Guest Writer - Gastautor - Gast Schrijver

Scaling Laws--Back to Basics

 

Scaling laws are extremely simple observations about how physics works
at different sizes. A well-known example is that a flea can jump dozens
of times its height, while an elephant can't jump at all. Scaling laws
tell us that this is a general rule: smaller things are less affected by
gravity. This essay explains how scaling laws work, shows how to use
them, and discusses the benefits of tinyness with regard to speed of
operation, power density, functional density, and efficiency--four very
important factors in the performance of any system.

Scaling laws provide a very simple, even simplistic approach to
understanding the nanoscale. Detailed engineering requires more
intricate calculations. But basic scaling law calculations, used with
appropriate care, can show why technology based on nanoscale devices is
expected to be extremely powerful by comparison with either biology or
modern engineering.

Let's start with a scaling-law analysis of muscles vs. gravity in
elephants and fleas. As a muscle shrinks, its strength decreases with
its cross-sectional area, which is proportional to length times length.
We write that in shorthand as “strength ~ L^2.” (If you aren't
comfortable with “proportional to,” just think “equals”: “strength = L
squared.”) But the weight of the muscle is proportional to its volume:
weight ~ L^3. This means that strength vs. weight, a crude indicator of
how high an organism can jump, is proportional to area divided by
volume, which is L^2 divided by L^3 or L^-1 (1/L). Strength-per-weight
gets ten times better when an organism gets ten times smaller. A
nanomachine, nearly a million times smaller than a flea, doesn't have to
worry about gravity at all. If the number after the L is positive, then
the quantity becomes larger or more important as size increases. If the
number is negative, as it is for strength-per-weight, then the quantity
becomes larger or more important as the system gets smaller.

Notice what just happened. Strength and mass are completely different
kinds of thing, and can't be directly compared. But they both affect
the performance of systems, and they both scale in predictable ways.
Scaling laws can compare the relative performance of systems at
different scales, and the technique works for any systems with the
relevant properties--the strength of a steel cable scales the same as a
muscle. Any property that can be summarized by a scaling factor, like
“weight ~ L^3,” can be used in this kind of calculation. And most
importantly, properties can be combined: just as strength and weight are
components of a useful strength-per-weight measure, other quantities
like power and volume can be combined to form useful measures like power
density.

An insect can move its legs back and forth far faster than an elephant.
The speed of a leg while it's moving may be about the same in each
animal, but the distance it has to travel is a lot less in the flea. So
frequency of operation ~ L^-1. A machine in a factory might join or cut
ten things per second. The fastest biochemical enzymes can perform
about a million chemical operations per second.

Power density is a very important aspect of machine performance. A
basic law of physics says that power is the same as force times speed.
And in these terms, force is basically the same as strength. Remember
that strength ~ L^2. And we're assuming speed is constant. So power ~
L^2: something 10 times as big will have 100 times as much power. But
volume ~ L^3, so power per volume or power density ~ L^-1. Suppose an
engine 10 cm on a side produces 1,000 watts of power. Then an engine 1
cm on a side should produce 10 watts of power: 1/100 of the
ten-times-larger engine. Then 1,000 1-cm engines would take the same
volume as one 10-cm engine, but produce 10,000 watts. So according to
scaling laws, by building 1,000 times as many parts, and making each
part 10 times smaller, you can get 10 times as much power out of the
same mass and volume of material. This makes sense--remember that
frequency of operation increases as size decreases, so the miniature
engines would run at ten times the RPM.

Notice that when the design was shrunk by a factor of 10, the number of
parts increased by a factor of 1,000. This is another scaling law:
functional density ~ L^-3. If you can build your parts nanoscale, a
million times smaller, then you can pack in a million, million, million,
or 10^18 more parts into the same volume. Even shrinking by a factor of
100, as in the difference between today's computer transistors and
molecular electronics, would allow you to cram a million times more
circuitry into the same volume. Of course, if each additional part
costs extra money, or if you have to repair the machines, then using
1,000 times as many parts for 10 times the performance is not worth
doing. But if the parts can be built using a massively parallel process
like chemistry, and if reliability is high and the design is
fault-tolerant so that the collection of parts will last for the life of
the product, then it may be very much worth doing--especially if the
design can be shrunk by a thousand or a million times.

An internal combustion engine cannot be shrunk very far. But there's
another kind of motor that can be shrunk all the way to nanometer scale.
Electrostatic forces--static cling--can make a motor turn. As the
motor shrinks, the power density increases; calculations show that a
nanoscale electrostatic motor may have a power density as high as a
million watts per cubic millimeter. And at such small scales, it would
not need high voltage to create a useful force.

Such high power density will not always be necessary. When the system
has more power than it needs, reducing the speed of operation (and thus
the power) can reduce the energy lost to friction, since frictional
losses increase with increased speed. The relationship varies, but is
usually at least linear--in other words, reducing the speed by a factor
of ten reduces the frictional energy loss by at least that much. A
large-scale system that is 90% efficient may become well over 99.9%
efficient when it is shrunk to nanoscale and its speed is reduced to
keep the power density and functional density constant.

Friction and wear are important factors in mechanical design. Friction
is proportional to force: friction ~ L^2. This implies that frictional
power is proportional to the total power used, regardless of scale. The
picture is less good for wear. Assuming unchanging pressure and speed,
the rate of erosion is independent of scale. However, the thickness
available to erode decreases as the system shrinks: wear life ~ L. A
nanoscale system plagued by conventional wear mechanisms might have a
lifetime of only a few seconds. Fortunately, a non-scaling mechanism
comes to the rescue here. Chemical covalent bonds are far stronger than
typical forces between sliding surfaces. As long as the surfaces are
built smooth, run at moderate speed, and can be kept perfectly clean,
there should be no wear, since there will never be a sufficient
concentration of heat or force to break any bonds. Calculations and
preliminary experiments have shown that some types of atomically precise
surfaces can have near-zero friction.

Of course, all this talk of shrinking systems should not obscure the
fact that many systems cannot be shrunk all the way to the nanoscale. A
new system design will have its own set of parameters, and may perform
better or worse than scaling laws would predict. But as a first
approximation, scaling laws show what we can expect once we develop the
ability to build nanoscale systems: performance vastly higher than we
can achieve with today's large-scale machines.

For more information on scaling laws and nanoscale systems, including
discussion of which laws are accurate at the nanoscale, see Nanosystems chapter 2, available online at
http://www.foresight.org/Nanosystems/toc.html#c2

 

Chris has studied nanotechnology and molecular manufacturing for more than a decade. After successful careers in software engineering and dyslexia correction, Chris co-founded the Center for Responsible Nanotechnology in 2002, where he is Director of Research. Chris holds an MS in Computer Science from Stanford University.

Copyright © 2004 Chris Phoenix

 

Chris Phoenix

CRN Director of Research