > 0. Considering that dTi(r)/ dr|r=0 =0, the surface temperature Ti(r) has a relative minimum value at r=0 and thus increases with distance from the top of the droplet. For β < 0, the resulting Marangoni flow is directed radially inward along the vapor-liquid interface. (2) if RN > R(θ), d2Ti(r)/dr2|r=0 < 0. Ti(r) has a relative maximum value at r=0 and decreases with distance from the droplet top, and the surface Marangoni flow reverses its direction. From the above derivation, it is obvious that the criterion given here can be easily changed to bePsuitable for the multilayer substrates by simply substituting RNi for RN, where RNi  hSikL/RkSi, hSi, and kSi are the thickness and the thermal conductivity for substrate layer i, respectively. To understand the result, it is reasonable to think in terms of the relative influences of the nonuniform evaporation rate and the nonuniform heat conduction path on the surface temperature profile. For large RN, hS/kS . h(r)/kL, and then Τi(r) ≈ Τ0 Qcond(r)hS/kS. Because hS/kS is constant, the surface temperature is then determined by the distribution of evaporation rate, and the drop will be coolest at the edge because of higher evaporation rate there. Contrarily, in the case of small RN, i.e., hS/kS , h(r)/kL, the surface temperature can be approximately described as Τi(r) ≈ Τ0 - Qcond(r)h(r)/kL. In such situation, the conduction path lengths may be dominant, and thus the top of the droplet ought to be coolest due to its longer conduction distance from the substrate. The function RN=R(θ) is plotted in Figure 4a. For values of RN above (below) the curve, the surface temperature decreases (increases) with distance from the top of the droplet, and the induced Marangoni flow is directed radially outward (inward) along the vapor-liquid interface. It can be deduced from the theory that, when the substrate is isothermal (i.e., RN = 0), the apex of the droplet ought to be coolest, which corroborates the conjecture by Hu et al.6 and Deegan et al.14 and the simulation works by Savino et al.1 and Girard et al.3 For a water droplet with a contact line radius of 1 mm on a glass substrate with a thickness of 0.15 mm (kR ≈ 1.58), we can obtain that the critical contact angle θCrit at which the surface temperature gradient reverses direction is about 9 by setting RN=R(θCrit)=0.095. This result is more consistent with the numerical works by Hu and Larson,5 who find that θCrit =14 (squares, Figure 4), than the theory by Ristenpart et al.,7 who predict that θCrit=31. 3.2. Comparison with the Theory by Ristenpart et al. Here, we will compare our theory with the one by Ristenpart et al.7 In our theory, if the value of hR is given, the critical contact angle can be determined only by the value of kR. The function kR=hR/R(θ) for a given value of hR is shown in Figure 4b. Similar to the theory by Ristenpart et al.,7 our theory indicates that the surface temperature increases (decreases) with distance from the top of the droplet for values of kR above (below) the curve, and also shows that a lower limit for kR exists below which the drop is Langmuir 2010, 26(3), 1918–1922

Figure 4. The directions of the surface temperature gradient and the induced surface Marangoni flow of evaporating droplets. Regions above and below the lines correspond to directions sketched in the respective insets. (a) The solid line is RN = R(θ). Open symbols: observed direction of surface Marangoni flow consistent with temperature decreasing with distance from the top. Filled symbols: observed direction of surface Marangoni flow consistent with temperature increasing with distance from the top. Diamonds: experimental observations for water on glass of thickness 1 mm. Triangles: experimental observations for water on glass of thickness 0.15 mm. Circles: experimental observations for isopropanol on glass of thickness 1 mm. Squares: numerical calculations by Hu and Larson for water on glass of thickness 0.15 mm.5 (b) The solid line is kR = hR/R(θ) for a given value of hR. The experimental results can not be depicted in one such figure because liquid droplets in the experiments have different values of hR.

warmest at the top (Figure 4b). The difference between these two theories is that the lower limit value for kR in our theory is not a constant as in the theory by Ristenpart et al., but dependent on the value of hR. Besides, unlike the prediction by Ristenpart et al., the theory here indicates that the upper limit for kR above which the drop is coldest at the top is nonexistent. This point of view is partly corroborated by our experiments which show that, even for isopropanol droplets on glass substrate (kR ≈ 7.1), inward surface temperature gradient and thus outward surface Marangoni flow appear (circles, Figure 4a). For an infinite thickness substrate, RN f ¥, and thus, according to our theory, the apex of the droplet will be warmest and the direction of the surface temperature gradient is independent of kR. Therefore, the criterion given here does not reduce to the one by Ristenpart et al. in the limit of infinite substrate thickness. The differences between these two theories are just because that our theory is devised only for finite DOI: 10.1021/la902666r

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thickness substrates and the theory by Ristenpart et al. only for infinite thickness substrates. It would be nice to develop also a model in the region near contact line in the spirit of Ristenpart et al. with a finite thickness substrate to compare with our predictions. 3.3. Experimental Corroboration. To further corroborate the theory, a series of experiments with deionized water and isopropanol droplets on glass substrates with thickness of 1 mm and 0.15 mm were performed. To trace the flow, 2 μm polystyrene fluorescent particles were suspended at concentration of about 0.1% (w/v) in the liquids. 4, 2, and 1 μL droplets of the liquids were deposited on the glass and allowed to dry, and the flow inside droplets was observed by a fluorescent microscope. The directions of the surface temperature gradient of the drops consistent with the observed Marangoni flow inside the drops are depicted in Figure 4a. It can be seen from Figure 4a that the observed directions of the surface temperature gradient of evaporating drops are well predicted by the criterion given in this paper. On Figure 4a, some data points stand out of the boundary curve. One possible reason for these inconsistent data may be the neglect of the buoyant convection of the liquid vapor. Another reason is possibly because of the uncertainty in determining the value of the contact line radius and the value of the contact angle in the experiments. For some water drops on glass substrates with almost the same contact angel, the same value of kR, and different values of hR, the directions of the surface temperature gradient are different. This indicates that the critical contact angle, at which the surface temperature gradient reverses its direction, depends not only on the relative thermal conductivities of the substrate and liquid, but also on the ratio of the substrate thickness to the contact-line radius of the droplet. For isopropanol droplets on glass substrate with infinite thickness (kR ≈ 7.1), according to the theory by Ristenpart et al., the drops should be always coldest at the top and surface Marangoni flows are always inward. However, this is not the case for finite thickness substrates. Our experiments shows that inward surface temperature gradient and thus outward

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surface Marangoni flow appear in the isopropanol droplets on glass substrate (circles, Figure 4a). In isopropanol droplets, the recirculatory flow appeared after tens of seconds of chaotic motion, which may result from Benard-Marangoni instability.7 Unlike isopropanol droplets, when an outward surface Marangoni flow was induced in water droplets, no recirculation occurred, i.e., all the liquid flowed outward toward the edge. It is consistent with the numerical result for water droplets by Hu and Larson.5 The reason is possibly because that the Marangoni number Ma in water is much smaller than in volatile liquids, and may be further reduced by surfactant contaminants.5,6

4. Conclusion We have performed an asymptotic analysis on the heat transfer in the central region of an evaporating drop on a finite thickness substrate. The surface temperature in such region was first obtained. Under the assumption that the temperature changes monotonously along the droplet surface, which was numerically verified by Hu et al.,5 a quantitative criterion was then obtained in the present paper to determine the direction of the surface temperature gradient and then the direction of the induced surface Marangoni flow. The asymptotic analysis showed that these two directions will reverse at a critical contact angle, which in turn depends not only on the relative thermal conductivities of the substrate and the liquid, but also on the ratio of the substrate thickness to the contact-line radius of the droplet. The theory is corroborated experimentally and numerically. Despite of its simple origin, the present theory may serve as a first attempt to interpret the influence of the substrate thickness on the direction of the surface temperature gradient of drying drops, and may provide a potential way to predict and control the flow field and the deposition of drying droplets. Acknowledgment. The work is financially supported by the National Key Basic Research Program of China and the Natural Science Foundation of China (Grant 50721004).

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