The dynamics of evaporation from a liquid surface

Chemical Physics Letters 513 (2011) 1–11

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FRONTIERS ARTICLE

The dynamics of evaporation from a liquid surface Olivia J. Maselli a,1, Jason R. Gascooke a,b, Warren D. Lawrance b, Mark A. Buntine a,c,⇑ a

Department of Chemistry, The University of Adelaide, Adelaide, SA 5005, Australia School of Chemical and Physical Sciences, The Flinders University of South Australia, G.P.O. Box 2100, Adelaide, SA 5001, Australia c Department of Chemistry, Curtin University, G.P.O. Box U1987, Perth, WA 6845, Australia b

a r t i c l e

i n f o

Article history: Available online 12 June 2011

a b s t r a c t We explore the collisional energy transfer dynamics of benzene molecules spontaneously evaporating from an in vacuo water–ethanol liquid beam. We find that rotations are cooled significantly more than the lowest-energy vibrational modes, while the rotational energy distributions are Boltzmann. Within experimental uncertainty, the rotational temperatures of vibrationally-excited evaporating molecules are the same as the ground state. Collision-induced gas phase energy transfer measurements reveal that benzene undergoes fast rotational relaxation, from which we deduce that the rotational temperature measured in the evaporation experiments (200–230 K) is an indication of the translational energy of the evaporate. Conversely, vibrational relaxation of the high frequency mode, m6, is very inefficient, suggesting that the m6 temperature (260–270 K) is an indication of the liquid surface temperature. Modelling of the relaxation dynamics by both ‘temperature gap’ and ‘Master Equation’ approaches indicates that the equivalent of 150–260 hard-sphere collisions occur during the transition from liquid to vacuum. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The molecular-level dynamics of transport across the liquid–vapour interface are critically important in a range of natural and technological environments, yet they have eluded determination. While one might have expected such a fundamental and ubiquitous process to be well understood in the early 21st century, our knowledge of evaporation at the molecular level is in its infancy. This situation prevails despite the thermodynamics of evaporation having been understood for over a century [1,2]. Developing a comprehensive molecular-level understanding of evaporative mass and energy transfer has a wide-ranging practical importance in, for example, better understanding atmospheric [3,4], chemical [5–9] and technological [10–16] processes. Inferences about the dynamical processes of surface desorption have been made using various techniques [17] including molecular scattering [18–21] and thermal desorption [22–26]. However, the great variety in the results implies that the nascent distribution of molecular energy is greatly dependent upon the nature of the surface from which the molecules are escaping. Some of the earliest studies involved probing evaporation from solid surfaces. Since the number of evaporating molecules is extremely low, the nascent distributions observed can be assumed to be a result of spontane⇑ Corresponding author at: Department of Chemistry, Curtin University, G.P.O. Box U1987, Perth, WA 6845, Australia. E-mail address: [email protected] (M.A. Buntine). 1 Present address: Department of Hydrologic Sciences, Dessert Research Institute, 2215 Raggio Parkway, Reno, NV 89512, USA. 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.06.010

ous evaporation. Spontaneous evaporation occurs when molecules are liberated from the condensed phase in the absence of external forces; all the energy to leave the surface is supplied either from the bulk phase beneath the evaporating molecule or by the molecule’s own internal energy [27]. Though the number of studies is few, the results from observations of spontaneous sublimation from a solid surface of the same material show some striking similarities. Klemperer’s 1962 studies on the sublimation of iodine [28] revealed that vibrational temperatures of subliming I2 were consistent with the bulk temperature. Nesbitt and co-workers [29] found that CO2 subliming from dry ice displayed Boltzmann distributions amongst the rotational and vibrational modes of the nascent sublimate with temperatures the same as that measured for the bulk. When the temperature of the dry ice film was raised, increasing the likelihood of gasphase collisions of the sublimate, the rotational distribution cooled to temperatures 0.8 times the measured surface temperature. Similarly, studies of the sublimation of NO from its bulk showed rotational distributions that described the surface temperature of 50 K [30]. Sadtchenko et al. [31] observed that D2O molecules sublimating from an in vacuo ice filament displayed a translational energy distribution whose temperature matched that of the bulk. These studies suggest that spontaneous sublimation of molecules from their solid phase produces gas phase molecules thermally equilibrated with the bulk. In contrast, experiments involving heterogeneous systems – where the thermally-desorbed molecule is different to the material of the surface – show different trends. Both theoretical and experimental studies show significant rotational cooling of molecules

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desorbed from solid surfaces [19,32,33]. For example, both Muhlhausen [19] and Cavanagh [22] observed cooler-than-surface-temperature rotational distributions for NO molecules desorbing from Ag(1 1 1) and Ru(0 0 1). In their review of the chemical dynamics of the gas-surface interface, Rettner et al. point out that gas molecules that are trapped and subsequently desorbed from the surface typically display a rotational energy distribution that is Boltzmann with a temperature that is less than the surface [34]. Theoretical calculations by Muhlhausen [19] confirm that the effect is due to strong rotation-to-translation coupling along the desorption coordinate. This occurs due to the presence of an energy barrier along the desorption coordinate, where internal energy must be partitioned into the translational coordinate if the molecule is to escape the surface. The desorption coordinate is typically assigned along the direction normal to the surface since it has been repeatedly shown that the flux of molecules evaporating from a surface, be it liquid or solid, is maximised along the surface normal [17,35– 37]. Theorists modelling collision-free molecular evaporation from the condensed phase typically describe the translational velocity distribution of escaping molecules as a half-range or modified Maxwellian distribution. Backward scattered molecules are not found in the vapour phase [38–40] since the surface escape requirement is to have a velocity along the surface normal. The presence of molecular collisions above the surface distort the nascent distributions toward a floating Maxwellian with a translational temperature cooler than the surface and with a net translational velocity away from it [36,41–46]. References to the substantial body of research performed on the translational velocities of desorbed molecules can be found in reviews such as those by Comsa [17] and Rettner [34]. In general, determining the translational, rotational and vibrational distributions of nascent molecules liberated from an equilibrated liquid surface can be confounded by the presence of a jacket of vapour molecules that sits above the surface, blurring the boundary between the isotropic liquid and vapour phases. The properties of energy transfer through this interface have been studied extensively by Molecular Dynamics (MD) and other theoretical methods [27,39,47–62]. In a recent review of simulations describing this interface, Garrett and co-workers [47] highlight the general conclusions that (i) the density transition from bulk liquid to water vapour occurs over a laterally-averaged distance of 0.3–0.6 nm, that is, over molecular length scales, and (ii) the interface is rough over these molecular length scales. Garrett’s simulations suggest that localised regions of the liquid–vapour interface are molecularly sharp, but as a whole, the interface appears as a rough outer layer consisting of molecules in direct contact with one another penetrating into the vacuum. X-ray scattering studies have measured liquid water surface roughness values that are in good agreement with the predictions from these MD simulations [63]. In general, the high vapour pressure of liquids enhances surface roughness and widens interface thickness; adding complexity to the evaporation processes. Interestingly, in the case of liquid sodium, which has a particularly low volatility, an experimental study of spontaneous emission of Na2 from its bulk liquid surface [35,37] revealed rotational and vibrational temperatures the same as the bulk phase. In summary, the investigations discussed above indicate that sublimation of molecules from homogenous solid surfaces liberates molecules with nascent rotational and vibrational temperatures equilibrated with the surface. A similar situation occurs following evaporation from homogenous liquid surfaces for molecules having very low volatilities. Sublimation from heterogeneous solid surfaces and from liquid surfaces where sample volatility is not particularly low is more complex, and results in varying amounts of translational, rotational and vibrational cooling. This

variation in the extent of cooling suggests that collisional energy transfer dynamics play a crucial role during the phase transition process, especially under conditions where the interfacial boundary is less well defined (on a molecular scale). In this Letter we explore the rotational and translational energy content of benzene molecules evaporating from an aqueous liquid surface and use the results to comment on the number of collisions experienced by a benzene molecule as it undergoes a transition from the condensed to the vapour phase. Collision-induced energy transfer that can occur between an evaporating molecule and the vapour phase prevents us from learning about the spontaneous mass transfer process since molecular collisions redistribute the evaporate’s nascent internal energy. The presence of vapour-phase collisions can tend to focus the molecular flux along the surface normal [17,24,35–37,40,43,44], remove molecular anisotropy [29,39], equilibrate the internal molecular degrees of freedom [42,43] and reduce the net evaporation flux by forcing some molecules to return back to the surface [27]. While temperature constraints on the bulk system can be used to control the number of vapour-phase collisions a molecule experiences as it escapes from a liquid surface, placing geometric constraints on the surface itself can allow for the study of spontaneous emission from more volatile liquids. An in vacuo cylindrical filament of water, for example, is able to satisfy the Knudsen condition for obtaining collision-free molecular flow from the liquid surface if the diameter of the filament is less than the estimated mean free path of the evaporating molecules [64–66]. Faubel et al. were able to use in vacuo liquid filaments, which we term ‘liquid microjets’ (LlJs), to determine translational velocity distributions of nascent, volatile liquids such as water [67] and carboxylic acids [68]. Using 5-lm diameter LlJs of pure water, it was found that translational velocities could be fitted with a floating Maxwellian distribution. The net bulk velocity was thought to be a result of only a few gas-phase collisions converting the internal energy of the evaporating molecules into translational energy away from the bulk. For water LlJs of 25-lm radius or greater, the velocity distribution observed was significantly narrower, which Faubel et al. concluded was due to significant translational cooling from collisions within the interface [68]. Augmenting the experimental studies, Molecular Dynamics (MD) simulations of evaporation have begun to provide insight into the mass and energy transfer across the liquid–vapour interface in unprecedented detail. Simulations have rapidly evolved to the stage where they are able to describe polyatomic liquid interfaces and have, within a rigid rotor approximation, included predictions of the rotational energy content of the evaporate [39,41]. Further extensions that explore the vibrational energy content of evaporating molecules have yet to be reported. This contribution builds upon the body of experimental research discussed above by quantifying the rotational and vibrational energy distributions of molecules spontaneously evaporating from an in vacuo LlJ. Specifically, we report the energy distributions for benzene molecules that have spontaneously evaporated from the surface of a 15-lm diameter aqueous LlJ and interpret the results in terms of collisional energy transfer. The robustness of the conclusions drawn from this study is underpinned by the use of two distinct (and unrelated) models to describe the collisional energy transfer dynamics. Both models yield, semi-quantitatively, the same conclusions. Benzene was chosen as the candidate with which to determine the internal energy distribution of molecules liberated from LlJ because its spectroscopy is well characterised [69–72]. The shape and intensity of a molecule’s rovibrational absorption contour is indicative of the temperature of the molecule’s rotational and vibrational motion, respectively. Temperature itself is essentially a description of the distribution of molecular energy within a

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particular degree of freedom. In this Letter we report the vibrational temperatures of several vibrational modes of benzene as well as the associated rotational temperatures. The results of complimentary gas-phase collision studies, and the modelling of collision numbers therein, show that the various rotational and vibrational motions of benzene relax at widely different rates and hence are sensitive to widely different collision number regimes. The observed temperatures of evaporating benzene can thus be correlated with the number of collisions the molecule experiences as it passes from the liquid into the vacuum.

Table 1 Rotational constants used in the simulation of the 610 , 621 =601 1120 , and 610 1611 =610 1111 absorption transitions. Constanta (cm1) Vibronic state

B

C

feff

00

0.18977b 0.18179b

0.09489b 0.09087b

– 0.5785b

2

0.189635d 0.18198(3)

0.09482d 0.09080(3)

0.588c 0.195(5)

0

0.18131(9)

0.09096(7)

0

1

161 l

1

0.189661d 0.18140(3)

0.094976e 0.09101(5)

0 0.479(9)

1

161 l

1

0.18136(4)

0.09087(7)

0.48(1)

1 1

0.18143(2)

0.09096(3)

0.48(3)

0.189640d 0.18116(9)

0.094897e 0.09084(1)

0 0.48(2)

61 61 62 l

62 l =112 161 61 l

2. Experimental details

61 l

1 1

The experimental apparatus and techniques have been given in detail in previous publications [73–75]. Details pertinent to the present experiments are as follows. All experiments were conducted using 103 M benzene (BDH, 99.7%) with ethanol (Ajax Finechem, 99.5%) in de-ionised water (25% v/v EtOH in H2O) as the solvent. The 7.5-lm radius liquid microjet was generated by flowing the solution through a tapered silica capillary (New Objective PicoTip emitter, uncoated SilicaTip) at high pressure (flow rate 0.25 mL/min). The liquid temperature just before entering the vacuum was measured by a thermocouple attached to the stainless steel tube housing the LlJ capillary. When the apparatus was operational, the thermocouple registered a temperature of 283 K, 10 K less than ambient. Benzene evaporate was ionised by 1 + 1 resonance-enhanced multiphoton ionisation (REMPI) via the 1 1 B2u A1g transition using a collimated and telescoped ultraviolet (UV) laser beam (10 lJ/pulse, 150 lm radius FWHM; 0.9  107 W cm2) propagating orthogonally to the liquid filament. The UV laser did not directly irradiate the LlJ, but ionised molecules that had moved a fixed distance away from the liquid filament. Typically, the UV laser was located 300 lm from the LlJ. The resultant benzene ions were injected into a reflectron timeof-flight (TOF) mass spectrometer with mass spectra collected over 50 laser shots at each UV laser wavelength. Typically, 2–4 consecutive wavelength scans (depending upon ion signal strength) were averaged to generate the spectra used for analysis. The rotational and vibrational energy content of the benzene molecules are reported in terms of rotational and vibrational mode-specific ‘temperatures’. Ground state rotational temperatures were determined by fitting the vibronic band contours with ones simulated from known spectroscopic constants and transition line strengths [71,76–81] using a Levenberg–Marquardt non-linear least squares algorithm [82] and assuming a Boltzmann distribution. For excited vibrational states where the rotational constants have not been previously reported, the rotational temperatures were determined by fits to the relevant ambient temperature contour. The temperatures of vibrational levels were determined by noting the change in the ratio of the integrated intensities of the relevant vibronic band to the 610 band from that found at (equilibrated) room temperature [83]. The spectral simulation algorithm was tested by fitting the room temperature 1 + 1 REMPI spectrum of benzene measured in a static cell. All spectroscopic constants used or determined in this study are reported in Tables 1 and 2. Full details regarding the spectroscopic constants determined in this study will be provided in a future publication [84]. The use of a solvent mixture potentially complicates the interpretation of the experimental results. The water–ethanol solvent mix was used for two reasons. Firstly, ethanol assists in dissolving the benzene into the highly polar water solvent. Secondly, as noted in previous reports from this laboratory, ethanol is used to prevent premature disintegration of the liquid filament [73–75,85]. Klein and co-workers have noted that segregation of water and ethanol leads to an enhanced ethanol concentration at the liquid surface

6 l 16 l 111 61 111

a All constants that are not reported in the literature were determined in this study by best-fit simulations of the rotational contours to the ambient temperature absorption spectra for 621 =601 1120 transitions or the ambient temperature 2D-LIF image for the 610 1611 =610 1111 transitions. One standard deviation in the error of the best-fit value is given in parentheses in units of the last digit. b Values were taken from Okruss [71]. c Value was taken from Weber [78]. d Values were taken from Hollenstein [77]. e Values calculated from the rotational defects reported by Jagod [81].

Table 2 Vibrational constants used in the simulation of the 610 , 621 =601 1120 , and 610 1611 =610 1111 absorption transitions.

Vibrational anharmonicity constants

Vibrational angular momentum anharmonicity constantsb

b

Constant

Valuea (cm1)

x006;6 x06;6 x0016;16 x016;16 x06;16 x06;11 g 006;6 g 06;6 g 0016;16 g 016;16 g 06;16 00 11

1.05

m 0

62 l =112 Fermi splittingc

0.7 -1.4 -1.22 0.45 4.5 0 1.7 0.52 0.88 2.09(4) 517.19(6) 5.92(6)

a Values with following parentheses are determined from this study. One standard deviation in the error of the best-fit value is given in parentheses in units of the last digit. b The anharmonic constants not determined from this study are taken from Atkinson [76]. c The value for the Fermi resonance energy splitting is caused by the interaction 0 of the 62 l and 112 vibronic levels in the B2u excited electronic state of benzene. Both the Fermi energy splitting and the frequency of mode 11, m0011 , were determined in this study by best-fit simulations of the rotational contours to ambient temperature absorption spectra.

and possibly exacerbates the surface roughness [86–88]. More recent water–alcohol simulations support these conclusions [89,90]. To explore the influence of the ethanol concentration on the measured energy distributions, comparisons were performed using a 10% v/v EtOH–H2O solvent mix. Within experimental uncertainty, there was no difference in the temperatures observed for benzene molecules evaporating from the 25% or 10% solvent mixes. Consistent with this, Zhang reported that adding small amounts of alcohol to a glycerol/water matrix had no effect on the observed vibrational temperatures of laser-desorbed benzimidazole [91]. For this reason, only the results for evaporation from a 25% ethanol LlJ are reported herein.

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3. Results and discussion The results of our earlier investigation of the rotational and vibrational energy content of benzene molecules evaporating from a water/ethanol LlJ have been presented previously [73–75]. The primary advances since this earlier work are (i) we are now able to extract vibrational temperatures for m11, providing a second vibrational mode from which to extract collision numbers, and (ii) we have undertaken more sophisticated modelling of the collisional energy transfer occurring as the benzene transits from the liquid surface to the vacuum. Both provide crucial tests of the robustness of our earlier conclusion that many hundreds of collisions are required to explain the vibrational temperatures observed. 3.1. Rotational temperatures The rotational temperatures measured for the benzene evaporate have been discussed previously [73–75]. The key results and conclusions from these earlier studies are:  The spectra are well fitted by a Boltzmann distribution of the rotational state population, demonstrating that the rotational motion of evaporating benzene is, within experimental uncertainty, at thermal equilibrium.  Within experimental uncertainty, the rotational temperature is the same for all vibrational levels.  Within error, the rotational temperatures appear unchanging for the range of surface residence times interrogated (i.e., increasing downstream distance from the nozzle orifice).  Within the 3r experimental error, where r represents the weighted error of the mean, there is no significant change in the rotational temperatures as a function of distance from the liquid filament, indicating that collisional energy transfer is complete 200 lm from the surface. All rotational temperatures observed fall within the range of 210–230 K (see Table 3). While we have not directly measured the surface temperature, there is evidence from other groups that it is significantly higher than the 210–230 K rotational temperatures we have measured. Cohen, Saykally and co-workers (C&S) have used Raman thermometry to investigate evaporative cooling rates of water and methanol from various sized LlJs into a vacuum [92–96]. They observed that the liquid surface temperatures decrease non-linearly with surface residence time, with the more volatile methanol cooling much more rapidly than water. Decreasing the diameter of the LlJ enhanced the amount of surface cooling observed. C&S’s results provide a means to estimate the surface temperature of the LlJ produced in our studies. The 15-lm diameter aqueous LlJs used here comprises 25% ethanol. While ethanol and water are totally miscible, MD simulations have shown that hydrophobic effects cause a slight increase in the concentration of ethanol at the solution surface [86–88,97]. Because of the presence of ethanol, it is ex-

Table 3 Temperatures of the different internal degrees of freedoms of benzene molecules evaporating from an in vacuo aqueous surface. Benzene internal energy mode

Temperature (K)

Rotation

210–230 230 251 256

m16 m11 m6

pected that the surface of the LlJs used here will cool slightly faster than the 21 lm water LlJs used by C&S, but slower than the same diameter methanol LlJs, since ethanol has only half the vapour pressure of methanol at 295 K.2 This leads us to expect a surface temperature range of 270–260 K over the 1–3 mm downstream distance (20–130 ls residence time) from the LlJ nozzle used in our studies. We measure the temperature of the bulk solution just prior to entering the vacuum to be 283 K. This correlates well with C&S’s earliest residence time measurements. The minor cooling of 10 K over this distance reported by C&S is consistent with our observation that the rotational temperatures are insensitive to liquid filament residence time. While Faubel took the 200 K translational temperature that he observed for water molecules evaporating from an in vacuo LlJ to be evidence of a supercooled liquid surface [67], a reinvestigation of this data by Sibold et al. [44] concluded that cooling of the translational motion is due to collisional energy transfer within the expanding vapour. The surface temperature needed to generate such a translational distribution is predicted to be closer to 300 K. If internal degrees of freedom are included in the analysis, Faubel predicts that the surface temperature will be lowered by 5%, dropping Sibold’s predicted surface temperature to 285 K. This value is consistent with Saykally’s results. The rotational temperatures observed here are consistently 50 K cooler than the predicted surface temperature. We attribute this to collisional cooling within the interface, analogous to the situation found by Sibold et al. [44] to explain Faubel’s [67] observations. It is well established by studies of gas-phase expansions [100,101] and supported by numerical investigations of evaporation [41] that gas-phase collisions cause the rotational temperature to rapidly converge toward the translational temperature. This efficient energy transfer is primarily due to the small energy gap between rotational levels. When molecules evaporate from the surface of an in vacuo cylindrical liquid surface they experience a molecular density profile in the vapour phase that is inversely proportional to the distance from the liquid filament [44,45]. As a result of the high molecular density within the interface, there are many low-velocity, multi-body collisions which are efficient in transferring energy between rotations and translation. Frezzotti’s numerical simulations of a rigid rotor evaporating into the gas phase show that rotations and translations rapidly converge over as few as 20 mean free paths [41]. It is thus not surprising that the benzene rotational temperature is similar to the translational temperatures reported by Faubel [67]. While it is clear that collisions are occurring during transport through the interface between the surface and vacuum, it is far from clear how many collisions are occurring, which is strongly related to the interface density profile. Because of the rapid convergence of translational and rotational temperatures, the rotational temperatures cannot provide insight into this. It is in this context that we consider the vibrational temperatures. 3.2. Vibrational temperatures Figure 1A shows the ambient-temperature benzene REMPI spectrum recorded in a gas cell at 295 K while Figure 1B shows the analogous spectrum for benzene molecules evaporating from the surface of an in vacuo LlJ. Each vibronic band is generated by an electronic transition originating from a different vibrational state. The 610 band originates in the ground state, 621 and 601 1120 are sequence bands which both start with one quantum of the in-plane (IP) ring mode m6, and 610 1611 and 610 1111 are sequence 2 At 298 K the relevant vapour pressures are: water (23.54 Torr), methanol (126.092 Torr) and ethanol (58.525 Torr). See: Ambrose et al. [98] and Bridgeman and Aldrich [99].

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bands starting from one quantum of the out-of-plane (OOP) vibrations m16 and m11, respectively. By monitoring the changes in intensities of the sequence bands relative to the 610 band we can follow how the population within the vibrational states is being redistributed with the change in conditions [83]. The ambient temperature spectrum is used as a reference since we can be confident the system is at equilibrium; which means that all the different vibrations will be at the same temperature of 295 K. The relative intensity of each band in this spectrum is thus due to the different Frank Condon (FC) factors for each of the transitions. The temperature, Tj, of the vibration, mi , in the system can thus be determined by the relationship

   Imi ;T j =Im0 ;T j E 1 1 ¼ exp   kB T j T 295 Imi ;T 295 =Im0 ;T 295

ð1Þ

where kB is Boltzmann’s constant, m0 represents the ground state and Imi ;T 295 is the integrated intensity of the band which is generated from a transition out of a state involving mi at room temperature (295 K). The integrated intensities of the sequence bands relative to 610 at 295 K are given in Figure 1A. Although the absorption band at 38 520 cm1 arises from overlap of 621 and 601 1120 , the initial vibrational level prior to laser excitation includes 61 in both cases, so the application of Eq. (1) to the overlapping absorption bands allows us to determine the degree of cooling for the m6 vibrational mode. Such analysis yields a m6 evaporative vibrational temperature of 256 ± 11 K. The uncertainty represents three times the weighted error of the mean, 3r. To lower transition energy, the series of peaks around 38 440 cm1 are assigned to the 610 1611 and 610 1111 vibronic transitions. Analysis of these overlapping bands to determine vibrational temperatures is less straightforward because the transitions arise

5

from molecules that are initially excited with either m11 or m16 in the ground electronic state. In our previous work we were unable to separate their respective contributions [74]. However, we have used two dimensional laser induced fluorescence (2D-LIF) [102] to separate these two bands in a room temperature sample to extract the relevant constants, as will be discussed in a future paper [84]. This information is used to perform a non-linear least squares fit of the intensity of the 610 1111 absorption band under the 610 1611 absorption band. This subsequently allows the determination of the vibrational temperatures of m11 and m16 independently. An analysis of the evaporation spectrum reveals a vibrational temperature of 230 ± 13 K for m16 and 251 ± 15 K for m11. Again, the uncertainty represents three times the weighted error of the mean. All modespecific temperatures of benzene evaporating from the LlJ surface determined in this study are reported in Table 3. Figure 2 shows that, within experimental error, all of the vibrational temperatures are unchanging as a function of both surface residence time and probing distance from the liquid surface (radial distance). As was observed for rotations, the vibrational cooling of benzene molecules evaporating from the LlJ surface is a result of collisional cooling as the molecules traverse the liquid–vacuum interface. However, unlike rotations, the different vibrational motions are cooled to different extents. This is not unexpected since the collisional deactivation rate of vibrational levels can vary significantly [103–109]. Computational studies by Bernshtein and Oref et al. [110] have shown that only the three lowest frequency benzene modes, m16, m6 and m11 have an appreciable vibrational relaxation cross-section. Of these three, m16 is predicted to have a relaxation cross-section an order of magnitude greater than the other two. This arises because m16 is the lowest frequency mode and its motion is OOP. Quantum calculations by Clary et al. [111] predict that the OOP nature of m11 increases its collisional

Figure 1. Benzene 1 + 1 REMPI absorption spectrum produced under various experimental conditions. (A) In an ambient temperature static cell. The integrated intensities of the hot bands relative to 610 are indicated under each band. (B) Evaporation of benzene from an in vacuo LlJ. The ionising laser was positioned 1 mm from the nozzle and 300 lm from the jet surface. (C) Benzene seeded into a free-jet expansion of nitrogen. The expansion conditions have been previously reported [74]. The intensity of all spectra at energies less than 38 535 cm1 have been enhanced ten-fold for clarity.

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Surface Residence time [ μs]

A

0

20

40

60

80

100

120

140

300

Vibrational Temperature [K]

ν 280

ν

260

ν

240 220 200 180 0.0

Vibrational Temperature [K]

B

0.5

1.0 1.5 2.0 2.5 Distance down the beam [mm]

3.0

3.5

pletely, eliminated [44,67,68]. The sharp boundary interface model used by Faubel and others to analyse evaporative translational energy distributions predicts that, at a liquid surface temperature of 295 K, water molecules evaporating from a 7.5 lm radius LlJ, as used in the experiments reported here, will undergo no more than 7 hard sphere gas phase collisions. As we demonstrate below, this small collision number is insufficient, by more than an order of magnitude, to cool m16 to the extent observed. Our data can be used to quantify the number of collisions the evaporating benzene molecules experience as they travel from the liquid surface into the vacuum. This is done by first observing and modelling the vibrational cooling that occurs when benzene is seeded in a free-jet expansion in order to extract the collisional transfer probabilities for the levels observed. The collisional transfer probabilities determined are used to model the vibrational cooling observed in the evaporation experiments and thereby quantify the number of collisions the molecule undergoes as it traverses the liquid interface. 3.4. Collisional energy transfer

300

ν

280

ν1

260

ν

240 220 200 180 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Radial Distance [mm] Figure 2. (A) Vibrational temperatures of benzene as a function of residence time in the vacuum. (B) Vibrational temperatures of the different vibrational modes as a function of distance from the liquid surface. The vibrational temperatures for m6, m16 and m11 are represented by the solid squares (j), open circles (s) and open squares (h), respectively. The error bars represent three times the weighted error of the mean, 3r.

relaxation cross-section so that, in spite of its higher frequency (674 cm1 compared with m6 = 608.3 cm1), its cross section is only slightly smaller than that for m6. On the basis of these calculations, we expect m16 to be cooled the most by collisions during evaporation while m11 would be cooled only slightly less than m6. Our data support this qualitative expectation.

3.3. Determination of collision numbers upon evaporation from an in vacuo liquid surface The important results to emerge from this study are that (i) the rotational and vibrational temperatures of the benzene evaporate are different, with the rotational temperature the same within experimental uncertainty for all vibrations observed and lower than any of the vibrational temperatures measured, and (ii) the vibrational temperatures of m6, m11 and m16 are different. This is only possible if there have been insufficient collisions to establish equilibrium between the rotational and various vibrational degrees of freedom as benzene passes from the liquid into the collision-free region of the vacuum. For sufficiently thin LlJs, gas phase collisions above the liquid surface are expected to be largely, but not com-

Collisional relaxation of electronically excited [103,112–120] and highly vibrationally excited [115,121,122] benzene has been studied in some detail, however studies performed in the low energy levels of the electronic ground state of benzene are scarce [123]. Lyman predicted that ground state vibrational relaxation rates for benzene are less than 1% of the gas-kinetic rate [123] whereas vibrational relaxation rates in the 1B2u excited electronic state of benzene are much larger than this [72]. In contrast, Knight and co-workers’ seminal study of p-difluorobenzene showed there is little difference between deactivation rates in the excited and ground electronic states of this small aromatic molecule [124]. Because the vibrational relaxation rates for the low-lying vibrational levels of benzene are not well established, we here estimate the rates of collision-induced energy transfer out of selected vibrational states in benzene’s electronic ground state. 3.5. Modelling collisional energy transfer rates The first step in extracting collision numbers from our evaporation data involved undertaking experiments to measure the rotational and vibrational cooling that occurs when benzene is seeded in a free-jet expansion in order to model the observed temperatures and extract collisional transfer probabilities. Initially, we attempted to replicate the collision partners of the LlJ evaporation experiments by expanding a room temperature benzene/water/ ethanol gas mix through the 15 lm LlJ nozzle into the vacuum to create a free-jet expansion [125,126]. However, the room-temperature vapour pressures of water and ethanol were insufficient to effect collisional cooling in the benzene and so N2 (15 kPa) was added as a carrier gas. Experiments with and without the water and ethanol present showed that within experimental error no differences in the benzene spectral profiles were observed. This indicates that water and ethanol do not have unusually large collisional relaxation efficiencies with benzene. We use the collisional relaxation efficiencies of nitrogen, determined as described below, to represent a lower limit to the efficiencies expected for a water– ethanol system. The relaxation efficiencies for N2 were determined from the 1 + 1 REMPI spectrum of benzene vapour cooled in a nitrogen (10 kPa benzene in 350 kPa N2) free-jet expansion. The spectrum is shown in Figure 1C. Consistent with previously reported trends [70,127–129], the temperatures 1 mm downstream from the nozzle orifice were determined to be 290:270:208:20 K for Tvib(m6):Tvib(m11):Tvib(m16):Trot. The rotational temperature quoted is that of the ground vibrational state. The rotational temperatures of

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3.6. Temperature gap model of collisional relaxation rates The TG model predicts that the rate of collisional energy transfer is linearly proportional to the extent a particular mode deviates from its equilibrium value. Adopting a similar approach, but expressing the energy transfer rate in terms of hard-sphere collision number Z rather than time, the TG model can be expressed as:

dT i ðZÞ ¼ ki1 ½T trans ðZÞ  T i ðZÞ dZ

ð2Þ

where ki1 represents the collisional energy transfer efficiency of the particular mode, i, into the surrounding levels, Ti(Z) is the temperature of the rotation or vibrational mode under investigation and Ttrans(Z) is the translational temperature of the colliding gas (as determined from the free-jet expansion modelling). The latter two parameters are expressed as a function of collision number. The mode-specific collisional energy transfer efficiencies (i.e., ki1) of benzene in nitrogen determined here are given in Table 4. The change in internal temperatures as a function of binary collision number that emerges from this modelling is illustrated in Figure 3. Rotational relaxation is found to occur at almost the hard-sphere collision rate; it is 200 times more efficient than relaxation of m16, 1000 times more efficient than m11 and 10,000 times more efficient than relaxation of m6. m16 relaxation is 50 times more efficient than relaxation of m6 and 5 times more efficient than m11.

Table 4 Mode-specific collisional energy transfer efficiencies (ki1) of benzene in nitrogen. Benzene internal energy mode

ki1 (collision1)

Rotation

0.9 4.3  103 9.0  105 8.1  104

m16 m6 m11

where, again, Z is the number of hard-sphere collisions, Ni is the population and gi the degeneracy of the state i, and kij is the rate constant describing the rate of population transfer between the states. The first term in Eq. (3) represents a gain in population for state wi by relaxation of the higher-energy wj state while the second term represents a loss in population for wi as its population is promoted to the wj state. Endoergic (up) transitions are related to exoergic (down) transitions by microscopic reversibility [133]. The modelling must potentially consider the possibility that a state may exchange its population with all neighbouring states, which can quickly involve many states. Levels above the highest state of interest, 111, must be included to account for population growth from these levels and endoergic transfer from the states of interest. Flynn notes that the number of rate constants needed to describe a system of n levels scales as nðn1Þ [106]. For example, 2 in the S0 electronic state of benzene there are nine vibrational states, including the ground state, with energy under 1300 cm1, yielding a total of 36 rate constants needed to describe the collisional energy rearrangement within this region. During the modelling, states above this energy were found to play an insignificant role in the collisional energy transfer. Stephenson and Rice [134] showed that in actuality only a small subset of the energeticallyaccessible levels are involved, though this varies with the vibrational level excited. In order to limit the number of rate constants in the analysis, several simplifications were made based on the

300

ν6 ν16

250

ν11

Temperature [K]

the hot and sequence bands were found to be the same within experimental uncertainty. It can be seen that m6 and m11 have undergone little cooling, m16 is moderately cooled, and the rotations are considerably cooled. Using established methods for characterising the physical properties of 3D free-jet expansions as a function of distance downstream from the nozzle orifice [100], we can quantify properties such as the total hard sphere collision number and translational temperature under our experimental conditions. Such modelling predicts that at a distance of 1 mm from the nozzle orifice 360 binary hard-sphere collisions have occurred, producing a translational temperature of 4 K.3 Thus, even after 360 gas-phase collisions, m6 and m11 are little cooled, m16 has undergone moderate cooling, while the rotations are substantially cooled, almost attaining the expected translational temperature of the expansion [100,130]. Two different kinetic models have been employed to describe the collisional relaxation observed in the free jet expansion experiments, and then used to model the evaporation experiments. The first is the simple, yet surprisingly accurate, temperature gap (TG) model that has been used by several groups [100,129,131,132] to model rotational and vibrational collisional energy transfer. The second is a Master Equation (ME) approach.

Rotation

200

Translation

150 100 50

3.7. Master equation model of collisional relaxation rates The ME is a more complex kinetic model involving a series of linearly coupled differential equations to describe the collision-induced flow of population amongst the selected vibrational states. Assuming a first order decay of population, the general rate equation which describes the exchange of population between lower and upper states wi and wj, respectively, as a function of collision number can be described as:

X d½Ni  X DEij ¼ kij g i ½Nj   kij g i ½N i  exp½  dZ kB T j;j–i j;j–i

ð3Þ

3 The parameters used in the 3D jet simulation are taken from Miller [101]; N2 Hard-sphere cross-section (r) 3.85 Å; Heat Capacity ratio (c) = 1.4.

0 0

50

100

150 200 250 No. of Collisions

300

350

Figure 3. Illustration of the collisional cooling experienced in the nitrogen free-jet expansion as a function of binary collision number. The vibrational temperatures for m6, m16 and m11 are represented by the solid squares (j), open circles (s) and open squares (h), respectively. The rotational and translational temperatures are represented by the overlapping solid grey and dashed black line, respectively. The translational temperature of benzene is determined by the expansion properties of the 3D jet, including the orifice diameter and the heat capacity of the carrier gas [101]. Under the experimental conditions reported here, the translational temperature asymptotes to a maximum of 360 binary collisions. The collisional cooling efficiencies of the other modes were varied until their temperature at 360 collisions was equal to that observed experimentally. The collisional cooling efficiencies required to achieve these temperatures are reported in Table 4.

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observed propensities [107,113,114,135]:

for

collisional

energy

transfer

(1) Allowed transitions involve no more than a two quanta change in vibration. (2) Only transitions that have a starting level with appreciable population at room temperature are included. (3) Transitions that involve the same quanta change of a particular vibration are described by the same rate constant. For example, the rate constants [61161 ? 161] and [61 ? 00] are set to be equal since they both describe a single quantum change in m6.

determined by the ME model that produce the best fit to the free jet temperatures are presented in Table 6. The results of the ME model demonstrate that the rate of individual collision-induced transitions may have little bearing upon the overall transfer of population out of the particular state, as determined by the TG model. For example, the transition 41 M 61 has a kij larger than the overall transfer rate out of 61 (Table 4). However, because the two states are close to isoenergetic, energy is taken out of the state as quickly as it is put in, and as a result, the 41 M 61 transition has no effective influence upon the rate at which m6 is seen to collisionally cool. 3.8. Estimate of collisional numbers upon evaporation

With these simplifications, the states included in our modelling of the collisional energy transfer involving 61, 161 and 111 are listed in Table 5. The population transfer pathways are illustrated in Figure 4. Application of the simplifications restricts the model to 17 transitions with 11 independent rate constants. By monitoring the change in state population with collision number we are able to determine the kij values needed to obtain the observed vibrational temperatures in the free-jet expansion. All rate constants were determined relative to the 61 M 00 collisional energy transfer efficiency. The rate constants

The collisional relaxation efficiencies determined from the modelling of the free jet temperatures can now be used to model the collisional energy transfer associated with evaporation of benzene from a water–ethanol LlJ. This, in turn, allows for an estimate of the number of collisions that are undergone as benzene crosses the liquid-vacuum interface. Two constraints are imposed in the analysis. First, since 360 hard-sphere collisions result in negligible cooling of m6 in the free jet, the evaporative m6 vibrational temperature of 256 K is taken to indicate the liquid filament surface

Table 5 Summary of possible levels to be included in the Master Equation (ME) model for determining relaxation rates in S0 benzene.a Initial

⁄ ⁄ ⁄ ⁄ ⁄

⁄ ⁄ ⁄ ⁄ ⁄

⁄ ⁄ ⁄ ⁄ ⁄ ⁄

⁄

Final b

State

Frequency cm

161 61 111 41 162 61161 62 61111 61 111 41 162 61161 62 61111 111 41 162 61161 62 61111 41 162 61161 62 61111 162 61161 62 61111 61161 62 61111 62 61111 61111

399 608 674 707 798 1007 1216 1282 608 674 707 798 1007 1216 1282 674 707 798 1007 1216 1282 707 798 1007 1216 1282 798 1007 1216 1282 1007 1216 1282 1216 1282 1282

1

gi 2 2 1 1 3 3 3 2 2 1 1 3 3 3 2 1 1 3 3 3 2 1 3 3 3 2 3 3 3 2 3 3 2 3 2 2

c

Relative population 0.29 0.11 0.04 0.03 0.07 0.02 0.01 0.00

d

State

Frequency

00 00 00 00 00 00 00 00 161 161 161 161 161 161 161 61 61 61 61 61 61 111 111 111 111 111 41 41 41 41 162 162 162 61161 61161 62

0 0 0 0 0 0 0 0 399 399 399 399 399 399 399 608 608 608 608 608 608 674 674 674 674 674 707 707 707 707 798 798 798 1007 1007 1216

b

1

cm

Transition energy defect

Quanta changee Dm

399 608 674 707 798 1007 1216 1282 209 275 308 399 608 817 883 66 99 190 399 608 674 33 124 33 542 608 91 300 509 575 209 418 484 209 575 66

1 1 1 1 2 2 2 2 2 2 2 1 1 3 3 2 2 3 1 1 1 2 3 3 3 1 3 3 3 3 2 4 4 2 2 2

gi 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3

D 1 O

D 1

D 1 O

1

a Rows indicated with an asterisk represent the transitions that are included in the model. Of the 36 possible transitions, only 17 are included; the others were discarded upon enforcement of the simplifications stated in the text. b Fundamental frequencies are taken from Atkinson and Parmenter [76], whilst overtone and combination frequencies are estimated by the summation of fundamental frequencies. c Degeneracies correlate will those quoted in Parmenter [114]. d Populations are relative to the vibrationless ground state at 295 K. e D, O, 1: these transitions are given equivalent rate constants, (kij), as they involve the same number of quanta change of a particular vibration. This leaves 11 independent rate constants to be calculated using the series of differential equations.

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1400

270 61 111

260

62

1200

T(ν6)

61 161

1000

162

T(ν11)

240 230 220

-1

Evib [cm ]

800

Temperature (K)

250

T(ν16)

41 111

600

61

210

Trot 200 400

161

0

50

100 150 200 No. of Collisions

250

300

Figure 5. Modelling the change in temperature of the different degrees of freedom of benzene as a function of binary, hard-sphere collision number. The results of the TG model are indicated using the solid lines whilst the ME results are indicated with the dashed lines. The vibrational temperatures for m6, m16 and m11 are represented by the solid squares (j), open circles (s) and open squares (h), respectively.

200

00

0

Figure 4. The 17 transitions included in the Master Equation kinetic model.

Table 6 Collision rate constants obtained using the Master Equation model. Transition

Energy defecta (cm1)

Multiple of 61 M 00 rateb

161 M 00, 162 M 161, 61161 M 61 61 M 00, 61161 M 161, 61111 M 111, 62 M 61 111 M 00, 61111 M 61 41 M 00 162 M 00 61 M 161 111 M 161 41 M 161 111 M 61 41 M 61 41 M 111

399 608

60 1

674 707 798 209 275 308 66 99 33

2 2  105 10 3  105 2  106 1  105 20 30 3

a Energy defects calculated from the frequencies reported by Atkinson and Parmenter [76]. b The kij value for the 61 M 00 transition was calculated to be 9  105 using the Temperature Gap model [74].

temperature. It will be recalled from our earlier discussion that on the basis of Cohen and Saykally’s work we expect a surface temperature in the range 260–270 K, and this is close to that expectation. Choosing a higher value for the surface temperature will lead to more collisions being required for vibrational cooling than are calculated using this value and hence the collision numbers we extract indicate the lower limits. Second, because the free-jet expansion results demonstrate that the rotational temperature of benzene closely follows the translational temperature, the evaporative rotational temperature of 206 K is taken to indicate the translational temperature of the benzene evaporate. This is the temperature towards which the collisions are cooling the rotational and vibrational distributions. This assumption is consistent with predictions arising from recent MD simulations which indicate that the translational and rotational temperatures of the evaporate rapidly converge [41]. Furthermore, MD simulations of the evaporation of water clusters into vacuum demonstrate that the

translational temperature of the evaporate tends toward 215– 220 K, close to the value suggested here, independent of the starting temperature [136]. These constraints mean that the number of collisions involved in evaporation from the LlJ is determined by matching the m16 and m11 temperatures calculated with those observed. The modespecific temperatures as a function of collision number determined using both models are presented in Figure 5. The collisional energy transfer efficiencies used are reported in Table 6. Not unexpectedly, the results of the two models differ slightly. Both models show the temperature of m6 to be almost independent of collision number, dropping by only a few degrees, as expected. The TG model, which is the only one of the two used to explore the issue of rotational relaxation, shows that the rotational temperature equilibrates to the translational temperature within a few collisions. Both models indicate that the vibrational temperatures continue to fall as collisions attempt to equilibrate vibrations with the translational and rotational temperature. The TG model finds m16 and m11 reach their observed temperatures after 185 and 220 binary, hard-sphere collisions, respectively. Similarly the ME model predicts the respective evaporation temperatures to be reached after 150 and 260 collisions. While the details are different, both models indicate that ca. 200 hard sphere collisions are required to explain the vibrational temperatures observed. This estimated collision number is sufficient to result in some cooling of the evaporate as it leaves the liquid surface, but is insufficient to equilibrate the various molecular degrees of freedom. The collision number required to explain the experimental observations is significantly larger than that proposed in earlier studies where a sharp boundary between the liquid and vapour phases was assumed [44,67,68,73,92,93,137].

4. Conclusions Having estimated the liquid surface temperature as 260 K, our experiments indicate that the benzene rotational temperature is cooled by 55 K during evaporation. Rotational cooling of water and methanol has been predicted in recent MD simulations of the evaporation process [39,41]. The extent of cooling predicted by these simulations is not as extensive as observed in the

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experimental studies reported here. Rotational cooling is generally efficient for polyatomics [138] and so the difference between our observations and the simulations are not likely to be a consequence of our use of benzene as a spy for behaviour at the water–ethanol liquid surface. This suggests that current MD simulations underestimate the extent of rotational cooling following evaporation. There are no simulation data yet available to compare to the observed vibrational cooling of 30 and 10 K for the m16 and m11 vibrational modes of benzene, respectively. Several simplifications and extrapolations were made in order to determine the collisional energy transfer rates report herein. These were calculated based upon the observed internal temperatures of benzene molecules seeded in a free-jet expansion. The collision numbers calculated are based on hard-sphere binary collisions; the collisional energy transfer rates in such an environment may differ from those experienced within the liquid–vapour interface. For example, through molecular simulation of the vapour–liquid equilibrium, Wang [139] shows that, although binary interactions are dominant, three-bodied interactions must be considered especially when modelling fluid mixtures. References [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]

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Olivia Maselli received her PhD in Physical Chemistry in 2010 from The University of Adelaide, Australia where she performed research into the dynamics and energetics of evaporating small molecule organics using molecular spectroscopy techniques. In 2011 Olivia joined the Desert Research Institute in Reno, NV as a Postdoctoral Research Fellow. Her current research involves the chemical characterisation of deep, Arctic and Antarctic ice cores. The focus on this research is to understand the role of anthropogenic pollutants on the past global climate.

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Jason Gascooke received his PhD from The Flinders University of South Australia in 2000 for work involving construction of an ion-imaging apparatus to measure energy releases during van der Waals molecule dissociation. This was followed by postdoctoral research at University of California, Berkeley for 3 years and then at The University of Adelaide. In 2008, Jason joined the Chemistry Department at Flinders University where his main interests are in experimental molecular dynamics, particularly that of evaporation and van der Waals molecule dissociation. He is also interested in developing the technique of 2-dimensional laser induced fluorescence as a spectroscopic tool.

Warren Lawrance undertook his PhD with Alan Knight at Griffith University, Brisbane, graduating in 1984. Following this he was a Miller Fellow with C. Bradley Moore at the University of California, Berkeley. He subsequently obtained a faculty position at Flinders University, Adelaide, where he has been since late 1985. He is an experimentalist whose research interests have largely focused on intra- and inter-molecular energy transfer in aromatics. Most recently this has involved studies of van der Waals molecule dissociation using velocity map imaging and two dimensional laser induced fluorescence.

Mark Buntine undertook undergraduate studies at Monash University in Melbourne, Australia, and his PhD with Richard Zare at Stanford University, graduating in 1992. He undertook postdoctoral studies with Mark Johnson at Yale University, before commencing an independent academic career at The University of Adelaide in 1994. He moved through the ranks to become a full professor in 2007. In 2009 Buntine moved to Curtin University in Perth where he has maintained an active research program focussed on the use of liquid beam methodologies to explore the molecular dynamics of liquid-vapor phase transitions via electronic spectroscopy.