Fast Drop Movements Resulting from the Phase Change on a Gradient Surface

See allHide authors and affiliations

Science  26 Jan 2001:
Vol. 291, Issue 5504, pp. 633-636
DOI: 10.1126/science.291.5504.633


The movement of liquid drops on a surface with a radial surface tension gradient is described here. When saturated steam passes over a colder hydrophobic substrate, numerous water droplets nucleate and grow by coalescence with the surrounding drops. The merging droplets exhibit two-dimensional random motion somewhat like the Brownian movements of colloidal particles. When a surface tension gradient is designed into the substrate surface, the random movements of droplets are biased toward the more wettable side of the surface. Powered by the energies of coalescence and collimated by the forces of the chemical gradient, small drops (0.1 to 0.3 millimeter) display speeds that are hundreds to thousands of times faster than those of typical Marangoni flows. This effect has implications for passively enhancing heat transfer in heat exchangers and heat pipes.

The movements of liquids resulting from unbalanced surface tension forces constitute an important surface phenomenon, known as the Marangoni effect (1). When regulated properly, these types of flows are of value in several industrial applications, such as the design and operation of microfluidic and integrated DNA analysis devices (2–4). Although the usual Marangoni motions are triggered by variations in temperature or composition on a liquid surface, a surface tension heterogeneity (5–7) on a solid substrate can also induce such motion. The typical speeds of these flows (speeds ranging from micrometers to millimeters per second) on a solid surface are too slow to have practical utility.

The main obstacle to drop motion on a solid surface arises from the hysteresis of contact angles that pin the drop edge. In order to surmount hysteresis, additional energy must be supplied to the drop while using the force arising from the gradient of surface tension to bias the drop motion. Here we report a new type of surface tension–guided flow, in which the drops move hundreds to thousand of times faster than the speeds of typical Marangoni flows. Furthermore, the experimental configurations used here provide a direct indication of how this effect can be harnessed to design improved heat exchangers. The central theme of these studies is that of rapid phase change, coupled with the fast removal of heat from steam condensing on a gradient surface.

The test surface for this study had a radially outward gradient of chemical composition that was prepared by diffusion-controlled silanization (8). A small drop (∼2 μl) of alkyltrichlorosilane [Cl3Si(CH2)nCH3,n = 8 to 14] was held about 2 mm above the center of a clean silicon surface. The silane evaporated from the drop and diffused radially while reacting with the silicon (Si/SiO2) surface. The central part of the silicon surface, closest to the drop, became maximally hydrophobic, with the contact angle of water about 100o, whereas its peripheral zone remained wettable by water with a near-zero contact angle. In air, small drops (1 to 2 mm) of water move on such a surface radially outward with typical speeds of 0.2 to 0.3 cm/s. However, when saturated steam (100°C) condenses on such a surface, even smaller drops (0.1 to 0.3 mm) attain speeds (9) that are two orders of magnitude higher than those observed under ambient conditions (Fig. 1) (10).

Figure 1

Video prints showing fast movements of water drops (indicated by the plume- and streaklike appearances) resulting from the condensation of steam on a silicon wafer possessing a radial gradient (1 cm diameter) of surface energy. (A) and (B) show that drops 1 and 3 leave their original positions to cross the edge of the gradient in about 0.033 s. These movements leave behind streaklike appearances, which are due to the flashes of light reflecting from the drops during fast movements. The central drop (2) does not initially move, as it is on the weakest part of the gradient. However, after 0.47s, as the drop becomes off-centered, it rapidly crosses the gradient (C to D) zone.

How do the drops acquire such high speeds? A loss of interfacial resistance by a hydrodynamically lubricating water film on the surface could speed up drop motion, but on a nonwettable surface, where the contact angle is nonzero, a liquid film breaks up to produce numerous disconnected droplets. Previous studies (11) detected only a molecular or submolecular film of adsorbed water between condensed droplets, which are unlikely to provide substantial hydrodynamic lubrication to the moving drops.

Other plausible scenarios are that the wetting hysteresis is bypassed under rapid condensation and/or that additional energy is supplied to the drop to surmount hysteresis. We examine these scenarios in some detail. When a liquid drop is placed on a surface with an energy gradient, the Laplace pressure within the drop attempts to equilibrate quickly (12, 13). Thus, the drop assumes the shape of a semispherical cap (14) to create nearly equal contact angles (θd) at edges A and B (Fig. 2). However, because θd is greater than the advancing contact angle at B (θaB), but less than the receding contact angle at A (θrA), the uncompensated wetting forces act at both A and B, which propel the whole drop toward the region of higher wettability. Typical speeds of such motion depend on the drop size and hysteresis but are in the range of a few millimeters per second. In the presence of fast condensation, however, the drop movement can be aided by the direct condensation of steam and by coalescence with other drops.

Figure 2

Schematics of a 1D wettability gradient of a surface. θa and θr represent the advancing and receding contact angles, respectively. In the absence of hysteresis, the driving force for the drop motion is provided by the difference of the equilibrium contact angles at points at B and A. Hysteresis reduces the driving force (dotted arrow). The upper right inset shows that additional driving force can be gained from coalescence with other droplets, which nucleate and grow ahead of the main drop.

Let us consider the case of small drops, which do not move on the surface because of hysteresis but grow by the condensation of steam. Condensation causes the contact angles at A and B to surpass θaA and θaB. However, as the Laplace pressure attempts to equilibrate within the drop, the dynamic contact angles at A and B now assume an average value of θaA and θaB, which is greater than θrA but smaller than θaA. Thus, the contact line at A neither advances nor recedes. However, as the dynamic contact angle at B is greater than θaB, it moves toward the region of higher wettability. The characteristic velocity of B by condensation (11) isV 1 ∼ 2k wΔT/(ρHR), wherek w is the thermal conductivity, ρ is the density, H is the heat of vaporization of water,R is the base radius of the drop, and ΔT is the degree of subcooling of the substrate surface from that of steam. For drops from 0.1 to 0.3 mm, the drop edge moves with a velocity of about 20 to 40 μm/s and catches up and coalesces with other droplets condensing ahead of it. At a high rate of nucleation, the rate of coalescence can be so fast that edge B advances far more rapidly thanV 1. θd at edge A thus continues to decrease until it becomes smaller than θrA, when the whole drop is dragged toward the more wettable part of the gradient.

A rough estimate of drop speeds (V) resulting from coalescence can be made by considering two drops of equal size of radiiD coalescing with each other, in which the drop edges move by a net distance of ∼0.37D. We consider a situation of low contact angle (θ ∼ 45°) so that the drop motion can be described approximately by lubrication theory (15). The energy budget of the process is the difference between the interfacial energies of the liquid drop before and after coalescence [∼0.1πD2γf(θ)], where γ is the surface tension of the drop andf(θ) = (2–3cosθ + cos3θ)/sin2θ. Considering that the time scale of coalescence τ = 0.37D/V is governed by the viscous drag forces at the contact line, the total energy dissipation is estimated to be 6ηV 2 Dατ/θ, where η is the viscosity of the liquid and α is a logarithmic factor (∼12) required to prevent divergence of shear stress at the contact line. Balancing of the two energies leads to the speed of coalescence approaching 1 m/s (16).

For saturated steam condensing on a gradient surface, these fast motions are biased toward the higher energy side of the gradient (17) and are facilitated by at least three factors. First, a drop always grows by condensation toward the more wettable part of the gradient and thus catches up with other drops condensing ahead of it. Second, the rate of nucleation increases with wettability; hence the probability of coalescence at edge B is higher than that at edge A (Fig. 2). Finally, as the drop starts moving, it grows in size by sweeping up other drops in its path and leaves behind a cleaner area for fresh condensation to begin. The drop gains more energy by coalescing with the larger drops ahead of it, not with the freshly nucleating drops behind it. The process thus autoaccelerates in order to produce a net movement of the drop toward the region of higher wettability.

The coupling between the surface tension gradient and the condensation of steam is evident in that the droplet speed slows down as soon as the steam flow is shut off (18). In the quiescent state, condensation continues at a slow rate and only slow motion of drops is observed. Occasionally, drops coalesce and the resultant drop moves toward the region of higher wettability (19). On a homogeneously hydrophobic substrate, the droplets grow and coalesce randomly. with the surrounding drops leading to a two-dimensional (2D) random walk (20). During the coalescence, however, drop edges retract with speeds comparable to the biased movements of condensing drops on a gradient surface.

Such coalescence-induced drop motion can be harnessed to enhance the performance of heat exchangers. In common heat exchangers, a metallic wall separates a cold water stream from a superheated vapor. Heat released from the condensation of steam is conducted through the metallic wall and transferred to the cold fluid. In practice, the heat flow is compromised by the thin insulating liquid film that accumulates on the steam side of the heat exchanger. Although various gravity-driven methods have been designed to remove the insulating water (21) from the surface of heat exchangers, there exists no good mechanism to remove water from the horizontal surfaces of heat exchangers and from those operating in microgravity situations. Here we describe a mechanism of water removal that takes advantage of the surface energy gradient.

Figure 3 provides the schematic of a heat exchanger made up of a cylindrical block (5 cm in diameter) of copper (22), the curved side of which is insulated with thick Teflon in order to ensure unidirectional (axial) heat conduction. The steam side of the copper block was prepared by polishing it to a mirror finish and then coating it with a thin film (∼100 nm) of silicon by vacuum deposition. A radially outward gradient of octyltrichlorosilane was then formed on this surface by means of diffusion-controlled silanization. Saturated steam (100°C) was passed over the gradient surface, while the other end of the block was cooled by cold water. As the condensing drops are continually removed by the surface tension gradient, the surface of the heat exchanger is renewed (23), thus enhancing heat flux through it. Figure 4 compares the heat transfer coefficients of the steam side of the heat exchanger under two situations: filmwise condensation on a horizontal unmodified surface and dropwise condensation on the gradient surface (24). The surface treatment enhances the heat transfer by a factor of 3 when the surface is subcooled (ΔT) from that of steam by about 20° and by a factor of 10 when subcooling is about 2°.

Figure 3

Schematics of a heat exchanger. The heat flux (J q) through the surface is estimated by knowing the conductivity (k cu) of the copper block and the temperature gradient (dT/dy) as follows: J q =k cu dT/dy. The temperature gradient within the block was measured with four thermocouples inserted radially in different positions of the block, from which the temperature of the copper-steam interface (T 0) was extrapolated. The difference between T 0 and that of steam (T s) gives the degree of subcooling of the surface.

Figure 4

The heat transfer coefficient of the copper block is enhanced with a chemical gradient. The heat transfer coefficient (Nu) is expressed in a dimensionless form as follows: Nu =J q D/(k wΔT). Here, J q is the heat flux, D is the diameter of the copper block, k w is the conductivity of water, and ΔT is the degree of subcooling of the condensed liquid. The closed circles (•) correspond to a siliconized copper surface, where steam condenses as a thin film. The open circles (O) correspond to the gradient surface, where steam condenses into drops that are removed by the surface tension gradient.

This methodology is also applicable on vertical surfaces, where the surface tension forces can be used in conjunction with gravity. This concept was demonstrated by preparing a 1D periodic gradient on a silicon wafer by diffusion-controlled silanization. Here the condensing drops are propelled from the hydrophobic zones and accumulate in the narrow hydrophilic channels to ultimately drain down because of gravity (25). A comparison has been made between such a drop removal mechanism and the usual method of promoting dropwise condensation by adding a surfactant (such as oleylamine) in the steam or hydrophobizing the surface by an organic coating (such as by silanization). Both the above methods yielded similar heat transfer coefficients [Nusselt number (Nu) ∼ 1300], whereas the gradient surface under similar operating conditions exhibited a higher heat transfer coefficient (Nu = 2750). Clearly, surface tension forces can enhance the effectiveness not only of horizontal heat exchangers but of vertical heat exchangers as well. A heat exchanger surface possessing a periodic gradient has the additional advantage that its size can be scaled up in both horizontal and vertical directions without compromising the benefits of the gradient.

The phenomenon described here could also be useful in other types of heat transfer problems involving two-phase flow, an example of which is capillary force–driven thermosyphoning in heat pipes. A capillary force–driven heat pipe (26) consists of a closed container that has a heating and a cooling section. A fluid is thermally evaporated in the heating section, the vapor of which flows to the cooling section and condenses. The condensed fluid returns to the heating section by capillary action. The circulation of fluid and the corresponding heat transfer thus created within the heat pipe have a wide range of applications, including temperature stabilization in aerospace situations, thermosyphoning in Rankine engines, removal of heat from integrated microelectronic chips and controlled cooling of human body parts during cryogenic surgery. The standard way of pumping a fluid from the condensation to the evaporation section uses a wicking material, which is placed against the inner wall of the heat pipes. It is possible that the capillary pumping inside a heat pipe can be facilitated by designing a surface tension gradient in its inner wall.


View Abstract

Navigate This Article