Rubber Motors

In 1967, when I was studying physics at UCLA, Dr. Fred Alan Wolf (currently known as Doctor Quantum) introduced the students in his class on thermodynamics to a weird property of rubber. Using a very simplistic model of a strand of rubber, he used the statistical thermodynamics that Maxwell and Boltzmann had pioneered to show how stretched rubber would behave like a pressurized gas, expanding and contracting as it was cooled and heated. Thus we have an analogue of Boyle’s law in strained rubber. In this case the length gained by stretching the rubber band stands analogous to the volume of the gas and the force needed to stretch the rubber band stands analogous to the pressure. So just as we have pV=nkT for a gas, we have Fx=nkT for a rubber band.

The property of hyper-elasticity, which we find in confined gases, gives us a mechanism by which we can manipulate heat to drive an engine, converting some of the heat into work. Certain solids, such as rubber, also possess the property of elasticity and can therefore also serve as heat manipulators in engines. In rubber we have a polymer chain that works as an entropic spring. If we stretch a rubber band, we reduce its entropy by reducing the number of microstates available to the monomers that comprise the band. Striving to return to the state of highest entropy, the band manifests a restoring force, which acts to return the band to its equilibrium (unstretched) state. Because it involves the manipulation of entropy the rubber band has an unusual relation with temperature. Heating a rubber band kept under constant tension causes the rubber band to contract and cooling it causes it to elongate. Conversely, stretching a tensioned rubber band generates heat and allowing it to relax again causes it to absorb heat.

To gain some idea of the magnitude of the effects we want to exploit consider the following. If we put a rubber band under 980 newtons (220 pounds) of tension (equivalent to the weight of 100 kilograms) at 293 K (68 F), then raising the temperature by 30 K (54 F) degrees increases the tension by 98 newtons (22 pounds) or decreases the elongation by ten percent.

With these simple facts in mind, we can now contemplate the design of several kinds of rubber motor.

The first kind consists of a rubber band stretched over two rollers that we can heat and cool. We use a motor to drive the cold roller and attach the hot roller to a generator. In this way we use the rubber band and the temperature difference between the rollers to amplify the power that we feed into the system with the motor. Under the same force, the contracted part of the rubber draws little power from the motor and the expanded part feeds more power into the generator, thereby converting the heat supplied to the rubber band into extra electricity.

Because we can make the rubber bands broad as well as long, we can use this kind of rubber motor in a solar-power setting. In this case instead of heating one of the rollers, we project solar heat directly onto the rubber band.

In designing this kind of system we must pay attention to the problem of mounting the rubber bands on the rollers when we first establish the device and when we must replace a rubber band. We want to have the simplest possible machine, so we want the rollers mounted on a solid foundation, not on one that can move in order to allow the rubber bands to be slipped over the rollers without tension and then the foundation jacked apart to apply tension to the rubber bands. In that case the rubber band must begin as a rubber strap that is subject to tension as it is mounted on the machine and its ends connected to make it into a band.

An obvious solution is to make the ends of the strap thin gradually so that when we overlap them they make a uniform thickness with the rest of the strap. Putting them under tension as we mount them on our motor, we use a clamp to hold the ends free of tension while we bind them together, either by causing the rubber strands to interpenetrate and crosslink or by applying a suitable adhesive. That, though, is a problem for the chemical engineers.

The second kind of rubber motor consists of a series of rubber bands connected at one end to a frame and at the other end to a crankshaft. On both sides of the motor we mount metal plates bent into a cycloidal cross section. Channels milled into the plates and then covered will carry hot fluid in one plate and cold fluid in the other. As the crankshaft turns, each rubber band will press against one plate and then the other. Because the crankshaft holds the rubber bands offset from one another, the motor will start running as soon as we begin pumping hot and cold fluid through the plates. This version is the rubber analogue of an automobile engine.

We have a third kind of rubber motor in an overbalanced wheel. In this version we have heavy long-handled hammers mounted on a horizontal axle through hinges. Rubber bands under tension connect the hammers, each to its neighbors, as far from the axle as possible. We now define a vertical plane that passes through the axle and we may put a barrier of some kind on that plane We blow hot air on one side of the plane and cold air on the other side, the rubber bands on the warm side contract and the rubber bands on the cool side expand, causing the hammers to shift and put more of their weight on the warm side, and the wheel turns.

This version resembles an all-too-common type of
perpetual-motion machine, but it is no such thing. This motor won’t run by
itself: it won’t create energy ex nihilo. In order to run this motor, we must
supply heat and a means to remove the excess, because this motor __does__
obey the laws of thermodynamics. We must keep one side of the wheel warmer than
the other: if the temperature across the wheel reaches equilibrium, the wheel
will stop turning.

I need to mention one caveat here for those who intend to try building rubber motors. Because stretched rubber oxidizes readily, these motors will need to have their rubber bands enclosed within oxygen free chambers. Either that or the builders must use silicone rubber, which won’t oxidize (it was invented to seal windows in spacecraft). Otherwise the life of a rubber motor will be relatively short.

The engines that I have described above seem impractical and likely will find no major applications in the real world. But one kind of rubber motor will likely play a major role in an important technology that we currently have under development. Imagine using rubber as a muscle in a robot or in a prosthetic arm or leg. If we use, instead of rubber bands, a bundle of thin rubber tubes, through which we can pump hot and cold fluids, then we have a rubber muscle. Its applications are limited only by human imagination.

We can also run the rubber motors in reverse, using them to move heat. We can thus imagine a rubber refrigerator. It would be a simple cycle of stretching the rubber band to make it give off heat, cooling it, then allowing it to relax and absorb heat, thereby cooling its immediate environment.

All of these rubber-based devices look impractical, but that’s because the rubber moves rather slowly. But imagine that at some time in the future nanotechnology is used to create rubber bands that are not at all random, that can exchange heat rapidly and move swiftly. Then what might we see?

Appendix: Rubber and Heat

If we want to understand the relation between rubber and heat, we need to employ a model of rubber that we can analyze through simple mathematical physics without producing an invalid result. As a polymer, rubber gives us an easy model to conceive: we need only imagine a molecular chain laid out along some arbitrary x-axis with some of its links in a kinked state and the others in an unkinked state. We further specify that we have attached one end of the chain to an immovable object and left the other end free to move. Thus we model a rubber band as a bundle of such chains and model a chain as a linear array of N links (monomers). We assume that the links exist in one of two states – unkinked (parallel to some arbitrarily chosen x-axis) and kinked (tilted away from the chosen x-axis).

The links are small enough and light enough that the normal thermal jitter in their environment makes them kink and unkink at random. The monomers that comprise a strand of real rubber meet that criterion closely enough that we can accept our model as valid. But now if we identify the kinked and unkinked states with steps taken to the left and to the right along a narrow sidewalk, then at any given instant our model strand of rubber also represents a linear graph of a single microstate in the Drunkard’s Walk. We want to know how many microstates (given arrays of kinked and unkinked links) constitute a given macrostate (a given length of the strand). We already know how to analyze the Drunkard’s Walk, so we can apply the results of that analysis to our model.

If our model has N links and if n of them are kinked up at a given instant, then we know that there are N!/n!(N-n)! different ways to manifest that description; that is, we know of N!/n!(N-n)! microstates that correspond to the macrostate of an N-link chain with n of its links kinked. If we let p represent the probability of a link being kinked and let q represent the probability of the link being unkinked (p+q=1, of course), then we can calculate the probability of the chain manifesting a given macrostate as

(Eq’n A-1)

Now let N become extremely large, on the order of Avogadro’s number or larger. In that case we can replace the expression on the right side of Equation A-1 with the equivalent Gaussian function,

(Eq’n A-2)

In going from the first line of that equation to the second I have assumed that p=q=1/2: in the absence of any information indicating the contrary, I can assume that any given link is as likely to be kinked as to be unkinked.

Next we use x_{0} to represent the difference
between the length contributed to the chain by a link when it’s kinked and when
it’s unkinked. Of course, in a strand of real rubber that difference varies from
link to link and from time to time, but in a strand made of a very large number
of links we can use an average and x_{0} represents that average. Now we
can calculate the length of our chain as

(Eq’n A-3)

in which C_{0} represents the length that the chain would have if all
of its links were kinked. Inspection of Equation A-2 tells us that the most
probable state in which we will find the chain is the one for which n=N/2. With
the chain in that macrostate we define the location of the free end of the chain
as x=0. Then for any other macrostate we have x=x_{0}(n-N/2). Using that
result, we can modify Equation A-2 to read

(Eq’n A-4)

Now we can calculate the entropy of the chain and then get into its actual thermodynamics. We start by recalling Boltzmann’s formula,

(Eq’n A-5)

in which S represents the entropy of a given macrostate and W represents the
number of microstates that comprise that macrostate. Because the number of
microstates stands in direct proportion to the probability of finding the system
in the macrostate they comprise (W=C_{1}P(N;n), in which C_{1}
represents a constant conversion factor), we can write the entropy as

(Eq’n A-6)

In order to understand how a stretched rubber band converts heat into work we must understand how the contributions to its total energy behave under conditions of constant temperature and constant elongation; that is, when dT=0 and dx=0. We gain that understanding through examination of the Helmholtz free energy associated with the rubber band. In differential form we have

(Eq’n A-7)

in which we have replaced the mechanical energy pdV of a confined gas with the mechanical energy Fdx of a stretched rubber band. We also have

(Eq’n A-8)

in which the coefficients of the differentials are determined at constant elongation and at constant temperature respectively. Comparing those two equations tells us that we can calculate the force in the rubber band as

(Eq’n A-9)

If the amount of heat in the rubber band is much greater than the amount of mechanical energy, then we have the approximation A=-ST. Making the appropriate substitution from Equation A-6, we calculate the force as

(Eq’n A-10)

That’s Hooke’s law with a spring constant that stands in direct proportion to the absolute temperature of the rubber. That formula lets us see easily how we can create a cycle of heating, contracting, cooling, and elongating a stretched rubber band to make it transform heat into work. And that cycle provides the foundation upon which we conceive the rubber motors described above.

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