Real-Time Pendant Drop Tensiometer Using Image Processing with Interfacial Area and Interfacial Tension Control Capabilities

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

181, 385–392 (1996)

0395

Real-Time Pendant Drop Tensiometer Using Image Processing with Interfacial Area and Interfacial Tension Control Capabilities G. FAOUR, M. GRIMALDI, J. RICHOU,

AND

A. BOIS 1

Laboratoire d’OptoElectronique, Universite´ de Toulon et du Var, Ifremer, Z.P. de Bre´gaillon, BP 330, 83506 La Seyne sur Mer, France Received May 8, 1995; accepted February 12, 1996

Among different methods used for the measurement of interfacial tension, the pendant drop is particularly well adapted to liquid–liquid interfaces. Using an integrated form of Laplace’s equation, we have developed an automatic pendant drop tensiometer which allows a real-time measurement (i.e., three to five measurements per second) with an accuracy less than 0.1%. The very new feature of this tensiometer is the possibility of controlling the interfacial area and the interfacial tension during an experiment for the study of (i) adsorption processes at constant area or tension and (ii) the dynamical properties of the Marangoni effect for adsorbed layers. q 1996 Academic Press, Inc. Key Words: interfacial tension; drop; adsorption; rheological properties.

I. INTRODUCTION

At the liquid–gas interface it is common to study the adsorption process of tensioactive molecules using the Wilhelmy plate method and the rheological properties of monolayers and adsorbed layers (Marangoni effect, dilatational viscosity, etc.) using a Langmuir trough. In contrast, for the liquid–liquid interface the usual Leconte du Nou¨y ring method is less reliable for studying absorption processes. Therefore, the pendant drop (1) or maximum bubble pressure (2) methods are preferable for such measurements. So far, the study of the dynamical properties of the Marangoni effect was not possible at the oil–water interface as it was for the air–water interface (3). The pendant drop method is based on the deformation of a drop depending on the interfacial tension. In 1806, Laplace (4) established the equation combining the curvature of the interface, the interfacial tension, and the hydrostatic pressure difference to describe the shape of interfaces;

F

Dp Å g

1

1 1 / r1 r2

G

,

[1]

To whom correspondence should be addressed.

where g is the interfacial tension; r1 , r2 are the radius of curvature in two perpendicular directions; and Dp is the hydrostatic pressure difference between both sides of the interface. It should be noted that the interfacial tension represents the energy of the intermolecular cohesion forces by unit area: g is in N/m or J/m 2 . The shape of the drop depends on the balance between the gravity forces, which tend to elongate the drop, and the interfacial tension, which minimizes the surface of the interface and tends to make the drop spherical. The profile of the drop is then directly related to the interfacial tension by means of Laplace’s equation [1]. As the drop is axisymmetric, the curvature can be obtained from the coordinates and the derivatives of the points of the profile (5). Equation [1] then becomes 1 d(x sin u ) 2 Å 0 cz, x dx b

where b is the radius of curvature at the apex (lower point of the drop), c is the capillary constant equal to g Dr / g, u is the angle of the tangent to the profile, Dr is the difference in density of the two liquids, g is the interfacial tension, and g is the acceleration of the gravity. In the past, the measurement of interfacial tension by the pendant drop method required the determination of two particular diameters on photographic records of the drop (1). Then, the value of the interfacial tension was obtained using the tables from Andreas et al. (6). When these two diameters were measured within 1 mm, the precision obtained was 0.01 mN/m for an interfacial tension of about 50 mN/m. However, with this method a real-time measurement of the interfacial tension evolution during an adsorption process was not possible. The recent development in digitized image processing made it possible to automate the measurement of the interfacial tension. These new techniques have been used by numerous authors (7–12). The most usual method was developed by Rotenberg (13): from an approximated initial value of g,

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[2]

0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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A( £i 0 x 2i zi ) / Bx 2i Å xi sin ui ,

[4]

with A Å c/2 Å g Dr /2g and B Å 1/b. The values of parameters £i and ui can be obtained by numerical integration and derivation, respectively, from the N points of the digitized profile. From xi , zi , ui , and £i we obtain a linear system of N equations with two unknown quantities A and B. This linear system of N equations can be solved by the least-squares method over all the points of the profile, giving two estimators to determine parameters A and B. Introducing in Eq. [4] the new variables si Å £i 0 x 2i zi ti Å x 2i ui Å xi sin ui FIG. 1. Shape of a pendant drop. A point Mi of the profile is defined by the coordinates xi , zi , the angle of the tangent ui , and the volume Vi . Point O, taken as the origin, is the apex of the drop.

and then minimizing over all the points of the profile by a quadratic error criterion, N0 1

eÅ

Laplace’s equation is numerically integrated by successive iterations using the Runge–Kutta method until the computed and the digitized profiles coincide. Nevertheless, this method requires a computational time of about 30 s for each measurement of the interfacial tension. Thus, the digitized images of the drop are stored in a video recorder (14, 15) and then processed after the end of the experiment. Therefore, there is no possibility to control the interfacial tension or the area of the drop during the experiment. We propose here a new method with a computational time of less than a second for the measurement of the interfacial tension with an accuracy of less than 0.1%. II. NUMERICAL METHOD

In this method an integrated form of Laplace’s equation over all the points of the profile (5) is used to determine the interfacial tension. As no successive iterations are needed, the computation time allows a real-time measurement. Equation [2] can be integrated by parts using the expression £ (z) Å * px 2 dz of the volume £ (z) Å £i included between the apex O and the point Mi of coordinates (xi , zi ) and tangent ui (see Fig. 1). This integration leads to an analytical expression of the volume £i , given by £i Å x 2i 0 2(x 2i /b 0 xi sin ui )/c.

[3]

Thus, we obtain

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∑ (Asi / Bti 0 ui ) 2 ,

iÅ0

we obtain the following system of two equations with two unknown quantities A and B: N0 1

N0 1

N0 1

iÅ0

iÅ0

iÅ0

A ∑ s 2i / B ∑ si ti Å ∑ si ui N0 1

N0 1

N0 1

iÅ0

iÅ0

iÅ0

A ∑ si ti / B ∑ t 2i Å ∑ ti ui .

[5]

The radius of curvature at the apex b and the interfacial tension g can be deduced from these two parameters using the relation b Å 1/B and g Å g Dr /2A. The main difficulty in this method is the determination of tangent ui for each point Mi of the profile. As this profile is digitized with an error of one pixel, a numerical derivation is very inaccurate to determine ui . This difficulty was overcome by the use of three different approximations: (i) near the apex, the profile is fitted by a circle, (ii) close to the tip where the drop ends, a conical fit is used, and (iii) a parabolic approximation over seven points taken every four pixels elsewhere. Using these approximations, the values of ui are obtained with an accuracy of {1 degree. III. EXPERIMENTAL SETUP

A schematic representation of the tensiometer is given in Fig. 2. The components of the system are optical bench (A); parallel red light source (B); aplanetic objective ( f Å 85

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FIG. 2. Scheme of the pendant drop tensiometer. (See Experimental Setup for details.)

mm) (C); black and white CCD camera (RTC 56470) with a 512 1 512 pixel sensor (D); Exmire 2.5-ml gas chromatography syringe (E); the pendant drop hanging on a stainless steel needle (internal diameter 2.04 mm) inside a HELLMA fluorimeter cell (10 1 10 1 45 mm, ref. 103) placed in a water-circulating thermostated jacket (F); two stepper motors for fast and slow motions of the syringe plunger (the slow motion is 0.84 mm per step of the motor) (G); an M:486 DX33 PC computer (H); keyboard (M); digitizing board (Imagine Technology PC Vision/ ) connected to the CCD camera (I); TV monitor (J); computer display (K); 68 HC11 micro controller connected to the computer by the RS 232 output driving the two stepper motors (L); for ascending drop, a J form needle and a (10 1 20 1 45 mm, ref. 103) HELLMA fluorimeter cell (N). As a pendant drop is sensitive to vibrations, this tensiometer was placed on a rigid table on a ground floor. The choice between pendant or ascending drops depends on the respective densities of the liquid of the drop and the surrounding liquid and on the needle, which must be wetted by the surrounding liquid. The tip of the needle must be circular with a polished surface perpendicular to the needle. Otherwise, the value of the measured interfacial tension changes when the needle is rotated. Such a machining of a stainless steel needle, 2 mm in diameter, is particularly difficult. The analog video signal is digitized by a trigger signal sent by the computer. This generates a 512 1 512 pixel image over 256 levels of gray in about 0.04 s in the image memory of the digitizing board. This digitized image corre-

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sponds to the meridian line, or profile, of the drop. It is highly contrasted as we use a parallel light source. The observed edge of the drop changes from the black gray level to the white gray level in 1 pixel. Thus, the threshold value is chosen as the mid-gray level between the two well-separated peaks (for white and black) found on the histogram of the gray levels of this image. This threshold value is sent to the LUT (Look Up Table) of the digitizing board, which returns a black and white image in a few 1/100ths of a second. A binary image is then obtained. The contour can then be extracted in two steps: (i) a search process of the drop in this image, (ii) followed by detection of the white to black transition by a local analysis along this contour. The profile of the drop is then represented by a set of coordinates xi , zi in pixels. The magnification of the optical system is determined with steel balls in place of the drop inside the cell filled with the surrounding liquid. The diameters of these balls were measured with an accuracy of 1/100 mm. The digitized image is then fitted by a circle: the magnification is the ratio of the digitized and real diameters. For example, a pixel typically corresponds to a distance of 0.02 to 0.03 mm, for a 3-mm-diameter oil drop in water. The measurement time is about 0.2 to 0.3 s, depending on the size of the drops. The computing software for the interfacial tension measurement was written in C. The 68 HC11 micro controller driving the two stepper motors was programmed in 6800 assembly language. This micro controller works in slave mode: the computer sends commands through the RS 232 interface defining the choice of the slow or fast motors, the direction of rotation, the speed, the number of steps, or the stop signal. The use of two motors connected to two different mechanical reduction assemblies was necessary to perform two kinds of displacements: the first very fast, and the second extremely slow and accurate. IV. ACCURACY OF THE METHOD

The accuracy and the reliability of this method were determined with photographs of drops and with real drops. Determination of the Accuracy from Photographic Drops At first, the determination of the accuracy of this method was done by comparing the values of the radius of curvature and of the interfacial tension obtained with computed profiles of pendant drops using the Runge–Kutta fourth-order algorithm. As this method did not take into account the errors of the digitizing system and of the algorithm of extraction of the contour, we have used photographic glass plates of real drops. This avoids all experimental difficulties such as cleanliness and purity of materials. These photographic drops were used as standard profiles of drops. For each

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TABLE 1 Mean Values and Standard Deviations for Photographic Drops g0 (mN/m) (reference value)

gm (mN/m) (mean value)

s (mN/m) (standard deviation)

50.34 39.8 19.84 11.83 9.53 7.31

50.22 39.75 19.81 11.82 9.527 7.31

0.1884 0.049 0.037 0.027 0.017 0.02

photographic drop, the interfacial tension was determined by the two-diameter method using a microcomparator with an accuracy better than 1 mm. The accuracy of these interfacial tensions taken as references was 0.01 mN/m. In Table 1, these reference values are compared with the computed values obtained from our method: the statistic mean values gm are in a good agreement with these reference values within the error corresponding to standard deviation s determined over 200 to 300 measurements. From the standard deviations obtained we can reasonably conclude that the determination of the interfacial tension is 0.4%. Therefore, our method is not as accurate as that using the Newton Raphson method, which is about 0.1% (11). Numerical Criterion for the Accuracy From the least mean square method used to compute the interfacial tension it is possible to determine the standard deviation of the two estimators A and B from which the interfacial tension g and radius of curvature at the apex b are computed. The confidence interval (16) for the interfacial tension obtained was typically 0.1 to 0.2 mN/m, which corresponds to an accuracy of 0.5 to 0.7%, which is in a good agreement with the data of Table 1. It should be noted that this error criterion can be used in real time although it is time consuming. Enhanced Precision Mode The study of the Gaussian distribution of the digitizing noise showed that the relative accuracy of this method could be greatly enhanced. Considering that several drop profiles recorded over a very short period are very close together, these profiles can be superposed in a unique profile which gives a better accuracy for the interfacial tension measurements. Table 2 compares the standard deviations for the basic algorithm and for five superposed profiles obtained with the photographic drops: these standard deviations are roughly divided by a factor of two, giving an accuracy of 0.2%. This algorithm is denoted ‘‘enhanced precision mode’’ from here onward. The measurement noise is sig-

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nificantly reduced as can be seen in Fig. 3 for a photographic drop ( g Å 50.34 mN/m). Fortunately, this algorithm can be used even for the relatively fast time evolution of the interfacial tension inducing a negligible error. Comparison with Known Systems Real drops were also used to determine the accuracy of our method for pure air–water and n-decane–water interfaces. Ultrapure water was obtained from a Millipore purification system (Milli-Q plus, Millipore, France). n-Decane was supplied by Prolabo (Paris, France) and repurified with active charcoal and adsorbing earth. At 187C, the measured value for the superficial tension at the air–water interface was g Å 73.4 { 0.1 mN/m, compared with the reference value of 73.4 mN/m (17). At 207C, for the n-decane–water interface, the interfacial tension was 52.01 { 0.04 mN/m, in good agreement with the value found by Aveyard (18). From the values of the interfacial tension versus time recorded with the n-decane–water interface the histogram in Fig. 4 was redrawn. The precision corresponding to the value of the standard deviation {0.044 mN/m is 0.08% using the enhanced precision mode with five profiles. V. APPLICATIONS

A. Adsorption of Tensioactive Materials It is well known that the adsorption of tensioactive material at the interface can be described by a diffusion process between the bulk phase and the subphase (a thin layer just beneath the interface) coupled with an energy barrier between this subphase and the interface (3, 19). This barrier corresponds to a penetration of the adsorbed layer and a change in conformation of the molecules (for protein and polymer the second process is apparently irreversible (20)). Regulation of the area. Adsorption kinetics are usually recorded at a constant area of the interface. However, we found that the area of a pendant drop increased when the TABLE 2 Comparison of the Standard Deviations for the Basic Algorithm and the Enhanced Precision Mode s (mN/m) (standard deviation) g0 (mN/m) (reference value)

Basic algorithm

Enhanced precision mode (5 profiles)

50.34 39.8 19.84 11.83 9.53 7.31

0.1884 0.049 0.037 0.027 0.017 0.02

0.0823 0.024 0.018 0.012 0.00782 0.00912

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FIG. 3. Recorded noise obtained from a photograph of a drop ( g Å 50.34 { 0.01 mN/m). (A) is the basic algorithm (one profile). The enhanced precision mode uses (B) two, (C) five, and (D) nine profiles superposed.

interfacial tension decreased, for a fixed volume of the drop. This change in area is related to the elongation of the drop profile. As it is possible to modify the volume of the drop during the experiment, we have realized a regulation for the area of the drop. The area of the interface is maintained constant by adjusting the volume of the drop ( the volume and area of a pendant drop vary following the relation V 1 / 3 É kA 1 / 2 ( 4 ) , with k close to 1 ) . Thus, the algorithm at first determines an initial ratio R Å Vi / A 3i / 2 and then computes the

correction of the volume at a time t by DV Å 1.5 RA(t) 1 /2 (A(t) 0 Ai ). This regulation is achieved via a PID (Proportional, Integral, Derivative). Figure 5 presents two adsorption kinetics for drops of 35 mm3 , the first at constant volume, and the second at constant area using this regulation. These two kinetics look very similar. For a larger drop, a volume of 40 mm3 (see Fig. 6), a slight increase of the interfacial tension is observed just before the drop detaches at constant volume. At constant area, these adsorption kinetics can be

FIG. 4. Histogram of the noise corresponding to the measurement of the interfacial tension of the n-decane–water interface for 220 s with the enhanced precision mode (five profiles).

FIG. 5. Kinetics of adsorption for BSA at the n-decane–water interface for two drops of volume à 35 mm3 with and without (arrow) regulation of the area of the interface.

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FIG. 6. Kinetics of adsorption for BSA at the n-decane–water interface for two drops of volume à 40 mm3 with (a) and without (b) regulation of the area of the interface: (A) interfacial tension; (B) interfacial area.

followed over a much longer period of time, as the drop remains on the tip of the needle. These two kinetics look different, and moreover, the detachment of the drop which usually occurs with a pendant drop can be eliminated. Other experiments have confirmed the great interest in this area regulation, in particular when there is a slight partial miscibility between the liquid of the drop and the surrounding liquid. Interfacial tension regulation. Adsorption kinetics are usually obtained from the time evolution of the interfacial tension (or dynamical interfacial tension). Unfortunately, the number of adsorbed molecules with time is needed to the determined energy barrier and diffusion constant. Several methods (that will not be discussed here) have been proposed: (i) interfacial equations of state, which are generally not precise, (ii) the Gibbs–Duhem equation extended to nonequilibrium (14), and (iii) the reduced diffusion constant including the energy barrier combined with an elasticity measurement (21). We suggest here a different experimental method which is very unusual: the changes in the interfacial area which are directly proportional to the amount of adsorbed molecules are recorded at constant interfacial tension. This method was used to follow the desorption (or dissolution) kinetics for monolayers at the air–water interface (22). From a theoretical point of view, the energy barrier of adsorption can be more easily determined with such experiments. Following Me´rigoux (23), we found that a quick increase of the volume of the drop during adsorption kinetics is followed by an increase of the interfacial tension as the surface concentration is lowered. Then, the interfacial tension decreases again in relation to the adsorption of new tensioactive molecules.

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Using the phenomenon described above, we were able to maintain the interfacial tension constant (24, 25) by servo control of the volume of the drop based on a PID, the correction needed for the volume being proportional to the variation of the interfacial tension. Figure 7 presents three adsorption kinetics at constant interfacial tension (the interfacial tension first decreases at constant area to a given interfacial tension, and then the servo control is activated). Comparing these three kinetics, one should note a large decrease in the slope of the area variations on curves a to c as the constant interfacial tension decreases. These variations in area are related to the surfactant flux. Thus, this method does not require the construction of an equation of state as is needed for constant area adsorption kinetics. However, flow convection may occur in this constant interfacial tension method. If the diffusive transport is limited to a very thin layer close to the interface we can probably neglect this flow convection; otherwise a comprehensive theory including energy barrier, diffusion, and flow convection variables should be written. Considering Fig. 7, flow convection may occur for curve a (as the time duration is less than 30 s). In contrast, for curves b and c the area evolutions are extremely slow so that flow convection can reasonably be neglected. The exponential behavior of these variations in area corresponds to the theoretical predictions for the adsorption process. Figure 8 shows the linear plot obtained when drawing the logarithm of area versus time for curve a of Fig. 7B. The standard deviation of the noise induced by the servo control is typically {0.2 mN/m. It should be noted that such a method with constant in-

FIG. 7. Kinetics of adsorption at constant interfacial tension for BSA at the n-decane–water interface. At constant area, the interfacial tension decreases, and then the regulation of tension is activated. (A) Interfacial tension; a, g Å 35.1 mN/m; b, g Å 29.1 mN/m; and c, g Å 25.4 mN/m. (B) Corresponding variations of the area of the interface.

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derived an enhanced precision mode giving an accuracy less than {0.08%. This tensiometer gives a real-time measurement of the interfacial tension (i.e., for each measurement, the value of the interfacial tension is computed, displayed, and stored). It should be noted that the use of a fast digitizing board and a computer can improve this measuring rate by a factor of four to six. Unlike other methods based on the Runge–Kutta integration of Laplace’s equation (13), this method makes it possible to control and modify the drop volume during the experiment in order to:

FIG. 8. Logarithm of the area for curve a of Fig. 7B ( s ) and the corresponding least-squares linear fit ( —).

terfacial tension was first developed for the study of enzymatic kinetics at the oil–water interface (26). These two capabilities (constant interfacial area and constant interfacial tension) fully illustrate the great interest in the real-time measurement which is feasible with our apparatus.

(i) maintain a constant interfacial area during adsorption kinetics (when the interfacial tension decreases, the area slightly increases as the drop is deformed—this feature can be used to avoid drop detachment from the needle), (ii) slowly compress the adsorbed layer at the interface, (iii) apply fast compressions to the adsorbed layer in a few tenths of a second and then follow the evolution of the interfacial tension with time. This new technique presents interesting new features to study the adsorption and the dynamical properties of adsorbed layers at the liquid–liquid interface using a pendant drop. ACKNOWLEDGMENTS

B. Dynamical Properties of Adsorbed Layers With this tensiometer, different strains can be applied to the adsorbed layer on the drop during an experiment. When the area is rapidly (in a few tenths of a second) or slowly (in 10 s or more) expanded (or reduced), the resulting interfacial tension variations are related to the dynamical properties of the adsorbed layer. This apparatus was used to compare the dynamical properties of mixed adsorbed layers of poly(vinyl acetate) and poly( D, L ) lactide at the dichloromethane–water and air– water interfaces (21), and to study the adsorption and the rheological properties of bovine serum albumin in the presence of poly( D, L ) lactide (27). These recently published results show that such a tensiometer can give new insight into the dynamical properties of adsorbed layers, as such experiments are now feasible at the liquid–liquid interface as they have been at the air–water interface. VI. CONCLUSION

Using an integrated form of Laplace’s equation, we have developed an automatic interfacial tensiometer by digitizing video images. This tensiometer allows up to three measures per second with a 486 DX 33 Mhz computer, with an accuracy of {0.5% in the determination of the interfacial tension. From the Gaussian distribution of the digitized noise, we

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The authors are indebted to Dr. M. Lin, IUSTI (Marseille), for the photographs of drops used to calibrate this apparatus, to Professor I. Panaiotov, Professor J. E. Proust, and Dr. F. Boury for helpful discussions on the rheological properties of adsorbed layers, and to Mr. M. Dufour for English corrections.

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23. Me´rigoux, R., and Taxy, M., Extrait de la revue de l’Institut Francais du Pe´trole et annales des combustibles liquides Vol. XVI, No. 2, February 1961. 24. Grimaldi, M., Bois, A., Nury, S., Rivie`re, C., Verger, R., and Richou, J., Congre`s OPTO 91, Paris, March 1991. 25. Grimaldi, M., Thesis, Universite´ de Toulon et du Var, June 1991. 26. Nury, S., Gaudry-Roland, N., Rivie`re, C., Gargouri, Y., Bois, A., Lin, M., Grimaldi, M., Richou, J., and Verger, R., Congre`s Bridge ‘‘Lipase: Structure, Mechanism and Genetic Engineering,’’ Braunschweig R. F. A., September 1990. 27. Boury, F., Ivanova, Tz., Panaiotov, I., Proust, J. E., Bois, A., and Richou, J., Langmuir 11, 1636 (1995).

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