Langmuir 1993,9, 3691-3694
3691
Surface Tension from Pendant Drop Curvature A. Ldpez de Ramos Department of Chemical Engineering, The University of Tulsa, Tulsa, Oklahoma 74104
R. A. Redner Department of Mathematical and Computer Sciences, The University of Tulsa, Tulsa, Oklahoma 74104
R. L. Cerro* Department of Chemical Engineering, The University of Tulsa, Tulsa, Oklahoma 74104 Received July 19,1993. In Final Form: October 14, 199P A basic property of the curvature of pendant drops and computer splines have been used to develop a simple and highly reliable method for the determination of interfacial tension of fluid-fluid interfaces using a drop shape technique. Solutions to the Laplace-Young equation for pendant drops predict a linearly varying curvature as a function of elevation. Plotting the mean curvature of the pendant drop versus elevation renders a straight line with slope a = -Ap glu, where Ap is the difference in densities of the two fluids, g is the acceleration of gravity, and u is the interfacial tension. Numerical differentiation of experimental data, however, is nearly impossible when any level of noise is present. Computer-generated spline functions were used to represent the experimental data to a prescribed degree of smoothness. In turn, derivatives of the spline functions provide an accurate and reliable way to determine the curvature of the drop image. The method has been tested with the computer-generated image of a water drop and with experimental data for five hydrocarbons as well as decyl alcohol.
Introduction Since the pioneering work of Worthington,l the drop shape technique has been favored by scientists as a way to determine the interfacial tension of fluid-fluid interfaces. Recently, this technique has received more attention as a consequence of the easy accessto video image analysis systems that allow computer capture of images and contours. Some of the drawbacks of these automated methods are the need to generate computer solutions of the Laplace-Young equation and the use of optimization techniques to generate a best fit between the computer solution and experimental data.2 The method proposed here takes advantage of a basic property of the curvature of pendant drops; the curvature decreases linearly with the ordinate. In fact, this observation is implicit in the words of Young3 describing the shape of capillary surfaces. Pujado et aL4 translated Young’s words into simpler terms: ”...the sum of the curvatures at a point on the surface is proportional to the ordinate of the point above the level of a flat,free surface, and therefore to the local hydrostatic pressure head.” This statement was formalized by Laplaces in what is now known as the LaplaceYoung equation:
Ro is the radius of curvature at the drop apex where y = 0, Ap = pfl - p a is the difference in density of the two
* To whom correspondence should be addressed.
Abstract published in Advance ACS Abstracts, December 1, 1993. (1) Worthington, A. M. Philos. Mag. 1885,19, 46. (2) Jenning, J. W.; Pallas, N. R. Langmuir 1988, 4, 960. (3)Young,T.Philos. Trans. R. SOC.London 1805,95, 65. (4) Pujado, P. R.;Huh,C.;Scriven, L. E. J. Colloid Interface Sci. 1972, 38,662. (5) Laplace,P.S.CelestialMechanical; SuplementtotheTenthBook, 1805, Translated by N. Bowditch 1839;reprinted by Chelsea: New York, 1966. Q
fluids, u is the interfacial tension, and H is the mean curvature of the free surface. Proper pendant drop surfaces are cylindrical surfaces, i.e. they have a vertical axis of symmetry and can be generated by the rotation of a curve in a 3D space. As a consequence of this property, only the contour of the surface in two dimensions is needed and the mean curvature is computed from the equation of a curve in a 2D space. If the data are given as an x-y table, the curvature can be alternatively computed assuming y ( x ) or x ( y ) (Scrivens) 2% = x ( 1
+y yx
y 2
+ ( 1 +Yy,2)3/2 xx -- x(1 +1x y
2-
xyy
(1
+ xy2)3’2
(2)
where y xand yxxare the first- and second-order derivatives of y with respect to x and xy and xyy are the first- and second-order derivatives of x with respect to y . Differentiation of experimental data introduces large numerical errors and these errors increase with the order of the derivative. Even when low levels of noise are present in experimental data, computation of the curvature of the pendant drop using eq 2 is not practical since the derivatives fluctuate wildly. Noise is unavoidable in experimental data and this is more so in data obtained using video images, because the number of pixels and consequently the resolution of a video screen is relative small, i.e. 480 X 480 pixels. An efficient way to compute derivatives of experimental data is to use spline functions. Computer-generated spline functions can be made to closely fit experimental data while retaining a prescribed degree of smoothness in the first and second derivatives. (6) Scriven,L E.Surface Geometry for Capillarity; ChEn 8104 clam notes; University of Minnesota, 1981.
0743-746319312409-3691$04.00/0 0 1993 American Chemical Society
L6pez de Ramos et al.
3692 Langmuir, Vol. 9, No. 12, 1993 High pressure cell
Computer-Generated Spline Functions Spline functions consist of a number of polynomials smoothlyjoined together at connecting points denoted as knots. Smoothness is assured at the connections by continuously matching the function and its derivatives up to a prescribed degree. A spline of order K with knots U(1) < U(2)< ... < U(m)is any function s ( x ) defined on [U(l), U(m)]which is a polynomial of degree K - 1 on each interval [ U(i),U(i+l)) for 1Ii < m such that s ( x ) is K- 2 times diffrentiable over the interval [U(l), U(m)). If s ( x ) is restricted to the interval [U(K),U(m-K+l)], then there are functions ( B i ( X ) ) i = l W K and constants (Ci)i=lWKSO that7
XYZ
4
Tubularbench
Vibration isolation feet
6
PC/Targa board
m-K
s(x) = C C i B i ( X )
(3)
i=l
The basis functions (Bi(x))i=lmK depend only on the location of the knots. The representation of the set of splines of order K over the knot sequence U(1) < U(2)< ... < U(m)is the basis of this method which uses a leastsquares method to enhance the closeness of the fit. Let ( ( ~ k , y ( x k ) ) ) k = 1be~ the set of points for an experimental pendant drop data profile. Once the order of the spline and the knots have been specified, the spline function is completely defined by the coefficients(Ci)itlWK. If the data set is free of noise, which will be unusual for experimental data, the coefficiets (Ci)i=lmKmay be computed using interpolation or a least-squares method that minimizes the deviation of the spline function from the data points. When the number of data points is small or noisy, as in the case of experimental data, a least-squares fit may be used and a penalty term can be added to further smooth out the fitting spline. The penalty term, I,in this case was defined on the basis of the second derivative since this is the derivative that has more incidence on the curvature and will be subject to larger error:
I=
p($s(x))
2
w(x) dx
(4)
where the weight function w(x) may be used to provide local control of the contribution of the second derivative to the penalty function. In addition, it is possible to assign a global weight to the contribution of the penalty function on the objective function, using a global parameter A, hereafter referred to as the smoothing factor. Let C denote the column vector of unknown coefficients C1, ..., C m ~ .Then E(C) can be defined as N
m-K
The set of coefficients(Ci)i=1WKthat minimize the objective function, E(C), must also satisfy the equation
(M + XS)C = R
(6)
where R is a column vector whose ith coordinate is (7) de Boor, C. A Practical Guide t o Splines; Springer-Verlag: New York, 1978.
Figure 1. Schematic of experimental apparatus.
(7)
M is a banded, (m- K ) by (m- K ) matrix whose ijth element is N
and S is a banded matrix whose ijth element is
sij = p3,”(x)Bi”(x)w(r)dx
(9)
When the number of data points is greater than the number of basis functions, the matrix M is almost always nonsingular and the system of equations has a unique solution. If X > 0 then M + AS is always invertible. Both M and S are banded matrices with K - 1bands above and below the diagonal. The resulting system of linear equations was solved using the Linpack routines. The system of equations can be generated in an analogous way using y as the independent variable. By use of y(x),the singularity in the curvature formula at the side of the drop is avoided, while using x ( y ) the singularityin the curvature formula at the bottom of the drop can be prevented. Pendant drop techniques require that the axis of the drop image be accuratelyaligned with one of the coordinate axis. Translations and rotations of coordinate points, necessary to assure symmetry,are done using the following expressions
Xi = ( X i , - Xo) cos 8 + ( Yiexp- Yo) sin 8 (loa)
yi= (Yiexp- Yo) cos 8 - (Xi,, - Xo) sin 8 (lob) where XOand YOare the components of the coordinate of the apex in the experimental data and 6 is the angle of rotation. Previous to the generation of the spline functions, a linear regression program, using the angle 8 as an adjustable parameter, minimizes the difference in ordinates from the right-hand side and left-hand side of the drop image. The position of the apex in turn, results from choosing the values of Xo and YOthat provide the best fit. Experimental Apparatus and Computer Processing of Experimental Data A sketch of the experimental apparatus is shown in Figure 1. Pendant dropsare created inside a low pressure cell usinga syringe and a needle truncated at a right angle. The size of the needle can be changed in order to create drops of different size. The enclosing cell is needed to avoid perturbations caused by the movement of air around the drop and also in order to use gases
Langmuir, Vol. 9, No.12, 1993 3693
Surface Tension from Pendant Drop Curvature
,
0.00
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Figure2. Watertheoreticalpendantdropcurvatureasafunction of y: (A) array YW;(b) array zW.
differentthan air. There is also ahigh pressure cell with sapphire windows,ratedup to 5000 psi4 that can be used for high-pressure experiments. The observationcellis mounted on an Orieltubular optical bench with vibration isolation legs and the entire setup reeta on a marblevibration isolationtable. The cellis illuminated from behind using a fiber optic lamp with a diffuser. A CCD video camera (CCD-72 from DAGE-MTI Inc.) is placed on a moving support on the tubular bench in front of the cell. A Nikkon 65" micro lens with a PK-13 extension ring is used for large drops and a D.O. Ind. Zoom 6OOO Microscopic is used for the smallest drops. The video image is recorded or directly fed to the computer using a TARGA-16 board in a 80846-33PC clone. Once the video image is stored in the computer, customdesigned computer programs are used to generatethe numerical output of the drop contour (Papadimitriou9. A contrast subroutine is used to eliminate dark spots or smears. An edge detection program is used to generate the edge of the drop, and an additional program saves the contour data as an z-y table. A modifled Sobel method was used for edge detection (Pallas and Harrisons). In general accuracy was within h1 pixel. Six liquidswere used to test the technique: n-decane (99+% , Aldrich), decyl alcohol (99+% ,Aldrich), 2,2,4trimethylpentane (99+%, Aldrich), heptane (99%,Aldrich), hexadecane (99%, Aldrich), and toluene (99% ,Aldrich). In all cases the gas phase was air at 25 OC and 1atm.
Results and Discussion To remove concerns on whether splines would introduce any bias in the computations of curvature from experimental data, a set of image points was generated using a computer solution to the Laplace-Young equation. This computer-generated data set, for a fluid with properties similar to water (densityp = 998 kg/mgandsurfacetension u = 0.072 N/m) was computed using double precision subroutines and its accuracy was estimated to be up to five digits. By contrast, the estimated precision for images generated from experimental data is between two and three digits. The data set had 500 points and the number of knots used was 70 with no smoothing factor. Figure 2 shows curvature as a function of elevation for the theoretical data set. Figure 2A shows how curvature computed using y ( x ) becomes unstable around the side of the drop. Figure 2B,on the other hand, shows curvature computed using x(y) and fluctuates wildly at the bottom of the drop. Even for a theoretical data set with no experimentalnoise, curvature cannot be reliably computed for the entire (8)Papadimitriou, A. An Fkperimental Study on Rheological,Electromechaaical and Wetting Behavior of Alumina-MetalOxides Suepeneione; Maeter Thesis,The University of Tulsa, 1991. (9) Pallas, N. R.; Harrison, Y. Colloids Surf. 1990, 43, 169-194.
Figure 3. Water theoreticalpendant drop curvature asa function of y: Overlap of parta A and B of Figure 2.
contour of the drop using a single definition of the mean curvature (eq 2). Figure 3 shows the combination of all data points &r the bottom and side fluctuations are eliminated. Figure 3 was generated by matching the two plots (Figure 2) around the point where dy/& = 1. The interfacial tension computed from this set is equal to u = 0.07203 f O.oooO1 N/m. If only the linear segments of Figure 2 are used to determine the interfacial tension, the computed value of u is exactly the same. As a consequence, there is no need to use the entire set of experimentalpoints; only the ones that fall within the linear range are needed. It is easy to run a sequence of linear regressions, rejecting the points that are outside a prescribed error from a straight line, until all points fall within prescribed limits. The use of the array x ( y ) is recommended over the array y(x) to calculate the curvature, since the array x ( y ) is a function (it does not have repeated values of the independent variable) and has the singularity in the curvature formula m). at the beginning of the data set (at y = 0, dx/dy Figure 4a shows the experimental data points for n-decane at 25 "Cin air at atmospheric pressure. Figure 4b shows the curvature of the drop image as a function of elevation for n-decane. In this case 16 knots and no smoothing factors were used. In all the experiments, 13 to 16knotswere usedand there was no need for asmoothing factor. The deviations from the straight line in Figure 4b are due to inherent errors of the spline functions at fitting boundary points. If these points are eliminated, about 70% of the data points still remain to be used in the linear regression. Figure 5 shows the effect on curvature of the number of knots used in the computation of the spline functions. Too few knots would propagate the end effects into the linear segment of the curvature. Too many knots would have a tendencyto follow closely the inherent noise present in the data. For data sets free of noise, as was the case of our computer-generated data set, a large number of knots can be used. The optimal number of knots is found by minimizing the deviation from unity of the correlation factor, R. The smoothness factor, A, is not needed when the number of knots is optimal. The effect of the smoothness factor on curvature is shown in Figure 6 for n-decane at three different values of A, O,lO-e, and 1W. A number of knots equal to 20 was used, not too far from the optimal number 16,in order to make this effect more visible. The interfacial tension determined using this figure is identical to the value determined using the optimum number of knots. This result confirms that when experimentaldata have a high level of noise, a proper combination of the number of knots and the smoothness factor would eventually bring the curvature into a straight
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Lbpez de Ramos et al.
3694 Langmuir, Vol. 9,No. 12, 1993 0.14
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Figure 6. Decane pendant drop curvature as a function of y for different smoothing factors (0,O;A, lE+,O,lE-3).
16
Table 1. Surface Tension Values from Pendant Drop Curvature (25 "C and 1 atm) surface tansion % deviation of (hO.1mN/m) spline with respect to Hanwn Hansen and and liquid spline Jasper Roderud Jasper Roderud
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0.3
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Figure 4. (a, top) Decane experimental pendant drop profile. (b, bottom) Decane pendant drop curvature BB a function of y (16 knots and no smoothing).
22.2 25.7 2,2,4-trimethyl- 18.1 pentane heptane 19.6 hexadecane 24.3 toluene 25.5
23.5 28.6 18.4
22.3 25.3 18.1
5 10 2
0 2 0
19.7 27.1 28.1
19.4 23.7 25.6
1 10
1 3 0
9
Figure 5. Decane pendant drop curvature BB a function of y for different knot numbers (0, 16 knots; A, 14 knots).
methods used for these sources were based on capillary rise. Hansen and Rpldsrudll solved the Laplace-Young equation numerically u s w a Runge-Kutta-Merson technique and correlated the ratio of the maximum diameter, DE,and the diameter at an elevation equal to the maximum diameter, Ds. The values of t9 = DS/DEare then correlated as a function of the interfacial tension. The agreement between our experimental results and others is very good, except perhaps with some values of Jasper."J Reproducibility was tested by computing the splines using the same drop image and by using two different drops of the same fluid. In the first case, errors reflect the reproducibility of the edge detection and spline fitting techniques. Typical error for this case was about 0.41 7%. In the second case, errors include also the reproducibility of the experimental technique and the effect of drop size. Typical error for the second case was 0.61 7% . The method presented here for the experimental determination of surface tension is simple and accurate. It has the additional appeal that it can be completely automated and will work even with noise levels that would preclude the use of other methods.
line, where the deviation of the second-order derivative from a mean value is very small. Table 1summarizes the experimental values of all six liquids tested and compares these interfacial tensions with results from two independent sources. Jasperlo collected results from several independent sources, but most of the
Acknowledgment. We thank the financial support of British Petroleum of America, the Officeof Research, and the office of the AssociateDean for Research of the College of Engineering of The University of Tulsa. A. L6pez de h o s wishes to thank Gran Mariscal de Ayacucho and the Universidad Sim6n Bolfvar (Venezuela) for providing partial doctoral research fellowships.
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(10) Jasper, J. J. Phye. Chem. Ref. Data 1971, I , 841.
(11) Hansen, F.K.;Raderud, G. J. Colloid Interface Sci. 1990,141,l.
