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Dec 4, 2014 - In the G phase, a mechanical stability theory for droplets at liquid surfaces indicates that droplet coalesce is primarily driven by sur...
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Liquid Droplet Coalescence and Fragmentation at the Aqueous−Air Surface Govind Paneru,† Bruce M. Law,*,† Koki Ibi,‡ Baku Ushijima,‡ Bret N. Flanders,† Makoto Aratono,‡ and Hiroki Matsubara‡ †

Physics Department, Kansas State University, 116 Cardwell Hall, Manhattan, Kansas 66506-2601, United States Department of Chemistry, Faculty of Sciences, Kyushu University, Higashi-ku, Hakozaki 6-10-1, Fukuoka 812-8581, Japan

‡

S Supporting Information *

ABSTRACT: For hexadecane oil droplets at an aqueous−air surface, the surface film in coexistence with the droplets exhibits two-dimensional gaseous (G), liquid (L), or solid (S) behavior depending upon the temperature and concentration of the cationic surfactant dodecyltrimethylammonium bromide. In the G (L) phase, oil droplets are observed to coalesce (fragment) as a function of time. In the coalescence region, droplets coalesce on all length scales, and the final state is a single oil droplet at the aqueous−air surface. The fragmentation regime is complex. Large oil droplets spread as oil films; hole nucleation breaks up this film into much smaller fluctuating and fragmenting or metastable droplets. Metastable droplets are small contact angle spherical caps and do not fluctuate in time; however, they are unstable over long time periods and eventually sink into the bulk water phase. Buoyancy forces provide a counterbalancing force where the net result is that small oil droplets (radius r < 80 μm) are mostly submerged in the bulk aqueous medium with only a small fraction protruding above the liquid surface. In the G phase, a mechanical stability theory for droplets at liquid surfaces indicates that droplet coalesce is primarily driven by surface tension effects. This theory, which only considers spherical cap shaped surface droplets, qualitatively suggests that in the L phase the sinking of metastable surface droplets into the bulk aqueous medium is driven by a negative line tension and a very small spreading coefficient.

1. INTRODUCTION The line tension or energy per unit length associated with a three-phase contact line is a subject of continuing interest. For small liquid droplets situated on a liquid or solid surface or for small solid spherical particles at a liquid surface, the line tension may cause the contact angle θ of the droplet or particle to deviate significantly from its macroscopic value θ∞ (characteristic of large droplets or particles at a surface).1−3 The line tension therefore plays a prominent role in numerous surface related processes including droplet nucleation,4 particle attachment at liquid surfaces,5−7 the dynamics of contact line spreading,8,9 and surface droplet10 and foam stability.11,12 Despite the importance that the line tension plays in surface related phenomena, both the magnitude and sign of the line tension are controversial. Theory predicts the line tension magnitude to be in the range ∼1−100 pN13−18 while computer simulations have found magnitudes ∼1−10 pN.19−22 Although a number of experiments find line tension magnitudes in agreement with theory and computer simulations, numerous other experiments have found line tension magnitudes which are orders of magnitude larger.2 Recently McBride and Law7 suggested that an atomic scale “point contact” contribution to the line tension, which will be very sensitive to atomic scale details of the three-phase contact line, may give rise to © 2014 American Chemical Society

anomalously large line tension magnitudes, especially if the droplet or particle surface composition is dissimilar to the surface upon which it resides. In this publication we are interested in yet another aspect of the line tension which has proven controversial, namely, the sign of the line tension and how this impacts droplet stability and behavior.23−27 Depending upon the surface potential in the vicinity of the three-phase contact line, the line tension can be either positive or negative,28 and in fact, both signs have been found experimentally and via computer simulations.2,15,20,29,30 There has been significant theoretical debate as to whether droplets, which possess a negative line tension, can be stable. Li and Steigmann23 suggested that droplets which possess a negative line tension are unstable, and the contact line must then become more corrugated. Numerous theoretical papers24−27 have subsequently examined how the sign of the line tension influences the stability of droplets. In recent experimental work, Matsubara, Takiue, Aratono, and coworkers31 (MTA) observed that surface droplets coalesce (fragment) when the line tension is positive (negative). This Received: June 30, 2014 Revised: November 30, 2014 Published: December 4, 2014 132

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finding indicates that the interrelationship between droplet stability and the sign of the line tension requires a closer examination. MTA (see ref 32 and references therein) examined the behavior of n-alkane oil droplets at the water−air surface in the presence of a cationic surfactant for a variety of n-alkane and surfactant chain lengths. Such systems are of particular interest to the oil industry for the purposes of tertiary oil recovery.33 Their best documented oil + surfactant system, which provides a generic example of the typical surface behavior, is a nhexadecane (C16) oil droplet at the air−water surface in the presence of the cationic surfactant dodecyltrimethylammonium bromide (DTAB). The oil droplet coexists with a twodimensional (2-D) surface film at the aqueous−air surface. This 2-D surface film exhibits a number of different surface behaviors as a function of surfactant concentration m and temperature T (Figure 1).34 At low surfactant concentration (m

and fragmentation in respectively the gaseous (G) and liquid (L) phase and (ii) provide a theoretical description that explains some of the observed behavior. This publication is set out as follows. Section 2 provides a theoretical description of spherical cap surface droplet stability and its interrelationship to the sign of the line tension. Experimental methods are provided in section 3. In section 4 a description of droplet coalescence as well as both thin film and droplet fragmentation in respectively the G and L phases is provided. The theory in section 2 provides a basis for understanding some of the experimental observations in the G and L phases. A summary of our droplet coalescence and fragmentation observations is provided in section 5 together with its interconnection to the theory in section 2.

2. THEORY OF SPHERICAL CAP SURFACE DROPLETS In this section we consider the conditions under which spherical cap surface droplets are stable or unstable. These considerations will provide theoretical insights into a number of experimental observations described in section 4. In order to simplify our calculations, we assume that the liquid−air surface is flat (Figure 2). This situation is

Figure 2. Schematic diagram of an oil droplet at the aqueous−air surface. The oil droplet has a lateral radius r, height h, contact angle θ, and oil−water radius of curvature R and area A.

approximately correct for oil droplets at the aqueous−air surface.35 The oil droplet has a contact angle θ, lateral radius r, thickness h, radius of curvature R, and oil−water surface area A. The three surface tensions are denoted γAW, γAO, and γOW, where superscript A = air, W = water, and O = oil. The threephase contact line, of circumference 2πr, has a line tension τ associated with it. The energy of the droplet, compared with a “bare” aqueous− air surface (e.g., the droplet is submerged below the liquid surface) is therefore given by

Figure 1. Surface phase diagram for the surface film which is in coexistence with a hexadecane oil droplet at the aqueous−air surface for the tertiary system hexdecane−water−DTAB. m is the concentration of the surfactant DTAB. In the various regions the surface film surrounding the oil droplet exhibits gaseous (G), liquid (L), or solid (S) behavior.

E = γ OWA + γ AOπr 2 + τ 2πr − γ AWπr 2

(1)

where, for simplicity, gravitational contributions have been neglected. A schematic of the droplet is shown in Figure 2. The oil droplet volume V and oil−water area A are given by37

≤ 0.75 mmol/kg) the film is a 2-D gas (G). At higher surfactant concentrations (m ≥ 0.75 mmol/kg) and low temperatures (T < 17.5 °C) the film is a 2-D bilayer solid (S) which melts to a 2D liquid (L) at high temperatures (T > 17.5 °C). The differing surface phases (i.e. S, L, or G) of the 2-D surface film influence the surface energies,35 line tension,36 and contact angle35 of the oil droplet. In all surface phases, the typical magnitude of the line tension is |τ| ∼ 10−50 pN. In the G and S phases, where the line tension is positive,36 oil droplets are observed to coalesce.32 In the L phase, where the line tension is negative, oil droplets are observed to fragment. The earlier observations of droplet coalescence and fragmentation31 provide only a cursory description of these phenomena. The purpose of this publication is twofold: (i) provide a fuller experimental description of droplet coalescence

V = πh2(3R − h)/3 = πh(h2 + 3r 2)/6

(2)

and A = 2πRh = π (h2 + r 2)

(3)

where R, r, and h are defined in Figure 2,and we have used the geometrical relationship R = (h2 + r2)/2h. Hence, eq 1 can be rewritten as E = −Seqπr 2 + γ OWπh2 + τ 2πr

(4)

where the equilibrium spreading coefficient is given by Seq = γ AW − (γ AO + γ OW ) 133

(5)

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and two droplets, each of energy E2, volume V/2, and radius r2. According to eq 4, therefore

This oil droplet at the aqueous−air surface is in mechanical equilibrium; therefore, dE/dr = 0 where for the short time scales appropriate for mechanical equilibrium the droplet volume V is constant, namely, dV = 0. One can then show that dE 4πh2r = 0 = −Seq 2πr − γ OW 2 + 2πτ dr h + r2

ΔE = −Seqπ (r12 − 2r2 2) + γ OWπ (h12 − 2h2 2) + τ 2π (r1 − 2r2)

(6)

(10)

Hence, two small droplets should coalesce into a larger droplet (Figure 3) provided that

By use of the geometrical relationships r/R = sin θ and h = r(1−cos θ)/sin θ as well as Young’s equation cos θ∞ = (γAW − γAO)/γOW, where θ∞ is the macroscopic contact angle (r → ∞), one can show that eq 6 reduces to the well-known modified Young’s equation1 τ cos θ = cos θ∞ − OW γ r (7)

d2E >0 dr 2

and

ΔE < 0

(11)

This equation describes how the droplet contact angle θ changes relative to the macroscopic contact angle θ∞ due to the presence of the line tension τ associated with the three-phase oil−water−air contact line. For oil droplets at the aqueous−air surface the contact angle θ is small35 or, equivalently, h ≪ r; this fact is readily deduced from the appearance of interference fringes within droplets. Thus, if this condition is combined with eq 6, then a useful expression for the droplet thickness is h ≈ r (τ − Seqr )/2γ OWr

(8)

An aspect of mechanical equilibrium which is often neglected, for liquid droplets at surfaces, is that the stability of these droplets is determined by the sign of the second energy derivative, namely ⎡ 2γ OWh2(5r 4 − h4) ⎤ d2E 2 = − + π S ⎢ ⎥ eq dr 2 (h2 − r 2)3 ⎣ ⎦ 2π [5τ − 6Seqr ] ≈ r

Figure 3. Time sequence of oil droplet coalescence at the aqueous−air surface in the G phase (DTAB concentration m = 0.400 mmol/kg) at room temperature. Scale bar = 100 μm. (9)

3. EXPERIMENTAL METHODS

In eq 9 the second (approximate) equality originates by using h ≪ r together with eq 8. The sign of d2E/dr2 (eq 9) determines the mechanical stability of a spherical cap shaped surface droplet. If d2E/dr2 > 0 (< 0) the droplet is mechanically stable (mechanically unstable). A number of limitations must be kept in mind when applying the current analysis in this section to liquid droplets at air− liquid surfaces. Our results are exact for a small spherical cap shaped droplet (less than the capillary length) on a flat surface (e.g., a liquid droplet on a solid surface) where, we believe, eq 9 is new. For an oil droplet at an aqueous−air surface, the oil−air surface of the droplet will exhibit some curvature and, in this case, the droplet shape is frequently determined numerically.38,39 In the L phase, as the air−liquid surface is approximately flat,35 the results in this section are expected to qualitatively describe this situation as well. A second caveat concerning application of eq 9 to spherical droplets in the L phase is that this approach underestimates the destabilizing effect of a negative line tension on the contact line, as it ignores the possibility that such a destabilization could lead to nonspherical droplet shapes.27 For such situations, one should therefore consider the much more complex second variation of the free energy27 when interpreting experimental data. For simplicity, we neglect these theoretical complexities here. To understand droplet coalescence, a second quantity must be considered, namely, the energy difference ΔE = E1 − 2E2 between a large droplet of energy E1, volume V, and radius r1

Materials and Material Parameters. The DTAB surfactant and hexadecane oil were purified following the procedure outlined earlier.36 Stock solutions were prepared in the G and L phases with respectively 0.400 and 0.898 mmol/kg of DTAB in 18 MΩ cm Milli-Q water. Typical values for relevant experimental parameters, measured in the G and L phases at T = 20 °C, are listed in Table 1. These values will be required when comparing droplet coalescence and fragmentation experiments with the theoretical considerations in section 2.

Table 1. Typical Experimental Parameters at 20 °C in the G and L Phases

a

phase

γOW (mN/m)

−Seq (mN/m)

τ (pN)

G L

50a 30a

4a 0 and Seq < 0 (Table 1), therefore d2E/dr2 > 0 (eq 9) and oil droplets are always mechanically stable. In this regime, from the values in Table 1, for all observable droplets (i.e., r > 1 μm) the spreading coefficient contribution dominates the line tension contribution as |Seqr| ≫ |τ|. Hence, from eq 8 h ≈ rΨ where Ψ = [−Seq/(2γOW)]1/2. Therefore, from eq 2 V ≈ Cr13 where the constant C = πΨ(3 + Ψ2)/6. As a consequence r2 =

Table 2. Optical Interference Fringe Interpretation film thickness, t

destructive interference

constructive interference

0, λ/2noil, λ/noil, ...

λ/4noil, 3λ/4noil, ...

is useful for interpreting the corresponding thickness t for both constructive and destructive interference fringes where the refractive index noil = 1.434. Liquid “petals” (or hills), spaced approximately 0.75 mm apart, are observed around the outer rim of this droplet (using the 1.25× objective). The fine petal structure, for closely spaced fringes, is only observable using the 10× objective (Figure 4b). An intensity plot (Figure 5) through one petal (dashed line in Figure 4a) provides an approximate indication of the petal and droplet structure. (Note: the petal is somewhat nonuniformly illuminated in Figure 4a; nevertheless, the interference fringe positioning can still be used to deduce petal structure.) The oil droplet in Figure 4a consists of a relatively uniform ∼80 nm thick oil film toward the center of the droplet which is surround by oil petals possessing a height of ∼320 nm (Figure 5). The oil film may thin in the vicinity of 135

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Figure 7. Linear growth of the nucleated hole diameter, for three different holes, where the growth velocity is ∼117 μm/s. Figure 5. Intensity plot through the petal structure in Figure 4a (dashed line). The constructive and destructive interference fringes indicate that the oil petal possesses an approximate height ∼320 nm while the central oil film possesses an approximate thickness of ∼80 nm.

unstable droplets will become clearer later in this section. The left sequence of six pictograms in Figure 8 documents the time evolution of a large droplet into two smaller droplets. This large droplet at time t = 0 s is composed of three petals separated by deep valleys. Holes nucleate in the valleys (t = 0.5 s) where these holes eventually coalesce into a single hole by t = 1 s. The droplet then consists of a ring of liquid (t = 1.5 s) which fragments into two droplets (t = 2.0 s). In the final pictogram, at t = 2.5 s, the left droplet with radius less than 100 μm is metastable and possesses a spherical cap shape, as indicated by the interference fringes. The interference fringes for the larger right droplet indicate that this droplet is nonspherical. It is shaped somewhat like a “rubber raft” with a thinner central base, which is surrounded by a thicker rim around its edge. Rubber raft shaped droplets are unstable. They spread, the central base thins, and holes nucleate in the thin central base. It is easy to distinguish between metastable and unstable droplets. Metastable droplets possess a spherical cap shape (Figure 8, right graph, lower inset) and remain unchanged over time spans of many minutes. Unstable droplets will either be petaled around the rim or possess a rubber raft shape; unstable droplets exhibit shape fluctuations or shape changes over time periods of a few seconds. The right graph in Figure 8 documents the average radii r ̅ of metastable (open circles) and unstable (crossed circles) droplets within the field of view of the 1.25× objective and how this droplet distribution evolves with time. From this graph, one readily deduces that over a ∼40 min time span large unstable droplets fragment into smaller metastable and unstable droplets. Typically, a droplet with a radius smaller (larger) than ∼100 μm is metastable (unstable). Actually, droplet radius does not completely demarcate unstable droplets from metastable droplets. Certainly it is true that large droplets, with radii larger than ∼100 μm, are always unstable, nonspherical in shape, and fluctuate or spread in time. Smaller droplets (r ̅ < 100 μm) are quite frequently metastable, possess a spherical cap shape, and nonfluctuating in time. However, not all smaller droplets are metastable; some smaller droplets are found to be unstable. This small droplet regime, with r ̅ < 100 μm, is discussed further at the end of this section. The line tension τ and spreading coefficient Seq undoubtedly play an important role in determining droplet and thin film behavior in the L phase. In prior work36 Takata et al. deduced the line tension by examining the variation in contact angle with metastable droplet size. For a DTAB concentration of m = 0.898 mmol/kg the line tension τ ∼ −55 pN. The value for Seq, given in Table 1 for the L phase, was obtained from individual surface tension measurements35 and therefore only provides an

an oil petal and nucleate holes at the base of the oil petal (Figure 4a). A movie in the Supporting Information documents how the droplet, shown in Figure 4a, evolved in time. The actual run time for this movie is approximately 5 min. It is observed in this movie that with increasing time adjacent petals may coalesce to form larger petals, the oil film spreads and thins, and holes may nucleate in this oil film (Figure 6). The

Figure 6. Oil film showing nucleated holes decorated with petals around the perimeter.

diameter of these nucleated holes increases linearly with time (Figure 7), in agreement with the expectations from dewetting,40 where the velocity for hole growth is ∼117 μm/ s. During this hole nucleation and growth process, material collects as petals around the periphery of each hole (Figure 6). Eventually, the nucleation of additional holes and the overlapping of expanding holes lead to the complete destruction of the oil film into much, much smaller oil droplets. The time evolution of these small oil droplets is documented in Figure 8. Depending upon the oil droplet radius, these droplets can be either metastable or unstable. The meaning of metastable, unstable, mechanically stable, and mechanically 136

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Figure 8. Left six pictograms: time evolution of an unstable droplet. Right graph: time evolution of a distribution of unstable (crossed circles, upper inset) and metastable (circles, lower inset) droplets. Scale bar = 100 μm.

approximate estimate for Seq. A more accurate estimate for Seq can be obtained by applying eq 6 to metastable droplets using the line tension τ and surface tension γOW values given in Table 1. Interference fringes within a droplet readily allow the droplet thickness h to be determined. Hence, from droplet thickness h and radii r data the spreading coefficient Seq ∼ −(5.6 ± 2.4) × 10−7 N/m is obtained using eq 6 at a DTAB concentration of m = 0.898 mmol/kg in the L phase. We now discuss the late time behavior of oil droplets in the L phase. Two hours after the commencement of the experiment, it is found that no oil droplets can be observed at the aqueous− air surface. Where have the oil droplets gone? There can only be a few potential explanations for the disappearance of oil droplets from the aqueous−air surface: (i) the oil droplets evaporate, (ii) the oil droplets fragment to a size below the diffraction limit of the microscope (i.e., r ≪ 1 μm), or (iii) oil droplets are mechanically unstable and sink into the bulk water phase. Evaporation is not thought to play a role in the disappearance of oil droplets because a macroscopic hexadecane droplet can be suspended at a water surface for many days without any observable decrease in volume (i.e., the evaporation rate of hexadecane is negligible). Experimental evidence indicates that oil droplets disappear from the aqueous−air surface because they sink below the aqueous−air surface into the bulk water medium. There are two differing modes for this disappearance. In the first mode, fluctuating droplets fragment into smaller and smaller droplets, and during this fragmentation process, some of the smallest droplets sink below the aqueous−air surface into the water phase. This droplet disappearance mode is documented in the movie “Disappearing fluctuating droplets” in the Supporting Information. This movie was acquired using the 1.25× objective and has an actual run time of ∼23 min. The second mode applies to quiescent droplets (i.e., nonfluctuating droplets) as well as metastable droplets which slowly sink below the aqueous−air surface. This disappearance mode is documented in the movie “Disappearing single droplet” in the Supporting Information, which was collected using the 10× objective and has an actual run time of 4 min. This movie follows the late time behavior of a single quiescent (i.e., nonfluctuating) droplet. Initially this droplet is unstable and possesses a nonspherical cap shape; however, this droplet is relatively quiescent and does not fluctuate rapidly in time. In the movie the droplet first rises and then sinks down at the surface, as indicated by the changing surface radius from an initial radius of ∼50 μm to a maximum radius of ∼90 μm. During the sinking phase this unstable droplet changes to a metastable droplet (as indicated by the appearance of spherical interference fringes) at a radius of ∼25 μm. This metastable droplet continues to sink below the

aqueous−air surface until; by the end of the movie, the droplet barely protrudes above the liquid surface. The disappearance of metastable droplets is qualitatively explained by eq 9. According to this equation, a droplet at the surface is mechanically stable (relative to a droplet of the same volume in the bulk liquid phase) if dE2/dr2 > 0 and mechanically unstable if dE2/dr2 < 0. Thus, dE2/dr2 = 0 defines the limit of mechanical stability with critical radius rcrit =

5τ ∼ 80 ± 30 μm 6Seq

(12)

above which spherical droplets are mechanically stable and below which spherical droplets are mechanically unstable. Thus, the disappearance of metastable droplets for r < 80 μm from the aqueous−air surface is explained by eq 12. And, in fact, very small droplets are expected to sink rapidly into the bulk liquid phase, as d2E/dr2 → −∞ (eq 9) in the limit r → 0. This tendency of metastable oil droplets to sink, due to surface tension and line tension effects (eq 12), is counteracted by buoyancy forces which cause oil droplets to rise (as hexadecane possesses a lower density than water where ρ(hexadecane) = 0.77 g/cm3). At equilibrium, these surface forces and buoyancy forces will balance, and most of the oil droplet (> 80%) will be submerged below the liquid surface with only a small fraction protruding above. These considerations are in reasonable agreement with the observations in the “Disappearing single droplet” movie in the Supporting Information. For droplet radii r > rcrit, spherical cap droplets are mechanically stable. However, although spherical cap droplets are mechanically stable in this radius regime, this does not necessarily imply that spherical cap droplets possess the lowest energy state. The experiments indicate that petaled and raft shaped droplets must possess a lower energy than spherical cap droplets of the same volume for r > rcrit.

5. SUMMARY In this publication we elucidate the coalescence and fragmentation behavior of hexadecane oil droplets at the air− liquid surface of DTAB−water mixtures in the gaseous (G) and liquid (L) phases, respectively. In the G phase, oil droplets are observed to coalesce on all length scales (Figure 3 and “Coalescence” movie in the Supporting Information) and the final state is a single large oil droplet at the aqueous−air surface. As the line tension τ > 0 and the spreading coefficient Seq < 0 in the G phase, all droplets are mechanically stable (eq 9), independent of droplet size. The coalescence behavior is primarily driven by a lowering of the droplet surface energy with increasing droplet size (eq 11), 137

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where the surface energy contribution (Seq) dominates the line tension contribution as the droplet size increases. In the L phase, large oil droplets are observed to form oil films decorated with petals around their rim (Figures 4 and 5). If the oil film is sufficiently thin in the vicinity of an oil petal, then holes may nucleate near the base of the oil petal (Figure 4a). With increasing time the oil film spreads, thins and holes nucleate in this oil film where petals form at the perimeters of all holes (Figure 6). The nucleated hole diameter grows linearly with time at a growth velocity of ∼117 μm/s (Figure 7). Eventually, adjacent nucleated holes grow to such a size that they overlap and lead to the complete fragmentation of the oil film into much smaller droplets. If the average radius of the droplet r ̅ ≥ 100 μm, the droplet is unstable and it fluctuates rapidly in time and eventually fragments into smaller droplets (Figure 8). Smaller droplets (r ̅ ≤ 100 μm) can be either unstable or metastable where metastable droplets possess a spherical cap shape, as indicated by their circular interference fringes. The “Fragmentation” movie in the Supporting Information depicts the spreading of a thin oil film and the nucleation of holes in this film where this film eventually completely fragments into much smaller unstable and metastable droplets. At late times, ∼2 h after the commencement of the experiment, oil droplets cannot be observed at the liquid surface. Surface tension and line tension forces cause small droplets (r < rcrit ∼ 80 μm (eq 12)) to sink into the water phase; however, buoyancy forces provide a counterbalancing force which causes these droplets to partially float. Thus, in the L phase, the final state is where all oil droplets are mostly submerged below the liquid surface with only a small fraction protrudes above (much like an iceberg). This final state is achieved either by droplets fluctuating and fragmenting and the smallest droplets sinking into the surface or by the slow sinking of quiescent droplets. These two droplet disappearance modes are documented in two movies in the Supporting Information. In a numerical study van Giessen, Bukman, and Widom24 examined the influence of a negative line tension on the behavior of large contact angle droplets (θ∞ = 120°). They found that the three-phase contact line was always stable, and they failed to find any fluctuations which would lower the free energy of this three-phase contact line. The current work indicates that the three-phase contact line becomes unstable for very small spreading coefficient (Seq ∼ −5 × 10−7 N/m) or, correspondingly, for very small contact angle (θ∞ ∼ 1° as cos θ∞ = 1 + Seq/γOW) in the presence of a negative line tension. The theoretical analysis in section 2 examines the mechanical stability of spherical cap shaped droplets at surfaces and does not consider nonspherical droplets; therefore, this analysis only provides a qualitative explanation for the sinking of droplets in the L phase. A more complete analysis should employ the second variational methods developed by Guzzardi, Rosso, and Virga27 for sessile droplets at surfaces where a negative line tension can destabilize the three-phase contact line and give rise to nonspherical surface droplets.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (B.M.L.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS B.M.L. thanks collaborators at Kyushu University for their kind hospitality. The authors thank Philip Hotz for assistance with preliminary experimental measurements.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Four movies entitled “Coalescence”, “Fragmentation”, “Disappearing fluctuating droplet”, and “Disappearing single droplet”. This material is available free of charge via the Internet at http://pubs.acs.org. 138

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dx.doi.org/10.1021/la502163e | Langmuir 2015, 31, 132−139