Coexistence and sudden entrapment between two dissimilar-miscible

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Coexistence and sudden entrapment between two dissimilar-miscible oil lenses Wei Sun, Qingyuan Ren, Zelin Wang, and Fuqian Yang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03724 • Publication Date (Web): 07 Jan 2019 Downloaded from http://pubs.acs.org on January 12, 2019

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Coexistence and sudden entrapment between two dissimilar-miscible oil lenses

Wei Sun a *, Qingyuan Ren a, Zelin Wang a, and Fuqian Yang b * a

College of Chemistry, Chemical Engineering and Environmental Engineering,

Liaoning Shihua University, No. 1 West Dandong Road, Fushun, Liaoning 113001, China [email protected] b

Materials Program, Department of Chemical and Materials Engineering

University of Kentucky, 177 F. Paul Anderson Tower, Lexington, KY 40506, United States [email protected]

Abstract The property of substrate is one of the important factors determining the interaction between two lenses (droplets). There likely exists different interaction between two dissimilar oil lenses (droplets) floating on the surface of a liquid phase from the interaction between two dissimilar oil droplets on rigid substrate, e.g. coalescence or coexistence. The interaction between two dissimilar oil lenses (droplets) is dependent on the intrinsic properties of both oil lenses (droplets) and external environmental factors. In this work, we investigate the contact interaction between two dissimilar-miscible oil lenses (toluene and silicone oil) on the surface of deionized water (DI water). The morphological evolution of two dissimilar-miscible oil lenses during the interaction under different experimental conditions is recorded and analyzed. The effects of the volume ratio of two dissimilar-miscible oil lenses, temperature of DI water, and viscosity of silicone oil on characteristic parameters are systematically studied. A sudden 1

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“entrapment” of a toluene lens into a silicone-oil lens occurs after a period of the “mass exchange” (coexistence) between these two oil lenses. Several characteristic parameters, including the duration of the “mass exchange”, critical sizes of the toluene lens at the onset of the entrapment and after the entrapment are found to be dependent on experimental conditions.

Keywords: Dissimilar oil lenses; water; entrapment; mass exchange.

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Introduction Understanding the interaction between liquid droplets is of scientific interest and practical importance in many applications, such as inkjet printing 1, microfluidics 2, and biomedical engineering 3. There have been extensive studies on the interactions of identical droplets 4 on solid (rigid) substrate for decades. Currently, there is of great interest in the study of the 5

interaction between dissimilar, miscible droplets phenomena, including coalescence chasing

6-11

due to the observation of interesting

, delayed coalescence

12, 13

, non-coalescence

14, 15

,

16, 17

, repelling 18, and self-propelling 19. These phenomena have stimulated research

to elucidate the mechanisms controlling the interaction between dissimilar, miscible droplets from different approaches of experiment 14, simulation 6, 18 and modeling 20, and dramatically broadened the practical applications of liquid droplets. For example, controllable manipulation of droplets have been achieved, and several simple devices in microfluidics have been developed 16, 21. It is known that a liquid droplet forms a circular lens on the surface of another immiscible liquid phase under certain conditions 22. The studies on the behavior of single liquid lens on the surface of another liquid phase have been focused on the geometry at equilibrium wetting/dewetting kinetics

24

, line tension

25

, wetting transition

droplets and surface aging 27. Torza and Mason

28

26

23

,

, evaporation of volatile

suggested the use of spreading coefficient

to predict different interactive modes for two immiscible liquid droplets on another immiscible liquid phase. They did not examine the effects of other important parameters, such as droplet volume, volume ratio, and viscosity. One important liquid-liquid system is “compound droplets”

29

for the important 3


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applications in many industries, such as biomedical, optical, petrochemical, etc.

30, 31

In the

“compound droplets” system, one of the droplets is “seated” on or surrounded by another immiscible droplet of similar volume

32

rather than on a bulk liquid phase. Most of the

researches on the “compound droplets” are focused on two-droplet systems. Iqbal et al.

33

extended the study on the “compound droplets” to a three-droplet system and observed the coalescent behavior of two identical water droplets floating on a immiscible oil droplet. They did not report the coalescence evolution of the droplets. The coalescence of liquid droplets, i.e. the merging of two or more liquid droplets, has been widely studied for two identical liquid droplets on solid substrate, and is of practical importance to the self-cleaning, microfluidics, and lab-on-chip devices

19

. The coalescent

behavior of liquid droplets are also investigated on liquid 34 or liquid-modified 35 surface with the study being generally limited to “two-identical-droplet” systems. Recently, we investigated the interaction of two identical oil lenses floating on the surface of deionized water (DI water) with the focus on the temporal evolution of the lenses during coalescence, and observed the presence of coalescent and non-coalescent states 36. In this work, we study the coalescent behavior of two dissimilar-miscible oil lenses (silicone oil and toluene) floating on the surface of DI water in contrast to traditional studies focusing on identical liquid droplets or on rigid substrate and the immiscible liquids used in the work of Torza and Mason

28

. The coalescence of the dissimilar-miscible liquid lenses

exhibits a more complex two-step phenomenon of sudden “entrapment” after a period of coexistence, which is significantly different from the direct coalescence or non-coalescence observed in our previous work

36

and has not been reported in literature. The coalescent 4


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process observed in this work can be referred to as a “swollen” or “entrapment” process rather than traditional fusion. The effects of viscosity, initial volume, volume ratio of two oil lenses, and sub-phase temperature on the behavior of the entrapment are studied. Dimensional analysis For the interaction between a toluene lens and a silicone oil lens floating on the surface of DI water, there are three characteristic parameters of “entrapment time”, ten, representing the time interval from the onset of the contact of two lenses to the onset of entrapment, “entrapment diameter”, D1, representing the contact diameter of the toluene lens at the onset of entrapment, and the ratio of the diameter of toluene lens after 10 s of the entrapment, D2, to the entrapment diameter, D1. Using dimensional analysis, we analyze the important factors that determine the “entrapment” behavior” of two dissimilar-miscible oil lenses. It is known that the coalescence of two dissimilar-miscible lenses (toluene and silicone oil) is dependent on the sizes, geometries, surface tensions, viscosities and other physical/chemical properties. The “entrapment time” of ten as a function of the independent variables can be expressed in as:

ten = ft (Vtol , Vs , ρ a , ρ w , ρtol , ρ s , γ aw , γ as , γ atol , γ stol ,η w ,ηtol ,η s , Dtol , Ds , Rtol )

(1)

where V is the initial volume of the oil lens in m3, ρ is the density of the material in kg/m3,

γ is interface tension in N/m, D is diffusion coefficient in m2/s, η is viscosity in Pa·s, and R is the evaporation rate of toluene in kg·m-2·s-1. The subscripts a, tol, s, and w denote air, toluene, silicone oil, and water, respectively. The aw, as, atol, and stol represent the interfaces of air/water, air/silicone oil, air/toluene, and silicone oil/toluene, respectively. Using the Π theorem yields

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= ten (

Vtol ρ a

γ aw

)1/2 Π t (

Vs ρ w ρtol ρ s γ a s γ atol γ stol , , , , , , ,ηw ,ηtol ,ηs , D tol , D s , Rtol ) Vtol ρ a ρ a ρ a γ a w γ a w γ a w

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(2)

with Vtd, ρa, and γaw as the fundamental quantities. The other parameters are

(ηw ,ηtol ,ηs ) = Vtol −1/6 ρ a −1/2γ a w −1/2 (η w ,ηtol ,η s )

(3)

( D tol , D s ) = Vtol −1/6γ a w −1/2 ρ a1/2 ( Dtol , Ds )

(4)

Rtol = Vtol −1/2 ρ a −1/2γ a w −1/2 Rtol

(5)

For a system with the phases of air (a), DI water (w) and toluene (tol), the parameters of

ρa, ρw, ρtol, γaw , γtol and Dtol are constants. We can simplify Eq. (2) as = ten (

Vtol ρ a

γ aw

)1/2 Π t (

Vs ρ s γ a s γ stol , , , ,ηw ,ηtol ,ηs , D s , Rtol ) Vtol ρ a γ a w γ a w

(6)

Similarly, we have the parameters of D1 and D2/D1 as

D1 Vtol1/3Π D1 ( =

Vs ρ s γ a s γ stol , , , ,ηw ,ηtol ,ηs , D s , Rtol ) Vtol ρ a γ a w γ a w

V ρ γ γ D1 = Π D1 / D2 ( s , s , a s , stol ,ηw ,ηtol ,ηs , D s , Rtol ) D2 Vtol ρ a γ a w γ a w

(7)

(8)

According to Eqs. (6)-(8), the coalescence between a toluene lens and a silicone-oil lens on the surface of DI water is mainly dependent on initial volumes, volume ratio, density, surface/interface tensions, viscosity, diffusion coefficient and evaporation rate. It needs to be pointed out that the parameters of surface/interface tensions, viscosity, diffusion coefficient and evaporation rate are temperature-dependent. Experimental details Non-volatile silicone oil and volatile toluene were used in this work to produce oil droplets/lenses on the surface of DI water. The silicone oil is highly soluble in toluene, i.e. silicone oil and toluene are miscible, while both silicone oil and toluene are practically 6


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insoluble in water. The dynamic viscosities of the silicone oils are 48, 484, and 970 mPa·s at 298 K, corresponding to kinematic viscosities of 50, 500, and 1000 cs, respectively. Table S1 in the supporting information lists the dynamic viscosities and surface tensions of the silicone oils and toluene. The interaction between a silicone oil lens and a toluene lens on the surface of DI water was studied in a ~25 mL glass beaker of an inner diameter of 37 mm, which was filled with ~20 mL DI water. A silicone oil droplet was first placed on the surface of DI water, and a toluene droplet was then placed on the surface of DI water about 2 seconds later with a distance of ~5 mm between the two droplets. The motion and morphologies of the two droplets on the surface of DI water were recorded by a digital, optical microscope (Dino-Lite AM4115ZT). The Image-pro plus software was used in the analysis of the optical images. For detailed information of the experimental procedure, including a schematic of experimental setup (Fig. S1), see the Supporting Information.

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Figure 1. Temporal evolution of a system consisting of a silicone oil lens of 100 cs in kinematic viscosity and a toluene lens floating on the surface of DI water with yellow dashed lines representing the contour of the silicone oil lens: (a) 0 s (onset of contact), (b) 10 s, (c) 150 s, (d) 212 s, (e) 214 s, and (f) 222 s (initial volume of silicone oil lens: 4 μL, and ratio of initial volume of silicone oil lens to toluene lens: 1:1, temperature of DI water: 298 K; the scale bars in all the figures represent 5 mm.) Results Entrapment process Figure 1 shows temporal evolution of a system consisting of a silicone oil lens of 100 cs in kinematic viscosity and a toluene lens on the surface of DI water at 298 K. The initial volume of the droplets for both lenses is 4 µL. After being placed on the surface of DI water, the oil droplets spread into circular lenses, as shown in Fig. 1a at the moment that the oil 8


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lenses initiated the contact. The silicone oil lens is slightly larger than the toluene lens at the onset of contact, whose sizes are dependent on the spreading time, surface tension, and viscosity. Note that the evaporation of toluene can also contribute to the size change of the toluene lens. After the initial contact, the relative motion between the two oil lenses leads to the formation of an interface between the silicone oil lens and the toluene, and introduces a dent onto the silicone oil lens on the interface with the toluene lens remaining circular. Such behavior can be attributed to the differences in the surface tensions and the sizes of the oil lenses. According to Table S1, toluene has a larger surface tension than silicone oil. It is more difficult to deform a toluene lens with a larger surface tension and a smaller size than a silicone oil lens with a smaller surface tension and a larger size. The interaction between the silicone oil lens and the toluene lens allows the toluene lens to continuously move towards the center of the silicone oil lens, and the size of the dent onto the silicone oil lens gradually increases with the increase of time (Fig. 1c–1e), leading to complete penetration of the toluene lens through the interface between the silicone oil lens and DI water into the silicone oil lens to form a core-shell structure with the toluene lens as core and the silicone oil lens as shell (Fig. 1f). The toluene lens is finally entrapped in the silicone oil lens. Such behavior is significantly different from the interaction of two dissimilar, miscible oil lenses on the surface of a solid (rigid) substrate, in which the lens of low surface tension “attacks” the one of high surface tension to form a liquid bridge between them during coalescence. There are three contact lines formed during the entrapment process: (a) the contact line 9


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between the toluene lens and DI water as presented by T-W, (b) the contact line between the silicone oil lens and water as presented by S-W, and (c) the contact line between the silicone oil lens and the toluene lens as presented by S-T. As shown in Fig. 1c-1f, the entrapment of the toluene lens into the silicone oil lens causes the shape change of the contact line of S-W, which becomes irregular. There exists mass transport across the S-T contact line during the entrapment, as shown in the Mov. S1 in the Supporting Information. The mass transport induces the formation of a layer of liquid beads and wave-like disturbances/structures, which separates the silicone oil lens from the toluene lens. The growth of the liquid beads and wave-like disturbances/structures eventually causes complete separation between the silicone oil lens and the toluene lens and the formation of “finger-like” structures connecting the silicone oil lens with the toluene lens. There is a critical moment, at which the toluene lens suddenly accelerates and moves into the silicone oil lens, as shown in Mov. S2 in the Supporting Information. This behavior is different from the coalescence of two dissimilar, miscible droplets on the surface of a solid (rigid) substrate

14

. It is the mass transport driven by the

concentration gradient and the reduction of surface energy that control the entrapment of the toluene lens in the silicone oil lens. After the entrapment, the system evolves into a “one-phase” system with a thick circular lens surrounded by a relatively thin oil film, which takes several seconds to reach a relatively stationary state. Note that the formed system is non-uniform and is at “dynamic” state, at which there still exist mass transport and the evaporation of toluene. After complete evaporation of toluene and the stop of the spreading of the silicone oil, the system finally reaches an equilibrium state. It is worth pointing out that 10


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the entrapment behavior is different from the engulfment of a liquid droplet into other immiscible liquid droplet, which is mainly controlled by surface tensions. Figure 2 schematically summaries the entrapment process due to the interaction between a silicone oil lens and a toluene lens floating on the surface of DI water.

Figure 2. Schematic of the entrapment process for the interaction between a toluene lens and a silicone-oil lens floating on the surface of DI water

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Figure 3. (a) Temporal variation of the diameter of the toluene lens for a two-lens system with a silicone-oil lens of 100 cs in kinematic viscosity and a toluene lens in the coexistence state and a single toluene lens with different initial volumes (ratio of the initial volumes of the toluene lens to the silicone-oil lens: 1:1, distance between two lenses: ~5 mm), (b) variation of the period in the “coexistence” state (entrapment time) with R, (c) variation of the diameter of the toluene lens at the onset of the entrapment, D1, with R, (d) variation of the volume ratio of D2/D1 with R (temperature of DI water: 298 K) Effect of volume ratio of silicone-oil lens to the toluene lens Figure 3a shows temporal variation of the diameter of toluene lenses for a two-lens system with a silicone oil lens of 100 cs in kinematic viscosity and a toluene lens in the coexistence state on the surface of DI water at 298 K. The ratio of the initial volumes of two oil lenses is 1:1. For comparison, the results for a single toluene lens, which has an initial 12


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volume of 2, 4, and 8 µL, respectively, are also included in Fig. 3a. For a single toluene lens on the surface of DI water, the toluene lens first experiences spreading and then “retracting” due to the loss of toluene through evaporation in accord with the observation by Sun and Yang

37

. For the two-lens system in the co-existence state, the silicone oil lens causes the

decrease of the spreading period and retracting rate of the toluene droplet. For the two-lens system with the droplet volume of ~8 μL, we note that the toluene lens experienced significant oscillation, corresponding to the “breathing behavior”, as shown in Mov. S3 in the Supporting Information. Such “breathing behavior” is due to the mass transport between the toluene lens and the silicone oil lens. Similar trend for the temporal variation of the size of the toluene lens in the two-lens system is also observed for DI water of different temperatures. The temperature of DI water does not alter the spreading-retracting behavior of the toluene lens, while the difference between the two-lens system and a single toluene lens decreases with the increase of the temperature of DI water due to the increase of the evaporation rate of toluene and the decrease of the viscosities with the increase of temperature. Figure 3b shows the effect of the volume ratio of toluene lens to the silicone oil lens, R, on the entrapment time of ten. The entrapment time decreases with the increase of the volume ratio, independent of the initial volume of the droplet. For the same volume ratio, the entrapment time increases with the increase of the volume of the silicone oil droplet, since a toluene droplet of large volume is needed accordingly. There is a less driving force for a larger toluene lens to move towards a silicone oil lens of the same size than that for a smaller toluene lens to move towards a silicone oil lens of the same size. Using a power relation between R and t (t � Rn), we obtain n of -0.66, -0.47 to -0.44 for the volumes of the silicone 13


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oil droplets of 8, 4, and 2 μL, respectively. The difference in the power index of n reveals the size effect of the oil lenses. Figure 3c shows the dependence of the diameter of D1 of the toluene lens at the onset of entrapment on the volume ratio of R for different volumes of the silicone-oil droplets. The diameter of D1 of the toluene lens at the onset of entrapment decreases with the increase of the volume ratio of R. This trend reveals the important role of the mass transport in controlling the entrapment of a toluene lens into a silicone-oil lens. For the same volume ratio of R, the larger the volume of the silicone-oil lens, the smaller is the resistance to the silicone-oil lens, and the larger is the size of the toluene lens at the moment of entrapment. Figure 3d shows the variation of the ratio of D2/D1 with the volume ratio of R. It is evident that the ratio of D2/D1 increases with the increase of the volume ratio of R. Such behavior is associated with the small resistance to the distortion of silicone-oil lens of large size in accord with the size effect of the silicone-oil lens shown in Fig. 3c. The effect of the volume ratio of R on D2 is shown in Fig. S3 in the Supporting Information. In general, D2 decreases with the increase of the volume ratio of R, and depends on the initial volume of silicone oil lens, temperature of DI water, and viscosity of silicone-oil lens. It is interesting to note that the temperature of DI water exhibits opposite trend to the viscosity, although increasing the temperature of DI water causes the decrease of viscosity. The different trend implies that the evaporation of toluene also plays an important role in the entrapment of the toluene lens.

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Figure 4. (a) Optical images of two-lens systems of a toluene lens and a silicone oil lens of 100 cs in kinematic viscosity at the onset of the entrapment and 10 s after the entrapment for DI water of different temperatures, (b) variation of the reciprocal of the entrapment time with the reciprocal of the temperature of DI water, (c) variation of the diameter, D1, of the toluene lens at the onset of the entrapment with temperature of DI water, and (d) variation of the ratio of D2/D1 with temperature of DI water (volume of silicone oil droplet: 4 μL; the scale bars in 15


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the figures represent 5 mm.) Effect of temperature of DI water Figure 4a shows optical images of two-lens systems just at the onset of entrapment and 10 s after the entrapment of a toluene lens into a silicone oil lens of 100 cs in kinematic viscosity for DI water of different temperatures. It is evident that there is no change in the shape of the toluene lens before and after the entrapment for all the temperatures used in this work. This result again reveals the important role of surface tension in controlling the morphological evolution of oil lenses and the interaction between dissimilar-miscible oil lenses. We note that increasing the temperature of DI water suppresses the spreading of the silicone oil lens, and reduces the length of the S-T contact line at the onset of the entrapment. This trend suggests that it requires less force to entrap a toluene lens onto a silicone oil lens at high temperatures. Using the optical images (movies), we determine the entrapment time of ten, the diameter of D1 of the toluene lens at the onset of the entrapment, and the ratio of D2/D1 as a function of the temperature of DI water for different volume ratios of R. Figure 4b-4d shows the effects of the temperature of DI water on the entrapment time, the diameter of the toluene lens at the onset of the entrapment, and the ratio of D2/D1, respectively. It is known that the Arrhenius relation of k=Aexp(-Ea/RT) can be applied to a thermally activated process, where k is the rate constant, T is absolute temperature, Ea is the activation energy, R is the gas constant, and A is a pre-exponential constant. For the entrapment process of a toluene lens into a silicone lens, the faster the entrapment, the smaller is the entrapment time. This trend suggests that the rate constant k can be proportional to the reciprocal of the entrapment time, i.e. k ∝ t-1 (s-1) or ln(t-1) ∝ T-1, if we can describe the “entrapment” process 16


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as a “first order” reaction. Using the Arrhenius relation to curve-fit the data given in Fig. 4b, the apparent activation energies for the entrapment of the two dissimilar, miscible oil lenses are obtained. The apparent activation energy increases from 11316 J/mol to 24042 J/mol with the increase of the volume ratio from 0.25 to 4.0. Such a difference likely suggests that there are three factors of the surface tension, mass transport and viscosity controlling the entrapment of the two dissimilar, miscible oil lenses. The larger the volume ratio of R, the larger is the energy barrier needed to overcome for the entrapment. According to Fig. 4c-4d for all the temperature used in this work, increasing the volume ratio of R causes the decrease of the diameter of D1 of the toluene lens at the onset of the entrapment, and the increase of the ratio of D2/D1. However, increasing the temperature of DI water leads to only slight increase of the diameter of D1 of the toluene lens at the onset of the entrapment, and the increase of the ratio of D2/D1. Such behavior can be attributed to the decrease of the viscosity of silicone oil, which reduces the resistance to the fluid flow inside the lenses and accelerates the entrapment process. Effect of viscosity of silicone oil It is known that viscosity determines the resistance to viscous flow, which plays an important role in the entrapment of a toluene lens onto a silicone oil lens. Figure 5a shows the comparison of the entrapment process of a toluene lens onto a silicone oil lens between two different systems; one has a kinematic viscosity of 100 cs, and the other has a kinematic viscosity of 1000 cs. For the system with a silicone oil lens of 100 cs in kinematic viscosity, it took about 1 s to complete the entrapment from the onset of the entrapment to the complete recovery of the toluene lens to a circular lens. At the moment for the sudden acceleration, 17


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there was about 2/3 of the toluene lens being encircled by the silicone oil lens, which was severely deformed. The toluene lens experienced small deformation during the entrapment. For the system with a silicone oil lens of 1000 cs in kinematic viscosity, it took about 8 s to finish the entrapment, which is much longer than that with a silicone oil lens of 100 cs in kinematic viscosity. At the moment for the onset of the entrapment, both the toluene lens and the silicone oil lens remained relatively circular, and there was only a small segment of the S-T contact line. During the entrapment, both the toluene lens and the silicone oil lens experienced large deformation with significant changes in the shapes. There are surface waves or wrinkles presented in both the toluene lens and the silicone oil lens similar to that with a silicone oil lens of 100 cs in kinematic viscosity. The size of the toluene lens entrapped in the silicone oil lens of 1000 cs in kinematic viscosity is larger than that entrapped in the silicone oil lens of 100 cs in kinematic viscosity for the same time of 10 s after the onset of the entrapment. All of these also confirm the important role of the viscosity of silicone oil in regulating the entrapment process.

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Figure 5. (a) Optical images of the two-lens system at different states of the entrapment with silicone oil lenses of 100 and 1000 cs in kinematic viscosity, respectively, (b) variation of the entrapment time, ten, with the viscosity of silicone oil, (c) variation of the diameter, D1, of the toluene lens at the onset of entrapment with the viscosity of silicone oil, and (d) variation of the ratio of D2/D1 with the viscosity of silicone oil (volume of silicone oil droplet: 4 μL, temperature of DI water: 298 K; the scale bars in the figures represent 5 mm.)

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Figure 5b-5d shows the effects of the viscosity of silicone oil on the entrapment time, the diameter of the toluene lens at the onset of the entrapment, and the ratio of D2/D1, respectively, for five different volume ratios of R. Increasing the viscosity of silicone oil reduces the entrapment time and increases the diameter of the toluene lens at the onset of the entrapment. Generally, increasing the viscosity of silicone oil leads to the increase of the ratio of D2/D1. It needs to be pointed out that there exist other entrapment processes, such as the “direct entrapment” mode, which is referred to as the immediate start of the entrapment process when the toluene lens starts to contact the silicone oil lens. Figure S4 and Mov. S4 in the Supporting Information demonstrate the entrapment processes with the “direct entrapment” mode. It is known that surfactant also plays an important role in the floating of liquid lenses 38. For the system of two dissimilar-miscible oil lenses floating on the surface of water, adding a small amount of surfactant in the water can cause the change to the interaction between the oil lenses, as shown in Fig. S5 in the Supporting Information. Prior to the contact, both the toluene and silicone-oil lenses exhibited larger initial sizes on the surface of DI water with 0.1 wt% SDBS (sodium dodecyl benzene sulfonate) than on the surface of DI water with other parameters remained the same. After a short contact (a few seconds), the two lenses separated from each other without the presence of coalescence and entrapment. Increasing the concentration of (SDBS) to 0.5 wt% and 1 wt% led to the rapid spreading of the silicone-oil lens to a thin film, which drove the toluene lens away.

20


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Mechanisms of entrapment There are extensive studies on the morphological evolution of a liquid lens on the surface of the other liquid. The formation of a stable floating lens usually follows the Neumann condition, which is conveniently evaluated by a negative spreading coefficient

39

. The

spreading coefficient is positive for the spreading of either toluene or silicone oil on the surface of DI water, indicating that the system studied in this work does not follow the Neumann condition. Also, the toluene-silicone oil system is experiencing dynamic process, such as the mass transport during the coalescence, which limits the use of the Neumann condition in the analysis. The dynamic process in the toluene-silicone oil system makes it difficult to quantitatively analyze the entrapment of a toluene lens into a silicon oil lens. It is known that the coalescence of two liquid lenses floating on another immiscible liquid phase causes the drainage of the thin film of the sub-phase sandwiched between the two lenses. The time to finish the drainage process is dependent on the materials properties and the interaction between lenses, and can fall into three categories of direct coalescence, delayed or non-coalescence

35

. The entrapment of a toluene lens into a silicone-oil lens

floating on the surface of water can be considered as a special type of coalescence, in which there exists a water film between the toluene lens and silicone oil lens, and its effect on the entrapment is likely limited. As shown in Fig. 1c and Movie S1 in the Supporting Information, mass transport occurs as the two oil lenses start to contact, suggesting rapid drainage of the water film and negligible “barrier” effect of the water film on the entrapment. However, there is a coexistence period usually prior to the onset of the entrapment, indicating the metastable state of the “toluene-silicone oil” system. 21


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The driving force for the entrapment is the difference of the total surface energy of the system before and after the entrapment. For a system with a driving force large enough to overcome the resistance to the entrapment, the toluene lens will be “entrapped” by the silicone oil lens, and the mass transport between the toluene and silicone oil lenses can accelerate the mixing between toluene and silicone oil and the entrapment. To examine the effect of the “mixing” of toluene and silicone oil on the entrapment, the entrapment experiments of two composite lenses with different volume ratios of toluene to silicone oil (95:5 and 5:95; 90:10 and 10:90; 85:15 and 15:85; 80:20 and 20:80) were performed. The experimental results shown in Fig. S6 of the Supporting Information reveal that the entrapment time of ten gradually decreases with the increase of the volume ratio of toluene droplet to silicone oil droplet (Fig. S7 in the Supporting Information). The two composite lenses with the volume ratios of 85:15 and 80:20 of toluene droplet to silicone oil droplet for the viscosity of silicone oil of 100 cs exhibit “direct entrapment” mode (also see Mov. S5 in the Supporting Information.) There are fewer disturbances to the morphology of the “coalescent” oil lens during the “direct entrapment”. The small entrapment time for the direct entrapment implies that there are larger driving forces for the entrapment of the two composite lenses of the volume fractions of toluene (0.2, 0.8) and (0.85, 0.15) than two pure oil lenses. The decrease of the apparent activation energy with the decrease of the volume ratio of R can be attributed to the stress-assisted rate process. The smaller the volume ratio of R, the larger is the driving force for the entrapment, and the smaller is the apparent activation energy. The stress accelerates the mass transport and reduces the energy barrier for the entrapment of 22


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a toluene lens into a silicon-oil lens. The temperature effect on the interaction between a toluene oil lens and a silicone oil lens is much more complex, since most material properties, including surface tension, viscosity, diffusion coefficient and evaporation rate, are temperature-dependent. The variations of these properties with temperature may have opposite effects on the entrapment process, even though the experimental results show the acceleration of the entrapment with the increase of temperature. Assume that a toluene lens is eventually entrapped in a silicone oil lens. As discussed above, the entire interactive process between a toluene lens and a silicone-oil lens floating on the surface of DI water includes a “coexistent” state and a “sudden entrapment” state. In the “coexistent” state, the toluene lens is gradually engulfed by the silicone-oil lens, and the “entrapment” state starts with a sudden acceleration of the toluene lens, which is completely “swollen” by the silicone-oil lens in a very short time period. Using a lumped-element model, the movement of the toluene lens prior to the entrapment can be approximately described by the motion of equation as

dx 2 m= Fsurf +s∆P − Fvis − s -Fvis − w -τ dt 2

(9)

where m is the mass of the toluene lens, x is the displacement of the toluene lens along the axis connecting the centers of the two lenses, t is time, s is the projection area of the toluene-silicon oil contact surface on the plane perpendicular to the x direction, Fsurf is the resultant surface force acting to the toluene lens, ∆P is the pressure difference between the toluene lens and the silicon oil lens, Fvis-s is the viscous resistance to the toluene lens from the silicon oil lens, Fvis-w is the viscous resistance to the toluene lens from DI water, and τ is the 23


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resultant line force (force per unit length) exerted on the toluene lens.

Figure 6. Schematic of the cross-section of the two-lens system in contact with the air/water phase, perpendicular to the surface of DI water. The surface tensions are represented by γ xy with xy being a/t (w/t), a/s (w/s), or t/s interface. Figure 6 depicts a typical cross-section of the system consisting a toluene (t) lens and a silicone-oil (s) lens in contact with the air phase (a) or water phase (w). For the t-s-a system, the component of the surface tension along the x-axis can be expressed as

= γ a γ as cos α + γ ts cos β -γ at cos ω

(10)

and the force on the t-s-a triple line (la) from the surface tensions can be calculated as

Fa = ∫ γ a dl

(11)

la

Similarly, we have the component of the surface tension along the x-axis and the force on the t-s-w triple line from the surface tensions as

= γ w γ ws cos α '+ γ ts cos β '-γ wt cos ω ' Fw = ∫ γ w dl

(12) (13)

lw

The resultant force from the surface tensions are

Fsurf= Fa + Fw

(14)

The coalescence of two liquid lenses floating on sub-liquid phase usually involves the distortion of the contact line (face) between the lenses, which is associated with the pressure 24


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difference between the lenses, i.e. the term of s∆P in Eq. (9). Also, there are viscous resistances to the motion of the toluene lens from the DI water and the silicon oil lens. The viscous resistance from the DI water is scaled as Fvis -w ≈ aη wv s with a being a geometric constant, ηw being the viscosity of the DI water, and v being the velocity of the toluene lens 36. Note that the mass transport between the two lenses suggests that the water thin film between the two lenses is drained rapidly. The effect of the water thin film on the coalescence between a toluene lens and a silicon oil lens is likely negligible. Similarly, the viscous resistance from the silicon oil lens can be scaled Fvis -s ≈ aη s v s with ηs as the viscosity of the silicone oil. Since the viscosity of silicone oil are two to three orders of magnitude larger than that of DI water, the viscous resistance is mainly from the silicone oil lens. It is known that the line tension, i.e. the term of τ in Eq. (9), plays a significant role in the spreading of a droplet on rigid substrate and sub-liquid phase

40-42

. For the liquid-liquid-gas

system, line tension normally exhibits a stabilizing effect on the liquid lens of small size, which assists the formation of a stable liquid lens under non-Neumann condition 39. For the oil lenses used in this work, the effect of the line tension on the motion and distortion might be negligible since the sizes of the lenses are in the order of millimeter. Note that the stabilizing effect of the line tension may contribute the long-time coexistence state prior to the “entrapment”. According to Eqs. (9) and (14), Fsurf is the main driving force for the motion of the toluene For a surface/interfacial tension of the order of 10 mN/m, a radius of an oil lens of the order of 1 mm, and the cosines in the range of -1~1, Fsurf can reach 10-5 N, and Fvis-s is in the order of 10-8 N ~10-9 N for silicone oil with viscosity in the order of 102~103 mPa·s, the 25


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velocity of the toluene lens in the order of 0.01 mm/s, and the interfacial area in the order of 1 mm2. It is evident that the maximum of Fsurf is three to four orders of the viscous resistance. Such a large difference between the maximum driving force of Fsurf and the viscous resistance suggests that sudden “entrapment” of a toluene lens into a silicon oil lens can occur when Fsurf >> Fvis-s. It needs to be emphasized that the driving force of Fsurf varies with the contact angles, as shown in Fig. 6, the contact angles and the driving force change with the motion of the toluene lens onto the silicon lens. For small driving force, the toluene lens is gradually engulfed by the silicone-oil lens; and for large driving force, the toluene lens is entrapped by the silicone-oil lens at a rapid speed. Note that the temperature dependence of surface tensions and viscosity implies the temperature dependence of Fsurf and the entrapment time. According to the relation of Fvis -s ≈ aη s v s , the viscous resistance is proportional to the viscosity of silicone oil. Increasing the viscosity of silicone oil leads to the increase of viscous resistance to the motion of toluene lens. In addition, there is a size effect on the motion of toluene lens in a silicone-oil lens. For silicone-oil lenses of same size, the larger the toluene lens, the larger is the resistance to the toluene lens. It is expected that the large resistance retards the sudden acceleration and increases the entrapment time of ten for the time period from the contact of the two lenses to the threshold of the sudden acceleration, as shown in Fig. 3b. It is known that the spreading of liquids is closed dependent on the viscosity of the liquid and local curvature/radius 43. There exists a characteristic time associated with the spreading of liquids on solid surface. Similarly, it is expected that there is a characteristic time 26


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associated with the “entrapment time” of ten. As discussed above, the dominant driving force of Fsurf is dependent on the surface/interface tensions, and the dominant resistance of Fvis-s to the motion of the toluene lens is proportional to the viscosity of silicon oil. Using dimensional analysis, we can obtain the characteristic time of τc as

τc ∝

ηs D1 γ max R

(15)

for the entrapment process, where γ max is defined as:

= γ max Max[|γ as + γ ts -γ at |,| γ ws + γ ts -γ wt |]

(16)

It is noteworthy pointing out that the mass transport during the contact interaction of two oil lenses also plays an important role in the motion of the toluene lens, since surface tensions, and local viscosity near the toluene/silicone-oil interface are dependent on local composition. The local mass transport can cause the change of local driving force and resistance on toluene lens as well as the characteristic time of τc. Conclusion The interaction of liquid droplets (lenses) on the surface of an immiscible liquid can exhibit different behavior from those on the surface of a solid. We have studied the coalescence between a toluene lens and a silicon-oil lens on the surface of DI water. A sudden entrapment of a toluene lens into a silicone-oil lens after a period of coexistence is observed, in contrast to the coalescence 14, coexistence 20, chasing 16 and self-propelling 44, 45 reported in the literature for two-droplet systems on solid surface. The effects of the volumes of oil droplets, volume ratio, temperature of DI water, and viscosity of silicone oil on the entrapment process have been systematically investigated. Several important parameters of the entrapment time, the diameter of the toluene lens at the onset of entrapment, and the 27


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diameter of the thick lens 10 s after entrapment have been proposed. There exists local mass transport across the interface between the toluene lens and the silicon-oil lens, which causes the changes in local compositions and the associated material properties. The experimental results demonstrate the possibilities of controlling the interaction between liquid lenses via the control of the important parameters. The results reported in this work likely have potential applications in a variety of fields. For example, we can alter the optical behavior of a single lens via the interaction between two or multiple lenses, and develop liquid-lens-based optical devices interaction

between

liquid

intelligent-microfluidic systems

lenses

can

31

. The diversity and controllability of the be

extended

to

the

development

of

46

. The “entrapment” behavior can also be used in the

pollution control for the collection of leaked oils. Acknowledgements WS is grateful for the financial support from the National Natural Science Foundation of China (No. 21805123), the Doctorial startup foundation of Liaoning (No. 20170520392) and General project of Liaoning Education Department (No. L2017LQN014) Supporting information description Experimental details, figures of the temporal evolution of diameters of the two lenses before the entrapment, figures of the variation of D2 with the volume ratio of the two lenses under different experimental conditions, optical images of the “direct entrapment” of the system of a toluene lens and a silicone oil lens (1000 cs), optical images of the temporal evolution of the system of a toluene lens and a silicone oil lens (100 cs) floating on the surface of DI water with the surfactant of 0.1 wt% SDBS, plot of the entrapment time vs the 28


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volume fraction of toluene in toluene-rich lens, optical images of the interaction of two composite lenses, movie of the mass exchange between the two lenses, movie of the typical entrapment process, movie of the “breathing” behavior of the toluene lens, movie of the “entrapment” process of toluene and high-viscous silicone oil, movie of the “direct entrapment” of the composite lenses.

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Table of Contents Graphic

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Coexistence and sudden entrapment between two dissimilar-miscible

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