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Langmuir 2008, 24, 1918-1923
Porous Media Characterization by the Two-Liquid Method: Effect of Dynamic Contact Angle and Inertia Becky Lavi,† Abraham Marmur,*,† and Joerg Bachmann‡ Department of Chemical Engineering, TechnionsIsrael Institute of Technology, 32000 Haifa, Israel, and Faculty of Natural Science, Institute of Soil Science, Leibniz UniVersity HannoVer, Herrenhaeuser Strasse 2, HannoVer, D-30419, Germany ReceiVed July 12, 2007. In Final Form: NoVember 9, 2007 The validity of using the Lucas-Washburn (LW) equation for porous media characterization by the two-liquid capillary penetration method was tested numerically and experimentally. A cylindrical capillary of known radius and contact angle was used as a model system for the tests. It was found that using the LW equation (i.e., ignoring inertia and dynamic contact angle effects) may lead to very erroneous assessment of the capillary radius and the equilibrium contact angle, for a relatively wide range of capillary radii and equilibrium contact angles. A correct assessment requires the application of a penetration kinetics equation that considers inertia and the dynamic contact angle.
Introduction The characterization of porous media and powders is of considerable importance for a variety of fields and applications, such as soil science,1 powder technology,2 suspensions and emulsions stability,3 and pharmaceutics.4 For applications that depend on the rate of capillary penetration into a porous medium, the concept of characterization usually implies the determination of an equivalent pore radius and an equivalent contact angle.5,6 Some methods characterize each of these properties separately: mercury porosimetry, for example, determines the distribution of pore sizes, whereas the Wilhemy plate method is used for contact angle assessment. A widespread method that measures both the equivalent radius and contact angle is the capillary penetration method,1,5-7 in which the rate of spontaneous penetration of a liquid into a porous medium is measured. The capillary penetration method is based on the assumption that an equivalent cylindrical capillary can be defined by equating the penetration rate into this capillary to that in the studied porous medium.2-10 Usually, the radius and contact angle of the equivalent capillary are calculated from the measured rate of penetration into the porous medium by using the LucasWashburn (LW)11,12 equation. However, these two parameters cannot be calculated from the results of a single experiment,5,6 and this problem is usually solved by the two-liquid method. * Corresponding author. Fax: 972-4-829-3088. E-mail: marmur@ technion.ac.il. † TechnionsIsrael Institute of Technology. ‡ Leibniz University Hannover. (1) Bachmann, J.; Woche, S. K.; Goebel, M. O.; Kirkham, M. B.; Horton, R. Water Resour. Res. 2003, 39 (12), 1353 ff. (2) Lazghab, M.; Saleh, K.; Pezron, I.; Guigon, P.; Komunjer, L. Powder Technol. 2005, 157, 79-91. (3) Cui, Z.-G.; Binks, B. P.; Clint, J. H. Langmuir 2005, 21, 8319-8325. (4) Buckton, G. Interfacial Phenomena in Drug DeliVery and Targeting; Harwood Academic Publishers, 1995. (5) Marmur, A.; Cohen, R. D. J. Colloid Interface Sci. 1997, 189, 299-304. (6) Marmur, A. Langmuir 2003, 19 (14), 5956-5959. (7) Van Oss, C. J.; Giese, R. F.; Li, Z.; Murphy, K.; Norris, J.; Chaudhury, M. K.; Good, R. J. J. Adhes. Sci. Technol. 1992, 6 (4), 413-428. (8) Siebold, A.; Walliser, A.; Nardin, M.; Oppliger, M.; Schultz, J. J. Colloid Interface Sci. 1997, 186, 60-70. (9) Marmur, A. In Contact Angle, Wetting and Adhesion; Mittal, K. L., Ed.; VSP BV: Utrecht, The Netherlands, 2003; Vol. 3, pp 373-383. (10) Labajos-Broncano, L.; Gonzalez-Martin, M. L.; Bruque, J. M. J. Colloid Interface Sci. 2003, 262, 171-178. (11) Lucas, R. Kolloid-Z. 1918, 23, 15. (12) Washburn, E. W. J. Chem. Soc. 1921, 119, 273-283.
First, the equivalent radius is determined by measuring the penetration rate using a liquid, which is known to have a zero contact angle with the medium (the “reference liquid”). Then, the contact angle of the liquid of interest is calculated from its measured penetration rate, using the equivalent radius (that was determined in the first experiment). For practical reasons, the experiments are performed by measuring the penetration rate in the vertical direction; however, most samples are sufficiently short to ignore the effect of gravity. It is well-known that the LW equation may lead to an oversimplified description of the penetration process, even in the case of a single cylindrical capillary.13 Two significant factors that are missing in the LW equation are inertia14-18 and the dynamic contact angle (DCA).19-22 Several papers23-26 considered the influence of both inertial and DCA effects on the penetration process in a capillary in general. The present paper focuses specifically on the effects of inertia and DCA on the two-liquid characterization method. It will be shown that ignoring inertia and the DCA may lead to very considerable deviations in the values of the equivalent radius and contact angle.
Theoretical Considerations Theoretical studies of the effects of inertia and DCA on the two-liquid characterization method were done by conducting (13) Marmur, A. In Modern Approach to Wettability: Theory and Applications; Schrader, M. E., Loeb, G., Eds.; Plenum Press: New York, 1992; pp 327-358. (14) Bosanquet, C. H. Philos. Mag., Ser. 6, 1923, 45, 525-531. (15) Szekely, J.; Neumann, A. W.; Chuang, Y. K. J. Colloid Interface Sci. 1971, 35 (2), 273-278. (16) Levine, S.; Reed, P.; Watson, E. J. In Colloid and Interface Science; Kerker, M., Ed.; Academic Press: New York, 1976; Vol. 3, p 403. (17) Letelier, M. F.; Leutheusser, H. J. J. Colloid Interface Sci. 1979, 72 (3), 465-470. (18) Quere, D.; Raphael, E.; Ollitrault, J.-Y. Langmuir 1999, 15, 3679-3682. (19) Joos, P.; Van Remoortere, P.; Bracke, M. J. Colloid Interface Sci. 1990, 136 (1), 189-197. (20) Friedman, S. P. J. Adhes. Sci. Technol. 1999, 13 (12), 1495-1518. (21) Siebold, A.; Nardin, M.; Schultz, J.; Walliser, A.; Oppliger, M. Colloids Surf., A 2000, 161, 81-87. (22) Martic, G.; Gentner, F.; Seveno, D.; Coulon, D.; De Coninck, J.; Blake, T. D. Langmuir 2002, 18, 7971-7976. (23) Martic, G.; De Coninck, J.; Blake, T. D. J. Colloid Interface Sci. 2003, 263, 213-216. (24) Stange, M.; Dreyer, M. E.; Rath, H. J. Phys. Fluids 2003, 15 (9), 25872601. (25) Fick, A. D.; Borhan, A. Ann. N.Y. Acad. Sci. 2006, 1077, 426-442. (26) Xiao, Y.; Yang, F.; Pitchumani, R. J. Colloid Interface Sci. 2006, 298, 880-888.
10.1021/la702090x CCC: $40.75 © 2008 American Chemical Society Published on Web 01/18/2008
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numerical simulations that imitate the actual characterization procedure. In these simulations, the actual measurement results were replaced with calculations using equations that do include the effects of inertia and DCA, whereas the interpretations were made using the LW equation (without including the additional effects). In order to study these effects in a well-defined system, the numerical simulations were conducted for a cylindrical capillary. This approach avoids dealing with the complexity of penetration into a real porous medium and directly studies the equivalent capillary system. Consequently, the deviations presented below may probably serve as lower bounds to the deviations expected in a real porous medium system. The influence of inertia on the penetration kinetics into a cylindrical capillary can be well described by the model proposed by Levine et al.16
dl Fr2 37 d2l 7 Fr2 dl 2 r dl rσl cos θe + ) l+ r 2+ + dt 8µl 36 dt 6 8µl dt 4l dt 4µl
(
)
()
(1)
where l is the length of the liquid column in the capillary, t is the time, F and µ are the density and the viscosity of the penetrating liquid, r is the capillary radius, σl is the liquid surface tension, and θe is the equilibrium contact angle (ECA) of the liquid with the capillary wall. In order to consider the DCA effects, θe in eq 1 should be replaced by θd, the DCA.
dl Fr2 37 d2l 7 Fr2 dl 2 r dl rσl cos θd + ) l+ r 2+ + dt 8µl 36 dt 6 8µl dt 4l dt 4µl
(
)
()
(2)
Among the models that exist in the literature for θd,19,27-29 the correlation that was developed by Cox28 has the simplest form. In addition, it was shown earlier30 that for the relevant ranges of velocities and contact angles the correlations of Cox,28 Blake and Haynes,27 and Shikhmurzaev29 similarly agree with experimental data. Therefore, the Cox equation was used in the present calculations:
(
)
µ dl θd ) θe3 + 9A σl dt
1/3
(3)
from eq 1 (or by replacing θd by θe in eq 4) and solving it to yield
l2 )
rσl cos θe t 2µ
(5)
As mentioned above, the numerical simulation imitated the procedure used in the two-liquid characterization method, for a cylindrical capillary system. First, eq 2 was solved for the system of a liquid with θe ) 0 (the “reference liquid”) and a given viscosity and a capillary of a given radius and length (the length chosen for the calculations was 10 mm, which is similar to a typical sample length in characterization experiments). These results were considered to simulate the experimental first stage. The calculated results of the kinetics of penetration were then fit to eq 5, in order to calculate the equivalent radius of the capillary. The equivalent radius was compared with the predetermined radius, and the relative deviation between them was calculated. The second stage of the simulation consisted of calculating the penetration rate of the liquid of interest (θe * 0) into the same capillary, again using eq 2. Then, the calculated results of the kinetics of penetration were fitted to eq 5, in order to calculate the equivalent ECA, using the already known equivalent radius. The equivalent ECA was compared with the predetermined value, and the relative deviation between them was calculated. In order to understand the relative contributions of inertia and DCA, the above two stages of the numerical simulation were repeated also with either eq 1 or eq 4 replacing eq 2, to separately elucidate the effects of inertia and DCA, respectively. Experimental Work
Here, A ≡ ln(R/s), R is the characteristic length of the system, and s is the slip length. In this case of capillary penetration the characteristic length, R, is the capillary radius. The slip length, s, is the distance from the capillary wall that defines the region where the continuum description breaks down.31 It is difficult to exactly calculate the value of the parameter A, but a rough approximation gave the value of about 14.32 An equation that considers only the effect of the DCA on the penetration kinetics can be obtained by omitting the inertia terms from eq 2 (the second, third, and forth terms on the left-hand side). The resulting equation becomes
dl rσl cos θd ) dt 4µl
Figure 1. Experimental system: (1) capillary, (2) liquid bath, (3) horizontal plate, (4) camera, (5) frame grabber equipped computer.
(4)
The LW equation can be obtained by omitting the inertia terms (27) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30 (3), 421423. (28) Cox, R. G. J. Fluid Mech. 1986, 168, 169-194. (29) Shikhmurzaev, Y. D. J. Fluid Mech. 1997, 334, 211-249. (30) Lavi, B.; Marmur, M. Colloids Surf., A 2004, 250, 409-414. (31) De Gennes, P. G.; Hua, X.; Levinson, P. J. Fluid Mech. 1990, 212, 5563. (32) Cazabat, A. M. AdV. Colloid Interface Sci. 1992, 42, 65-87.
In order to validate the theory used in the simulations of the characterization process the capillary penetration method was experimentally tested. First, the validity of eq 2 was experimentally confirmed. Then, the characterization procedure was applied to cylindrical capillaries of known radius and contact angle, in order to compare their derived equivalent values with the actual ones. The experimental system is described in Figure 1. The capillary (1) is mounted above a liquid bath (2), which is placed on a horizontal plate (3). The capillaries used were glass 0.25 µL micropipettes, supplied by Drummond Scientific Company. The internal radius of the capillaries, as stated by Drummond Scientific Company, was 0.04953 ( 5 × 10-4 mm. The experiment was performed by lowering the capillary until it just touched the surface of the liquid bath; the rise of the liquid in the capillary was then recorded by a Pulnix CCD camera (4) and captured as a sequence of frames by a Media Cybernetics, Pro-Series capture kit, frame grabber (5). The grabbed frames were analyzed with an image analysis software (Media Cybernetics, Image Pro Plus 4), to find out the liquid height as a function of time. The liquids used are listed in Table 1, along with their relevant physical properties; all of them were of analytical grade. The room temperature was kept at 24 ( 1 °C. The ECAs of the various liquids with the capillary wall, which are listed in Table 1, were estimated by measuring the equilibrium height of the liquids in a capillary. For this purpose, 5 µL micropipettes of 0.233 ( 2.3 × 10-3 mm internal radius were used. These bigger
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Figure 2. Capillary rise experimental results of the liquids listed in Table 1. 0, PDMS 1000; 9, PDMS 100; 2, cycloheptanol; [, squalane. For each liquid, the solid line is the best fit of eq 2 and the dashed line is the LW equation. Table 1. Physical Parameters of the Liquids Used for the Capillary Rise Experiments surface tension mN/m
liquid PDMS 100 (Aldrich)a PDMS 1000 (Aldrich)a squalane (Acros) cycloheptanol (Fluka) a
viscosity mPa‚s
density kg/m3
contact angle with capillary walls (deg)
21
96
960
0
21.2
970
970
0
34
34.1
800
41.5
37
53.6
959
38
Polydimethylsiloxane.
Table 2. Ranges of Values of the Parameters Used to Solve Eqs 1, 2, and 4 parameter
range of values
capillary radius liquid viscosity liquid surface tension ECA length of penetration
1-200 µm 1-100 mPa‚s 20-73 mN/m 0-87° 10 mm
capillaries were needed since in the 0.25 µL micropipettes the liquids rise to the end of the capillary before they reach their equilibrium heights. The ECA values are also listed in Table 1. The reported values are the averages of three measurements. The results for PDMS did not show any variation, while for the other liquids the variation did not exceed 2°.
Results and Discussion Equation 2 was used to simulate the characterization experiments, under the assumption that it describes the penetration kinetics accurately. In order to validate this assumption, it was tested by conducting capillary rise experiments. The results of these experiments are presented in Figure 2 as the squared height of the liquid column in the capillary versus time. Although the experiments were performed in a capillary rise mode, gravity can be neglected in the calculations, since the measured length of penetration (around 5 mm) is small relatively to the expected equilibrium height of penetration (order of magnitude of 100 mm). Figure 2 also presents the best fit of eq 2 for each of the experimental curves and the appropriate LW equation. Equation 2 had to be fitted to the experimental data since the parameter A, as explained above, cannot be directly assessed. However, for calculating the LW kinetics, the actual radius of the capillary and the contact angles of the liquids were used. As can be seen in Figure 2, the fit of eq 2 to the experimental data of all the liquids is excellent. The fitted values of the parameter A ranged from 11 to 16, in good agreement with expected values.32 On the other
Figure 3. Kinetics of penetration of water into a capillary. Comparison between eqs 2 (dashed lines) and 5 (solid lines). Thin lines, θe ) 0°; thick lines, θe ) 45°. (a) Capillary radius is equal 10 µm. (b) Capillary radius is equal 50 µm.
hand, the LW equation curves for each liquid fall far away from the experimental curves. These results justify the use of eq 2 as a reliable model for the kinetics of penetration. The disagreement between experimental data for the kinetics of capillary penetration and the LW equation was also demonstrated by others.19,21,33 Prior to the discussion of the errors that may be associated with the use of the LW equation in the twoliquid characterization method, it is worthwhile to directly demonstrate the possible deviations between the predictions of eqs 2 and 5, which represent the extreme cases. Figure 3 compares the theoretical predictions of these equations for the penetration of water into a capillary. The kinetics of penetration was calculated for two capillary radii, 10 and 50 µm, and for two ECAs, 0° and 45°. It can be seen that for the larger capillary (radius of 50 µm), the deviations are very meaningful. For the smaller capillary (radius of 10 µm) the effects of the DCA and inertia are meaningful for the case of an ECA of 0°. Thus, for pores of radii equal to or greater than about 10 µm it is expected that the characterization based on the LW equation will lead to meaningful deviations. It should be emphasized that the effects of inertia and the DCA are strongest at the initial stages of penetration, when the velocity is highest; therefore, they affect the later stages of penetration as well. The following detailed calculations indeed demonstrate that this is the case. As described above, the numerical simulations of the characterization process were done in two steps, as in the actual two-liquid characterization method. In the first step, the equivalent radius was elucidated by assuming the liquid to have a zero ECA. In the second step, the equivalent ECA for the liquid of interest was calculated, using the equivalent radius received in the first step. The penetration kinetics was simulated by eq 2, and the elucidation of the equivalent radius and equivalent contact angle was done by fitting the LW equation, eq 5, as is done in practice. Table 2 summarizes the ranges of the relevant parameters used in the calculations, which cover most of the practically (33) Hamraoui, A.; Thuresson, K.; Nylander, T.; Yaminsky, V. J. Colloid Interface Sci. 2000, 226, 199-204.
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Figure 5. Relative deviations between the equivalent and the actual ECA. Calculations are for a penetration length of 10 mm, σl ) 45 mN/m, F ) 1000 kg/m3, and A ) 14. The actual ECAs used in the calculation were 20° (bold line), 40° (regular line), and 70° (thin line). The viscosities of the reference liquid were 1 mPa‚s (solid line), 10 mPa‚s (dashed line), and 100 mPa‚s (dotted line). (a) Test liquid viscosity equals 1 mPa‚s. (b) Test liquid viscosity equals 100 mPa‚s.
Figure 4. Relative deviation between the equivalent and the actual capillary radius. Calculations were done for a capillary length of 10 mm, σl ) 20 mN/m, F ) 1000 kg/m3, and A ) 14. The viscosities of the reference liquid were 1 mPa‚s (solid line), 10 mPa‚s (dashed line), and 100 mPa‚s (dotted line). (a) Accounting for both inertia and DCA influences by solving eq 2. (b) Accounting for the DCA influence only by solving eq 4. (c) Accounting for inertia influence only by solving eq 1.
important values. While choosing the values of the liquid surface tension and ECA for the calculations, it was ensured that the corresponding solid surface tension had a reasonable value; in addition, it was ascertained that this value could be the same in the reference and test liquid simulations. The results of the numerical simulations of the characterization process are presented as the relative deviations between the radii or ECAs extracted by fitting the LW equation to the solution of eq 2 and the actual radii or ECAs used to solve eq 2. In order to demonstrate the relative contributions of inertia and the DCA to these deviations, the simulations were in some cases performed also by using eq 1 or 4, respectively, instead of eq 2. The relative deviations in the equivalent capillary radius, using reference liquids (surface tension of 20 mN/m, ECA of 0°) with various viscosities, are presented in Figure 4 for situations that involve inertia and DCA effects (Figure 4a), DCA effects only (Figure 4b), and inertia effects only (Figure 4c). The abscissa in Figure 4 is the actual radius that was used to solve eqs 1, 2, or 4. The ordinate presents the relative deviation between the actual radius and the radius extracted by the best fit of eq 5 to the solution of eqs 1, 2, or 4. Figure 4 shows that the deviation of the equivalent radius from the actual radius is significant even for relatively small radii. As expected, the deviation is higher as the actual radius is larger and may reach about 50% at the high end of the radii of interest. When both inertia and the DCA effects are taken into account, the deviations are, in general, higher than those for each
effect separately, depending on the liquid viscosity. The influence of viscosity is significant only for relatively large radii. When only the DCA effect on the LW equation is tested, the viscosity of the reference liquid has no influence on the deviations in the radius; therefore, only one curve is shown in Figure 4b. The contribution of inertia is considerably affected by the liquid viscosity, as can be seen in Figure 4c, for viscosities that are below 10 mPa‚s. It is interesting to notice that the deviations in the three cases (with or without DCA and inertia effects) may be of the same order of magnitude. This indicates that the deviations due to the individual effects are not necessarily additive. This can be explained by the fact that the DCA and inertia may have opposing effects. Inertia tends to accelerate the process and raise the velocity of penetration. Higher velocities lead to higher DCAs, which, in turn, tend to lower the velocity of penetration. In addition, it should be remembered that the best fit of the LW equation is independently achieved in each case so that additivity is not necessarily expected. The second step in the simulation is the determination of the equivalent ECA, using the equivalent radius calculated from the simulations with the reference liquid. Again, this was done by the following procedure: (a) solving eq 2 as a simulation of the kinetics of penetration, (b) fitting eq 5, the LW equation, to the results, in order to get the product r cos θe, and (c) using the equivalent radius calculated in the first step to get the equivalent ECA. These calculations were repeated for each of the equivalent radii received for the various reference liquids. Examples for the relative deviations in the ECA obtained by considering both the DCA and inertia are given in Figure 5 for several actual ECAs, several reference and test liquid viscosities, and test liquid surface tension of 45 mN/m. Figure 5 shows that there is a large influence of the actual capillary radius and ECA on the relative deviations in the equivalent ECA. The most significant deviations (up to 100%) are for low and medium ECAs. The deviations are less than 10% for high ECAs. In addition, strong effects of the viscosity also exist. It is interesting to notice that in some cases the
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Table 3. Equivalent Capillary Radii and Equivalent ECAs Calculated from the Fit of the LW Equation to the Experimental Data reference liquid
equivalent radius [mm] (relative deviation)
test liquid
equivalent ECA [deg] (relative deviation)
PDMS 100
0.03789 (23.5%)
squalane cycloheptanol
34.5 (16.8%) 40.0 (11.2%)
PDMS 1000
0.03528 (28.8%)
squalane cycloheptanol
28.0 (32.4%) 34.9 (3.15%)
deviations for high viscosity are higher than for low viscosity. In addition, some of the curves exhibit maxima and minima. It should be noted that this behavior is observed only when either the reference liquid or test liquid or both have viscosities that are lower than 10 mPa‚s. Examining the effects of the DCA only, by performing the same procedure using eq 4, revealed no such extremum points; however, calculations using eq 1 did show extremum points. Therefore, these phenomena can be attributed to the influence of inertia. Similar calculations were done for other surface tensions of the test liquid, and the results were qualitatively similar to those presented in Figure 5. Figure 6 shows a more general picture of the influence of the various parameters on the relative deviations in the ECA when both inertia and DCA are considered. This figure presents “phase diagrams” that divide the range of parameters into zones of high (>20%), medium (>10% and
