Analysis of and methods for void-free liquid filling of blind microchambers in centrifugal microfluidics

Abstract

Centrifugal microfluidics are widely used in point-of-care testing applications. Blind microchambers, microchambers that have only one access point by which to interact with an external environment, are commonly used in centrifugal microfluidic chips. However, achieving void-free liquid filling of blind microchambers poses a significant challenge as the injection of liquid and the exhausting of air occurs simultaneously and thus interference leads to incomplete liquid filling with the presence of residual bubbles. To resolve this issue, we propose a strategy for achieving void-free liquid filling of blind microchambers by designing a tapered microchannel to modify the gas–liquid two-phase flow pattern, effectively preventing bubble formation. The liquid–gas two-phase flow pattern is analysed, and the corresponding inference is verified via high-speed camera analysis. According to the theoretical and experimental findings, tapered designs are implemented to the branch channels connected to the blind microchambers. By using tapered designs, the fluid velocity increases, leading to the transitions from Taylor flow to annular flow, thereby avoiding bubble generation during liquid injection. Our work reveals a mechanism that offers a simple path to achieve void-free liquid filling of blind microchambers in centrifugal microfluidics, without the need for complex surface treatments or external forces, and therefore has the potential to benefit the microfluidics community.

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Introduction

Centrifugal microfluidic chips, in which the transport of a liquid between various microchambers is achieved by centrifugal force, have drawn great interest in recent years. By using tailored designs, the liquid in the sample injection chambers or the microfluidic channels can be automatically driven into a group of reaction chambers, making centrifugal microfluidic chips particularly suitable for high throughput screening and analysis of biomarkers such as nucleic acids and so on.^1–4 Moreover, the on-demand control of the liquid offers the possibility to integrate multiple processing steps on a single chip, offering advantages in terms of low cost, time and ease of operation.^5–9 Furthermore, the integrated chips enable superior sealing of the samples to prevent potential contamination during reagent transfer, thereby enhancing their isolation ability for on-site pathogen detection.^10–14 As a result, centrifugal microfluidics have been widely applied in various point-of-care testing scenarios, such as in hospitals, disease control centers, customs and so on.^15–20

The key to parallel isolated sample analysis on centrifugal microfluidic chips is injecting the liquid into individual microchambers. In general, the microchambers can be categorized into two types depending on the structure: the through-type which has at least two channels connected to the microchamber, and the blind type which has only one channel connected.

The through-type microchambers consist of both an inlet and outlet. The liquid filling process for this type of microchamber involves the liquid entering into the microchamber through the inlet, meanwhile the air is expelled through the outlet. This structure helps to prevent the formation of bubbles inside the microchambers. However, the multiple access channels compromise the isolation integrity of the chamber, and thus additional sealing measures are required during the chip's operation to prevent liquid cross-contamination between neighbouring microchambers. For instance, Podbiel et al. proposed a sealing method in which a second immiscible liquid is introduced as a sealant after the chamber has been filled with the first type of liquid, ensuring the independence of each microchamber to prevent cross-contamination.²¹ Focke et al. suggested using hydrophobic membranes and adhesive tapes to seal the inlet and exhaust ports of centrifugal microfluidic chips to reduce the risk of DNA contamination.²²

In blind microchambers, the inlet and outlet share a single channel. In this case, liquid enters the microchamber and gas inside the microchamber exits through the same channel, forming a gas–liquid two-phase flow. Compared to microchambers with both inlets and outlets, the blind microchamber structure provides better sealing performance without the need for additional sealing treatments, thus avoiding cross-contamination issues. Under similar conditions, a blind microchamber structure simplifies the operation of the microfluidic chip and enhances detection efficiency. However, to function properly, achieving void-free liquid filling in blind microchamber structures is a critical step which remains a challenge. Firstly, the gas flow may interfere with the liquid filling process, preventing smooth expulsion. Secondly, gas may be trapped in the microchambers and form dead-end areas, leading to inconsistencies between different chambers.²³ Additionally, the trapped bubbles may significantly impact the performance of the microfluidic chips. Schlosser et al. pointed out that optical detection in microfluidic systems may fail due to light reflection by trapped bubbles.²⁴ Liu et al. highlighted that the presence of air bubbles in microreactors is one of the main causes of PCR failures. At temperatures around 95 °C during DNA denaturation, bubbles can expand rapidly, and even in strongly sealed chambers, they may eject the PCR sample from the chamber.²⁵

To overcome the difficulties of void-free liquid filling in blind microchambers, several methods have been proposed. One approach is to introduce a type of gas with high liquid solubility into the microchamber before liquid filling. Zengerle et al. proposed the CO₂ pre-charging method, which prevents bubble encapsulation due to CO₂'s high solubility in water, ensuring bubble-free filling.²⁶ However, this method is limited to specific gas/liquid combinations where the gas has high solubility in the liquid. Additionally, it requires the pre-filling of the gas before liquid injection, which adds an extra operational step. Another technique involves evacuating the gas within the microchamber by creating a vacuum environment. Yu et al. utilized thin PDMS membranes that are gas permeable to make microfluidic chips. By applying a vacuum, gas is expelled from closed chambers and dead-end channels, facilitating rapid filling of complex structures. This method is particularly effective for filling small, intricate structures and dead-end channels without the need for vent holes.²⁷ However, this approach imposes additional material and structural requirements on the microchamber, with a potential risk of liquid leakage during the filling process. However, a multi-layered structural design can be used. For example, Steinert et al. proposed a two-level microplate structure consisting of two capillary channel layers with the upper layer being wider than the lower one. Due to the capillary barrier effect, liquid is injected through the lower layer into the blind end of the channel, while gas is expelled through the upper layer. At the blind end, the liquid is drawn into the upper layer by capillary forces and fills the structure in the opposite direction, preventing bubble formation.²⁸ However, this method requires a complex fabrication process and high precision, making it unsuitable for mass production. In addition, Xing et al. used an interim chamber connected with each microchamber via centrifugal valves to assist the liquid filling. The liquid is firstly injected into the interim chambers easily with no bubble since the interim chambers have wide openings connected to the main channel. Then by increasing the centrifugal speed, the liquid in the interim chamber is forced to pass through the valve between interim chamber and microchamber, expelling gas from the microchamber and completing the liquid filling.²⁹ While this design along with the two-step filling process ensures bubble-free filling, it significantly complicates the structure of the microfluidic chip, making it less suitable for high density array-based chips with multiple reaction chambers.

In this study, we thoroughly investigate the liquid filling process for blind microchambers in centrifugal microfluidics and propose an effective strategy to achieve void-free liquid filling without introducing extra processes. Based on experimental investigations using a high-speed camera, the effects of Coriolis force directions on the liquid filling and the reasons for residual gas bubbles are analysed. Additionally, the influence of the gas–liquid two-phase flow regime transition on the filling process is examined. Based on the analysis, we utilize the Venturi effect and propose a tapered microchannel structure that can not only increase the fluid velocity to maintain an ideal gas–liquid two-phase flow regime but also can reduce the pressure at the chamber inlet, facilitating the smooth expulsion of gas bubbles from the chamber.^30,31 Based on the transition conditions of the gas–liquid two-phase flow, the tapered microchannel structure with a 1/2 inlet-to-outlet width ratio is selected for its ability to maintain a relatively stable two-phase flow.³² Centrifugal liquid injection experiments show that, compared to a microchannel structure with a uniform diameter, the tapered microchannel structure reduces the time required for liquid filling by 60.57%, improves the stability of fluid velocity in the branch channel by 66.78% during the liquid filling process, and achieves void-free liquid filling. This tapered channel design eliminates the need for multiple functional zones or additional sealing operations, allowing one-step fabrication via micro milling or hot embossing during microfluidic chip production while avoiding the complex requirements for processing methods and alignment precision inherent in multilayer microfluidic structures. This simple yet highly effective design imposes no negative constraints on the applicability of microfluidic chips, demonstrating significant advantages in microarray-based detection,^33–37 PCR,^38–41 and cell culture^42–46 applications without requiring subsequent bubble removal procedures.

Materials and methods

Microfluidic chip

To investigate the liquid filling process in blind microchambers, we designed and fabricated a testing centrifugal microfluidic chip. Microfluidic channels were made on a piece of PMMA sheet using micro milling which has a dimensional tolerance of ±10 μm. Fig. 1A shows the photograph of the fabricated chip with 96 blind microchambers. The blind microchamber has a radius of 1580 μm, and a height of 600 μm. Twelve blind microchambers are connected to a main channel via individual branch channels. The main channels have a rectangular cross-sectional width of 700 μm and height of 200 μm. Each main channel is connected to a sample injection reservoir. In our study, we investigated multiple different designs of the branch channels, four representative structures are shown in the inset of Fig. 1A. The first two types of microchambers have branch channels with a uniform width of 700 μm, and the center of the microchambers are either on the left side of the branch channel or on the right side. The other two types of branch channels have a tapered shape. The initial width is 900 μm and the decreases to 450 μm. Channel IV differs from channel III by having chamfers with radii of 1 mm and 0.5 mm.

Fig. 1 Microfluidic chip structure and experimental conditions. A) Photos of the overall centrifugal microfluidic chip and the detailed designs of the blind microchambers. B) Experimental setup to observe the centrifugation process using a high-speed camera.

Experimental setup

To examine the liquid filling process, a testing and recording system was built, as shown in Fig. 1B. A dye solution was prepared by adding amaranth red dye into deionized water at a concentration of 1 mg mL⁻¹. Then, 300 μL of the dye solution was injected into the sample injection reservoirs of the microfluidic chip using a micropipette. After that, the microfluidic chip was placed in a centrifugal device, and a high-speed camera was fixed above the centrifugal device using a tripod. The light source was adjusted to the optimal angle and the centrifuge was started to begin the experiment. The centrifuge parameters were set to 1200 r min⁻¹ for 1 min, and the high-speed camera records at a frame rate of 27 000 fps. The liquid filling experiments were performed under room temperature (22 ± 1 °C). Under these conditions, the fluid exhibited a dynamic viscosity of 0.955 cP, and a density difference of 995.8 kg m⁻³, with a liquid-microchannel contact angle of 70 ± 2°.

To ensure statistical reliability, the experimental protocol included 10 full device replicates, each device contained 12 blind microchambers with specific structures. The filling quality was quantitatively evaluated by microcavity inspection, where any instance exhibiting bubble occupation exceeding 5% of the chamber volume was classified as a failure for void-free filling. The filling success rates for each device were calculated by dividing the number of void-free filling microchambers to the total number of microchambers. Statistical data was obtained by averaging the success rates of 10 devices.

In this study, the gas/liquid content was quantified through image analysis using the ImageJ software (version 1.53t). The phase boundaries between the gas and liquid were delineated manually to calculate the volume fractions, which were determined as the ratio of segmented phase pixels to total valid pixels. The manual delineation was independently performed by five researchers, with the average values adopted to minimize subjective bias.

Theoretical analysis

Pumping forces applied on liquid plugs

Without rotation, the liquid is stored in the reservoir at the top of the microchannel and stops at the entrance of the microchannel. After starting centrifugation, the liquid overcomes the resistance and enters the microchannel under the action of centrifugal force. The flow of liquid in the microchannels is typically very complex, especially in complex microfluidic systems, where the flow may involve different flow regimes such as laminar flow, turbulence, and gas–liquid two-phase flow. These flow regimes make force analysis more complicated. In this paper, we divide the liquid in the microchannels into many sections of liquid plugs, therefore the focus can be placed directly on the stability and motion of each liquid plug after force application, making the analysis simple. The continuous liquid can be treated as a series of segmented liquid plugs, each representing multiple continuous, relatively stable liquid blocks formed under the influence of fluid flow and external centrifugal forces.

The movement of the fluid plugs in the centrifugal microfluidic systems is dominated by the centrifugal induced force. From the perspective of a non-inertial rotating frame rotating with [small omega, Greek, vector], the liquid traveling with a velocity u⃑ along the channel is affected by the centrifugal force f⃑_ω (perpendicular to the direction of u⃑) and Coriolis force f⃑_c (parallel to the direction of u⃑). Thus, a fluid element with a mass density of ρ that rotates at an angular velocity of ω at a distance of r from the central axis is subjected to a centrifugal force as described by following fomula.⁴⁷

f⃑_ω = −ρ[small omega, Greek, vector] × ([small omega, Greek, vector] × r⃑) (1)

In addition, the Coriolis force induces the transversal flow inside the microchannel relative to the axial down channel flow. The Coriolis force is written as follows.

f⃑_c = −2ρ[small omega, Greek, vector] × u⃑ (2)

Physical effects relevant to the liquid filling process

When driving the fluid into the microchambers, two effects need to be considered: air compression and Rayleigh–Taylor instability.⁴⁸

Firstly, the air in the microchambers is compressed during liquid filling. When the liquid moves forward in both the main channel and the branch channel, thus occupying more room, or when it enters the blind microchamber, it compresses the air inside. The compression of the air in the channel causes an increase in air pressure Δp_air in the receiving microchambers.⁴⁹ The pressure generated by the compression of the air within the microchamber will resist the liquid's entry. Therefore, when designing the microchannel structure, the interaction between the liquid pressure and the air pressure must be considered to prevent the trapping of air bubbles within the microchamber.

Secondly, a fluid instability phenomenon, referred to as Rayleigh–Taylor instability, that is caused by density differences and disturbances of the two fluid at the gas–liquid interface needs to be considered.⁵⁰ Under the action of high centrifugal force, the interface between the liquid and air becomes unstable, and the critical wavenumber ε for interface instability at centrifugal frequency is given by eqn (3).

where λ_crit is the critical length of the interface, serving as a fundamental parameter that characterizes the relative importance between the surface tension and body forces in the system, w_ρ represents the density difference between the two fluids which serves as the driving factor of the Rayleigh–Taylor instability. This term quantifies the buoyancy force arising from the density stratification under gravitational or equivalent acceleration, σ represents the surface tension coefficient.

The Rayleigh–Taylor instability typically leads to amplified interface fluctuations, which can affect the stability, distribution, and flow characteristics of the fluid. In the microfluidic chip discussed in this study, to enhance the stability of the gas–liquid interface and avoid Rayleigh–Taylor instability, the approach primarily involves adjusting the fluid flow rate and optimizing the channel design.

Bubble formation in gas–liquid two-phase flow

Previous investigations on gas–liquid two-phase flow regime transitions have primarily relied on extensive experimental data, leading to empirical flow regime transition maps.³¹ For microchannels with hydraulic diameters exceeding 100 μm, inertial forces dominate the fluid dynamics. The primary gas–liquid flow regimes in microchannels can be divided into four types, bubbly flow, Taylor (slug) flow, Taylor-annular flow, and annular flow, as depicted in Fig. 2. In bubbly flow, the characteristic diameter of the bubbles is smaller than the channel width, whereas in Taylor flow, the bubble diameter exceeds the channel width. These two flow regimes typically occur at low gas and liquid velocities. When the fluid velocity in the channel keeps increasing, the increased gas void fraction leads to the merging of neighbouring bubbles, thus forming a Taylor-annular pattern. In this pattern, a gas core is surrounded by a liquid film where large-amplitude solitary waves appear, and small bubbles are sometimes present in the liquid. As the fluid velocity increases further, the long waves disappear, and an annular flow pattern is established. In this flow regime, the gas and liquid phases remain relatively independent, making it more suitable to exhaust all gas in blind microchambers like those studied in this work. To maintain this specific flow regime, it is essential to sustain a certain flow velocity with minimal fluctuations.

Fig. 2 The four main forms of gas–liquid two-phase flow in microchannels and their effects on bubble formation. A) Bubbly flow refers to the formation of a large number of small gas bubbles within the liquid. B) Taylor flow refers to the alternating formation of gas bubble segments and liquid segments, with gas bubbles moving through the liquid. C) In Taylor-annular flow, the gas bubbles take the shape of a thin annular layer, with the gas occupying the core of the channel and the liquid forming a thin film surrounding the gas core, the diameter of the gas core is uneven. D) The characteristics of annular flow are similar to those of Taylor-annular flow, but the diameter of the gas core in annular flow is uniform.

However, for microchannels with hydraulic diameters below 100 μm, surface tension becomes the dominant governing force for fluid motion. The enhanced Laplace pressure and wettability significantly affects bubble/droplet deformation during flow while substantially increasing flow resistance.⁵¹ These conditions adversely affect phase separation in gas–liquid two-phase flows and preclude the formation of desired annular flow regimes. Consequently, our study exhibits inherent limitations for microchannels with hydraulic diameters below 100 μm. The present investigation of flow regime transitions in gas–liquid two-phase systems is therefore specifically confined to microchannels exceeding 100 μm in hydraulic diameter, where inertial forces dominate and annular flow formation becomes favourable.

To achieve void-free liquid filling, it is required that the gas–liquid two-phase flow in the microchannel is maintained within Taylor-annular or annular flow regimes. The gas–liquid two-phase flow in microchannels exhibits a variety of complex patterns under different flow rates, channel designs, fluid properties, and external conditions. This study focuses on how changes in channel dimensions affect the fluid flow rate, the transition of flow patterns in the gas–liquid two-phase flow, and the impact of the gas–liquid two-phase flow on the gap-free liquid filling process in the blind microchamber.

Venturi effect and tapered structure design

From the Venturi effect, it is known that when confined flow passes through a reduced cross-sectional area, the fluid exhibits an increase in velocity, which is inversely proportional to the cross-sectional area.

The relationship between the fluid velocity and cross-sectional area in the inlet section and the throat of the tube can be expressed as follows.

S₁V₁ = S₂V₂ = Q (4)

Here, S₁ and V₁ represent the cross-sectional area and average velocity of the fluid in the inlet section, while S₂ and V₂ denote the cross-sectional area and average velocity of the fluid at the throat, and Q represents the flow rate of the fluid in the tube.

The relationship between the fluid velocity and pressure in the inlet section and at the throat can be expressed as follows.

Here, V₁ and P₁ represent the average velocity and average pressure of the fluid at the inlet section, while V₂ and P₂ denote the average velocity and average pressure of the fluid at the throat, and ρ represents the fluid density.

According to the formula derived from the Venturi effect, if one wishes to increase the fluid velocity in the channel while reducing the fluid pressure at the outlet of the branch channel without changing the total flow rate, a tapered channel can be considered. In this paper, the angle between the right edge of the branch channel (with a 60° structure) and the main channel will be changed from 60° to 75°. The ratio of inlet width D1 to outlet width D2 will be set to 2 : 1, 3 : 1, 4 : 1, and 5 : 1 to investigate the effect of different inlet and outlet width ratios on liquid filling results.

Simulation modeling

The simulation of the liquid filling process was conducted by using COMSOL Multiphysics. The liquid filling process involves the interaction between the gas and liquid phases. Two-phase flow and level-set modules are employed to simulate the liquid filling process in a microchamber. Based on the fluid flow direction and microchannel dimensions observed in experiments, a two-dimensional geometric model of the microchannel is established in numerical simulations to reduce computational complexity and time.

In the initial flow field, region (1) is designated as the liquid phase, while region (2) is designated as the gas phase (as shown in Fig. 3). Since the liquid enters and the gas exits through the same opening in the microchannel, the left boundary of the reservoir is set as an open boundary. The microchannel walls are defined as wettability boundary conditions, with a contact angle set to π/2. The slip length of the wetting wall is described using the minimum element length factor, with a numerical value of 1. Both the gas and liquid phases within the microchannel are initially at rest, with zero initial velocity and pressure. The liquid is driven by centrifugal force, which is applied as a body force in the X-direction based on eqn (1). The Coriolis force is incorporated as a body force in the Y-direction based on eqn (4). To simulate the acceleration phase of the centrifugal system, a ramp function with a slope of 0.5 and a truncation value of 1 is applied to ω before adding the body force. Based on the experimental conditions, ω is set to 40π rad s⁻¹. A standardized 1D monitoring cross-section was implemented at the branch channel inlet to acquire the transient fluid velocity data with temporal resolution during the 7 s transient study output duration.

Fig. 3 The microchannel geometry, domain and boundary settings in the simulation model.

Detailed information of the simulation is described in the ESI.†

Results and discussion

Liquid filling process and effects of the Coriolis force

To investigate the liquid filling process of the blind microchamber, the force analysis primarily focuses on the forces acting on the liquid at the junction between the main and branch channels. The branch channels can be either at the right or left side of the main channel, and thus the force experienced by the liquid in the microchannel differs at the same rotational frequency. If the rotation direction of the centrifugal system is set as clockwise, there are two different cases. The detailed analysis and experimental data are presented as follows.

When the blind microchamber is located to the right side of the main channel, a force diagram of the liquid plug in the microchannel and the corresponding data are shown in Fig. 4. Once the rotational frequency of the centrifugal system stabilizes, the liquid plug in the microchannel is primarily subjected to four forces: the centrifugal force f⃑_ω generated by the rotation of the centrifugal system, the transverse Coriolis force f⃑_c, and the expansion forces p⃑_a and p⃑_b generated by the compression of air inside the main channel and the blind microchamber, respectively. The directions of these forces are outward along the main and branch channels (as shown in Fig. 4A).

Fig. 4 The liquid filling process when the direction of the Coriolis force points towards the blind microchamber. A) Force analysis of liquid flow in the microchannels. B) Liquid filling process observed using a high-speed camera reveals different two-phase flow patterns leading to either incomplete liquid filling or void-free liquid filling.

As the direction of the liquid flow in the microchannel aligns with the direction of the Coriolis force, the liquid plug is subjected to the combined effect of the centrifugal and Coriolis forces, resulting in the acceleration and stabilization of the liquid plug. The air expansion forces in the two channels act as resistance to the liquid. In this case, the liquid filling process can be categorized into two stages as learned from the experimental data.

In the first stage, the gas–liquid two-phase flow is established. Once centrifugation begins, the fluid rapidly fills the main channel, at the same time a part of the branch channel is occupied by the liquid when t = 0.367 s. With the expulsion of gas, phase separation occurs in the main channel. Due to the alignment of the Coriolis force with the chamber direction, the gas phase is expelled upwards to the left side of the main channel, and the liquid phase moves along the right side, as indicated by the image taken at t = 0.793 s. Thus, the flow in the main channel presents as Taylor-annular or annular flow. While in the branch channel, the flow is bubbly flow since the velocity of liquid is less than in the main channel. This process continues until the amount of gas to be expelled in the main channel decreases, then the phase separation in the main channel halts, and the two-phase flow transforms to Taylor flow, as indicated by the image taken when t = 0.894 s. In this stage, only a small amount of fluid is present in the branch channel and the blind microchamber, and the flow pattern in the branch channel is bubbly flow.

After the main channel reaches a balanced state, the liquid starts to be injected into the blind microchambers in a relatively large amount. Then comes the second stage. In this stage, the flow pattern in the branch channels can be different. If the conditions of the flow in the branch channel are within the Taylor flow regime during the liquid filling process, trapped gas bubbles are likely to form in the microchamber, as indicated by the image taken when t = 1.843 s, shown in the inset of Fig. 4B-I. This leads to failure in the sample injection, causing the blind microchamber to be incompletely filled, as shown in Fig. 4B-III. Conversely, if the two-phase flow in the branch channel is in the annular flow regime, the air inside the blind microchamber can be expelled completely. This ensures void-free liquid filling (as shown in the inset of Fig. 4B-II and -IV).

The analysis and results when the microchamber is located on the left side of the main channel are shown in Fig. 5.

Fig. 5 The liquid filling process when the direction of the Coriolis force is opposite to the direction of the blind microchamber. A) Force analysis of liquid flow in the microchannels. B) Liquid filling process observed using a high-speed camera reveals different two-phase flow patterns leading to either incomplete liquid filling or void-free liquid filling.

Similarly, when the rotational frequency of the centrifugal system stabilizes, the liquid plug in the microchannel is primarily subjected to four forces: centrifugal force, the Coriolis force, and the air expansion forces in both directions. While in this case, the centrifugal force assists the liquid in entering the microchamber, the Coriolis force and the air expansion forces in both channels act as resistance to the liquid entering the microchamber (as shown in Fig. 5A), causing deceleration or stagnation of the plug, and possibly even leading to instability or breakup of the liquid plug. In detail, the liquid filling process can be also divided into two stages.

Initially, upon start of centrifugation, the fluid rapidly enters the main channel when t = 0.367 s. Liquid also enters a section of the branch channels, and thus there is air in the branch channel and the blind microchambers. Only a small amount of fluid is present in the channel and the blind microchambers are expelled simultaneously. In the branch channel, the flow is bubbly flow. These bubbles from the branch channels enter the main channel and merge with those bubbles in the main channel, and thus in the main channel no phase separation occurs and the two-phase flow behaves according to the Taylor flow characteristics, with large bubbles moving upwards. The results are shown by the images taken when t = 0.793 s. Additionally, as the direction of the Coriolis force is opposite to that of the blind microchamber, the Coriolis force does not aid fluid entry into the branch channel. During the upward movement of the large bubble in the main channel, the upward gas pressure at the junction between the branch and main channels changes the direction of some of the liquid flow, causing liquid to enter the branch channel, as indicated by the images taken when t = 0.894 s. After this stage, the blind microchambers are partially filled with liquid, leaving part of the room occupied by the compressed air.

In the second stage, the state of the two-phase flow in the main channel changes as the amount of gas remaining in the main channel decreases. If the flow state remains in the Taylor flow regime, large gas bubbles may get trapped at the entrance of the branch channel (as shown by the inset of Fig. 5B-I), preventing gas from being efficiently expelled from the chamber and leading to sample injection failure (as shown by the inset of Fig. 5B-III). Conversely, if phase separation occurs in the main channel, the liquid phase moves downward along the right-side wall due to the Coriolis force, while the air phase locates near the left wall and moves upwards. When the air plug passes through the branch channel entrance, the upward gas pressure at the junction of the branch and main channels still allows some liquid to enter the branch channel, as indicated by the inset of Fig. 5B-II. With efficient expulsion of gas, gap-free liquid filling is achieved (as shown in the inset of Fig. 5B-IV).

According to the analysis and experimental data, when the direction of the Coriolis force aligns with the direction of the chamber, it helps to yield liquid injection. To achieve better liquid filling results, the centrifugal method with the Coriolis force direction aligned with the blind microchamber direction is employed in the subsequent experimental studies in this paper.

Effects of blind microchamber position

We also investigated the liquid filling process when the relative positions of the blind microchambers and branch channels are different. Simulation and experimental studies were conducted on structures with branch-to-main channel angles of 75° and 60°, the results are shown in Fig. 6.

Fig. 6 Simulated and experimental exploration of the chamber position and velocity of the liquid, the colormap in the figure represents the phase fraction, with blue corresponding to air and red denoting liquid. Simulation and experimental results of liquid filling in the 75° and 60° structures when the centers of the microchambers are on the left side A) and right side B) of the branch channels. C) The process of gas discharge when the center of the microchamber is located on the left and right sides of the branch channel. D) Comparison of the liquid filling process observed via simulation and a high-speed camera. E) The flow velocity curve of the fluid in the branch channel of 60° structure when the center of the microchamber is located on the right side of the branch channel.

For the case where the microchamber center is located on the left side of the inlet branch channel, as shown in Fig. 6A, it is noted that the complete liquid filling process fails, with only a small portion of liquid entering several chambers. When the microchamber's center is located on the right side of the inlet branch channel, the results indicate that void-free liquid filling happens for a majority of the blind microchambers. However, residual bubbles may still remain (as shown in Fig. 6B).

This happens because both the main channel and the branch channel are simultaneously filled with liquid during the first stage of the filling process. For the case that the microchamber's center is on the left, the liquid prevents most of the gas in the chamber from being expelled (as shown in Fig. 6C), as a consequence, the filling process cannot be completed at the second stage. Conversely, when the center of microchamber is on the right side of the inlet channel, although the chamber is also half-filled, the branch channel remains unblocked, allowing the remaining gas in the chamber to be expelled.

Therefore, for optimized structural design, a configuration with the microchamber's center on the right side of the inlet branch channel should be used.

Residual bubbles in the blind microchamber

To investigate the cause of residual bubbles, we take the structure where the microchamber center is located on the right side of the inlet branch channel with a branch-to-main channel angle of 60° (hereafter referred to as the “60° structure”) as an example. By analysing the simulation and experimental results, it is noted that the filling process can be divided into five steps (as shown in Fig. 6D). In steps 1 through 3, the two-phase flow pattern in the branch channel remains in the Taylor flow regime. During steps 3 to 4, the flow pattern in the branch channel transitions from Taylor flow to annular flow, which is a critical factor in determining whether the filling process is successful or not. If the flow pattern transition happens, the filling process can be completed, whereas residual bubbles will form in the chamber.

High-speed camera experiments reveal that the success of the flow pattern transition from step 3 to step 4 is quite random. As discussed earlier, the primary factor influencing the gas–liquid two-phase flow pattern transition within the microchannel is the fluid velocity in the microchannel. The velocity can be used as an indicator to predict the possibility of successful void-free liquid filling. Using the COMSOL software, the fluid velocity at the entrance of the branch channel at different times is plotted, as shown in Fig. 6E. According to the fluid velocity curve, the presence of Taylor flow in the microchannel causes significant fluid flow oscillations, reflected as noticeable but irregular fluctuations on the velocity curve. The random fluctuation is due to the influence of Rayleigh–Taylor instability, i.e., the gas–liquid interface cannot maintain a stable state, preventing the fluid from sustaining a stable flow at a sufficient velocity and resulting in the failure of flow pattern transition.

Performance of the tapered structure

According to the Venturi effect formula, a tapered structure can increase the fluid velocity within the channel without altering the total flow rate, and therefore has the potential to improve the successful transition from Taylor flow to annular flow. We designed tapered structures for the branch channels with inlet-to-outlet width ratios of 2 : 1, 3 : 1, 4 : 1, and 5 : 1, respectively. The obtained simulation results of the filling process and the fluid velocity curves within the branch channel of the tapered structure are shown in Fig. 7A and Table 1.

Fig. 7 Verification of the performance of the tapered structures, the colormap in the figure represents the phase fraction, with blue corresponding to air and red denoting liquid. A) Simulation results of the tapered structures with different inlet-to-outlet width ratios. B) Liquid filling process of the tapered 2 : 1 structure. C) Liquid filling performance of the tapered 2 : 1 structure.

Table 1 Comparison of the four inlet-to-outlet width ratios in terms of void-free liquid filling

	2 : 1	3 : 1	4 : 1	5 : 1
Average velocity	117.867 mm s^−1	58.843 mm s^−1	43.956 mm s^−1	71.275 mm s^−1
Dispersion coefficient	0.18	0.56	2.73	2.85

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From the simulations of various taper ratios, it is observed that only the 2 : 1 structure shows no characteristics of Taylor flow, reducing the number of bubbles in the branch channel. By examining the velocity curves, it is found that although the peak fluid velocity in the branch channel increases with the inlet-to-outlet width ratio (in accordance with the Venturi effect), the 2 : 1 structure has the smallest dispersion coefficient among the four curves (as shown in Table 1). The dispersion coefficient is defined as the ratio of the standard deviation to the mean and can be expressed as follows.

Here, V_s represents the dispersion coefficient, while σ represents the standard deviation, and X̄ represents the mean. A lower dispersion coefficient signifies a smoother and more stable curve. The dispersion coefficient of the four structures, in terms of void-free liquid filling, indicates that the velocity of fluid is most stable for the 2 : 1 structure, with minimal influence from Rayleigh–Taylor instability, which is beneficial for maintaining stable flow. Therefore, the optimal taper ratio for the structure is set to 2 : 1 for the inlet-to-outlet width.

Subsequently, a comparison between the simulation and experimental results in terms of the liquid filling processes for the tapered 2 : 1 structure was conducted and the results are shown in Fig. 7B and C.

The filling process for the optimized structure can be also divided into five steps. Observations from high-speed camera footage reveal that, throughout steps 1 to 5, no clear Taylor flow characteristics appear in the branch channel. The two-phase flow in the images resembles annular flow with fewer bubbles, allowing smooth gas expulsion. The overall filling performance for multiple blind microchambers is also improved, with no noticeable bubbles appearing (as shown in Fig. 7C).

The influence of the branch channel length was also investigated, as shown in Fig. S1 and Table S1 in the ESI.† Generally, the results demonstrate that the length of the tapered branch channels does not significantly alter the fluid velocity, and thus does not impair the void-free liquid filling performance.

Effects of rounded-inlet and validation

We further investigated the effects of the rounded-inlet and compared the performance of three structures including the 60° structure, the tapered structure with a 2 : 1 ratio and the rounded-inlet tapered structure with a 2 : 1 ratio. The results are shown in Fig. 8 and Table 2.

Fig. 8 Comparison of the liquid filling performance of three different inlet structures. A) The liquid filling process observed using a high-speed camera. B) The fluid velocity curves in the branch channel for the three structures. C) Comparison of the liquid filling performance of the three structures (single chamber). D) Comparison of the liquid filling performance of the three structures (multiple chambers).

Table 2 Comparison of parameter related to liquid filling performance of three different inlet structures

	60° structure	Converging structure	Rounded-inlet converging structure
Time required to complete filling	5.15 s	4.58 s	2.03 s
Average velocity	36.522 mm s^−1	117.867 mm s^−1	113.515 mm s^−1
Dispersion coefficient	0.66	0.18	0.22

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In the 60° structure, the two-phase flow pattern in the branch channel exhibits characteristics of Taylor flow, and the Taylor-to-annular flow transition is incomplete, resulting in bubbles in the chamber after filling. In the tapered structure, the two-phase flow pattern in the branch channel aligns with the characteristics of annular flow, and the filling process proceeds as expected. In the rounded-inlet tapered structure, the two-phase flow in the branch channel also exhibits annular flow characteristics. This structure achieves the fastest filling speed among the three. The chambers in the first two structures are still half-filled at 1.9 s, while the chambers in the rounded tapered structure are fully filled without any bubbles. This indicates that the rounded tapered structure further reduces fluid resistance, promoting both filling and gas expulsion.

Using the COMSOL software, the fluid velocity curves in the branch channels of the three structures were obtained, and the key data are summarized in Table 2 and Fig. 8B. Combined with the experimental results, it is noted that, compared to the 60° structure, the tapered structure requires less time to complete filling, achieves a higher average fluid velocity during filling, and demonstrates a more stable fluid velocity curve.

The liquid filling results for a single blind chamber and multiple blind chambers are shown in Fig. 8C and D. In the 60° structure, large residual bubbles can be found and there is a high possibility that the blind microchambers are not filled completely. The tapered structure and rounded-inlet tapered structure significantly reduce the quantity of residual bubbles. As far as we are concerned, no bubbles are found in either the microchambers or the branch channels in the centrifugal microfluidic chips with blind microchambers designed with rounder-inlet tapered structures. Reproducibility test results of the three structures demonstrated liquid filling success rates of 71.67% ± 1.52% for the 60° structure, 95.83% ± 0.58% for the converging structure, and 99.17% ± 0.19% for the rounded-inlet converging structure. The data is presented as mean ± standard error.

Conclusions

This study proposes a strategy for achieving void-free liquid filling in blind microchambers through the use of tapered structures. This void-free liquid filling is accomplished by altering the flow regime of the gas–liquid two-phase flow in the microchannel during the filling process. Experimental results show that compared with the uniform diameter microchannel structure, the tapered microchannel structure reduces the time required for liquid filling by 60.57%, improves the stability of fluid velocity in the branch channel by 66.78% during the liquid filling process, and achieves void-free liquid filling. Furthermore, this strategy achieves void-free liquid filling solely by modifying the microchannel structure, without the need for complex surface treatments or external forces. This approach has great potential to inspire the community and offer valuable guidance in the design of centrifugal microfluidic chips.

Data availability

The data that support the findings of this study are available on request from the corresponding author, M. W. and J. L. upon reasonable request.

Author contributions

Weiyao Ni conducted the chip design and experiments, and wrote the manuscript; Yi Gao and Enming Cui designed the experimental devices; Yifei Li, Yangyang Wang and Yi Li provided suggestions on data analysis; Yahua Liu provided the experimental equipment and site; Mengxi Wu performed the review and editing; Junshan Liu supervised the work.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (U23A20362).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5lc00323g

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