© 2002 by CRC Press LLC 4.7 Molecular-Based Models In the continuum models discussed thus far, the macroscopic fluid properties are the dependent variables, while the independent variables are the three spatial coordinates and time. The molecular models recognize the fluid as a myriad of discrete particles: molecules, atoms, ions and electrons. The goal here is to determine the position, velocity and state of all particles at all times. The molecular approach is either deterministic or probabilistic (refer to Figure 4.1). Provided that there is a sufficient number of micro- scopic particles within the smallest significant volume of a flow, the macroscopic properties at any location in the flow can then be computed from the discrete-particle information by a suitable averaging or weighted averaging process. This section discusses molecular-based models and their relation to the continuum models previously considered. The most fundamental of the molecular models is a deterministic one. The motion of the molecules is governed by the laws of classical mechanics, although, at the expense of greatly complicating the problem, the laws of quantum mechanics can also be considered in special circumstances. The modern molecular dynamics (MD) computer simulations have been pioneered by Alder and Wainwright (1957; 1958; 1970) and reviewed by Ciccotti and Hoover (1986), Allen and Tildesley (1987), Haile (1993) and Koplik and Banavar (1995). The simulation begins with a set of N molecules in a region of space, each assigned a random velocity corresponding to a Boltzmann distribution at the temperature of interest. The interaction between the particles is prescribed typically in the form of a two-body potential energy, and the time evolution of the molecular positions is determined by integrating Newton’s equations of motion. Because MD is based on the most basic set of equations, it is valid in principle for any flow extent and any range of parameters. The method is straightforward in principle but has two hurdles: choosing a proper and convenient potential for particular fluid and solid combinations and the colossal computer resources required to simulate a reasonable flowfield extent. For purists, the former difficulty is a sticky one. There is no totally rational methodology by which a convenient potential can be chosen. Part of the art of MD is to pick an appropriate potential and validate the simulation results with experiments or other analytical/computational results. A commonly used potential between two molecules is the generalized Lennard–Jones 6–12 potential, to be used in the following section and further discussed in the section following that. The second difficulty, and by far the most serious limitation of molecular dynamics simulations, is the number of molecules N that can realistically be modeled on a digital computer. Because the compu- tation of an element of trajectory for any particular molecule requires consideration of all other molecules as potential collision partners, the amount of computation required by the MD method is proportional to N 2 . Some saving in computer time can be achieved by cutting off the weak tail of the potential (see Figure 4.11) at, say, r c = 2.5 σ and shifting the potential by a linear term in r so that the force goes smoothly to zero at the cutoff. As a result, only nearby molecules are treated as potential collision partners, and the computation time for N molecules no longer scales with N 2 . The state of the art of molecular dynamics simulations in the early 2000s is such that with a few hours of CPU time general-purpose supercomputers can handle around 100,000 molecules. At enormous expense, the fastest parallel machine available can simulate around 10 million particles. Because of the extreme diminution of molecular scales, the above translates into regions of liquid flow of about 0.02 µm (200 Å) in linear size, over time intervals of around 0.001 µs, enough for continuum behavior to set in for simple molecules. To simulate 1 s of real time for complex molecular interactions (e.g., including vibration modes, reorientation of polymer molecules, collision of colloidal particles) requires unrealistic CPU time measured in hundreds of years. Molecular dynamics simulations are highly inefficient for dilute gases where the molecular interac- tions are infrequent. The simulations are more suited for dense gases and liquids. Clearly, molecular dynamics simulations are reserved for situations where the continuum approach or the statistical methods are inadequate to compute from first principles important flow quantities. Slip-boundary conditions for liquid flows in extremely small devices are such a case, as will be discussed in the following section. © 2002 by CRC Press LLC 4.7 Molecular-Based Models In the continuum models discussed thus far, the macroscopic fluid properties are the dependent variables, while the independent variables are the three spatial coordinates and time. The molecular models recognize the fluid as a myriad of discrete particles: molecules, atoms, ions and electrons. The goal here is to determine the position, velocity and state of all particles at all times. The molecular approach is either deterministic or probabilistic (refer to Figure 4.1). Provided that there is a sufficient number of micro- scopic particles within the smallest significant volume of a flow, the macroscopic properties at any location in the flow can then be computed from the discrete-particle information by a suitable averaging or weighted averaging process. This section discusses molecular-based models and their relation to the continuum models previously considered. The most fundamental of the molecular models is a deterministic one. The motion of the molecules is governed by the laws of classical mechanics, although, at the expense of greatly complicating the problem, the laws of quantum mechanics can also be considered in special circumstances. The modern molecular dynamics (MD) computer simulations have been pioneered by Alder and Wainwright (1957; 1958; 1970) and reviewed by Ciccotti and Hoover (1986), Allen and Tildesley (1987), Haile (1993) and Koplik and Banavar (1995). The simulation begins with a set of N molecules in a region of space, each assigned a random velocity corresponding to a Boltzmann distribution at the temperature of interest. The interaction between the particles is prescribed typically in the form of a two-body potential energy, and the time evolution of the molecular positions is determined by integrating Newton’s equations of motion. Because MD is based on the most basic set of equations, it is valid in principle for any flow extent and any range of parameters. The method is straightforward in principle but has two hurdles: choosing a proper and convenient potential for particular fluid and solid combinations and the colossal computer resources required to simulate a reasonable flowfield extent. For purists, the former difficulty is a sticky one. There is no totally rational methodology by which a convenient potential can be chosen. Part of the art of MD is to pick an appropriate potential and validate the simulation results with experiments or other analytical/computational results. A commonly used potential between two molecules is the generalized Lennard–Jones 6–12 potential, to be used in the following section and further discussed in the section following that. The second difficulty, and by far the most serious limitation of molecular dynamics simulations, is the number of molecules N that can realistically be modeled on a digital computer. Because the compu- tation of an element of trajectory for any particular molecule requires consideration of all other molecules as potential collision partners, the amount of computation required by the MD method is proportional to N 2 . Some saving in computer time can be achieved by cutting off the weak tail of the potential (see Figure 4.11) at, say, r c = 2.5 σ and shifting the potential by a linear term in r so that the force goes smoothly to zero at the cutoff. As a result, only nearby molecules are treated as potential collision partners, and the computation time for N molecules no longer scales with N 2 . The state of the art of molecular dynamics simulations in the early 2000s is such that with a few hours of CPU time general-purpose supercomputers can handle around 100,000 molecules. At enormous expense, the fastest parallel machine available can simulate around 10 million particles. Because of the extreme diminution of molecular scales, the above translates into regions of liquid flow of about 0.02 µm (200 Å) in linear size, over time intervals of around 0.001 µs, enough for continuum behavior to set in for simple molecules. To simulate 1 s of real time for complex molecular interactions (e.g., including vibration modes, reorientation of polymer molecules, collision of colloidal particles) requires unrealistic CPU time measured in hundreds of years. Molecular dynamics simulations are highly inefficient for dilute gases where the molecular interac- tions are infrequent. The simulations are more suited for dense gases and liquids. Clearly, molecular dynamics simulations are reserved for situations where the continuum approach or the statistical methods are inadequate to compute from first principles important flow quantities. Slip-boundary conditions for liquid flows in extremely small devices are such a case, as will be discussed in the following section. © 2002 by CRC Press LLC 5 Integrated Simulation for MEMS: Coupling Flow-Structure- Thermal-Electrical Domains 5.1 Abstract 5.2 Introduction Full-System Simulation • Computational Complexity of MEMS Flows • Coupled-Domain Problems • A Prototype Problem 5.3 Coupled Circuit-Device Simulation 5.4 Overview of Simulators The Circuit Simulator: SPICE3 • The Fluid Simulator: N εκ T α r • Formulation for Flow-Structure Interactions • Grid Velocity Algorithm • The Structural Simulator • Differences between Circuit, Fluid and Solid Simulators 5.5 Circuit-Microfluidic Device Simulation Software Integration • Lumped Element and Compact Models for Devices 5.6 Demonstrations of the Integrated Simulation Approach Microfluidic System Description • SPICE3– N εκ T α r Integration • Simulation Results 5.7 Summary and Discussion Acknowledgments 5.1 Abstract Full-system simulation of microelectromechanical systems (MEMS) involves coupling of many diverse pro- cesses with disparate spatial and temporal scales. In its simplest form, all elements in a MEMS device are represented as equivalent analog circuits so that robust simulators such as SPICE can solve for the entire system. However, devices, and especially fluidic devices, do not usually have such equivalent analogs and may require full simulation of individual components. This is particularly true for new designs, which often involve unfamiliar physics; such lumped models and continuum approximations are inappropriate in this case. In this chapter, we address such issues for an integrated simulation approach for MEMS with Robert M. Kirby Brown University George Em Karniadakis Brown University Oleg Mikulchenko Oregon State University Kartikeya Mayaram Oregon State University © 2002 by CRC Press LLC 6 Liquid Flows in Microchannels 6.1 Introduction Unique Aspects of Liquids in Microchannels • Continuum Hydrodynamics of Pressure Driven Flow in Channels • Hydraulic Diameter • Flow in Round Capillaries • Entrance Length Development • Transition to Turbulent Flow • Noncircular Channels 6.2 Experimental Studies of Flow Through Microchannels Proposed Explanations for Measured Behavior • Measurements of Velocity in Microchannels • Nonlinear Channels • Capacitive Effects 6.3 Electrokinetics Background Electrical Double Layers • EOF with Finite EDL • Thin EDL Electro-Osmotic Flow • Electrophoresis • Similarity between Electric and Velocity Fields for Electro-Osmosis and Electrophoresis • Electrokinetic Microchips • Engineering Considerations: Flowrate and Pressure of Electro-Osmotic Flow • Electrical Analogy and Microfluidic Networks • Practical Considerations 6.4 Summary and Conclusions Nomenclature 6.1 Introduction Nominally, microchannels can be defined as channels whose dimensions are less than 1 mm and greater than 1 µ m. Above 1 mm the flow exhibits behavior that is the same as most macroscopic flows. Below 1 µ m the flow is better characterized as nanoscopic. Currently, most microchannels fall into the range of 30 to 300 µ m. Microchannels can be fabricated in many materials—glass, polymers, silicon, metals— using various processes including surface micromachining, bulk micromachining, molding, embossing and conventional machining with microcutters. These methods and the characteristics of the resulting flow channels are discussed elsewhere in this handbook. Microchannels offer advantages due to their high surface-to-volume ratio and their small volumes. The large surface-to-volume ratio leads to a high rate of heat and mass transfer, making microdevices excellent tools for compact heat exchangers. For example, the device in Figure 6.1 is a cross-flow heat exchanger constructed from a stack of 50 14-mm × 14-mm foils, each containing 34 200- µ m-wide × 100- µ m-deep channels machined into the 200- µ m-thick stainless steel foils by the process of direct, high-precision mechanical micromachining [Brandner et al., 2000; Schaller et al., 1999]. The direction of flow in adjacent foils is alternated 90 ° , and the foils are attached by means of diffusion bonding to create a stack of cross-flow heat exchangers capable of transferring 10 kW at a temperature difference of Kendra V. Sharp University of Illinois at Urbana–Champaign Ronald J. Adrian University of Illinois at Urbana–Champaign Juan G. Santiago Stanford University Joshua I. Molho Stanford University . discusses molecular-based models and their relation to the continuum models previously considered. The most fundamental of the molecular models is a deterministic one. The motion of the molecules is. discusses molecular-based models and their relation to the continuum models previously considered. The most fundamental of the molecular models is a deterministic one. The motion of the molecules is. containing 34 20 0- µ m- wide × 10 0- µ m- deep channels machined into the 20 0- µ m- thick stainless steel foils by the process of direct, high-precision mechanical micromachining [Brandner