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Mechanosynthesis

8.1. Overview

Mechanosynthesis and nanomechanical design are interdependent subjects. Advanced mechanosynthesis will employ advanced nanomachines, but advanced nanomachines will themselves be products of advanced mechanosynthesis. This circular relationship must be broken both in exposition and (eventually) in technology development. In development, the circle can be broken by using either conventional synthesis or noneutactic mechanosynthesis to construct first-generation nanomachines, as discussed in Chapters 15 and 16 of Part III. In the following exposition, the circle is broken as follows:

(1) This chapter shows that a wide range of diamondoid structures can be made, provided that accurately controlled mechanical motions can somehow be imposed as a boundary condition in molecular systems;

(2) Part II then shows how diamondoid structures can serve as components of nanomachines able to power and guide accurately controlled mechanical motions, including motions of the sort assumed in this chapter.

8.1.1. Mechanochemistry: terms and concepts

The term mechanochemistry was coined by Ostwald decades ago to describe " aa branch of chemistry dealing with the chemical and physico-chemical changes of substances of all states of aggregation due to the influence of mechanical energy" (Heinicke, 1984); it has more recently been used to describe both the conversion of mechanical energy to chemical energy in polymers (Parker, 1984) and the conversion of chemical energy to mechanical energy in biological molecular motors; proponents of the first use (Casale and Porter, 1978) urge that processes of the second sort be termed "chemomechanical" in accord with the direction of energy flow. As used here, mechanochemistry refers to processes in which mechanical motions control chemical reactions, guiding the molecular encounter and often providing { }^{\circ}activation energy; this is consistent with Ostwald's broad definition. Since mechanochemical systems can approach thermodynamic reversibility (Section 8.5), the direction of conversion between chemical and mechanical energy cannot be used as a defining characteristic. The present usage is thus related to but substantially distinct from those just cited. The term mechanosynthesis is here introduced to describe mechanochemical processes that perform steps in the construction of a complex molecular product.

Mechanochemistry, in the present sense, has features in common with reactions in organic crystals (McBride et al., 1986) and in enzymatic active sites (Creighton, 1984). In both crystals and enzymes, the environment surrounding a molecule can apply substantial stresses to it, or to a reaction intermediate, and can constrain reaction pathways so as to produce a specific product. In crystals, these forces limit motion (McBride et al., 1986). Mechanochemical systems of the sort contemplated here, in contrast, characteristically transport reactive moieties over substantial distances, induce substantial displacements in the course of a reaction, and supplement local thermal and molecular potential energies with work done by external devices.

Mechanochemistry plays several roles in the systems described in Part II. These include the transformation of small molecules, generation of mechanical power, preparation of bound reactive moieties, and mechanosynthesis of complex structures using such moieties. Although the discussion in the present cilapter centers on reagent preparation and mechanosynthesis, the principles described are of general applicability.

8.1.2. Scope and approach

Like solution-phase chemistry, { }^{\circ}machine-phase chemistry embraces an enormous range of possible reactions. Its potential breadth places the field beyond any hope of thorough exploration at present, and beyond any possibility of summarizing in the space of a chapter. Nonetheless, an attempt must be made to delineate the general capabilities and limitations of mechanochemical processes. How is this problem to be approached? In the absence of opportunities for immediate experimentation, two options suggest themselves:

The first approach relies on detailed theoretical modeling of mechanochemical processes. This yields quantitative descriptions of the sort familiar to those schooled in physics and engineering; several examples are presented in Section 8.5. Theoretical modeling suffers from several difficulties and limitations, however. Chemical reactions cannot be modeled using a molecular mechanics PES, but require either a specialized PES (e.g., Sections 8.5.3, 8.5.4) or the use of aba b initio molecular orbital methods to calculate the energies of a set of molecular configurations (e.g., Section 8.5.4b). These quantum calculations have been expensive, and are of limited accuracy.1 Available techniques in computational quantum chemistry (to say nothing of future techniques) can be used to provide an extensive and detailed understanding of the capabilities of mechanosynthesis, but this will require considerable time, effort, and expense. Today, a survey of the capabilities of mechanosynthesis must rely on other methods.

The second approach begins by surveying the demonstrated capabilities of solution-phase organic synthesis, then examines how the conditions of mechanochemistry both add capabilities and impose constraints relative to this model. The resulting picture of the capabilities of mechanosynthesis thus is linked to experimental results in chemistry, rather than being directly founded on theory. This picture contains fewer quantitative descriptions than a physicist or an engineer may expect. If, however, we think that chemists are competent in chemistry (as their achievements show), then we have reason to assume that their methodologies and modes of explanation are well suited to their subject matter. Organic synthesis is a notoriously qualitative field [it is "very little dependent on mathematical calculations," (Hendrickson, 1990)], and much of the discussion in Sections 8.3 and 8.4, taking organic synthesis as its point of departure, accordingly represents a substantial departure from the style of exposition in the preceding chapters. Like much of chemistry, it relies heavily on example and analogy, reasoning from experimental results on model systems supplemented by the general principles of chemical kinetics and thermodynamics.

The following sections approach this topic from several directions: Section 8.2 provides a brief overview of the nature and achievements of solution-phase organic synthesis, offering a perspective from which to judge the expected capabilities of mechanosynthesis. Section 8.3 compares solution-phase synthesis and mechanosynthesis, examining their relative strengths and limitations. This section describes quantitative criteria for reaction reliability in light of the anticipated requirements of molecular manufacturing, and provides a preliminary description of the role of mechanical forces in chemical reactions. Section 8.4 provides an overview of several classes of reactive species from a mechanosynthetic perspective; Section 8.5 then examines several mechanochemical processes in more detail, focusing in particular on the exemplary processes of tensile bond cleavage and hydrogen abstraction. Finally, Section 8.6 presents several approaches to the mechanosynthesis of diamondoid structures, drawing on the preceding sections and on studies of reaction mechanisms in the chemical vapor deposition of diamond.

8.2. Perspectives on solution-phase organic synthesis

8.2.1. The scale and scope of chemistry

The advent of organic synthesis commonly is dated from the 1828 synthesis of urea by Wöhler, at a time when "chemists were just beginning to speculate about the arrangement of atoms within organic molecules" (Brooke,1985). By the late 1800 s, the basic concepts of atoms and molecular geometry were well established (except in the view of ardent Positivists), synthesis was subject to rational planning, and synthetic organic chemicals had become a major industry.

Today, a single nationally based professional group, the American Chemical Society, has 150000\sim 150000 members. One year's output of Chemical Abstracts now occupies 2 m\sim 2 \mathrm{~m} of shelf space, containing abstracts of 5×105\sim 5 \times 10^{5} articles; the volumes published to date occupy 30 m\sim 30 \mathrm{~m} of shelf space, with the indexes occupying 30 m\sim 30 \mathrm{~m} more. After 110 years of publication, Beilsteins Handbuch der Organischen Chemie occupies 20 m\sim 20 \mathrm{~m} of shelf space with over 300 volumes. The chemists' view of the current capabilities of organic synthesis is suggested by the observation that "the problem of synthesis design is simply that the number of possible synthetic routes to any target molecule of interest is enormous" (Hendrickson, 1990). This author estimates tens of millions of paths for a typical five-step synthesis, with vastly differing yields. A further indication is the barely cautious warning that "the idea persists that we can synthesize anything. The major flaw in this view is that it fails to recognize effectiveness and practicality" (Trost, 1985).

During the first century of organic synthesis, chemists proceeded with no guidance from quantum mechanics, relying instead on a growing set of concepts regarding molecular geometry and thermodynamics, together with rules of thumb for estimating the energy differences between similar molecular configurations and the rate differences between similar reactions. These concepts and rules grew from efforts to rationalize experimental results; they are still being extended and remain central to organic synthesis to this day. Many reactions are known to yield particular results; somewhat fewer can be described in mechanistic detail (i.e., which atoms go where as the system passes through its transition state); far fewer have been modeled at the level of potential energy surfaces and molecular dynamics. Detailed theory thus describes only a fraction of the synthetic operations available to the modern chemist.

8.2.2. The prominence of qualitative results in organic synthesis

Organic synthesis is fundamentally qualitative, in much the same sense that topology (as distinct from, say, analytic geometry) is fundamentally qualitative. In synthesis, the chemist seeks to construct molecules with a particular pattern of bonding that can be described (give or take discrete, stereochemical differences) in topological terms. The discovery that substances of type A can be transformed into substances of type B is inherently qualitative and often of great value. The development of such useful species and processes as Grignard reagents, Wittig reactions, and Sharpless epoxidations are examples of contributions of this kind.

In describing an organic synthesis, numbers are used to specify temperatures, quantities of material, and characteristics such as melting points, refractive indexes, and so forth, but these numbers play a peripheral role. One can frequently examine several chapters of a monograph on chemistry while encountering no numbers save for dimensionless integers, such as those used to specify molecular structures and charge states. To be useful in experimental work, most measurements need only discriminate among different molecular species, and accurate discrimination need not require great precision. Many theoretical contributions of great fame and value take the form of nonquantitative rules (some subject to violation) that are expressed in terms of structure and geometry; examples include Markownikoff's rule for regioselectivity of addition to alkenes, the Alder rule for the production of isomers in Diels-Alder reactions, and the Woodward-Hoffmann rules regarding orbital symmetry in { }^{\circ}cycloaddition processes.

Returning to the picture of potential energy surfaces in configuration space, the discovery of a new kind of reaction corresponds to the discovery of a set of similar cols between potential wells corresponding to similar molecular species. In standard chemistry, useful cols must be low enough to be thermally accessible at moderate temperatures and other cols must be relatively inaccessible (or harmless to traverse). The quantitative results of greatest interest in organic synthesis relate to the height and width of the cols (which determine reaction kinetics) and the height and width of the potential wells (which determine reaction equilibria).

8.2.3. A survey of synthetic achievements

No simple first-principles analysis leads from elementary facts regarding molecular physics to conclusions regarding the range of structures that can be made by diffusive, solution-phase chemical processes. Accordingly, one must examine the actual capabilities that chemists have developed. These include the synthesis of:

  • Highly strained molecules, such as cyclopropane rings 8.1, with 6060^{\circ} bond angles, rather than the optimal 109.5109.5^{\circ}; cubane 8.2 , with a total strain energy of 1.15aJ\sim 1.15 \mathrm{aJ} (Eaton and Castaldi, 1985), about twice the typical CC\mathrm{C}-\mathrm{C} bond energy; and [1.1.1]-propellane 8.3 (Wiberg and Walker, 1982), in which the bonding configuration at two of the carbon atoms is inverted relative to the normal tetrahedral geometry-all four bonds are in one hemisphere.

8.1

8.2

8.3

  • Highly symmetrical molecules, such as dodecahedrane 8.4 (Ternansky et al., 1982) and cyclo[18]carbon 8.5 (Diederich et al., 1989).

8.4

8.5

  • Molecules that spontaneously decompose to free highly reactive species, such as carbenes (species containing divalent carbon), Eq. (8.1), and even free carbon atoms (Shevlin and Wolf, 1970), Eq. (8.2). (These species ordinarily exist only briefly before reacting.)
  • Some six hundred organic structures, containing a few tens to a few hundred atoms, identical to molecules synthesized by molecular machinery in living systems (Corey and Cheng, 1989). Some examples include 8.6 [anno- tinine (Wiesner et al., 1969)], 8.7 [bleomycin A2\mathrm{A}_{2} (Aoyagi et al., 1982)], and 8.8 [endiandric acid C (Nicolaou et al., 1982)]

  • Chains of a hundred or more monomers, joined in a precise sequence to build biologically active protein and DNA molecules containing on the order of 1000 precisely bonded atoms (Caruthers, 1985; Kent, 1988).
  • Molecules designed to self-assemble into larger structures (Cram, 1988; Lehn, 1988; Rebek, 1987), and even to catalyze the synthesis of copies of themselves (Tjivikua et al., 1990).

8.3. Solution-phase synthesis and mechanosynthesis

8.3.1. Analytical approach

This section uses the demonstrated mechanisms and capabilities of solutionphase organic synthesis as a basis for understanding the capabilities of mechanosynthesis. It proceeds by examining differences:

  • The solution-phase conditions and unit operations excluded by mechanosynthetic constraints.
  • The new conditions and unit operations provided by mechanosynthesis.
  • The additional system-level capabilities implied by these differences.
  • The additional constraints that must be satisfied to exploit these systemlevel capabilities.

Taken together with the known capabilities of diffusive organic synthesis, this set of differences gives substantial insight into the capabilities of mechanosynthesis.

a. A terminological note: exothermic vs. exoergic. In solution-phase chemistry, when molecular potential energy is transformed into mechanical energy, this promptly appears as heat. Reactions that reduce potential energy are accordingly termed exothermic; those that increase it, endothermic. In mecha- nochemical systems, however, potential energy released by a reaction can often be stored elsewhere in the mechanical system, either as potential energy or as kinetic energy in the form of orderly motion. Accordingly, the more general terms exoergic and endoergic are appropriate. Changes in free energy determine the direction of a reaction, and since changes in entropy are usually small in mechanosynthetic steps (which typically transform one highly constrained structure into another), changes in free energy and in potential energy are often almost equal. Under these conditions, exoergic reactions can produce both heat and work, and most endoergic reactions are driven more by work than by heat.

b. Mechanochemical parameters. Two major themes in the following sections are reaction rates and reaction reliability in mechanochemical systems. Useful variables for an approximate analysis include the actuation time tact t_{\text {act }}, during which motions carry the system toward and then away from a reactive geometry; the transformation time ttrans t_{\text {trans }}, during which the geometry of the system permits a reaction to occur; the reaction rate kreact k_{\text {react }}, which measures the probability per unit time of a reactive transformation during ttrans t_{\text {trans }}, given that a reaction has not yet occurred; the intersystem crossing rate kisc k_{\text {isc }}, which measures the rate of the electronic intersystem crossing "reaction"; and the error rate kerr k_{\text {err }}, which measures the rate of unwanted reactions. These parameters will be related to the probability of an error in a mechanochemical operation, PerrP_{\mathrm{err}}. It will usually be assumed that tact 106 s,ttrans 107 st_{\text {act }} \approx 10^{-6} \mathrm{~s}, t_{\text {trans }} \approx 10^{-7} \mathrm{~s}, and that each contribution to PerrP_{\mathrm{err}} is to be 1015\leq 10^{-15} (see Sections 8.3.4c to 8.3.4f).

8.3.2. Basic constraints imposed by mechanosynthesis

a. Loss of natural parallelism. Solution-phase synthesis often yields macroscopic quantities of product. In replacing a macroscopic volume of reagent solution with a single device, the number of reacting entities is reduced by a factor on the rough order of 102310^{-23}. To compensate for this difference requires many mechanosynthetic devices, each operating at a high frequency. Some consequences of this constraint are discussed in Section 8.3.4c.

b. Limitations on reagents and products. Mechanical control of reactions typically requires that reagent moieties be bound to extended "handle" structures; control of products typically requires that they, too, be bound, excluding the use or production of small, freely moving species. This constraint is less severe than it might seem, however, because free and bound species can often participate in similar reactions. For example, although an SN2\mathrm{S}_{\mathrm{N}} 2 reaction producing a free ion [Eq. (8.3)] would cause unacceptable disorder, (8.4) is permissible.

In reality, an ionic process like reaction (8.3) would require a favorable electrostatic environment, and that environment could be used to bind the product ion. In water, small ions (from Li+\mathrm{Li}^{+}to I\mathrm{I}^{-}) are found in solvent cages, and are bound with energies 330maJ\geq 330 \mathrm{maJ} relative to vacuum (Bockris and Reddy, 1970a). In condensed-phase acid reactions " H+\mathrm{H}^{+}" is always bound. Reactions characteristic of small free { }^{\circ}radicals (e.g., H,F,OH\mathrm{H}, \mathrm{F}, \mathrm{OH} ) can frequently be made to proceed by concerted mechanisms in which the radicals are never free. Binding of such species usually lessens reactivity, but Section 8.3.3 describes several effective techniques for speeding reactions in mechanochemical systems. Purely { }^{\circ}steric confinement sometimes is feasible, and has little effect on the electronic structure of the reagent.

The requirement for handles (of any kind) places significant steric constraints on feasible reactions. Small, diffusing species can reach many sites (e.g., deep in pockets on a surface) that would be inaccessible to a mechanically positioned reagent moiety. In planning the synthesis of an object, geometric constraints are thus of increased importance. Finally, for reaction processes to be reliable, reagents must have adequate stability against unimolecular decomposition reactions (Section 8.3.4f).

c. Lack of true solvation. Fully { }^{\circ}eutactic systems must lack true solvation, since solvents are not eutactic. As discussed in Section 8.3.3b, however, solventlike eutactic environments can be superior to true solvents in promoting reaction speed and specificity. Mechanical transport can likewise substitute for diffusive transport; note that this sidesteps the constraint that reactants and products be soluble.

d. Limited usefulness of true photochemistry. Although similar results can be achieved using other sources of electronic excitation, photochemical reactions in a strict sense require photons. Optical wavelengths, however, are measured in hundreds of nanometers, and a cubic-wavelength block accordingly contains millions of cubic nanometers. This mismatch relative to the molecular scale limits the utility of photochemistry in mechanosynthesis (but note Section 8.3.3d).

e. Limitation to moderate temperatures. Mechanosynthetic systems are sensitive to damage from thermomechanical degradation (Section 6.4), and the single-point failure assumption (Section 6.7) is adopted here. With careful design, reliable operation at substantially elevated temperatures may be feasible. Nonetheless, temperatures like those used in melt-processing of typical ceramics are almost surely infeasible. The present work assumes that temperatures remain near 300 K300 \mathrm{~K}, thereby eliminating from consideration a large class of reactions that require elevated temperatures to proceed at an acceptable rate. This constraint could be partially evaded by exploiting highly localized, nonthermal kinetic energy sources to provide activation energy (e.g., impacts driven by stored elastic energy), but this approach is not pursued here.

8.3.3. Basic capabilities provided by mechanosynthesis

a. Large effective concentrations. In a solution-phase reaction A+BC\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}, etc., the reaction frequency of molecules of type AA is directly proportional to the concentration of molecules of type B (so long as molecules of B rarely interact with one another). In bulk chemistry, concentrations are commonly stated in moles per liter (M)(\mathbf{M}); here, they are stated as a molecular number density (1 nm31.66M)\left(1 \mathrm{~nm}^{-3} \approx 1.66 \mathrm{M}\right). The concentration of water molecules in liquid water is 33 nm3\sim 33 \mathrm{~nm}^{-3}.

Because their moieties are held in proximity to one another, mechanosynthetic reactions resemble intramolecular reactions; these, in turn, often resemble intermolecular reactions, save for a covalent link between reacting moieties. From the rate for an intramolecular reaction (in s1\mathrm{s}^{-1} ) and the rate constant for an analogous intermolecular reaction (in nm3 s1\mathrm{nm}^{3} \mathrm{~s}^{-1} ), one can compute the effective concentration of a moiety BB with respect to a moiety AA in the intramolecular process by computing the (hypothetical) concentration of B-like molecules (in nm3\mathrm{nm}^{-3} ) that would subject an A-like molecule to the same reaction frequency in the intermolecular process. The concept of effective concentration, although of no intrinsic importance to mechanosynthesis, is quite useful in estimating machine-phase reaction rates from solution-phase data.

Experiment shows that effective concentrations for reactions between moieties on small molecules are not limited to 10 nm3\sim 10 \mathrm{~nm}^{-3} (typical of immersion in a liquid); they can be as large as 103\sim 10^{3} to 1010 nm310^{10} \mathrm{~nm}^{-3} (Creighton, 1984). Two examples cited by Creighton are:

The occurrence of large magnitudes can readily be understood: In the probability-gas picture of classical transition state theory (Section 6.2.2), the rate of transitions is proportional to the density of the gas. For concreteness and ease of visualization, consider an intramolecular reaction in which a single atom is transferred (e.g., atom abstraction). The concentration of an atom at the bottom of the potential well defining its initial state is just the reciprocal of the effective volume; for example, using Eq. (6.5) and a stiffness of 20 N/m20 \mathrm{~N} / \mathrm{m} in each of three dimensions (vs. 30 N/m\sim 30 \mathrm{~N} / \mathrm{m} typical of bond angle-bending; Section 3.3.2b) yields a number density of 2×104 nm3\sim 2 \times 10^{4} \mathrm{~nm}^{-3}; this local density is unrelated to (and far larger than) the mean number-density of nuclei in solids at ordinary pressures. Further, many reactions require a relatively precise orientation of the reagent moieties; structural constraints in intramolecular reactions can greatly increase the concentration of properly oriented moieties, multiplying the effective concentration by a large factor. (Intramolecular reactions can also be accelerated by intermoietal strain; this can, however, be viewed as a violation of the intramolecular-intermolecular analogy that motivates the notion of effective concentration, and is more closely analogous to the piezochemical effects described in Section 8.3.3c.)

Binding of reagents so as to ensure high effective concentrations is a major mechanism of enzymatic catalysis (Creighton, 1984). Mechanochemical processes can likewise present reagent moieties to one another in favorable orientations and positions, producing high effective concentrations. Relative to intermolecular reactions in solution (with a typical concentration of 1 nm3\sim 1 \mathrm{~nm}^{-3} ), this mechanism should routinely yield rate accelerations on the order of 10410^{4} or more.

b. Eutactic "solvation." Where reactions involve substantial changes in charge separation, solvent effects can alter reaction rates by factors in excess of 10510^{5}. Relative to vacuum or nonpolar environments, a polar solvent greatly decreases the energetic cost of creating dipoles or separated ions by orienting its molecules in a manner that decreases electrostatic field strengths. The feasible reduction in free energy is, however, lessened by the entropic cost of orienting the solvent molecules. A preorganized structure (Cram, 1986), in contrast, can equal the electrostatic effects of a good solvent without imposing entropic costs on the transition state.

Indeed, a preorganized environment can yield larger electrostatic effects. The solvent structure at the transition state of a solution-phase reaction minimizes the free energy of the transition-state system as a whole, including that of the solvent. A different organization of solvent molecules would minimize the free energy of the transition-state reactants alone; this organization typically is more highly ordered (imposing additional entropic costs) and more highly polarized (imposing substantial energetic costs). Preorganized environments can be constructed to mimic this hypothetical, high-free-energy solvent state, paying the free-energy cost in advance and thereby outperforming a good solvent in promoting the desired reaction (a principle exploited by enzymes).

The magnitude of these effects can be roughly estimated. Ionic reactions commonly involve charge displacements on the order of one electron-charge and one bond length, or 2.5×1029Cm\sim 2.5 \times 10^{-29} \mathrm{C} \cdot \mathrm{m}. Fields of 2×109 V/m2 \times 10^{9} \mathrm{~V} / \mathrm{m} (e.g., 1 V1 \mathrm{~V} across 0.5 nm0.5 \mathrm{~nm} ) are easily achieved in a polarized environment; contact potentials in metallic systems (Section 11.7.2a) can produce fields of this magnitude. The energy difference associated with this field and charge displacement, 50maJ\sim 50 \mathrm{maJ}, gives a rough measure of the effect that local electrostatics can have on the energies of transition and product states in ionic reactions. This energy suffices to change rates and equilibria by factors of >105>10^{5} at 300 K300 \mathrm{~K}. The electrostatic energy difference between two orientations of an HCN\mathrm{HCN} molecule adjacent to a pair of oriented HCN molecules (Figure 8.1 ) is 110maJ\sim 110 \mathrm{maJ}, an energy differential sufficient to change a rate or equilibrium by a factor >1010>10^{10}.

Enzymatic active sites often resemble preorganized solvent structures, reducing the energy of a transition state. Mechanochemical systems based on a more general class of molecular structures can equal or exceed enzymes in this regard. The requirement that all structures be well bound, prohibiting true solvation (Section 8.3.2c), does not appear to be a significant sacrifice.

As enzymes show, molecular flexibility can be achieved without freely moving small molecules. In a mechanochemical context, however, it frequently is desirable to provide relatively rigid support for a reagent structure; one strategy is to surround the structure with a nonbonded (hence solventlike) but rigid shell.

Figure 8.1. Dipole-dipole interaction geometries and energies. In (a), the two parallel HCN\mathrm{HCN} molecules (modeling bound nitrile moieties) have been held fixed at a separation of 0.3 nm0.3 \mathrm{~nm} while the energy of the third is minimized. In (b), the orientation of the third has been reversed. The increase in the MM2/CSC electrostatic (and total) energies from (a) to (b) is 110maJ\sim 110 \mathrm{maJ}.

c. Mechanical forces. The term piezochemistry is in general use to describe solution-phase chemical processes in which mechanical pressure modifies chemical reactivity.2 Derived from the Greek piezein ("to press"), the term is here adopted to refer to a wider range of machine-phase chemical processes in which time-dependent mechanical forces (not necessarily a homogeneous, isotropic, slowly varying pressure) modify chemical reactivity.

In conventional piezochemistry, pressures accessible in commercially-available laboratory equipment (e.g., 0.1 to 2GPa2 \mathrm{GPa} ) frequently have substantial effects on reaction rates and equilibria. Transition states in solution-phase chemistry are characterized by an activation volume ΔV\Delta V^{\ddagger} defined by Eyring in terms of a constant-temperature partial derivative:

ΔV=kT(plnkreact )T\begin{equation*} \Delta V^{\ddagger}=-k T\left(\frac{\partial}{\partial p} \ln k_{\text {react }}\right)_{T} \tag{8.7} \end{equation*}

where kreact k_{\text {react }} is the rate constant for the reaction (Section 6.2.1) and pp is the pressure. Typical values of ΔV\Delta V^{\ddagger} are in the range of -0.01 to 0.10 nm3-0.10 \mathrm{~nm}^{3}; values can be positive, for example, in fragmentation reactions (Isaacs and George, 1987; Jenner, 1985). Many reactions have negative volumes of activation because their transition states combine bond making with bond breaking, and because the shortening of nonbonded distances by partial bond formation exceeds the lengthening of bonded distances by partial bond breakage. The relatively compact transition state is then favored by increased pressure. 3

In the (poor) approximation that ΔV\Delta V^{\ddagger} is independent of pressure, the rate increases exponentially with pressure. In reality, increasing pressures alter ΔV\Delta V^{\ddagger} by altering the potential energy surface. Equation (8.7) and moderate-pressure values of ΔV\Delta V^{\ddagger} give only a rough guide to the magnitudes of the effects that can be expected in higher-pressure piezochemical processes. With this assumption, a pressure of 2GPa2 \mathrm{GPa} applied to a reacting system with ΔV=0.02 nm3\Delta V^{\ddagger}=-0.02 \mathrm{~nm}^{3} results in a rate increase of 1.6×1041.6 \times 10^{4} at 300 K300 \mathrm{~K}.

Nanomechanical mechanosynthetic devices can be built of diamond and diamondoid structures, and in the continuum approximation (Chapter 2), stress is a scale-independent parameter. Accordingly, mechanosynthetic devices can apply pressures equaling those in macroscale diamond-anvil pressure cells. These reach 550GPa\geq 550 \mathrm{GPa} (Mao et al., 1989; Xu et al., 1986), corresponding to a 30nN\geq 30 \mathrm{nN} compressive load per bond in a (111) diamond plane (neglecting the change in areal bond density with pressure). The effects of such pressures on bonding are substantial: at 150GPa,H2\sim 150 \mathrm{GPa}, \mathrm{H}_{2} becomes metallic (Hemley and Mao, 1990), as does CsI at 110GPa\sim 110 \mathrm{GPa} (Mao et al., 1989); xenon has likewise been metallized. Even a lesser pressure of 50GPa50 \mathrm{GPa} ( 3nN/\sim 3 \mathrm{nN} / bond) can have large chemical effects. In the constant ΔV\Delta V^{\ddagger} approximation, p=50GPap=50 \mathrm{GPa} and ΔV=0.01 nm3\Delta V^{\ddagger}=-0.01 \mathrm{~nm}^{3} would yield a 500maJ500 \mathrm{maJ} reduction in activation energy (and a physically unrealistic >1052>10^{52} speedup); effects in real systems are smaller, but still large. The change in free energy resulting from high pressures can greatly exceed the change resulting from high temperatures (within the conventional laboratory range).

Mechanically applied energy (unlike random thermal vibration) is subject to precise control. This discussion has been cast in terms of pressure and volume because it draws on experiments and theories applicable to solution-phase processes. In mechanochemistry, however, force and displacement are more useful variables, and (for a particular mode of displacement) an activation length

Δ=kT(Flnkreact )T\begin{equation*} \Delta \ell^{\ddagger}=-k T\left(\frac{\partial}{\partial F} \ln k_{\text {react }}\right)_{T} \tag{8.8} \end{equation*}

can be defined in terms of the applied force FF. A typical magnitude for Δ\Delta \ell^{\ddagger} is 0.1 nm\sim-0.1 \mathrm{~nm}, and feasible compressive loads can extend to >5nN>5 \mathrm{nN}, which would once again yield a characteristic energy of 500maJ\sim 500 \mathrm{maJ}. (Section 8.5.2a and 8.5.4 develop more realistic estimates for certain systems.)

Mechanical instabilities can limit compressive loads: for example, if a singleatom tip is pressed against another atom, it has a tendency to slip sideways ("down off the hill") unless this is resisted by an adequate transverse stiffness. The negative stiffness associated with this instability has a magnitude

ks, instab =Fcomprr1+r2\begin{equation*} k_{\mathrm{s}, \text { instab }}=-\frac{F_{\mathrm{compr}}}{r_{1}+r_{2}} \tag{8.9} \end{equation*}

in the (conservative) model of hard spheres with radii r1r_{1} and r2r_{2}. A typical transverse stiffness ksk_{\mathrm{s} \perp} for an unloaded tip is 20 N/m20 \mathrm{~N} / \mathrm{m} (characteristic of angle bending for a single bond; Section 3.3.2b), and a typical atomic radius for a nonbonded contact under substantial loads is 0.1 nm\sim 0.1 \mathrm{~nm} (Figure 3.9), hence the condition

ks, instab +ks>0\begin{equation*} k_{\mathrm{s}, \text { instab }}+k_{\mathrm{s} \perp}>0 \tag{8.10} \end{equation*}

permits Fcompr 4nNF_{\text {compr }} \approx 4 \mathrm{nN}. A load of this magnitude suffices to store >100maJ>100 \mathrm{maJ} of potential energy in overlap repulsion between two unreactive atoms (Figure 3.8), and can store additional energy in the more-compliant deformation modes that lead toward a chemical reaction.

Unlike forces resulting from hydrostatic pressure, forces applied by mechanochemical devices can be highly anisotropic and inhomogeneous on a molecular scale: large loads (including tension, shear, and torsion) can be applied to specific atoms and bonds in a controlled manner. As noted in Section 6.4.4, these forces can cleave otherwise-stable bonds. Further, steric difficulties can be reduced by molecular compression and deformation in conventional piezochemistry (Jenner, 1985). Under mechanochemical conditions, larger effects can be obtained which can significantly offset steric difficulties posed by the mechanosynthetic requirement for bound reagents [for comparison, at 4GPa4 \mathrm{GPa} the atomic number density in liquid cyclohexane is increased by a factor of 1.5\sim 1.5 (Gray, 1972)]. In general, the availability of controlled forces of bond-breaking magnitude permits piezochemical modulation of reactions greatly exceeding that seen in solution-phase chemistry or in the comparatively low-strength, low-stiffness environment of an enzymatic active site. Section 8.5 discusses piezochemical effects in further detail.

d. Localized electrochemistry, "photochemistry." Mechanochemical systems can exploit nonmechanical energy sources, for example, through electrochemistry and energy transfer via electronic excitations. Both of these mechanisms can be controlled more precisely in a mechanochemical environment than in solution or solution-surface systems.Electrochemistry finds significant use in organic synthesis (Kyriacou, 1981). Electrostatic potentials and tunneling rates can vary sharply on a molecular scale (Bockris and Reddy, 1970b), permitting molecular-scale localization of electrochemical activity. Accordingly, electrochemical processes are well suited to exploitation in a mechanochemical context; they are also subject to modulation by piezochemical means (Swaddle, 1986). In electrochemical cells, pyridine can withstand an electrode potential of 3.3 V3.3 \mathrm{~V} without reaction and tetrahydrofuran, 3.2 V-3.2 \mathrm{~V}, both with respect to a (catalytically active) platinum electrode (Kyriacou, 1981); these potentials correspond to energy differences with a magnitude >500maJ>500 \mathrm{maJ} per unit charge. In field-ion microscopes (which provide one model for an electrode surface), electric fields can reach 50 V/nm\sim 50 \mathrm{~V} / \mathrm{nm} (Nanis, 1984). Chapter 11 will discuss certain aspects of electrical engineering on a nanometer scale, but electrochemical processes, despite their undoubted utility, are not exploited in the following analysis.

Direct photochemistry suffers from problems of localization (which do not preclude its use), but photochemical effects can often be achieved by nonphotochemical means. Photochemical processes begin with the electronic excitation of a molecule by a photon, but this energy often can migrate from molecule to molecule as a discrete exciton before inducing a chemical reaction, and this process is highly sensitive to molecular structures and positions. Accordingly, the transfer of excitons can provide a better-controlled means for achieving photochemical ends; potential applications are, however, neglected in this volume.

e. Broadened options for catalysis. The structural requirements for mechanochemical reagents discussed in Section 8.3.2b are satisfied by many catalytic structures: some are parts of solid surfaces already, and others (e.g., many homogeneous transition metal catalysts) have analogues that can be covalently anchored to a larger structure. The remarks of Section 8.3.2b apply to small catalytic species (e.g., hydrogen and hydroxide ions).

Aside from regeneration treatments (which are infrequent on a molecular time scale), conventional catalysts operate under steady-state conditions. In a typical catalytic cycle, reagents are bound to form a complex, the complex rearranges, and a product departs, all in the same medium at constant pressure, temperature, and so forth. If any transition state in this sequence of steps is too high in energy, its inaccessibility will block the reaction. If any intermediate state is too low in energy, its stability will block the reaction. If any feasible alternative reaction (with any reagent or contaminant in the diffusing mixture) leads to a stable complex, the catalyst will be poisoned. Many successful catalysts have been developed, but the above conditions are stringent, requiring a delicately balanced energy profile across a sequence of steps (Crabtree, 1987).

The range of feasible catalytic processes is broadened by the opportunities for control in mechanochemical processes. The elementary reaction in a catalytic cycle can occur under distinct conditions, lessening the requirement for delicate compromises to avoid large energy barriers or wells. As discussed in Section 8.5.10c8.5 .10 \mathrm{c}, mechanochemical catalysts can be subjected to manipulations that modulate bond and transition state energies, typically by many times kT300k T_{300}. Finally, comprehensive control of the molecular environment enables the designer to prevent many unwanted reactions (Section 8.3.3f), permitting the use of more reactive species (Section 8.3.3 g8.3 .3 \mathrm{~g} ).

f. Avoidance of competing reactions. In diffusive synthesis, achieving 95%95 \% yield in each of a long series of steps would ordinarily be considered excellent. At the end of a 100 -step process, however, the net product would be 0.6%\sim 0.6 \%; and at the end of a 2000 -step process, 1043%\sim 10^{-43} \%. A million tons of starting reagents would then reliably yield zero molecules of the desired product. At equilibrium, a reaction with ΔF145maJ\Delta \mathscr{F} \leq-145 \mathrm{maJ} at 300 K300 \mathrm{~K} leaves less than 101510^{-15} of the starting molecules unreacted. An energy difference of this magnitude is not uncommon, and a series of reactions with this yield would permit over 101010^{10} sequential steps with high overall yield, in the absence of side reactions.

The complexity of the structures that can be built up by diffusive synthesis is limited not by an inability to add molecular fragments to a structure, but by the difficulty of avoiding mistaken additions. This problem is substantial, even in 100 -atom structures. Moreover, as structures grow larger and more complex, they tend to have increasing numbers of functional groups having similar or identical electronic and steric properties (on a local scale). Reliably directing a conventional reagent to a specific functional group becomes increasingly difficult, and ultimately impossible.4

In a well-designed eutactic mechanochemical system, unplanned molecular encounters do not occur, and most unwanted reactions accordingly are precluded. One class of exceptions consists of reactions analogous to unimolecular fragmentation and rearrangement; these instabilities are discussed in Section 6.4, and are discussed further in connection with reagent moieties in Sections 8.4 and 8.5. The other class of exceptions consists of reactions that occur in place of desired reactions; these can be termed misreactions.

A typical mechanosynthetic step involves the mechanically guided approach of a reagent moiety to a target structure, followed by its reaction at a site on that structure. In general, unguided reactions would be possible at several alternative sites, each separated from the target site by some distance (properly, a distance in configuration space). At one extreme, the alternative sites are separated by a distance sufficient to make an unwanted encounter in the guided case energetically infeasible (e.g., requiring that the mechanical system either break a strong bond or undergo an elastic deformation with a large energy cost). At the other extreme, the potential energy surface is such that passage through a single transition state leads to a branching valley, and then to two distinct potential wells, only one of which corresponds to the desired product; in this circumstance, unwanted reactions would be unavoidable. In intermediate cases, transition states leading to desired and undesired products are separated by intermediate distances, and the mechanical stiffness of the guiding mechanism imposes a significant energy cost on the unwanted transition state, relative to the unguided case.

Considering the approach of a reagent moiety to a target structure in threedimensional space, a reaction pathway can be characterized by the trajectory of some atom in the moiety (such as an atom that participates in the formation of a bond to the surface). Inspection of familiar chemical reactions concurs with elementary expectations in suggesting that reaction pathways leading to alternative products commonly are characterized by trajectories that differ by a bond length (dt0.15 nm)\left(d_{\mathrm{t}} \approx 0.15 \mathrm{~nm}\right) or more at the competing transition states (for example, see Section 8.5.5a). This is not universally true, but it is the norm; accordingly, avoidance of the exceptional situations imposes only modest limitations on the available sequences of chemical reactions. Where reagents are selective, or have reactivities with a strong orientational dependence, alternative trajectories usually are separated by far greater distances.

A reasonable stiffness for the displacement of an atom in a mechanically guided reagent moiety (measured with respect to a reasonably rigid or wellsupported target structure) is 20 N/m20 \mathrm{~N} / \mathrm{m} (comparable to the 30 N/m\sim 30 \mathrm{~N} / \mathrm{m} transverse stiffness of a carbon atom with respect to an sp3s p^{3} carbon, Section 3.3.2b). Note that the mechanical constraints on a reactive moiety can in many instances include not only the stiffness of its covalent framework, but forces resulting from an unreactive, closely packed surrounding structure that substantially blocks motion in undesired directions (see Figure 8.2). Using the sinusoidal worst-case decoupling model of Section 6.3.3d to estimate error rates with nonsinusoidal potentials, the condition Perr1015P_{\mathrm{err}} \leq 10^{-15} requires that the elastic energy difference between the two locations be 180maJ\geq \sim 180 \mathrm{maJ} at 300 K300 \mathrm{~K}. With ks=20 N/mk_{\mathrm{s}}=20 \mathrm{~N} / \mathrm{m}, this condition is satisfied at transition-state separations dt0.135 nmd_{\mathrm{t}} \geq 0.135 \mathrm{~nm}. Accordingly, with modest constraints on the chemistry of the reacting species, suppression of

(a)

(b)

(c)

(d)

(e)

Figure 8.2. Models of a reagent moiety of low intrinsic stiffness with varying degrees of support from surrounding nonbonded contacts (MM2/CSC potential). Each model includes an { }^{\circ}alkyne moiety representing (for example) an alkynyl radical of the sort that might be used as a hydrogen { }^{\circ}abstraction tool (Section 8.5.4c). In (a), the moiety is supported by an adamantyl group; the bending stiffness of the moiety at its most remote carbon atom, taking the hydrogen atoms attached to the six-membered ring opposite the moiety as fixed, is 6 N/m\sim 6 \mathrm{~N} / \mathrm{m}. In (b), a more crowded environment increases the stiffness to 11 N/m\sim 11 \mathrm{~N} / \mathrm{m} (relative to those hydrogen atoms not either bonded to or in contact with the alkyne moiety); (c) replaces H\mathrm{H} atoms with Cl\mathrm{Cl}, increasing the stiffness to 20 N/m\sim 20 \mathrm{~N} / \mathrm{m}; greater stiffness could be achieved by adding structures outside the Cl\mathrm{Cl} ring that press its atoms inward. Structure (d) surrounds the alkyne with a ring of oxygenlinked carbon atoms, yielding a calculated stiffness of 65 N/m\sim 65 \mathrm{~N} / \mathrm{m}. Structure (e), however, represents a possible failure mode of (d) in which a rearrangement has cleaved six CO\mathrm{C}-\mathrm{O} bonds, yielding six aldehyde groups. Although this process is quite exoergic, the transition state may be effectively inaccessible at room temperature.

unwanted reactions by factors of better than 101510^{-15} should be routinely achievable. With ks=30 N/mk_{\mathrm{s}}=30 \mathrm{~N} / \mathrm{m} and dt0.15 nmd_{\mathrm{t}} \geq 0.15 \mathrm{~nm}, this model yields Perr 1027P_{\text {err }} \leq 10^{-27}. (Although the differing ratios of accessibility between alternative transition points at different times during the approach complicate the situation, the essential conclusions remain unchanged.)

Section 8.5.5a describes a specific, relatively challenging case: discriminating between the ends of the double bond in an alkene during radical addition. It concludes that error rates 1015\leq 10^{-15} are achievable with stiffnesses of 10 N/m\sim 10 \mathrm{~N} / \mathrm{m}, or 6 N/m\leq 6 \mathrm{~N} / \mathrm{m} using a trajectory-biasing technique.

g. Reliable control of highly reactive reagents. It might seem that the most useful reagents would be those (e.g., strong free radicals, carbenes) that can react with many other structures. In solution-phase chemistry, however, unwanted reactions are the chief limit to synthesis, and reagents are prized less for their reactivity than for their selectivity. The ideal reagent in solution-phase synthesis is inert in all but a few circumstances, and it need not react swiftly when it reacts at all.

In mechanosynthesis, however, selectivity based on the local steric and electronic properties of the reagents themselves can be replaced by nearly perfect specificity based on positional control of reagent moieties by a surrounding mechanical system. Accordingly, highly reactive reagents gain utility, including

Table 8.1. Comparison of solution-phase synthesis and mechanosynthesis (a briefer, more general comparison appears in Table 1.2).

CharacteristicSolution-phase synthesisMechanosynthesis
ParallelismNaturalRequires many devices
Reagent structureUnconstrained (but soluble)Bound to "handle"
Electrostatic
environment
Control of solvent
dielectric constant
Control of
dielectric constant, fields
ElectrochemistryUsefulUseful, localized
PhotochemistryUsefulPoorly localized
(or exciton mediated)
TemperaturesCan be highMust be moderate
Max. effective
concentrations
100 nm3\sim 100 \mathrm{~nm}^{-3}>109 nm3>10^{9} \mathrm{~nm}^{-3}
Available pressuresUp to 2GPa\sim 2 \mathrm{GPa} (routinely)>500GPa>500 \mathrm{GPa}
Control of forcesMagnitude of
uniform pressure
Location, magnitude,
direction, shear, torque
Positional controlNoneAll three degrees
of freedom
Orientational controlNoneAll three degrees
of freedom
Reagent requirementsSelectivityStability and reactivity
Reaction site selectivitySteric, electronic influencesDirect positional control
Max. synthesis complexity100\sim 100 to 1000 steps5>1010>10^{10} steps

reagents that are highly unstable to intermolecular reactions among molecules of the same type (e.g., benzynes, reactive dienes, and other fratricidal molecules). The use of reagents with increased reactivity can increase reaction frequencies, adding to (or, more properly, multiplying together with) the influences of high effective concentration, well-designed eutactic "solvation," and applied force.

8.3.4. Preview: molecular manufacturing and reliability constraints

The capabilities of mechanosynthetic processes can be gauged by comparing the limitations discussed in Section 8.3.2 with the strengths discussed in Section 8.3.3, taking the capabilities of diffusive synthesis as a baseline. This comparison, summarized in Table 8.1, suggests that mechanosynthetic operations are (overall) more capable than diffusive operations. These capabilities must, however, be judged in light of their proposed applications. This section previews proposed molecular manufacturing systems and discusses their associated speed and reliability requirements. Molecular manufacturing systems are examined more thoroughly in Chapters 13 and 14 after the analysis of nanomechanical components and systems summarized in Chapters 9 through 12.

a. Molecular manufacturing approaches. Mechanochemical devices are of interest in this volume chiefly as components of mechanosynthetic systems capable of building large, complex structures, including systems of molecular machinery. The necessary positioning and manipulation of reagent moieties can be achieved in any of several ways, but these can be roughly divided into molecular mill approaches and molecular manipulator approaches. Molecular mills perform simple repetitive motions; molecular manipulators enable complex, programmable motions (see Chapters 13 and 14). Mills are well suited to the task of preparing reagent moieties for manipulators, but can also produce final products directly. In all cases considered, mechanical transport mechanisms replace diffusion.

b. Reagent preparation vs. application. Product synthesis can use reagent moieties that have been prepared from other reagent moieties, starting ultimately with simple feedstock molecules. The preparative steps can occur in an environment significantly different from that of the final synthetic steps.

In reagent preparation, the entire surrounding structure can be tailored to facilitate the desired transformation. In this respect, the reaction environment can resemble that of an enzymatic active site, but with the option of exploiting a wider range of structures, more active reagents, piezochemical processes, and so forth. The freedom to tailor the entire reaction environment during reagent preparation is a consequence of the small size and consequent steric exposure of the structures being manipulated.

In the final synthetic step, however, one reacting surface must be a feasible intermediate stage in constructing the product, and a prepared reagent must be applied to that surface.6 Construction strategies can be chosen to facilitate the sequence of synthetic reactions, but the freedom to tailor the entire environment solely to facilitate a single reaction is not available. In these reagent application steps, highly reactive moieties are of increased utility and moieties compatible with supporting structures of low steric bulk are desirable. These remarks apply both to manipulator-based systems and to the reagent-application stages of a mill-style system engaged in direct synthesis of complex products.

c. Reaction cycle times. One measure of the productivity of a manufacturing system is the time tprod t_{\text {prod }} required for it to make a quantity of product equaling its own mass. Mill-style systems are anticipated to contain 106\sim 10^{6} atoms per processing unit, with each unit responsible for converting a stream of input molecules into a stream of reagent moieties which are incorporated into identical sites in a stream of product structures. Manipulator-style systems are anticipated to contain 108\sim 10^{8} atoms per manipulator, with each unit responsible for performing a series of mechanosynthetic operations on a single product structure. Each processing unit and manipulator operates on products at some frequency fsynth f_{\text {synth }}. A set of devices will transfer a mean number of atoms nsynth n_{\text {synth }} per operation; in any one operation, the number transferred may be positive or negative (e.g., in abstraction reaction to prepare a radical site on a workpiece), but a typical value is nsynth 1n_{\text {synth }} \approx 1. Accordingly, if fsynth =103 Hz,tprod =103 sf_{\text {synth }}=10^{3} \mathrm{~Hz}, t_{\text {prod }}=10^{3} \mathrm{~s} for a mill-based system and 105 s10^{5} \mathrm{~s} for a manipulator-based system; for fsynth =106 Hzf_{\text {synth }}=10^{6} \mathrm{~Hz}, the corresponding values of tprod t_{\text {prod }} are 1 and 100 s100 \mathrm{~s} respectively. Values of fsynth f_{\text {synth }} \geq 103 Hz10^{3} \mathrm{~Hz} are acceptable for many practical applications, and 106 Hz\sim 10^{6} \mathrm{~Hz} is used as a reference value in the systems analysis of Chapter 14.

The exemplar calculations of the following sections assume that the time available for a reaction ttrans =107 st_{\text {trans }}=10^{-7} \mathrm{~s}. This is compatible with tact 106 st_{\text {act }} \approx 10^{-6} \mathrm{~s}, or with fsynth 106 Hzf_{\text {synth }} \approx 10^{6} \mathrm{~Hz}. These times are all long compared to the characteristic time scale of molecular vibrations, 1013 s\sim 10^{-13} \mathrm{~s}, but require reactions with low energy barriers.

d. Constraints on misreaction rates. In molecular manufacturing, two basic classes of error are (1) those that damage one product structure (fabrication errors), and (2) those that damage the manufacturing mechanism (destructive errors). Overall reliability can be increased in each instance by dividing a system (product or manufacturing) into smaller modules and replacing those that fail. Where fabrication errors occur, the module being made can be discarded before being incorporated into a product system. Where destructive errors occur, the manufacturing module can be stopped and its function taken over by one or more backup modules (Section 13.3.6). Simplicity of design favors the use of relatively large modules, which in turn requires good reliability. For simplicity, the systems analysis of Chapter 14 assumes that all fabrication errors are destructive, placing further demands on reliability.

For many purposes, 10810^{8}-atom modules can be considered large. Fabrication errors at a rate of 1010\leq 10^{-10} per reagent application operation, resulting in a fabrication failure rate of 102\leq 10^{-2}, can thus be considered low. If we demand that each module in a manufacturing system process 102\geq 10^{2} times its own mass before failing (and assume nsynth 1n_{\text {synth }} \approx 1 ), then a destructive-error rate of 1010\leq 10^{-10} per reagent application operation is again acceptable. The following calculations, however, take as a requirement that both classes of error occur at rates 1015\leq 10^{-15} per mechanosynthetic operation; Section 8.3.3f describes conditions on mechanical stiffness and transition-state separation that can satisfy this constraint. An error rate of 101510^{-15} is compatible with the assumptions made in Chapters 13 and 14; higher error rates could be tolerated at the cost of increased design complexity.

e. Meeting constraints on omitted reactions in a single trial. Section 8.3.3f describes conditions for keeping the probability of misreactions Perr=1015\leq P_{\mathrm{err}}=10^{-15}; a related but distinct problem is to ensure that the desired reaction occurs, with a probability of omission (failure to react) Perr\leq P_{\mathrm{err}}. As also noted in Section 8.3.3f, a reaction with ΔF145maJ\Delta \mathscr{F} \leq-145 \mathrm{maJ} at 300 K300 \mathrm{~K} (at the time of kinetic decoupling, in terms of the approximation discussed in Section 6.3.3) can proceed with an equilibrium probability of remaining in the starting state <1015<10^{-15}. For a strongly exoergic reaction characterized by a rate constant kreact k_{\text {react }}, the probability of omission in a single trial falls to <1015<10^{-15} when the available reaction time ttrans 35/kreact t_{\text {trans }} \geq 35 / k_{\text {react }}. A one-dimensional model based on transition state theory (Section 6.2.2), together with the bounds just described, yields a bound on the allowable { }^{\circ}barrier height for the reaction

ΔVkTln(treact fTSTln(Perr ))\begin{equation*} \Delta \mathscr{V}^{\ddagger} \leq k T \ln \left(\frac{t_{\text {react }} f_{\mathrm{TST}}}{-\ln \left(P_{\text {err }}\right)}\right) \tag{8.11} \end{equation*}

where fTST f_{\text {TST }} is the transition state theory frequency factor; for mechanochemical reactions with relatively rigid, well-aligned reagent moieties, a reasonable value is fTST 1012 s1f_{\text {TST }} \geq 10^{12} \mathrm{~s}^{-1}. At 300 K300 \mathrm{~K}, with treact =107t_{\text {react }}=10^{-7} and Perr =1015,ΔV33maJP_{\text {err }}=10^{-15}, \Delta \mathscr{V}^{\ddagger} \leq 33 \mathrm{maJ} is acceptable.

How does this compare to typical solution-phase reactions? In the laboratory, characteristic reaction times (the reciprocals of the reaction rates) vary widely, from <109<10^{-9} to >106 s>10^{6} \mathrm{~s}; a not unusual reaction time in organic synthesis is 103 s\sim 10^{3} \mathrm{~s} at a reactant concentration of 1 nm3\sim 1 \mathrm{~nm}^{-3}. Relative to this, achieving a reliable reaction in ttrans =107 st_{\text {trans }}=10^{-7} \mathrm{~s} requires a speedup of 3×1011\sim 3 \times 10^{11}. Increased effective concentration owing to mechanical positioning can easily provide a speedup of >3×104>3 \times 10^{4} (Section 8.3.3a). Achieving the remaining factor of 10710^{7} requires that the energy barrier be lowered by 70\sim 70 maJ. Earlier sections have shown that electrostatic effects in eutactic environments can exceed 100maJ100 \mathrm{maJ}, and that (crudely estimated) piezochemical effects can exceed 500maJ500 \mathrm{maJ}; shifts from lessactive to more-active reagents can likewise have large effects. Achieving adequate reaction speeds does not appear to be a severe constraint. The duration of ttrans t_{\text {trans }} can be increased by 102\sim 10^{2} without undue cost by mechanisms that prolong the encounter process (Section 13.3.1).

Intersystem crossing from the { }^{\circ}singlet state to a low-lying { }^{\circ}triplet can cause errors, including omitted reactions and misreactions. Singlet transition-state geometries often resemble triplet equilibrium geometries (Salem and Rowland, 1972), with corresponding reductions in singlet-triplet energy gaps. To ensure reliability, it is sufficient to ensure either (1) that the singlet-triplet gap ΔVs,t\Delta \mathcal{V}_{\mathrm{s}, \mathrm{t}} always exceeds 145maJ[kTln(1015)145 \mathrm{maJ}\left[\approx k T \ln \left(10^{15}\right)\right. at 300 K300 \mathrm{~K} ], or (2) that as the ΔVs,t\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} increases, it exceeds 145maJ145 \mathrm{maJ} at some time t0t_{0} (prior to the time t1t_{1} at which the geometry is no longer suitable for correct bond formation), and that the integral of the intersystem crossing rate kisc k_{\text {isc }} meets the condition that

t0t1kisc(t)dtln(Perr)\begin{equation*} \int_{t_{0}}^{t_{1}} k_{\mathrm{isc}}(t) d t \geq \ln \left(P_{\mathrm{err}}\right) \tag{8.12} \end{equation*}

Meeting a somewhat weaker but more complex condition would also suffice.

f. Avoiding omitted reactions through conditional repetition. The conditions described in Section 8.3.4e were chosen to assure a reliable reaction in a single trial, but this is unnecessary. A conditional repetition process uses the outcome of a measurement process (Chapter 11) to determine whether an operation is repeated. For example, consider a reaction process in which the reaction rate and time ensure equilibrium, but in which ΔF=0\Delta \mathscr{F}=0 rather than 145maJ-145 \mathrm{maJ}. This process has a .5 probability of success in any single trial. If a series of trials is terminated whenever success is achieved, then the mean number of trials required to achieve success is 2 , and after 50 trials the probability of failure by omission is <1015<10^{-15}. A similar process with an exoergicity of 25maJ25 \mathrm{maJ} would (at 300 K300 \mathrm{~K} ) have a .9976 probability of success in a given trial, requiring a sequence of 6 conditional repetitions to achieve Perr1015P_{\mathrm{err}} \leq 10^{-15}, with the mean number of trials before success 1.002\approx 1.002.

Conditional repetition can be viewed as a variation on the chemical strategy of driving a reaction to completion by steadily removing product from a reacting mixture. A simple and efficient approach to implementing conditional repetition in a molecular mill is presented in Section 13.3.1c.

g. Requirements for reagent stability. To meet the reliability objectives stated in Section 8.3.4d, reagent instability must not cause errors at a rate greater than the rate of misreactions. If the mean time between reactions is 100tact=104 s\leq 100 t_{\mathrm{act}}=10^{-4} \mathrm{~s}, and the frequency factor for the instability is 1013 Hz\leq 10^{13} \mathrm{~Hz}, then limiting the instability-induced error rate per reaction to 1015\leq 10^{-15} requires a barrier height 230maJ\geq 230 \mathrm{maJ}.

Many potential rearrangement and dissociation reactions can be suppressed in a mechanochemical context by surrounding the reagent moiety with a suitably structured environment. For example, nonbonded interactions that block motions on the pathway to a rearrangement before the structure reaches an alternative, deeper potential well can suppress the instability. Solid cage structures that block the escape of a molecular fragment can likewise suppress dissociative instabilities.

Since each of the failure mechanisms discussed in this and the previous sections is exponentially dependent on energy parameters, exceeding the specified objectives yields large improvements in reliability. For example, with a barrier height 275maJ\geq 275 \mathrm{maJ} the error rate per reaction is 1020\sim 10^{-20}, and sets of 104\sim 10^{4} reagent moieties have a mean time to failure (via this instability) of 10,000\sim 10,000 years.

8.3.5. Summary of the comparison

Solution-phase chemistry has enabled the synthesis of small molecules with an extraordinary range of structures (Section 8.2), but has not yet succeeded in constructing large structures while maintaining eutactic control. The specific comparisons made in Sections 8.3.2-8.3.4 support some general conclusions regarding the relative capabilities of mechanosynthesis:

a. Versatility of reactions. Relative to diffusive synthesis, mechanosynthesis imposes several significant constraints on the kinds of reagents that can be effectively employed. It requires that reagent moieties be bound, which can reduce reactivity and impose steric constraints. It requires that reactions be fast at room temperature, limiting the magnitude of acceptable activation energies. It requires that reagent moieties have substantial stability against unimolecular decomposition reactions, precluding the use of some reagents that are acceptable in the diffusive synthesis of small molecules.

Offsetting these limitations, however, are several advantages. Fundamentally, mechanochemical processes permit the control of more degrees of freedom than do comparable solution-phase processes; these degrees of freedom include molecular positions, orientations, force, and torques. As a consequence, highly reactive moieties can be guided with precision, enabling the exploitation of reagents that are too indiscriminate for widespread use in solution-phase synthesis. Bound reagent moieties are subject to mechanical manipulation, enabling piezochemical effects to speed reactions and overcome substantial steric barriers through localized compression and deformation of molecular structures. Finally, entirely new modes of reaction become available when reagent moieties can be subjected to forces of bond-breaking magnitude. Overall, these gains in versatility appear to exceed the losses, and hence the range of local structural features that can be constructed by mechanosynthesis should equal or exceed the range feasible with diffusive synthesis.

b. Specificity of reactions. In diffusive synthesis, most reactions entail substantial rates of misreaction, and the probability of a misreaction during any given step tends to increase with the size of the product structure. Experience suggests that the cumulative probability of error becomes intolerable for product structures of more than a few hundred to a few thousand atoms.

In mechanosynthesis, reliable exclusion of misreactions can be achieved given (1) a sufficient distance between alternative transition states, and (2) a sufficient mechanical stiffness resisting relative displacements of the reagent moieties. Distances on the order of a bond length combined with stiffnesses comparable to those of bond angle-bending yield Perr<1015P_{\mathrm{err}}<10^{-15}. Error rates (per step) are independent of the size and complexity of the product structure, given that the product is either stiff or well supported.

c. Synthetic capabilities. Within the constraints required to achieve reliable, specific reactions, the versatility of the set of chemical transformations available in mechanosynthesis can be expected to equal or exceed the versatility of the set of transformations available in solution-phase synthesis. This versatility is sufficient to suggest that most kinetically stable substructures will prove susceptible to construction (the challenging class of diamondlike structures is considered in more detail in Section 8.6). Transformations that (1) satisfy these reliability constraints, and (2) yield kinetically stable substructures can then be composed into long sequences that maintain eutactic control and yield product structures of 101010^{10} or more atoms.

8.4. Reactive species

8.4.1. Overview

This section examines several classes of reagents from a mechanosynthetic perspective. Numerous classes are omitted, and those included are discussed only briefly. As noted in Section 8.2, chemistry is a vast subject; introductory textbooks on organic chemistry commonly exceed 1000 pages.

Polycyclic, broadly diamondoid structures are the products of greatest interest in the present context, hence this discussion focuses on the formation of carbon-carbon bonds; much of what is said is applicable to analogous nitrogen- and oxygen-containing compounds. As Section 8.3.3 g8.3 .3 \mathrm{~g} indicates, highly reactive species are of particular interest. Many of the following species would be regarded as reaction intermediates (rather than reagents) in solution-phase chemistry.

8.4.2. Ionic species

Although ions are important in many solution-phase chemical systems, they are uncommon in the gas phase at room temperature; the electrostatic energy of an ion in vacuum can reduce stability by 500maJ\geq 500 \mathrm{maJ}, favoring neutralization. In millstyle reagent preparation, reactions can occur in an electrostatically tailored environment; ionic species then can be as stable as those in solution and desired transformations can be driven by local electric fields. Accordingly, the utility of ionic species in these environments is, if anything, enhanced. In manipulatorbased reagent application, however, it may frequently be desirable to expose reagent moieties on a tip moving through open space toward a product structure that is not tailored for favorable electrostatics. The electrostatic energy of ionic species is then high (making solution precedents inapplicable), and the utility of ionic species in manipulator-based operations may be relatively limited.7

a. Rearrangements and neutralization. Ionic species vary in their susceptibility to unimolecular rearrangement. Carbonium{ }^{\circ} \mathrm{Carbonium} ions, for example, are prone to 1,2-shifts with low [or zero (Hehre et al., 1986)] barrier energies:

A reagent moiety in which this process can occur will likely fail to meet the stability criterion for use in molecular manufacturing processes. If, however, each of the R-groups is part of an extended rigid system, this rearrangement is mechanically infeasible, as in Eq. (8.14). Rearrangement could also be prevented by steric constraints from a surrounding matrix, or (since the rearrangement involves charge migration) by local electrostatics.

{ }^{\circ}Carbanions, having filled { }^{\circ}orbitals, are somewhat less prone to rearrangement (Bates and Ogle, 1983) and can more readily meet stability criteria. Again, structural, steric, and electrostatic characteristics can in many instances be used to suppress unwanted rearrangements.

Charge neutralization provides another, nonlocal failure mechanism. To avoid this requires a design discipline that takes account of the ionization energies and electron affinities of all sites within a reasonable tunneling distance (several nanometers) of the ionic site, ensuring that charge neutralization is energetically unfavorable by an ample margin (e.g., 145maJ\geq 145 \mathrm{maJ} ).

8.4.3. Unsaturated hydrocarbons

In diffusive chemistry, { }^{\circ}unsaturated hydrocarbons (alkenes and alkynes) find extensive use in the construction of carbon frameworks. Their reactions characteristically redistribute electrons from relatively high-energy pi{ }^{\circ} \mathrm{pi} bonds to relatively low-energy { }^{\circ}sigma bonds, thereby increasing the number of covalent linkages in the system while reducing the bond order of existing linkages. This process has a strongly negative ΔV\Delta V^{\ddagger} and ΔVreact \Delta V_{\text {react }}, since the increase in interatomic separation resulting from a reduction in bond order is outweighed by the decrease resulting from the conversion of a nonbonded to a bonded interaction. Accordingly, these reactions are subject to strong piezochemical effects. Typical examples are Diels-Alder reactions such as Eq. (8.15),

which have ΔV\Delta V^{\ddagger} in the range of 0.05\sim-0.05 to 0.07 nm3-0.07 \mathrm{~nm}^{3}.

Unsaturated hydrocarbons undergo useful reactions with ionic species, and their reactions with other reagents are touched on in the following sections. Sections 8.5.5-8.5.9 describe certain classes of reactions in somewhat more detail.

From the perspective of solution-phase chemistry, mechanosynthesis has a greater freedom to exploit reactions involving strained (and therefore more reactive) alkenes and alkynes. The cyclic and polycyclic frameworks necessary to enforce strain will be commonplace, hence use of strained species can be routine; their use is desirable and practical for reasons discussed in Sections 8.3.3 g8.3 .3 \mathrm{~g} and 8.3.4c. Planar alkenes have minimal energy, but molecules as highly pyramidalized as cubene 8.9 (Eaton and Maggini, 1988) and as highly twisted as adamantene 8.10 (Carey and Sundberg, 1983a) have been synthesized.

8.9

8.10

The reduced bonding overlap in these species (zero for 8.10) makes the energetic penalty for unpairing of the pi electrons small enough to permit them to engage in diradical-like reactions under mechanochemical conditions.

Alkynes are of lowest energy when linear, but such highly reactive species as benzyne 8.11, cyclopentyne 8.12, and acenaphthyne 8.13, have been synthesized (Levin, 1985).

8.11

8.12

8.13

Structures 8.11 to 8.13 have short lifetimes in solution owing to intermolecular reactions, but all have mechanically anchored analogues that are kinetically stable in eutactic environments but highly reactive in synthetic applications. Again, one pi bond is sufficiently weak to permit diradical-like reactivity under suitable conditions (Levin, 1985).

Also of high energy are allenes 8.14, cumulenes 8.15, and polyynes 8.16 (Patai, 1980).

Cumulenes and polyynes are of particular interest in building diamondoid structures consisting predominantly of carbon (Section 8.6); they bring no unnecessary atoms into the reaction.

Unsaturated hydrocarbons are prone to various rearrangements, subject to constraints of geometry, bond energy, and orbital symmetry. Those shown here are stable, however, and (as with carbonium ions) mechanical constraints from a surrounding structure can inhibit many rearrangements of otherwise-unstable structural moieties.

8.4.4. Carbon radicals

Free radicals result when a covalent bond is broken in a manner that leaves one of the bonding electrons with each fragment. Radicals thus have an unpaired electron spin and (in the approximation that all other electrons remain perfectly paired) a half-occupied orbital. Radicals can be stabilized by delocalization, for example in pi systems, but most are highly reactive.

a. Reactions. Among the characteristic reactions of radicals are abstraction, in which the radical encounters a molecule and removes an atom (e.g., hydrogen), leaving a radical site behind,

addition to an unsaturated hydrocarbon (here, a reactive pyramidalized species), generating an adjacent radical site on the target structure,

and radical coupling, the inverse of bond cleavage.

Radical addition and coupling have significant values of ΔV\Delta V^{\ddagger} [for addition, 0.025 nm3\sim-0.025 \mathrm{~nm}^{3} (Jenner, 1985)] and ΔVreact \Delta V_{\text {react }}. Abstraction reactions often have a smaller ΔV\Delta V^{\ddagger}, and no significant ΔVreact \Delta V_{\text {react }}. Their susceptibility to piezochemical acceleration is analyzed in Section 8.5.4.

b. Radical coupling and intersystem crossing. Electron spin complicates radical coupling and related reactions. Bond formation demands that opposed spins be paired (a singlet state), but two radicals may instead have aligned spins (a triplet state), placing them on a repulsive PES. Bond formation then requires an electronic transition (triplet \rightarrow singlet intersystem crossing). As the radicals approach, the gap between the triplet and singlet state energies grows, but this decreases the rate of intersystem crossing. In delocalized systems, bond forma- tion can occur without intersystem crossing, at the energetic cost of placing some other portion of the system into a triplet state. If intersystem crossing is required during the transformation time, however, then a condition like Eq. (8.12) must be met, but with t1t_{1} representing the time by which bond formation must have occurred (if, that is, the system is to operate correctly). The condition that ΔVs,t145maJ\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} \geq 145 \mathrm{maJ} imposes a significant constraint because kisc k_{\text {isc }} varies inversely with the electronic energy difference ΔVisc \Delta \mathcal{V}_{\text {isc }}, which (in the absence of mechanical relaxation) would equal the difference in equilibrium energies ΔVs,t\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}}, and will frequently be of a similar magnitude.

Values of kisc k_{\text {isc }} for radical pairs in close proximity vary widely, and intramolecular radical pairs (diradicals) provide a model (Salem and Rowland, 1972). For 1,3 and 1,4 diradicals, kisc k_{\text {isc }} has been estimated to be comparable that of the S1T1S_{1} \rightarrow T_{1} intersystem crossing in { }^{\circ}aromatic molecules (Salem and Rowland, 1972), where kisc k_{\text {isc }} commonly is between 10610^{6} and 108 s110^{8} \mathrm{~s}^{-1} (Cowan and Drisko, 1976; Salem and Rowland, 1972). Experiments with 1,3 diradicals in cyclic hydrocarbon structures found more favorable values of kisc k_{\text {isc }}, ranging from 107\sim 10^{7} to >1010 s1>10^{10} \mathrm{~s}^{-1} (Adam et al., 1987); the differences were attributed to conformational effects on orbital orientation, confirming rules proposed in Salem and Rowland (1972).

The presence of high- ZZ atoms relaxes the spin restrictions on intersystem crossing by increasing spin-orbit coupling (Cowan and Drisko, 1976; Salem and Rowland, 1972). In an aromatic-molecule model, kisc k_{\text {isc }} for the S1T1S_{1} \rightarrow T_{1} transition in naphthalene is increased by a factor of 50\sim 50 by changing the solvent from ethanol to propyl iodide, and bonded heavy atoms have a larger effect: kisc k_{\text {isc }} increases from 6×106 s1\sim 6 \times 10^{6} \mathrm{~s}^{-1} in 1-fluoronaphthalene to >6×109 s1>6 \times 10^{9} \mathrm{~s}^{-1} in 1-iodonaphthalene (Cowan and Drisko, 1976). The energy-gap condition is met at ordinary temperatures in this system: ΔVisc \Delta \mathscr{V}_{\text {isc }} for 1-iodonaphthalene is 209maJ209 \mathrm{maJ} (Wayne, 1988).

From these examples, it is reasonable to expect that, in the absence of special adverse circumstances, the inclusion of high- ZZ atoms bonded in close proximity to radical sites can be used to ensure values of kisc 109k_{\text {isc }} \geq 10^{9} in radical coupling processes where ΔVs,t145maJ\Delta \mathcal{V}_{\mathrm{s}, \mathrm{t}} \geq 145 \mathrm{maJ}. This is consistent with Perr<1015P_{\mathrm{err}}<10^{-15} and ttrans <107 st_{\text {trans }}<10^{-7} \mathrm{~s}. Failure to achieve intersystem crossing rates of this magnitude would increase the required value of ttrans t_{\text {trans }} for a particular operation, but would have only a modest effect on processing rates in a system as a whole (Section 8.3.4c), and no effect on the set of feasible transformations.

c. Types of radicals. Carbon radicals can broadly be divided into pi radicals (e.g., 8.17) and sigma radicals (e.g., 8.18 to 8.20), depending on the hybridization of the radical orbital. Of these, sigma radicals are higher in energy and hence more reactive; examples include radicals at pyramidalized sp3s p^{3} carbon (e.g., the 1-adamantyl radical) 8.18, aryl radicals 8.19, and the alkynyl radical 8.20.

8.17

8.18

8.19

8.20

The alkynyl radical forms the strongest bonds to hydrogen and has excellent steric properties. AbA b initio calculations on the ethynyl radical (HCC)), however,

Figure 8.3. A structure with a sterically exposed alkynyl carbon (here in a model alkyne group) having MM2/CSC stiffnesses of 4.5 and 21 N/m21 \mathrm{~N} / \mathrm{m} in orthogonal bending directions.

predict a low-energy electronic transition, A2ΠX2ΣA^{2} \Pi \leftarrow X^{2} \Sigma, with an energy of only 40maJ\sim 40 \mathrm{maJ} (Fogarasi and Boggs, 1983). If alkynyl radicals have a similar state at a similar energy, they will have a significant probability of being found in the wrong electronic state. Interconversion, however, requires no intersystem crossing and should be fast compared to typical values of ttrans t_{\text {trans }}. Figure 8.3 illustrates an exposed alkyne stiffened by nonbonded contacts.

d. Radical rearrangement. Radicals are much less prone to rearrangement than are carbonium ions. Intramolecular abstraction and addition reactions of radicals are common, where they are mechanically feasible, but shifts analogous to that shown in Eq. (8.13) are almost unknown unless the migrating group is capable of substantial electron delocalization (e.g., an aryl group).

8.4.5. Carbenes

Carbenes are divalent carbon species, formally the result of breaking two covalent bonds. The two nonbonding electrons in a carbene can be in either a singlet or a triplet state; the unpaired electrons in the triplet species behave much like those in radicals. Some carbenes are ground-state singlets in which the singlettriplet energy gap ΔVs,t145maJ\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} \geq 145 \mathrm{maJ}, making the probability of occupancy of the triplet state 1015\leq 10^{-15} at 300 K300 \mathrm{~K}. Singlet carbenes can react directly to form singlet ground-state molecules; to achieve analogous results with triplet carbenes requires intersystem crossing (Section 8.4.4b).

a. Carbene reactions. Carbenes are of broad synthetic utility.8 They can undergo addition to double bonds, yielding cyclopropanes

insertion into CH\mathrm{C}-\mathrm{H} bonds,

and coupling (Neidlein et al., 1986)

These reactions frequently proceed with energy barriers of 20maJ\leq 20 \mathrm{maJ}; many have a barrier of zero (Eisenthal et al., 1984; Moss, 1989). Coupling of a singlet carbene and a radical

should likewise proceed with little or no barrier.

b. Singlet and triplet carbenes. The prototypical carbene is methylene, 8.21, a ground-state triplet with ΔVs,t=63maJ\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}}=-63 \mathrm{maJ} (Schaefer, 1986).

Adamantylidene 8.22\mathbf{8 . 2 2} is thought to be a ground-state triplet (Moss and Chang, 1981). Carbenes with better steric properties (i.e., with supporting structures occupying a smaller solid angle) tend to be ground-state singlets: reducing the bond angle at the carbene stabilizes the singlet state, as does a double bond. In cyclopropenylidene 8.23 the predicted value of ΔVs,t\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} is 490maJ\sim 490 \mathrm{maJ} (Lee et al., 1985); in vinylidene 8.24, a prototype for alkylidenecarbenes 8.25, ΔVs,t\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} \approx 320maJ320 \mathrm{maJ} (Davis et al., 1977); and in cumulenylidenecarbenes such as 8.26, ΔVs,t\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} >400maJ>400 \mathrm{maJ} (based on studies of odd-numbered carbon chains by Weltner and van Zee, 1989). These gaps are consistent with reliable singlet behavior in a molecular manufacturing context. Since singlet states of carbenes are appreciably more polar than triplets, singlet states can be significantly stabilized by a suitable electrostatic environment. For diphenylcarbene, experiment indicates that the shift from a nonpolar solvent to the highly polar acetonitrile increases ΔVs,t\Delta \mathscr{V}_{\mathrm{s}, \mathrm{t}} by 10maJ\sim 10 \mathrm{maJ} (Eisenthal et al., 1984). A preorganized environment (Section 8.3.3b) having a polarization greater than that induced in a solvent by the singlet dipole (and doing so without imposing an entropic cost) can increase the stabilization. Substituents including N, O, F, and Cl\mathrm{Cl} also tend to stabilize the singlet state.

c. Carbene rearrangements. Carbenes have a substantial tendency to rearrange; alkylcarbenes, for example, readily transform into alkenes:

Among unsaturated carbenes, alkadienylidenecarbenes and cumulenylidenecarbenes have no available local rearrangements at the carbene center, and although vinylidene itself readily transforms to ethyne [indeed, it may not be a potential energy minimum (Hehre et al., 1986)], reaction (8.24) is not observed (Sasaki et al., 1983), and related cyclic species, as in reaction (8.25), may also be stable against this process.

8.4.6. Organometallic reagents

Various reagents with metal-carbon bonds are used in organic synthesis; this section briefly discusses only a few classes.

a. Grignard and organolithium reagents. Grignard reagents (RMgX)(\mathrm{RMgX}) and organolithium reagents ( RLi\mathrm{RLi} ) find extensive use, providing weakly bonded carbon atoms with a high electron density (Carey and Sundberg, 1983b). As is common with organometallic species, their metal orbitals can accept electron pairs from coordinated molecules; this provides options for improving stability and mechanical manipulability. For example, in an ether solution a typical Grignard reagent structure is 8.27\mathbf{8 . 2 7}; organometallic species such as 8.28\mathbf{8 . 2 8} to 8.30\mathbf{8 . 3 0} are typical of species that might be used in mechanosynthesis.

8.27

8.29

8.28

8.30

b. Transition-metal complexes. Complexes containing a dd-block transitionmetal atom exhibit versatile chemistry; such complexes are prominent in catalysis, including reactions that make and break carbon-carbon bonds. The presence of accessible dd orbitals in addition to the pp orbitals available in first-row elements changes chemical interactions in several useful ways: the orbital-symmetry constraints of reactions among first-row elements are relaxed, and bonded structures can have six or more ligands (rather than four); further, the relatively long bonds (typical MC\mathrm{M}-\mathrm{C} lengths are 0.19\sim 0.19 to 0.24 nm\sim 0.24 \mathrm{~nm} ) reduce steric congestion, thereby facilitating multicomponent interactions. (Longer bonds result in coordination shells with areas 1.6\sim 1.6 to 2.6 times larger than those of first-row atoms). Many transition-metal complexes readily change their coordination number and oxidation state in the course of chemical reactions. Electronic differences among transition metals are large; complexation further increases their diversity. Clusters containing multiple metal atoms eventually resemble metal surfaces, which also find widespread use in catalysis.

Transition metals in bulky complexes appear more useful in reagent preparation and small-molecule processing than in sterically constrained reagentapplication operations. In enfolded sites, ligand arrangements can be determined by mechanical constraints in the surrounding structure and placed under piezochemical control. Further discussion of the mechanochemical utility of these species is deferred to Section 8.5.10.

8.5. Forcible mechanochemical processes

8.5.1. Overview

Section 8.3.3 delineates some fundamental characteristics of mechanochemical processes, giving special attention to the use of mechanical force. Section 8.4 describes various reactions and reactive species, weighing their utility in a mechanochemical context. The present section examines a selected set of forcible mechanochemical processes in more detail. It starts by expanding on the discussion of piezochemistry in Section 8.3.3c, introducing the issue of thermodynamic reversibility. Tensile bond cleavage and hydrogen abstraction are then presented as model reactions and examined in quantitative detail. Several other reactions (involving alkene, alkyne, radical, carbene, and transition-metal species) are considered, building on results from the cleavage and abstraction models.

8.5.2. General considerations

a. Force and activation energy. Forces in piezochemical processes alter the reaction PES, reducing the activation energy; in some instances, they can eliminate energy barriers entirely, thereby merging initially distinct states. Section 8.3.4e calculates that barrier reductions of 70maJ\sim 70 \mathrm{maJ} can convert reactions that take 103 s10^{3} \mathrm{~s} in solution into reactions that complete reliably in 107 s10^{-7} \mathrm{~s}. How much force is required to have such an effect?

Initial motion along a reaction coordinate can resemble the stretching of bonds or the compression of nonbonded contacts. As these motions continue, the resisting forces increase, but (usually) not so rapidly or so far as they would in simple bond cleavage or in the compression of an unreactive molecular substance. The potential energy curve instead levels off, passes through a transition state, and falls into another well. Since the energy stored in a given degree of freedom by a given force is proportional to compliance, the energy stored by a force applied through an unreactive bonded or nonbonded interaction is usually lower than that stored in a reactive system.

As shown by Figure 3.8, a compressive load of 2\sim 2 to 3nN3 \mathrm{nN} stores 70maJ70 \mathrm{maJ} in a nonbonded interaction in the MM2 model; the 30nN\sim 30 \mathrm{nN} per-bond compressive load in a diamond anvil cell (Section 8.3.3c) is an order of magnitude larger. The energy stored in bonds is more variable, but for a CC\mathrm{C}-\mathrm{C} bond in the Morse model (Table 3.8), a tensile force of 5nN\sim 5 \mathrm{nN} stores 70maJ70 \mathrm{maJ}.

b. Applied forces and energy dissipation. When actuation times are relatively long ( 106 s)\left.\sim 10^{-6} \mathrm{~s}\right), acoustic radiation from time-varying forces (Section 7.2 ) is minimal, as are free-energy losses resulting from potential-well compression (Section 7.5), given reasonable values for critical stiffnesses. Likewise, with small displacements (1 nm)(\sim 1 \mathrm{~nm}) and low speeds (103 m/s)\left(\sim 10^{-3} \mathrm{~m} / \mathrm{s}\right), phonon-scattering losses (Section 7.3) are small. In an elementary reaction process, the most significant potential sources of dissipation are transitions among time-dependent wells (Section 7.6).

Although the issue is distinct from the basic qualitative question of mechanosynthesis (i.e., what can be synthesized?), minimizing energy losses is of practical interest. Losses can broadly be divided into three classes: (1) those that are many times kTk T, resulting from the merging of an occupied high-energy well with an unoccupied low-energy well; (2) those on the order of kTk T, resulting from the merging of wells of similar energy; and (3) those of negligible magnitude, resulting from the merging of a low-energy, occupied well with a high-energy, unoccupied well. The simplest way to achieve high reaction reliability is to follow route (1), dissipating 145maJ\geq 145 \mathrm{maJ} per operation. During forcible mechanochemical processes, however, it will in many instances be possible first to follow route (2) or (3) to a state in which the wells are merged (or rapidly equilibrating over a low barrier), then to use piezochemical effects to transform the PES to a type (1) surface before separation. This yields a process with reliability characteristic of (1), but with energy dissipation characteristic of (2) or (3). Systems capable of altering relative well depths by 150maJ\geq 150 \mathrm{maJ} in mid-transformation can achieve error rates <1015<10^{-15} with an energy dissipation <0.1kT300<0.1 k T_{300}. Opportunities for this sort of control are discussed in several of the following sections.

8.5.3. Tensile bond cleavage

Cleavage of a bond by tensile stress is perhaps the simplest mechanochemical process, providing an instance of the conversion of mechanical energy to chemical energy and illustrating the relationship between stiffness and thermodynamic reversibility. Further, tensile bond cleavage plays a role in several of the mechanosynthetic processes described in later sections.

As Figure 6.11 suggests, a typical CC\mathrm{C}-\mathrm{C} bond has a relatively large strength. As Table 3.8 shows, ksk_{\mathrm{s}} for such a bond is lower than that for bonds to several other first-row elements, but higher than that for bonds to second-row elements. In many practical applications, the bond to be cleaved is of lower strength and stiffness than a typical CC\mathrm{C}-\mathrm{C} bond. Cleavage of the CC\mathrm{C}-\mathrm{C} bond will be considered in some detail, however, and can serve as a basis for comparison to other bond cleavage processes.

a. Load and strength. The 300 K300 \mathrm{~K} bond-lifetime curves in Figure 6.11 indicate the tensile loads required to effect rapid bond cleavage. To achieve a level of reliability characterized by Perr P_{\text {err }} requires a barrier meeting the condition given by Eq. (8.11); for a C-C bond in this model, achieving Perr1015P_{\mathrm{err}} \leq 10^{-15} within ttrans =t_{\text {trans }}= 107 s10^{-7} \mathrm{~s} requires a tensile load of 4.2nN\sim 4.2 \mathrm{nN}. The Morse potential underestimates bond tensile strengths, but the problem of achieving sufficient tensile loads for rapid bond cleavage essentially pits the strength of one bond against that of others, hence errors in estimated strengths approximately cancel.

Figure 8.4. Bond angle in a distorted tetrahedral geometry.

As long as a carbon atom occupies a site with tetrahedral symmetry, straining one bond to the theoretical zero-Kelvin, zero-tunneling breaking point necessarily does the same to the rest. To concentrate a larger load on one bond requires that the angle θbond \theta_{\text {bond }} be increased (Figure 8.4), thereby increasing the alignment of the back bonds with the axis of stress and reducing their loads. Figure 8.5 illustrates a structure that has a geometry of this sort when at equilibrium without load. With load, however, even an initially tetrahedral geometry distorts in the desired fashion. Increase of θbond \theta_{\text {bond }} from 109.5109.5^{\circ} to 115115^{\circ} reduces back-bond tensile stresses to 0.79 of their undistorted-geometry values (neglecting the favorable contributions made by angle-bending forces in typical structures). Breaking of a back bond under these conditions would of necessity be a thermally activated process, and the energy barrier for breaking more than one bond at a time would be prohibitive. Moreover, in the structures considered here, breaking of a single bond is strongly resisted by angle-bending restoring forces from the remaining bonds (a0.1 nm(\mathrm{a} \sim 0.1 \mathrm{~nm}, bond-breaking deformation is associated with an angle-strain energy of 300maJ\sim 300 \mathrm{maJ} ); as has been discussed, this solid-cage effect invalidates the model used in Section 6.4.4a and strongly stabilizes structures. These stress and energy differences are more than adequate to ensure a>1015a>10^{15} difference in rates of bond cleavage.

b. Stiffness requirements for low-dissipation cleavage. Transitions between time-dependent potential wells can cause energy dissipation (Section 7.6.2), and distinct wells are of necessity separated by regions of negative stiffness in the potential energy surface. The potential energy for a pair of atoms undergoing bond cleavage can be described as the sum of the bond energy and the elastic

Figure 8.5. A structure having a surface carbon atom with a significantly nontetrahedral geometry (θbond =116.6\left(\theta_{\text {bond }}=116.6^{\circ}\right. ). In this structure, ksz=225 N/mk_{\mathrm{sz}}=225 \mathrm{~N} / \mathrm{m} (MM2/CSC) for vertical displacement of the central surface carbon atom with respect to the lattice-terminating hydrogens below (shown in ruled shading); with an approximate correction for compliance of a surrounding diamondoid structure (Section 8.5.3d), ksz190 N/mk_{\mathrm{sz}} \approx 190 \mathrm{~N} / \mathrm{m}.

Structural diagram:

Mechanical model:

Figure 8.6. Diagrams illustrating tensile bond cleavage and the corresponding coordinates. The lengths d0d_{0} and r0r_{0}, corresponding to unstrained springs and an unstrained bond, determine the values of the coordinates Δd=dd0\Delta d=d-d_{0} and Δr=rr0\Delta r=r-r_{0}.

deformation energy of the structures in which the atoms are embedded:

Vcleave (Δr,Δd)=Vbond (Δr)+12ks, struct (ΔdΔr)2\begin{equation*} \mathscr{V}_{\text {cleave }}(\Delta r, \Delta d)=\mathscr{V}_{\text {bond }}(\Delta r)+\frac{1}{2} k_{\mathrm{s}, \text { struct }}(\Delta d-\Delta r)^{2} \tag{8.26} \end{equation*}

where Figure 8.6 and its caption describe Δd\Delta d and Δr\Delta r, and the function Vbond (Δr)\mathcal{V}_{\text {bond }}(\Delta r) is the bond potential energy. The elastic deformation energy (neglecting modes orthogonal to the reaction coordinate) is a function of the displacement between the bonded atoms and their equilibrium positions with respect to the supporting structure, and corresponds to some positive stiffness ks, struct k_{\mathrm{s}, \text { struct }}. The bond energy function, however, has a negative stiffness in the bond-breaking separation range (Section 3.3.3a). The Morse potential predicts an extreme value of 0.125ks-0.125 k_{\mathrm{s}}, where ksk_{\mathrm{s}} is the stretching stiffness of the bond at its equilibrium length; the Lippincott potential predicts negative stiffnesses of greater magnitude ( -115 rather than 55 N/m-55 \mathrm{~N} / \mathrm{m}, for a standard CC\mathrm{C}-\mathrm{C} bond), and hence is more conservative in the present context. It is adopted in the following analysis.

Figure 8.7 illustrates potential energy curves as a function of Δr\Delta r for various values of Δd\Delta d for one set of model parameters. As can be seen, a steady increase

Figure 8.7. Potential energy as a function of CC\mathrm{CC} distance, for several values of support separation. Stiffness of support =50 N/m=50 \mathrm{~N} / \mathrm{m}.

Figure 8.8. Barrier heights vs. stiffness, for various bonds placed under a tensile load which equalizes the well depths (bond parameters from Table 3.8).

in Δd\Delta d causes the evolution of the system from a single well, to a pair of wells, to a single well again; larger values of ks,structk_{\mathrm{s}, \mathrm{struct}} first reduce and then eliminate the barriers.

At finite temperatures and modest speeds, transitions can occur without causing substantial dissipation, so long as good equilibration occurs between the old well and the new before the energy of the new well has fallen substantially below that of the old. This requires that, during the time when the wells are of nearly equal energy, the mean interval between transitions be short compared to the time required for significant changes in relative well depth to occur. A transition rate of 109 s110^{9} \mathrm{~s}^{-1} ensures low dissipation (small compared to kTk T ) when the characteristic time for the evolution of the wells is 107 s\sim 10^{-7} \mathrm{~s}. Assuming a frequency factor of 1013 s110^{13} \mathrm{~s}^{-1}, this is achieved for barrier heights 38maJ\leq 38 \mathrm{maJ} at 300 K300 \mathrm{~K}.

For estimating energy dissipation, a conservative measure of the barrier height for the process as a whole is the height when the two wells are of equal depth, ΔV=\Delta \mathcal{V}_{=}^{\ddagger} (assuming substantial values of ks, struct k_{\mathrm{s}, \text { struct }}, to limit the entropic differences between the two wells). Figure 8.8 plots ΔV=\Delta \mathscr{V}_{=}^{\ddagger} as a function of ks, struct k_{\mathrm{s}, \text { struct }} for several bond types. For processes in which ks, struct >90 N/mk_{\mathrm{s}, \text { struct }}>90 \mathrm{~N} / \mathrm{m}, and characteristic times are 107 s\geq 10^{-7} \mathrm{~s}, dissipation is small relative to kTk T for all the single bonds shown; for standard CC\mathrm{C}-\mathrm{C} bonds, ks, struct >60 N/mk_{\mathrm{s}, \text { struct }}>60 \mathrm{~N} / \mathrm{m} is sufficient.

c. Spin, dissipation, and reversibility. In the absence of intersystem crossing, bond cleavage yields a singlet diradical. In a well-separated diradical, however, the singlet-triplet energy gap approaches zero, and thermal excitation soon populates the triplet state. If bond cleavage is fast compared to intersystem crossing, this equilibration process results in ΔF=ln(2)kT\Delta \mathscr{F}=-\ln (2) k T (corresponding to the loss of one bit of information). Conversely, if intersystem crossing is fast, the thermal population of the (repulsive) triplet state results in a reduction of the mean-force bond potential energy during cleavage, and no significant dissipation. Note that slow intersystem crossing can cause large energy dissipation in mechanically forced radical coupling, even when the reverse process has a dissipation <kT<k T.

d. Atomic stiffness estimation. In the linear, continuum approximation, the zz-axis deformation of a surface at a radius rr from a zz-axis point load is 1/r\propto 1 / r (Timoshenko and Goodier, 1951). Accordingly, most of the compliance associated with displacement of an atom on a surface results from the compliance of the portion of the structure within a few bond radii.

A carbon atom on a hydrogenated diamond (111) surface can be taken as a model for sp3s p^{3} carbons on the surfaces of diamondoid structures. The MM2 model value for the zz-axis stiffness kszk_{\mathrm{sz}} of such a carbon atom on a semi-infinite lattice can be accurately approximated by measuring the stiffness in a series of approximately hemispherical, diamondlike clusters of increasing radius (Figure 8.9), holding the lattice-terminating hydrogen atoms fixed. In the continuum model, the compliance of the region outside a hemisphere is 1/r\propto 1 / r; treating the number of carbon atoms as a measure of r3r^{3} and fitting the four cluster-stiffness values with this model yields ksz=153±2 N/mk_{\mathrm{sz}}=153 \pm 2 \mathrm{~N} / \mathrm{m}.

This model holds for small displacements, but at the larger displacements corresponding to peak forces in the bond-cleavage process, bond angles and lengths are significantly distorted; this affects stiffness. Examination of the energy as a function of zz-axis displacements shows that the decrease in bondstretching stiffness resulting from tension is, under loads in the range of interest, more than offset by the increase in kszk_{\mathrm{sz}} resulting from changes in bond angle. Under tensile loads of 3 to 7nN7 \mathrm{nN}, the stiffness is increased by a factor of 1.05\sim 1.05. The MM2 model is known to have unrealistically low bond-bending stiffnesses (Section 3.3.2g); increasing these stiffness values by a factor of 1.49 (to approximate MM3 results) increases kszk_{\mathrm{sz}} by a factor of 1.14 , yielding an overall estimate of ksz183 N/mk_{\mathrm{sz}} \approx 183 \mathrm{~N} / \mathrm{m}.

Since the compliances of the two bonded atoms are additive, ks, struct =ksz/2k_{\mathrm{s}, \text { struct }}=k_{\mathrm{sz}} / 2 \approx 90 N/m90 \mathrm{~N} / \mathrm{m}. This significantly exceeds the required stiffness for low-dissipation cleavage of all the single bonds in Figure 8.8, save for OO\mathrm{O}-\mathrm{O}. For the CC\mathrm{C}-\mathrm{C} bond, available stiffness exceeds the requirement by a factor of 1.5 ; accordingly, lowdissipation bond cleavage should be a feasible process in a broad range of circumstances.

Figure 8.9. A series of structures modeling a diamond (111) surface, with corresponding values of kszk_{\mathrm{sz}} (MM2/CSC) for vertical displacement of the central surface carbon atom with respect to the lattice-terminating hydrogens below (shown in ruled shading). Note that this carbon atom has poor steric exposure, but that a better-exposed carbon atom need not sacrifice stiffness (e.g., Figure 8.5).

8.5.4. Abstraction

Abstraction reactions can prepare radicals for use either as tools or as activated workpiece sites. Although various species are subject to abstraction, the present discussion focuses on hydrogen. Most exoergic or energetically neutral hydrogen abstraction reactions have activation energies 100maJ\leq 100 \mathrm{maJ} (Bérces and Márta, 1988). The abstraction reaction

H3C+HCH3H3CH+CH3\begin{equation*} \mathrm{H}_{3} \mathrm{C} \cdot+\mathrm{HCH}_{3} \longrightarrow \mathrm{H}_{3} \mathrm{CH}+\cdot \mathrm{CH}_{3} \tag{8.27} \end{equation*}

can serve as a model for reactions involving more complex hydrocarbons, including diamondoid moieties. The energy barrier for this process is substantial, 100\sim 100 maJ (Wildman, 1986), making it representative of relatively difficult abstraction reactions.

a. Abstraction in the extended LEPS model. The piezochemistry of abstraction reactions can be modeled using the extended LEPS potential, Eq. (3.23). This three-body potential automatically fits the bond energies, lengths, and vibrational frequencies of each of the three possible pairwise associations of atoms. In a symmetrical process such as Eq. (8.27), the two independent Sato parameters can be used to fit the barrier height and the CC separation, r(CC)r(C C), at the linear, three-body transition state [calculated to be 0.2669 nm0.2669 \mathrm{~nm} for reaction (8.27) using high-order ab initio methods (Wildman, 1986)]. Fitting a LEPS function to these values using parameters based on standard bond lengths and energies for methane and ethane (Kerr, 1990) and bond-stretching stiffnesses from MM2 (Table 3.8), yields Sato parameters of 0.132 for the two CH\mathrm{CH} interactions and -0.061 for the CC interaction. (The pure LEPS potential predicts deflection of the hydrogen atom from the axis at small CC separations, but a modest angle-bending stiffness suffices to stabilize a linear geometry.)

With this function in hand, it is straightforward to evaluate the energy barrier ΔV\Delta^{\mathscr{V}} as a function of the compressive load Fcompr F_{\text {compr }}, as shown in Figure 8.10. In

Figure 8.10. Barrier height for abstraction of hydrogen from methane by methyl, plotted as a function of the compressive load applied to the carbon atoms (linear geometry). This function is based on an extended LEPS potential fitted to aba b initio transition state parameters for an unloaded system.

this model, ΔV=0\Delta \mathscr{V}^{\ddagger}=0 at Fcompr 3.6nNF_{\text {compr }} \approx 3.6 \mathrm{nN}, and r(CC)0.242 nmr(\mathrm{CC}) \approx 0.242 \mathrm{~nm}; this load is well within the achievable range (Section 8.3.3c). At ordinary temperatures, however, there is little practical reason to drive ΔV\Delta \mathscr{V}^{\ddagger} to zero (indeed, with tunneling, there is little reason to do so even at cryogenic temperatures). The barrier is reduced to 2kT3002 k T_{300} at a load of 2nN2 \mathrm{nN}, with r(CC)0.251 nmr(C C) \approx 0.251 \mathrm{~nm} at the transition state. At a load of 1nN,ΔV29maJ1 \mathrm{nN}, \Delta^{\mathscr{V}} \approx 29 \mathrm{maJ}, and r(CC)0.258 nmr(\mathrm{CC}) \approx 0.258 \mathrm{~nm}. Assuming a frequency factor of 1013 s110^{13} \mathrm{~s}^{-1}, this barrier would be consistent with an omitted-reaction probability 1015\leq 10^{-15} in a mechanochemical system with a transformation time of 4×109 s\sim 4 \times 10^{-9} \mathrm{~s} (if, that is, the reaction were also sufficiently exoergic). These conditions are consistent with achieving high reaction reliability via the conditional-repetition mechanism (Section 8.3.4f).

b. Exoergic abstraction reactions. A reliable single-step reaction must meet stringent exoergicity requirements (Section 8.3.4e) at the time of kinetic decoupling. The energy differences in Eqs. (8.28) and 8.30) are consistent with Perr<P_{\mathrm{err}}< 101510^{-15} without using cycles of repetition and measurement, while for Eq. (8.29), Perr1013P_{\mathrm{err}} \approx 10^{-13} :

The energies in these reactions are computed from differences in CH\mathrm{C}-\mathrm{H} bond strengths (Kerr, 1990). See also Musgrave et al. (1992).

Figure 8.11 compares potential energy curves under compressive loads for the methyl-methane abstraction reaction (discussed previously) to a set of similar curves for a strongly exoergic reaction: with greater exoergicity, the barrier tends to disappear under a substantially smaller load.

c. Hydrogen abstraction tools. The large CH\mathrm{C}-\mathrm{H} bond strength of alkynes ( 915maJ915 \mathrm{maJ} ) enables alkynyl radicals to abstract hydrogen atoms from most exposed nonalkyne sites with a single-step Perr<1015P_{\mathrm{err}}<10^{-15}. As shown in Figure 8.2, alkyne moieties can be buttressed by nonbonded contacts to increase their stiffness, greatly reducing the probability of reacting with sites as much as a bond length from the target (Section 8.3.3f). In a typical situation, a supporting structure need only have good stiffness in resisting displacements in one direction, and can permit larger excursions in directions that lack nearby reactive sites. Figure 8.3 illustrates an alkyne moiety that achieves greater steric exposure through selective buttressing.

Moieties with CH\mathrm{C}-\mathrm{H} bond energies between those of alkynes and aryls can abstract hydrogen atoms from most sp3s p^{3} carbon sites with good reliability. Strained alkenyl radicals such as 8.31\mathbf{8 . 3 1} or 8.32\mathbf{8 . 3 2} should bind hydrogen with the

Figure 8.11. Potential energy surfaces for abstraction reactions under various compressive loads, plotted as a function of a reaction coordinate, the ratio of a CH\mathrm{CH} distance to the (variable, optimized) CC distance. The curves in the upper panel are for the abstraction of hydrogen from methane by methyl, based on the extended LEPS model described in the text. The lower curves are representative of low-barrier exoergic processes (bond lengths and energies fit the abstraction of H\mathrm{H} from methane by ethynyl, but all Sato parameters are an arbitrary 0.15 .)

necessary energy ( 790maJ\geq 790 \mathrm{maJ}, vs. 737maJ\sim 737 \mathrm{maJ} for ethenyl).

These structures can exhibit good stiffness without buttressing from nonbonded contacts. Their steric properties are desirable, and their supporting structures strongly suppress bond-cleavage instabilities that would otherwise be promoted by the radical site.

d. Hydrogen donation tools. Structures capable of resonant stabilization of a radical permit easy abstraction of a hydrogen; these include cyclopentadiene, shown in Eq. (8.31), and various other unsaturated systems. Some atoms bind hydrogen weakly, including tin and many other metals. Moieties like these, with CH\mathrm{C}-\mathrm{H} bond energies less than 530maJ\sim 530 \mathrm{maJ}, can donate hydrogen atoms reliably to carbon radicals where the product is an ordinary sp3s p^{3} structure. Section 8.6 accordingly assumes that the abstraction and donation of hydrogen atoms can be performed at will on diamondoid structures.

8.5.5. Alkene and alkyne radical additions

The addition of radicals to alkenes and alkynes often is exoergic by 160\sim 160 to 190maJ190 \mathrm{maJ}; this is sufficient to ensure reaction reliability. Because radical addition converts a (long) nonbonded contact into a (short) bond, compressive loads can couple directly to the reaction coordinate, reducing the activation energy and driving the reaction in the forward direction. Activation energies for alkyl additions to unsaturated species are typically 30 to 60maJ60 \mathrm{maJ} (Kerr, 1973). Since this barrier is lower than in the methyl-methane hydrogen abstraction reaction (Section 8.5.4), and since piezochemical effects are expected to be larger, modest loads ( 1nN\sim 1 \mathrm{nN} ) can be expected to reduce the energy barrier sufficiently to permit fast, reliable reactions with ttrans =107 st_{\text {trans }}=10^{-7} \mathrm{~s}. A semiempirical study (using the MNDO method) of the addition of the phenyl radical to ethene yielded a substantial overestimate of the barrier height ( 98maJ\sim 98 \mathrm{maJ} ) and a maximal repulsive force along the reaction coordinate of 1.5nN\sim 1.5 \mathrm{nN} (Arnaud and Subra, 1982).

In these reactions, the radical created at the adjacent carbon destabilizes the newly formed CC\mathrm{C}-\mathrm{C} bond, permitting the addition reaction to be reversed. The energy of the destabilized bond ( 160 to 190maJ190 \mathrm{maJ} ), together with the barrier on the path between bonding and dissociation ( 30\sim 30 to 60maJ60 \mathrm{maJ}, above), makes many such bonds acceptable with respect to the stability requirements for reagents and reaction intermediates (barriers 230maJ\geq 230 \mathrm{maJ}, Section 8.3.4 g8.3 .4 \mathrm{~g} ). Stability problems in other structures can often be remedied either by satisfying the radical (e.g., by hydrogen donation) or by mechanically stabilizing the newly added moiety using other bonds or steric interactions.

The energy dissipation caused by radical addition (and its inverse) depends on an interplay of stiffness and reaction PES like that examined in Section 8.5.3 for bond cleavage. Obtaining the necessary PES data would make an interesting aba b initio study.

a. Stiffness and misreaction rates in radical addition. Radical addition reactions provide a convenient example of a process having two similar, closely spaced transition states. Structure 8.33 illustrates a transition state in the attack of an aryl radical on an alkene, with asterisks marking equivalent, alternative locations for the position of the attacking radical. Although transition states can be arbitrarily close together, the separation of the pair of locations on a single side of the alkene (moderately more than a bond length) is toward the low end of the range of anticipated typical cases, presenting a relatively challenging situation for achieving reliable positional control.

Quantum calculations can be used to estimate the transition-state geometry for the addition of a phenyl radical to ethene (Arnaud and Subra, 1982), yielding lengths of 0.135 nm0.135 \mathrm{~nm} for the breaking double bond and 0.225 nm0.225 \mathrm{~nm} for the bond undergoing formation, with an angle of 95.495.4^{\circ} between them. This yields a distance between alternative positions for the attacking radical of 0.177 nm0.177 \mathrm{~nm}. (In the full configuration-space picture, the distance between transition states is increased by contributions from other displacements, including differences in the angle of the attacking radical.)

At this distance, an elastic energy difference of >180maJ>180 \mathrm{maJ} can be achieved with a stiffness of 12 N/m\sim 12 \mathrm{~N} / \mathrm{m}; a stiffness in this range can effectively suppress misreactions (rates 1015\leq 10^{-15} ). Because the most probable misreactions result from a displacement in a particular direction, however, an advantage can be gained by biasing the equilibrium position of the attacking moiety in the opposite direction during the approach to the reaction site. For example, with a stiffness of 6 N/m6 \mathrm{~N} / \mathrm{m}, a 0.1 nm0.1 \mathrm{~nm} bias of this sort results in an elastic energy difference of 200\sim 200 maJ between the two transition state geometries, while raising the energy of the favored transition by only 25maJ25 \mathrm{maJ}. Accordingly, stiffnesses in the 5 to 10 N/m10 \mathrm{~N} / \mathrm{m} range should prove adequate to suppress misreactions with high reliability in a wide range of synthetic operations.

8.5.6. Pi-bond torsion

Rotation of one of the methylene groups of ethene by 9090^{\circ} breaks the pi bond, yielding a diradical. Analogous structures and transformations can be used to modulate the strength of adjacent sigma bonds. The transition (a) \rightarrow (b) in Figure 8.12 represents an RC\mathrm{R}-\mathrm{C} bond cleavage yielding a twisted alkene; (c) \rightarrow (d) represents an RC\mathrm{R}-\mathrm{C} cleavage yielding a planar alkene. (Figure 8.13 illustrates a specific structure with good steric properties.) The differences in alkene energy around the illustrated cycle can be used to estimate the difference in R-group bond energies between (a) and (b). The (a) \rightarrow (c) transition involves torsion of a single bond that links centers of roughly twofold and threefold symmetry; the resulting energy difference is usually small in the absence of substantial steric interference between the end groups. The difference in R-group bond energies between (a) and (c) thus approximates the energy difference between twisted and planar alkenes (b) and (d). The analogous energy difference is 453maJ453 \mathrm{maJ} for ethene (Ichikawa et al., 1985). Alternatively, one can assume that the R-group bond energy in a structure like (a) is unaffected by the presence of an adjacent twisted radical (i.e., would be unchanged if the radical site were hydrogenated), then use the bond weakening caused by an unconstrained radical to estimate the energy difference; from thermochemical data (Kerr, 1990), this difference is 410\sim 410 to 440maJ440 \mathrm{maJ} for hydrocarbon R-groups. Thus, the bond energy of a typical R-group in (a), 550\sim 550 to 650maJ650 \mathrm{maJ}, drops dramatically when the structure is twisted to the configuration of (c). Indeed, elimination of a more weakly bonded group at this site can become exoergic.

A typical reaction cycle for a mechanism of this kind could involve (1) abstraction of a relatively tightly bound moiety by a radical site like that in (b), yielding a structure like (a); (2) torsion to a state like (c) weakening the new bond; and (3) abstraction of the moiety by another radical, delivering it to a more weakly bound (i.e., high-energy) site and leaving a structure like (d). If

Figure 8.12. Modulation of bond strength by pi-bond torsion; see Section 8.5.6 for discussion.

steps 1 and 2 must be exoergic by 145maJ145 \mathrm{maJ}, then the net increase in the energy of the transferred moiety can be 120\geq 120 maJ. Moreover, by modulation of the torsion angle during (rather than between) reaction steps, the transition states for steps 1 and 2 can be approached forcibly under conditions that make the transitions endoergic or isoergic, with separation under exoergic conditions. This meets the conditions stated in Section 8.5.2b for a low-dissipation process, and the isolated twisted-alkene state (b) never occurs. Alternatively, using the conditional repetition approach (Section 8.3.4f) to remove exoergicity requirements, the net increase in the energy of the transferred atom can be >400maJ>400 \mathrm{maJ}.

Note that abstraction of a moiety by a radical to yield an alkene resembles radical coupling: it requires spin pairing, raising questions of intersystem crossing rates. As discussed in Section 8.4.4b, kisc k_{\text {isc }} is usually adequate in the presence of a suitably coupled high- ZZ atom. Bismuth (Z=83)(Z=83), with its ability to form three (albeit weak) covalent bonds, is a candidate for inclusion at a nearby site in the supporting diamondoid structure.

Figure 8.13. A structure suitable for imposing torsion on an alkene while maintaining good steric exposure at one of the carbon sites (pyramidalization of the radical site increases the change in energy).

Mechanochemical processes involving pi-bond torsion appear to have broad applications. For example, reactions of dienes (e.g., Diels-Alder and related reactions) can be accelerated by torsions that weaken the initial pair of double bonds. In the reverse direction, the sigma bonds resulting from the reaction can (for reactions yielding noncyclic products) later be cleaved with the aid of radicals generated by torsion of the new double bond. Some other systems having a single pi bond can likewise be manipulated by mechanical torsion; these include some metal-ligand and boron-nitrogen double bonds. In the latter instance, torsion results in a lone pair and an empty orbital, causing a shift in electron density.

8.5.7. Radical displacements

Various mechanochemical processes are analogous to pi-bond torsion in that they modulate the strength of a sigma bond by altering the availability of a radical able to form a competing pi bond. For example, the addition of a radical to an alkyne can facilitate an abstraction reaction as shown in Figure 8.14.

A process of this sort could employ a weakly bonding moiety (or one with modulable bonding, as in Section 8.5.6) as the attacking radical R\mathrm{R}, generating a strong alkynyl radical by tensile bond cleavage [step (d) \rightarrow (e)]. A mechanochemical cycle based on these steps can first use an alkynyl radical to abstract a tightly bound atom from one location, then deliver the atom to a moderately bound site R\mathbf{R}^{\prime}, and finally regenerate the original alkynyl radical by cleaving a weak bond to R\mathrm{R}. Given the strong weakening of the bond to hydrogen in (b), each of these steps should proceed rapidly and reliably under moderate mechanical loads. [Note that step (b) \rightarrow (c) requires spin pairing.] Analogous displacement operations can be used to regenerate alkene- and aryl-derived sigma radicals (e.g., 8.19, 8.31, 8.32).

8.5.8. Carbene additions and insertions

As noted in Section 8.4.5a, the standard carbene addition and insertion reactions have low barriers. The changes in geometry resulting from these reactions show that mechanical loads can be directly coupled to the reaction coordinate, producing strong piezochemical effects; the increase in bond number points to the same conclusion. The insertion of carbenes into CC\mathrm{C}-\mathrm{C} bonds, although analogous to metal insertion reactions such as Eq. (8.35), has not been observed in solution-phase chemistry; the ubiquitous presence of alternative reaction pathways with less steric hindrance, lower energy barriers, or greater exoergicity is presumably responsible, since this reaction is exoergic and permitted by orbital symmetry. Section 8.6.4c8.6 .4 \mathrm{c} presents a reaction step which assumes that positional control and mechanical forces can effect carbene insertion into a strained, sterically exposed CC\mathrm{C}-\mathrm{C} bond.

Since bond-forming carbene reactions are highly exoergic, single-step reactions can be highly reliable (assuming, as always, that conditions and mechanical constraints are chosen to exclude access to transition states leading to unwanted products). Fast, reliable reactions involving triplet carbenes can require fast

Figure 8.14. Removal of a tightly bound alkynyl hydrogen atom, facilitated by the addition of a radical. A supporting structure provides nonbonded contacts to one side of the alkyne; this enables the application of force to accelerate the radical addition step, (a)(b)(a) \rightarrow(b). See Section 8.5.7 for discussion.

intersystem crossing (Section 8.4.5). The potential for low-dissipation carbene reactions is at present unclear.

8.5.9. Alkene and alkyne cycloadditions

{ }^{\circ}Cycloaddition reactions can find broad use in mechanosynthesis, as they have in solution-phase synthesis. The [4 + 2] Diels-Alder reactions [e.g., Eq. (8.15)] form two bonds and a ring simultaneously, and, as has been mentioned (Section 8.4.3), they have large negative volumes of activation at moderate pressures (0.05\left(\sim-0.05\right. to 0.07 nm3)\left.-0.07 \mathrm{~nm}^{3}\right). These reactions are more sensitive to piezochemical effects than are abstraction reactions (Section 8.5.4). Energy barriers vary widely, from <60<60 to >130>130 maJ. Reaction (8.15) proceeds at low rates at ordinary temperatures and pressures; piezochemical rate accelerations under modest loads should make them consistent with ttrans =107 st_{\text {trans }}=10^{-7} \mathrm{~s}. Under these conditions, alkenes can be replaced with less-reactive alkynes, yielding less saturated products. Exoergicity [ 280maJ\sim 280 \mathrm{maJ} for Eq. (8.15)] is sufficient to yield stable intermediate products (Section 8.3.4 g8.3 .4 \mathrm{~g} ) in a reliable process.

Cycloaddition reactions are subject to strong orbital-symmetry effects. The [2+2][2+2] cycloaddition reaction

is termed thermally forbidden (according to the Woodward-Hoffmann rules), because the highest-energy occupied orbital on one molecule fails to mesh properly with the lowest-energy unoccupied orbital on the other (the unoccupied orbital presents two lobes of opposite sign, divided by a node; the occupied orbital has no such node, and creating the required node is akin to breaking a bond). But, as with well-observed "forbidden" spectral lines, the ban is not complete. Piezochemical techniques (including pi-bond torsion and direct compression) can force the formation of one of the new sigma bonds, even at the expense of breaking both pi bonds, thereby creating a pair of radicals that can then combine to form the second sigma bond:

This reaction can also proceed through a polar, asymmetric intermediate, which is likewise subject to strong piezochemical effects. For the dimerization of ethene, the activation energy is 305maJ\sim 305 \mathrm{maJ} (Huisgen, 1977).

8.5.10. Transition-metal reactions

Section 8.4.6 surveys some of the general advantages of transition-metal compounds as intermediates in mechanosynthetic processes. The present section describes some reactions and mechanochemical issues in more detail.

a. Reactions involving transition metals, carbon, and hydrogen. Transition metals participate in a wide variety of reactions, including many that make or break carbon-carbon and carbon-hydrogen bonds:

LnMHHLnMHH=LnMHHLnMRHR=LnMRH=LnMRLnMRRLnMRRLnMR\begin{aligned} & \mathrm{L}_{n} \mathrm{M}_{\mathrm{H}}^{\mathrm{H}} \rightleftharpoons \mathrm{L}_{n} \mathrm{M}_{\mathrm{H}}^{\mathrm{H}}=\mathrm{L}_{\mathrm{n}} \mathrm{M}_{\mathrm{H}}^{\mathrm{H}} \\ & \mathrm{L}_{n} M_{R} \mathrm{H}_{\mathrm{R}}=\mathrm{L}_{n} \mathrm{M}_{\mathrm{R}}^{\mathrm{H}}=\mathrm{L}_{n} \mathrm{M}_{R}^{{ }^{\prime}} \\ & L_{n} M \underset{R^{\prime}}{R} \rightleftharpoons L_{n} M_{R^{\prime}}^{R} \rightleftharpoons L_{n} M_{R^{\prime}}^{\prime} \end{aligned}

All illustrated states and transformations in each sequence have been observed, though not necessarily in a single complex (Crabtree, 1987).

The complexes in Eqs. (8.33)-(8.35) have single bonds between metal and carbon, but double-bonded species (metal carbene complexes and alkylidenes 8.34) and triple-bonded species (metal alkylidynes 8.35) are also known (Nugent and Mayer, 1988).

Species such as metal alkylidene carbenes, 8.36, and metal alkylidyne radicals, 8.37, can presumably exist (given stable ligands and a suitable eutactic environment) and may be of considerable use in synthesis. (Note that high- ZZ transition metals can accelerate intersystem crossing in reactions in which they participate.)

Among the reactions of metal carbene complexes are the following (Dötz et al., 1983; Hehre et al., 1986; Masters, 1981):

(Again, all illustrated states and transformations in each sequence have been observed, though not necessarily in a single complex.) Transition metals can also serve as radical leaving groups in SH2\mathrm{S}_{\mathrm{H}} 2 reactions at sp3s p^{3} carbon atoms, transferring alkyl groups to radicals (Johnson, 1983).

b. Ligands suitable for mechanochemistry. The reactivity of a transition metal atom is strongly affected by the electronic and steric properties of its ligands. These can modify the charge on the metal, the electron densities and energies of various orbitals, and the room available for a new ligand. A ligand can be displaced by a new ligand, or can react with it and dissociate to form a product.

In a molecular manufacturing context, reagent stability is commonly adequate if all components are bound with energies 230maJ\geq 230 \mathrm{maJ} (Section 8.3.4). Typical M-C bond strengths (Crabtree, 1987) are 210\sim 210 to 450maJ450 \mathrm{maJ} (vs. 550maJ\sim 550 \mathrm{maJ} for typical CC\mathrm{C}-\mathrm{C} bonds); the stronger bonds have adequate stability by themselves, and weaker bonds are acceptable if incorporated into a stabilizing cyclic structure. Stabilization by cyclic structures will often be necessary to prevent unwanted rearrangements in the ligand shell. Since MH\mathbf{M}-\mathbf{H} bonds are estimated to have strengths 100\sim 100 to 200 maJ greater than MCM-C bonds (Crabtree, 1987), their strengths should be adequate in the absence of special destabilizing circumstances. { }^{\circ}Electronegative ligands such as F\mathrm{F} and Cl\mathrm{Cl} should also be relatively stable, particularly in structures that lack adjacent sites resembling anion solvation shells in polar solvents.

The stability and manipulability of ligand structures in a mechanochemical context are greatest when they are bound to strong supporting structures. This is easily arranged for a wide variety of ligands having metal-bonded carbon, nitrogen, oxygen, phosphorus, or sulfur atoms. Of these, phosphorus (in the form of tertiary phosphines, PR3\mathrm{PR}_{3} ) has been of particular importance in transition-metal chemistry. Carbon monoxide, another common ligand, lacks an attachment point for such a handle and may accordingly be of limited use. Isonitriles, however, are formally { }^{\circ}isoelectronic to CO\mathrm{CO} at the coordinating carbon atom; they exhibit broadly similar chemistry (Candlin et al., 1968) and can be attached to rigid, extended RR groups (e.g., 8.42). (Many ligands not mentioned here also have useful properties.)

Ligand supporting structures can maintain substantial strength and stiffness while occupying a reasonably small solid angle. This enables several ligands to be subjected to simultaneous, independent mechanochemical manipulation. The following structures provide one family of examples:

is 115 N/m\sim 115 \mathrm{~N} / \mathrm{m} for extension and 20 N/m\sim 20 \mathrm{~N} / \mathrm{m} for bending, both measured for displacements of the carbon atom at the tip relative to the bounding hydrogen atoms at the base.

Alternatively, several ligand moieties can be integral parts of a diamondoid structure, as in the following two views of a bound metal atom with two of six octahedral coordination sites exposed:

8.44\mathbf{8 . 4 4}

The metal-nitrogen bond lengths shown in this structure are appropriate for octahedrally coordinated cobalt.

Single ligands mounted on independently manipulable tips represent one extreme of mobility; multiple ligand moieties in a single rigid structure represent another. An intermediate class incorporates several ligand moieties into a single structure, subjecting them to substantial relative motion by elastic deformation of that structure. This can ensure large interligand stiffnesses, facilitating low-dissipation processes.

c. Mechanically driven processes. Low-stiffness, low-strength bonds are more readily subject to mechanochemical manipulation: both the required force and (for low-dissipation processes) the required stiffness of the surrounding structure are smaller. The stretching frequencies of MH\mathrm{M}-\mathrm{H} bonds (Crabtree, 1987) imply stiffnesses between 130 and 225 N/m,0.25225 \mathrm{~N} / \mathrm{m}, \sim 0.25 to 0.50 times the stiffness of a typical CH\mathrm{C}-\mathrm{H} or CC\mathrm{C}-\mathrm{C} bond; the stiffness of MC\mathrm{M}-\mathrm{C} bonds (which are longer and of lower energy) should likewise be low. Given that bonds as stiff as C-C can be cleaved in a low-dissipation process (Section 8.5.3), a wide range of mechanochemical processes involving transition metals can presumably be carried out in a positive-stiffness, zero-barrier manner (or, with similar effect, in a manner encountering only small regions of negative stiffness, and hence only low barriers).

In solution-phase chemistry, catalysts capable of inserting metals into { }^{\circ}alkane CH\mathrm{C}-\mathrm{H} bonds, Eq. (8.35), have been unstable, either attacking their ligands or the solvent, and insertion into CC\mathrm{C}-\mathrm{C} bonds [Eq. (8.36)] has required a strained reagent (Crabtree, 1987). With an expanded choice of ligands and the elimination of accessible solvent molecules, the first problem should be avoidable. Further, when loads of bond-breaking magnitude can be applied between a transitionmetal atom and a potential reagent, intrinsic strain is presumably no longer required in the reagent. Configurations like 8.44 seem well suited for insertion into a sterically exposed bond. Since metal insertion in the above instances has the effect of replacing a strong, stiff bond with two weaker, more compliant bonds, it can facilitate further mechanochemical operations.

Transition-metal complexes with large coordination numbers lend themselves to bond modulation based on manipulation of steric crowding. In octahedral complexes, for example, the metal atom can be anchored by (say) three ligands while two of the remaining ligands are rotated or displaced to modulate overlap repulsion on the sixth. By analogy with processes observed in solution, the introduction of new ligands can be used to expel existing ligands by a combination of steric and electronic effects. Mechanochemical processes can forcibly introduce ligands almost regardless of their chemical affinity for the metal, driving the expulsion of other, relatively tightly bound ligands. Conversely, such processes can remove ligands (having suitable "handles"), even when they are themselves tightly bound. Again, the presence of multiple other ligands to anchor the metal atom facilitates such manipulations.

More subtly (and conventionally), the binding of a ligand in a square or octahedral complex can be strongly affected by the electronic properties of the ligand on the opposite side (the trans-effect); changes in this trans ligand can alter reaction rates by a factor of 104\sim 10^{4}, suggesting changes in transition-state energy of 40maJ\sim 40 \mathrm{maJ} (Masters, 1981). Accordingly, mechanical substitution or other alteration of ligands (e.g., double-bond torsions) should be effective in modulating bonding at trans sites in mechanochemical reaction cycles.

In summary, although transition metals are of only moderate interest as parts of nanomechanical products, they are of considerable interest as components of tools in mechanochemical systems for building those products. Their comparatively soft interactions, relaxed electronic constraints, and numerous manipulable degrees of freedom suit them for the preparation and recycling of reagent moieties, and (where steric constraints can be met) for direct use as reagents in product synthesis. In light of the broad capabilities of other reagents under mechanochemical conditions, the use of transition-metal reagents is unlikely to expand the range of structures that can be built; it is, however, likely to expand greatly the range of structures that can be built with low dissipation, in a nearly thermodynamically reversible fashion.

8.6. Mechanosynthesis of diamondoid structures

Fundamental physical considerations (strength, stiffness, feature size) favor the widespread use of diamondoid structures in nanomechanical systems. In chemical terms, diamondoid structures comprise a wide range of polycyclic organic molecules consisting of fused, conformationally rigid cages. This section considers the synthesis of such structures by mechanochemical means, based on reagents and processes of sorts described in the preceding section, and using diamond itself as an example of a target for synthesis.

8.6.1. Why examine the synthesis of diamond?

Diamond is an important product in its own right, but here serves chiefly as a test case in exploring the feasibility of more general synthesis capabilities. It is impractical at present to examine in detail the synthesis of numerous large-scale structures. Accordingly, it is important to choose a few appropriately challenging models.

Diamond has several advantages in this regard, as can be seen by a series of comparisons. Synthetic challenges often center around the framework of a molecule, and diamond is pure framework. In general, higher { }^{\circ}valence and participation in more rings makes an atom more difficult to bond correctly. At one extreme is hydrogen placement on a surface; at the other is the formation of multiple rings through tetravalent atoms. (Divalent and trivalent atoms such as oxygen and nitrogen are intermediate cases.) Solid silicon and germanium present the same topological challenges as diamond, but atoms lower in the periodic table are more readily subject to mechanochemical manipulation owing to their larger sizes and lower bond strengths and stiffnesses. Thus, a structure built entirely of rings of sp3s p^{3} carbon atoms appears to maximize the basic challenges of bond formation, and diamond is such a structure. Further, diamond has the highest atom and covalent-bond density of any well-characterized material at ordinary pressures, maximizing problems of steric congestion. Although diamond is a relatively low-energy structure, lacking significant strain or unusual bonds, existing achievements (Section 8.2.3) suggest that the latter features need not be barriers to synthesis, even in solution-phase processes.

Finally, diamond is a simple and regular example of a diamondoid structure. Accordingly, the description of a small synthetic cycle can suffice to describe the synthesis of an indefinitely large object; this avoids the dilemma of choosing between (1) syntheses too complex to describe in the available space, and (2) syntheses that might in some way be limited to small structures.

8.6.2. Why examine multiple synthesis strategies?

The identification of several distinct ways to synthesize a particular structure suggests that ways can be found to synthesize different but similar structures. Identifying multiple syntheses for diamond provides this sort of evidence

Figure 8.15. A hydrogen-terminated diamond (111) surface.

regarding the synthesis of other diamondoids. Further, diamond itself has several sterically and electronically distinct surfaces on which construction can proceed, and sites on these surfaces can serve as models for the diverse local structures arising in the synthesis of less regular diamondoids. Accordingly, the following sections survey several quite different techniques, not to buttress the case for diamond synthesis (a process already known in the laboratory), but to explore the capabilities of mechanosynthesis for building broadly diamondlike structures by using diamond as an example.

8.6.3. Diamond surfaces

Corners and other exposed sites pose fewer steric problems than sites at steps in the middle of planar surfaces. Section 8.6 .4 considers a set of reaction cycles at such sites on low-index diamond surfaces; the present section introduces the surfaces themselves. (In the diagrams here and in the following section, structures are truncated without indicating bonds to missing atoms.)

a. The (111) surface. When prepared by standard grinding procedures, a closely packed diamond (111) surface is hydrogen terminated (Pate, 1986), Figure 8.15. When heated sufficiently to remove most of the hydrogen (1200 to 1300 K1300 \mathrm{~K} ), this surface reconstructs into a structure with (2×1)(2 \times 1) symmetry (Hamza et al., 1988); the Pandey pi-bonded chain model for this surface, Figure 8.16(b), has received considerable support (Kubiak and Kolasinsky, 1989; Vanderbilt and Louie, 1985). One calculation (Vanderbilt and Louie,1985) yielded an exoergicity of 50maJ\sim 50 \mathrm{maJ} per surface atom for this reconstruction, driven by the conversion of radical electrons to bonding electrons (but offset by strain energy).

It is not presently known whether a bare diamond (111) surface, Figure 8.16(a), spontaneously transforms to the (2×1)(2 \times 1) structure at room temperature. Calculations for the similar reconstruction of a silicon (111) surface (Northrup and Cohen, 1982) suggest an energy barrier of 5maJ\leq 5 \mathrm{maJ} per surface atom. This energy barrier, however, cannot be equated with an energy barrier for the

(a)

Figure 8.16. A bare, unreconstructed diamond (111) surface (a) and the Pandey (2×1)(2 \times 1) reconstruction (b)(b).

Figure 8.17. An unreconstructed diamond (110) surface.

nucleation of a (2×1)(2 \times 1) domain on an unreconstructed (1×1)(1 \times 1) surface, since nucleation requires the simultaneous motion of a number of atoms and stores deformation energy in the domain boundary. Whatever the true situation, since hydrogenation is known to suppress the (2×1)(2 \times 1) reconstruction, and since donating and abstracting hydrogen atoms from (111) surface sites is straightforward, an unreconstructed (111) surface can be maintained during construction.

b. The (110) surface. Termination of the bulk structure leads to a surface like that shown in Figure 8.17. The bonded chains along the surface resemble those formed in the (2×1)(2 \times 1) reconstruction of a (111) surface, but with the pi bonds subject to greater torsion and pyramidalization. Although this geometry generates no radicals, the pi bonds are quite weak, and hence the pp-orbital electrons can exhibit radical-like reactivity (Section 8.4.3).

c. The (100) surface. Simple truncation of the bulk structure would yield a surface like that shown in Figure 8.18(a), covered with carbene sites. Displacement of surface atoms to form pi bonded pairs yields the stable reconstruction shown in Figure 8.18(b) (Verwoerd, 1981). The resulting surface alkene moieties are strongly pyramidalized and under substantial tension, increasing their reactivity.

d. Some chemical observations. The diversity of surface structures possible on a single bulk structure, diamond, shows how a diamondoid structure (like most complex molecules) can be built up through intermediates having widely varying chemical properties. Further, the differing strained-alkene moieties found on (110) and (100) surfaces show that stable diamondoid intermediates can have highly reactive surfaces, facilitating synthetic operations. Finally, the (possible) instability of a bare, radical-dense, unreconstructed (111) surface illustrates how requirements for temporary, stabilizing additions (such as bondterminating hydrogens) can arise.

8.6.4. Stepwise synthesis processes

Section 8.6.5 describes synthesis strategies that exploit the regular structure of diamond by laying down reactive molecular strands. Synthesis based on the mechanical placement of small molecular fragments, in contrast, suggests how

(b)

Figure 8.18. A diamond (100) surface, without (a) and with (b) reconstruction.

specific irregular structures might be synthesized, and thus provides a point of departure for considering the synthesis of a broad class of diamondoids.

a. Existing models of diamond synthesis. Models of synthesis via small molecular fragments have been developed to explain the low-pressure synthesis of diamond under nonequilibrium conditions in a high-temperature hydrocarbon gas, a process of increasing technological importance. Two models have been advanced and subjected to studies using semiempirical quantum mechanics. Both propose mechanisms for the growth of diamond on (111) surfaces, one based on a cationic process involving methyl groups (Tsuda et al., 1986) and one based on the addition of ethyne (Huang et al., 1988). Of these, the ethyne process appears more plausible and more directly relevant to feasible mechanosynthetic processes (regardless of its frequency during diamond growth from the gas phase).

Figure 8.19 (based on illustrations in Huang et al., 1988) shows the addition of two ethyne molecules to a hydrocarbon molecule which, with suitable positional constraints, was used to model a step on a diamond (111) surface. The overall set of calculations used MNDO and consumed 35\sim 35 hours of CPU time on a Cray XMP/48. The reaction mechanism originally proposed (Frenklach and Spear, 1988) for the transformations 242 \rightarrow 4 and 464 \rightarrow 6 involved multiple steps, later shown to be concerted (steps 3 and 5). The initiation step involves abstraction of a hydrogen atom by atomic hydrogen (12(1 \rightarrow 2; process not shown), and has a significant energy barrier. Steps 3 and 5 were calculated to proceed without energy barriers, suggesting that any energy barriers that actually occur are unlikely to be large.

Figure 8.19. A sequence of reaction steps for the addition of ethyne to a compound serving as a model of a step on a hydrogen-terminated diamond (111) surface. Redrawn from Huang et al. (1988); see Section 8.6.4a. Bonds in the process of breakage and formation are shown in black; radical sites are marked with dots.

In an analogous mechanosynthetic process, ethyne could be replaced by an alkyne moiety bonded to a structure serving as a handle. Step 5 liberates a free hydrogen atom, which is unacceptable in a eutactic environment, but a related approach (Section 8.6.4b) avoids this loss of control.

b. Mechanosynthesis on (111). Step 1 of Figure 8.20 illustrates a kink site in a step on a hydrogenated diamond (111) surface; one bond results from a reconstruction, and two hydrogen atoms have been removed to prepare radical sites. In step 2 of the proposed synthetic cycle, an alkyne reagent moiety such as 8.45\mathbf{8 . 4 5}

is applied to one of the radical sites, resulting in a transition structure analogous to step 3 of Figure 8.19. The chief differences are that insertion occurs into a strained CC\mathrm{C}-\mathrm{C} bond rather than an unstrained CH\mathrm{C}-\mathrm{H} bond, and that large mechanical forces can (optionally) be applied to drive the insertion process. In step 3, tensile bond cleavage occurs (use of Si or another non-first-row atom reduces the mechanical strength of this bond), and a bond forms to the remaining prepared radical. Deposition of a hydrogen at the newly generated radical site and abstraction of two hydrogens further along the surface step then completes the cycle (not shown), generating a kink site like that in step 1, but with the diamond lattice extended by two atoms. Save for accommodating boundary conditions at the edge of a finite (111) surface, this sequence of operations (like those in the following sections) suffices to build an indefinitely large volume of diamond lattice.

c. Mechanosynthesis on (110). Figure 8.21 illustrates the extension of a pi bonded chain along a groove in a diamond (110) surface, using an alkylidenecarbene reagent moiety such as 8.46\mathbf{8 . 4 6}.

Figure 8.20. A sequence of reaction steps for the addition of a pair of carbon atoms from an alkynyl moiety to a kink site in a step on a hydrogen-terminated diamond (111) surface (see Section 8.6.4b).

Figure 8.21. A sequence of reaction steps for the addition of a carbon atom from an alkylidenecarbene moiety to a pi-bonded chain on a diamond (110) surface (see Section 8.6.4c).

Step 1 illustrates the starting state; step 2 illustrates a transitional state in the insertion of the carbene into a strained CC\mathrm{C}-\mathrm{C} bond. This process takes advantage of mechanical constraints to prevent the addition of the carbene into the adjacent double bond to form a cyclopropane moiety. Instead, electron density is accepted from the terminal atom of the pi bonded chain into the empty pp orbital of the carbene carbon, developing one of the desired bonds, while electron density is donated from the sigma orbital of the carbene carbon to form the other desired bond (shown in step 3). Substantial forces ( 4nN\geq 4 \mathrm{nN}; Section 8.3.3c) can be applied to drive this process, limited chiefly by mechanical instabilities.

Step 4 illustrates the use of torsion to break a pi bond, thereby facilitating tensile bond cleavage, step 5 . The final state, step 6, resembles that in step 1 save for the extension of the lattice by one atom. A further cycle (restoring the state of step 1 exactly, save for the addition of two atoms) would be almost identical, except that the equivalent of step 5 involves attack by a newly forming radical on a weak pi bond, rather than its combination with an existing radical.

d. Mechanosynthesis on (100). Figure 8.22 illustrates a synthetic cycle on diamond (100) in which reactions occur on a series of relatively independent rows of pairs of dimers. In step 1 of Figure 8.22, a strained cycloalkyne reagent moiety such as 8.47

Figure 8.22. A sequence of reaction steps for the addition of pairs of carbon atoms from a strained alkyne moiety to a row of dimers on a diamond (100) surface (see Section 8.6.4d).

is applied. (The division of the supporting structure into blocks is intended to suggest opportunities for modulating bond strength by control of torsional deformations; alternative structures could serve the same role, providing weak bonds for later cleavage.) The reaction in step 1 (promoted by the nearly diradical character of the strained alkyne and by applied mechanical force) is formally a [4+2][4+2] cycloaddition process, provided that the resulting pair of radicals is regarded as forming a highly elongated pi bond. Step 2 is then formally a (thermally forbidden) [2+2][2+2] cycloaddition process, but the small energy difference between the bonding and antibonding orbitals in the "pi bond" provides a low-lying orbital of the correct symmetry for bonding, hence the forbiddenness should be weak. Moderate mechanical loads should suffice to overcome the associated energy barrier. Step 4 is an endoergic retro-Diels-Alder reaction yielding a high-energy, highly pyramidalized alkene moiety (which is, however, less pyramidalized than cubene; Section 8.4.3). The energy for this process is supplied by mechanical work.

Step 5 represents the state of the row after bridging dimers have been deposited at all sites. The resulting three-membered rings are analogous to epoxide structures in a model of an oxidized diamond (100) (2×1)(2 \times 1) half-monolayer surface (Badziag and Verwoerd, 1987).

Steps 6-8 represent a cycle in which dimers are sequentially inserted along the row: In step 6 , a dimer has already been added to the right, cleaving the strained rings and generating two radical sites adjacent to a cleft in the surface. Step 7 illustrates the bonds that undergo formation and cleavage as a result of the mechanical insertion of a strained alkyne into the cleft. The nature of the transition state will depend on the spin state of the radical pair and orbital symmetry considerations. The large exoergicity for this process can be seen from a comparison of the bonds lost (two pi bonds in the strained alkene and two strained sigma bonds in the three-membered rings) with the bonds formed (four almost strain-free sigma bonds). The geometry of the cleft permits the application of large loads without mechanical instability, hence large energy barriers to bond formation (in the unloaded state) are acceptable. Afterward, the transition to step 8 (equivalent to step 6, but displaced) is achieved by a retro-Diels-Alder reaction like that in step 4.

8.6.5. Strand deposition processes

Stepwise synthetic processes like those suggested in Section 8.6 .4 need not be applied in regular cycles to build up a regular structure, but could instead be orchestrated to build up diamondoid structures tailored for specific purposes. Where diamond itself is the target, synthesis can take direct advantage of structural regularities.

a. Cumulene strands. Cumulene strands, 8.48, are high-energy, pure-carbon structures that represent promising precursors in the mechanosynthesis of diamond.

[=C=C=]n[=\mathrm{C}=\mathrm{C}=]_{\mathrm{n}} 8.488.48

Figure 8.23 represents a pair of similar reaction processes on a dehydrogenated step on a diamond (111) surface and on a groove in a (110) surface: in each, a pi-bonded chain is extended by the formation of additional bonds to a cumulene strand in an exoergic, largely self-aligning process. Substantial forces can be applied through nonbonded interactions (which can also be used to constrain strand motions). The reaction on a hydrogenated (111) surface may

Figure 8.23. Reaction processes bonding cumulene strands to (left) a dehydrogenated step on a hydrogenated diamond (111) surface and (right) a groove on a diamond (110) surface (see Section 8.6.5a).

require a series of stepwise reactions both to dehydrogenate atoms in the plane to be covered and to hydrogenate new atoms in the plane being constructed; the (110) reaction has no such requirement.

b. Hexagonal diamond from hexagonal strands. The unsaturated, hexagonal, columnar structure 8.49 can be regarded as a tightly rolled tube of graphite; it can made from a { }^{\circ}saturated structure by abstraction of all hydrogens. Like a cumulene strand, 8.48 , this is a pure-carbon structure; owing to pyramidalization and torsion of pi systems, it is also relatively high in energy.

8.49

A semiempirical quantum chemistry study using AM1 on the model structure 8.50 (R. Merkle)

yielded bond lengths of 0.1384 nm\sim 0.1384 \mathrm{~nm} for the three central, axially aligned bonds, and 0.1510 nm\sim 0.1510 \mathrm{~nm} for the twelve adjacent bonds. These values are close to standard values for pure double and pure single bonds ( 0.1337 and 0.1541 nm0.1541 \mathrm{~nm} ), hence structure 8.49, representing the hexagonal column as a network of strained alkenes, provides a good description.

Figure 8.24 represents a (100) surface of hexagonal diamond, bounded by similar strained alkenes. Figure 8.25 illustrates a reaction in which a hexagonal column bonds to a groove adjacent to a step on that surface; this can be regarded as proceeding by the attack of a strand radical on a surface alkene, generating a surface radical, which then attacks the next strand alkene, and so forth. Thus, each row of alkenes undergoes a process directly analogous to the free-radical chain polymerization that yields polyethylene, save for the greater reactivity of the participating alkenes and the presence of mechanical forces tending to force each radical addition.

8.6.6. Cluster-based strategies

Syntheses based on cumulene and hexagonal-column strands suggest the feasibility of synthesizing diamondoid solids using reactive molecular fragments of intermediate size (e.g., 10 to 30 atoms). Fragments of this size can be strongly

Figure 8.24. A (100) surface of hexagonal diamond.

Figure 8.25. A reaction process bonding a tube formed of alkenes to a (100) surface of hexagonal diamond (see Section 8.6.5b).

convex, relaxing steric constraints in their synthesis. Containing tens of atoms, they can embody significant structural complexity and deliver that preformed complexity to a workpiece. By incorporating unsaturated structures, radicals, carbenes, and the like, they can form dense arrays of bonds to a complementary surface. Fullerenes are examples of unsaturated carbon clusters, and C60\mathrm{C}_{60} has been crushed to diamond at room temperature by anisotropic compressive loads of 20GPa\sim 20 \mathrm{GPa} (Regueiro et al., 1992), suggesting the feasibility of mechanosynthesis of diamond and diamondoid structures from fullerene precursors.

This approach is a form of the familiar chemical strategy of convergent synthesis. Further, the contemplated size range of these fragments is familiar in organic synthesis today; their relatively high reactivity could be achieved (starting with more conventional structures) by a series of mechanochemically guided abstraction reactions in the protection of a eutactic environment.

8.6.7. Toward less diamondlike diamondoids

Members of the broad class of diamondoids can differ from diamond both in patterns of bonding and in elemental composition. The differing bonding patterns created during intermediate stages of the diamond syntheses proposed in this section, together with general experience in chemistry, suggests that nondiamond structures are readily accessible. Deviations from the diamond pattern usually reduce the overall number density of atoms, thereby reducing steric congestion and (all else being equal) facilitating synthesis.

Regarding differences in elemental composition, it is significant that the classes of reagent species exploited in this section (unsaturated hydrocarbons, carbon radicals, and carbenes) have analogues among other chemical elements. For example, most other elements of structural interest (N, O, Si, P, S), all can form double (and sometimes triple) bonds [e.g., C=N,CN,C=O,C=S\mathrm{C}=\mathrm{N}, \mathrm{C} \equiv \mathrm{N}, \mathrm{C}=\mathrm{O}, \mathrm{C}=\mathrm{S}; and the more exotic examples Si=Si\mathrm{Si}=\mathrm{Si} and C=Si\mathrm{C}=\mathrm{Si} (Raabe and Michl, 1989), and C=P\mathrm{C}=\mathrm{P} and CP\mathrm{C} \equiv \mathrm{P} (Corbridge, 1990)]. All these elements can host radical sites. Silicon can form a carbenelike divalent species, silene (Raabe and Michl, 1989). Fluorine, chlorine, and bromine can, like hydrogen, participate in abstraction reactions. The focus on hydrocarbon structures in this chapter has been driven more by limits on time and page space than by limits on chemistry.

The diversity of feasible syntheses for a challenging test-case, diamond, suggests that most reasonably stable diamondoid structures will prove susceptible to mechanosynthesis. Part II of this volume proceeds on this assumption.

8.6.8. Mechanosynthesis of nondiamondoid structures

As we move away from diamondoids, while remaining within the class of reasonably stable structures, synthesis appears to grow easier for reasons of the sort discussed in Section 8.6.1. Within the class of covalent structures, reduced rigidity might hamper mechanosynthesis, but the feasibility of using bound intermediates (using either covalent bonds or nonbonded interactions) can largely compensate. Further, the synthesis of flexible covalent structures is in the mainstream of existing chemical achievement.

Noncovalent solids present different challenges. In general, however, the ability to transfer atoms one at a time or in small clusters, and to perform piezochemical manipulation on a growing surface, can provide broad control. Accordingly, it seems reasonable to assume that most reasonably stable structures-diamondoid or not-will prove susceptible to mechanosynthesis. Aside from considering the use of metallic conductors (Section 11.6.1), the balance of this volume will neglect noncovalent structures.

8.7. Conclusions

The achievements of solution-phase synthesis show that a remarkably wide range of molecular structures can be built, despite the absence of the standard basis for construction, that is, the ability to move and position parts to guide assembly. A comparison reveals several limitations of mechanosynthesis relative to solution-phase synthesis, but displays a more-than-compensating set of strengths. Chief among these strengths are (1) the ability to achieve reactions at specific sites while avoiding them elsewhere by exploiting direct positional control of reagent moieties, (2) as a consequence of this, the ability to apply highly reactive moieties, such as radicals and carbenes, with great specificity, and (3) the ability to accelerate reactions by the application of localized forces of bondbreaking magnitude. Misreaction probabilities are exponentially dependent on relative energies, and with reasonable barrier heights can be <1015<10^{-15}; omittedreaction probabilities are exponentially dependent on repetition and relative well depths, and can likewise be low. It appears that the required conditions for high reliability do not excessively constrain the set of feasible synthetic operations. As a consequence, precise structures having >1012>10^{12} atoms can be made in good yield.

A review of reactive species used in solution-phase chemistry suggests that unsaturated hydrocarbons, carbon radicals, carbenes, and transition-metal complexes (among others) are attractive in a mechanosynthetic context. A consideration of the utility of these species when used in conjunction with positional control and mechanical force indicates the feasibility of a wide range of useful transformations, some of which can be performed with energy dissipation <kT<k T.

Finally, the mechanosynthetic production of diamondoid structures has been examined, using several proposed syntheses of diamond as an example. The diversity of feasible reactions (and the availability of analogous reactions involving elements other than C\mathrm{C} and H\mathrm{H} ) indicates the feasibility of constructing a wide range of diamondoid and other structures. Accordingly, Part II of this volume proceeds on the assumption that a mature mechanosynthetic technology can manufacture most reasonably stable covalent structures.

Some open problems. Mechanosynthesis presents numerous open theoretical problems. The analysis in this chapter indicates the feasibility of a wide range of processes, but it describes only a few in substantial structural and energetic detail. The broader application of aba b initio molecular orbital methods to potential mechanosynthetic reactions is a high priority. Research is needed to identify and characterize reaction pathways in mechanosynthesis of diamond and diamondoid structures (including the verification, modification, or rejection of the processes suggested in Section 8.6). Similar studies are needed to describe steps in reagent preparation. A major challenge in this work will be the identification and characterization of potential misreaction pathways to determine whether the proposed reaction can be made reliable enough for molecular manufacturing. Semiempirical molecular orbital methods can assist in the exploration of relevant potential energy surfaces. (Note that mechanosynthetic processes can in practice be far less sensitive to errors in PES contours than are solution-phase chemical processes; Section 4.4.)

A long-term objective in this area is to compile a large and well-characterized set of mechanosynthetic reactions suitable for the synthesis of diamondoid structures. This, in turn, can eventually provide the basis for software able to perform retrosynthetic analysis, accepting a component design as an input and (at least in favorable instances) producing a description of a corresponding mechanosynthetic process.

Footnotes

  1. Note, however, that stiff positional control of reactants can make predictions of the ratios of reaction rates among competing reaction pathways quite insensitive even to large errors in predicted transition-state energies (Section 4.4.3c). Further, the equilibrium results of highly exoergic reactions can be predicted with confidence despite substantial errors in calculating the energies of the reactants and products. Thus, the results of many mechanosynthetic reactions can be predicted with ample confidence despite errors in calculated energies that would be unacceptable in predicting the results of typical solution-phase reactions.

  2. Pressure in piezochemistry plays a fundamentally different role from pressure in gasphase reactions. In the gas phase, so-called pressure effects on kinetics and equilibria have no direct relationship to the applied force per unit area, being mediated entirely by changes in molecular number density and resulting changes in collision frequencies. In the gas phase, the PES describing an elementary reaction process is independent of pressure, since each collision occurs (locally) in vacuum, free of applied forces. Likewise, so-called pressure effects in liquid and solid-surface environments exposed to reactive gases usually result more from changes in molecular number density than from piezochemical modifications to the PES of the reaction. (Section 6.3.1 outlines the concept of a PES dependent on boundary conditions, such as pressure.)

  3. † In polar, solution-phase systems, the effects of solvent reorganization can be large, and bond-forming but charge-neutralizing reactions can have positive volumes of activation: the polar solvent, freed of strong fields, relaxes and expands.

  4. Note, however, that convergent, "structure-directed" synthesis strategies (Ashton et al., 1989) can loosen this constraint by combining larger fragments in a manner analogous to biological self-assembly. Distinctive features of larger-scale structures then guide the reactions, with no obvious bound on the feasible complexity of the products.

  5. Values in this range are seldom achieved.

  6. Note, however, that a convergent synthesis can reduce the asymmetry of the reagent application process; see Section 8.6.6.

  7. Strongly polar tip structures may mitigate this problem.

  8. "It is no exaggeration to claim a major role for carbenes in the modern chemist's attitude that he can very probably make anything he wants." (Baron et al., 1973)