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Molecular Sorting, Processing, and Assembly

13.1. Overview

Nanomechanical mechanisms and processes have natural applications to molecular manufacturing. Previous chapters show that nanometer-scale structural components can have diverse shapes and good stiffness (Chapter 9); that nanometer-scale gears, bearings, chain drives, cams, and the like are feasible (Chapter 10); that mechanical and electromechanical components (Chapter 11) can be combined to build systems of power-driven machinery capable of executing complex, programmed patterns of motion (Chapter 12); and that mechanical manipulation of reactive moieties can be used to direct a broad range of synthetic operations with reliable positional control (Chapter 8). The present chapter considers how these and related capabilities can be combined to acquire, order, and process feedstock molecules, first preparing reagent moieties and then applying them to workpieces to build up complex structures.

This process begins with small molecules in a liquid and ends with complex structures in an evacuated region. Liquid solutions are a natural starting point for molecular manufacturing: they are dense, thoroughly disordered, and can carry a wide variety of molecules. A vacuum environment is a natural destination: the use of vacuum avoids both viscous drag and mechanical interference from stray molecules, while minimizing unwanted chemical reactions. These characteristics improve the performance of molecular manufacturing systems and simplify their design and analysis. (For many applications, of course, the manufacturing system must ultimately deliver its product to a nonvacuum external environment.) To accomplish these steps, the required processes include:

  • Acquiring and ordering molecules from solution
  • Transforming streams of molecules into streams of reagent moieties
  • Assembling reagent moieties to form complex molecular objects

Analyzing these processes requires attention to:

  • Assembly processes
  • Materials transport
  • Cycle times
  • Energy requirements
  • Error rates
  • Error sensitivities

This chapter addresses these issues in several contexts. Section 13.2 describes the sorting and ordering of molecules to produce nearly deterministic streams of molecules of single type starting from a disordered external solution. Section 13.3 describes the transformation of streams of input molecules into streams of bound reagent moieties suitable for use in molecular manufacturing processes, adopting a molecular mill approach. Section 13.4 describes programmable molecular manipulators for combining reagent moieties (and larger reagent structures) to build complex products. The next chapter builds on these results to describe systems capable of assembling macroscale products.

13.2. Sorting and ordering molecules

Nearly deterministic molecular manufacturing processes require well-ordered inputs, derived ultimately from a disordered feedstock. In designing a system, the contents of this feedstock can, within broad limits, be chosen for convenience. For concreteness, this section considers liquid feedstock solutions that can be prepared today: moderately pure mixtures consisting of small, relatively stable molecules of selected kinds in chosen proportions. (A typical small molecule, for present purposes, is one containing 20 or fewer atoms each of which has a degree of surface exposure equal to or greater than that of the central carbon atom in propane.) An input port to a manufacturing system can be presented with a feedstock consisting predominantly of a single chemical species (with trace contaminants) or with a mixture of species each present in substantial concentrations (again with trace contaminants).

The major species in a feedstock mixture can be chosen to serve as convenient fuels or raw materials; the trace contaminants must be assumed to be diverse. It is this diversity that poses the major challenge in analysis, forcing consideration of a set of molecules with a wide range of properties, and (in the early stages of processing) too large to be exhaustively described.

In constructing nanomechanical systems, only relatively exotic devices require isotopic refinement (e.g., for the construction of small, high-speed rotors with a precisely balanced mass distribution). Neither devices of this kind nor isotopic refinement are considered here.

This section first describes a class of mechanisms capable of selectively binding and transporting chemical species from a feedstock solution. It then discusses the combination of such mechanisms to build purification systems able to deliver streams of molecules that are (with high reliability) of a single type. Finally, it discusses the conditions that must be met in order to capture and orient single molecules from such a stream in a reliable manner, providing a eutactic input stream for subsequent processing.

13.2.1. Modulated receptors for selective transport

a. Basic concepts. Molecular biology provides numerous examples of systems that selectively bind molecules from solution (e.g., antibodies, enzymes, and various signal-molecule { }^{\circ}receptors) and of systems that transport bound molecules against concentration gradients (e.g., active-transport systems in cell membranes). Nanomechanical systems can perform similar operations. The analysis of purification operations is fundamentally different from the analysis of processes in a eutactic environment, in that the population of impurity molecules is not specified. Accordingly, portions of this analysis are cast in terms of general properties of molecular shapes, although practical systems are likely to gain great advantages from the exploitation of other properties of particular molecules of interest.

Figure 13.1 illustrates a class of mechanisms capable of using mechanical power to transport molecules of a particular type from regions of lower to higher free energy, and from regions of lower to higher purity (see caption for description). Analogous mechanisms include devices with differing geometries (e.g., cam surfaces to the side of the rotor, rather than in the hub) and differing mechanisms for modulating the binding interaction (e.g., elastic deformation of the receptor, alteration of local charge distributions). A slightly more elaborate class of mechanism, motivated by specificity concerns, is discussed in Section 13.2.1d. The general capabilities of such devices can be analyzed using a combination of principles from statistical mechanics and experimental observations from molecular biology.

The capabilities of general structures can equal or exceed those of antibodies, about which much is known from experimentation. As noted in Section 9.5.4, antibodies can be developed that will bind any one of a wide range of small molecules with small { }^{\circ}dissociation constants [typical values of KdK_{\mathrm{d}} are 10510^{-5} to 1011 nm310^{-11} \mathrm{~nm}^{-3} (Alberts et al., 1989)] and strong specificity (relatively poor binding of all but very similar molecules). The probability that a receptor will be found occupied by a molecule of the desired type is related to the concentration of a bindable molecule (ligand), cligand c_{\text {ligand }}, and KdK_{\mathrm{d}} by the expression

Poccupied =cligand KdPunoccupied \begin{equation*} P_{\text {occupied }}=\frac{c_{\text {ligand }}}{K_{\mathrm{d}}} P_{\text {unoccupied }} \tag{13.1} \end{equation*}

Figure 13.1. A sorting rotor based on modulated receptors. In this approach (illustrated schematically), a cam surface modulates the position of a set of radial rods. In the binding position (mapping the illustration onto a 12-hour clock dial, 10:00), the rods form the bottom of a site adapted to bind molecules of the desired type. Between 10:00 and 2:00, the receptors undergo transport to the interior, driven by shaft power (coupling not shown). Between 2:00 and 4:00, the molecules are forcibly ejected by the rods, which are thrust outward by the cam surface. Between 4:00 and 8:00, the sites, now blocked and incapable of transporting molecules, undergo transport to the exterior. Between 8:00 and 10:00, the rods retract, regenerating an active receptor. Section 13.2.1c discusses receptor properties; Section 13.2.1e discusses energy dissipation.

Values of KdK_{\mathrm{d}} depend on the interaction energies of the ligand and receptor, the ligand and solvent, and the solvent and receptor. Among differing solvent systems in which the interactions of the solvent with the ligands and the receptor are relatively weak and nonspecific, KdK_{\mathrm{d}} will be relatively solvent independent.

The number of atoms required to equal the basic structural diversity of antibodies was estimated on combinatoric grounds to be <35<35 (Section 9.5.4). On geometric grounds, however, a receptor that encloses a molecule of substantial size (e.g., 0.9 nm\sim 0.9 \mathrm{~nm} in diameter) with a layer of equal thickness must have an excluded-volume diameter of 2.7 nm\sim 2.7 \mathrm{~nm}, and contain 103\sim 10^{3} atoms. (The diameter of an active, Fv fragment of an antibody provides a highly conservative upper bound of 4 nm\sim 4 \mathrm{~nm} on receptor size.) Allowing 2.7 nm2.7 \mathrm{~nm} of circumference per receptor, and 12 receptors per rotor yields a rotor diameter of 10 nm\sim 10 \mathrm{~nm}. With a rotor thickness of 2.7 nm2.7 \mathrm{~nm}, the total number of atoms in the rotor system is 2×104\sim 2 \times 10^{4}.

Modulated receptor mechanisms can be applied either to concentrate a molecular species from a relatively dilute feedstock solution, or (in a multistage cascade, Section 13.2.2) to purify a stream of molecules by excluding impurities. For use in concentration, receptors having KdK_{\mathrm{d}} considerably less than the actual solution concentration are of value, since these can (given reasonable specificity) deliver a product stream dominated by the desired species in a single operation. Since diffusion-controlled rate constants for binding of small molecules are typically 109 s1 nm3\sim 10^{9} \mathrm{~s}^{-1} \mathrm{~nm}^{-3} (Creighton, 1984), and since several tens of molecular species can be simultaneously present in solution at concentrations 0.1 nm3\geq 0.1 \mathrm{~nm}^{-3}, 100\sim 100 encounters occur in a 10610^{-6} second exposure time, ensuring equilibration. With Kd103 nm3K_{\mathrm{d}} \leq 10^{-3} \mathrm{~nm}^{-3}, the probability of receptor occupancy is .99\geq \sim .99.

Delivering the contents of these receptors to a compartment containing a high concentration of product molecules reduces entropy, requiring a work of concentration amounting to 20maJ\sim 20 \mathrm{maJ} per molecule at 300 K300 \mathrm{~K}. This can readily be provided by coupling to a mechanical drive mechanism (details of such drive mechanisms are not considered here; they can be built up from gears, bearings, springs, cams, and motors of the sorts described in Chapters 10 and 11). With the application of greater drive forces, a modulated receptor mechanism can easily pump against pressures of several gigapascal (i.e., several nanonewtons per square nanometer), at an energy cost on the order of 100maJ100 \mathrm{maJ} per molecule. Where concentrations and pressures differ little from one side to the other, as in the later stages of purification cascades (Section 13.2.2), the input work required can be quite small.

b. Diamondoid vs. protein structures for receptors. Diamondoid structures have advantages over proteins in their feasible affinities and specificities. In general, stiffer structures permit greater specificity. At one extreme, a sufficiently soft receptor can exhibit solventlike specificity, that is, very little. (Receptors containing flexible, oligomeric chains can provide mechanically modulated, broad-spectrum affinity.) Proteins can exhibit substantial stiffness, enabling strong specificity (Creighton, 1984). Diamondoid structures can exhibit stiffness greater by several orders of magnitude, permitting significant improvements in specificity, particularly where exclusion of a particular species depends on overlap forces resulting from a bad geometric fit.

Diamondoid structures have a greater atom number density, leading to stronger van der Waals attractions and hence to greater binding energies, which can contribute to greater affinities. Further, the greater design-stage flexibility of general structures will often permit better matching of ligand and receptor, with respect to both geometry and electrostatics.

c. General considerations in designing for specificity. Receptor affinity (that is, 1/Kd1 / K_{\mathrm{d}} ) and specificity (roughly, differences in affinity for different molecular species) are related but distinct properties. In a practical system, what matters is the ability to discriminate between the desired molecule and the competitors actually present at that location in the process, not specificity for the desired molecule with respect to all possible competitors. Accordingly, the optimal receptor structure may differ at different stages in a cascade.

A high binding affinity for the preferred ligand can be a poor design choice if it leads to slow dissociation kinetics for a competitor, even if the differential in affinities remains large. Under these circumstances, the occupancy of receptors in transit through the barrier is controlled by competitive binding kinetics, rather than by equilibria; this typically results in poor discrimination. This problem does not arise in systems with Kd>103 nm3K_{\mathrm{d}}>\sim 10^{-3} \mathrm{~nm}^{-3} and having 106 s\sim 10^{-6} \mathrm{~s} receptor exposure times.

If two similar molecules can compete for the same receptor, then (neglecting differences in solvation) the ratios of their dissociation constants are directly related to the differences in their free energies of binding in vacuum and (neglecting pVp V effects and differences in entropy) are approximately related to the differences in their potential energies of binding in vacuum

Kd,1/Kd,2exp[(ΔF1ΔF2)/kT]exp[(ΔV1ΔV2)/kT]\begin{equation*} K_{\mathrm{d}, 1} / K_{\mathrm{d}, 2} \approx \exp \left[\left(\Delta \mathscr{F}_{1}-\Delta \mathscr{F}_{2}\right) / k T\right] \approx \exp \left[\left(\Delta \mathscr{V}_{1}-\Delta \mathscr{V}_{2}\right) / k T\right] \tag{13.2} \end{equation*}

The potential energy differences in Eq. (13.2) can readily be estimated by energy minimization in a molecular mechanics model; evaluation of the free energy differences in Eq. (13.2) requires calculations that either explore a region of the PES (Mitchell and McCammon, 1991; Straatsma and McCammon, 1991) or perform an integration over a region (see Section 4.3.2).

Consider a receptor that, unlike those in Figure 13.1, entirely surrounds its ligand (to permit the entrance and exit of ligands, the receptor must have at least one moving part). Assume that the receptor conforms closely to the contours of the preferred ligand, with the interfacial atoms placed near their minimum-energy distances, or somewhat closer. Alternative ligands can now be classified into three categories based on their size and shape: (1) those that resemble the preferred ligand, but occupy a strictly smaller volume in some conformation (e.g., having a similar structure, but with H\mathrm{H} substituted for CH3\mathrm{CH}_{3} ); (2) those that resemble the preferred ligand, but occupy a strictly larger volume in all conformations (e.g., by having the reverse substitution); and (3) those of all other shapes. Ligands in category 1 typically bind more weakly, owing to loss of van der Waals attraction energy; those in category 2 bind more weakly (or are effectively excluded) owing to overlap repulsion; those in category 3 suffer both difficulties simultaneously, since they can neither fill the receptor nor avoid overlap.

Discrimination against ligands in category 1 increases with the Hamaker constant (Section 3.5.1a) of the receptor structure; this constant is larger for typical diamondoid structures than for proteins. Even for proteins, however, the discrimination against ligands that lack a CH3\mathrm{CH}_{3} group (putting H\mathrm{H} in its place) can be 24maJ\sim 24 \mathrm{maJ} (Creighton, 1984), altering binding affinity by a factor of 0.003\sim 0.003 at 300 K300 \mathrm{~K}. (To model binding of molecules in ordinary solvents, this energy difference has been reduced by subtracting contributions from hydrophobic interactions.)

Discrimination against ligands in categories 2 and 3 can be maximized by using a stiff receptor structure that surrounds the preferred ligand at less than the minimum-energy distance. The use of such tight-receptor structures sacrifices 'binding energy in exchange for a particular kind of specificity, and may reduce discrimination against ligands in category 1 . One of the smaller increments in steric bulk occurs between structures in which CH\mathrm{CH} replaces N\mathrm{N}, shifting regions of similar overlap energy outward by 0.09 nm\sim 0.09 \mathrm{~nm} over a small region. If the initial separation between an N\mathrm{N} atom in a ligand and an atom of the receptor is chosen to be 0.3 nm\sim 0.3 \mathrm{~nm}, computational experiments in the MM2 approximation indicate that the adverse overlap energy experienced by the competitor can be 45maJ\sim 45 \mathrm{maJ}, sufficient to alter affinities by a factor 5×104\sim 5 \times 10^{4} at 300 K300 \mathrm{~K}, according to Eq. (13.2). (Note that in the present application, preferred ligand molecules can be selected with ease of purification as a criterion, hence structures presenting unusual difficulties can be avoided during the design stage.)

This example-discriminating between two species differing by the substitution of CH\mathrm{CH} for N\mathrm{N}-also illustrates the utility of nongeometric mechanisms for molecular discrimination. Only small differences in shape arise from replacing a lone pair with a bond to hydrogen, but the resulting structures differ greatly in their electronic properties. In this instance, the nitrogen lone pair (but not the CH)\mathrm{CH}) can participate in the formation of a dipolar bond or a hydrogen bond. Geometrically, such bonds permit a closer contact between the N\mathrm{N} and the interface of the receptor. Energetically, they permit stronger binding. In general, different species with similar shapes have dissimilar electronic properties, permitting nongeometric mechanisms for discrimination.

d. A class of tight-receptor mechanism. A tight-receptor mechanism can be implemented as shown in Figure 13.2. In this paired-rotor approach, all competing molecular species are in fast equilibrium with the fully exposed receptors on the left, but only the preferred species (and strictly smaller ligands) can pass through the fully enclosed position between the two rotors without encountering a large barrier caused by overlap repulsion. According to the estimate made in Section 13.2.1c, barriers >30>30 maJ should be easily implemented. Tuning of the rotor potential energy function can effectively eliminate barriers when passing the preferred ligand (the common case for a purification process).

The paired-rotor, tight-receptor mechanism requires a compliant drive mechanism to operate properly. With an excessively stiff drive mechanism, even bulky competing ligands would routinely be forced through the gap (which must permit their passage in order to avoid failure by mechanical obstruction, regardless of drive stiffness). With a sufficiently compliant drive mechanism, however, the rotor moves (on short time scales) as if in free rotational diffusion. The rotational relaxation time for a rotor this size is <107 s<10^{-7} \mathrm{~s}, assuming a mean viscosity

Figure 13.2. A sorting rotor as in Figure 13.1, but with an auxiliary rotor that forces bound molecules to pass through a totally enclosed state, effectively excluding molecular species that are unable to fit within a volume of defined size and shape (for discussion, see Section 13.2.1d). The alignment of the primary and auxiliary rotors can be ensured by means of a geared interface (not shown).

of the surrounding medium like that of water (Creighton, 1984). Accordingly, in a system transferring receptors at mean intervals of 106 s10^{-6} \mathrm{~s}, a bulky competing ligand momentarily blocked by a barrier has ample time to be carried backward by several receptor diameters and escape into the surrounding solution. A nonlinear compliance can provide a small driving force (e.g., 1012 N\sim 10^{-12} \mathrm{~N}, measured at the rotor rim) with negligible stiffness over a certain angular range of motion (e.g., several radians), while providing a strong, stiff constraint that prevents larger excursions of the rotor relative to an underlying drive mechanism of ordinary stiffness. This appears to meet the relevant constraints.

Mechanisms of this sort can strongly discriminate against ligands unable to fit within the space occupied by the preferred ligand. This capability is exploited in the analysis of staged cascades for purification of input streams.

e. Energy dissipation. Properly designed sorting rotor systems can approach thermodynamic reversibility in the limit of slow motion when handling fluids of nearly pure composition (otherwise, entropies of mixture can be significant). Aside from speed-dependent (chiefly viscous) drag mechanisms, free energy is dissipated when potential wells in disequilibrium are merged (Section 7.6.2). Well merging occurs in a system like that of Figure 13.1 or 13.2 whenever a receptor is exposed to a molecular reservoir; to avoid energy dissipation during this process, the wells must, in each configuration as they are exposed, have a probability of occupancy that is precisely the probability of occupancy that they would have after a long, equilibrating period of exposure in that configuration. This condition is automatically achieved for well-blocked receptors in the outbound transportation process: their probability of occupancy is zero when exposed to the interior reservoir, and remains zero when they are exposed to the exterior reservoir. This condition can also be achieved for receptors during the inbound transportation process between two reservoirs containing known concentrations: a receptor with a particular configuration has a certain probability of occupancy as it leaves the external reservoir, and would (if unchanged) have a higher probability of occupancy on exposure to the internal, higher-concentration reservoir. A downward modulation of the affinity of the receptor during the inbound transportation process, however, can make these probabilities correspond. This can be achieved by adjusting the shape of the cam surface.

Viscous drag can be estimated from rotor dimensions, given some choice of effective viscosity for the surrounding solution environments.1 With the dimensions described in Section 13.2.1a, the wetted area per rotor is 50 nm2\sim 50 \mathrm{~nm}^{2}, and the characteristic length for shear (which is subject to considerable design-stage control) is 2.7 nm\sim 2.7 \mathrm{~nm}. At a rim speed of 0.0027 m/s(1060.0027 \mathrm{~m} / \mathrm{s}\left(10^{6}\right. receptor-sites per second )), the drag power in a fluid of viscosity η=103 Ns/m2\eta=10^{-3} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2} is 1016 W\sim 10^{-16} \mathrm{~W}, or 0.1maJ\sim 0.1 \mathrm{maJ} per receptor transferred. Sliding-interface drag in associated bearings can be made negligible in comparison.

The energy dissipation per operation can be reduced somewhat by using thick rotors with several rows of receptors, thereby increasing the characteristic scale length of fluid shear and reducing fluid structure effects. Dissipation can be reduced to a greater extent by increasing the number of rotors (or the number of rows of receptors per rotor) and slowing the rotor speeds in proportion. If a single-row rotor has 2×104\sim 2 \times 10^{4} atoms (Section 13.2.1a), and a rotor with its housing and prorated share of the drive system contains 105\sim 10^{5} atoms, then at 10610^{6} operations per second, it processes its own mass in <0.1 s<0.1 \mathrm{~s}. A considerably lower frequency would remain compatible with high productivity.

f. Damage mechanisms and lifetime. Modulated receptors for concentrating and purifying molecules differ from other mechanisms considered in this volume in that they are, by the nature of their task, exposed to an ill-defined chemical environment. Their susceptibility to damage caused by chemical reactions varies with structure and environment, and can most easily be estimated by analogy to other ill-defined chemical systems.

The surface reactivities of diamondoid structures (if designed with low surface strain, etc.) resemble the reactivities of smaller organic molecules. The stability and lifetime of solvent molecules in relatively pure, unreactive solutions can thus provide a model for the stability and lifetime of small patches of surface structure, including receptors, exposed to a reasonably pure, unreactive feedstock solution. This suggests reasonably long operational lifetimes.

Damage resulting from trace quantities of highly reactive contaminants can be minimized by flowing feedstock solutions past surfaces bearing bound moieties resembling those used on critical rotor surfaces, but selected for higher reactivity. Sacrificial moieties of this sort can combine with and neutralize many reactive species, including free radicals.

Proteins in living systems provide a model for molecular machines in a relatively complex, chemically aggressive environment. Metabolic enzymes can have lifetimes of several days (Creighton, 1984), despite the relative fragility of protein structures. Lifetimes of diamondoid sorting rotors of greater stability in a more benign environment should be longer, but even a one-day lifetime is sufficient for a device to process 106\sim 10^{6} times its own mass before requiring replacement. (Issues of redundancy and system reliability are addressed in more detail in Chapter 14.)

13.2.2. Cascades of modulated receptors

To achieve low error rates in a molecular manufacturing process requires an input stream with a low fraction of contaminant molecules. For example, if a device is to perform 10610^{6} operations per second for several decades without an error, the fraction of contaminant molecules must be held to 1015\leq 10^{-15}. This degree of purity can readily be achieved (for a wide range of reasonably stable feedstock molecules) by using a cascade of modulated receptors.

Multi-stage cascade systems are ubiquitous in chemical engineering practice, with implementations including leaching systems, distillation columns, extraction columns, and the like. The analysis of cascades is in general complex, and is described in books such as (McCabe and Smith, 1976). The following section presents an analysis of a simple case that illustrates the essential properties of cascades for the present application.

a. A model of staged cascade systems. Cascades (see Figure 13.3) combine multiple separation stages in a counterflow system. For simplicity, consider a system with NN reservoirs of fluid containing predominantly molecules of a single type (the desired product), but contaminated by small concentrations c0,c1c_{0}, c_{1}, c2,cN1c_{2}, \ldots c_{N-1} of impurity molecules (measuring concentrations in molecule fractions). Each reservoir nn (save the last) is linked to reservoir n+1n+1 by an inbound flow (from the source to the destination) carrying fin f_{\text {in }} molecules per second, and each save the first is linked to reservoir n1n-1 by an outbound flow carrying fout f_{\text {out }} molecules per second. (This is a simple instance of a flow pattern that permits mass balance in each reservoir.) Each flow is the product of a selective transport mechanism characterized by a differential affinity for impurity and product molecules; the inbound stream from compartment nn thus has an impurity concentration cnαin c_{n} \alpha_{\text {in }}, and the outbound stream has an impurity concentration cnαout c_{n} \alpha_{\text {out }}. The condition for mass balance for impurities in reservoir nn is then

cn1αin fin +cn+1αout fout (cnαin fin +cnαout fout )=0\begin{equation*} c_{n-1} \alpha_{\text {in }} f_{\text {in }}+c_{n+1} \alpha_{\text {out }} f_{\text {out }}-\left(c_{n} \alpha_{\text {in }} f_{\text {in }}+c_{n} \alpha_{\text {out }} f_{\text {out }}\right)=0 \tag{13.3} \end{equation*}

The assumption of stage-independent values for fin ,fout ,αin f_{\text {in }}, f_{\text {out }}, \alpha_{\text {in }}, and αout \alpha_{\text {out }}, together with a constant value of c0c_{0}, leads to a steady-state solution in which cnc_{n} declines exponentially with nn. Convenient variables are then

fout /fin =frat ;αout /αin =αrat ;cn/cn1=cn+1/cn=R\begin{equation*} f_{\text {out }} / f_{\text {in }}=f_{\text {rat }} ; \quad \alpha_{\text {out }} / \alpha_{\text {in }}=\alpha_{\text {rat }} ; \quad c_{n} / c_{n-1}=c_{n+1} / c_{n}=R \tag{13.4} \end{equation*}

where the value of RR, which defines the rate of the exponential decline, is determined by

R=(frat αrat )1\begin{equation*} R=\left(f_{\text {rat }} \alpha_{\text {rat }}\right)^{-1} \tag{13.5} \end{equation*}

Figure 13.3. Schematic diagram of a portion of a staged-cascade process (Section 13.2.2a).

The final impurity concentration is then

cN=c0RN\begin{equation*} c_{N}=c_{0} R^{N} \tag{13.6} \end{equation*}

which for R1\mathrm{R} \ll 1 can be extremely small for modest values of NN. Although real systems will never correspond precisely to this model, it can be used to estimate the relationships among receptor properties, system size, relative flow rates, and product purity. Figure 13.4 illustrates a modulated-receptor implementation of a cascade system.

b. Purification using staged cascades of modulated receptors. For concreteness, consider a system that must deliver a stream of product molecules with a contaminant fraction 1015\leq 10^{-15}, starting with a feedstock mixture containing

Figure 13.4. Schematic diagram of a staged-cascade process based on sorting rotors like those of Figure 13.1. (Note the inversion of the cams in the lower rotors.)

product molecules at a fractional concentration of 102(0.1 nm3)10^{-2}\left(\sim 0.1 \mathrm{~nm}^{-3}\right). For generality, assume that only the universally available geometric properties of molecules are exploited to achieve receptor specificity, neglecting the use of electrostatics, hydrogen bonds, and so forth (all of which play prominent roles in binding by proteins). Although the staged-cascade model of Section 13.2.2a assumes that the selectivity factor αrat \alpha_{\text {rat }} is a constant, the presence of a mixture of impurities complicates this picture, motivating the use of different receptor structures and specificities at different stages.

Receptors for the inward transport mechanism of the first stage are best designed for high product affinity in the presence of a dominant background of quite different molecules. Analogies with antibodies (Section 13.2.1a) suggest that an inward-transport rotor can deliver a product stream with impurity fractions of 104\sim 10^{-4} to 10910^{-9}, depending on affinities, specificities, and the concentrations of the effectively competing ligands. The value of αrat \alpha_{\text {rat }} for the transport stage depends on the specificity of the outbound mechanism. If this is completely unselective, and if frat =0.1f_{\text {rat }}=0.1 (corresponding to a modest outbound flow rate), then the value of RR for this stage can range from 103\sim 10^{-3} to 10810^{-8}, yielding impurity fractions of similar numerical magnitudes.

Receptors for the subsequent purification stages should usually be designed with lower product affinities (Section 13.2.1c), to ensure equilibrium rather than kinetic control of transport selectivity. With tight-receptor mechanisms (Section 13.2.1d), the concentrations of competing ligands having excess bulk is reduced rapidly, with feasible values of R5×103R \geq \sim 5 \times 10^{3}, given frat =0.1f_{\text {rat }}=0.1 (Section 13.2.1c). Ligands substantially smaller than the product molecule are effectively discriminated against by loss of van der Waals attraction energy.

After several stages of this sort, the chief impurities will typically be molecules that resemble the product, save for lower steric bulk in some region. The number of molecular types that can stand in this relationship to an already-small product molecule are few. Low values of RR can be ensured in this circumstance by use of several selective, tight-receptor mechanisms in the outbound direction, each adapted to bind a specific impurity species.

With transport mechanisms adapted to each stage as just discussed, values of αrat 104\alpha_{\text {rat }} \geq 10^{4} should be readily achievable throughout, hence analyses using R=R= 10310^{-3} will yield conservative results. Accordingly, producing a stream with an impurity fraction 1015\leq 10^{-15}, starting with a feedstock with an impurity fraction .99\sim .99, should require a number of stages N5N \leq 5.

c. Overall performance. To construct a system with relatively large throughput, many sorting rotors can be used in parallel, each with multiple rows of modulated receptors; the number of rotors in a set can then be adapted to the magnitude of the stream of molecules to be handled, and operating rates can be equalized. Parallel systems lend themselves to the construction of redundant architectures in which the useful system lifetime is comparable to the mean lifetime of the individual rotors. As noted in Section 13.2.1e, rotors with multiple receptor rows can also reduce the viscous drag loss per receptor.

The rotors handling the inward flow streams are the dominant source of energy dissipation, assuming low values of frat f_{\text {rat }} as above. For N=5N=5 and capacities of 106\sim 10^{6} molecules per second per row, losses from this mechanism amount to 0.5maJ\sim 0.5 \mathrm{maJ} per molecule delivered, using the parameters assumed in Section 13.2.1e. This is small compared to the useful work performed in molecular concentration ( 20maJ\sim 20 \mathrm{maJ} per molecule, Section 13.2.1a); losses from entropy of mixing are greater than this, amounting to 10%\sim 10 \% of the useful work (assuming frat =f_{\text {rat }}= 0.1). With the estimated per-rotor masses (etc.) assumed in Section 13.2.1a, a purification system can deliver a quantity of product equaling its own mass in <0.5 s<0.5 \mathrm{~s}.

d. Handling harder cases. Separation on the basis of shape will fail (or, more properly, require many steps) if shapes are nearly identical. Among small molecules (defined at the beginning of Section 13.2) that consist of the elements of greatest interest (Figure 1.3), differences in atomic valence, bond length, and nonbonded radius guarantee substantial differences in shape. Among molecules containing transition or lanthanide metals, however, these differences can become quite subtle. In these instances, alternative strategies relying more heavily on chemical and electronic properties (e.g., differential complexation of ionic species, as in chromatography) may be desirable; laboratory experience demonstrates that these elements can be separated in this way. In manufacturing systems consuming the mix of elements expected to be typical, separation of these more difficult elements will be a quantitatively small portion of the overall processing task, having little impact on such system parameters as mass, energy consumption, and productivity.

13.2.3. Ordered input streams

Sorting processes of the sort described in Section 13.2.2 can keep a reservoir filled with molecules that are reliably of a single type. To provide deterministic inputs to a molecular manufacturing process, these molecules must be transformed into a stream of bound moieties (e.g., in a molecular mill of the sort described in Section 13.3) such that each site in the stream is reliability occupied. This could be accomplished by combining an unreliable mechanism for ordering and reactively binding molecules with a mechanism for probing and sorting the resulting structures; this section will instead describe approaches capable of achieving reliability directly.

a. Steps in ordering and input. One approach to providing ordered inputs comprises the following steps:

  1. Expose a receptor to a pure liquid under conditions that ensure reliable occupancy of that site by a bound molecule.
  2. Expose the bound molecule to a reagent moiety (on a separate transport mechanism) under conditions that ensure the reliable formation of a specific covalent bond.
  3. Transfer the molecule to the other transport system as a covalently bound moiety, emptying the receptor for reuse.

This approach to molecular ordering is illustrated in Figure 13.5, with transfer to a reagent belt mechanism like those discussed in Section 13.3.1. In a mechanism of this sort, step 3 is a natural consequence of the pattern of motion, leaving steps 1 and 2 as the primary design problems.

b. Ensuring reliable receptor occupancy. A receptor will be occupied reliably (Pempty 1015)\left(P_{\text {empty }} \leq 10^{-15}\right) if the free energy of binding has a magnitude greater than 143maJ\sim 143 \mathrm{maJ}. The loss of entropy that occurs when a molecule is bound from solution is comparable to the loss of entropy in freezing to a crystal; for small organic molecules, a typical value is 6×1023 J/K6 \times 10^{-23} \mathrm{~J} / \mathrm{K}, hence the required potential energy of binding is ΔVbinding 161maJ\Delta V_{\text {binding }} \approx-161 \mathrm{maJ}.

Two sources of driving force can be applied to increase ΔVbinding \Delta \mathcal{V}_{\text {binding }} : attractive intermolecular forces to reduce the energy of the bound molecule, and hydrostatic pressure to increase the Gibbs free energy of the unbound molecules. In general, feasible binding energies increase in magnitude with increasing molecular size, polarizability, and dipole moments. Feasible hydrostatic pressures (i.e., those that do not cause the liquid to solidify) tend to decrease with the same variables. The magnitudes of feasible binding energies can be evaluated using molecular mechanics methods; the magnitudes of feasible hydrostatic pressures aiea_{i} e indicated by data on liquids at high pressures.

Figure 13.6 shows a model receptor for a small nonpolar molecule, ethyne; molecular mechanics calculations indicate ΔVbinding =97maJ\Delta \mathscr{V}_{\text {binding }}=-97 \mathrm{maJ} (this example is chosen to illustrate receptor principles with a simple, symmetrical structure, not necessarily to recommend ethyne as a feedstock). A larger model (including more atomic layers) would increase van der Waals attractions, increasing the magnitude of the binding energy. Achieving ΔGbinding =161maJ\Delta \mathscr{G}_{\text {binding }}=-161 \mathrm{maJ} requires an increment of 64maJ64 \mathrm{maJ}; this can be supplied by hydrostatic pressure.

Many fluids of small organic molecules remain liquid at pressures of 1\sim 1 to 2GPa2 \mathrm{GPa} at 300 K300 \mathrm{~K} (e.g., methanol, >3GPa>3 \mathrm{GPa}; ethanol, >2GPa>2 \mathrm{GPa}; acetone and carbon

Figure 13.5. Schematic diagram of a mechanism for removing molecules from a liquid phase and covalently binding them to a moving belt (of the sort described in Section 13.3). This diagram omits mechanisms for modulating the properties of the receptor; these would be necessary in order to approach thermodynamic reversibility (exposure of an empty receptor as shown would be inherently dissipative). Generalized mechanisms for modulating the receptor can also relax constraints on reaction geometry and conditions (for example) by forcing compression and motion of the confined molecule.

Figure 13.6. MM2 model of an ethyne receptor.

disulfide, >1GPa>1 \mathrm{GPa} ), and in this pressure range, typical organic liquids have a volume per molecule 0.75\sim 0.75 that at atmospheric pressure (Gray, 1972). The volume per molecule of ethyne (low-pressure liquid density =621 kg/m3=621 \mathrm{~kg} / \mathrm{m}^{3} ) is accordingly 5×1029 m3/\sim 5 \times 10^{-29} \mathrm{~m}^{3} / molecule in the pressure range of interest. Supplying an energy increment of 64maJ64 \mathrm{maJ} per molecule thus requires a pressure of 1.3GPa\sim 1.3 \mathrm{GPa}, at which liquids consisting of small, nonpolar organic molecules like ethyne are still far from solidification.2 Larger molecules require less pressure for a given energy, and require less energy to make up for deficits in the interaction energy with the receptor. Conversely, smaller molecules require more pressure and a larger energy increment, but tolerate greater pressures without solidification. The range of feedstock molecules that can be ordered in this manner is large.

c. Bonding to a molecule in a receptor. For suitably chosen feedstock molecules and receptor designs, it can be assumed that the molecule is either well oriented, or that patterns of symmetry and reactivity make molecular orientation unimportant. The properties of the receptor itself need not constrain the reaction environment: the bound molecule is confined by moving under the capping surface (Figure 13.5), and can then be subjected to direct, forcible mechanical manipulation. Such processes can move it to a new location characterized by high free energy and stiff intermolecular contacts.

The reaction that transfers the molecule must take the form of an addition (unless means are provided for handling displaced groups). For unsaturated molecules, radical additions or Diels-Alder cycloadditions are candidates. For saturated molecules, additions to transition metals [e.g., Eq. (8.34)] are candidates. To overcome activation energy barriers, large piezochemical forces can be applied. Strategies for forming strong bonds with little energy dissipation are available (Section 8.5.6). The principles underlying these processes are the same as those in the mechanosynthetic processes discussed in Chapter 8. In working with small molecules, however, it is difficult to provide stiff restoring forces that resist motion of the molecule in the direction of the attacking moiety. This limitation may, for example, preclude low energy dissipation in radical coupling reactions involving small molecules (see the related discussion in Section 8.5.3b).

As is true throughout the design of materials input subsystems, any candidate feedstock molecule that presents an intractable problem in some regard can be discarded in favor of an alternative. Feedstock molecules can be chosen to provide the necessary elemental inputs in convenient forms.

13.3. Transformation and assembly with molecular mills3

Chapter 8 assumes that accurately controlled mechanical motions can be imposed as a boundary condition in molecular systems, implying control of positions, trajectories, and forces during chemical transformations. This section describes mechanisms that can provide such control using systems of belts and rollers; Section 13.4 describes more flexible (and less efficient) mechanisms based on programmable manipulators.

13.3.1. Reactive encounters using belt and roller systems

Systems of belts and rollers can be used to implement mill-style molecular processing systems. Belts moving over rollers and along guides find widespread use in macroscale manufacturing: they transmit power and transport materials and parts. They can serve this function in nanoscale systems as well. With the help of auxiliary mechanisms-such as moving parts in { }^{\circ}reagent devices, rollers and cam surfaces to drive those parts, devices for transfer to and from other transport systems-molecular mills built in this fashion can perform a wide range of mechanochemical operations within a larger manufacturing system.

a. Simple encounters. The simplest reactive encounters involve direct contact and separation, a motion which can be provided by a device like that shown in Figure 13.7(a). An encounter of this sort can transfer a group through an abstraction reaction (Section 8.5.4) or through a more complex chemical mechanism (e.g., those illustrated in Figures 8.12 and 8.14). Among the mechanical parameters subject to design control are the peak compressive load (which can be >10nN>10 \mathrm{nN}, Section 8.3.3c8.3 .3 \mathrm{c} ), the stiffness of reagent moieties relative to one another along the line of motion ( 90 N/m\sim 90 \mathrm{~N} / \mathrm{m} with sufficiently stiff bearings; see Section 8.5.3d), and the relative stiffness of the moietal supporting structures perpendicular to this line (with pairs of reagent devices having sterically complementary interfaces placed under compressive load, this can easily exceed 50 N/m50 \mathrm{~N} / \mathrm{m} ). An encounter of this sort can be performed with a belt-belt contact, a belt-roller contact, or a roller-roller contact (only belt-belt contacts are illustrated in Figure 13.7).

The reaction time depends on belt speed, moietal trajectories (determined by the belt and wheel geometries), and the range of positions regarded as comprising the encounter state. Assuming that the trajectories are characterized by simple circular motions of radii r1r_{1} and r2r_{2} at a speed vv with a positional tolerance of ±δ\pm \delta (assumed small), the reaction time is

ttrans2v(r11+r21)1cos1(12δ(r11+r21))4vδ(r11+r21)1\begin{equation*} t_{\mathrm{trans}} \approx \frac{2}{v}\left(r_{1}^{-1}+r_{2}^{-1}\right)^{-1} \cos ^{-1}\left(1-2 \delta\left(r_{1}^{-1}+r_{2}^{-1}\right)\right) \approx \frac{4}{v} \sqrt{\delta\left(r_{1}^{-1}+r_{2}^{-1}\right)^{-1}} \tag{13.7} \end{equation*}

For v=0.005 m/s,r1=r2=10 nmv=0.005 \mathrm{~m} / \mathrm{s}, r_{1}=r_{2}=10 \mathrm{~nm}, and δ=0.01 nm,ttrans 1.8×107 s\delta=0.01 \mathrm{~nm}, t_{\text {trans }} \approx 1.8 \times 10^{-7} \mathrm{~s}, moderately greater than the typical value assumed in Chapter 8. Fast, simple-encounter reactions could be performed with higher speeds and smaller mechanisms. Reaction times can be indefinitely prolonged by placing belts in contact and passing them between a pair of backing surfaces, as in Figure 13.7(b).

b. Complex encounters. In complex reactions (e.g., exploiting pi-bond torsion, Section 8.5.6), a device must drive several coordinated, properly sequenced

(b)

50 nm\sim 50 \mathrm{~nm}

(c) three outputs ( 1/31 / 3 frequency)

(d)

100000\sim 100000 atoms per roller assembly

Figure 13.7. Schematic diagrams of reactive encounter mechanisms, providing for simple contact and separation (a), prolonged contact with opportunity for substantial cam-driven manipulation (b), and transformations of moiety transit frequency (c) and speed (d).

motions. This can be accomplished by elaborating the structure of the reagent device to include moving parts actuated by auxiliary rollers or cam surfaces (Chapter 10), while extending the encounter time (if need be) as suggested in Figure 13.7(b).

This approach enables thorough control of molecular trajectories and reaction environments. As the two reagent devices come together, they can entirely surround the pair of moieties with a eutactic environment. Multinanometerscale reagent devices containing multiple moving parts can manipulate this environment, forcing the moieties and adjacent structures to execute motions driven by stiff mechanisms capable of exerting large forces. Cam surfaces adjacent to the belts can interact with one or more protruding cam followers on the device to guide and power an arbitrarily long sequence of arbitrarily complex motions. Relatively small and simple mechanisms in this class are capable of exploiting pi-bond torsion effects, performing three-moiety encounters such as regeneration of alkynyl radicals assisted by radical displacement (Section 8.5.7), and so forth.

A complex encounter mechanism can also exploit nonbonded interactions to force the transfer of a group. For example, a bulky structure joined to another by a single bond can be mechanically clamped and then pulled to force cleavage of the bond and creation of a pair of radicals.

c. Conditionally repeated encounters. Section 8.3.4f describes the use of conditional repetition to convert a reaction that on the basis of single-encounter thermodynamics and kinetics has a probability of failure PP into a reaction process with a smaller probability of failure PnP^{n}, where nn is the potential number of encounters. Conditional repetition can both increase reliability and (by permitting the use of less exoergic reactions) decrease energy dissipation.

Figure 13.8 schematically illustrates a reagent device incorporating a probe and a cam follower; the specific geometry and kinematics represent only one embodiment of a broad class of analogous mechanisms. For successful operation, the probability of the probe moving to its extended position [a state like (d)] must be negligible (e.g., 1015\leq 10^{-15} ) while the transferable group is still present. The driving force for this motion can be small (e.g., 0.1nN\sim 0.1 \mathrm{nN} ), and the formation of a bond between the probe and the site exposed by transfer of the group is among the acceptable outcomes, so long as that bond can be broken by the forcible retraction of the probe.

Figure 13.9 illustrates the use of a reagent device like that in Figure 13.8 to perform a conditional-repetition process, the net effect of which is to attempt the transfer reaction if and only if the group has not yet been transferred. As illustrated, the blockage of further attempts after a successful transfer can be direct, by blocking the reactive site; the motion of the probe structure could equally well achieve the same result though indirect mechanical linkages. Note that analogous mechanisms can be devised that make reaction attempts contingent on the absence of a group, hence the probe assembly need not be on the source side of a reactive encounter system.

When the cam follower passes through a location where two cam grooves merge, it undergoes a well merging process of the sort analyzed in Section 7.6. In any particular mechanism, the wells being merged have fixed probabilities of (a)

(b)

steric probe blocked (c)

steric probe retracted, group transferred (d)

steric probe unblocked

10000\sim 10000 atoms

Figure 13.8. Schematic diagram of a reagent device for implementing a conditionalrepetition strategy to increase reaction reliability. In (a), the reagent moiety with its transferable group is exposed; the steric probe can be placed at any desired distance at this time. In (b), the probe is blocked by the transferable group and prevented from moving past it by a large (e.g., 150maJ150 \mathrm{maJ} ) energy barrier; the bias force need not be large. In (c), a successful reaction has transferred the group, removing the barrier. In (d), the bias force has driven the probe past its previous stopping point.

Figure 13.9. Schematic diagram of a conditional-repetition system. Each small diagram below (numbered 1 to 10) represents the state of a mechanism like that illustrated in Figure 13.8 at a different time as it moves along one of the five possible paths through the system. The block above represents a structure with a network of cam grooves, shown from behind (with supporting material cut away). The cam-follower protrusions on the sliding mechanisms are (in the assembled configuration) constrained to follow the grooves. When a groove branches, the path taken is determined by the presence or absence of the transferable group at times 2,4,6,82,4,6,8, and 10 (see Figure 13.8). So long as the group is present, the cam follower remains in the upper groove. Once it is absent (in the illustrated sequence, at time 6), the bias force moves the follower to the lower groove. The position of the probe then prevents further reactive encounters, as discussed in Section 13.3.1c. Several reactive devices can simultaneously be in transit through a cam mechanism of this sort, following different paths.

occupancy. If the sizes and depths of these wells are designed such that Eq. (7.73) holds, then the merging wells are in effect preequilibrated, and there is no fixed lower bound on the energy dissipation. Under these conditions, the work of compression done in merging wells directly increases the free energy of the product by reducing its entropy (i.e., increasing reaction reliability).

13.3.2. Interfacing mechanisms

a. Frequency and speed. System-level flexibility requires several kinds of transformations among streams of reagent moieties in molecular mill systems. Where several consumer subsystems require inputs of moieties at low frequencies, but a producer subsystem can efficiently produce them at high frequencies (or vice versa), it is desirable to transform frequencies by splitting and merging streams. Figure 13.7(c) diagrams how this can be done.

Differing subsystems can have widely differing optimal speeds of belt motion. Figure 13.7(d) diagrams how sites on a belt moving at one linear speed can be placed into smooth, nonsliding contact with sites on a belt moving at another linear speed while transferring materials.

b. Transfer to pallets, nondeterministic interfaces. In subsystems built using devices of the sorts described thus far, the rate of motion and frequency of operation at any given point fully determine the rates and frequencies at every other point (save for transient fluctuations). In building larger systems, it is desirable to surround subsystems of this sort with interfaces permitting greater flexibility of operation. In particular, this is required for fault-tolerant operation, which in turn is required in building relatively large and durable systems.

Transfer of a stream of moieties or product structures to a stream of pallets, moving along a track but not linked to form a belt, can provide the necessary flexibility. In regions where pallets are close packed, a stream of pallets can be treated like a moving belt, and similar transfer operations can be performed. In other regions, pallets can be more or less closely spaced (permitting the accumulation and use of buffer stocks), and the tracks bearing them can split or merge.

Two classes of track junction are useful in the design of fault-tolerant systems. These are fair, nonblocking merging junctions, and fair, nonblocking distribution junctions. A merging junction (1) accepts at least two input streams of pallets and produces a single stream of output pallets that includes contributions from both inputs (i.e., it is fair), and (2) continues to produce a stream of output pallets so long as one input stream has not failed (i.e., stopping inputs does not block the node). Similarly, a distribution junction (1) accepts a single input stream and produces one or more output streams, distributing the inputs across all the outputs, and (2) continues to distribute inputs to an output so long as one output stream remains unblocked. A failed junction of either kind is assumed to block one or more of its inputs; in the worst case, failure blocks all inputs (and thus outputs) by blocking all motion through the junction. These simple abstract behaviors can be implemented with active mechanisms, and perhaps without them.

A network of mechanosynthetic devices linked in this manner can behave as a demand-driven supply system. The combination of applied drive forces and mechanochemically derived forces from exoergic reactions can bias devices toward outputting product, but their output can be blocked when space for more product becomes unavailable. Production is then paced by the rate of removal of products at the end of the network, and no computational synchronization of production with demand is necessary. This somewhat resembles justin-time delivery systems in conventional manufacturing, in which empty containers are the signal to produce more parts, but it more nearly resembles the delivery of water by a pipe, in which removal of material at one end creates a pressure gradient that moves material throughout the length.

13.3.3. Reagent preparation

Molecular mills can be used to transform low-entropy streams of feedstock molecules (Section 13.2.3) into low-entropy streams of reagent moieties covalently bound to reagent devices. Each of the group-transfer reactions in a process of this sort can occur in an enclosing environment designed to facilitate that reaction, to minimize { }^{\circ}misreactions, and to minimize energy dissipation. Some of the feasible reactions and associated mechanochemical manipulations are surveyed in Chapter 8.

A typical input molecule might be ethyne; a typical reagent moiety might be a strained alkyne like 8.43. The intermediate stages in this process might include two radical additions (Section 8.5.5) and hydrogen abstractions (Section 8.5.4), followed by the regeneration of the abstraction tools in a process that ends with the elimination of molecular hydrogen from a transition metal, Eq. (8.33). Each of these steps (save the last) requires an encounter between two reactive groups, and each results in the transfer of some number of atoms between one site and the other. This example suggests that cycles that prepare reagent devices by charging them with fresh reagent moieties may typically require on the order of 10 reactive encounters.

13.3.4. Reagent application

Molecular mills can be used to apply reagent devices to workpieces. Although programmable manipulators can do this with greater flexibility, producing many different products using one mechanism, mills can produce a single product with greater efficiency. Architectures for flexible manufacturing of macroscale products will likely include both mill-style and manipulator-style assembly mechanisms (Section 13.4).

Reagent application closely parallels reagent preparation, except that the workpiece-side structure in each reactive encounter is subject to the constraint that it be an intermediate in the construction of a desired device. This precludes thorough adaptation of the reaction environment for the purpose of facilitating reliable, low-dissipation operations. Nonetheless, the freedom to choose designs that can in fact be manufactured and synthetic sequences that generate relatively favorable intermediate structures provides considerable latitude for avoiding unfavorable reaction environments.

13.3.5. Size and mass estimates

The masses of components in a mill-style processing mechanism are significant both in estimating system productivity and in estimating radiation-damage lifetimes. As suggested by the discussion in Section 13.3.1, the structure (and hence the mass) of reactive encounter mechanisms can vary widely; to make a rough estimate of the typical system mass on a per-mechanism basis, a standard density can be assumed for all components (here, 2500 kg/m32500 \mathrm{~kg} / \mathrm{m}^{3} or 125\sim 125 atoms /nm3/ \mathrm{nm}^{3} ), and reference sizes can be allocated to provide for a moderately complex process and for transport of moieties over a moderately long distance between processes.

a. Component sizes for the encounter mechanism. Roller pairs and backing-surface pairs can have comparable masses; the roller option is considered here. A radius of 5 nm5 \mathrm{~nm} and a mean thickness of 2 nm2 \mathrm{~nm} provide ample room for a stiff bearing of the sort described in the sample calculations of Section 10.4.6 and shown in Figure 10.17. A pair of rollers, including axles, has a volume of 310 nm3\sim 310 \mathrm{~nm}^{3}.

A moderately complex encounter process can be driven by a mechanism occupying a volume 4 nm4 \mathrm{~nm} on a side, or 64 nm364 \mathrm{~nm}^{3} (four times the volume of the exemplar interlock mechanism described in Section 12.3.3). This mechanism can be formed by the mating of reagent devices from each of two facing belts, each of which can then be 4×4×24 \times 4 \times 2 nanometers in size. An auxiliary cam surface of substantial complexity can fit within a space 4×2×104 \times 2 \times 10 nanometers in size, adding 80 nm380 \mathrm{~nm}^{3}.

These estimates are appropriate for reagent preparation processes, where only small moieties are handled. They can serve reasonably well for reagent application processes, so long as the workpiece is limited to a diameter of several nanometers.

b. Component sizes for associated structure and transport mechanisms. A mean belt length of 20 roller-radii (=100 nm)(=100 \mathrm{~nm}) per roller provides for substantial flexibility of system layout. If each belt is a strip with a cross sectional area of 0.5 nm20.5 \mathrm{~nm}^{2}, then the volume is 50 nm350 \mathrm{~nm}^{3} plus that of one or more reagent-device pairs. If each belt is a close-packed chain of 4×4×24 \times 4 \times 2 nanometer reagent devices, the volume of belt structure per roller-pair is 1600 nm31600 \mathrm{~nm}^{3}. In the latter case, there are 50 reagent moieties per reactive encounter mechanism.

In a reasonably densely packed system, the distances between encounter mechanisms can be short; they can be stacked in parallel in near contact. A supporting structure with a mass equal to that of five struts of 4 nm24 \mathrm{~nm}^{2} cross sectional area, each 15 nm15 \mathrm{~nm} ( =3=3 roller radii) in length seems ample; this amounts to 300 nm3300 \mathrm{~nm}^{3} of structural material per roller pair.

c. Totals (per reactive encounter mechanism). The total volume, mass, and atom-count per reagent device depend substantially on the choice of reagentdevice density along the belts. At the upper extreme, where the belt consists of a chain of close-packed devices, the numbers are 2300 nm3,5.7×1021 kg\sim 2300 \mathrm{~nm}^{3}, \sim 5.7 \times 10^{-21} \mathrm{~kg}, and 2.9×105\sim 2.9 \times 10^{5} atoms. Systems of this sort have relatively high throughput and energy efficiency, and will serve as the basis for later calculations unless otherwise indicated. At the lower extreme, with only one pair of reagent devices per roller pair, the numbers are 800 nm3,2.0×1021 kg\sim 800 \mathrm{~nm}^{3}, \sim 2.0 \times 10^{-21} \mathrm{~kg}, and 1.0×105\sim 1.0 \times 10^{5} atoms. Systems of this sort have lower throughput and efficiency, but less mass per kind of operation performed (i.e., greater versatility per unit mass). For comparison, 2.0×1021 kg2.0 \times 10^{-21} \mathrm{~kg} is 40\sim 40 times the mass of many enzyme molecules.

These estimates are substantially larger than would be appropriate for a simple encounter mechanism (Section 13.3.1a) and substantially smaller than would be appropriate for a conditionally repeated complex-encounter mechanism (Sections 13.3.1b and 13.3.1c), but should not grossly underestimate the size and mass required per encounter mechanism in a mill-style processing system handling small reagent moieties. Assuming an arbitrarily chosen filled-volume fraction for the more massive system of 10%\sim 10 \% yields a crude estimate of the total volume per reactive encounter mechanism, 2×1023 m3\sim 2 \times 10^{-23} \mathrm{~m}^{3}.

d. Total mass of a reagent processing system. As noted in Section 13.3.3, systems for transforming a stream of small feedstock molecules into a stream of small reagent moieties can be expected to require 10\sim 10 reactive encounters. Combining this estimate with those made in Section 13.3.5c yields 5.7×1020\sim 5.7 \times 10^{-20} and 2.0×1020 kg\sim 2.0 \times 10^{-20} \mathrm{~kg}, respectively, or 2.9×106\sim 2.9 \times 10^{6} and 1.0×106\sim 1.0 \times 10^{6} atoms. These estimates indicate that high-throughput reagent processing systems can produce their own mass in deliverable moieties in 3 s\sim 3 \mathrm{~s}. Since reagent application adds a single step to a sequence of 10\sim 10, estimates of the mass and productivity of a system producing delivered moieties are little changed.

13.3.6. Error rates and fail-stop systems

Errors in molecular mills can arise from damage to the mechanisms of the mill, from the instability of reagent moieties, and from failed reactions (i.e., { }^{\circ}omitted reactions and { }^{\circ}misreactions). Sections 8.3.3f8.3 .3 \mathrm{f} and 8.3.4( dg)8.3 .4(\mathrm{~d}-\mathrm{g}) discuss conditions that suffice to ensure that errors resulting from misreactions, omitted reactions, and reagent instabilities occur at rates 1015\leq 10^{-15} per operation.4 The large stiffnesses feasible in mill mechanisms can help in meeting these conditions. Even with the exploitation of diamondoid, eutactic environments and stiff mechanochemical processes, however, these conditions exclude many potential reagents and transformations from use in reliable mechanosynthesis. It will here be assumed that choices compatible with these constraints are made, and that error rates in molecular mill systems are accordingly dominated by radiation damage. The following sections consider the consequences of damage, and describe how those consequences can, at the subsystem level, be made to result in a simple cessation of activity.

a. Consequences of damage. In a molecular mill, significant damage can be defined as damage that causes a failed reaction. In extreme instances, this may occur because damage has physically blocked the motion of the mill. In marginal instances, this may occur because damage has altered component geometries enough that misalignments increase the error rate.

In the former case, damage stops the mechanism directly; in all other cases, the mechanism continues to move for some time after the failed reaction. After a failed reaction, however, one or more of the departing product moieties will have an incorrect structure. An incorrect structure will usually cause failures in further reactive encounters, causing a cascade of damage transmitted from reagent devices to other reagent devices. Accordingly, significant damage will rapidly result in additional damage.

A damage cascade could be prevented by measurement and correction systems following each reactive encounter mechanism, but the complexity of doing so in general appears excessive. Conditional-repetition mechanisms (Sections 8.3.4f and 13.3.1c) are formally members of this class, but they correct only a single, well-defined kind of error: an omitted reaction.

b. Error detection and fail-stop subsystems. After a reaction, a processing system can gauge the shape of each moiety and product using a measurement mechanism based on a roller with a shape complementary to that of the correct structure. If the actual structure has a protrusion where the correct structure would not, then the roller (if it has a small contact area) is typically forced outward by a distance comparable to the size of the protrusion. Note that failed reactions will usually be signaled by a protrusion on one of the output structures: the atoms that enter the reactive encounter must leave again, and a misplaced atom will typically occupy space that would otherwise be empty (if the space is internal, detection of the event must be indirect). A single gauge mechanism can test a reagent device or a substantial fraction of the area of a workpiece. Gauge mechanisms can reliably discriminate between correct and incorrect structures (Sections 11.2.3 and 11.2.4), using iterated probing if necessary. Detection of an error can trigger a mechanism that blocks further activity in a molecular mill subsystem. (Mobile debris species must be bound or confined.)

The number of gauge mechanisms necessary to detect an error cascade in a large molecular mill subsystem is a small fraction of the number of reactive encounter devices, provided that an error can safely be allowed to remain undetected for many operation cycle-times. If products from each subsystem must transit a relatively long belt before being delivered to a consumer subsystem, then a relatively tardy detection process can prevent the delivery of mismanufactured product structures. The resulting fail-stop subsystems make convenient building blocks in the design of redundant, fault-tolerant manufacturing systems (Section 14.3.3).

c. Acceptable subsystem masses. Redundant, fault-tolerant systems can be constructed from sets of subsystem units having identical functions, if these are linked in such a way that the essential input-output properties of each set are unchanged when one or more of its units fails. With cumulative damage (no repair is assumed here), system lifetime remains finite, limited by the lifetimes of the units.

It is convenient to consider units with a .01 probability of failure in a 10-year period. Assuming that failure is dominated by radiation damage, Eq. (6.54) implies that this degree of reliability can be achieved so long as the mass is 2×1018 kg\leq 2 \times 10^{-18} \mathrm{~kg}, assuming a background radiation level of 0.5rad/0.5 \mathrm{rad} / year. This mass corresponds to that of a solid block of material 100 nm\sim 100 \mathrm{~nm} on a side, or to 350\sim 350 reactive-encounter mechanisms.

13.3.7. Estimates of energy dissipation

Energy dissipation in mechanochemical processes results from both mechanical and chemical steps. Mechanical energy dissipation results from the motions involved in transport and moietal manipulation; chemical energy dissipation results from the chemical transformations themselves (e.g., from merging shallow occupied wells with deep unoccupied wells, Section 7.6.4).

a. Energy dissipation caused by mechanical operations. Low energy dissipation is of greatest importance in systems designed for high throughput, and such systems will typically have belts with closely spaced reagent devices. Adopting a 4 nm4 \mathrm{~nm} spacing, a mechanism that delivers 10610^{6} moieties per second requires a 0.004 m/s0.004 \mathrm{~m} / \mathrm{s} belt speed; this speed is assumed throughout this section. Assuming a 5 nm5 \mathrm{~nm} roller radius (as in Section 13.3.5a), a 2 nm2 \mathrm{~nm} bearing radius and length, and 1000 N/m1000 \mathrm{~N} / \mathrm{m} bearing stiffness (as in the examples of Section 10.4.6) yields an estimated power dissipation per roller of 7×1020 W\sim 7 \times 10^{-20} \mathrm{~W}, using the more adverse parameters for band-stiffness scattering. With 20\sim 20 rollers in the system, this corresponds to 1.4×1024 J\sim 1.4 \times 10^{-24} \mathrm{~J} per delivered moiety. This is negligible in comparison both to chemical energies and to kT300k T_{300}.

Dissipation is larger where belts pass between backing surfaces. With assumptions analogous to those made for the bearing interface, a 16 nm216 \mathrm{~nm}^{2} reagent device sliding over a surface at 0.004 m/s0.004 \mathrm{~m} / \mathrm{s} with an interfacial stiffness of 1000 N/m1000 \mathrm{~N} / \mathrm{m} dissipates 1019 W\sim 10^{-19} \mathrm{~W}, or 1024 J\sim 10^{-24} \mathrm{~J} per pair sliding a distance of 20 nm20 \mathrm{~nm}. This remains quite small.

The motion of cam followers through cam grooves causes phonon scattering by the mechanism described in Section 7.3.4. Scaling from the sample calculation in that section yields an estimated energy dissipation of 1027 J\sim 10^{-27} \mathrm{~J} per nanometer traveled. Thermoelastic damping [Eq. (6.50)] is of the same order, assuming forces of 1nN\sim 1 \mathrm{nN} applied to areas of 1 nm2\sim 1 \mathrm{~nm}^{2}. Since these values are 106kT300\sim 10^{-6} k T_{300} per nanometer of sliding motion, extensive use of cams is compatible with low energy dissipation per operation.

Reactive encounters can dissipate free energy through nonisothermal compression of potential wells (Section 7.5). With good design practice, typical values at the speeds assumed should commonly be 0.1maJ\leq 0.1 \mathrm{maJ} per cycle. Dissipation from this source can easily exceed that from all other mechanical mechanisms combined. A reasonable estimate of the total energy dissipated by mechanical operations in a ten-step mechanochemical process is therefore 1maJ\sim 1 \mathrm{maJ} per moiety delivered; if nonisothermal compression losses are minimized, the energy dissipated can be reduced.

b. Energy dissipation caused by chemical transformations. A generous estimate of the energy dissipated in a series of chemical transformations can be made by assuming that each operation is as exoergic as the combustion of carbon, 650maJ\sim 650 \mathrm{maJ} per carbon atom. If each atom passes through ten such steps on the way to its final state, then the manufacture of a carbon-rich diamondoid product dissipates 3×108 J/kg\sim 3 \times 10^{8} \mathrm{~J} / \mathrm{kg} in the form of waste heat. A less generous but still naively high estimate can be made by instead assuming that each chemical transformation must be sufficiently exoergic to make it highly reliable ( PerrP_{\mathrm{err}} 1015\leq 10^{-15} ); this would require the dissipation of 145maJ\sim 145 \mathrm{maJ} per step, yielding an estimate of 7×107 J/kg\sim 7 \times 10^{7} \mathrm{~J} / \mathrm{kg}. A more sophisticated estimate requires attention to ways in which good design practice can minimize energy dissipation. (Note that the naive estimate may in fact be reasonable for systems where low energy dissipation is not a design objective.)

As discussed in Section 8.5.2b, reliable mechanochemical operations can in some instances approach thermodynamical reversibility in the limit of slow motion. At 10610^{6} operations per second, many processes in this class dissipate energies in the range characteristic of mechanical motions, as discussed in Section 13.3.7a. In comparison to 145maJ145 \mathrm{maJ}, these energies are negligible. Accordingly, if one assumes the development of a set of mechanochemical processes capable of transforming feedstock molecules into complex product structures using only reliable, nearly reversible steps, then the energy dissipated in millstyle manufacturing processes could approach zero, and in practice be 1maJ\sim 1 \mathrm{maJ} per atom, or 105 J/kg\sim 10^{5} \mathrm{~J} / \mathrm{kg}. (The conversion of a disordered feedstock into an ordered product structure reduces entropy, producing waste heat without corresponding energy dissipation; see Section 14.4.8.) The conditions for combining reliability and near reversibility are, however, quite stringent: reagent moieties must on encounter have structures favoring the initial structure, then be transformed smoothly into structures that, during separation, favor the product state by 145maJ\sim 145 \mathrm{maJ} (to meet the reliability standards assumed in the present chapter).

The availability of conditional-repetition processes (Sections 8.3.4f, 13.3.1c) substantially loosens these constraints. A reliable, low-dissipation process can be implemented if the conditions of the preceding paragraph are met, but with a modulation in relative energies not of 145maJ\sim 145 \mathrm{maJ}, but of (5+145/N)maJ\sim(5+145 / N) \mathrm{maJ}, where NN is the maximum number of available repetitions, and 5maJ5 \mathrm{maJ} is the energy bias favoring the initial structure. Under these conditions, the energy dissipated can be 1maJ\sim 1 \mathrm{maJ} per operation. For N=10N=10, the required energy difference ( 20maJ\sim 20 \mathrm{maJ} ) can be induced by relatively modest manipulations of bond strain, overlap energy, electrostatics, or electronic structure. Further, a simple conditional-repetition process with N=10N=10 can achieve high reliability with an exoergicity (and energy dissipation) of 15maJ\sim 15 \mathrm{maJ} per operation, without need for well-depth manipulations.

In reagent preparation, the feasibility of tailoring the entire reaction environment should commonly permit the use of low-dissipation operations; but in reagent application, the prospects are less clear. Group-transfer reactions often proceed through a single transition state, without the formation and subsequent cleavage of stable bonds linking the reagent device to the workpiece. In reactions of this sort, it should commonly be feasible to choose a group-transfer reagent having bond strengths that make the reaction only moderately exoergic; in addition, it should commonly be feasible to modulate bond strengths so as to permit a close approach to thermodynamic reversibility. In these instances, reagent application causes an energy dissipation of 15\sim 15 and 1\sim 1 maJ, respectively.

Some reagents, however, form strong bonds to a workpiece without simultaneously sacrificing bonds to their carrier. These must pass though a second, bond-cleaving transition. Whether the energy liberated in the initial bond formation appears as heat or work depends on details of the reaction PES and the reagent device design. In particular, the net exoergicity of bond formation can often be reduced to low values by imposing mechanical constraints that force a temporarily unfavorable geometry on the product (e.g., large angle-bending strains in new single bonds, or large torsional strains in new double bonds). This tends to increase transition state energies, but large piezochemical forces can be used to reduce energy barriers (Sections 8.3.3c and 8.5) and drive the system along the reaction coordinate. In a suitably designed reagent device, the resulting strain energy can be relieved smoothly as cam-controlled motions modify the geometry of the surrounding structure, thus avoiding the production of thermal vibrational energy. The example of radical coupling (aside from spin-pairing considerations, the time reversal of homolytic bond cleavage, Section 8.5.3b) shows that a stiff mechanism can form highly exoergic bonds ( 500maJ\sim 500 \mathrm{maJ} ) without substantial energy dissipation, even without exploiting the strained-intermediate strategy just described.

In light of these diverse strategies for reducing the required reaction exoergicity far below 145maJ145 \mathrm{maJ}, and for reducing the energy dissipation far below the exoergicity, it seems that mechanosynthetic processes can be quite efficient. In estimating the energy dissipation of systems engineered for good efficiency, it seems reasonable to assume that nearly reversible pathways can be followed in reagent preparation, and that most reagent application steps are either nearly reversible or dissipate an amount of energy characteristic of a moderately exoergic reaction. In this picture, many reactions dissipate 1maJ\sim 1 \mathrm{maJ}, some dissipate 15maJ\sim 15 \mathrm{maJ}, and a few (owing to synthetic constraints requiring energetic reagents and precluding energy-recovery strategies) dissipate 100maJ\sim 100 \mathrm{maJ} or more. If the mean energy dissipated per operation is 30maJ\sim 30 \mathrm{maJ}, and the mean number of atoms transferred is 1\sim 1, then the energy dissipated per kilogram of product is 1.5×106 J\sim 1.5 \times 10^{6} \mathrm{~J}. If this estimate should prove to be wrong, the only conclusions affected are those regarding energy consumption and waste heat generation in manufacturing processes.

13.3.8. Mechanochemical power generation

If a mechanochemical process is nearly thermodynamically reversible and is exoergic, then it converts chemical potential energy into mechanical work. Accordingly, mechanochemical systems can be designed to serve as sources of power; via electrostatic generators, they can provide electrical power. (Mechanochemical processes involving charge transfer can provide electrical power more directly, but are not considered here.)

The flexibility of transition-metal chemistry (Section 8.5.10) strongly suggests that a wide variety of reactions among small molecules can be carried out in a nearly reversible fashion via intermediate transition-metal complexes, including the reaction

2H2+O22H2O\begin{equation*} 2 \mathrm{H}_{2}+\mathrm{O}_{2} \rightarrow 2 \mathrm{H}_{2} \mathrm{O} \tag{13.8} \end{equation*}

which yields 475maJ\sim 475 \mathrm{maJ} per molecule of product (delivered to the liquid phase). If conducted at a rate of 10610^{6} product molecules per second with a mechanochemical system of the size and mass described in Section 13.3.5, the resulting power density relative to the mechanochemical apparatus itself is 8×106 W/kg\sim 8 \times 10^{6} \mathrm{~W} / \mathrm{kg}, or 2×109 W/m3\sim 2 \times 10^{9} \mathrm{~W} / \mathrm{m}^{3}. An overall efficiency >.99>.99 is suggested by the energy dissipation estimates of Section 13.3.7. Similar results can presumably be obtained for the oxidation of small organic molecules.

In molecular manufacturing, the conversion of typical hydrogen-rich, smallmolecule feedstocks into diamondoid structures liberates excess hydrogen. If this is combined with atmospheric oxygen, the yield of H2O\mathrm{H}_{2} \mathrm{O} typically is on the order of one molecule per carbon atom in the product structure, and the associated net exoergicity is 400maJ\sim 400 \mathrm{maJ} per atom. This surplus energy can be made to appear as mechanical or electrical power.

13.4. Assembly operations using molecular manipulators

It is sometimes necessary to use a system to build other systems of equal or greater complexity (if this were not done, an infinite regress would make molecular manufacturing impossible). This can be achieved by applying reagent moieties using a multiple-degree-of-freedom positioning mechanism, controlled either by a local nanocomputer or by a more remote macroscale computer: a single mechanism of this kind can perform an arbitrarily complex sequence of operations, with no simple bound on the complexity of the resulting products. Whether directly or indirectly, mill-style systems are likely to be implemented using manipulator-style systems.

Industrial robots are examples of multiple-degree-of-freedom positioning mechanisms. With the availability of nanometer-scale digital logic systems, motors, gears, bearings, and so forth, analogous designs are feasible on a smaller scale (e.g., 100 nm100 \mathrm{~nm} ). Aside from differences of scale and component properties, molecular manipulators differ from macroscale devices in that they must maintain positional accuracy despite thermal excitation. This problem can be minimized either (1) by operation at reduced temperatures, which receives no further attention here; or (2) by the use of a stiff mechanism, as described in Section 13.4.1; or (3) by use of local nonbonded contacts to align reagent devices to workpieces immediately before reaction, as discussed in Section 13.4.2.

Molecular manipulators typically execute many internal motions (e.g., rotations of drive shafts and gears, displacements of logic rods) per reagent moiety applied. Accordingly, their peak operating frequencies are reduced relative to mill-style systems, and their energy dissipation per operation (at a given frequency) is greater. Peak operating frequencies can exceed 106 s110^{6} \mathrm{~s}^{-1}, although at a substantial (many kT300k T_{300} per operation) penalty in energy consumption. Productivity on a per-unit-mass basis likewise is less than for a mill-style system, but still many orders of magnitude greater than that of conventional macroscale manufacturing systems.

13.4.1. A bounded-continuum design for a stiff manipulator

A sufficiently stiff manipulator can position a reagent moiety relative to a workpiece without needing to exploit interactions with the workpiece to ensure accurate alignment; this simplifies analysis. Completely general positioning requires control of six degrees of freedom (termed six axes): x,yx, y, and zz; and roll, pitch, and yaw. These functions could be divided among multiple loosely coupled mechanisms. For example, control of xx and yy could be provided by a mechanism moving the workpiece, with control of zz provided by a mechanism moving the reagent device, and control of the roll, pitch, and yaw orientation of the reagent moiety could be provided by a mechanism that sets and locks the geometry of an adjustable reagent device before loading it into the zz-axis mechanism. The present section, however, instead describes an integrated, armlike mechanism providing six-axis control. The range of choices in manipulator design is enormous; the specific set of choices explored here is intended to demonstrate lower bounds on capabilities without attempting to present an optimal design. In particular, little effort has been made to minimize energy dissipation, to maximize speed, or to simplify kinematics.

Section 8.5.5a concludes that positioning stiffnesses (for a reagent moiety with respect to a reaction site) of 5 to 10 N/m10 \mathrm{~N} / \mathrm{m} should be adequate to permit a wide range of reliable synthetic operations. The stiffness of an arm is best analyzed in terms of a series of (additive) compliances. A reasonable design goal is to keep the arm compliance small compared to 0.1 m/N0.1 \mathrm{~m} / \mathrm{N} (e.g., 0.04 m/N)0.04 \mathrm{~m} / \mathrm{N}), thereby allowing most of the permissible compliance to be allocated to other parts of the system, such as the workpiece and the reagent moiety itself. This goal is, however, not trivial to meet: the scaling law of Eq. (2.3) suggests that a 100 nm100 \mathrm{~nm} long arm with a compliance of 0.04 m/N0.04 \mathrm{~m} / \mathrm{N} corresponds to a meter long arm with a compliance of 4×109 m/N4 \times 10^{-9} \mathrm{~m} / \mathrm{N}. This is quite stiff compared to present practice in industrial robotics, motivating a clean-slate design rather than an attempt to scale and adapt existing designs.

a. General structure and kinematics. A hollow tubular shape can combine low bending compliance with internal space for drive mechanisms. Motion requires joints, which add compliance. A telescoping joint with a threaded interface can combine good structural continuity with adjustable length, adding little compliance. Canted rotary joints (as in some experimental robot arms and hardshell space suits) can permit angular motion of the arm with respect to the base and of the working end with respect to the arm, while again maintaining good structural continuity. The helical threads of the telescoping joint and the circular threads in the bearings of the rotary joints can form high-shear-stiffness interfaces like those in Section 10.4.7. Figure 13.10 illustrates the external form and range of motion of a device with these characteristics; it has a maximum extended length (with tool holder) of 100 nm100 \mathrm{~nm}, a diameter (save for the telescoping section) of 30 nm30 \mathrm{~nm}, and a typical wall thickness of 3.5 nm3.5 \mathrm{~nm}.

A drive system forces the tubular structural segments to rotate with respect to one another, and thus to rotate with respect to the local axis of the arm. In some instances (chiefly the two joints closest to the base, J1 and J2 in Figure 13.11) these rotational motions must be accomplished while maintaining high torsional stiffness. The approach chosen here exploits toroidal worm drives (Section 11.3.2) driven by mechanisms in a flexible core structure.

A stiff arm of this design has substantial dynamic friction, yielding relatively poor trade-offs between speed and energy dissipation. Speed and efficiency can be substantially improved if the arm need not move through a large distance between operations. Accordingly, the present design reserves a substantial volume of space for a transportation pathway from the base to the tip, permitting reagent moieties to be interchanged without arm motion by means of an internal conveyor mechanism.

b. Power supply and control. The power supply and control system is outside the arm itself, and hence is not subject to tight geometric constraints. It need only be assumed that shaft power is available, and that a control mechanism can engage and disengage smaller drive shafts (using clutches), causing them to turn by a set number of rotations between locked states. Reasonable implementations can be devised that use gear trains to produce an odometerlike encoding of the number of shaft rotations and use a corresponding device to encode a representation of the desired number of shaft rotations (this device can be written to like a register); a match between the two can be made to trigger a motion causing the disengagement of a clutch and the locking of the corresponding drive shaft. To initiate a motion, a digital logic device stores a suitable number in the register, then engages a clutch to cause shaft rotation in the appropriate direction.5

Drive shafts undergo longitudinal displacements as the arm moves (most obviously, when the telescoping joint J4\mathrm{J} 4 is actuated). A coupling that permits these motions while applying torque (e.g., a drive nut engaging longitudinal

Figure 13.10. External shape and range of motion of a stiff arm design compatible with implementation on a 100 nm100 \mathrm{~nm} scale (internal mechanisms are described in Section 13.4.1 and Figures 13.11-13.). System compliance is analyzed in Section 13.4.1e.

Figure 13.11. Cross section of a stiff manipulator arm and identification of parts (schematic); see also Figure 13.12.

grooves on a shaft segment) can interface these shafts to drive and control mechanisms with fixed positions.

c. Drive shafts, gears, and the core structure. Figure 13.11 shows a cross section of an arm, including all the moving parts required for arm motion save for three drive shafts (and gears) not in the plane of the section. As shown, the shafts have diameters of 1.5 nm1.5 \mathrm{~nm}; the compliance analysis of Section 13.4.1e assumes the use of a material with a modulus 0.1\sim 0.1 that of diamond, lowering the

Figure 13.12. Cross section A-A of arm in Figure 13.11, perpendicular to the axis.

stresses caused by shaft bending. Surface strains in the shafts are well within the acceptable range even of high-modulus diamondoid structures. (Jointed and telescoping structures for drive shafts are also feasible.) One shaft, ending at joint J1\mathrm{J} 1, need not be flexible and is located outside the core structure.

The remaining shafts are threaded through bearing apertures in core plates, and are free to rotate and to slide longitudinally with respect to these plates. The rims of the core plates interlock with drive rings that in turn interlock with joints in the arm, forcing coplanarity between each core plate and its corresponding joint. Both segments linked by the joint can rotate with respect to the core plate.

Core plates are linked by core bellows segments. These flex to accommodate the angle and offset between the planes of the joints; the angle and offset have fixed magnitudes and relative orientation, but they can rotate with respect to the core structure. The core bellows segments link the core plates in a manner that gives the core structure substantial torsional stiffness. (Metal bellows often are used to transmit torque between misaligned shafts in macroscale engineering practice, despite the lower elastic limit and susceptibility to fatigue damage characteristic of macroscale metal structures.) Additional structures resembling the core plates could be introduced to further constrain and shape the paths of the bellows and drive shafts.

Within the envelope of the core bellows segments and core plates is a region with a diameter of 7 nm7 \mathrm{~nm} that plays no role in the arm structure or in driving its motions. This region is reserved for a mechanism to transport reagent moieties (or reactive clusters) to the tip of the arm, where a larger region is reserved for a mechanism to position and lock these moieties or clusters into position for use. An articulated conveyor system based on belts, rollers, and sliding-interface guideways can serve the transport function. Paths like those allocated for drive shafts can carry several shafts for actuating the reagent positioning and locking mechanism, or (in related applications) for transmitting signals from a sensor at the tip of the arm to a computational system outside the arm.

Drive shafts end in drive gears that engage the inner surface of a drive ring. As a drive shaft rotates, the drive gear forces rotation of the drive ring with respect to the core plate, which is itself restrained from substantial rotation about its axis by the torsional stiffness of the core bellows segments.

d. Joints and joint drive mechanisms. The internal structure of the arm design is illustrated in Figure 13.11, including the components of the six joints. The joint nearest the tip of the arm, J6\mathrm{J} 6, plays no role in positioning and transverse compliance, affecting only the rotational angle of the tool. Its stiffness requirements are slight, hence it can be directly geared to a drive shaft.

The other rotary joints, J1, J2, J3\mathrm{J} 1, \mathrm{~J} 2, \mathrm{~J} 3, and J5\mathrm{J} 5 are implemented as toroidal worm drives (Section 11.3.2). Harmonic drives (Section 11.3.1) offer an alternative to the toroidal worm drive, but if repulsive interactions are used to implement their teeth, the large and time-varying radial forces applied by the wave generator are likely to result in larger energy dissipation at the interface between the wave generator and the flexspline than at the corresponding interface between the threaded drive tube and the triple-threaded torus. Further, the large asymmetrical loads would require additional structural analysis. The speed ratios and stiffnesses of these joints are examined in Section 11.3.2.

Figure 13.13. Structure and kinematics of a stiff telescoping joint. The drive shaft and attached drive gear rotate about their shared axis, forcing rotation of the drive ring about the shared axis of the core plate and the other illustrated components. The drive ring engages several transmission gears mounted in the torsional-stiffening sleeve. This sleeve shares longitudinally grooved interfaces with the upper and lower tube segments, preventing these segments and the torsional-stiffening sleeve from rotating with respect to one another, while permitting longitudinal displacements. The transmission gears engage an extension of the telescoping screw-sleeve, thereby coupling its rotational motion to that of the drive shaft. Rotation of the screw-sleeve drives longitudinal, telescoping motions of the tube segments via left- and right-handed helically threaded interfaces. The complexity of this scheme results from the placement of the screw sleeve outside the tube, which in turn is motivated by space limitations inside the tube and by advantages in bending stiffness resulting from a greater radius.

The telescoping joint assembly, J4 (including threaded interfaces J4a and J4b), can be directly geared to a drive shaft, since the thread angles (with a thread pitch of 0.5 nm0.5 \mathrm{~nm} ) provide a speed reduction that provides ample stiffness [by Eq. (13.9)]. The structure and kinematics of this joint are described by Figure 13.13.

e. Estimated arm compliance. The compliance of the arm (as in most beamlike structures) is greatest in bending modes. This section will estimate compliance in arms with the tip pointing upward or inward (relative to the orientation of Figure 13.10) in the worst-case geometry of full J4 extension and maximum bending (as in Figure 13.14). The compliance is then greatest for displacements perpendicular to the direction of bending (i.e., perpendicular to the plane of Figure 13.14). Table 13.1 summarizes the main contributions to total arm compliance in this geometry, and the physical assumptions behind the calculated contributions from structures outside the core and drive system.

The compliance contributed by the core and drive system can be reduced by using toroidal worm drives having large speed-reduction ratios. The compliance of the arm can be made almost independent of the core and drive compliances by exploiting the relationship

Cs,2=(v2/v1)2Cs,1\begin{equation*} C_{\mathrm{s}, 2}=\left(v_{2} / v_{1}\right)^{2} C_{\mathrm{s}, 1} \tag{13.9} \end{equation*}

where Cs,1C_{\mathrm{s}, 1} and Cs,2C_{\mathrm{s}, 2} are the compliance of the system measured with respect to displacements at points 1 and 2 , and v1v_{1} and v2v_{2} are the speeds of those points as

Figure 13.14. Cross section of a stiff manipulator arm, showing its range of motion (schematic).

the system executes some motion. (This expression assumes no compliance between points 1 and 2, i.e., that their velocities and displacements remain in a fixed ratio.) A mechanism providing a large speed-reduction ratio between the motion of the surface of the drive gear and the motion of the driven segment can tolerate large drive-shaft and power-supply compliances.

Drive shafts with a diameter of 1.5 nm1.5 \mathrm{~nm} and a shear modulus of 5×1010 N/m25 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2} will have a torsional compliance (for surface displacements) of 6 m/N\sim 6 \mathrm{~m} / \mathrm{N} at a distance of 100 nm100 \mathrm{~nm} from a torsionally stiff locking mechanism. Torsional compliance in the core bellows structure adds directly to the drive rod compliance, but this increment can be made comparatively negligible.

For J6, direct gearing of the joint to the drive gear gives a standard deviation in joint angle of 0.02\sim 0.02 radians. This is orthogonal to the other compliances discussed here, and is negligibly small for most mechanochemical operations involving small reagent moieties.

For J4, if a drive gear is coupled to rotation of the threaded outer sleeve with a gear ratio of unity, it contributes a compliance in manipulator extension of 0.018 m/N\sim 0.018 \mathrm{~m} / \mathrm{N}, assuming a threaded interface with a radius of 15 nm15 \mathrm{~nm} and threads of 0.5 nm0.5 \mathrm{~nm} pitch. This is orthogonal to the bending compliances considered elsewhere in this section.

In J2, J3\mathrm{J} 2, \mathrm{~J} 3, and J5\mathrm{J} 5, drive-rod compliance contributes to transverse-bending compliance. In each, the difference in radius between the geared inner surface and threaded outer surface of the drive ring contributes a speed increase of 1.8\sim 1.8, 1.5\sim 1.5, and 1.8\sim 1.8 respectively. The respective drive-rod lengths are 25,35\sim 25, \sim 35, and 75 nm\sim 75 \mathrm{~nm}. The corresponding contribution of each of these joints to the transverse compliance at the tip can be limited to 0.001 m/N\leq 0.001 \mathrm{~m} / \mathrm{N} apiece if the speed-reduction ratios of the toroidal drive assemblies are 190,70 , and 65 , respectively, measuring ratios from the speed across the threaded interface of the drive ring to the

Table 13.1. Major contributions to transverse compliance of a manipulator.

SourceCoordinateMagnitude
(m/mN)(\mathrm{m} / \mathrm{mN})
Multiplier a^{\mathrm{a}}Contribution
(m/mN)(\mathrm{m} / \mathrm{mN})
Tube bending b^{\mathrm{b}}End displacement13.60113.6
J1 rocking c{ }^{\mathrm{c}}Max. local stretch0.05593.0
J2 rockingMax. local stretch0.05452.3
J3 rockingMax. local stretch0.05321.6
J4a rockingMax. local stretch0.05191.0
J4b rockingMax. local stretch0.0590.5
J5 rockingMax. local stretch0.0540.2
J6 rocking { }^{\text {d }}Max. local stretch0.0730.2
J1 torsion e{ }^{\mathrm{e}}Max. local shear0.31257.8
J2 torsionMax. local shear0.3172.2
J3 torsionMax. local shear0.3110.3
J5 torsionMax. local shear0.310.50.2
S1 torsion f{ }^{\mathrm{f}}Max. local shear0.11252.8
Drive system(( see Section 13.4.1e)
Total4.0

a Values of (v2/v1)2\left(v_{2} / v_{1}\right)^{2}, as in Eq. (13.9).

b Tube bending: manipulator structure modeled as a tube of modulus E=1012 N/m2E=10^{12} \mathrm{~N} / \mathrm{m}^{2}, length =100 nm=100 \mathrm{~nm}, inner radius =11.5 nm=11.5 \mathrm{~nm} and outer radius =15 nm=15 \mathrm{~nm} (based on excluded volume), with a 0.1 nm0.1 \mathrm{~nm} surface correction for structural vs. excluded volume (Section 9.4.2).

c{ }^{\mathrm{c}} Rocking (joints J1, J2, J3, J4a,J4 b, J5\mathrm{J} 1, \mathrm{~J} 2, \mathrm{~J} 3, \mathrm{~J} 4 \mathrm{a}, \mathrm{J} 4 \mathrm{~b}, \mathrm{~J} 5 ): compliance is measured with respect to the maximum stretching deformation across the joint (stretching varies from positive to negative around each ring). A bearing interface patterned on Figure 10.17, with a width of 2 nm2 \mathrm{~nm} and a radius of 13 nm13 \mathrm{~nm}, provides 650\sim 650 close interatomic contacts. Assuming a mean stiffness of 10 N/m10 \mathrm{~N} / \mathrm{m} per contact, the rocking compliance is 0.00003 m/N\sim 0.00003 \mathrm{~m} / \mathrm{N}. An additional compliance of 0.00002 m/N\sim 0.00002 \mathrm{~m} / \mathrm{N} results from the joint-associated thinning of the tube wall thickness. Each multiplier is the square of the ratio of the distance to tip and the radius of joint.

d Rocking (joint J6): As for other joints, but with a correction for smaller radius.

e Torsion (joints J1, J2, J3, J5\mathrm{J} 1, \mathrm{~J} 2, \mathrm{~J} 3, \mathrm{~J} 5 ): In these joints, torsional deformations resulting from shear across the threaded torus of the drive mechanism can result in tip motions (with differing lever arms). Threaded toruses with circumference =80 nm=80 \mathrm{~nm} and minor radius == 1 nm1 \mathrm{~nm} can have 640\sim 640 close interatomic contacts with threads on each of the two adjacent structural (or base) segments. A mean stiffness of 10 N/m10 \mathrm{~N} / \mathrm{m} per contact yields the stated compliance for the two interfaces taken together.

f Torsion (segment S1): Like J1 torsion, this contributes to tip motion. This segment is modeled as a tube (with dimensions as above) of length 15 nm15 \mathrm{~nm} and shear modulus G=G= 5×1011 N/m25 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}.

speed of the driven segment. (Toroidal worm drives of this size can provide ratios >3000>3000.) In turning a tube segment through one full rotation, even the J3\mathrm{J} 3 and J5\mathrm{J} 5 drive shafts undergo >700>700 rotations.

For J1\mathrm{J} 1, no flexible drive rods are needed, hence the drive system compliance can be limited to a small value. Taking 0.1 m/N0.1 \mathrm{~m} / \mathrm{N} as a readily achievable target, the contribution to the tip compliance can be limited to 0.001 m/N\leq 0.001 \mathrm{~m} / \mathrm{N} with a speed- reduction ratio of 50 . Drive system contributions to transverse compliance then total 0.004 m/N\leq 0.004 \mathrm{~m} / \mathrm{N}.

The total arm compliance estimated in Table 13.1 is 0.04 m/N0.04 \mathrm{~m} / \mathrm{N}, corresponding to a stiffness of 25 N/m25 \mathrm{~N} / \mathrm{m}. This permits the use of moderately compliant workpieces and reagent moieties (0.1 m/N)(\sim 0.1 \mathrm{~m} / \mathrm{N}) while keeping error rates 1015\leq 10^{-15} in directing reactions to chemically equivalent sites separated by a single bond length (Section 8.5.5a). This stiffness also permits the application of 1nN1 \mathrm{nN} forces with an elastic deflection in the arm of only 0.04 nm0.04 \mathrm{~nm}. (Elastic deflection under a planned load need not contribute to positional error.)

f. Speed, productivity, and magnitude of power dissipation. To estimate the power dissipation of the manipulator system on a per-operation basis requires a model of a typical motion, which in turn requires consideration of a typical sequence of operations in the synthesis of diamondoid structures. To build such a model, let us assume a synthetic strategy based on the transfer of small moieties to a surface. The characteristic scale length of required motions can be taken as the 0.25 nm0.25 \mathrm{~nm} spacing of the bonds perpendicular to a diamond (111) surface. Efficiency favors small motions between operations. Consider an operation at a typical point (a), with subsequent operations required both at a set of 6\sim 6 neighboring points (b) and at a set of more distant points (c). After completing an operation at point (a), a large motion would be required if and only if synthetic constraints somehow demanded that each of the operations in set (b) be delayed until after executing an operation at a point (d) in set (c). Save in rare or contrived circumstances, this somewhat unusual constraint could arise only if the point (d) were adjacent to a point in set (b), thereby permitting a substantial electronic or steric interaction between them. Accordingly, the required motion distance should often be one scale-length (0.25 nm)(0.25 \mathrm{~nm}), occasionally be two (0.5 nm)(0.5 \mathrm{~nm}), and only rarely be greater. Treating 0.5 nm0.5 \mathrm{~nm} as a typical displacement distance between operations should yield conservative estimates of energy dissipation.The present arm design, however, causes substantial tip displacement in the course of executing a rotation in joints J3\mathrm{J} 3 and J5\mathrm{J} 5, which are provided chiefly to control tip orientation. Assuming that such motions are commonplace and result in tip displacements of 10 nm\sim 10 \mathrm{~nm}, it is conservative to assume that joints J1\mathrm{J} 1 and J2\mathrm{J} 2 will commonly have to execute motions able to cause comparable but compensating tip displacements. If such motions are performed in 106 s\sim 10^{-6} \mathrm{~s}, the speed of the fastest sliding motion (that of the J2\mathrm{J} 2 drive ring) is 1 m/s\sim 1 \mathrm{~m} / \mathrm{s}. Assuming that phonon drag at sliding interfaces (which scales with speed squared) is the dominant power-dissipation mechanism, motion in this joint can be expected to account for a substantial fraction of the total power dissipated.

To limit shear displacements of a threaded drive ring with respect to the triple-threaded torus of the worm drive, an interfacial shear stiffness of 100 N/m\sim 100 \mathrm{~N} / \mathrm{m} is ample; the interfacial compressive stiffness can be several times this value. The corresponding values of the phonon transmission coefficient, Ttrans T_{\text {trans }} are 104\sim 10^{-4} [from Eq. (7.41), assuming diamondlike materials]. Applying assumptions like those used in the bearing analyses of Section 10.4.6 to a sliding-interface area of 200 nm2\sim 200 \mathrm{~nm}^{2} yields an estimated energy dissipation per motion of 10maJ\sim 10 \mathrm{maJ}. An estimate of 100maJ\sim 100 \mathrm{maJ} per motion for the arm as a whole therefore

Figure 13.15. A stiffer set of base joints for a manipulator arm.

appears generous. Note, however, that this estimate does not include energy dissipated in the power and control system, and is rather crude.

If an arm emplaces 1\sim 1 atom per motion, a dissipation of 100maJ\sim 100 \mathrm{maJ} is significant on the chemical energy scale. If an arm emplaces a cluster of 2 nm\sim 2 \mathrm{~nm} size, containing 1000\sim 1000 atoms, the energy dissipated per atom is a trivial 0.1maJ\sim 0.1 \mathrm{maJ}.

These calculations indicate that arms performing 10610^{6} operations per second need not incur excessive penalties from friction (although the energy cost is large compared to that of a mill-style mechanism). The total number of atoms in a mechanism of this size is 5×106\sim 5 \times 10^{6} (neglecting the base, power and control system, etc.); accordingly, the time required for a mechanism of this type to perform the motions necessary to build a product structure of similar mass and complexity is 5 s\sim 5 \mathrm{~s}. This suggests that the use of manipulator mechanisms need not preclude high overall system productivity.

g. Toward better designs. The design illustrated by Figure 13.11 is far from optimal. Although maximizing stiffness is a major design objective, and the major sources of compliance are in the region from joints J1\mathrm{J} 1 to J3\mathrm{J} 3, these joints and tube segments are no stiffer than the less critical structures further along the arm. A design based on a Stewart platform (Figures 16.4 and 16.5 illustrate the general structure and kinematics) might provide superior stiffness.

Minimizing energy dissipation is another major design objective. Increasing joint stiffness (as in Figure 13.15) and using several drive shafts for critical joints can lower the magnitude of required speed-reduction ratios, decreasing the sliding speeds in interfaces and lowering dissipation.

Other major improvements can doubtless be made. The present design exercise merely shows that suboptimal designs are adequate to enable reliable, reasonably efficient molecular manipulation.

13.4.2. Self-aligning tips and compliant manipulators

A stiff manipulator can be used to apply small, simple reagent devices to workpieces without any special adaptation to the workpiece structure (aside from choice of moiety, position, orientation, force, etc.). At the cost of greater tip complexity, reactive encounters can exploit nonbonded interactions to achieve reliable alignment, permitting substantially greater arm compliance while maintaining high reaction reliability.

The intermediate states in the construction of a diamondoid structure can be limited to surface irregularities of atomic height on a nanometer length scale in the vicinity of each reaction site. To perform a reaction at such a site, a tip can be

1

2

Figure 13.16. Schematic diagram showing alignment of a reagent moiety with respect to a workpiece aided by sterically complementary probes. The diagrams illustrate (1) a reagent device approaching the workpiece, (2) probes in contact with distinctive features (bumps and hollows) of the workpiece, bringing the reagent device into alignment before contact, (3) the reactive encounter, and (4) separation after reaction and transfer of a moiety. Note that the number of probes can be indefinitely multiplied, and that each complementary contact can increase alignment stiffness and specificity.

configured to have regions of specific steric complementarity with the workpiece surface, bringing the tip into stiff alignment (compliance 0.01 m/N\leq 0.01 \mathrm{~m} / \mathrm{N} ) with the surface before the final reactive encounter (Figure 13.16). The required control of tip geometry can be achieved either by adding adjustable probes to the end of the arm (e.g., configured by geared mechanisms driven by rotating shafts), or by attaching probe structures of suitable geometry (selected from a diverse set) to the reagent device beforehand. Multiple-position detent mechanisms (Section 10.9.2) can be exploited to provide the necessary steric diversity.

13.4.3. Error rates and sensitivities

Manipulator-style mechanisms will tend to have lower stiffness and hence higher error rates than those of mill-style mechanisms. Nonetheless, the feasible stiffness permits errors stemming from thermal excitation to be kept to negligible levels (1015)\left(\ll 10^{-15}\right), given suitable choices of reagent, reaction site, and encounter geometry (Sections 8.3.3f and 13.4.1e). Self-aligning tips (Section 13.4.2) can increase stiffness, broadening the range of arm structures and reaction conditions that are compatible with low error rates.

Manipulators under programmable control can more easily be designed for fault tolerance than can mill systems. With the ability to vary the sequence of reagent devices and motions, and with the ability to make these sequences contingent on the results of measurements of the workpiece structure, it is feasible to continue operations after discarding damaged workpieces and tools. In some instances, flaws could be corrected, but this capability is not assumed in the following analyses.

13.4.4. Larger manipulator mechanisms

Within the size range where gravitational effects are negligible,6 the design of manipulator arms becomes easier with increasing size. Assuming motions of constant speed (as in Section 2.3.2), structural stiffness scales with the characteristic dimension LL, and joint stiffnesses and power dissipations scale as L2L^{2}. Since the mass being delivered commonly scales as L3L^{3}, and the time for its delivery scales as LL, the energy per unit mass delivered is a constant. In practice, some of the increase in stiffness resulting from larger scale can be sacrificed in exchange for decreased friction, decreasing energy dissipated per unit mass. This is consistent with industrial experience, in which the energy required to move a macroscopic object over a short distance is trivial in comparison with its thermal or chemical energy content. Accordingly, the feasibility of stiff manipulator arms of sizes intermediate between 100 nm100 \mathrm{~nm} and macroscopic dimensions is assumed in the following sections without presenting additional design and analysis.

13.5. Conclusions

Nanomechanical systems can be used to acquire feedstock molecules from solution, sorting and ordering them to provide a flow of input materials to essentially deterministic molecule-processing systems. Exploiting mechanochemical processes, these systems can convert feedstock molecules into reactive moieties of the sorts discussed in Chapter 8. These moieties, in turn, can be applied to workpieces in complex patterns to build up complex structures.

Each of these operations can be made sufficiently tolerant of thermal excitation that damage to manufacturing mechanisms and product structures occurs chiefly as a result of ambient ionizing radiation. Molecular sorting operations can be quite efficient, with energy dissipation <1maJ<1 \mathrm{maJ} per molecule. Assembly operations using mills can be energetically efficient, with the energy dissipated in many operations being <1maJ<1 \mathrm{maJ}, and an estimated mean dissipation for a mix of efficient and inefficient operations being 30maJ\sim 30 \mathrm{maJ} per operation. The efficiency of manipulator motions is expected to depend heavily on the energy dissipated by their control mechanisms.

Each system has been characterized at a frequency of 10610^{6} operations per second. At this frequency, the sorting, processing, and manipulation subsystems each process a quantity of reagent moieties equaling its own mass within a few seconds. The next chapter describes how fast, reliable processing units like these can be combined with larger-scale assembly systems to build molecular manufacturing systems capable of delivering macroscopic products.

Some open problems. As in the previous three chapters, the task of expanding the range of well-characterized, atomically detailed models for devices defines a large set of open problems. Modulated receptors can readily be designed within the constraints of existing software and computers. A more detailed understanding of purification processes could be developed by choosing a small feedstock molecule, compiling an exhaustive list of similar molecules of smaller size, and characterizing the interactions of each with feedstock receptor structures of several designs.

Many of the problems of greatest interest in mill and manipulator design depend on the prior characterization of a compatible set of mechanochemical processes (as discussed at the close of Chapter 8); the discussion in this chapter highlights the interest in identifying sets of nearly thermodynamically reversible processes that can combine fuel and oxygen molecules. Various mechanical components of mill and manipulator systems (providing transport, power supply, alignment, error detection) are independent of the specifics of the mechanochemical process being performed, hence their detailed design can proceed with existing knowledge (subject, as always, to limitations in the accuracy of molecular mechanics methods).

Footnotes

  1. At this size scale, liquid structure effects can be important. Designs having small liquid-filled gaps with adverse geometrical and surface properties could produce solidlike ordering among small feedstock molecules (Section 11.4.1a). At ordinary temperatures, however, different choices of geometry and surface structure should avoid this problem, but bulk-phase viscosity values should be taken as no more than a rough guide to liquid behavior near surfaces.

  2. The chemical stability of liquid ethyne itself at this pressure (in small volumes of pure substance, bounded by inert walls) is an open question; polymerization is exoergic, but the low-pressure activation energy is large. Sample calculations in Chapter 14 assume the use of acetone as a carbon source. Various strategies can enable the use of reactive molecules like ethyne as feedstocks; for example, they could be made to form bound complexes in an early, low-pressure stage, permitting later purification and orientation stages to handle a less reactive structure.

  3. mill nn : Any of various machines for making something by some action done again and again. (Webster's New World Dictionary)

  4. An error rate of 101510^{-15} corresponds to a mean time to failure of 3000\sim 3000 years for a single device working at 10410^{4} operations per second. The failure rate of a mill module might be dominated by one device with an error rate of 101510^{-15}, while a hundred others work with error rates of 1018\leq 10^{-18} (a factor of 10310^{3} in failure rate corresponds to a difference of <30maJ<30 \mathrm{maJ} in barrier height). The net mean time to failure then is still 3000\sim 3000 years.

  5. The ratios of the times and energies consumed by a typical nanomechanical computation operation to the times and energies consumed by a typical robotic arm motion are enormously greater than the corresponding ratios for microprocessor computation and macromechanical robotic arm motions. J. S. Hall has noted that this favors a shift away from moment-by-moment computer control of arm trajectories; the approach described here reflects this preference.

  6. Note that the arm discussed in this Section 13.4 can easily apply 1nN1 \mathrm{nN} forces, enabling it to lift 1010 kg10^{-10} \mathrm{~kg} against terrestrial gravity. This is 109\sim 10^{9} times the mass of the arm itself.