Equilibrium morphology of gas-liquid Janus droplets:A numerical analysis of buoy

Shaobin Zhang ,Helen Yan ,Yuhao Geng ,Ke Wang ,Jianhong Xu ,*

1 The State Key Lab of Chemical Engineering,Department of Chemical Engineering,Tsinghua University,Beijing 100084,China

2 College of Chemistry,University of California,Berkeley,CA 94728,USA

3 State Key Laboratory of Industrial Vent Gas Reuse,The Southwest Research&Design Institute of the Chemical Industry,Chengdu 610225,China

Keywords:Micro fluidics Gas-liquid-liquid emulsion Janus droplets

ABSTRACT In this article,a theoretical model for predicting the equilibrium morphology of gas-liquid Janus droplets was built.Based on this model,the effects of bubble radius and volume ratio on morphology change was systematically studied.The increase of bubble radius causes the two parts(bubble and oil drop)in Janus droplets tend to merge while the impactofvolume ratio is complicated.When volume ratio increases,these two parts firstly tend to merge,then gradually separate.The accuracy of this model was verified by experimental results.

1.Introduction

Emulsions are attracting extensive interests because of their potential applications in various fields[1-6].Most of these researches are focused on micro fluidic methods because micro fluidics permits better control of the experimental conditions[7,8]than other methods.Up to now,double emulsions are stillthe research focus within applications for emulsions,and a large number of investigations have been reported[9-13].

According to the value of spreading coefficients[14-17],there are three typical structures for double emulsions:core-shell,Janus,and totally separated.For different morphologies,there are different applications such as capsules for core-shell[16-19].It is worth noting that Janus droplets are attracting more attention because of their unique anisotropic properties,which make them suitable for use in numerous applications[20-24].For Janus droplets,the structure can be further adjusted as long as the spreading coefficients are in the exact range for Janus morphology[25].Depending on the separation degree of two components,Janus droplets can be categorized into several types such as“perfect Janus emulsions”,“Dumbbell Janus emulsions”,etc.[26].Also,Janus morphology could have an important effect on their promising application.For example,the two parts in “perfect Janus droplet”are hemispherical,and they constitute a spherical Janus droplet,which has the potential to be used in display screen[27].

The equilibrium morphology of Janus droplets prepared by micro fluidic methods is a result of force balance with three interfacial tensions.Some researchers have made a detailed analysis about equilibrium topology of Janus droplets.According to their models[28-30]for liquid-liquid Janus droplets,there are only two factors in fluencing the equilibrium morphology:interfacial tensions and two-phase volume ratio.Buoyancy and gravitational effects were neglected in these examinations.However,this theoretical model may be not suitable for gas-liquid Janus droplets because gas-liquid density difference is far greater than liquid-liquid density difference.Thus,it is necessary to make modifications to the previous model and build a new one that takes the effect of buoyancy into account.

In this article,we demonstrated that buoyancy need to be considered when the radius of microbubble is sufficiently large(N 300 μm).Hence,we built a new model to predict the equilibrium morphology of gas-liquid Janus droplets by making modifications to the liquid-liquid model.Furthermore,the effects of volume ratio and microbubble radius on the morphology of gas-liquid Janus droplets were investigated.Finally,the theory model was verified by experimental results.The micro fluidic device used is the same as the one(see supplementary material)we used in our previous work[31].

Fig.1.The relationship between Bond number and characteristic length.

2.Fundamental Background

2.1.Preliminary calculation

Local Bond number[32](Bo= Δρg l2/γ)can be used to assess the buoyancy effect,where Δρ,g,and l represent density difference,gravitational constant,and the characteristic length of three phase contact line.

There are three phases in our experiments:air,oil,and water(A,O,and W represent air,oil,and water respectively).To simplify the calculation,we take the density of water(ρW)as the density difference between air and water(ρAW).The interfacial tension between oil and water(γOW)was used for calculation as it is the least of three interfacial tensions.Thus,Bo= ρWg l2/γOW.The value of(γOW)adopted is 10 mN·m-1,which is representative in experimental systems.As shown in Fig.1,the buoyancy effectcan'tbe neglected when the characteristic length l is larger than 300 μm.At that time,Bond number is larger than 0.1,which means buoyancy is at the comparable magnitude order with interfacial tension.Thus,it is meaningful to investigate the buoyancy effect on the equilibrium morphology when microbubble grows to a certain size(N300 μm).

2.2.Theoretical model

Generally,the equilibrium morphology for gas-liquid Janus droplet is determined by two factors:the force equilibrium between interfacial tensions and buoyancy and the volume ratio of the two phases in the Janus droplet.However,buoyancy in this case is in fluenced by many parameters(see Eqs.(4)-(6))such as the radius of the bubble in the Janus droplet,which makes the actual situation more complicated.

Fig.2 demonstrates the force analysis and geometry analysis of a Janus droplet that is in equilibrium morphology(A,O,and W represent air,oil,and water respectively.)All the interfaces are assumed as sphere caps according to the references.

For force analysis:

WhereγAWrepresents the interfacial tension between airand water,the same goes for γOWand γAO.FBis the buoyancy,and Lcis the perimeter of contact line.θ1is the angle between γAWand vertical axis.θ2is the angle between γOWand vertical axis.θ3is the angle between γAOand vertical axis.

Where rAWrepresents bubble radius,the same goes for rOWand rOA.

In Eq.(4),ρOWrepresents the density difference between oil and water.VOand VArepresent the volume of oil part(oil drop,red)and gas part(bubble,green),and they can be obtained by combining the spherical canopy volume formula and the geometry relationship.

Fig.2.Force analysis(left)and geometry analysis(right)for a Janus droplet.The upper part represents bubble while the lower part represents oil droplet.

Fig.3.The effects of volume ratio and bubble radius on buoyancy.

For volume ratio:

For the radius[28-30]:

The relationship between θ1,θ2,θ3,and μ,η,ε are:

With the above equations,at a given microbubble radius(rAW),it is sufficient to calculate the equilibrium morphology of gas-liquid Janus droplet.We used MATLAB to solve the above equations,and a numerical solution could be obtained.(The MATLAB code is provided in supplementary materials).

3.Results and Discussion

3.1.The parameters determining buoyancy

From Eqs.(4)-(6),we can see that buoyancy is determined by rAW,rOA,rOW,μ,η,ε,ρOW,ρA.For a given system,γAW,γOW,γAO,ρOW,ρAare determined.Along with above equations,we can get the value of buoyancy as long as rAWand q are known.Fig.3 shows that buoyancy increases with bubble radius.In comparison,buoyancy decreases when volume ratio increases,and the amplitude is limited.

3.2.The effect of bubble radius on equilibrium morphology

From above results,increasing bubble radius leads to the increase of buoyancy,which will cause a change in equilibrium morphology.To illustrate this effect,we chose an experimental system(air,water+0.06 wt% fluoro-surfactant,and 1,6-hexanediol diacrylate)with γAW=29.56 mN·m-1,γOW=10.01 mN·m-1,γAO=28.53 mN·m-1as an example.The volume ratio is fixed to 1.

In Fig.4,the morphologies obtained from the model considering buoyancy are compared with the morphology obtained from previous liquid-liquid model.There is visible difference between these two morphologies when bubble radius is larger than 300 μm,which means that buoyancy effectcannotbe neglected atthis time.The resultis consistent with that of preliminary calculation.The difference is more significant when bubble radius becomes larger.Furthermore,we can see that the coverage of bubble increases when bubble radius increases,which means the two parts in Janus droplets get closer with each other.

To make a quantitative analysis of this morphology change,we defined the angle between γAWand γOWas α,and α can be considered as an indicator of the separation degree ofbubble and oil drop i.e.the morphology change(the bigger α,the smaller separation degree).The relationship betweenαand bubble radius was investigated,which is shown in Fig.5(a).We can see that α increases with the increase of bubble radius,which means the separation degree decreases when bubble radius increases.And this phenomenon is in agreement with the increasing coverage.Fig.5(b)demonstrates that both θ1and θ2decrease when radius increases,which can account for the increasing bubble α (α =180- θ1- θ2).

Through qualitative and quantitative analysis,we could draw the conclusion that the buoyancy effect on the equilibrium morphology of gas-liquid Janus droplets will become apparent when bubble radius increases to a certain extent.Bigger bubble radius makes the bubble and the oil drop in the gas-liquid Janus droplet become closerto each other.

3.3.The effect of volume ratio on equilibrium morphology

The same interfacial data was used,and bubble radius was fixed to 500 μm when volume ratio was changed.Fig.6 shows that volume ratio has a big impact on the equilibrium morphology of gas-liquid Janus droplets,which is similar to that of the effect of volume ratio on morphology in liquid-liquid model.

Fig.7 illustrates that volume ratio has a complicated impact on α.With the increase of volume ratio,α firstly increases to the maximum value,and then gradually decreases.This is because the changes for θ1andθ2are complex.In general,θ1decreases with the increase ofvolume ratio while θ2rises with increasing volume ratio.However,the rates of change for these two angles vary when volume ratio is changed.At the beginning,the reduction rate of θ1is bigger than the increase rate ofθ2,thusαincreases when volume ratio becomes larger.When volume ratio increases to a certain extent,the reduction rate for θ1is smaller than the increase rate for θ2,which leads to a smaller α.

Fig.4.Morphological comparison between two models.

Fig.5.(a)The effectofbubble radius onα.The dash line represents the value ofαfor liquid-liquid model,which is constantatgiven interfacialdata.(b)The effectofbubble radius onθ1 andθ2.

Interestingly,for previous model,α will not change with the change of volume ratio.In liquid-liquid model,α is only determined by three interfacialtension relationship,thus α is a constant for a known system.When buoyancy is considered,volume ratio is also needed to get the value of α.So volume ratio plays a role in determining α as well.

3.4.The effect of interfacial tensions on equilibrium morphology

For different systems,γAW,γOW,rAOare different.This variation not only in fluences the values of interfacial tensions in force balance,but also in fluences other parameters in those equations.To investigate this effect,two realistic systems(one has already been shown above,the other consists of air,water+0.04 wt% fluoro-surfactant+0.06 wt%SDS,and tetradecane with γAW=28.62 mN·m-1,γOW=6.94 mN·m-1,γAO=26.98 mN·m-1)were used.Bubble radius and volume ratio was fixed to 500 μm and 1 respectively.

We changed the value of γAOin firstsystem to 28.53,while the value ofγOWin second system was adjusted to 6.94(stillin the range for Janus morphology).Then we could use these four sets of data(shown in Table 1)to research how interfacial tensions in fluence the equilibrium morphology.By comparing system 1 and system 2,we could see that separation degree increases when γAOincreases.Similar analyses show that γAWhas negative effect on separation degree while γOWhas positive effect on separation degree.

Fig.6.Equilibrium morphology at different volume ratio.

Fig.7.(a)The effect of volume ratio on α.(b)The effect of volume ratio on θ1 and θ2.

Table 1 Interfacial tensions for different systems

3.5.Verification by experimental results

To verify the accuracy ofgas-liquid Janus modelused in this article,a series ofexperiments were performed.We chose system4 in Table 1 for experiments.At different flow rates,gas-oil Janus droplets were obtained using a coaxial micro fluidic device through one-step approach.Then these Janus droplets were transferred to an observation cell and observed by a microscope.Side view of gas-liquid Janus droplets can be obtained,thus we could measure the realistic angle between γAWand γOW(termed as α′).By comparing α′with α,we can assess the accuracy of model.

Fig.8 shows the side view of Janus droplets.When bubble radius is 250 μm,the value of α′is well consistent with that of α (103.10°vs 107.29°).Meanwhile,the value of α′is also well consistentwith the calculation result of previous model(103.10°vs 105.69°).For our model,the good consistency can still be seen(112.46°vs 114.00°and 126.25°vs 120.10°)when bubble radius becomes larger(549 μm and 704 μm).However,there is noticeable difference between the value of α′and the figure obtained from previous model(112.46°vs 105.69°and 126.25°vs 105.69°),and this difference willbecome larger when bubble radius increases.This phenomenon shows that buoyancy cannot be neglected when bubble radius grows to a certain extent.

A bigger difference(～6°)was observed when bubble radius is 704 μm.The reason for this is that bubble has been deformed(as seen in Fig.8),and the deformation can cause an impact on the equilibrium morphology,which made the value of α′larger than it should be.However,the relative error between α and α′is approximately 5%,which is still accurate enough,so this model can be used to predict the equilibrium morphology of gas-liquid Janus droplet.We will make a thorough study about the deformation in our future work.

4.Conclusions

In summary,we built a theoretical model used to calculate the equilibrium morphology of gas-liquid Janus droplets by taking buoyancy effect into account.Then we investigated how bubble radius,volume ratio and interfacial tensions in fluence the equilibrium morphology.The results show that bigger bubble radius leads to a smaller separation degree in gas-liquid Janus dropletwhile volume ratio and interfacial tensions have complicated in fluence on that of Janus droplet.Finally,we verified this theory model with experiments,which shows that calculation results are in agreement with experiment results.Thus,this model can be used to predict the equilibrium morphology of gas-liquid Janus droplets.

Nomenclature

Bo the local Bond number

FBthe buoyancy,N

g the gravitational constant,m·s-2

Lcthe perimeter of contact line,m

l the characteristic length of three phase contact line,m

O oil

q the volume ratio

rAWthe radius of oil drop,m

rOAthe radius of virtual water drop,m

rOWthe bubble radius,m

VAthe volume of bubble,m3

VOthe volume of oil drop,m3

W water

α the angle between γAWand γOW,(°)

α′ the realistic angle between γAWand γOW,(°)

γ the interfacial tension,N·m-1

γAOthe interfacial tension between air and oil,N·m-1

γAWthe interfacial tension between air and water,N·m-1

γOWthe interfacial tension between oil and water,N·m-1

ε the angle between rOAand the central axis,(°)

η the angle between rAWand the central axis,(°)

θ1the angle between γAWand vertical axis,(°)

θ2the angle between γOWand vertical axis,(°)

θ3the angle between γAOand vertical axis,(°)

Fig.8.Janus droplets with different equilibrium morphology.

μ the angle between rOWand the central axis,(°)

ρAthe density of air,kg·m-3

ρAWthe density difference between water and air,kg·m-3

ρOthe density of oil,kg·m-3

ρOWthe density difference between water and oil,kg·m-3

ρWthe density of water,kg·m-3

Δρ the density difference,kg·m-3

Supplementary Material

Supplementary data to this article can be found online athttps://doi.org/10.1016/j.cjche.2018.06.007.