Recently, our direct numerical simulations [Duan et al., Phys. Fluids 36, 033112 (2024)] showed that fluid elasticity affects the extension length and pinch-off time of the droplet formation process, thus changing the flow pattern. However, the effect of fluid elasticity on the morphology and properties of polymeric droplets is not yet fully understood. In this work, by analyzing the stretched state of the polymer macromolecule and the velocity distribution of the flow process, we find that the increase in fluid elasticity (characterized by the relaxation time) inhibits the contraction of the dispersed phase during droplet pinching and resists the effect of surface tension after droplet generation, which significantly affects the droplet geometry, volume, and generation frequency. The results demonstrate that the length and volume of polymeric droplets increase with the relaxation time of the polymer fluid, while the generation frequency decreases. Meanwhile, the effects of polymer viscosity and the superficial velocity ratio of the continuous to the dispersed phase on the droplets' morphology are investigated. The semi-empirical models for the length, volume, and generation frequency of polymeric droplets are developed for the first time by considering the elastic interaction. The purpose of our work is to provide a better understanding and experimental guidance for controlling the parameters of polymeric droplets with viscoelasticity of different shapes and sizes.

In the past two decades, droplet microfluidics technology has flourished through a wide range of work and has been used as the basis for chemical and biological applications.1,2 In these applications, it is crucial to have precise control over droplet characteristics, including shape, size, and generation frequency.3 Numerous studies have demonstrated that the flow rate ratio of the two phases, the viscosity ratio, and the channel structure are the key parameters for regulating the characteristics of Newtonian droplets with simple rheological behavior.4–6 Nevertheless, the mechanisms that affect the characteristics of droplets with complex rheological properties in microchannels are still relatively unknown.7 

To expand the application field of droplet microfluidics, researchers have investigated the generation process of non-Newtonian fluid droplets.8–11 Generally, polymeric droplets are increasingly interesting because of their widespread utilization in chemistry, biomedicine, and materials engineering.12–14 Polymeric droplets typically exhibit complex viscoelasticity due to the microstructure of the fluid within the microchannel being easily distorted.15 Du et al.16 discovered that within flow-focused microchannels, the elasticity of the dispersed phase is a critical factor in determining the droplet breakup mechanism. Changes in this elasticity lead to variations in the liquid–liquid interfacial properties, influencing how droplets form and separate within the microfluidic environment. Wang et al.17 focused on the formation characteristics of polymeric droplets with shear-thinning behavior in T-junction parallel microchannels. Polymeric droplets generated under the same conditions have a larger size and a smaller droplet generation frequency compared to Newtonian droplets. However, accurately determining experimentally the parameters affecting droplet characteristics and providing information on the local flow field of the droplets remains challenging.18 

High-fidelity numerical simulations are expected to complement the aspects of physical phenomena that are not available from experiments.19–21 Nooranidoost et al.22 numerically studied the effect of fluid viscoelasticity on droplet formation and dynamics in an axisymmetric flow-focusing structure. The results demonstrated that viscoelasticity is a critical parameter influencing droplet properties by significantly delaying droplet formation. The numerical investigation by Sontti and Atta23 found that the power-law and consistency indexes play a key role in the length, volume, and deformation index of droplets generated in a non-Newtonian fluid. Nema et al.24 conducted a numerical study on the deformation and migration of viscoelastic droplets in ratchet microchannels. The study revealed that the presence of a ratchet interface leads to a significant stress distribution in the viscoelastic droplets, which, in turn, affects the residence time and shape of the droplets in the microchannel. Unfortunately, the characteristics of polymeric droplets under different fluid elasticities have not yet been investigated in depth.

Recently, we explored the effect of fluid elasticity on the flow pattern of the polymeric droplet generation process based on a three-dimensional direct numerical simulation (DNS) approach.25 By utilizing the volume of fluid (VOF) method and an adaptive mesh refinement scheme, we obtained three typical flow patterns, namely, threading, jetting, and dripping flow, under different elasticities and viscosities of polymer fluids. The variation of the elastic action strength at different polymer fluid viscosities affects the extension length and pinch-off time of the droplet formation process, which, in turn, alters the flow pattern. However, the effect of fluid elasticity on the morphology and properties of polymeric droplets is not yet fully understood. Currently, there is a lack of predictive correlations for the characteristics of viscoelastic polymeric droplets, such as length, volume, and frequency, which limits their practical applications.

In this Letter, we extend our research to study the effect of fluid elasticity on the characteristics of polymeric droplets in capillary microchannels. The exponential Phan–Thien–Tanner (PTT) constitutive model26 is utilized to describe the rheological behavior of polymeric fluids. The rheological parameters and boundary conditions used in this simulation are consistent with our previous work because the simulation results of the viscoelastic two-phase solver have been validated with experimental results in our earlier work.

Our previous work details the current governing equations and numerical schemes for the viscoelastic two-phase solver (rheoInterFoam) in OpenFOAM.25 To stabilize the simulations, we employed both diffusion technique and the log-conformation formulation. It is important to note that in the exponential PTT model, the polymeric stress tensor (τ) is related to the conformation tensor (c). This can be expressed as
τ = η p λ ( 1 ζ ) ( c I ) ,
(1)
where ηp, I, and ζ are the polymeric viscosity, the identity tensor, and the constant model parameter that considers non-affine polymeric chain deformation, respectively. The solver is improved by using the natural logarithmic form of the conformation tensor c, Θ = ln ( c ), which provides excellent robustness to highly elastic flows. The natural logarithm of the conformational tensor tr( Θ) is utilized to obtain an impression of the average strain of the polymer molecule.

This research explores the impact of fluid elasticity, as controlled by polymer relaxation time (λ), on the properties of polymeric droplets, such as length (l), deformation index (DI), volume (V), and generation frequency (f), while also considering the effects of polymer viscosity (ηd) and the superficial velocity ratio of the continuous to the dispersed phase (Uc/Ud) on the droplets' morphology. l and V are obtained using the ParaView post-processing tool, and f is evaluated based on the time required to form two consecutive droplets in a steady state. Meanwhile, the deformation index (DI) of the droplet is calculated using the equation DI = (l − h)/(l + h), where a DI value of 0 signifies a droplet with perfect sphericity.

The morphology of the droplet plays a crucial role in practical application, which affects the property and quality of the product.13,27 Figure 1 illustrates how the geometry of polymeric droplets evolves under different conditions, including relaxation times, viscosities of the dispersed phase, and ratios of superficial velocities. The droplet size is larger for higher relaxation times, while the curvature of the droplet nose slightly decreases. This is mainly due to the fact that under the shear action of the continuous phase, droplets with low relaxation times are in a tight state, while longer relaxation times cause the droplets to be relatively relaxed. Meanwhile, the size of the polymeric droplet decreases as the dispersed viscosity rises. The higher dispersed viscosity prolongs the pinch-off time of the droplet,28 which enhances the shear action of the continuous phase to which the droplet is subjected during droplet formation, resulting in a smaller droplet size. Furthermore, it is worth noting that the superficial velocity ratio exhibits a negative correlation with the droplet size. An elevation of the dispersed phase velocity shifts the occurrence of droplet pinch-offs from the reactor junction to downstream,29 resulting in the formation of larger droplets.

FIG. 1.

(a) Schematic diagram of the cross-junction microchannel and variation of droplet geometry at (b) different relaxation times (ηd = 0.05 kg·m−1·s−1, Uc/Ud = 12), (c) dispersed phase viscosities (λ = 0.02 s, Uc/Ud = 12), and (d) superficial velocity ratios (λ = 0.02 s, ηd = 0.01 kg·m−1·s−1).

FIG. 1.

(a) Schematic diagram of the cross-junction microchannel and variation of droplet geometry at (b) different relaxation times (ηd = 0.05 kg·m−1·s−1, Uc/Ud = 12), (c) dispersed phase viscosities (λ = 0.02 s, Uc/Ud = 12), and (d) superficial velocity ratios (λ = 0.02 s, ηd = 0.01 kg·m−1·s−1).

Close modal

The droplet geometry is dictated by the microscopic stretch state of the polymer, quantitatively captured by the conformational tensor, which is integral in assessing polymer macromolecular structure and determining droplet morphology.30–32  Figure 2 demonstrates the distributions of the trace of the natural logarithm of the conformation tensor Θ of the dispersed phase in a vertical cutting plane. As shown in Fig. 2(a), the relaxation time has a significant impact on the tr( Θ) of the polymeric droplets. The value of tr( Θ) increases as the relaxation time rises and the polymer molecules deform to a greater extent. In Fig. 2(b), the region of high tr( Θ) is reduced by the increase in dispersed phase viscosity for the same relaxation time. This phenomenon is primarily caused by the interaction of viscous and elastic forces. When the dispersed phase has low viscosity, the elastic effect becomes more prominent, enhancing the deformation degree of the polymer macromolecules. Moreover, Fig. 2(c) exhibits the effect of superficial velocity ratios on tr( Θ). An augmented superficial velocity of the dispersed phase leads to a discernible stretching of the droplet, which extends the high tr( Θ) zone within it.

FIG. 2.

Distributions of the trace of the natural logarithm of the conformation tensor Θ of the dispersed phase in a vertical cutting plane: (a) effect of relaxation times (ηd = 0.05 kg·m−1·s−1, Uc/Ud = 12), (b) effect of dispersed phase viscosities (λ = 0.02 s, Uc/Ud = 12), and (c) effect of superficial velocity ratios (λ = 0.02 s, ηd = 0.01 kg·m−1·s−1).

FIG. 2.

Distributions of the trace of the natural logarithm of the conformation tensor Θ of the dispersed phase in a vertical cutting plane: (a) effect of relaxation times (ηd = 0.05 kg·m−1·s−1, Uc/Ud = 12), (b) effect of dispersed phase viscosities (λ = 0.02 s, Uc/Ud = 12), and (c) effect of superficial velocity ratios (λ = 0.02 s, ηd = 0.01 kg·m−1·s−1).

Close modal

The axial velocity (Ux) distributions of the continuous and dispersed phases are given in Fig. 3. Figures 3(a) and 3(b) illustrate that with an increase in relaxation time and a corresponding decrease in the viscosity of the dispersed phase, there is a significant elevation in Ux within the central region of the droplet. This phenomenon can be attributed to the enhanced elasticity, which inhibits the breakup of the dispersed phase,25 consequently resulting in larger droplet formation. Additionally, the thinning of the liquid film minimizes leakage of the continuous phase fluid from the interstitial “gutters” between the droplet and the channel walls.33 This not only elevates the internal velocity of the droplet but also amplifies the radial Ux gradient within it. Regions of high velocity are also frequently associated with substantial deformation of the polymer, as depicted in Figs. 2(a) and 2(b). Figure 3(c) shows a remarkable enhancement of the droplet velocity as the superficial velocity ratio falls. Moreover, the difference in curvature of the droplets from nose to tail creates a pressure gradient between the droplets, which alters the velocity of the continuous phase.6,34

FIG. 3.

Distributions of the velocity vector field in a vertical cutting plane: (a) effect of relaxation times (ηd = 0.05 kg·m−1·s−1, Uc/Ud = 12), (b) effect of dispersed phase viscosities (λ = 0.02 s, Uc/Ud = 12), and (c) effect of superficial velocity ratios (λ = 0.02 s, ηd = 0.01 kg·m−1·s−1). Note that the velocity magnitude of Ux is shown inside the droplet.

FIG. 3.

Distributions of the velocity vector field in a vertical cutting plane: (a) effect of relaxation times (ηd = 0.05 kg·m−1·s−1, Uc/Ud = 12), (b) effect of dispersed phase viscosities (λ = 0.02 s, Uc/Ud = 12), and (c) effect of superficial velocity ratios (λ = 0.02 s, ηd = 0.01 kg·m−1·s−1). Note that the velocity magnitude of Ux is shown inside the droplet.

Close modal

Figures 4(a) and 4(b) illustrate the effect of relaxation time on polymeric droplet length for different dispersed phase viscosities and superficial velocity ratios. Note that at low superficial velocity ratios, threading flow occurs at low relaxation times, and no droplets are produced, resulting in no data points. This is because the elastic effect is small under these conditions and is not sufficient to resist the stretching of the dispersed phase fluid downstream of the reactor.25 An increase in relaxation time leads to an increase in polymeric droplet length. Liu et al.35 reported similar results for the formation of viscoelastic droplets in a step-emulsification microdevice. The dependence of droplet length on relaxation time is attributed to the restraining effect of elasticity on the contraction process of the dispersed phase during droplet pinching and the resistance of elasticity to surface tension after droplet generation. In addition, the viscosity of the dispersed phase and the superficial velocity ratio are inversely related to the polymeric droplet length for the same relaxation time. Figure 4(c) shows that the deformation index of droplets gradually increases with the rise of the relaxation time at low viscosity. For droplets at higher viscosity, the increase in viscous action weakens the elastic action of the polymeric droplets. As a result, the height of the droplet is greater than its length, resulting in a negative deformation index. A negative correlation between dispersed phase viscosity and deformation index was also observed by Sontti and Atta.6 For high dispersed phase viscosity, at lower relaxation times, droplet shape is governed by the equilibrium between viscous forces, elastic forces, and surface tension. Initially, viscous forces keep the shape near its original, while elastic forces and surface tension act toward a more spherical shape, affecting DI fluctuations. With increased relaxation times, enhancing elasticity, microchannel geometry—especially radial width—becomes crucial, forcing axial elongation due to limited height growth, thus affecting DI. In Fig. 4(d), it can be seen that the larger velocity of the dispersed phase leads to a larger deformation index at superficial velocity ratios. This causes the droplet to change shape from nearly spherical to slug.

FIG. 4.

Effect of relaxation time on polymeric droplet length for (a) different dispersed phase viscosities and (b) superficial velocity ratios. Effect of relaxation time on polymeric droplet deformation index for (c) different dispersed phase viscosities and (d) superficial velocity ratios. Effect of relaxation time on droplet volume for (e) different dispersed phase viscosities and (f) superficial velocity ratios. Effect of relaxation time on droplet generation frequency for (g) different dispersed phase viscosities and (h) superficial velocity ratios.

FIG. 4.

Effect of relaxation time on polymeric droplet length for (a) different dispersed phase viscosities and (b) superficial velocity ratios. Effect of relaxation time on polymeric droplet deformation index for (c) different dispersed phase viscosities and (d) superficial velocity ratios. Effect of relaxation time on droplet volume for (e) different dispersed phase viscosities and (f) superficial velocity ratios. Effect of relaxation time on droplet generation frequency for (g) different dispersed phase viscosities and (h) superficial velocity ratios.

Close modal

The droplet volume and generation frequency are essential features in the design of microfluidic systems.11, Figure 4(e) depicts that an increase in relaxation time results in an expansion of the droplet volume, while an increase in viscosity leads to a decrease in droplet volume. The breakup mode of the droplet is governed by the competition between elastic and viscous effects, which, in turn, affects the droplet volume.22 At high viscosities, viscous effects dominate, which extends the stretching length of the polymer fluid. This means that smaller volume droplets are formed downstream of the reactor. On the other hand, the strain rate at the moment of droplet breakup is large, and polymer fluid with high relaxation times hinders droplet generation due to elastic effects, resulting in larger droplets. Moreover, the droplet volume is significantly affected by superficial velocity ratios, as illustrated in Fig. 4(f). At a constant superficial velocity of the continuous phase, the volume of the droplets increases as the velocity of the dispersed phase increases. This is the result of a significant enhancement of the dispersed phase inertial force to a level sufficient to resist the continuous phase shear stress.23 As shown in Figs. 4(g) and 4(h), an increase in relaxation time reduces the droplet generation frequency. With greater relaxation time, the enhancement of the elastic force resists the droplet pinch-off process. Therefore, the droplet generation time is slowed down, and the generation frequency is diminished. This phenomenon agrees with the experimental conclusion that elasticity inhibits the droplet pinch-off process.36 Furthermore, it is evident from Fig. 4(g) that an increase in the dispersed phase viscosity, at the same relaxation time, improves the droplet generation frequency. The combination of the shear action of the continuous phase and the viscosity increment results in the formation of smaller droplets, which ultimately increases the droplet generation frequency, although the increased dispersed phase viscosity elevates drag. The results of the simulations agree with previous studies obtained experimentally.7,17 In Fig. 4(h), it can be seen that the droplet generation frequency is greater as the superficial velocity ratios decrease. The main reason is that as the dispersed phase flow rate increases, the time it takes for the dispersed phase fluid to penetrate shortens.37 

The recently proposed semi-empirical model establishes a relationship between the length of a non-Newtonian droplet and various key parameters, including the two-phase flux ratio, the viscosity ratio, and the continuous-phase capillary number.17 These correlations serve as a valuable tool for predicting droplet morphology with complex rheological properties, which is critical for advancing the understanding and design of multiphase flow systems.38 Here, we employed a semi-empirical correlation equation considering the Weissenberg number Wi, the dispersed phase capillary number Cad, and the superficial velocity ratios to predict the droplet size and generation frequency. By fitting the data to the operating conditions, the droplet dimensionless length, volume, and generation frequency are predicted, as shown in the following equations:
l w d = 1.33 W i 0.02 C a d 0.05 ( U c U d ) 0.24 ( R 2 = 0.98 ) ,
(2)
V w d 3 = 0.57 W i 0.04 C a d 0.13 ( U c U d ) 0.33 ( R 2 = 0.98 ) ,
(3)
f = Q d V = U d 0.57 w d W i 0.04 C a d 0.13 ( U c U d ) 0.33 ,
(4)
where wd = 100 μm is the width of the dispersed phase inlet of the microreactor and Wi = λUd/wd ranges from 0.002 to 160. Cad = Udηd/γ ranges from 0.0027 to 0.027, where γ = 0.036 kg·s−2 is the interface tension coefficient. Uc/Ud ranges from 1.5 to 12. Meanwhile, the prediction model's errors are displayed in Fig. 5. The proposed correlations are found to be reliable, as the predicted polymeric droplet lengths and volumes are within 10% error, and the predicted generation frequencies are within 20% error.
FIG. 5.

Prediction of (a) droplet length, (b) droplet volume, and (c) droplet generation frequency.

FIG. 5.

Prediction of (a) droplet length, (b) droplet volume, and (c) droplet generation frequency.

Close modal

In summary, we use direct numerical simulations (DNS) to explore the impact of fluid elasticity, characterized by polymer relaxation time, on the properties of polymeric droplets such as length, deformation index, volume, and generation frequency, while also considering the effects of polymer viscosity and the superficial velocity ratio of the continuous to the dispersed phase on the droplets' morphology. By analyzing the stretched state of the polymer macromolecule and velocity distribution of the flow process, we find that the increase in fluid elasticity inhibits the contraction of the dispersed phase during droplet pinching and resists the effect of surface tension after droplet generation, which significantly affects the droplet geometry, volume, and generation frequency. The results show that the greater the relaxation time of the polymer fluid, the larger the length and volume of the polymeric droplets and the lower the generation frequency. Meanwhile, it is worth noting that the polymer viscosity exhibits an inverse relationship with the droplet size, while it is directly proportional to the generation frequency. On the other hand, the superficial velocity ratio of the continuous to the dispersed phase shows an inverse relationship with both the droplet size and the generation frequency. Finally, semi-empirical models are developed, considering the elastic interaction, to characterize the length, volume, and generation frequency of polymeric droplets. Our work aims to present a better comprehension and experimental guidance for controlling the parameters of polymeric droplets with viscoelasticity of different shapes and sizes.

We gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 52306203) and QCY Innovative and Entrepreneurial Talent Programme of Shaanxi Province (No. QCYRCXM-2022-134).

The authors have no conflicts to disclose.

Lian Duan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Wenjun Yuan: Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Nanjing Hao: Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). Mei Mei: Investigation (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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