Langmuir 1998, 14, 3937-3943
3937
Capillary Flow in Irregular Surface Grooves R. R. Rye,* F. G. Yost, and E. J. O’Toole Sandia National Laboratories, Albuquerque, New Mexico 87185-1411 Received November 10, 1997. In Final Form: April 13, 1998 1-Heptanol flow in irregularly shaped surface grooves in Pd-coated Cu is shown to be an example of Poiseuille flow with simple Washburn kinetics of the form z2 ) C(γ/µ)t, where γ is the liquid surface tension, µ is the viscosity, and C is a function of the groove dimensions and the contact angle θ. A shape independent expression is derived for the geometric factor, C(S,w,θ) ) (S cos(θ) - w)/4π, where w is the width of the groove at the surface and S is the arc length, or total length of groove surface in a plane perpendicular to the groove axis. This expression is general for any groove shape and reduces to the form derived previously for V-shaped grooves. Along with scanning electron microscopy, three different techniques, stylus profilometry, laser profilometry, and optical interferometry, were used to characterize the groove geometry, especially to determine S and w. While reasonable agreement is obtained between literature values of γ/µ and values obtained from the experimental kinetics, the main conclusion is that measurement of the groove dimensions is the main limitation to experimental verification of the form of C and to the use of the kinetics of groove flow as an absolute measure of the factor γ/µ. However, we show that if C is calibrated for a specific groove with a known liquid, the kinetics of capillary flow in open surface grooves furnishes a simple, easily applied method for measurement of the surface tension-to-viscosity ratio, γ/µ.
Introduction Washburn1
showed in 1921 that flow of liquid in a cylindrical capillary followed a very simple kinetic relationship, z2 ) kt, where z is the length of liquid flowing in the capillary; k is given by (γr cos(θ))/2µ, where γ is the surface tension, µ is the viscosity, θ is the equilibrium contact angle, and r is the capillary radius. This relationship is well justified experimentally,2 although Szekely, Neumann, and Chuang3 have pointed out that dz/dt f ∞ as t f 0. This stems from the steady-state approximation (Poiseuille flow) in the derivation of the classic Washburn equation. Research on wetting of rough surfaces and infiltration of porous media by liquids goes back at least 60 years.4,5 Comparatively, flow into porous media has received considerably more attention. Models of wetting kinetics on rough surfaces have largely been empirical or at best thermodynamic in origin. Conceptually, a rough surface can be thought of as a two-dimensional network of connected, open capillaries, and considerable progress has been made in studying the unit process of flow through straight, systematically varied V-shaped grooves using organic alcohols6,7 and liquid Sn/Pb solder8,9 as the flowing liquid. The utilization of open surface grooves to study capillary flow allows measurement of surface tension and viscosity for liquids, such as solder, that wet opaque materials used to form capillaries. For a number of different alcohols and V-groove shapes, the major conclusion was that both a detailed similarity solution10 and simple Washburn approaches yielded the same form, z2 ) (Cγ/µ)t, with the factor C containing the contact angle, (1) Washburn, E. W. Phys. Rev. 1921, 17, 273. (2) Fisher, L. R.; Lark, P. D. J. Colloid Interface Sci. 1979, 69, 486. (3) Szekely, J.; Neumann, A. W.; Chuang, Y. K. J. Colloid Interface Sci. 1971, 35, 273. (4) Parker, E. R.; Smoluchowski, R. Trans. ASM 1944, 35, 362. (5) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (6) Mann, J. A., Jr.; Romero, L.; Rye, R. R.; Yost, F. G. Phys. Rev. E 1995, 52, 3967. (7) Rye, R. R.; Mann, J A., Jr.; Yost, F. G. Langmuir 1995, 12, 555. (8) Rye, R.; Yost, F. G.; Mann, J. A., Jr. Langmuir 1996, 12, 4625. (9) Yost, F. G.; Rye, R. R.; Mann, J. A., Jr. Acta Mater. 1997, 45, 5337. (10) Romero, L.; Yost, F. G. J. Fluid Mech. 1996, 322, 109.
groove size, and geometry terms. For various assumptions about the liquid shape, the differences between models were as small or smaller than that between experiment and the various kinetic formulations. While the alcohol flow was linear when plotted as z2 versus time and scaled properly with γ/µ, observable curvature was apparent at short times in the flow of solder. This nonlinearity at short time was interpreted as due to a relaxation time for evolution of the static advancing contact angle. With use of the exponential decay θt ) θ0(1 - exp(-t/τ)), suggested by Newman,11 excellent agreement was obtained between experiment and theory for both the initial and longer term flow behavior. While study of such ideally shaped surface grooves is a necessary first step toward modeling of a rough surface as a connected network of open capillaries, one cannot expect that such networks are composed of ideal V-shaped capillaries. Here we explore the case of less ideal shaped open surface capillaries such as can be produced in the average laboratory or machine shop. Our main conclusion is that a modified form of the classic Washburn equation is adequate to explain the flow observed in such grooves. For irregularly shaped surface grooves, C is derived as a function of the arc length S, the groove width w, and the contact angle θ, C(S,w,θ) ) (S cos(θ) - w)/4π. As indicated in Figure 1A for a cross section perpendicular to the groove axis, w is the measured width at the surface and S is the total arc length that defines the groove profile. This is a general expression appropriate to any groove profile and reduces to the expression derived previously for V-shaped grooves. We show, however, that the major limitation in the interpretation, or experimental justification for the form of C, is accurate measurement of capillary shapes having dimensions on the order of tens of micrometers. Experimental Section The overall substrate dimensions were 2.5 cm × 2.5 cm × 0.3 cm. The Cu blanks were first faced on both sides to provide a parallel work surface. A reservoir having dimensions of 0.158 cm diameter × 0.061 cm deep was then created in one corner of (11) Newman, S. J. Colloid Interface. Sci. 1968, 29, 209.
S0743-7463(97)01224-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/05/1998
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Figure 1. (A) Schematic representation of groove spreading illustrating the geometric parameters. (B) Representative single video frame obtained during capillary flow in surface grooves. the substrate (see Figure 1) using a ball end mill. Two different carbide-tipped tools were used to create the V-grooves used in this study. The first tool used was a threading tool, which was purchased with a preground 60° carbide tip. The second tool was similar but with the tip ground to approximate a 30° V-shape. Both tools were examined optically for imperfections and were subsequently hand-polished to the desired sharp, pointed shape. Both were further modified by turning their square shanks into round shanks so they could be attached to a milling machine by means of a collet. With the collet rigidly locked in position, to prevent any rotation or side-to-side movement, the milling machine table was positioned so that a cutting tool was directly over the reservoir. A tool was then lowered into the reservoir 0.089 mm, after which the milling machine table was translated so that a groove was pressed into the soft Cu surface using the relief side of the cutting tip, not the cutting side. This operation is called breaching, as opposed to machining. The tool was replaced and the substrate repositioned as required so that a second groove could be breached in each substrate. After careful preparation of the substrate grooves, the edges of the newly formed grooves were seen to have a raised burr. This raised burr is a direct result of the pressing action used to create the grooves and was subsequently removed (by polishing) with a 6 µm diamond paste applied to a hard polishing cloth. The hard cloth was used to prevent any rounding of the groove edges. The result of this polishing process was a more accurately defined V-groove shape, relative to the polished surface. Cu substrates used in this study were sputter coated with 2.3 µm of Pd. Prior to Pd deposition the Cu samples were subjected to TCE (trichloroethane) vapor degrease, acetone ultrasonic clean, isopropyl alcohol (IPA) rinse, and a nitrogen blow dry. Since characterization of the groove dimensions and geometry is critical to the analysis of flow kinetics, we have extensively studied this using the following four techniques: scanning electron microscopy (SEM) (JEOL model JSM-6300V); stylus profilometry (Dektak 8000); laser profilometry (Focodyn); optical profilometry (Wyco). We will now discuss each of these methods in detail. SEM gives a very accurate representation of the groove shape and has excellent depth of field. However, accurate
Figure 2. SEM micrographs of two groove shapes taken prior to the final polishing step: groove A (produced by the 30° tool) with an approximate parabolic shape and groove B (produced by the 60° tool) with an appreciable flat-bottomed shape. measurements of groove dimensions are very hard to obtain with SEM, except possibly for the groove width at the surface. SEM micrographs, taken before the final polishing step, of grooves A and B used in this study are shown in Figure 2. Groove A was formed with the 30° carbide tool while groove B was produced with the 60° tool. What is immediately obvious is that neither groove is truly V-shaped; the narrow groove A is better approximated by a parabola while the wider groove B is more flat bottomed. The Dektak 8000 stylus profilometer (Sloan Technology) works by tracking the vertical movement of the stylus (electrically), and then converting the acquired electric signal into a digital format that is stored in memory. The fundamental limitation of the stylus method is the convolution of the stylus tip shape with the shape being measured. The cone angle of the Dektak 8000 is on the order of 30°. Therefore, when measuring grooves having included angles of 30°, or less, the measured profile will be the cone angle of the stylus tip. With grooves having included angles greater than 30°, one should rigorously deconvolute the tip shape from the measured shape, but we have shown in the past that the Dektak method will give an adequate measurement of 60° grooves.7 The Focodyn laser profilometer (Feinpru¨f Perthen GmbH) works on the principle of dynamic focusing. The profilometer objective lens focuses the laser beam so that the focal point lies 0.9 mm below the aperture of the measuring arm. The returning beam is split to form two arms of an interferometer. The beam detector is part of a control loop that readjusts the measuring
Capillary Flow in Irregular Surface Grooves
Langmuir, Vol. 14, No. 14, 1998 3939 the contrast between the 1-heptanol/coumarin solution and the substrate. The surface tension and viscosity values shown in Table 2 were obtained from the ref 12. These will be discussed in more detail later. Video equipment with a time code generator board was used to record, at a framing speed of 30 frames/s, a complete flow experiment. Prior to the placement of the 1-heptanol, the videotape recorder was started and set to record. All flow events from introduction of the source drop to completion of flow were recorded. For calibration purposes, several seconds of video were recorded using a 10 mm by 10 mm optical calibration reticule located at the sample position. A computer equipped with a digital frame grabber board and image analysis software (Image Pro Plus) was used to collect digital “snapshots”. One example of a single video frame is given in Figure 1B. The digital images were then analyzed using an image analysis program (Optimas) to obtain pairs of (z,t) data. The estimated error in measurement of z is approximately 3-5%. This error is due to the subjectivity involved in the placement of the measurement cursor when using the image analysis software.
Capillary Flow Modeling As shown previously,7 the simplest analysis starts with the change in surface energy E as the liquid moves through a surface groove
∆E ) (γsl - γs)∆Asl + γl∆Al ) γl[∆Al - ∆Asl cos(θ)] (1)
Figure 3. Illustration of instabilities that occur in optical measurements of profiles of grooves with steep slopes: upper curve, Focodyn optical profile; lower image, SEM micrograph. arm continuously so that the distance of 0.9 mm to the surface remains constant. An inductive path measuring system converts the motion of the measurement arm into an electrical signal. This signal is then converted into a (x,y) data set that represents the surface shape. Steep vertical profiles and sharp profile edges are a known limitation of this method. Sharp profile edges are interpreted by the detector as a secondary light source and steep vertical profiles cause little, or no light, to be seen by the detector. Both cases can cause the measured profile to deviate from the real profile. A particularly clear example of this is shown in Figure 3 where a direct comparison is given between a SEM micrograph and a Focodyn profile for the same groove. The Wyco, RST Plus, optical surface profilometer (Wyco Corp.) uses white light as the source in an interferometer and measures the degree of fringe modulation, or coherence, present. The peak of the coherence function provides a measure of surface geometry. This method exhibits measurement inaccuracies similar to those found with the Focodyn instrument. These measurement limitations are also due to steep inclination and sharp edges. As previously,7 1-heptanol was used after adding a small amount of the fluorescent dye coumarin to provide contrast. Irradiation of the 1-heptanol/coumarin solution with an ultraviolet (UV) light source greatly increased the contrast between the fluorescing solution in the grooves and the substrate. All alcohol experiments were performed at 20 °C nominal in a room air environment. Prior to performing the study (and between subsequent runs) each substrate was scrubbed with soapy water, rinsed in deionized water, ultrasonically cleaned in methanol, rinsed in IPA, and blown dry with nitrogen. All room lights were turned off during the wetting experiments to help increase
where the γs refer to surface tension (solid-liquid, solid, and liquid), the As are the corresponding surface areas, and θ is the static advancing contact angle. The changing solid-liquid surface area is given by S∆z - b′, where S is the arc length defining the groove profile, Figure 1, and z is the length of the liquid in the groove. The S∆z term comes from assuming that the groove is filled completely to the length z, an assumption that overcounts the area due to the experimentally observed gradual transition in the threshold region between filled and unfilled regions of the groove.6 b′ reflects this overcount. Since the threshold shape changes with z,6 b′ should strictly be a function of z; however, we have shown that the length of threshold region is small compared to z and can be treated as a small constant correction.7 Similarly, the change in liquid surface area is given by w∆z - b, where w is the width of the groove. Here the liquid surface is assumed flat, but previous studies7 have shown that the assumed shape of liquid surface has little effect on the results. Classically, force is given by the negative gradient of the total mechanical or capillary energy, which with the areas in eq 1 expressed in terms of z yields
Fγ ) -
dE ) γl[S cos(θ) - w] dz
(2)
The volume flow rate for Poiseuille flow in cylindrical capillaries has the form
q)
A2 πr4 ∆P ) ∆P 8µz 8πµz
(3)
where µ is the liquid viscosity, r is the capillary radius, and A is the capillary cross-sectional area. Whereas eq 3 was derived for circular cross sections, we assume it valid for arbitrary open capillary shapes. The volume flow rate can also be written as (12) Beilstein Handbook of Organic Chemistry (STN on line); Beilstein Institute for Organic Chemistry: Frankfurt/Main, Germany.
3940 Langmuir, Vol. 14, No. 14, 1998
q)A
Rye et al.
dz dt
(4)
Solving eqs 3 and 4 for ∆P, the force per unit area, yields the viscous dissipation force
Fµ ) A∆P ) 8πµz
dz dt
(5)
At this point one can include inertial terms3 and solve the differential equation or simply assume steady state and equate the surface tension driving force, eq 2, to the viscous dissipation force, eq 5. In recent work9 we have shown that the inclusion of the inertial term gave rise to unrealistic results, and since it has been shown7 that many approaches, including a highly detailed similarity solution,10 yield experimentally indistinguishable results, we will take a simple steady-state approach. Equating eqs 2 and 5 yields
[
]
γ S cos(θ) - w dz2 γ ) f(S,w,θ) ) dt µ µ 4π
(6)
If a V-shaped groove is assumed with S and w expressed in terms of groove depth and groove angle, eq 6 reduces to the same equation derived previously for the specific case of V-grooves.7 To include nonlinear effects at very short times, we assume that these are due to the necessary evolution of the contact angle to the steady-state value. Following ref 8 we include this by assuming f(S,w,θ(t)) ) f(S,w,θ0)[1 - exp(-t/τ)]. Substituting this relationship into eq 6 and integrating yields
{ [
( )]}
t t) τ t γ S cos(θ) - w t - τ 1 - exp µ 4π τ
z2 ) k t - τ 1 - exp -
[
]{ [
( )]}
t (7)
The general applicability of eq 7 stems directly from its foundation in changes of wetted surface area, regardless of groove shape, and the use of the standard Poiseuille equation expressed in terms of cross-sectional areas regardless of specific shape. The exponential terms, which are discussed in more detail in refs 8 and 9, are necessary only for the initial stages of flow. For t/τ large, eq 7 reduces to the simple steady state form, z2 ) kt. Results and Interpretation The critical factor for validation and use of eq 7 is characterization of groove shape and dimensions. Figure 2 shows SEM micrographs of two grooves on the same sample taken immediately after fabrication of the grooves and before final polishing to remove surface upheaval produced at the groove edges. To illustrate the groove shape, the micrographs were taken at an angle relative to the surface normal and at the point where the individual grooves intersected the common source liquid (i.e., z ) 0). Groove A has a roughly parabolic shape while groove B is relatively more flat bottomed, and both contain very tiny striations along the length of the groove. While the SEM micrographs are useful for assessing the general groove shape, accurate measurements of the arc length and groove width are necessary. For the two grooves in Figure 2, representative groove profiles obtained using the Focodyn, Dektak and Wyco techniques are shown in Figure 4. It is immediately obvious that the Dektak profiles deviate significantly from the other profiles and from the shape seen in the SEM micrographs, Figure 2.
Figure 4. Profiles for the grooves shown in Figure 2 obtained using the Dektak (solid line), the Wyco (dashed line), and Focodyn (dotted line) profile measurement techniques.
This reflects a convolution of the ∼30° cone angle of the Dektak stylus tip and the actual groove shape. The remaining Wyco and Focodyn optical methods appear to be in reasonable agreement with each other and with the SEM micrographs in Figure 2. The results for measurements at three points along each of the grooves by all three techniques are summarized in Table 1. For each profile such as in Figure 4, the approximate points at each groove edge, where the groove wall intersects the flat surface, were identified (indicated for one case by the arrows in Figure 4). The groove width, w, was taken as the distance between these points, and arc length S was taken as the sum of the distances between all measured coordinates within these limits. To obtain a measure of the reproducibility of the measurements, each of the three Focodyn measurements for width, depth, and arc length were averaged; the percent variation given throughout the table for width, depth, arc length, and the difference S-w represent the percent variation from these average values. All three methods are in very good agreement as to the groove width and depth but considerably more disagreement occurs for the measurement of the arc length. What is somewhat surprising is that despite the obvious measurement distortion in the case of the Dektak method, the measured values are in reasonable agreement with the Focodyn technique. The problem for use in eq 7, however, is that these variations can be enhanced when the difference is taken between the experimental measurements of S cos(θ) and w. The kinetic data for capillary flow of 1-heptanol in grooves A and B are plotted in Figure 5 as the square of the flow distance, z, versus time. Except for a slight initial curvature discussed above, the data are linear, consistent with Poiseuille flow and a simple Washburn model. The solid line through the points is the result of a nonlinear least-squares fit of the experimental data to eq 7. Equation 7 defines the fitting parameter k as (γ/µ)[(S cos(θ) - w)/ 4π]. The values for γ/µ listed in Table 1 were calculated from this relationship for a measured contact angle of zero.7 In this case the percentage variation is defined relative to the literature value of 387 cm/s reported in ref 12. While there is reasonable precision in the measured
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Langmuir, Vol. 14, No. 14, 1998 3941
Table 1. Variation of Groove Dimensionsa Focodyn laser profilometer sample A1
width (µm)
175.6 -1.0% A2 178.9 +0.85% A3 177.7 +0.17% mean 177.4 B1 233.6 -1.1% B2 231.4 -2.0% B3 243.7 +3.2% mean 236.2
depth (µm)
arc (µm)
S-w (µm)
127.8 -1.0% 130.7 +1.2% 128.8 -0.23% 129.1 117.8 -1.6% 118.9 -0.72% 122.6 +2.4% 119.8
385.3 -2.6% 408.2 +3.2% 393.4 -0.56% 395.6 430.1 -1.7% 431.2 -1.5% 451.4 +3.2% 437.6
209.7 -3.9% 229.3 +5.1% 215.7 -1.1% 218.2 196.5 -2.4% 199.8 -7.53% 207.7 +3.2%
Dektak stylus profilometer γ/µ
width (µm)
depth (µm) 131.6 +1.5% 130.6 +0.77%
Wyco
arc (µm)
S-w (µm)
γ/µ
width (µm)
depth (µm)
326.1 -9.5% 326.8 -9.3%
146.0 -33% 145.3 -33.%
692.9 +79% 696.2 +80%
171.9 -3.1% 176.0 -0.79%
128.8 -0.23% 128.8 -0.23%
239.7 +1.5% 232.1 -1.8%
120.6 +7.0% 117.4 -2.0%
482.4 +25% 441.2 +14% 469.0 +21%
180.1 +1.5% 181.5 +2.3%
648.5 +68% 637.7 +65% 613.5 +59%
225.8 121.4 352.4 -4.4% +0.66% -12 241.1 125.8+4.3% 371.9 +2.1% -6.7%
126.6 1006 -37.% +160% 130.8 974.2 -35% +152%
arc (µm) 339.8 -14.1% 343.3 -13.%
371.7 -15% 382.6 -13
S-w (µm)
γ/µ
167.9 602.5 -23% +56% 167.3 604.7 -23% +56%
132.0 965.3 -34% +149% 150.5 846.7 -25% +119%
a For γ/µ the percent variation is relative to the literature value of 387 cm/s from ref 12. The remaining percent variation is relative to the average of the corresponding measurements of the same groove using the Focodyn instrument.
Figure 5. Plot of the square of the 1-heptanol spreading distance versus time obtained from individual video frames as illustrated in Figure 1A. Same grooves as in Figures 2 and 3.
groove dimensions, the values of γ/µ derived using eq 7 are larger than the literature values in the present study for all grooves and all profile measurement techniques. Discussion One explanation for the larger derived value of γ/µ in Table 1 is that the addition of a small amount of coumarin dye increased the surface tension and we are in fact measuring the correct value. However, this is inconsistent with previous work for a series of alcohols flowing in a set of V-grooves with different groove angles. These data for 1-heptanol, 1,4-butanediol, cyclohexanol, 1-butanol, 2-octanol, and diethylene glycol7 are summarized in Table 2. For V-shaped grooves, the limiting slope in eq 7 reduces to h0(γ/µ)f(R,θ), where h0 is the groove depth and R is the groove angle. For the specific case of V-shaped geometry, the factor h0f(R,θ) corresponds to the general geometric factor C(S,w,θ) in eq 7. For what was nominally a 60° V-groove in copper, Poiseuille flow occurred as can be seen from the linear plot of z2 versus t for 1-heptanol given in
Figure 6. These grooves were designed to be 60° V-grooves but experimentally determined to have a somewhat larger angles. Table 2 column 1 gives the alcohol, column 2 the literature values of γ/µ,12 column 3 the experimentally observed rates (the slopes of plots such as Figure 6), column 4 the measured groove depths in micrometers, column 5 the calculated values of f(R,θ), and column 6 the values of γ/µ derived from the data and model. As with the current data, the values of γ/µ derived from the 60° groove data are systematically larger than the literature values, but the 30° groove data systematically gives smaller values than the literature values. This would suggest that these systematic errors result from problems in determining the geometric terms h0f(R,θ) (equivalent for V-grooves to the C(S,w,θ) term in eq 7). This is reasonable given the problems reported above in measurement of groove dimensions. One way around this is to use the measured kinetics of a known liquid standard to calibrate the groove geometry. The use of calibration with a known liquid is a common practice in the use of capillary rise to measure surface tension. Calibration is necessary because of the same problems with measurement of the capillary dimensions. As Adamson points out,13 the most accurate measure of surface tension results from the capillary rise technique, but only because the theory has been worked out in considerable detail and the experimental details can be closely controlled. Although capillary rise and capillary kinetics differ in that the former is an equilibrium measurement of surface tension and the latter a kinetic measurement of the surface tension-to-viscosity ratio, they share a common problem of dimensional measurements. For the data in Table 2, we use the 1-heptanol data as our calibration liquid, a choice requiring an accurate value for γ/µ. Beilstein12 reports two values of γ at 20 °C that differ by only 0.28 dyn/cm; the average value is 27.11 dyn/cm. While only a single value of 0.0700 P at 20 °C is reported for the viscosity, extensive data are listed for the viscosity at different temperatures, and this value is fully consistent with these data. Thus, it is reasonable to assume a value of γ/µ of 387 cm/s derived from these values. With the 1-heptanol data in Table 2 and 387 cm/s for the value of γ/µ, a value for the groove depth, h0, was determined for each of the grooves: 30°, 72µ; 60°, 117µ; 90°, 60µ. Using these calibrated groove depths instead of (13) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons: New York, 1990.
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Table 2. Experimental Kinetic Data for Capillary Flow in V-Grooves from Reference 7 alcohol
γ/µ (cm/s) literature
exptl rate (cm2/s)
1,4-butanediol cyclohexanol 1-butanol 2-octanol diethylene glycol 1-heptanol
50.9 46.1 825 302 124 387
0.0353 0.0353 0.679 0.271 0.0859 0.297
1,4-butanediol cyclohexanol 1-butanol 2-octanol diethylene glycol 1-heptanol
50.9 46.1 825 302 124 387
0.0324 0.040 0.741 0.329 0.0515 0.349
1,4-butanediol cyclohexanol 1-butanol 2-octanol diethylene glycol 1-heptanol
50.9 46.1 825 302 124 387
0.0 0.010 0.169 0.0795 0.0924
f(R,θ) calculated
γ/µ model
γ/µ calib
30° Grooves 98.7 98.7 98.7 98.7 98.7 98.7
0.0847 0.106 0.106 0.106 0.0784 0.106
42.2 (-17%) 33.7 (-23%) 649 (-21%) 259 (-14%) 111 (-10%) 284 (-27%)
57.6 (+13%) 46.0 (-0.2%) 885 (+7.2%) 353 (+17%) 151 (+22%)
60° Grooves 91.0 91.0 91.0 91.0 91.0 91.0
0.0511 0.0756 0.0756 0.0767 0.0441 0.0767
69.7 (+37%) 58.1 (+26%) 1077 (+31%) 471 (+56%) 128 (+3.2%) 500 (+29%)
53.9 (+5.9%) 45.0 (-2.4%) 833 (+1.0%) 365 (+21%) 99.3 (-20%)
90° Grooves 47.8 47.8 47.8 47.8 47.8 47.8
0.0 0.0376 0.0376 0.0396 0.0 0.0396
0.0 55.6 (+21%) 940 (+14%) 420 (+39%) 0.0 488 (+26%)
0.0 44.1 (-4.3%) 745 (-9.7%) 333 (+10%) 0.0
depth (µm)
Figure 6. 1-Heptanol spreading in regular V-shaped grooves from ref 7.
the measured values in Table 2, the values of γ/µ for the remaining five alcohols were derived. These are contained in column 7 along with the percent deviation from the literature value. Calibration with 1-heptanol as a standard leads to a marked improvement in the derived value of γ/µ for 1,4-butanediol, cyclohexanol, and 1-butanol in all three grooves while the values for 2-octanol and diethylene glycol differ by ∼20% from the literature. Of course, the comparison in Table 2 is to literature values which themselves contain experimental error. For example, the ratio γ/µ ) 302 cm/s in Table 2 for 2-octanol is based on an average of three values in Beilstein for the surface tension at 20 °C (26.52, 23.87, and 22.97 dyn/cm) and two values for the viscosity (0.0819 and 0.0800 P).12 This represents good precision for the viscosity, but there is appreciable scatter in the surface tension values. If instead of an average, the largest γ value is used, the literature value in Table 2 becomes 328 cm/s for a difference of only -8% for the 30° groove with similar
improvement seen for the 60° and 90° grooves. A similar difference between literature and calibrated values occurs in the case of diethylene glycol. Little can be said about accuracy since Beilstein reports only single values at 20 °C for both γ and µ.12 However, since the Beilstein value for the surface tension of 48.50 dyn/cm at 25 °C is over 4 dyn/cm larger than the values at both 20 and 30 °C12 and Adamson lists a value at 20 °C of 30.9 dyn/cm,13 one can only conclude that similar uncertainties exist for diethylene glycol as to the more accurate value. Thus, one must conclude that within the limitations of the literature values, calibration of the groove geometry with a known liquid allows accurate determination of γ/µ for other liquids. The obvious candidate for such a calibration liquid is water, for which γ and µ are well-known. What is clear from the data presented here and from the previous data is that universal Washburn behavior is obtained for all capillary surface grooves, regardless of groove shape. In unpublished work, we have obtained similar kinetics for grooves with half-circular and square profiles.14 While a form of the Washburn relationship has been derived that is appropriate for irregularly shaped grooves, it can only be experimentally verified to the extent that z2 ) C(γ/µ)t, where all geometry dependent terms are contained in the factor C. The problem is not measurement of the kinetics from experimental data such as shown in Figures 5 and 6; these measurements can be made accurately. The problem is one of measuring the relevant dimensions of surface grooves with sizes of the order of tens of micrometers. Interpretation of flow in terms of changes in the wetted surface area for an irregular groove places more stringent requirements on the arc length measurement. There are striations in the groove surfaces that become obvious in SEM micrographs. While these striations would mean that the actual arc length is larger than that measured, and the resultant γ/µ value smaller, we have no way of evaluating the effect of these striations since the measurement techniques cannot reliably replicate the groove shape. Control of the experimental aspects, of course, is a necessary prerequisite to the validation of any theoretical development. However, complete validation of a theoretical description is not necessary for applications. What has been established is that capillary kinetics in open (14) Rye, R. R., Emerson, J. A. O’Toole, E. J. Unpublished ressults.
Capillary Flow in Irregular Surface Grooves
surface capillaries is an example of Washburn flow with kinetics given by the form z2 ) C(γ/µ)t, where the constant C is only a function of the geometric parameters of the groove. Further evaluation of the form of the geometric factor C will depend on better characterization of the grooves. The most promising technique for this is the use of etched Si surfaces. Due to the large differences in the etch rate of different crystal faces of Si, smooth, reproducible V-grooves can be obtained by etching; the groove angle, for example, is determined by the angle of intersection of two crystallographic planes. Gerdes and Stro¨m produced such grooves for spreading experiments.15 The experimental technique described above is a simple method for obtaining the ratio of surface tension to viscosity in a single measurement and is the only one we know of. Certainly it is possible to measure these quantities separately to obtain the ratio, but the above test is more direct and provides an independent measure that can be used for validation. The test is also useful for measuring this ratio in systems that are opaque, such as liquid solder in copper tubing. For these systems surface tension and viscosity can again be measured separately, but the rate of flow into an open or porous structure is often of most interest and a direct measurement of the ratio is very useful. (15) Gerdes, S.; Stro¨m, G. Colloids Surf. A 1996, 116, 135.
Langmuir, Vol. 14, No. 14, 1998 3943
Summary For flow of 1-heptanol in irregularly shaped grooves produced in copper surfaces (Pd coated), the following conclusions can be drawn. (1) Except for a slight initial curvature, the data are linear when plotted as z2 versus t, indicative of Poiseuille flow with simple Washburn kinetics of the form z2 ) C(γ/ µ)t, where the factor C is a function of the groove dimensions. (2) In our treatment, the factor C has the form C(S,w,θ) ) (S cos(θ) - w)/4π, where S is the arc defining the groove shape and w is the groove width, which is applicable to all groove shapes. (3) With suitable calibration of the geometric factor C, the kinetics of flow in open surface capillary grooves can be used to measure the surface tension-to-viscosity ratio, γ/µ. (4) Experimental verification of the factor C is limited by the ability to measure detailed groove geometry. Acknowledgment. This work was supported by the United States Department of Energy under Contract DEAC04-94AL85000. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy. LA9712247
