Scaling Laws—Back to Basics

The following is #7 in a series of Science & Technology Essays from CRN:

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 1018 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, so 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.

Chris Phoenix, CRN Director of Research