# Scaling Law I. BioSensing & BioMEMS 530/ Jeff Wang Johns Hopkins University 1

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1 Scaling Law I Jeff Wang Johns Hopkins University 1

2 Why scaling is important at the micro and nano scale Micro and nano devices are orders of magnitude smaller than macro counterparts As the sizes shrink, different physical forces become more important or less important depending on their nature These changes dictate how the devoice must be built and what forces it will use to operate. This issue is called scaling. Jeff Wang Johns Hopkins University 2

3 Size and Shape In nature, the differences in sizes (for animals and plants) are also accompanied by the difference in shapes Dik-Dik (5 Kg) Dog Buffalo (500 Kg) Rhino Elephant Jeff Wang Johns Hopkins University 3

4 Allometric Formula (Power Laws) a y bx (power-law formulas) simply compare the relation of two measurements, ignoring all the complexities and subtleties of detailed changes in form. x and y are relative sizes of two parts in an organism a and b are constants What is the scaling law for the leg size (d) relative to the body size (L) (i.e. aspect ratio d/l)? Jeff Wang Johns Hopkins University 4

5 Isometry (geometric similarity) Size change in geometrically similar living bodies has numerous effects and profound biological significance. Some of them are dependent on the length dimension (l), but most of them are dependent on the length dimension square (l 2 ), and a few are dependent on the length dimension cube (l 3 ) Length (l) the most obvious things that vary linearly are height or length (e.g. Height in trees, leg length in running animals) Surface tension is powerful to keep a small wax-coated paper clip floating at the surface of a dish of water, but a larger paper clip sinks. A water strider depend on the forces of surface tension to support its weight. Small insects of a suitable design can walk on water; while a large insect of the exactly the same design could not. Jeff Wang Johns Hopkins University 5

6 Length Square (l 2 ) The exchange of gases (e.g. oxygen and carbon dioxide) occurs through surfaces of the lungs, or gills or leaves (in plants) In animal, the food is assimilated through the surface of the gut Heat loss takes place through the skin The strength of a bone or a muscle, or a insert structure of a tree is a function of its crosssectional area In locomotion, the drag of a swimming animal is, under certain conditions, proportional to its wetted surface area Length Cube (l 3 ) The most significant factor is mass Energy generation of an organism The larger the animal or plant, the greater the significance of gravity The buoyancy scales as l 3 means that a scaled-up floating body floats just as well as a smaller one. Most mammals float with about the same fraction of their bulk out of the water. At the right is a floating sea otter, left are hippos Jeff Wang Johns Hopkins University 6

7 Why are there no mice is the arctic? Mice and other smaller animals, which have a high surface-to-volume ratio, often struggle to retain sufficient heat. Jeff Wang Johns Hopkins University 7

8 Example 1: Racing Shells (the dependence of velocity on the length of the racing shell?) b l l/b ~ constant boat mass per oarsman~ constant Water drag : >90 % by skin drag at this speed (another energy loss is wave generation) P req = Drag Velocity = k 1 (l 2 V 2 ) V P avail = k 2 (# of oarsmen,n) l 3 V l 1/3 n 1/9 Jeff Wang Johns Hopkins University 8

9 Average speed a 2000-m course for rolling shells seating one, two, four, and eight oarsmen. Data from 1964, 1968 Olympics, and 1970 World Rowing Championships. Jeff Wang Johns Hopkins University 9

10 Example 2: Weight Lifting (What s the dependence of the maximum weight a human can lift on his/her body weight?) According to the rule of isometry, for geometrically similar animals of the same mass density The body mass m l 3 ; The surface area l 2 ; Weight surface area (mass) 2/3 World weight-lifting record. The weight lifted (w T ) is precisely proportional to the 0.67 power of body weight Jeff Wang Johns Hopkins University 10

11 Microbial Movement Small -> Avoid most problems related to gravity and inertial (New problems:) Molecular cohesion (including surface tension) Low Reynolds numbers A world of low Reynolds number Re ~ Inertial force Viscous force (e.g. Re (bacteria) ~ 10-6 ) Stopping Distance The distance of gliding after propulsion stops Pelican vs. Fly fishing The stopping distance (gliding distance) is ~ 1/18 (Re D) D: diameter of an object e.g. a 2 m bacteria at Re ~10-6, the stopping distance is m. Clearly, bacteria don t move with the thrust-and-glide motions, characteristics of fish, some of which glide more than five body lengths between propulsive strokes. Jeff Wang Johns Hopkins University 11

12 Reversibility Video Another import thing about swimming at low Reynolds number is reversibility. Reversibility of a low Reynolds-number flow. The narrow gap between the outer and the inner cylinders are filled glycerin. A blob of dyed glycerin is introduced. When the inner cylinder is turned in a direction the dye is mixed. When the cylinder is turned in the opposite direction the dye goes back to its original position. If a ciliated protozoan (~ m) makes a motion with its cilia and then reverses those motion exactly, what happens? How does a small organism swim? Jeff Wang Johns Hopkins University 12

13 Ciliary Propulsion Mechanism : like a flexible oar Organisms using ciliary propulsion with size 20 m 2 mm Regardless of organism size (including lung), typical length of cilia : 5-12 m at Hz Speed : ~ 1 mm/s Swimming at the the low Reynolds number. A flexible oar remain rigid on the thrust stroke but bends significantly on the return stroke. Jeff Wang Johns Hopkins University 13

14 Flagellar Propulsion Mechanism : flagella function by means of the propagated waves generated 2-D wave : like a wavy line (generated by beating the flagellum) 3-D wave : like a corkscrew (generated by rotating the flagella, acting like a rotating propeller ) Flagella : typically m in diameter, m in length Speed : 0.1 to 0.2 mm/s, regardless of their size, which is only about 1/10 the swimming speed of ciliates. Human sperm cells. Each consists of a head attached to a whiplike tail, the flagellum. They swim by beating their tails from side to side. Motions of sperm cell from a sea urchin Jeff Wang Johns Hopkins University 14

15 Insect Flight The flight muscles of insects are among the most active animal tissues. (transport of oxygen is through the air-filled tubes, or tracheae, instead of blood) As small as 1 g with a wingspan of ~ 200 m. The Reynolds numbers of hovering insects : ~ 1 to 10 4 Separation bubble The bubble acts if it were part of the wing, guiding the air around it in the proper way to reduce drag and maintain lift For airplane wing, it is ineffective when Re is below 10 5 Dragonfly wing : (two horrible-looking) pleats Increase Strength (like the corrugation design in a piece of cardboard); poor aerodynamic properties? Lift-to-drag ratio The ratio of aircraft wing at the low Re : ~2 (it s ~ 30 at high Re) The ratio of dragonfly wing at the low Re: 12 Jeff Wang Johns Hopkins University 15

16 A World Governed by Molecular Adhesion Large organisms ruled by inertial and gravity Miniature organism force and attraction between molecules ( a variety of phenomena including surface tension ) rule Surface Tension The molecules on the surface tend to be pulled away from the surface, and therefore work must have been done to bring the molecules from the body of the liquid to the surface. liquid air liquid Surface tension occurs The work required per unit area to bring molecules to the surface (I.e., to create a new surface) is called surface tension. Surface : work/area it s dimension :force/length Young & Lapalce equation 1 1 P ( ) R 1 R 2 P: pressure difference :surface tension R: radius of a surface Jeff Wang Johns Hopkins University 16

17 Surface Tension and Gravity Design of a liquid-storage tank Not required to endure bending stresses, only tensile stresses Uniform tension per length no one region is likely to rupture than another Size of a droplet d F surface tension W D 3 d D/d (1/d) 2/3 D Jeff Wang Johns Hopkins University 17

18 Cold Add detergent Hot Which way the boat moves? Surface tension can be adjusted by 1. Surfactants : surface tension is reduced with the addition of surfactant 2. Temperature: surface tension decreases as temp increases 3. Electric potential : a potential can alter (mainly decrease ) surface tension Jeff Wang Johns Hopkins University 18

19 Comparison of Inertial force and Cohesion Inertial L 3 Cohesion L 1 (surface tension and molecular adhesion) Beetle or Gecko on the Wall The importance of gravity and cohesion are roughly equal at the size of ~ 1mm (e.g. fly can walk on wall) A beetle has hairlike protuberances that allows close contact with rough surfaces The toe pad of a gecko is constructed of a series of waves. Each wave is made of of tufts, like the tufts of a carpet Jeff Wang Johns Hopkins University 19

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