Breakup dynamics of slender droplet formation in shear-thinning fluids in flow-focusing devices

Author’s Accepted Manuscript Breakup dynamics of slender droplet formation in shear-thinning fluids in flow-focusing devices Taotao Fu, Youguang Ma, Huai Z. Li

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S0009-2509(16)00030-0 http://dx.doi.org/10.1016/j.ces.2015.12.031 CES12753

To appear in: Chemical Engineering Science Received date: 21 July 2015 Revised date: 16 October 2015 Accepted date: 31 December 2015 Cite this article as: Taotao Fu, Youguang Ma and Huai Z. Li, Breakup dynamics of slender droplet formation in shear-thinning fluids in flow-focusing devices, Chemical Engineering Science, http://dx.doi.org/10.1016/j.ces.2015.12.031 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Breakup dynamics of slender droplet formation in shear-thinning fluids in flow-focusing devices Taotao Fua,b*, Youguang Maa* and Huai Z. Lib a

State Key Laboratory of Chemical Engineering, Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China

b

Laboratory of Reactions and Process Engineering, University of Lorraine, CNRS, 1, rue Grandville, BP 20451, 54001 Nancy cedex, France * Corresponding authors: [email protected] (T. Fu) ; [email protected] (Y. Ma)

Abstract This work reports the breakup dynamics for slender droplet formation in shear-thinning fluids in flow-focusing devices consisted of respectively 600 μm and 400 μm square microchannels. Silicon oil droplets are generated in non-Newtonian shear-thinning fluids - polyacrylamide (PAAm) aqueous solutions. The thinning of the thread of the dispersed phase during droplet formation can be characterized by a power-law relationship with the remaining time before the final pinch-off, with an exponent of 1/3 in the non-universal collapse stage, followed by an exponent of 1 in the final universal pinch-off stage. The dependence of the pre-factors on the viscosity ratio of both phases and the capillary number is discussed. A scaling law is proposed to predict the size of droplets formed in shear-thinning fluids in such devices by taking into account the rheological property of the non-Newtonian fluids as well as the hydrodynamics.

Keywords: microfluidics, complex fluids, droplet, breakup, pinch-off, hydrodynamics

1

Introduction Over the last decades, microfluidics has gained increasing importance in a wide range of academic investigations and industrial applications (Juarez and Arratia, 2011; Whitesides, 2006). Droplets are frequently encountered in the applications of microfluidics and have received much attention in recent years (Baroud et al., 2010; Bremond and Bibette, 2012; Christopher and Anna, 2007). They are usually generated by either a flow-focusing device or a T-junction based on cross-flowing rupture technique either by active or passive methods (Anna et al., 2003; Carrier et al., 2014; Thorsen et al., 2001). For the passive method, the droplet size is controlled by various parameters, such as the flow rates and viscosities of both phases, the channel size, and the interface tension between two phases (Christopher and Anna, 2007; Garstecki et al., 2006). Recently, several studies have been performed on the effects of liquid viscosities on droplet formation in microfluidic devices, which are quite important for the application of microfluidic technique in chemical engineering, polymer engineering, food engineering, and biological engineering. For example, for droplets formed at a microfluidic T-junction in the dripping regime, the size of droplets decreases with the increase of the viscosity of the continuous phase, as the pinch-off process is accelerated by the augmented drag forces with high viscosities of the continuous phase (Husny and Cooper-White, 2006). In the squeezing regime, the size of droplets is not relevant to the viscosity of the continuous phase (Garstecki et al., 2006). In the transition regime, the experimental results indicate that the droplet formation is not affected by the viscosity ratio of dispersed/continuous phases when its value is less than 1/50 (Christopher et al., 2008). In addition, many active methods, such as magnetic, thermal, electric and surface acoustic wave (SAW), are also used to provide better control on the droplet size (Nguyen et al., 2007; Schmid 2

and Franke, 2013; Tan et al., 2008, 2011, 2014a, 2014b), with the advantage of non-invasion towards the fluid flow in microchannels. For example, the droplet size can be well controlled by an ac voltage applied around a flow-focusing junction, and the electrodes do not contact with the fluids in the microchannel to avoid electro-chemical effects (Tan et al., 2014a). The droplet can either be formed in the dripping regime at low voltages or be generated in the jetting regime at high voltages. The droplet size can be scaled with an effective capillary number taking into account the Maxwell stress in the dripping regime, which explains the dependence of the droplet size on the applied voltage. While in the jetting regime, the stability of the droplet generation is related to the fluid conductivity and the applied field frequency (Tan et al., 2014a).

Most of previous works concern the droplet formation in Newtonian fluids with low viscosities (Christopher et al., 2008; Fu et al., 2012b; Garstecki et al., 2006), while many fluids for industrial applications are likely to exhibit complex behaviors, such as high viscosity, and non-Newtonian properties (Arratia et al., 2009; Arratia et al., 2008; Ingremeau and Kellay, 2013; Juarez and Arratia, 2011; Nghe et al., 2011; Qiu et al., 2010; Sang et al., 2009; Steinhaus et al., 2007). For example, the microfluidics technique has been employed as a promising tool to generate multiple droplets that contain polymeric shells (Abate et al., 2011). Up to now, only a few studies treat with the droplet formation in highly viscous or non-Newtonian fluids in both T-junctions and flow-focusing junctions (Derzsi et al., 2013; Qiu et al., 2010; Sang et al., 2009). The special character for non-Newtonian fluids is that their viscosities are dependent of the shear rates, which are varied with the dynamical evolution of the liquid-liquid interface during droplet formation in confined spaces. For droplets formed in non-Newtonian fluids in a T-junction, the numerical results indicate that the rheology of 3

the continuous fluid affects significantly the formation process and the size of droplets (Qiu et al., 2010). Droplets are formed according to two different breakup mechanisms: dripping and jetting regime, characterized by the capillary number (Qiu et al., 2010). The droplet size decreases with the increase of the apparent viscosity of the continuous phase and can be predicted by an analytical model based on the force balance for droplets formed in non-Newtonian fluids in a T-junction (Sang et al., 2009). The effect of the elasticity of the continuous phase on the droplet formation in flow-focusing devices has been studied recently to show that the elasticity reduces slightly the size of the formed droplet, in comparison with the Newtonian focusing liquid (Derzsi et al., 2013). However, the distribution of droplet sizes depends on the rheological properties of the continuous phase for droplets formed in non-Newtonian focusing liquids in such devices (Derzsi et al., 2013).

In comparison with the production of droplets in Newtonian fluids, the information on the generation of droplets in non-Newtonian fluids in microfluidic devices is still quite limited. To the best of our knowledge, the dynamics of droplet formation in non-Newtonian fluids in flow-focusing devices is far from being fully understood. The present work focuses on the dynamics of oil droplet formation in shear-thinning non-Newtonian fluids within flow-focusing devices. Silicone oil droplets are often applied in therapeutic protein products, especially in pre-filled syringes or double-chamber cartridges, retina surgery, oil-in-water (O/W) emulsions, and even used as intraocular tamponade and microreactors (Barca et al., 2014; Elvira et al., 2013; Umbanhowar et al. 2000).

Experimental Section The microfluidic flow-focusing devices are fabricated in a plate (45 mm × 27.5 mm × 2 mm) of polymethyl methacrylate (PMMA) by precision milling and sealed with another thin PMMA plate by 4

screws, with which, the channel has not been observed to be deformed to change the demension. It consists of four microchannels with square cross sections of width Wc which meet at the crossing section (Wc = 400 μm and 600 μm, designated as +400 and +600, respectively). The dispersed phase is introduced at a volumetric flow rate Qo from the main channel, and the continuous phase is introduced from the side channels at a rate of Qw/2 (Figure 1). Stainless steel tubes (di = 1 mm) are used to connect the inlets and outlet of the microchannel to tygon tubes (ID = 1.02 mm), which are employed to connect the microchannel device with liquid supplies. Liquids are delivered at constant flow rates by using syringe pumps (Harvard Apparatus, PHD 22/2000, USA). All of our experiments are conducted at room temperature and atmospheric pressure. A high-speed camera CamRecord600 (Optronis GMBH, Germany) equipped with a microscopic magnification lens is set above the channel to capture images. In this work, the frame rate is 1000 fps (frames per second). The spatial resolution of the images was up to 2 μm/pixel. The images are processed by a Matlab program to obtain the quantitative features for the droplet formation, with an error of ± 2 μm (1 pixel) for the length scale. A cold fiber light (Jeulin S.A., France) is placed at the other side of the device to illuminate it. A micro-Particle Image Velocimetry (micro-PIV) system (Dantec Dynamics, Denmark) is used to obtain velocity fields in the continuous phase around the thread of the dispersed phase during the droplet formation.

Silicone oils [polydimethylsiloxane (PDMS)] (HS10, HS100) (Dow Corning 200, France) are used as the dispersed phase (Table 1). Two different concentrations of polyacrylamide (PAAm) (SNF Floerger, France) (0.1 and 1 wt %) in demineralized water are employed as the continuous phase. They behave as shear-thinning non-Newtonian fluids and can be characterized by the power-law 5

model in the range of the shear rates (1 - 600 s-1) corresponding to the volumetric flow rates of the fluids: h = KDn-1 , where η and D are the viscosity and shear rate, respectively; K is the consistency and n the flow index. A Rheometric Fluid Spectrometer RFS II (Rheometric Scientific, USA) is employed to measure the rheological property of the fluids as shown in Figure 2. The values of K and n for the PAAm solutions are gathered in Table 2. Interfacial tensions (surface tensions) of the fluids are measured using a tensiometer by the pendant drop technique on a Tracker apparatus (I.T. Concept, France), as shown in Tables 1-3.

Results and Discussion A typical process for droplet formation in the dripping regime is illustrated in Figure 3. The evolution of the droplet formation can be described as follows: after the detachment of the formed droplet, the thread of the dispersed phase retracts (0-20 ms); the thread of the dispersed phase expands in both the axial and radial directions to form a slender one (20-55 ms); the slender thread is then stretched mainly in the axial direction (55-100 ms); after then, a neck is formed and the thread of the dispersed phase contracts around the neck (100-125 ms); finally, the thread of dispersed phase pinches off quickly to form a new droplet (126-131 ms). Unlike droplets formed in Newtonian fluids with low viscosities in such devices (Fu et al., 2012b), a cusp is observed for both the rear of the just formed droplet and the tip of the just pinched-off dispersed thread (131 ms), and the thread of the dispersed phase is narrower than the channel throughout the entire formation process. The differences can be attributed to the high pressure around the thread of the dispersed phase, which focuses on the thread of the dispersed phase to thin it in the radial direction, and to the high viscous forces around the thread of the dispersed phase owing to the high viscosity of the PAAm solutions, 6

which stretches the thread of the dispersed phase in the axial direction (Fu et al., 2011; Fu et al., 2012a; Xu and Nakajima, 2004). The high pressure around the thread of the dispersed phase can be understood by the fact that the liquid pressure is proportional to the product of the flow rate and the resistance according to the Hagen-Poiseuille law, and for a single liquid phase flow in a channel the resistance is proportional to the product of the liquid viscosity and the length of the channel. Thus, the high viscosity of the PAAm solutions used results in great resistance that leads to high pressure around the thread of the dispersed phase when other parameters were kept constant. In addition, owing the blocking of the continuous phase by the dispersed phase at the cross-junction, the pressure can be also increased in the continuous phase around the thread of the dispersed phase to drive it, which is called as a blocking-pinching mechanism (Guillot and Colin, 2005; Garstecki et al., 2005; Guillot et al., 2007).

The dynamical evolution of the thread neck of the dispersed phase can give information to the pinch-off mechanism of the dispersed phase for the droplet formation (Dollet et al., 2008; Eggers and Villermaux, 2008; van Hoeve et al., 2011). The evolution of the minimum width Wm of the thread of the dispersed phase during droplet formation in a flow-focusing device is plotted as shown in Figure 4a. After the stretching stage, the thread of the dispersed phase collapses at a gradually increased speed, followed by a fast pinch-off stage. As shown in Figure 4, the fast pinch-off stage and the collapse stage are distinguished by the different thinning rate of the minimum width of the thread for the dispersed phase. The relationship between the minimum width of the thread of the dispersed phase Wm and the remaining time before the final breakup (T - t) can be characterized by a power-law function as shown in Figure 4b: Wm = B(T - t )a , with an exponent of a1 = 0.30 and a 7

pre-factor B1 = 64 for the collapse stage and an exponent of a 2 = 0.99 and a pre-factor B2 = 22 for the final pinch-off stage, where T is the period for the droplet formation and t the time. The final pinch-off time is defined as the moment when the neck of the thread of the dispersed phase reaches its critical width Wm=0 and breaks. From the images captured by the high speed camera, it should be found that this moment occurs between the last frame before the actual final pinch-off (130ms in Fig. 3) and the first frame after the pinch-off (131ms in Fig. 3). Therefore, the time of the final pinch-off a

was estimated by assuming that the collapse exhibits power law behavior with Wm µ (T - t ) , where the exponent and the period T are a priori unknown, similarly as done in van Hoeve et al., (2011). From the best fit to the data, we obtained the exact pinch-off moment and thereby the period for droplet formation, with the maximum systematic error is equal to the time between two frames (usually less than it as shown for example, the error bars for time in Fig. 4b). With this method, we a

obtained the Wm = B (T - t )

relationship for various experimental conditions as shown in Figure 5a.

In this figure, the error bars for time are not shown to illustrate clearly the experimental data. By using this relationship, we can also obtain

Wm Wc = A1,2 ( (T - t ) Tc )

a1,2

in the following part to

plot Fig. 6 & 7, without the error bars shown in the figures. Moreover, the cross-over point for the two stages is observed at about (T - t )o = 4.7 ms and Wo=101.8 µm before the final breakup as shown in Fig.5a. The collapse stage is controlled by a flow-driven mechanism that is governed by the surface tension of liquid viscosity (which can be explained by the filling effect as discussed in van Hoeve et al., 2011), and the pinch-off stage is by a self-similar surface-tension-driven mechanism that is governed by the balance between viscous forces of both phases and surface tension at the liquid-liquid interface (Eggers and Villermaux, 2008), as reported previously for bubble and droplet formation in Newtonian fluids in a similar device (Dollet et al., 2008; Fu et al., 2010a; Fu et al., 8

2012b; Garstecki et al., 2005). Guillot et al., (2007, 2008) analyzed the effect of the confinement on the stability of a jet in a cylindrical/square/rectangular geometry at low Reynolds number, and they related the transition between the droplet regime and the jet regime to the absolute-convective transition of the Rayleigh plateau instability. For the nonsqueezed jet in square/rectangular geometry, where the continuous phase completely wets the channel walls and where the dispersed phase is not squeezed by the geometric confinement, the presence of the corners between the dispersed thread and the channel walls allow the instabilities to develop at a finite rate as the continuous phase can be transferred through these corners. They also showed that decreasing the size of the jet radius decreases the interfacial area and energy cost, which promotes the instability (Guillot et al., 2008). Thus, the thinning of the thread of the dispersed phase is accelerated owing to the promoted instability with the decreasing of the thread width. After the cross-over point of the width of the thread of the dispersed phase, the thinning is accelerated even greatly as it is driven by the interfacial tension to experience the capillary instability finally, also known as the Rayleigh-Plateau instability (Rayleigh, 1879; Taylor, 1934). Therefore, the diminishing speed of the thread in the final pinch-off stage is faster than that in the collapse stage, i.e. a 2 > a1 . This phenomenon is similar to the breakup dynamics for bubble formation in a flow-focusing device. van Hoeve et al. (2011) found two stages for the breakup of the gaseous thread during bubble formation in a flow-focusing device: the neck collapses with a scaling exponent of 1/3 driven by a filling effect, followed by a 2/5 scaling liquid inertia driven stage. Lu et al. (2014b) observed that the exponent is less than 1/3 in the beginning of the breakup and increases to 0.5 approaching to the final pinch-off, for the breakup of the gaseous thread for bubble formation in a flow-focusing device with cross-junctions. These studies suggested that the transition of the exponent is the result of the interface rearrangement of the gas-liquid 9

interface during its evolution in the axial and radial directions accompanying the varying of the dominated forces.

Similar phenomena are found for a wide range of experimental conditions as shown in Figure 5a. For the collapse stage, a1 is approaching 1/3, ranging from 0.19 to 0.35 and is sensitive to the initial conditions in the experiments. This verifies that the collapse of the thread of the dispersed phase is driven by the flow-driven mechanism. Dollet et al. (2008) found that in the ‘final’ stage of the collapse for bubble formation in a flow-focusing device the radius of the neck scales with the remaining time with a 1/3 power-law, which suggested that gas inertia and the Bernoulli suction effect become important. However, van Hoeve et al. (2011) revealed that during the collapse, the flow of gas reverses and accelerates towards its maximum velocity at the moment of pinch-off. But the resulting decrease in pressure due to Bernoulli suction, as too low to account for the accelerated collapse. They proposed that the 1/3 scaling law can be explained by the filling effect of the liquid on the gas in the neck, in which a pressure driven flow through a thin liquid layer surrounding the neck causes it to thin with an exponent of 1/3. The liquid flow rate in the film between the gas-liquid interface and the channel walls can be described by using Darcy’s law for pressure driven flow in porous media, which is related to the pressure drop that can be correlated to the capillary pressure. The gas flow rate at the gaseous neck region can be expressed through the volume in the neck region. The gas in the neck should be filled by the liquid to lead to the balance between the gas flow rate and liquid flow rate in the neck region to lead to the 1/3 power law. Furthermore, as shown in Figure 5a, the exponent a1 seems to approach to 0.33 with the decrease of Qo Qw (with the increase of Qw at a fixed Qo) for a certain liquid-liquid system in a flow-focusing device as shown in Fig. 5b. This 10

phenomenon is similar to Fig. 3b in Lu et al., (2014b) for bubble formation in low-viscosity liquid at a flow-focusing device, signifying that the flow-driven effect is more predominated at high flow rates of the continuous phase. As shown in Fig. 5c, The exponent

a1 seems to increase with the

increase of Reynolds number for a certain liquid-liquid system in a flow-focusing device, and this effect is true for both low and high concentrations of PAAm solution except that the Reynolds number falls into lower ranges for the high concentration of PAAm solution (1% ). This phenomenon suggests that the flow-driven effect is more predominated at high Reynolds numbers for a certain liquid-liquid system in a flow-focusing device. This stage is non-universal and depends on the initial conditions. For the final pinch-off of the thread of the dispersed phase, a 2 » 1 . This exponent is not affected by the initial conditions, suggesting a universal self-similar surface-tension-driven mechanism for the breakup of the thread of the dispersed phase in highly viscous non-Newtonian fluids. In this stage, the viscous force balances the surface tension force in Stokes regime as the Reynolds numbers are quite small, i.e. Reo £ 0.018 , Rew £ 2.13 (Table 4) (Cohen et al., 1999). The Reynolds number for both phases are defined as Reo = rouoWc mo and Rew = rwuwWc hw , where uo = Qo Wc2 , r o , m o are the mean velocity, density and viscosity of the dispersed phase

respectively; uw = Qw Wc2 , r w , h w are the mean velocity, density and viscosity of the continuous phase respectively, with hw = KDw n-1 , and

Dw = uw W c . The present result of

power-law relation is reminiscent of drops formed in quiescent immiscible liquids in conventional columns at macroscale (Lister and Stone, 1998). For drop formed in potential flow regime in conventional large-scale devices (Lister and Stone, 1998) under dominant inertia conditions, the exponent is 2/3; while viscosity is relevant (Lister and Stone, 1998), the exponent is close to 1. It is worth noting that a very short-lived intermediate transient viscous regime exists for HS10 droplets 11

formed in 1.0% PAAm solutions at high flow rates of the continuous phase (240 µL/min and 480 μL/min) as experimental data illustrated within the circles as shown in Fig. 5, which is observed also numerically for the breakup of liquid filament during droplet formation in conventional large-scale devices recently (Castrejón-Pita et al., 2015). However, this period is very short compared to the durations for the other stages during droplet formation in this study. For example, the approximate durations for the collapse stage, the transient viscous stage, and the final pinch-off stage for HS10 droplets formed in 1.0% PAAm solutions at high flow rates of the continuous phase are respectively 126/133.45, 2/133.45, 5.45/133.45 for 240 µL/min, and 120/128.6, 2/128.6, 6.6/128.6 for 480 μL/min. Moreover, the amount of time spent by filaments in each regime remains unexplored (Castrejón-Pita et al., 2015). It should be also noted that the elastic effect of the PAAm solutions on the droplet formation can be neglected as the PAAm solutions have viscous fluid-like behavior for small Deborah number De (De <<1), and the De is defined as the ratio of the characteristic time of fluid to the characteristic time of process (Frank and Li, 2006). The relaxation time is the typical time-scale for the isolated PAAm particle – water system to eventually reach equilibrium irrespective of its initial state, and depends in detail on the nature of the inter-particle interactions. The flow time of process is the time for the fluid flow required to traverse a distance of characteristic length for the microchannel (usually taken as the width of the microchannel) at a certain mean velocity. For example, the relaxation time is about 10 ms for 1% aqueous PAAm solution, and the typical flow time is about 200 ms (0.4 mm/2 mm/s) in the present experiment, leading to De = 1/20. Moreover, the axisymmetric thread should also inhibit the elastic effect.

We also find that the curves for Wm ~ (T - t ) have similar trend as shown in Figure 5, which 12

triggers us to construct a universal function for Wm ~ (T - t ) . The minimum width of the thread of the dispersed phase Wm is non-dimensioned by the width of the microchannel Wc, and the remaining time (T - t ) by a capillary time Tc ( Tc = rWc3 g ) (Basaran, 2002): Wm Wc = A1,2 ( ( T - t ) Tc )

a1,2

,

where the subscripts 1 and 2 denote the collapse stage and the final pinch-off stage, respectively. The prefactors A1 , A2 and the exponents a1,2 thus represent the demensionless thinning speed for the thread of the dispersed phase, that are determined by the driven mechanisms for the thinning process. For the collapse stage, A1 can be correlated with the capillary number Caw’ and the viscosity ratio of both phases mo h '

w

(

by using multiple regression method: A1 = 0.034Caw' -0.44 mo hw'

)

-0.14

(Fig. 6,

with an averaged relative standard deviation of 18.44%), where Caw' = uwhw' g is the local capillary number

of

the

continuous

phase,

and

hw' = KDw' n-1

,

Dw' = 3 u 'w (Wc - Wo )

,

uw' = Qw (Wc2 - πWo2 4 ) are respectively the viscosity, and the shear rate and the mean velocity of , the continuous phase when the width of the thread of the dispersed phase reaches the transition one Wo. It is plausible to explain this scaling law by the argument that the collapse stage for the breakup of the thread of the dispersed phase is predominated by the flow-driven mechanism, so that the thinning is associated to the flow rates, viscosity and surface tension of fluids in terms of dimensionless groups Caw' and mo h ' . The absolute value of the exponent -0.44 for Caw' is w

larger than that of -0.14 for mo h '

w

demonstrates that the effect of the Caw' , representing the

relative importance of viscous forces to surface tension forces, is pronounced on the thinning compared to the mo h '

w

representing the relative importance of viscosity for the dispersed phase to

the continuous phase. It should be also pointed out that the flow rate of the dispersed phase is missing in this expression for A1, which effect on the thinning is reflected by the influence on a1 . Nevertheless, a solid theoretical model for A1 needs to be explored further (Fu et al., 2010a, 2012a; 13

Dollet et al., 2008; van Hoeve et al., 2011). In comparison with bubble formation in PAAm solutions in flow-focusing devices (Fig. 7 in Fu et al., 2012a), A1 = 0.08Caw' -0.39 is obtained, signifying similar physics at play. While in the bubble case, mo h ' is so small that can be ignored in the w

expression. It is also noteworthy that the transition width of the thread of the dispersed phase is used as a typical scale to obtain the shear rate for estimating the viscosity of the continuous phase, which is also dependent of initial conditions.

For the fast pinch-off stage, A2 is supposed to be A2 = H ( mo hw' )

g mo Wc Tc

, where H ( mo hw' ) is a

universal function related to the viscosity ratio of both phases that needs to be determined as suggested by the dimensional analysis (Cohen et al., 1999; Cohen and Nagel, 2001). This suggests that if droplet profiles near rupture are rescaled by g mo (T - t ) , they should collapse onto a universal curve so that the profiles may be self-similar near pinch-off. In the present experiment,

H (mo hw' ) = 0.0075(mo hw' )-0.50 for +600 µm (with an averaged relative standard deviation of 9.52%), and H (mo hw' ) = 0.0061(mo hw' )-0.52 for +400 µm (with an averaged relative standard deviation of 1.54%) as shown in Fig. 7, with mo hw' Î ( 0.2,0.9 ) , Wm Î ( 0,100 ) μm and ,

(T - t ) Î ( 0,10) ms .

g é 30 11 ù Î m.s -1 , , mo êë103 103 úû

The small deviation for the pre-factors for the two

devices stems probably from the typical scale Wo used for obtaining the shear rate. The law for the exponent is consistent with previous results as: H (mo hw ) µ (mo hw )-0.53±0.05 , for mo hw Î ( 0.1,10 ) (Cohen et al., 1999). This demonstrates that the final pinch-off is universal and driven by a surface-tension mechanism. For the prefactor, In Cohen et al., (1999, 2001), an example was presented

to

show

the

thinning

of

the 14

thread

of

the

dispersed

phase

as

Wm = ( 0.031 ± 0.008)

g 30mN.m-1 g » » 0.03m.s -1 , Wm Î ( 0,30 ) μm , (T - t ) , for mo hw = 1 , mo 1000mPa.s mo

(T - t ) Î ( 0, 40) ms .

In fact, H ( mo hw' ) =

A2 Wc Tc

g mo

=

Wm Wc Wc Tc Wm (T - t ) = represents the g mo (T - t ) Tc g mo

ratio of the thinning speed of the thread of the dispersed phase to the speed

g , i.e., a capillary mo

number based on the thinning speed of the thread of the dispersed phase. From this study and that in Cohen et al., (1999, 2001), we can conclude, at least, that the magnitude of the thinning speed of the thread of the dispersed phase for the two cases is comparable, by taking into account the prefactors obtained - the viscosity ratio of the dispersed phase to continuous phase, and the

g . However, mo

Cohen et al., (1999, 2001) also pointed out that the solid simple expression for the viscosity ratio dependence of the parameters characterizing the shape of the liquid-liquid interface around the pinch-off area is missing, and another dimensionless group such the Reynolds number of the flows needs to be determined. This maybe another reason for the slight deviation of the prefacor (0.0075 vs. 0.0061) between the two microchannels used in this study. Furthermore, it should be pointed out that the viscosity of the continuous phase we calculated in this case is based on a mean shear rate by using a characterized scale ̢ the width of the thread of the dispersed phase. As a matter of fact, there is definitely a distribution in the shear rate in the continuous phase (both in the liquid-liquid interface and in the bulk, as well as in three dimension) around the thread of the dispersed phase according to the velocity distribution in the continuous phase that, of course, leads to a viscosity distribution in the continuous phase – the shear thinning fluid used. In addition, the velocity distribution and thereby the shear rate and viscosity distributions of the shear thinning fluid is time-dependent (see Fig. 10 in Fu et al. 2012a for example - viscosity distribution around the gaseous 15

thread during bubble formation in shear-thinning fluids that as used in this study). Keeping this in mind, the viscosity ratio mo hw' is thus a space- and time-dependent variable, making it more difficult to attain an exact expression for A1 , A2 .

The transition width Wo of the dispersed phase is determined definitely by the joint action of the collapse stage and the pinch-off stage. Thus, the above-mentioned parameters predominating the variation of the width of the thread of the dispersed phase for the two stages (i.e. Wm Wc = A1,2 ( (T - t ) Tc )

a1,2

) were used to predict the transition width Wo by using multiple

regression method. As shown in Fig. 8, the dimensionless transition width Wo of the dispersed phase from

the

collapse

Wo Wc = ( Qo Qw )

0.12

stage

Caw' -0.18 ( mo hw' )

to 0.054

the

pinch-off

stage

is

expressed

as

, by taking into account both the hydrodynamics of the

fluids and the physical properties of two phases. This relationship is understandable that the increase of Qo Qw mo hw' and the decrease of Caw' lead to the increase of dimensionless transition width , , of the dispersed phase from the collapse stage and the pinch-off stage, that’s to say, the collapse stage gives way to the pinch-off stage at greater transition width of the dispersed phase at such situations owing to the greater resistance of the dispersed phase to the continuous phase. It is also noting that the Wo Wc is predominated by Qo Qw

,

and Caw' rather than mo hw' from the exponents

expressed in this equation, which can be understood by the fact that the viscosity of the continuous phase is coupled with the local shear rate that is determined by the flow rate of the continuous phase and the liquid-liquid interface, and the latter one is predominated by the Qo Qw and Caw' . Besides , this consideration, the effect of the viscosity of the continuous phase on this expression may also be reflected in terms of Caw' . 16

To reveal the breakup mechanism for droplet formation in flow-focusing devices, the velocity distributions in the continuous phase around the thread of the dispersed phase are determined by the micro-PIV technique as shown in Fig. 9. The flow-driven mechanism is approved by Fig. 9a-e, as the continuous phase flows downstream in these situations. Furthermore, the surface tension-driven mechanism is by Fig. 9f as the continuous phase around the neck of the dispersed thread moves towards the fast pinching thread, owing to the rapid contracting of the dispersed thread driven by the surface tension. This mechanism is demonstrated clearly in Fig. 10, captured for the moment just before the final pinch-off of the thread of the dispersed phase. A reversal flow around the neck of the thread of the dispersed phase is observed owing to the pressure differences between the place at a downstream position (Fig. 10b) and that around the neck of the dispersed phase, owing to the fast contracting of the neck of the dispersed phase. This phenomenon is also observed for the breakup of bubble formation in a low-viscosity liquid in a microfluidic T-junction (van Steijn et al., 2009), and in a non-Newtonian fluid (PAAm solution) in a flow-focusing device (Fu et al., 2012a). Furthermore, a recirculation zone is noticeably present in the continuous phase at the corner of the introducing channel as shown in Fig. 9f & 10a, owing to the interplay of the thread of the dispersed phase and the continuous phase with high viscosity at the cross-junction.

For the size of droplets formed in non-Newtonian fluids in flow-focusing devices, we find that the droplet size decreases with the increase of the flow rate of the continuous phase as shown in Fig. 11 and Fig. 12a & b. A dimensionless number is defined to quantify the droplet size as r/Wc. r is the equivalent radius of the droplet r = ( 3Vb 4p ) , and Vb is the droplet volume and is calculated 13

17

directly from the captured images by Vb = p ( wb 2 ) L - ( 4 3) p ( wb 2 ) , where L is the droplet 2

3

length, and wb the droplet width as shown in Fig. 11. Furthermore, the droplet size augments with the increase of the flow rate of the dispersed phase and the channel size as shown in Fig. 12a & b, and with the decrease of the concentration of the PAAm solutions, owing to the reduced viscosity of the continuous phase at the same shear rate with lower concentration of the PAAm solutions. It is found that the influence of the viscosity of the dispersed phase on the droplet size can be negligible under the same flow rates of both phases as shown in the inset of the Fig. 12c, in which, the viscosity of the dispersed phase changed almost ten-fold ( mo Î (11,103) mPa.s ) with the dimensionless Ohnesorge number Oho of the dispersed phase within Oho Î ( 0.086,0.79 ) . The Ohnesorge number is defined as

Oho =mo

roWcg which represents the ratio of internal viscosity dissipation to the surface tension ,

energy. This phenomenon is similar to the previous literature regarding to the droplet formation in co-flowing highly viscous fluids via a glass nozzle with 0.02 mm of the inner diameter in a flow channel of 20 mm in height and 2.5 mm in width (Cramer et al., 2004). It is noteworthy that the viscosity of the continuous shear-thinning fluid could be varied with the bubble size as the shear rate varied with the change of the flow field in the continuous phase around the droplet even at the same flow rates of both phases employed, this effect seems to be more pronounced than that of the viscosity of the dispersed phase on the droplet size as shown in Figure 12c. The dimensionless size of formed droplets r/Wc can be scaled with the flow rate ratio of both phases as a power-law relationship as illustrated in the insets in Fig. 12a & b & c as well as Fig. 13a, which coincides with the power-law scaling proposed for flow-driven breakup mechanism of droplets in microfluidic flow-focusing devices (Fu et al., 2012b; Garstecki et al., 2005). Also, it is found that the dimensionless size of formed droplets r/Wc can be scaled with the capillary number Caw with a 18

power-law relationship as shown in Fig. 13, which supports the surface-tension-driven mechanism for the pinch-off of the thread for the dispersed phase for droplet formation in flow-focusing devices (Fu et al., 2012b; Umbanhowar et al., 2000). Overall, a scaling law is proposed by taking into account the flow rates of both phases and the capillary number as r Wc = 0.44(Qo Qw )0.18 Caw'' -0.15

,

where Caw'' = hw'' uw'' g is the local capillary number of the continuous phase around the forming droplet, and hw'' = KDw'' n-1 , Dw'' = 3uw''

(

)

2Wc - Wb , uw'' = Qw (Wc2 - pWb2 4 ) are the characteristic

viscosity of the non-Newtonian fluids, the representative shear rate (Arratia et al., 2008) and the mean velocity of the continuous phase around the droplet. This expression can be used to predict the size of formed droplets in flow-focusing devices within a wide range of experimental conditions gathered in Table 4. It is worth noting that the form of our scaling law for the droplet size is similar to those proposed for the formation of both bubbles and droplets in Newtonian or non-Newtonian fluids with low and high viscosities in various microfluidic devices, but with different pre-factors and exponents as shown in Table 5 (Christopher et al., 2008; Cubaud and Mason, 2008; Fu et al., 2010b, 2012a; Fu et al., 2012b; Lu et al., 2014a; Tan et al., 2008; Xu et al., 2008). These scaling laws suggest a coupling rule of squeezing mechanism and shearing mechanism for bubble or droplet formation in flow-focusing devices (Fu et al., 2012b; Lu et al., 2014a). Furthermore, our results suggest that the key point for the scaling law for droplet formation in non-Newtonian fluids is to define a suitable characteristic shear rate to obtain the apparent viscosity of the continuous phase, i.e., the non-Newtonian fluids.

It should be noted that, the findings – scaling laws, in the present study for surfactant free system needs to be checked to see if it is valid still or not for surfactant involved systems. The surfactants 19

are usually added to the system in microfluidics applications, to facilitate the controlled droplet generation and to prevent the droplet coalescence in the collection sections. For a tentative study, we found a 1/3 scaling law for the thinning of the thread of the dispersed phase during the generation of silicon oil droplet in a surfactant sodium dodecyl sulfate (SDS) covered polyethylene oxide (PEO) aqueous solutions (Zhang et al., 2015).

Nevertheless, the scaling laws we found paves the way for

further experimental and theoretical studies on breakup dynamics and mechanism for droplet formation in surfactant involved systems in microfluidic applications. In addition, we also found recently that the generation of the water-89.5%glycerol highly viscous droplet in highly viscous silicon oils in a flow-focusing device experiences also two distinct thinning stages, due to the disparity of the strain field at the point of the detachment, which results in the symmetric and asymmetric breakup of the thread of the dispersed phase in a flow-focusing junction (Du et al., 2015). Moreover, we did not observe droplet coalescence phenomenon in the downstream straight channel, as the oil droplets move separately by PAAm plugs in the simple and single channel employed. Nevertheless, droplets (and bubbles) can coalesce in surfactant free systems in microchannels with specific geometries such as expansion and T-junction, where they have the chance to touch (Bremond et al., 2008; Fu et al., 2015; Jose and Cubaud, 2012; Wu et al., 2014).

Conclusions In summary, this study shows that the rheological property of the continuous phase affects the breakup dynamics for droplet formation by varying the local viscosity of the continuous phase around the dispersed phase, owing to the variation of the shear rate accompanying the dynamical evolution of the liquid-liquid interface in confined spaces. This study demonstrates that the breakup 20

process for the droplet formation in shear-thinning fluids in flow-focusing devices can be categorized into a non-universal collapse stage followed by a universal fast pinch-off stage. The collapse stage is dependent of initial conditions; while the pinch-off stage is not relevant to initial conditions. The minimum width of the neck of the dispersed thread and the remaining time before the final breakup can be scaled by power-law relationships, and the exponent in the final pinch-off is about 1, greater than 1/3 for the collapse stage. While the pre-factor A2 in the final pinch-off can be correlated with

g Tc ( moWc ) . The pre-factor A1 in the collapse stage can be correlated with the capillary number Caw' and the viscosity ratio of both phases mo h ' . A scaling law is proposed to predict the w

transition of the minimum width of the dispersed thread from the collapse stage to the final pinch-off stage. The droplet size is scaled with the flow rate ratio of both phases Qo Qw and the capillary number Caw'' , with a suitable characteristic shear rate for estimating the viscosity of the continuous shear-thinning fluids. This work extends previous studies on breakup dynamics for droplet formation in Newtonian fluids to non-Newtonian fluids in flow-focusing devices (Fu et al., 2012b). The scaling law for the transition between the collapse stage and the pinch-off stage reveals that this breakup process for droplet formation in shear-thinning fluids in such devices is controllable. Therefore, a general scaling law can be also proposed for the size of formed droplets in shear-thinning fluids in flow-focusing devices, by taking into account the hydrodynamics of fluid flow and the physical property of the fluids. This work paves the way for the application of droplet-based microfluidics technique in the engineering involving complex fluids with non-Newtonian rheology, such as double emulsions with non-Newtonian fluids used as the middle phase, and multiphase micro-reactors involving complex fluids (Derzsi et al., 2013; Elvira et al., 2013; Nghe et al., 2011).

21

Acknowledgments The financial supports for this project from the National Natural Science Foundation of China (21276175), the aid of Opening Project of State Key Laboratory of Chemical Engineering (SKL-ChE-13T04) and the Program of Introducing Talents of Discipline to Universities (B06006) are gratefully acknowledged. T. Fu appreciates the financial assistance from both the China Scholarship Council and the French Embassy in China. We thank the anonymous referees for their valuable comments that have helped improve this paper substantially.

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Tan, J., Xu, J.H., Li, S.W., Luo, G.S., 2008. Drop dispenser in a cross-junction microfluidic device: Scaling and mechanism of break-up. Chemical Engineering Journal 136, 306-311. Tan, S.H., Semin, B., Baret, J.-C., 2014a. Microfluidic flow-focusing in ac electric fields. Lab on a Chip 14, 1099-1106. Tan, S.H., Maes, F., Semin, B., Vrignon, J., Baret, J.-C., 2014b. The Microfluidic Jukebox. Scientific Reports 4, 4787. Tan, S.H., Nguyen, N.T., 2011. Generation and manipulation of monodispersed ferrofluid emulsions: The effect of a uniform magnetic field in flow-focusing and T-junction configurations. Physical Review E 84, 036317 Tan, S.-H., Sohel Murshed, S.M., Nguyen, N.-T., Wong, T.N., Yobas, L., 2008. Thermally controlled droplet formation in flow focusing geometry: formation regimes and effect of nanoparticle suspension. Journal of Physics D: Applied Physics 41, 165501. Taylor, G.I., 1934. The formation of emulsions in definable fields of flow. Proceedings of the Royal Society of London. Series A 146, 501-523. Thorsen, T., Roberts, R.W., Arnold, F.H., Quake, S.R., 2001. Dynamic pattern formation in a vesicle-generating microfluidic device. Physical Review Letters 86, 4163-4166. Umbanhowar, P.B., Prasad, V., Weitz, D.A., 2000. Monodisperse emulsion generation via drop break off in a coflowing stream. Langmuir 16, 347-351. van Hoeve, W., Dollet, B., Versluis, M., Lohse, D., 2011. Microbubble formation and pinch-off scaling exponent in flow-focusing devices. Physics of Fluids 23, 092001. van Steijn, V., Kleijn, C.R., Kreutzer, M.T., 2009. Flows around confined bubbles and their importance in triggering pinch-off. Physical Review Letters 103, 214501. 27

Whitesides, G.M., 2006. The origins and the future of microfluidics. Nature 442, 368-373. Wu, Y., Fu, T., Zhu, C., Ma, Y., Li, H.Z., 2014. Bubble coalescence at a microfluidic T-junction convergence: from colliding to squeezing. Microfluidics and Nanofluidics 16, 275-286. Xu, J.H., Li, S.W., Tan, J., Luo, G.S., 2008. Correlation of droplet formation in T-junction microfluidic devices:from squeezing to dripping. Microfluidics and Nanofluidics 5, 711-717. Xu, Q., Nakajima, M., 2004. The generation of highly monodisperse droplets through the breakup of hydrodynamically focused microthread in a microfluidic device. Applied Physics Letters 85, 3726-3728. Zhang, Q., Fu, T., Zhu, C., Ma, Y,, 2015. Formation and size prediction of slug droplet in viscoelastic

fluid

in

flow-focusing

microchannel.

CIESC

Journal,

DOI

˖

10.11949/j.issn.10438-11157.20151056.

Caption of figures Figure 1. The photograph of the microfluidic flow-focusing device with the same size for all of the square channels. The dispersed silicone oil phase is introduced from the main channal with a volumetric flow rate of Qo, and the continuous phase - 0.1 % PAAm solution is fed from the lateral channels with a volumetric flow rate of Qw/2. wc = 400 μm.

Figure 2. The rheological behavior of PAAm solutions used.

Figure 3. Evolution of droplet formation in a microfluidic flow-focusing device. Dispersed phase: HS10, continuous phase: 0.1% PAAm solution. Qo =30 µL/min. Qw = 240µL/min. Microfluidic 28

device: +400. T = 131 ms.

Figure 4. (a) Evolution of the minimum width of the dispersed thread Wm. The inset pictures represent the evolution of the thread of the dispersed phase. The minimum width of the dispersed thread Wm is shown on the bottom inset on the left side. The dashed line characterizes the transition from the collapse stage to the fast pinch-off stage. (b) Relationship between the minimum width of the dispersed thread Wm and the remaining time before final pinch-off (T - t). T is the period for the droplet formation, t the time. Dispersed phase: HS10, continuous phase: 0.1% PAAm solution. Qo = 30 µL/min, Qw= 240 µL/min. Microfluidic device: +400. T = 131 ms.

Figure 5. (a) Relationship between the minimum width of the dispersed thread Wm and the remaining time before final breakup (T - t) under various experimental conditions. The open experimental data represent: dispersed phase - HS10, continuous phase - 0.1% PAAm solution, +600; the solid experimental data represent: dispersed phase - HS10, continuous phase - 1% PAAm solution, +600; the half up - solid experimental data represent: dispersed phase - HS100, continuous phase - 1% PAAm solution, +600; the half down - solid experimental data represent: dispersed phase - HS10, continuous phase - 0.1% PAAm solution, +400.

The experimental data in circles represent the

range of the transient regime for HS10 droplets formed in 1.0% PAAm solutions at high flow rates of the continuous phase (240 ­L/min and 480 μL/min). (b) Relationship between a1 and Qo Qw . (c) Relationship between a1 and Re w .

' Figure 6. Relationship between A1 and the local capillary number Caw and the viscosity ratio

29

mo h 'w for the dispersed phase to the continuous phase.

' Figure 7. Relationship between A2 and the viscosity ratio mo h w for the dispersed phase to the

continuous phase. Continuous phase: 0.1% PAAm solution.

' Figure 8. Relationship between Wo/Wc and the local capillary number Caw , the flow rate ratio

Qo Qw and the viscosity ratio mo h ' w for the dispersed phase to the continuous phase.

Figure 9. Velocity distributions in the continuous phase around the dispersed thread. Dispersed phase: HS10, continuous phase: 1% PAAm solution. Qo = 10 µL/min, Qw= 20 µL/min. +400. (a) t ms, (b) t + 133.33 ms, (c) t + 266.67 ms, (d) t + 400 ms, (e) t + 800 ms, (f) t + 2133.33 ms.

Figure 10. The reversal flow around the dispersed thread in the continuous phase and the formed recirculation zone in the liquid introducing channel. Dispersed phase: HS10, continuous phase: 1% PAAm solution. Qo = 10 µL/min, Qw= 20 µL/min. +400.

Figure 11. Effects of the flow rate of the continuous phase on the droplet size. Dispersed phase: HS10. Continuous phase: 0.1% PAAm solution. wc = 400 μm. Qo = 30 μL/min. (a) Qw = 60 μL/min; (b) Qw = 120 μL/min; (c) Qw = 240 μL/min; (d) Qw = 320 μL/min.

Figure 12. Effects on the droplet size. (a) the flow rates of both phases; (b) the concentration of the continuous phase and the size of the microchannel; (c) the viscosity of the dispersed phase. The 30

insets: scaling for the dimensionless droplet size r/Wc with the flow rate ratio of both phases Qo/Qw.

'' Figure 13. The scaling for the dimensionless droplet size r/Wc with the capillary number Caw . (a) '' '' r/Wc ~ Caw and Qo/Qw; (b) r/Wc ~ Caw for various flow rates of the dispersed phase; (c) r/Wc ~

Caw'' for various concentrations of the continuous phase and sizes of the microchannel.

Caption of tables Table 1. Physical properties of the dispersed silicone oil phases.

Table 2. Physical properties of PAAm solutions used.

Table 3. Interfacial tension g between the dispersed silicone oil phase and the continuous PAAm solutions (unite: mN·m-1).

Table 4. Experimental conditions for the droplet formation in non-Newtonian fluids in flow-focusing devices.

Table 5. Scaling laws for the size of bubbles and droplets formed in microfluidic devices in literature.

31

Table 1. Physical properties of the dispersed silicone oil phases.

Fluid

Viscosity μo,

Density ρ, kg·m-3

Surface tension γ, mN·m-1

mPa·s HS10

920

11

19.2

HS100

960

103

20.2

Table 2. Physical properties of PAAm solutions used.

Density, ρ

Surface tension, γ

Fluid

Consistency, K (Pa sn)

Flow index, n

(kg/m3)

(mN/m)

0.1 % PAAm

1000

71

0.34

0.49

1 % PAAm

1000

67.7

7.91

0.29

Table 3. Interfacial tension γ between the dispersed silicone oil phase and the continuous PAAm solutions (unite: mN·m-1).

Fluid

HS10

HS100

0.1 % PAAm

29.1

31.3

1 % PAAm

29.6

29.5

32

1

100

Min

Max

Qo (µL/min)

2000

1

Qw (µL/min)

571.8

6.2

Dw (s-1)

2.16

0.013

h w (Pa s)

33

0.824

0.005

mo h w

0.223

0.00187

Caw

0.018

5 × 10-5

3 × 10-5 2.13

Cao

Rew

Table 4. Experimental conditions for the droplet formation in non-Newtonian fluids in flow-focusing devices.

0.23

0.0026

Reo

1.12

0.34

r/wc

Droplet

Droplet

Droplet

flow-focusing,

square

Wc: 250;

μd: 0.5-8; μc: 1-10

0.82-50

μd: 50-1214; μc:

Wc: 100

flow-focusing,

0.92-10.18

600

square

μd: 11, 103; μc:

(0.10-1.00%)

solutions

Wc: 400,

600

Wc: 400, μd: 11-103; μc: PAAm

Viscosity (mPa.s)

flow-focusing,

square

flow-focusing,

size (μm)

geometry

droplet

Droplet

Channel

Channel

Bubble or

0.23

0.14

Cac -0.42

Cac -0.19

34

L Wc = 1.59 (Qd Qc )

(L <2.5h)

Cac -0.19

Cac -0.20

0.19

0.20

L Wc = 0.5 (1 + Qd Qc )

( L>2.5h)

L Wc = 0.0022 (1 + Qd Qc ) Cac -1

(L/Wc>2.35)

L Wc = 0.30 (Qd Qc )

(L/Wc<2.35)

L Wc = 0.72 (Qd Qc )

r Wc = 0.44(Qd Qc )0.18 Ca''-w0.15

Scaling laws

Cac<0.01

10-5
0.001
0.00187< Ca '' w <0.223

Capillary numbers

Table 5. Scaling laws for the size of bubbles and droplets formed in microfluidic devices in literature.

(Tan et al.,

Mason, 2008)

(Cubaud and

2012b)

(Fu et al.,

work

present

References

10.84

h: 200

rectangular

h: 50

rectangular

μd: 0.54; μc: 0.92–

μd: 1; μc: 6-350

Wc: 150;

T-junction,

Wc: 300;

0.92-10.18

h: 40

rectangular

T-junction,

μd: 0.018; μc:

(0.10-1.25%)

solutions

Wc: 120;

600

device

μd: 0.018; μc: PAAm

μd: 0.018; μc: 5-400

T-junction,

Wc: 400,

Wc: 600

flow-focusing

device

flow-focusing

h: 200

Subscripts: d – dispersed phase; c – continuous phase

Droplet

Droplet

Bubble

Bubble

Bubble

rectangular

c

2

d

0.18

Cac -0.25

35

1/3

Qc ) , Cac -1 3

Wc = 0.26 (Qd Qc )

(W h ) ~ (Q

0.5

Cac-0.33

Cac -0.29

0.27(1-n )

0.52

L Wc = 0.75 (Qd Qc ) Cac -0.2

Vb

( Lwb )

r Wc » 0.41(Qd Qc )

Vb Wc 3 = 1.12 (Qd Qc )

0.002
0.01
0.0058
0.008
0.00065
al., 2008)

(Xu et

al., 2008)

(Christopher et

2010b)

(Fu et al.,

2012a)

(Fu et al.,

2014a)

(Lu et al.,

2008)

36

Figure 1

37

Figure 2

38

Figure 3

39

Figure 4

100

(a)

(b)

0.4

1

10

Wm (mm) 0.2 0.0

0.3

a1

0.33

1

0.2

40

Qo/Qw

10

(T - t) (ms)

0.4

100

0.6

30/900, 100, 0.26, 24, 0.99 100/400, 135, 0.23, 34, 1.01 100/2000, 75, 0.33, 24, 0.94 30/60, 91, 0.23, 17, 1.01 30/120, 78, 0.28, 12, 1.04 30/240, 71, 0.28, 7, 1.05 30/480, 50, 0.34, 5.7, 0.95 30/120, 44, 0.35, 4.0, 0.96 30/60, 103, 0.19, 31, 1.02 30/120, 85, 0.22, 22, 0.99

Qo/Qw (mL/min), B1, a1, B2, a2

0.4

(c)

0.2

0.3

a1

0.33

0.01

41

Figure 5

0.1

Rew

1

HS10-PAAm 0.1%, +600, Qo=30 mL/min HS10-PAAm 0.1%, +600, Qo=100 mL/min HS10-PAAm 1%, +600, Qo=30 mL/min HS100-PAAm 1%, +600, Qo=30 mL/min HS10-PAAm 0.1%, +400, Qo=30 mL/min

A1 0.10

0.15

0.20

0.25

0.30

0.35

3

4

6

Figure 6

42

7

(mo/h¢w)

-0.44

Ca¢w

5 -0.14

8

9

10

+600, HS10-PAAm0.1% +600, HS10-PAAm1% +600, HS100-PAAm1% +400, HS10-PAAm0.1%

1

0.034

A2moWc/(gTc) 0.008

0.009

0.010

0.011

0.012

0.013

0.3

0.5

43

0.6

mo/h¢w

Figure 7

0.4

+600 +400

0.7

A2=0.0061(mo/h¢w)

0.8

0.9

gTc/(moW c)

gTc/(moW c)

-0.52

-0.50

HS10-PAAm0.1% A2=0.0075(mo/h¢w)

Wo/Wc 0.1 0.8

0.2

0.3

Ca¢w

44

0.15

(mo/h¢w)

1.6 -0.18

Figure 8

(Qo/Qw)

0.12

1.2

1

+600, HS10-PAAm0.1% +600, HS10-PAAm1% +600, HS100-PAAm1% +400, HS10-PAAm0.1%

0.054

2.0

45

46

Figure 9

47

Figure 10

48

Figure 11

49

(c) 0.6

0.4

0.5

r/Wc

mo/h¢¢w

50

Figure 12

0.01

+600, 1%PAAm, Qo=10 mL/min

HS100, Oho=0.79

HS10,Oho=0.086

0.6 0.11

0.4

0.1

0.14

0.8

Qo/Qw

r/W c=0.56(Qo/Qw)

r/W c=0.56(Qo/Qw)

0.4

r/Wc

0.8

1.0

(a)

(b)

0.5

0.6

0.7

0.8

0.9

1.0

0.4

0.6

r/Wc

r/Wc

-0.67

0.4

0.010

Ca¢¢w

51

+400, HS10-0.1%PAAm

0.005

Qo/Qw

-0.30

r/W c=0.14Ca¢¢w

r/W c=0.03Ca¢¢w

0.0

+600, HS10-0.1%PAAm Qo=100 mL/min r/W c~Qo/Qw

0.04 0.28

0.8

0.8

1.0

0.4

0.6

0.015

Qo=10 mL/min

Qo=1mL/min

r/W c~Ca¢¢w

r/W c=0.96(Qo/Qw)

Ca¢¢w -0.70

r/W c=0.05Ca¢¢w

0.02

r/Wc

(c)

r/Wc 0.2

0.4

0.6

0.8

1.0

1.2

0.05

Ca¢¢w

52

Figure 13

0.10

0.15

-0.91

r/W c=0.014Ca¢¢w

-0.64

r/W c=0.05Ca¢¢w

Qo=30 mL/min

+600, HS10-0.1% PAAm +600, HS10-1% PAAm +400, HS10-0.1% PAAm

0.20

-0.70

r/W c=0.12Ca¢¢w

0.25

53

5. A scaling law is proposed to predict the size of droplets

54

4. The thinning of the thread of the dispersed phase can be characterized by a power-law relationship

3. The rheological property of the continuous phase affects the breakup dynamics for droplet formation

2. The breakup dynamics of liquid-liquid interface is highlighted

1. Droplet formation in shear-thinning fluids is studied

Highlights: