INTRODUCTION
Droplet-based microfluidic system
Microfluidics, derived from the terms "Micro" and "Fluidics," involves the manipulation of fluids and gases at microliter or nanoliter scales, enabling high-performance biochemical analysis This technology offers numerous advantages, including precise control, rapid processing, and minimal sample and reagent requirements Microfluidics has been instrumental in the development of Lab-on-a-Chip and micro total analysis systems across various fields such as pharmaceuticals, biomedicine, chemistry, and life sciences In microfluidic systems, laminar flow is utilized under the assumption of no slip boundary conditions, allowing for accurate chemical manipulation Initially used for droplet production in materials processing, droplet-based microfluidic devices have evolved to facilitate chemical and biological analyses, with droplets functioning as micro-reactors of small volumes.
Contraction microchannel in microfluidic system
Droplet-based microfluidic devices often utilize contraction microchannels to create extensional flow with high strain rates, enabling various applications For example, Anna et al demonstrated the generation of water droplets suspended in an oil continuous phase using contraction microchannels Zhu et al conducted experiments on droplet breakup within expansion-contraction microchannels Additionally, hyperbolic contraction microchannels allow for the continuous control and stretching of large DNA molecules, facilitating optical detection and gene analysis Furthermore, the rheological properties of polymeric materials can be effectively measured through the use of contraction microchannels.
Dynamics of droplet in contraction microchannel
Recent investigations into droplet dynamics within contraction microchannels have utilized numerical methods, experiments, and theoretical analyses Most prior numerical studies have focused on two-dimensional scenarios, such as examining the effects of viscoelasticity on droplets and the surrounding medium in a 5:1:5 planar contraction-expansion microchannel through finite element methods Additionally, the entrance effects of contraction geometry and rheological properties on droplet behavior have been analyzed using numerical techniques Furthermore, both numerical and experimental approaches have been employed to assess the impact of shear and elongation on droplet deformation in hyperbolic convergent channels.
Research on droplet dynamics in microchannels has highlighted various influential parameters, including capillary number (Ca), Reynolds number (Re), Weber number (We), and viscosity ratio (λ) Studies by Christafakis and Tsangaris, as well as Harvie et al., have focused on these factors in both two-dimensional and axisymmetric microfluidic contractions However, three-dimensional numerical investigations remain limited, with Zhang et al examining droplet deformation in various three-dimensional contractions Experimental research is also sparse, with Mulligan and Rothstein exploring droplet behavior in planar hyperbolic contractions, while Chio et al investigated the effects of transient pressure on clogging Additionally, Faustino et al analyzed red blood cell deformability in hyperbolic microchannels, and Carvalho et al proposed using GUVs to simulate blood rheology On the theoretical front, Jensen et al conducted research on large wetting bubbles in contraction microchannels to minimize clogging pressure.
Droplet dynamics in extensional flow
The dynamics of droplets in microfluidic systems are influenced by the type of flow, specifically extension or shear Planar extensional flow is commonly used to characterize droplet dynamics, particularly in applications related to materials processing and microfluidics This flow type is essential for analyzing emulsions, polymers, and measuring droplet viscosity through deformation In droplet-based microfluidic systems, extensional flow facilitates the generation, trapping, mixing, and manipulation of small liquid droplets Additionally, research has shown that planar extensional flow can effectively trap and manipulate cells over extended periods, as well as assess cellular mechanical behavior Devices like microfluidic cross-slot systems are utilized to explore the dynamics of vesicles, and planar extensional flow has been employed to quantitatively evaluate mechanical damage to cells in bioreactor environments.
Numerous experimental, numerical, and theoretical studies have been conducted on droplet dynamics, with Taylor being one of the first to present findings on deformation and breakup under shear and extensional flow Building on Taylor's work, subsequent research has explored droplet behavior across various flow conditions, providing detailed insights into three-dimensional droplet shapes.
The study of droplet dynamics in polymer processing encompasses both steady and transient states, highlighting the non-Newtonian behavior of droplets and their surrounding medium due to complex rheological properties Research has focused on how these properties influence droplet deformation and breakup, with particular attention to the effects of droplet position and varying flow rates in axisymmetric extensional flow, which contribute to asymmetric breakup phenomena.
To investigate droplet deformation in extensional flow, several theories have been proposed Taylor introduced a small deformation theory to predict droplet behavior at steady state under low capillary number flow Subsequently, Cox and Barthès-Biesel along with Acrivos explored theoretical models for the transient behavior of droplets Additionally, an approximate theory was developed to describe the breakup of slender droplets under significant deformation A phenomenological model was also presented to characterize droplet deformation in three-dimensional shapes for arbitrary flow conditions.
Problem statement
Recent studies indicate that practical microfluidic systems typically feature microchannels with a rectangular cross-section, highlighting the need for a deeper understanding of droplet dynamics within these channels However, previous research on droplet behavior in contracting microchannels has primarily focused on two-dimensional models, necessitating further investigation for a comprehensive analysis of droplet movement.
The dynamics of droplet behavior in contraction microchannels require thorough investigation through a three-dimensional model, as existing literature lacks comprehensive guidelines for their design This study aims to explore droplet dynamics in detail within contraction microchannels using advanced three-dimensional numerical simulations and theoretical modeling.
In droplet-based microfluidic systems, dynamics of droplet in microfluidic systems is determined by the strength of the flow type which is extension or shear
Due to the high computational costs associated with three-dimensional simulations of microfluidic systems, there is a need for simplified theoretical models to predict droplet deformation Previous research has shown that existing models are often complex This study aims to introduce a simplified model that describes droplet deformation by drawing an analogy between droplet dynamics and a damped spring-mass system The model incorporates external and damping forces to analyze droplet behavior at low Reynolds and capillary numbers Its accuracy has been validated through extensive computational simulations, and the study also proposes a theoretical estimation of the critical capillary number for droplet breakup.
Dissertation overview
This dissertation is structured into six chapters, beginning with an introduction to related works, motivations, and objectives in Chapter 1 Chapter 2 outlines the study's problem description, while Chapter 3 offers a brief overview of the Taylor analogy and details the proposed model Chapter 4 focuses on the three-dimensional computational model and its validation The findings and discussions are presented in Chapter 5, culminating in conclusions and recommendations in Chapter 6 The research framework is illustrated in Figure 1.1.
Fig 1.1 Overview of the dissertation
PROBLEM DESCRIPTION
Problem description of contraction microchannel
Figure 2.1 illustrates a contraction microchannel geometry featuring a droplet with a diameter D suspended in a fluid medium Initially, the droplet is positioned in a larger microchannel measuring L i in length and 2D in width It then moves into a contraction microchannel that is 15D long and W wide To accurately capture the droplet dynamics while minimizing the influence of the outlet boundary, a contraction microchannel length of 15D was selected for this study The overall microchannel depth is set at 2.5D to ensure that the effects of the top and bottom walls on droplet behavior are negligible.
The contraction level, represented by the dimensionless number C, is defined as C=D/W to analyze the impact of microchannel width on droplet dynamics In this context, the initial droplet diameter D consistently exceeds the microchannel width W, with contraction values ranging from 1.11 to 2.5 To optimize computation time, a symmetric model, which represents a quarter of the complete three-dimensional geometry, was utilized, as illustrated in Fig 2.1 (a).
Fig 2.1 Geometry of the contraction microchannel: (a) a full geometry and symmetric domain for computational model, which is illustrated by the grey color and (b) a view from top of the contraction microchannel [52].
Problem description for proposed model
This study focuses on droplet dynamics in microfluidic systems by examining planar extensional flow through a proposed theoretical model and simulation data As depicted in Fig 2.2, a droplet with radius R is suspended in a medium fluid that experiences this type of flow, highlighting the relevant velocity field for analysis.
10 extensional flow is expressed by Equation (2.1) where 𝜀̇ is extension rate and velocity components in X, Y and Z directions are termed V X , V Y and V Z , respectively
The droplet's velocity components are defined as \( v_X = \dot{\epsilon} X \), \( v_Y = -\dot{\epsilon} Y \), and \( v_Z = 0 \) Figure 2.2(a) illustrates the droplet on the XY plane, while a magnified view is provided in Figure 2.2(b) The deformation parameters of the droplet are characterized by L and B, where L represents the half-length of the droplet in the X-direction.
In the context of droplet dynamics, B represents the half breadth of the droplet in the Y-direction, while the displacement of the droplet's equator in the X-direction is expressed as x = L – B It is assumed that the droplet maintains an ellipsoidal shape throughout its deformation, which is quantified by a dimensionless parameter D, as outlined in Equation (2.2) [37,48].
Fig 2.2 Illustration of (a) a droplet suspending in a planar extensional flow and
(b) a description of the droplet magnified at XY plane [53]
Dimensionless numbers
The viscosities of the droplet and medium are represented by μ_d and μ_m, while their densities are denoted as ρ_d and ρ_m The surface tension coefficient between the droplet and medium phases is indicated by σ Droplet dynamics are analyzed using dimensionless parameters, with the capillary number (Ca) defined as Ca = μ_m ε̇ R / σ, where the extension rate ε̇ is expressed as ε̇ = v / R, with v representing a characteristic velocity Two distinct characteristic velocities are employed to define capillary numbers based on the droplet's position in the contraction microchannel The first, Ca_I, is defined as Ca_I = μ_m σ v_i for larger microchannels of length L_i, using the inlet velocity (v_i) as the characteristic velocity In contrast, the capillary number Ca_II, defined as Ca_II = μ_m σ v_c, utilizes the average velocity (v_c) in the contraction microchannel, specifically within a length of 15D.
The values of viscosity, velocity, and interfacial tension can be derived from the definitions of the capillary number The Reynolds number (Re) is calculated using the formula Re = 𝜌 𝑚 𝜀̇𝑅² / 𝜇 𝑚 A viscosity ratio (λ) is defined as λ = 𝜇 𝑑 / 𝜇 𝑚, with a specific value of 0.15 utilized to analyze droplet dynamics in a contraction microchannel The proposed model, based on the Taylor analogy, explores droplet behavior across a broad spectrum of viscosity ratios and capillary numbers Additionally, the density ratio (κ) is set to unity in this study, defined as κ = 𝜌 𝑑 / 𝜌 𝑚.
TAYLOR ANALOGY MODELING
Damped spring-mass model
A basic oscillatory system is made up of a mass, a spring, and a damper The damped spring-mass model is represented by the equation x = (F - d*x')/m, where x denotes the spring's displacement, m represents the mass, F indicates the external force applied, k is the spring coefficient, and d signifies the damping coefficient.
In a damped spring-mass system, the motion can be categorized into three distinct cases based on the damping ratio ξ, defined as ξ = 2𝑚√𝑘/𝑚𝑑 When ξ = 1, the system experiences critical damping, where any slight reduction in damping force results in oscillatory motion For ξ > 1, the system is classified as overdamped, characterized by a large damping coefficient d relative to the spring constant k Conversely, when 0 < ξ < 1, the system is underdamped, indicating a small damping coefficient compared to the spring constant The solutions for these scenarios are represented by Equations (3.2), (3.3), and (3.4), with the displacement x of the spring being non-dimensionalized by setting 𝑦 = 𝑥/𝑅, and defining parameters such as 𝑟₁ = −2𝑚𝑑 + √((2𝑚𝑑)² − 𝑚𝑘) and 𝑟₂ = −2𝑚𝑑 − √((2𝑚𝑑)² − 𝑚𝑘), along with α = 2𝑚𝑑.
𝜔 = √ 𝑚 𝑘 − ( 2𝑚 𝑑 ) 2 , and b 1, b 2 are constants defined based on initial conditions [56]
It can be seen that the displacement at steady behavior is given as y (𝑡 → ∞) = 𝑅 1 𝐹 𝑘
𝑦(𝑡) = 𝑅 1 𝐹 𝑘 + 𝑏 1 𝑒 𝑟 1 𝑡 + 𝑏 2 𝑡𝑒 𝑟 1 𝑡 , when ξ = 1 (3.2) 𝑦(𝑡) = 𝑅 1 𝐹 𝑘 + 𝑏 1 𝑒 𝑟 1 𝑡 + 𝑏 2 𝑒 𝑟 2 𝑡 , when ξ > 1 (3.3) 𝑦(𝑡) = 𝑅 1 𝐹 𝑘 + 𝑒 −𝛼𝑡 (𝑏 1 cos 𝜔𝑡 + 𝑏 2 sin 𝜔𝑡), when 0 < ξ < 1 (3.4)
Taylor analogy breakup (TAB) model
Taylor analogy breakup (TAB) model was developed to depict the droplet breakup in a spray model This TAB model was found to have advantages in terms of simplicity and accuracy, so it has been used in several applications [57-61]
According to the TAB model, the displacement x in Equation (3.1) corresponds to the displacement of the droplet equator x described in Fig 2.2(b), m is the mass of the droplet, F is the pressure drag force, k is the surface tension component, d is the viscosity component More specifically, the physical coefficients in Equation (3.1) can be expressed as Equations (3.5), (3.6), and (3.7), where C F , C k , and C d are the dimensionless constants and v is the relative velocity between the droplet and the medium
In the engine sprays, the surface tension is more dominant than the viscosity, it means the damping ratio ξ is less than unity, thus it can be regarded as an underdamped case Thus, the solution of Equation (3.1) in the TAB model can be written as Equation (3.8)
𝐶 𝑘We)} sin 𝜔𝑡] (3.8) where We is the Weber number defined as We= 𝜌 𝑚 𝜎 𝑣 2 𝑅 , y 0 and 𝑦̇ 0 are initial displacement and velocity, respectively, which are assumed to be zero in the TAB model [55]
Proposed model
This section introduces a theoretical model that utilizes the Taylor analogy to explain droplet dynamics in a low Reynolds number regime It proposes theoretical formulations for the external force (F) and the damping coefficient (d) as outlined in Equation (3.1), while maintaining the surface tension force component consistent with the TAB model presented in Equation (3.6) Notably, the influence of the Reynolds number is not considered in these theoretical models.
In the low Reynolds number regime, viscous friction primarily governs the drag experienced by liquid droplets The viscous drag force acting on a droplet is defined by Equation (3.9) [62] Furthermore, this drag force varies with the droplet's shape, which undergoes deformation in this context, leading to the formulation of an external force as expressed in the relevant equation.
(3.10) where C 1 is a constant The mass of droplet, m is given as 𝑚 = 𝜌 𝑑 4 3 𝜋𝑅 3 Therefore, F/m is given as Equation (3.11) It can be noted that the present model including Equations (3.6) and (3.11) assumes at low Ca and low Re regimes
In the TAB model, the damping coefficient (d) primarily considers the viscosity of the droplet, as the viscosity of air is negligible However, this study examines the effects of both droplet and medium viscosities.
16 are dominant and they should be considered in viscosity effect Hence, the component d/m is empirically proposed as Equation (3.12), where Q is a constant
Finally, by substituting Equations (3.6), (3.11), (3.12) into Equation (3.1) and by using a dimensionless displacement as 𝑦 = 𝑥/𝑅, the Equation (3.1) can be non- dimensionalized as Equation (3.13)
In this study, viscosity plays a more significant role than surface tension, resulting in a damping ratio ξ greater than one, which allows the system to function in an overdamped state The initial conditions for position (y₀) and velocity (𝑦̇₀) are both set to zero The solution to Equation (3.13) is presented in Equation (3.14), where 𝑦ₛ represents the steady-state displacement of the droplet's equator, as defined in Equation (3.15) Additionally, r₁ and r₂ are defined in Section 3.1, while Ca is characterized as Ca = 𝜇ₘ𝑣/𝜎 [52].
The proposed model's performance was validated through a comparison with existing experimental data from the literature [37,40] The evaluation of droplet deformation, represented by D f in Equation (2.2), reveals that the model relies solely on a single one-dimensional displacement parameter, y.
The cross-sectional area of the droplet at the XY plane remains constant during deformation, an assumption that holds true for low Capillary number flow Consequently, the droplet deformation parameter Df can be reformulated as shown in Equation (3.16) The steady-state droplet deformation, occurring at infinite time, is represented as Ds.
Condition for droplet breakup
This study primarily focuses on the dynamics of droplet breakup, aiming to determine the critical capillary number and droplet deformation, which depend on the viscosity ratio A droplet will break when the dimensionless displacement \( y_s \) exceeds a critical coefficient \( C_b \), as outlined in inequation (3.17) Consequently, the critical capillary number \( C_{ac} \) for droplet breakup can be derived, as shown in Equation (3.18).
COMPUTATIONAL MODEL AND VALIDATION
Computational model and methods
Stokes flow, derived from the Navier-Stokes equations, is characterized by negligible inertial forces and is essential for understanding droplet dynamics in microfluidic systems, where laminar flow predominates In this context, both the droplet and medium phases are treated as incompressible Newtonian liquids, with the dynamics described by conservation of momentum and mass To accurately capture the interface between the droplet and the surrounding medium, a volume of fluid (VOF) model is employed, while the continuum surface force model addresses surface tension as a body force Additionally, a no-slip condition is applied at the microchannel wall, and the droplet is considered non-wetting, resulting in a contact angle of 180 degrees between the droplet and the wall.
ANSYS Fluent is utilized for the numerical simulation of droplet deformation within contraction microchannels and planar extensional flows The solution methods incorporate a coupled scheme for pressure-velocity, a second-order upwind scheme for the momentum conservation equation, the PRESTO! scheme for pressure interpolation, and the Geo-Reconstruct scheme for interface interpolation For time discretization, a variable time step method is implemented, with a Courant number of 0.05 to accurately capture transient behavior, while a larger Courant number of 0.25 is suitable for recording droplet deformation at steady state.
Computational domain of contraction microchannel
To minimize computational costs, a symmetric contraction microchannel simulation is represented by a grey domain, as illustrated in Fig 2.1(a) The computational domain is uniformly discretized using hexahedral elements with a size of W/30 Due to a lack of experimental data for validation, the computational model was verified against a T-junction microchannel, as depicted in Fig 4.1 This study aimed to determine the model's ability to capture various droplet generation regimes and included a quantitative analysis of droplet length along the microchannel Fig 4.2 displays three droplet generation regimes linked to different velocities of the dispersed and continuous phases, while Fig 4.3 illustrates droplet length as a function of capillary number for two flow rates Both qualitative and quantitative assessments indicate that the simulation results align well with existing experimental data.
The schematic diagram of the T-junction used for validation is illustrated in Figure 4.1, featuring both a full geometry view and a side view The dimensions are measured in micrometers, with Wc representing the inlet width for the continuous phase and Wd denoting the inlet width for the dispersed phase.
= W c = W d ) and L T is droplet length in the downstream
The study presents three regimes of droplet generation, as illustrated in Fig 4.2, comparing experimental results from Li et al (2012) with current simulations The parameters include continuous phase inlet velocities (v ct) and dispersed phase inlet velocities (v d) at various rates: (a) v ct = 0.83 mm/s and v d = 0.69 mm/s, (b) v ct = 3.47 mm/s and v d = 0.28 mm/s, (c) v ct = 0 mm/s and v d = 5.0 mm/s, and (d) v ct = 0 mm/s and v d = 0 mm/s These findings highlight the relationship between inlet velocities and droplet formation dynamics.
The dimensionless droplet length varies with the capillary number (Ca) for two distinct flow rates of the dispersed phase: 8.06 μL/h and 20 μL/h Specifically, Li et al.'s first experiment and the corresponding simulation utilized a dispersed phase flow rate of 8.06 μL/h, while their second experiment and simulation employed a flow rate of 20 μL/h.
Computational domain for the proposed model
In the study of droplet dynamics under planar extensional flow, a cubic computational domain is utilized To optimize computational efficiency, a symmetric model representing one-eighth of the complete three-dimensional geometry is employed, as illustrated in Fig 4.4 The boundary conditions are defined by surfaces 1, 2, and 3.
In this study, a velocity field defined by Equation (2.1) is applied to three surfaces, while the remaining three surfaces are treated as symmetric boundaries, as illustrated in Fig 4.4 The domain's edge length is set to be six times the radius of the droplet.
The boundary effect of the computational domain on droplet dynamics is negligible when using a discretized mesh of 100×100×100 uniform hexahedral elements Mesh convergence tests were conducted with four different mesh sizes: 40×40×40, 75×75×75, 100×100×100, and 150×150×150 The tests evaluated droplet deformation at steady state and transient behavior, as illustrated in Fig 4.5, confirming that the 100×100×100 mesh provides a reasonable solution while minimizing computational costs.
Fig 4.4 A one-eighth of the full model used for the computational domain in planar extensional flow [53]
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Fig 4.5 Mesh convergence test for λ=1 and Ca=0.067; (a) steady state, (b) transient behavior of the droplet deformation
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