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New Laser Technique for Automatic Interfacial Tension Measurements: Laser Scanning Drop Shape Analysis (LASDA) Armin Semmler,†,‡ Reinhold Ferstl,§ and Hans-Helmut Kohler*,† University of Regensburg, Institute of Physical and Macromolecular Chemistry, D-93040 Regensburg, Germany, and D. & R. Ferstl GmbH, D-93155 Hemau, Germany Received December 15, 1995. In Final Form: April 22, 1996X Laser scanners are widely used both in science and technology for precision measurements, e.g. in mechanical product control. Mostly they are used for the one-dimensional analysis of rigid bodies, allowing a precision of 0.2 µm. We have developed a new laser scanning apparatus which yields the coordinates of a pendant drop with an accuracy of 1-2 µm. The new tensiometer can be operated from a PC with special software that allows completely automated measurement and simultaneous calculation of interfacial tension. Using an interrupt technique, temperature control of the whole system and the mechanical control of two dispense systems are managed at the same time. The value of interfacial tension σ is obtained within 3 s (using a 60 MHz Pentium PC) by numerical integration (Implicit Euler Method) of the Bashforth Adams differential equations with about 100 drop coordinates. Fitting (Downhill Simplex Method) is carried out by variation of three parameters. The new tensiometer automatically yields the dynamic interfacial tension. The LASDA device includes two autocalibrating systems to achieve measurements without any external calibration. The laser scanner serves as a kind of ‘magic eye’, controlling drop formation and break off as well as drop vibrations. Time-dependent changes are permanently shown on the measurement screen. The user software includes all features necessary for scientific and routine measurements; data administration and analysis, graphics, statistics, data export, and color printing. Due to special measuring programs the application range of LADSA is very wide: normal surface tension determination, dynamic measurements, relaxation studies, high-temperature/pressure measurements, etc. Examples of interfacial tension measurements are shown.
Introduction Exact measurement of interfacial tension of fluid interfaces and the determination of contact angles are of essential interest for a wide range of scientific and technological fields. Interfacial tension is the most important experimental parameter describing the structure and the thermodynamic state of fluid interfaces. Many efforts have been made to develop various techniques for measuring interfacial tension. For detailed reviews see Padday,1 Ambwani and Fort,2 and Neumann and Good.3 During the last two decades, methods working with bubbles or drops (maximum bubble pressure, drop volume technique, shape analysis of axisymmetric drops) have been widely used. The determination of the interfacial tension from the profile of drops, either sessile or pendant, is an old method, with roots lying in the late 19th century.4 This absolute method is based on exact physical theory without approximations and corrections. Although there are other advantages, such as the independence of wetting, this method did not find much application, mainly for two reasons: It was hardly possible * To whom correspondence should be addressed. E-mail:
[email protected]. † University of Regensburg. ‡ E-mail:
[email protected]. § D. & R. Ferstl GmbH. X Abstract published in Advance ACS Abstracts, July 1, 1996. (1) Padday, J. F. In Surface and Colloid Science; Matijevic´, E., Eirich, F. R., Eds.; Wiley-Interscience: New York, 1969; Chapter: Surface Tension, Vol. 1. (2) Ambwani, D. S.; Fort, T., Jr. In Surface and Colloid Science; Good, R. J., Stromberg, R. S., Eds.; Plenum Press: New York, 1979; Chapter 3. (3) Neumann, A. W.; Good, R. J. In Surface and Colloid Science; Good, R. J., Stromberg, R. S., Eds.; Plenum Press: New York, 1979; Chapter 2. (4) Bashforth, F.; Adams, J. C. An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid; University Press: Cambridge, 1883.
S0743-7463(95)01543-5 CCC: $12.00
to get the drop coordinates with sufficient accuracy and, without computers, the numerical calculations were laborious. Thus, different methods of approximation were developed,5-9 all of them with reduced accuracy. Since 1962, computers are used for numerical calculations.10,11 A collection of tables and solutions for the determination of interfacial tension from drops and bubbles is found in Hartland and Hartley.12 In the early 80s, there was a renaissance, evoked by the work of Girault13 and the group of A. W. Neumann.14 With an instrumental setup, mainly consisting of a light source and a CCD video camera, they determined the interfacial tension from a video picture using image processing with frame-grabbing hardware and software. Up to now, several other groups contributed improvements and modifications,15-21 although some limitations remain (see Summary and Outlook). We have developed a fast pendant drop tensiometer, employing a two-dimensional laser scanner. Laser scanners of various types are widely (5) Andreas, J. M.; Hauser, E. A. J. Phys. Chem. 1938, 42, 1001. (6) Stauffer, C. E. J. Phys. Chem. 1965, 69, 1933. (7) Fordham, S. Proc. R. Soc. London 1948, A194, 1. (8) Mills, O. S. Br. J. Appl. Phys. 1952, 3, 358. (9) Misak, M. D. J. Colloid Interface Sci. 1968, 27, 141. (10) Staicopolus, D. N. J. Colloid Sci. 1962, 17, 439. (11) Butler, J. N.; Bloom, B. H. Surface Sci. 1966, 4, 1. (12) Hartland, S.; Hartley, R. W. Axisymmetric FluidsLiquid Interfaces; Elsevier: Amsterdam, 1976. (13) Girault, H. H. J.; Schiffrin, D. J.; Smith, B. D. V. J. Colloid Interface Sci. 1984, 101, 257. (14) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (15) Bathia, Q. S.; Chen, J.-K.; Koberstein, J. T.; Sohn, J. E.; Emerson, J. A. J. Colloid Interface Sci. 1987, 106, 55. (16) Anastasiadis, S. H.; Chen, J.-K.; Koberstein, J. T.; Siegel, A. F.; Sohn, J. E.; Emerson, J. A. J. Colloid Interface Sci. 1987, 119, 55. (17) Jennings, J. W., Jr.; Pallas, N. R. Langmuir 1988, 4, 959. (18) Pallas, N. R.; Harrison, Y. Colloids Surf. 1990, 43, 169. (19) Hansen, F. K.; Rødsrud, G. J. Colloid Interface Sci. 1991, 141, 1. (20) Hansen, F. K. J. Colloid Interface Sci. 1993, 160, 209. (21) Lin, S.-Y.; Hwang, H.-F. Langmuir 1994, 10, 4703.
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With the “shape factor” denoted by
β)
∆Fgb2 σ
(3)
we get the dimensionless Bashforth Adams form of eq 2
dφ sin φ + ) 2 + βzj dsj xj
(4)
which describes the shape of the drop in terms of φ, s, and β. The bar indicates a length normalized to the radius b. Using sj as the independent variable, we obtain the following system of differential equations for xj, zj, and φ:
Figure 1. Geometrical relations of a pendant drop hanging at the tip of a capillary.
used both in science and technology for high-precision measurements; e.g., one-dimensional scanners in mechanical product control of rigid bodies commonly have a precision of 0.2 µm. Three-dimensional scanners have been used in surface analysis and in shape and color detection of complex real-world objects. Our laser scanner tensiometer yields the coordinates of a drop with an accuracy of 1-2 µm and does not need image processing. The drop coordinates are determined in real time. It may be operated from a standard PC platform with special software to allow for fully-automated measuring and simultaneous calculation of the interfacial tension. We shall designate our system by the acronym LASDA (laser scanning drop shape analysis). Fundamentals of Drop Shape Analysis Equations. The pressure difference across a curved interface is given by the classical Laplace equation
(
∆p ) σ
)
1 1 + R1 R2
(1)
where ∆p ) pressure difference (inside minus outside) across the interface, σ ) interfacial tension, and R1,2 ) principal radii of curvature. We regard a drop with ideal rotational symmetry with respect to the z-axis (Figure 1). The following derivations are for the case of a pendant drop, but the differential equations are also valid for a sessile drop. More complete derivations can be found in ref 12. We locate the origin of the coordinate system at the apex of the drop. At this point, R1 and R2 are equal and denoted by b; s is the arc length measured from the origin to a point on the surface of the drop. With the geometrical relations R1-1 ) dφ/ds and R2-1 ) sin φ/x, eq 1 becomes
σ
2σ sin φ ) + - ∆Fgz (dφ ds x ) b
(2)
where ∆F is the density difference between the two phases and ∆Fgz the deviation of ∆F from its value at z ) 0.
dxj ) cos φ dsj
(5a)
dzj ) sin φ dsj
(5b)
sin φ 1 dφ ) 2 + βzj ) dsj xj R h1
(5c)
where eqs 5a and b are geometrical relations and eq 5c is identical with eq 4. The boundary conditions are x(0) ) z(0) ) φ(0) ) 0. To calculate σ from eq 3, we need b and β. One cannot solve these equations analytically, so numerical procedures must be applied. Numerical ProceduressIntegration and Fitting. Simultaneous integration using the implicit Euler method (first used in ref 14) is started at the origin with estimated values of b and β (see Appendix 1). The calculated profile is fitted to the measured values by variation of the three parameters b, β, and z0. The latter represents the vertical displacement between the laser and the drop coordinate system (the horizontal values of the two coordinate systems are the same; see also Principle of Measurement and Accuracy). Fitting (see Appendix 2) is carried out using the Downhill Simplex Method,22,23 which has the following advantages: it does not need derivatives, it is robust and fast, and it allows one to fit an n-dimensional system simultaneously. As in any multidimensional minimization routine, there are cases, in which the routine may fail to locate the minimum exactly, resulting in a large scatter of the calculated values of interfacial tension. This will be the case when the shape factor β is close to 0, i.e. when the shape of the drop is nearly spherical. Such accuracy problems are of minor importance in the range of ‘normal measurements’ (0-100 mN m-1). Quite generally, one could minimize these difficulties by choosing another method, such as the Powell routine,23 a direction set method for multidimensional problems. The best solution would be a combination of both methods. Before entering into the numerical routines, the measured drop profile values pass a filter (this routine is part of the measurement software) that eliminates coordinates with an error greater than 10 µm. The remaining data are used to calculate a preliminary value of the interfacial tension, as described above. After that, in order to achieve highest precision, very small errors ranging from 3 to 10 µm are recognized and removed by a secondary level fitting routine (Contributed by Gerd Pelg, University of Regensburg, Germany), leading to a more accurate value of the interfacial tension. As a consequence, scattering of different scans of the same drop is reduced. This performance increase exceeds the extension in calculation (22) Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308. (23) Press, W. H.; Teukolsky, S. A.; Vetterling, V. T.; Flannery, B. P. Numerical Recipes in C, The Art of Scientific Computing; Cambridge University Press: Cambridge, MA, 1992; Chapter 10, pp 408/735.
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Figure 2. Essential components of the LASDA measurement system.
time of ∼25%. The errors can be attributed to dust particles (l/v measurements) or inhomogeneities in the lighter liquid, which may arise from temperature gradients (l/l measurements). In general, the numerical errors in interfacial tension due to rounding etc. are smaller than 0.01 mN m-1 and, thus, do not significantly alter the precision of the technique. Instrumental HardwaresOptomechanical Principle Essential ComponentssSchematic Setup. Figure 2 shows the essential components of the LASDA measurement system: the tensiometer with its main part, the laser scanner. The dispense system with a microliter syringe is mounted on the measurement chamber. A highresolution stepping motor drive permits the drop volume to be controlled in steps of 0.125 µL. The dispense system can also be operated with a 3/2 valve, making it possible to change the sample without dismounting the syringe. The processor thermostat controls the temperature in the chamber to within (0.1 °C (in the range 10-70 °C). All components are controlled by the LASDA software running on a PC. Principle of Measurement and Accuracy. The Gaussian laser beam (red diode laser, 670 nm) is focused on the middle plane of the drop (diameter ∼ 100 µm). It is displaced horizontally by a high-speed rotating optomechanical system (f ) 80 Hz). Simultaneously the resulting ‘laser line’ is displaced in direction z by a linear lifting device. The horizontal displacement is achieved using a glass body of a special shape, acting as a planparallel sheet. Thus, the parallel displacment δ is given by the Snells Law:
[
δ ) d sin 1 -
]
cos (N - sin2 )1/2 2
(6)
where d is the thickness of the sheet, the angle of incidence, and N the ratio of the refraction index of glass over the refraction index of air. The optoelectronical resolution is about 0.4 µm in both axes. The scan range is about 18 mm × 14 mm. Despite the rapid rate of the scanning device, vibrations are practically not transferred to the drop because the chamber and the dispense system are mechanically separated from the scanner. All components are made of selected damping materials (e.g. cast iron, special polymers). Thus, the
device can be run on a common laboratory table, or, in buildings with stronger vibrations, on a damped table (see also SoftwaresAutomatic Measuring). When the laser beam passes the edge of the drop, there is a significant change in photodetector current. The electronic device determines the time the laser spot needs to go from one drop edge to the other in the horizontal direction. This time is translated into a “distance” using a high-precision calibration window of known dimensions which is part of the optical system of the scanner. This distance is twice the absolute value of the x-coordinate of the profile. The corresponding z-value is obtained from an electronic counter connected to the linear lifting device. The edge detection procedure used in the LASDA tensiometer is different from the procedures used in common, commercial one-dimensional laser scanners working with the precondition of a spherically or elliptically symmetrical laser spot moving in a direction perpendicular to the straight edge of the body analyzed. The edge corresponds to 50% of the laser intensity (middle point of the spot) and can be easily obtained from the zero of the second derivative of the detector current with respect to time (maximum gradient method). This gradient method can also be applied to a pendant drop as long as the laser spot diameter is very small compared to the curvature of the drop (virtually straight edge) and the laser passes the edge at an angle distinctly different from zero. The latter condition, however, is violated near the apex of the drop, where the angle approaches 0. The maximum gradient of the detector signal then no longer coincides with 50% of the laser intensity. Therefore, we have developed a special electronic circuit (permanent offset correction), setting the trigger point to exactly 50% of the actual laser intensity (even if there are intensity fluctuations). In l/l measurements, it may happen that the lighter liquid shows significant absorption. Even in these cases of reduced intensity, the offset correction supplies coordinates of high accuracy provided that the absorption is less than 70% (see below and Autocalibrating Systems in the SoftwaresAutomatic Measuring section). The drop coordinates of each scan (only a few bytes of data!) are filtered (see Numerical RoutinessIntegration and Fitting) and then stored in a virtual first in first out register. The numerical routines mentioned in Fundamentals of Drop Shape Analysis (see also Appendices 1
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Table 1. Technical Data feature
value
additional remarks
resolution measurement range scan time/profile max. scan rate coordinates/profile calculation time/profile
0.4 µm 0-100 mN m-1 0.3-2.0 s 1.5 scans/s 150 max. 3 s or less
max. absorption
70%
PC, minimal configuration
i 80486/50 MHz
both axes can be changed variable drop diameter 2 mm adjustable dependent on PC, number of coordinates lighter phase, l/l measurements 4 MB RAM, 1 MB VGA RAM, DOS 5
and 2) calculate the interfacial tension and deliver this value, together with that of time, to the visualization routines. Thus, at every moment of the measurement, the value of the time-dependent interfacial tension can be observed on the screen. To check the accuracy of our instrument, we have performed tests using mechanical standards: high-precision metal rods (diameters 2-7.6 mm, error (0.2 µm) and glass spheres (diameter 2-5.4 mm, error (1 µm). In more than 90% of all cases the diameters of the rods, measured in both axes, deviated by less than (0.6 µm from the nominal value. The spheres with diameter r were used for testing edge detection along a drop and for checking the numerical routines (with β ) 0 and b ) r). The measured values of r differed from the real mean value by less than 1 µm, the quoted error of the radius of the sphere itself. The shape factor obtained was scattering around 0 in the range 10-5 > β > -10-5. Compared with the ideal curve of a circle the measured coordinates did not show any significant distortion through the whole scan area. Further details about the measurement principle are described in ref 24. The technical data are given in Table 1. Measurement Chamber. Figure 3 shows the measurement chamber which can be used for various kinds of measurements up to 70 °C. It allows automated l/v as well as l/l measurements. It is made of Plexiglass with windows of high-precision optical glass. There are two heating elements. In the middle, there is a block of Teflon with a drainage and a mounting support for cuvettes (l/l measurements). A capillary and a temperature sensor are shown. The chamber is gastight up to ∼3 atm. For measurements up to 200 °C, the Plexiglas can be replaced by glass or stainless steel. The LASDA instrument can be run with other chambers (for even much higher temperatures/pressures), as long as the laser beam can pass through the chamber. SoftwaresAutomatic Measuring Running a pendant drop tensiometer based on a CCD camera normally requires a lot of manual work (forming the drop, taking the pictures, frame grabbing, numerical calculation of interfacial tension). In order to optimize this procedure, we have developed a software package which allows easy, fast, and automatic operation and at the same time high-precision measurements. The software package can be roughly divided into two parts, the measurement software and the user software. The first consists of subroutines responsible for different special tasks, such as process control and autocalibration, while the second is a comfortable interface for running the instrument. Measurement Software. The software is programmed in assembler language and C. We use a highly sophis(24) Ferstl, R.; Semmler, A. German Patent DE 44 04 276, 1995.
ticated interrupt technique to achieve automatic measurement with a common PC. There is no need for additional hardware. The computer manages the following processes simultaneously: scanning, control of the dispense system(s), temperature control of the chamber, numerical calculations, and on-line presentation of timedependent interfacial tension on the measurement screen. This software concept allows the user to implement new measurement programs by upgrading the software without any changes in the hardware equipment. User Software. A multiwindow display characterizes the user software (see Figure 4). After the parameters and setups are fixed, personal measuring programs can be defined and performed automatically. Apart from the main menu, the software provides three different screens: The dynamic working measurement screen offers permanent control of all relevant functions. Data managing has several search and sorting algorithms. Hence every user gets a quick overview about and fast access to earlier measurements. Any data file can be analyzed graphically on the third screen, which provides statistics, mathematical overlay functions, data export, and colored printing. Special Process Controls‘Magic Eye’. Generally, it is not easy to form drops of reproducible volume and quality. This is especially true for l/l interfacial tension measurements. We found that a high degree of automation increases the performance and the reliability of pendant drop measurements. So we have developed a special process control strategy based on the laser scanner. Using special control software, the scanner serves as a kind of ‘magic eye’. It is employed for smooth and automatic measurement, for drop formation, drop break off, and automatic cleaning processes between different drops. For every single drop, optimal speed parameters for the drop formation are calculated from the values of the drop volume and the diameter of the capillary (which is detected by the magic eye). Another important function is the detection of drop vibrations. Immediately after drop formation, the laser is positioned at the apex of the drop. Vibrations in the horizontal and vertical directions with amplitudes in the micrometer range are detected. Normally, scanning is only started if the drop has calmed down. If the drop is too large (for a given interfacial tension) and therefore keeps on vibrating for a period of about 30 s, the magic eye reports this condition to the user. Due to the cooperation of the magic eye and the dispense system, drop volumes can be produced with an accuracy of parts of 1 µl. Autocalibrating Systems. The LASDA tensiometer consists of mechanical and optical parts of highest precision. To avoid calibration problems, usually a major problem of pendant drop tensiometers, we have developed two autocalibrating systems. One manages the offset correction due to absorption in l/l measurements (see also Principle of Measurement and Accuracy); the other, working in the background, provides an internal absolute calibration. Every single scan of a drop is controlled via the software using a special mechanical calibration window.24 Normally, there is no need for any external calibration, but we have developed an automatic setup calibration which can also be used for checking the system, if desired. Fields of Application The measurements presented below are just a few examples of the wide field of applications of pendant drop tensiometers, especially of LASDA. In another paper, we will treat in more detail experimental topics related to LASDA (capillaries, drop size, temperature control, clean-
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Figure 3. Chamber for measurements up to 70 °C.
ing of substances and equipment, etc.) and demonstrate various measurements. Test Liquids: Water and Ethanol. The standard substances pure water and ethanol are well suited for testing the instrument. The measurements were done at (25.0 ( 0.1) °C in saturated air. Figure 5 shows a measurement of 25 different water drops, each scanned about 50 times, i.e. ∼1200 data points. We used ultrapure water (Milli-Q 185 Plus, Millipore). The mean value of the interfacial tension, 72.14 mN m-1, is very close to the literature values obtained from other drop shape measurements (72.10 and 71.99 mN m-1)17,18 and also to values obtained with the Wilhelmy plate method (71.98 and 72.05 mN m-1)26,27 and the capillary rise method (72.04 mN m-1).26 Note the low scattering yielding a small standard deviation of the measurement of a single drop (SD ∼ 0.05 mN m-1). In order to check liquids with comparatively low interfacial tension, we used absolute ethanol (p.a., from Riedel-de-Hae¨n, D-Seelze). Figure 6 illustrates the measurement against a saturated vapor of ethanol. Here, we measured 20 drops, each scanned about 40 times. In this case, the mean value 22.18 mN m-1 deviates significantly from the literature value of 22.39 mN m-1.25 We have never found such high values, which could be due to water contamination. The standard deviation of the total measurement, 0.03 mN m-1, is very small. l/l Interfacial Tension. The high level of automation and the efficiency of the magic eye make it relatively easy to perform l/l measurements. Using a high-precision dye laser cuvette, we are able to keep LASDA’s high precision even in l/l measurements. The autocalibrating system allows working with up to 70% absorption by the higher liquid (the heavier one, of course, is free to absorb 100% of the laser intensity). In Figure 7 the interfacial tension for an aqueous solution of anionic surfactant (c ) 3.3 × 10-2 mol L-1) against hexane is shown (with kind permission of Klaus Lunkenheimer, Max-Planck-Institut fu¨r Kolloid und Grenzfla¨chenforschung, D-Berlin). The surfactant, sodium decanesulfonate, is of surface chemical pure quality (The substance was purified by Anke Goebel and Klaus (25) Jasper, J. J. J. Phys. Chem. Ref. Data 1972, 1, 852. (26) Pallas, N. R.; Pethica, B. A. Colloids Surf. 1983, 6, 221. (27) Harkins, W. D. In Physical Methods of Organic Chemistry; Weissberger, A., Ed.; Interscience: New York, 1949; Part 1, Vol. 1, p 1483.
Lunkenheimer.);28 hexane p.a. (from Merck, D-Darmstadt) was chromatographically purified by Anke Goebel (MaxPlanck-Institut, D-Berlin). In addition to the good reproducibility, SD ) 0.015 mN m-1 (beginning at 20 s), the small changes in interfacial tension also suggest that the substances were very clean. Time-Dependent Measurements. LASDA is very useful for long-term time-dependent measurements. As an example we observed the decreasing interfacial tension of an aqueous solution of an enzyme (Lipase, from wheat germ, purity > 95%, Sigma, St. Louis), c ) 6.8 mg L-1, against commercial sunflower oil for a period of 24 min (Figure 8). A high degree of reproducibility was obtained. Application Range. The range of applicability is wide. LASDA is best suited for long-term, time-dependent measurements and especially for l/l measurements. The pendant drop arrangement is also amenable for investigations under extreme conditions (high T/p measurements). LASDA is also ideally suited for industrial, online process control. The software can be easily modified to fit special requirements. Another possible application would be as a control purification instrument, e.g. employed to automate the surfactant purification process developed by K. Lunkenheimer.28 Summary and Outlook To determine the interfacial tension from drop profiles, the drop coordinates must be determined with high accuracy. Our laser scanner provides these coordinates with high precision in real time with the numerical calculations done simultaneously. The tensiometer runs without external calibrations. Every measurement performed with LASDA provides the time dependence of the interfacial tension. Drop shape analysis has some obvious advantages. It is based on an exact physical theory without approximations or corrections. The method is independent of contact angle and wetting phenomena. A small amount of sample is sufficient for a measurement which meets the requirements of (industrial) research: testing of potential new products as well as expensive or rare substances. In addition, drop shape analysis is the only method which can be easily adapted to high-T/p measurements. (28) Lunkenheimer, K.; Pergande, H. J.; Kru¨ger, H. Rev. Sci. Instrum. 1987, 58, 2313.
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Figure 5. About 1200 data points of ultrapure water (Milli-Q 185 Plus, Millipore) measured at (25.0 ( 0.1) °C against saturated air. Mean ) 72.14 mN m-1, SD ) 0.06 mN m-1 (total measurement).
Figure 6. 800 data points of absolute ethanol, measured at (20.0 ( 0.1) °C against ethanol-saturated air. Mean ) 22.18 mN m-1, SD ) 0.03 mN m-1.
Figure 7. Aqueous solution of sodium decanesulfonate, c ) 3.3 × 10-2 mol L-1, measured at 22 °C against hexane. SD ) 0.015 mN m-1 (for t > 20 s).
Figure 4. Essential screens of the software: main menu, measurement screen, data managing, and data analysis.
Our LASDA device has additional advantages over previous drop shape devices. Interfacial tension is determined with high speed on a common PC without image processing. The numerical calculations are reduced to a minimum. In contrast, the CCD video camera based instruments currently available require some level of manual operation from the user to obtain a single value of the interfacial tension from a video picture. Magnification factors and x,y-offsets also mean that these current methods have to deal with five or six fit parameters. Thus, the calculation of interfacial tension is more complicated and requires more time than with a laser scanner. Frame grabbing is an additional time-consuming factor. Although these disadvantages can be overcome using fast hardware, there are some other shortcomings resulting from misalignment of essential components such as light source, calibration grids, and CCD camera plus macroscope. The optomechanical parts of our LASDA tensiometer are manufactured with high precision. The error
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With the linear approximation of the right-hand side, this yields (F ) F(X))
∆X 1 ) F + Fx ∆X ∆sj 2
(A1-5)
(E - 21 ∆sjF )∆X ) ∆sjF
(A1-6)
or
x
Figure 8. Aqueous solution of Lipase (from Sigma, St. Louis), c ) 6.8 mg L-1, measured at 22 °C against commercial sunflower oil.
coming from misalignments is negligibly small; moreover, there is an internal autocalibrating device.24 Thus, the LASDA tensiometer offers a high-precision device with a high degree of automation and a powerful software package. It will prove useful in equilibrium and dynamic interfacial tension measurements, especially in l/l systems, including relaxation studies of adsorption and of surface reactions. It may also be employed in applications requiring on-line process control. The scanner, acting as a magic eye, opens a new field of automation of pendant drop tensiometers. The possibility of working under extreme conditions offers another scope of applications, e.g. the fields of molten polymers, salts, ceramics, and metals. Acknowledgment. The development of LASDA is a cooperation of the group of H.-H. Kohler, University of Regensburg, and the D. & R. Ferstl GmbH, D-Hemau. The authors want to thank Dietmar Ferstl (D. & R. Ferstl GmbH) and Gerd Pelg (University of Regensburg) for valuable contributions (software). Financial support by the Deutsche Forschungsgemeinschaft (DFG) is greatfully acknowledged. Appendix 1. Numerical Integration The Euler method is a one-step algorithm with firstorder accuracy. Its explicit form (forward Euler) tends to become unstable.23 Therefore, we use implicit or backward Euler, which is an absolutely stable algorithm. If X(sj) is given by
()
xj X ) zj φ
where E is the unitary matrix. Fx is given by
Fx )
(
)
∂f1/∂xj ∂f1/∂zj ∂f1/∂φ ∂ F(X) ) ∂f2/∂xj ∂f2/∂zj ∂f2/∂φ ) ∂X ∂f3/∂xj ∂f3/∂zj ∂f3/∂φ
(
0 0 -sin φ 0 0 cos φ sin φ/xj2 β -cos φ/xj
)
(A1-7)
Thus, eq A1-6 may be rewritten as
(
1
)( ) )
1 0 /2∆sj sin φ ∆xj -1/2∆sj cos φ 0 1 ∆zj ) -∆sj sin φ/2xj2 -∆sjβ/2 1 + ∆sj cos φ/2xj ∆φ
(
cos φ sin φ ∆sj (A1-8) 2 + βzj - sin φ/xj
With obvious abbreviations this gives
( )( ) ( ) z1 1 0 c ∆xj 0 1 f ∆zj ) z2 z3 g h i ∆φ h
(A1-9)
which is solved by
∆xj ) z1 - ∆φc
(A1-10a)
∆zj ) z2 - ∆φf
(A1-10b)
(A1-1)
we can rewrite the system of differential equations given by eqs 5a,b,c in the following way:
X′ ) F(X)
(A1-2)
∆φ )
z3 - gz1 - hz2 i - gc - hf
(A1-10c)
where the prime denotes derivation with respect to sj and
( )(
f1(X) cos φ F(X) ) f2(X) ) sin φ 2 + βzj - sin φ/xj f3(X)
)
where the expression for ∆φ consists only of known values.
(A1-3)
We approximate eq A1-2 in the neighborhood of sj ) sj0 by
1 ∆X ) F X + ∆X ∆sj 2
(
)
(A1-4)
where X ) X(sj0), ∆X ) X(sj0 + ∆sj) - X(sj0), and F ) F(X(sj0)).
The starting point for the numerical integration is the apex of the drop (xj0 ) zj0 ) φ0 ) 0). We start with values slightly different from 0 to avoid division by 0. The step size ∆sj is kept constant during integration (and fitting). Once a single theoretical profile with initial values of β and b has been calculated, a comparison between this profile and the set of measured values is made (see Appendix 2). Finally, after a certain number of iterations, the values of β and b are accurate enough to be used for the calculation of the interfacial tension from eq 3.
4172 Langmuir, Vol. 12, No. 17, 1996
Semmler et al.
Appendix 2. Fitting For the purpose of fitting, the measured coordinates xexp and zexp are transformed as follows: i i
xexp i b
(A2-1)
- z0 zexp i b
(A2-2)
Figure 9. Schematic drawing illustrating least square fitting.
where z0 is the vertical displacement of the coordinate system. The transformation is very simple compared to the CCD camera method.14 We use least square fitting; thus, for m data points the objective function is given as
parameters: β, b, and z0. Thus, there is a threedimensional system to be minimized, described by
xji ) zji )
m
Q(u) )
m
Q)
rj2i ∑ i)1
where
(
u1 ) β u ) u2 ) b/fb u3 ) z0/fz
(A2-4)
The value of (xjsi, zjsi) is obtained by interpolation between two calculated points. Therefore, 2
rji =
[(zjj - zji)(xjj - xjj+1) - (xjj - xji)(zjj - zjj+1)]2 (xjj - xjj+1)2 + (zjj - zjj+1)2
(A2-6)
(A2-3)
From Figure 9 we obtain
jri2 ) (xjsi - xji)2 + (zjsi - zji)2
rji2(u) ∑ i)1
(A2-5)
As described in the section Fundamentals of Drop Shape Analysis, the LASDA method needs only three fit
)
(A2-7)
fb and fz are normalization constants. Minimization is done by the Downhill Simplex Method. This method has a geometrical naturalness, which makes it easy to illustrate the algorithm, especially for the case of three fit parameters. See ref 23 for a detailed description. LA9515433
