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The Effect of Scaling

There is a mismatch between our macroscopic mechanical sense and what is observed in the millimetre range. Things appear to behave different when they are small. It is hard to understand that and ant can lift masses of his own body weight. This can be physically explained by the concept of scaling1).

Fig. 1: Deflection of a mass on a cantilever beam

An example is shown in figure 1. A cubic mass of size $R^{3}$ is suspended by a beam of length $L$, width $b$ and thickness $d$. Due to gravity, the beam deflects and the mass displaces by a distance $\Delta y$. The mass of the cube is given by $m = R^{3} \rho$ with $\rho$ the density of the cube material (we neglect the mass of the beam for simplicity). The spring constant of the beam is given by2)

\begin{equation} k = \frac{Ebd^{3}}{4L^{3}} \label{eq:SpringConstant} \end{equation}

with $E$ the Young’s modulus. The force on the spring is equal to the force induced by gravity $F = k\cdot\Delta y = m \cdot g$ resulting into

\begin{equation} \Delta y = \frac{mg}{k} = \frac{4g \rho}{E} \frac{R^{3}L^{3}}{bd^{3}} \end{equation}

with $g$ the gravity constant. When reducing the size of the structure of figure 1, which means that each linear dimension is decreased by a factor of for example $0.1$, the mass reduces by the power of three and the spring constant by the power of one. The result is that the deflection $\Delta y$ does not reduce by a factor of $0.1$, but by $0.1^{2}$. Apparently, small structures appear to be stiffer. This is the reason that the body of a butterfly can be supported by legs which are relatively thin with respect to the legs of larger animals.

A more general overview of scaling can be given by considering a certain relevant length $S$ in physical structures. This length can be the length of an arm, the distance of an air-gap or the thickness of a membrane. As seen in the example above, masses of objects are related to volumes and therefore proportional to $S^{3}$. Bending strengths of beams are proportional to $S$ as we have seen from Eq. \eqref{eq:SpringConstant}. The pulling strengths of beams and muscles are commonly proportional to their cross section S2.

Capacitive transducers have the advantage that when reducing the air-gap, the capacitance and plate-to-plate force increases as $S^{-1}$. Very suitable for miniaturisation, even when realising that the capacitance decreases with $S^{2}$ as a function of the plate size. An example is the capacitive microphone which needs an electret at normal scale to supply the hundreds of volts of biasing voltage. At micron scale the operational point can be biased from a simple low-voltage power supply.

For magnetic transducers the situation is worse3). Assuming a constant current density through a wire or coil, the magnetic flux is proportional to $S^{3}$. For permanent magnets the scaling is proportional to the volume $S^{2}$.

More scaling rates are summarised in table 14). From top to bottom the phenomena are more dominant at a larger scale. MEMS devices are typically characterised by phenomena in the $S^{2}$ and lower domains. This means that magnetism, inertial forces and masses are of lesser significance than surface tension, electrostatic forces, friction and diffusion.

Physical phenomenon Scales with size $S$
Capacitor electric field $S^{-1}$
Time $S^{0}$
Van der Waal’s forces $S^{1/4}$
Diffusion $S^{1/2}$
Size/velocity $S^{1}$
Bending stiffness $S^{1}$
Surface $S^{2}$
Thermal loss $S^{2}$
(Muscle) strength $S^{2}$
Electrostatic force $S^{2}$
Friction $S^{2}$
Volume/Mass $S^{3}$
Inertia $S^{3}$
Magnetism $S^{3}$
Tab. 1: Scaling laws

However, table 1 does not say exactly at what dimension a certain phenomenon becomes prevailing. This depends on all specific geometries and material properties. For certain phenomena there are some guidelines. In the field of microfluidics, the dimensionless Reynolds number is an indicator of whether flow is laminar or turbulent5). For large Reynolds numbers, convective and inertial forces dominate as we are used to with large objects in water. For small Reynolds numbers, on the other hand, viscosity is so large that transport of, for example, heat in the medium depends on diffusion rather than on convection. This will be the case in channels in the micron range.

Although time does not scale with size as a first order approximation, a smaller device will have a larger throughput. Microsystems will be faster in their response and consume less analyte in case of chemical systems. Diffusion based transport enables quick responses without the need for mechanical convective systems. This has resulted in static micromixers without moving parts6). Due to the fast diffusion processes, thermal and electrochemical actuation are new options for the creation of mechanical actions7). Another example of an application profiting from fast diffusion due to downscaling is the amperometric ultramicroelectrode8).

Sensor Technology TOC

These are the chapters for the Sensor Technology course:

1) , 5)
Marc Madou, Fundamentals of microfabrication, the science of miniaturization, second edition (CRC Press, 2002)
2) , 3)
James M. Gere and Stephen P. Timoshenko, Mechanics of Materials, 3rd edition (PWS-KENT Publishing Company), 1990
Note that some scaling powers are arguable depending on the configuration. For example, a force due to a magnetic flux can be said to scale with $S^{3}$, although the magnetic force between two current carrying wires scales with $S^{4}$
H. Möbius, W. Ehrfeld, V. Hessel and Th. Richter, Sensor controlled processes in chemical microreactors, The 8th International Conference on Solid-State Sensors and Actuators, Transducers ’95 and Eurosensors IX, Stockholm, Sweden, June 25 - 29, 199, p. 775 - 778
Cristina Neagu, A medical microactuator based on an electrochemical principle, Thesis University of Twente, 1998
Werner E. Morf and Nicolaas F. de Rooij, Performance of amperometric sensors based on multiple microelectrode arrays, Eurosensors X, Leuven, Belgium, September 8-11, 1996
theory/sensor_technology/st9_scaling.txt · Last modified: 2017/10/10 18:38 by glangereis