6.1 Introduction to Scaling Scaling theory is a value guide to what may work and what may not work when we start to design the world of micro.

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1 Chapter 6 Scaing Laws in Miniaturization 6. Introduction to Scaing Scaing theory is a vaue guide to what may work and what may not work when we start to design the word of micro. Three genera scae sizes: (a) Astronomica ( 天體的 ) objects; (b) Macro-objects; (c) micro-objects. - Things effective at one of these scae sizes often are insignificant at another scae size. - Exampes: Gravitationa forces dominate on an astronomica scae (e.g., the earth

moves around the sun), but not on smaer scaes. Macro-sized motors use magnetic forces for actuation, but micro-sized ones usuay use eectrostatic fieds instead of magnetic.

2 moves around the sun), but not on smaer scaes. Macro-sized motors use magnetic forces for actuation, but micro-sized ones usuay use eectrostatic fieds instead of magnetic. (Reference: MEMS Handbook, edited by Mohamed Gad-e-Hak, CRC Press) Two types of scaing aws:. The first type: depends on the size of physica objects.. The second type: invoves both the size and materia properties of the system. 6. Scaing in Geometry Surface and voume are two physica quantities that are frequenty invoved in micro-device design. - Voume: reated to the mass and weight of a device, which are reated to both mechanica and therma inertia. (therma inertia: reated to the heat capacity of a soid, which is a measure of how fast we can heat or coo a soid. important in designing a therma actuator) - Surface: reated to pressure and the buoyant forces in fuid mechanics, as we as heat absorption or dissipation by a soid in convective heat transfer. Surface to voume ration (S/V ratio) - S ; V - S / V - As the size decreases, its S/V ratio increases. - Exampes S/V ratio of an eephant ( -4 ) vs. of a dragonfy ( - )

3 An eephant and a fea have ces of about the same size. Too arge a ce wi not have enough surface for substance exchanges with its surroundings to support the active metaboism ( 新陳代謝 ) within, uness it is highy eongated ike a vertebrate nerve ce, increasing the S/V ratio. (Biochemistry by Mathews et a.) igure.8: Range of sizes of objects studied by biochemists and bioogists. (Biochemistry by Mathews et a.) Eukaryotes - Organisms whose ces are compartmentaized by interna ceuar membranes to produce

4 a nuceus and organees. Lipids - 脂質, 脂 6. Scaing in Rigid-Body Dynamics 6.. Scaing in Dynamic orces at t v s + (6.) By etting, v t s a (6.4) ) )( ( t t sm Ma (6.5) 6.. The Trimmer orce Scaing Vector Trimmer (989) proposed a unique matrix to represent force scaing with reative parameters of acceeration a, time t, and power density P/V. orce scaing factor: 4 ] [ (6.6) Acceeration a: from Eq. (6.5), 4 ] [ ] ][ [ a (6.7) Time t: from Eq. (6.5), ]) ][ ][ ([ sm t (6.8) 4

5 Power Density P/V : from Eq. (6.5), W s Since P, thus t t P s (6.9) V tv P V ([ [ ][.5 ][ ][ ]) [ ] ].5.5 (6.) 6.4 Scaing in Eectrostatic orces In ig. 6.4, the eectric potentia energy induced in the parae pates is: U CV ε rε WL d V (.7) Breakdown votage - The votage required to initiate discharge. - or d > µm, V (see ig. 6.5) 5

6 ( )( )( )( )( ) U (6.) - A factor of decrease in inear dimension wi decrease the potentia energy by a factor of. In ig. 6.6, the eectrostatic forces are, U d -,, and ( ε ε WLV d r d (.8) W L U ε rε LV W d (.) U ε rε WV L d (.) d W L ) - A times reduction in the pate sizes means a times decrease in the induced eectrostatic forces. 6.5 Scaing in Eectromagnetic orces In this section, it is shown that eectromagnetic actuation is not scaed down 6

7 neary as favoraby as eectrostatic forces. - The eectromagnetic forces can be induced in a conductor or a conducting oop in a magnetic fied B by passing current i in the conductor. - The eectromotive force (emf) is the force that drives the eectrons - through the conductor. φ U (6.) L where φ is the magnetic fux, and L is the inductance. - Since φ Li, Li U (6.4) - The induced eectromagnetic force woud be U x - or constant current case, φ cons tan t i cons tan t (6.5a) U (6.5b) x i L x Since i and L / x, 4 - If times reduction in size () Eectromagnetic force:, times reduction Comparison: Eectrostatic force: ony times reduction Concusion: Eectromagnetic force is ess favorabe in scae-down than Eectromagnetic force. 6.6 Scaing in Eectricity Exampes: Microsystem actuation by eectrostatic, piezoeectric, and therma resistance heating. Eectric Resistance: L A R ρ (6.8) where ρ, L, and A are the resistivity, ength, and cross-sectiona area, respectivey. Eectric Power Loss: P V R (6.9) 7

8 where V is the appied votage Eectric fied energy density: u ε E (6.) where the dieectric permittivity ε, and the eectric fied E. Exampe: or a system that carries its own power, the avaiabe power E av. P. (6.) E av - That is, a times reduction of eads to times greater power oss due to the resistance increase. - Disadvantage of scaing down of power suppy systems. 6.7 Scaing in uid Mechanics In ig. 6.7, moving the top pate to the right induces the motion of the fuid. - Newtonian fow: dθ τ, or dt dθ dv τ µ µ dt dy where τ: shear stress; μ: coefficient of viscosity ( 黏滯性 ); dθ/dt: strain rate; V: fuid veocity. - τ Thus, µ R s (6.) where R s V max /h - Rate of voumetric fuid fow: Q A s V ave (6.) where A s : cross-sectiona area for the fow; V ave : average veocity of the fuid. 8

9 Renods number: ρvl Re µ where ρ: fuid density; V & L: characteristic veocity and ength scaes of the fow. - Re (inertia forces)/(viscous force) - Macro fows: high inertia forces high Re turbuence fow - Micro fows: high viscosity ow Re aminar fow p.s.: () turbuence fow: fuctuating and agitated; () aminar fow: smooth and steady; () transition from aminar to turbuent: ~ 5 (from Micromachines: A New Era in Mechanica Engineering, by Iwao ujimasa, Oxford University Press, 996) In ig. 6.8, with the pressure drop ΔP over the ength L, the rate of voumetric fow of the fuid is (Hagen-Poiseuie aw), πa 4 P Q (6.4) 8µ L 9

10 Q - With V ave π a, 8µ VaveL P (6.5) a 4 - Thus, Q a ; P a L 6.8 Scaing in Heat Transfer 6.8. Scaing in Heat Conduction ( 傳導 ) Scaing of Heat ux Heat conduction in soid is governed by the ourier aw, T ( x, y, z, t) q x k x where q x : heat fux aong the x axis; k: therma conductivity of the soid; T(x,y,z,t): temperature fied. Rate of heat conduction: or soids in meso- and microscaes, ( )( ) Q T Q qa ka (6.7) x That is, reduction in size eads to the decrease of tota heat fow. Scaing in Submicrometer Regime In the submicrometer regime, the therma conductivity is, k cvλ (6.8) where c, V, and λ are specific heat, moecuar veocity, and average mean free path, respectivey. - Thus, Q ( )( ) (6.9) - A reduction in size of woud ead to a reduction of tota heat fow by.

11 Scaing in Effect of Heat Conduction in Soids of Meso- and Micro-scaes A dimensioness number, caed the ourier number, is used to determine the time increments in a transient heat conduction anaysis. αt L - where α: therma diffusivity of the materia, and t: time for heat to fow across the characteristic ength L. - t L α 6.8. Scaing in Heat Convection ( 對流 ) Heat transfer in fuid is in the mode of convection (Newton s cooing aw), Q qa ha T (6.) where Q: tota heat fow between two pates; q: heat fux; A: cross-sectiona area for the heat fow; h: heat transfer coefficient; two points. T : temperature difference between these - h: depends primariy on the fuid veocity, which does not pay a significant roe in the scaing of the heat fow. - Thus, in meso- and micro-regimes, Q A or the cases in which gases pass in narrow channes at submicro-meter scae, The cassica heat transfer theories based on continuum fuids break down. The seemingy convective heat transfer has in fact become conduction of heat among the gas moecues as the effect of the boundary ayer becomes a dominant factor. In ig. 6.9, H < 7λ where λ 65nm for gases, and. μm for iquids.

12 λ ρ k cvλ V 8kT πm where T: mean temperature of the gas; and m: moecuar weight of the gas. Effective heat fux: k T q eff (6.4) H + ε where T : temperature difference between two pates; ε: depends on the gases entrapped between two pates,.4λ<ε<.9λfor air, O, N, CO, methane, and He, and ε.7λwith H>7λfor H.

Elements of Kinetic Theory

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