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Filling of liquid samples is realized in a microfluidic device with applications including analytical systems, biomedical devices, and systems for fundamental research. The filling of a disk-shaped polydimethylsiloxane (PDMS) microchamber by liquid is analyzed with reference to microstructures with inlets and outlets. The microstructures are fabricated using a PDMS molding process with an SU-8 mold. During the filling, the motion of the gas-liquid interface is determined by the competition among inertia, adhesion, and surface tension. A single ramp model with velocity-dependent contact angles is implemented for the accurate calculation of surface tension forces in a three-dimensional volume-of-fluid based model. The effects of the parameters of this functional form are investigated. The influences of non-dimensional parameters, such as the Reynolds number and the Weber number, both determined by the inlet velocity, on the flow characteristics are also examined. An oxygen-plasma-treated PDMS substrate is utilized, and the microstructure is modified to be hydrophilic. Flow experiments are conducted into both hydrophilic and hydrophobic PDMS microstructures. Under a hydrophobic wall condition, numerical simulations with imposed boundary conditions of static and dynamic contact angles can successfully predict the moving of the meniscus compared with experimental measurements. However, for a hydrophilic wall, accurate agreement between numerical and experimental results is obvious as the dynamic contact angles were implemented.

During the last twenty years, microfluidic devices have been developed for a broad range of applications in bio-analysis and chemical processing technologies. The original fabrication techniques of the microfluidic devices were derived from the semiconductor industry and most devices adopted silicon or glass as the substrates. These techniques were thought to be intricate and expensive. In recently years, polymeric materials have been extensively used for the fabrication of the microfluidic devices. Among these materials, poly(dimethylsiloxane) (PDMS) receives a lot of attention due to its optical transparency and biocompatibility and is utilized as the main substrate for various microfluidic systems.

Liquid filling is commonly encountered in the operation of the microfluidic chip. Chips utilize lengthy microchannels to deliver liquid solution or air bubbles in liquids from one place to another or bring different liquid fluids into microchambers [

Pressure, high electric fields and gravity are usually exploited to drive the flow of liquid in microchannels. However, surface tension, which is typically negligible in macro-scale systems, is also a force responsible for a variety of physical phenomena involving small volumes of liquid. Molecules in the liquid state exert strong attractive forces on each other. Surface tension, a cohesive force, arises from attractive forces among the molecules in a fluid, whereas an adhesive force to the wall acts on the fluid interface at the contact point with the wall. Yang

The advantages of utilizing the surface tension effect in microsystems include lower power consumption, ease of fabrication and no active fluidic component. A variety of applications for surface tension exist in microelectromechanical systems (MEMS), including hydrophobic valves [

Experimental and numerical studies of surface tension-driven flow in microchannels have been performed. Tseng

The simulation of surface tension-driven flow in microchannels has been attracting a lot of attention. Results were generated by assuming a constant value of the contact angle, which means that only the static contact angles were applied in all previous computational research. The contact angle is the angle between the free liquid surface and the confining wall, and is a result of the competition between surface tension and the adhesive force. When the competing forces are in balance, the contact line,

In numerical treatment of moving contact lines, contact angles are required and applied as a boundary condition at the contact line. However, the dynamic contact angle of the contact line depends on the velocity of the moving contact line. A mean-field free-energy lattice Boltzmann model was used to solve the moving contact line problem of liquid-vapor interfaces and the dynamics of the wetting and movement of a three-phase contact line confined between two superhydrophobic surfaces [

Understanding the liquid filling process is significant to design microfluidic chips. Successful filling by the liquid flows of microchannels or microchambers is not as easy as that of a macro-scale system because the surface effects are significant [

This work numerically establishes the filling processes of the liquid in the disk-shaped microstructures by considering microchambers with inlets and outlets fabricated using MEMS technologies. Three-dimensional governing equations are adopted to simulate the fluid flows inside the microfluidic devices. A general functional form with velocity-dependent contact angles is implemented for the accurate calculation of surface tension forces in a volume-of-fluid based model. The effects of the parameters of the functional form are investigated. Finally, the microfluidic devices are fabricated in PDMS substrates simply from a mold. Due to the hydrophobicity of the PDMS, the oxygen plasma treatment method is used to modify the surface of PDMS microchannels to be hydrophilic. Flow experiments are conducted into both hydrophilic and hydrophobic PDMS microstructures and the results compared with those of numerical simulations.

The liquid filling processes inside an oval or a circular disk-shaped microchamber are accepted to be governed by the continuity and the Navier-Stokes equations subject to appropriate boundary and initial conditions. Numerical simulations are conducted to elucidate the motion of a liquid flow inside the microchannels and the oval disk-shaped microchamber.

Assuming that the fluid obeys the linear Newtonian friction law and neglecting the compressibility, the governing equations comprise conservation of mass and momentum, and a scale function, F, of the liquid volume fraction inside a computational cell. The conservation equations of mass and momentum are solved to yield the motion and the shape of the gas-liquid interface. The continuity equation can be expressed as follows:

The momentum equation for a continuum is the analogue of Newton’s second law for a point mass. The momentum principle states that the rate of change of linear momentum of material in a region equals the sum of the forces on that region. Two such forces may be included: body forces, which act on the bulk of the material in the region, and surface forces, which act on the boundary surface. The momentum equation,

The orientation of the interface near the contact line reflects the contact angle, which is the angle between the normal to the liquid-solid interface and the normal to the solid surface at the contact line. A general correlation for the contact angle as a function of contact line velocity is not easy to be obtained. van Mourik

The slope-intercept form and the relationship between the contact angle and the contact line velocity.

_{d}_{t}_{s}_{lc}_{uc}_{d}_{uc}_{s}_{t}^{3} + _{t}^{2} + _{t}

A finite volume approach is adopted using computational fluid dynamics (CFD) software ESI-CFD (V2006, CFD Research Corporation, Huntsville, AL, USA) to model the progress of a gas-liquid free surface. The method is comprised of the volume-of-fluid (VOF) method [

This equation must be solved together with Equations (1) and (2) to achieve computational coupling between the velocity field solution and the liquid distribution. In the numerical simulation, the surface tension at the free surface is modeled with a localized volume force

A three-dimensional time-variant fluid field is adopted to specify the flow characteristics of microsystems. The classification of the VOF method as a volume tracking method follows directly from the use of

This work considers a configuration of the flow systems. It comprises inlet/outlet microchannels and a circular or an oval disk-shaped microchamber with an adhesive wall surface. The grid systems in the computational domain are selected to ensure the orthogonality, the smoothness and the low aspect ratios that prevent numerical divergence, shown in ^{4} to 1.6033 × 10^{5} are used and the two finest meshes give a negligible relative difference in their corresponding values which indicates they are mesh-independent. The values of maximum velocity inside the microchannel at two different times for the five mesh densities are also expressed in ^{4} has been chosen for further investigation since the maximum velocities inside the microchannel are almost the same and the numerical results are grid-independent.

The grid systems of the computation domain for (

The shapes of the liquid fronts at times of (^{4}, Grid2: 3.0720 × 10^{4}, Grid3: 5.6000 × 10^{4}, Grid4: 9.6768 × 10^{4}, and Grid5: 1.6033 × 10^{5}.

The analysis of the grid size independence.

Number of nodes | Maximum velocity at 0.02 s | Relative difference in maximum velocity at 0.02 s (%) | Maximum velocity at 0.04 s | Relative difference in maximum velocity at 0.04 s (%) |
---|---|---|---|---|

Grid1: 1.2096 × 10^{4} |
0.305065 | - | 0.311528 | - |

Grid2: 3.0720 × 10^{4} |
0.195608 | 55.97 | 0.195607 | 59.26 |

Grid3: 5.6000 × 10^{4} |
0.200395 | 2.39 | 0.200394 | 2.39 |

Grid4: 9.6768 × 10^{4} |
0.203124 | 1.34 | 0.203124 | 1.34 |

Grid5: 1.6033 × 10^{5} |
0.204233 | 0.54 | 0.204233 | 0.54 |

For the purpose of experimental characterization of the filling process of the microchamber, the microchannels and microchamber in poly(dimethylsiloxane) (PDMS) substrate are covered with a PDMS or a glass plate. The flow device from PDMS is fabricated by a replica molding method. Initially, a silicon wafer is cleaned and dehydrated on a hotplate. A thick film is fabricated by spin coating negative photoresist (SU-8) onto a silicon wafer. The resist is then soft baked on a level hotplate. The pattern is fabricated by photolithography using a photo mask. After development, the master is washed and baked to fix the photoresist. A wafer with patterned SU-8 is then obtained. Once the mold is complete, the wafer is rinsed in deionized (DI) water and dried with nitrogen. The height of the positive patterns on the molding masters is 100 μm when measured with a surface profiler. The PDMS prepolymer mixture which is thoroughly mixed with the base solution and curing agent using a 10:1 weight ratio is degassed with a mechanical vacuum pump to remove air bubbles. After pouring the PDMS prepolymer mixture onto the wafer, micro-structures are fabricated using a PDMS replica molding process. The PDMS is then cured in an oven and the replicas are peeled off from the mold. The inlet and outlet holes are then drilled. Methanol is used as a surfactant to prevent two oxygen-plasma-treated PDMS replicas from being irreversibly bonded when aligned improperly. A glass slide is also used to bond the PDMS replica. After O_{2} plasma treatment and bonding, the designed microstructure, which consists of two microchannels and a microchamber has been fabricated.

Due to the high hydrophobicity of PDMS, it is difficult to transport aqueous fluid on the PDMS surface without any external power. One of the advantages of fabricated PDMS microfluidic devices is easy visualization, so PDMS is still widely used as the most important substrate for microfluidic systems. The O_{2} plasma treatment method is often used to modify the surface of PDMS microchannels to be hydrophilic [_{2} plasma. In this work, the time-dependent change of the contact angle can be observed clearly after surface treatment. The contact angle of unmodified PDMS substrates remains at around 100°–112° from our measurements. According to the literature, the contact angle of pure PDMS is greater than 105°. However, the contact angle of PDMS with surface modification decreases from 112° to 10°. And it increases to 20° about 60 min after surface modification.

The measurement of the static contact angle of water on the surface-modified PDMS surfaces.

The microchamber is designed to study the effects of various forces such as inertia force, adhesion force and surface tension on the filling characteristics in a chamber. For the filling experiment in pressure-driven flows, fluid is injected into the microchannel using a syringe pump (Programmable Syringe Pump, KD Scientific, Holliston, MA, USA) at preset constant flow rates. The experimental setup for testing the performance of the fabricated microchamber is described as follows. One syringe is loaded with DI water. The working fluid first enters the inlet channel, flows through the microchamber, and finally exits through an outlet channel. Several experiments are performed with various flow rates for different microchambers to investigate the effects of operational and geometrical parameters on filling performance. The filling patterns are dynamically recorded using a computer system captured by a high-speed camera at a magnification of 40× with a graphic grabber system (VCD-Gear TV Plus) at rates of 125, 250, 500, and 1000 fps. The rate depends on the flow inlet velocity.

In this section, two configurations of the flow network system are considered. The microchannels are 100 μm wide, 100 μm deep, and the circular disk-shaped microchamber has a diameter of 800 μm. These dimensions correspond to angles between the microchannel and the microchamber at the intersection of 138.6°. The microchannels are 100 μm wide in the oval disk-shaped microchamber for 700 μm and 250 μm of the semi-major and semi-minor axes. These dimensions correspond to angles between the microchannel and the microchamber at the intersection of 111.9°. The chambers may have hydrophilic or hydrophobic surfaces. The contact angle of the air-water interface on the solid surface is specified to simulate filling under various surface conditions and to elucidate the mechanism by which the front changes in the chamber. In the following, the effects of the various parameters on the motion and the shape of the air-water interface in a circular disk-shaped chamber are studied first. Then, the comparisons between numerical and experimental results are shown. The above values of the parameters are used unless otherwise stated.

In an effort to understand the surface tension flows inside microfluidic chips, the changes in the shape of the liquid front are used to evaluate the liquid filling processes. _{inlet}W_{inlet }^{−2}, 0.77 × 10^{−2}, 0.85 × 10^{−2}, 1.54 × 10^{−2}, 2.72 × 10^{−2}, 4.06 × 10^{−2}, 4.62 × 10^{−2}, 5.37 × 10^{−2}, 6.16 × 10^{−2}, 6.24 × 10^{−2}, and 6.43 × 10^{−2} s, respectively. The liquid flows forward and approaches the entrance of the chamber. Regarding the contact angle of 80°, the liquid has a concave interface with air. When the interface reaches the sharp corner, the meniscus near this corner stops moving forward and remains at this point until the bulk of the flow has advanced sufficiently, allowing the meniscus to bulge sufficiently over the corner so that the contact angle at the edge reaches 80° (marked by a blue dotted circle). The adhesive force between the liquid and the solid wall is established to create a capillary pressure barrier that stops the flow. An abrupt enlargement in the cross-sectional area for the contact angle of 80° makes the pressure in the liquid become negative. Then, the capillary gating effect can be seen. At this point, the angle of the liquid meniscus must change to adopt the equilibrium contact angle at the slanted walls. This side of the meniscus descends down the side of the chamber and along the surface. The other side of the meniscus keeps moving forward until it reaches the entrance of the chamber, because the wall of the inlet channel is tangential to the wall of the chamber and the inertia force continues to drive the liquid flow forward. Then, the flow leaves the inlet channel, wetting the side surface. This change in the angle between the microchannels and the microchamber at the intersection increases the gas-liquid area to more than the solid-liquid area for a given volume change, resulting in a negative opposing pressure. The surface tension of the meniscus and the liquid pumping force overcome this capillary pressure barrier that develops when the cross section of the channel changes abruptly, establishing a positive pressure that pulls the liquid into the microchamber. We is equal to 0.0138, so the surface tension dominates the flow mechanism. As the liquid flows into the chamber, the flow front becomes convex because of the inertia of the fluid flow. The curvature of the wall of the chamber changes the shape of the front such that it becomes flat when the liquid occupies almost one-third of the chamber volume. Then the shape becomes concave. The variations in the shape of the front are related directly to the inertia force and the properties of the wall. When the liquid meniscus reaches the outlet channel, one side of the liquid meniscus reaches the sharp corner, and it stops moving forward (marked by a red dashed circle). The other side of the meniscus continues to flow through the chamber. When the angle of the liquid meniscus changes to the equilibrium contact angle at the sharp corner, the gas-liquid interface exits through the outlet channel. The results indicate that the changed angles at the entrance and the exit of the chamber are closely related to the flow progression when geometric changes occur at the connections between the chamber and the channels. Finally, the liquid completes the filling process.

Filling process for inlet velocity of 0.1 m/s and _{s}

Re increases with the inlet velocity and the inertia force becomes dominant, as presented in ^{−3}, 3.34 × 10^{−3}, 5.17 × 10^{−3}, 7.02 × 10^{−3}, 9.29 × 10^{−3} and 12.47 × 10^{−3} s, respectively, at an inlet velocity of 0.5 m/s. As the free surface reaches the sharp corner at the entrance of the chamber, the gas-liquid interface near this corner stops moving. The other side of the free surface keeps moving. The inertial force causes the meniscus to move faster; it overcomes the flow resistance at the corner easily and drives the fluid flowing into the chamber. The strong inertial force causes the interface to remain convex for almost two-thirds of the chamber volume. In

Filling process for inlet velocity of 0.5 m/s and _{s}

When the liquid flows through the microstructure, a precise agreement between simulated data and experimental results is complicated [_{t}_{s}_{lc}_{uc}^{2}/m^{2}, ^{3}/m^{3}, ^{2}/m^{2}, and _{t}_{t}^{−3}, 2.61 × 10^{−3}, 10.51 × 10^{−3}, and 12.57 × 10^{−3} s, respectively. The difference is almost negligible at the four times. It means that the liquid filling processes inside our microstructures are approximately the same among three functional forms with specific parameters. The overall computational time for predicting the filling processes in our system by using the single ramp model, the stretched hyperbolic tangent model and the quadratic model is about 36 h, 51 h, and 53 h per case, respectively, using a computer with Intel^{®} Core™2 Duo CPU T8300 and 2G RAM. However, the computational time for predicting the liquid filling process in our system is the shortest by using the single ramp model. Thus the single ramp model is chosen to prevent a lot of additional computational time in the following simulations.

Filling process for three models: (

_{s}_{lc}_{uc}_{t}_{t}_{t}_{d}_{t}

Filling process for the single ramp model when _{s}_{lc}_{uc}

The effect of the parameters of the single ramp function at an inlet flow velocity of 0.25 m/s is studied and demonstrated in ^{−2}, 0.30 × 10^{−2}, 0.73 × 10^{−2}, 0.94 × 10^{−2}, 1.19 × 10^{−2}, and 1.33 × 10^{−2} s, respectively. For a hydrophilic wall in case (a), the flow front keeps moving into the chamber without stopping because of surface tension and inertia. When the surface tension dominates, the fluid flows into the chamber, wetting the side surface, and the effect of a capillary pressure barrier does not remain obvious. When the liquid moves into the chamber, the increase in the cross-sectional area of the chamber reduces the inertia. The interface stops moving at the sharp corner near the exit of the chamber and its angle changes to the contact angle. The interface retains its concave shape throughout the filling process. Given a hydrophobic wall for case (c), the front is convex as the liquid flows through the chamber without wetting the side surface. After the meniscus reaches the sharp corner near the inlet or the outlet channel, it remains at the corners until the bulk of the flow has sufficiently advanced. The liquid flow is still driven by the inertia, so the meniscus bulges over the whole chamber. The wave front becomes attached to the wall before the meniscus stretches across the whole chamber. The meniscus bulges sufficiently over the sharp corner near the outlet channel so the contact angle at the edge reaches imposed conditions.

Filling process for various _{s}_{lc}_{uc}_{s}_{lc}_{uc}_{s}_{lc}_{uc}_{s}_{lc}_{uc}

From the previous work [_{s}_{lc}_{uc}_{t}_{t}^{−2}, 0.30 × 10^{−2}, 0.73 × 10^{−2}, 0.98 × 10^{−2}, 1.19 × 10^{−2}, and 1.34 × 10^{−2} s, respectively. When _{t}_{t}_{t}

Finally, experimental and numerical studies of surface tension-driven flow in our surface-modified microstructures are performed. Because of the hydrophobicity of the PDMS, an oxygen plasma treatment is used to modify the surface of the PDMS substrate. Then the surface of PDMS microstructures can be modified to be hydrophilic. Thus flow experiments are conducted into both hydrophilic and hydrophobic PDMS microstructures. In ^{−1}, 4.78 × 10^{−1}, 6.24 × 10^{−1}, and 8.52 × 10^{−1} s, respectively. The results are confirmed by numerical simulation and experimental interface locations. The numerical results are also compared with experimental measurements and reveal similar filling processes. The parameters of the functional form of the dynamic contact angle for each case are (_{s}_{uc}_{lc}

A glass slide is also used to bond the PDMS replica. After O_{2} plasma treatment and bonding, the designed microstructure has been fabricated. This chip is also almost completely transparent in the visible spectrum. ^{−1}, 3.62 × 10^{−1}, 4.29 × 10^{−1}, 5.63 × 10^{−1}, and 6.91 × 10^{−1} s, respectively. And it can be seen that the static contact angle remains the similar value at 40 min to 100 min after the oxygen treatment process. A flow hindrance effect with regard to the surface tension for a pressure-driven flow can cause the contact angle to range from 90° to 180°. The assumed hydrophilic and hydrophobic wall conditions for cases (a) and (b), respectively, cannot be used for accurate calculation of surface tension forces. The flow front shapes show similar results only between case (c) and (d) experimental interface locations. Results show that numerical simulations with a single ramp functional model can successfully predict the moving of the meniscus compared with experimental measurements.

Filling process for various _{s}_{lc}_{uc}_{s}_{lc}_{uc}_{s}_{lc}_{uc}

Computational results corresponding to experimental measurements. (_{s}_{lc}_{uc}_{s}_{lc}_{uc}_{s}_{lc}_{uc}

Computational results corresponding to experimental measurements. (_{s}_{lc}_{uc}_{s}_{lc}_{uc}_{s}_{lc}_{uc}

In this work, the filling processes of the liquid in a disk-shaped microfludic device are numerically studied using microchambers with inlets and outlets fabricated by standard MEMS technology. A three dimensional computational model is proposed to trace the gas-liquid interface. Three different models of the contact angle variation with contact line velocity are employed on predictions of the liquid filling processes. The single ramp model is chosen to prevent a lot of additional computational time in our simulations. By utilizing the dynamic contact angle model, the simulations compared with the previous experimental results [

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC 102-2313-B-020-011-. Daryl Switak is appreciated for his editorial assistance.

Jyh Jian Chen is a person who makes substantial contributions to conception and design, and acquisition of data, participates in drafting the article, and gives final approval of the version to be submitted. Shih Chuan Liao is a person who provides numerical simulation, and makes analysis and interpretation of data. Mao Hsun Liu is a person who provides experimental measurement, and makes analysis and interpretation of data. Jenn Der Lin is a person who makes analysis and interpretation of data, and participates in drafting the article. Tsung Sheng Sheu is a person who provides experimental measurement, and gives final approval of the version to be submitted. Ming Miao Jr. is a person who provides numerical simulation, and gives final approval of the version to be submitted.

The authors declare there are no conflicts of interest.