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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
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