Determination of the Evaporation Coefficient of D 2 O

: The evaporation rate of D 2 O has been determined by Raman thermometry of a droplet train (12-15 µ m diameter) injected into vacuum (~10 -5 torr). The cooling rate measured as a function of time in vacuum was fit to a model that accounts for temperature gradients between the surface and the core of the droplets, yielding an evaporation coefficient ( γ e ) of 0.57 ± 0.06. This is nearly identical to that found for H 2 O (0.62 ± 0.09) using the same experimental method and model, and indicates the existence of a kinetic barrier to evaporation. The application of a recently developed transition-state theory (TST) model suggests that the kinetic barrier is due to librational and hindered translational motions at the liquid surface, and that the lack of an isotope effect is due to competing energetic and entropic factors. The implications of these results for cloud and aerosol particles in the atmosphere are discussed.


Abstract:
The evaporation rate of D 2 O has been determined by Raman thermometry of a droplet train (12-15 µm diameter) injected into vacuum (~10 -5 torr). The cooling rate measured as a function of time in vacuum was fit to a model that accounts for temperature gradients between the surface and the core of the droplets, yielding an evaporation coefficient (γ e ) of 0.57 ± 0.06. This is nearly identical to that found for H 2 O (0.62 ± 0.09) using the same experimental method and model, and indicates the existence of a kinetic barrier to evaporation. The application of a recently developed transitionstate theory (TST) model suggests that the kinetic barrier is due to librational and hindered translational motions at the liquid surface, and that the lack of an isotope effect is due to competing energetic and entropic factors. The implications of these results for cloud and aerosol particles in the atmosphere are discussed.

Introduction:
The evaporation and condensation rates of liquid water are of fundamental importance to many chemical, biological, and atmospheric processes. In particular, the formation and growth rates of cloud and aerosol particles are, in principle, sensitive to both kinetic and thermodynamic variables . Clouds and aerosols have a cooling effect on the earth's atmosphere due to scattering of solar radiation, although some aerosols (black carbon) have a warming effect (IPCC, 2007). Current cloud models vary widely in their predictions for the radiative effects of anthropogenic emissions that affect the number and composition of particles on which cloud droplets condense (CCN) (IPCC, 2007;Laaksonen et al., 2005;McComiskey and Feingold, 2008;Lohmann et al., 2007). This variation is in part due to differing values for water evaporation and condensation kinetics and their relation to particle growth rates in these models (Laaksonen et al., 2005). Direct measurements of the microscopic rates of evaporation and condensation of pure water vary over three orders of magnitude, although recent measurements have narrowed the range to between 0.05 -1 times the gas kinetic limit (Eames et al., 1997;Marek and Straub, 2001;Davidovits et al., 2006). Some of the variation in older literature is likely due to impurities in or on the surface of the water samples used in the experiments; we note this fact hints that impurities will be important determinants of evaporation and condensation rates in mixed systems, a notion supported by field measurements of droplet growth rates (Feingold and Chuang, 2002;Ruehl et al., 2008). It is generally accepted that condensation and evaporation occurring faster than 10% of the gas kinetic limit results in thermodynamic control over droplet growth while slower rates result in kinetic control over these growth rates (Chuang et al., 1997;Laaksonen et al., 2005).
where p sat is the saturation vapor pressure. Since the activity of the pure liquid is unity, this expression for the evaporation rate holds at all vapor pressures. However, not all substances evaporate at the maximum rate (McFeely and Somorjai, 1972;Eames et al., 1997). Deviations from the maximum rate are treated by introducing the evaporation coefficient (γ e ) and the condensation coefficient, alternatively referred to as the mass accommodation coefficient (α m ):

= =
(4) The evaporation and mass accommodation coefficients, which have values between zero and one, must be equal due to detailed balance at equilibrium. The equality holds for non-equilibrium systems as long as the velocity distribution in the gas phase does not deviate significantly from a Boltzmann distribution, because a Boltzmann distribution is assumed in the derivation of Eq. (1). Indeed, theoretical treatments of condensation have shown little sensitivity to gas phase speeds selected from the Boltzmann distribution at room temperature, although the condensation probability lowers for very high speeds (~1000 m/s) (Chakraborty and Zachariah, 2008).
In addition to insuring the absence of impurities, measuring γ e for H 2 O is challenging because the high vapor pressure makes it difficult to observe evaporation or condensation in isolation without significant contributions from the opposing term. In addition, knowledge of the liquid surface temperature is required, and evaporation results in cooling of the surface by as much as 3-4 K relative to the bulk (Ward and Stanga, 2001).
Our own experiments have made use of liquid jets and droplet streams with high vacuum (10 -4 torr) maintained around the fluid, such that evaporation occurs with negligible accompanying condensation. This greatly simplifies the interpretation as compared with many other recent experiments. The liquid jets and droplets also provide a renewing surface, minimizing contamination issues. Measurements of isotopic ratios in evaporation between 264 and 295 K showed that γ e < 1 and that it varied with the H/D ratio in the liquid (Cappa et al., 2005). Using Raman thermometry we derived a precise value of γ e from the temperature change associated evaporation of pure H 2 O, yielding a value of 0.62 ± 0.09 over a temperature range of 245 -295 K (Smith et al., 2006). We interpreted the results of our prior experiments using a transition-state theory (TST) model of liquid water evaporation (Cappa et al., 2007). The calculations indicated that the evaporation rate is primarily influenced by the intermolecular hindered translational and librational motions of molecules at the liquid surface.
Here we describe the extension of our previous studies to droplet train measurements of γ e of pure D 2 O. These measurements serve as a further test of the microscopic theory of evaporation and of the reproducibility and precision of the methodology, providing a firm basis for future studies of the evaporation from mixtures of water with salts, oils or surfactants.

Method:
Evaporation rates from liquid D 2 O in vacuum were determined by measuring the temperature change of evaporating droplets using Raman thermometry. The evaporation rate is deduced from the cooling rate and the well known heat of vaporization. The droplets were formed with a vibrating orifice aerosol generator (VOAG). The D 2 O used in this study was obtained from Cambridge Isotope Laboratories, with a stated purity of 99.9%.
The Raman spectroscopy apparatus has been described in detail previously (Smith et al., 2006) and a schematic is given in Fig. 1. Briefly, a syringe pump (Teledyne ISCO Model 260D) is used to force the liquid through a fused silica orifice (2.5 -4 µm radius) mounted on a piezoelectric ceramic. The silica orifice is generated by pulling 100 µm ID silica tubing to the desired size with a commercial CO 2 laser micropipette puller. The piezo is driven with a 0-20 V square wave at 200 -1000 kHz to generate a uniform droplet train with a spread in radius of less than 0.1 µm (Sayer et al., 2003). The radii of the droplets produced is calculated from the liquid flowrate and the oscillation frequency (Smith et al., 2006).
The VOAG is mounted on a bellows attached to the top of a 7 cm cubical vacuum chamber pumped by a 110 liter/second turbomolecular pump. The VOAG as attached to an XYZ manipulator to allow positioning of the droplet stream. Pressures in the chamber during experiments were lower than 5 × 10 -4 torr. At these pressures, heat transfer from the walls of the chamber to the droplets is negligible. Viewports on the chamber allow the introduction of the 514.5 nm line from an argon ion laser operating at ~250 mW or less, which is focused onto the droplet train. The laser power is sufficiently low to avoid heating the droplets or otherwise affecting their evaporation (Sayer et al., 2003;Smith and Baker, 1981). Raman scatter from the droplets is filtered and collected at 90 degrees through a fiber-optic cable and routed to a spectrometer with a liquid nitrogen cooled CCD camera. After the droplets leave the interaction volume, they enter a liquid nitrogen trap located ~ 50 cm from the nozzle. To ensure that the droplets are uniform in size, a photodiode is placed in the path of the laser, after it has crossed the droplet train. The photodiode signal and the modulation frequency are monitored with an oscilloscope. As a droplet passes through the laser beam, there is a dip in signal on the photodiode; the oscillation frequency is tuned until the signal is sinusoidal in nature, indicating the formation of uniform droplets as described above.
The OD-stretching region of the Raman spectrum (2150-2800 cm -1 ) is used to determine the temperature of the droplets in a manner similar to that reported by Smith et al. (2006). Calibration curves were collected using both the thermostated nozzle technique described by Smith et al., and by measuring the total Raman scatter from liquid D 2 O in a cuvette over a similar temperature range (0-50 °C). The cuvette method was found to be more consistent (< 2% deviation) compared to the jet method (~ 5% deviation) and has the additional advantage of requiring significantly smaller amounts of liquid. Calibrations taken using the cuvette method were used for the bulk of the data.
Examples of the spectra used to generate the temperature calibrations are shown in Fig.   2a, and Fig. 2b shows one of the calibration curves.
Measurements were taken as a function of distance from the VOAG nozzle, which we converted to the residence time in the vacuum using the velocity of the droplet train. This velocity is calculated from the liquid flowrate and the orifice size. As described previously, the orifice size is determined by measuring the liquid jet diameter immediately after the nozzle using Mie scattering with the VOAG turned off (Cappa et al., 2005). The initial temperature of the droplets was determined by collecting the Raman spectrum of the droplet train in ambient air, where evaporative cooling is minimal (Smith et al., 2006).

Results and Analysis:
Measurements for eight different droplet sizes with radii in the range between 5.3 µm and 8.1 µm were performed. Data were collected as a function of residence time in the vacuum chamber for maximum times ranging from 435 µs to 1117 µs, with most around 600 µs. This corresponds to a temperature range from 295 K to as low as 255 K.
That our measurements are taken in the free evaporation regime was verified by calculating the average number of collisions experienced by an evaporating molecule as it leaves the droplet surface and reaches an infinite distance: where r 0 is the droplet radius, is the mean free path of the vapor, and coll d is the collision diameter (2.6 × 10 -10 m). For D 2 O, the vapor pressure predicted at a distance of 1 mm from the nozzle, where the first data point is taken, is ~7.5 torr. This corresponds to a mean free path of ~ 12 µm; thus molecules evaporating from droplets with radii less than 12 µm experience less than one collision in the vapor phase on average and condensation may be neglected (Smith et al., 2006).
We model the observed cooling numerically in the same manner as in our study of H 2 O (Smith et al., 2006). We divide the droplet into concentric spherical shells, and considering evaporation from the outermost shell. Using Eq. (3), the cooling rate of the outermost shell is expressed as where A is the surface area of the outermost shell (= 2 0 4 r π ), p sat is the satsuration vapor pressure, m is the molecular mass, k is Boltzmann's constant, T is the temperature of the outermost shell, where r 0 and r 1 are the outer and inner radii of the outermost shell of the droplet. The only tunable parameter is γ e . Thermal diffusion between adjacent shells is modeled as where κ is the thermal conductivity (0.595 W/m*K at 298 K for D 2 O), A is the surface area of the shell and dT/dr is the temperature difference between the two adjacent shells; we do not assume instantaneous thermal equilibrium. As the outermost shell evaporates, mass loss due to evaporation is taken into account and the droplet and all the shells are re-sized accordingly at each time step (10 -10 s). The droplet radius typically decreases by ~5% over the duration of a measurement. The temperature gradient and the volumeaveraged temperature of the entire droplet are calculated at each time step. As the entire droplet resides within the laser focal volume, we interpret the observed temperature as the volume-averaged temperature. We determine γ e by fitting the observations to the model represented by Eq. (7). We have considered three variations on the form of γ e , one where γ e is a constant with temperature and two where γ e is allowed to vary with temperature.
An example calculation is shown in Fig. 3. The figure shows the calculated temperatures for each of 20 spherical shells as a function of time, as well as the volumeaveraged temperature for the entire 6.65 µm droplet. The magnitude of the surface-bulk temperature difference in the droplet, defined as the difference between the outer shell temperature and the volume-averaged temperature of the droplet, is also shown. The maximum difference is 3 K, which occurs during the first 100 µs and thereafter drops below 1 K. Tests of the numerical accuracy of the calculations show that 20 shells are sufficient to converge the calculations.
There are a few assumptions made in the modeling procedure, but these have little effect on the model results. First, we assume that there is no re-condensation, only evaporation. The small amount of condensation resulting from evaporating molecules impinging on adjacent droplets in the droplet train. This effect has been quantified previously and leads to an underestimate of γ e of less than 0.01 (Smith et al., 2006).
Second, we assume that the liquid surface in our vacuum system can be directly compared with the liquid surface in an equilibrium system. On sufficiently short timescales, it is possible that the rapid evaporation in the absence of condensation could affect the liquid surface structure and thus the evaporation dynamics. However, if one assumes that γ e = 1 (i.e. the maximum evaporation rate) and considers evaporation from a 1 nm square patch of liquid surface, which is larger than the water-water correlation length (Head-Gordon and Hura, 2002), Eq. (3) yields an evaporation rate of one evaporating molecule every 10 ns. (Note: We presented a similar argument in (Smith et al., 2006) but incorrectly reported a 10nm square instead of a 1nm square.) Molecular simulations suggest that the timescale for reorganization of the liquid water surface is on the order of a few picoseconds (Garrett et al., 2006). Therefore, evaporation events are too rare to perturb the liquid surface structure, and the surface should be the same under vacuum as it is in equilibrium. Lastly, the model treats some parameters as constant, such as density and thermal conductivity, which in fact vary with temperature. For completeness, a temperature-dependent equation for the density of D 2 O from Kell (Kell, 1967) and an empirical temperature-and density-dependent equation for the thermal conductivity of D 2 O from the International Association for the Properties of Water and Steam (Kestin, 2007) were both incorporated into the model. The inclusion of these temperature dependent values changed γ e by less than 0.1%.
We tune γ e in this model to fit observations of a 6.65 µm droplet train, shown in The reported error is primarily experimental and is likely associated with determining the temperature from the Raman spectrum; we are only able to obtain the temperature with a precision of ± 2 K. This limitation arises from the calibration curves used. Other possible sources of experimental error are small; shape oscillations in the droplets produced by the VOAG are expected to decay after a few microseconds, and are therefore negligible on the > 500 µs timescale of the measurements (Weierstall et al., 2008;Devarakonda et al., 1998). Other effects of the droplet production, such as rotation of the droplets, should be minimal and are not expected to affect the evaporation.
The theoretical cooling curve for γ e = 1, also shown in Fig. 4a, yields significantly more cooling for a given interaction time than was observed. For comparison, we have also included the model results using the temperature dependent γ e from Li et al, which increases from 0.17 at 295 K to 0.32 at 258 K (Li et al., 2001). This predicts significantly less cooling than was observed. We have also fit two different temperature dependent functions to this data set, as shown in Fig. 4b. These functions represent the maximum positive and negative temperature dependences that are still consistent with the observations. First, we use the functional form of the temperature dependence described based on calculations of the absolute evaporation rates of the different isotopes as a function of deuterium mole fraction in the liquid (Fig. 5 of that study), although the relative evaporation rates of the pure liquids was not explicitly calculated (Cappa et al., 2007 where J e,max is the maximum evaporation rate from Eq. (2), * Q and s Q are the partition functions of the transition state and the liquid surface species, respectively, and a E ∆ is the difference in activation energies between the two isotopes (Cappa et al., 2007). The suband superscripts H and D refer to H 2 O and D 2 O, respectively. By calculating the ratio r γ we avoid several assumptions associated with calculating the absolute evaporation rates of the isotopes, such as knowledge of the transition state area and the absolute magnitude of the activation energy (Cappa et al., 2007).
The observed ratio from experiment, r γ ,exp , is 1.09 ± 0.18. The calculated ratio r γ from Eq. (9) was found to vary between 0.90 and 1.08 at 295 K, depending on the specific choices of * Q , s Q , and a E ∆ given in Cappa et al. (Cappa et al., 2007). The smallest isotope effect (r γ = 1.02) was found using the values derived from the "scaled" surface frequencies. These intermolecular translational and librational frequencies of the liquid surface species were scaled by a factor of 4 / 3 relative to the bulk frequencies to better approximate the looser binding at the surface (Smith et al., 2003).  (Ward and Stanga, 2001). Assuming that the higher temperature in the vapor is due to an energetic barrier to evaporation, and using the specific heat capacity of the vapor (C p = 37.47 J mol -1 K -1 ), the barrier size is predicted to be very small (~300 J/mol) for H 2 O. Our data provide a weak constraint on the size of the barrier. The analysis above suggested the energetic barrier is less than 1.8 kJ/mol for D 2 O.
The apparent lack of a temperature dependence to γ e observed in this study and that of Smith et al. (Smith et al., 2006) may appear to be in contrast to an earlier study by Cappa et al. (Cappa et al., 2005) wherein a stronger temperature dependence to γ e was suggested. As noted in that work, and discussed in a later publication (Cappa et al., 2007), the prediction involved several assumptions and high uncertainty. However, the relative evaporation rates of the isotopic species in the mixtures measured in that study are expected to display a temperature dependence.
Our measurements of γ e for D 2 O and H 2 O can be compared with other recent measurements. Our value of γ e falls within the range of, but has much higher precision than, recent measurements by Winkler et al. Winkler et al., 2006) and Voigtlander et al (Voigtlander et al., 2007), both of which were condensation studies. for uptake on NaCl particles. Li et al. determined that γ e has an inverse temperature dependence, with γ e increasing from 0.17 ± 0.03 at 285 K to 0.32 ± 0.04 at 258 K (Li et al., 2001). Similarly, a group from the Polish Academy of Sciences (Jakubczyk et al., 2007;Zientara et al., 2008) observed γ e to increase from 0.13 at 293.1 K to 0.18 at 273.1 K. It is important to note that in all of these studies, the γ e values determined are higher than the 0.1 threshold below which cloud formation becomes kinetically controlled.
While attempts have been made to reconcile some of these experiments (Davidovits et al., 2004) there is not yet a satisfactory explanation for the observed differences. A recent study by Fukuta and Myers (Fukuta and Myers, 2007) highlights the "moving boundary effect" which can occur during modeling of evaporation or condensation. They report that incorrectly accounting for the shifting liquid-vapor boundary as a droplet grows or shrinks can lead to errors in the calculated evaporation or condensation coefficients as large as several percent. In our case, the droplet radius shrinks by ~5% over the duration of a measurement, but the moving boundary effect is explicitly accounted for within the model by resizing the droplet at every timestep. Zientara et al.
argue that even larger corrections may be necessary in many cases due to thermal effusion near the droplet surface (Zientara et al., 2008). Many models of condensation and evaporation treat the vapor using the framework of diffusion, but Zientara et al. argue that at distances below the mean free path of the vapor, a droplet must be modeled as evaporating and condensing through vacuum. In certain cases, while the droplet would be considered to be quasi-stationary in the diffusion framework, thermal effusion near the surface can significantly change the temperature of the surface of the droplet from what would be predicted by diffusion, dependent on the different timescales of various experiments (Zientara et al., 2008). In our experiments, the vapor is negligible and our model explicitly accounts for the cooling of the surface of the droplets. Therefore such a temperature jump cannot explain the difference between our measured values and those of Li et al. and Zientara et al. In a previous work (Cappa et al., 2005) we discussed a possible source of discrepancy between our experiments and those of Li et al., claiming that our formulations of γ e and α m are different, with γ e ~ (1 -α m ). We now recognize that this prior analysis was incorrect and resulted from equating two different rates in formulations of the evaporation and mass accommodation processes which are not equivalent.
Reanalysis of the different formulations of γ e and α m indicates that the formulations are equivalent, and consistent with the definitions in Eqs. (3) and (4) (MacMullin, 1969;Hornung and Giauque, 1955;Horita et al., 1992;Shmulovich et al., 1999), but the kinetic effects on the evaporation coefficient are unknown. It has been argued that surface active solutes can lower the evaporation rate considerably, possibly leading to large changes in γ e (Marek and Straub, 2001). Many studies of the effect of surfactants on water evaporation indicate a dramatic lowering of the evaporation rate upon sufficient surface coverage by surfactant molecules (Lunkenheimer and Zembala, 1997;Rusdi and Moroi, 2004;Seaver et al., 1992). Additionally, recent molecular dynamics (MD) simulations of condensation onto an aerosol particle coated with organics show a significantly lowered condensation probability (Chakraborty and Zachariah, 2008). Given that CCN are often composed of inorganic solutes as well as of organics which might partition to the surface, measurements of γ e on these more complex mixtures will be important to improving microscopic models of cloud droplet growth rates, number and size. Indeed, a recent study sampling atmospheric aerosol from several different sites found that cloud droplet growth rates were often consistent with values of γ e less than 0.1 (Ruehl et al., 2008), providing additional motivation for continued development of methods capable of precise determination of γ e .

Conclusions:
We have determined the evaporation coefficient of D 2 O to be 0.57 ± 0.06 using Raman thermometry measurements on droplets undergoing evaporation in the absence of condensation. This value is the same, within experimental error, as that previously measured for H 2 O (Smith et al., 2006). Thus, γ e for pure water is less than unity, but is not small enough to have a significant impact on models for cloud formation and aerosol growth rates. A TST model for water evaporation (Cappa et al., 2007) is consistent with both the H 2 O and D 2 O observations and indicates that the lack of an isotope effect is due to competing energetic and entropic effects. Further understanding of the source and magnitude of these effects and how they are affected by the presence of salts, oils and surfactants will likely be important to understanding evaporation and condensation in mixed phase systems.   shell. Absolute temperature is on the left axis and the magnitude of the temperature difference is on the right axis. The inset is an enlarged image of the first 50 µs to depict the shell temperatures more clearly. The volume-averaged temperature (black line) is the output that is fit to the experimental data. Note that the thermal gradient quickly drops below 1 K. Both temperature-dependent fits were tuned to yield the strongest temperature dependence that agrees with the observations within experimental error.