reading

Chapter 4: Dynamics of Capillary Flow in a Channel with Constrictions and Enlargements

Appendix 8.1 Calculation of the Laplace Pressure for a Droplet on a Horizontal Cylindrical Wire

Appendix 8.4 Transition Between “All Wetted” and “All But Corners” Cases

Publishers at Scrivener
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Phillip Carmical (pcarmical@scrivenerpublishing.com)

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A book is the result of many converging efforts. I first would like to thank my co-authors, Ken and Erwin, to whom I am deeply indebted for the achievement of this book. Ken for his help with his key program Surface Evolver and his rigor in the scientific approach. Erwin for his flair in the development and design of new microfluidic devices and his vision of the future of point-of-care systems.

I also acquired considerable knowledge by participating to the development of point-of-care and home care devices with the Leti-Avalun joint program. In particular I would like to thank Myriam Cubizolles, Patrick Pouteau, Vincent Poher, Gwenola Sabatte, and Anne-Gaëlle Bourdat for turning my attention to the exciting interactions between physics and biology.

I am grateful to Professors Sophie Cribier at UPMC, Christian Frétigny at ESPCI, and to Lea Di Cioccio for their support for my researches.

I am also grateful to my colleagues who have contributed with photographs, sketches and discussions: David Gosselin, Maxime Huet, Giacomo Groplerro di Troppenburg, and Noemie Villard.

Many demonstrations figuring in this book have been done with devices fabricated for this purpose. I thank François Boizot, Catherine Pudda and Nicolas Verplanck for their involvement in the fabrication of these devices.

I would like to thank my company for having given me encouragements and support for this project, especially my management, Guillaume Delapierre, Fabrice Navarro, Claude Vauchier and Daniel Velou.

I am grateful to Martin Scrivener, my editor, for his patience during the long time it took us to write this book.

Finally I thank my children Erwin, Linda and Rosanne, and my grandchildren Noam and Eden (even if they are quite young to appreciate this book!), for constantly encouraging me during this work.

I would like to thank my Ph.D. advisor Fred Almgren for introducing me to the mathematics of soap films and liquid surfaces and his support and encouragement during the early development of my Surface Evolver. I would also like to thank the Geometry Center at the University of Minnesota for numerous summer visits and a sabbatical, which greatly contributed to Evolver development.

I would like to thank my postdoctoral mentor, Prof. David Beebe, for his support and guidance during my work in the area of open microfluidics as well as my colleagues Prof. Ashleigh Theberge and Dr. Ben Casavant. Many thanks too to my colleagues and friends in the laboratory of Prof. Beebe at the University of Wisconsin - Madison for inspiring conversations and exciting research that has lead to the development of open microfluidic systems. Finally, I would like to thank my wife Sanitta Thongpang for her support through highs and lows.

In 2012, Ken and Jean produced the book The Physics of Microdroplets. The aim of the book was to present the behavior of droplets in the many different configurations that occur in microsystems. It encompassed the behavior of sessile droplets on inhomogeneous substrates, droplets electrically actuated, interaction of droplets with interfaces, droplets in two-phase flows, and the use of droplets to align objects. The book was not strictly a story of droplets, as an introductory chapter to “open microfluidics” was also included. At that time, microflows with open boundaries, i.e. liquid-air interfaces, were starting to interest scientists working in space exploration as well as in biotechnology, biology and energy domains.

The interest in open microflows has continued to grow. Biologists especially have found that the accessibility provided by flow with open boundaries was extremely useful. Reagents can be easily added using a pipette–the fundamental tool of biologists–and fluid can be retrieved the same way. For biotechnological uses, open systems have the great advantage of simplicity of fabrication: they can be easily milled or molded, and they can be assembled together. They are also compatible with the new techniques of 3D printing. This simplicity is associated with low cost, which is required for the development of point-of-care and home-care devices–devices that can be used directly by a patient at home or at the doctor’s office. Open systems can also be straightforwardly converted into closed or partly closed systems by covering with thin plastic films, now currently available commercially, bringing a new versatility to microfluidic devices. From an energy standpoint, open interfaces allow for evaporation, and so cooling of microsystems can be easily performed using open microflows. Finally, open microfluidics, or open fluidics, is omnipresent in space applications where weight is the enemy: the removal of solid channel walls is a definite advantage.

Hence, using the same methodology as that of our first book, we decided that a continuation we call “open microfluidics” was opportune, due to the fast developments of this type of microfluidics. Erwin joined us in this enterprise, bringing the experience of a developer and pioneer of “suspended microfluidics”, a particular form of open microfluidics.

Due to the openness of some flow boundaries, the driving pressure must be small or even zero, else the fluid would overflow, and capillarity is the basis of the actuation of the fluids in open microfluidics. Open microfluidics is indissolubly linked to capillarity, as will appear in the following chapters.

In this new book, we tried to merge theoretical developments, numerical approaches–principally with the software Surface Evolver, stretching its application with care to microflows dominated by surface tension–and experimental examples, in order to give the reader the widest possible view of “open microfluidics”.

In the spirit of continuity of our approach with that of the previous book, we are happy that our former publisher Martin Scrivener continued to have confidence in us and the book. The Evolver files corresponding to the examples and problems of this book are available for the reader at the internet address http://www.susqu.edu/brakke/openmicrofluidics.

We hope that our work will be useful to boost the developments of microfluidic systems and that this book will find an echo in the micro and nanotechnology world.

Jean Berthier, CEA-Leti, University of Grenoble, France
Kenneth A. Brakke, Susquehanna University, PA, USA
Erwin Berthier, Department of Biomedical Engineering,
University of Wisconsin Madison,
USA

Readers of this book are entitled to access all the Surface Evolver datafiles used in production of this book at http://www.susqu.edu/brakke/openmicrofluidics. There are also several animations, and an interactive app that does the phase diagrams in chapter 8.

Open Microfluidics

Microfluidics is a relatively new scientific domain. Nevertheless its evolution has been extremely fast. Even if solutions for microelectronics [1-4] and outer space [5-8] have contributed to the development of microfluidics since the mid-1950s, it is now mainly biotechnology that boosts microfluidics and contributes to making it a growing scientific domain.

The goal of biotechnology is the fabrication of highly sophisticated tools to assist biologists in their research, automate and increase the efficiency of biology and medicine, and furnish solutions for the discovery of new drugs in pharmacology. At its beginning, biotechnology followed an engineering approach, due to the necessary physical development of the techniques. Progressively it has shifted to a biology-oriented field, in order to be closer to the needs of biology and medicine. [9].

These tools have first targeted with success genomics and DNA recognition. For example, many different solutions for sequencing DNA and biorecognition have been developed [10-14]. Amplification of DNA strands by massively parallel PCRs (polymerase chain reaction) is probably one of the greatest achievements of biotechnology [15-17]. Then, owing to its immense potentialities, biotechnology extended its applications to protein analysis [19] and cell studies [19-24]. In particular, cell study, which includes cellular culture, cellular communication, cell migration, stem cell differentiation, cellular mechanics, etc., is now of utmost importance for the pharmaceutical industry [25-27].

Because biological targets are nearly always immersed in liquids, biotechnology heavily relies on microfluidics. However, the particularity of microfluidics for biotechnology is that it is seldom a stand-alone field. Microfluidic solutions for biotechnology most of the time require a multiphase approach (figure 1).

Figure 1 Biotechnology is a composite science in which microfluidics is a fundamental subdomain.

Biology, material science and chemistry are closely linked to the achievements of biotechnological systems. Clearly, systems for biology require adequate microfluidics to transport, concentrate, and sort the biochemical targets or the biologic objects. They need adapted materials, whose microfabrication is not too complicated and not too expensive, and the required chemical species and reagents for the completion of the biological processes. A good biotechnological solution is like a puzzle that incorporates compatible and adapted sub-solutions in each of these subdomains. Due to the evolution of biotechnology, different microfluidic solutions have been successively developed, which are shown in figure 2.

Figure 2 The main categories of microfluidics and their applications. Inertial microfluidics [97], reprinted with permission ©2008 ACS; paper-based microfluidics, reprinted with permission by Albert Folch, University of Washington, and from [96], reprinted with permission ©2011 ACS; rail based microfluidics [93], reprinted with permission ©2005 ACS; suspended microfluidics [74], reprinted with permission ©2013 PNAS; digital microfluidics, two phase flows and encapsulation, courtesy CEA-Leti; emulsion [38], reprinted with permission ©2003 AIP.

The first microfluidic solutions were based on closed or confined microflows. The fluidic network is a transposition to the microscale of conventional flow systems. These microfluidic solutions, usually called Lab-on-a-Chip (LOC), have the great advantage of using smaller volumes of samples and costly reagents than macroscale devices. Also, their sensitivity is higher, and operating time much shorter, due to the reduced dimensions of such systems. These flows are driven by pumps or syringes external to the chip itself and many different types of valves have been developed [28-30]. These devices are mostly used in laboratories, owing to the need of auxiliary external systems, such as pumps, multiple syringes systems, reservoirs, etc. Systems based on closed microfluidics have had great success and accomplished many important achievements, such as massively parallel DNA amplification [31,32], and the study of stem cell behavior [33-35].

In order to further reduce sample and reagent volumes, it was found that droplets could be used as vessels to perform the desired processes. The term droplet microfluidics is used to characterize such systems. The volumes used in such systems can be very small, on the order of a few nanoliters. Two different approaches depending on the targeted applications have been followed: first, a two-phase approach where the sample and reagents (usually aqueous liquids) are transported by an immiscible fluid (usually an organic liquid, such as mineral oil) in a larger network [36-40]. Second, a digital microfluidic approach, where droplets are moved one by one or in parallel on a patterned substrate by electrical (electrowetting and EWOD) or acoustic (SAW) methods [41-45].

Recently, the need for portable systems has appeared. This need is linked to the development of point-of-care (POC) and home-care medicine, where user-friendly, portable, and low-cost systems can be used at the doctor’s office or directly by the patients themselves to monitor their health, or detect bacteria and viruses from a blood prick [46-50]. Contrary to the conventional microfluidic solutions presented above, the requirement for portability and low cost is associated to the development of passive or nearly passive solutions, where external auxiliary systems are absent, except perhaps the energy of a mobile phone or a compact transportable energy source. Obviously, capillarity is the solution for moving liquids under these conditions [51]. In a capillary solution, the energy required for the motion of the fluids is the surface energy of the walls, which is built in at the moment of fabrication, or by appropriate functionalization of the walls [46,52].

Such portable systems arewell-adapted, for example, to blood monitoring [53,54]. Human blood contains a bounty of information on human health: from the numerous metabolites contained in the plasma, such as glucose, cholesterol, and thyroidal hormones, to the bacteria and viruses transported by blood cells, and circulating tumor cells characteristics of cancerous attack [55-58]. Moreover, cell count, coagulation time, hemoglobin and fibrinogen levels are of great importance for health monitoring [59,60].

Capillary flow in cylindrical tubes was first studied by Bell, Cameron, Lucas, Washburn and Rideal in the 1910s [61-64]. With the development of new biological solutions for point-of-care and home care systems, studies on capillary flows have seen a revival. The first capillary systems to have been developed are fully closed rectangular channels; new functionalities such as trigger valves and capillary pumps have been invented to enhance the potentialities of such devices [65-66].

Still more recently, it became apparent that direct accessibility to biological systems would be a great advantage [67]. Open systems, i.e. microfluidic systems with open boundaries, bring the advantages of accessibility: Addition of reagents, pipetting for the addition or retrieval of biologic liquids or objects, and human interventions on the system can then all be easily performed [52]. Also, optical observation is facilitated. Finally, these systems have the ability to eliminate air bubbles, which are a serious drawback in many closed systems. All these aspects contribute to making open capillary systems an interesting choice for POC and home-care systems, under the condition that the limit of detection (LOD) and scalability are sufficient.

Probing biological systems locally in an open space can lead to new insight and breakthroughs. Living matter likes surfaces. Substrates that are functionalized for biological applications are increasingly used and also commercially available. Microfluidics should be able to interact with such substrates in the open space, essentially in their native state, which will facilitate the study of biological samples. To succeed in these endeavors, microfluidics needs to eliminate one of their major constraints: the walls.

These arguments have led to the development of capillary systems where some boundaries are open, i.e. in contact with the surrounding air. The names of open microfluidics, or open-surface microfluidics, or open-space microfluidics have emerged.

In fact, the domain of open microfluidics covers many different situations. Open capillarity has many different aspects, from the propagation of capillary filaments in corners [5,6,69-71], to the spontaneous capillary flow in open U and V-grooves [71-73], to suspended capillary flows [74,75], and to paper-based and thread-based microfluidics [76-80]. A panel of the different open-surface microfluidic configurations is shown in figure 3. In this book, electrowetting, capillary self-alignment and capillary rise are not treated extensively, as they are already widely documented in the literature [81,82].

Figure 3 The main categories of microfluidics and their applications.

The first chapter of this book is dedicated to the theoretical approach to spontaneous capillary flow (SCF). Using the Gibbs free energy [83], it is shown that the condition to obtain SCF in an open or closed, composite or not, flow channel is that the equivalent Cassie angle defined in a cross-section is less than 90° It demonstrates that SCF occurrence depends only on the geometry and the contact angles [84]. Next, the dynamics of capillary flows are presented. It is shown that, except for a very small length at the channel entrance where inertial effects appear [85], the viscous regime defined by the Lucas-Washburn-Rideal (LWR) model can be transposed to arbitrary cross-sectional channels, if precautions are taken [62-64,86]. Finally, the question of the dynamic contact angle is investigated. It is shown that an advancing contact angle only concerns essentially the entrance to the capillary channel [87-89].

The second chapter presents an oft-encountered feature in modern capillary microsystems: capillary filaments. The physics of these filaments was first investigated by Concus and Finn [5,6] in the context of spacecraft studies. These filaments may form in sharp corners or in cracks, and can extend endlessly as long as there is liquid available [82]. In capillary systems, these filaments may flow alone, or with the bulk of the liquid [69-71]. The different flow regimes in rectangular open channels are presented. Next, it is shown that the SCF condition in sharp V-grooves, deduced from the theory of the preceding chapter, reduces to the Concus-Finn condition [73,84]. Finally, the formation of filaments in different geometries is theoretically and numerically investigated.

Rectangular, open microchannels, for simplicity called U-grooves in this book, are probably the most common open microfluidic devices, due to their easy fabrication. It suffices to mill a plastic plate to obtain such channels. The study of spontaneous capillary flow in such channels is the subject of chapters 3 and 4. In chapter 3, the conditions for SCF in the geometry of U-grooves are presented. Different geometries are investigated: straight, turning U-grooves, and U-grooves of varying cross-section (figure 4).

Figure 4 Different geometries of U-grooves. A: SCF passing through multiple cylindrical chambers, from an inlet port (right) to an outlet port (left); B: parallel SCFs from an inlet port to multiple outlet ports; C: multiple microgrooves in parallel; D: winding U-groove with cylindrical wells; E: SCF filling of a cylindrical cavity; F: Concus-Finn filaments in an open cylinder; G: Concus-Finn filaments in a varying cross-section U-groove. Photographs: J. Berthier, N. Villard, D. Gosselin (CEA-Leti).

In chapter 4, dynamical considerations on the capillary flow in U-grooves are presented [72,90,91]. The concept of flow resistor is developed. It is shown that the concepts of trigger valves, capillary pumps, and flow resistor, transposed from closed capillary systems [65,66] to open channels, may still be valid if precautions are taken.

Suspended microfluidics has very recently appeared in the literature [74,75,92]. It is the subject of chapter 5. By definition, suspended microflows are flows in channels devoid of ceilings and floors. Spontaneous flow conditions for different types of suspended microflows are given. Suspended microfluidics brings additional accessibility to open biotechnological systems, and is the source of new applications. Especially, applications to suspended flows of liquid polymers are presented.

Chapter 6 presents new developments for rail-based microflows [93]. Rail-based microfluidics is, in principle, similar to suspended microfluidics. In such systems, the liquid flows between two horizontal rails, top and bottom, and the flow has open boundaries on both sides. At first sight, it is similar to suspended microfluidics, with a 90° rotation. However, it is very different from suspended microfluidics when considering the concepts of microfluidic networks. Such networks are not compatible with suspended geometries. SCF conditions in rail geometries are detailed in the chapter. Different rails morphologies are investigated.

Chapter 7 presents the development of paper-based microfluidics. Paper-based systems were first proposed long ago by Yagoda in the year 1937 [94]. They have recently seen a considerable revival with the developments of paper-strips and µPADs (micro paper-based analytical-devices). Strips are narrow bands of cellulose fiber where the liquid wicks the fibers and progresses in one direction and where the reaction zones are regions placed perpendicularly to the flow (figure 5). µPADs are two-dimensional planar devices where reaction zones are placed at the extremity of branches [76]. It appears that the solutions provided by labs-on-paper are very promising and have a large scope of applications [95]. In this chapter the principles, designs, detection methods and fabrication processes of paper-based devices and labs-on-paper are presented.

Figure 5 A: Sketch of paper strips. B: Photograph of a µPAD. From [98], reprinted with permission ©2014 Springer. C: Close up of a thread showing the fiber bundle. From [78], reprinted with permission ©2010 ACS.

The final chapter of the book, chapter 8, is dedicated to thread-based microfluidics. It is a very new domain, which has recently seen new developments [79,80]. The concept of thread-based microfluidics is the use of fibers to guide and transport liquids. The particular physics of fiber wicking is developed in the chapter, and applications to smart bandages are presented.

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Chapter 1

Theory of Spontaneous Capillary Flows

1.1 Introduction

By definition, spontaneous capillary flow (SCF) is a capillary flow occurring spontaneously under the action of a negative Laplace pressure at its front end while the rear end is at zero (relative) pressure.

Sometimes SCF can be a drawback in microsystems, other times it can be purposely used for fluid actuation. Let us first give two examples of “involuntary” drawbacks of SCF experimentally observed in microsystems.

The first example is the spontaneous absorption of liquid in cracks. It is a well-established fact–when manipulating liquid drops, or small amounts of liquids–that these liquids are absorbed by cracks, i.e. the cracks are easily wetted as soon as the liquid is slightly wetting, even if the contact angle is close to 90°. Figure 1.1 shows a droplet on a cracked dielectric substrate. In this particular case, the droplet does not penetrate the very hydrophobic cracks; however, as soon as the substrate is slightly wetting, the liquid flows by capillarity into the cracks.

Figure 1.1 A: A sessile droplet placed over a dielectric which has suffered breakdown; B: Evolver modeling of a droplet on a crack in the substrate; C: Surface Evolver modeling of a droplet located above two cracks in a solid substrate [1].

The second example pertains to the observation of uncontrolled spreading of liquid glue during the sealing of a cover of a microfluidic system, lab-on-a-chip, MEMS (micro-electromechanical system), or NEMS (nano-electrical-micro system). Glue is often used in microfabrication to seal the microdevice. The other alternative is the use of direct bonding, which requires restrictive conditions, such as very smooth, hydrophilic surfaces, resistance to heating (annealing) above 200°C, and small “void” chamber volumes in order that a large percentage of the cover surface is in contact with the substrate. So, in many cases, microsystems are sealed by a cover glued on top of the etched substrate. If the liquid glue contacts a micro- or nano-groove, SCF drives the glue inside the groove, even though its viscosity is of the order of 10 to 30 times that of water. After polymerization, there is a degradation of the system due to this undesirable spreading of liquid glue (figure 1.2).

Figure 1.2 A: View of a groove/trench in the microfabrication process of a NEMS (Nano-Electrical-Micro-System); B: perspective view of the grooves that can be found in NEMS; C: liquid glue has invaded the groove and has been polymerized (photo CEA-LETI).

Figure 1.3 shows the spreading of liquid glue in a groove similar to that of figure 1.3, calculated with the software Surface Evolver [1]. Depending on the volume of liquid available and the volume of the groove, the liquid can all be absorbed by the groove.

Figure 1.3 Spreading of liquid in an asymmetric groove (Evolver calculation).

Conversely, SCF can be of great help to move liquids. For example, in spacecraft, liquids in tanks are moved along “vanes,” which are wetting structures guiding the capillary flows, as shown in figure 1.4 [2].

Figure 1.4 Evolver calculation of SCF with different types of vanes. A: one foil perpendicular to a substrate; B: two foils and a substrate; C: SCF guided by a foil parallel to the substrate.

SCFs are also important in biotechnological systems. Historically, two main approaches have been followed. The first one is capillary wicking by paper fibers, leading to what is now called µPADs–for micro paper analytical devices [3]. These devices are convenient for doing fast and easy-to-use tests for metabolites in human plasma, such as glucose, cholesterol, thyroid hormones, and now even viruses. This approach involves the particular physics of wetting and will be presented in chapter 9. The second approach uses capillary flows in channels and has received the name of “Capillarics” [4-6]. Such microfluidics networks can perform many microfluidic functions, and they are beginning to be used for POC (point-of-care) and home care systems with the same aims as µPADs. Figure 1.5 shows a paper-based capillary system for the extraction of plasma from whole blood, and a typical capillary microfluidic network.

Figure 1.5 A: Paper-based device for the detection of glucose (courtesy CEA-Leti); B: capillary network.

In this chapter, we conduct a theoretical and numerical study of spontaneous capillary flows. First, we follow a static or quasi-static approach to determine if a SCF can occur depending on the geometry of the channel and the wetting properties of the liquid. The general condition for SCF deriving from Gibbs free energy is given [7-9]. Second, a dynamic approach is made, using the well-known Lucas-Washburn-Rideal (LWR) law as the basis of the study [10-13]. Finally, we deal with the much debated question of the dynamic contact angle [14-18].

1.2 Quasi-static Approach to SCF

In this section, we present a theoretical approach that assumes that the advancing interface adopts immediately, at each time step of its motion, a static shape–i.e. satisfying the properties of a minimal energy surface–even if the interface is not at rest. Assuming that the upstream condition is zero pressure (large reservoir), if the front end of the flow has a negative Laplace pressure, the flow will continue; if it has a zero Laplace pressure (zero mean curvature), the flow is stopped; if it has a positive Laplace pressure, the flow will recede.