Introduction

In this chapter, the focus is on presenting the fundamentals of molecular simulations, particularly molecular dynamics (MD) and its derivatives, particularly nonequilibrium molecular dynamics (NEMD), or MD in a system that is exposed to an external field. The number of molecular simulation tools has grown substantially over the last few years, and it would be impossible to mention all the applications that have appeared in the literature. MD simulations have been used in a variety of ways; four of the most important ways in the context of this text that such calculations have been used are as follows:

• Calculating transport properties of new fluids and mixtures for which such properties have not been measured or as a test of the accuracy of such experimental measurements

• Predicting the equilibrium conformation of complex biomolecules and polymers

• Verifying fluid dynamics boundary conditions (Koplik et al., 1989)

• Predicting the flow of pure fluids and complex mixtures in very small channels and circular pores, where the continuum approximation may break down and where direct experimental measurements are not available

Indeed, it is the last class of problems that is of particular interest in this chapter. It is certainly to be noted that while restricted to very short length and time scales, molecular simulations provide information that cannot be found any other way.

As has been seen throughout this book, ionic and biomolecular transport devices are now being used for drug development and delivery, single molecule manipulation, detection and transport, and rapid molecular analysis.

Micro- and nanofluidics

Analyzing and computing fluid flow at small scales is becoming increasingly important because of the emergence of new technologies such as the ability to construct microelectromechanical systems (MEMS). These systems may be used for drug delivery and its control; DNA and protein manipulation and transport; and the desire to manufacture laboratories on a microchip for rapid molecular analysis, requiring the modeling of flows on a length scale approaching molecular dimensions. On these small scales, new flow features appear that are not seen in macroscale flows.

Because of the large surface-to-volume ratio in nanochannels, surface properties become enormously important. Because the pressure drop Δp ˜ 1/h 3, it is prohibitively large for a nanoscale channel. Thus fluid, biomaterials such as proteins, and other colloidal particles are most often transported electrokinetically, and the art of designing micro- and nanodevices requires a significant amount of knowledge of fluid flow and mass transfer (biofluids are multicomponent mixtures) and often heat transfer, electrokinetics, electrochemistry, and molecular biology. To efficiently manufacture laboratories on a microchip, the analysis and computation of flows on a length scale approaching molecular dimensions, the nanoscale, are required.

The common thread is micro- and nanofluidics. Thus micro- and nanofluidics play the role of unifying the fields of fluid mechanics, heat and mass transfer, electrostatics and electrodynamics, electrochemistry, and molecular biology. In particular, nanofluidics opens the door to uncovering the structure and conformation of biomaterials, such as proteins, through molecular simulation.

Syllabus: Introduction to Micro- and Nanofluidics for Engineers and Physical Scientists (Three hour lecture, 1 hour lab)

Syllabus for a 4 hour (3 hour lecture, 1 hour lab) semester course 15 weeks long, assuming 30 lectures of length 1 hour 12 minutes, two lectures per week, and does not include exam periods. The lab meets once per week, with two lectures on fabrication and experimental methods – the content of lecture 6a. If the class meets three times per week, prorate lectures to 45.

Objective: The objective of this course is to introduce students to the basic physical foundations of incompressible fluid mechanics appropriate to micro and nanosize conduits. The course will emphasize the fundamental principles involved in the formulation and solution of problems in fluid mechanics and mass transfer for pressure-driven and electrically driven motions of biofluids and individual components such as ions. On completion of the course, the student should be able to extract from a raw physical situation the essential principles from which a useful fluid mechanical model may be developed.

The lab will provide students with an introduction to the design, fabrication, and testing of microfluidic and nanofluidic devices and how practical devices can be analyzed theoretically.

The book is meant to be used as a text for an interdisciplinary course in micro- and nanofluidics that includes the study of ionic and biomolecular transport at the advanced undergraduate and beginning graduate levels. The rationale for this book is that most, if not all, problems in the twenty-first century are interdisciplinary in nature, yet no textbooks address the topics required for investigating problems that cut across disciplines in engineering, the physical sciences, and mathematics. The closest approach to this concept is in the several texts that address the thermal sciences at a strictly undergraduate level (Moran et al., 2003). Another set of texts addresses problems in applied mathematics applicable to engineering problems generally at the advanced graduate level (Bird et al., 2002). Still another set of texts under the general area of biophysics links the mathematics and biological sciences, again most often at the advanced graduate level (Murray, 2001, 2003). In contrast, this book aims at the advanced undergraduate and beginning graduate student pool.

A number of other related books are on the market, but all are monographs directed at the senior graduate student (Karniadakis et al., 2005; Masliyah and Bhattacharjee, 2006; Tabeling, 2005; Liou & Fang, 2006; Nguyen & Wereley, 2002; Bruus, 2008; Kirby, 2010; Chang and Yeo, 2010). All these texts emphasize the unique features of transport at the micro- and nanoscale, of which there are many.

Introduction

As discussed in Chapter 1, many of the current or planned microdevices contain rectangular channels arranged in an array; such channel systems are termed nanopore membranes. For example, the nanopore membranes depicted in Figure 1.1 contain up to 30,000 channels, depending on the channel height, which is the nanoscale dimension. Scanning electron microscope (SEM) images of nanopore membranes fabricated by Dr. Shuvo Roy's group at the Cleveland Clinic1 are depicted in Figure 4.1. You will recognize Figure 4.1(a) as Figure 1.1(b), and is repeated for clarity. The nanopore membrane is fabricated so as to contain channels of a uniform nanoscale height. They fabricated membranes with a variety of pore sizes, ranging from h = 6.3 nm to h = 91 nm, and there are 11, 500 pores per membrane. Similar membranes are now being used in the development of an artificial kidney (Fissell, 2006). Obviously, it would be difficult to model the flow in such an array from first principles.

There are at least two ways to overcome this problem: (1) treating the membrane as a porous media problem, the membrane being characterized by a permeability and a porosity, and (2) investigating the flow in each channel separately, assuming that the flow behaves similarly in each individual channel. It is the latter approach that is explored here.

Introduction

As has been mentioned in Chapter 2, applications of micro- and nanofluidic analyses include drug delivery and its control, DNA manipulation and transport, protein separations, rapid molecular analysis, renal assist devices, biochemical sensing, and cancer treatment. In the present chapter, several of these applications are described, and modeling tools are developed using the ideas developed in the preceding chapters.

Generally, many of the biomedical devices being tested for the preceding aplications are essentially synthetic micro- or nanopore membranes consisting of an array of channels, much as in Figure 1.1. For example, these devices can be used to deliver insulin to a diabetes patient, as needed, in a highly controlled manner. They have also been considered for use as a renal assist device (RAD) whose purpose is to replace some of the functions of the kidney; this device is discussed later. DNA has been shown to be able to navigate through approximately cylindrical nanopores, and this problem is also discussed in this chapter; the in vivo transport of DNA through a nanopore has been demonstrated by Bayley and Cremer (2001).

More general applications of nanotechnology and additional information on some of the applications described may be found in the short monograph edited by Malsch (2005) and in the monograph by Ciofalo et al.

Introduction

In the previous chapter, the types of steady viscous flows common to micro- and nanofluidics were presented. Of particular importance is the Poiseuille flow and flow past a sphere. In the present chapter, classical solutions for heat transport and mass transport of uncharged species that may occur in parallel with the Poiseuille flow are described. This chapter will lead into a discussion of electrostatics and electrochemistry, concepts that are so important in micro- and nanofluidics.

Both heat and mass transfer are discussed in this chapter. This is because of the similarity in the forms of the governing equations and boundary conditions and the similar nature of the transport properties: the thermal conductivity and the mass diffusivity or diffusion coefficient. Indeed, heat and mass transfer often occur simultaneously (Figure 5.1). Both heat and mass transfer can be grouped into two modes: conduction heat transfer, diffusion mass transfer, and convective heat and mass transfer. A third mode of heat transfer, radiation heat transfer, is not discussed.

Fortunately, most liquid flows in micro- and nanochannels do not encounter large temperature and concentration gradients so that in general, the fluid properties remain relatively constant. Thus the governing equations can usually be decoupled to some extent, even at Reynolds numbers not normally viewed as being small.

Several basic problems of heat transfer in channels are considered before presenting the formal definition of the fully developed temperature distribution. Unlike the formal definition of fully developed fluid flow, the solution for the fully developed temperature distribution depends on the boundary conditions on the temperature at the walls of a tube or channel.

Introduction

A working definition of the field of electrochemistry is the study of charged chemical and biological material. The field of electrochemistry appears to have begun over 200 years ago, with the discovery by Galvani that a frog's limbs responded to the touching of its nerves by a scalpel with a severe twitch when the scalpel was electrically charged by a nearby source (Bockris & Reddy, 1998).

Much of electrochemistry involves the study of the properties of ionic solutions first studied by P. Debye and E. Hückel in the 1920s. The modern field of electrochemistry is separated into two related but distinct aspects: ionic solutions and electrode kinetics. Of course, in the problems of interest in this book, the two are related; however, Bockris and Reddy (1998) point out that the two fields today are significantly different, with the field of what they call “electrodics” branching out to applications in the automobile industry, the study of batteries, and fuel cells.

In this chapter, the essential principles of electrochemistry are presented, enabling a thorough understanding of the chemical principles involved in the study of electrokinetic phenomena to follow. Many biofluids are aqueous mixtures of electrolytes, which are species that are either positively or negatively charged. The mixtures of interest are aqueous and are usually dilute in the electrolyte species. An example of a common electrolyte mixture is phosphate buffered saline (PBS), which contains five different ionic species in an aqueous solution.

Introduction

It may be said that at some level cells are the building blocks of all living things. The most basic of all organisms are made up of single cells, and higher organisms, such as animals and humans, are made of a large number of cells all arranged in a specific way.

Because the nanoscale is the scale of biology, there has been an explosion of new knowledge and new ideas in the biological sciences. This has been the result of significant advances in static measurement techniques such as the surface force apparatus (SFA), the atomic force microscope (AFM), and the scanning tunneling microscope (STM). Moreover, optical techniques can be employed for single molecule detection and analysis. With explosive improvements in computer architecture, molecular dynamics (MD) and its derivatives (nonequilibrium molecular dynamics, NEMD) are now being employed to study the motion of ions and proteins in a flowing solution and the conformation of proteins and other macromolecules in a static medium.

The links between nanotechnology and biology are numerous. Entire devices for the rapid analysis of proteins, drug delivery, biochemical sensing, and other applications are being developed at a rapid pace. Entire books are being published on biomedical applications at the nanoscale (Malsch, 2005; Ciofalo et al., 1999). Artificial and implantable pumping systems employing nanopore membranes are being developed for filtering proteins and ions for use as a renal assist device (Fissell, 2006) (see Figure 4.1).

Introduction

It was shown in Chapter 2 that as the length scale of microchannels approaches the nanoscale, pressure-driven flow becomes increasingly difficult to achieve. The precise cutoff depends on the desired flow rate, but for a volume flow rate of Q = 1 µL/min, electro-osmotic flow (EOF) becomes significantly more efficient around an individual channel height of h = 40 nm. Using the material studied in Chapters 4–8, we can now investigate flows of ionic and biomolecular species in micro- and nanochannels.

Electrokinetic phenomena are generally grouped into four classes:

1. 1. Electro-osmosis (EOF): the bulk motion of a fluid caused by an electric field

2. 2. Electrophoresis: the motion of a charged particle in an otherwise motionless fluid or the motion of a charged particle relative to a bulk motion

3. 3. Streaming potential or streaming current: the potential induced by a pressure gradient at zero current flow of an electrolyte mixture

4. 4. Sedimentation potential: the electric field induced when charged particles move relative to a liquid under a gravitational or centrifugal or other force field

Note that electrophoresis and sedimentation potential refer to particles and electro-osmosis and streaming potential refer to electrolyte solutions. The first three of these are by far the most important for our applications, and in this chapter, we discuss the phenomenon of electro-osmosis first, followed by a discussion of electrophoresis and streaming potential.

Introduction

Electrostatics and electrodynamics are essential features of micro- and nanofluidics. At the macroscale pumps, compressors and other fluid-handling devices move fluid using a pressure difference, as was noted in Chapter 4. However, as was demonstrated in Chapter 1, this is often not feasible at the micro- and nanoscale because very large pressure drops on the order of atmospheres are required for flows through channels under about 0.1 µm, channels that are often used in applications.

The objective of this chapter is to introduce, in a general way, the concept of an electric field at the advanced undergraduate level, typically associated with the material presented in an introductory physics class for physics majors; for those students already familiar with the principles of electrostatics, this chapter may be scanned or skipped.

In this chapter, the basics of electrostatics are presented, paying particular attention to the electric field associated with one or more charged spheres, planes, or cylindrical surfaces. In the course of discussion, the concept of a surface charge density is defined, which is a concept central to electrochemistry and micro- and nanofluidics. This is followed by a discussion of the fundamental law of Gauss relating the electric field to the charge density of a given medium. The concept of electrical potential is then introduced, and the similarities between the velocity potential of fluid mechanics are noted.

Next, the concept of an electric dipole is introduced, followed by the derivation of the Poisson equation of electrostatics, which is solved for several different situations.

Introduction

Mechanical engineers design new products for consumer use: engines for automobiles, airplanes, and other devices; cars; air-conditioners; heat pumps; compressors; fans; hair dryers; and all sorts of other products. Increasingly, mechanical and chemical engineers are involved in the design of biomedical devices for drug delivery systems, biochemical sensing, and rapid molecular analysis. In this chapter, the basic numerical techniques used to provide design and performance criteria of these devices are described. A simplified view of a general design process is depicted in Figure 10.1.

In this chapter, we shift gears a bit and discuss some basic concepts associated with numerical methods.1 These methods are required when no simple analytical solution is possible. What is meant by the term analytical is that no solution can be found in terms of simple functional forms such as the polynomial, trigonometric, exponential, logarithmic, or hyperbolic functions. In the following, much attention is focused on the basic methods required to solve a nonlinear ordinary differential equation. This requires several different capabilities:

• Numerical differentiation

• Solving sets of linear(ized) equations

• Numerical integration

There are many situations for which numerical methods are required in micro- and nanofluidics. For example, determining the dependence of the ζ potential on pH in Chapter 7 requires a numerical zero-finding technique. The non-linear Poisson equation for the electrical potential discussed in the preceding chapter requires a numerical solution for the potential. In general, the solution of the potential equation for multicomponent and multivalent mistures requires a numerical solution.

Introduction

Why study perturbation techniques? The answer lies in the fact that we can simplify the problem significantly if we study the structure of the equation. In some cases, analytical solutions may be obtained, which eliminates any need for computational work. More commonly, the application of these techniques results in a problem that may be solved numerically much easier than the original problem.

Perturbation methods had been known since the days of the ancient astronomers to predict the trajectories of planets subject to small disturbances. The modern concept of regular and singular perturbation problems was developed in the 1960s and refined over the 1970s in fluid mechanics to investigate the high Reynolds number flow past a flat plate, the boundary layer. The function of a boundary layer is to bring the velocity to zero on a solid body. Because the boundary layer is very thin, of O(Re −1/2), the velocity will vary rapidly across the layer, making numerical solutions very difficult, if not impossible. In the intervening years, boundary layer–type behavior, the rapid variation of any quantity over a short distance or period of time, has been used in a wide variety of disciplines to describe such behavior.

There are a number of books on this subject, and these appear in the bibliography; they include Kevorkian and Cole (1981, 1996), VanDyke (1975), Nayfeh (1973), Bellman (1964), Bender and Orszag (1999), Holmes (1995), Hinchcliffe (2003), and Howison (2005). The book by VanDyke (1975) is specifically oriented toward fluid mechanics, whereas the others are more general treatments.

Introduction

In the first two chapters, the field of micro- and nanofluidics was introduced in a general framework, describing multiple applications as well as the scientific issues associated with fluid flow at small length scales. In Chapter 2, the fundamental transport properties involved in mathematical description of fluid flow, heat and mass transfer, and electrostatics were introduced. In this chapter, we use these concepts to derive, from first principles, the governing equations necessary for analyzing micro- and nanofluidic phenomena. For the purposes of this chapter, it will be assumed that the properties of the fluid medium are constant.

Microfluidic devices are being used for rapid and continuous purification of proteins; a sketch of such a device is shown in Figure 3.1. The device addresses the need for high-throughput purification of very small amounts of proteins and enzymes from the carrier fluid. The term protein purification refers to a series of operations meant to isolate a single protein or enzyme in a complicated mixture. Here the microfluidic transport processes involve mass transport of a relatively large number of species with the target molecules present in as little as microgram per liter concentrations. This device can purify a sample in a short period of time and does not require a large amount of sample. The means of developing a model for such a device is discussed in this chapter.

The governing equations of fluid motion on the macroscale are the (incompressible) Navier–Stokes equations. These equations, along with conservation of mass, or the continuity equation, enable the calculation of, in the general three-dimensional case, the three velocity components and the pressure.