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The capillary outward flow inside pinned drying droplets
International Journal of Heat and Mass Transfer 83 (2015) 307–310
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
The capillary outward flow inside pinned drying droplets Doudou Huang a, Liran Ma b, Xuefeng Xu a,⇑ a b
School of Technology, Beijing Forestry University, Beijing 100083, China State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 2 June 2014 Received in revised form 27 November 2014 Accepted 3 December 2014
Keywords: Evaporating droplet Flow field Marangoni effect Particle deposition
a b s t r a c t In this paper, a theoretical model is introduced and an analytical expression is obtained for the heightaveraged velocity of the outward flow inside thin drying droplets for the case the convective Marangoni flow exists. The velocity is also measured by tracking the motion of fluorescent microspheres in the drying droplets. The experimental observations show that a region of constant height-averaged velocity exists and the reciprocal of the velocity decreases nearly linearly with time over the pinning stage. Such variances of the velocity are different from the previous studies and are well explained by the present model. Furthermore, analytical expressions are established for the growth rate of the deposit ring and for the total ring mass upon drying. The results are likely to enrich the knowledge about the evaporating droplets and may provide a potential means to predict and control the evaporation-driven deposition and assembly of colloids and other materials. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction A drying droplet often leaves solute particles on the substrate, forming various deposition patterns [1–7]. This phenomenon has been used as the basis of a wide range of industrial and scientific processes such as DNA/RNA mapping [8] and ink-jet printing [9]. Controlling the distribution of the deposition is a vital problem in the processes. During drying, the accumulation of the solid solute at the droplet edge will pin the contact line [2,10]. A capillary outward flow will be induced inside the pinned droplet, and can carry almost all the dispersed solute to the edge to form a ring-like deposit [1,2]. Deegan et al. [1,2] analyzed the height-averaged velocity u dðp2hÞ=ð2p2hÞ , where d of the outward flow and showed that u is the radial distance from the droplet edge and h is the contact angle. Hu and Larson [11] found that, within a sessile droplet, a convective Marangoni flow can be generated which deposits the solute particles preferentially at the droplet center rather than at the edge [3,5,6,12]. Xu et al. [13–15] further showed that both the capillary outward flow and the convective Marangoni flow may appear inside a pinned droplet (also see the inlets of Figs. 1 and 2). While the outward flow carries the solute particles toward the contact line and creates the ring-like deposit, the convective ⇑ Corresponding author at: Box 8#, Beijing Forestry University, No. 35 Tsinghua East Road, Haidian District, Beijing 100083, China. Tel.: +86 10 62338153; fax: +86 10 62338142. E-mail address: [email protected] (X. Xu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.12.018 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
flow directs the solute inward and produces a more uniform deposition. Despite the extensive investigations on the flow field of drying droplets [1–4,11–17], detailed observation and analysis of the outward flow and of its effects on the particle deposition are still lacking. In the present paper, a theoretical model of the outward flow is introduced for the case the convective Marangoni flow exists, and an analytical expression is obtained for the height-averaged velocity of the outward flow. By using fluorescent colloidal particles as tracer particles, the outward flow has been measured and found consistent with the theoretical predictions. Furthermore, the influences of the contact line pinning and the Marangoni effect on the deposition distribution are discussed.
2. Theory Here, we consider a small, pinned, and slowly evaporating liquid droplet with contact angle of h and contact-line radius of R resting on a flat substrate (Fig. 1). The evaporation flux along the droplet surface can be well approximated by a simple form k JðrÞ ¼ J 0 ð1 r2 =R2 Þ , where r is the distance from the axis of the droplet, and k = (1/2 h/p) [1,2,18]. From Fig. 1, it can be seen that the evaporation loss from the droplet surface beyond the stagnation point is compensated by the outward flow. Thus, the height-averaged velocity of the outward flow in the region within the stagnation point can be expressed as
308
D. Huang et al. / International Journal of Heat and Mass Transfer 83 (2015) 307–310
Fig. 1. Sketch of the flow structure inside a pinned droplet resting on a flat substrate. A cylindrical coordinate system with radial coordinate r and axial coordinate z is chosen.
ðt; rÞ ¼ u
_ mðtÞ 2pqrHðr; tÞ
ð1Þ
where q is the density of the liquid, H(r, t) is the height of the out_ ward flow at the radial position r, and mðtÞ is the evaporation rate from the surface beyond the stagnation point, which is
r 2 k _ 2prJ0 1 mðtÞ ¼ R RL Z
R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 @hðr; tÞ 1þ dr @r
_ ~L ~L2 Þ1KðhÞ MðtÞð2
ð2Þ
where h(r, t) is the height of the drop surface, L(t) is the radial distance between the stagnation point and the contact line, ~L ¼ LðtÞ=R, _ MðtÞ is the evaporation rate from the whole droplet surface, and KðhÞ ¼ 0:2239ðh p=4Þ2 þ 0:3619 [18]. For thin droplets, the height of the outward flow at the stagnation point HðtÞ ¼ LðtÞ tanðhðtÞÞ LðtÞhðtÞ. Because L is often much shorter than R, the height-averaged velocity at the stagnation point can be approximated by
ðtÞ u
_ mðtÞ 2pqRLðtÞhðtÞ
ð3Þ
_ is almost For the thin pinned droplets, the evaporation rate MðtÞ a constant, and as a result, the contact angle varies nearly linearly with the time [2,16–18]. Thus,
qðV 0 V d Þ pqR3 ðh0 hd Þ _ MðtÞ ¼ td 4td
ð4Þ
Fig. 2. Schematic diagram of the experimental apparatus for observing the outward flow.
glass slice and are allowed to dry in a large enclosure. The room temperature is about 27 °C and the relative humidity is about 20%. During drying, an Olympus BX51 microscope (100) and an Andor DU897 CCD camera are used to observe and record the tracer particles at a time interval of about 0.5 s. In order not to alter significantly the viscosity and the flow field of the fluid, the suspension is dilute (the volume fraction is about 104) and thus the inter-particle interactions can be neglected. The inertia effect is also insignificant for the micro-scale particles. During observation, the Brownian drift of the 1-lm microspheres is about 2 lm, which yields a velocity variance of about 0.5 lm s1 [19].
and
hðtÞ ¼ ðh0 hd Þðt d tÞ=td þ hd
ð5Þ
where V0 is the initial droplet volume, h0 is the initial contact angle, Vd and hd are the droplet volume and the contact angle respectively when depinning of the contact line occurs, and td is the time at which the depinning occurs. Considering that KðhÞ 1=2 for the thin droplets, and inserting Eqs. (2), (4), (5) into Eq. (3), we have
ðtÞ u
1=2 2 Rðh0 hd Þ 1 ~L 8½ðh0 hd Þðt d tÞ þ t d hd
ð6Þ
and
ðtÞ1 u
8 ð2=~L 1Þ
1=2
R
ðt d tÞ þ
t d hd ðh0 hd Þ
ð7Þ
3. Experiments In the experiments, de-ionized water droplets dispersed with 1-lm fluorescent microspheres are deposited on the surface of a
4. Results and discussion 4.1. Spatial variation of the outward flow The experimental observations show that the outward flow inside the pinned drying droplets can be divided into two parts, namely, an accelerated flow region which is adjacent to the contact line and a constant flow region which locates farther away from the contact line (Fig. 3). This velocity distribution can be explained by the shape of the outward flow. In the region beyond the stagnation point, the outward flow region is wedge-shaped (Fig. 1). According to the theory of Deegan et al. [1,2], the height-averaged velocity in such region will increase as the contact line is approached. Contrarily, the height of the outward flow is almost constant in the region within the stagnation point [14], which results in a nearly constant height-averaged velocity in such region (see Eq. (1)). From Fig. 3, the distance L can be estimated from the position of the dividing point between these two regions. This yields a value of about 25 lm, which is consistent with the observed value of about 20 lm (see Fig. 2).
D. Huang et al. / International Journal of Heat and Mass Transfer 83 (2015) 307–310
309
gan et al. [1,2], the present observations show that a constant velocity region of the outward flow exists. This difference is just are not the same. In the Deegan because that the definitions of u is calculated by integrating the radial velocity comet al.’s theory u ponent of the liquid with respect to z from 0 to the drop height h. is computed by integrating from However, in the present model, u 0 to the height H of the outward flow (see Fig. 1). Deegan et al. [1,2] predicted that the velocity of the outward flow diverges at the end of the drying. In the present paper, the Eq. (6) shows that the velocity remains finite at the depinning time. However, if the contact line is pinned over the whole drying time (i.e., hd = 0 and td = tf where tf is the total drying time), Eq. (6) will ðtÞ ðtf tÞ1 , which is the same as in the theory by reduce to u Deegan et al. 4.4. Influences of contact line pinning and Marangoni effect on deposition distribution vs the distance to the contact line d. The Fig. 3. The height-averaged velocity u volume of the droplets is about 1 lL. The initial contact angle is about 19°, 22°, and 24° for droplet A, B, and C respectively. The solid lines are the best linear fits to the data points within the stagnation point.
4.2. Temporal variation of the outward flow 1 is to good The reciprocal of the height-averaged velocity u approximation linear in time (Fig. 4). This means that the prefactor of (td t) in Eq. (7) is almost a constant, which in turn indicates that the value of L is nearly time-independent over the pinning stage. By substituting the measured values of R, td, and h0 into Eq. (7), and then comparing with the fitted linear function in Fig. 4, the distance L and the contact angle hd can be calculated. The computed L are 24, 10, and 17 lm for droplet A, B, and C, respectively, which are close to the values obtained in Section 4.1. The computed hd are 2.97°, 5.74° and 4.62° for droplet A, B, and C, respectively, which are also consistent with the value of 2–4° measured by Hu and Larson [18]. These consistencies may corroborate the validity of the present model. 4.3. Comparison with the theory by Deegan et al. Different from the power law increase of the height-averaged velocity with decreasing distance to the contact line given by Dee-
When pinned, the rate of increase in the ring mass is approximately proportional to the evaporation rate from the droplet surface beyond the stagnation point and can be expressed as 1=2 _ r ¼ m0 ð1 hd =h0 Þð2~L ~L2 Þ =td m
ð8Þ
where m0 is the total mass of solute particle present initially in the droplet. Therefore, the total mass of the deposit ring after the droplet is dried is 1=2 _ r t d ¼ m0 ð1 hd =h0 Þð2~L ~L2 Þ mrt ¼ m
ð9Þ
Eq. (9) indicates that the contact line pinning and the Marangoni effect have significant influences on the growth of ring. A strong contact line pinning and a weak Marangoni effect will result in a dense ring-like deposit along the droplet edge, and contrarily, a weak contact line pinning and a strong Marangoni effect will generate a more uniform deposit. 5. Conclusions In summary, we have investigated experimentally and theoretically the outward flow inside pinned liquid droplets for the case the convective Marangoni flow exists. The experimental observations showed that a constant height-averaged velocity region of the outward flow exists and the reciprocal of the velocity is nearly linear in time over the pinning stage. An analytical model for the outward flow has also been proposed to explain the variations of the velocity. Further analysis on the growth of the deposit ring indicated that both the contact line pinning and the Marangoni effect have significant influences on the distribution of the deposit. Conflict of interest None declared. Acknowledgments The work is financially supported by the National Natural Science Foundation of China (Grant No. 51275050), the Program for New Century Excellent Talents in University (Grant No. NCET-120786), and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20120014120017). References
1 vs the time to depinning, Fig. 4. The reciprocal of the height-averaged velocity u td t. The solid lines are the best fit to a linear function. The deviation of the data points from the line may result from the Brownian motion and the positioning error.
[1] R.D. Deegan et al., Capillary flow as the cause of ring stains from dried liquid drops, Nature 389 (1997) 827–829. [2] R.D. Deegan et al., Contact line deposits in an evaporating drop, Phys. Rev. E 62 (2000) 756–765.
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D. Huang et al. / International Journal of Heat and Mass Transfer 83 (2015) 307–310
[3] W.D. Ristenpart, P.G. Kim, C. Domingues, J. Wan, H.A. Stone, Influence of substrate conductivity on circulation reversal in evaporating drops, Phys. Rev. Lett. 99 (2007) 234502. [4] G. Berteloot et al., Evaporation of a sessile droplet: inside the coffee stain, J. Colloid Interface Sci. 370 (2012) 155–161. [5] Y.F. Li, Y.J. Sheng, H.K. Tsao, Evaporation stains: suppressing the coffee-ring effect by contact angle hysteresis, Langmuir 29 (2013) 7802–7811. [6] Y.C. Hu, Q. Zhou, H.M. Ye, Y.F. Wang, L.S. Cui, Peculiar surface profile of poly(ethylene oxide) film with ring-like nucleation distribution induced by Marangoni flow effect, Colloids Surf. A: Physicochem. Eng. 428 (2013) 39–46. [7] A. Crivoi, F. Duan, Amplifying and attenuating the coffee-ring effect in drying sessile nanofluid droplets, Phys. Rev. E 87 (2013) 042303. [8] J.P. Jing et al., Automated high resolution optical mapping using arrayed, fluidfixed DNA molecules, Proc. Natl. Acad. Sci. U.S.A. 95 (1998) 8046–8051. [9] H. Sirringhaus et al., High-resolution inkjet printing of all-polymer transistor circuits, Science 290 (2000) 2123–2126. [10] Y.F. Li, Y.J. Sheng, H.K. Tsao, Solute concentration-dependent contact angle hysteresis and evaporation stains, Langmuir 30 (2014) 7716–7723.
[11] H. Hu, R.G. Larson, Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet, Langmuir 21 (2005) 3972–3980. [12] K. Zhang, L.R. Ma, X.F. Xu, J.B. Luo, D. Guo, Temperature distribution along the surface of evaporating droplets, Phys. Rev. E 89 (2014) 032404. [13] X.F. Xu, J.B. Luo, D. Guo, Criterion for reversal of thermal Marangoni flow in drying drops, Langmuir 26 (2010) 1918–1922. [14] X.F. Xu, J.B. Luo, Marangoni flow in an evaporating water droplet, Appl. Phys. Lett. 91 (2007) 124102. [15] X.F. Xu, J.B. Luo, D. Guo, Radial-velocity profile along the surface of evaporating liquid droplets, Soft Matter 8 (2012) 5797–5803. [16] Y. Hamamoto, J.R.E. Christy, K. Sefiane, Order-of-magnitude increase in flow velocity driven by mass conservation during the evaporation of sessile drops, Phys. Rev. E 83 (2011) 051602. [17] F. Girard, M. Antoni, K. Sefiane, On the effect of Marangoni flow on evaporation rates of heated water drops, Langmuir 24 (2008) 9207–9210. [18] H. Hu, R.G. Larson, Evaporation of a sessile droplet on a substrate, J. Phys. Chem. B 106 (2002) 1334–1344. [19] X.F. Xu, J.B. Luo, J. Yan, A PIV system for two-phase flow with nanoparticles, Int. J. Surf. Sci. Eng. 2 (2008) 168–175.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
The capillary outward flow inside pinned drying droplets Doudou Huang a, Liran Ma b, Xuefeng Xu a,⇑ a b
School of Technology, Beijing Forestry University, Beijing 100083, China State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 2 June 2014 Received in revised form 27 November 2014 Accepted 3 December 2014
Keywords: Evaporating droplet Flow field Marangoni effect Particle deposition
a b s t r a c t In this paper, a theoretical model is introduced and an analytical expression is obtained for the heightaveraged velocity of the outward flow inside thin drying droplets for the case the convective Marangoni flow exists. The velocity is also measured by tracking the motion of fluorescent microspheres in the drying droplets. The experimental observations show that a region of constant height-averaged velocity exists and the reciprocal of the velocity decreases nearly linearly with time over the pinning stage. Such variances of the velocity are different from the previous studies and are well explained by the present model. Furthermore, analytical expressions are established for the growth rate of the deposit ring and for the total ring mass upon drying. The results are likely to enrich the knowledge about the evaporating droplets and may provide a potential means to predict and control the evaporation-driven deposition and assembly of colloids and other materials. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction A drying droplet often leaves solute particles on the substrate, forming various deposition patterns [1–7]. This phenomenon has been used as the basis of a wide range of industrial and scientific processes such as DNA/RNA mapping [8] and ink-jet printing [9]. Controlling the distribution of the deposition is a vital problem in the processes. During drying, the accumulation of the solid solute at the droplet edge will pin the contact line [2,10]. A capillary outward flow will be induced inside the pinned droplet, and can carry almost all the dispersed solute to the edge to form a ring-like deposit [1,2]. Deegan et al. [1,2] analyzed the height-averaged velocity u dðp2hÞ=ð2p2hÞ , where d of the outward flow and showed that u is the radial distance from the droplet edge and h is the contact angle. Hu and Larson [11] found that, within a sessile droplet, a convective Marangoni flow can be generated which deposits the solute particles preferentially at the droplet center rather than at the edge [3,5,6,12]. Xu et al. [13–15] further showed that both the capillary outward flow and the convective Marangoni flow may appear inside a pinned droplet (also see the inlets of Figs. 1 and 2). While the outward flow carries the solute particles toward the contact line and creates the ring-like deposit, the convective ⇑ Corresponding author at: Box 8#, Beijing Forestry University, No. 35 Tsinghua East Road, Haidian District, Beijing 100083, China. Tel.: +86 10 62338153; fax: +86 10 62338142. E-mail address: [email protected] (X. Xu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.12.018 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.
flow directs the solute inward and produces a more uniform deposition. Despite the extensive investigations on the flow field of drying droplets [1–4,11–17], detailed observation and analysis of the outward flow and of its effects on the particle deposition are still lacking. In the present paper, a theoretical model of the outward flow is introduced for the case the convective Marangoni flow exists, and an analytical expression is obtained for the height-averaged velocity of the outward flow. By using fluorescent colloidal particles as tracer particles, the outward flow has been measured and found consistent with the theoretical predictions. Furthermore, the influences of the contact line pinning and the Marangoni effect on the deposition distribution are discussed.
2. Theory Here, we consider a small, pinned, and slowly evaporating liquid droplet with contact angle of h and contact-line radius of R resting on a flat substrate (Fig. 1). The evaporation flux along the droplet surface can be well approximated by a simple form k JðrÞ ¼ J 0 ð1 r2 =R2 Þ , where r is the distance from the axis of the droplet, and k = (1/2 h/p) [1,2,18]. From Fig. 1, it can be seen that the evaporation loss from the droplet surface beyond the stagnation point is compensated by the outward flow. Thus, the height-averaged velocity of the outward flow in the region within the stagnation point can be expressed as
308
D. Huang et al. / International Journal of Heat and Mass Transfer 83 (2015) 307–310
Fig. 1. Sketch of the flow structure inside a pinned droplet resting on a flat substrate. A cylindrical coordinate system with radial coordinate r and axial coordinate z is chosen.
ðt; rÞ ¼ u
_ mðtÞ 2pqrHðr; tÞ
ð1Þ
where q is the density of the liquid, H(r, t) is the height of the out_ ward flow at the radial position r, and mðtÞ is the evaporation rate from the surface beyond the stagnation point, which is
r 2 k _ 2prJ0 1 mðtÞ ¼ R RL Z
R
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 @hðr; tÞ 1þ dr @r
_ ~L ~L2 Þ1KðhÞ MðtÞð2
ð2Þ
where h(r, t) is the height of the drop surface, L(t) is the radial distance between the stagnation point and the contact line, ~L ¼ LðtÞ=R, _ MðtÞ is the evaporation rate from the whole droplet surface, and KðhÞ ¼ 0:2239ðh p=4Þ2 þ 0:3619 [18]. For thin droplets, the height of the outward flow at the stagnation point HðtÞ ¼ LðtÞ tanðhðtÞÞ LðtÞhðtÞ. Because L is often much shorter than R, the height-averaged velocity at the stagnation point can be approximated by
ðtÞ u
_ mðtÞ 2pqRLðtÞhðtÞ
ð3Þ
_ is almost For the thin pinned droplets, the evaporation rate MðtÞ a constant, and as a result, the contact angle varies nearly linearly with the time [2,16–18]. Thus,
qðV 0 V d Þ pqR3 ðh0 hd Þ _ MðtÞ ¼ td 4td
ð4Þ
Fig. 2. Schematic diagram of the experimental apparatus for observing the outward flow.
glass slice and are allowed to dry in a large enclosure. The room temperature is about 27 °C and the relative humidity is about 20%. During drying, an Olympus BX51 microscope (100) and an Andor DU897 CCD camera are used to observe and record the tracer particles at a time interval of about 0.5 s. In order not to alter significantly the viscosity and the flow field of the fluid, the suspension is dilute (the volume fraction is about 104) and thus the inter-particle interactions can be neglected. The inertia effect is also insignificant for the micro-scale particles. During observation, the Brownian drift of the 1-lm microspheres is about 2 lm, which yields a velocity variance of about 0.5 lm s1 [19].
and
hðtÞ ¼ ðh0 hd Þðt d tÞ=td þ hd
ð5Þ
where V0 is the initial droplet volume, h0 is the initial contact angle, Vd and hd are the droplet volume and the contact angle respectively when depinning of the contact line occurs, and td is the time at which the depinning occurs. Considering that KðhÞ 1=2 for the thin droplets, and inserting Eqs. (2), (4), (5) into Eq. (3), we have
ðtÞ u
1=2 2 Rðh0 hd Þ 1 ~L 8½ðh0 hd Þðt d tÞ þ t d hd
ð6Þ
and
ðtÞ1 u
8 ð2=~L 1Þ
1=2
R
ðt d tÞ þ
t d hd ðh0 hd Þ
ð7Þ
3. Experiments In the experiments, de-ionized water droplets dispersed with 1-lm fluorescent microspheres are deposited on the surface of a
4. Results and discussion 4.1. Spatial variation of the outward flow The experimental observations show that the outward flow inside the pinned drying droplets can be divided into two parts, namely, an accelerated flow region which is adjacent to the contact line and a constant flow region which locates farther away from the contact line (Fig. 3). This velocity distribution can be explained by the shape of the outward flow. In the region beyond the stagnation point, the outward flow region is wedge-shaped (Fig. 1). According to the theory of Deegan et al. [1,2], the height-averaged velocity in such region will increase as the contact line is approached. Contrarily, the height of the outward flow is almost constant in the region within the stagnation point [14], which results in a nearly constant height-averaged velocity in such region (see Eq. (1)). From Fig. 3, the distance L can be estimated from the position of the dividing point between these two regions. This yields a value of about 25 lm, which is consistent with the observed value of about 20 lm (see Fig. 2).
D. Huang et al. / International Journal of Heat and Mass Transfer 83 (2015) 307–310
309
gan et al. [1,2], the present observations show that a constant velocity region of the outward flow exists. This difference is just are not the same. In the Deegan because that the definitions of u is calculated by integrating the radial velocity comet al.’s theory u ponent of the liquid with respect to z from 0 to the drop height h. is computed by integrating from However, in the present model, u 0 to the height H of the outward flow (see Fig. 1). Deegan et al. [1,2] predicted that the velocity of the outward flow diverges at the end of the drying. In the present paper, the Eq. (6) shows that the velocity remains finite at the depinning time. However, if the contact line is pinned over the whole drying time (i.e., hd = 0 and td = tf where tf is the total drying time), Eq. (6) will ðtÞ ðtf tÞ1 , which is the same as in the theory by reduce to u Deegan et al. 4.4. Influences of contact line pinning and Marangoni effect on deposition distribution vs the distance to the contact line d. The Fig. 3. The height-averaged velocity u volume of the droplets is about 1 lL. The initial contact angle is about 19°, 22°, and 24° for droplet A, B, and C respectively. The solid lines are the best linear fits to the data points within the stagnation point.
4.2. Temporal variation of the outward flow 1 is to good The reciprocal of the height-averaged velocity u approximation linear in time (Fig. 4). This means that the prefactor of (td t) in Eq. (7) is almost a constant, which in turn indicates that the value of L is nearly time-independent over the pinning stage. By substituting the measured values of R, td, and h0 into Eq. (7), and then comparing with the fitted linear function in Fig. 4, the distance L and the contact angle hd can be calculated. The computed L are 24, 10, and 17 lm for droplet A, B, and C, respectively, which are close to the values obtained in Section 4.1. The computed hd are 2.97°, 5.74° and 4.62° for droplet A, B, and C, respectively, which are also consistent with the value of 2–4° measured by Hu and Larson [18]. These consistencies may corroborate the validity of the present model. 4.3. Comparison with the theory by Deegan et al. Different from the power law increase of the height-averaged velocity with decreasing distance to the contact line given by Dee-
When pinned, the rate of increase in the ring mass is approximately proportional to the evaporation rate from the droplet surface beyond the stagnation point and can be expressed as 1=2 _ r ¼ m0 ð1 hd =h0 Þð2~L ~L2 Þ =td m
ð8Þ
where m0 is the total mass of solute particle present initially in the droplet. Therefore, the total mass of the deposit ring after the droplet is dried is 1=2 _ r t d ¼ m0 ð1 hd =h0 Þð2~L ~L2 Þ mrt ¼ m
ð9Þ
Eq. (9) indicates that the contact line pinning and the Marangoni effect have significant influences on the growth of ring. A strong contact line pinning and a weak Marangoni effect will result in a dense ring-like deposit along the droplet edge, and contrarily, a weak contact line pinning and a strong Marangoni effect will generate a more uniform deposit. 5. Conclusions In summary, we have investigated experimentally and theoretically the outward flow inside pinned liquid droplets for the case the convective Marangoni flow exists. The experimental observations showed that a constant height-averaged velocity region of the outward flow exists and the reciprocal of the velocity is nearly linear in time over the pinning stage. An analytical model for the outward flow has also been proposed to explain the variations of the velocity. Further analysis on the growth of the deposit ring indicated that both the contact line pinning and the Marangoni effect have significant influences on the distribution of the deposit. Conflict of interest None declared. Acknowledgments The work is financially supported by the National Natural Science Foundation of China (Grant No. 51275050), the Program for New Century Excellent Talents in University (Grant No. NCET-120786), and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20120014120017). References
1 vs the time to depinning, Fig. 4. The reciprocal of the height-averaged velocity u td t. The solid lines are the best fit to a linear function. The deviation of the data points from the line may result from the Brownian motion and the positioning error.
[1] R.D. Deegan et al., Capillary flow as the cause of ring stains from dried liquid drops, Nature 389 (1997) 827–829. [2] R.D. Deegan et al., Contact line deposits in an evaporating drop, Phys. Rev. E 62 (2000) 756–765.
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