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Determining liquid–liquid interfacial tension from a submerged meniscus
Colloids and Surfaces A: Physicochem. Eng. Aspects 459 (2014) 267–273
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Determining liquid–liquid interfacial tension from a submerged meniscus Anita Hyde, Chi Phan ∗ , Gordon Ingram Department of Chemical Engineering, Curtin University, GPO Box U1987, Perth, WA 6845, Australia
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Liquid–liquid interfacial tension was determined from submerged holm meridian. • The interfacial tension was obtained successfully for four oil-water interfaces • The method is applicable for a density difference as low as 10 kg/m3 .
a r t i c l e
i n f o
Article history: Received 21 March 2014 Received in revised form 9 July 2014 Accepted 11 July 2014 Available online 22 July 2014 Keywords: Interfacial tension Optical tensiometry Young–Laplace equation Digital image analysis
a b s t r a c t Liquid–liquid interfacial tension plays a crucial role in multiphase systems in the chemical industry. The available measurement methods for liquid–liquid interfacial tension are poorly suited to low bond number systems, which are often found in industrial processes. This study developed and verified a new method of calculating the interfacial tension of liquid–liquid systems by using the “submerged holm” meniscus. The holm meridian was experimentally formed around a solid object submerged at the interface. A program was developed in MATLAB to calculate the interfacial tension from the submerged holm meridian. The interfacial tension calculated by the new method was found to be consistent with available data for multiple oil–water systems. This is the first time the submerged holm meniscus has been used successfully for determining interfacial tension. More importantly, the method is applicable to liquid–liquid systems with a small density difference between the two phases. As a demonstration, the interfacial tension of silicone oil (1000 cP) – water was measured, where the difference in density was less than 30 kg/m3 (3%). The method is potentially suitable for processes involving hazardous or unstable chemicals, elevated pressure or temperature. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Liquid–liquid interfacial tension is an important physical parameter affecting multiphase systems throughout a range of processes, such as separation and emulsification. Consequently, it impacts through the chemical industry, from food to cosmetics and chemical processing. In the literature, there are a number of methods for measuring interfacial tension as reviewed by [1]. However,
∗ Corresponding author. Tel. +61 892667571. E-mail address: [email protected] (C. Phan). http://dx.doi.org/10.1016/j.colsurfa.2014.07.016 0927-7757/© 2014 Elsevier B.V. All rights reserved.
most of these methods are only applicable to gas–liquid systems. For example, the common tensiometric methods based on force measurement, such as the Du Noüy ring and the Wilhelmy plate, are not practical for analysis of liquid–liquid systems. The measurement of liquid–liquid interfacial tension relies predominantly on optical analysis, such as the pendant drop, sessile drop and spinning drop techniques. The pendant drop method, which employs Axisymmetric Drop Shape Analysis (ADSA), has remained the most practical method for the tensiometric analysis of liquid–liquid systems. ADSA methods make use of specialized analysis software and high-resolution images to match experimental drop profiles with
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solutions to the Young–Laplace equation of capillarity. This typically involves a combination of numerical integration, for solving the Young–Laplace equations, with multivariate optimization to find the best-fit parameters defining the drop boundary. The optimal fitting parameters are then used to determine the interfacial tension. Several variations on ADSA programs have been developed to facilitate the analysis of pendant and sessile drops and their phase-inverted counterparts: emergent and constrained bubbles. These are presented in several papers [2–6]. While the ADSA technique can be applied to both gas–liquid and liquid–liquid interfaces, the ability to form and maintain the droplet or bubble remains a physical limitation of the method, particularly for systems requiring a long equilibration time. Additionally, droplets are typically formed and maintained by a micro-pump. For oil–water measurements, the fluid phase in the droplet is usually the oil due to its non-transparency. Pumping and maintaining an adequately sized droplet is extremely difficult for some viscous oils, such as crude oils. Additionally, in the case of certain oil–water systems, particularly where the density difference is small, the interfacial tension dominates the gravitation effect, leading to near-spherical drops. These systems are characterized by a low Bond number, which often results in large errors in the estimation of interfacial tension [7]. Recently, magnetic resonance images, instead of optical images, were combined with ADSA to overcome the oils’ non-transparency and allow the oil to be used as the bulk fluid [8]. Nevertheless, the method is expensive and requires very strong magnetic fields. The spinning drop method is another technique strongly dependant on interfacial distortion, although caused by centrifugal instead of gravitational forces. As with ADSA, the technique is restricted to high Bond number systems [9] due to a practical limit on the spinning speed, less than 20,000 rpm. Although the method can measure systems with interfacial tensions below 0.1 mN/m [10], it requires a significant density difference between the two fluids. As before, the method requires depositing a small oil droplet by a syringe, which is impractical for highly viscous oils. For systems with a density difference of less than 5%, it can be extremely difficult to determine a reliable interfacial tension using these methods. However, such systems are common in many chemical and petroleum processes. A recent paper [11] describes a method capable of determining the interfacial tension of silicone oil (20 cS) (PDMS) and water, for which the density difference is 3–4%. In brief, the method avoids the use of the Young–Laplace equation by basing the calculation on the force balance acting on the bubble cap. While this method appears more reliable than pendant or sessile drop techniques for the measurement of systems with a low density difference, it instead relies on highly accurate pressure measurements. Additionally, this method is still affected by the limits to precision which constrain all methods based on image analysis. In this paper, we present a method that exploits the principles of ADSA to calculate the interfacial tension of low Bond number systems using a simple experimental method that is easily applied even to viscous oils. The pendant and sessile meridians are two of the four gravitational shapes classified by [12], each associated with a phase-inverted counterpart. The other two meridian shapes, liquid bridges and holms, have not been widely used so far for surface tension analysis. A recent addition to the ADSA family, ADSA-NA (No Apex) [6] can be used with fluid bridges as well as sessile drops. However, to the best of the authors’ knowledge, there has been no method to date which uses the holm meridian for the calculation of interfacial tension. The present study developed and verified a method for calculating the interfacial tension of liquid–liquid systems by using the submerged holm meridian. The meniscus was formed around a solid sphere partially submerged at the fluid interface and could be easily produced and maintained for a range of
liquid–liquid systems. In an adaptation of the ADSA methodology, a program was developed in MATLAB to numerically fit the theoretical Young–Laplace curves to the experimental interfacial profile. Ultimately, this study aimed to develop a new and effective method of determining interfacial tension for low Bond number liquid–liquid systems. 2. Theoretical 2.1. Fundamentals of interfacial deformation Capillary systems such as pendant or sessile drops are axisymmetric fluid bodies where surface curvature changes with vertical position due to gravitational effects. A series of eight interfacial configurations have been identified [12]. All eight meridians are described by the Young–Laplace equation, which can be expressed as a system of three ordinary differential equations: sin d + = 2(H − Y ) X dS
(1a)
dX = cos dS
(1b)
dY = sin dS
(1c)
In Eq. (1) above, S is the dimensionless distance along the drop surface and the meridian angle, , is measured from the horizontal plane. The ‘contact angle’ is defined as the meridian angle at the point where the meridian contacts the supporting object. For pendant drops, this would be the capillary tube. For the case we present of the submerged holm, it is the angle at which the holm contacts the solid sphere. The shape factor, H, describes the shape and curvature of the bubble or droplet, while the parameter takes on different values depending on the drop type (pendant drop, sessile drop, holm, etc.). The coordinates X and Y are the dimensionless radial and vertical coordinates, normalized by the capillary length g · /. The interfacial deformation is characterized by the a= Bond number, which is the ratio of gravitational to surface forces. In Eq. (2), is the difference in density between the two fluids, g is the gravitational acceleration, R is the characteristic length of the system and is the interfacial tension. Bo =
gR2
(2)
Like the coordinate system, the shape factor H is typically based on the capillary constant or Bond number, although a range of
Fig. 1. Theoretical configurations observed for axisymmetric fluid–fluid interfaces in a gravitational field: (a) pendant drop with shape factor 0.5, solid line; and 0.2, dashed line. (b) Submerged holm meridian with shape factor 0.020, solid line and 0.015, dashed line. The curves were generated as solutions to the Young–Laplace equation by numerical integration using the shape factors given above. While (b) was integrated from the sphere side (left extreme: contact angle 130◦ ) as we use in our method, it has been annotated to show the ‘shape factor’ X* used in the common method of integration from the boundary condition at the horizontal asymptote (right extreme). Note that the vertical axis direction of (b) has been reversed, as the depth below the undisturbed interface is of interest.
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Fig. 2. Submerged holm meridian formed by a fluid interface interacting with an axially symmetric object: (a) a cylinder and (b) a sphere. Also shown in (a) is an unbounded fluid interface where the holm tends to a horizontal asymptote far from the object, whereas in (b) wall effects cause the interface to rise (or fall) close to the edges of the container. Both curves are derived from the Young–Laplace equation. The distance between the sphere and wall in (b) is small enough, compared to the capillary length, that the meridian does not reach an ‘unbounded’ (horizontal) state within the confines of the contained. As a result, effects from both the sphere and the wall clearly act on the interface. For comparison, integration of (b) is completed using the best-fit parameters derived for the hexane–water interface presented in this paper.
Table 1 Two specific cases of the Young–Laplace equation. Case
Computational range
Pendant drop Submerged holm
+1 H = 0 H = 0
0◦ < < 180◦ 180◦ < < 360◦ 0 < < 180◦
As shown in [12] This paper
modifications have been presented throughout the literature. This makes it possible to determine the interfacial tension based on the shape factor that describes the curve, and is the basis of all ADSAtype techniques. In Eq. (1), Boucher defined the shape factor as H2 = 2Bo , where Bo is the Bond number evaluated using the radius of curvature at the drop apex as the characteristic length of the system. While the pendant drop (Fig. 1a) has been used extensively in the literature, this study focuses on the submerged holm, shown in Fig. 1b. The corresponding values for these two shapes are given in Table 1. Note that the computational range for for the submerged holm assumes the starting position at the horizontal asymptote, as given in Boucher’s initial publication. This method requires an estimate of the coordinates where the meridian reaches
its asymptote. In contrast, this paper presents a method to integrate the holm from the opposite end where the meridian contacts the solid. The pendant drop method relies on distortion of the droplet to produce an accurate result. High Bond number systems typically display significant deformation, such as an elongated droplet with a pronounced neck at the top. It was found [3] that accurate data points around the necked area of the droplet significantly improve the accuracy of the ADSA method. In contrast, analysis of liquid–liquid systems is hindered by the small density difference between the two fluids, which leads to low Bond numbers and little deformation. The dominance of interfacial tension produces almost spherical drops. As a result, the accuracy of ADSA methods can be significantly reduced. 2.2. The holm meridian The holm meridian describes the meniscus formed around a stationary solid object located at a fluid–fluid interface (Fig. 1b). The interface will tend toward a flat surface once it extends sufficiently far from the object, giving the boundary conditions Y → 0 and X→ ∞
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as the meridian angle tends to 0◦ at the horizontal asymptote (note that we have reversed the direction of the Y axis for the submerged holm, such that Y increases with depth). The point X* is defined as the point where = * → 180◦ . Note that * is not exactly 180◦ as that position is not unique, being part of the asymptote, and hence is not suitable for integration. X* was typically used to replace the shape factor and distinguish a particular curve. The Young–Laplace equation for the holm meridian, which holds for any submerged object symmetrical around the vertical axis, is shown in its reduced form below [13] (the case at X = 0, required for the purposes of integration, was determined using l’Hopital’s rule):
⎧ ⎨ Y − sin , for X =/ 0 d X = dS ⎩ −Y,
(3)
for X = 0
Eq. (3) can be solved numerically with equations Eqs. (1b) and (1c), and the initial conditions at S = 0: X = X0 , Y = Y0 and = 0 . The dimensional coordinate system is normalized by the capillary length, giving X = x/a and Y = y/a. In previous studies [12], the numerical prediction of the submerged holm meridian started at the radial coordinate, X* , the point at which the holm’s horizontal asymptote is effectively reached (shown in Fig. 1b). However, experimental determination of X* is difficult as a sufficiently large fluid body is typically not used. This complicates the process of matching the experimental and theoretical positions of the fluid interface. The present work overcame this problem by starting the integration of the Young–Laplace equations at the other end, where the liquid–liquid interface is in contact with the solid. This avoids use of the conditions at the asymptote, X = X* . The initial point can be easily estimated using image analysis and subsequently tuned during optimization. Using this method, modeling the unbounded fluid surface, where the interface is almost horizontal, is not required. A characteristic feature of the holm is that the theoretical curves do not reach the vertical axis, X = 0, shown in Fig. 1b. Instead, ◦ they have a necking point at which the meridian angle is 90 (dX/dY = 0) and X is minimized. In most literature, however, the holm meridian has been typically formed around a cylinder (Fig. 2a) or submerged plate, although note that a holm formed around a plate is not axially symmetric. In these scenarios, the meridian angle cannot exceed 90◦ and necking of the curve is not realized. An alternative scenario is shown in Fig. 2b, where the holm meridian is formed around a spherical object [14]. 3. Experimental Analysis was undertaken of hexane–water, hexadecane–water, dodecane–water and air–water systems. High-resolution images were obtained with an optical resolution of 1 micron per pixel. Examples of the raw images are shown in Fig. 3 (reflections within the oil phase are blackened for clarity). As the focus of the study was to show comparability with existing techniques, rather than reproducing the surface tension of purified systems, the oils were used without purification. Deionized water was used for all experiments.
Fig. 3. Three examples of the holm meridian: (a) water–hexadecane where the distance between the sphere (6.35 mm) and the cell wall (cell size 55 mm) greatly exceeds the capillary length. In this situation, the holm reaches an asymptote as expected. (b) water–hexadecane where the distance between the sphere (11.11 mm) and the cell wall (cell size 33 mm) is comparitavely small. The holm does not reach an asymptote. Instead, the contour of the entire holm is affected by the Young–Laplace equation. Both oil–water systems show smooth, continuous transitions from the sphere to the holm. (c) water–air interfaces, showing the sharp contact angle and discontinuity at the point of contact. The blur (reflection) visible at the air–water interface was noted with all images taken of the gas–liquid system and proved detrimental to fitting. (Sphere size 11.11 mm, cell size 33 mm).
The densities were taken from literature values and confirmed using a commercial density meter (Anton Paar). Each oil–water pair was also analyzed using the pendant drop method as a benchmark for the surface tension values. Pendant drop images were taken using a 1.2 mm outer diameter nozzle, with the water pumped into the oil layer. The pendant drop method was implemented using commercial software [15] and a second program using the pendant drop methodology implemented by the authors in the MATLAB programming environment, as indicated in the results shown in Table 2. The results presented are the mean and 95% confidence interval associated with the interfacial tension measurement of multiple, non-consecutive images for each system. Additionally, the method was applied to the measurement of a silicone oil (PDMS, 1000 cP)–water interface.
Table 2 Comparison of the results obtained using the new method and commercial pendant drop software for two water–oil systems. The average and 95% confidence interval is calculated over at least five images. System
kg/m3
Method
Interfacial tension (Average) mN/m
Confidence interval (95%) mN/m
Hexane–water
342.0
Dodecane–water
249.4
Hexadecane–water
224.8
Holm meridian Pendant drop [15] Holm meridian Pendant drop Holm meridian Pendant drop Holm meridian
38.2 40.7 45.6 45.7 31.5 31.0 37.7
±1.3 ±2.0 ±1.2 ±0.7 ±0.4 ±0.3 ±1.2
Silicone oil–water
24.7
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The holm meridians were formed within small cubic cells, with sides 3 cm or 5 cm in length, around a hydrophobic (Teflon) sphere supported at the interface by a thin Teflon line. The sphere, of known diameter, was used for image scaling, the sphere radius being used as the characteristic length of the system. A range of sphere sizes were used, from 6.35 to 12.70 mm diameter, with a tolerance of 0.1 mm. All measurements were done at room temperature (approximately 18 ◦ C), with the actual temperature used to determine the fluid densities from literature values. Unlike forming droplets for measurement, the experimental apparatus did not require a micro-pump and was easy to clean. Moreover, the holm meridian was stable for long periods of time, with no risk of collapse. The sphere position was adjusted vertically, with an emphasis on achieving a necking point. Several images were captured, with different sphere positions and extents of curvature of the holm, for each system. It is noteworthy that the necking was obtained for the oil–water interfaces but not for the air–water interface. Two issues were noted. Firstly, the reflection and blurring shown in Fig. 3c was present on all of the air–water images, leaving little of the holm available for fitting. Blurred areas were avoided due to uncertainty as to the position of the edge in the images. Secondly, the contact angle was much sharper on the air–water interface, leading to significantly less curvature of the holm. There was generally significant error between subsequent images of the air–water system, seemingly reminiscent of the fitting issues that ADSA encounters when fitting spherical drops. Consequently, the air–water interface, which was not the focus of this study as several techniques already exist which can measure it accurately, was not pursued further for modeling.
4. Modeling
271
from multiple starting points, with multiple assumed surface tensions. The fitting procedure included the following steps: 1. A starting point (x0 , y0 )0 was defined at the point where the detected edge of the holm was separated by more than 5 pixels from the sphere surface. It follows that the depression of the holm at the point of contact with the sphere is Y0 − Yh , where Y0 = y0 /a. 2. The starting contact angle, 0 , was estimated from the tangent of a polynomial fitted to the curved section of the holm (50–100 pixels) and allowed to vary by ±10% during optimization. The polynomial took the form x = C0 + C1 y + C2 y2 where (x, y) are the coordinates (in pixels) of the detected edge. 3. A constrained multivariate solver was used to optimize three variables: the shape factor, a; vertical position of the origin, y0 ; and the initial contact angle, 0 . Integration of the Young–Laplace equations produced the theoretical curve, which was matched to the detected edge. Numerical optimization was used to minimize the difference in the positions of the curves. 4. Uncertainty around the exact location of the edge was reduced by running the optimization repeatedly at different points around the initial starting point. For example, an image may be analyzed at each the first 10 coordinate points along the holm. 5. At each starting point, fitting was conducted 3 times, each time using a random selection of 150 coordinate points, assuming different values of the initial interfacial tension. The use of multiple initial estimates for different runs ensures that the fitting converges on the global minimum of the error function, rather than being caught in local minima. 6. The interfacial tension was calculated from the optimized parameters, (a).
4.1. Image analysis and boundary detection
5. Results and discussion
Edge detection was accomplished using the Canny edge detection algorithm with a Gaussian filter, supported by the edge function in MATLAB’s Image Analysis Toolbox. Canny edge detection has been widely reported as the most appropriate edge detector for use with ADSA-type image analysis, in particular for its strength in picking weak edges from images where noise is present [16]. Canny edge detection requires two threshold parameters, one for strong and one for weak edges. The global image threshold, T, was calculated using Otsu’s method (supported by MATLAB’s threshold function). MATLAB’s recommended low and high thresholds were used, which are based on the intensity gradients in the image [17].
The interfacial tensions of three unpurified oils–hexane, hexadecane and dodecane–with water were determined using the new method (holm meridian) and the pendant drop method. The results were found to agree well between the two methods (Table 2). The reported values are averages taken over multiple images, along with the 95% confidence intervals calculated using the Student’s t distribution. The authors note that, as the oils were not purified prior to use, the values reported indicate the actual oil–water interfacial tension, rather the interfacial tension of the pure oils, and serve to demonstrate the agreement between values determined using the different methods. The program developed in this study was able to successfully fit the Young–Laplace equation to a fluid–fluid interface intersecting a solid sphere. Fitting was achieved for oil–water interfaces with varying degrees of curvature and a range of contact angles. Analysis of the same oil using the pendant drop method and commercial software [15] returned a consistent result. However, as expected, the interfacial tensions of the unpurified oils were significantly lower than those for purified oils. Several papers have been published on the effects of contaminants such as salts [18], organic molecules [19] and surfactants [20] in lowering interfacial tension. Depending on the size of the sphere relative to the cell size, the bounded oil–water interface did not always tend to a horizontal plane, instead sloping up toward the walls of the container (Fig. 3). Where the sphere diameter was large when compared to the cell size, there was no evidence of the horizontal asymptote expected with the theoretical holm meridian in an unbounded fluid. The Young–Laplace equation generated from the best-fit parameters was consistent with the image of the holm as it rose to meet the wall of the container. In this region, the deformation of the fluid–fluid
4.2. Optimization to determine best-fit parameters for the theoretical profile The origin of the coordinate space used in the Young–Laplace equation is defined from the height of the undisturbed fluid–fluid interface far from the object, in line with the center of the immersed object. The position of the vertical axis can be easily determined by fitting a circle to the profile of the submerged sphere and determining the coordinates of the center. As it is not possible to ascertain the horizontal axis position from an image of a meridian in a bounded fluid, such as within a small glass cell, Yh was taken as a parameter for optimization and determined numerically. The initial estimate for the capillary constant was taken from an estimated surface tension and was constrained to be non-negative during the optimization process. An accurate estimate was not required as the program appeared quite capable of converging from significant distances. Additionally, the optimization procedure was undertaken
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Fig. 4. Calculated values for interfacial tension from four distinct sphere sizes. The scatter around the overall average interfacial tension shows no particular trend with the change in sphere size, demonstrating that the method is applicable using a variety of cell and ball sizes.
interface came from both the submerged sphere and the wall. The theoretical curve was found to follow the shape of the observed profile if extended even if only the first (curved) section of the meridian was used as an input to the program. As in these instances the horizontal asymptote is not present, it would not be possible to estimate the fitting based on method used by Boucher [12]. In order to test the effect of the wall on the calculation of interfacial tension, the interfacial tension of hexadecane (unpurified) with water was calculated from twelve images, using four different sized spheres. The larger (5 cm) cell was used, with sphere diameters of 7.14 mm, 9.54 mm, 11.11 mm and 12.70 mm. The interfacial tension results were found the be consistent within the range of sphere sizes. There was no obvious trend in the deviation from the overall mean value, as can be seen from Fig. 4. The calcuated interfacial tension (31.5 ± 0.4 mN/m) was comparable with the results calculated using the pendant drop method (31.0 ± 0.3 mN/m). The new method has a number of advantages over the existing methods. Forming the meridian around a solid sphere located at the fluid–fluid interface allowed a relatively simple and stable experimental setup. In addition, the ability to move the sphere vertically at the interface made it possible to create images with a very pronounced necking region. The shape factor of pendant drops is strongly affected by drop deformation, particularly necking. As a result, drops with pronounced necks tend to give more reliable results using ADSA techniques [7]. A similar effect is proposed for the new method presented in this study. The fitting is improved where the curvature around the holm is more pronounced, ideally with an initial contact angle in excess of 90◦ . While the ADSA pendant drop software is limited in its ability to analyze near-spherical drops, a characteristic of low Bond number systems, the new method is capable of analyzing these systems, with potential to analyze systems with very small differences in density between the two fluids. Indeed, this new method may work better as the densities become closer and surface tension effects dominate the meridian shape. Fig. 5 predicts the theoretical profiles for oil–water systems with the same interfacial tension and contact angle, but assuming oil densities from 400 to 990 kg/m3 . The length of the curved portion of the holm increases with increasing oil density. At the most extreme condition, where the density difference approaches 10 kg/m3 (1%), the new method
remains practical, whereas measurement of such a system is not possible with either the pendant drop or spinning drop method. This could allow measurement of water–oil interfacial tensions for crude oils with densities in the 850–1000 kg/m3 range, as found in the petroleum industry [21]. Conversely, the holm rises rapidly to an asymptote when the Bond number is large (the two lower profiles in Fig. 5, which have been trimmed at the start of the asymptote). As a result, there is a shorter length of the holm suitable for fitting and a potential limit to the effectiveness of the new method for high Bo systems. In these instances, one might consider using a smaller sphere to reduce the Bond number and capillary length. A recent paper [11] presents a method suitable for the analysis of low Bond number systems. The applicability of this method was demonstrated by determining the silicone oil–water interfacial tension (PDMS, 20 cS). The spherical drop shape encountered with low Bond number systems tends to introduce significant errors when attempting to fit the Young–Laplace equation for both pendant and sessile drops. In [11], the force balance over the drop cap is used rather than the Young–Laplace equations, allowing the analysis of spherical drops. However, this method also requires a highly accurate pressure reading as an input to the force balance, introducing a different source of error to the method on top of the issues with accuracy inherent in all methods relying on precise image analysis. Furthermore, the method still requires pumping to form the drop, hence does not address the issues faced when dealing with highly viscous or difficult-to-clean fluids. Measurement of the silicone–oil water interface shows that this method, using the holm meridian, can be applied to highly viscous fluids with very small density differences. The calculated interfacial tension of 37 ± 1.2 mN/m, for silicone oil (PDMS, 1000 cS). Due to the extremely small density difference between the two fluids, it is not possible to determine the interfacial tension of silicone oil–water systems using conventional methods. Consequently, very little data is available for this system. Using the force-balance method [11], the interfacial tension of a less viscous oil of 20 cS was 35 mN/m at 18 ◦ C. A scatter of ±1.1 mN/m is reported, which is comparable to the 95% confidence interval of ±1.2 mN/m found using the method presented in this paper. An interfacial tension of 34.4 mN/m was found by A. Juel (unpublished, cited in [11]) for PMDS (1000 cS) at 25 ◦ C.
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Fig. 5. Theoretical oil–water interface profiles for different oil densities (oil ) with other parameters remaining constant (x0 , y0 as shown, 0 = 134◦ , Rsphere = 5.57 mm, = 41.9 mN/m, water = 997 kg/m3 ). The curvature becomes more pronounced as the density difference is reduced and the Bond number decreases.
The method is also more practical than the spinning drop in measuring dynamic tension of oil–water interfaces [22]. The spinning drop is restricted to low tension values, approximately 20 mN/m, and short spinning times to control the heat released by spinning. The new method, in contrast, can be applied at a wider range of tensions and over a longer time. The long measurement time is important for industrial systems, in which the tension is dominated by naturally occurring surfactants with adsorption time in the order of hours [23,24]. Finally, the cell in this method can be easily pressurized, which is infeasible with many of the existing methods, to replicate the elevated pressures in industrial processes. 6. Conclusion The new method was found to be effective in fitting images of oil–water interfaces that are in contact with an immersed solid sphere. The experimental setup was simple and stable, and the method returned consistent results for the liquid–liquid interfacial tension when analyzing multiple images. The submerged holm meridian is generally easier to form and more stable to maintain than a pendant drop for liquid–liquid systems. The experiments allowed the vertical position of the sphere to be adjusted to form a more pronounced curvature in the holm, leading to more reliable results. Analysis of hexane–water, hexadecane–water and dodecane– water systems gave consistent values for the interfacial tension that were in good agreement with values produced by the pendant drop method. This is the first time the submerged shape has been used successfully for determining interfacial tension. More significantly, the new method is well-suited to the analysis of liquid–liquid systems and capable of analyzing systems with a small density difference. The analysis of a liquid–liquid system with a density difference as low as 30 kg/m3 (3%) was found to confer well with the few published results. Hence, the new method could facilitate interfacial analysis in many industrial systems, where neither the pendant drop nor spinning drop methods are practical. It can be used for analysis of extremely viscous oils, where viscosity precludes pumping, and does not rely on highly sensitive pressure measurements. This method may facilitate the measurement of systems involving hazardous or unstable chemicals, elevated pressures or high temperatures. References [1] C. Chang, E. Franses, Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms, Colloids Surf. A 100 (1995) 1–45.
[2] F.K. Hansen, G. Rodsrud, Surface tension by pendant drop: a fast standard instrument using computer image analysis, J. Colloid Interface Sci. 41 (1) (1990) 1–9. [3] P. Cheng, A. Neumann, Computational evaluation of axisymmetric drop shape analysis-profile (ADSA-P), Colloids Surf. 62 (4) (1992) 297–305. [4] O. del Rio, A. Neumann, Axisymmetric drop shape analysis: computational methods for the measurement of interfacial properties from the shape and dimensions of pendant and sessile drops, J. Colloid Interface Sci. 196 (2) (1997) 136–147. [5] S.M.I. Saad, Z. Policova, E.J. Acosta, A.W. Neumann, Axisymmetric drop shape analysis-constrained sessile drop (ADSA-CSD): a film balance technique for high collapse pressures, Langmuir 24 (19) (2008) 10843–10850. [6] A. Kalantarian, Development of Axisymmetric Drop Shape Analysis – No Apex (ADSA-NA), University of Toronto, 2011, Ph.D. thesis. [7] A.W. Neumann, R. David, Y. Zuo, Applied surface thermodynamics, in: Surfactant Science Series, 2nd ed., CRC Press, Boca Raton, 2011. [8] R. Hussain, S.J. Vogt, A. Honari, K.G. Hollingsworth, A.J. Sederman, J. Mitchell, M.L. Johns, Interfacial tension measurements using MRI drop shape analysis, Langmuir 30 (6) (2014) 1566–1572. [9] B. Vonnegut, Rotating bubble method for the determination of surface and interfacial tensions, Rev. Sci. Instrum. 13 (1) (1942) 6–9. [10] S. Deng, R. Bai, J.P. Chen, G. Yu, Z. Jiang, F. Zhou, Effects of alkaline/surfactant/polymer on stability of oil droplets in produced water from ASP flooding, Colloids Surf. A Physicochem. Eng. Aspects 211 (23) (2002) 275–284. [11] F. Peters, D. Arabali, Interfacial tension between oil and water measured with a modified contour method, Colloids Surf. A Physicochem. Eng. Aspects 426 (0) (2013) 1–5. [12] E.A. Boucher, Capillary phenomena: properties of systems with fluid/fluid interfaces, Rep. Prog. Phys. 43 (4) (1980) 497. [13] C. Huh, L. Scriven, Shapes of axisymmetric fluid interfaces of unbounded extent, J. Colloid Interface Sci. 30 (3) (1969) 232–337. [14] E. Boucher, T. Jones, Capillary phenomena (part 18), J. Chem. Soc. Faraday Trans. 78 (5) (1982) 1499–1506. [15] Y.Y. Zuo, C. Do, A.W. Neumann, Automatic measurement of surface tension from noisy images using a component labeling method, Colloids Surf. A Physicochem. Eng. Aspects 299 (13) (2007) 109–116. [16] A. Kalantarian, S.M.I. Saad, A.W. Neumann, Accuracy of surface tension measurement from drop shapes: the role of image analysis, Adv. Colloid Interface Sci. 199–200 (2013) 15–22. [17] The MathWorks Inc., Mathworks Documentation Centre, 2013. [18] E.R.A. Lima, B.M. de Melo, L.T. Baptista, M.L.L. Paredes, Specific ion effects on the interfacial tension of water/hydrocarbon systems, Braz. J. Chem. Eng. 30 (2013) 55–62. [19] S. Azizian, K. Kashimoto, T. Matsuda, H. Matsubara, T. Takiue, M. Aratono, Interfacial tension studies of crown ethers at air/water and hexane/water interfaces, J. Colloid Interface Sci. 316 (1) (2007) 25–30. [20] L. Liggieri, F. Ravera, A. Passerone, Equilibrium interfacial tension of hexane/water plus Triton X-100, J. Colloid Interface Sci. 169 (1) (1995) 238–240. [21] Z. Wang, B. Hollebone, M. Fingas, B. Fieldhouse, L. Sigouin, M. Landriault, P. Smith, J. Noonan, G. Thouin, J. Weaver, Characteristics of Spilled Oils, Fuels, and Petroleum Products: 1. Composition and Properties of Selected Oils, Tech. Rep., 2003. [22] R.P. Borwankar, D.T. Wasan, Dynamic interfacial tensions in acidic crude oil/caustic systems. Part 1. A chemical diffusion-kinetic model, AIChE J. 32 (3) (1986) 455–466. [23] A. Gaonkar, Interfacial tensions of vegetable oil/water systems: effect of oil purification, J. Am. Oil Chem. Soc. 66 (8) (1989) 1090–1092. [24] S. Poteau, J.-F. Argillier, D. Langevin, F. Pincet, E. Perez, Influence of pH on stability and dynamic properties of asphaltenes and other amphiphilic molecules at the oil–water interface, Energy Fuels 19 (4) (2005) 1337–1341.
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Determining liquid–liquid interfacial tension from a submerged meniscus Anita Hyde, Chi Phan ∗ , Gordon Ingram Department of Chemical Engineering, Curtin University, GPO Box U1987, Perth, WA 6845, Australia
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Liquid–liquid interfacial tension was determined from submerged holm meridian. • The interfacial tension was obtained successfully for four oil-water interfaces • The method is applicable for a density difference as low as 10 kg/m3 .
a r t i c l e
i n f o
Article history: Received 21 March 2014 Received in revised form 9 July 2014 Accepted 11 July 2014 Available online 22 July 2014 Keywords: Interfacial tension Optical tensiometry Young–Laplace equation Digital image analysis
a b s t r a c t Liquid–liquid interfacial tension plays a crucial role in multiphase systems in the chemical industry. The available measurement methods for liquid–liquid interfacial tension are poorly suited to low bond number systems, which are often found in industrial processes. This study developed and verified a new method of calculating the interfacial tension of liquid–liquid systems by using the “submerged holm” meniscus. The holm meridian was experimentally formed around a solid object submerged at the interface. A program was developed in MATLAB to calculate the interfacial tension from the submerged holm meridian. The interfacial tension calculated by the new method was found to be consistent with available data for multiple oil–water systems. This is the first time the submerged holm meniscus has been used successfully for determining interfacial tension. More importantly, the method is applicable to liquid–liquid systems with a small density difference between the two phases. As a demonstration, the interfacial tension of silicone oil (1000 cP) – water was measured, where the difference in density was less than 30 kg/m3 (3%). The method is potentially suitable for processes involving hazardous or unstable chemicals, elevated pressure or temperature. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Liquid–liquid interfacial tension is an important physical parameter affecting multiphase systems throughout a range of processes, such as separation and emulsification. Consequently, it impacts through the chemical industry, from food to cosmetics and chemical processing. In the literature, there are a number of methods for measuring interfacial tension as reviewed by [1]. However,
∗ Corresponding author. Tel. +61 892667571. E-mail address: [email protected] (C. Phan). http://dx.doi.org/10.1016/j.colsurfa.2014.07.016 0927-7757/© 2014 Elsevier B.V. All rights reserved.
most of these methods are only applicable to gas–liquid systems. For example, the common tensiometric methods based on force measurement, such as the Du Noüy ring and the Wilhelmy plate, are not practical for analysis of liquid–liquid systems. The measurement of liquid–liquid interfacial tension relies predominantly on optical analysis, such as the pendant drop, sessile drop and spinning drop techniques. The pendant drop method, which employs Axisymmetric Drop Shape Analysis (ADSA), has remained the most practical method for the tensiometric analysis of liquid–liquid systems. ADSA methods make use of specialized analysis software and high-resolution images to match experimental drop profiles with
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solutions to the Young–Laplace equation of capillarity. This typically involves a combination of numerical integration, for solving the Young–Laplace equations, with multivariate optimization to find the best-fit parameters defining the drop boundary. The optimal fitting parameters are then used to determine the interfacial tension. Several variations on ADSA programs have been developed to facilitate the analysis of pendant and sessile drops and their phase-inverted counterparts: emergent and constrained bubbles. These are presented in several papers [2–6]. While the ADSA technique can be applied to both gas–liquid and liquid–liquid interfaces, the ability to form and maintain the droplet or bubble remains a physical limitation of the method, particularly for systems requiring a long equilibration time. Additionally, droplets are typically formed and maintained by a micro-pump. For oil–water measurements, the fluid phase in the droplet is usually the oil due to its non-transparency. Pumping and maintaining an adequately sized droplet is extremely difficult for some viscous oils, such as crude oils. Additionally, in the case of certain oil–water systems, particularly where the density difference is small, the interfacial tension dominates the gravitation effect, leading to near-spherical drops. These systems are characterized by a low Bond number, which often results in large errors in the estimation of interfacial tension [7]. Recently, magnetic resonance images, instead of optical images, were combined with ADSA to overcome the oils’ non-transparency and allow the oil to be used as the bulk fluid [8]. Nevertheless, the method is expensive and requires very strong magnetic fields. The spinning drop method is another technique strongly dependant on interfacial distortion, although caused by centrifugal instead of gravitational forces. As with ADSA, the technique is restricted to high Bond number systems [9] due to a practical limit on the spinning speed, less than 20,000 rpm. Although the method can measure systems with interfacial tensions below 0.1 mN/m [10], it requires a significant density difference between the two fluids. As before, the method requires depositing a small oil droplet by a syringe, which is impractical for highly viscous oils. For systems with a density difference of less than 5%, it can be extremely difficult to determine a reliable interfacial tension using these methods. However, such systems are common in many chemical and petroleum processes. A recent paper [11] describes a method capable of determining the interfacial tension of silicone oil (20 cS) (PDMS) and water, for which the density difference is 3–4%. In brief, the method avoids the use of the Young–Laplace equation by basing the calculation on the force balance acting on the bubble cap. While this method appears more reliable than pendant or sessile drop techniques for the measurement of systems with a low density difference, it instead relies on highly accurate pressure measurements. Additionally, this method is still affected by the limits to precision which constrain all methods based on image analysis. In this paper, we present a method that exploits the principles of ADSA to calculate the interfacial tension of low Bond number systems using a simple experimental method that is easily applied even to viscous oils. The pendant and sessile meridians are two of the four gravitational shapes classified by [12], each associated with a phase-inverted counterpart. The other two meridian shapes, liquid bridges and holms, have not been widely used so far for surface tension analysis. A recent addition to the ADSA family, ADSA-NA (No Apex) [6] can be used with fluid bridges as well as sessile drops. However, to the best of the authors’ knowledge, there has been no method to date which uses the holm meridian for the calculation of interfacial tension. The present study developed and verified a method for calculating the interfacial tension of liquid–liquid systems by using the submerged holm meridian. The meniscus was formed around a solid sphere partially submerged at the fluid interface and could be easily produced and maintained for a range of
liquid–liquid systems. In an adaptation of the ADSA methodology, a program was developed in MATLAB to numerically fit the theoretical Young–Laplace curves to the experimental interfacial profile. Ultimately, this study aimed to develop a new and effective method of determining interfacial tension for low Bond number liquid–liquid systems. 2. Theoretical 2.1. Fundamentals of interfacial deformation Capillary systems such as pendant or sessile drops are axisymmetric fluid bodies where surface curvature changes with vertical position due to gravitational effects. A series of eight interfacial configurations have been identified [12]. All eight meridians are described by the Young–Laplace equation, which can be expressed as a system of three ordinary differential equations: sin d + = 2(H − Y ) X dS
(1a)
dX = cos dS
(1b)
dY = sin dS
(1c)
In Eq. (1) above, S is the dimensionless distance along the drop surface and the meridian angle, , is measured from the horizontal plane. The ‘contact angle’ is defined as the meridian angle at the point where the meridian contacts the supporting object. For pendant drops, this would be the capillary tube. For the case we present of the submerged holm, it is the angle at which the holm contacts the solid sphere. The shape factor, H, describes the shape and curvature of the bubble or droplet, while the parameter takes on different values depending on the drop type (pendant drop, sessile drop, holm, etc.). The coordinates X and Y are the dimensionless radial and vertical coordinates, normalized by the capillary length g · /. The interfacial deformation is characterized by the a= Bond number, which is the ratio of gravitational to surface forces. In Eq. (2), is the difference in density between the two fluids, g is the gravitational acceleration, R is the characteristic length of the system and is the interfacial tension. Bo =
gR2
(2)
Like the coordinate system, the shape factor H is typically based on the capillary constant or Bond number, although a range of
Fig. 1. Theoretical configurations observed for axisymmetric fluid–fluid interfaces in a gravitational field: (a) pendant drop with shape factor 0.5, solid line; and 0.2, dashed line. (b) Submerged holm meridian with shape factor 0.020, solid line and 0.015, dashed line. The curves were generated as solutions to the Young–Laplace equation by numerical integration using the shape factors given above. While (b) was integrated from the sphere side (left extreme: contact angle 130◦ ) as we use in our method, it has been annotated to show the ‘shape factor’ X* used in the common method of integration from the boundary condition at the horizontal asymptote (right extreme). Note that the vertical axis direction of (b) has been reversed, as the depth below the undisturbed interface is of interest.
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Fig. 2. Submerged holm meridian formed by a fluid interface interacting with an axially symmetric object: (a) a cylinder and (b) a sphere. Also shown in (a) is an unbounded fluid interface where the holm tends to a horizontal asymptote far from the object, whereas in (b) wall effects cause the interface to rise (or fall) close to the edges of the container. Both curves are derived from the Young–Laplace equation. The distance between the sphere and wall in (b) is small enough, compared to the capillary length, that the meridian does not reach an ‘unbounded’ (horizontal) state within the confines of the contained. As a result, effects from both the sphere and the wall clearly act on the interface. For comparison, integration of (b) is completed using the best-fit parameters derived for the hexane–water interface presented in this paper.
Table 1 Two specific cases of the Young–Laplace equation. Case
Computational range
Pendant drop Submerged holm
+1 H = 0 H = 0
0◦ < < 180◦ 180◦ < < 360◦ 0 < < 180◦
As shown in [12] This paper
modifications have been presented throughout the literature. This makes it possible to determine the interfacial tension based on the shape factor that describes the curve, and is the basis of all ADSAtype techniques. In Eq. (1), Boucher defined the shape factor as H2 = 2Bo , where Bo is the Bond number evaluated using the radius of curvature at the drop apex as the characteristic length of the system. While the pendant drop (Fig. 1a) has been used extensively in the literature, this study focuses on the submerged holm, shown in Fig. 1b. The corresponding values for these two shapes are given in Table 1. Note that the computational range for for the submerged holm assumes the starting position at the horizontal asymptote, as given in Boucher’s initial publication. This method requires an estimate of the coordinates where the meridian reaches
its asymptote. In contrast, this paper presents a method to integrate the holm from the opposite end where the meridian contacts the solid. The pendant drop method relies on distortion of the droplet to produce an accurate result. High Bond number systems typically display significant deformation, such as an elongated droplet with a pronounced neck at the top. It was found [3] that accurate data points around the necked area of the droplet significantly improve the accuracy of the ADSA method. In contrast, analysis of liquid–liquid systems is hindered by the small density difference between the two fluids, which leads to low Bond numbers and little deformation. The dominance of interfacial tension produces almost spherical drops. As a result, the accuracy of ADSA methods can be significantly reduced. 2.2. The holm meridian The holm meridian describes the meniscus formed around a stationary solid object located at a fluid–fluid interface (Fig. 1b). The interface will tend toward a flat surface once it extends sufficiently far from the object, giving the boundary conditions Y → 0 and X→ ∞
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as the meridian angle tends to 0◦ at the horizontal asymptote (note that we have reversed the direction of the Y axis for the submerged holm, such that Y increases with depth). The point X* is defined as the point where = * → 180◦ . Note that * is not exactly 180◦ as that position is not unique, being part of the asymptote, and hence is not suitable for integration. X* was typically used to replace the shape factor and distinguish a particular curve. The Young–Laplace equation for the holm meridian, which holds for any submerged object symmetrical around the vertical axis, is shown in its reduced form below [13] (the case at X = 0, required for the purposes of integration, was determined using l’Hopital’s rule):
⎧ ⎨ Y − sin , for X =/ 0 d X = dS ⎩ −Y,
(3)
for X = 0
Eq. (3) can be solved numerically with equations Eqs. (1b) and (1c), and the initial conditions at S = 0: X = X0 , Y = Y0 and = 0 . The dimensional coordinate system is normalized by the capillary length, giving X = x/a and Y = y/a. In previous studies [12], the numerical prediction of the submerged holm meridian started at the radial coordinate, X* , the point at which the holm’s horizontal asymptote is effectively reached (shown in Fig. 1b). However, experimental determination of X* is difficult as a sufficiently large fluid body is typically not used. This complicates the process of matching the experimental and theoretical positions of the fluid interface. The present work overcame this problem by starting the integration of the Young–Laplace equations at the other end, where the liquid–liquid interface is in contact with the solid. This avoids use of the conditions at the asymptote, X = X* . The initial point can be easily estimated using image analysis and subsequently tuned during optimization. Using this method, modeling the unbounded fluid surface, where the interface is almost horizontal, is not required. A characteristic feature of the holm is that the theoretical curves do not reach the vertical axis, X = 0, shown in Fig. 1b. Instead, ◦ they have a necking point at which the meridian angle is 90 (dX/dY = 0) and X is minimized. In most literature, however, the holm meridian has been typically formed around a cylinder (Fig. 2a) or submerged plate, although note that a holm formed around a plate is not axially symmetric. In these scenarios, the meridian angle cannot exceed 90◦ and necking of the curve is not realized. An alternative scenario is shown in Fig. 2b, where the holm meridian is formed around a spherical object [14]. 3. Experimental Analysis was undertaken of hexane–water, hexadecane–water, dodecane–water and air–water systems. High-resolution images were obtained with an optical resolution of 1 micron per pixel. Examples of the raw images are shown in Fig. 3 (reflections within the oil phase are blackened for clarity). As the focus of the study was to show comparability with existing techniques, rather than reproducing the surface tension of purified systems, the oils were used without purification. Deionized water was used for all experiments.
Fig. 3. Three examples of the holm meridian: (a) water–hexadecane where the distance between the sphere (6.35 mm) and the cell wall (cell size 55 mm) greatly exceeds the capillary length. In this situation, the holm reaches an asymptote as expected. (b) water–hexadecane where the distance between the sphere (11.11 mm) and the cell wall (cell size 33 mm) is comparitavely small. The holm does not reach an asymptote. Instead, the contour of the entire holm is affected by the Young–Laplace equation. Both oil–water systems show smooth, continuous transitions from the sphere to the holm. (c) water–air interfaces, showing the sharp contact angle and discontinuity at the point of contact. The blur (reflection) visible at the air–water interface was noted with all images taken of the gas–liquid system and proved detrimental to fitting. (Sphere size 11.11 mm, cell size 33 mm).
The densities were taken from literature values and confirmed using a commercial density meter (Anton Paar). Each oil–water pair was also analyzed using the pendant drop method as a benchmark for the surface tension values. Pendant drop images were taken using a 1.2 mm outer diameter nozzle, with the water pumped into the oil layer. The pendant drop method was implemented using commercial software [15] and a second program using the pendant drop methodology implemented by the authors in the MATLAB programming environment, as indicated in the results shown in Table 2. The results presented are the mean and 95% confidence interval associated with the interfacial tension measurement of multiple, non-consecutive images for each system. Additionally, the method was applied to the measurement of a silicone oil (PDMS, 1000 cP)–water interface.
Table 2 Comparison of the results obtained using the new method and commercial pendant drop software for two water–oil systems. The average and 95% confidence interval is calculated over at least five images. System
kg/m3
Method
Interfacial tension (Average) mN/m
Confidence interval (95%) mN/m
Hexane–water
342.0
Dodecane–water
249.4
Hexadecane–water
224.8
Holm meridian Pendant drop [15] Holm meridian Pendant drop Holm meridian Pendant drop Holm meridian
38.2 40.7 45.6 45.7 31.5 31.0 37.7
±1.3 ±2.0 ±1.2 ±0.7 ±0.4 ±0.3 ±1.2
Silicone oil–water
24.7
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The holm meridians were formed within small cubic cells, with sides 3 cm or 5 cm in length, around a hydrophobic (Teflon) sphere supported at the interface by a thin Teflon line. The sphere, of known diameter, was used for image scaling, the sphere radius being used as the characteristic length of the system. A range of sphere sizes were used, from 6.35 to 12.70 mm diameter, with a tolerance of 0.1 mm. All measurements were done at room temperature (approximately 18 ◦ C), with the actual temperature used to determine the fluid densities from literature values. Unlike forming droplets for measurement, the experimental apparatus did not require a micro-pump and was easy to clean. Moreover, the holm meridian was stable for long periods of time, with no risk of collapse. The sphere position was adjusted vertically, with an emphasis on achieving a necking point. Several images were captured, with different sphere positions and extents of curvature of the holm, for each system. It is noteworthy that the necking was obtained for the oil–water interfaces but not for the air–water interface. Two issues were noted. Firstly, the reflection and blurring shown in Fig. 3c was present on all of the air–water images, leaving little of the holm available for fitting. Blurred areas were avoided due to uncertainty as to the position of the edge in the images. Secondly, the contact angle was much sharper on the air–water interface, leading to significantly less curvature of the holm. There was generally significant error between subsequent images of the air–water system, seemingly reminiscent of the fitting issues that ADSA encounters when fitting spherical drops. Consequently, the air–water interface, which was not the focus of this study as several techniques already exist which can measure it accurately, was not pursued further for modeling.
4. Modeling
271
from multiple starting points, with multiple assumed surface tensions. The fitting procedure included the following steps: 1. A starting point (x0 , y0 )0 was defined at the point where the detected edge of the holm was separated by more than 5 pixels from the sphere surface. It follows that the depression of the holm at the point of contact with the sphere is Y0 − Yh , where Y0 = y0 /a. 2. The starting contact angle, 0 , was estimated from the tangent of a polynomial fitted to the curved section of the holm (50–100 pixels) and allowed to vary by ±10% during optimization. The polynomial took the form x = C0 + C1 y + C2 y2 where (x, y) are the coordinates (in pixels) of the detected edge. 3. A constrained multivariate solver was used to optimize three variables: the shape factor, a; vertical position of the origin, y0 ; and the initial contact angle, 0 . Integration of the Young–Laplace equations produced the theoretical curve, which was matched to the detected edge. Numerical optimization was used to minimize the difference in the positions of the curves. 4. Uncertainty around the exact location of the edge was reduced by running the optimization repeatedly at different points around the initial starting point. For example, an image may be analyzed at each the first 10 coordinate points along the holm. 5. At each starting point, fitting was conducted 3 times, each time using a random selection of 150 coordinate points, assuming different values of the initial interfacial tension. The use of multiple initial estimates for different runs ensures that the fitting converges on the global minimum of the error function, rather than being caught in local minima. 6. The interfacial tension was calculated from the optimized parameters, (a).
4.1. Image analysis and boundary detection
5. Results and discussion
Edge detection was accomplished using the Canny edge detection algorithm with a Gaussian filter, supported by the edge function in MATLAB’s Image Analysis Toolbox. Canny edge detection has been widely reported as the most appropriate edge detector for use with ADSA-type image analysis, in particular for its strength in picking weak edges from images where noise is present [16]. Canny edge detection requires two threshold parameters, one for strong and one for weak edges. The global image threshold, T, was calculated using Otsu’s method (supported by MATLAB’s threshold function). MATLAB’s recommended low and high thresholds were used, which are based on the intensity gradients in the image [17].
The interfacial tensions of three unpurified oils–hexane, hexadecane and dodecane–with water were determined using the new method (holm meridian) and the pendant drop method. The results were found to agree well between the two methods (Table 2). The reported values are averages taken over multiple images, along with the 95% confidence intervals calculated using the Student’s t distribution. The authors note that, as the oils were not purified prior to use, the values reported indicate the actual oil–water interfacial tension, rather the interfacial tension of the pure oils, and serve to demonstrate the agreement between values determined using the different methods. The program developed in this study was able to successfully fit the Young–Laplace equation to a fluid–fluid interface intersecting a solid sphere. Fitting was achieved for oil–water interfaces with varying degrees of curvature and a range of contact angles. Analysis of the same oil using the pendant drop method and commercial software [15] returned a consistent result. However, as expected, the interfacial tensions of the unpurified oils were significantly lower than those for purified oils. Several papers have been published on the effects of contaminants such as salts [18], organic molecules [19] and surfactants [20] in lowering interfacial tension. Depending on the size of the sphere relative to the cell size, the bounded oil–water interface did not always tend to a horizontal plane, instead sloping up toward the walls of the container (Fig. 3). Where the sphere diameter was large when compared to the cell size, there was no evidence of the horizontal asymptote expected with the theoretical holm meridian in an unbounded fluid. The Young–Laplace equation generated from the best-fit parameters was consistent with the image of the holm as it rose to meet the wall of the container. In this region, the deformation of the fluid–fluid
4.2. Optimization to determine best-fit parameters for the theoretical profile The origin of the coordinate space used in the Young–Laplace equation is defined from the height of the undisturbed fluid–fluid interface far from the object, in line with the center of the immersed object. The position of the vertical axis can be easily determined by fitting a circle to the profile of the submerged sphere and determining the coordinates of the center. As it is not possible to ascertain the horizontal axis position from an image of a meridian in a bounded fluid, such as within a small glass cell, Yh was taken as a parameter for optimization and determined numerically. The initial estimate for the capillary constant was taken from an estimated surface tension and was constrained to be non-negative during the optimization process. An accurate estimate was not required as the program appeared quite capable of converging from significant distances. Additionally, the optimization procedure was undertaken
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Fig. 4. Calculated values for interfacial tension from four distinct sphere sizes. The scatter around the overall average interfacial tension shows no particular trend with the change in sphere size, demonstrating that the method is applicable using a variety of cell and ball sizes.
interface came from both the submerged sphere and the wall. The theoretical curve was found to follow the shape of the observed profile if extended even if only the first (curved) section of the meridian was used as an input to the program. As in these instances the horizontal asymptote is not present, it would not be possible to estimate the fitting based on method used by Boucher [12]. In order to test the effect of the wall on the calculation of interfacial tension, the interfacial tension of hexadecane (unpurified) with water was calculated from twelve images, using four different sized spheres. The larger (5 cm) cell was used, with sphere diameters of 7.14 mm, 9.54 mm, 11.11 mm and 12.70 mm. The interfacial tension results were found the be consistent within the range of sphere sizes. There was no obvious trend in the deviation from the overall mean value, as can be seen from Fig. 4. The calcuated interfacial tension (31.5 ± 0.4 mN/m) was comparable with the results calculated using the pendant drop method (31.0 ± 0.3 mN/m). The new method has a number of advantages over the existing methods. Forming the meridian around a solid sphere located at the fluid–fluid interface allowed a relatively simple and stable experimental setup. In addition, the ability to move the sphere vertically at the interface made it possible to create images with a very pronounced necking region. The shape factor of pendant drops is strongly affected by drop deformation, particularly necking. As a result, drops with pronounced necks tend to give more reliable results using ADSA techniques [7]. A similar effect is proposed for the new method presented in this study. The fitting is improved where the curvature around the holm is more pronounced, ideally with an initial contact angle in excess of 90◦ . While the ADSA pendant drop software is limited in its ability to analyze near-spherical drops, a characteristic of low Bond number systems, the new method is capable of analyzing these systems, with potential to analyze systems with very small differences in density between the two fluids. Indeed, this new method may work better as the densities become closer and surface tension effects dominate the meridian shape. Fig. 5 predicts the theoretical profiles for oil–water systems with the same interfacial tension and contact angle, but assuming oil densities from 400 to 990 kg/m3 . The length of the curved portion of the holm increases with increasing oil density. At the most extreme condition, where the density difference approaches 10 kg/m3 (1%), the new method
remains practical, whereas measurement of such a system is not possible with either the pendant drop or spinning drop method. This could allow measurement of water–oil interfacial tensions for crude oils with densities in the 850–1000 kg/m3 range, as found in the petroleum industry [21]. Conversely, the holm rises rapidly to an asymptote when the Bond number is large (the two lower profiles in Fig. 5, which have been trimmed at the start of the asymptote). As a result, there is a shorter length of the holm suitable for fitting and a potential limit to the effectiveness of the new method for high Bo systems. In these instances, one might consider using a smaller sphere to reduce the Bond number and capillary length. A recent paper [11] presents a method suitable for the analysis of low Bond number systems. The applicability of this method was demonstrated by determining the silicone oil–water interfacial tension (PDMS, 20 cS). The spherical drop shape encountered with low Bond number systems tends to introduce significant errors when attempting to fit the Young–Laplace equation for both pendant and sessile drops. In [11], the force balance over the drop cap is used rather than the Young–Laplace equations, allowing the analysis of spherical drops. However, this method also requires a highly accurate pressure reading as an input to the force balance, introducing a different source of error to the method on top of the issues with accuracy inherent in all methods relying on precise image analysis. Furthermore, the method still requires pumping to form the drop, hence does not address the issues faced when dealing with highly viscous or difficult-to-clean fluids. Measurement of the silicone–oil water interface shows that this method, using the holm meridian, can be applied to highly viscous fluids with very small density differences. The calculated interfacial tension of 37 ± 1.2 mN/m, for silicone oil (PDMS, 1000 cS). Due to the extremely small density difference between the two fluids, it is not possible to determine the interfacial tension of silicone oil–water systems using conventional methods. Consequently, very little data is available for this system. Using the force-balance method [11], the interfacial tension of a less viscous oil of 20 cS was 35 mN/m at 18 ◦ C. A scatter of ±1.1 mN/m is reported, which is comparable to the 95% confidence interval of ±1.2 mN/m found using the method presented in this paper. An interfacial tension of 34.4 mN/m was found by A. Juel (unpublished, cited in [11]) for PMDS (1000 cS) at 25 ◦ C.
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Fig. 5. Theoretical oil–water interface profiles for different oil densities (oil ) with other parameters remaining constant (x0 , y0 as shown, 0 = 134◦ , Rsphere = 5.57 mm, = 41.9 mN/m, water = 997 kg/m3 ). The curvature becomes more pronounced as the density difference is reduced and the Bond number decreases.
The method is also more practical than the spinning drop in measuring dynamic tension of oil–water interfaces [22]. The spinning drop is restricted to low tension values, approximately 20 mN/m, and short spinning times to control the heat released by spinning. The new method, in contrast, can be applied at a wider range of tensions and over a longer time. The long measurement time is important for industrial systems, in which the tension is dominated by naturally occurring surfactants with adsorption time in the order of hours [23,24]. Finally, the cell in this method can be easily pressurized, which is infeasible with many of the existing methods, to replicate the elevated pressures in industrial processes. 6. Conclusion The new method was found to be effective in fitting images of oil–water interfaces that are in contact with an immersed solid sphere. The experimental setup was simple and stable, and the method returned consistent results for the liquid–liquid interfacial tension when analyzing multiple images. The submerged holm meridian is generally easier to form and more stable to maintain than a pendant drop for liquid–liquid systems. The experiments allowed the vertical position of the sphere to be adjusted to form a more pronounced curvature in the holm, leading to more reliable results. Analysis of hexane–water, hexadecane–water and dodecane– water systems gave consistent values for the interfacial tension that were in good agreement with values produced by the pendant drop method. This is the first time the submerged shape has been used successfully for determining interfacial tension. More significantly, the new method is well-suited to the analysis of liquid–liquid systems and capable of analyzing systems with a small density difference. The analysis of a liquid–liquid system with a density difference as low as 30 kg/m3 (3%) was found to confer well with the few published results. Hence, the new method could facilitate interfacial analysis in many industrial systems, where neither the pendant drop nor spinning drop methods are practical. It can be used for analysis of extremely viscous oils, where viscosity precludes pumping, and does not rely on highly sensitive pressure measurements. This method may facilitate the measurement of systems involving hazardous or unstable chemicals, elevated pressures or high temperatures. References [1] C. Chang, E. Franses, Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms, Colloids Surf. A 100 (1995) 1–45.
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