Scaling law Professor Huang

Maximum animal size limit As objects shrink, the ratio of surface area to volume increases, rendering surface forces more important. Land animals fighting the resistance of gravity : African elephant at 3.8m For creatures of the sea without gravity barrier: a whale with 20m long

Minimum animal size limit The heat loss from a living creature is proportional to its surface area (l2), and the rate of compensating heat generation through eating is proportional to its volume (l3). A warm-blooded animal can not be smaller than a shrew or a hummingbird. Insects has no problems of heat loss by being cold-blooded.

Scaling and surface tension Capillary forces are caused by surface tension. The mass of a liquid in a capillary tube, and hence the weight, scales l3 and decreases more rapidly than the surface tension, which scales with l, as the system becomes smaller. That is why it is more difficult to empty liquids from a capillary than to spill coffee from a cup.

Walking on water A 10mg mosquito-sized insect needs a 1mm of total edge to be supported by surface tension A 60-kg man need 8000m of foot edge.

The effort of jump Effort E ~ m x h, m(mass), h(height)

Scaling and Diffusion For a spherical molecule, the diffusion coeff., D(m2/s), is given by: D=kT/(6 πηr) η:absolute viscosity (kg/m.s) r: hydradynamic radius According to random walk equation, the diffusion length, x, of a molecule in solution is given by: x=(2Dτ)0.5 or τ=x2/(2D) A small drop liquid can help diffusion

Mixing Mixing is very fast at the micro level and should allow for reaction times to be determined by inherent kinetics rather than the time it takes for reactant species to meet in solution. Mixing small amounts of fluid in a large set of these parallel micro reactors leads to a much higher mixing and reaction efficiency than in a big reactor.

In nature, only the smallest animals rely on diffusion for transport; Animals made up of more than a few cells cannot rely on diffusion anymore to move materials within themselves. They augment transport with hearts, blood vessels, pumped lungs, digestive tubes, etc.

Scaling of Minimal Analytical Sample Size V=1/(aNC) a: sensor efficiency between 0 and 1 N: 6.02x1023 mol-1 C: concentration of analyte (moles/L) log V+ logC = log(aN) = constant

Fundamental frequency f0=t(E/ρ)0.5/(4πL2) L=100,W=3,t=0.1, Si ,12KHz L=0.1,W=0.01, t=0.01, 1.2GHz In NEMS, mechanical devices almost as fast as today’s electronic devices

Quality factor Q=ω0M/b=(KM/b)0.5 b: damping For an electrical system Q= ω0L/R =(L/C)0.5/R A poly-SiC lateral resonator at <0.01mTorr produce Qs>100,000 In NEMS, Qs=1000-100000 in moderate vacuum

Trimmer’s vertical branket notation An important bottleneck in the development of microactuators in general is that resistive forces, such as viscous drag(l2) and surface tension(l), exceed motive forces [mass(l3)xsurface(l2)] in the micro domain. Miniaturization engineers look for driving forces that scale more advantagously such as electrostatics(l2) and capillary force(l). What we need is bigger net force when miniaturization since the loss mechanisms may scale in the same way or become even more important in the micro domain.

Electrostatics E=-CV2/2 = -εwvV2/(2d) E~l0l1l1 (l1)2 / l1 = l3 Fx=-∂F/∂x= l3 / l1 = l2