Langmuir 1996, 12, 4625-4627
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Wetting Kinetics in Surface Capillary Grooves R. R. Rye,* F. G. Yost, and J. A. Mann, Jr.† Sandia National Laboratories, Albuquerque, New Mexico 87185-1411, and Case Western Reserve University, Cleveland, Ohio 44106 Received May 28, 1996. In Final Form: July 29, 1996X For V-shaped surface grooves in copper, we have obtained the capillary driven flow kinetics for two liquids: unreactive 1-heptanol and eutectic Sn/Pb solder, which is known to react with copper. We show experimentally that the flow of both liquids in these grooves follows the classical Washburn kinetics, i.e., a Poiseuille flow process, modified to include a dynamic contact angle. Because no subsidiary processes are necessary to fit our data, we propose that in this geometry capillary driven solder flow is too rapid for reaction to provide an appreciable effect. Thus, to observe the effects of Sn/Cu reaction kinetics, the flow rate must be decreased, which the present experiments allow through redesign of the groove geometry and size.
Droplet spreading of simple liquids on smooth, inert substrates has been modeled as a Poiseuille flow process by many authors.1-3 However, for systems that undergo reaction and other subsidiary processes, this model seems to fall short.4,5 We have previously shown6 that the flow of a series of organic alcohols in V-shaped surface grooves in copper follows the classical Washburn kinetics,7 a Poiseuille process, in which the square of the flow distance is linear in time. In contrast to the alcohols which are unreactive toward Cu, the classical Sn/Pb solder alloys are known to chemically react with Cu, leading to formation of Sn/Cu intermetallic compounds.8 In this letter we directly compare the capillary flow processes for one nonreactive alcohol (1-heptanol) and eutectic Sn/Pb solder in the same V-grooves and conclude that the only modification necessary of the classical Washburn kinetics is the inclusion of an initial time dependent contact angle in the case of solder flow. Only the static, equilibrium contact angle is necessary to describe alcohol flow, although an indication (but unmeasurable) of curvature is apparent in the initial points. The upper portion of Figure 1 illustrates the ideal groove geometry used in these experiments and defines the various parameters used in modeling. The experimental capillaries were produced6,9 with tool steel ground and polished to the desired groove angle β; after grooving, the copper samples were repolished. The lower portion of Figure 1 shows the profile (solid curve) of such a groove obtained using a Dektak-8000 profilometer. Reasonable triangular grooves were always obtained with only a slight asymmetry and rounding of the groove bottom. Spreading was recorded with a CCD camera; individual video frames were used to measure the spreading distance with time6 to a precision better than 1%. The alcohol was doped with coumarin dye and illuminated with an * To whom correspondence should be addressed. † Case Western Reserve University. X Abstract published in Advance ACS Abstracts, September 1, 1996. (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Tanner, L. H. J. Phys. D 1979, 12, 1473. (3) Neogi, P.; Miller, C. A. J. Colloid Interface Sci. 1982, 86, 525. (4) Ambrose, J. C.; Nicholas, M. G.; Stoneham, A. M. Acta Metall. Mater. 1992, 40, 2483. (5) Ambrose, J. C.; Nicholas, M. G.; Stoneham, A. M. Acta Metall. Mater. 1993, 41, 2395. (6) Rye, R. R.; Mann, J. A., Jr.; Yost, F. G. Langmuir 1995, 12, 555. (7) Washburn, E. W. Phys. Rev. 1921, 17, 273. (8) Romig, A. D., Jr.; Chang, Y. A.; Stephens, J. J.; Marcotte, V.; Lea, C.; Frear, D. R. In Solder Mechanics: A State of the Art Assessment; Frear, D. R., Jones, W. B., Kinsman, K. R., Eds.; TMS: Warrendale, PA, 1991. (9) Mann, J. A., Jr.; Romero, L.; Rye, R. R.; Yost, F. G. Phys. Rev. E 1995, 52, 3967.
S0743-7463(96)00520-3 CCC: $12.00
Figure 1. Groove geometry: (upper curve) ideal geometry and parameter definition; (lower curve) profilometer measure of experimental groove profile (solid curve) and quenched liquid Sn/Pb solder.
ultraviolet lamp for enhanced contrast. For solder experiments, the Cu sample was heated to 200 °C and coated with a mild flux, and the flow was started by placing an 18 mg solder ball in the reservoir; see Figure 2. Note that solder is flowing in three separate grooves having different groove angles and emanating from the same source drop in the lower right corner. Image analysis software and the time stamp in the lower portion of the image allowed measurement of the distance between the edge of the source drop and the front of the liquid, z, at a known time. Occasionally solder flow was frozen rapidly by a liquid nitrogen quench. The result is shown by the scanning electron micrograph in the lower portion of Figure 2. The liquid solder clearly fills the groove to the edge and forms a concave liquid surface. The flow kinetics for 1-heptanol are shown plotted in Figure 3 as z2 vs t; detailed studies of alcohol flow are contained in ref 6. The basic Washburn model6 for © 1996 American Chemical Society
4626 Langmuir, Vol. 12, No. 20, 1996
Letters
Figure 3. Square of the spreading distance plotted vs time for 1-heptanol flow: (points) experimental data; (solid curve) least squares fit.
Figure 2. Solder flow in V-grooves: (upper image) individual video image during flow in three grooves with 30°, 60°, and 90° groove angles; (lower image) scanning electron micrograph of solder flow quenched with liquid nitrogen.
capillary flow is
2h0γf(R,θ0) ) 8πµz
dz dt
(1)
where γ is the liquid surface tension, µ is the viscosity, and the remaining terms are defined in Figure 1. Integration gives
z2 ) kt, where k )
γh0 f(R,θ0) 2πµ
(2)
Several models lead to slight variations in the form of f(R,θ0), but all require R > θ for spontaneous flow and all are indistinguishable within experimental uncertainty.6 The solid line in Figure 3 is a least squares fit of eq 2 to 1-heptanol flow data that yielded a correlation coefficient of 0.9993. Assuming6 a flat liquid surface for simplicity, f(R,θ0) is given by (cos(θ0) - cos(R))/sin(R). Using this relationship and measured values of R ) 54.8°, h0 ) 103µ, and θ0 ) 0°, the experimental value of k yields 489 cm/s for γ/µ, a value which differs by only 6% from literature values.6 Thus, to a high degree of accuracy, the capillary flow of alcohols6 in these narrow triangular grooves is well represented by a simple Washburn model. For fast flow of solder in the same grooves, one modification of the basic Washburn model is required.
Figure 4. Square of the spreading distance vs time for liquid Sn/Pb solder flow: (points) experimental data; (solid line) least squares fit after deleting the initial nine points; (dashed line) nonlinear regression fit to all the data.
There is curvature in plots of z2 vs t (Figure 4), but the asymptotic, longer time behavior is linear (solid line) and shows the scaling expected from the Washburn model, eq 2. If inertia terms, as reviewed by Marmur,10 are included, the simple Washburn model is modified to include exponential terms and the result gives excellent mathametical fits to the experimental data.11 But in our case this approach leads to physically unrealistic fitting parameters (i.e., density of ∼3800 g/cm3). A more detailed discussion of these points has been submitted for publication.11 A physically realistic model invokes the dynamic (10) Marmur, A. Modern Approaches to Wettability; Schrader, M. E., Loeb, G. I., Eds.; Plenum Press: New York, 1992. (11) Yost, F. G.; Rye, R. R.; Mann, J. A., Jr. Acta Materialia, submitted for publication.
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Langmuir, Vol. 12, No. 20, 1996 4627
contact angle θ(t). At t ) 0 the contact angle must start with a large value and evolve with time to the equilibrium value. Since all our previous modeling has established R - θ0 > 0 for spontaneous flow, the natural limits are θ(0) ) R and θ(∞) ) θ0. An exponential decay function suggested by Newman12 (see also ref 13) is used to modify the f(R,θ0) term in eq 1,
( τt)]
[
f(R,θ0,t) ) f(R,θ0) 1 - exp -
(3)
where τ is interpreted as a time constant associated with the approach of the dynamic contact angle to θ0. We realize that this approach to the data analysis is ad hoc, but a full fluid flow analysis that includes other effects is beyond the scope of this work. Our effort was directed at determining whether the Washburn equation was adequate. Combining eqs 1 and 3 and integrating yields
{ [
( τt)]}
z2 ) k t - τ 1 - exp -
(4)
where k is defined in eq 2. The dotted curve in Figure 4 is the result of fitting the experimental points in Figure 4 to obtain k ) 10.7 cm2/s and τ ) 0.275 s. This value of k and an assumed contact angle of 25° for solder/copper yields γ/µ ) 16 100 cm/s. We estimate from the literature14,15 that the surface tension of the solder/flux interface (12) Newman, S. J. Colloid Interface Sci. 1968, 29, 209-213. (13) Cherry, B. W.; Holmes, C. M. J. Colloid Interface Sci. 1969, 29, 174. (14) White, D. W. G. Met. Trans. 1971, 2, 3067.
is ∼400 mJ/m2 and from the work of Ejima et al.16 that the viscosity is 2.7 mPa. These values yield a surface tension to viscosity ratio of 14 810 cm/s, only 8% smaller. Thus, even for reactive liquid metal solder flow that is sufficiently fast (Figure 4), only Poiseuille flow with a dynamic contact angle is required to describe the experimental data. Similar behavior can be seen in the alcohol data, but the time constant is so small compared to the total flow time that accurate measurements cannot be made. The large value of γ/µ, and the resulting large flow rate, for solder over 1-heptanol results primarily from a large γ for the liquid metal; we expect the time constant for approach of θ(t) to θ0 to scale with viscosity. Experimentally, one requires measurably large values of (z,t) in the region where curvature occurs; this requires large k (or large γ/µ) for z and large τ for t. Through the use of open surface grooves one can systematically vary liquid/solid material combinations and parameters such as groove geometry and size to probe such nonlinear behavior as well as subsidiary effects including reactions at the threephase contact line. Acknowledgment. This work was performed at Sandia National Laboratories, which is supported by the Department of Energy under contract number DEAC04-94AL85000. LA9605201 (15) Howie, F. H.; Hondros, E. D. J. Mater. Sci. 1982, 17, 1434. (16) Ejima, T.; Sato, Y.; Yamamura, T.; Hayashi, A.; Yamazaki, T. J. Jpn. Inst. Met. 1990, 54, 1005.
