Chemical Physics Letters 508 (2011) 134–138

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Chemical Physics Letters

journal homepage: www.elsevier .com/locate /cplet t

Morphology tunable pinning force and evaporation modes of water dropletson PDMS spherical cap micron-arrays

Chi Zhang, Xiaolong Zhu, Luwei Zhou ⇑State Key Laboratory of Surface Physics and Department of Physics, Fudan University, 200433 Shanghai, China

a r t i c l e i n f o

Article history:Received 13 March 2011In final form 7 April 2011Available online 12 April 2011

0009-2614/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cplett.2011.04.041

⇑ Corresponding author.E-mail address: lwzhou@fudan.edu.cn (L. Zhou).

a b s t r a c t

It is well known that three modes (the constant contact radius mode, the constant contact angle modeand the mixed mode) are existed during the droplet evaporation process. However, the mechanisms ofthese modes and their transitions are still in dispute. In this paper, pinning force is introduced and for-mulated to give a quantitative explanation to the evaporation modes and their transitions. The effect ofsurface morphology on the pinning force is discussed using our experiments of water droplets evaporat-ing and contact line receding on the surfaces covered by PDMS (polydimethylsiloxane) spherical caparrays with gradually increasing height.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Droplet evaporation has various applications in nanostructurefabrications [1,2], inkjet printing [3] and meteorology [4]. It is ob-served that there are three modes of evaporation [5]: the constantcontact radius mode (CCR), the constant contact angle mode (CCA)and the mixed mode. In the CCR mode, droplets usually evaporatewith constant contact radius and diminishing contact angle,whereas in the CCA mode, with diminishing contact radius andconstant contact angle. In the mixed mode, the contact radiusand the contact angle decrease simultaneously and it usually hap-pen at the end of evaporation. Since the different evaporationmodes were first observed in 1977 [5], a lot of efforts have beenmade to explain the observed phenomena theoretically and exper-imentally [6–18].

On the other hand, it was reported that sessile droplets displaysome hysteresis: the droplets cannot advance if the contact angle his smaller than a critical value ha and cannot recede if h is largerthan hr [19], and this hysteresis was proved to be a possible reasonof the evaporation modes [20]. Such a hysteresis is resulted fromchemical heterogeneity and physical roughness of the surface[21]. Experimental observations and theoretical explanations[22–27] were made to explain the hysteresis and it was suggestedthat the most important parameter was the force per unit length topin the contact line [24].

Although a lot of work have been done, the mechanism of dif-ferent evaporation modes and the reason of the pinning–shrinking(PS) transition (from the CCR to the other phases) are not com-pletely clear. Few attempts have been made to relate the evapora-tion modes to the morphology of substrates. In this paper, we

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introduce and formulate the pinning force to help us explain thedifferent evaporation modes and the PS transition. To understandhow the pinning force and the evaporation modes are affected bythe surface morphology, experimental data of water droplets evap-orating and contact line receding on the surface covered by PDMSspherical cap arrays with gradually increasing height are presentedand analyzed.

2. Theory

In this section, pinning force is introduced to establish a modelto explain the mechanism of the PS transition during the evapora-tion. Consider a small sessile water droplet. When the droplet issmaller than its capillary constant, the effect of gravity is negligibleand the droplet forms a shape of spherical cap on the substrate, asillustrated in Figure 1A. The initial contact angle hi of a liquid drop-let on a rough substrate can be described by the Cassie–Baxterequation [28] as

cos hi ¼ �ð1� /Þ þ r//csv � cls

clvð1Þ

where / is the fraction of the projected area of the wetted solidarea, and r/, the roughness factor (the ratio of actual area and theprojected area) of the wetted solid area. cls; csv ; clv stand for the sur-face tensions of the liquid–solid, solid–vapor and liquid–vaporinterfaces, respectively. When / ¼ 1, Eq. (1) degenerates to theWenzel equation [29]; when / ¼ 1 and r/ ¼ 1, to the Young’sequation.

During the evaporation process, the contact angle would main-tain h0 if the substrate surface was absolutely smooth and homoge-neous. In a real system, however, this condition cannot be satisfied,and a pinning force which is caused by heterogeneity will be re-acted to the contact line leading to the hysteresis [25]. If Fp

Figure 1. (A) An illustrative graph of a water droplet of spherical cap shape on thesubstrate and the resolution of force about the contact line. b and h are the contactradius and the contact angle, respectively. (B) Side view of a droplet on roughnesssolid, showing the effect of the local tilt surface on the apparent contact angle h. (C)Top view of the contact line following the rim of a substrate cap. The dash arrowsshow the local force provided by the rim and the solid arrow shows the averageforce �f .

C. Zhang et al. / Chemical Physics Letters 508 (2011) 134–138 135

(has a threshold of Fpc) is denoted as a pinning force applied by thesubstrate surface to a unit length of a contact line (see Figure 1(A))and introduced to Eq. (1), the total force of the contact line per unitlength can be written as

F ¼ ðð1� /Þclv þ clv cos h� r//ðcsv � clsÞÞ � Fp ð2Þ

Assuming the evaporation as a quasi-static process, Eq. (2) shouldalways equal to 0. That is

ðð1� /Þclv þ clv cos h� r//ðcsv � clsÞÞ ¼ Fp ð3Þ

At the beginning of the evaporation, the contact line is pinned bythe pinning force, forcing the evaporation goes to the CCR mode.At this stage, the values of both sides of Eq. (3) gradually increaseuntil Fp reaches its threshold Fpc . Then, to keep the left side of Eq.(3) equal to Fpc , h should maintain a certain value. So the contactline depins and the process comes to the CCA mode. The contact linewill shrink and the contact angle will remain constant hc as de-scribed by the following equation

cos hc ¼ �ð1� /Þ þ r//csv � cls

clvþ Fpc

clv¼ cosðhiÞ þ

Fpc

clvð4Þ

where hi is the initial angle as described by Eq. (1).One may find that the most important parameter is the thresh-

old of the pinning force which we may call ‘the critical pinningforce’. One natural question is ‘how does the critical pinning forceaffected by the morphology of the substrate surface’. Consider thecontact line on a local tilt surface as shown in Figure 1B. The appar-ent contact angle h can be written as

h ¼ h0 � a ð5Þ

where h0 is the microscopic contact angle which is the same as thecontact angle of the droplet on a flat surface, and a is the angle be-tween the surface and the horizontal line. Affected by the local tiltsurface, the apparent contact angle is smaller than h0. This is equiv-alent to a force f (per unit length) dragging the contact line andleading to the hysteresis.

f ¼ clvðcosðhÞ � cosðh0ÞÞ ¼ clvðcosðh0 � aÞ � cosðh0ÞÞ ð6Þ

It is reported that the contact line of the droplet is following therims of the substrate patterns [30] as shown in Figure 1B in our sys-tem. Hence, the average force �f (per unit length) provided by onesubstrate cap can be written as

�f ¼ 2p

f ð7Þ

The roughness is consisted by rims of the substrate caps, and thecontribution of the roughness is depend on the linear fraction /l

of the substrate caps along the contact line. From geometry, a canbe expressed as

a ¼ arccos 1� hr

� �ð8Þ

and /l can be simplified as

/l ¼br¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

hr� ðh

rÞ2

rð9Þ

where r;h, and b are the radius, height and ground area radius of thesubstrate caps, respectively. The surface of our substrate (PDMS) isa kind of polymer, containing roughness in molecular scale, leadingto the hysteresis which is denoted as F0. This molecular scale rough-ness is the reason for the pinning force in the flat substrate. Withabove discussion, the total hysteresis of the substrate surface, thecritical pinning force in other word, can be described as

Fpc ¼ /l�f þ F0 ð10Þ

If the feature of the substrate material is obtained (that means theinitial contact angle h0 and hysteresis F0 of the flat substrate, S0),the critical pinning forces of substrate S1–S5 can be predicted fromEqs. (5)–(10).

3. Experimental section

The evaporation experiments were carried out on PDMS(polydimethylsiloxane) surfaces with microstructures, therebybeing transparent and ultrahydrophobic. The precisely controllablemicrostructures were realized via a soft lithography techniqueusing self-assembly colloidal crystals and a PDMS elastomer [31].First, a colloidal monolayer of latex spheres was assembled on goldcoated electrodes. Molds with tunable morphology can be obtainedthrough controllable electrochemical deposition and spheres dis-solution processes [32]. Then, PDMS prepolymer was dropped ontothese molds. In order to impregnate in the interspaces of themicromolds completely, the molds with pre-casted prepolymerwere treated under vacuum (0.1 MPa) for 1 h to remove theentrained gas before they were cured in a conventional dryingoven at 65 �C for 6 h. The PDMS samples can then be peeled offthe molds easily [33].

To focus on the effect of morphology avoiding any effect due tochemical difference, six substrates with the same material but dif-ferent morphology were fabricated. The surfaces morphology ofsubstrates S0–S5 were spherical caps with different cap heightsranging from 0 to r (1.5 lm) with a step of 0.2r as shown inFigure 2 (h � 0:2r;h � r and large area). Intuitively, the critical pin-ning forces would be gradually increased from substrates S0–S5.

The droplets of deionized water were produced by deliveringeach of them from a capillary tube of 0.3 mm in diameter ontothe PDMS surfaces, and the radii of the droplets were about2 mm. During the evaporation, the substrates and the dropletswere put on a platform and the experimental setup was in stillair of 25 �C with relative humidity about 30%. In the receding anglemeasurement, the contact line was moved by using a capillary tosuck water from the droplet.

Figure 2. SEM images of PDMS spherical cap arrays. (A) h � 0:2r, where h is theheight of the spherical caps, (B) h � r and (C) triangular lattice with large area. Scalebar: 5 lm.

A

B

Figure 3. Contact radii and angles of water droplets during evaporation. The curveswere shifted vertically to be seen clearly. The color symbols stand for the graduallyincreasing heights of the substrate spherical caps form 0 (black closed squares) to r(olive open triangles). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

136 C. Zhang et al. / Chemical Physics Letters 508 (2011) 134–138

A 16-bit resolution microscope (VHX-1000, Keyence Co., Japan)with resolution of 1600 � 1200 pixels were used in our experi-ments to monitor and record the shape (side view) of the waterdroplets. The camera took 4 pictures per minute during the wholeevaporation process and 15 pictures per second while the contactline are receding. In the evaporation measurements, since thewater droplets maintain shapes of spherical cap, we need onlymeasure their heights and the contact radii in the snap photos,and the contact angles can be easily obtained from geometry.The uncertainty of the PS transition angle comes from two parts.The first part is from the uncertainty of determining the interfacesin the snap photos (brings error around 1� of the contact angle).The second part is from the fluctuation of the experimental valuesof the constant contact angle (substrates S0–S2: 2.5�, 1.4� and 0.6�,respectively), or the uncertainty of the PS transition point (sub-strates S3–S5: around 2�, 2� and 3�, respectively). In the recedingangle measurement, the droplet was forced to deform by the cap-illarity. Instead of measuring the contact radius and height, we di-rectly measured the receding angle manually. The uncertainty of

the receding angle is resulted from the same reasons as the PS tran-sition angle, only greater in this case.

4. Results and discussion

In our experiments, the mixed mode only occurs at the very endof the evaporation, therefore we consider only the earlier stages ofthe droplets evaporating with either constant contact radius orconstant contact angle. Figure 3 shows the contact radii and thecontact angles of the water droplets on gradually heightened PDMSspherical caps during evaporations. It is easy to see that dropletsbehave distinguishingly on substrates with different morphology.The initial contact angle gradually increases, as shown in Table 1,while the height of substrate spherical caps increases. The relationof the surface morphology and the initial contact angle was welldiscussed in previous work [34]. In this paper, we will focus onthe evaporation modes and the PS transition. To understand whythe droplets have distinguishing behaviors on different substrates,four pairs of data lines (for substrates S0, S2, S4 and S5 with heightof substrate spherical caps of 0, 0.4r, 0.8r and r, respectively) arepicked out from Figure 3 to form Figure 4. We can find three evap-oration stages: stage I with constant contact radius and diminish-ing contact angle, stage II with constant contact angle and

Table 1Initial angles, PS transition angles, receding angles and their corresponding experimental pinning forces of water droplets on PDMS spherical caparrays of different cap heights.

Substratescap heights

Initial angle(�)

Evaporation (PS transition) Receding

Angle (�) Force ðN m�1Þ Angle (�) Force ðN m�1Þ

0 102.6 76 ± 3 0.0331 ± 0.0037 72 ± 4 0.0380 ± 0.00480.2r 105.7 72 ± 2 0.0417 ± 0.0024 70 ± 4 0.0441 ± 0.00470.4r 116.6 69 ± 1 0.0580 ± 0.0012 65 ± 5 0.0627 ± 0.00570.6r 121.2 64 ± 3 0.0689 ± 0.0034 61 ± 4 0.0722 ± 0.00440.8r 127.7 61 ± 3 0.0789 ± 0.0033 58 ± 4 0.0822 ± 0.0043r 131.1 55 ± 4 0.0886 ± 0.0041 53 ± 5 0.0907 ± 0.0050

A

C

B

D

Figure 4. The contact radii (blue open) and the contact angles (black closed) of water droplets evaporating on PDMS substrates with spherical cap arrays of different capheights (A 0, B 0.4r, C 0.8r, D r). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

C. Zhang et al. / Chemical Physics Letters 508 (2011) 134–138 137

diminishing contact radius, and stage III with simultaneouslydecreasing contact radius and contact angle.

In stage I (CCR), at the beginning of the evaporation process,

Fp < Fpc ð11Þ

meaning that the pinning force is strong enough to pin the contactline. The contact angle h has to diminish as the volume of the drop-let gradually decreases during the evaporation. Meanwhile, the pin-ning force gradually increases from 0 to Fpc as the contact anglediminishes.

At the end of stage I, when the contact angle reaches the criticalvalue hc (see Eq. (4) and Table 1), Eq. (11) is no longer satisfied.Thus, the contact line depins, and the droplet begins to shrink. Dri-ven by the minimum free energy principle, the contact angle re-mains a constant value of h ¼ hc , and the evaporation entersstage II (CCA) as the evaporation goes on. The experimental criticalpinning force can be obtained from Eq. (4) as

Fpc ¼ clvðcos hc � cos hiÞ ð12Þ

Obviously, the increasing height of substrate spherical caps en-hances the pinning force (see Table 1), validating our formerenvision.

At the end of the evaporation, the droplet will come to stage IIIwith both contact radius and contact angle diminishing simulta-neously. The most possible reason of this mixed mode was pro-vided by Kim et al. [15]. Here, the discussion will focus on theevaporation modes and the PS transition. As illustrated in Figures4A and 4B, governed by Eq. 4, the increase of the critical pinningforce will push the PS transition point to the right handside ofthe time scale. If a pinning force provided by the substrate is strongenough, the PS transition point could be pushed to the right of theII–III transition point. In this case, stage II will be skipped and theevaporation process will directly go from stage I to stage III asshown in Figure 4C and D.

Figure 5. Measured critical pinning forces of water droplets evaporating (redcircles) and contact line receding (gray squares) on the PDMS substrates withspherical cap arrays of different cap heights and their prediction (blue line)calculated from Eqs. (5)–(10). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

138 C. Zhang et al. / Chemical Physics Letters 508 (2011) 134–138

Table 1 shows the PS transition angle, the receding angle andtheir experimental critical pinning force (see Eq. (12)) of waterdroplets on different substrates. Based on the above explanation,the receding angle should be the same as the PS transition anglesince the reason of contact angle hysteresis is also the pinningforce as in the CCR evaporation mode. From the experiments, theinput parameters h0 and F0 are obtained (h0 ¼ 102:6� andF0 ¼ 0:0331 N m�1, see Table 1). As mentioned in the theory sec-tion, the prediction of the critical pinning force can be solved asshown in Figure 5 (blue line) from Eqs. (5)–(10). The red circlesand gray squares are the experimental values (see Table 1) of thecritical pinning forces in case of droplet evaporation and contactline receding, respectively. The experimental data are in goodagreement with the prediction in the theory section.

5. Conclusions

In this paper, we discussed the mechanism of different evapora-tion modes and PS transition. To clarify this problem, we estab-lished a model based on the pinning force of the contact line anddiscussed the effect of the morphology of the substrate surface.The pinning force can pin the contact line and force the dropletto evaporate with CCR mode before the contact angle reaches a cer-tain value. At the PS transition point, the contact line begins toshrink, and the evaporation goes into the CCA mode since the pin-ning force is no longer strong enough. To understand the effect ofthe substrate morphology and to verify our model, experiments ofwater droplets evaporating and contact line receding on PDMSspherical cap arrays with different heights were conducted. The

experiments presented distinct evaporation properties which werein good agreement with our model.

Acknowledgement

The authors thank Dr. Fang Haiping, Dr. Chen Wei, Dr. Shi Lei,Huang Junying and Zheng Jie for fruitful discussions. This work issupported by the National Natural Science Foundation of Chinaunder Grants No. 10974030 and No. 10574027, and by the Sci-ence and Technology Commission of Shanghai Municipality(09DZ1976602) and the ‘973’ Grant of Ministry of Science andTechnology of China (2006CB504509).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.cplett.2011.04.041.

References

[1] T.P. Bigioni, X.-M. Lin, T.T. Nguyen, E.I. Corwin, T.A. Witten, H.M. Jaeger, Nat.Mater. 5 (4) (2006) 265.

[2] D. Xia, S.R.J. Brueck, Nano Letters 4 (7) (2004) 1295.[3] J. Park, J. Moon, Langmuir 22 (8) (2006) 3506.[4] C.D. Whiteman, Mountain Meteorology: Fundamentals and Applications,

Oxford University Press, New York, 2000.[5] R.G. Picknett, R. Bexon, J. Colloid Interface Sci. 61 (2) (1977) 336.[6] K.S. Birdi, D.T. Vu, A. Winter, J. Phys. Chem. 93 (9) (1989) 3702.[7] C.W. Extrand, Langmuir 22 (20) (2006) 8431.[8] S.M. Rowan, M.I. Newton, G. McHale, J. Phys. Chem. 99 (35) (1995) 13268.[9] G. McHale, S.M. Rowan, M.I. Newton, M.K. Banerjee, J. Phys. Chem. B 102 (11)

(1998) 1964.[10] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Nature

389 (6653) (1997) 827.[11] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Phys.

Rev. E 62 (1) (2000) 756.[12] E. Kissa, J. Amer. Oil Chemists’ Soc. 72 (7) (1995) 793.[13] A.J. Petsi, V.N. Burganos, Phys. Rev. E 73 (4) (2006) 041201.[14] C. Bourges-Monnier, M.E.R. Shanahan, Langmuir 11 (7) (1995) 2820.[15] J.H. Kim, S.I. Ahn, J.H. Kim, W.C. Zin, Langmuir 23 (11) (2007) 6163.[16] X. Fang, B. Li, E. Petersen, Y. Ji, J.C. Sokolov, M.H. Rafailovich, J. Phys. Chem. B

109 (43) (2005) 20554.[17] H. Hu, R.G. Larson, J. Phys. Chem. B 106 (6) (2002) 1334.[18] W. Ervina, T.H. Michael, AIChE J. 54 (9) (2008) 2250.[19] R.H. Detree.[20] D.M. Soolaman, H. Yu, J. Phys. Chem. B 109 (2005) 17967.[21] P.G. de Gennes, Rev. Mod. Phys. 57 (3) (1985) 827.[22] D. Bonn, H. Kellay, G.H. Wegdam, Phys. Rev. Lett. 69 (13) (1992) 1975.[23] E. Rolley, C. Guthmann, Phys. Rev. Lett. 98 (16) (2007) 166105.[24] D. Bonn, J. Eggers, J. Indekeu, J. Meunier, E. Rolley, Rev. Mod. Phys. 81 (2)

(2009) 739.[25] Y.L. Chen, C.A. Helm, J.N. Isrealachvili, J. Phys. Chem. 95 (1991) 10736.[26] C.W. Extrand, Langmuir 18 (2002) 7991.[27] L. Gao, T.J. McCarthy, Langmuir 22 (2006) 6234.[28] A.B.D. Cassie, S. Baxter, Trans. Faraday Soc. 40 (1944) 546.[29] R.N. Wenzel, Ind. Eng. Chem. 28 (1936) 988.[30] F. Birembaut, N. Perney, K. Pechstedt, P.N. Bartlett, A.E. Russell, J.J. Baumberg,

Small 4 (12) (2008) 2140.[31] X. Zhu, L. Shi, X. Liu, J. Zi, Z. Wang, Nano Research 3 (11) (2010) 807.[32] M.E. Abdelsalam, P.N. Bartlett, T. Kelf, J. Baumberg, Langmuir 21 (2005) 1753.[33] X. Zhu, C. Zhang, L. Shi, X. Liu, J. Zi, L. Zhou, Unpublished.[34] A. Marmur, Langmuir 19 (2003) 8343.