Article pubs.acs.org/Langmuir
Measurement of Capillary Radius and Contact Angle within Porous Media Saitej Ravi, Ramanathan Dharmarajan, and Saeed Moghaddam* Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida 32611, United States ABSTRACT: The pore radius (i.e., capillary radius) and contact angle determine the capillary pressure generated in a porous medium. The most common method to determine these two parameters is through measurement of the capillary pressure generated by a reference liquid (i.e., a liquid with near-zero contact angle) and a test liquid. The rate of rise technique, commonly used to determine the capillary pressure, results in significant uncertainties. In this study, we utilize a recently developed technique for independently measuring the capillary pressure and permeability to determine the equivalent minimum capillary radii and contact angle of water within micropillar wick structures. In this method, the experimentally measured dryout threshold of a wick structure at different wicking lengths is fit to Darcy’s law to extract the maximum capillary pressure generated by the test liquid. The equivalent minimum capillary radii of different wick geometries are determined by measuring the maximum capillary pressures generated using n-hexane as the working fluid. It is found that the equivalent minimum capillary radius is dependent on the diameter of pillars and the spacing between pillars. The equivalent capillary radii of micropillar wicks determined using the new method are found to be up to 7 times greater than the current geometry-based first-order estimates. The contact angle subtended by water at the walls of the micropillars is determined by measuring the capillary pressure generated by water within the arrays and the measured capillary radii for the different geometries. This mean contact angle of water is determined to be 54.7°.
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with time. The rate of liquid progression is fit to theoretical models based on solutions to the continuity and momentum equations. When the effects of gravity and evaporation are neglected, the model is similar to the Lucas−Washburn equation,13,14 and the fitting yields the product of the capillary pressure and the permeability of the medium. The capillary pressure generated in the medium can be calculated by using a theoretical model relating the geometry of the medium to its permeability. If the effects of gravity are included in the theoretical model, both the capillary pressure and the permeability of the medium can be independently determined.15−17 While the simplicity of the rate of rise method has resulted in widespread use, the high sensitivity of this method to errors in experimental data makes it unsuitable for precise measurements of the capillary radius and contact angle. For example, in a study by Adkins and Dykhuizen,18 an error of 4 mm in the position of the moving liquid front caused an 80% change in the permeability value. Such high sensitivities to the liquid front position imply that only approximate values of capillary pressure (capillary radius and contact angle) and permeability can be estimated from the rate of rise method. The accuracy of the rate of rise method can be improved by
INTRODUCTION The propagation of liquids through a porous medium occurs due to the action of a capillary force generated within the pores of the medium. The interactions between the phases at the triple-phase boundary and the pore size (capillary radius) in the solid phase determine the magnitude of this force. The interaction at the triple-phase boundary manifests itself as the solid−liquid contact angle. Accurate estimates of the capillary radius and contact angle are fundamental to understanding the behavior of systems such as groundwater filtration,1−3 pharmaceutical drug release,4 passive two-phase devices,5,6 mineral flotation,7,8 etc. The equivalent capillary radius (Rc) of a medium can be calculated by measuring the capillary pressure generated within the medium by a liquid of known contact angle. Generally, the liquid used for this measurement completely wets the solid surface (contact angle, θ ∼ 0°). The Rc of the medium is then calculated from the capillary pressure using the Young−Laplace equation: Pc =
2σ cos θ Rc
(1)
The most widely used method to characterize the capillary pressure of porous media is the rate of rise (fall) method,9−12 where the rise (fall) of liquid in a porous medium is recorded © 2015 American Chemical Society
Received: August 20, 2015 Revised: November 3, 2015 Published: November 5, 2015 12954
DOI: 10.1021/acs.langmuir.5b03113 Langmuir 2015, 31, 12954−12959
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Figure 1. (a) Scanning electron micrograph of the micropillar array in device A, (b) typical image of a device with the wick structure and etched ruler (the evaporator region is highlighted in red), (c) image of the reverse side of a device consisting of a thin-film heater and embedded temperature sensors, and (d) image of a thin-film temperature sensor within the evaporator.
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measuring the change in weight of the porous medium with respect to time (as the liquid rises through the medium) instead of the position of the liquid front.19 An alternate method for measuring the capillary pressure of porous media is the bubble point method, which estimates the capillary pressure by measuring the minimum pressure required to force an air bubble through a porous medium saturated with the test liquid. Since the smallest pores generate the maximum capillary pressure in a medium while air flows through the largest pores, this method only measures the size of largest pores in the medium. An added disadvantage of this method is its infeasibility in measuring the capillary radii of media bounded by a solid wall on one side, such as etched microstructures or nanowires grown on a planar substrate. The capillary pressure characterization methods described above have also been used to measure contact angles of liquids within porous media.19 This is because the common measurement techniques (such as the sessile drop method,20,21 the captive bubble method,22 the axisymmetric droplet shape analysis method,23−27 the Wilhelmy plate method,20 etc.) are unsuitable due to liquid imbibition into the media. First, the equivalent capillary radius of the medium (Rc in eq 1) is calculated from eq 1 using the capillary pressure generated by a completely wetting liquid (contact angle, θ = 0°). Subsequently, the contact angles of other test liquids are calculated from eq 1 using the capillary pressures generated by these liquids and the estimated value of Rc. However, the accuracy of the contact angle values obtained using this method depends on the accuracy of the previously discussed capillary pressure measurement techniques. In earlier studies,28−30 we introduced a new method to simultaneously and independently measure the capillary pressure and permeability of a porous medium. This method was used to characterize the capillary transport of micropillar wicks etched on silicon substrates. In the current study, we demonstrate the implementation of this method to measure the equivalent minimum capillary radius and the in-pore contact angle of a porous medium. In the following section, brief descriptions of this experimental technique and the devices tested in this study are provided. The minimum equivalent capillary radii of different micropillar geometries are calculated by measuring the maximum capillary pressures generated by nhexane within these geometries. n-Hexane is used as the reference liquid because of its low surface energy and near zero contact angle.17,31 The capillary radii of the different devices are then used to determine the contact angle of water within each device.
EXPERIMENTAL PROCEDURE
Device Fabrication. The porous media investigated in this study were composed of arrays of micropillars fabricated on silicon substrates. Each device measured 6 cm by 1 cm (Figure 1b) and was composed of an array of micropillars on one side and a platinum thin film heater with embedded temperature sensors on the other side (Figure 1c,d). The evaporator (0.5 cm by 1 cm) was aligned along one end of the device. Micropillar arrays were etched on the other side of the wafer through a deep reactive ion etching process (Figure 1a). The wettability of the etched surface was enhanced by coating the surface with 2000 Å of silicon oxide (plasma-enhanced chemical vapor deposition). A detailed description of the fabrication process can be found in our earlier study.28 The geometric dimensions of the devices tested in this study are summarized in Table 1. Device C was selected
Table 1. Geometric Dimensions of the Devices Characterized in This Study device
diameter, d (μm)
spacing, w (μm)
height, h (μm)
A B C D E F G H I
15 15 15 15 15 19 22 40 42
15 23 30 30 30 30 30 30 48
100 100 100 85 150 100 100 100 100
as the base geometry, and the geometry of each device differed from that of device C in only one dimension. These geometries were selected in order to study the individual effect of each dimension on the capillary radius. Test Methods. The devices were tested in a 6 in. stainless steel, vacuum compatible chamber (Figure 2) capable of maintaining saturated conditions. The top flange of the chamber included a translatable arm to adjust the device position with respect to the liquid pool. An electrical feedthrough was also included in this flange to connect the heaters and sensors on the devices to an external data acquisition system. A thermoelectric cooler, attached to one of the flanges, functioned as a condenser. Devices were mounted on a polycarbonate holder attached at the bottom of the translatable arm. The chamber was connected to a vacuum pump and pumped down for 12 h before distilling the test liquid (n-hexane or deionized water) into the chamber. This method ensured that the chamber was at saturated conditions with respect to the liquid (saturation temperature of 24 °C during experiments). Once the desired volume of liquid was distilled into the chamber, the device was partially lowered into the liquid. The wicking length of the device (L, height between top of device and liquid reservoir surface) was measured using a ruler etched adjacent to the micropillar array (Figure 12955
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heat loss was calculated to be 0.26 W, which was less than 3% of the total heat input at a wicking length of 2 cm for all devices (with water as the working fluid). Therefore, conduction losses through the wick were neglected in further analyses. Liquid flow through the wick structures was modeled using Darcy’s law (eq 2). The dryout threshold of the wick structures was related to the mass flow rate using the latent heat of vaporization of the liquid (eq 3). The experimental data on the dryout threshold vs wicking length was fit to eq 3 (with the mass flow rate from eq 2), while considering the capillary pressure (Pc) and the permeability (K) as the unknown parameters. ṁ =
ρKA ⎛ Pc − ρgL ⎞ ⎜ ⎟ ⎠ μ ⎝ L
Q = mh ̇ fg
(2) (3)
In these equations, ρ is the density of the test liquid, μ is the viscosity of the test liquid, A represents the flow cross-sectional area, g is the acceleration due to gravity, Q is the dryout threshold, and hfg is the latent heat of vaporization of the test liquid. The outcome of the Darcy fits were verified by comparing the results to those obtained from the solution to the Brinkman equation for flow through porous media (eq 4).32 Liquid transport through the medium was assumed as one-dimensional and steady. A no-slip boundary condition was applied at the lower wall (y = 0), and zero shear stress was assumed at the liquid−vapor interface (y = h).
Figure 2. (a) Image of the test chamber showing translatable arm [A], pressure sensor [B], electrical feedthrough [C], fluid and vacuum port [D], and glass viewport [E]. (b) Images of devices installed inside the chamber as seen through the glass viewport; the top and bottom images show a long wicking length and a short wicking length test configuration, respectively. 2b). The power input to the heater was increased in incremental steps, with the system allowed to reach a steady state (fluctuations in temperature below 0.1 °C) after each step. A sharp increase in the evaporator temperature (measured using the embedded temperature sensors) indicated the onset of dryout. At this point, the power to the heater was turned off. The device was then lowered by 0.5 cm into the liquid, and the dryout threshold at the new wicking length was measured. Data were gathered in this manner up to a wicking length of 2 cm. At each wicking length, the dryout threshold was recorded at least three times. Additional details on the test methods can be found in our previous study.28 Measurement Uncertainty and Data Analysis. The maximum uncertainties in reading the liquid height, voltage, and current were ±0.05 cm, 0.03 V, and 0.005 A, respectively. Based on the uncertainties in measuring the current and voltage, the maximum error in the input power to the heater was found to be ±0.32 W. The maximum heat loss due to conduction (heat transfer due to conduction along the porous medium) occurred at a wicking length of 2 cm. At dryout in this configuration, the maximum temperature difference between the evaporator and the pool of water was measured to be 7 °C. Using the thermal conductivity of silicon, the conduction
μ dP d2u = − u + μeff 2 dx K dy
(4)
The value of μeff was assumed as μ/ϕ, where ϕ represents the porosity of the medium, as per Ochoa-Tapia and Whitaker.33 The mass flow rate of liquid obtained from eq 4 with the appropriate boundary conditions is
⎛ ⎛ ϕ ⎞ ⎞ ⎛ ⎛ ϕ ⎞ ⎞⎤ ⎛ Pc − ρgL ⎞⎡ ⎜ ⎟⎢tanh⎜ ⎜ ⎟ h⎟ − ⎜ ⎜ ⎟ h⎟⎥ ⎜ ⎝ ⎠ ⎟ ⎜ ⎝ ⎠ ⎟ ⎠⎢⎣ L μ(ϕ/K )3/2 ⎝ ⎝ K ⎠ ⎝ K ⎠⎥⎦ (5) Here, h represents the height of the pillars and W represents the width of the micropillar array. The dryout threshold corresponding to the Brinkman equation was calculated using eq 3 with the mass flow rate from eq 5. ṁ = −
ρWϕ
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RESULTS AND DISCUSSION The dryout threshold data with n-hexane and water as the working fluids are plotted in Figure 3. The data for each device
Figure 3. Experimental results of the thermal tests on devices A−I using (a) n-hexane and (b) deionized water as the working fluids. The error bars in the figure correspond to one standard deviation about the mean calculated from replicates. The line plots from the data fitting process using Darcy’s law (dashed lines) and Brinkman’s equation (solid lines) are also included for each data set. 12956
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recede into the wick.34 Just prior to dryout, the menisci in the evaporator recede to the base of the micropillars, resulting in the minimum capillary radius (and, hence, maximum capillary pressure). Therefore, the values reported in Table 2 correspond to the maximum capillary pressure and are greater than the capillary pressure generated under adiabatic conditions in the arrays. The minimum capillary radii (Rc,min) of the different geometries were calculated from the maximum capillary pressures generated by n-hexane (Table 2) and the Young− Laplace equation (eq 1). These calculations were based on the assumptions of a near-zero contact angle between n-hexane and silicon oxide17,31 and no interdependence between the Rc,min of a geometry and the contact angle of the test liquid. The individual effects of the array dimensions on the Rc,min can be inferred from Figure 4. At constant pillar diameter and spacing
are represented as the mean dryout threshold plotted at different wicking lengths; the error bars represent one standard deviation about the mean. While the smaller spacing between the pillars in device A generates a large capillary pressure, viscous losses due to low permeability result in lower dryout thresholds than the other geometries. The increase in the height of pillars from device C to E results in greater dryout threshold values because of the increase in permeability. The magnitude of this increase in dryout threshold is greater with water as the working fluid. Device I, which is based on the optimized geometry obtained in our previous study,30 exhibits the best thermal performance with n-hexane and is comparable to that of device E with water as the working fluid. The data plotted in Figure 3 were fit to eqs 2 and 3 based on the approach described in our previous study.28 The basic fitting algorithm from our earlier study was simplified by fitting the mean dryout thresholds instead of all data replicates to eqs 2 and 3. This approach improves the accuracy of the fitting, as evidenced by lower sum of the squares of error and higher adjusted R-square values when compared to the basic algorithm. Liquid properties at a saturation temperature of 24 °C were used in the fitting equations. The experimental data for each device was fit to Darcy’s law (eqs 2 and 3) and Brinkman’s equation (eqs 3 and 5), plotted as dashed and solid lines respectively in Figure 3. The Darcy and Brinkman line plots for each data set coincide and exhibit identical “goodness of fit” parameters. Since the capillary pressure and permeability are unknown in the fitting equation (Darcy or Brinkman), each data set yields a unique combination of these two parameters. The subject of the permeability of micropillar arrays was discussed and analyzed in detail in a previous study.28 The capillary pressures obtained from the Darcy and Brinkman fits for a given test liquid and device are identical, serving as a validation of the methodology and the measurements. The capillary pressures generated by n-hexane and water in each device are summarized in Table 2. It must be noted that the shape of the meniscus and the effective capillary radius are influenced by liquid evaporation and the thermophysical properties of the liquid. In the present study, the effect of temperature on the capillary pressure (and effective capillary radius) can be neglected due to the small changes in temperature. As the heat input to the evaporator increases, evaporation causes the local menisci to
Figure 4. Effect of each geometric dimension of the micropillar arrays on the minimum capillary radius of the micropillar array. The error bars represent the standard error.
(devices C, D, and E), the height of the pillar has a negligible impact on Rc,minincreasing the pillar height from 85 to 150 μm changes Rc,min by less than 10%. This result indicates that the curvature and shape of the liquid meniscus are independent of the pillar height. The minimum capillary radius increases linearly with the spacing between the pillars (Figure 4). However, the magnitude of Rc,min is much greater than the spacing; geometry A with pillars spaced 15 μm apart has a Rc,min of 50 μm while geometry C with a pillar spacing of 30 μm has a Rc,min of 113 μm. This is contrary to the first-order approximations used by Ding et al.35 and Hale et al.,36 who approximate the capillary radius as half the pillar spacing. The measured values indicate that approximating Rc,min as the distance between pillars located along the diagonal of a square unit cell could also lead to significant errors in estimating the capillary pressure. Finally, increasing the diameter of the pillar 2.6 times (devices C, F, G, and H) decreases Rc,min by a factor of 40%. Such a strong effect of the diameter on Rc,min at a constant spacing can be attributed to the effect of contact line pinning along the circumference of the pillars. The individual effects of pillar diameter and spacing on Rc,min suggest that the capillary radius is determined by the interaction
Table 2. Capillary Pressures Generated by the Different Geometries with n-Hexane and Water as the Working Fluidsa capillary pressure (Pa) device
n-hexane
water
A B C D E F G H I
711.5 428.3 319.4 345.5 324.5 350.7 498.7 562.5 346.3
1709.0 845.4 823.7 776.4 874.6 1097.0 750.1 940.7 809.3
a
Identical values of capillary pressure were obtained from Darcy’s Law (eqs 2 and 3) and Brinkman equation (eqs 3 and 5) for each geometry and liquid. 12957
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The difference between the contact angles on a flat surface and within the arrays can be attributed to two factors: (a) roughness caused by the presence of scallops (formed during deep reactive ion etching) on the pillar walls and (b) the mean contact angle obtained from the data fits accounts for dryout occurring at different temperatures and different wicking lengths. The measured contact angle lies within the range of values reported by Siebold et al. for water within a porous medium consisting of silicon oxide powder.19
of both these parameters and not by the spacing alone. This is evident when comparing the Rc,min values of devices C, E, and I (Table 3)despite increasing the pillar diameter 3-fold and the Table 3. Experimentally Measured Capillary Radii for Devices C, E, and I and the Values Estimated from Different Models from the Literature capillary radius (μm) device
diameter, d (μm)
spacing, w (μm)
this study
w/235,36
ref 37
ref 38
C E I
15 15 42
30 30 48
113.1 111.3 104.3
15.0 15.0 24.0
46.7 48.1 54.4
85.9 85.9 122.7
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CONCLUSION Accurate modeling and prediction of capillary-driven flow through porous media can be accomplished only through precise measurements of the contact angle and the equivalent capillary radius. In this study, we demonstrated the implementation of a technique, introduced in our previous study,28 to measure the equivalent minimum capillary radius (Rc,min) of a porous medium and the solid−liquid contact angle subtended by a test liquid within the pores. In this technique, the dryout threshold of a porous medium is measured at different wicking lengths with a reference liquid (n-hexane in this study) and the test liquid (water in this study). The maximum capillary pressure and the permeability of the medium are determined by modeling the experimental data using Darcy’s law. Considering micropillar arrays as porous media, the dryout thresholds of n-hexane and water were measured at different wicking lengths. Assuming a near-zero contact angle, the Rc,min of each geometry was determined from the maximum capillary pressure generated by n-hexane. Contrary to certain assumptions in literature, the minimum capillary radius was found to be up to 7 times greater than half the spacing between adjacent pillars. At constant pillar spacing, the pillar diameter had a significant impact on the Rc,min. The height of the pillars did not affect the Rc,min of the arrays. The minimum capillary radii data were then used to measure the contact angles subtended by water within the arrays. The mean contact angle of water within the micropillar arrays was 54.7°. The differences between the contact angles within the arrays and those on a flat surface can be attributed to the effect of surface roughness and the effect of the evaporator temperature during dryout.
spacing by 2.6 times, the Rc,min of device I is less than that of devices C and E. Table 3 also includes the theoretical capillary radii calculated using various literature models to facilitate a comparison with the experimental data. The capillary radii estimated by these models correspond to the equilibrium capillary radii and are lower than the experimentally measured Rc,min for all device geometries (which determine the maximum capillary pressure of a given wick). The first model is based on the first-order approximation proposed by Ding et al.35 and Hale et al.36 discussed in the previous paragraph. The experimentally calculated Rc,min are much greater (up to 7 times) than those calculated using this particular model. This is because the complex shape and curvature of the liquid meniscus are not determined by just the pillar edge-to-edge spacing but by the spatial arrangement of the four pillars forming a unit cell along with the diameter of each individual pillar. Therefore, an Rc,min based only on the spacing between adjacent pillars can significantly overestimate the capillary pressure, leading to very large errors in (a) modeling and predicting liquid transport in capillary-driven flows in wick structures and (b) permeability values calculated from the rate of rise experiments. The remaining two theoretical capillary radii in Table 3 (Xiao et al.37 and Srivastava et al.38) are based on capillary pressure models developed for micropillar arrays. These models relate the geometry of a micropillar array to the equilibrium capillary pressure (pressure generated under adiabatic conditions) generated by the liquid within the array. The capillary radii values of these models were calculated by equating the theoretical capillary pressure generated by n-hexane to the Young−Laplace equation (eq 1) with a contact angle of 0°. The capillary radii values calculated from the Xiao et al.37 model are about 50% lower than the experimentally measured Rc,min values. While the Srivastava et al.38 capillary radii are greater than those calculated from the Xiao et al.37 model, they differ from the experimentally measured Rc,min values and do not capture the interaction effects between the pillar diameter and spacing. The values calculated using this model are less than the measured quantities for devices C and E and greater than the measured value for device I. The contact angles of water (θw) for devices A−I are computed from the Young−Laplace equation (eq 1), the Rc,min values and the maximum capillary pressures generated by water. Based on these values, the mean contact angle for water, averaged across all devices, is 54.7°. This value corresponds to the contact angle formed by the liquid meniscus at dryout. The measured contact angle differs by less than 15% from the contact angle of 47.7° (averaged over nine different measurements) exhibited by water on a smooth PECVD oxide surface.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: saeedmog@ufl.edu (S.M.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors gratefully acknowledge the support of DARPA and Dr. Bar-Cohen, program manager. Fabrication of the devices was conducted in the Nanoscale Research Facility (NRF) at the University of Florida.
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