PHYSICAL REVIEW E 91 , 013007 (2015)
Surface tension profiles in vertical soap films N. Aclami’ and H. Caps' GRASP, Departement de Physique B5, Universite de Liege, R4000 Liege, Belgium (Received 31 January 2014; revised manuscript received 21 October 2014; published 9 January 2015) Surface tension profiles in vertical soap films are experimentally investigated. Measurements are performed by introducing deformable elastic objets in the films. The shape adopted by those objects once set in the film is related to the surface tension value at a given vertical position by numerically solving the adapted elasticity equations. We show that the observed dependency of the surface tension versus the vertical position is predicted by simple modeling that takes into account the mechanical equilibrium of the films coupled to previous thickness measurements. DOI: 10.1103/PhysRevE.91.013007
PACS number(s): 68.03.Cd, 68.15.+ e, 46.25.- y , 46.35,+z
L INTRODUCTION
Anyone who has ever looked carefully at a soap film by simply pulling a frame out of soapy water had seen that it exhibits horizontal interference fringes [1,2] (see Fig. 1). As the film has just been pulled out of the soapy water, all the fringes seem to be vertically equally separated. After a few seconds, the interfringe grows from the bottom to the top of the film. This fact attests to both the existence of a nonuniform thickness profile in the soap film and also to the evolution of that profile with time. From a mechanical point of view, this thickness profile implies that (i) as proposed by Gibbs in 1878 [3], for any value H of the height above the bottom edge of the frame, the weight of the part of the film which lies beneath H must be counterbalanced by the local surface tension forces and (ii) since the fringe pattern in the film stretches from bottom to top, implying that the thickness gradient in the film gets smaller as H increases, the sustaining surface tension profile must both increase and saturate with H. Due to their highly interfacial nature, soap films have been in the center of numerous studies during the past few decades [1,4-13], Early works by Mysels and co-workers [1,13,14] showed that their behaviors strongly depend on parameters such as viscosity [13] but also on the chemicals used to produce them. Specific surfactant-linked phenomena such as marginal regeneration [1,7] have also been noticed to account in phenomena such as drainage, leading to unexpected film lifetimes [1], Those studies of isolated soap films have been widely used to model global behaviors of more complex systems such as foams [15]. Theoretical expressions based on surfactant molecule thermodynamics have been proposed in order to describe the evolution of both the thickness and the surface tension profiles versus H [4,16]. To our knowledge, no experiment has yet been proposed in order to confirm those theoretical trends. The main reason for this is that the usual sur face tension sensors, such as Whilhelmy plates, are unadapted to the case of soap films. In fact, due to their wettability, these sensors are prone to induce local strong deformations (i.e., menisci and related marginal regeneration) of the film, which make these intrusive measurements perturbative with respect to the natural state of the soap film.
*[email protected] f [email protected] 1539-3755/2015/91(1 )/013007(7)
Recent studies on soft elastic objects have shown that considerable deformations of those objects can be induced by capillary constraints [17-24]. The case of the surface pressure linked to a surfactant monolayer leading to deformations [2,18,19,25] can be used either to characterize the evolution of the object shapes knowing the surface pressure or to determine the surface pressure leading to an observed deformation. In this paper, we show how such elasto-capillary effects can be used to probe the surface tension profiles present in vertical maintained soap films. Figure 1 shows a soft rectangle which has been plugged into a vertical soap film. When no soap film is present inside the rectangle, the latter inflates due to the surface tension forces applied by the outer film interfaces. Solving the elasticity equations, written in this particular case, would allow one to obtain a theoretical link between the deflections experienced by the objects and the surface tension value at the corresponding H. The experimental surface tension profiles can then later be deduced from a combination of a simple model for the mechanical equilibrium and thickness measurements in the film.
II. EXPERIMENTAL SETUP AND MATERIALS A. Soap films
Our soap films are built from solutions made of 3% of a SLES + CAPB mixture described in [26] plus 0.3% glycerol and double-distilled water. This solution leads to a typical density of pi = 1000 kg/m 3 and a typical viscosity of q = 1.09 x 10-3 Pas. The surface tension y0 of the solution has been determined to be 29.8 ± 0.2 mN/m with a pending drop method. We suppress the temporal evolution of the thickness profile of our films by feeding them using the setup sketched in Fig. 2 [27-29], A flow made of the soapy solution is injected at a constant flow rate Q from both sides in a slit pipe, the slot pointing upward. When the solution comes out of the slot, it follows the edges of the pipe to reach the top of the film, which lies beneath the pipe. By doing so, it is possible to suppress the gravitational drainage and to build soap films which can last for hours [29], Choosing the flow rate carefully allows us to fix the thickness profile of the film in time. The thickness dependency versus the vertical coordinate H (see Fig. 2) has been determined by using infrared absorption in [29]. The idea of this measurement is to use the absorption of infrared rays by the water molecules of the soap film to
013007-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 013007 (2015)
N. ADAMI AND H. CAPS
FIG. 1. Illustrative picture of an elastic rectangle plugged into a vertical soap film, taken at an angle of 60° with respect to the normal to the film, so that both the inflated rectangle and the interference pattern are visible. Note that the deflection measurements (see text for details) are performed with a normal incidence with respect to the film interfaces, as illustrated in Fig. 2.
FIG. 3. Left: Geometrical characteristics of the rectangles used to perform surface tension measurements in soap films (/ = 24 mm, w = 12 mm, £ = 3 mm, and e = 0.8 mm). Right: Typical shape adopted by rectangles when introduced in the soap films.
B. Elasto-capillary probe
determine its thickness through a Beer-Lambert law. In fact, if the absorption coefficient of the infrared rays which goes through the film is known, the infrared intensity observed behind the soap film is linked to the infrared intensity before the film as 7, = 70 exp(—fie), where ji is the absorption coefficient of water corresponding to the infrared rays used to perform the experiment, e is the thickness of the film, and 7, and 70 are the infrared intensities of those rays after and before they passed through the soap film, respectively. The lateral edges of the frame sustaining the film are made of stainless steel rods. All the films described in the present study are 150 x 150 mm2.
The elastic objects used to perform surface tension mea surements are vinyl polysiloxane rectangles as presented on Fig. 3 [18,19]. They are made of long flexible lateral arms of length l and rigid top and bottom arms of height f , the global width of the rectangle being w. The cross section of the lateral arms is a square of size e. Those probes are maintained in the films at a given height thanks to a needle going through the upper horizontal arm of the rectangle. If no soap film is present inside the rectangle, the surface tension forces acting on the lateral arms will cause their bending, inflating the rectangle (see Figs. 1, 3, and 4). The amplitude of the corresponding maximal deflection 5 results in a balance between interfacial, gravitational, and bending energies. The bending energy B of a flexible arm is related to its bending stiffness as E I0, where E is the Young modulus of the polymer and 70 = e£3/12 is the arm quadratic momentum, with § being the width of the arm. In the particular case of the lateral arm, this expression reduces to e4/12, since they are designed so that e = §. The width f of the top and bottom edges of the rectangle is chosen so that only the lateral arms can bend once the object is set in the film (i.e., the bending energies of the top and bottom arms are too large to be balanced by the interfacial energy of the
FIG. 2. (Color online) Left: Sketch of the experimental setup. The slit pipe, the soap film, the camera, and the deflected object are represented. Right: Schematic of the bottom meniscus linking the film to the bottom of the frame (not to scale). See next sections for details.
FIG. 4. Left: Superposition of images of a rectangle when set inside and outside the soap film, as used to measure S versus 77. Right: Enlarged view of the bottom part. The dashed lines indicate the position of the bottom part of the rectangle with and without the soap film inside the rectangle, illustrating the existence of /'.
013007-2
SURFACE TENSION PROFILES IN VERTICAL SOAP FILMS
film). The Young modulus and the density of the rectangles are, respectively, 0.3 MPa and 1023 kg/m 3. Before designing the elastic probes, it is instructive to compare the bending and interfacial energies involved in our experiments by order-of-magnitude calculations. Figure 4 shows a superposition of two pictures of a rectangle in a soap film when the soap film is present both inside and outside the rectangle (undeformed) and when the inner soap film is burst (inflated case). A careful look at the bottom edge of the rectangle allows us to see that it rises up a quantity /' after the inner film burst, due to the stretching applied by the outer film. This effect involves an increase of the potential gravitational energy of both the bottom part of the rectangle and the meniscus linking this arm to the film. As one can see on Fig. 4, the inner film burst leads to an /' value that is weak (~ 1 0 -4 m) compared to 3 (~ 10-3 m), this latter being relatively weak compared to Z(~ 10-2 m). Thus, from a geometrical point of view, /' can be estimated as 32/Z. If the polymer density is denoted ps , the energy balance linked to the inflated rectangle then reads „ 82 ( E I 0 \ 2y8l ~ Y y - p - + Y W +Ps'Vsg + pLVLgj,
( 1)
where the left-hand side represents the interfacial energy linked to the rectangle inflation. The first term of the right-hand side represents the bending energy of the lateral arm, the second one is the interfacial energy linked to the bottom part uprise, while the third and the fourth terms stand for the potential energies linked to the bottom part of the rectangle and its linking meniscus, respectively. Introducing orders of magnitude in Eq. (1) leads to an estimate for the surface tension of y ~ 10 mN/m, which is typical for a water-air interface. Equation (1) allows us to emphasize the basic consequence of the absence of soap film inside the rectangle, and it represents the energy balance after the inner film burst. The idea of the surface tension measurement we want to perform is to move the rectangle along the vertical coordinate H in order to see the fluctuations of 8 with H. It is then interesting to wonder what are the magnitudes of 8 and V which are linked to an increase in H. From the thickness measurements (see [29]) and the mechanical equilibrium in the film, we can estimate the surface tension variation to be Ay ~ 0.2 mN/m if H is increased or decreased by 1 cm (i.e., the magnitude of the vertical surface tension gradient in our films is of the order of 20 mN/m2). One can note that the y variation for a 1-cm increase in H is two orders of magnitude smaller than the typical y value obtained from Eq. (1). We can also note from Eq. (1) that the bending energy is weaker than the other terms. Since the curvature linked to the lateral arm is expressed as 8/ l 2 and since the potential energy increments are proportional to
Surface tension profiles in vertical soap films N. Aclami’ and H. Caps' GRASP, Departement de Physique B5, Universite de Liege, R4000 Liege, Belgium (Received 31 January 2014; revised manuscript received 21 October 2014; published 9 January 2015) Surface tension profiles in vertical soap films are experimentally investigated. Measurements are performed by introducing deformable elastic objets in the films. The shape adopted by those objects once set in the film is related to the surface tension value at a given vertical position by numerically solving the adapted elasticity equations. We show that the observed dependency of the surface tension versus the vertical position is predicted by simple modeling that takes into account the mechanical equilibrium of the films coupled to previous thickness measurements. DOI: 10.1103/PhysRevE.91.013007
PACS number(s): 68.03.Cd, 68.15.+ e, 46.25.- y , 46.35,+z
L INTRODUCTION
Anyone who has ever looked carefully at a soap film by simply pulling a frame out of soapy water had seen that it exhibits horizontal interference fringes [1,2] (see Fig. 1). As the film has just been pulled out of the soapy water, all the fringes seem to be vertically equally separated. After a few seconds, the interfringe grows from the bottom to the top of the film. This fact attests to both the existence of a nonuniform thickness profile in the soap film and also to the evolution of that profile with time. From a mechanical point of view, this thickness profile implies that (i) as proposed by Gibbs in 1878 [3], for any value H of the height above the bottom edge of the frame, the weight of the part of the film which lies beneath H must be counterbalanced by the local surface tension forces and (ii) since the fringe pattern in the film stretches from bottom to top, implying that the thickness gradient in the film gets smaller as H increases, the sustaining surface tension profile must both increase and saturate with H. Due to their highly interfacial nature, soap films have been in the center of numerous studies during the past few decades [1,4-13], Early works by Mysels and co-workers [1,13,14] showed that their behaviors strongly depend on parameters such as viscosity [13] but also on the chemicals used to produce them. Specific surfactant-linked phenomena such as marginal regeneration [1,7] have also been noticed to account in phenomena such as drainage, leading to unexpected film lifetimes [1], Those studies of isolated soap films have been widely used to model global behaviors of more complex systems such as foams [15]. Theoretical expressions based on surfactant molecule thermodynamics have been proposed in order to describe the evolution of both the thickness and the surface tension profiles versus H [4,16]. To our knowledge, no experiment has yet been proposed in order to confirm those theoretical trends. The main reason for this is that the usual sur face tension sensors, such as Whilhelmy plates, are unadapted to the case of soap films. In fact, due to their wettability, these sensors are prone to induce local strong deformations (i.e., menisci and related marginal regeneration) of the film, which make these intrusive measurements perturbative with respect to the natural state of the soap film.
*[email protected] f [email protected] 1539-3755/2015/91(1 )/013007(7)
Recent studies on soft elastic objects have shown that considerable deformations of those objects can be induced by capillary constraints [17-24]. The case of the surface pressure linked to a surfactant monolayer leading to deformations [2,18,19,25] can be used either to characterize the evolution of the object shapes knowing the surface pressure or to determine the surface pressure leading to an observed deformation. In this paper, we show how such elasto-capillary effects can be used to probe the surface tension profiles present in vertical maintained soap films. Figure 1 shows a soft rectangle which has been plugged into a vertical soap film. When no soap film is present inside the rectangle, the latter inflates due to the surface tension forces applied by the outer film interfaces. Solving the elasticity equations, written in this particular case, would allow one to obtain a theoretical link between the deflections experienced by the objects and the surface tension value at the corresponding H. The experimental surface tension profiles can then later be deduced from a combination of a simple model for the mechanical equilibrium and thickness measurements in the film.
II. EXPERIMENTAL SETUP AND MATERIALS A. Soap films
Our soap films are built from solutions made of 3% of a SLES + CAPB mixture described in [26] plus 0.3% glycerol and double-distilled water. This solution leads to a typical density of pi = 1000 kg/m 3 and a typical viscosity of q = 1.09 x 10-3 Pas. The surface tension y0 of the solution has been determined to be 29.8 ± 0.2 mN/m with a pending drop method. We suppress the temporal evolution of the thickness profile of our films by feeding them using the setup sketched in Fig. 2 [27-29], A flow made of the soapy solution is injected at a constant flow rate Q from both sides in a slit pipe, the slot pointing upward. When the solution comes out of the slot, it follows the edges of the pipe to reach the top of the film, which lies beneath the pipe. By doing so, it is possible to suppress the gravitational drainage and to build soap films which can last for hours [29], Choosing the flow rate carefully allows us to fix the thickness profile of the film in time. The thickness dependency versus the vertical coordinate H (see Fig. 2) has been determined by using infrared absorption in [29]. The idea of this measurement is to use the absorption of infrared rays by the water molecules of the soap film to
013007-1
©2015 American Physical Society
PHYSICAL REVIEW E 91, 013007 (2015)
N. ADAMI AND H. CAPS
FIG. 1. Illustrative picture of an elastic rectangle plugged into a vertical soap film, taken at an angle of 60° with respect to the normal to the film, so that both the inflated rectangle and the interference pattern are visible. Note that the deflection measurements (see text for details) are performed with a normal incidence with respect to the film interfaces, as illustrated in Fig. 2.
FIG. 3. Left: Geometrical characteristics of the rectangles used to perform surface tension measurements in soap films (/ = 24 mm, w = 12 mm, £ = 3 mm, and e = 0.8 mm). Right: Typical shape adopted by rectangles when introduced in the soap films.
B. Elasto-capillary probe
determine its thickness through a Beer-Lambert law. In fact, if the absorption coefficient of the infrared rays which goes through the film is known, the infrared intensity observed behind the soap film is linked to the infrared intensity before the film as 7, = 70 exp(—fie), where ji is the absorption coefficient of water corresponding to the infrared rays used to perform the experiment, e is the thickness of the film, and 7, and 70 are the infrared intensities of those rays after and before they passed through the soap film, respectively. The lateral edges of the frame sustaining the film are made of stainless steel rods. All the films described in the present study are 150 x 150 mm2.
The elastic objects used to perform surface tension mea surements are vinyl polysiloxane rectangles as presented on Fig. 3 [18,19]. They are made of long flexible lateral arms of length l and rigid top and bottom arms of height f , the global width of the rectangle being w. The cross section of the lateral arms is a square of size e. Those probes are maintained in the films at a given height thanks to a needle going through the upper horizontal arm of the rectangle. If no soap film is present inside the rectangle, the surface tension forces acting on the lateral arms will cause their bending, inflating the rectangle (see Figs. 1, 3, and 4). The amplitude of the corresponding maximal deflection 5 results in a balance between interfacial, gravitational, and bending energies. The bending energy B of a flexible arm is related to its bending stiffness as E I0, where E is the Young modulus of the polymer and 70 = e£3/12 is the arm quadratic momentum, with § being the width of the arm. In the particular case of the lateral arm, this expression reduces to e4/12, since they are designed so that e = §. The width f of the top and bottom edges of the rectangle is chosen so that only the lateral arms can bend once the object is set in the film (i.e., the bending energies of the top and bottom arms are too large to be balanced by the interfacial energy of the
FIG. 2. (Color online) Left: Sketch of the experimental setup. The slit pipe, the soap film, the camera, and the deflected object are represented. Right: Schematic of the bottom meniscus linking the film to the bottom of the frame (not to scale). See next sections for details.
FIG. 4. Left: Superposition of images of a rectangle when set inside and outside the soap film, as used to measure S versus 77. Right: Enlarged view of the bottom part. The dashed lines indicate the position of the bottom part of the rectangle with and without the soap film inside the rectangle, illustrating the existence of /'.
013007-2
SURFACE TENSION PROFILES IN VERTICAL SOAP FILMS
film). The Young modulus and the density of the rectangles are, respectively, 0.3 MPa and 1023 kg/m 3. Before designing the elastic probes, it is instructive to compare the bending and interfacial energies involved in our experiments by order-of-magnitude calculations. Figure 4 shows a superposition of two pictures of a rectangle in a soap film when the soap film is present both inside and outside the rectangle (undeformed) and when the inner soap film is burst (inflated case). A careful look at the bottom edge of the rectangle allows us to see that it rises up a quantity /' after the inner film burst, due to the stretching applied by the outer film. This effect involves an increase of the potential gravitational energy of both the bottom part of the rectangle and the meniscus linking this arm to the film. As one can see on Fig. 4, the inner film burst leads to an /' value that is weak (~ 1 0 -4 m) compared to 3 (~ 10-3 m), this latter being relatively weak compared to Z(~ 10-2 m). Thus, from a geometrical point of view, /' can be estimated as 32/Z. If the polymer density is denoted ps , the energy balance linked to the inflated rectangle then reads „ 82 ( E I 0 \ 2y8l ~ Y y - p - + Y W +Ps'Vsg + pLVLgj,
( 1)
where the left-hand side represents the interfacial energy linked to the rectangle inflation. The first term of the right-hand side represents the bending energy of the lateral arm, the second one is the interfacial energy linked to the bottom part uprise, while the third and the fourth terms stand for the potential energies linked to the bottom part of the rectangle and its linking meniscus, respectively. Introducing orders of magnitude in Eq. (1) leads to an estimate for the surface tension of y ~ 10 mN/m, which is typical for a water-air interface. Equation (1) allows us to emphasize the basic consequence of the absence of soap film inside the rectangle, and it represents the energy balance after the inner film burst. The idea of the surface tension measurement we want to perform is to move the rectangle along the vertical coordinate H in order to see the fluctuations of 8 with H. It is then interesting to wonder what are the magnitudes of 8 and V which are linked to an increase in H. From the thickness measurements (see [29]) and the mechanical equilibrium in the film, we can estimate the surface tension variation to be Ay ~ 0.2 mN/m if H is increased or decreased by 1 cm (i.e., the magnitude of the vertical surface tension gradient in our films is of the order of 20 mN/m2). One can note that the y variation for a 1-cm increase in H is two orders of magnitude smaller than the typical y value obtained from Eq. (1). We can also note from Eq. (1) that the bending energy is weaker than the other terms. Since the curvature linked to the lateral arm is expressed as 8/ l 2 and since the potential energy increments are proportional to
