Article pubs.acs.org/Langmuir
Flotation of Denser Liquid Drops on Lighter Liquids in Non-Neumann Condition: Role of Line Tension D. George, S. Damodara, R. Iqbal, and A. K. Sen* Indian Institute of Technology Madras, Chennai 600036, India S Supporting Information *
ABSTRACT: Flotation of a denser liquid drop on lighter liquid has been explained earlier via the Neumann triangle. We demonstrate the flotation of a denser liquid (water) drop on a lighter liquid in a pair that does not satisfy the Neumann triangle. We attribute this newly studied phenomenon to the role of line tension τ which prevents the water droplet from complete engulfment. A simple model is used to explain the underlying physics and to obtain critical line tension value for stable flotation. We establish line tension values for different liquids with water and show possible heterogeneous nucleation that contributes toward the variance of line tension values.
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INTRODUCTION Line tension in surface thermodynamics was first introduced by Gibbs in the 1870s1 by treating the contact line, where surface discontinuities meet, analogous to surface tension. However, despite the efforts of multiple research groups, unlike surface tension, both theoretical and experimental studies of line tension of various combinations of liquids remain incomplete and riddled with contradictions.2 Experimental values of line tensions ranging from 10−11 to 10−5 J/m, both positive and negative values, have been reported2 as opposed to the theoretical predictions which is of the order of 10−12 J/m. If we were to consider the higher range of line tension values in literature (∼10−5 J/m), a contact radius of ≤10−3 m may be sufficient for line tension effects to have a significant influence in various systems. It has been reported that the line tension can play a significant role in many processes including heterogeneous nucleation, wettability of patterned surfaces,3−5 droplet and solid flotation, droplet coalescence, and biological processes.6 Among these, many studies can be found on the flotation of liquid lenses or particles on liquid surfaces owing to its potential applications in ore separation. There are multiple methods that have been used to study the magnitude of line tension including techniques involving liquid lenses on liquid surfaces,7−11 solid particles or liquid marbles on liquid surfaces,12−15 sessile droplets on solid surfaces,16,17 nucleated droplets or bubbles on solid surfaces,18 droplet emulsions,19 and solid and hollow cones.20,21 Measurements which include solids face larger experimental hurdles, as they demand careful surface preparation.17 On the other hand, liquid surfaces inherently possess molecular smoothness. The most frequently used study in liquid−liquid−gas systems is based on the shape of a liquid lens formed when a lighter liquid is placed on another liquid and the interfacial tensions of the liquids with © XXXX American Chemical Society
each other and air satisfies the Neumann triangle. The difference between the expected macroscopic contact angle and the experimentally obtained value is attributed to the influence of line tension which either reduces or increases (depending on sign) the length of the three phase contact line.22 Feasibility of liquid lens formation between three fluids is generally being explained with the help of the Neumann condition, which says that, for all the possible combinations, interfacial tension at one interface should be less than the sum of the other two interfacial tension values in the liquid lens system (for example, γwa < γoa + γwo)). This criterion is represented in a more convenient way by many researchers by using spreading coefficient (S = γwa − (γoa + γwo)). Extending from Neumann’s criteria, for a set of liquid to form a stable lens, each of the possible spreading coefficients of the lens system has to be negative. For a set of liquid having negative spreading coefficient, flotation of liquid drop is possible, irrespective of its density values, given the size of the droplet is sufficiently small.23 Even though there exists experimental proof that a heavier liquid lens can float on top of a lighter liquid,24,25 the line tension measurements so far have been confined to experiments involving lighter liquid lens on top of heavy liquids. In these reported cases, the flotation of the heavier liquid drop on oil is achieved because the interfacial tension values satisfied the Neumann triangle. However, the studies dismissed the effect of line tension since the size of the contact line is too large so line tension has no impact. Received: July 26, 2016 Revised: September 15, 2016
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DOI: 10.1021/acs.langmuir.6b02771 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir In this work, we demonstrate the conspicuous effect of line tension by achieving the formation of a heavier liquid lens on a lighter liquid despite Neumann triangle not being satisfied. DI water is used as the denser liquid on top of lighter liquids (toluene, mesitylene and xylene). These sets of liquids are chosen since their interfacial tension values with DI water are not complying with the Neumann conditions. The surface tension value of water (γwa) is larger than the sum of the surface tension of lighter liquid (either toluene or mesitylene or xylene) and the interfacial tension between the lighter liquid and water (γoa + γwo). As a result of this violation of the Neumann construction, the water droplet is expected to be completely engulfed inside the lighter liquid and subsequently sink to the bottom as it is denser than the lighter liquid. However, it is observed that the water droplet stays suspended at the oil−air interface with a small radius of contact, which is only due to the manifestation of line tension as the interfacial tension values and the density of water set an unfavorable condition for flotation. Here, the surface tension equivalent of line tension approaches a magnitude which is comparable to the individual interfacial tension values. Thus, in order to consider the impact of line tension, the Neumann triangle is corrected and the liquid satisfies the Neumann quadrilateral.26,27 This flotation behavior of small water droplets on less dense liquids is leveraged to obtain the magnitude of line tension.
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-wo =
y2
∫y
1
-wa =
∫y
y2
⎛ dx ⎞ ⎜π(Po − Pw)x wo2 + 2πx woγwo 1 + wo ⎟dx dx ⎠ ⎝ π(Pa − Pw)x wa 2 + 2πx waγwa 1 +
1
dx wa dx dx
The extremum of the function in eq 1 can be found by taking variation, which results in the following vectorial relationship:28 τ⃗ ⎯→ γoa + ⎯→ γ⎯wo + ⎯→ γwa + = 0 r
(2)
This indicates that one needs to correct the Neumann triangle to a quadrilateral (Figure 1) by considering line tension in order to completely evaluate the formation of a lens at the interface. Resolving eq 2 along horizontal and vertical directions results in the following equations, τ γwo cos α + γwa cos β + = γoa cos ϵ rc (3)
γwo sin α − γwa sin β = 0
(4)
The vertical component of surface tension γoa will result in the following equation,
2πγoarc sin ϵ = Fm − Fb
(5)
where Fm is the weight of the droplet, Fb is the buoyancy force, and ϵ is the angle made by the fluid 1−fluid 2 interface with the horizontal. Equation 5 can be further expanded by rewriting the volume in terms of rc, α, and β as follows:
THEORY
The feasibility of lens formation for a set of liquids which does not satisfy Neumann triangle can be found by obtaining the extremum of the free energy - of the liquid lens system. Expression for free energy of a liquid lens system with given pressures inside fluid 1 (Pa), fluid 2 (Po), and fluid 3 (Pw), interfacial tension values γoa, γwo, and γwa at two phase interface surfaces, line tension value τ at the three phase interface of radius rc, and locus of the x coordinate of interface as
sin ϵ ≈
rc 2(f (α)(ρw − ρo )g + f (β)ρw g ) 2γoa
(6)
where f (α) =
cos3 α − 3cos α + 2 3sin 3 α
f (β) =
cos3 β − 3cos β + 2 3sin 3 β
and ρo and ρw are density of fluid 2 and 3, respectively. Hence for a small value of rc, the sin ϵ becomes sufficiently small (which is 0°) is possible for the values of τ* > 1 for all lighter liquids. However, for each lens system, there exists a critical line tension τ* value above which α = 0 = β and complete spreading occurs.
Figure 3. Variation of angles α and β with line tension τ̅ at a constant volume of droplet.
for all positive τ.̅ This implies that for a fixed volume of droplet, as long as the τ̅ is positive, it is theoretically possible for a water droplet to float on oil if the τ/rc value is sufficiently large. This is possible even if τ has only a small value given that the rc is sufficiently small to give high τ/rc. The values of the change in Gibbs energy of the system (ΔG) will further shed light on the stability of a hanging droplet. The difference in Gibbs energy of a system with a lens of water, having a surface area AA in contact with oil and area AB in contact with air, on the oil surface as compared to that of the system when the drop of the same volume with a radius (R) is present in air is obtained as follows” ΔG = (γowAA + γwaAB + 2πrcτ ) − 4πR2γwa − πrc 2γoa
(18)
where AA = 2πrc2(1 − cos α)/sin2 α and AB = 2πrc2(1 − cos β)/sin2 β. Here, the areas of the parts of the lens sharing its interface with oil and air are obtained using a spherical cap approximation. We verified that the sign of ΔG is negative for the line tension values for which the lens formation occurs and hence the system is thermodynamically stable. For example, for a line tension value of τ ∼ 10−7 N we estimated the value of ΔG ∼ − 10−9 Nm.
Figure 2. Variation of angles α and β with line tension τ* for mesitylene, xylene, and toluene at rc = 0. 1 mm. C
DOI: 10.1021/acs.langmuir.6b02771 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir
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images obtained were analyzed using ImageJ to obtain information regarding the contact angle of the submerged part of the droplet (α), the radius of the interface (rc), and the radius (R) of the drop. Dropsnake method31 was used to find the contact angle of the floating drop. The reflection of the drop was used to identify the contact points which were typically within 0.01 mm, which is the contact diameter as shown in Figure 5a. These values of rc from side view were validated
MATERIALS AND METHODS
Experiments were conducted by placing a water drop on a large reservoir of lighter liquids as the primary liquid. The primary liquids were selected such that these are less dense than water and the interfacial tension values do not satisfy the Neumann triangle with water. Olive oil, hexane, mesiylene, xylene, toluene, and cyclohexanol satisfied these criteria. Among these liquids, it is reported that olive oil contains linoleic acid, a water-soluble surfactant, that has the potential to reduce the effective surface tension of water.24 To prevent the impact of such natural surfactants which are constituents of several natural oils, we considered only pure hydrocarbons in our analysis. The liquids were procured from Sigma/Sigma-Aldrich (Bangalore, India), and the properties of the liquids used are presented in Table 1. The values of surface tension and interfacial tension were found using a Du Nuoy ring method (Sigma 701 force tensiometer, Biolin Scientific). The surface tension of water was measured to be 72.1 mN/ m. Water droplets of varying volumes were placed on a primary liquid filled in a square optical glass cuvette of dimensions 12.5 × 12.5 × 45 mm3 at 298 K. A conventional pipet assisted dispensing cannot be employed as the minimum possible volume with the pipet is limited. A liquid specific approach was followed in order to dispense smaller quantity of water drops into the primary fluid well. Colorless and additive free pipet tips were used in its pristine condition to transfer the water droplets into the medium. Use of a colored pipet tip or a tip containing other additives may affect the properties of water as the surfactant and other chemicals present in it may leach into the water.30 A small drop of water is obtained on the tip surface by dipping the tip inside a water droplet and rapidly retracting it. The water droplet on the polypropylene surface is pinned and this pinning can be removed by introducing a thin film of oil between water and polypropylene. As the tip is brought inside the primary liquid surface a spontaneous thin film is introduced between water and the polypropylene. This results in depinning of the water droplet. Eventually the droplet starts moving and finally gets transported to the primary liquid surface to avoid undesirable momentum as shown in Figure 4a (also in Video S1 in the Supporting Information).
Figure 5. (a) Representative figure showing measurement of contact radius rc. (b) (i) Side view of the floating droplet. (ii) Top view of the droplet. Interface is highlighted using a white line. with a top view of the interface (Figure 5b). The volume of the droplet was approximated by that of a sphere of the observed diameter. As mentioned earlier (see eq 3), it is assumed that gravity does not affect the angle between the oil-air interface and the horizontal (ϵ). The validity of the assumption that ϵ → 0° was checked by simulating the effect of gravity on the liquid lens system using Surface Evolver32 with modified interfacial tension value γoa
modified
(
= γoa −
τ rc
) and the
assumption was found to hold true. In the limiting case, if we consider the largest droplet that floats in our experiments, from eq 3, it is expected to have the maximum ϵ for a case where rc is small; that is, when τ* = 1 (Figure 2). However, from the simulations it is observed that the angle ϵ ≈ 0° (i.e., interface remains flat) even for the largest droplet floated during experiment at τ* = 1, as shown in Figure 6.
Figure 4. (a) (i) Tip containing water droplet is being brought toward the oil-air interface. (ii) The water droplet is being detached and is moving toward the equilibrium position. (iii) Water droplet at equilibrium. (b) Schematic of the experimental setup for measurement.
Figure 6. Sliced view of Surface Evolver simulation of water drop on mesitylene with (a) τ* = 1 and (b) τ* > 1.
Also, the validity of approximating the droplet volume to a sphere of the observed diameter was analyzed by finding the maximum error associated with it. For a fixed angle α, the maximum percentage error caused by this approximation can be found by comparing the volume of a complete sphere of radius R and that of the spherical cap of angle α and radius R, which can be expressed as (cos3(180 − α) − 3cos(180 − α) + 2)/4 × 100. This shows that for the lowest angle α observed (155°), the error caused by this assumption is
