Measuring the Heat Capacity of Large Isolated Molecules via Gas

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Measuring the Heat Capacity of Large Isolated Molecules via Gas-Phase Collisions: C 60

Rongping Deng, and Olof Echt J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp512276e • Publication Date (Web): 16 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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The Journal of Physical Chemistry

Measuring the Heat Capacity of Large Isolated Molecules via Gas-Phase Collisions: C60 Rongping Deng,† Olof Echt,‡,* † ‡

Department of Physics and Astronomy, Beloit College, Beloit, WI 53511 Department of Physics, University of New Hampshire, Durham, NH 03824

Rongping Deng: [email protected] * Corresponding Author: O. Echt. E-mail: [email protected]. Phone: +1 603 862 3548. Fax: +01 603 862 2998. ABSTRACT We present a novel method to measure the heat capacity of isolated molecules or clusters. Neutral molecules emerge from an effusive source at known temperature. They are heated in a single sticking collision with an atomic ion of known kinetic energy; the breakdown curve of the adduct ion is measured as a function of collision energy. The procedure is repeated for different source temperatures. The heat capacity of the neutral molecule equals the change in the breakdown energy divided by the change in source temperature. The method avoids potential sources of systematic errors inherent to other approaches that involve multiple collisions. The accuracy of the method is demonstrated by colliding an effusive beam of C60 with a monoenergetic beam of Na+ which produces endohedral Na@C60+. The value obtained for the heat capacity of C60 at 535 ± 35 °C agrees with theoretical ones within the experimental uncertainty of 11 %. 1. INTRODUCTION Among the very first properties of atomic clusters to attract interest were thermodynamic quantities. More than a century ago Pawlow discussed the dependence of the vapor pressure on droplet size; he observed that the melting temperature of phenyl salicylate (Salol) decreased by 1.1 ºC when the grain size was reduced to 2 µm.1 Seven decades later a dramatic drop in the melting temperature of supported, nanometer-sized gold clusters from 1340 to 300 K was reported by Buffat and Borel.2 Phase transitions featured prominently in an early review devoted to atomic clusters, written by Gil Stein.3 The following years saw a surge in the development of new methods to synthesize and investigate free, isolated atomic clusters4-7 but it took much longer until methods were developed to investigate the thermal properties of isolated clusters, including their heat capacity. The main challenge lies in measuring the temperature. It is possible to thermalize neutral or charged clusters in a heat bath at a well-defined temperature T but a measurement of the heat capacity requires a second measurement of the temperature after adding a known amount of heat to the cluster. For several years, therefore, computer experiments were pretty much the only way to unravel the fascinating thermodynamic properties of isolated clusters (see, for example, recent reviews 8-10 and references therein). In the late nineties Haberland and co-workers pioneered a method that avoids the need to determine the temperature of the clusters after adding a defined amount of heat.11 They thermalized cluster ions in helium gas (the heat bath) at temperature T1, added a well-defined energy E1 by photon absorption, and monitored the rate of unimolecular dissociation or, equivalently, the pattern of fragment ions that resulted from dissociation within a fixed time window. The experiment was then repeated at a higher heat bath temperature T2 and the excitation energy reduced until, at E2, the same fragment pattern appeared.12 The heat capacity follows from C = -(E2 - E1)/(T2 - T1).

C60SpecHeat 150215 submitted.doc

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The method has been successfully applied to measure caloric curves of sodium cluster cations; the size dependences of the melting temperature and latent heat have been deduced.11,13,14 A variant of the method has been developed by Jarrold and coworkers.15 The basic idea is the same but excitation is achieved by multiple collisions in a cell filled with helium; the excitation energy is varied by changing the kinetic energy of the ions before they are injected into the cell. The method has been applied to study melting of size-selected charged clusters of aluminum, gallium and tin.16-18 A third approach has been designed by Zamith, Labastie, L'Hermite and others.19 It, too, relies on collisions of accelerated cluster ions but instead of fragmentation, cluster growth due to sticking collisions is monitored. For example, sodium cluster ions are passed through a cell filled with sodium vapor, or water cluster ions through a cell filled with water vapor.19-21 Still another variant has recently been demonstrated by Schmidt and von Issendorff.22 The energy transferred to the cluster in the collision cell is varied by changing the pressure, hence the number of collisions, rather than the injection energy. These various approaches have uncovered a range of fascinating, unexpected discoveries including strong irregularities in the size dependence of the melting temperature and latent heat for sodium11,23 and aluminum clusters,17 backbending in the caloric curve of Na147+,13 and melting temperatures of gallium clusters that exceed the bulk value.18 However, in spite of the high precision that these methods have reached they may involve significant systematic errors. The Bloomington15 and the Toulose group19 have critically assessed their methods; the latter group concludes that heat capacities might be over- or underestimated by a factor two. Several factors contribute. For example, cross sections depend on cluster shape, and the amount of energy transfer may depend on the phase of the cluster. A constant systematic error in heat capacities should not affect the values of melting temperatures extracted from these data but two recent experiments on protonated water cluster ions differ in Tmelt by about 15 %.21,22 We present an alternative approach whose accuracy is not compromised by assumptions concerning collisional energy transfer. The approach is demonstrated by measuring the heat capacity of C60. As in the other approaches the molecules emerge from a source with a well-defined temperature TC60 (in our case a Knudsen cell) but they are excited by a single sticking collision with an ion (sodium). The ion yield of the adduct Na@C60+ (which is endohedral, but this is of no relevance to the data analysis) initially rises linearly above the insertion threshold with increasing collision energy Ecm (measured in the center-of-mass system). Ultimately, however, its yield will decrease because its dissociation rate will increase as its vibrational excitation energy E* increases.24 E* is the sum of three terms, E* = Ecm + Evib + Eendo (1) where Evib is the vibrational energy of a canonical ensemble of C60 at source temperature TC60 and Eendo is the binding energy of Na+ in the endohedral Na@C60+. The measurement is repeated at a different source temperature, and the data are modeled. The change in Evib that compensates the change in temperature is deduced; the heat capacity follows from C = ∆Evib/∆TC60 . (2) Possible sources of systematic errors will be discussed. 2. EXPERIMENT Details of the experimental approach have been described previously.24,25 In short, a pulsed beam of sodium ions with a tunable kinetic energy collides with an effusive beam of C60 in the ion source of a time-of-flight (TOF) mass spectrometer. Product ions that form as a result of collisions are accelerated by pulsed extraction potentials towards a detector at the end of a drift tube. The mass spectra reflect the ion distribution 9 to 11 µs after formation of product ions. Sodium ions are extracted from a thermionic source that consists of a resistively heated cartridge capped with a 0.25" diameter porous tungsten plug coated with sodium-doped aluminasilicate glass. C60SpecHeat 150215 submitted.doc


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This source provides ions with a very small kinetic energy spread of about 2 kBT = 0.3 eV.26,27 Mass spectra indicate that contributions from ions other than sodium are below 1 %.28 The sodium ions are accelerated, sent through an ion gate that consists of three closely spaced nickel grids, decelerated to an energy Elab (measured in the lab system) and focused to a spot of about 5 mm diameter at the center of a Wiley-McLaren-type ion source where the ions collide with the C60 beam at 90°. The electric field in the collision region is zero during the duration of the sodium ion pulse. The experiment is run at a repetition rate of 6 kHz. The sodium ion current is measured before and after a mass spectrum is recorded by moving an electrode into the collision region. In dc mode, the ion current amounts to a few hundred nA; results presented here are corrected for minor variations in the metal ion current. C60 (SES Research, purity 99 %) is vaporized from a Knudsen cell whose temperature is stabilized to better than 1 ºC. The C60 flux is monitored by a quartz microbalance; data are corrected for slight changes in the flux. The number density of C60 in the collision zone is on the order of 106 cm-3. The probability that a C60 molecule collides with a sodium ion is 10-7; the probability of multiple collisions is therefore negligible. 3. RESULTS AND DISCUSSION Representative TOF mass spectra, recorded at a C60 source temperature of 570 ºC and four different collision energies Elab (in the lab system) are displayed in Fig. 1. At a collision energy of 35 eV (Fig. 1a) a weak ion peak due to C60Na+ appears. The signal grows as the energy is raised to 40 eV; a new peak appears that we assign to the fragment ion NaC58+, formed by the reaction NaC60+ → NaC58+ + C2 (3) + Although the structure of C60Na (endohedral versus exohedral) is irrelevant to the present work it is clear that the ion must be very stable in order to survive for ≈10 µs at this collision energy. An activation energy of Ea = 10.17 eV ± 0.02 eV (statistical error) was derived in our earlier work for reaction (3).24 Fig. 1 shows an increase of the C60Na+ yield as the collision energy is raised to 40 eV and a dramatic decrease between 45 and 50 eV (Fig. 1b through 1d). At the same time the yield of the fragment ion NaC58+ increases strongly. We should mention that, in principle, C60+ might contribute to the signal that we assign to Na@C58+. The mass resolution is not sufficient to distinguish between these ions whose mass (for the main, pure 12C isotopologue) differs by 1 u. C60+ might form via dissociation of NaC60+ into Na + C60+ or by thermionic emission of the excited C60 as a result of a nonsticking collision with Na+ but these are minor channels at best.24 More importantly, the ambiguity is of no concern here because the specific heat of C60 will be derived solely from the energy dependence of the C60Na+ ion yield. Many additional mass spectra have been recorded. The yield of the adduct ion extracted from these spectra is displayed in Fig. 2 versus collision energy Elab. Also shown is the collision energy Ecm in the center-of-mass system (upper abscissa). Three such data sets have been recorded at different source temperatures TC60. The data show the linear rise above threshold mentioned earlier. A maximum is reached around 41 eV followed by a steep drop whose position depends on TC60. The higher the C60 temperature the lower the collision energy at which the NaC60+ yield breaks down. The data presented in Fig. 2 have been analyzed as described elsewhere;24 here we merely present a brief synopsis. The measured NaC60+ ion yield Y is assumed to be a product of a linear term that accounts for the energy dependence of the insertion cross section29 and a term that describes the survival probability of the excited adduct, Y = B ⋅ ( Ecm − Ethr ) ⋅ exp ( −ktmax ) . (4)



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B is the product of several poorly known quantities that include the C60 number density in the molecular beam, detection efficiency etc but it does not depend on the collision energy Ecm. Ethr is the insertion threshold energy determined earlier (29.6 eV in the cm system24), tmax ≈ 10 µs the time that the ion needs to survive in order to contribute to the mass spectrum, and k the unimolecular dissociation coefficient that we model with an Arrhenius law  E  k = ν o exp − a  (5)  k BT  The relation between the temperature T and the excitation energy E* that characterizes the microcanonical ensemble of NaC60+ (eq. 1) involves a finite heat bath correction to the caloric curve E(T)30-32 Ea 2 . (6) 2 12(C − k B )T Eqs. 5 and 6 provide the link between the experimental ion yield (eq. 4) and the caloric curve Evib(TC60) (eq. 1). The equations involve several other quantities, namely B, Ethr, tmax Eendo, νo, and Ea.33 Only two parameters, B and Evib, are varied to fit the data (the data sets displayed in Fig. 2 were divided by their B-values, therefore the rising parts coincide). The other quantities are either reasonably well known from our previous work (Ethr, tmax Eendo) or they have been determined self consistently (νo and Ea).24 The essential point is that they do not depend on the excitation energy E*. However, several quantities exhibit significant spreads. The kinetic energy distribution of the sodium ions emitted from a thermionic source has a width of about 0.3 eV;26,27 the vibrational energy distribution of C60 emerging from the Knudsen cell is about three times larger. The overall energy spread34 is taken into account by numerical integration; a common value that best describes all three data sets is used when fitting the parameters and B and Evib. Also taken into account is the spread of tmax. Radiative cooling is included as well; for ions in the breakdown region it amounts to about 1.5 eV within 10 µs.35 The procedure may seem convoluted but the conversion from microcanonical to canonical temperature is unavoidable for any experiment that involves clusters that are not in contact with a heat bath at the time of dissociation. For the purpose of determining heat capacities one could merely define an empirical relation between the ion yield and Ecm in order to quantify the fragmentation pattern and its shift with temperature TC60 but it is gratifying to see that a realistic model reproduces the observed shape of the ion yield. Obviously we cannot determine the absolute value of Evib without precisely knowing Eendo, νo, and Ea. Instead, the value of Evib at 500 ºC is estimated; all that matters is the change in Evib as TC60 is changed. The temperature dependence of ∆Evib, referenced to Evib(500 ºC),36 is plotted in Fig. 3. Fitting a straight line to the data one obtains a heat capacity of E ∗ = E (T ) − k BT + 1 E a +

C=

∆Evib = 12.6 ± 1.4 meV ∆TC 60

(7)

This is an average value for the temperatures explored here, 500 °C < TC60 < 570 °C. For comparison, the classical value of the vibrational heat capacity of C60 in the harmonic approximation (equipartition theorem) equals 174 k = 15.0 meV/K. Several authors have computed the caloric curve of C60.37-41 At low temperature they differ considerably depending on the set of vibrational frequencies that were used. Kolodney et al. computed the density of states in the harmonic approximation using a combination of theoretical and experimental vibrational frequencies weighted with the well-known degeneracies.39 We have used the same approach but used the vibrational frequencies computed by Jishi et al..42 At 1000 K we derive a heat capacity of 13.2 meV/K which compares reasonably well with 13.8 meV/K reported by Kolodney et al..39 At 808 K, the average temperature of our C60SpecHeat 150215 submitted.doc


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measurement, we compute a heat capacity of 13.0 meV/K. In an earlier publication we had computed the caloric curve from vibrational frequencies published by Schettino et al.43 which resulted in C = 11.7 meV/K at 808 K.41 The excellent agreement of our experimental value (12.6 ± 1.4 meV/K) with these theoretical values attests to the accuracy of the approach. Thanks to its simplicity the method reduces systematic errors. Errors would still occur if the adduct ion that is formed in a sticking collision would suffer one or more non-sticking collisions either before or after its formation. In our setup the probability for this to happen is 10-7. The probability would increase if one were to improve the sensitivity of the method by increasing the ion current. For example, one could use liquid metal ion sources which combine long-term stability with narrow energy distributions.27 Second, the kinetic energy of the atomic ion might be in error due to contact potentials and space charge effects. However, all that is needed are energy differences, i.e. one merely requires that the energy scale remains linear. Third, radiative cooling is not insignificant; it amounts to about 1.5 eV during the relevant time window of 10 µs.35 However, even if we completely ignore radiative cooling the breakdown energy changes by only 0.05 eV. The dependence of this change on oven temperature, from 500 and 570 ºC, is much less than 0.05 eV; the resulting systematic error in the heat capacity is on the order of 1 %. Experimental methods have their strengths and limitations. The heat capacity C reported here is an average over a rather wide temperature interval; the approach would not be useful to measure latent heats. In the present situation, however, a calculation of the heat capacity based on published vibrational frequencies shows that C will increase by only 0.67 meV/K from 500 to 570 ºC, only half the uncertainty of our result. The precision of our method is low because it lacks the amplification factor that is common to other methods.15 That factor arises from the low efficiency (about 5 %) with which kinetic energy is converted to internal energy in non-sticking collisions. On the other hand, knowing that the efficiency equals 100 % for a sticking collision eliminates a major source of systematic error. The method has other limitations but they are common to other approaches used so far (if not always fully recognized). First, radiative cooling (or heating) may change the vibrational energy distribution of the clusters on their path from the heat bath to the point of excitation.44 Even if the degree of cooling is small it will preferentially deplete the high-energy tail of the distribution which is the one that contributes most to the fragmentation pattern. Second, the vibrational energy distribution after excitation will not resemble that of a canonical ensemble; it will be narrower or broader depending on the excitation process. Again, this will distort the high-energy wing. It may not be sufficient to merely consider the average of the energy that is being added. Third, the method demonstrated here involves collisions between a neutral cluster with atomic ions. Applying our method to a beam of neutral clusters Xn that contains a range of sizes (n-values) would not work because one couldn’t distinguish between a fragment ion that results from a sticking collision of Na+ with Xn+1 with subsequent fragmentation into XnNa+, and a primary adduct XnNa+. However, variants of the method are conceivable. For example, one could thermalize Xn+ in a helium cell, extract, accelerate and size-select the ions and collide them with a beam or stagnant gas of heavy neutral atoms or strongly bound molecules such as C60. A measurement of the breakdown energy of XnC60+ as a function of the temperature of the helium bath would reveal the heat capacity of the cluster ions. The simplicity of the approach presented here would be sacrificed though. CONCLUSIONS We have presented a novel method to measure heat capacities of isolated molecules or clusters. It has been demonstrated for neutral C60. Like other methods it starts with the extraction of molecules from a heat bath at well defined temperature. Unlike other methods it adds energy to the molecule in a single C60SpecHeat 150215 submitted.doc


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sticking collision. The change in collision energy that is necessary to mitigate the effect of changing the heat bath temperature is determined. The ratio of the two changes, -∆E/∆T, equals the heat capacity. The approach greatly reduces systematic errors because no assumptions have to be made about the efficiency of energy transfer in non-sticking collisions, or the probability of non-sticking versus sticking collisions. On the flip side the method lacks precision; it is less suitable to measure latent heats. The method may prove useful to calibrate the energy scale of other methods that offer higher precision. ACKNOWLEDGEMENTS This work was supported by NSF under Grant PHY-9507959



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REFERENCES (1) Pawlow, P. Über die Abhängigkeit des Schmelzpunktes von der Oberflächenenergie eines Festen Körpers (the Dependency of the Melting Point on the Surface Energy of a Solid Body). Z. Phys. Chemie 1909, 65, 545-548. (2) Buffat, P.; Borel, J. P. Size Effect on Melting Temperature of Gold Particles. Phys. Rev. A 1976, 13, 2287-2298. (3) Stein, G. D. Atoms and Molecules in Small Aggregates - the Fifth State of Matter. Phys. Teach. 1979, 17, 503-512. (4) Clusters of Atoms and Molecules I: Theory, Experiment, and Clusters of Atoms; Haberland, H., Ed.; Springer Series in Chemical Physics, Vol. 52, Springer, 1994. (5) Clusters of Atoms and Molecules II: Solvation and Chemistry of Free Clusters, and Embedded, Supported and Compressed Clusters; Haberland, H., Ed.; Springer Series in Chemical Physics, Vol. 56, Springer, 1994. (6) Johnston, R. L. Atomic and Molecular Clusters; Taylor & Francis, 2002. (7) Alonso, J. A. Structure and Properties of Atomic Nanoclusters, 2nd ed.; Imperial College Press, 2011. (8) Dong, Y.; Springborg, M.; Pang, Y.; Morillon, F. M. Analyzing the Properties of Clusters: Structural Similarity and Heat Capacity. Comput. Theor. Chem. 2013, 1021, 16-25. (9) Li, Z. H.; Truhlar, D. G. Nanosolids, Slushes, and Nanoliquids: Characterization of Nanophases in Metal Clusters and Nanoparticles. J. Am. Chem. Soc. 2008, 130, 12698-12711. (10) Baletto, F.; Ferrando, R. Structural Properties of Nanoclusters: Energetic, Thermodynamic, and Kinetic Effects. Rev. Mod. Phys. 2005, 77, 371. (11) Schmidt, M.; Kusche, R.; von Issendorff, B.; Haberland, H. Irregular Variations in the Melting Point of Size-Selected Atomic Clusters. Nature 1998, 393, 238-240. (12) In practice, excitation was done by multiphoton excitation at fixed wavelength. The temperature was increased until the absorption of one fewer photon produced the same fragmentation pattern. (13) Schmidt, M.; Kusche, R.; Hippler, T.; Donges, J.; Kronmüller, W.; von Issendorff, B.; Haberland, H. Negative Heat Capacity for a Cluster of 147 Sodium Atoms. Phys. Rev. Lett. 2001, 86, 1191-1194. (14) Hock, C.; Bartels, C.; Strassburg, S.; Schmidt, M.; Haberland, H.; von Issendorff, B.; Aguado, A. Premelting and Postmelting in Clusters. Phys. Rev. Lett. 2009, 102, 043401. (15) Neal, C. M.; Starace, A. K.; Jarrold, M. F. Ion Calorimetry: Using Mass Spectrometry to Measure Melting Points. J. Am. Soc. Mass Spectrom. 2007, 18, 74-81. (16) Breaux, G. A.; Benirschke, R. C.; Sugai, T.; Kinnear, B. S.; Jarrold, M. F. Hot and Solid Gallium Clusters: Too Small to Melt. Phys. Rev. Lett. 2003, 9121, 5508. (17) Aguado, A.; Jarrold, M. F. Melting and Freezing of Metal Clusters. Ann. Rev. Phys. Chem. 2011, 62, 151-172. (18) Pyfer, K. L.; Kafader, J. O.; Yalamanchali, A.; Jarrold, M. F. Melting of Size-Selected Gallium Clusters with 60-183 Atoms. J. Phys. Chem. A 2014, 118, 4900-4906. (19) Chirot, F.; Feiden, P.; Zamith, S.; Labastie, P.; L'Hermite, J. M. A Novel Experimental Method for the Measurement of the Caloric Curves of Clusters. J. Chem. Phys. 2008, 129, 164514. (20) Zamith, S.; Labastie, P.; L'Hermite, J. M. Heat Capacities of Mass Selected Deprotonated Water Clusters. J. Chem. Phys. 2013, 138, 034304. (21) Boulon, J.; Braud, I.; Zamith, S.; Labastie, P.; L'Hermite, J. M. Experimental Nanocalorimetry of Protonated and Deprotonated Water Clusters. J. Chem. Phys. 2014, 140, 164305. (22) Schmidt, M.; von Issendorff, B. Gas-Phase Calorimetry of Protonated Water Clusters. J. Chem. Phys. 2012, 136, 164307. C60SpecHeat 150215 submitted.doc


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(23) Schmidt, M.; Haberland, H. Phase Transitions in Clusters. Comptes Rendus Physique 2002, 3, 327-340. (24) Deng, R.; Echt, O. Collisional Formation and Dissociation of Na@C60+. Chem. Phys. Lett. 2002, 353, 11-18. (25) Deng, R.; Echt, O. Hyperthermal Collisions of Atomic Clusters and Fullerenes. Int. J. Mass Spectrom. 2004, 233, 1-12. (26) Alton, G. D. Sources of Low-Charge-State Positive-Ion Beams. In Experimental Methods in the Physical Sciences; Dunning, F. B., Hulet, R. G., Eds.29A, Academic Press: San Diego, 1995, pp 69-168. (27) Handbook of Ion Sources; Wolf, B. H., Ed.; Handbook of Ion Sources, Vol., CRC Press: Boca Raton, 1995. (28) Deng, R.; Clegg, A.; Echt, O. Mass Spectrometric Study of K+ + C60 Collisions. Int. J. Mass Spectrom. 2003, 223-224, 695-701. (29) Bernshtein, V.; Oref, I. Endohedral Formation, Energy Transfer, and Dissociation in Collisions between Li+ and C60. J. Chem. Phys. 1998, 109, 9811-9819. (30) Klots, C. E. Systematics of Evaporation. Z. Phys. D 1991, 20, 105-109. (31) Andersen, J. U.; Bonderup, E.; Hansen, K. On the Concept of Temperature for a Small Isolated System. J. Chem. Phys. 2001, 114, 6518-6525. (32) Hansen, K. Statistical Physics of Nanoparticles in the Gas Phase; Springer, 2012; Vol. 73. (33) Furthermore, eq. 6 involves the heat capacity C of Na@C60+. The term presents a very minor correction that amounts to 0.04 % of E* in the breakdown region; an estimate of C is sufficient (see ref. 24 for details). (34) Another mechanism, the kinetic energy distribution of the thermal C60 beam, contributes much less to the broadening of the E* distribution thanks to the crossed-beam geometry. (35) Andersen, J. U.; Bonderup, E. Classical Dielectric Models of Fullerenes and Estimation of Heat Radiation. Eur. Phys. J. D 2000, 11, 413-434. (36) Although one may argue that the value at the reference point (500 ºC) would have zero error we include an error because the fitting procedure involves uncertainties for each of the three data sets. (37) Wurz, P.; Lykke, K. R. Multiphoton Excitation, Dissociation, and Ionization of C60. J. Phys. Chem. 1992, 96, 10129-10139. (38) Andersen, J. U.; Brink, C.; Hvelplund, P.; Larsson, M. O.; Nielsen, B. B.; Shen, H. Radiative Cooling of C60. Phys. Rev. Lett. 1996, 77, 3991-3994. (39) Kolodney, E.; Tsipinyuk, B.; Budrevich, A. The Thermal-Energy Dependence (10-20 eV) of Electron-Impact Induced Fragmentation of C60 in Molecular-Beams - Experiment and ModelCalculations. J. Chem. Phys. 1995, 102, 9263-9275. (40) Vostrikov, A. A.; Agarkov, A. A.; Dubov, D. Y. Kinetics of Radiation Cooling of Fullerenes. Tech. Phys. 2000, 45, 915-921. (41) Echt, O.; Yao, S.; Deng, R.; Hansen, K. Vibrational Energy Dependence of the Triplet Lifetime in Isolated, Photoexcited C60. J. Phys. Chem. A 2004, 108, 6944-6952. (42) Jishi, R. A.; Mirie, R. M.; Dresselhaus, M. S. Force-Constant Model for the Vibrational Modes in C60. Phys. Rev. B 1992, 45, 13685. (43) Schettino, V.; Pagliai, M.; Ciabini, L.; Cardini, G. The Vibrational Spectrum of Fullerene C60. J. Phys. Chem. A 2001, 105, 11192-11196. (44) Andersen, J. U.; Gottrup, C.; Hansen, K.; Hvelplund, P.; Larsson, M. O. Radiative Cooling of Fullerene Anions in a Storage Ring. Eur. Phys. J. D 2001, 17, 189-204.



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FIGURE CAPTIONS Figure 1 Time-of-flight mass spectra of ions formed by gas-phase collisions between Na+ and C60. C60 was vaporized at 570 ºC; the kinetic energy of sodium ions in the lab reference frame is indicated in the Figure. Figure 2 The ion yield of NaC60+ versus collision energy (Elab and Ecm, bottom and top axes) for three different source temperatures TC60. Dashed lines represent least-squares fits of eq. 4 to the data.

200

Figure 3 Change of the vibrational energy Evib of C60 with source temperature TC60.36 The dashed line represents a linear fit to the data; its slope equals the heat capacity.

+

C60 + Na

100

a) 35 eV

+

200

b) 40 eV NaC60+

100

Ion yield (arb. units)

0

NaC60

+

NaC58

c) 45 eV

NaC60+

200

100

200

+

NaC58

d) 50 eV +

NaC56

NaC60+ 100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8000

8500

Time-of-flight (arb. units)

Figure 1



9


30

35

Collision energy Ecm (eV) 40

45

50

+

NaC60

55

500 ºC 535 ºC 570 ºC

Ion yield (arb. units)

1

0 30

35

40

45

50

55

Collision energy Elab (eV)

∆E/∆T = 12.6 ± 1.4 meV/K

0.5

∆Evib (eV)

1.0

Figure 2

0.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 10

500

520

540

560

TC60 (°C)

Figure 3

TOC



10

Measuring the Heat Capacity of Large Isolated Molecules via Gas

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