Freezing of Heavy Water (D2O) Nanodroplets

Article pubs.acs.org/JPCA

Freezing of Heavy Water (D2O) Nanodroplets Ashutosh Bhabhe,† Harshad Pathak,† and Barbara E. Wyslouzil*,†,‡ †

William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, Ohio 43210, United States ‡ Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: We follow the freezing of heavy water (D2O) nanodroplets formed in a supersonic nozzle apparatus using position resolved pressure trace measurements, Fourier transform infrared spectroscopy, and small-angle X-ray scattering. For these 3−9 nm radii droplets, freezing starts between 223 and 225 K, at volume based ice nucleation rates Jice,V on the order of 1023 cm−3 s−1 or surface based ice nucleation rates Jice,S on the order of 1016 cm−2 s−1. The temperatures corresponding to the onset of D2O ice nucleation are higher than those reported for H2O by Manka et al. [Manka, A.; Pathak, H.; Tanimura, S.; Wölk, J.; Strey, R.; Wyslouzil, B. E. Phys. Chem. Chem. Phys. 2012, 14, 4505]. Although the values of Jice,S scale somewhat better with droplet size than values of Jice,V, the data are not accurate enough to state that nucleation is surface initiated. Finally, using current estimates of the thermophysical properties of D2O and the theoretical framework presented by Murray et al. [Murray, B. J.; Broadley, S. L.; Wilson, T. W.; Bull, S. J.; Wills, R. H.; Christenson, H. K.; Murray, E. J. Phys. Chem. Chem. Phys. 2010, 12, 10380], we find that the theoretical ice nucleation rates are within 3 orders of magnitude of the measured rates over an ∼15 K temperature range.

I. INTRODUCTION The crystallization of ice from the supercooled liquid or solid amorphous phases of water occurs during cloud formation in the upper atmosphere,1−3 in interstellar space,4−8 and in cryopreservation media.9,10 Although H2O crystallization from supercooled liquid and amorphous solids has been extensively investigated experimentally at temperatures above the homogeneous nucleation limit for bulk water, TH ≈ 235 K,11−22 it remains extremely challenging to measure crystallization rates over the entire temperature range of interest and to interpret the results to extract quantitative ice nucleation rates. In particular, in the range of 150 < T/K < 235, crystal nucleation rates are above ∼1010 cm−3 s−1, and even micrometer sized liquid samples and thin amorphous films crystallize extremely rapidly. Very recently, the development of the monatomic water (mW) potential23 has led to a number of insightful studies examining the crystallization of water near and below 235 K, both in bulk samples24−26 and in nanodroplets.27,28 The results suggest that the ice that forms from supercooled water is a mixture of cubic and hexagonal layers that has been referred to as stacking-disorderd ice,29 hybrid ice I,27 or “ice Ic”.30 Even the critical clusters, the first fragments of the emerging phase, are not purely cubic or hexagonal, nor are they necessarily compact structures.25,26 In general, however, the overall picture arising from modeling24−28 supports the data emerging from lab experiments.29,30 © 2013 American Chemical Society

One way to study the liquid water−ice phase transition under more extreme conditions, i.e., at temperatures below TH, is to generate nanometer size droplets by vapor to liquid condensation at temperatures below TH in a supersonic flow. Here, the small volume of each droplet reduces the probability of nucleation while the high flow velocities make it possible to resolve processes on the microsecond time scale. Nucleating droplets at temperatures below TH is an experimental analogue to instantaneously reducing the temperature of a droplet to the desired value in a simulation, in the sense that both strategies avoid ice formation at higher temperatures. We recently used this approach to measure the nucleation kinetics of H2O at temperatures ranging from 202 to 215 K.31 In this paper we investigate ice nucleation in supercooled D2O nanodroplets. We follow the freezing process by combining the results from three separate experimental techniques: pressure trace measurements (PTM), small angle X-ray scattering (SAXS), and Fourier transform infrared spectroscopy (FTIR). These techniques provide complementary information regarding the properties of the flowing gas mixture, the properties of the aerosol and the state of the condensate as a function of position z or, equivalently, time t. We compare the current results to our previous experiments with H2O and to the limited D2O ice nucleation data13,18 Received: January 3, 2013 Revised: May 22, 2013 Published: June 13, 2013 5472

dx.doi.org/10.1021/jp400070v | J. Phys. Chem. A 2013, 117, 5472−5482

The Journal of Physical Chemistry A

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subtracted 1-D spectra are fit assuming that the aerosol follows a Schulz distribution of polydisperse spheres36 to yield the mean droplet radius ⟨r⟩, width of the distribution δ, and the particle number density N. We use the iterative analysis procedure outlined in Manka et al.31 to determine the remaining flow variables, temperature T, density ρ, velocity u, area ratio A/A*, and condensate mass fraction g, in a manner that is consistent with both the PTM and the SAXS data. First, we estimate the area ratio A/A* of the expansion from the dry pressure trace measurement. For the condensing flow experiments we then estimate the other flow variables by solving the diabatic flow equations37 using A/A* from the dry trace and p from the wet trace as input. Next, the values of T and ρ are used to calculate more accurate estimates for g based on analysis of the SAXS measurements (gSAXS). We combine the values of g estimated from PTM, gPTM, for the initial stages of condensation with gSAXS values from intermediate and later stages of condensation and fit these to a sigmoidal equation to yield gfit. We solve the diabatic flow equations again but now use p and gfit as the input variables. The new values of T and ρ are then used to update the values of gSAXS, and the process is repeated until the solution converges to within 0.5% of the results from the previous iteration. Typically, only 2−3 iterations are required. To follow freezing, we measure the position resolved IR aerosol absorption spectra using an IR beam generated by a Perkin-Elmer spectrum 100 FTIR and detected with a liquidN2-cooled MCT detector. The absorption spectra are measured at wave numbers ν between 450 and 4000 cm−1 at a resolution of 4 cm−1. Scattering is negligible in our experiments because the droplet size is ∼2 orders of magnitude smaller than the wavelength of light. At every position we measure the transmitted intensity through the nozzle for carrier gas alone (the “empty nozzle”), Ie(ν), and for the sample (carrier gas + condensable vapor + frozen/unfrozen droplets), Is(v). The number of scans taken, typically 32 or 64, is increased as the volume fraction of the aerosol decreases. The Perkin-Elmer Spectrum software version 6.3.4 is used to obtain the sample absorbance spectrum, −log[Is(υ)/Ie(υ)]. This software employs a built-in routine to suppress interference from atmospheric CO2 and H2O vapor present in the unpurged beam path. The sample spectra are corrected for baseline errors, and normalized by the path length of the beam l = 12.6 mm and the molar concentration of the condensate to yield the molar absorptivity of the aerosol εa (in m2/mol):

available in the literature. Stöckel et al.18 reported that for a fixed nucleation rate D2O nucleates more easily than H2O, i.e., D2O nucleates at a lower degree of supercooling, ΔT = Tm − T, where Tm is the equilibrium melting temperature. This is also the case in our experiments when ΔT is less than ∼60 K. Following the work of Murray et al.,32 we extract an estimate for the interfacial surface energy σsl between the supercooled D2O and D2O ice from the Stöckel et al.18 data at 242 K and estimate the expected D2O ice nucleation rates over the extended temperature range. Although Murray et al.’s theoretical approach is not entirely consistent with the hybrid ice picture arising from simulations, and involves extrapolating the interfacial energies to low temperatures in an ad hoc manner, it provides a convenient framework to qualitatively compare the current D2O ice nucleation rates to our recently published data for H2O. Finally, although scaling suggests that the surface nucleation rates for D2O are more consistent between different experiments than the volume based nucleation rates, uncertainty in the experiments makes it difficult to state which process dominates. Thus, as suggested by Sigbjornsen and Signorell33 we report both volume and surface based ice nucleation rates. The paper is organized as follows. We briefly describe the experimental setup and techniques in section II. We then present our experimental results and compare them to data available in the literature, to the predictions of the classical nucleation theory (CNT) and to the H2O ice freezing data of Manka et al.31 in section III. Finally, in section IV we summarize the work and present our conclusions.

II. EXPERIMENTAL SECTION A. Materials. Nitrogen (N2; Airgas and Praxair) had a minimum purity of 99.99%, and D2O (Cambridge Isotope Laboratories) had at least 99.9% D substitution. The thermophysical properties of N2 and D2O are summarized in the Supporting Information. B. Experimental Setup and Techniques. Experiments were conducted using the supersonic nozzle setup described in detail in Laksmono et al.34 and Manka et al.31 Very briefly, a carrier gas−condensable vapor mixture of known composition flows continuously through a Laval nozzle with flat side walls. The flow is characterized using pressure trace measurements (PTM), the aerosol size distributions are measured using smallangle X-ray scattering (SAXS), and the fraction of condensate in the liquid and crystalline states is determined using FTIR. The nozzle characteristics are summarized in Table 1.

⎛ I (v) ⎞ M D2O εa(v) = −log⎜ s ⎟ ⎝ Ie(v) ⎠ lρg

Table 1. Summary of the Characteristics of the Nozzles Including the Window Type and Linear Expansion Rates d(A/A*)/dz nozzle

window

linear expansion rate (cm−1)

H (H2) C3 (C2)

CaF2 (mica) CaF2 (mica)

0.057 0.079

(1)

where MD2O is the molecular weight of D2O, and ρ and g come from the integrated PTM/SAXS analysis. The FTIR spectra are used to calculate the solid fraction of the aerosol as it crystallizes. In particular, once we establish the aerosol molar absorptivities of the coldest liquid εa,liquid(ν) and the completely frozen εa,solid(ν) aerosol, the fraction of solid (ice) in the mixed state aerosol is determined by fitting the molar absorptivity of the mixed state aerosol εa,mixed(ν) using

In a PTM, the static pressure p is measured as a function of location z along the nozzle axis using a movable static pressure probe. Pressure measurements are made for both the pure carrier gas (dry trace) and for the carrier gas−vapor mixtures (wet trace). Position resolved SAXS measurements are performed using the 12-ID_C beamline35 at the Advanced Photon Source, Argonne National Lab at a sample to detector distance of 0.851 m. The radially averaged, background

εa,mixed(ν) = Fsolidεa,solid(ν) + Fliquidεa,liquid(ν)

(2)

where Fsolid and Fliquid are the fractions of solid and liquid in the “mixed” aerosol. If the following assumptions hold: 5473



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(1) Only one nucleation event occurs per droplet. (2) The time required for a droplet to freeze is very short compared to the time required for the entire aerosol to freeze. (3) The spectra at the exit correspond to a completely frozen aerosol. then the fraction of solid in the aerosol equals the fraction of droplets that have frozen, and the latter can be used to extract the ice nucleation rates. Bartell and Chushak16 tested the first two assumptions using the criterion proposed by Kashciev et al.38 For their H2O ice nucleation experiments in supersonic flows they found that both assumptions were reasonable. Since our conditions are comparable to those used by Bartell and Chushak16 they should also be valid for our experiments. The validity of assumption 3 is discussed in section III.C.

III. RESULTS AND DISCUSSION Experiments with condensable vapor stagnation pressures, pv0 = 0.494 and 1.005 kPa, were carried out in nozzles H and H2 at a stagnation temperature T0 = 25 °C and a stagnation pressure p0 = 60 kPa, and the experiments at pv0 = 0.346 kPa used nozzles C3 and C2 with T0 = 35 °C and p0 = 30 kPa. The carrier gas was N2 in all experiments. A. Analysis of PTM + SAXS. The pressure measurements for the D2O experiments are presented in Figure 1a. The solid lines correspond to the isentropic expansions of a noncondensing gas mixture in nozzles H (black) and C3 (gray). For the isentropic expansions, the difference in p/p0 reflects the difference in expansion rate between the nozzles, and the lower expansion rate in nozzle H results in a higher pressure ratio (p/ p0) at the nozzle exit. For the condensing flow traces (short dashed, dashed-dotted, and long dashed lines), as pv0 increases the onset of vapor−liquid nucleation occurs at higher p/p0 and T and there is a stronger deviation from the corresponding isentropic expansion because more heat is released into the flow. The lines in Figure 1b illustrate the corresponding gas mixture temperatures that were determined using the iterative procedure described in section II.B, and the vertical arrows in Figure 1b indicate where FTIR measurements first detect D2O ice. In some cases, ice nucleation starts while the drops are still growing rapidly and, thus, the temperature of the droplets is higher than that of the surrounding carrier gas. The symbols in Figure 1b correspond to the droplet temperatures Td estimated from an energy balance on the droplets.39 In the most extreme case, for pv0 = 0.346 kPa, Td is estimated to be ∼15 K higher than the temperature of the gas mixture when ice is first detected. As in our H2O ice nucleation studies,31 even when droplet growth is essentially complete prior to freezing, as is the case when pv0= 1.005 kPa, not enough heat is released to the flow by crystallization to produce a second “bump” in the pressure/temperature curves. Figure 2a summarizes the position resolved mean droplet radii, ⟨r⟩ and the data show that ⟨r⟩ increases with the amount of condensable entering the system. Comparing the results of Figures 1b and 2a, it appears that the larger droplets may start to freeze at slightly higher Td than the smallest droplets. Furthermore, even for the highest pv0, where rapid droplet growth has stopped well before crystallization starts, ⟨r⟩ is not obviously affected by freezing. This is consistent with a negligible density change between liquid and crystalline D2O in this temperature range, and is in good agreement with the observation of Manka et al.31 for H2O. Although ⟨r⟩ does not

Figure 1. (a) Pressure profiles for the three stagnation partial pressures (pv0) of D2O investigated (short dashed, dashed-dotted, and long dashed lines). The difference in the expansion rates of the two nozzles is reflected in the isentropic expansions in the two nozzles (solid lines). (b) The corresponding temperature estimates use the same line styles as in (a) for experiments with and without condensable. The symbols show the estimated droplet temperatures. The horizontal arrows point to the appropriate temperature axis. The vertical arrows indicate the location of the onset of ice nucleation.

appear to change due to freezing, in some experiments ⟨r⟩ does change as the particles continue to grow. This is particularly true for the experiments with pv0 = 0.494 and 0.346 kPa. In Figure 2b we show the position resolved values of g derived using PTM alone (gPTM, dashed-dotted line) and SAXS alone (gSAXS, symbols). The mass fraction of condensable entering the nozzle, ginf, is indicated by the gray dashed line. In the region immediately downstream of the onset of nucleation, there is good agreement between gPTM and gSAXS, and in the region of rapid growth and further downstream gSAXS captures the behavior more accurately, approaching ginf near the nozzle exit. The difference between gPTM and gSAXS arises because PTM analysis alone cannot account for changes to the effective area ratio of the expansion that occurs when condensation compresses the boundary layers that grow along the nozzle walls.40 Finally, gfit corresponds to a Weibull 5 parameter sigmoidal curve that interpolates between and smoothes the appropriate gPTM and gSAXS data. The gfit values are used in the iterative data analysis procedure described in section II.B. 5474



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Figure 2. (a) Position resolved average droplet radii ⟨r⟩ increase as the available condensable increases. For the experiment conducted at pv0 = 0.494 (0.346) kPa, droplet size increases by 12% (30%) between the start of freezing and the nozzle exit. (b) Position resolved measurements of the condensate mass fraction g from PTM, gPTM, agree well with g from SAXS gSAXS during the early stages of condensation. Further downstream, however, gPTM and gSAXS diverge and gSAXS is more reliable (see main text). A Weibull 5 parameter fit (solid line) (gfit) allows for a smooth transition between gPTM, upstream and at onset of nucleation, to the gSAXS data further downstream. Arrows in both panels a and b indicate the approximate location of the onset of D2O ice nucleation.

Figure 3. D2O spectra downstream of the onset of condensation are shown for pv0 = 1.005 kPa. The position in the nozzle z and the estimated droplet temperatures Td are both given in the legend. Under these conditions, Td differs from T by less than 1 K and we assume Td = T. In the range of 2300 < ν (cm−1) < 2395, the measured spectra are influenced by interference from ambient CO2. (a) The dashed and dotted lines are the aerosol spectra calculated using the optical properties reported by Bertie et al.42 at 295 K and Max and Chapados43 at 297 ± 2 K. The similarity between the measured and calculated spectra suggests the aerosol is comprised of liquid droplets. (b) Position resolved FTIR spectra at and downstream of the onset of freezing are shown for the same experimental conditions as (a). The spectra at z = 5.28 cm and z = 6.88 cm represent the coldest liquid and most highly frozen aerosol, respectively. The peak positions are consistent with literature values. The lack of overlap between the spectra near the nozzle exit suggests that the aerosol is not completely frozen. The absorption data from Schaff and Roberts44 (solid line, T = 175 K) is included here, but quantitative comparison is not possible because the absorption values are reported in arbitrary units.

B. Analysis of FTIR Spectra. Since neither PTM nor SAXS are able to detect or quantify the liquid−solid phase transition,31 we use FTIR spectroscopy to do so. We measure position resolved IR spectra and focus our analysis on the O−D stretch region of the condensed phase, 2200 < ν/cm−1 < 2700, where the signal is strong and the peak location and peak intensity change significantly upon freezing. Figure 3a,b presents the aerosol molar absorptivities measured for the experiment at the highest partial pressure of D2O (pv0 = 1.005 kPa) and, correspondingly, the experiments with the largest droplets. We distinguish between the various D2O phases by the appearance of distinctive features in the O−D stretch region located at the wavenumbers noted in Table 2. Figure 3a illustrates the spectra measured downstream of condensation and upstream of freezing. As Td decreases, the feature in the liquid spectra at ν ≈ 2500 cm−1 moves to lower wavenumbers and is accompanied by an increase in εa. This shift is qualitatively consistent with the observations of Manka et

Table 2. Wavenumbers Corresponding to the Distinctive Features Observed in the O−D Stretch Region for Each Phase As Measured in Our Experiments and Reported in the Literaturea absorption features in the O−D stretch region (ν, cm−1) phase

experiments

literature

refs

liquid solid

∼2490, 2530 2499, 2450

∼2400, 2504, ∼2600 2484, 2430, 2353

41 44 and 45

a

The absorption features that correspond to those observed in our experiments, are denoted in bold in the list of literature values.

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al.31 for liquid H2O and can be explained by an increase in the strength of the hydrogen bonding environment as the temperature decreases. The spectra shown by the dashed and dotted lines are the expected room temperature aerosol molar absorptivities calculated using

ature as shown in Figure 4. For the liquid, we only considered spectra measured for droplets that were no longer growing

εa,calc(ν) =

⎞ M D2O 6n(ν)k(ν) 6πν ⎛ ⎜ ⎟ ln(10) ⎝ (n2(ν) − k 2(ν) + 2)2 − (2n(ν)k(ν))2 ⎠ ρD O 2 (3)

where ρD2O is the density of liquid D2O and the real n(ν) and imaginary k(ν) components of the index of refraction are those measured at 295 K by Bertie et al.42 and at 297.15 ± 2 K by Max and Chapados,43 respectively. The similarities in the features of the O−D stretch region between the measured and calculated spectra suggest that for this experiment the droplets are in the liquid state at temperatures as low as Td ≈ 227 K. Figure 3b shows the spectra measured at and downstream of the onset of freezing as well as the spectra of an annealed D2O ice film measured by Schaff and Roberts44 at T = 175 K. In their experiments at 160 K (175 K) Schaff and Roberts45,44 observed features in D2O ice spectra at ν ≈ 2484 (2485), 2430 (2432) and 2353 (2345) cm−1. At the higher temperatures present in our measurements we see features at ν ≈ 2499 and 2450 cm−1, that correspond to the two higher wavenumber features reported by Schaff and Roberts.45,44 Interference from atmospheric CO2 makes it difficult to discern the feature at the lowest wavenumber. The ice spectra measured in our experiments are also broader and less peaked than those reported by Schaff and Roberts.45,44 This is a consequence both of the difference in temperature, features in the ice spectra sharpen at lower temperatures, and of the fact that the absorption spectra from nanodroplets reflects not only the contribution from a crystalline core but also the contribution from the less organized surface and subsurface layers. The latter phenomenon was modeled by Buch et al.46 to explain the IR spectra measured for 1.4 and 6 nm radius H2O ice nanocrystals47,48 and is consistent with the thin layer of disordered ice/liquid observed at the surface of water nanoparticles in mW droplet modeling studies.27 Finally, in Figure 3b, the isosbestic point observed near ν ≈ 2530 cm−1 is consistent with that observed by Millo et al.49 at ν ≈ 2475 cm−1 when they followed the melting of D2O ice between 263 and 275.97 K. In Figure 3b the spectra at z = 5.28 and 6.88 cm correspond to the coldest liquid and the aerosol containing the largest fraction of ice, respectively. In our initial analysis of this experiment, these spectra are taken as εa,liquid(ν) and εa,solid(ν), respectively, and are used to calculate Fliquid and Fsolid for the intermediate “mixed state” aerosol spectra via eq 2. It is clear from Figure 3a that both the peak intensity and the peak positions of the observed features in the O−D stretch region of the measured IR spectra for the liquid are sensitive to temperature. In the current analysis we do not consider the temperature dependence of the intensity of the absorption by the droplets both because this is difficult to quantify and, for H2O, was estimated to only change the final nucleation rate by ∼30%.31 We can, however, greatly improve the fit to the mixed state spectra if we account for the temperature dependence of the peak position. To do so, we plot the peak position in the O−D stretch region of the spectra for both liquid (circles) and frozen (diamonds) aerosols as a function of droplet temper-

Figure 4. Peak positions for the liquid and frozen aerosol spectra shift toward lower wave numbers as the temperature decreases. The error bars represent the uncertainty in locating the peak position because the peak is rather broad.

rapidly, and so Td is very close to T. For the solid we considered spectra where we estimated that more than 80% of the condensate was frozen and thus the position of the strongest peak should primarily reflect the position of the pure ice peak. The slopes of the linear fits, shown by dashed lines in Figure 4, are used to shift the liquid (Δνliquid) and solid aerosol spectra (Δνsolid) as follows: Δνliquid = 0.6890(Tmixed − Tliquid)

(4a)

Δνsolid = 0.5620(Tmixed − Tsolid)·

(4b)

Here Tmixed is the temperature of the mixed state aerosol, Tliquid is the temperature of the coldest liquid droplets, and Tsolid is the temperature of the solid particles. All temperatures refer to the droplet/particle temperatures. We fit the “mixed” spectra as a linear combination of the temperature corrected solid and liquid molar absorptivities using least-squares regression and determined the best fit parameters Fice and Fliquid. Figure 5 compares the “mixed” (dashed line) and “calculated” (gray solid line) spectra with Fice = 0.356 and Fliquid = 0.642. There was generally good agreement between the measured and fitted spectra, and thus, we used this approach to estimate Fice for all of the measured spectra. The solid fractions for all three experiments are plotted as a function of droplet temperature in Figure 6. The onset of ice nucleation is delayed somewhat to lower temperatures as the droplet size decreases, in a manner consistent with but not as dramatic as the changes observed by Manka et al.31 This size dependence reflects the stochastic nature of nucleation, i.e. the fact that the probability of nucleation depends on the droplet volume50,51 as well as the degree of supercooling of the liquid and the cooling rate. Beyond the onset temperature for ice nucleation, there is a rapid increase in the fraction of ice in the aerosol. Although overall mass balance, i.e. Fice + Fliquid = 1, was not imposed as a constraint during the fit, Figure 6 shows that 5476



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the nucleation rate is not a strong function of temperature. Under these assumptions, the time dependence of the fraction of frozen droplets in the aerosol Fice can be written as Fice(t ) = 0,

t < t0

= 1 − exp( −B(t − t0)), t ≥ t0

(5)

where the parameter t0 is the time corresponding to the onset of ice nucleation. If we assume ice nucleation occurs throughout the volume of the droplets, B = Jice,VV where Jice,V is the volume based nucleation rate and V = (4π/3)⟨r3⟩ is the average volume of a droplet. The functional form of eq 5 is also valid for surface dominated nucleation if B = Jice,SS where Jice,S is the surface-based nucleation rate and S = 4π⟨r2⟩ is the average surface area of a droplet. The fits (solid lines) of eq 5 to the experimentally estimated solid fractions (symbols) are shown in Figure 7. From the fit Figure 5. Spectra of the “mixed” aerosol are fit using the temperature corrected liquid and solid reference spectra.

Figure 7. Fraction of ice as a function of time for all of the investigated experimental conditions. The aerosol was assumed to be fully frozen at the nozzle exit. The solid lines are fits of eq 5 assuming a monodisperse aerosol distribution. The monodisperse fits agree well with the experimental estimates of the solid fraction near the onset of freezing, but do not adequately capture the trend near the nozzle exit.

parameter t0 we can determine the conditions in the flow, the droplet size, and droplet temperature corresponding to the onset of ice nucleation. The temperature at the onset of freezing Ton,ice is taken as Td(t0) and the fit parameter B yields Jice,V or Jice,S. Table 3 summarizes the onset conditions and ice nucleation rates calculated this way. Figure 7 shows that there is good agreement between the fits and experimental measurements in the early stages of crystallization for all experimental

Figure 6. Once freezing begins the fraction of ice increases rapidly as the temperature decreases. The onset of ice nucleation occurs at a slightly lower temperature as the droplet radii decrease. The overall mass balance of the linear regression fit is always within 10% of Fice + Fliquid = 1.

mass balance was satisfied to within 10% and generally within 6%. The deviations from mass balance are largest for experiments with the biggest temperature difference between the liquid and solid reference spectra. This behavior may be a consequence of our inability to accurate model changes in absorption intensity with temperature. For pv0 = 1.005 and 0.346 kPa, Fice is still increasing near the nozzle exit, especially compared to the behavior of Fice for pv0 = 0.494 kPa, emphasizing that in the former cases the aerosol is unlikely to be fully frozen at the nozzle exit. C. D2O Ice Nucleation Rates. To calculate the liquid− solid nucleation rate from the fraction of ice data, we first assume that the droplets are essentially monodisperse and that

Table 3. Droplet Temperatures Corresponding to the Onset of Freezing Ton,ice, the Fit Parameter B (s−1), and the Volume Jice,V or Surface Jice,S Ice Nucleation Rates Found by Fitting Fice to eq 5 Summarized As a Function of the Droplet Radiusa ⟨r⟩ (nm) ∼3.1 ∼5.2 ∼9.0 a

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Ton,ice (K) 222.3 225.1 224.7

B (s‑1)

Jice,V (cm‑3 s‑1)

Jice,S (cm‑2 s‑1)

43 000 39 200 80 000

3.6 × 10 6.8 × 1022 2.7 × 1022

3.7 × 1016 1.2 × 1016 7.9 × 1015

23

The aerosol was assumed to be completely frozen at the nozzle exit. dx.doi.org/10.1021/jp400070v | J. Phys. Chem. A 2013, 117, 5472−5482


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conditions. Further downstream, however, the fits from eq 5 do not capture the behavior as accurately as we expect based on the H2O ice nucleation measurements of Manka et al.31 The lack of fit most likely reflects the fact that changes in the spectra near the nozzle exit suggest that the aerosol is not yet fully frozen. Figure 3b illustrates the case for the experiment with pv0 = 1.005 kPa. Nevertheless, we can still estimate Fice and Fliquid, if we assume that x is the fraction of the aerosol at the nozzle exit that is frozen and calculate a new reference spectrum for the solid using εa,solid(ν) =

respectively. In light of these observations, all further analyses are based on the Fice,exit values reported in Table 4. We next investigated the importance of incorporating aerosol polydispersity into the analysis. To include the polydispersity of the aerosol, Sigburjörnsson and Signorell33 suggest that the experimental data should be fit to the following expression: Fice(t ) = 1 ⎡ ⎤ − ⎣⎢ f (r , t )V (r ) exp[− (Jice,V (T )V (r ) + Jice,S (T )S(r ))(t − t0)] dr ⎦⎥

∫

⎡ ⎤ ⎣⎢ f (r , t )V (r ) dr ⎦⎥,

∫

εexit(ν) − (1 − x)εa,liquid(ν)

where εexit(ν) is the molar absorptivity of the aerosol at the exit. In deriving εa,solid(ν) we corrected εa,liquid(ν) to Td at the exit. We then recalculate the solid fractions and the fit parameters for different values of x. As illustrated in Figure 8 for pv0 = 0.346

Figure 8. Fits of eq 5 to experimental Fice values improve when we assume the fraction of aerosol frozen at the exit of the nozzle is less than 100%. For pv0 = 0.346 kPa, the best fit corresponds to ∼80% of the aerosol frozen at the exit.

Figure 9. Values of Fice predicted by eqs 5 and 7 for pv0 = 0.346 kPa and Fice,exit = 80%. For the same values of B, the predictions that incorporate polydispersity lie distinctly above the predictions that assume a monodisperse aerosol. This is the most severe case observed because in this experiment the droplets are simultaneously freezing and growing. Reducing B by ∼30% essentially reproduces the monodisperse fit suggesting that the uncertainty in the reported rates is not strongly affected by polydispersity.

kPa, as x decreases the agreement between the fit and the experimental data improves. From Figure 8 the there is better agreement when we assume that ∼80% of the aerosol is frozen at the exit. Applying the same analysis to the experiments with pv0 = 0.494 and 1.005 kPa, good agreement corresponds to ∼95% and 80% of aerosol frozen at the exit, respectively. The estimates for Jice,V and Jice,S based on the revised analysis are summarized in Table 4. In all cases the rates decreased but are still within a factor of 2 of our original estimates. In contrast, the values of t0 and Ton,ice change by less than 1% and 0.5%

pv0 = 0.346 kPa. In contrast to the H2O ice data,31 we observe that the new values of Fice lie distinctly above the original monodisperse fit. Reducing the value of B by ∼30% in eq 7, however, yields values of Fice very close to our original monodisperse fit. Thus, the nucleation rates are not very sensitive to polydispersity even in this extreme case where ⟨r⟩ increased by ∼30% as the aerosol froze. Following the work of Kuhn et al.,52 we also investigated whether our data support surface initiated freezing as the dominant phenomenon in our nanometer sized droplets. This subject is controversial because many other H2O freezing experiments with micrometer sized droplets18,19 find no evidence for surface initiated freezing. MD simulations by Johnston and Molinero27 with a monatomic water model (mW) find no evidence for surface freezing in nanometer sized droplets, whereas Vrbka and Jungwirth’s53 simulations of water slabs using a six-site interaction potential for water54 found that

Table 4. Fraction of Aerosol Frozen at the Exit of the Nozzle Fice,exit, Droplet Temperatures Corresponding to the Onset of Freezing Ton,ice, Fit Parameter B (s−1) Assuming a Monodisperse Aerosol, and the Volume and Surface Ice Nucleation Rates Determined Summarized As a Function of the Droplet Radius r (nm)

Fice,exit

Ton,ice (K)

B (s−1)

Jice,V (cm‑3 s‑1)

Jice,S (cm‑2 s‑1)

∼3.1 ∼5.2 ∼9.0

∼0.8 ∼0.95 ∼0.8

223.4 225.4 224.8

27500 35500 48600

2.3 × 1023 6.1 × 1022 1.6 × 1022

2.3 × 1016 1.1 × 1016 4.8 × 1015

(7)

Here f(r, t) is the time dependent particle size distribution calculated using the Schulz distribution of polydisperse spheres. To test the sensitivity of Fice to polydispersity we recalculated Fice using eq 7 and the values of B = Jice,VV and t0 derived from the fit to eq 5, assuming Jice,S = 0 and accounting for the change in particle size of the evolving aerosol. The recalculated values of Fice (dashed line) at Fice,exit = 80% are plotted in Figure 9 for

(6)

x

t ≥ t0

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freezing almost always started in the subsurface, i.e., within ∼1 nm of the surface. In our previous experiments,31 we could not discern whether freezing in nanometer sized H2O droplets was initiated at the surface, in part because the analysis based on fitting all of the data meant that there was no overlap in temperature between the nucleation rates measured for different droplet size. In order to measure nucleation rates as a function of temperature for different droplet sizes we examine the change in Fice(t) between successive points and calculate the nucleation rate for t ≥ t0 as55 Jice,V (t1)V = Jice,S (t1)S = B(t1) =

Figure 10 confirms our assumption that the temperature dependence of the ice nucleation rates is rather weak since under our experimental conditions even Jice,V changes by at most an order of magnitude over the ∼15 K temperature range. Given the uncertainty in the data, the weak maximum near 216 K should not be interpreted as an indication that these experiments are close to the cross over point between crystallization controlled by nucleation and crystallization controlled by growth. Further experiments, where droplet formation occurs at lower temperatures, are required to establish whether this is the case. In Figure 11, we compare our values of Jice,V from Table 4 to the literature data available for H2O and D2O freezing,

⎛ 1 − Fice(t1) ⎞ 1 ln⎜ ⎟ t 2 − t1 ⎝ 1 − Fice(t 2) ⎠ (8)

where the subscripts 1 and 2 correspond to consecutive SAXS measurements. We evaluated eq 8 using the estimates of Fice based on the values of Fice,exit in Table 4 and determined Jice,V(t) and Jice,S(t) from B(t) using the average droplet size. The temperature assigned to each nucleation rate corresponds to T(t1). When we plot Jice,V and Jice,S as a function of droplet temperature in Figure 10, Jice,V increases systematically with

Figure 11. Nucleation data Jice,V for both H2O and D2O reported by Taborek,13 Stöckel et al.18 and Manka et al.31 and the current work as a function of Td. Theoretical nucleation rate estimates from Murray et al.32 for H2O and from the current work for D2O are also shown. Both theoretical estimates use scaling parameter n = 0.97 for the temperature dependence of the solid−liquid interfacial tension.

considering only experiments that studied both substances. Figure 11 illustrates that the freezing data for D2O are at higher temperatures than the data for H2O at the same nucleation rate reflecting, in part, the fact that at 1 atm the equilibrium melting point of D2O is higher than that of H2O. Surprisingly, for the current D2O experiments there is less spread in the temperatures at which the smallest droplets start to freeze relative to the largest, than for H2O. For the smallest droplets, the differences between the H2O and D2O experiments largely reflect the differences in operating conditions between the two experiments. These differences, in turn, affect the temperature history of the droplets and, therefore, the temperature at which freezing can be initiated. In particular, for the r = ∼3 nm droplets, the H2O experiments were conducted with high pressure (p0 = 60 kPa) Ar as the carrier gas, H2O droplet formation from the vapor phase occurred at ∼198 K and, the temperature of the droplets that we could observe, Td, was estimated to be less than ∼207 K. In contrast, the D2O experiments used low pressure (p0 = 30 kPa) N2, D2O droplet formation occurred at ∼205 K, and the temperature of the droplets we could observe is estimated to reach as high as 229 K. The higher temperature of the D2O droplets is entirely consistent the lower carrier gas pressure used in these experiments, i.e. the gas mixture temperature increases more dramatically for the same latent heat release, and the droplets

Figure 10. Volume (open symbols) and surface (filled symbols) based nucleation rates are plotted as function of droplet temperature Td. Values of Jice,S are a weaker function of the droplet size than Jice,V. Error bars are shown for one data set and reflect our estimates for the uncertainties that arise from determining the reference spectra and from fitting the intermediate spectra.

decreasing droplet size, whereas Jice,S is slightly more consistent between experiments. Our result is consistent with that reported by Kuhn et al.52 who found that for their smallest droplets, Jice,V values increased with decreasing particle size while Jice,S was relatively size independent. This lead them to conclude that surface nucleation dominated for droplets below 5 μm. We note, however, that in our experiments, where 3 nm < ⟨r⟩ < 9 nm, it becomes difficult to distinguish between “surface/subsurface” and “bulk” nucleation since the outer 1 nm constitutes from 70% to 30% of the volume of the droplet. Furthermore, the uncertainty introduced by experiments where the aerosol leaving the nozzle is not fully frozen, complicates the error analysis required to determine whether the differences observed in Figure 10 are significant. 5479



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Figure 12. Comparison of the nucleation rate data for the crystallization of H2O and D2O from supercooled liquid droplets and amorphous films. The nucleation rate estimates based on CNT follow the experimental trends qualitatively but not always quantitatively.

method to estimate the temperature dependence of σsl as follows:

are cooled less effective by the carrier gas. Thus, the smallest H2O droplets could not freeze at a temperature close to that of the large H2O droplets (215 K) because the former never existed under those conditions. In contrast, there is significant overlap in the temperatures reached by the smallest and largest D2O droplets in the current experiments and, thus, the smallest droplets can start to freeze at temperatures close to those of the largest droplet. Finally, the fact that ice nucleation in the 5 nm droplets appears to start at a temperature slightly (∼ 0.7 K) above that for the 9 nm droplets most likely reflects experimental scatter about the true answer, and the increased sensitivity of the nucleation rate to temperature as temperature increases. Further experiments, using operating conditions similar to those of Manka et al., are required to extend the range of D2O ice nucleation rate measurements to lower temperatures. To calculate the theoretical nucleation rates, we followed the approach suggested by Murray et al.32 and use the following expression: ⎛ 16πσsl 3vm 2 ⎞ ⎟ Jice,V = A exp⎜ − 3 2 ⎝ 3(kT ) (ln Si) ⎠

⎛ T ⎞n σsl(T ) = σsl(T0)⎜ ⎟ ⎝ T0 ⎠

where the factor n ranges from 0.3 to 0.97 and is chosen without theoretical consideration in order to provide the best fit joining the nucleation rates measured in μm sized droplets to those measured in nm sized droplets. Here we use Murray et al.’s32 choice for H2O ice, n = 0.97, to be consistent with the work of Manka et al.31 To determine σsl(T0) and A(T0) we plotted Stöckel et al.’s18 D2O nucleation rate data as ln Jice,V versus T−3(ln S)−2, and a straight line fit yields σsl(T0) = 20.57 mJ/m2 and ln A(T0) = 68.8 or A(T0) = 7.44 × 1029 cm−3 s−1 for T0 = 242 K. The agreement between the theoretical predictions and our experimental measurements for D2O is not quite as good as that observed for H2O. Some of the disagreement may be due to uncertainty in the thermophysical properties for D2O, in particular the vapor pressure over supercooled D2O cubic ice, the choice of n = 0.97 as a temperature scaling parameter for σsl, and the value of ΔGh→c, the Gibbs free energy change for the transformation of hexagonal to cubic ice of D2O that we have assumed to be the same as that for H2O for lack of any other information. It may also be due to the uncertainty of about a factor of 50 that Stöckel et.al18 report for their measurements of Jice,V that translates directly into an uncertainty in A(T0). In their analysis of H2O and D2O ice nucleation rates, Stöckel et al.18 report that D2O nucleates more easily than H2O in the sense that the degree of supercooling, ΔT = Tm − T, required to initiate crystallization is lower for D2O than H2O. We also observe that D2O ice nucleation starts at smaller values of ΔT than H2O at least up to ΔT ≈ 60 K. At higher degrees of supercooling we would expect this relationship to fail since eventually the nucleation rates begin to decrease with the theoretical rate curves crossing near ∼185 K. We then reanalyzed the data of Manka et al.31 using eq 8 to obtain nucleation rates over the entire temperature range. Figure 12 compares Jice,V, calculated this way with data on crystallization of supercooled liquid droplets13,18 for both the isotopes of water as well as rates derived from the crystallization kinetics of films of amorphous H2O.20,22 As expected,

(9)

where vm is the molecular volume of D2O ice, σsl is the interfacial tension between D2O ice and supercooled D2O liquid, Si is the supersaturation with respect to cubic D2O ice, and k is the Boltzmann constant. Furthermore, the preexponential factor A, given by A=

2(σslkT )0.5 vm 5/3η

(10)

where η is the shear viscosity of supercooled water, is approximated as A(T ) ≈ A(T0)

η(T 0) η(T )

(12)

(11)

and the values of η are calculated by extrapolating the correlation of Vedamuthu et al.56 The most critical property in the estimation of the nucleation rates is σsl, whose temperature dependence is not well-known. Huang and Bartell55 and Murray et al.32 both use an ad hoc scaling 5480



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nucleation rates increase rapidly as we approach and enter “no man’s land” from both the supercooled liquid and amorphous solid sides. The nucleation rates calculated using classical nucleation theory (CNT) (solid black and gray lines for H2O and D2O, respectively) and the procedure outlined by Murray et al.,32 describe the experimental trends qualitatively across the entire temperature range but not entirely quantitatively for either isotope. Experiments that extend both the range of droplet sizes investigated and the temperatures experienced by the droplets are required to obtain a better understanding of the crystallization process. Reliable methods to transform the crystallization kinetics measured in amorphous solid films57,58 to “bulk” nucleation rates would also be extremely helpful.

1033439. Use of the Advanced Photon source at Argonne National Laboratories was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE AC02 06CH11357. We thank S. Tanimura for the initial pressure trace and SAXS measurements and H. Laksmono, A. Manka, D. Bergmann, J. Wölk, K. Mullick, S. Seifert, and R. Winnans for their help in the SAXS experiments.

■

IV. SUMMARY AND CONCLUSIONS We followed the freezing of nanometer sized D2O droplets in our supersonic nozzle apparatus using three experimental techniques, PTM, SAXS, and FTIR. The complementary information lets us characterize the flowing gas mixture and the aerosol in detail. The current D2O experiments were carried out at three condensable vapor stagnation pressures in nozzles with differing expansion rates. On the basis of the measured FTIR spectra, we observe that under some experimental conditions supercooled D2O droplets begin to freeze even as the droplets continue to grow rapidly. For the smallest droplets, droplet temperatures are estimated to exceed the temperature of the carrier gas temperature by up to ∼15 K and hence the onset of ice nucleation is referenced to the droplet temperature. The onset of ice nucleation is observed at temperatures between 223 and 225 K with Jice,V and Jice,S values ranging from 1.6 × 1022 to 2.3 × 1023 cm−3 s−1 and 4.8 × 1015 to 2.3 × 1016 cm−2 s−1, respectively, for droplet radii ranging between 9 and 3 nm. As expected, the smallest droplets freeze at lower droplet temperatures and at higher nucleation rates than the largest droplets. Although there is some evidence that Jice,S scales better with aerosol surface area than Jice,V scales with volume, in our nanometer sized droplets the outer 1 nm constitutes between 30% and 70% of the volume making the distinction between surface, or near surface, and volume based nucleation almost moot. Finally, following the work of Murray et al.,32 we calculate theoretical nucleation rates for D2O freezing using the same scaling parameter for σsl that Murray suggested for H2O. Despite the assumptions required to estimate the thermophysical properties of D2O at these low temperatures and our choice of n, the agreement between our experimental data and the theoretical predictions is still within 3 orders of magnitude.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information as discussed in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant Nos. CHE-0911144, CHE-1213959 and CBET5481



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Freezing of Heavy Water (D2O) Nanodroplets

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