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The Flow of Liquids in Surface Grooves R. R. Rye,*,† J. A. Mann, Jr.,‡ and F. G. Yost† Sandia National Laboratories, Albuquerque, New Mexico 87185, and Case Western Reserve University, Cleveland, Ohio 44106 Received February 9, 1995. In Final Form: October 4, 1995X We have obtained detailed capillary kinetic data for flow of a series of alcohols with various surface tension to viscosity ratios, γ/µ, spreading in open V-shaped grooves cut in Cu with three different groove angles. The location of the three-phase contact line, z, with time always follows the formula z2 ) K(R,θ)[γh0/µ]t where R is related to the included groove angle β (R ) 90 - β/2), θ is the contact angle, and h0 is the groove depth. Two theoretical models which assume Poiseuille flow and static advancing contact angles were tested against the experimental data. One is a detailed hydrodynamic model with the basic driving force resulting from the pressure drop across a curved interface. The second depends on the total interfacial energy change, independent of the shape of the liquid interface. Both agree with the experimental data. In agreement with experiment, both models predict that the rate approaches zero as R f θ, and both require R - θ > 0. Both, including a physically unrealistic approximation by a cylindrical capillary, correctly scale the experimental data. Both predict numerical values in general agreement with experiment and with each other. Differentiation between the models is possible only in the K(R,θ) term which is shown to be only weakly dependent on the range of R,θ values studied. In the threshold region where the transition occurs between filled and empty regions of the groove, the liquid height decreases linearly with distance, within experimental limitations, and forms an angle which roughly scales as the contact angle for a significant fraction of the threshold region. On the basis of the present detailed experimental data for both kinetics and threshold profile, the differences between experiment and theory and between the theoretical models are insufficient to allow a clear choice between the models.
Introduction Research on the wetting of rough surfaces and the infiltration of porous media by liquids goes back at least 60 years.1,2 This long term interest is a direct result of the many related technological applications. 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. While remarkable wetting rates have been observed3 on rough surfaces, no detailed model of this kinetic behavior is known to us. Conceptually a rough surface can be thought of as a three-dimensional network of connected open capillaries that would have measurable root-mean-square (rms) amplitude, rms slope, etc. As such, a rough surface can be viewed as a network of contiguous valleys through which liquid is drawn by capillary forces. It would be desirable to relate these simple measures of surface roughness to actual wetting behavior so that surfaces could be engineered for specific applications. But rather than constructing a treatment of wetting on rough surfaces based on a random network of connected capillaries, as a first step the approach taken here is to treat, both experimentally and theoretically, the unit process of flow through surface cusps or “valleys” produced by straight, systematically varied V-shaped grooves. The advantage of this approach, besides simplicity, is that the groove depth and groove slope (relative to the horizontal) can be compared with the rms roughness and slope spectrum of the rough surfaces. Treatments of the kinetics of capillary flow in closed capillaries have been studied for over 50 years. For horizontal cylindrical capillaries, Washburn4 and Rideal5 †
Sandia National Laboratories. Case Western Reserve University. X Abstract published in Advance ACS Abstracts, December 15, 1995. ‡
(1) Parker, E. R.; Smoluchowski, R. Trans. ASM 1944, 35, 362. (2) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988. (3) Yost, F. G.; Michael, J. R.; Eisenmann, E. T. Acta Metall. Mater. 1995, 43, 299. (4) Washburn, E. W. Phys. Rev. 1921, 17, 273.
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using slightly different approaches were able to show that the length of liquid column, z, entering the cylindrical capillary followed relatively simple kinetics, z2 ) cos(θ)[γr/µ]t where γ is the liquid surface tension, r is the capillary radius, θ is the contact angle, and µ is the bulk liquid viscosity. Despite some problems the basic Washburn equation has remained the basis of nearly all treatments of capillary dynamics. Mathematically, Szekely et al.6 point out that the Washburn equation predicts that d(z)/d(t) f ∞ as t f 0. This has lead several authors6,7 to derive alternate relationships. However, as Szekely et al. point out, the Washburn equation and alternative solutions should be identical for virtually all practical systems. In a very detailed experimental paper, Fisher and Lark8 reported exact agreement with the Washburn equation for cyclohexane using a wide range of capillary diameters, but a slight deviation in the case of water for which they resorted to a nonchanging contact angle of 30°; even for water, however, the basic z ∝ t1/2 dependence predicted by the Washburn equation was observed. The same basic Washburn approach has been applied to capillary spreading into the two-dimensional capillary formed between flat plates,9 and to square capillaries.10 Even in the case of radial flow into a flat two-dimensional slit from a point source9 where Washburn kinetics was not followed over the entire range, the kinetics follow an approximate t1/2 dependence after a short induction period. In contrast to flow in closed capillaries, we report here on the flow kinetics of organic alcohols containing a small amount of a fluorescent dye in open triangular surface grooves in copper. A preliminary report of this work and the theoretical basis has been given previously.11 Experimentally we show, for a wide range of surface tension (5) Rideal, E. K. Phil. Mag. 1921, 44, 1152. (6) Szekely, J.; Neumann, A. W.; Chuang, Y. K. J. Colloid Interface Sci. 1971, 35, 273. (7) Good, R. J. J. Colloid Interface Sci. 1973, 42, 473. (8) Fisher, L. R.; Lark, P. D. J. Colloid Interface Sci. 1979, 69, 486. (9) Marmur, A. J. Colloid Interface Sci. 1988, 124, 301. (10) Lenormand, R.; Zarcone, C. In 59th Annual Technical Conference and Exhibition; Society of Petroleum Engineering: Richardson, TX, 1984.
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Figure 1. Schematic representation of the groove spreading experiment giving the experimental and geometric parameters.
to viscosity ratios (γ/µ), groove angles (R) and groove depths (h0), that the spreading length, z, of liquid in open surface V-grooves follows a square root of time dependence with the same basic form as for closed cylindrical capillaries: z2 ) K(R,θ)[γh0/µ]t. Scaling the slopes of linear plots of z vs t1/2 with the basic Washburn parameters (γh0/µ)1/2 accounts for the majority of the difference between the different alcohol spreading systems; the experimental kinetics are only weakly dependent on the K(R,θ) term for the range of R,θ values studied. Two theoretical models were compared to these experimental kinetics. One is a detailed similarity solution12 based on a model using the condition of mass balance, the Laplace equation, static contact angles, and Poiseuille flow, which through its dependence on the Laplace equation requires a curved liquid surface at the threshold of the liquid flowing down the column. The second, more similar to the original Washburn approach,4 is based on static contact angles, Poiseuille flow, and the total interfacial energy change as the liquid flow down the groove, a model which allows capillary flow independent of the shape of the liquid surface. Both approaches yield the same basic relationship z2 ) K(R,θ)[γh0/µ]t for the spreading kinetics, but with different forms for the geometry term K(R,θ). Thus, testing of these theoretical formulations is reduced to comparing experimental and theoretical values for K(R,θ). Such comparison, however, based on both scaling and numerical values yields no significant differentiation between experiment and theory or between theories. We concentrate in this report on alcohol spreading obtained with sufficient accuracy to establish the approach and the extent of the theoretical/ experimental agreement. In future publications we will consider the spreading of liquid Sn/Pb solder in the same Cu grooves13 and the spreading of the same alcohols and sodium napthalenide etching solutions in comparable grooves in Teflon14 to explore the material dependence of these spreading kinetics. Experimental Section V-grooves, Figure 1, were cut in six different 2.5 cm × 2.5 cm × 0.3 cm pieces of polished Cu with carefully ground and polished (11) Mann, J. A., Jr.; Romero, L.; Rye, R. R.; Yost, F. G. Phys. Rev. E 1995, 52, 3967. (12) Romero, L.; Yost, F. G. J. Fluid Mech., submitted for publication. Preprints are available from the authors. (13) Yost, F. G.; Rye, R. R.; Mann, J. A., Jr. The flow of liquid Sn/Pd solder in V-grooves in Copper. Manuscript in preparation. (14) Rye, R. R.; Yost, F. G.; Mann, J. A., Jr. The flow of alcohols in V-grooves in Teflon. Manuscript in preparation. (15) Beilstein Handbook of Organic Chemistry (STN on line); Beilstein Institute for Organic Chemistry: Frankfurt/Mann, Germany.
Figure 2. Profilometer traces of representative V-grooves (negative excursions) and dental impressions of the corresponding grooves (positive excursions). machine cutting tools having included angles of 30°, 60°, and 90°. Each hard tool was weighted for pressing into the soft Cu, positioned, and then drawn across the surface in a single pass such that each sample contained one example of each groove angle intersecting at a common point. The effect was more of pressing the image of the nonrotating, hard cutting tool into the soft Cu than of cutting. Since the pressing action resulted in surface eruption of Cu at the edges of the groove, all samples were polished subsequent to groove cutting to produce more accurate triangular grooves relative to the polished surface. The depths of the resulting grooves were determined by two techniques: a Dektak-8000 profilometer and a WYCO (MHT-II) vertical scanning interferometer. The interferometer is an optical instrument which produces a two-dimensional view of a segment of the groove with an associated depth scale. The scanning needle of the Dektak profilometer had a cone angle of 60° and could be expected to yield accurate profiles and depths for only the 90° grooves. The recorded Dektak data is a convolution of the actual groove shape and the probe tip shape. To obtain accurate groove depths for the 30° and 60° grooves, dental impression material (hydrophilic vinyl polysiloxane, Dentsply International, Milford, DE) was used to produce images of the grooves for profilometry. Profilometer traces for representative 30°, 60°, and 90° grooves and their dental images are given in Figure 2, and the groove depths obtained by both methods for all samples are given in Table 1. As expected, the 90° groove and its dental image give nearly identical measures of both shape and depth; the groove depths obtained by the two methods differ by less than 5%. For the 60° grooves, profilometry gives approximately the same depths as interferometry, but the trace of the sides of the groove are effectively linear suggesting that the side wall shape is an artifact resulting from the use of a profilometer needle with a cone angle of 60°. For the 30° grooves there is considerably larger variation, in some cases, in the groove depth, Table 1, between the Dektak and WYCO values and a greater distortion of the dental image shape, Figure 2, suggesting that the WYCO optical data may be more accurate for the narrow 30° grooves. As a result the groove depths given in Table 2 for use in data reduction refer to an average value of the Dektak and WYCO data for the 60° and 90° grooves but to only the WYCO data for the 30° grooves. Both the Dektak data and the WYCO data show that the grooves are reasonably accurate triangular grooves but with a slight asymmetry due to either a slight asymmetry in grinding the cutting tools or vertical positioning and a rounding of the groove bottom which is obvious in the case of the 90° grooves. While we will continue to differentiate between experimental results in terms of 30°, 60°, and 90° grooves, it is clear from the curves in Figure 2 that there is considerable deviation from the shape of the tool used to create the grooves: the grooves show
The Flow of Liquids in Surface Grooves
Langmuir, Vol. 12, No. 2, 1996 557
Table 1. Groove Depths and Angles β Obtained from a Combination of Scanning Profilometry and Interferometry for Six Polished Cu Samples 30° grooves groove depth, µm
60° grooves groove depth, µm
90° grooves groove depth, µm
Cu sample
Dektak
WYCO
groove angle β, deg
Dektak
WYCO
groove angle β, deg
Dektak
WYCO
groove angle β, deg
1 2 3 4 5 6
89.1 72.6 66.6 110.0 95.3 100.6
85.7 92.9 74.4 98.7 98.6 98.3
42 50 49 45 50 40
58.8 56.9 57.4 89.0 83.1 74.9
50.7 53.3 57.7 92.8 89.9 85.5
71 79 69 77 79 71
33.2 40.4 37.7 47.4 45.0 41.3
32.5 40.4 39.6 48.2 45.9 42.5
122 119 109 124 125 119
Table 2. Experimental Parameters and Experimental and Calculated Kinetic Rate Data for Sample 4a K(R,θ)1/2 liquid
γ/µ, cm/s
contact angle θ, deg
meas groove angle, deg
groove height, µm
exptl rate, cm/s1/2
theory exptl
eq 20
eq 11
eq 13
0.356 (0.343) 0.369 (0.372) 0.369 (0.372) 0.363 (0.369) 0.347 (0.335) 0.363 (0.369)
0.282 (0.311) 0.301 (0.330) 0.301 (0.330) 0.297 (0.327) 0.273 (0.302) 0.297 (0.327)
0.291 (0.319) 0.325 (0.348) 0.325 (0.348) 0.326 (0.350) 0.280 (0.309) 0.326 (0.350)
1,4-butanediol cyclohexanol 1-butanol 2-octanol diethylene glycol 1-heptanol
59.5 58.2 941 408 162 489
29 6 6 0 for flow to occur, and the experimental and calculated values of K(R,θ) are similar. 3. Only Poiseuille flow and static advancing contact angles are assumed. 4. Detailed geometry and contact angle together determine whether flow does or does not occur but play only a minor role in the kinetics once flow occurs. 5. On the basis of the present experimental data, no distinction can be made between the theoretical models. Even the simple Washburn-based approaches contain the essential surface chemistry and predict, within experimental capability, the detailed kinetics, but only the approach of Romero and Yost has the refinement potential to calculate more accurately the profile shape. Acknowledgment. The authors wish to thank the referee for pointing out that the groove angles in Figure 2 were significantly greater than the tool angles of 30°, 60°, and 90°. R.R. wishes to thank his dentist G. M. Yarbrough, DDS, for loan of the dental impression equipment. This work was performed at Sandia National Laboratories, which is supported by the Department of Energy under Contract Number DE-AC04-94AL85000. LA9500989
