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Force Measurements for the Movement of a Water Drop on a Surface with a Surface Tension Gradient Hitoshi Suda* and Satoshi Yamada Department of Biological Science and Technology, Tokai University, 317 Nishino, Numazu, Shizuoka 410-0321, Japan Received August 15, 2002. In Final Form: November 20, 2002 On a solid surface with a surface tension gradient, a liquid drop is driven toward the more wettable end. To investigate the mechanism for the movement of the drop, we first measured directly the driving force of the drop, using a flexible glass microneedle. We determined that the driving force results from an imbalance of surface tension acting on the liquid-solid contact line on the two opposite sides of the drop. The hydrodynamic force, which is conventionally used as a balance force against the driving force, was not compatible with our results. Here, we show that a balanced drag force with the driving force has to be reconstructed, using a new concept of solidlike friction.
I. Introduction The movement of liquid drops on a gradient of surface free energy is a well-known autonomous transport system that automatically moves without an external force.1-5 This phenomenon is obviously important in the design and operation of microfluidic devices6,7 as well as in understanding the interface phenomena of friction8,9 and biological problems10 such as cell motility, muscle contraction,11 and organelle transport. The driving force in this movement system is generally believed to be the difference of wettability acting on the front and rear ends of the drop. The equation is approximately described by the unbalanced Young force (FY):3
FY ) LγLV(cos θa - cos θr)
(1)
where γLV is the surface tension of the liquid at the liquidvapor interface, L is the length between the front and rear ends of the drop, and θa and θr represent the advancing and receding contact angles for the drop, respectively. The contact angle θ and the length L depend on the position of the drop on the solid surface. The equation of hydrodynamic friction force (FH) for the drop is given as follows by assuming that the drop profile is circular.1
FH = 3πηRV ln
( ) xmax xmin
(2)
where η is the viscosity of the liquid, R is the base radius of the drop, V is the drop velocity, and xmin and xmax are two cutoff lengths, the first being the molecular dimension and the second on the order of the drop radius. If we put xmax ) 1 mm, xmin ) 0.1 nm, V ) 1 mm/s, and η ) 1 cp, * To whom correspondence should be addressed. Phone +8155-968-1111. Fax: +81-55-968-1156. E-mail:
[email protected]. (1) Brochard, F. Langmuir 1989, 5, 432. (2) Chaudhury, M. K.; Whitesides, G. M. Science 1992, 256, 1539. (3) Lee, S.-W.; Laibinis, P. E. J. Am. Chem. Soc. 2000, 122, 5395. (4) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Science 2001, 291, 633. (5) Daniel, S.; Chaudhury, M. K. Langmuir 2002, 18, 3404. (6) Gau, H.; Herminghaus, S.; Lentz, P.; Lipowsky, R. Science 1999, 283, 46. (7) Gallard, B. S.; et al. Science 1999, 283, 57. (8) Chaudhury, M. K. J. Phys. Chem. B 1999, 103, 6562. (9) Ghatak, A.; Vorvolakos, K.; She, H.; Malotky, D. L.; Chaudhury, M. K. J. Phys. Chem. B 2000, 104, 4018. (10) Suda, H. Langmuir 2001, 17, 6045. (11) Suda, H.; Ishikawa, A. Biochem. Biophys. Res. Commun. 1997, 237, 427.
then FH is 0.015 dyn. On the other hand, if we substitute γLV ) 73 dyn/cm, L ) 1 mm, and cos θa - cos θr ) 0.06 into eq 1, then FY ) 0.43 dyn. According to this estimation, the magnitude of expected hydrodynamic drag force is at least about 30 times less than that of the driving force. This result suggests that the driving force is not able to fully counteract the hydrodynamic friction. Indeed, we experimentally determined using a direct force detection method that the estimation was correct. Thus, we need another friction instead of the hydrodynamic friction as a drag force to explain our experimental results. To test whether eq 1 is really acceptable, we measured directly the driving force by using a flexible glass microneedle. The second aim in this study is to quantify the relationship between the velocity and the driving force. The dynamics of the drop are usually explained by a force balance between the driving and the hydrodynamic drag forces on the moving drop.1 Recently, Lee and Laibinis3 reported that the velocity was proportional to the unbalanced Young force and that this relation could be explained by the conventional interpretation. On the other hand, Daniel and Chaudhury5 showed that the velocity of the drop was proportional to the radius of the drop, but it became zero at the finite value of the radius. The latter group’s result is partially inconsistent with that of the former group. In the present paper, we show that the velocity behaves as an exponential of the driving force. Our results partially contradict those of the two groups. The reason for the difference will be discussed in detail later. II. Experimental Methods and Results In our experiment, the method was basically the same as that of Chaudhury and Whitesides2 except for the use of a glass slide (Matsunami, No.1, S-7213) instead of a silicon wafer as a substrate. The surface tension gradient was generated by exposing glass slides to the diffusing front of a decyltrichlorosilane vapor, Cl3Si(CH2)9CH3. The slides were housed in a plastic case to protect from disturbance of air. The slides were cleaned by immersion in KOH-saturated ethanol. After that, the substrate surface was thoroughly washed with pure water and supersonic cleaning. Then, the surface was dried on a clean bench. The glass microneedle was made from a glass capillary (Drummond Scientific Company, 10 µL) by using a pipet puller (Narishige, Model PB-7, Japan). The stiffness of the microneedle was determined by the method of Ka-
10.1021/la0264163 CCC: $25.00 © 2003 American Chemical Society Published on Web 12/31/2002
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Figure 1. Typical example for the direct measurement of the driving force of a water drop. We measured the driving force by using a glass microneedle. Open circles denote the force FN produced by the deflection of the microneedle, while crosses represent the calculated driving force on the basis of eq 1, FY. The abscissa Dc denotes the position of the top of the drop on the substrate, while at this time the position of the needle is represented at that of the drop. The arrow shows the pausing state of the drop. The inset indicates a real image of force detection. The stiffness of the microneedle was 4.25 dyn mm-1, and the applied volume of the drop was 2.5 µL.
mimura.12
The force, FN, was detected by the glass microneedle and was found by multiplying the deflection of microneedle by its stiffness. Also, we measured the speed of the drop as a function of the tilt of the substrate. The tilt was controlled using a hinge attached to the reverse side of the glass slide. The contact angle, the speed, and the deflection of the needle were analyzed on a computer by using images recorded by an S-VHS video recorder through a CCD camera, using a video-capturing system. All experiments were performed at 20-25 °C. Figure 1 shows a typical example of direct measurement of driving force in the movement of the water drop on the surface gradient substrate. The glass microneedle was inserted into the water drop as seen in the inset of Figure 1. When the inserted needle moved with the drop on the surface gradient substrate, the needle gradually deflected with the drop. Since the tip of microneedle was thin, that passed through the droplet in some region. The drop and the needle stopped soon at a balance point of both forces. When the glass needle was removed from the drop at this pausing state, the drop started to move again. Such a situation is shown in the figure. The driving force of the drop was derived from the opposing force of the deflected needle at this standstill position. At this position, the force by the needle was very close to the unbalanced Young force given by eq 1. Accordingly, this fact suggests that eq 1 is suitable to express the driving force. To investigate an effect of external load, we sloped the glass base from the horizontal plane to 70°, where the hydrophobic end was lowered relative to the hydrophilic end. The weight of the drop itself became equal to the external load (Fg) against the movement. A drop of water (2 µL) placed at the hydrophobic end ran uphill toward the more wettable top. As shown in Figure 2, the relationship for V and Fg was not linear, but rather the (12) Kamimura, S.; Takahashi, K. Nature 1981, 293, 566.
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Figure 2. Gravity dependence of the uphill water drop. The measured velocity (O) was averaged at 1.5-2.0 mm on the substrate. The volume of the drop was 2 µL. The nonlinear least-squares fitting equation was given by V ) 1.12 sinh(1.86(1.70 - Fg)) and r ) 0.975, where Fg represents mg sin θ, m is the mass of drop, g is the gravitational constant, and mg ) 2 dyn in this case.
speed exponentially decreased with increasing tilt of the substrate from a horizontal plane. We investigated the relation between the velocity and the driving force in order to verify whether eq 5 is correct. In Figure 3, two experimental results are shown. As soon as we placed a drop on a substrate, it moved toward the higher wettability. A slight pulsation was observed at both ends, so the velocity was defined as the time change of the position Dc for the top of the drop. As seen in Figure 3, the velocity increased exponentially as the driving force increased. We can exclude the possibility that the velocity is proportional to the driving force, whereas we cannot exclude the possibility that it becomes zero at a finite value of force. We could not conduct an experiment at extremely low velocities to give an observation time of 1 h, because we could not neglect the evaporation of water, even for a period of just 3 min. So we could not eliminate the possibility of V ) 0 at FY * 0. III. New Friction Theory and Discussion Here, we propose a friction theory to explain our experimental results. Let us assume that the contact surface between a drop and a substrate is composed of N bonds. In our model, which is named the discrete model, the water layers of the drop in the vicinity of the substrate were not regarded as a hydrodynamic continuum but rather an assembly of N small solidlike domains. Their domains were also assumed to have the same average area and shape and to coordinate mutually. In this paper, we present the discrete model against the conventional continuum theory. This means that each domain behaves like a solid. The mean velocity (V) of drop may be given as follows.13
V ) l(k+ - k-)
(3)
where l is the mean distance between nearest neighbor domains, and k+ and k- represent the dissociation rate
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Langmuir, Vol. 19, No. 3, 2003 531
be the same as that of the forward motion, and k0 represents the spontaneous dissociation constant without any force. The intrinsic bond energy is included in k0.10 Also β-1 ) kBT, where kB is the Boltzmann constant and T is the absolute temperature. Accordingly, in the bias mode of fX (*0), the following equation is derived from eqs 3 and 4.
V ) 2lk0 sinh(βfd)
(5)
This equation is revealed in various fields8-10,13,14 and should be generally accepted as a new law of friction. If the externally applied load (FE) per drop acts in parallel along the movement, then f must be replaced with (FY FE)/N. Therefore, the force-velocity relation of the drop is rewritten by
V ) 2lk0 sinh[(FY - FE)]
Figure 3. Velocities vs driving forces. The results of two different experiments are shown, where the substrates were different. (A) The applied volume of the drop was 2 µL. The experiments were carried out repeatedly three times. The inset is the normal plot. The solid lines are the fitting curves that were calculated on the basis of eq 5: V ) 0.0164 sinh(4.84FY), r ) 0.924. The error bars are the standard deviation. The average of the force was calculated from all forces that corresponded to the arranged velocity at the step of 0.05 mm/s. (B) To widen the dynamic range of speeds, we used 2-4 µL as the applied volume. The experimental data fit well with V ) 0.0646 sinh(3.71FY) and r ) 0.825.
constants of forward and backward motion of each binding domain, respectively. As f ()FY/N) is the average external driving force per bond, the energy barriers for the forward and backward motions are biased by -fX, where X represents the reaction coordinate of the potential on the substrate. Thus, the dissociation rate constants13 are given by
k+ ) k0 exp(βfd) and k- ) k0 exp(-βfd)
(4)
where the d-value of the backward motion is assumed to
(6)
where ) βd/N. In Figure 2, the velocity was shown to be nonlinear against the external load. The data fit well with eq 6, where we replaced FE with Fg. In fact, as the tilt increased, the shape of the drop changed and the driving force was influenced by the tilt (data not shown). Thus, although we should not analyze simply with eq 6, the concave behavior seems to rule out the hydrodynamic effect. In Figure 3, we also showed that the velocity increased exponentially as the driving force increased. The experimental data were fitted well with eq 5. The decay constant ()βd/N) in eq 6 is characteristic of the movement system. From the determination of the fitting parameter, we obtained βd/N ) 3.7-11 dyn-1. On the basis of this value, let us estimate the number and the size of the domains for the discrete model. If the interface or the bound water layer between the water drop and the substrate is assumed to be composed of N domains, then N ≈ L2/4d2, where domains are assumed to have the same diameter 2d. Taking into account L ) 1-2.5 mm as the size of drop, we may obtain d ≈ 0.07-0.2 µm and N ≈ (2.7-120) × 107. Although this estimation is rough, the size of the solidlike domain that probably exists is evaluated to be on the order of 0.1 µm. Our results indicate clearly that a slight change of force causes a big change of velocity. This implies that the adhesion force is essentially dominant against the velocity of the movement. If the separation force is externally acting between a drop and a substrate, it is predicted that the sliding velocity of the drop increases remarkably in comparison to the case of the absence of separation force. The fast movement (∼1 m/s) of the drop4 in a saturated steam appears to be due to such an explanation, because probably the adhesion force between the water drop and the silicon wafer or the intermolecular force was remarkably weakened in the saturated steam. At the transition state, the hydrodynamic friction apparently balances with the driving force. In eq 5, this corresponds to the decrease of intrinsic bond energy that is revealed in k0. Even if the adhesion force between two surfaces decreased only several times, the velocity would become hundreds to thousands times faster. Indeed, this effect seems to be revealed in the experimental results of Figure 3, where we replace the driving force with the adhesion force in this interpretation. In the present work, experimental results could be explained well by our new friction theory. LA0264163 (13) Suda, H. J. Jpn. Soc. Tribol. 2001, 46, 324. (14) Heslot, F.; Baumberger, T.; Perrin, B.; Caroli, B.; Caroli, C. Phys. Rev. E 1994, 49, 4973.
