10942
Langmuir 2008, 24, 10942-10949
New Development in Processing Pendant Droplet Tensiometry Data Y. Leong Yeow,*,† Christopher J. Pepperell,† Firdaus M. Sabturani,† and Yee-Kwong Leong‡ Department of Chemical and Biomolecular Engineering, The UniVersity of Melbourne, Victoria 3010, Australia, and School of Mechanical Engineering, The UniVersity of Western Australia, Crawley Western Australia 6009, Australia ReceiVed May 26, 2008. ReVised Manuscript ReceiVed July 3, 2008 A database linking the dimensionless volume of pendant droplets Vpen and the dimensionless volume of the spherical caps at the apex of the droplets Vcap has been constructed from the governing equations of pendant droplet tensiometry. The Bond number Bo that relates surface tension to gravitational body force appears as an independent parameter in this database. Computing Vpen and Vcap from the measured profile of a droplet and making use of the database allow the prevailing Bo to be determined and surface tension to be calculated. This new way of converting measured profiles into surface tension has a number of advantages, such as reliability and simplicity, compared to existing methods. These are demonstrated by applying the new method to a number of measured profile data taken from the literature.
1. Introduction Pendant droplet tensiometry is a widely adopted method for measuring surface and interfacial tension.1-5 In a typical pendant droplet tensiometer of today, a CCD camera interfaced with a computer-based data acquisition system is used to capture the image of an equilibrium droplet. Edge-detecting software is then used to convert this image into a large set of closely spaced droplet height versus radius (z, r) data points. This automated drop shape tracing technique has sparked off the development of new computational schemes that can be used to convert the large droplet profile data set into surface or interfacial tension. For example, in a currently widely adopted scheme, the governing equations of the pendant droplet1
dr ) cosφ ds dz ) sinφ ds dφ ∆Fg 2 sinφ ) ×zds rApex γ r
(1)
dV ) πr2 sinφ ds is solved numerically to yield a computed profile. In eq 1 the tangent to the droplet profile φ and the droplet volume V, as well as z and r, are regarded as functions of the arc length s along the droplet surface. All these variables are measured from the apex of the droplet. γ and ∆Fg are the surface tension and gravitational body force, respectively. The unknown γ is determined by repeated integration of eq 1, while it is being adjusted iteratively so that the computed profile closely matches the measured drop shape. This is a nontrivial computational * Corresponding author. E-mail:
[email protected]. Fax: 61 3 8344 4153. † The University of Melbourne. ‡ The University of Western Australia. (1) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; John Wiley: New York, 1997. (2) Padday, J. F. Surf. Colloid Sci. 1969, 1, 101. (3) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (4) Lin, S.-Y.; Chen, L.-J.; Xyu, J.-W.; Wang, W.-J. Langmuir 1995, 11, 4159. (5) Zhou, Y. Z.; Gaydos, J. J. Adhesion 2004, 80, 1017.
problem. Hoorfar and Neumann6 examined in great detail many of the recent hardware, software, and computational techniques used to obtain, compute, and match droplet profiles and they came to the conclusion that even with the latest techniques and instruments the resulting surface tension is still not as reliable as desired. This is a reflection of the inherently difficult nature of the pendant drop tensiometry problem. Instead of solving eq 1, an alternative approach is to compute the mean curvature of the droplet surface directly from the measured profile. γ is then deduced from the variation of the computed mean curvature with droplet height z.7 In such an approach the mean curvature is typically obtained by fitting spline or other forms of curve through the measured profile and differentiating the fitted curve twice. As differentiation is an ill-posed operation in that it amplifies the noise in the data, the surface tension obtained this way is very sensitive not only to the noise level and its distribution in the data but also to the assumed functional form of the curve used to fit the measured profile.8 Recently, Yeow et al. applied Tikhonov regularization to extract the mean curvature from the measured profile.9,10 Tikhonov regularization has the advantage that its built-in regularization parameter is able to keep noise amplification under control and it does not require the functional representation of the droplet profile to be specified. However, this method is unable to cope with the apex or the equatorial plane of the droplet where either dr/dz or dz/dr becomes singular. As a consequence, a small neighborhood of these regions has to be excluded. Tikhonov regularization is also considered by some as mathematically too complex for routine use in the laboratory. This paper describes an entirely different way of obtaining surface tension from measured droplet profiles. The method does not involve the noise-amplifying operation of differentiating measured droplet profiles. It also does not require repeated numerical solution of eq 1 and iterative adjustment of γ. The (6) Hoorfar, M.; Neumann, A. W. AdV. Colloid Interface Sci. 2006, 121, 25– 49. (7) de Ramos, A. L.; Redner, R. A.; Cerro, R. L. Langmuir 1993, 9, 3691. (8) Engl, H. W.; Hanke, M.; Neubauer, A. Regularization of InVerse Problems; Kluwer: Dordrecht, 2000. (9) Yeow, Y. L.; McGowan, K. L.; Ong, S. I.; Leong, Y. K. Langmuir 2005, 21, 11241–11250. (10) Yeow, Y. L.; Pepperell, C. J.; Sabturani, F. M.; Leong, Y. K. Colloids Surf., A 2008, 315, 136–146.
10.1021/la801598w CCC: $40.75 2008 American Chemical Society Published on Web 08/27/2008
Processing Pendant Droplet Tensiometry Data
Langmuir, Vol. 24, No. 19, 2008 10943
Figure 2. Constant Bo curves in the Vpen-Rapex plane. Bo ) 0.1 (topmost), 0.2, 0.3,...0.7, 0.8 (lowest) on these curves. Figure 1. Pendant droplet. Volume of whole droplet ) Vpen, volume of spherical cap ) Vcap, and common radius of Vpen and Vcap ) length scale ) rscale.
method is applicable to droplet profiles that take the classical pendant shape, i.e., droplets that exhibit an equatorial plane. It is not applicable to spherical cap droplets. The method is based on a set of precomputed dimensionless relationships between Vpen, the volume of a pendant droplet, and Vcap, the volume of the spherical cap that has the same radius as the pendant droplet and forms the apex of the pendant droplet. The common radius of Vpen and Vcap becomes the natural length scale of the problem and will be denoted by rscale. Figure 1 shows diagrammatically the definitions of the two volumes and that of the length scale. For a given fluid, Vpen and Vcap clearly depend on rscale as well as on ∆Fg and γ. Except for the simple limiting case of γ f ∞ where the pendant droplet takes on the shape of a truncated sphere, there are no simple expressions for Vpen and Vcap and no analytical expression relating these two volumes. For a general droplet these have to be evaluated numerically. In the present investigation the numerical relationship between Vpen and Vcap for different γ (more correctly, its dimensionless equivalent) is computed once and for all and stored as a database to be retrieved when required.
2. Construction and Properties of (Wcap, Wpen, Bo) Relationship The starting point for the construction of the general relationship between Vcap and Vpen is the dimensionless equivalent of eq 1
dR ) cos φ dS dZ ) sin φ dS dφ sin φ 2 - Bo × Z ) dS RApex R
(2)
dV ) πR2 sin φ dS Upper case symbols are used here to represent dimensionless quantities. For example, the dimensionless arc length S ≡ s/rscale and the dimensionless volume V ≡ V/(rscale)3 etc. The dimensionless Bond number, Bo ≡ ∆Fgrscale2/γ, arises naturally in the dimensionless equations. The boundary conditions associated with eq 2 are R ) Z ) φ ) V ) 0 at S ) 0 (the apex). From the way the problem has been made dimensionless, the integration of eq 2 starts from S ) 0 and stops at the point S ) Send where R(Send) ) 1. Send can correspond physically to the attachment point of the pendant droplet to the capillary of the tensiometer. In which case rscale can also be identified with the radius of the
capillary. But, as will be shown subsequently, this does not have to be so in general. The Rapex in eq 1 is the dimensionless radius of curvature at S ) 0. In tensiometry measurements, the droplet volume is, within limits, an adjustable parameter. In the solution of eq 2 this role is taken over by Rapex, which can be adjusted to vary the volume of the droplet V(Send). From eq 2 it can be seen that Bo is another independent parameter of the problem and can be regarded as a replacement for the second adjustable physical parameter, viz. the capillary radius. The nonlinear nature of eq 2 means that for appropriate combinations of Bo and Rapex it is possible to find multiple Send that satisfy the condition R(Send) ) 1. The smallest Send corresponds to a spherical cap droplet and the next smallest Send corresponds to the classical pendant droplet that exhibits an equatorial plane. All subsequent larger Send are associated with droplet profiles that exhibit a “waist”. Such droplets are physically unstable and are not observed in the laboratory.11 Thus, for any Bo, the extension of the Rapex versus V curve beyond this point is not relevant to standard pendant droplet tensiometry. Standard Runge-Kutta type integration procedure implemented in many scientific computing software is applied to integrate eq 2.12 For a given combination of Bo and Rapex only a very limited amount of iterative computation is required to locate the two Send of practical interest. This task is performed automatically by standard numerical procedures for solving algebraic equations found in most scientific computing software. No numerical convergence or other difficulties were encountered in the present investigation. Selected examples of the variation of Rapex with the dimensionless droplet volume V and Bo are shown in Figure 2. These are for Bo ) 0.1 (topmost), 0.2, 0.3...0.0.8 (lowest). There are several hundred points on each of these constant Bo curves so that together they trace out a smooth curve in the V-Rapex plane. Fine details, such as the minimum point in some of the constant Bo curves, do not show up clearly in a figure of this size. Similar curves have also been constructed for closely spaced Bo in the range 0.01 e Bo e 0.83. These are the Bo of practical relevance in pendant droplet tensiometry. The range of Rapex that gives rise to stable pendant droplets depends on Bo but falls within the general range of 0.85-4.8. In the construction of the constant Bo curves, the range of Rapex of interest is divided into a large number of points, with ∆Rapex as small as 0.001 toward the lower limit of Rapex, to ensure that the resulting curves have the degree of accuracy and resolution required in the determination of γ. All the constant Bo curves have the same general shape. See Figure 2. All of them exhibit a minimum in Rapex in the general neighborhood of V ) 2.5. To the left of the minimum point the (11) Schulkes, R. M. S. M. J. Fluid Mech. 1994, 278, 83–100. (12) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes, 4th ed.; Cambridge University Press: Cambridge, 2007.
10944 Langmuir, Vol. 24, No. 19, 2008
droplet takes the shape of a spherical cap; i.e., V there represents the dimensionless analog of the Vcap referred to in Figure 1. As the volume is increased, it is accompanied by a very steep reduction in Rapex. Beyond the minimum Rapex, the droplet changes over to the classical pendant shape and V here becomes the dimensionless analog of Vpen referred to in Figure 1. Here Rapex increases gradually from the minimum value with further increase in V and attains a maximum. At this point the pendant droplet starts to develop a “waist”. As mentioned above, such a droplet is unstable so the mapping of the constant Bo curves is terminated at this point. From Figure 2 it can be seen that the length of the constant Bo curves to the right of the minimum point decreases rapidly with increasing Bo. For example, for Bo ) 0.1 the righthand branch extends to Vpen as large as 47.63 (not shown) whereas for Bo ) 0.8 the curve terminates at around 4.483. This is in agreement with the practical observations that experimental conditions leading to large Bo, because of low surface tension or large capillary radius, do not support the formation of large stable pendant droplet. Computations show that the constant Bo curve, for Bo > 0.845 (approximately), does not extend beyond the minimum point. This implies that stable pendant droplets become unobservable under these conditions. From the above discussion and from Figure 2 it is clear that, for a specified Bo, there is a well-defined relationship between the volume of a pendant droplet Vpen and the volume of its constituent spherical cap Vcap. The present method of obtaining surface tension is based directly on this key property of pendant droplets. The (Vpen, Vcap) pair, for selected Bo, are shown in Table 1. Examples of (Vpen, Vcap) relationship are also presented graphically in Figure 3 as constant Bo curves in the Vpen-Vcap plane. All these results together with additional closely spaced (Vpen, Vcap) points for Bo in the range 0.01 < Bo < 0.8 (in step of ∆Bo ) 0.005) have been computed and organized as a database on a computer.
3. Treatment of Tensiometry Data and Results From the measured profile of a pendant droplet it is relatively simple to compute its volume Vpen and that of the spherical cap Vcap associated with it. Their common radius rscale can also be extracted directly from the measured profile. These then allow an experimental (Vpen, Vcap) point to be computed. As mentioned above, rscale can be taken to be the radius of the capillary the pendant droplet is attached to. In that case the (Vpen, Vcap) point is associated with the entire pendant droplet. However, from the same measured droplet profile, it is just as simple to compute the volume Vpen* of the pendant droplet at any height z* less than its full height. Associated with this partial pendant droplet volume, the corresponding Vcap* and rscale* can also be obtained. This then allows another experimental (Vpen, Vcap) point to be identified. Proceeding this way, a curve showing the variation of Vcap with Vpen and another showing the variation of rscale with Vpen can be constructed from a single measured droplet profile. As Bo ≡ ∆Fgrscale2/γ, it therefore varies along the experimental Vpen-Vcap curve. In the key step of the present method of processing pendant droplet tensiometry data, this variation of Bo along the experimental Vpen-Vcap curve is determined by locating the intersections of the experimental curve with the precomputed constant Bo curves in the Vpen-Vcap plane, such as those in Figure 3. From this Bo and the known rscale, a surface tension γ ) ∆Fgrscale2/Bo can therefore be calculated for any point on the experimental Vpen-Vcap curve. Thus, from a single measured droplet profile, it is possible to extract a sequence of estimates
Yeow et al.
of the surface tension. Provided there are no significant errors in the measured profile, these estimates should remain relatively close to one another. In fact, their variations can be taken as an indication of the quality of the experimentally measured droplet profile. This way of extracting surface tension from pendant droplet profiles will now be demonstrated by a number of data sets taken from the literature. Simulated Droplet. Figure 4a shows the computed droplet profile for water (∆F ) 998.2 kg m-3, assumed γ ) 72.79 mN m-1). There are 101 points in this computer-generated data set. It is the same set used by Yeow et al. to assess the performance of their Tikhonov regularization procedures.9,10 Except in their investigation, because of the complexity of their procedures, they have left out a significant number of these computed data points. For the purpose of computing Vpen and the associated Vcap and rscale, a polynomial curve of the form
r ) √z(a0 + a1z + a2z2...anzn)
(3)
is used to represent the discrete data points in Figure 4a (and also for all the other droplet profiles investigated below). The best-fit eq 3, with n ) 12, is shown as a light continuous curve in Figure 4a. The z term is introduced to take into account the singularity of dr/dz at z ) 0. The fitted curve is used only to evaluate Vpen, Vcap, and rscale, which does not require it to be differentiated. Consequently, serious noise amplification has been avoided. Provided thez term is included, the computed volumes and rscale are not very sensitive to changes in the form of the expression used to represent the discrete data. Typically, 8 < n < 14, the choice depends on the number of experimentally measured (z, r) data points. To generate a sequence of Vpen as described above, the height of the pendant droplet in Figure 4a, between its equatorial plane and its full height, is divided arbitrarily into 151 regularly spaced points. The resulting dimensionless experimental Vpen-Vcap curve and the corresponding Vpen-rscale curve are shown as continuous curves in Figures 4b and 4c, respectively. The shapes of these curves are typical of that observed in all the other pendant droplets investigated. Figure 4d is an enlargement of Figure 4b. Selected constant Bo curves, retrieved from the precomputed database, are shown as lighter curves in the same figure. For ease of identification, their intersections with the experimental Vpen-Vcap curve have been marked out. These intersections can, in principle, be read off directly from Figure 4d, but this is unlikely to have the accuracy required. Instead, they were obtained, to a high degree of accuracy, by solving numerically the algebraic expressions used to represent locally the experimental Vpen-Vcap curve and the precomputed constant Bo curve over the small neighborhood of each intersection point. There are several ways of performing this operation, all of which are simple to carry out particularly if performed using any of the scientific computing software currently on the market.12 The intersections, for selected Bo, are summarized in Table 2. The rscale at the intersections and the surface tension given by γ ) ∆Fg(rscale)2/Bo at these points are also shown in Table 2. The results in the final row of this table (and also in similar tables for all the other cases considered in this investigation) were obtained via a slightly different route and will be discussed separately in the next section. The values of γ in Table 2 show clearly that the experimental Vpan-Vcap curve together with the precomputed constant Bo curves has succeeded in retrieving, to a very high degree of accuracy, the surface tension used to generate the simulated droplet profile in the first place. While it can be argued that this is to be expected as there is no noise in the data, it should be pointed out that the
Processing Pendant Droplet Tensiometry Data
Langmuir, Vol. 24, No. 19, 2008 10945
Table 1. Selected (Vpen, Vcap, Bo) Relationships (a) Bo ) 0.01
Bo ) 0.02
Bo ) 0.04
Bo ) 0.06
Bo ) 0.08
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
2.09966 15.1776 28.2555 41.3334 54.4113 67.4892 80.5672 93.6451 106.723 119.801 132.879 145.957 159.035 172.113 185.19 198.268 211.346 224.424 237.502 250.58 263.658 276.736 289.814 302.892 315.97 329.048 342.125 355.203 368.281 381.359 394.437 407.515 420.593 433.671 446.749 459.827 472.905 485.983 499.06 512.138 525.216
2.09966 0.602193 0.464852 0.401462 0.362628 0.335553 0.315211 0.299159 0.286048 0.275061 0.265668 0.257512 0.250336 0.243956 0.238232 0.233056 0.228344 0.224031 0.220062 0.216393 0.212986 0.209814 0.206848 0.204069 0.201456 0.198994 0.196669 0.194469 0.192382 0.190401 0.188515 0.186719 0.185004 0.183367 0.181801 0.180302 0.178865 0.177489 0.176169 0.174906 0.173705
2.10499 8.44848 14.792 21.1355 27.4789 33.8224 40.1659 46.5094 52.8529 59.1964 65.5399 71.8834 78.2269 84.5704 90.9139 97.2573 103.601 109.944 116.288 122.631 128.975 135.318 141.662 148.005 154.349 160.692 167.036 173.379 179.723 186.066 192.41 198.753 205.097 211.44 217.784 224.127 230.471 236.814 243.158 249.501 255.845
2.10499 0.799854 0.614432 0.528448 0.475998 0.439609 0.412391 0.391 0.37359 0.359046 0.346649 0.335911 0.326488 0.318128 0.310644 0.303891 0.297755 0.292149 0.286999 0.282246 0.277842 0.273747 0.269926 0.26635 0.262994 0.259837 0.256861 0.25405 0.251389 0.248866 0.24647 0.244191 0.242022 0.239955 0.237983 0.236101 0.234303 0.232588 0.230954 0.229405 0.227981
2.11582 5.13889 8.16195 11.185 14.2081 17.2312 20.2542 23.2773 26.3003 29.3234 32.3465 35.3695 38.3926 41.4157 44.4387 47.4618 50.4849 53.5079 56.531 59.5541 62.5771 65.6002 68.6233 71.6463 74.6694 77.6925 80.7155 83.7386 86.7617 89.7847 92.8078 95.8309 98.8539 101.877 104.9 107.923 110.946 113.969 116.992 120.015 123.038
2.11582 1.06567 0.824692 0.7081 0.636353 0.586535 0.549335 0.520176 0.49651 0.476799 0.460046 0.445576 0.432911 0.421705 0.411697 0.402688 0.394524 0.387081 0.380259 0.373978 0.368172 0.362784 0.357768 0.353084 0.348699 0.344583 0.340713 0.337065 0.333621 0.330364 0.32728 0.324357 0.321583 0.318948 0.316446 0.314068 0.311811 0.309671 0.307653 0.305766 0.30408
2.12692 4.09827 6.06963 8.04098 10.0123 11.9837 13.955 15.9264 17.8978 19.8691 21.8405 23.8118 25.7832 27.7545 29.7259 31.6972 33.6686 35.6399 37.6113 39.5827 41.554 43.5254 45.4967 47.4681 49.4394 51.4108 53.3821 55.3535 57.3249 59.2962 61.2676 63.2389 65.2103 67.1816 69.153 71.1243 73.0957 75.067 77.0384 79.0098 80.9811
2.12692 1.24434 0.977819 0.841908 0.756768 0.697226 0.652634 0.617648 0.589258 0.565628 0.545564 0.528255 0.513126 0.499757 0.487835 0.47712 0.467423 0.458596 0.450518 0.443093 0.436239 0.429889 0.423987 0.418486 0.413345 0.408529 0.404007 0.399755 0.395749 0.39197 0.388401 0.385027 0.381835 0.378815 0.375958 0.373257 0.370709 0.368316 0.366087 0.364058 0.362418
2.13828 3.59297 5.04767 6.50237 7.95706 9.41176 10.8665 12.3212 13.7758 15.2305 16.6852 18.1399 19.5946 21.0493 22.504 23.9587 25.4134 26.8681 28.3228 29.7775 31.2322 32.6869 34.1416 35.5963 37.051 38.5057 39.9604 41.4151 42.8698 44.3245 45.7792 47.2339 48.6885 50.1432 51.5979 53.0526 54.5073 55.962 57.4167 58.8714 60.3261
2.13828 1.37773 1.10113 0.952639 0.857644 0.790533 0.740007 0.700256 0.667954 0.641052 0.618209 0.598507 0.581296 0.566098 0.552556 0.540395 0.5294 0.519402 0.510263 0.501871 0.494134 0.486976 0.480332 0.474147 0.468375 0.462977 0.457918 0.453168 0.448702 0.444498 0.440536 0.436801 0.433279 0.429957 0.426828 0.423887 0.421131 0.418568 0.416221 0.414157 0.41289
(b) Bo ) 0.1
Bo ) 0.2
Bo ) 0.4
Bo ) 0.6
Bo ) 0.8
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
2.14992 3.2869 4.42387 5.56085 6.69782 7.8348 8.97177 10.1087 11.2457 12.3827 13.5197 14.6566 15.7936 16.9306 18.0676 19.2045 20.3415 21.4785 22.6155 23.7524 24.8894 26.0264 27.1634 28.3003
2.14992 1.48655 1.20865 1.0519 0.949312 0.875955 0.820341 0.776401 0.740604 0.710745 0.685368 0.663471 0.64434 0.627449 0.612402 0.598895 0.586691 0.575598 0.565466 0.55617 0.547607 0.539691 0.53235 0.525524
2.2129 2.73693 3.26096 3.78498 4.30901 4.83303 5.35706 5.88108 6.40511 6.92913 7.45316 7.97719 8.50121 9.02524 9.54926 10.0733 10.5973 11.1213 11.6454 12.1694 12.6934 13.2174 13.7415 14.2655
2.2129 1.8182 1.57907 1.41777 1.30096 1.21203 1.14178 1.08468 1.03723 0.997079 0.9626 0.932626 0.906297 0.882962 0.862122 0.843387 0.826448 0.811054 0.797002 0.784124 0.772282 0.761358 0.751255 0.741889
2.37148 2.59418 2.81688 3.03958 3.26228 3.48497 3.70767 3.93037 4.15307 4.37577 4.59846 4.82116 5.04386 5.26656 5.48926 5.71195 5.93465 6.15735 6.38005 6.60275 6.82544 7.04814 7.27084 7.49354
2.37148 2.1757 2.0221 1.89835 1.79648 1.71112 1.63853 1.57604 1.52167 1.47392 1.43166 1.394 1.36023 1.32979 1.30223 1.27717 1.25429 1.23335 1.21412 1.19642 1.18009 1.16501 1.15105 1.13812
2.6116 2.72648 2.84137 2.95625 3.07113 3.18602 3.3009 3.41578 3.53066 3.64555 3.76043 3.87531 3.9902 4.10508 4.21996 4.33484 4.44973 4.56461 4.67949 4.79438 4.90926 5.02414 5.13903 5.25391
2.6116 2.50436 2.41051 2.32771 2.25413 2.18834 2.1292 2.07578 2.02732 1.98318 1.94286 1.9059 1.87194 1.84065 1.81177 1.78507 1.76035 1.73742 1.71613 1.69636 1.67799 1.66091 1.64504 1.63029
3.15521 3.18841 3.22161 3.25481 3.288 3.3212 3.3544 3.3876 3.4208 3.454 3.4872 3.5204 3.5536 3.5868 3.62 3.6532 3.6864 3.71959 3.75279 3.78599 3.81919 3.85239 3.88559 3.91879
3.15521 3.1229 3.09219 3.06299 3.03522 3.00878 2.98358 2.95955 2.93662 2.91474 2.89384 2.87388 2.85481 2.83659 2.81919 2.80257 2.78669 2.77154 2.75708 2.74329 2.73014 2.71762 2.70571 2.69439
10946 Langmuir, Vol. 24, No. 19, 2008
Yeow et al. Table 1. Continued (b)
Bo ) 0.1
Bo ) 0.2
Bo ) 0.4
Bo ) 0.6
Bo ) 0.8
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
Vpen
Vcap
29.4373 30.5743 31.7113 32.8482 33.9852 35.1222 36.2592 37.3961 38.5331 39.6701 40.8071 41.944 43.081 44.218 45.355 46.4919 47.6289
0.51916 0.513215 0.507651 0.502434 0.497536 0.492933 0.488604 0.48453 0.480697 0.477092 0.473706 0.470535 0.467579 0.464845 0.46236 0.460196 0.458671
14.7895 15.3135 15.8376 16.3616 16.8856 17.4096 17.9337 18.4577 18.9817 19.5057 20.0298 20.5538 21.0778 21.6018 22.1259 22.6499 23.1739
0.73319 0.725096 0.717555 0.710521 0.703958 0.697831 0.692112 0.68678 0.681814 0.677203 0.672938 0.669017 0.665449 0.662255 0.659485 0.657243 0.655801
7.71624 7.93893 8.16163 8.38433 8.60703 8.82973 9.05242 9.27512 9.49782 9.72052 9.94322 10.1659 10.3886 10.6113 10.834 11.0567 11.2794
1.12614 1.11503 1.10472 1.09517 1.08632 1.07814 1.0706 1.06367 1.05733 1.05157 1.04638 1.04178 1.03778 1.03441 1.03173 1.02983 1.02889
5.36879 5.48367 5.59856 5.71344 5.82832 5.94321 6.05809 6.17297 6.28786 6.40274 6.51762 6.6325 6.74739 6.86227 6.97715 7.09204 7.20692
1.6166 1.60391 1.59216 1.58132 1.57134 1.56219 1.55384 1.54628 1.5395 1.53349 1.52824 1.52377 1.5201 1.51726 1.51528 1.51423 1.51419
3.95199 3.98519 4.01839 4.05159 4.08479 4.11798 4.15118 4.18438 4.21758 4.25078 4.28398 4.31718 4.35038 4.38358 4.41678 4.44998 4.48318
2.68365 2.67347 2.66384 2.65475 2.6462 2.63817 2.63065 2.62365 2.61715 2.61116 2.60567 2.60069 2.59621 2.59223 2.58878 2.58584 2.58342
methods based on Tikhonov regularization reported by Yeow et al.9,10 were unable to achieve a comparable level of accuracy with the same set of noise-free data. This is because their methods include the noise-amplifying step of differentiating the droplet profile, which has amplified the unavoidable numerical noise in the computation. The simplicity of the calculations that led to the multiple estimates of surface tension estimates should also be noted. Simulated Droplet with Added Noise. To test the performance of the present procedure in coping with the unavoidable noise in experimental droplet profiles, it is applied to the simulated droplet data in Figure 4a but with 1.5% random noise added to it. The resulting noisy profile is shown in Figure 5a where the data points have been joined up to reveal, even with an imposed noise level as low as 1.5%, how jagged the profile has become. The best-fit eq 3 used in the computation of Vpen, Vcap, and rscale is shown as a lighter curve on the same figure. This curve has to be thickened for it to show up over the imposed noise. Following the same steps as described in the previous subsection, the resulting experimental Vpen-Vcap curve is shown in Figure 5b. The general shape of this curve is similar to that in Figure 4b and remains featureless. The intersections of this curve with the precomputed constant Bo curves together with other associated information and the resulting surface tension are summarized in Table 3. As expected, there are now significant variations in γ along the experimental Vpen-Vcap curve, particularly the 94.76 mN m-1 in the last row of Table 3. This exceedingly large γ has been excluded in the evaluation of the average surface tension and the cause of this erroneous result will be discussed later. It is noted that the average surface tension
of 71.23 mN m-1 differed from that in Table 2 by approximately 2.2%sa difference comparable with the level of the imposed noise. This demonstrates the ability of this new procedure in coping with random noise. The performance of the present method
Figure 4. Simulated droplet. (a) Droplet profile. (2) Simulated profile, light continuous curve: best-fit eq 3. (b) Vpen-Vcap based on droplet profile in (a). (c) Vpen-rscale based on droplet profile in (a). (d) Enlarged Vpen-Vcap plot. Dark continuous curve Vpen-Vcap based on (a); lighter curves: precomputed constant Bo curves, Bo ) 0.75 (lowest), 0.8, 0.85, 0.9, 0.95 (topmost). (b) Intersection points. Table 2. Results for Simulated Droplet
Figure 3. Constant Bo curves in the Vpen-Vcap plane. Bo ) 0.1 (lowest), 0.2, 0.3,...0.7, 0.8 (topmost) on these curves.
Bo
Vpen
Vcap
rscale (mm)
γ (mN m-1)
0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.125 0.15 0.191049
20.1095 18.019 16.2459 14.7268 13.4124 12.2648 11.2553 7.61171 5.315 2.20691
0.56928 0.596898 0.624741 0.652893 0.681464 0.710576 0.740329 0.903768 1.11207 2.20691
0.0721325 0.0746641 0.0771137 0.0794869 0.817907 0.84032 0.862144 0.963914 1.05591 1.19165
72.7718 72.7716 72.7733 72.7732 72.7721 72.7721 72.7711 72.7723 72.7715 72.7696
Processing Pendant Droplet Tensiometry Data
Langmuir, Vol. 24, No. 19, 2008 10947 Table 4. Results for Experimental Water Droplet13
Figure 5. Simulated droplet with 1.5% imposed random noise. (a) Droplet profile. Dark continuous curve: noisy simulated profile; light continuous curve: best-fit eq 3. (b) Vpen-Vcap based on droplet profile in (a). Table 3. Results for Simulated Droplet with Random Noise Bo
Vpen
Vcap
rscale (mm)
γ (mN m-1)
0.075 0.08 0.085 0.09 0.095 0.1 0.125 0.15 0.146198
16.9041 15.7769 14.7612 13.8343 12.982 12.1916 8.95331 6.68399 2.17798
0.611965 0.632053 0.652277 0.672823 0.693816 0.715417 0.836243 0.981362 2.17798
0.760627 0.777171 0.793353 0.809323 0.825158 0.840955 0.920226 0.996164 1.18957 average γ
75.5235 73.9166 72.4959 71.2527 70.1698 69.2381 66.3252 64.7695 94.7628a 71.2251
a
Excluded.
Bo
Vpen
Vcap
rscale (mm)
γ (mN m-1)
0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.347097
19.9003 15.4102 12.1813 9.75121 7.85153 6.31057 5.0063 3.81722 2.32412
0.628383 0.703291 0.783843 0.873319 0.976572 1.1021 1.2675 1.52367 2.32412
1.05723 1.142 1.22309 1.30159 1.37828 1.4539 1.529 1.60415 1.67104 average γ
72.9536 72.962 73.2298 73.7162 74.3938 75.2546 76.2943 77.5184 78.7632 75.0095
in Table 4. While there are significant variations in the surface tension shown in the last column, they cover the expected range. The average value of 75.01 mN m-1 is in acceptable agreement with the 71.79 mN m-1< γ < 72.40 mN m-1 reported by Jennings and Pallas13 and γ ) 73.95 mN m-1 reported by Yeow et al.9 n-Decane Droplet. de Ramos et al.7 used locally fitted splines to evaluate the derivatives of droplet profiles and obtained the surface tension from the variation of the mean curvature with droplet height. These authors used the profile of an n-decane droplet to demonstrate the performance of their technique. This profile, reconstructed by automatic scanning and digitizing from their Figure 4, is reproduced as discrete points in Figure 7a. As scanned, this profile has 587 data points. Only every fifth of the scanned points are shown in Figure 7a and used in the present analysis. de Ramos et al. did not include the origin in their diagram. This has been added in Figure 7a. The resulting experimental Vpen-Vcap curve for this droplet is shown in Figure 7b and the corresponding results in Table 5. Again, whereas there are variations in the individual surface tensions from different points on the experimental Vpen-Vcap curve, the average γ of 23.88 mN m-1 is definitely in good agreement with the result of 22.2 mN m-1 < γ < 23.5 mN m-1 reported by de Ramos et al.7 and that of 22.39 mN m-1 < γ < 23.37 mN m-1 by Yeow et al.9
Figure 6. Experimental water droplet. (a) Droplet profile. Discrete points: measured profile data13; continuous curve: best-fit eq 3. (b) Vpen-Vcap based on droplet profile in (a).
for this particular case again represents a significant improvement over that of Tikhonov regularization computation as reported by Yeow et al.9,10 Water Droplet in Saturated Nitrogen. Figure 6a shows the droplet profile data of Jennings and Pallas13 for a water droplet at 22 °C suspended in a nitrogen atmosphere saturated with water vapor. The LHS and RHS profiles, reported separately by these authors, have been combined and treated as a single set. The combined set, with only 28 data points, is extremely small compared to the large profile data set now routinely generated by CCD images. As there are perceptible differences between the LHS and RHS profiles, additional noise will be introduced by treating them as a single set. The best-fit droplet profile, based on eq 3, used in the computation of Vpen, Vcap, etc. is shown as a continuous curve in Figure 6a. As in the previous examples, the experimental Vpen-Vcap curve based on the data of Jennings and Pallas13 is shown in Figure 6b and the information extracted from the intersections of this curve with the precomputed constant Bo curves are summarized (13) Jennings, J. W.; Pallas, N. R. Langmuir 1988, 4, 959–967.
Figure 7. n-Decane droplet. (a) Droplet profile. Discrete points: measured profile data7; continuous curve: best-fit eq 3. (b) Vpen-Vcap based on droplet profile in (a). Table 5. Results for n-Decane Droplet7 Bo
Vpen
Vcap
rscale (mm)
γ (mN m-1)
0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.478896
11.6095 9.542 7.92758 6.75773 5.89611 5.20381 4.56858 3.79085 2.45227
0.886123 0.968416 1.06104 1.15807 1.2581 1.36622 1.49647 1.71303 2.45227
0.994927 1.04279 1.08857 1.12752 1.15955 1.18719 1.2136 1.24585 1.28301 average γ
23.5821 23.9131 24.1972 24.2292 24.0238 23.7013 23.3918 23.354 24.5666 23.8843
10948 Langmuir, Vol. 24, No. 19, 2008
Yeow et al.
Figure 8. Ethanol droplet. (a) Droplet profile. Discrete points: measured profile data14; continuous curve: best-fit eq 3. (b) Vpen-Vcap based on droplet profile in (a).
Figure 9. Variation of Bond number BoMaxcap with volume VMaxcap of the largest spherical cap droplet.
Table 6. Results for Ethanol Droplet14
as required. The (Vpen, Vcap) pairs as tabulated in Tables 1 and 2 cover the requirements of most surface tension measurements. As γ is very sensitive to small changes in Bo, manual interpolation of the data in these tables is unlikely to have the precision required; systematic computer-based interpolation is recommended. As mentioned above, additional closely spaced constant Bo curves, with intervals as small as ∆Bo ) 0.005, have been included in the database of (Vpen, Vcap). These additional data points greatly simplify and enhance the accuracy of the determination of the intersection points such as those shown on Figure 4d. All the computations in this investigation were performed to 12 or more significant figures, even though only 5-6 figures are reported here. Most existing methods of handling pendant droplet data yield only a single surface tension from a measured droplet profile. The present method routinely yields multiple estimates of the surface tension from a single measured profile. The number of estimates depends on the number of constant Bo curves available and hence the number of intersections with the experimental Vpen-Vcap curve. With the database used in this investigation, it is a relatively simple task to obtain 6-8 estimates from a single measured profile. Except for a noise-free profile, there will be an unavoidable spread in these estimates. Since the evaluation of Vpen and Vcap involves only numerical integration of the measured profile, which unlike differentiation, tends to even out random errors in the experimental data, the computation leading from the measured profile to the experimental Vpen-Vcap curve and then to the surface tension is unlikely to be a significant contributor to the observed spread in the surface tension. This has been confirmed by the results for the noise-free droplet data. The observed spread should be regarded as an indicator of the quality of the measured profile. The multiple estimates of the surface tension from a single droplet allow the effects of the unavoidable noise in the data to be ameliorated. This too has been confirmed by the results reported in this paper. While the present method indeed represents a new development in pendant droplet tensiometry, it can be related distantly to an earlier method of estimating surface tension that is now seldom used. In this superseded method, for a given pendant droplet, the surface tension is computed from the volume VMaxcap of the spherical cap with the maximum rscale ) rMax, i.e., the radius at the equatorial plane.1 From their definition, at this point VMaxcap ) Vpen ) Vcap can be computed. This corresponds to the unique minimum point on each of the constant Bo curves on the V-Rapex plane in Figure 2 and to the left-hand end point of each of the constant Bo curves on the Vpen-Vcap plane in Figure 3. Figure 9 is a plot of the variation of this special Bond number BoMaxcap against VMaxcap. Using the rMax and VMaxcap deduced from a measured pendant droplet profile, the BoMaxcap and hence γ ) ∆Fg(rMax)2/BoMaxcap can be obtained directly. The γ obtained
-1
Bo
Vpen
Vcap
rscale (mm)
γ (mN m )
0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.48119
13.7207 11.8671 9.70902 8.12252 6.77323 5.59513 4.83626 4.29202 3.74966 2.45484
0.821 0.880632 0.962848 1.05133 1.15695 1.29002 1.42003 1.5517 1.72552 2.45484
0.90047 0.937786 0.984332 1.02831 1.07223 1.11473 1.14508 1.16797 1.19015 1.22345 average γ
22.8306 22.6985 23.084 23.3934 23.7387 24.0541 23.8886 23.4725 23.0897 24.0862 23.4336
Ethanol Droplet. Dingle et al. applied Galerkin FEM to compute the profile of pendant droplets and obtained the surface tension by performing nonlinear regression computation between the computed and measured profiles.14 The experimental data of an ethanol droplet used by these authors to demonstrate their procedure are shown in Figure 8a. The data points from their LHS and RHS profiles have again been combined into a single set and processed together. The best-fit eq 3 used to describe this set of data is shown as a lighter curve on Figure 8a. As in the earlier cases, the experimental Vpen-Vcap curve for this droplet is shown in Figure 8b and the results following from this curve are summarized in Table 6. The surface tension, as expected, shows some variations but the average value of γ ) 23.43 mN m-1 is again in very good agreement with the value 22.62 mN m-1 reported by Dingle et al.14 and 23.40 and 23.63 mN m-1 by Yeow et al.9,10
4. Discussion This paper introduces a method of processing pendant droplet data that is significantly different from existing methods. This new method does not require the noise-amplifying operation of differentiating the measured profile or the numerically sensitive step of matching the computed profile with the measured profile. These two properties of the method are mainly responsible for its enhanced reliability as demonstrated by the examples considered. Just as important as its reliability is the simplicity of the present method. It also does not require the usual repeated solution of the governing differential equation followed by nonlinear regression of the computed and measured profiles. Most of the computational effort of the present method is in the construction of the Vpen-Vcap curves for a set of closely spaced Bo values. In dimensionless form these results are universal and they only need to be computed once and stored on computer to be recalled (14) Dingle, N. M.; Tjiptowidjojo, K.; Basaran, O. A.; Harris, M. T. J. Colloid Interface Sci. 2005, 186, 647–660.
Processing Pendant Droplet Tensiometry Data
this way is included in the last row of Tables 2–6. With the exception of that in Table 3, the γ values obtained via this route blend in well with the other estimates on the same table. The extremely high surface tension of 94.76 mN m-1 in Table 3 is clearly in error. The factors leading to this totally unexpected result can be identified by comparing the last row in Table 2 with that in Table 3. Because of the imposed random noise, the profiles given by the best-fit eq 3 in Figures 4a and 5a are slightly different with the former being better in agreement with the exact profile given by the solution of the Laplace-Young equation. This difference shows up in the difference in the locations of the equatorial plane. As a consequence, the VMaxcap and rMax in the last row of Tables 2 and 3 are also different. But their differences are consistent with the average magnitude of the imposed random noise of 1.5%. The two Bond numbers BoMaxcap, based on the two VMaxcap, given by the special relationship shown in Figure 9, are 0.1910 and 0.1462, respectively. The latter is too small by approximately 23.5%, a much larger error than the imposed random noise. It is this inherent sensitivity of BoMaxcap, particularly for low Bond number, to small changes in the VMaxcap obtained from a measured droplet profile that is responsible for the large error associated with the surface tension of 94.76 mN m-1. Because of the slowly changing radius near the equatorial plane, the determination of the exact location of this plane and hence of a reliable VMaxcap for a set of noisy profile data is particularly difficult. The consequential error in VMaxcap and the magnified
Langmuir, Vol. 24, No. 19, 2008 10949
error in BoMaxcap and in γ are in some ways unavoidable. It is therefore not surprising that this single-point method based on VMaxcap, in spite of its great simplicity, has essentially gone into disuse. It is interesting to note that when the erroneous surface tension in Table 3 is included in the averaging process, the resulting average γ of 73.58 mN m-1 may be acceptable for certain practical situations. This further demonstrates the advantages of having multiple estimates of the surface tension from a single experimental pendant droplet profile.
5. Conclusions The experimental Vpen-Vcap curve together with the precomputed constant Bo curves provides a reliable means of computing the surface tension from measured pendant droplet profiles. A key attraction of this method is its simplicity. The method has the added advantage that it routinely generates multiple estimates of the surface tension from a single measured profile. These estimates give an explicit indication of the quality of the measured droplet profile data and their average improves the reliability of the resulting surface tension. Acknowledgment. The authors gratefully acknowledge the pendant droplet profile data provided by Prof. Michael Harris and Dr. Nicole Dingle. LA801598W
