Forces Acting on a Single Particle in an Evaporating Sessile Droplet

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Anal. Chem. 2010, 82, 784–788

Letters to Analytical Chemistry Forces Acting on a Single Particle in an Evaporating Sessile Droplet on a Hydrophilic Surface Jung-yeul Jung,† Young Won Kim,‡ Jung Yul Yoo,‡ Junemo Koo,† and Yong Tae Kang*,† Department of Mechanical Engineering, Kyung Hee University, Gyeonggi 446-701, Korea, and School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Korea The evaporating sessile droplet of a mono/didisperse colloid on a plate is a very useful and handy technique in micro/nano/bioapplications to separate, pattern, and control the particles. Although the fundamental nature of the evaporation phenomena and its applications have been extensively proposed, the crucial forces affecting a single particle motion in an evaporating droplet are not reported yet. To elucidate the impact of various forces including the drag, electrostatic, van der Waals, and surface tension forces on the particle motion in suspension, the magnitudes of them are compared using the scale analysis. In the early stage of the evaporation, in which the contact line is fixed, the motion of a single particle suspended in liquid are mostly affected by drag force. Later, with the incidence of the contact line recession, the surface tension force takes over the control of the single particle motion. As a droplet of the particle suspension dries out, the suspended particles move to the plate/liquid/air interface by the liquid flow induced by evaporation. The application examples of the evaporating droplets include optical mapping of DNA molecules, making the patterns of nanoparticles and separation of microparticles and biocells.1-3 With the use of the hydrophobic patterned surface on the hydrophilic plate, the nanoparticles in an evaporating colloidal liquid could be patterned by hydrophobicity on the plate.2 It is well-known that the liquid in an evaporating colloidal droplet on the hydrophilic surface with a fixed contact line must flow radially outward for the contact line to maintain its position. In the theoretical aspects, Deegan et al.4 proposed the capillary flow mechanism for contact-line deposition. They reported that the * To whom correspondence should be addressed. E-mail: [email protected]. † Kyung Hee University. ‡ Seoul National University. (1) Jing, J. P.; Reed, J.; Huang, J.; Hu, X. H.; Clarke, V.; Edington, J.; Housman, D.; Anantharaman, T. S.; Huff, E. J.; Mishra, B.; Porter, B.; Shenker, A.; Wolfson, E.; Hiort, C.; Kantor, R.; Aston, C.; Schwartz, D. C. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 8046–8051. (2) Fan, F. Q.; Stebe, K. J. Langmuir 2004, 20, 3062–3067. (3) Jung, J. Y.; Kwak, H. Y. Anal. Chem. 2007, 79, 5087–5092. (4) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827–829.

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deposition could be predicted and controlled without knowing the chemical natures of the host liquid, suspended particle, or substrate. Various previous results3-7 are in good agreement with the model of Deegan et al.4 Recently, Jung and his colleagues3,5 reported the separation of microparticles and biological cells and the inward motion of the microparticles (i.e., 5 µm diameter polystyrene beads (PSBs)) in the didisperse colloidal droplet on the hydrophilic plate. With the use of the dielectrophoretic (DEP) force generated by the patterned gold electrode on the silicon dioxide surface and the drag force induced by the outward flow of host liquid, microparticles and biological cells such as red blood cells (RBC) and Escherichia coli were separated.3 Microparticles (i.e., 5.0 µm diameter) in the dilute didisperse colloidal of suspended PSBs of 0.5 and 5.0 µm in diameter moved to the center of the droplet at the final stage of drying the droplet on the hydrophilic surface.5 In this study, we performed the scale analysis of the forces acting on a single particle in an evaporating droplet on a hydrophilic surface, where the roles of the forces including the van der Waals, electrostatic, and surface tension forces have not been analyzed in any previous studies to the best of our knowledge. The van der Waals and electrostatic forces do affect the colloidal systems, but most previous studies on the evaporating droplet did not consider the effects. Furthermore, the surface tension force acting on the particles is dominant, one for the case of a single particle in the dilute colloidal droplet on the hydrophilic surface. From the scale analysis, the dominant factors for the separation, detection, and patterning of the micro/nanoparticles and biocells using the evaporating colloidal droplet on the hydrophilic surface are elucidated. THEORETICAL BACKGROUND Drag Force. There can be two kinds of fluid flow affecting on motion of the particle in an evaporating sessile droplet. First is the capillary flow because the effective fluid flow toward the rim of the droplet in an evaporating droplet on the hydrophilic surface (5) Jung, J. Y.; Kim, Y. W.; Yoo, J. Y. Anal. Chem. 2009, 81, 8256–8259. (6) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Phys. Rev. E 2000, 62, 756–765. (7) Hu, H.; Larson, R. G. J. Phys. Chem. B 2002, 106, 1334–1344. 10.1021/ac902288z  2010 American Chemical Society Published on Web 01/12/2010

maintains its triple contact line among substrate, liquid, and air (i.e., three-phase boundary), and this flow is dominant in the water droplet system.4 The second one is related to the thermocapillary and thermophoresis flow due to the Marangoni effect.6,8-10 For water droplets, the Marangoni flow was not observed or very weak as compared to values reported in previous studies.4,9,11 So it could be negligible because we used the pure water-based and diluted (i.e., 0.005 vol %) colloid droplet. At an early stage of the evaporating sessile droplet on the hydrophilic surface, most particles in an evaporating droplet are imposed by the drag force due to the fluid flow. Therefore, the so-called “coffee stain” can remain after the colloidal droplet dries out.4 Considering the balance between the evaporation rate and convection in the droplet, a relationship between the heightaveraged velocity and the local evaporating rate can be written as4,7 Fl

1 ∂(rvh) ∂h +F ) -J(r, t) ∂t r ∂r

(1)

where Fl is the liquid density, h is the droplet height, t is the time, r is the droplet radius, v is the radial velocity, and J(r,t) is the local evaporation rate. The radial and vertical velocities of the particles inside the evaporating droplet can be obtained with appropriate boundary conditions. A detailed treatment including the boundary conditions and discussion on the above equation are given in Hu and Larson.7 The drag force acting on the particle inside the evaporating droplet can also be obtained from the Stokes’ law as follows12 FD ) 6πRηv

(2)

where R is the particle radius and η is the dynamic viscosity of liquid. van der Waals Force. In the real systems, the van der Waals force exerts between particles or between particles and substrate, which is one of the dominant factors affecting colloidal stability. Hamaker13 calculated the potential energy concerning the pairwise summation of all the microscopic relevant intermolecular interactions. Equation 3 represents the Hamaker theory for the van der Waals force, which can be used to estimate the magnitude of the van der Waals force between a particle and substrate surface. Considering the contribution of the extended contact area between a particle and the surface from the particle deformation14 and the effect of retardation,15 the van der Waals force can be expressed as16-18 (8) Maillard, M.; Motte, L.; Ngo, A. T.; Pileni, M. P. J. Phys. Chem. B 2000, 104, 11871–11877. (9) Hu, H.; Larson, R. G. J. Phys. Chem. B 2006, 110, 7090–7094. (10) Kim, I.; Kihm, K. D. Langmuir 2009, 25, 1881–1884. (11) Savino, R.; Paterna, D.; Favaloro, N. J. Thermophys. Heat Trans. 2002, 16, 562–574. (12) Batchelor, G. K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, U.K., 1967. (13) Hamaker, H. C. Physica 1937, 4, 1058–1072. (14) Visser, J. Part. Sci. Technol. 1995, 13, 169–196. (15) Gregory, J. J. Colloid Interface Sci. 1981, 83, 138–145. (16) Xu, K.; Vos, R.; Vereecke, G.; Doumen, G.; Fyen, W.; Mertens, P. W.; Heyns, M. M.; Vinckier, C.; Fransaer, J. J. Vac. Sci. Technol., B 2004, 22, 2844– 2852. (17) Hogg, R.; Healy, T. W.; Fuersten, Dw. Trans. Faraday Soc. 1966, 62, 1638– 1651.

FVDW ) -

A132R 6z

2

(

1+

a2 Rz

)(

1 1 + 14(z/λret)

)

(3)

where A132 is the Hamaker constant of particle 1 on a substrate 2 in a medium 3 (A132 = {(A11)1/2 - (A33)1/2}{(A22)1/2 - (A33)1/2}), where Aii is the interaction constant of two bodies of material i.19 The Hamaker constants are typically on the order of 10-20 to 10-19 J. z is the particle-surface closet-separation distance () 0.4 nm),19 λret is the characteristic wavelength of interaction (∼10 nm),15,20 and a is the deformation induced contract radius, which was modified to take into account the surface energy effect from the Hertz equation21 as follows; a)

[ KR {F + 3πRW

132

}]

+ √6πW132F + (3πRW132)2

1/3

(4) where F is the load (i.e., body force), W132 is the work of adhesion of a particle 1 on a plate 2 in a medium 3 (W132 ) γ13 + γ23 γ12, where γ is the interfacial energy), and K is the composite modulus of the particle/plate system, which can be expressed as21

K)

(

2 1 - µ22 4 1 - µ1 + 3 E1 E2

)

-1

(5)

where µ is the Poisson ratio and E is the Young’s modulus for the particle and substrate. Electrostatic Force. Most particles and substrate immersed in liquid are electrically charged and surrounded by electrical double layers (EDLs). Therefore, there should be an electrostatic force between particles and substrate in an evaporating colloidal droplet on the plate. On the basis of the surface-charge model,16-18 the electrostatic force can be expressed as FE )

2πεκR [2Ψ2 exp(-κz) + 1 - exp(-2κz) (Ψ22 + Ψ12) exp(-2κz)]

(6)

where ε is the permittivity of the medium (7 × 10-10 F m-1 for water), κ is the Debye-Huckel parameter (which is the reciprocal of EDL thickness), and Ψ is the surface potential of particle 1 or plate 2, which can be measured by light scattering equipment (i.e., ELS-8000, Otsuka Electronic, Japan). Surface Tension Force. The surface tension force decreases to the order of 101 while the inertia force decreases to the order of 103 with decreasing the scale by the scaling law. In the microscale world, the surface tension force becomes more important than the body force while the surface tension force is usually neglected in the macroscale world. With the liquid evaporation of the colloidal droplet on a hydrophilic surface, the contact angle exceeds the critical angle so that (18) Kar, G.; Chander, S.; Mika, T. S. J. Colloid Interface Sci. 1973, 44, 347– 355. (19) Visser, J. Adv. Colloid Interface Sci. 1972, 3, 331–363. (20) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, U.K., 1991. (21) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301–313.

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Figure 1. Measured droplet shape (A and C) and calculated droplet one and radial velocity in the droplet (B and D). Parts A and B are at t ) 0.5 s and parts C and D are at t ) 215 s.

the thin core liquid region begins to recede from the contact line.22 Therefore, the liquid/air/plate interface continues to move toward the center of a droplet, and the droplet eventually dries out. In the mean time, the liquid-gas interface exerts the surface tension force on the particles trapped in the rim of the droplet. If a particle is surrounded by a liquid layer, the surface tension force can be expressed as FST ) 2πRσ cos θ

(7)

where σ is the surface tension of the liquid and θ is the contact angle of droplet () 2 tan-1(h/r)). At the depinning (i.e., about t ) 215 s in this study), the contact angle is ∼22° obtained from Figure 1. Although we cannot observe the actual surrounding perimeter of the particle, the relationship between the contact angle and the surface tension force can be estimated using eq 7 (see Figure 6 in ref 5 for the schematic diagram). Static Coefficient of Friction. During the evaporation of the colloid droplet on the hydrophilic surface, the particles in the droplet are advected to the rim of droplet by the fluid flow. Most particles in the droplet are sedimented during the drying process, especially, the microparticles such as 5 µm diameter PSBs are sedimented in the early stage of the drying process. The time required (τ) for the sedimentation of a particle in a dilute aqueous suspension is given by23-25

τ)

9hη 2R2(Fp - Fl)g

(8)

where g is the acceleration due to gravity and Fp is the particle density. In this study, most 5 µm diameter PSBs sedimented on the substrate within ∼140 s of drying time, which is in good agreement with the estimated maximum time of 153 s for the particles suspended near maximum height (i.e., 418 µm, center of droplet as shown in Figure 1). (22) Chon, C. H.; Paik, S.; Tipton, J. B.; Kihm, K. D. Langmuir 2007, 23, 2953– 2960. (23) Batchelor, G. K. J. Fluid Mech. 1972, 52, 245–268. (24) Xu, W.; Nikolov, A.; Wasan, D. T. J. Colloid Interface Sci. 1998, 197, 160– 169. (25) Dokou, E.; Barteau, M. A.; Wagner, N. J.; Lenhoff, A. M. J. Colloid Interface Sci. 2001, 240, 9–16.

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When a particle begins to move on the substrate under an external force (i.e., drag force in this study), the static coefficient of friction (f) is given by26 f ) FD /Fa

(9)

where Fa is the total adhesive force, which is given by26 Fa ) Fg + FE + FVDW

(10)

where Fg is the force due to gravity. RESULTS AND DISCUSSION General Colloid-Droplet Behavior on the Hydrophilic Surfaces. The behaviors of evaporating droplet on hydrophilic surfaces have been widely investigated.3,4,6,7 Deegan et al.4 introduced the evaporative droplet including micro/nanoparticles on the hydrophilic surfaces. It is known that the liquid in an evaporating colloidal droplet with a fixed contact line must flow radially outward due to the difference of the local evaporation rate.3,4,6,7 Particles dispersed in the evaporating droplet on the solid substrate move to the boundary of the droplet. Considering the evaporation and the mass balance in the droplet, a relationship between the height-averaged velocity and the local evaporating rate can be written as eq 1.4,7 Figure 1 shows the photographs and simulated velocity field in a colloidal droplet (where 0.5 and 5 µm diameter PSBs were suspended) on a hydrophilic surface (i.e., on a glass slide). The unsteady simulation is performed based on the theoretical equations developed by Deegan et al. and Hu and Larson.4,7 In the starting phases of evaporation (i.e., at 0.5 s), the simulated radius and height of the droplet are in good agreement with the measured ones with 98% confidence level as shown in Figure 1A,B. Just before the incidence of the boundary depinning (i.e., at 215 s), the estimated values are in good agreement with the measured ones with a 95% confidence level as shown in Figure 1C,D. From the validation above, the considered evaporative colloidal droplet system, which will be introduced in the next section, is believed describe the pure waterbased colloid sessile droplet on the hydrophilic surface.4 From the velocity information based on the simulation results, the drag force acting on the particle can be estimated using eq 2. (26) Tohru, N.; Yasuo, K.; Teiji, F. Part. Part. Syst. Charact. 1989, 6, 69–73.

Figure 2. Time-lapse photos of particles moving in the didisperse colloidal droplet on the glass slide. Time sequences are (A) 1, (B) 346, (C) 384, and (D) 414 s. The bright items are 5 µm diameter PSBs excited by 488 nm light, while 0.5 µm diameter ones appear gray.

Particle Behavior in a Didisperse Evaporating Droplet. In the previous experiments,5 it was observed that the particles of both small and large modes initially moved and deposited in the boundary rim. Then on the incidence of the contact line depinning, while the particles in the smaller mode stayed in the original position of the triple interface and those in the larger mode started to migrate to the center of the droplet with the motion of the contact line.3 Figure 2 shows the bottom view of particles behavior in the didisperse colloidal droplet on the slide glass. The mixture used in the present study consists of 0.005 vol % of 0.5 µm diameter and 0.005 vol % of 5 µm diameter PSBs. The bright items are the 5 µm diameter PSBs excited by 488 nm light, while the 0.5 µm diameter PSBs appear gray. Most 5 µm PSBs moved to the droplet center. At a relatively high volume fraction of 5.0 µm diameter PSBs (i.e., 0.05 vol %), the mixed particles are not easily separated (see Figure 3 in ref 5), which is attributed to the increase of various forces such as the electrostatic and van der Waals forces due to the aggregation of the particles.5 Forces Acting on Particle. Friction, drag, and surface tension forces can be obtained using eqs 1, 3, and 8, respectively. The radial velocity in eq 2 is obtained from the validated simulation results. The radial velocity is in the range between v = ∼1.5 µm/s initially and ∼10.0 µm/s near the incidence of the contact line depinning as shown in parts B and D of Figure 1. The interfacial energy of the system in eq 4 can be expressed as

W132 ) γ13 + γ23 - γ12

(11)

where the interfacial energies between the solid and the liquid γ13 and γ23 are equivalent to the surface tension force, which is readily available. In contrast, γ12 is the interfacial energy between the solid particle and the substrate. It is a function of the contact radius induced by the deformation, the hardness

Figure 3. The variation of the surface tension force acting on a single 5 µm diameter PSB depending on the contact angles of an evaporating sessile droplet on the hydrophilic surface.

of the substrate, H, and the particle radius, which is expressed as27 γ12 ) Ha2 /2R

(12)

In this study, the load is so small that the contact radius induced by deformation a is very small, making the square of it negligible. Therefore the interfacial energy between the PSB and the glass slide also becomes negligible. As an example, the interfacial energy between polystyrene and water γ13 is ∼0.022 N/m at 300 K, while the interfacial energy of 5 µm diameter PSB on glass substrate γ12 is ∼0.000 94 N/m at 300 K in the case of a 1 nm contact radius at worst. Figure 3 shows the variation of surface tension force acting on a 5 µm diameter PSB depending on the contact angle from 0 to 22° (i.e., the depinning contact angle in this study). The surface tension force decreases with increasing contact angle because the (27) Demejo, L. P.; Rimai, D. S.; Chen, J. H.; Bowen, R. In Particles On Surfaces: Detection, Adhesion, and Removal; Mittal, K. L., Ed.; Marcel Dekker, Inc.: New York, 1994; pp 33-45.

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Figure 4. The estimated forces depending on particle size acting on a single PSB under ambient conditions: (A) the drag force and (B) the drag, van der Waals, electrostatic, and surface tension forces.

surrounding perimeter decreases with increasing the contact angle. As the three-phases boundary recedes at the final stage of evaporation, the contact angle decreases.28 Therefore, the surface tension force will increase as drying time goes by after the threephases boundary depins. Figure 4 shows the estimated forces depending on the particle size acting on a single PSB under ambient conditions. The analyses were carried out before and at the moment of the receding three-phase boundary. In the early stage of droplet evaporation (i.e., before receding boundary), the drag force (on the order of 10-12 N) is the dominant factor governing the particle motion in the droplet because most particles move toward three-phase boundaries. So there is no surface tension force when the contact line is pinned. However, the effect of the surface tension force, which is actually the required energy to extend the surface between liquid and solid, is relatively large when the contact line is depinned (i.e., at the receding boundary), so it becomes important when the contact line recedes. As the contact line recedes, the drag force becomes negligible because the surface tension force becomes greater than the drag one as shown in Figure 4. Static Coefficient of Friction. The static coefficient of friction was obtained by using eq 2 and the experimental and analytical results. The estimated one is ∼0.03 × 10-4, which is very small compared to 0.0206 obtained by Niida et al.26 Although the theoretical values of the van der Waals and electrostatic forces obtained by previous study are comparable to ones estimated in this study, the drag forces to begin to move the particle are very different between previous and present studies. Niida et al. used the critical velocity instead of the average velocity, while we used the average velocity obtained from experimental results. (28) Shin, D. H.; Lee, S. H.; Jung, J. Y.; Yoo, J. Y. Microelectron. Eng. 2009, 86, 1350–1353.

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CONCLUSIONS Understanding single particle motion in an evaporating colloid sessile droplet on plate is very important in the micro/nano/bio applications. In this study, the motion of particles in a dilute didisperse colloid solution containing 0.5 and 5.0 µm diameter PSBs on the hydrophilic surface is investigated, while the droplet evaporates out. The 0.5 µm diameter PSBs moved to the rim and stuck near the fixed triple contact line among the droplet, air, and substrate, while the 5.0 µm diameter PSBs moved toward the center of the droplet. To elucidate the impact of various forces including the drag, electrostatic, van der Waals, and surface tension forces on a single 5 µm diameter PSB motion in suspension, the magnitudes of them are compared using the scale analysis. In the early stage of the evaporation, when the contact line is fixed, the motion of a single particle suspended in liquid is mostly affected by the drag force. At the incidence of the contact line recession, most particles lie on the substrate and partly submerge in the liquid layer and the surface tension force takes over the control of the motion of single particle. The static coefficient of friction is obtained ∼0.03 × 10-4 by using experimental and analytical results, which is very small compared to previous results. The observations and analyses above provide the necessary foundation for separating, sensing the biocells, and patterning the micro/nanoparticles and the biocells using the evaporating droplet on a hydrophilic plate. ACKNOWLEDGMENT This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) (Grant No. R01-2008-000-20458-0(2008)). Received for review October 9, 2009. Accepted January 7, 2010. AC902288Z