1.1 What is Microfluidics?

Microfluidics is the science of fluids on the micro- and nanometer scale. Academically, it is a subdiscipline of fluid mechanics, as the fundamental equations describing the physics of fluids at larger length scales are identical to the equations underlying microfluidics. However, compared to classical fluid mechanics, the length scales in microfluidic systems are significantly smaller. There is no clear prerequisite as to when a system can be considered as being microfluidic. In general, if one of the characteristic length scales, e.g., the height or width of a fluid system features dimensions in the micrometer range or below, such a system can be considered a microfluidic system. At these length scales, the fluid physics is different from what we are used to seeing in larger scale systems. As an example, microfluidic flows are usually strictly laminar. Laminar flow is rare in macroscopic systems. At an example, consider a river or even a seashore. The water in these systems usually flows under turbulent flow conditions. This is something we rarely see in microfluidic systems.

Another important aspect of microfluidic flows is effects of surface tension. Surface tension is usually an effect not very important for macroscopic fluidic systems. However, in microfluidics, gravitational forces are usually negligible due to the fact that the amount of liquids used is so small. Surface tension becomes predominant, e.g., in systems which transport liquids in the form of single droplets on open surfaces or within closed channels.

As we will see, the fundamental laws of fluid mechanics are valid for microfluidics as well. However, due to the fact that effects such as fluid turbulence, gravity, and the like can often be neglected, the equations describing microfluidic systems are often significantly simplified versions of the equations of fluid physics. This makes microfluidic an attractive discipline for the study of the properties and dynamics of fluid flow where effects, such as, diffusion, can be studied at very high resolution.

1.2 A Brief History of Microfluidics

The development of microfluidics is very closely related to the history of classical microstructure technology. The latter is mainly based on the huge scientific and commercial success of microelectronics that revolutionized electrical engineering and electronics. The term microelectromechanical system (MEMS) was originally defined at the beginning of the 1980s and describes systems that feature both electrically and mechanically actuated or interrogated components. The term was later complemented by the aspect of optical component integration, for which the term microoptoelectromechanical system (MOEMS) was introduced. The first real systems for which the term MEMS is justified were presented by Petersen et al. in 1982 [1] and later by Chen et al. [2] in 1984. These systems consisted mainly of a mechanically movable mass manufactured alongside electronic circuitry on a chip which was used to measure acceleration. It seems like a straightforward implementation of a very basic MEMS system: the electronic components could be produced in complementary metal oxide semiconductor (CMOS) technology which also allowed the creation of bulk mechanical structures such as the acceleratable masses required for these accelerometers. The mass could be made movable by means of sacrificial layers that were etched during manufacturing. Interrogating the position of this mass by means of the electronic circuitry allowed for the creation of the first MEMS systems. As of today, the most commonly used material for semiconductors is still silicon for which decades of extensive research and process optimization have resulted in highly scalable and reproducible processes. The early MEMS systems have profited extensively from this experience which explains why silicon (and its close relative, glass) are still frequently used materials. This also explains the fact that the early microfluidic systems were almost exclusively produced in silicon by means of photolithographically structured resist layers serving as a mask for wet etching, both of which were among the most frequently used processes in MEMS technology at that time. How closely related microfluidic is to MEMS can be seen when looking at early reviews of microfluidic systems and processing techniques, e.g., the review by Gravesen et al. written in 1993 [3]. The manufacturing techniques used were almost exclusively established MEMS processes such as lithography and wet or dry etching.

In contrast to the early MEMS accelerometers, the first microfluidic systems predicted the variety that would later become one of the most important characteristics of the community. Three early contributions are worth noting here.

1.2.4 Microfiuidics Today

Today, the microfluidic community is a diverse scientific amalgamation of various disciplines ranging from physics, engineering, material sciences, all the way to biology, biochemistry, and even information technology. Since 1990 the number of papers and patents on microfluidics has increased steadily as the community has grown (see Fig. 1.1). There are several noteworthy annual meetings of the community such as the International Conference on Miniaturized Systems for Chemistry and Life Sciences (μTAS) and the Microfluidics, BioMEMS, and Medical Microsystems.

1.3 Commercial Aspects

There have been many studies and reports about the market potential and the most important potential applications for microfluidics. One of the most commonly cited sources of the market potential of MEMS is published annually by Yole Développement (www.yole.fr). The most recent report, dating from 2015, estimates the global MEMS market to double between 2014 and 2020, reaching a worldwide market volume of 22 000MUS$. The share of microfluidic devices is estimated to be around a fifth of this volume [9].

Because the use of microfluidics in inkjet printers has been such a commercial success, the community has often sought the “killer application”, i.e., a device mainly based on microfluidic concepts and principles that would yield highly selling products. However, until now, such an application has not been found. Despite this, microfluidics is considered to be the key discipline for the development of laboratory test instruments, as well as home care diagnostics. The reason why the technology has not yet found the widespread applications for which it is doubtlessly suitable is difficult to find. This has been noted repeatedly in the literature, see for example, the recent comment by Whitesides [10]. An excellent series of articles has been published over the last two years by Becker, who focused on commercialization aspects of microfluidic devices and tried to elaborate why there are still so few successfully commercialized microfluidic products. The series discusses the question of whether or not there actually is a “killer application” [11, 12], the factors influencing the manufacturing cost and therefore the industrial producibility of a device [13], the importance of intellectual property [14], and the need for (or the lack) of standardization [15]. It also includes a detailed discussion of the industrial requirements for successful device commercialization [16, 17]. However microfluidics promises such immense advantages, e.g., low sample volume consumption and fast chemical reactions due to short diffusion lengths, strictly laminar flow, etc. that it is one of the most promising evolving technological fields.

1.4 About This Book

The aim of this book is to provide a general and easy-to-follow introduction to the fundamentals and the mathematics of microfluidics. One of the main advantages of microfluidics is the fact that it is comparatively easy to derive theoretical models for experimental data. However, microfluidics is still fluid physics. Thus, students eager to learn about the fundamentals of microfluidics will very quickly find themselves faced with differential equations, vector analysis, thermodynamics, and engineering mathematics. I have found many students struggling with this wide choice of academic disciplines. Some students may even discover that their respective curriculum did not include sufficiently detailed courses on fluid physics, engineering mathematics, or thermodynamics. Thus, some of the most important fundamental concepts of microfluidics will remain a black box for them.

This book intends to cover microfluidics as a multi-disciplinary topic. It contains large introductory sections which will reintroduce and revise most concepts required for understanding the fluid mechanics of microfluidics. The book takes time deriving and explaining the equations. It is not the intention to simply list the most important ones. There are excellent textbooks in the literature which summarize the most important equations of microfluidics and which may serve as a quick lookup if a specific equation is sought. This book is different. It will not only list the equations, it will derive them. It will explain them. And it will show practical examples using the Algebra package Maple which helps in visualizing the equations and their significance. Some of the equations can even be solved and displayed with Microsoft Excel. As it turns out, this is surprisingly easy.

The book comes with a set of Maple worksheets which can be used to experiment with the equations. It will allow students and researchers to quickly adapt the equations to a specific application. Examples include diffusion, dispersion, pressure drop, velocity distribution, and similar effects.

The book is intended for a broad readership. It will start explaining all fundamental concepts required to understand later chapters. You only remember Fourier or Taylor series expansion from the early days of your study? You have never heard of Laplace transforms, operators, or vector calculus? You have never attended a lecture on thermodynamics? Do not worry – this is not a prerequisite for understanding this book. This book will start from scratch, even allowing students who may have had little to no engineering mathematics in their curriculum to understand the mathematical tools required for solving the seemingly complex equations of microfluidics. As we will see, microfluidics is not very complex and, after a proper introduction, most of the tools can be understood and applied quite readily.

1.5 Structure of This Book

This book is divided into four parts. The first part is entitled “Fundamentals” and details fundamentals in engineering mechanics, vector analysis, thermodynamics, as well the tools for solving differential equations. This section may serve as repetition for some readers, others may wish to study it in detail in order to be able to follow the derivations made at a later stage of this book.

The second part is entitled “Bulk Fluid Flows” and describes the phenomena of fluid flows without taking into account the effects of surface tension effects. This section will introduce and derive the fundamental equations of fluid mechanics and describes some of the most relevant flow scenarios and their underlying equations.

The third part of this book is entitled “Fluid Surface Effects”. It describes the physics of effects which are due to or experienced at fluid surfaces. Besides capillarity and wetting, there are numerous important effects related to surface tension, e.g., the instability of fluid jets.

The fourth part of this book is entitled “Numerics” and details numerical methods and their applicability to fluid physics and microfluidic flows. In this part, the most important numerical methods are introduced and explained using examples from microfluidics. As a final project, a three-dimensional numerical solver will be implemented from scratch, which can be used to solve complex flow problems.