SSurface tension of flowing soap films
Aakash Sane, Shreyas Mandre, and Ildoo Kim ∗ School of Engineering, Brown University, Providence, RI 02912, USA (Dated: November 22, 2017)
Abstract
The surface tension of flowing soap films is measured with respect to the film thickness and theconcentration of soap solution. We perform this measurement by measuring the curvature of thenylon wires that bound the soap film channel and use the measured curvature to parametrize therelation between the surface tension and the tension of the wire. We find the surface tension of oursoap films increases when the film is relatively thin or made of soap solution of low concentration,otherwise it approaches an asymptotic value 30 mN/m. A simple adsorption model with only twoparameters describes our observations reasonably well. With our measurements, we are also ableto estimate Gibbs elasticity for our soap film. ∗ [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] N ov . INTRODUCTION Soap film channels have been used as a model system for two-dimensional flow [1–8] formany years, however, their recent exploitation is expanding in the third dimension. Forexample, impingement of solid objects [9–11], liquid drops [12–14], and gaseous jets [15]on soap films have been investigated. The deformation of the soap film accompanies thecreation of extra surface area, which sometimes takes a substantial portion of the system’senergy. In this regard, it is critically important to know the surface tension and elasticityof the soap film in order to make the best energetic analysis. However our understanding ofthis property is lacking.The lack of measurement of the surface tension is due to the technical difficulties thatarise from the fact that all existing techniques are not applicable to the soap films. Sometechniques (e.g. pendant drop methods) require the use of a bulk of liquid, whose surfacetension is not necessarily equal to that of a soap film, and others (e.g. Du Noüy ring;[16]; or Wilhemly plate method) are intrusive and rely on the formation of menisci, whichsignificantly alter the local surface chemistry. Recent measurement [17] using a deformableobject inserted into soap films is also not suitable if the flow speed of the film is comparableto the Marangoni wave speed [18].In this work, we present a simple method to measure the surface tension of the flowingsoap film. Our method is applicable to the common setup of a flowing soap film channel[19] without any additional instrumentation except a camera. We use the fact that thoseconventional soap films consist of two flexible nylon wires serving as two-dimensional channelwalls. These flexible wires are tensed by hanging a known weight and curve inwards by thesurface tension of the soap film. The measurement of the surface tension is made possibleby measuring the bending curvature and the applied tension.We present our measurement of the surface tension of soap films of different thicknessand concentration of soap solution. The thickness of the film is varied from 1 to 10 µ m , andthree different soap concentrations, 0.5%, 1%, and 2%, are used. Our measurements showthat if the soap film is thick or made from 2% solution, then it possesses a surface tensionof 30 mN/m. However, if the film is relatively thin or made from 0.5% or 1% solutions, itssurface tension has a greater value.Our observation shows the apparent equivalence between the thinning and the dilution of2he soap film. The surface tension σ depends on the surface concentration Γ of surfactants,i.e., σ = σ − α Γ , where σ is the surface tension of pure liquid and α is the proportionalityconstant [20, 21]. The fact that the thinner films possess higher σ implies a smaller valueof Γ than thicker films. Dilution of the soap concentration causes the equivalent effect thatincreases σ .We rationalize these observation using a quantitative theory based on the conservationequation for surfactants and the Langmuir’s adsorption isotherm. First, in a patch of a soapfilm, the total number of surfactants is the sum of surfactants at the interfaces and those inthe bulk between interfaces. This can be written in the form of the conservation law c = c + 2Γ h , (1)where c is the concentration of the soap solution and c is the concentration of interstitialsurfactants. Second, using the Langmuir’s adsorption isotherm, the relation between Γ and c is formulated with two parameters Γ ∞ and c ∗ Γ = Γ ∞ c ∗ /c . (2)Physically, Γ ∞ is the concentration of surfactants when the surface is fully occupied bysurfactants, and c ∗ is the value of c when Γ = Γ ∞ / . Using our measurement of σ withrespect to c and h , we estimate these two parameters Γ ∞ and c ∗ . Our analysis shows thatthis two-parameter model describes our system reasonably well. The two parameter modelalso enables us to estimate the Gibbs elasticity for our soap film setup. II. EXPERIMENTAL METHODA. Soap film setup
We carry out the experiments using a standard soap film setup, which is introduced anddiscussed in published work [18, 19]. The main component of the standard setup is a pairof nylon wires (WL and WR in Fig. 1(a)) that are connected to a soap solution reservoir atthe top (RT) and to a suspended weight mg at the bottom. The suspended weight createstension in the wires which causes them to remain vertical. To create a soap film, we opena needle valve (V) to allow soapy water to flow along two nylon wires as they are initiallyabreast, and then we pull them apart from each other by using four auxiliary wires. The3 igure 1. ( a ) The main component of the standard soap film channel setup is two flexible nylonwires (WL and WR) that are anchored at four points, A, B, C, and D. In the presence of a soapfilm, wires are bent toward each other due to the surface tension. ( b ) A length element of wire ds ismade tight by T and bent by σ . The factor 2 of σ reflects that the soap film has two surfaces. ( c )Cross section of the film in y - z plane. In the control volume shown by dashed rectangle, pressure atpoint F is different from the pressure at point G due to the curvature in z direction. However, thispressure difference is canceled out with the effect of the menisci. This can be shown by extendingthe control volume to point F (cid:48) where the film is flat. auxiliary wires provide support so that the main wires are fixed at four points: A, B, C,and D. We set the coordinate system such that ˆ x is longitudinal and ˆ y is transverse to theflow with the origin O being the centre of the soap film. The soap film channel is carefullytuned to be symmetric about both the axes.We prepare solutions of commercial dish soap (Dawn, P&G) in distilled water at threedifferent concentrations. The soap concentration c is defined as the volume fraction of thesoap, with c = φ ofthe channel is varied from 0.26 to 1.55 cm / s by adjusting the needle valve (V).The whole channel is 1.8 m long, and the length L between two fixed points A and B is1.2 m and is equal to the distance between C and D, i.e., L ≡ AB = CD = 1 . . Likewise,the reference width of the channel W is defined as W ≡ AC = BD and varied from 6 cm to14 cm. We use a mass of 400 g for the hanging weight at the bottom, therefore a tension of4 u ( c m / s ) (a) u by particle trackingq/h(y) by fringe counting 10 -2 -1 q (cm /s)10 h ( µ m ) Figure 2. ( a ) The flow speed profile across y at q = 0 .
33 cm / s . The open circles are the directmeasurement of u ( y ) using the particle tracking, and the closed squares are the calculation of q/h ( y ) , where q is fixed and h ( y ) is measured by counting fringes of equal thicknesses. ( b ) Therelation between h and q . We find that h = h ( q/q ) . , with q = 0 . / s and h = 4 . µ m . T is O (10 ) N, the surface tension σ is O (10 − ) N/m, and the lengthof the soap film L is O (1) m. The channel width is then contracted slightly, by 1% of itsoriginal width. We tune the weight to be small enough to substantiate the width contractionbut yet to minimize the distortion of the channel that may disturb the flow.In developing a quantitative theory, it is necessary to find the thickness of the soap film.We define the mean thickness of the film as h ≡ q/u t , where q ≡ φ/W is the flux per unitwidth and u t is the terminal velocity of the flow. We find that q is independent of y . Usingthe particle tracking velocimetry, we find that the profile of u ( y ) is parabolic as seen in Fig.2(a). This profile matches well with an independent measurement q/h ( y ) counting fringesof equal thicknesses under a monochromatic illumination. Also, q only weakly depends on x . Even though the channel width depends on x , its variation is limited to a few percentand smaller than 5% uncertainty of φ .We observe that both h and u t are increased by increasing q , namely h ∝ q . and u t ∝ q . . Here, the terminal velocity u t is measured at y = 0 and approximately 0.7 m5way from the top of the channel where it becomes independent of x . This gives a relationof h to q as delineated in Fig. 2(b), h = h ( q/q ) . , (3)where q = 0 . / s and h = 4 . µ m . The power exponent 0.75 in Eq. (3) is somewhathigher than 0.6 reported in literature [22]. Currently, no theory is available to take accountof this scaling relation. However, we find that our empirical relation between q and h isclear and reproducible; in the rest of the paper, we use Eq. (3) to estimate h . In our rangeof φ and W , q is varied from 0.02 to 0.4 cm / s , and h is varied from 1 to 12 µ m . B. Relation of the surface tension to the curvature
When a soap film is formed, the surface tension of the fluid pulls the wires (WL and WR)toward each other and creates curvature. In the following calculation, we consider the forcebalance on a length element ds of the wire as seen in Fig. 1(b).In the tangential direction, the forces acting on ds are the following: the tension T ofthe wire, the gravitational force, and the viscous force between the wire and the flow. Herewe assume that the deflection of the wire is so small that the wire is nearly vertical. Thisassumption allows us to approximate the gravitational force to be in the tangential direction.When the wire is stationary, ∂T∂x + ρπ d g − µ ∂u∂y πd = 0 , (4)where the diameter of the wire d = 0 .
038 cm , the density of the wire ρ = 1 .
15 g / cm , andthe viscosity of the soap solution µ (cid:39) . × − Pa · s . Considering that du/dy ≈ s − is usually reported in soap films [22], we find that ∂T /∂x ≈ − N/m. We conclude that T = mg/ .
96 N at any location of the wire.In the normal direction, the surface tension σ exerts to bend the wires, and the tension T resists as seen in Fig. 1(b). Using the Euler-Bernoulli equation, EI ∂ y∂x − σ + T κ = 0 , (5)where E ≈ Pa is the elastic modulus of the wire, I = π ( d/ / (cid:39) × − m is thesecond moment of area, and κ = dθ/ds ≈ d y/dx (cid:39) (30 m) − is the curvature. Comparing6he relative importance of the first term to the second term as EIκ/ (2 σL ) ≈ − and tothe third term as EI/ ( T L ) ≈ − , we conclude that the bending stiffness of the wire isnegligible. Eq. (5) is then simplified to σ = T κ mg ∂ y∂x , (6)which provides us the means to measure σ by characterizing the local curvature of the wire.We also examine the effect of the Laplace pressure due to the curvature in z direction.Figure 1(c) shows the cross section of the film in y - z plane. As shown in Fig. 2(a), the filmis thicker near the wires, and there is non-zero curvature in z direction. In a control volumeenclosed by dashed rectangle in Fig. 1(c), the pressure at point F is smaller than the pressureat G, and exerts an additional force to bend the wire. However, the curvature also reducesthe component of surface tension and increases the area of the control surface. Because theLaplace pressure is related to the curvature, these effects cancel each other exactly. Thisbecomes clear when we extend the control volume to point F (cid:48) (shown in dash-dotted line)where the film is flat, and the net force to the control volume is zero except σ . Combinedwith the observation that there is no flow in the y direction in the soap film, it is inferredthat Eq. (6) holds irrespective of the control volume chosen. C. Measurements
A digital camera is set for visual observation of the soap film. A conventional digitalsingle lens reflection camera (Nikon D90) is placed approximately 3.5 m away from the soapfilm. The camera’s optical axis is aligned with z axis of our coordinate system so thatthe centre of the image is the centre of the soap film channel (the origin in the coordinatesystem). The entire soap film is captured in a single frame.The captured digital image of the soap film channel is processed using our in-housecode for the digitization and regression analysis. Each photo is converted to a set of ( x, y ) coordinates of points on the wire, and more than 3000 data points are acquired per wire.We then fit the digitized data to the polynomial function y = (cid:80) a n x n . Inspection ofmultiple cases shows that a n for n > is negligible, and we conclude that the use of thequadratic regression model y = a x + a x + a is sufficient to fit our data. The coefficients a , a , and a are determined by least-square method, and the curvature κ is calculated by7aking a second derivative, as κ = ∂ y/∂x = 2 a . Then, σ = mga / according to Eq. (6). III. RESULTS AND DISCUSSIONA. Surface tension
In Fig. 3(a), our measurements of σ are presented with respect to q , ranging from 0.02to 0.4 cm / s , for three different c = q is sufficiently large thusthe film is sufficiently thick, σ approaches an asymptotic value σ ∞ = 30 mN / m . This valueof σ ∞ equals to the surface tension of bulk solution; in a separate measurement of σ ofsoap solutions in Petri dishes using du Nuöy ring method, we find that σ = 30 mN / m if c > . . This observation implies that in the asymptotic regime, the surface of the soapfilm is fully covered with surfactants and the surface concentration Γ approaches a constantvalue Γ ∞ as σ also approaches σ ∞ . Using the asymptotic properties, the proportionalityconstant in the assumed linear relation σ = σ − α Γ can be obtained as α = ( σ − σ ∞ ) / Γ ∞ .The required thickness to possess σ ∞ is inversely proportional to c . We find that h =3 . µ m ( q = 0 .
08 cm / s ) is required for c = 2% , and h = 7 . µ m ( q = 0 . / s ) is requiredfor c = 1% to possess σ ∞ . By extrapolating data points, h (cid:39) µ m ( q = 0 . / s ) isestimated to be required for c = 0 . .The inverse proportionality between c and h shows the apparent equivalence betweenthinning and dilution. Our measurement in Fig. 3(a) indicates that there are two ways todecrease Γ in soap films; one could decrease Γ by reducing c (dilution), or by reducing h (thinning). If we consider a patch of a soap film, thinning does not change the volume ofthe patch but introduces fresh interface and reduces the number density of surfactants inthe patch. The dilution results in the reduction of the total number of surfactants in thepatch and Γ is reduced.To accommodate our observation quantitatively, we present a self-consistent model ofthe surface tension by using the conservation equation (or as we call it the surface dilutionequation ) in Eq. (1) and the Langmuir adsorption equation in Eq. (2). The basic featureof the Langmuir adsorption equation is that Γ increases with c for low concentration, andthen reaches an asymptotic value Γ ∞ . This behaviour qualitatively agrees with that of soapsolutions; when c is small, Γ and c are roughly proportional to each other, for asymptotically8 .0 0.1 0.2 0.3 0.4q (cm /s)303540 σ ( m N / m ) Figure 3. ( a ) The surface tension σ measured using Eq. (6). When c is high, σ is independent of h . However, when c is low, σ increases as h increases. ( b ) Data points from various experimentalconditions all collapse into a single curve that represents the equilibrium between Γ and c . It isinferred that the main surfactant of our soap films are in equilibrium. large c , Γ reaches a constant asymptotic value [23]. Leaving two parameters Γ ∞ and c ∗ tobe determined later, we solve for Γ using Eqs. (1) and (2), Γ = Γ ∞ − Γ (cid:48) ( h, c ; Γ ∞ , c ∗ b ) , (7)where Γ (cid:48) ( h, c ; Γ ∞ , c ∗ b ) ≡ [( c h + c ∗ h − ∞ ) /
16 + Γ ∞ c ∗ h/ / − ( c h + c ∗ h − ∞ ) / . Thefunction Γ (cid:48) is positive-valued and approaches zero as h → ∞ . Then the surface tension σ is σ = σ − α Γ = σ ∞ + ( σ − σ ∞ ) Γ (cid:48) ( h, c )Γ ∞ , (8)where σ ∞ ≡ σ − α Γ ∞ equals the experimentally observed value 30 mN/m.We now determine two parameters Γ ∞ and c ∗ by regression. Experimental data in Fig.3(a) are provided as an input to the model in Eq. (8), and the model is iteratively solved tominimize the mean squared residual of the fitting. The analysis yields that Γ ∞ = 0 . · µ m and c ∗ = 0 . as the best fitting of our data. In Fig. 3(a), the calculation of the modelusing the obtained parameters are displayed as solid lines. The grey shades around thecurve shows ±
10% uncertainty of the calculation. By substituting Γ ∞ = 0 . · µ m and c ∗ = 0 . in Eqs. (7) and (1), Γ and c are calculated for each measured datum. Fig. 3(b)9hows σ plotted with respect to such calculated c , and our measurement from three differentconcentrations collapses into a single curve. The black line in the figure is the Langmuiradsorption in Eq. (2) and represents the relation between Γ and c in equilibrium.Two important implications about our soap films are learned. First, the primary surface-active component of our soap films approach equilibrium. The commercial dish soap we useis a mixture of several species of surfactants with different properties. However, it is shownin Fig. 3(b) that the data points from various experimental conditions all collapse into theequilibrium relation, and this collapse strongly indicates that Γ and c of the main surfactantsare in equilibrium in our soap films. Further, we estimate the diffusive time scale across thethickness τ d (cid:39) h / D ≈ D (cid:39) × − cm / s [20]. The flow speed u t ∼ u t τ d , is less than 0.1 m. Considering that our soap film has an initial expansion zone that isapproximately 0.3 m long, it is evident that the equilibrium between the surface and the bulkmust have been reached even in the thickest film. Second, however, other than the primarysurfactants, there may be other, perhaps heavier, components of the soap that remain outof equilibrium in the soap film setup. This is suuported by our independent surface tensionmeasurements of soap solutions, which show further reduction in the value of the surfacetension than that predicted by the Langmuir isotherm. B. Gibbs elasticity
Elasticity is related to σ and is pertinent for the energetic analysis of soap films. Theelasticity of a soap film quantifies the stability of soap film under disturbance and is definedas the change in σ per the fractional change in the surface area, i.e., E = Adσ/dA , where A is the area of a soap film [24]. The measurement of σ with respect to h allows us to calculatethe Gibbs elasticity of soap films. Using the incompressibility ln A = − ln h , the elasticity is E = − dσd ln h . (9)Depending on the time scale of the disturbance, there are two elasticities of soap films, theMarangoni and the Gibbs elasticity. The Marangoni elasticity applies to the scenario inwhich a soap film is stretched suddenly, i.e. the time scale of the disturbance τ is shorterthan the diffusive time scale τ d , so that the interstitial surfactants do not have time to diffuse10o the surfaces. Then the local decrease in Γ will increase σ , producing Marangoni stress thatrecovers the soap film back to the previous equilibrium state. Otherwise, τ > τ d , the Gibbselasticity emerges when a patch of a soap film is stretched slowly so that the interstitialsurfactants diffuse to the surfaces and a new thermodynamic equilibrium between Γ and c is reached. In current study, we measure σ with respect to h while Γ and c of the mainsurfactants remain in equilibrium, and therefore the Gibbs elasticity can be measured.Our firsthand inspection of σ with respect to ln h suggests that E G is proportional to h − / for a given c . From the observation, we make an ansatz that helps us to pick E G fromthe noisy data such that E G ( c , h ) (cid:39) E ( c ) h − / , (10)where E is the concentration dependent parameter. Integration of Eq. (9) using Eq. (10)gives that σ = σ ∞ + 23 E h − / . (11)Measurement of E G is possible by fitting the data using Eq. (11). We get E = 10 . for soapfilms made of 2% soap solution, E = 22 . for 1% soap films, and E = 246 [mN/m · µ m / ]for 0.5%. All cases, σ ∞ = 30 ± / m from the fitting. In the range of h where Eq. (10)is valid, our measurement of E G is displayed in Fig. 4.In Fig. 4, we compare our measurement with the values in the literature. Measurementby Prins et al. [25] reports that E G is 14 to 29 mN/m for 4 mM, and 1.1 to 0.45 mN/mfor 15 mM solution of sodium dodecyl sulfate (SDS)[26]. According to the manufacturer’schemical safety data sheet, the main ingredient of the soap that we used is also SDS, andthe concentration of SDS in 0.5% solution is estimated roughly ± mM. The comparisonshows that our measurement of E G is in rough agreement with the previous reports andsupports our measurements.The curves in Fig. 4 are the estimation of E G based on the Langmuir adsorption throughdirect differentiation of Eq. (8). For all c , our physical model underestimate the elasticity.The discrepancy between the measurement and the model shows the complex fluid natureof the soap films. The Langmuir adsorption is a simple and idealized model in which aconstant adsorption and desorption rate is assumed. However, in soap films, the interplaybetween the surface chemistry and the hydrodynamics complicates the overall dynamics,and a complete description of its physical property requires further thorough consideration.11 h (µm) -1 E G ( m N / m ) (0.5% solution) (1% solution) (2% solution) Prins et al. (4 mM)Prins et al. (15 mM) Figure 4. Gibbs elasticity is measured by extracting − dσ/d ln h from Fig. 3 with Eq. (10). Themeasured values are consistent with the reported values in literatures [25] but greater than theprediction of Langmuir adsorption model in Eq. (2). Finally, we remark that in the range that we can conveniently establish a flowing soapfilm, the Gibbs elasticity is smaller than the Marangoni elasticity. The Marangoni elasticityis reported to be 22 mN/m in soap films made of 1 to 4% soap solution in the thicknessrange between 4 and 11 µ m [18]. Even though a soap film requires both elasticities in orderto last for a long time, only those with sufficient Marangoni elasticity can be establishedat the first place. Also, two elasticities are complementary; the Gibbs elasticity diminishesas Γ → Γ ∞ , the Marangoni elasticity diminishes as Γ → . Considering that the fastdisturbance is commonly encountered in a usual lab environment, soap films with greaterMarangoni elasticity is expected to be observed more frequently. IV. SUMMARY
We presented a non-intrusive method to measure the surface tension of a flowing soapfilm setup. The method is applicable to standard soap film setups whose main componentis thin flexible wires serving as channel walls. When a soap film is formed between wires,the wires are bent toward each other by the action of the surface tension. We showed thatthe bending curvature is determined by the relative strength of the surface tension to the12ension of the wire.Using the presented protocol, the surface tension was measured by probing the bendingcurvature of soap films made from different conditions. Our measurements show that a soapfilm has a surface tension of 30 mN/m if its thickness is relatively thick or if it is made ofsoap solutions of higher concentrations. Otherwise, the surface tension deviates from theasymptotic value 30 mN/m and increases. Two distinct physical processes, thinning of soapfilm and dilution of soap solution, yield the same consequences that increases the surfacetension.We demonstrated a theoretical model using surfactant conservation and Langmuir ad-sorption isotherm. These two equations can be solved for an analytic solution, and twoparameters of the model determined by matching the model and experimental data. Theseparameters are in agreement with other independent measurements.Lastly, using our measurement of the surface tension with respect to the thickness of thefilm, we estimated the Gibbs elasticity. In our experimental range, the Gibbs elasticity isgreater when the film is made of dilute soap solution or when the film is thinner. In oursetup, the Marangoni elasticity is bigger than the Gibbs elasticity.
ACKNOWLEDGEMENT
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