Coalescence dynamics of two droplets of different viscosities in T-junction microchannel with a funnel-typed expansion chamber

Weixi Guo,Chunying Zhu,Taotao Fu,Youguang Ma

State Key Laboratory of Chemical Engineering,School of Chemical Engineering and Technology,Tianjin University,Tianjin 300072,China

Keywords:Microchannel Droplet Coalescence Viscosity ratio Critical capillary number

ABSTRACT The coalescence behavior of two droplets with different viscosities in the funnel-typed expansion chamber in T-junction microchannel was investigated experimentally and compared with droplet coalescence of the same viscosity.Four types of coalescence regimes were observed:contact non-coalescence,squeeze non-coalescence,two-droplet coalescence and pinch-off coalescence.For droplet coalescence of different viscosities,the operating range of non-coalescence becomes narrowed compared to the droplet coalescence of same viscosity,and it shrinks with increasing viscosity ratio η of two droplets,indicating that the difference in the viscosity of two droplets is conducive to coalescence,especially when 1＜η＜6.Furthermore,the influences of viscosity ratio and droplet size on the film drainage time(Tdr)and critical capillary number (Cac) were studied systematically.It was found that the film drainage time declined with the increase of average droplet size,which abided by power-law relation with the size difference and viscosity ratio of the two droplets:Tdr～ (ld)0.25±0.04 and Tdr～ (η)-0.1±0.02.For droplet coalescence of same viscosity,the relation of critical capillary number with two-phase viscosity ratio and dimensionless droplet size is Cac=0.48λ0.26 l-2.64,while for droplet coalescence of different viscosities,the scaling of critical capillary number with dimensionless average droplet size,dimensionless droplet size difference and viscosity ratio of two droplets is Cac=

1.Introduction

In the past 20 years,the modularization,miniaturization and integration of the process engineering have attracted increasing attention.Microfluidic equipment and its application in microchemical have achieved great advancement [1].In industry,the microfluidic technology has been widely used in emulsification,drug encapsulation,material synthesis,crystallization and chemical reactions due to its high efficiency of mass and heat transfer,easy control,environmental protection and high process safety[2–8].Liquid-liquid two-phase flow in microchannels is an important branch for the research and application of microfluidic technology.The droplet coalescence of different viscosities could control quantitatively the processes of chemical reaction [9],mixing[10]and mass transfer[11],such as biochemical reaction based on microdroplets including the synthesis process of nanoions,which requires the coalescence of two droplets with different properties to achieve good mixing effect.Therefore,the fully understanding on the coalescence of two droplet with different viscosities is a basic requirement for industrial application.

The theory of liquid film draining is generally used to explain the droplet coalescence mechanism,which includes four basic processes:(1) droplets capture or positioning;(2) two droplets approach each other accompanied with collision and deformation;(3) the continuous phase liquid film between droplets is discharged;(4) the interface of droplets is broken and fused [12].Chesters[13]proposed the correlations of liquid film drainage time for two identical droplets based on three interface models:fixed,partially moved and fully moved.Klaseboer et al.[14]investigated the liquid film draining process of two head collision droplets and established a model based on the lubrication theory,in which ttypical=was defined to describe liquid film drainage time,here R represents the radius of spherical droplet and V is the droplet velocity.Subsequently,Yeo et al.[15] analyzed the influence of various parameters on draining kinetics by establishing a mathematical model,they perceived that the liquid film drainage time was inversely proportional to the two-phase viscosity ratio.

Logically,the liquid film drainage rate depends on many factors,such as the impact velocity and droplet viscosity,which could be characterized by capillary number.Stone et al.[16] found that the critical capillary number is the key parameter to determine whether coalescence would occur.Hu et al.[17] thought that the droplet coalescence depends on whether the liquid film is drained to the minimum thickness within the range in which the vander Waals force could play attractive role.When the initial offset and viscosity ratio of two droplets are given,the minimum liquid film thickness hinges only on capillary number.Christopher et al.[18]conducted an experimental investigation on the coalescence of identical droplet at a microfluidic T-junction,the results indicated that when the two-phase viscosity ratio was less than 1.3,the critical capillary number and viscosity ratio showed a power-law relation with exponent 0.49 ± 0.04.Liu et al.[19,20] found that the critical capillary number for droplet coalescence decreased with the increase of the intersection angle of the channel,and the power-law exponent between the critical capillary number and viscosity ratio is -0.75.Recently,Ma et al.[21]studied the coalescence process of two identical droplets in the symmetrical Tjunction with expanded convergence,and they found that the relation between the critical capillary number and two-phase viscosity ratio could be expressed by the power-law equation as:Cac=0.004η-0.518.

In practical application,the coalescence of two droplets with different viscosities is more required for chemical or biochemical reactions compared to the droplet coalescence with same viscosity.In this study,the T-junction microchannel with a funnel-typed expansion chamber was used to investigate droplet coalescence.The expansion structure has been proven to greatly extend the operating range and improve coalescence efficiency [22].Moreover,Ionic liquid is a type of the most promising catalytic system and reaction medium solvent in green chemistry,especially,it could be recycled [23].However,the higher viscosity,larger density and lower interfacial tension of ionic liquid could remarkably affect the droplet coalescence dynamic behavior,therefore,it is of significant importance to investigate droplet coalescence of ionic liquid system in order to intensify mixing and reaction process.In this study,a high-speed digital camera was used to study the coalescence of ionic liquid ([BMIM][PF6]) and glycerol-water solution droplets with different mass concentrations,including droplet coalescence of different viscosities and same viscosity.The influences of two-phase flow rate,dispersed phase viscosity and two-droplet viscosity ratio on the transition of coalescence regimes were investigated.Furthermore,the effects of droplet size and viscosity ratio on the liquid film drainage time and critical capillary number were also studied systematically.

2.Experimental

The experimental setup used in the experiment is sketched in Fig.1,including microfluidic device,fluid control system and image capture and acquisition system.The high-speed camera(FASTCAM SA1.1,Photron,Japan) was utilized to capture the process image of droplet coalescence.The microchannel was connected to the sampling system through an austenitic stainless steel pipe interface and a polyvinyl-chloride pipeline.Two precise syringe pumps were used to drive ionic liquid ([BMIM][PF6]) and glycerol-water solution into horizontally placed microchannel,respectively.The 12VDC halogen lamp was applied as the light source to obtain the best shooting effect by adjusting the exposure time,shooting range and light intensity.The recording rate used in this experiment was 1000 fps (frames per second).For each flow rate,the image was recorded after the system had stabilized at least 5 minutes.

The microfluidic device is illustrated in Fig.2.The microchannel was fabricated in two polymethyl methacrylate (PMMA) plates,one of them was engraved on the microchannel by a precision milling machine and the other was extruded and packaged with bolts.The cross section of microchannel was 400 μm(wide) × 400 μm (deep) and the fabrication error was about 5 μm.The microchannel included two T-junction droplet generating structures and a T-junction droplet coalescence structure with a funnel-typed expansion chamber.The continuous phase([BMIM][PF6])was introduced from the main channel with the flow rate of Qcand divided into two tributaries,and the dispersed phase(glycerol-water solution) was introduced from the two side channels with the flow rate of Qdwhich was pinched off by the continuous phase at the first T-junction to generate droplets.The droplets continued to flow downstream into the funnel-typed expansion chamber at the second T-junction to coalesce.In this experiment,the requirements for droplet synchronization are as follows:(1)the droplets are generated synchronously at the two T-junction droplet generating structures;(2) the wall roughness and the corner angles of the two branch channels are completely the same,so that the channel will not cause different effects on the flow process of droplets.In this case,the droplets will reach the entrance of the funnel-typed expansion chamber synchronously.The size parameters of the expansion chamber are marked in Fig.2.Wsis defined as the channel width,L is the droplet length,and l=L/Wsis the dimensionless droplet length.

Fig.1.Schematic diagram of experimental setup.

Fig.2.Schematic diagram of microfluidic device (unit:mm).

The continuous phase was an ionic liquid ([BMIM][PF6]),and the dispersed phase was glycerol-water solution with mass concentration of glycerol of 0%,10%,30%,50% and 70%,respectively.Glycerol-water solutions of different concentrations were mainly used to study the effect of dispersed phase viscosity on droplet coalescence.Due to the variation of interfacial tension was only 2.4%,thus the change of interfacial tension with concentration was negligible.The fluid densities were measured by a vibrating tuber density meter (Anton Paar,Australia).The liquid–liquid interfacial tension of two phases was measured by the pendant drop method with a tensiometer (OCAH200,Data Physics instruments GmbH,Germany)and a fully automatic Ubbelohde viscometer(iVisc,LAUDA,Germany)was used to measure the viscosities of the fluids.In the measurement process,each sample was measured 3 times and the average value was taken.The experimental temperature was 298.15 K and the pressure was 1.013×105Pa.During the experiment,the ranges of the continuous phase flow rate Qcand dispersed phase flow rate Qdwere:0.4＜Qc＜2.0 ml.h-1and 0.2＜Qd＜5.0 ml.h-1,respectively.The physical properties of liquids are listed in Table 1.

Table 1Physical properties of liquids used in the experiment

3.Results and Discussion

3.1.Droplet coalescence regime

3.1.1.Coalescence regimes of two droplets of same viscosity

When the viscosity of two droplets was the same,four kinds of droplet coalescence regimes were observed in the experiment:contact non-coalescence,squeeze non-coalescence,two-droplet coalescence and pinch-off coalescence,as shown in Fig.3.

(a) Contact non-coalescence:the process of contact noncoalescence is shown in Fig.3(a).In this regime,the droplet sizeswere small,the velocities of the droplets reduced when they reached the funnel-typed expansion chamber,then they contacted each other (290 ms).There was almost no deformation during the flow in the expansion chamber,while a relative rotation occurred at the shrinkage (630 ms).Finally,the two droplets entered the downstream channel in sequence (950 ms).

(b) Squeeze non-coalescence:the process of squeeze noncoalescence is shown in Fig.3(b).The droplet size would be gradually increased with the increase of dispersed phase flow rate[24].After two droplets came into contact,they would enter the funneltyped expansion chamber and deform due to the squeezing effect(310 ms),but the contact time was not long enough to reach the liquid film drainage time,thus no coalescence occurred.Then the two droplets flowed out the expansion chamber through the shrinkage section (640 ms).

(c)Two-droplet coalescence:the process of two-droplet coalescence is shown in Fig.3(c).In this regime,the droplet size was further increased with the increase of flow rate.Either the droplets arrived at the T-junction simultaneously,or the former droplet entered the funnel-typed expansion chamber and slowed down,waiting for the latter droplet to enter until they contact each other(170 ms).The liquid film was gradually drained,and eventually the two droplets coalesced (260 ms) and flowed out the expansion chamber into the downstream channel (500 ms).

(d) Pinch-off coalescence:the process of pinch-off coalescence is shown in Fig.3(d).When the dispersed phase flow rate was much large than the continuous phase flow rate,overlong droplets would be generated.The head of latter droplet has entered the expansion chamber before the former droplet completely flowed out,and the latter droplet would squeeze the former one (80 ms)until a smaller droplet was generated by pinching off (100 ms).Finally,the latter droplet coalesced with the generated daughter droplet (140 ms) and flowed out the expansion chamber.

Fig.3.Coalescence types of droplets with same viscosity observed in the funnel-typed expansion chamber of T-junction.(The dispersed phase is pure water)(a)Contact noncoalescence,Qc=1.6 ml.h-1,Qd=0.2 ml.h-1;(b)Squeeze non-coalescence,Qc=2 ml.h-1,Qd=0.6 ml.h-1;(c)Two-droplet coalescence,Qc=2 ml.h-1,Qd=1 ml.h-1;(d)Pinchoff coalescence,Qc=1.2 ml.h-1,Qd=3.4 ml.h-1.

As shown in Fig.3(b) and 3(c),the squeeze non-coalescence occurred at Qc=2 ml.h-1,Qd=0.6 ml.h-1,while the two-droplet coalescence occurred when Qc=2 ml.h-1,Qd=1 ml.h-1,indicating that the total flow rate of two-droplet coalescence was greater than that of squeeze non-coalescence.Liu et al.[19] found that the liquid film drainage time decreased with increasing total flow rate,when the total flow rate was large,the contact time required for coalescence decreased accordingly.Moreover,comparing Fig.3(b)with(c),it could be found that the droplet size in Fig.3(b)was smaller than that in Fig.3(c),and the liquid film drainage time decreased with increasing droplet size in this experiment,therefore,the coalescence could occur at shorter contact time under the condition of Fig.3(c).

Fig.4 shows the flow patterns of five mass concentrations of glycerol-water solutions.For Fig.4(a)-4(e),the x-axis is dispersed phase flow rate Qdand the y-axis is continuous phase flow rate Qc.For Fig.4(f),the x-axis is the Reynolds number Redof dispersed phase and the y-axis is the Reynolds number Recof continuous phase,here Red=ρudWs/μdand Rec=ρucWs/μc,respectively.Under the same Qc,with the increase of Qd,the flow pattern changes from contact non-coalescence to squeeze non-coalescence,two-droplet coalescence and pinch-off coalescence in sequence.When the dispersed phase viscosity is low(Fig.4(a),4(b)),the operating range of squeeze non-coalescence is very narrow,indicating that droplets are apt to coalescence at low viscosity.The increase of Qcreduces the contact time and increases the difficulty of coalescence,accordingly the operating range of non-coalescence is gradually expanded with increasing Qc.Comparing Fig.4(a) -(e),it is found that as the dispersed phase viscosity increases,the operating range of two-droplet coalescence significantly reduces and the operating range of non-coalescence is expanded,indicating that the increase in viscosity is not conducive to coalescence.Since the droplet size increases with the rise of dispersed phase viscosity [24],pinch-off coalescence tends to occur at a smaller Qdwhen the dispersed phase viscosity is larger.

In this study,the variation of interfacial tension with viscosity is negligible,and evolution of coalescence regime is mainly controlled by viscous force and inertial force.Reynolds number represents the relative importance of viscous force to inertial force.When Redis large,the inertial force plays a dominant role in the coalescence process,thus coalescence is easy to occur for larger droplet.When Redis small,the droplet coalescence is controlled by the viscous force,the higher viscous force is not conducive to coalescence.As the continuous phase viscosity is fixed,hence,Reconly reflects the inertial effect of continuous phase.The higher Recmeans larger continuous phase flow rate,which reduces the contact time and expands the operating range of noncoalescence.Fig.4(f) shows the transition line between different flow patterns.The transition line between contact noncoalescence and squeeze non-coalescence is Rec=0.0187Red+2.29 × 10-3,the transition line between squeeze non-coalescence and two-droplet coalescence is Rec=0.0156Red+5.71 × 10-4,and the transition line between two-droplet coalescence and pinch-off coalescence is Rec=0.00273Red+1.22 × 10-4.

3.1.2.Coalescence regimes of two droplets of different viscosities

For the coalescence of two droplets with different viscosities,only three kinds of droplet coalescence regimes were observed:contact non-coalescence,two-droplet coalescence and pinch-off coalescence,as shown in Fig.5.

(a) Contact non-coalescence:as shown in Fig.5(a).In this regime,the droplet size was small and there was a certain time difference when they arrived at the T-junction (0 ms).The former droplet firstly entered the funnel-typed expansion chamber and its velocity slowed down owing to the increase of cross-section area of channel,it could wait for a while and then contact with the latter droplet in the expansion chamber (330 ms).However,the contact time is insufficient for the liquid film drainage,thus the coalescence did not occur (470 ms).Afterwards,the two droplets flowed into the downstream channel in sequence (730 ms).

Fig.4.The flow pattern map of droplet in the funnel-typed expansion chamber of T-junction.(Droplets with same viscosity).(a)Water;(b)10%Gly;(c)30%Gly;(d)50%Gly;(e) 70% Gly;(f) Rec-Red.

Fig.5.Coalescence types of droplets with different viscosities observed in the funnel-typed expansion chamber of T-junction.(The dispersed phase is pure water(left) and 30% glycerol (right)) (a) Contact non-coalescence,Qc=2 ml.h-1,Qd=0.4 ml.h-1;(b) Two-droplet coalescence,Qc=1.2 ml.h-1,Qd=1.2 ml.h-1;(c) Pinch-off coalescence,Qc=1.2 ml.h-1,Qd=3.0 ml.h-1.

(b)Two-droplet coalescence:as shown in Fig.5(b).The head of two droplets contacted at T-junction (170 ms),and the liquid film was gradually drained until the interface became fused (200 ms).Due to the different viscosities of two droplets,an obvious mixing process could be seen after coalescence(370 ms),and then the coalesced droplet flowed out the expansion chamber from the shrinkage mouth (690 ms).

(c) Pinch-off coalescence:as shown in Fig.5(c).The pinch-off coalescence process was similar to the coalescence of droplets with same viscosity.Due to the squeezing effect of latter droplet(240 ms),the former droplet would be pinched off to generate a small daughter droplet(270 ms).Since the droplet viscosities were different,the interfaces of two droplets were still obvious after coalescence(310 ms).Gradually,the coalesced droplet would completely fuse.

Fig.6 shows the flow patterns of droplets with different viscosities.η=μd1/μd2is defined as the viscosity ratio of two droplets,here μd1is the dispersed phase viscosity with higher viscosity,and μd2is the dispersed phase viscosity with lower viscosity.It could be found that when η＞1,the operating range of noncoalescence of droplet coalescence of different viscosities is smaller than that of the coalescence of droplets with same viscosity,the flow regime of squeeze non-coalescence disappears,and the operating range of non-coalescence decreases with the increase of η,indicating that the difference in viscosity of two droplets is conducive to coalescence.In light of the Marangoni effect,when the interfacial tension increases in one direction,a force against it would be generated [25].As interfacial tension increased slightly with increasing droplet viscosity,the local interfacial tension would be increased when the heads of two droplets with different viscosities contacts with each other,while the interfacial tension of untouched part is relatively low.The force stemming from interfacial tension gradient facilitates the liquid film drainage and thereby accelerates coalescence.

Fig.6.The flow pattern map of droplet in the funnel-typed expansion chamber of T-junction.(Droplets with different viscosities).The solid lines represent the flow pattern transition line of two droplets with different viscosities coalesced.The dash lines represent the flow pattern transition line of two pure water droplets coalesced.(a)Water,10% Gly,η=1.27;(b) Water,30% Gly,η=2.64;(c) Water,50% Gly,η=5.40;(d) Water,70% Gly,η=20.45.

Comparing Fig.6(a)-6(d),it could be found that when η gradually increases,the pinch-off coalescence occurs under smaller flow rate of dispersed phase,and the transition line between twodroplet coalescence and pinch-off coalescence of droplets with different viscosities deviates gradually from that of droplets with same viscosity.The occurrence of pinch-off coalescence is primarily resulted from the formation of overlong droplets.The droplet size increases with the increase of viscosity,which facilitates the appearance of pinch-off coalescence.For the coalescence of two droplets with different viscosities,the operating range of twodroplet coalescence reduces,implying that the larger viscosity difference is not conducive to the two-droplet coalescence.

3.2.Liquid film drainage time

3.2.1.Liquid film drainage time of droplet coalescence of same viscosity

Fig.7 shows the influences of various factors on the liquid film drainage time for the coalescence of two droplets with same viscosity.It could be seen from Fig.7(a)that as the dimensionless droplet size increases,the liquid film drainage time decreases.The increase of droplet size reduces the space between the droplet and the channel wall,which obstructs continuous phase fluid to flow downstream and accordingly leads to the accumulated pressure of the upstream continuous phase,thus the process of droplet coalescence is accelerated [21].Moreover,the increase of droplet size could increase the contact area between two droplets,which is in favor of droplet coalescence.The droplet coalescence needs to destroy the stability of the droplet interface.When the surface of the droplet is spherical,its interface area is the smallest,correspondingly,the surface energy is the smallest,in this situation,the interface is the most stable [26].The increase of the contact area between two droplets would change the shape of the droplet,and the larger contact area of the droplets would force them to become more flattened,thereby cause in an increase of surface energy and the non-stability of droplet interface.Therefore,the larger contact area between two droplets is in favor of droplet coalescence.Meanwhile,It could also be found from Fig.7(b) that the liquid film drainage time increases with the increase of twophase viscosity ratio defined as λ=μd/μc.The viscosity is the dominated force controlling the liquid film drainage time[27],and the increase of λ increases the initial liquid film thickness[15],thus the increase in λ would prolong the liquid film drainage time.The liquid film drainage time varies linearly with two-phase viscosity ratio,as μcis a fixed value,therefore,the liquid film drainage timer relies linearly on the dispersed phase viscosity.This result is also consistent with the correlation equation of liquid film drainage time proposed by Chesters [13].

Fig.7.The effects of dimensionless droplet size and two-phase viscosity ratio on the film drainage time.(Droplets with same viscosity).(a)Dimensionless droplet size l;(b)Viscosity ratio of dispersed phase to continuous phase λ.

3.2.2.Liquid film drainage time of droplets coalescence of different viscosities

Fig.8 shows the effects of different factors on the liquid film drainage time for the coalescence of two droplets with different viscosities.Due to the viscosity difference of two droplets,the sizes of the two droplets are different under the same flow rate (Fig.5).To facilitate description,the dimensionless droplet average size ls(Fig.8(a)) and the dimensionless droplet size difference ld(Fig.8(b)) are adopted to investigate the effect of droplet size on liquid film drainage time,here ls=(L1+L2)/2Wsand ld=(L1-L2)/Ws,respectively.The viscosity difference of two droplets is reflected by viscosity ratio η (Fig.8(c)).

Fig.8.The effects of dimensionless droplet size and two-droplet viscosity ratio on the film drainage time.(Droplets with different viscosities).(a)Dimensionless average size of two droplets ls;(b) Dimensionless droplet size difference ld;(c) Viscosity ratio of two droplets η.

It could be seen from Fig.8(a) and 8(b) that the liquid film drainage time decreases with the increase of droplet average size,but increases with the increase of droplet size difference,and Tdr～(ld)0.25±0.04.The increase of droplet size elevates the accumulated pressure of the upstream continuous phase and enlarges the contact area of two droplets,consequently,accelerating the coalescence of two droplets.However,the increase of droplet size would cause a time difference of two droplets arriving at the entrance of the funnel-typed expansion chamber,thus increasing the difficulty of coalescence.Moreover,it could be found from Fig.8(c)that the liquid film drainage time decreases exponentially with the increase of two-droplet viscosity ratio,and Tdr～(η)-0.1±0.02.When the viscosity difference of two droplets increases,the local interfacial tension would be increased significantly,which could promote the liquid film drainage and accelerate coalescence.However,when η is greater than 6,the declining trend of liquid film drainage time slows down,indicating that the effect on the liquid film drainage time is weak at larger η,and the effect is more obvious when 1＜η＜6.

3.3.Critical capillary number

Theoretically,the process of droplet coalescence is jointly affected by viscous force,interfacial tension and inertial force.Stone et al.[16] found that the critical capillary number Cacis a key parameter to determine whether coalescence occurs or not.Therefore,Cacis also utilized to analyze quantitatively the occurrence of droplet coalescence in this study,when Ca＞Cac,the coalescence would take place.

3.3.1.Critical capillary number of droplets coalescence of same viscosity

Fig.9 shows the scaling of the critical capillary number with the dimensionless droplet size and the two-phase viscosity ratio for two droplets with same viscosity.As Fig.9(a) shows,the critical capillary number decreases with the increase of droplet size,indicating that for larger droplet,the coalescence could occur even though the capillary number is smaller,thus the operating range of coalescence becomes widened.As shown in Fig.9(b),the critical capillary number increases with the increase of two-phase viscosity ratio.As the dispersed phase viscosity increases,the difficulty degree of coalescence would rise,accordingly,the critical capillary number for coalescence increases.By fitting the dimension droplet size and the two-phase viscosity ratio as the function of critical capillary number using least square method,a correlation is obtained:Cac=0.48λ0.26l-2.64.The comparisons of experimental Cacwith the predicted values are shown in Fig.9(c).It could be seen that the prediction is in good agreement with the experiment under experimental condition.

Fig.9.Variation of critical capillary number with dimensionless droplet size and two-phase viscosity ratio.(Droplets with same viscosity).(a)Critical capillary number Cac versus dimensionless droplet size l;(b) Critical capillary number Cac versus viscosity ratio λ;(c) Comparisons of experimental Cac with the predicted results of correlation.

The obtained results are different with the previous results.The main reasons for this inconsistency are as follows:(1)the physical properties of liquids are different.Most of the previous experiments were aimed at systems with low continuous phase viscosity.For example,Liu et al.[19,20]used silicone oil with a mass fraction of 10%,20% as continuous phase,which viscosity was below 20 mPa.s.Christopher et al.[18] used a silicone oil which the viscosity was 100 mPa.s as continuous phase.The continuous phase ionic liquid used in this experiment has a higher viscosity of251.76 mPa.s.(2) the channel structures used are different.Previous experiments were mostly carried out in conventional Tjunction microchannels,while the channel structure used in this experiment is a T-junction microchannel with a funnel-typed expansion chamber.The experimental results show that the change of the droplet size has a greater effect on coalescence,while the change of the dispersed phase viscosity has slight effect on coalescence in this structure.

3.3.2.Critical capillary number of droplets coalescence of different viscosities

Fig.10 shows the variation of the critical capillary number with the dimensionless average size of two droplets,dimensionless droplet size difference and viscosity ratio η of two droplets.It could be seen from Fig.10(a) and (b) that the critical capillary number decreases with the increase of average size of two droplets and increases with increasing droplet size difference.The increase of the size difference of two droplets would lead to a time difference between the two droplets arriving at the entrance of the expansion chamber,which shortens the contact time of the two droplets,and thereby increases the difficulty degree of coalescence,consequently,the critical capillary number increases.As shown in Fig.10(c),the critical capillary number decreases with the increase of two-droplet viscosity ratio,indicating that the increase of viscosity difference of two droplets could facilitate droplet coalescence,especially when the viscosity ratio is close to 1.When η＞6,the declining trend of the critical capillary number obviously slows down.Moreover,by fitting the dimension droplet average size,the dimensionless droplet size difference and the twodroplet viscosity ratio with the critical capillary number using least square method,the following correlation is obtained:Cac=0.11-.The comparisons of experimental Cacwith the predicted values are shown in Fig.10(d).It could be found that the correlation has considerable prediction performance.

Fig.10.Variation of critical capillary number with dimensionless droplet size and two-droplet viscosity ratio.(Droplets with different viscosities).(a)Dimensionless average size of two droplets ls;(b) Dimensionless droplet size difference ld;(c) Viscosity ratio of two droplets η;(d) Comparisons of experimental Cac with the predicted results by fitting equation.

From the relationship between the critical capillary number to the two-droplet viscosity ratio and the droplet size,it could be found that the droplet coalescence of different viscosities was slightly affected by the dispersed phase viscosity,but was strongly affected by the droplet size.When droplets with different viscosities are come into contact,the local interfacial tension would increase,and the larger droplets would tend to become flat due to more contacting area,thus the interfacial energy would also be higher.According to the Marangoni effect,the gradient of the interfacial tension at the droplet interface would generate a force opposite to the direction of the interfacial tension increasing,which is consistent with the direction of liquid film drainage,this is conducive to droplets coalescence.Therefore,in the coalescence process of the droplets with different viscosities,the interfacial tension is the dominant force,while the effect of viscous force is almost negligible.

4.Conclusions

The effect of the viscosity difference of two droplets on their coalescence was investigated systematically,and the coalescence dynamics was studied.Four coalescence regimes were observed:contact non-coalescence,squeeze non-coalescence,two-droplet coalescence and pinch-off coalescence.With the increase of dispersed phase viscosity,the operating range of coalescence significantly declines,indicating that the increase in the dispersed phase viscosity is not conducive to coalescence.For the coalescence of two droplets with different viscosities,the operating range of non-coalescence decreased with the increase of twodroplet viscosity ratio η,implying that the difference in the viscosity of two droplets could facilitate coalescence.When the viscosity of two droplets was same,the liquid film drainage time decreased with the increase of droplet size,but it linearly increases with dispersed phase viscosity.When the viscosity of two droplets is different,the liquid film drainage time decreases with increasing average droplet size,while increases with the droplet size difference and two-droplet viscosity ratio.The power-law relation was Tdr～(ld)0.25±0.04and Tdr～(η)-0.1±0.02,respectively.Furthermore,the effect of capillary number on droplet coalescence was also explored.The relation of critical capillary number to the droplet size and two-phase viscosity ratio for the coalescence of two droplets with same viscosity was Cac=0.48λ0.26l-2.64.While for the coalescence of two droplets with different viscosities,the scaling of critical capillary number with average droplet size,droplet size difference and two-droplet viscosity ratio is Cac=.This study is helpful for the manipulation of droplet coalescence of different viscosities in the microchannel.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Natural Science Foundation of China(92034303,91834303 and 21776200),and thanks for the aid of Program of Introducing Talents of Discipline to Universities (BP0618007).