What can we learn from scaling?Christian Schonenberger

Institute of Physics and Center of Nanoscience (NCCR),University of Basel, Klingelbergstr. 82, Switzerland

(Dated: October 22nd, 2002)

I. INTRODUCTION

Any physical system obeys some laws, the accelerationof a body is determined by the its mass and the forceacting on the body. The corresponding law, New-tons law, applies to many systems of dierent size,i.e. to the motion of planets within our stellar sys-tem, to human beings, to a tennis ball, and withinsome limitations, even to the motion of a collection ofatoms. Not every law remains valid over a large spanof length scales. Nonetheless, one can learn a lot byasking how dierent quantities scale, i.e. how a qual-ity such as the mass, velocity, etc. depends on the sizeof the object under consideration. As we go form themacroworld, where physical laws are known, down tothe nanoworld or even atomic scale, the scaling ap-proach is useful to understand the very new problemswe may be facing at the smaller size regime. Here, Iwould like to illustrate scaling using examples primar-ily from mechanics.

II. DEFINITION OF LENGTH SCALING

Consider a ping-pong ball having diameter d and atennis ball having diameter D. Assume that the ten-nis ball is 2-times bigger than the ping-pong ball, sothat D = 2d. We may now ask, for example, howmuch bigger the volume of the tennis ball to that ofthe ping-pong ball is. Sure, it is 8-times bigger, or 23.If the tennis ball were k times larger in its linear di-mension, the volume would be k3 larger. We say thatvolume scales as the third power of its linear dimen-sion. In scaling we compare always similar lookingobjects, i.e. both the tennis ball and the ping-pongball are spheres. To retain the similarity all lengthsare scaled, either blown up or squeezed down, simi-larly. In the following we will be using the letter L todenote a characteristic length of the object, for exam-ple the diameter or radius of the balls. We then saythat volume scales as L3. We use the notation:

volume = V L3 (1)Similarly, we would write for the surface area

area = A L2 (2)

III. THE HUMAN AS A MACHINE

How does the power of a human scale with its size?

As a reminder power P is energy E per time t. Theavailable energy in our body scales with its volume.However, our body cannot deliver power that scales asL3, since this would contradict some basic laws fromthermodynamics. Any machine operating close tothermodynamic equilibrium has an eciency smallerthan one. Consequently, some of the stored energyhas to go into heat and cannot be used to performwork. So our body can heat up for some time, butthis cannot go on for ever. If we go jogging we rstheat up to reach some equilibrium. From now on,the heat generated in our body must leave our body.Since this heat release is via the skin, it scales as thesurface. Therefore the (equlibrium) power of a human(animal) scales as:

power human L2 (3)I note, that the peak power can scale with the L3, butonly for a relatively short instance.

How often do we have to eat or drink?

The amount we eat scales with the volume, i.e. thesize of our stomach. Therefore, the added amount ofenergy Efood scales with the volume too. After a bigdinner we have to digest for a period Tdigest. SinceEfood = PTdigest, we nd for the scaling of Tdigest:

period without supplied food L (4)We may also talk about the rate (or frequency) ofeating, which is the inverse of the respective time:

rate of eating L1 (5)

We can learn quite a lot from these simple equations.For example, an animal living in the desert has to waitfor a long time to nd food, hence, it is better big!

If we compare a baby with an adult, the baby neces-sarily needs to eat more often (has a larger frequencyof eating). That is why parents need to get up during

2the night to feed their babies. A y is in search offood all the time.

If we look into the interior of our body, their or mil-lions of nanomchines (proteins) working. It is obviousthat these nanomachines can only function if food isaround all the time. They are imbedded in a soup offood, called ATP.

IV. FORCE AND ACCELERATION

The gravitational force on earth is given by FG = mg,where g is the gravitational acceleration and taken tobe a constant. Therefore,

weight L3 (6)

However, most forces do scale dierently, not withthe third power of L. The reason is that the size ofstructures, for example the steel bars in a bridge, thediameter of our legs, are optimized according to thestrength of the material. A bone (or a piece of steel)can sustain forces up to a maximum stress. Stress isdened as the force per area. Therefore, structuralforces scale (for constant stress) with the area (thecross section):

structural force L2 (7)

mgmg/4

d

Have a look at the gure. Taking four legs, each onehas to carry a force of mg/4, where m is the mass ofthe animal. Since stress is taken to be constant, wehave mg/d2 = const. Therefore:

leg diameter L3/2

Again, we can learn a lot, both for the large and smallscale: big animals have over-proportional legs, smallones have very thin legs.

Going down to the ultimate smallness, i.e. molecu-lar biology, the concept of legs does not make senseanymore. As you know, cells have no legs.

What about acceleration. Galilei found that all ob-jects accelerate exactly identical. Hence, for gravita-tional forces, the acceleration a is scale independent.This is not true for structural forces. For example, ifyou ask how fast you can accelerate (or be acceleratedin e.g. an accident) without breaking your bones, theanswer is:

maximum acceleration L1 (8)

As a consequence, small objects can accelerate fast,large ones cannot. This is the reason why we havediculties to hit a y. Because of its small size the yaccelerates much too fast for us to react.

V. HOW FAST: SPEED AND FREQUENCY

Simple mechanics tells us that the frequency of anoscillator is given by f =

k/m, where k is the

spring constant. This equation does not yet help ustoo much, because we do not know the scaling of thespring. If we add n similar springs with spring con-stant k in series, the eective new spring constant isreduced to k/n. In contrast, if we add the springs inparallel, the new spring constant is enlarged to nk. Apiece of exible material can be considered as a col-lections of springs. If we think in terms of a rod thenumber of springs in series scales as the length of therod. The number of springs in parallel, on the otherhand, scales as the cross section, i.e. as L2. The ef-fective spring constant of a three-dimensional objectscales as L2/L:

spring constant L (9)

It follows from this equation that the frequency scalesaccording to:

frequency L1 (10)

Since the frequency is the inverse of a characteristictime (for a harmonic oscillator the period of motion),the velocity v scales as v a/f , where a is the accel-eration and f the frequency. Taking what we knowfrom before (equ. 8), we are let to conclude that:

velocity L0 (11)

(I note here, that this equation is only valid, if there is no

friction. We will have a look at friction below.)

It is a good moment to check consistency. Energy isgiven by work, which is force times distance. If youlook at the rate of energy/work per unit time, we getwhat we call power. Hence, power equals force timesvelocity: P = Fv. As F L2 and v L0, we ndP L2 as before.

3We learn that the dynamics, determined by the fre-quency, is fast on the small scale, but net velocitiesare scale independent provided there is no friction.We know from our own experience, that this scaling iswrong if friction is determining the motion. A featherfalls much slower in air than a bottle of wine (if un-fortunately it slips out of our hands).

VI. FRICTION

In Physics-I we learn that the friction force FR isgiven by FR = FG, where is the coecient offriction (a material parameter). This law predictsFR L3. Though this law is correct, it fails on thesmall scale (micrometer and nanometer). Friction isdue to the adhesion between bodies in contact. Ittherefore should be determined by the area of con-tact. The crucial point to realize is that all bodiesrest on three legs, even if there are four as in case ofa table. Hence, friction is determined by a few pointswhere contact is established. Using continuum me-chanics, one can show that the area of contact scaleswith the body weight. This is the origin of the abovelaw. If we focus on a single contact, or assume thatthe whole surface is in true contact, which is possibleby using a lubricant, the friction force scales with thearea of contact:

contact friction L2 (12)

(This scaling law can also be obtained by saying that shear

stress should be scale-independent. Since shear stress is

given be the shear force - the frictional force - divided by

the area of contact, Fshear L2.)

Consider now a viscous uid in which the object isembedded. Then, basic physics laws due to Newtontell that the frictional force is determined by the ve-locity gradient. More explicitly, shear stress is pro-portional to the velocity gradient. Hence, a constantshear stress, typical velocities scale as:

velocity with friction L1 (13)

This is indeed dierent to the scale-independence ofvelocity without friction.

If the movement of objects is determined by friction,the velocities are getting small in small objects. Weknow that very small grains of sand, or better dust,can stay in air for a long time. Their velocity relativeto the surrounding air is virtually zero. Dust from avolcano eruption can be spread over a whole continent.

VII. POWER OF A GENERIC MEN-MADEMACHINE

The following gure represents our generic machine.Water streams through a pipe with cross-section A at

v

h

velocity v. For the density of water we use the symbol. Per unit time, the pipe delivers mass, determinedby Av. This mass falls down, attracted by earthgravitational acceleration g, to keep the rotating wheelin motion. This wheel can deliver mechanical power,which is given by the eciency of the conversion (aconstant) times the gravitational energy, determinedby the height h. Hence, we have P = ghm = ghAv L3, provided v can be taken as constant. Therefore,

machine power, no friction L3 (14)

This is equation holds if friction can be neglected. Un-der this assumption we have learned that the velocityis scale-independent. Usually, this is true for large ma-chines, the ones we human tend to build. It is moreconvenient to divide the power by the volume of themachine, i.e. to look at the power density:

machine power density at no friction L0 (15)

Hence, the power density is scale-independent. Note,however, if it comes to the exchange of energy, thesituation is dierent, as I have tried to explain in therst section. Exchange of energy (and information)can at most scale with L2 (the area).

Let us look at small machines, where friction sets thelimit on velocities. Since v L1, we have:

machine power density, friction limited L1 (16)

The whole molecular bio-machinery is small (actuallyvery small) and functions in a liquid environment inwhich velocities are limited by friction. Hence, thisequation predicts that bio-machines must be ine-cient. This appears as a contradiction, because weare taught that biology has had millions of years toperfect itself!?

4VIII. POWER OF A MOLECULARBIO-MACHINE

Molecular motors work dierent to men-made ma-chines. They operate close to thermal equilibrium.As already mentioned, food in the form of APT isaround each bio-machine in excess. The time neededto capture the food is determined by the thermal dif-fusion time d. If the protein (the molecular machine)is of linear size L, the diusion time scales as:

diusion time L2 (17)

The rate of uptake of APT is not only determined bythe respective time scale but also by the number ofAPTs that can bind to the protein. The latter scaleswith the surface, i.e. with L2, too. The power istherefore, scale invariant:

power density molecular machines L3 (18)

This is the solution of the above problem. Nature hasfound a way to design machines with very large powerdensities at a small scale. This scaling-law predicts,that the smaller the better. Indeed, bio-machines arevery small proteins.

Very impressive is an absolute comparison. A men-made machine typically delivers a peak power of or-der 1MW/m3. A motor protein can deliver a forceof order 1 nN at a velocity of 10m/s. This yieldsa power of 10 fW. Taking for the volume a size of10 nm, yields for the power density the amazing valueof 10GW/m3. (I hope, I have not done anythingwrong). This number reects what potentially couldbe the peak power. This is however (may be alsofortunately) never reached. The equilibrium powerof a human amounts to approximately 1 kW/m3 andpeak powers (for a very short instant) come close to1MW/m3.

IX. EXERCISE: JUMP HEIGHT

Try to nd the scaling-law for the jump height againstgravitational acceleration. Do this in an explicit man-ner using d and l to denote the diameter and lengthof the legs, and V to denote the volume of the body.You should arrive at the following expression:

jump height ld2

V L0 (19)

Hence, the jump height is scale invariant. A grasshop-per can jump as high as we can do. Sure, such state-ments have to be taken with caution. A beetle cangenerally not jump as high.

X. EXERCISE: VELOCITY AGAINSTGRAVITATION

The question is: Who can run faster uphill, a small ora large person? Assume that we are asking not for thepeak speed, i.e. the one that is possible for a short pe-riod, but the steady-state velocity uphill. To answerthis question, you have to express the power at a givenvertical velocity for a body of mass m against gravi-tational acceleration g. Since we ask for the steady-state, the power scales as L2. The answer to thequestion is:

running velocity against gravitation L1 (20)

XI. COMPARISON

The following table compares scaling for the mechan-ical power:

power densities scaling

equilibrium L2

peak, no friction L0

peak, (viscous) friction L1

peak, molecular machines L3

The table illustrates that scaling exponents depend onthe underlying physics and can vary quite a lot.

XII. A BIT ELECTRONICS

Scaling laws are particularly important in the eldof integrated circuit (IC) technology. As you know,each generation of a memory chip contains more bits.This is only possible, if the size of the electrical el-ements (transistors, capacitors and resistors) scaledown. How they scale determines design rules.

We have started the mechanical part above, by notingthat the scaling of structural forces is determined bythe strength of the material. The same is true in elec-tronics, in which the electrical eld strength has torespect a maximum value beyond which there is whatengineers call a breakdown. Hence, the electrical eldE is taken to be scale-independent:

electric eld L0 (21)Because voltage V is eld times length:

voltage L1 (22)

5The mechanical force F acting, for example on a ca-pacitor plate, is given by the eld times the area ofthe plate:

force L2 (23)Next, we look at Ohms law for the resistance R =l/A, where is the specic resistivity (a materialconstant), l the length of the resistor and A its cross-section. Therefore:

resistance L1 (24)Ohms law says that the current I is given by V/R:

current L2 (25)For the electrical power P = IV we nd:

power L3 (26)

These equations are very nice, because both the cur-rent density (current per unit area) and the powerdensity (power per unit volume) are scale invariant,provided the assumption of constant electrical eldholds. If these equations would hold circuit engineer-ing of new smaller chips would be simple. However,all scaling laws have limitations. Today, one of theforemost problem in this eld of research is the scal-ing of the resistance. Microfabricated interconnectsare very thin, so that the wire thickness can hardlybe scaled anymore. Then, R L0 and I L. Conse-quently, the current density will scale as L1 whichis disastrous. If the current density increases beyondsome threshold the wires are destroyed!

There are many more problems of this kind. In partic-ular quantum mechanical tunneling sets limits on thethickness of gate oxides. Yet another problem is foundin the electrical charge, which is used to store the in-formation. Charge scales as L3. Hence, it rapidlygets very small. If downscaling will go on, a storagecapacitor may store only one single electron. At thisultimate limit, scaling must break down.

Without going into details, the future of electronicsmust consider the quantized nature of matter, the factthat charge is a multiple of single electrons and thefact that localization (to store some charge at one spotin space) is made impossible by quantum-mechanicaltunneling.

At the atomic scale, physical laws are very dierent,sometime surprisingly dierent. I mention only tworecent phenomena:

A guitar string vibrates with some tone. If we increasethe tension, the tone gets higher. If we do the sameon a string of atoms, i.e. four atoms in a row, theoutcome is reversed. If you pull on the string, thevibration frequency is decreased!Ohms lay tells that the electrical resistance increaseswith length. How about single atoms. The resistanceof a single atom can be the same as that for twoatoms in series. The two atom resistance is not twicethe one of a single atom!

XIII. CONCLUDING REMARKS

This lecture serves to demonstrate that scaling-lawscan be used to understand the distinct dierence inbehavior of similar objects at dierent size. We haveused simple physical laws from classical physics. Wemay use these laws to extrapolate from macrophysicsto microphysics. We should however never forget, thatall laws have a range of validity. Scaling laws are greatto get some insight, but should always be used withcaution. In the eld of nanoscience, the breakdown ofclassical scaling is set by quantum mechanics and thegranularity of matter (single electron, single atom),so that the laws of large numbers (statistical physics)cannot be used anymore. Actually, the breakdown ofclassical physics is what makes nanophysics an excit-ing area of research for physicists. Nanophysics couldactually be dened as such.

Electronic address: Christian.Schoenenberger@unibas.ch;www.unibas.ch/phys-meso