Casing an assembler

by

Ralph C. Merkle
Xerox PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
merkle@xerox.com
www.merkle.com

Copyright 1998 by Xerox Corporation. All Rights Reserved.

This paper has been accepted for publication in the Special Issue of Nanotechnology devoted to the The Sixth Foresight Conference On Molecular Nanotechnology.

This rough draft is available on the web at http://www.zyvex.com/nanotech/casing.html

WARNING: if "103" looks like "103" then your browser does not support standard conventions for superscripts and subscripts. This will lead to confusion in the following text, particularly in exponents.


Abstract

Assemblers have been proposed as general purpose manufacturing devices, able to build a wide range of useful products as well as copies of themselves. Most proposals for assemblers involve an inside and an outside, separated by a barrier or casing. The outside is a relatively uncontrolled environment (a liquid solvent with dissolved feedstock molecules and impurities) while the inside environment is both better controlled (no impurities or undesired contaminants) and unreactive (a vacuum, or a noble gas such as neon).

A graphite casing would be strong, flexible, relatively unreactive, and largely impenetrable. The use of a graphite casing raises a number of design issues: how to manufacture the new casing within the confines of the old casing, what shape the casing should have and what maintains that shape, the conditions under which the casing might fail, how to eject the new casing into the outside solvent, etc.

We also discuss an improved method of using acoustic signals to power and control the assembler. In particular, the (possibly bulky) pressure-actuated pistons that were located internally to the casing in previous proposals are moved outside the casing. This lets us use pressurized neon internally to maintain the shape of the casing, much as the shape of an inflated balloon is maintained by internal air pressure. It also makes it easier to use larger pistons which can be driven by smaller pressure changes.

Introduction

In preparing for battle I have always found that plans are useless, but planning is indispensable.
Dwight D. Eisenhower
A previous paper (Merkle, 1996, Design considerations for an assembler) considered the major design issues in a simple assembler. This paper continues that analysis in greater depth for a particular component: the external casing of an assembler.

While we will not repeat the background material from that previous paper, the following paragraphs provide a brief overview as context for the present paper.

To simplify the design issues the focus of this and related papers is on hydrocarbon assemblers. These form a subset of the more general class of molecular manufacturing systems called assemblers. The set of possible assemblers is in turn a proper subset of all possible molecular manufacturing systems. One non-assembler based molecular manufacturing architecture is convergent assembly (Drexler, 1992; Merkle 1997a). Within the class of assembler based systems, most are more complex than the present proposal and are able to fabricate a wide range of structures using most of the elements in the periodic table (Drexler, 1992; Merkle, 1992). In contrast, hydrocarbon assemblers can make only stiff hydrocarbons, greatly simplifying the chemical reactions involved in the synthesis of this class of structures (Merkle, 1996b, A proposed "metabolism" for a hydrocarbon assembler). This, however, imposes the constraint that the structural components of the assembler must be made from hydrogen and carbon (the rest of the 100 or so elements in the periodic table being excluded -- though a very few additional elements are included for catalytic purpuses: e.g., a transition metal such as platinum; and silicon and tin to permit creation of radicals with graded reactivities).

The present paper changes some of the system assumptions made previously (Merkle, 1996). In particular, earlier papers assumed that acoustic control of the mechanisms inside the assembler was done by permitting pressure waves to pass from the exterior of the assembler to the interior, and that internally to the assembler these pressure waves were converted to mechanical motion to control the various mechanisms in the assembler. This assumption requires both a method for equilibrating the pressure between the interior and the exterior of the assembler, and further that the interior of the assembler house many (likely a few dozen) pressure actuated pistons. The reliable operation of these pistons in the face of thermal noise requires a certain minimal energy (i.e., deltaP x deltaV >> kT) which in turn implies that the pistons must be relatively large, that the change in pressure be relatively large, or both.

While these constraints can be met, it would clearly be advantageous to reduce the magnitude of the required pressure changes and the bulk of the internal mechanisms in the assembler which respond to those pressure changes. This can be done by using two externally located pressure actuated pistons of larger size, and demultiplexing the resulting signals in order to control the tens of actuators used internally in the assembler. Transfer of signals from the demultiplexor to the internal devices can be done using control cables consisting of a flexible rod inside a flexible sheath. Pulling on the rod (and simultaneously pushing on the sheath) produces a force that can be transmitted along the length of the cable. The concept should be familiar to anyone who has used a derailleur or brake cable on a bicycle. A particularly simple design would be a polyyne rod (a carbyne rod or long purely linear carbon polymer) encased in a (9,0) buckytube sheath. Pulling on one end of the polyyne rod would transmit force to its opposite end through the buckytube. Forces of a few nanonewtons can be transmitted this way. As the interior of the buckytube is relatively inert (Mauser et al., 1997) the polyyne should not react with its sheathing buckytube.

As the orginal design concept used many internal pistons to provide force actuation of mechanical components, and as control cables can equally provide force actuation, this change requires only that the control cables be strung in such a way that they do not interfere with the normal operations of the assembler.

This design eliminates the need to equalize the external and internal pressure. As a consequence, the internal pressure can be increased sufficiently to insure that the flexible casing maintains its shape. No additional supports or structural elements are needed to insure that the casing does not sag or collapse in an undesired way. (Recall that some of the proposed molecular tools (Merkle, 1996b; Brenner et al. 1996; Musgrave et al. 1991; Sinnott et al. 1994; Merkle 1997d) are highly reactive, and that it is therefore essential to insure that the casing does not contact them).

The benefits of this design are:

On the down side, this design requires the use of a demultiplexor capable of converting the force from two pistons into corresponding force on any one of several molecular control cables, and stringing those cables in a way which does not interfere with assembler operation. Neither requirement is prohibitive. Many macroscopic robotic devices use cables to control their motion. There are many designs for mechanical computers, and Drexler (Drexler, 1992) has provided a number of designs for molecular mechanical computers. The design of a mechanical demultiplexor will be substantially easier. The detailed design of a molecular mechanical demultiplexor and its simulation using molecular dynamics would be an excellent target for additional research.

Design issue: appropriate pressure

A small, capped buckytube about 8 nm in size
The most obvious design issue in this approach is the internal pressure required to keep the casing (a.k.a. a large buckytube) from collapsing. While sufficiently small single walled buckytubes in vacuum will maintain their tubular shape, larger tubes (with a radius larger than about 2.7 nm) are more stable when collapsed (Chopra et al., 1995) because of attractive van der Waals forces between the two half-sheets of the collapsed tube. This is true even when there is no external pressure, which (when present) makes the collapsed state even more energetically favorable than the inflated state. By inflating the buckytube with, e.g., neon gas at a sufficient pressure, this collapse can be prevented.

As the external pressure is varying, an internal pressure sufficient to prevent collapse of the casing when the external pressure is at a minimum might be insufficient when the external pressure is at a maximum. In addition, the casing might lose shape even if it does not collapse. Thus, the internal pressure must be elevated above the maximum external pressure by some safety margin.

At too high a pressure, the casing will burst. As a graphitic casing is quite strong, this pressure will be well above the pressure needed to maintain the shape of the casing.

The bursting pressure of a cylindrical casing

We first consider the bursting pressure of a cylindrical casing, ignoring the contributions of the end caps (we will later note that the bursting pressure of a sphere is higher than that of a cylinder of similar radius, so we expect that the bursting pressure of the cylinder plus appropriately designed end caps will reduce to the bursting pressure of the cylinder). Before analyzing the case of a graphite casing, we first consider a more general casing using a continuum model. Notationally, the casing has length l, radius r, and thickness t all in meters; and breaking stress s in Pa (Pascals, or N/m2). We assume that t is small compared with r. The casing encloses a gas at some pressure pinternal Pa, and outside the casing there is some pressure pexternal Pa. We let p = pinternal - pexternal Pa.

If the pressure p is too high, the casing will burst. We want to know pburst, the pressure at which this will occur.

Consider a plane that bisects the casing along its length. The force acting to pull the two halves of the casing apart is just the cross sectional area of the bisecting plane times the pressure:

f = 2 r l p

The stress this creates in the casing is just the force divided by the area of the bisecting plane that intersects the skin of the casing:

stress = f / ( 2 t l )

Which means that:

stress = 2 r l p / ( 2 t l )

or:

stress = r p / t

When the stress exceeds the strength of the material the casing will burst, so the critical pressure is:

pburst = s t / r

In words, this says that the bursting pressure is proportional to the strength of the material from which we make the casing and the thickness of the casing, and is inversely proportional to the radius of the casing.

If we apply a similar analysis to a sphere we arrive at a formula which is generally similar but which differs by a factor of two: pburst sphere = 2 s t / r. That is, the bursting pressure for a sphere is twice that for a cylinder, provided that the strength and thickness of the material are similar in both. We therefore expect that failure of the cylinder will occur somewhere along its length, rather than at the end caps.

With appropriate parameters, the continuum model can be used to analyze the case of a graphite casing. Properly used, continuum models can yield good approximations to molecular behavior when the molecular structure is reasonably well described by the continuum description (Drexler, 1992). This is the case in the simple graphite sheet under consideration here.

If we assume the strength of the casing material s = 2 x 1010 Pa (which is ~2% of the in-plain elastic modulus of a perfect graphite sheet (Kelly, 1981), a very conservative estimate); that the radius of the casing is 100 nm (10-7 meters); that the thickness of the casing is 0.335 x 10-9 m (the distance between adjacent sheets in graphite, (Kelly, 1981)); then we conclude that the bursting pressure is ~0.67 x 108 Pa, or ~660 atmospheres.

The conclusion here is valid for a perfect sheet of graphite. In a eutactic environment (Drexler, 1992) a perfect sheet of graphite with some tens of millions of atom in it will suffer a remarkably low failure rate from background radiation, while other failure mechanisms can be reduced below the radiation induced failure rates. However, the external environment for the casing is not eutactic so we must consider the possibility that a failure from this source might be of significance. As buckytubes have proven remarkably resilient in "normal" environments it seems unlikely that an external feedstock solution which is very pure by present standards would induce a failure in the graphite casing. (In the unlikely event that this assumption proved false we could either modify the external feedstock solution so that it was more benign, or adopt a stronger material for the casing, or both).

Conclusion: a buckytube 100 nm in radius should be able to contain a pressure of ~ 0.6 108 Pa (~600 atmospheres) or more. More detailed analysis could be used to more exactly determine the maximum pressure that could be used, but this approximate value is sufficient for our present purposes.

Minimum pressure required to inflate the casing

For buckytubes of relatively large radius (100 nm qualifies), there will be some minimal pressure required to overcome the van der Waals attractive force that would otherwise collapse the tube into a long sheet. To determine this critical pressure, we consider a partially inflated tube rolled up like a party blowout (or sleeping bag).

For the collapsed portion of the tube (assuming that it is rolled up, and hence that each graphite surface is exposed to two neighboring surfaces), the energy will be:

Ecollapsed = 2 pi r Ev deltal

where deltal is the infinitesimal length of tube under consideration and Ev is the energy of two graphite sheets held together by the van der Waals forces between them. For graphite, an approximation for Ev is 0.25 J/m2 (derived by using HyperChem's MM+ potential energy function and computing the attractive force between small finite graphite sheets). This value is reasonably close to the experimental value (Kelly, 1981) of 0.234 J/m2 (which in principle includes more than just the energy from adjacent graphite sheets, though in practice it is dominated by this energy). Ev will vary depending on the nature of the material, and can be deliberately influenced by design. For example, bonding "spacers" to the graphite sheets to hold them apart would effectively reduce Ev, and hence the pressure required to inflate the tube. We will not pursue this possibility further here.

For the inflated portion of the tube, the energy per unit length (from compressed gas in the tube) is:

Einflated = P V = p pi r2 deltal

If P V is high enough, then the energy in the compressed gas will be greater than the energy produced by the attraction of the graphite sheets, leading to inflation of the tube:

If p pi r2 deltal > 2 pi r Ev deltal then the tube will inflate

or:

If p r > 2 Ev then the tube will inflate

For r = 100 nm and Ev = 0.25 J/m2, we find that p = 5 x 106 Pa or about 50 atmospheres.

This analysis is similar to the analysis by (Yakobson et al., 1996), though it differs in that we assume the tube is rolled up (hence bringing together about twice the surface area and introducing a factor of two additional van der Waals energy) and that, because the size of tube we are considering is reasonably large, we neglect the strain energy.

This assumes we are rolling up an infinitely long tube, and therefore that collapse of a small section of the tube will not change the pressure in the remaining section. For relatively short tubes (the case we are considering) the collapse of a section of the tube will increase the pressure in the remainder of the tube, implying that a somewhat lower initial pressure would be adequate to keep the tube inflated. In addition, we are neglecting the energy required to bend the graphite sheet. While not a major concern for the size considered here (r ~ 100 nm), this will become increasingly significant for smaller values of r. For sufficiently small r, the buckytube will not collapse at all, even when p = 0.

These and other approximations don't change the basically qualitative conclusion that a pressure somewhere between 5 x 106 and 6 x 107 Pa (50 to 600 atmospheres) should be sufficient to inflate a buckytube of reasonable size (r ~ 100 nm) but not rupture it.

The volume of our casing is:

pi r2 l

For l = 1,000 nm and r = 100 nm, the enclosed volume is ~31,000,000 nm3.

P V = n kT

and the minimum pressure of 5 x 106 Pa, implies we will need about 37,000,000 neon atoms to inflate the casing at room temperature (k = 1.38 x 10-23 and T = 300 Kelvins).

The surface area of the casing (neglecting the end caps) is:

2 pi r l

where l is the length. Our 1,000 nm long graphite tube therefore has a total surface area of ~630,000 nm2, which implies ~25,000,000 carbon atoms in the casing (graphite has ~39 carbon atoms / nm2).

Putting the casing inside itself

If the old assembler is to build at least one new assembler, the casing must fit inside itself. When the casing is flattened, it forms a rectangle that is pi r l, or 314 nm x 1,000 nm. This rectangle consists of two layers of graphite, which is approximately 0.68 nm thick (note: while this is based on the approximate spacing of the graphite sheets in bulk graphite, small samples of graphite will have a different spacing because of reduced van der Waals forces pressing the sheets together. The error introduced by this approximation does not significantly alter the conclusions reached here). The volume occupied by the collapsed casing is therefore 210,000 nm3, or 0.7% of the total volume of the inflated casing. The collapsed casing can be folded or rolled in various ways to form a more compact shape.

A simple shape for the collapsed casing would be a rolled up cylinder 310 nm in length with a radius (from the formula for the volume of a cylinder and the known volume of the collapsed casing) of 15 nm. This will easily fit lengthwise in the inflated casing. (This neglects the resistance of graphite to bending, which implies that the center of the cylinder will have a small empty region because the curvature of the graphite in that region is higher than is energetically feasible. For the sizes considered here, this should not qualitatively influence the conclusion).

If we were to fold the flat collapsed casing in half along its long axis (the fold would be 1,000 nm in length) prior to rolling it up, the rolled up casing would have a length of 157 nm and a radius of 21 nm. This would let the rolled-up casing fit into the inflated casing cross-wise.

The casing must be manufactured within the confines of the inflated casing, hence it must be at least partially collapsed during manufacture. Extruding the new casing along its long axis during the manufacturing process while simultaneously rolling it up permits the entire operation to take place in a volume that is a modest fraction of the size of the inflated casing. If the casing is extruded in the folded-over shape described above (reminiscent of a rolled up sleeping bag), then the extrusion process can take place at one end of the inflated casing. This is convenient if the manufacturing component of the assembler is located at one end of the inflated casing.

Getting the new assembler out

In this article we have divided the assembler into the casing and the manufacturing component. The manufacturing component is largely defined as everything that is not the casing and includes the two pistons, demultiplexor, positional device(s) (robotic arm(s)) (Merkle, 1993; Merkle, 1997c), neon intake, binding sites for the feedstock molecules and the "vitamin" (Merkle, 1997b), input processing (conversion of input feedstock molecules into molecular tools) (Merkle, 1997d), etc. The problem of getting the new assembler out of the old assembler can therefore be described as: build the new casing, build the new manufacturing component, put them together, push the new assembler out of the casing, and inflate it.

While there are several ways of getting the new assembler out of the old casing while preserving the internal environment of the assembler (and preventing the entry of unwanted contaminants), a simpler approach is to have the old assembler manufacture two new assemblers and then simply rupture the old casing. While this process is both wasteful of material (the old casing and the old manufacturing component are both sacrificed) and time (a complete replication cycle requires the manufacture of two new assemblers by the old assembler rather than just one, approximately doubling the replication time) it is somewhat simpler to design.

As the old casing is a cylinder, opening one end of the cylinder is probably the simplest way of permitting the new assemblers to get out. As the new assemblers inflate, they will occupy greater volume and be pushed out of the old casing. There are various detailed concerns about the exact way in which this process occurs (will the neon intake remain unblocked by the old casing? Will a new casing "jam" during the inflation process because it is positioned incorrectly (crosswise)? Will the old casing adhere to one of the new casings, and if so what damage might be done?) but should any of these concerns prove to be a problem a mechanism to push the new assemblers out of the old casing could be adopted. For example, the old casing could have a piston at one end which, driven by gas pressure, would push everything out of the old casing once the opposite end of the casing was opened.

Closing the open end of the casing

If the casing is a simple cylinder, closed at one end, then something must block the open end during the manufacturing cycle. As the van der Waals forces acting on a graphite sheet are significant, this force might be adequate. We explore this possibility here.

If we assume that the open end of the casing is blocked by a simple disk-shaped plug, then the internal pressure of the neon gas in the casing will push on the plug with a force of:

f = 2 pi r2 p

For a pressure of 5 x 106 Pa and a radius of 100 nm, this gives a force of ~314 nanonewtons.

The circumference of the tube is just 2 pi r or ~628 nm. If the open end of the tube is simply slipped between two graphite sheets (attached to the disk shaped plug) then van der Waals forces will tend to hold the casing between the sheets and hence attached to the plug. The energy of the overlapping graphite sheets will be:

Eoverlap = circumference x depth of overlap

= 2 pi r d 2 Ev

where Ev = 0.25 J/m2 and d is the depth of overlap of the casing and the graphite sheets.

The force exerted will depend on the derivative of the energy with respect to the overlap distance, or:

Foverlap = 4 pi r Ev

The ratio of the gas-generated force pushing the plug out and the van der Waals force holding the plug in is:

2 pi r2 p / ( 4 pi r Ev )

or

r p / ( 2 Ev )

Which, for our example, is:

100 x 10-9 x 5 x 106 / ( 2 x 0.25 )

or about 1. For values greater than 1, the plug will be pushed out, while for values less than 1 the plug will be held in place. For the particular values selected, the ratio of force pushing the plug out to the force pushing the plug in is roughly balanced at ~314 nanonewtons. As a consequence, we will likely need to adopt some additional measure to insure that the plug remains attached to the end of the casing during normal operation. One simple method would be to put a few holes near the open end of the casing of perhaps one nanometer in diameter. Rods thrust through the holes would then provide additional force to prevent the casing from slipping away from the plug. (This approach is conceptually similar to the use of rivets at the macro scale). Removal of the rods would then free the plug from the casing.

It is also worth noting that the plug, because it must be housed inside the external casing during construction, should have a diameter smaller than that of the external casing. This implies that during construction the new casing (being attached to the new plug inside the external casing) will need to taper inwards to attach to the new plug. As this taper must be temporary (we must be able to eject the plug-and-uninflated-casing assembly from the end of the inflated casing), the end of the casing will be folded over in some fashion when it is attached to the plug.

Casing movement caused by pressure changes

In the present proposal there is no need for the active manufacturing component (and in particular, the tip of the robotic arm) to come into contact with the inflated casing, except where the casing is anchored to the end plug. This implies that movements of the casing caused by external pressure changes will not cause errors during the manufacture of a new assembler, because the external casing does not touch either the new assembler or the system manufacturing it.

There are, however, many alternative architectures which would benefit from the ability of the active manufacturing component to interact directly with the inflated casing, and in particular for the robotic arm to modify the casing or attach components to it. To what extent will the pressure changes in the external environment cause the casing to move? (A second question (which we will not address) is the extent to which such motion might actually impede modifications to the casing, given that the movements are quite predictable).

The greatest movement will be the extension and contraction of the casing along its 1,000 nm length. The stiffness of the casing with respect to this motion depends on the cross sectional area of the skin of the casing with respect to a slice perpendicular to its long axis. This cross sectional area is:

a = 2 pi r t

The change in force along the axis of the casing caused by external pressure changes is:

f = pi r2 deltap

The modulus of graphite with respect to stretching along the axis is ~0.686 x 1012 Pa (Drexler, 1992). The percentage stretch will be:

stretch = f / ( a modulus )

or:

stretch = pi r2 deltap / ( 2 pi r t modulus )

or:

stretch = r deltap / ( 2 t modulus )

For a change in pressure of 2 x 105 Pa (~2 atmospheres) this gives:

stretch = 100 x 10-9 m x 2 x 105 Pa / ( 2 x 0.335 x 10-9 m x 0.686 x 1012 Pa )

or a percentage stretch of ~4.4 x 10-5. For a 1,000 nm casing, this implies an absolute movement of 0.044 nm (.44 angstroms).

To put this in perspective, our 0.044 nm movement is about 29% of the length of a typical ~0.15 nm carbon-carbon bond. As this is the worst case movement the active robotic arm should usually be able to precisely interact with the casing in a direct and simple manner. If the robotic arm has a range of motion of about 100 nm, then, for regions within reach of the robotic arm, movements induced by external pressure changes will be some ten times smaller: about 0.0044 nm or ~3% of a carbon-carbon bond length. Larger pressure changes (several atmospheres) will cause larger movements. For any specific proposal that involves interactions between the robotic arm and the inflated casing it will be necessary to determine the pressure-induced movements to verify they don't cause problems. As the pressure induced movements are predictable, the resulting interactions between the casing and the robotic arm can be fully analyzed.

The size of the driving pistons

The external pistons used to provide both control information and power to the assembler need to be large enough to function reliably in the face of thermal noise. An adequate margin would be 100 kT. As it is sometimes convenient to include various ratchets and other mechanisms that might require greater energy to drive, we adopt a more generous margin of 200 kT. At room temperature, this will be about 8.3 x 10-19 J. If we assume the external pressure changes are about 100,000 Pa (one atmosphere), then each piston must have a volume of 8,300 nm3. This can be provided by a piston that is 12 nm in radius and 20 nm in length.

If we use two pistons, one with a higher threshold pressure and the second with a lower threshold pressure, then we can selectively address either the higher or lower pressure piston by adjusting the pressure of the feedstock solution. If the lower pressure piston cycles the demultiplexor through its possible outputs (selecting which among the many output lines is to be driven), while the higher pressure piston drives the currently selected output, then we can select and drive the outputs of the demultiplexor with any desired signal pattern using only two pistons.

If the moving piston is 12 nm in radius, 10 nm in length and made of diamond, it will have a mass of ~1.6 x 10-20 kg. (Note that the 10 nm length of the piston itself is different from the 20 nm distance that it moves). A 100,000 Pa pressure on the face of the piston will produce a force of about 45 piconewtons. This is much smaller than the maximum force that a polyyne rod can tolerate without breaking, which is in excess of ~6 nanonewtons (Drexler, 1988). From f = m a we infer that the acceleration is ~2.8 x 109 m/s2. From d = 1/2 a t2, we infer that the time for the piston to move 20 nm is ~3.8 nanoseconds.

The amplitude of an acoustic wave that has traveled some fixed distance has a gaussian fall off with increasing frequency. As a consequence, higher frequency operation becomes rapidly unattractive. While the piston described above should be able to respond to frequencies up to 100 megahertz, a 10 megahertz sound wave that has traveled 0.01 m (1 centimeter) in a "typical" organic liquid solvent has already been attenuated 10% (Drexler, 1992). Increasing the frequency to 100 megahertz would reduce the amplitude to low levels and result in substantial heating of the solvent. We therefore conclude that operating frequencies of 10 megahertz should be feasible and that substantially higher frequency operation will be severely limited by attenuation of the acoustic signal through the feedstock solution.

Control cables

The mechanical operations of the assembler will be controlled by many actuators, the actuators in their turn being controlled by molecular control cables using polyyne rods in buckytube sheaths. The speed of operation should not be greatly influenced by the mass of the polyyne rods. The C-C single bond in polyyne is 0.1377 nm, the C#C triple bond is 0.1192 nm (Lide, 1995), and a single carbon atom has a typical mass of ~2 x 10-26 kg. A polyyne rod therefore has a mass of ~1.6 x 10-16 kg/m. A 1000 nm polyyne rod has a mass of 1.6 x 10-22 kg, much smaller than the mass of the moving piston. The need to move the polyyne rod attached to the piston as well as the piston itself should have little impact on the piston's frequency response. We conclude that the piston and control cable can respond to acoustic signals in the 100 megahertz range and below, and are not the limiting factor in the operating frequency.

A second concern is the stiffness of the polyyne rods. If we pull on one end of a polyyne rod, to what extent will the force be transmitted to the other end and to what extent will the force be lost in stretching the cable? This question was considered in (Drexler, 1988).

The stretching stiffness of the C#C triple bond is approximately 1,560 N/m and the C-C single bond in polyyne is about 824 N/m (Drexler, 1988). The stiffness of the (9,0) buckytube sheath will be significantly greater than that of the polyyne rod, so we will neglect it. In the following discussion it will be more convenient to use compliance (the reciprocal of stiffness). For springs connected in series, compliance is additive. For example, if a 10 m spring has a compliance of 1 m/N, then pulling on it with a force of 1 Newton will stretch the spring one meter. If two such springs are connected in series, pulling on the combined spring will apply a force of one Newton to each spring, each spring will stretch one meter, and the combined spring will stretch 2 meters. The combined spring has a compliance of 2 m/N.

The compliance of two bonds in a polyyne rod will therefore be ( 1/1,560 + 1/824 ), or 0.00185 m/N (which corresponds to a stiffness of ~540 N/m). The length of the two bonds is 0.1377 nm + 0.1192 nm = 0.2569 nm. The compliance per unit length is thus 7.22 x 106 N-1. The compliance of a 1,000 nm polyyne rod is just compliance per unit length x length, or 7.22 m/N (a stiffness of 0.139 N/m).

If we were to pull on a 1,000 nm polyyne rod with a force of 45 piconewtons (the force from one of our pistons, computed above) then it would stretch the cable by only 0.32 nm. As our piston moves a total of 20 nm, our 1,000 nm polyyne rod has been stretched only ~1.6 percent of the total motion, which is quite acceptable.

Thermal noise will also cause uncertainty in the position of the end of the polyyne rod. The relationship between positional uncertainty, temperature and stiffness is:

sigma2 = kb T / ks

where sigma is the mean positional error, kb is Boltzmann's constant (~1.38 x 10-23), T is the temperature in Kelvins, and ks is the stiffness in N/m. (This equation is derived and discussed further in (Drexler, 1992)).

At room temperature, the mean positional error caused by thermal noise in the position of the end of our 1,000 nm polyyne rod is 0.17 nm.

An additional factor that contributes somewhat to the effective compliance of our polyyne rod in its (9,0) buckytube sheath is the fact that the cable will not be centered in the sheath, particularly when the polyyne rod is under tension and the sheath is curving. The force restoring the cable to the center of the sheath increases as the length of the cable increases, and so for a 1,000 nm cable this restoring force will be quite large. For a cable that has been curved into a circle, we note that shortening the polyyne rod by 1 nm will move it 1 / ( 2 pi ) ~ 0.16 nm closer to the sheath. This large motion (on the molecular scale) against such a large stiffness will be greatly resisted, and the contribution to total compliance from this mechanism will be significantly smaller than the contribution from the stretching compliance of the polyyne rod.

Time required to replicate

If we assume that the manufacture of two assemblers will require on the order of one billion atoms, and further that on average the addition of one atom to an assembler will require 1,000 pressure cycles, then we require a total of 1012 (one trillion) pressure cycles to complete one replication cycle. At 10 megahertz, this is about 28 hours. While this is obviously a very crude estimate, it indicates that the total replication time using this approach need not be unreasonable. However, it also shows that the slow data rate of acoustic signaling is a major limiting factor in achieving a more rapid replication time. Clearly, the better the data compression and decompression methods used to reduce the total number of bits that must be transmitted to the assembler, the more rapidly the assembler could replicate. This suggests that the simple demultiplexor proposed here might advantageously be replaced with a more sophisticated (albeit more complex) instruction decode unit. Eventually, this direction of research leads to an on-board computer able to interpret much more compact representations of the instructions and greatly reduces the amount of data that must be transmitted to the assembler.

Benefits of two band signaling

This proposal uses a two-band signaling approach, hence each piston requires its own pressure range of about 1 atmosphere. To maintain a positive pressure at all times, the unmodulated pressure should be in excess of 1 atmosphere. An operating pressure of 2 atmospheres, with excursions to 1 atmosphere to operate the low pressure piston and excursions to 3 atmospheres to operate the high pressure piston should suffice for reliable operation. The maximum pressure is significantly smaller than that proposed by (Drexler, 1992) for two major reasons: first, by adopting a two-band signaling system (instead of a many band signaling system) we reduce the number of pressure ranges that must be accomodated from many (perhaps 20 or more) to only two. Second, by reducing the number of pistons and locating them externally, we can increase their size and hence their sensitivity to pressure changes. Fewer narrower pressure ranges reduces the maximum pressure that is required.

On the down side, to address a specific output of the demultiplexor we must cycle through other (undesired) outputs until the desired output is selected, which is additional overhead not required if one output among many is directly addressed by adjusting the pressure. If the low pressure (selection) piston is used to cycle through the demultiplexor outputs, then every time we cycle the selection piston we are performing an operation that is not required in a scheme that uses many different pistons with many different pressure thresholds, and addresses those pistons by directly adjusting the pressure. This inefficient use of time is not a major problem, as it seems likely that we will cycle the high pressure (data) piston many times once a specific output has been selected by cycling the selection piston. The ratio of data to selection bits is likely to be high. Further, if we wish to cycle all or many of the demultiplexor outputs to perform a complex operation, then we will only need to cycle the selection piston a few times before we select an output to which we we wish to transmit many bits by cycling the data piston. In the extreme case, if we wish to send data to all outputs, then we will cycle the selection piston only once between bursts of data. The pattern of acoustic transmissions will look like: cycle the selection piston once; cycle the data piston many times to transmit data; cycle the selection piston once to select the next output; cycle the data piston many times; cycle the selection piston once to select the next output; ....

General architectural comments

It should be clearly understood that the architecture proposed here is not "the" design for an assembler, nor even a design able to economically manufacture a wide range of stiff hydrocarbons. It suffers from severe limitations imposed by the basic constraint that the design be very simple. Its one redeeming feature is the ability to fabricate a wide range of stiff hydrocarbons, providing us with an entrée into the world of molecular manufacturing.

This design wastes energy with awesome prodigality. Each basic operation, during which on average a single atom might be added to the structure being manufactured, requires ~1,000 pulses. Each pulse drives a piston which uses ~200 kT or ~10-18 J/pulse, implying an energy cost of ~10-15 J/atom even before taking into account acoustic energy not absorbed by a piston but simply dissipated in the feedstock solution or elsewhere. To put this in perspective, one kilogram of carbon is about 5 x 1025 carbon atoms, which would require 5 x 1010 joules to manufacture using this approach. At a cost of ~0.1 dollars per kilowatt hour, or 28 nanodollars/J, the energy costs for a kilogram of product will exceed $1,000. While present manufacturing costs for the active region of a computer chip (the thin layer at the top of the chip which actually does the computing) exceed $10,000,000/kg (showing that even at a manufacturing cost of several times $1,000/kg there would be many products which, by present standards, would be well worth the expense), it is clear that energy costs should be reduced by adopting a more energy efficient design.

While not quite as wasteful of material as of energy, this design still creates and throws away the external casing and other components of the old assembler during each replication cycle. Again, a better design is clearly feasible.

The use of acoustic signaling for detailed control of even the most minute actions of the assembler imposes a major bandwidth burden. This, coupled with the highly inefficient encoding of control information, results in a replication period in excess of one day. As the use of higher frequencies (substantially above 10 megahertz) is limited by severe attenuation problems, it seems clear that if acoustic signaling of such large amounts of data is to be retained the encoding of information should be drastically improved.

The waste of energy and time can both be addressed by using an on-board computer. Such a computer could be given relatively high-level instructions (reducing by several orders of magnitude the total bandwidth required during a replication period, simultaneously reducing the horrendous energy waste caused by inefficient acoustic signaling and the excessive replication time). The waste of material can be addressed by changing the method of releasing new assemblers from the external casing.

Again, the only justification for such a poor design is its relative simplicity. Today, we are unable to build any assembler. We are, therefore, searching among the space of simpler designs for systems which might be easier to make. The present design is not among the simplest because it is constrained by the requirement that it be able to fabricate stiff hydrocarbons. Designs that might be appropriate targets for direct synthesis with existing technologies are unlikely to be able to make diamond, but will probably work with molecular building blocks whose assembly is simpler and which impose fewer constraints on the environment (e.g., building blocks which can be assembled in a standard solution environment (Krummenacker, 1994)). The present design is neither the next experimental target nor the final destination. It is a way point along the route from present capabilities to the future.

Conclusions

Schematic depiction of a simple hydrocarbon assembler. Two new HC assemblers have just been completed but not yet released. Their casings are shown rolled up, and the two pressure actuated pistons are shown as inactive (in the fully "in" position). The output of the demultiplexor is indicated, but the control lines from it have not been shown going to the various controlled components of the assembler to avoid confusion in the drawing.

Assemblers have been proposed as a way to make a very wide range of structures. The hydrocarbon assembler focuses on the much narrower range of structures defined by the class of stiff hydrocarbons. Synthetic reactions that have been proposed to flexibly synthesize a wide range of stiff hydrocarbons use highly reactive molecular tools (radicals, carbenes) and require an inert environment. A small inert environment can be provided inside a casing made by inflating a large buckytube (on the order of 100 nm in radius and 1,000 nm long) with an inert gas such as neon. This volume should be adequate to allow the manufacture inside such a casing of more hydrocarbon assemblers. The manufacture of an uninflated casing inside an inflated casing raises various issues, including the means by which the new casing is released from the old casing, the geometry of the new casing during its synthesis, the method by which the shape of the casing is maintained, etc. These issues were discussed in the present paper.

In addition, the use of acoustic control to provide both power and information to the assembler during its operation requires the use of appropriate acoustic sensors. Two relatively large (~104 nm3) externally located pistons operating a demultiplexor can provide multiple control signals while requiring a relatively modest operating pressure (a few atmospheres).

References



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