Casimir Chemistry D

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9-1-2009 Casimir chemistry D. P. Sheehan University of San Diego, [email protected]

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This Article is brought to you for free and open access by the Department of Physics and Biophysics at Digital USD. It has been accepted for inclusion in Physics and Biophysics: Faculty Publications by an authorized administrator of Digital USD. For more information, please contact [email protected]. THE JOURNAL OF CHEMICAL PHYSICS 131, 104706 ͑2009͒

Casimir chemistry ͒ D. P. Sheehana Department of Physics, University of San Diego, San Diego, California 92110, USA ͑Received 6 April 2009; accepted 26 June 2009; published online 14 September 2009͒

It is shown that, at the nanoscale, the Casimir effect can be used to mechanically tune critical aspects of chemical reactions ͑e.g., energies, equilibrium constants, activation energies, transition states, reaction rates͒ by varying the spacing and composition of reaction vessel boundaries. This suggests new modalities for catalysts, nanoscale chemical manufacturing, chemical-mechanical engines, and biochemical processes in organisms. © 2009 American Institute of Physics. ͓doi:10.1063/1.3224158͔

I. INTRODUCTION experimental and theoretical evidence abounds for the exis- tence of zero point energy, there is no convincing evidence Electromagnetic zero point fluctuations1–3 are integral to for its nonconservative extraction. a broad spectrum of phenomena, including the Casimir 4 5 effect, spontaneous emission of radiation, the Davies– II. THEORY Unruh effect,6,7 the Lamb shift,8,9 the Casimir–Polder effect,10 van der Waals forces,11–13 the anomalous magnetic The Casimir effect is the pressure ͑positive or negative͒ moment of the electron,14 adhesion ͑stiction͒, friction, wet- exerted on surfaces due to the exclusion of zero point modes ting, gas-liquid condensation, interactions of colloids and between them. ͑Alternatively, it can be explained in terms of aerosols, cellular biology, laser linewidths, and the stability van der Waals forces, as correlated electric and magnetic of atoms.15 field fluctuations in the media of the affected structures. Here ͒ The Casimir effect, originally investigated in the context we adopt the zero point model. In its original formulation, 4 the Casimir effect referred to interactions between perfectly of colloids, has been studied extensively both theoretically 4 and experimentally, and now finds wide application across electrically conducting parallel plates; however, it was soon 3,13 generalized to interactions between arbitrary dielectric chemistry, physics, engineering, and biology. Until now, 20 its connections with atomic and molecular systems have surfaces. This so-called Lifshitz formalism, with its various technical refinements,21 is generally considered valid and has been confined primarily to physical and optical phenomena, 22–25 such as the Casimir–Polder effect,10 latent heats of been verified by numerous experiments. 16 17 18 The full Lifshitz formalism is analytically intractable for vaporization, surface tension, and atomic spectroscopy. all but the most idealized cases, e.g., parallel plates, spheres, This article examines how the Casimir effect can bear on or cylinders. For this study, which seeks merely to illuminate chemical phenomena at nanometer distance scales. Specifi- new facets of the Casimir effect, it will suffice to use ap- cally, it is shown that, by suitably engineering and manipu- proximate formulas that capture its essential features. The lating the boundaries of micro- or nanoscopic reaction ves- discussion will be restricted to parallel plate geometry, al- sels, the Casimir effect can shift chemical reaction energies though interesting effects can arise in more complex geom- and equilibria and alter activation energies, transition states, etries ͑e.g., spherical, cylindrical, corrugated surfaces͒. For and reaction rates. As a result, Casimir processes could hold details, the reader is directed to more complete applications for micro- and nanoscopic chemical reactors, 3,13 treatments. catalysts, and chemical-mechanical motors, as well as for the machinery of living cells. The remainder of this paper is organized as follows. In A. Parallel plate Casimir effect Sec. II relevant fundamentals of the Casimir effect are sum- Consider two thick ͑semi-infinite͒, parallel dielectric marized, followed in Sec. III by a thought experiment dem- plates26 composed of media ␣ and ␤ ͓Fig. 1͑a͔͒, narrowly onstrating the Casimir chemical effect. In Sec. IV several separated by distance z in a medium m. ͑Here the medium potential applications are explored; directions for future re- will be exclusively vacuum.͒ To first order, the Casimir in- search are suggested in Sec. V. teraction energy per unit area is given by13 It is maintained throughout that zero point operations are ͑ ͒ H␤m␣ z conservative; that is, no cyclic process can be used to extract E␤ ␣ =− , ͑1͒ m ␲ 2 net energy from the zero point field. This is in contrast to a 12 z minority but vocal viewpoint that posits that net energy can where H␤ ␣ is the Hamaker coefficient pertinent to materials 19 m be extracted cyclically from zero point field. While ␤, ␣, and m. Although H is a variable function of z,itis usually given in terms of its zero-separation value. As evi- ͒ a Electronic mail: [email protected]. Telephone: 619-260-4095. dent in Table I, H can vary over roughly two orders of mag-

TABLE I. Select Hamaker coefﬁcients H ͑in zJ͒ for near-zero plate separation across vacuum and water ͑Ref. 13͒.

Substance Vacuum Water

Ag 200–500 100–400 Cu 400 300 Diamond ͑IIa͒ 296 138 ͑ ͒ TiO2 rutile 181 60 ͑ ͒ SiO2 quartz 66 1.6 PMMA 58 1.5 Polystyrene 79 13

͑ ͒ ͑ ͒ In Eq. 2 , Rn z is a relativistic retardation correction factor that accounts for the finite speed of light and its effect on the correlation between field fluctuations in the plates’ media. If a mode period is less than or equal to the light Ӎ travel time in the gap, then Rn 1; it falls to zero as the light travel time becomes large compared with the mode period. For a vacuum gap z and for mode wavelength ␭, one has ͑ Ӷ␭͒Ӎ ͑ ϳ␭͒Ӎ / ͑ ӷ␭͒→ Rn z 1, Rn z 1 2, and Rn z 0. In effect, ͑ ͒ Rn z offsets the dominance of short wavelength modes aris- ing from the Matsubara summation. As rough rules of thumb concerning the Casimir effect: ͑i͒ at a given plate separation the dominantly contributing modes have wavelengths less than or comparable to the gap distance; ͑ii͒ a mode and its effects penetrate roughly one wavelength into a medium; ͑iii͒ Casimir forces see into a medium roughly the distance of separation between the ob- jects ͑or in the case of good conductors, the electronic FIG. 1. Semi-infinite dielectric slabs ␣ and ␤ separated by medium m. screening distance͒; and ͑iv͒ the more disparate the values of ␣-plate is stationary and ␤-plate is movable in z direction. ͑a͒ Bare dielectric ⑀ plates and ͑b͒ ␣-plate coated with thin layer of species A. between interacting media, the more negative the Casimir energy. Our analysis will concentrate on small distances of a few nanometers or less, so the most relevant frequencies will nitude depending on the interacting substances and the inter- be in the UV portion of the spectrum. vening dielectric m. It can be calculated fairly precisely from Combining Eqs. ͑1͒ and ͑2͒, the Casimir interaction en- experimental values of the media’s frequency-dependent di- ergy between two parallel plates composed of materials ␣ electric response functions. and ␤ and separated by a distance z in medium m ͓Fig. 1͑a͔͒ For the case of semi-infinite parallel plates at nonzero can be written as temperature, the Hamaker coefficient is given by ͓Fig. 1͑a͔͒27 ϱ kT ϱ Ӎ ⌬ ⌬ ͑ ͒ ͑ ͒ 3 E␣m␤ − ͚Ј ␣m ␤mRn z . 4 Ӎ ⌬ ⌬ ͑ ͒ ͑ ͒ 8␲z2 H␤m␣ kT͚Ј ␣m ␤mRn z . 2 n=0 2 n=0 For our interests, a slightly more complicated system is Here the subscripts ␤m␣, ␣m, ␤m refer to the interfaces called for, one having an additional thin layer of species A on between media ␣, ␤, and m, kT is the thermal energy, and plate ␣ ͑thickness t ͒, as portrayed in Fig. 1͑b͒. In this case, ⌬ A ␣m,␤m are defined as the Casimir interaction energy is the sum of two parts, corresponding to two pairs of interacting interfaces, ⑀ ⑀ ⑀ ⑀ ␣ − m ␤ − m ⌬␣ ϵ , ⌬␤ ϵ . ͑3͒ m ⑀ ⑀ m ⑀ ⑀ ͑ ͒ ␣ + m ␤ + m E␤mA␣ z;tA = E␤m/Am + E␤m/␣A ͑ ͒ ͑ ͒ ⑀ H␤m/Am z H␤m/␣A z + tA Here i are the frequency-dependent dielectric susceptibili- =− − . ͑5͒ ␣ ␤ ⑀ ϵ⑀ 12␲z2 12␲͑z + t ͒2 ties of , , and m; for vacuum m vacuum=1. In the com- A ͚ϱ putation of H, the sum n=0 extends over the Matsubara sam- ͑ / ͒ ͑ Here, pling frequencies, fn = kT h n n=1,2,3...; h is Planck’s constant͒, a set of discrete ͑positive͒ imaginary frequencies ϱ 3 that span the frequency range appropriate to the Casimir ef- ͑ ͒Ӎ ⌬ ⌬ ͑ ͒͑͒ H␤m/Am z kT͚Ј ␤m AmRn z 6 fect, in increments of the thermal energy kT. For most appli- 2 n=0 cations, pertinent frequencies range from the IR into the deep UV. and 104706-3 Casimir chemistry J. Chem. Phys. 131, 104706 ͑2009͒

͑AB͒ might involve, for example, chemical or hydrogen bonds, changes in material phase, or molecular conformational rearrangements. ͑AB can also be shorthand for more complex reactions involving multiple species.͒ To begin, it is presumed that surfaces ␣ and ␤ are chemically and catalytically inert with respect to A and B, aside from un- avoidable physisorption due to van der Waals interactions. In fact, the A-layer need not even be in contact with ␣; it can be freestanding. For the cycle depicted in Fig. 2, the abscissa corresponds to the plate separation ͑z͒ and the ordinate to the surface energy density ͑J/m2͒ of the entire two-plate system including substance A. The cycle is performed quasistatically and is taken to be thermodynamically reversible. In the initial state of the cycle, the movable plate is out, coated with substance A. This four-step cycle, consisting of two mechanical steps involving Casimir energy and two chemical steps involving chemical energy, proceeds as follows. → ␤ Step 1 2: Plate is moved from zout inward to zin, FIG. 2. Casimir surface energy density vs plate separation, pertinent to releasing mechanical Casimir energy via the Casimir force, plates in Fig. 1͑b͒ for reaction AB. Chemical reaction energy varies with E =E␤ ␣͑z ;t ͒, as defined in Eq. ͑5͒. plate separation due to Casimir interaction. 1 mA in A Step 2→3: Substance A is quasistatically transformed into substance B, releasing chemical energy E =E ϱ 2 chem 3 ͑A→B;z ͒. ͑By caveat, this reaction is taken to be ͑ ͒Ӎ ⌬ ⌬ ͑ ͒ ͑ ͒ in H␤m/␣A z + tA kT͚Ј ␤m ␣ARn z + tA , 7 ͒ 2 exothermic. n=0 → ␤ Step 3 4: Plate is returned to zout, requiring the where ⌬s are defined similarly as in Eq. ͑3͒. The first term on input of mechanical work against the Casimir force, ͑ ͒ ͑ ͒ the right-hand side of Eq. 5 gives the interaction energy of E3 =−E␤mB␣ zin;tA . ␤ with respect to A, while the second term gives it for ␤ with Step 4→1: Substance B is quasistatically transformed ␣ respect to .Asz becomes comparable to and less than tA, back into A, requiring chemical energy E4 =Echem ͑ ͒ ͑ → ͒ the first term on the right-hand side of Eq. 5 increasingly B A;zout . This step completes the cycle. dominates, retrieving Eq. ͑4͒ in the limit t ӷz. The first term A Since the cycle is closed and since energy is a state in Eq. ͑5͒, which gives the interaction energy of species function, the net energy change around the cycle is zero; A with the ␤-plate, will be the primary contributor to the ͑i.e., ͛dE=0=E +E +E +E ͒. Now, because the Hamaker Casimir chemical effect. 1 2 3 4 coefficients for A and B are distinct with respect to ␤ and ␣, The analysis thus far assumes a z−2 dependence for the so too must be their interaction energies; i.e., E E . If this surface energy density, but the actual situation is less clear 1 3 is so, then the conservation of energy constraint requires that cut.3,13 Indeed, for zഛ2 nm the energy density can be taken E E . In other words, if the Casimir interaction energies to vary as Eϳz−2; however as z approaches 50 nm, the 2 4 are different for substances A and B, the chemical energy for z-dependence changes to Eϳz−3 and remains so out to about the reaction AB must depend on the relative position and 103 nm, after which it returns to Eϳz−2. Unless otherwise composition of the reaction plates. noted, this discussion focuses on zഛ2 nm, so Eqs. ͑1͒–͑5͒ At first sight, this conclusion is jarring. Normally, the hold well. energy of a chemical reaction is considered independent of the geometry of its containment vessel, e.g., whether the re- III. CASIMIR CHEMICAL EFFECT action is carried out in a test tube or in an Erlenmeyer flask. We now introduce the central result of this study: the While this is generally true at macroscopic scales, this cycle effect of zero point fluctuations ͑Casimir effect͒ on chemical demonstrates otherwise for the nanoscale; indeed, for sepa- reaction energies. This can be demonstrated through a simple rations less than about 100 nm, zero point modes ͑Casimir chemical-mechanical cycle ͑Fig. 2͒, based on the system effect͒ can be decisive. This is the principal result of this shown in Fig. 1͑b͒. study; the remainder will explore its implications. Consider two parallel plates, whose lateral dimensions are large compared with their initial separation in vacuum, zout. Let zout be sufficiently large that the Casimir interaction ͑ Ͼ 3 ͒ A. Chemical implications between the two plates is negligible zout 10 nm . The left- hand plate ͑material ␣͒ is stationary, while the right-hand The primary result can be extended to related cases. In ͑ ␤͒ ͑⌬ ϵ͉ ͉͒ plate material is inwardly movable from zout to zin. The Fig. 2, the reaction energy Echem EA −EB is larger at ␣ ͑ stationary plate is thinly coated with substance A thick- zout than at zin, owing to the stronger Casimir interaction of A ͒ ␤ ␤ ͑ ͒ ness tA , which can convert into substance B. The reaction with than for B with . Figure 3 a depicts the obvious 104706-4 D. P. Sheehan J. Chem. Phys. 131, 104706 ͑2009͒

FIG. 4. Chemical energy vs reaction coordinate q for reaction AB. ͑a͒ Comparison between no-Casimir interaction case ͑i͒ and Casimir interaction case ͑ii͒. Greater canonical force indicated for case ͑ii͒. ͑b͒ System accom- modation to Casimir interaction for exothermic reaction: extension in q and reduction in Eact.

FIG. 3. Variations on Fig. 2: Casimir surface energy density vs plate sepa- scape ͓Fig. 4͑a͔͒, with energy plotted versus reaction coordi- ration for reaction AB. ͑a͒ Converse of energy density vs plate separation ͑ ͒ nate q. Here case ͑i͒ refers to a fiduciary case where no relationship in Fig. 2. b Inversion of surface energy density with zin and zout. Casimir forces act. As expected for an exothermic reaction A B, A can proceed over the activation energy barrier Eact, releasing energy E −E ϵ⌬E . On the other hand, if the ⌬ A B chem converse situation: the greater Echem now occurs at zin ⌬ reaction energy Echem is increased, holding the reaction co- rather than at zout. Again, thermodynamic steps 1–4 above ordinate fixed ͓Fig. 4͑a͒, case ͑ii͔͒, the canonical force on the -q͒ is commensurately increased. Subץ/Eץ−apply. system ͑F ϵ ͑ ͒ q For the exothermic reaction A B in Figs. 2 and 3 a , jected to this greater force, the system should respond to B is energetically favored over A as a product, regardless diminish it. As shown in case ͑ii͒ of Fig. 4͑b͒, this reduction of z. In principle, however, Casimir interaction energies can be accomplished either by stretching the reaction coor- ͑ ͒ can be sufficiently strong to invert this order. In Fig. 3 b , dinate between reactant and product or by reducing the acti- as before, substance B is energetically preferred over vation energy. Both changes lessen the energy gradient, i.e., substance A at zout. Now, however, owing to the large the canonical force, on the system. Changes in Eact are sig- and disproportionate Casimir interaction energy of A nificant since they can exponentially affect reaction rates versus B, at zin substance A is energetically preferred over through the Ahhrenius relation. ͑ ͒ B. The crossover point zcross , where A and B are ener- Additionally, it is conceivable that the Casimir interac- getically equal, is given approximately by zcross tion energy could alter chemical pathways, for instance, by Ӎͱ͑ / ␲͉͓͒͑ ͒/͑ ͑ ͒ ͑ ͔͉͒͒ ͑ ͒ 1 12 HA −HB EA zout −EB zout , where Eq. 1 changing the relative energetic stabilities between possible has been used with zero-separation values for H. reaction products, as suggested in Fig. 5 for the reactions Insofar as the energetic stability translates to thermody- A→B versus A→C. Here, at z , substance A might pref- ⌬ out namic stability, a Casimir-induced change in Echem should be reflected as a change in a reaction’s chemical equilibrium ͓ constant Keq. For the remaining discussion, “energetically more stable ͑preferred͒” will be taken to be roughly equiva- lent to “thermodynamically more stable ͑preferred͒.”͔ For ϵ͓ ͔/͓ ͔ ͓ ⌬ / ͔ the reaction A B, one has Keq B A =exp − G RT , where ⌬G is the molar Gibbs energy and R the gas constant. Although Echem is clearly not G, it can act as a rough proxy, ignoring entropic effects. If so, it may be possible to vary ⌬G ⌬ through Echemical=E2 −E4 so as to tune the equilibrium ratio of products and reactants for a given reaction at the nanoscale through proper choice of substrates and geometries. In this sense, the equilibrium constant becomes a variable. ͑Of course, the usual notion of equilibrium constant can be re- trieved by including plate separation and Casimir interaction energy as part of the formal physical specification of the system’s state. Here it is left out to highlight the novel role played by the Casimir effect.͒ ⌬ Figures 2 and 3 indicate that reaction energy Echem and the equilibrium constant Keq can be tuned by the Casimir interaction, but they are silent about either the activation energy or chemical pathway for a given reaction. It is likely FIG. 5. Redirection of chemical pathway due to Casimir interaction. At z that these too should be affected by Casimir interactions. out species C is thermodynamically most favored, while at zin species B is most Consider a slice of a hypothetical potential energy land- favored. 104706-5 Casimir chemistry J. Chem. Phys. 131, 104706 ͑2009͒ erentially convert to C rather than into B owing to the former’s greater energetic stability, whereas at zin the situation might be reversed since B is now energetically favored over C. The Hammond–Leffler hypothesis28,29 supports these conclusions, positing that the transition state for a reaction should resemble the species nearest to it in energy. With Casimir interactions, where relative energies between products and reactants can be appreciably shifted ͓or even reversed, as in Fig. 3͑b͔͒, the transition state should likewise be shifted to resemble more its energetically nearest species. Furthermore, if the transition state is altered, so too might the pathway through the reaction’s potential energy landscape. It follows that the activation energy and reaction rate should FIG. 6. Casimir energy per molecule ͑␩͒ vs molecular areal footprint ͑S͒ for likewise be affected. ⌬H=200 zJ and ⌬H=20 zJ. Energy benchmarks and characteristic molecu- In summary, it appears likely that Casimir interactions lar sizes are labeled. cannot only directly affect energies and chemical equilibria 1 of surface reactions, but also, via the Hammond–Leffler hy- interaction energy to be roughly 2 zJ or roughly 2 kT at tem- pothesis, could also affect reaction pathways, activation en- perature T=300 K. This is thermally appreciable and enough ergies, and reaction rates. to perhaps drive conformational changes ͑e.g., a gauche to anti conformation in a sterically hindered organic molecule͒, but little else. ͑Or, if the A-layer is freestanding rather than ␣ 1 deposited on the -plate, then this 2 kT might be manifested B. Effect magnitude as thermal energy, locally heating the sample.͒ For larger The magnitude of the Casimir chemical effect can be molecules, with footprints on the order of 30–50 atoms ͑ϳ ϫ −18 2͒ estimated based on realistic physical parameters. Referring to 3 10 m , the Casimir energy can be 80 zJ, roughly ͑ ͒ 20kT, comparable to that of a hydrogen bond. Still larger Fig. 1 b , let substance A have layer thickness tA =1 nm, molecules, such as typical proteins ͑Sϳ10−17–10−16 m2͒, the and at zin=0.5 nm let Hamaker coefficients be H␤m/Am ͑ ͒ ͑ ͒Ӎ Casimir energy can be comparable to that of one to perhaps zin=0.5 nm =H␤m/␣A zin=0.5 nm 250 zJ, which are ap- 30 several chemical bonds. ͑Of course, if the excess Casimir propriate to substance A being electrically conductive. interaction energy was not uniformly distributed over the Again let medium m be vacuum. Let substance B, into which surface, but instead concentrated in spots, then in principle, A can convert ͑AB͒, have Hamaker coefficients energy densities sufficient for chemical bonds might be at- H␤ / ͑z +t ͒=H␤ /␣ ͑z +t ͒Ӎ50 zJ, appropriate if B is m Bm in A m B in A tained even for small molecules.͒ nonconducting. Given these parameters, the difference in Casimir energy can be understood in terms of pres- surface reaction energy density ⌬E between plate chem sure-volume work. Just as gas pressure can provide the en- separations z and z can be shown to be roughly ⌬E in out Cas ergy for bulk chemical transformations, so too can Casimir ϵ͉⌬E ͑z ͒−⌬E ͑z ͉͒Ӎ2ϫ10−2 J/m2. The maximum chem out chem in pressure. The magnitude of the Casimir pressure between Casimir interaction energy densities are about two orders of suitable dielectrics can be inferred from Eq. ͑1͒ to be magnitude less than the energy densities necessary to reform -z=−H/6␲z3. Across pure vacuum at a plate sepaץ/Eץ−=P all chemical bonds on a surface. ͑This assumes 2 bonds/ ration of z=2 nm, this pressure can exceed 100 atms. This atom, 5 eV/bond, and 1019 atoms/m2.͒ This suggests that Casimir surface energy density is comparable to that of this effect should be most important either for reactions in pressure-volume surface work density for a displacement of which reactants and products are nearly energetically degen- 100 nm under 1 atm of pressure. erate or for reactions of relatively large molecules having In standard gas pressure-volume work scenarios, pres- specifically targetable reaction channels. sure differentials are understood at the microscopic level as The physical manifestation of this Casimir chemical en- due to differential momentum flux densities from gas mol- ergy should depend critically on the surface areal footprints ecules. As shown by Milonni et al.,31,3 Casimir pressure can ͑m2͒ of the participating molecules. ͑Let other physical prop- be understood analogously as due to the differential momen- erties of A and B be identical ͑e.g., mass density, bulk modu- lus͒, so as not to otherwise affect the energetics.͒ Let ϵ ␩ TABLE II. Benchmark energies for representative physical chemical phe- SA =SB S be the areal footprints of A and B and let be the ⌬ ͑␩͒ nomena in various energy units. average ECas per molecule. In Fig. 6 is plotted log versus log͑S͒. In order to interpret this plot quantitatively, Table Energy type kT zJ eV kcal/mole II presents several benchmark energies and energy scales. ͑ ͒ For reference, energy benchmarks are included in Fig. 6,as Thermal 300 K 1 4 0.026 0.5 well as characteristic size scales for molecular classes. Conformational 2 8 0.05 1 Physisorption 2 8 0.05 1 Several observations can be made. Taking the surface Hydrogen bond 20 80 0.5 10 footprint of an atom or small molecule on a surface to be Chemical bond 200 800 5 100 roughly 10−19 m2, Eq. ͑5͒ predicts the maximum Casimir 104706-6 D. P. Sheehan J. Chem. Phys. 131, 104706 ͑2009͒ tum flux densities of virtual photons between the interior and exterior surfaces of closely spaced plates. As such, Casimir pressure-volume work is not mysterious; rather, it is the expected outcome of virtual photons whose mode densities have been constrained by the boundary conditions imposed by the confining dielectric plates. How Casimir interaction energy is expressed—as conformational changes, chemical bonds, changes in bulk physical properties, or simply as excess heat—should depend on the reaction channels available in the system. Molecules ex- hibiting robust zero point effects are likely to have relatively large areal footprints since Casimir energy scales linearly with area, but they should also be relatively flat since the majority of this energy will be distributed within about a wavelength of the substrate’s surface. Second, because Casimir-induced changes in energy are derived from changes in the relative differences in dielectric susceptibilities ͓Eq. ͑3͔͒, it would seem to be advantageous that the dielectric responses be distributed over large distances in the candidate molecules, perhaps via delocalized electrons. This suggests that highly conjugated or aromatic organics or polymers might be good Casimir candidates, such as derivatives of polyacetylene, polyaniline, polypyrrole, or polyphenylene. Additionally, it would seem advantageous that small changes in molecular structure give rise to large changes in suscepti- bility, as seen in the electronic behavior of molecular reso- FIG. 7. Casimir catalysts. ͑a͒ Nanocatalyst particle with large plate separa- 32 tion: species B preferred. ͑b͒ Nanocatalyst particle with small plate separa- nant tunneling diodes or in shape-sensitive biomolecules tion: species A preferred. such as proteins.33,34 Detailed consideration of Casimir chemical candidates is beyond the scope of this paper. It is emphasized that Casimir chemical effects are not transition states and reaction rates, while not being consumed due to standard surface chemical potentials; they are distinct in the reactions. The Casimir systems considered thus far in at least two important ways. First, surface potentials in- satisfy these criteria; however, they can also apparently volve quantum and Coulombic interactions among surface change reaction energies and equilibrium constants, setting atoms and electrons, whereas Casimir effects are attributable them apart in fundamental ways. For this discussion they to virtual electromagnetic modes of the vacuum interacting will be called Casimir catalysts. with the bulk medium. Second, compared with usual chemi- On a purely academic level, it might be argued that cal surface potentials,35 Casimir effects are long range. They Casimir catalysts are not possible because they shift chemi- should to be manifest out to the long wavelength limit of the cal equilibria, whereas it is normally understood that a cata- Casimir force, roughly a few microns, which is two to three lytic shifting of equilibrium can lead to a violation of the orders of magnitude greater than the typical range of surface second law of thermodynamics. This objection is not valid chemical potentials. Admittedly though, the most dramatic here because any exess in chemical energy resulting from a Casimir effects operate out to only a few nanometers, less Casimir cycle ͑e.g., Figs. 2 and 3͒ can be traced to net work than an order of magnitude greater distance than standard performed against Casimir force, not supplied by a heat bath; surface potentials. In this respect, Casimir energy can be thus, the second law is not challenged. taken as secondary to surface potential, especially given the One can envision tuning the size and structure and com- larger energies typically associated with the latter. position of Casimir catalysts to optimize the output of a particular product or intermediate in a reaction or to suppress IV. APPLICATIONS unwanted side products. Figures 7͑a͒ and 7͑b͒ depict two possible structures. ͓Figure 3͑b͒ gives the energy density– Casimir chemical energy effects could be expressed gap width diagram pertaining to them.͔ The catalyst particle through a variety of applications. Here we briefly consider a ͑ ͒ few possibilities. in Fig. 7 a has a relatively wide gap such that substance B is energetically ͑thermodynamically͒ favored over substance A, whereas in the case of a narrow gap ͓Fig. 7͑b͔͒, substance A A. Casimir Catalysts is favored over substance B. In principle, both A and B could Catalysts form the backbone of the world’s chemical in- be present side by side if, for instance, one inner surface of dustry, contributing to the manufacture of roughly 90% of the catalyst particle was corrugated. commercial chemical products. As enzymes, abzymes, and This physical situation can be complicated by electro- ribozymes, they render life possible. Hallmarks of traditional static potentials between the plate materials owing to differ- catalysts are the abilities to lower activation energies, modify ences in their chemical potentials. ͑For semiconductors these 104706-7 Casimir chemistry J. Chem. Phys. 131, 104706 ͑2009͒

FIG. 8. Zero point molecular forge. ͑a͒ Casimir surface energy density vs plate separation for reaction AB. ͑b͒ Chemical energy vs reaction coordinate for various plate separations. Activation energy barrier decreases with decreasing z. are called built-in potentials, while for metals they are contact potentials.͒ These would be expressed at the ␣-␤ mate- FIG. 9. Operation of zero point molecular forge. ͑a͒ Ribbon B in gap with rial interface and also across the vacuum gap where they ␤ ͑ ͒ ␤ ͑ ͒ ␤ -plate retracted out to zout. b -plate at zin forging B into A. c -plate could appear as large electric fields. Such intrinsically biased retracting, forged A pulled out, and new charge of B inserted. electrocapacitive catalytic effects have been discussed elsewhere.36 Here they can be simply avoided by creating the nanocatalyst particles out of a single material, say ␤. The activation energy barrier back to B. The net result of the zero point effects would remain, subject to their redefinition cycle is to convert ͑forge͒ low-energy B into high-energy A. with ␣=␤ in Eqs. ͑1͒–͑5͒. Of course, to satisfy energy conservation, this net increase in In summary, static arrangements of nanoscopically chemical energy is supplied by the net mechanical work spaced plates should favor particular molecular species, al- done against the Casimir force. ͑Notice that Step 1→2 pro- lowing new types of nanocatalysts. vides less work output than Step 3→4 requires.͒ This Casimir work can be likened to standard pressure- B. Casimir molecular forge volume work that might be performed to forge, say, a low- energy elemental form of carbon ͑graphite͒ into a higher- The idea of Casimir catalyst can be extended to the dy- energy metastable form ͑diamond͒. In the present case namic case. Because the thermodynamic stability of chemi- ͑B→A͒, however, the forge pressure is distinct in two ways: cal species can depend on plate separation z, it should be ͑a͒ its sign is usually negative owing to the typically attrac- possible to convert one chemical species into another ͑e.g., A tive nature of the Casimir force; and ͑b͒ it is not conveyed by into B͒ simply by moving the Casimir reaction plates with a material substance ͑gas, fluid, solid, or even real photons͒, respect to each other. This dynamic process will be called but rather by virtual photons of the quantum vacuum. Thus, Casimir molecular forging. this zero point process offers another way to drive chemical To concretize these ideas, consider the thermodynamic reactions without necessitating material intermediates to cycle in Fig. 8͑a͒, the reverse of the cycle in Fig. 3͑b͒.Atz out transmit pressure. Classical electromagnetic analogs would species B is thermodynamically favored, while at zin the reverse is true. Figure 8͑b͒ plots chemical energy versus reac- be the pressures exerted on free charges and currents by electric and magnetic fields. tion coordinate q. The activation energies at zout and zcrit are sufficiently high that the thermally driven reaction rate is Let us examine the molecular forge more systematically. ͑ ͒ negligible on the timescale of the forge’s mechanical motion, In Fig. 9 a , the chemical-mechanical cycle begins with a while at z the activation energy is sufficiently low that it ribbon of species B pulled into place between the stationary in ␣ ␤ ͑ ␣ can easily be surmounted thermally on these timescales. ͑Re- -plate and the movable -plate. In principle, the -plate is call from Sec. III A that Casimir interactions can affect both unnecessary since the B-ribbon can be freestanding.͒ In Fig. activation energy and reaction rates.͒ 9͑b͒, the ␤-plate is brought down toward the ribbon, this → ͑ ͒ Figure 9 depicts a molecular forge in various stages of corresponding to Step 1 2 in Fig. 8 a .Atzin, species B ␤ ͓ → ͑ ͒ operation. Let the -plate begin at zout and let A and B be at thermally converts into species A Step 2 3 in Fig. 8 a , thermodynamic equilibrium; i.e., B is in excess over A. As based on the energetics shown in Fig. 8͑b͔͒. Finally, in Fig. the plates are brought together, the energy of A decreases 9͑c͒, the ␤-plate is retracted against the increased Casimir more rapidly than B and, furthermore, the activation energy force ͓Step 3→4, Fig. 8͑a͔͒, and the ribbon is pulled out barrier falls. By the time the plate reaches zin, species B has from between the plates, after which a new charge of B is thermally converted largely into A. Now, as the plate is re- brought into the gap to complete the cycle. As the net result, turned to zout, species A is unable to thermally surmount its a length of ribbon B has been chemically forged into species 104706-8 D. P. Sheehan J. Chem. Phys. 131, 104706 ͑2009͒

FIG. 11. Streamlined Casimir two-cycle: surface energy density vs plate separation for net reaction B→D. ͑See Fig. 10.͒͑a͒ Energy bridging reaction ͑A→C͒ absent. ͑b͒ Rotatable dual surface ͑␤/␥͒ movable plate for molecular forge, pertinent to ͑a͒.

⌬ chemical energy change for this double cycle is Echem,total ⌬ ⌬ FIG. 10. Casimir two-cycle: surface energy density vs plate separation for = EI + EII. In principle, an arbitrary number of Casimir net reaction B→D. Cycle I ͑Steps 1→4 for reaction B→A͒ connected to cycles can be linked to build up chemical energy. The cycle II ͑Steps 5→8 for reaction C→D͒ by bridging reaction ͑Step 4→5 maximum energy gain, however, is limited ultimately by → ͒ for reaction A C . differences in dielectric susceptibilities between interacting materials. A. Again, note that Casimir forging does not require direct ␣ ␤ contact between the reacting ribbon and either plate or . D. Casimir chemical engines In summary, it seems possible to nanomechanically forge chemicals via a work cycle involving the Casimir Chemically driven nanoscopic machines and motors ͑en- force, where zero point fluctuations act as the working fluid. gines͒ take many of forms. The first, most efficient, and most This represents a novel nanoscale technique to convert me- complex molecular machines are biological, e.g., actin- chanical energy directly into chemical energy. The current art myosin, kinesin, and flagellar motors, powered by adenosine 33,34 of micro- and nanoelectromechanical systems ͑MEMS and triphosphate ͑ATP͒. Manmade molecular machines— NEMS͒ appears adequate to allow tests of this concept.37,38 motors, rotors, ratchets, shuttles, switches, and tweezers— Perhaps such “virtual” chemical processes will become com- have been synthesized to be powered by electric currents, monplace in future nanoscale chemistry. gradients in temperature and chemical potentials, light, or other reactive molecules.37 NEMS are solid-state devices powered almost exclusively by electric fields.38,39 Their C. Casimir multicycles chemical powering is indirect, via batteries or fuel cells. Ca- It appears possible to use multiple, interlinked Casimir simir effects might offer an additional route to direct chemi- cycles to bootstrap upward in energy, beyond the limitations cal drive at the nanoscale. Additionally, it does not require of a single cycle. Consider the multistep chemical pathway direct physical contact between a device’s moving parts, a depicted in Fig. 10, whereby B is converted through two leading cause of damage and failure in standard NEMS and Casimir cycles into high-energy product D ͑B→A→C MEMS. →D͒. Two independent cycles ͑I and II͒ are linked by a Just as an electric motor run in reverse acts as an elec- bridging reaction A→C, perhaps driven by an external reac- trical generator, so too should the Casimir molecular forge tant. The total chemical energy increase over the two-cycle ͑Sec. IV B͒ run in reverse act as a molecular motor, trans- ⌬ consists of the energy of the bridging reaction Ebridge plus forming chemical energy into mechanical motion. Again zero ⌬ ⌬ those of the two Casimir cycles EI,II, namely, Echem,total point radiation acts as the working fluid in a thermodynamic ⌬ ⌬ ⌬ = EI + EII+ Ebridge. As before, the chemical energies cycle. A traditional thermodynamic analog would be a steam ⌬ ͑ ͒ EI,II are provided by net mechanical work against the engine driven by chemical reactions burning coal , with Casimir force performed by the ␤-plate. steam as the working fluid, operating between hot and cold An even more streamlined version of this bootstrap pro- heat reservoirs. Here the hot and cold reservoirs are replaced cess can be imagined ͑Fig. 11͒. Here, the reaction A→B by UV ͑hot͒ and IR ͑cold͒ vacuum fluctuations that predomi- → D is carried out via two Casimir cycles without recourse nate at zin and zout, respectively. Purely photonic heat engines to a bridging reaction ͓Fig. 11͑a͔͒. Here the reaction plate is have been examined by others;40,41 some incorporate zero double-sided ͓Fig. 11͑b͔͒. Plate-␤ is used for cycle I, then point fluctuations.42 The present proposal is unique in linking flipped over at Step 4 to expose plate-␥ for cycle II. ͑Clearly, chemical changes to changes in the zero point field. this requires that the Casimir interaction energy of species A Consider Fig. 12, a modified version of the dielectric with plate media ␤ and ␥ be suitably disparate.͒ The net plates in Fig. 1͑b͒. The stationary ␣-plate is unchanged, 104706-9 Casimir chemistry J. Chem. Phys. 131, 104706 ͑2009͒

FIG. 13. Zero point engine. ͑a͒ Chemical energy vs reaction coordinate at ͑ ͒ zout and zin pertinent to cycle in Fig. 2. b Traditional thermodynamic P-V work cycle: Casimir force vs plate separation. Area enclosed by cycle is FIG. 12. Mechanically driven zero point engine with alternating A-B ribbon ␤ ⌬ ideal work output. -plate oscillation energy provided by ECas. drive. ␤-plate oscillation energy provided by forcibly pulling ribbon through gap. ͓For self-propelling ribbon, the upper precursor section ͑B-ribbon͒ must be pure A-ribbon.͔ spring. Substance A remains stable against thermal conversion into B by a high activation energy barrier. → while the movable ␤-plate ͑mass m␤͒ is attached to a spring Step 2 3: At zin the activation energy barrier is lower ͑ ͒ → 43 spring constant ks and acts as a piston. The surface layer of so that the reaction A B proceeds rapidly. Substance B’s substance A is replaced by a long, flat translatable ribbon smaller, attractive Casimir force is overcome by the retract- extending beyond the gap, consisting of alternating sections ing force of the spring, accelerating the piston outwardly ͑to of substances A and B, having different Hamaker coeffi- the right͒. → cients. For the moment, A and B are unable to chemically Step 3 4: The piston moves from zin to zout, relaxing interconvert. the spring, and overshooting its equilibrium position. Let the Casimir interaction diagram for A and B be the Step 4→1: The newly created B-section of the ribbon is 44 same as in Fig. 2, where A exhibits stronger Casimir inter- forced out of the gap by a new section of A that replaces it. actions than B. It is easy to show that, if the ribbon is forc- The piston returns to its initial configuration, and the cycle ibly pulled through the gap at a linear rate commensurate repeats. with the natural mechanical oscillation frequency of the This cycle can be recast as a standard thermodynamic ␤-plate, a resonant oscillation of the piston can be excited, engine work cycle. In Fig. 13͑b͒ is plotted the Casimir force z͒ versus gap width z. The cycle proceeds asץ/Eץ−␻ ͱ / ͑F ϵ with frequency res= ks m␤. The mechanical energy for the Casimir oscillation derives from the work necessary to pull the ribbon above in Fig. 2. The area enclosed by the cycle is the work through the gap against the restoring Casimir force. Like the performed in an ideal cycle. Note that this cycle is essentially oscillation of a child’s swing, pumped by leg and arm mo- the reverse of the Casimir molecular forge of Sec. IV B. tion, the forced synchronous insertion of low- and high- The power of an individual Casimir engine should be Hamaker sections of ribbon into the gap pull and release the tiny, but its power density could be substantial. Taking piston, subject to the restoring force of the spring. Specifi- the surface energy density of the piston plate to be ␳ ϳ −2 / 2 cally, high-Hamaker substance A moves through the gap dur- E 10 J m and its operating frequency to be typical of 9 ing the instroke, pulling the ␤-plate piston to the left and radio frequency NEMS ͑f ϳ10 Hz͒, its power density stretching the spring, while low-Hamaker substance B moves would be appreciable: Pϵ␳f Ӎ107 W/m2. through during the outstroke, releasing the piston and allow- In summary, Casimir effects might be used to help drive ing the spring to retract it. nanoscopic chemical-mechanical engines. They are currently This oscillation can be driven purely chemically—as ap- under active study and will be reported upon in greater depth posed to mechanically by forcibly pulling the ribbon—if the in future communications. reaction ͑A→B͒ is allowed to proceed. In this case, let a ribbon of pure A move through the gap subject to the Casimir E. Casimir biochemistry cycle in Fig. 2 and the reaction energy diagram in Fig. 13͑a͒. ͑ ͒ As before in Fig. 8 b ,atzout the activation energy bar- Life can be described as a complex, autocatalytic net- rier cannot be surmounted thermally on the timescale of the work of chemical reactions guided by nanoscopic molecular ͑␶ϳͱ / ͒ 45,33,34 piston’s motion m␤ ks , whereas at zin the barrier is machines. Casimir chemical effects could conceivably low enough to be quickly surmounted thermally. Beginning play a role in some of these machines. at point 1 in Fig. 2, with the piston moving inwardly and the Insofar as many of the micro- and nanoscopic structures spring unstretched, the Casimir engine cycle proceeds as naturally present in cells seem conducive to the Casimir ef- follows. fect ͑e.g., biplanar cell membranes, spherical vesicles, cylin- Step 1→2: With A in the gap, the piston is pulled in- drical microtubules46͒; and inasmuch as many biochemical wardly ͑to the left͒ by the Casimir force, stretching the reactions turn on energetic changes comparable to those pre- 104706-10 D. P. Sheehan J. Chem. Phys. 131, 104706 ͑2009͒ dicted for Casimir chemistry ͑e.g., conformational and steric From a broader perspective, perhaps it should not be effects, hydrogen bonding͒; and since Life seems compelled surprising that the Casimir force should have chemical rami- to exploit any possible process in service of natural selection, fications. After all, it is intimately related to the van der it is predicted that Casimir chemistry will be discovered at Waals force, which has deep connections in chemistry. The the cellular level, most likely in the context of reactions on van der Waals interaction is a principal long-range “bond” the boundaries of some of the above cellular structures. In- between gas molecules, and it can be key in liquids, as evi- deed, some of the most consequential processes in denced by surface tension. It bears strongly on the melting biochemistry—e.g., protein folding, DNA coiling, drug points of nonpolar solids ͑e.g., paraffin͒ and, as noted above, binding—rely on some of the weakest and most subtle it can be paramount to sensitive biochemical reactions. Still, forces, including ones already directly linked to vacuum the Casimir effect’s chemical implications appear not to have fluctuations, e.g., van der Waals forces. Perhaps this is not been examined carefully before, and deeper exploration surprising; after all, subtlety and sensitivity hold sharp the might reveal new applications as well as theoretical research razor’s edge between order and chaos, upon which Life must directions. tread. In summary, a new chemical effect is proposed based on the Casimir effect whereby fundamental aspects of chemical reactions—e.g., reaction energy, equilibrium constants, acti- V. FUTURE DIRECTIONS vation energies, and reaction pathways—can be tuned. These In this paper, ideas for Casimir chemistry have been Casimir chemical effects might inspire new applications, in- sketched in broad strokes, overlooking many details. At the cluding nanocatalysts, Casimir molecular forges, and en- most basic level, the effects of intervening dielectric other gines. It is speculated that these zero point effects might than vacuum have not been pursued, although many, if not already operate in cellular processes. most, real industrial and biological processes are likely to involve water or other solvents. Thus far, only negative pres- ACKNOWLEDGMENTS sure Casimir effects have been addressed, whereas both theory and experiment confirm that positive ͑repulsive͒ The author is grateful to Dr. T. Herrinton, Dr. P. Greene, 26,47,48 Casimir effects are also possible and could give rise to Dr. E. J. Page, Dr. J. P. Sullivan, Dr. D. C. Cole, and Dr. P. novel applications, like push-pull motors. Furthermore, only W. Milonni for illuminating conversations, and to C. L. parallel plate geometry has been considered, although other Proctor, J. K. Ramis, and S. H. Nogami for the initial impe- geometries ͑e.g., spherical, cylindrical, elliptical, corrugated͒ tus to explore this subject. The author also thanks the anony- are likely to be pertinent elsewhere. The many possible in- mous reviewer for helpful suggestions. This work was con- teractions and feedbacks between Casimir energy and sur- ducted with the support of a University of San Diego faculty face chemical or electrostatic potentials should also be reassign time award ͑2008͒. This work was performed in part explored. at the Center for Integrated Nanotechnologies, Office of Because the maximum Casimir interaction energy densi- Basic Energy Sciences user facility, at Los Alamos National ties are relatively low ͑ϳ10−2 J/m2͒, channeling this energy Laboratory ͑Contract No. DE-AC52-06NA25396͒ and into high-energy reactions would seem to require specialized Sandia National Laboratories ͑Contract No. DE-AC04- molecules and reaction channels that “focus” this energy and 94AL85000͒. This work is dedicated to the memories of Dr. preclude its loss via perhaps more ubiquitous low-energy William F. Sheehan and Patrick C. 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