Micromechanics of engineered and biological systems

G K ANANTHASURESH1 and SUMAN CHAKRABORTY2 1Mechanical Engineering, Indian Institute of Science, Bangalore, India. e-mail: [email protected] 2Mechanical Engineering, Indian Institute of Technology, Kharagpur, India. e-mail: [email protected]

This review paper is concerned with the mechanics issues at the micron scale. The topics consi- dered here are microsystems (microelectromechanical systems – MEMS), microfluidic devices, and biological cells. Mechanics is important for all the three fields. Micromechanics of solids is impor- tant for microsystems while microfluid mechanics is essential for microfluidics. Understanding the mechanics of cells requires both and much more. We highlight here some of the major advances and important challenges in all the three fields.

1. Introduction phenomena and systems – engineered or biological. Another critical aspect that ties the three topics of This review paper focuses on three aspects of this paper together is microfabrication and micro- mechanics of systems of small size: microsystems machinery. Without the ability to make intricate (microelectromechanical systems – MEMS – as structures of small size, to actuate and to control they are popularly known), microfluidics, and bio- them, these devices would not have come into exis- logical cells. The three topics are substantially tence. Without the micromachinery, which includes different from each other but there is significant sensors and actuators, quantitative investigation of interplay among them. The critical dimension of the behaviour of cells is unthinkable. To make it micron size is of course common to all three. How- all work, we need an understanding of mechanics ever, when we consider devices and systems, the issues and that is the focus of this paper. size sometimes tends to be much bigger than a micron; and sometimes it can be much smaller than a micron going down to a nanometer when 1.1 Microsystems we try to understand a phenomenon or to make an idea work in a device. The bigger than micron Here, we use the word ‘microsystem’ rather than size mainly comes because of microsystems. While ‘MEMS’ because the latter is restrictive as it the critical dimensions of micromachined sensors, only conveys mechanical and electrical aspects. actuators, and devices are indeed of micron size, Microsystems are good examples of integrated their packaged size tends to be much bigger. Cells engineered systems of small size. Although this are of micron size but their mechano-transduction field started in early 1970s, its formal identity came takes place because of entities much smaller than only in late 1980s [1]. In the two recent decades a micron – down to proteins of nanometer size. of its intense activity and growth, the emphasis Tissues are necessarily bigger than cells and they has been on sensors and actuators and some inte- exhibit behaviour that can be understood from grated systems. Undoubtedly, rapid advances in the investigations on the single cells but not all microfrabrication techniques for silicon and other of it. So, it suffices to say that they are all small materials have played a major role in the develop- and hence different from the traditional ‘macro’ ment of the microsystems field. It is also pertinent

Keywords. Microsystems; MEMS; microfluidics; cell mechanics.

83 84 G K ANANTHASURESH AND SUMAN CHAKRABORTY to note that mechanics too has played an equally deformable structure is released by dissolving a important role. The origin of much of the diffi- sacrificial (fugitive) layer that holds the releasable culty in simulating and designing a microsystems structure until the end of the process. The surface component or device lies in two things: coupling tension forces arising from the liquid in the narrow with other energy domains and scaling effects. gaps is large enough to pull the deformable struc- There is not a useful microsystems device that ture down and make it stick to the substrate. Mois- operates without involving more than one energy ture and electrical charge accumulation also lead domain. A simple beam-based switch needs actu- to stiction. Narrow gaps also lead to another com- ation to work. If this actuation is electrostatic plication because the fluid (mostly air surrounding force, coupling between elastic structures and elec- the device) causes squeezed film effects [4]. In gen- trostatic field ensues. A diaphragm-based micro- eral, flow behaviour is intrinsically coupled with pump involves a strong coupling between solid the microsystems area. More details on microflu- and fluid mechanics fields and also couples with idics follow next. a field associated with what actuates it. Often the coupling between the fields is strong and, in 1.2 Microfluidics some cases, unprecedented at the macroscale. It is important to note that mechanics couples with The ever-increasing possibilities of creating struc- other energy domains in all these problems. Con- tures and patterns on small length scales with sequently, new approaches – both analytical and ultra-tunable precision have triggered a wide numerical – are warranted to efficiently simulate or range of scientific investigations involving the design microsystems devices. A key component of transportation and manipulation of fluids in this effort is the need for model reduction. As mul- miniaturized devices and systems. These types tiple energy domains are solved simultaneously to of investigations, broadly classified under the simulate the behaviour of an integrated microsys- theme of microfluidics (concerning flow length tems device, usually requiring the solution of two scales over micrometer ranges) or nanofluidics or more coupled partial differential equations in (concerning flow length scales over nanometer spatial and temporal domains, the computational ranges), have rekindled interests in a classical complexity tends to be very high. The degrees of area of fluid dynamics: low-Reynolds-number flows, freedom involved will be large too. The results with a fusion and agglomeration of concepts of such simulations are useful to see if a device from molecular physics, surface chemistry, and works or not but they are not necessarily amenable life sciences. As a commonly acceptable metric, for gaining insight and for designing efficiently. microfluidics refers to the control of small volumes −15 −6 Hence, microsystems researchers often resort to of liquids (fL–μL, 10 –10 L). Traditionally, model reduction. It can be done analytically in silicon micromachining methods have been used some cases [2] but not always. Therefore, tech- to fabricate micron-scale channels from silicon niques for model reduction are developed. Some of and glass for transporting these ultra-small fluidic these are available today in microsystems design volumes. Other types of materials (such as poly- software but there is a lot more to be desired in this dimethylsiloxane (PDMS), polymethyl mythacry- direction. late (PMMA)) are also being extensively used. Scaling effects are many as we move down to The latter systems offer potential advantages of the micro size. Although continuum mechanics faster design times, low cost, the ability to fabricate assumptions do not break down at this scale for nanoscale features, and the possibilities of obtain- deformable solids, a number of problems come to ing deformable shapes. the fore because of the small size. Residual stresses The principal components of the present day and the stress gradients are common in micro- microfluidics are lab-on-a-chip devices, which are fabrication processes. These tend to be very high nothing but miniaturized analytical laboratories, as compared with the stiffness of the microme- typically consisting of a network of microchannels chanical structures. Hence, unintended deforma- and reservoirs on small glass or plastic chips. tions occur if the effects of residual stresses are Additionally, these chips may contain electrodes, not properly accounted for. The inhomogeneity of sensors and electrical circuits, for performing the residual stresses is hard to characterize and control desired protocols. Such devices are capable of in a microfabrication process. The same is true of duplicating some specialized functionalities as their anisotropic properties of materials. There is often larger-sized counterparts in the macroscopic world, disagreement on basic properties such as Young’s such as clinical diagnostics, DNA scanning, cel- modulus when it is determined with different lular analysis, and separation processes. How- methods [3]. Stiction is another major problem ever, with a dramatically reduced sample volume in deformable structures with features separated requirement, the lab-on-a-chip concept leads to by narrow gaps. This is especially so when a much shorter reaction and analysis times, higher MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 85 throughput, automation, design flexibility, and first time and coined the term ‘cell’. Since then portability. Modern developments in the design mechanics and biology have moved on more or less and utilization of microfluidic devices have found separately but are now coming together to enrich many applications, ranging from the life sciences each other. It is also pertinent to note that Francis industries for pharmaceuticals and biomedicine Crick,theco-discovererofthedoublehelixstruc- (drug design, delivery and detection, diagnostic ture of DNA also has the credit of performing devices) to industrial applications of combinatorial one of the earliest mechanics experiments on cells. synthesis (such as rapid chemical analyses and Thus, there have been instances where researchers high-throughput screening). In other branches of saw synergy between biology and mechanics. medicine, new paradigms for noninvasive diagnos- Application of mechanics to biology and the tics and surgery (possibly implanted or ingested) development of mechanics for biology have been have also been possible through the develop- active at the macroscale for skeletal and bone ments of microdevices. Such miniaturized micro- mechanics, locomotion studies, hemodynamics total-analysis systems (μTAS) are advantageous and blood circulation in living systems, muscle in many respects, particularly attributed to the behaviour, and others [5,6]. But biomechanics, in fact that smaller devices are lower in cost, are the context of this article, can be described as a less energy and material consuming, and are mini- field that involves the application, development, mally invasive and conveniently disposable without and extension of mechanics applied to biological contamination. Furthermore, sophisticated micro problems at the molecular, cell, and tissue levels. analytical systems can be built using individual We restrict our discussion to cells here because of micro-chip mono-blocks, thereby integrating dif- the focus on the micromechanics. ferent complex functionalities together on a small Investigations into the mechanics at the cellular scale in a coherent manner. In the non-biological level help uncover the fundamental ways in which sector, there are numerous applications as well, cells function. The mechanical properties and the including electronic chip cooling and inkjet print- forces have been shown to play important roles ing. Inevitably, all these devices would necessitate in cell behaviour and responses. Examples cover the handling and processing of small amounts of both healthy and disease-affected cells. It has been fluids, and this is where fluid mechanics of small shown that mechanical stretching of transmem- scale systems comes into play. brane proteins enhances self-assembly and orga- Fluid mechanics at the small scale is often nization [7]. It is also shown that striation of intriguing. It is a vast area of research by itself. muscle cells requires a substrate of moderate stiff- In this paper, we will by no means attempt in ness: too soft or too hard substrates are ineffective detailing with all the related issues, but would [8]. Another study has found that there is signifi- emphasize on a few salient features that may hold cant difference in the stiffness of healthy red blood the key towards unveiling several apparently non- cells and the malaria-affected ones [9]. In develop- trivial facets of this extremely interesting topic. mental biology, the application of controlled forces There is, of course, a common synergy behind the on growing embryos is shown to help in under- central theme of our discussions: as the length standing how cells are differentiated at different scales become smaller, surface effects tend to domi- stages [10]. nate. Thus, inertia forces turn out to be negligi- The sensing of mechanical forces and characteri- ble in comparison to electrostatic, electrodynamic, zation of mechanical properties at the micro and viscous, or capillary effects. The later effects, there- nanoscales is ably supported by the rapidly grow- fore, may strongly influence the flow behaviour, in ing cell manipulation techniques. There have been a manner much more emphatic than in classical good reviews of a variety of techniques for effect- macro-flows. ing and sensing forces up to femto (10−15)N Solid mechanics of microsystems and fluid and sensing displacements up to a billionth of mechanics at the microscale arise in interesting a metre [6,11–12]. The theoretical and practical and complex ways when we consider biology at the bases for these tools lie in the fields of micro and microscale. We discuss this next. nanomechanical systems. Thus, the field of micro- systems is inextricably tied to the field of cellular 1.3 Biomechanics of cells mechanics.

Mechanics and cellular biology were intimately coupled with each other at the inception of both 2. Computational mechanics in of these fields: it was the same Robert Hooke who microsystems hypothesized the fundamental law of mechanics – the Hooke’s law – by relating stress and strain in Continuum theory of solid mechanics mostly a solid and also observed the cork cells for the remains unchallenged at the sizes of microsystems. 86 G K ANANTHASURESH AND SUMAN CHAKRABORTY

This is especially true for the elastic regime describing the coupled electrostatic-mechanical in which constitutive stress-strain relationship is and electrothermal-mechanical effects. linear and is far from permanent deformation and failure. This is because elastically deformable 2.1 Coupled electrostatic-elastic simulations structures in microsystems devices have to operate reversibly and repetitively. Nonlinearities in struc- A parallel-plate capacitor suffices to explain the tural behaviour such as large displacements and coupled nature of the electrostatic forces and rotations, contact, and stress-stiffening are present. elastic behaviour of a deforming body. Imagine But none of these pose as significant a problem that one plate of the capacitor is fixed while in simulating these devices as the coupling of the other end is made translatable by restrain- the elastic domain with other force fields does. ing it with a linear spring of spring constant, k. The phenomena that do not matter at the Upon applying an electric potential V between the macroscale become significant at the micro sizes. plates, the attractive electrostatic force acts on the The result of the coupling and scaling effects movable plate and is given by εAV 2/(2g2)where: is seen in electrostatically actuated microme- ε is the permittivity of the medium between the chanical structures. Here, the structural deforma- plates, A and g are the area of overlap and the tion interacts nonlinearly with the static electric gap of separation between the plates. Since the field ensuing between electrical conductors and gap changes as the plate moves, the displacement, dielectrics. As has been argued well in the litera- x, is to be shown in the expression for the elec- ture, electrostatic force scales vary favourably trostatic force. The displaced linear spring has a at the microscale and therefore numerous micro- force of kx. As can be seen in this simple example, systems devices use this. Many such devices are the equation equilibrium between the electrosta- 2 2 commercially available and yet not all prob- tic force, εAV /(2(g0 − x) ), and the elastic force, lems associated with their simulation are com- kx, is nonlinear. Here, we have used g0 to indi- pletely resolved. We consider them in the next cate the gap before the voltage is applied. This section. nonlinearity is so severe that catastrophic collapse, Other widely used actuation techniques in called electrostatic pull-in, occurs after the applied microsystems devices are magnetic and electro- electric potential crosses a certain threshold appro- thermal. Magnetic actuation, although thought priately called the pull-in voltage.Atthatcriti- to be unfavorably scalable to microsystems at cal voltage, the moving plate collapses onto the the inception of the field, has now come back. fixed plate. Furthermore, if there is a thin dielec- The issue of requiring large currents to gener- tric layer above the fixed plate, the plate remains in ate significant forces has been solved by using that pulled-in position until the voltage is lowered external magnetic field generated by powerful down to a pull-up voltage. When the displace- permanent magnets and, in some cases, having ment of the plate is plotted against the applied wound coils to get large magnetic fields in narrow voltage, we see an apparent hysteresis [13]. This areas. Fortunately, the coupling here is not strong simple phenomenon becomes increasingly complex enough to warrant any special techniques that as we move to 2D and 3D continua with involved are different from their macro-scale equivalents. geometry such as what layered or anisotropically Electro-thermal actuation, however, is different. etched silicon micromechanical structures would If it is generating out-of-plane deformations, analy- have. sis of bimorph-type actuation is sufficient and is The equations governing the general 3D bodies not involved. When it comes to in-plane actuation, in which coupled electrostatic and elastic fields sequential electrical, thermal, and elastic domain exist are shown for a general geometry in figure 1. coupling needs to be solved. Among other things, The electrostatic force, being a force that acts temperature-dependent mechanical properties on the surface, appears as the coupling term in matter in this case. We consider this later in this the elastic deformation equation. This can be section. seen in the equations in figure 1. What might Modeling dissipation in mechanical structures not be immediately apparent is how the elas- is still a current issue in the dynamic simula- tic structure influences the electric field. This tion of micromechanics. Fluids squeezed in narrow arises because the deformation of the elastic gaps, thermoelastic damping, and surface energy structure changes the surface charge distribu- are to be carefully accounted for. The need for tion, ψs. Hence, the electrostatic problem has careful modeling of dissipation arises because to be solved for the new deformed geometries of the desire to achieve high quality factors, of the conductors and dielectrics. This force is i.e., acutely sharp resonance peak in the fre- once again applied to the elastic structure to quency response of the microscale dynamic obtain its deformation, and then the process is systems. It is also discussed in this section after repeated until both the governing equations are MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 87

Figure 1. Schematic portrayal of a general electrostatically actuated micromechanical structure. solved self-consistently. The process is iterative and we show the infinite domain as a sufficiently large requires repeated analysis of the two domains to bounding box at the boundaries of which the elec- get a converged solution. Note that the re-meshing tric field is presumed to be sufficiently low and for the electrostatic calculation is necessary when- all the charges reside only on the surfaces of the ever the deformed geometry of the conductors and conductors. Instead, if we consider the integral dielectrics changes. All this takes considerable com- form of the governing equations of the electrosta- putation time and today’s microsystems simula- tic problem, the problem size reduces consider- tion software take a few hours to compute the ably as only the surfaces of the conductors need pull-in voltage for practically useful microsystem to be meshed in the boundary element method. components. The computational algorithm switches between What is described above is a simple relaxa- BEM (for the electrostatic problem) and FEM (for tion approach to the simultaneous solution of the the elastic deformation problem) in a staggered coupled equations. By taking the sensitivities of manner. the quantities in one field with respect to the quan- Instead of the staggered approach, recently, a tities in the other, a Newton’s algorithm is often monolithic approach to a simultaneous solution of used to increase the speed of computation [14]. the two governing equations is proposed. The key A combination of finite element and boundary ele- to this new method is the application of electro- ment methods (FEM and BEM) is also used to static force not as a surface force but as a body solve this problem. This is because the electrosta- force by using Maxwell’s stress tensor [15–18]. This tics problem is an infinite domain problem. That is given by σM = ε(∇V ∇V − 0.5(∇V ·∇V )I)and is, the electrostatic equation is to be solved over hence ∇·σM applies at each point of the 3D the infinite domain from which the space occu- continuum as a body force. As a result, there pied by the conductors is subtracted. In figure 1, is no need for re-meshing in each iteration if 88 G K ANANTHASURESH AND SUMAN CHAKRABORTY the electrical permittivity is smoothly interpolated of insufficient knowledge about the heat transfer in the entire region of the empty space, coefficients for convection and shape factors for dielectrics, and conductors within the bounding radiation. One needs to model the fluid sur- box shown in figure 1. This approach increases rounding the deforming solid to obtain the heat the computational efficiency. Furthermore, it is transfer coefficients. But most often, microsystems useful in optimization-based design and can researchers are content with borrowing empirical handle dielectric materials embedded within the formula from the macro-scale literature. Because of domain. the narrow gaps involved in micromechanical struc- The nonlinear coupling between electrostatic tures, one cannot be sure if conduction through and elastic domains also gives a rich dynamic air is to be modeled or convection. This issue is behaviour wherein chaotic motion can also be also not given importance in many a thermal sim- observed [19]. There are no reports of chaotic ulation. Convection through vertical surfaces of behaviour effectively used in microsystem device essentially thin microsystem components is some- but it does become an important issue in the oper- times significant. Improper modeling of convection ation of the device and in its control. The chaotic is known to give qualitatively different deforma- response aside, simulating the nonlinear behaviour tion patterns from those reality [23]. Hence, using has important practical consequences in tunable a 3D model in numerical computations is imper- capacitors [20], mechanical filters whose frequen- ative. When the elastic structure touches itself cies can be adjusted [21], radio frequency switches, or another structure, as it happens in switches etc. Here, for example, in the case of switches one and other microsystems devices, thermal bound- would need to compute the time for pull-in and ary conditions change suddenly. This causes addi- pull-up. A simple structure such as a cantilever too tional complications. The basic issue of the type of shows involved behaviour in terms of its pull-in boundary for the interface between the microsys- geometries. With a voltage higher than the pull-in tem component and its substrate is debatable. We voltage, the beam does not just touch the electri- normally assume that there is a large thermal mass cally insulated ground substrate but zips onto it at that interface and impose Dirichlet (fixed tem- under certain circumstances. When it pulls up too, perature) boundary condition. But it will not be it can undergo different transformational states true if there are many such components located among zipping, tip-contact, and complete pulled close to each other. Then, we may need a zero heat up configurations [22]. flux or a mixed boundary condition. On top of all The coupled simulation of electrostatically actu- of these, the knowledge of the temperature depen- ated micromechanical structures also needs to dence of material properties is neither accurate nor consider the residual stresses and their gradients complete. (which is not so difficult), fluidic effects, and Owing to all the above issues, electro-thermal- thermoelastic damping. The latter two are dis- elastic analysis is not straightforward. Here, there cussed in subsection 2.3. are no unresolved conceptual issues but the numeri- cal solution involves a lot of computation. Hence, 2.2 Electro-thermal-elastic simulations there is a need for fast algorithms and better modeling capabilities to account for convection Many transduction mechanisms in microsystems and radiation with few degrees of freedom and require electric current to be passed through them. without having to model the fluid around the This means that there will be Joule heating solid. and the ensuing thermally-induced deformations. Since most of the properties vary with tempera- ture, it causes additional difficulties in simulation. 2.3 Dissipation in microsystems Thermal forces can also be advantageous and can be used for actuation. Indeed, thermal force is quite Micromechanical structures are essentially dyna- large as compared with electrostatic force for the mic systems because they move. Due to their inter- same applied voltage. nal structure and external environment, they do Simulation of electro-thermal microactuators dissipate considerable energy that is put into them. requires a sequential or simultaneous analysis of Many dissipation mechanisms exist and almost three energy domains: electrical, thermal, and all of them become important for microsystems elastic. The governing equations of electrical and, more importantly, nanoelectromechanical sys- current and thermal conduction are both diffu- tems (NEMS). Modeling dissipation is impor- sion equations. Solving them sequentially will suf- tant in these systems because of the desire to fice if electrical conductivity dependence on the achieve high quality factors (a consequence of min- temperature can be neglected. Thermal analysis, imal losses) and the demonstrated possibilities of by itself, is complicated at the microscale because achieving them through miniaturized mechanical MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 89 structures. The other reasons are achieving accu- direction across the cross-section of the beam) heat rate predictive capability for the component-level conduction equation, thermoelastic damping can and system-level behaviour of the devices. Due to be included in numerical simulation. the lack of accurate methods to estimate energy   dissipation mechanisms, the experimentally deter- ∂2w ∂2 ∂2w ρA + EI + EαIT = F (x, t) mined quality factors have large discrepancy with ∂t2 ∂x2 ∂x2 the simulated values. It is thus currently an active area of research. Dissipation mechanisms in microsystems can be ∂T ∂T EαT0 ∂w = κ + y , classified into five types: bulk or internal damp- ∂t ∂y ρCp ∂t ing, surface loss mechanisms, viscous losses to the surroundings, anchor losses, and package-related where losses. Internal damping may be due to defects [24], phonon-mediated dissipation among different ρ =massdensity internal vibration modes [25], and thermoelastic A = area of cross-section damping. Of these three, thermoelastic damping, w = transverse displacement of the beam a special situation of the second, has received con- E = Young’s modulus siderable attention in recent years. Thermoelastic dissipation of energy is a con- F = any other external force sequence of irreversible heat flow between high T = temperature and low temperature regions of a deformed struc- y = distance in the depth direction across the ture. In a micromechanical structure, local thermal cross-section of the beam gradients arise because of the mechanical strain: κ = thermal diffusivity regions with compressive strain get hotter because Cp = specific heat at constant pressure of increased internal vibrations while tensile strain I = moment of inertia of the beam regions become relatively colder. This was first  explained by Zener [26] for thin beams and his IT = A yT dA = thermal contribution to the approximation to the quality factor is quite good beam’s inertia. even in light of recent experimental data. But it is not accurate enough when it comes to micro and However, many assumptions still remain in the nanomechanical vibrating structures where there analysis and numerical simulations carried out is persistent effort to increase the quality factor. so far. For example, only a beam analysis is Therefore, Lifshitz and Roukes [27] set aside the carried out as opposed to continuum analysis. assumptions made by Zener, simultaneously solved Temperature gradients are considered only in the the temperature profile, and obtained a slightly depth direction and not along the beam. While better estimate of the quality factor but never- the relaxation of these assumptions takes us to theless exact solution under the small-amplitude more accurate estimates of the quality factor, the harmonic excitation and other assumptions made numerical complexity continues to increase. Note by them. According to Zener, more than 98.6% that the elastic, electrostatic (or other if the actu- of dissipation occurs through the first mode of ation is not electrostatic), and thermal conduction flexural vibration of the harmonically excited equations need to be solved for obtaining proper beam. Recently, it was found that when a micro dynamic response of a micromechanical vibrating or nanomechanical beam is excited arbitrarily and structure. The effect of convection and radiation through large motions (as it happens in electrosta- and relaxation time constants at those length scales tically actuated structures), more than one mode is yet to be ascertained. needs to be considered. And more importantly, Surface loss mechanisms depend on surface modal solution for the temperature profile is not roughness, adsorbed particles, and monolayers accurate and only a complete numerical solution [28]. What might have been second order effects matches the experimentally obtained quality factor at the macroscale turn out to be non-negligible at data [19]. the microscale. Another, more dominant, dissipa- It is rather interesting that only a single material tion mechanism is viscous damping. Not all micro- property, the linear thermal expansion coefficient systems devices are vacuum-packaged. In some, α, controls thermoelastic damping. For slender the cost is prohibitive and some others, some structures undergoing transverse flexural vibra- amount of damping is necessary to prevent sus- tions, Euler–Bernoulli beam modeling is deemed tained oscillations so as to increase the bandwidth to be adequate at present. So, by adding an and resolution of the device. When there is air, it extra term to the well-known beam equations gets squeezed in narrow gaps that are prevalent and solving a one-dimensional (in the depth in micromechanical structures. This necessitates 90 G K ANANTHASURESH AND SUMAN CHAKRABORTY the invoking of isothermal compressible Reynolds for given specifications. Mathematically, the syn- equation, which is used widely in lubrication thesis problem is posed as an inverse problem of theory. Much work is done on this topic and [2] analysis. Most algebraic or differential equations may be consulted for a good review. At one simpli- are amenable for this albeit not without difficulties. stic level, an extra term may be added to the beam Optimization is a useful tool for synthesis. Many equation above or one can consider the full numeri- optimal synthesis methods have been developed for cal solution of the Reynolds equation. Microsystem microsystems [30]. Among these, topology optimi- simulation software can handle this equation simul- zation methods can address the problem at a more taneously with the elastic field equation. However, abstract level than the others. complications do exist when a vertically vibrating Design for a microsystem, or for that matter structure has perforations for which additional most other engineering systems, is the process of pipe flow needs to be considered [4]. determining the sub-systems or components and Anchors, places where micromechanical struc- their geometrical form and materials. Topology tures attach to their substrates, are not as rigid optimization has the capability to do this at the as they are supposed to be. Hence, dissipation component level for specified requirements posed in of energy takes place through these anchors too. the form of an objective function and constraints. Another recent observation is that additional dis- And, it can do so with minimal input from the sipation routes arise because of packaging of the designers. The input required is often the size and microsystem components wherein the diced chip, shape of the space in which the design ought to the die, is attached to a packaging substrate [29]. fit and the quantitative specifications of what that Here, a rigid die attachment can minimize the dis- component should do. Topology optimization, in sipation but compromises the vibration isolation this sense, is simply the problem of distributing a of the device. Hence, careful design is necessary limited amount of material into a larger domain for to achieve a trade-off. The anchor and package given quantitative specifications [30,31]. related losses are slightly more traditional pheno- Material distribution within a domain can be mena as compared with the other dissipation seen as a binary optimization problem of having mechanisms. Apparently, everything matters when material at a point or not. Since such a problem is desired quality factor is exceeding one million! not amenable for efficient computation, gradient- While this section considered the analysis (simu- based optimizations methods are used by trans- lation) aspects, the next section takes a design per- forming it into a continuous problem. For this, the spective. 0–1 nature of the material distribution becomes a continuous function ρ(x), which is defined at every point x of the domain and is constrained to be 3. Mechanics-based design perspective between 0 and 1. ρ(x) can be thought of as a in microsystems fictitious density and it works well for problems with mechanical loads. But the mechanics does Microfabrication is an expensive proposition and get complicated when we consider coupled prob- it is also time-consuming. One cannot rely upon lems. For example, in coupled electrostatic-elastic trial-and-error methods to conceive a design intui- problem, we have forces acting on the boundaries. tively and/or by using prior experience and then But we do not know boundaries within the domain make it and test to see if it is good enough. apriori. Rather the regions with zero value of The same applies to simulation-based design ρ(x) become holes and boundaries get determined. because coupled simulation too is computation- Here, we need a way of transforming the bound- ally expensive. Hence, trial-and-error design is not ary force into a body force such that the force acts recommended nor is viable. In order to ease this everywhere with varying magnitude depending on problem, two techniques – both rooted in mechan- the material state (0 or 1) of that point. Keeping ics – are often used. They are synthesis tech- this in mind, we had mentioned using the con- niques and reduced-order modeling, which we dis- cept of Maxwell’s stress tensor [15] in section 2.1. cuss next. This ensures that the force acts everywhere and the material is everywhere, although it is negligi- 3.1 Synthesis of microsystems bly small within the holes and bulky interiors and is present only at the boundaries, as it should be. Synthesis, simply put, is the inverse of analysis. Similar situations arise in the case of piezoelec- In analysis, when a design is on hand, we simu- tric and thermal actuation problems and in each late (analyse) it to see how well it meets the func- case they are dealt with differently so that suitable tional and performance specifications. In synthesis, modifications are made to the mechanics problems on the other hand, we come up with a design [32,33]. MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 91

Imagine the problem of modeling convection While lumped models are constructed analyti- in a thermally actuated microsystems component. cally with an understanding of the physics of the If the topology is not known, that is, the number, phenomena involved, numerical methods can also size, and shape of the holes are not known, how be used for generating reduced-order models. The would one model convection, which is essentially main idea here is to obtain a reasonably accurate a surface phenomenon? Here too, one has to system of equations in as few degrees of freedom as smoothen the problem to have convection ever- possible. The data used here can come from exten- where – even in the bulky inside – but with negli- sive finite element solutions of partial and ordinary gibly small magnitude so that gradients can be differential equations or from real experiments. The taken [33]. task then is to reduce the large dof of a finite While topology optimization can give geo- element model to only a few. A natural thing to do metric forms for given specifications, the obtained here is to use eigenmodes [37] and it is common in topology may not always be manufacturable. mechanics. Since in any system, the response would This is because microfabrication techniques are not have large contributions from only a few modes, adequately versatile to give any shape to a com- the method of superposition of a few selected eigen- ponent. Incorporating the manufacturing consi- modes works and it is amenable for increasing the derations into the topology optimization is a need accuracy by taking more and more modes into of the hour at present. Some efforts are underway account. However, the eigenmode method works for in this direction. In some cases, the manufacturing only small displacements of the structure. Further- restrictions are included explicitly as constraints more, in a coupled system of equations, computing [34] while in others, they are taken into account in the forces acting on the mechanical structures can- defining the design variables for topology optimi- not easily be done in the reduced system of equa- zation [35,36]. The aim of these methods is to auto- tions. Coupled electrostatic-elastic problem is a mate the synthesis of microsystems. An alternative good example. Here, in order to avoid the computa- to reduce the computational burden on a designer tion of the capacitance in those few degrees of free- is to use reduced-order models. dom special attention is required. One method is to use Pad`e-type approximations in which the capaci- 3.2 Reduced-order modeling tance is fit to a rational polynomial based on the full-computation of the entire system of equations The behaviour of microsystems is mostly governed done a priori [38]. The other, more efficient, method by partial differential equations that couple with is to use Karuhunen–Loeve series [39]. A recent each other. A discretized partial differential equa- monograph contains a succinct description of the tion involves many degrees of freedom (dof), which model order reduction methods applied to electro- easily runs into a few thousands even in a moder- thermal-elastic micromechanical structures [40]. ately complex problem. The accuracy is also pro- Some of these methods, which are applicable portional to the number of dof: usually, the more for other types of structures also, include: con- the number of dof, the better the accuracy. The trol theory methods wherein non-observable and essence of reduced-order modeling is to use as few non-controllable freedoms are suppressed; Krylov dof as possible without overly compromising on the subspace methods where reduced models can be accuracy. constructed using Lanczos or Arnoldi processes; One simple but effective technique used often and Guan reduction methods where the internal microsystems is the lumped-modeling [2]. Here, an freedoms are expressed in terms of the freedoms energetically correct model is used with as few vari- of the external physical nodes of the finite element ables as possible. For example, a mass attached mesh. Micromechanical simulation software allows to a spring can capture the elastic kinetic and today automatic reduced-order model extraction potential energies stored in a large deformable but the accuracy of these processes needs further structure. Likewise, a simple parallel-plate capac- improvement. itor for electrostatic energy, and an inductor for We have discussed the role of mechanics in simu- magnetic energy, and so on. This concept is com- lation and design in the previous and current monly used in microsystem design and is well artic- sections. We next discuss how mechanics is affected ulated in [2]. But determining the values of the by or affects materials and microfabrication. lumped parameters (the spring constant, capaci- tance, inductance, etc.) is far from trivial and it can be done easily only for linearized (small displace- 4. Materials, microfabrication and ment) equations. Even then, it is useful because micromechanics the lumped models indicate how the response of a component depends on material properties and Microsystems behaviour is largely determined by geometry. their mechanical properties even though they 92 G K ANANTHASURESH AND SUMAN CHAKRABORTY operate in multiple physical energy domains. The variability of up to 30% between different methods properties are intrinsically related to the fabri- used to measure it. Making tensile test speci- cation techniques used to make them. Thus, mens and measuring their elongation using opti- materials, microfabrication, and mechanics are cal methods [44], nano-indentation methods [45], dependent on each other and influence the feasibi- Raman spectroscopy [46], etc., are used. But pre- lity of miniaturization and the performance of the ferred method seems to be on-chip test struc- miniaturized system. tures that can be co-located with the devices being made [47]. This helps one to know the properties 4.1 Material properties issues that take into account process variabilities across a batch and even across the wafer. One of the popular In 1987, Senturia wrote a paper with the title, methods is to measure the electrostatic pull-in volt- ‘Can we design microrobotic devices without age of cantilever and fixed-fixed beams and thereby knowing their mechanical material properties?’ estimate the Young’s modulus. By using beams of [41]. Provocative as the title may be, it indeed cap- varying widths, Poisson’s ratio can also be esti- tures the essence of the problem: knowing the prop- mated in this manner [47]. Measurement of shear erties is all too important in simulation and design. moduli and thermal expansion coefficient are less Materials aspects in microsystems were compre- understood [3]. New measurement techniques and hensively reviewed by Spearing in 2000 [42] and standardized techniques for them are necessary. again revisited in 2007 [3]. The basic scaling law This is especially needed because not only silicon of having, at the microscale, increased surface area but also polymers and ceramics are increasingly as compared with volume has many consequences being used in microsystems today. Next, we move for mechanics of solids. Friction forces have a much on to the issue of limitations of microfabrication larger effect on the mechanical behaviour to the techniques and how they affect the geometry of fab- extent that microsystem designers avoid as much ricated structures from the mechanical behaviour as possible the use of parts that rub against each viewpoint. other. Instead, compliant (i.e., elastically flexible) structures are used [43]. Mechanical strength is enhanced at the 4.2 Geometry and microfabrication microscale because plastic flow that occurs because of dislocation motion is restricted as the Relative tolerance in microfabricated features is layers become very thin. Increased strength also not very good; when one wants a beam of 1 micron means that materials can sustain large residual wide, the variability in it can be significant [48]. stresses as compared with macro-mechanical struc- Undercutting in etching, mask misalignment, unin- tures. Interfacial strength becomes a bottleneck tended anisotropic or isotropic etching, etc., lead in microsystems, which are mostly heterogeneous to complex geometrical features that are not gen- layered structures. Fatigue failure is also of poten- erally considered in the design process. Further- tial concern in microsystems devices because they more, since all lithography-based processes can undergo cyclic stresses a large number of times anchor a structure only at the substrate on the (note that one of the implications of the scaling law bottom side, boundary conditions are not always is that the natural frequencies are extremely high very clear. A cantilever beam is not really a straight at the microscale). For example, micro-mirrors one as assumed at the macroscale; it is often a used in projection displays – successful commer- bent compliant support. Because of the conformal cial products today – undergo trillions of cycles nature of the deposition processes, the geometry of in their lifetime. Yet, fatigue is not deemed to be the curved portion of the beams is often unclear. a major cause for concern in them. Indeed, there But small variations in this geometry can have a is a debate as to whether cyclic fatigue really significant influence on the mechanical behaviour. occurs at that scale. Because of the high purity of Today’s microsystem simulation software includes the microfabricated materials, crack initiation and a module to generate the solid model for ana- growth tend not be very high. However, strength lysis by graphically emulating the microfabrication considerations remain important in the design of processes. But their abilities are far from perfect. microsystems components where residual stresses, Hence, a design based on such modeling tends bond strength between layers, friction and wear, not be completely accurate. Ironically, the emu- etc., play a significant role. lated geometry can only be done after the design is Reliable and consistent measurement of mecha- completed rather than before. This problem needs nical properties of microsystems components to be addressed in the design stage by properly remains a challenge. A common and the most accounting for the microfabircation considerations. essential property such as the Young’s modulus, Some work is underway in this direction in an as recently as a decade ago, showed considerable automated procedure called topology optimization MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 93

Figure 2. Optimal structural topology optimization for surface-micromachined structures that illustrates the importance of taking the mciromachined geometry in design. (a) Problem specification for a stiff structure constrained on the left edge and with a point load, (b) optimal design without any manufacturing considerations, (c) optimal design for a chosen micromachining process, (d) the layered geometry of (c) with sacrificial layers also shown, (e) optimal design for another chosen micromachining process, (f) the layered geometry of (e) with sacrificial layers also shown. Compromise in the strain energy (SE), the measure of stiffness, can be noticed in the figure for obeying the manufacturing consideration.

[30,49]. Here, as shown in figure 2, the mask lay- transport system. For a vast majority of systems, outs themselves are used as the design variables so the task will boil down eventually to the design of a that the actual geometry is taken to build a model turbulent flow network, keeping in mind that large which is optimized [36,50]. Even here, the lack of Reynolds numbers are common to the traditional predictability of exact geometry remains a concern engineering conduits in most practical applica- and needs to be investigated further. tions. On the other hand, the flow in microfluidic As we have discussed so far, dealing with solids devices is commonly laminar (because of low has considerable challenges in microsystems even Reynolds number effects), which implies certain though the scaling effects on solids is largely advantageous and disadvantageous features simul- unchanged up to micron scale. It is not so in the taneously acting in tandem. Apparent advantages case of microfluidics, as discussed next. are the facts that for inexperienced analysts the laminar flow analysis may appear to be deceptively simpler than turbulent flows, whereas a well-known 5. Traditional vs. microfluid mechanics: disadvantage is that rapid mixing is difficult to general consequences in internal flows achieve over diffusion dominated laminar regimes. Not only that, many aspects of a classical flu- In traditional systems on macroscopic scales, fluids idic actuation in the laminar regime may appear are typically handled in large vessels and ducts to be prohibitive towards their implementations (pipes). Although in many ways the fundamental and extensions in the micro-systems, such as the equations describing the physics and chemistry of unfavorable scaling of the frictional energy dissi- such larger-scale processes are still similar as in pation in inverse proportion of the fourth power microfluidics, some effects being important on a of the hydraulic diameter, for a given flow rate. macroscopic scale become unimportant on smaller Furthermore, several other critical features predo- scales, whereas other effects that can largely be minate in a small scale, for example the capillary neglected macroscopically turn out to be domi- effects. While the capillary pressure (difference in nant in microfluidics. We may briefly delineate pressure across the meniscus demarcating a two- this issue by referring to a simple instance from fluid system in a capillary) scales inversely with traditional thermo-fluid sciences. Let us consider the capillary diameter and hence may be insignif- an engineer who is designing a macroscopic fluidic icant for larger diameter pipes, those may indeed 94 G K ANANTHASURESH AND SUMAN CHAKRABORTY bear significant consequences with regard to the gradient creates the driving surface tension force), control of fluidic transport in smaller scale systems. one can also exploit the effects of electrocapil- Neglecting the surface tension effects, thus, may larity or electrowetting, in which electric potentials give rise to significant errors in the analysis of a can be employed to alter the surface tension and microfluidic system, whereas similar effects might thereby cause a fluid motion. The first comprehen- not be so dominant over systems of larger extents. sive investigations on electrocapillary phenomena Surface tension force by itself may give rise to were performed by Lippman, way back in 1875 [53]. several other interesting phenomena in microscale In Lippman’s experimental apparatus, the inter- internal flows, which may not otherwise be impor- facial tension modulation due to electrical effects tant in their macroscale counterparts. From clas- was observed through a capillary rise phenom- sical fluid dynamics, it is well known that internal enon, and hence, was later termed as electrocapil- flows are typically associated with an entrance larity [54]. A decisive advantage of electrocapillary region followed by a fully developed regime. actuation, in comparison to its thermal counterpart In microscale liquid flows, a third regime (known is the speed with which electrical potentials can as meniscus traction regime) also comes into play, be applied and regulated, with possible characteri- so as to match the velocity profiles with those stic timescales of less than a few milliseconds. obtained in the vicinity of the capillary meniscus. Furthermore, electrocapillary based microactua- The later regime is attributed with a dynami- tors consume much less power as compared to the cally evolving interplay of adhesion and cohesion typical thermocapillary microdevices. Thus, com- forces, consistent with the meniscus topography pared to the thermocapillary flows, electrocapil- [51]. Fluid dynamic influences of the meniscus trac- lary flows are much more energy efficient, with a tion regime may be further complicated by the fact much faster speed of operation (speeds more than that the velocity of liquid meniscus itself may be 100 mm/s have been successfully achieved by elec- related to its surface energy. In simplistic terms, trowetting action, in contrast to a typical speed immediately after a layer of fluid molecules is of only about 1 mm/s in thermocapillary flows). adsorbed onto the microchannel surface, the next The electrocapillary principles are all based on the set of fluid molecules may sail more easily over fact that the surface tension occurs to be a strong the same, giving rise to a dynamic evolution of the function of the electrical potential acting across an contact angle. Physically, since the points on the interface. As a consequence, the apparent contact interfacial lines arrive at the contact line within a angle, θ, gets modified from its value at zero volt- finite time span, an effective slip law needs to be cV 2 age, θ0, in the following form: cos θ =cosθ0 + , posed for relieving a force singularity condition by 2γlv where c is the capacitance of the interface per unit ensuring that a finite force is necessary to move the area and V is the interface potential. A physi- contact lines of a fluid. This effectively imposes a cal interpretation to the voltage-dependent con- dynamical constraint on the contact angle θ,asa function of velocity of the contact line. From fluid tact angle modification term can be provided as dynamics perspective, this departure of contact follows. Upon connecting the initially uncharged angle from its static equilibrium value is mainly droplet to the power source (battery), a charge δQ due to viscous bending of the interface near the flows from the battery to the droplet and to the contact lines, in local equilibrium with the dynami- electrode. The resultant work done on the droplet- cally evolving cohesive and adhesive forces, and electrode capacitor (of total capacitance C)isgiven need to be aptly considered for modeling the per- by δWdroplet = VQδQ,whereVQ is the potential of tinent capillary dynamics [52]. the capacitor on being energized with an instan- taneous charge of Q.Since VQ = Q/C, the total 1 2 6. Surface tension gradients as work done is given by δWdroplet = 2 CVB ,where flow actuators VB is the potential of the power supply (battery). The incremental work done on the battery, on the − Fundamental principles associated with surface other hand, is given by δWB = VBδQB = VB( δQ). tension driven flows rely on the motion induced by Since the battery voltage is a constant, we have, − − 2 the variation of surface tension from point to point δWB = VB δQB = CVB .Thus,thenetwork on a mobile phase boundary, giving rise to tangen- done on the battery-electrode system is given − 1 2 tial strains on the same. Such tangential strains at by δWdroplet + δWB = 2 CVB .Inaphysical the interface are inevitably accompanied by a fluid sense, electrowetting decreases the effective con- motion, so as to maintain the balance of surface tact angle, which is driven by the energy gain upon tension and viscous forces. Fluid motion induced redistributing the charge from the battery to the by the tangential gradients of surface tension is droplet. This reduction of apparent contact angle classically known as the Marangoni effect. Simi- is fundamentally related to the fact that a minimi- lar to the thermocapillarity (where temperature zation of the free energy requires a maximization MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 95 of the capacitance. Applying a potential between the nanoparticles to the surrounding liquid. This the droplet and the electrode, therefore, would significantly accelerates the evaporation process tend to spread the droplet as much as possible, at the liquid–air interface forming the meniscus. in an effort to increase the capacitance. In even Simultaneously, the original liquid contact line is more simple terms, because of the applied elec- pinned and the liquid lost in evaporation is repleni- tric field, opposite charge densities are induced shed from the interior region. The vapour in the at the fluid-solid interface and in the substrate colder air adjacent to the substrate-liquid interface electrode. The electrostatic attraction between condenses almost immediately after the evapora- the aqueous solution and the electrode enhances tion. As a consequence, droplets form very close the coverage of the surface by the aqueous to the liquid–air interface. These droplets then solution, thereby inducing an enhanced wetting coalesce with each other and grow into the larger effect. ones. These larger droplets eventually merge with Researchers have attempted in altering the the original liquid body and extend its contact surface tension forces in small scale systems by line (see figure 3a for a schematic illustration). optical means as well, unveiling a promising and The nanoparticles are drawn to the modified con- fascinating area of opto-microfluidics. One way of tact line, by a combined advective-diffusive trans- achieving the same has been demonstrated through port. A continuous flow augmentation through this a coating of the microfluidic substrate with photo- mechanism may be expected, if the optics is also sensitive surface films (such as TiO2), and shin- translated in tune with the dynamic evolution of ing the resultant surface with a light irradiation the liquid–vapour interface. Previous observational of matching wavelength (for instance, UV light for studies have demonstrated that the droplet coa- TiO2 that has a band gap energy of 3.2 eV), as a lescence can facilitate flow significantly, and holds result of which electron-hole pairs are formed [55]. the potential of altering the contact line dynamics These pairs can then reduce and oxidize the species to a significant extent, leading to a micro-capillary that are adsorbed on the surface. Recombination motion. of the electron and hole is a competing reaction. Using optofluidics, various micro-devices have Based on the relative rates of oxidation and reduc- recently been designed, tested, and demonstrated. tion, an excess or depletion of charge can be gen- For example, we may refer to the schematic of erated at the surface. The consequent change in an optofluidic valve developed by our group, as surface energy due to the direct absorption of pho- depicted in figure 3b. This device operates on the tonic energy by the optically tunable semiconduc- basis that the upper inner surface of a T-junction tor film may create local gradients in the surface is coated with a hydrophobic metal oxide semi- tension forces (altering the state from a hydropho- conductor film. Depending on the side to which we bic to a hydrophilic one, for example), and hence may want to direct a droplet waiting at the junc- may actuate a fluidic motion on the microchannel tion of the T, we may shed UV light on the inner substrate. It also needs to be noted in this context surface of either the left or the right branch. The that the micro-flow velocities may change only on exposed side becomes hydrophilic because of opto- the part of the surface of TiO2 that is irradiated capillary actuation, so that the droplet moves pre- with UV light. Therefore, a desired spatial resolu- ferentially towards that direction. This movement, tion of fluidic control may be obtained since various in terms of speed and directionality may be con- patterns of TiO2 can be designed on the channel trolled at will, by selectively switching the light surfaces being exposed to the fluid. This would source on and off and by translating the optical allow a local control of the flow velocities, without system. any fringing effects. Not only that, the light can be strategically switched on or off, allowing the effects to be temporally controlled as well. 7. Interfacial boundary condition: In order to modulate the bulk flow features Slip vs. no-slip with light actuation, photosensitive nanoparticles (such as gold nanocrescent particles) may also be 7.1 General considerations dispersed in the fluid [56]. Without any photo- actuation, the liquid on a hydrophobic surface It is at an interface between solid and fluid where remains stationary. Because of the ‘coffee-ring’ microfluidics turns truly complex: near the con- effect, the local concentration of the nanopar- tact, the physical properties of the fluid are not ticles near the liquid–air interface is higher than the same as in the bulk and the influence of that of the interior. When a focused light illumi- molecular scale interactions tend to become pre- nates the nanoparticles near the liquid–air inter- dominant. The celebrated no-slip boundary condi- face, heat is almost instantaneously (within a tion, implying zero relative velocity between the few nanoseconds) generated and transferred from fluid and the solid, may tend to get severely 96 G K ANANTHASURESH AND SUMAN CHAKRABORTY

Figure 3. (a) Optofluidic manipulation through evaporation-condensation augmented through photo-sensitive nanopar- ticles; courtesy G L Liu et al. Nature Mat. (2006), (b) Tunable optofluidic droplet valve developed in our research group.

Figure 4. Concept of slip parameterized by the slip length, Ls. challenged at many instances involving interesting from the pertinent surface area to volume ratios. microflows. The no-slip assumption at the inter- For traditional devices, this may be of the order of face is an idealized paradigm, assuming moder- 1m−1, as compared to as low as 106 m−1 for typical ately strong attractive forces between fluid and microdevices with characteristic length scales of wall. At the micro and nanoscale, new phenomena the order of 1 μm [57]. This profoundly affects the arise because surface forces, and all types of micro mass, momentum and energy transport; and leads effects, ignored at the macroscale, become impor- to additional effects like slip flow, rarefaction, com- tant, these include: surface tension, air bubbles, pressibility, etc. The later effects are particularly liquid evaporation, porosity, osmotic effects, van more perceptible and prominent in gases than in der Waals forces, electrostastic forces, etc., result- liquids, primarily attributable to the significant ing in possible true or apparent slips. In the case characteristic alterations in the density of gases of slip, a slip length, Ls, must be introduced. Slip with changes in pressure as compared to liquids. length is defined as the distance behind the inter- The recent spurt in the development of MEMS face at which the liquid velocity extrapolates to and NEMS has raised several interesting questions zero (see figure 4). with regard to the suitability of the traditionally accepted thermo-fluidic modeling considerations 7.2 Gas flows for microscale gas flows. Central to these consider- ations, the applicability of the classical continuum The primary distinction between fluid flows in the hypothesis has often been seriously debated, which micro-scale and the conventional devices originates is based on the following two important postulates: MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 97

(a) there are sufficiently large numbers of molecules slip velocity is proportional to the local Knudsen in chosen elemental volumes so that statistical number. The contribution of the stream-wise tem- uncertainties with regard to their respective posi- perature gradients associated with the thermal tions and velocities do not perceptibly influence creep phenomenon was subsequently added to the the averaged fluid/flow property predictions, as Maxwell-slip conditions, with a consideration that well as the predictions in the local gradients of the slip velocity is simultaneously caused by the properties through well-known rules of differential velocity gradient normal to the wall and a tem- calculus, and (b) the system is not significantly perature gradient tangential to the wall. Identical deviated from local thermodynamic equilibrium. considerations were also invoked by Smoluchowski When gas molecules collide with a solid bound- for modeling the temperature jump boundary con- ary, those are temporarily adsorbed on the wall ditions at the walls [59]. and are subsequently ejected. This allows a partial Several slip-based models have been proposed transfer of momentum and energy of the walls to in the literature for modeling microscale gas the gas molecules. This process slows down the flows, following up the seminal contributions from fluid and renders the tangential component of the Maxwell. However, the very fact that the stan- fluid velocity equal to the corresponding compo- dard continuum considerations may still succeed to nent of the boundary velocity. However, this con- work at all regions except at the solid boundary sideration remains valid only if the fluid adjacent started giving rise to several interesting questions. to the solid wall is in thermodynamic equilibrium. It could eventually be inferred that the concerned Deviation from thermodynamic equilibrium may discrepancies may not originate from the fact that result in a ‘slip’ between fluid and the solid bound- the fundamental conservation principles cease to ary in small channels where the mean free path work for microscale gas flows. Rather, these may may be of comparable order as that of the channel be attributable to an apparent abstraction from dimension. This phenomenon may be more aggra- the seemingly non-trivial dependences of the fluid vated by the presence of strong local gradients of flow characteristics on the pertinent molecular level temperature and/or density, because of which the interaction mechanisms with characteristic sharp molecules tending to ‘slip’ on the walls experience gradients adjacent to the confining boundaries. a net driving force. Such phenomena are usually A common consensus is that it is not necessarily termed as ‘thermophoresis’ and ‘diffusophoresis’, that the continuum conservation equations tend to respectively. The extent of this deviation from the get invalid in these cases, but the gradient terms in classical no-slip scenario is not merely dictated these equations might become incapable of captur- by the mean free path (λ)inanabsolutesense, ing strong local gradients of fluid properties adja- but also its comparability with the characteristic cent to the solid substrate. system length scale (L) that describes the rela- Recently, it has been suggested by several tive importance of rarefaction in the system. The researchers [60,61] that the constitutive assump- ratio of these two, known as the Knudsen number tion in-built with the standard forms of the (Kn = λ/L), appears to the single important deci- Navier–Stokes–Fourier (NSF) equations need to sive parameter that determines the applicability of be extended, to render them applicable for fluid a particular flow modeling strategy as against the flows with strong local gradients of density and extent of rarefaction of the flow medium. When Kn temperature. These modifications have originally exceeds the order of 10−1, an apparent transition been motivated from a thoughtful continuum- of even the bulk flow paradigm from a continuum based re-interpretation of experimentally-observed to a free molecular one may occur, which becomes thermophoretic motion, the hierarchical reorder- increasingly more significant as Kn is progressively ing of the Burnett terms in the Chapman–Enskog increased to higher orders. expansion of the viscous stress tensor, and the Several authors have theoretically addressed the velocity/thermal creep coefficients introduced by phenomenon of enhanced mass flow rates as a con- Maxwell in an effort to account for non-zero slip sequence of this apparent deviation from no-slip velocities and temperature jumps at the fluid– condition, by employing the continuum equations solid interface (irrespective of the fact that the in conjunction with the application of the Maxwell continuum hypothesis might otherwise be satisfied slip boundary condition [8]. Maxwell [58] originally in the bulk). Brenner [60] first pointed out that the introduced a first-order slip model, which assumes Navier–Stokes equations, in their traditional forms, that the slip velocity is equal to the velocity at a are incomplete and provided extensions based on mean free path away from the wall. The extent of the concept of volume diffusion. In a strict sense, slip was modeled to be proportional to the local the concept of continuum is an abstraction that mean free path, for partially specular and partially does not exist in reality, because of the presence of diffuse walls. When appropriately normalized, the discrete material entities (such as molecules) that Maxwell slip velocity condition shows that the have empty spaces separating them. Such empty 98 G K ANANTHASURESH AND SUMAN CHAKRABORTY spaces are heavily interlinked with the extent of diffusive flux can be manifested by the Lagrangian rarefaction or compression in the system. As a velocity of a passive and neutrally-buoyant non- consequence, the presence of strong gradients of Brownian tracer particle introduced into the flow, density may pose certain ambiguities in mathe- relative to the advective fluid motion. Accordingly, matically defining the ‘velocity’ of the continuum the net ‘velocity’ may be considered as a sum- at a point. In an effort to avoid such ambigui- mation of the advective and the diffusive compo- T D T ties, one may refer to fluxes, instead of velocities. nents, to yield [60,61] Ui = Ui + ui ,whereUi In that interpretation, the term ‘velocity’ that is the total velocity, Ui is the advective compo- D is commonly employed to describe the conserva- nent of velocity and ui is the diffusion velocity. tion of fluid mass (continuity equation) essentially The explanation of the mass diffusion phenomenon describes a normalized flux density of a system by kinetic theory is immediately obvious: a net of particles with a fixed mass and identity. With flow may result in the direction in which gradients respect to an Eulerian control volume, the same of the thermodynamic properties are present, in can be interpreted as an advective flux density consequence of difference in the thermal velocity of of mass across an element of the pertinent con- molecules at different locations [62]. trol surface. Superimposed on this advective flux, one may also have a phoretic or diffusive trans- 7.3 Slip boundary condition for liquids? port of mass across the control surface, which may be spontaneously actuated by local gradients of Because of sufficient intermolecular forces of density and/or temperature. Physically, this dif- attraction between the molecules of the solid sur- fusive flux can be manifested by the Lagrangian face and a dense medium such as the liquid, velocity of a passive and neutrally-buoyant non- it is expected that the liquid molecules would Brownian tracer particle introduced into the flow, remain stationary relative to the solid bound- relative to the advective fluid motion. Because of ary at their points of contact. Only at very high such diffusive transport of fluid particles, one may shear rates (typically realizable only in extremely have a net diffusive transport of mass across the narrow confinements of size roughly a few mole- control surface, without violating the constraining cular diameters), the straining may be sufficient requirements of the conservation of mass from a in moving the fluid molecules adhering to the continuum perspective, in a sense that individual solid by overcoming the van der Waals forces of molecules are free to enter and leave a material attraction. Another theory argues that the no- spacewithaninvariantmass. slip boundary condition arises due to microscopic Based on the local gradient-driven diffusive boundary roughness, since fluid elements may get transport considerations, researchers have deve- locally trapped within the surface asperities. If the loped modified constitutive relationships based fluid is a liquid then it may not be possible for the on the velocity of a material frame that advects molecules to escape from that trapping, because of with the fluid flow, and therefore, is a representa- an otherwise compact molecular packing. Following tive of the advective flux density of the local flow. this argument, it may be conjectured that a mole- This ‘velocity’ invariably features in the calcula- cularly smooth boundary would allow the liquid tions of the material or total derivatives appearing to slip, because of the non-existence of the surface in the pertinent continuum conservation equations. asperity barriers. In a strict sense, however, this is not the true physi- One of the important consequences of the inter- cal velocity of fluid flow, but merely a description facial phenomenon in the context of liquid flows in of a normalized flux density that advectively trans- microchannels may be appreciated by referring to ports mass, momentum or energy within the fluid the apparently anomalous predictions from simple envisaged as a continuum. Superimposed on this pressure drop – flow rate characterizing experi- advective flux, one can also have a phoretic or dif- ments. Numerous experimental investigations [63] fusive transport of mass across the control surface, have been executed by the researchers in this which may be actuated by local gradients of den- regard to pinpoint several significant (though, not sity and/or temperature, independent of action of necessarily systematic and repetitive) deviations in any equivalent macroscopically-resolvable imposed the frictional behaviour of pressure-driven liquid pressure gradient. A classical example of phoretic flows through microchannels, in perspective of the transport is the phenomenon of thermophoresis, in classical theory that has been emphasized over which heat-conducting, force-free and torque-free the years as an inevitable ritual. These studies tracer particles (typically, spherical) move from have almost routinely exhibited an increase of fric- hotter to colder regions in the fluid (usually, a gas), tion factor with increase in effective surface rough- against an externally imposed temperature gradi- ness, characterized through suitable parameters. ent, without necessitating the aid of any exter- Researchers have also proposed semi-empirical nally imposed pressure gradient. Physically, this models based on the concept of the so-called MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 99

condition, which, for decades, had marked the saga of innumerable successes in reproducing the essen- tial features of fluid flow within the continuum regime. However, because of the lack of precise understanding of the relative competing influences of friction enhancing and friction reducing effects, the fundamentals of frictional characteristics in pressure-driven liquid microflows remained to be poorly understood for a long time. A series of recent investigations by our research group have attempted to resolve the apparent anomaly of ‘reduced’ fluid friction while con- fronting with ‘rough’ solid surfaces in narrow confinements, by appealing to the thermo- fluidics over experimentally-resolvable spatio- temporal scales. In simple terms, it has thus been recently established that we may design rough sur- faces to make them behave in the smoothest pos- sible way in terms of fluid friction. We discovered from our studies that specially designed tiny water- transport channels (or pores) may achieve this apparently impossible task by a simple mechanism [67]. Confining rough surfaces made of hydrophobic (water-disliking) materials may trigger the forma- tion of nano-scale bubbles adhering to the walls in tiny channels (see figure 5a). This incipient vapour layer acts as an effective smoothening blanket, by disallowing the liquid on the top of it to be directly exposed to the rough surface asperities. In such cases, the liquid is not likely to feel the presence of the wall directly and may smoothly sail over the intervening vapour layer shield. Thus, instead of ‘sticking’ to a rough channel surface, the liquid may effectively ‘slip’ on the same (see figure 5b). While the microscopic roughness of the solid surface impede the motion of the adher- ing fluid promoting the stick-flow, the incipient vapour layers formed on the solid surface tend to augment the level of slippage. Thus, deliberately designed rough surfaces may lead to an appar- ently frictionless flow behaviour. This may essen- tially not be due to an explicit effect of the surface roughness, but an implicit effect of the same that Figure 5. (a) Nanobubbles on a microchannel substrate, leads the inception of tiny vapour bubbles of few (b) apparent slip due to nanobubble formation. nanometer sizes close to the channel walls [67]. Nanobubbles may be nucleated when the driving force required to minimize the area of liquid– ‘roughness viscosity’ [64], ‘porous medium layer’ vapour interface is smaller than the forces that pin [65], etc., in an effort to fit the theoretical pre- the contact line of the substrate. In such cases, dictions with the observed experimental data. On the hydrophobic surface itself acts like a nucle- the other hand, in sharp contrast, a number of ation site. Hydrophobic units are not thermody- research studies have revealed reductions [66] in namically favored to form hydrogen bonds with friction for pressure-driven liquid microflows, in water molecules. Hence, these give rise to excluded comparison to the classical theory of Poiseullian volume regions encompassing the locations charac- flows, despite encountering rough microchannel- terized with sharply diminishing number density of walls. Such apparently contradictory observations water molecules. Close to small hydrophobic units, tended to threaten the amazing acceptability of water molecules can structurally change and reor- the paradigm of the celebrated ‘no-slip’ boundary ganize without sacrificing their hydrogen bonds. 100 G K ANANTHASURESH AND SUMAN CHAKRABORTY

However, close to larger hydrophobic units, per- Intensified research investigations on confine- sistence of a hydrogen bond network is virtually ment induced hydrophobic interactions and the impossible. The loss of hydrogen bonds near the consequent smoothening or roughening effects have hydrophobic surfaces effectively expels liquid water apparently redefined the physical understanding of to move away, thereby forming thin vapour lay- the underlying complex thermo-physical systems, ers. Such interfacial fluctuations may destabilize raising further unresolved issues in understand- the liquid further away from the solid walls, lead- ing the science of fluid-solid interactions at small ing to a pressure imbalance. This effectively gives scales. Utilizing some of the recent advances in rise to an attractive potential between the two sur- this regard and addressing some of the concerned faces [67]. Further, loss of hydrogen bonds close ‘open’ questions in the near future, researchers may to any such hydrophobic surface effectively repels potentially develop engineered rough surfaces with liquid molecules, thereby favoring the formation of triggered hydrophobic interactions and consequent thin vapour layers. In confined fluids, long-ranged inception of friction-reducing vapour phases [69] so interactions may also trigger separation-induced as to achieve a super-fluidic transport over small phase transitions. Such separation-induced cavita- scales. Narrow confinements capable of mimicking tion physically originates from an increase in the the selective and rapid fluidic transport attainable local molecular field due to the replacement of in biological cellular channels but designed on the polarizable fluids by solid walls. basis of such newly-discovered surface roughness- Conditions in which liquid layers sail smoothly hydrophobicity coupling would open up a wide over thin vapour layers covering the solid bound- range of new applications, such as transdermal aries may be termed as ‘apparent slip’, since the no- rapid drug delivery systems, selective chemical slip boundary condition still remains a valid propo- sensing and mixing in nano-scales and several other sition at the walls (see figure 5a). It is only the lab-on-a-chip based applications. apparent inability to capture and resolve the steep velocity gradients within the ultra-thin vapour lay- ers that prompts an analyzer to extrapolate the 8. Electrokinetics in microfluidics velocity profiles obtained in the liquid layer above the vapour blanket, to mark an apparent devia- A majority of microfluidic functions required in tion from the no-slip boundary condition at the various lab-on-a-chip devices includes pumping, wall. Interestingly and counter-intuitively, sponta- mixing, thermal cycling, dispensing and separat- neous roughening effects on liquid flows occurring ing. Many of these processes are executed elec- over smooth bubble layers have also been recently trokinetically, implying that electrical fields are observed, thereby raising serious doubts on the uni- employed in manipulating the relative movements versatility of the above-mentioned physical conjec- between the various particles, solid, and fluid ture. Influences of stochasticities on this amazing phases. physical behaviour have also been seriously ques- Length scale matching is important for efficient tioned, although they have by no means been well- momentum and energy transfer for controlling the resolved. motion of desired fluids and molecules. Most of the Keeping in view of the disparate physical man- biological objects of interest, such as DNA, pro- ners in which a pressure-driven liquid microflow teins, and cells, have a characteristic length from could be influenced by the intrinsic surface con- nanometer to micrometer. Electrokinetics is espe- ditions, the above considerations have been fur- cially effective in this domain by taking advan- ther extended by us to develop a generalized model tage of the small length scales. With the develop- of fluid friction for pressure driven liquid flows ment of MEMS fabrication technology, integration through microchannels. This theory has been based of micro or nanoscale electrodes in fluidic device is on a broad consensus that has been achieved a relatively simple procedure. Therefore, electroki- regarding the patterns of influence of the sur- netic forces are ideal for manipulating biological face conditions on the resultant magnitude of fric- objects and performing fluidic operations. By uti- tion factor (which, effectively, is a non-dimensional lizing electrokinetic effects, fluid flow can be manip- pressure drop). Considering the stochastic nature ulated, in general, using the following methodolo- of many input model parameters, as attributed to gies. In this context, it is important to keep in mind the uncertainties in prevailing surface conditions, that a general expression for electrokinetic body numerous random experiments have been per- force on a continuum, per unit volume, may be formed with chosen ranges of these data, to come written as up with a generalized formulation of friction factor   for pressure-driven liquid microflows through par- 1 1 ∂ε F = ρe E − E • E∇ε + ∇ E • E ρ , allel plate channels [68], replacing the use of the  2 2 ∂ρ celebrated Poiseullian formalisms, instead. ∇•(εE) MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 101

the ions, wall and the adjacent dipoles. From the compact layer to the electrically neutral bulk liquid, the net charge density gradually reduces to zero. The layer of mobile ions beyond the Stern layer is called the Guoy–Chapman layer, or the diffused layer of the EDL. These two layers are separated by a shear plane. The potential at this shear plane is known as the zeta potential (ζ). The thickness of the EDL is known as the Debye length (λ), which is the length from the shear plane over which the EDL potential reduces to (1/e)ofζ. The Debye length solely depends on the properties of the liquid and not on any properties of the surface. For example, for a√ 1 : 1 electrolyte of concentration C,1/κ =0.304/ C nm (where C is in molar units). This is estimated by tak- Figure 6. Charge and potential distribution within the ing T = 298 K,ε =78.5 × 8.85 × 10−12 C2/N.m2, EDL. −19 −23 e =1.602 × 10 C, kB =1.38 × 10 J/K and n∞ = 1000 N, where N is the Avogadro number. Thus, for a 10−4 M NaCl solution, 1/κ ≈ 30 nm and where E is the local electric field, ε is the local ≈ permittivity of the medium, and ρ is the fluid den- for 1 M of the same solution 1/κ 30 nm. Physi- sity. In the foregoing discussions, we will consider cally, as the bulk ionic concentration increases, two fundamental examples concerning electroki- more counterions are attracted to the region netic effects, namely, electroosmosis and streaming close to the charged surface to neutralize the sur- potential. face charge. As a result, the EDL thickness is reduced and the EDL appears to be somewhat ‘compressed’. If a potential is applied along the 8.1 Electroosmosis channel, the diffuse electrical double layer will move due to the electrostatic force. Due to the When a solid is in contact with an electrolyte, the cohesive nature of the hydrogen bonding of water chemical state of the surface is generally altered, molecules, the entire aqueous solution is effectively for example, by ionization of covalently bound sur- pulled. Since this pull depends on the local charge face groups or by ion adsorption. As a result, density, it effectively becomes zero outside the the surface inherits a charge while counterions EDL. Thus, there is no effective velocity gradient are released into the liquid. For instance, com- outside the EDL, and the velocity profile remains mon glass, SiOH, in the presence of H2O, ion- − virtually uniform across the remaining part of the izes to produce charged surface groups SiO and microchannel cross-section, typically reminiscent release of a proton. At equilibrium, a balance of a plug like flow. between electrostatic interactions and thermal agi- tation generates a charge density profile. The liquid is electrically neutral, but for a charged layer adja- 8.2 Streaming potential cent to the boundary, which bears a charge locally equal in amplitude and opposite in sign to the It is interesting to note that even in a pure bound charge on the surface. This charged layer pressure-driven flow occurring through a narrow is commonly known as the electric double layer fluidic confinement, the ions in the mobile part of (EDL) [70]. the EDL are transported towards the down-stream A schematic depicting the charge and the poten- end with the liquid motion. This causes an elec- tial distribution within an EDL is depicted in trical current, known as the streaming current, to figure 6. Immediately next to the charged sur- flow in the direction of the imposed fluid motion. face, a layer of immobilized counterions is present, However, the resultant accumulation of ions in which is known as the compact layer or the Stern the downstream section of the channel sets up its layer or the Helmholtz layer. This layer is about a own induced electrical field, known as the stream- few Angstroms in thickness, and hence the poten- ing potential [71]. This field, in turn, generates a tial distribution within the same can be taken current to flow back against the direction of the as almost linear. The charge and the potential pressure-driven flow. This so-called conduction cur- distributions within this layer are strong func- rent balances the streaming current at steady state, tions of the geometrical characteristics of the ions so that the net electrical current becomes zero, con- and the short-range interaction forces between sistent with a pure pressure-driven flow condition. 102 G K ANANTHASURESH AND SUMAN CHAKRABORTY

Researchers have utilized the generation of stream- channel surface, very much typical to such tiny ing potential to convert hydrostatic pressure differ- pores, may in turn amplify this tendency of slip- ences into useful electrical energy, to characterize page to a large extent, by pumping the layer of interfacial charge of organic thin films, to measure fluid even more effectively along with the mov- wall charge inversion in presence of multivalent ions able charges [72]. Based on this novel conjecture, in a nanochannel, to analyze ion transport through we may design miniaturized super-fluidic systems nanoporous membranes, to design efficient nano- with an unimaginably high rate of liquid pump- fluidic batteries, etc. as some of the technologically ing, with the employment of very low flow-pumping relevant applications. pressure-gradient. The streaming current and streaming poten- The engineered microsystems of solids and fluids tial in a pressure-driven flow can drive an exter- are complex enough as discussed until now in this nal load and therefore may represent a means of paper. However, this complexity pales in compari- directly converting hydraulic energy into electri- son when we consider biological microsystems – the cal power through exploitation of micro/nanoscale cells. We will briefly consider them next. flow physics. The results are more favourably scal- able for nanochannels than microchannels, pri- 9. Mechanical models of cells marily because of the fact that the extent of zone in which a net volumetric charge density Cells are soft and wet. So are tissues, which exists in a nanochannel is much more than that are collections of cells and inanimate extracellular in a microchannel, when normalized with respect matrix secreted by cells. The behaviour of cells to their respective hydraulic radii. The notion of and tissues cannot be understood only from the employing such electrokinetic effects in an energy deterministic approaches of classical mechanics. conversion device is not new, but implementation Just as engineered microsystems have coupled of this concept towards making practical engineer- energy domains, biological microsystems have even ing devices have not been far from being elusive, stronger coupling with other domains, which is except for isolated test cases. Even in such cases, not fully understood yet. This is a field that is the energy conversion efficiency has been found still in its nascent stage of development with a to be quite poor (about 5%), primarily attribut- lot of challenges ahead but with many promising able to a huge flow-pumping power necessary approaches that are pursued today. The descrip- to overcome strong frictional resistances against tion that follows is neither exhaustive nor is it pressure-driven microflows and nanoflows. Indeed, universally agreed upon; it is an emerging field of high energy-conversion efficiency and high out- study. put power are the fundamental requirements for such devices to be practical and capable of solv- 9.1 Biophysical approaches ing some burning global problems in the field of energy science and engineering. Exploration of this Just like engineered microsystems do, cells also nascent and novel technology for generating electri- attach themselves to their environment and to cal energy directly from a hydraulic form (instead each other, move about, and deform. This is only of going through other mechanical/coupling losses) one side of the coin. The flip side is that cells in miniaturized devices without necessitating any also grow, re-organize themselves, sense their envi- fuels, thus, appears to be very much pertinent ronment, transduce the mechanical signals they and appealing in the global energy crisis scenario, receive into chemical signals, and induce a variety and indeed a very attractive proposition. Efforts, of biochemical processes. No engineered material, however, need to be directed to make the scheme active or smart as they are called, are capable energy efficient and viable through exploration of of this complex behaviour in such a deceptively some additional facets of interfacial phenomena simple system like a cell. The crucial fact that over small scales. cells are living entities brings in another dimen- Based on our recent discovery that one may clev- sion that their properties continuously change due erly design deliberate rough hydrophobic surfaces to their interactions with their environment. This (which are common artifacts of the micro/nano- is a rather unusual feature when we compare it fabrication processes) to make them behave in with engineered systems. Another equally impor- the smoothest possible way in terms of result- tant feature is that thermal fluctuations have sig- ing in ultra-low flow resistances, one may realize nificant effect on cells. Hence, statistical mechanics a significant rate of ‘apparently slipping’ flow on plays an important role in addition to classical utilizing only a meager applied pressure gradient, mechanics. by exploiting the surface roughness as a bless- There is no agreement among different models ing in disguise. The spontaneous formation of an suggested for cells’ mechanical behaviour. The rea- electrically charged layer (EDL) adhering to the sons for this may be that there are a number of cell MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 103

Figure 7. Rough estimates of forces experienced by biomolecules [78]. types, the experimental data is usually obtained interior of the cells remain in deep energy wells in vitro rather than in vivo, cells reorganize unless disturbed by large fluctuating forces. The themselves over spatial and temporal scales com- fluctuation depends on the temperature. Thus, the parable to that of the time-resolution of the experi- behaviour can vary from flow-like response to stiff mental techniques used, and cells consist of several solids. vastly different entities within them. The cell mem- The mechanical behaviour of the cells is brane, made up of lipid molecules arranged in also explained using the concept of tensegrity a bilayer due to hydrophobic-hydrophilic effects, structures. Tensegrity structures, proposed by is embedded with bulky transmembrane proteins. Buckminster Fuller, are stabilized assemblies of The cell membrane behaves both as a solid and cable-like elements that are in tension and tube- a fluid. It not only provides protection or demar- like elements in compression. Thus, unlike brick cation boundary to enclose the cell but also and masonry structures that carry mostly com- helps in mechano-sensing through ion channels and pression, tension gives the structural integrity to other mechanisms. Cytoskeleton that provides the tensegrity structures. Ingber [75] argued that actin shape and size to the cells is a complex struc- filaments in the cytoskeleton are in tension while ture in itself as it is made hierarchically with microtubules are in compression. This is shown actin filaments, intermediate filaments, and micro- to be true in experiments because actin filaments tubules. Many theories are proposed to understand are curved when left alone whereas microtubules cytoskeleton’s mechanical behaviour. The study are straight. But the reverse is true inside the of mechanical behaviour of cell nucleus has only cell: the actin filaments are taut and tubules begun now. There are a large number of other (because of compression-induced buckling) are molecules within the cytoplasm that makes up the curved. There is also other circumstantial evidence interior of the cell apart from the nucleus. A com- to the tensegrity theory. The membrane tension ment made by Crick and Hughes in their seminal and the effects of extra cellular matrix as well as paper of 1950 (wherein they studied, reportedly for the dynamic behaviour can also be explained with the first time, the physical properties of cytoplasm the tensegrity model. Conveyance of mechanical of chick fibroblasts) is recalled today by researchers and chemical signals through the tensegrity struc- in this field: “...If we were compelled to suggest ture of the cytoskeleton remains to be explored a model, we would propose Mother’s Work Basket further. — a jumble of beads and buttons of all shapes There are also arguments against treating cyto- and sizes, with pins and threads for good measure, plasm as an aqueous medium. It is argued that all jostling about and held together by ‘colloidal when a membrane ruptured accidentally in the forces’.” A quick glance at the modeling attempts normal wear and tear of the cells or intentionally to understand such a system is pertinent at this during experiments, the cells should not survive point to appreciate the level of difficulty in propos- [76]. Therefore, the cytoplasm is to be treated as ing mechanical models for cells. a gel. The gel can undergo a phase transition to Based on the experimental measurements of dif- a solid state. The phase transition can explain the ferent types of cells using different techniques, motility and various ways in which cells sense and Fredberg and Babry [73] propose Sollich’s [74] respond. This view leaves it open for application model of soft glassy material for cytoskeletons of the sol-gel theories developed by engineers for of cells. In this theory, the constituents of the non-living matters of that kind. 104 G K ANANTHASURESH AND SUMAN CHAKRABORTY

A bottom-up approach to modeling cells is also 9.2 Continuum biomechanics pursued. Here, the behaviour of single filaments is studied and using that the behaviour of a network Understanding the behaviour of the soft matter of filaments is deduced. Figure 7 shows approxi- of cells and tissues warrants substantial extension mate values of forces acting on biomolecules. Cova- of classical mechanics that is primarily developed lent forces keep the molecules intact and uncut as for engineering materials. Just as mechanics they change their shapes. Long molecules undergo is based on fundamental postulates based on a little stretching, some twisting, and mostly bend- which predictive theories can be deduced, bio- ing. These are often modeled using the simple mechanics too aims to elucidate the responses theories of strength of materials although the forces of the biological matter using analytical deduc- acting on them are vastly different from that of tion. However, that level of an understanding is the macroscale flexible beams. As can be seen in not yet available for biological matter at dif- figure 7, viscous, thermal, electrostatic and van ferent size scales. Hence, sometimes an alter- der Waals forces dominate. Analysis of the defor- nate term ‘mechanobiology’ is used to describe mation of individual molecules requires statistical an inductive approach wherein particular obser- mechanics treatment known as freely jointed chain, vations are used to formulate theories for under- worm-like chain, etc., [77]. Bending and twisting lying mechanisms [83,84]. In spite of the fact stiffnesses of the single molecules are estimated that biological matter behaves significantly differ- using entropic theories. Two and three dimensional ently under different conditions, the classical the- networks are also considered in a similar spirit to ories of continuum mechanics have helped, with deduce equivalent elastic moduli [77]. some extensions, in understanding some aspects Some others have used continuum models for of biomechanics of tissues, bone, etc. The tools understanding the experimental observations. Con- employed in these studies are finite elasticity, tinuum models are applicable only when the membrane mechanics, viscoelasticity, mixture the- observed displacements are much larger than the ory, growth and remodeling, and thermomechan- smallest features of the constituent parts. Given its ics. The latter highlight the fact that the elasticity occasional fluid-like behaviour, viscoelastic models of soft matter is primarily entropic rather than are often used. Indeed, finite element simulations energetic [85,86]. have been attempted for understanding the defor- Recent thinking in continuum mechanics appre- mation behaviour of erythrocytes (red blood cell) ciates the ability of biomaterials to grow and [79]. Such simulations and models have a number of remodel their internal constitution in response to limitations in that they only qualitatively capture subtle changes in their surroundings due to dis- the behaviour and that too with debatable fidelity ease, injury, etc. [83]. Interesting questions have [80]. This is because, as noted above, cells behave been raised as to whether stress, strain, strain both like solids and liquids and hence the con- rate or a similar mathematical measure of con- tinuum treatment should take that into account. tinuum mechanics can explain complex ways in In this line of thought, some multi-phasic con- which cells can sense subtle mechanical stimuli. tinuum theories are also put forward [80]. Here, the How cells sense these mathematically defined solid and fluid behaviours are simultaneously con- quantities is unclear and warrants further stud- sidered with appropriate momentum balance terms ies at the molecular level. The issue here is to added to the equilibrium equations. In addition to relate conformational changes in the biomolecules any body force that may be present (which is often to changes in bulk mechanical behaviour. The negligibly small in cells) in the equilibrium equa- continuum mechanics of development of tissue, tion, the momentum of the fluid flowing past the growth, and remodeling, damage and healing, also solid phase is added to the divergence of the solid need substantial extensions. The role of experi- state stress; similarly, the momentum term is sub- mentation is equally important [87] in paving the tracted from the stress in the fluid phase. Consti- way for these studies. Overall, continuum biome- tutive relations are to be written for both the solid chanics is an interdisciplinary field that requires and fluid phases. The basis for this type of model- close collaboration among biologists, biochemists, ing of cells comes from the continuum theories biophysicists, engineers, mathematicians, and developed for hydrated soft tissues [81]. Indeed, clinicians. two phases are not enough when the ionic phase is also considered. Hence, triphasic and multi-phasic models are also considered [82]. The challenges in 10. Experimental tools for this approach are in estimating the constitutive bio-micromechanics properties that reflect reality and the complexity in the numerical solution of the coupled multi-phasic Measuring displacements and forces acting on the equations. cells or exerted by them in their native aqueous MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 105

Figure 8. Cellular biomechanics experimental techniques. medium are important in biomechanical studies. end. The range of forces that can be applied by As discussed above, cells are small, soft, and wet. varying suction pressures is from tens of pN to tens This is in contrast to the macroscale objects and of μN. This technique helps deduce the mechanical hard (mostly silicon, its compounds, and metals) properties of the cell membrane rather than the objects at the microscale. So, an entirely new set cytoplasm because it is the membrane that gets of measurement techniques are needed. Many are stretched. developed and are continually improved. We con- Attaching particles, usually beads coated with sider some of them below. antibodies or other biomolecules for inducing adhe- sion to cells, is another technique for applying 10.1 Metrology for biomechanics forces on cells. If the particles are magnetic, exter- nal magnetic field is varied to stretch or twist the It is estimated that a typical cell experiences forces cells. These are known as magnetic tweezers [90] or in the range of 1 pN to 10 nN from its environ- magnetic twist cytometry [91]. Electric fields can ment. The displacements caused due to the forces also be used by choosing particles of appropriate are of the order of a few nm to a few μm. Today, dielectric constant. If the particles have appropri- there are tools that can probe groups of cells or iso- lated single cells in these ranges of forces and dis- ate refractive index, lasers are used to apply radi- placements. Figure 8 shows a large subset of such ation pressure and thereby stretch or move the techniques. Most of these can measure elastic and cells. This is known as laser tweezers technique [92]. viscoelastic properties while some can only mea- Lasers can also be used without attaching particles. sure viscous properties. One such technique is an optical stretcher where For groups of cells, the most common technique two de-focused lasers create statically balanced is culturing them on a membrane and stretching stretching on a cell [93]. the membrane. The cells adhering to the membrane Atomic force microscope (AFM) is a widely stretch, which is measured optically. The strain used technique. It contains a sharp (10–50 nm dia- induced in this way can be uni-axial or bi-axial meter), usually silicon-made, tip mounted on a can- and is up to 30% [88]. Sedimentation is a tech- tilever beam. An AFM, which is usually used to nique wherein sufficiently small granules are dif- imaging, can also be used to apply forces on cells. fused through a cell to measure the viscosity of Resolution of the forces that can be applied on a cytoplasm [89]. Hydrostatic pressure is applied on cell depends on the stiffness of the cantilever, which the cells using compressed air or cells are grown typically ranges from few hundredths to tens of on a porous surface to apply pressure to measure N/m. This is usually of the order of pN. Spatial the effect of mechanical stress on the cell response. resolution, which is determined from the lateral In order to study the effect of shear stress on deformation of the cells, is of the order of 100 nm. cells, a fluid is passed over the cells. While the cell An AFM measures the local mechanical proper- aggregates indicate the average effect of mechani- ties rather than the bulk stiffness of the cells or cal stimuli, single cell studies help understand the the stiffness distribution inside them. Flowing a biomechanics of the cells more specifically for fun- fluid over to apply shear stress can also be done on damental insights. single cells. This is useful to endothelial cells. Many The most common technique of inducing force/ other techniques are reported for mechanical prob- displacement on a cell is using micropipettes, which ing of cells. Good overviews of these can be found are simply sharp glass needles of diameters as small in [6,12,88]. Apart from these, there is another per- as a fraction of a micron. Singles cells can be sucked spective called bio-micromanipulation. We discuss inside the pipettes by applying vacuum at the other this next. 106 G K ANANTHASURESH AND SUMAN CHAKRABORTY

Figure 9. Micromanipulation for mechanical characterization of cells. (a) the setup, (b) compliant polydimethylsiloxane (PDMS) grippers of varying sizes, the smaller being 6 mm × 6mm× 1 mm, (c) a zebrafish embryo grasped between the jaws of the compliant gripper, and (d) the close-up view of the gripper.

10.2 Bio-micromanipulation measure their displacements and from those, the forces are computed by solving an inverse problem Manipulation in robotics and mechanisms in elasticity [97]. Thus, this technique enables us to literature means changing the pose of an object or compute the bulk stiffness of a single cell because changing the shape of an object. While microma- the displacement is known because of visual mea- nipulation retains this connotation, it also implies surement and the force is known because of the certain kinds of ‘cell surgery’. That is, injecting computation. into a cell, pronuclei formation by exchanging The mechanical characterization using micro- nucleus between two cells, in vitro fertilization, grippers has an advantage over all other tech- etc. [94]. niques: it enables us to estimate the material Interestingly, using microgrippers for mechani- property distribution inside the cells. This is pos- cal characterization of cells has not received much sible by solving a second inverse problem wherein attention. Holding a cell between two plates to the boundary forces and displacements are used to push, pull, or roll it, is reported [95]. Complete estimate the stiffness distribution of the interior of characterization, bulk and intrinsic property dis- a body presumed to be elastic [98]. This requires tribution inside the cells, might be possible if a several sets of boundary force-displacement data microgripper is used to grasp and manipulate a [99]. For this, it is necessary to manipulate the cell. Figure 9 shows a biomanipulation setup using cell by rolling, twisting, and re-grasping differently. which a cell can be grasped and manipulated This is possible with compliant grippers. This work using a compliant gripper [50]. Compliant gripper in progress by our group [98] and will be reported is a special type of deformable structure known in the future. as compliant mechanism. These can be designed for any prescribed manipulation task, for different 11. Closure materials and sizes, and for requisite stiffness. The polydimethylsiloxane (PDMS) grippers shown in An overview of mechanics of microsystems, micro- figure 9b have stiffness as low as 1 N/m and can fluidics, and biological cells is presented in this in fact be designed for lower stiffness than this. paper. We will give a summary of each of them The gripper can be immersed in aqueous medium below. because specific points in it are moved mechani- Microsystems: Microsystems (or MEMS as it is cally using a probe attached to a micropositioner. popularly known) is a mature field today. Its main The forces applied on the gripper during manipula- emphasis is on using solids and deforming them tion are measured using vision-based force-sensing using a variety of force fields to conceive novel sen- technique [50,96]. Here, suitable number of points sors, actuators, and systems. Many of them are on the compliant gripper are monitored visually to available commercially today. In this paper, we MICROMECHANICS OF ENGINEERED AND BIOLOGICAL SYSTEMS 107 highlighted the role of mechanics in simulation and considerably different from those of the traditional design of microsystems. Even though the behaviour mechanics. They call for statistical mechanics and of solids at the micron scale does not deviate much more fundamental physical biophysical approaches from the assumptions made by continuum mechan- vis-`a-vis classical mechanics and continuum ics, modeling of microsystems remains challeng- mechanics. The modeling work in this nascent ing. This is due to the coupling, often strong and field of cellular biomechanics is going in tandem nonlinear, among various energy domains involved. with the experimental techniques. Hence, we took This leads to the challenging problem of solv- a quick look at that aspect too in this paper. ing coupled partial differential equations involv- In summary, this paper provides a broad-brush ing a large number of degrees of freedom. We summary of the state of the three topics as they explained this using the examples of electrostatic- stand today while identifying gaps and challenges elastic and electro-thermal-elastic microsystems. that lie ahead. It is by no means exhaustive. We also noted that dissipation is an important and as-yet incompletely understood issue in microsys- References tems. We then considered two important aspects: synthesis and reduced-order modeling of microsys- [1] Trimmer W S 1997 (ed.) Micromechanics and MEMS: tems. 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