Following heterogeneous nucleation of CO₂ on H₂O ice nanoparticles with microsecond resolution

Abstract

Heterogeneous nucleation of CO₂ onto H₂O ice particles may play an important role in proposed innovative CO₂ capture technologies, as well as in the formation of Martian clouds. In this work we follow the nucleation/condensation of CO₂/H₂O gas mixtures with microsecond resolution in supersonic Laval nozzles using pressure trace measurement (PTM) and small angle X-ray scattering (SAXS). The latent heat release detected by the PTM reveals that the first phase transition in the expanding CO₂/H₂O mixture is the formation of H₂O ice particles by the homogeneous nucleation/condensation and freezing of H₂O. This is followed by the heterogeneous nucleation and growth of CO₂ on the H₂O ice particles. The onset conditions for heterogeneous nucleation, i.e. the partial pressure of CO₂ and temperature from PTM and the radius of gyration of the H₂O ice particles from SAXS, were determined in the temperature range 124 to 146 K and for particles with radii of gyration in the range of 2.1 to 4.3 nm. The onset conditions suggest that the heterogeneous nucleation of CO₂ may start from the supercooled liquid phase under our conditions. Downstream of the onset point, the partial pressure of CO₂ and temperature rapidly approach the vapor–solid equilibrium line of CO₂, demonstrating that even if CO₂ condensation is initiated by heterogeneous nucleation of the liquid phase, it proceeds via growth of the solid.

---

1. Introduction

Heterogeneous nucleation of carbon dioxide (CO₂) onto pre-existing, frozen H₂O particles may play an important role in innovative technological processes. In a proposed low-temperature CO₂ capture technology, for example, flue gas cools as it expands in a supersonic nozzle driving CO₂ deposition, and the CO₂ particles generated are subsequently separated from the gas by centrifugal forces.^1,2 Under the conditions where the CO₂ capture technology is to be applied, however, H₂O vapor is also present.^2,3 As illustrated in Fig. 1, the low vapor pressure of H₂O relative to CO₂ suggests that even with reasonable levels of H₂O vapor removal from the gas stream, pure H₂O will still condense in the supersonic nozzle at much higher temperatures than pure CO₂ does. Based on this phase diagram, we expect that condensation from a CO₂/H₂O mixture expanding in a supersonic flow is initiated by the homogeneous nucleation of H₂O. The H₂O droplets grow and solidify as the temperature drops during the continued expansion, and finally CO₂ condenses/deposits heterogeneously on the H₂O ice particles. Although this intuitive picture is appealing, there are no experimental results that confirm the prediction, or report the degree of CO₂ saturation required to drive heterogeneous nucleation under the rapid cooling conditions characteristic of supersonic flows.

Fig. 1 Phase diagram and onset conditions for unary homogeneous nucleation of CO₂ or H₂O.^4–8 Partial pressures of condensable species in isentropic flow (non-condensing flow) under a set of conditions in this study are also shown. These conditions are, for example, close to the typical conditions of the flue gas of an iron production process (Corex process),⁹ temperature 313 K, pressure 3.5 atm, CO₂ 0.24–0.30 mol mol⁻¹, and H₂O 0.01 mol mol⁻¹.

Earlier studies of heterogeneous nucleation of CO₂ on H₂O ice have largely been motivated by the fact that this process is a possible pathway for the formation of CO₂ clouds in the Martian atmosphere.^10,11 Glandorf et al. monitored the deposition of CO₂ on flat H₂O ice films under conditions that mimicked the Martian atmosphere (T = 130.2–140.0 K) using transmission Fourier transform infrared (FTIR) spectroscopy.¹⁰ They reported that heterogeneous nucleation, at a rate of ∼1 cm⁻² s⁻¹, required a saturation with respect to CO₂ solid of ∼1.34 and that the contact parameter between solid CO₂ and solid H₂O was 0.95. A contact parameter less than unity, i.e. a contact angle larger than zero, is consistent with the results of the temperature programmed desorption (TPD) measurements for CO₂ on H₂O ice surfaces of Noble et al.,¹² that suggested CO₂ deposited on H₂O ice preferentially forms clusters (or islands). Other FTIR studies involving CO₂/H₂O aerosols^11,13,14 focused on the structure and structural evolution of the particles rather than characterizing the conditions required to initiate heterogeneous nucleation. They also reported that heterogeneous nucleation of CO₂ onto H₂O ice particles yielded an architecture consisting of CO₂ patches on the underlying ice rather than a perfectly coated core–shell structure. These experimental results stand in contrast to recent classical molecular dynamics simulations of the deposition of CO₂ onto H₂O ice that showed the CO₂ molecules fully wet the H₂O ice surface.¹⁵

To the best of our knowledge there are no experiments that have directly followed the heterogeneous nucleation of CO₂ onto H₂O ice particles. Here we report our efforts to study this phenomenon by conducting experiments in our supersonic nozzle apparatus. Under steady flow conditions we can follow the progression of nucleation/condensation with a time resolution better than several μs, by changing the measurement position from upstream to downstream in the nozzle. Static pressure trace measurements (PTM) of the supersonic flow of CO₂/H₂O mixtures in nitrogen (N₂) gas provide the gas temperature, and the heat release due to CO₂/H₂O phase changes. We find which species is involved in the detected phase change by investigating the dependence of the heat release on the composition of the CO₂/H₂O gas mixture, and we determine the onset conditions (partial pressure of the condensable species and temperature) for that phase change. From the size of the condensed particles determined by small angle X-ray scattering (SAXS) measurements, we examine how the onset conditions for heterogeneous nucleation depend on the size of the seed particle. The structure of the resultant particles is the subject of ongoing research and is not discussed here.

2. Experimental

2.1. Flow system and nozzles

Two modifications were made to our previous experimental setup¹⁶ in order to supply the condensable species, CO₂ and H₂O, as shown in Fig. 2. The first change is the addition of a CO₂ cylinder that is connected to the main flow through a mass flow controller (Type 1559A, 151.1 standard liter per min (SLM); MKS). Although the pressurized CO₂ liquid in the cylinder (∼60 atm) is at room temperature, the temperature of the CO₂ gas drops significantly as it expands adiabatically across the regulator. An inline heater was therefore added to the exit of the CO₂ gas cylinder to prevent the deposition of solid CO₂ in the regulator.

Fig. 2 Experimental set up with a supersonic Laval nozzle. The inset at lower right illustrates the set up for pressure trace measurement.

The second modification is that we pressurized the H₂O (liquid) bottle by connecting it to the flow tube just upstream of the vapor generator. In the current experiments the pressure at the system near the water injection point is around 2.8 atm and this change ensures that the pressures at the inlet and outlet of the peristaltic pump are almost the same, thereby stabilizing the H₂O flow rate.

When the system is running, two high-pressure liquid N₂ Dewars (#1 and #2) connected to inline electrical heaters provide up to 20 mol min⁻¹ of room temperature carrier gas. The flow rate of N₂ from the Dewars is controlled by mass flow controllers (Type 1559A, 200 and 400 SLM; MKS), each calibrated to an accuracy of 1% of reading. Gas from Dewar #2 is heated to about 40–50 °C before entering the vapor generator. Here, the condensable liquid (H₂O) is fed into the system using a peristaltic pump, and part of the N₂ disperses that liquid into a fine spray. The remaining N₂ provides the energy to evaporate the droplets and further dilute the condensable vapor. The flow from the vapor generator merges with the main N₂/CO₂ flow supplied by Dewar #1 and the CO₂ cylinder. The mixed gas flows through a heat exchanger and enters the plenum where the final temperature adjustment is made. The stagnation temperature T₀ is measured in the plenum, using a high accuracy platinum resistance thermometer (RTD). The stagnation pressure p₀ is determined by measuring the static pressure in the region of the nozzle with constant cross sectional area, and correcting for the velocity of the gas. After leaving the plenum, the gas flows through the nozzle and is discharged to the atmosphere through 75 mm tubing by two rotary vane vacuum pumps (Type R5 RA0305D, 0.1 m³ s⁻¹ displacement for each pump; Busch).

For all of the experiments conducted here the desired stagnation temperature T₀ was 15.0 °C (288.2 ± 0.05 K) or 35.0 °C (308.2 ± 0.05 K), and the stagnation pressure p₀ was set to 2.00 atm (202.6 ± 0.4 kPa). When the flow rate of N₂ exceeds the total capacity of the two mass flow controllers, 200 + 400 = 600 SLM, the third liquid N₂ Dewar was connected to the main flow via a mass flow controller (Type 1559A, 215.8 SLM; MKS) in a manner similar to Dewar #1. The H₂O liquid used in this study was deionized water with a resistivity greater than 10 MΩ cm or distilled water with a resistivity of 1 MΩ cm. Preliminary experiments confirmed that CO₂ gases with purities of 99.9% and 99.99% gave the same experimental results. Thus, the lower purity CO₂ (99.9%) was used in most experiments in this study.

We used conventional Laval nozzles with rectangular cross sections. Two nozzles, Nozzle T1 and T3, characterized by different nominal expansion rates and flat-plate side walls were used for the PTM. Two additional nozzles, Nozzle T1_mica and Nozzle T3_mica, with the same nominal expansion rates as Nozzle T1 and Nozzle T3, respectively, but with mica windows in the sidewalls, were used for SAXS experiments. Further details regarding the dimensions of the nozzles are given in Appendix 1.

2.2. Pressure trace measurement

In the pressure trace measurement (PTM), we measure the static pressure along the length of the nozzle using a movable static pressure probe (see the inset at lower right in Fig. 2). The probe is a 0.92 mm outside diameter stainless steel tube sealed at the upstream tip and with 2 equally spaced holes drilled ∼310 mm downstream from the tip.

We also measured the static pressure at the physical throat, where the cross section of the nozzle reaches a minimum, through a small hole (0.34 mm diameter) in the shaped nozzle block in Fig. 14(b). By comparing the pressure at the physical throat with the pressure profile measured by the pressure probe, the distance between the physical throat and the effective throat can be determined. The latter is defined as the position where the effective flow area reaches a minimum and, the pressure ratio (≡pressure at the effective throat/stagnation pressure) takes on the value, p*/p₀ = {2/(γ + 1)}^γ/(γ−1) for constant specific heat ratio, γ.¹⁷ When the heat capacity is strongly temperature dependent, as is the case for CO₂, the correct value of p*/p₀ is determined by using the integration scheme detailed in the Appendix of ref. 18. The distance between the physical throat and the effective throat is required in order to combine the data measured by PTM and SAXS, because the pressure profile is measured as a function of the distance from the effective throat, while the spectra are measured as a function of the distance from the physical throat. For the remainder of this paper the term “throat” will mean the effective throat unless otherwise indicated. The stagnation conditions for the PTMs are summarized in Table 1 together with those for the SAXS measurements.

Table 1 A summary of the experimental stagnation conditions including stagnation pressure, p₀, stagnation temperature, T₀, weight fraction of CO₂ (H₂O), w_CO2 (w_H2O), and mole fraction of CO₂ {H₂O}, (y_CO2)₀ {(y_H2O)₀}. The values in the parentheses are the conditions in SAXS measurements

Nozzle	p0 (kPa)	T0 (K)	wCO2 (−)	wH2O (−)	(yCO2)0 (−)	(yH2O)0 (−)	Symbol
T1 (T1_mica)	202.6	288.2	0 (0)	0.0017 (0.0017)	0 (0)	0.0026 (0.0027)	T1,20
			0.055 (0.056)	0.0016 (0.0017)	0.036 (0.037)	0.0026 (0.0027)	T1,21
			0.109 (0.109)	0.0016 (0.0016)	0.072 (0.072)	0.0026 (0.0027)	T1,22
			0.212 (0.210)	0.0016 (0.0016)	0.146 (0.144)	0.0027 (0.0026)	T1,23
			0.309 (0.281)	0.0016 (0.0015)	0.222 (0.200)	0.0027 (0.0027)	T1,24
			0.210	0	0.145	0	T1,03
T3 (T3_mica)	202.6	288.2	0.037	0.00060	0.024	0.0010	T3,10
			0.053 (0.054)	0.00064 (0.00062)	0.035 (0.035)	0.0010 (0.0010)	T3,11
			0.211 (0.210)	0.00057 (0.00061)	0.146 (0.145)	0.0010 (0.0010)	T3,13
		308.2	0.054 (0.054)	0.0017 (0.0016)	0.035 (0.035)	0.0026 (0.0026)	T3,21
			0.106 (0.106)	0.0017 (0.0016)	0.070 (0.070)	0.0027 (0.0027)	T3,22
			0.211 (0.210)	0.0016 (0.0016)	0.145 (0.145)	0.0026 (0.0027)	T3,23
			0.346 (0.343)	0.0015 (0.0015)	0.252 (0.250)	0.0026 (0.0026)	T3,24
			0.054 (0.054)	0.0032 (0.0032)	0.035 (0.035)	0.0051 (0.0050)	T3,31
			0.104 (0.106)	0.0031 (0.0031)	0.069 (0.070)	0.0049 (0.0050)	T3,32
			0.211 (0.210)	0.0031 (0.0030)	0.145 (0.144)	0.0051 (0.0050)	T3,33
			0.344 (0.343)	0.0028 (0.0028)	0.250 (0.250)	0.0050 (0.0050)	T3,34

---

2.3. Small angle X-ray scattering measurement

The SAXS measurements were performed using the 12-ID_C beamline at the Advanced Photon Source (APS), Argonne National Laboratories, Argonne, IL. In the SAXS measurements, we used a 0.2 × 0.2 mm² beam of 12 keV (λ = 0.10 nm) X-rays with a wavelength spread Δλ/λ = 10⁻⁴. The X-rays were detected by a two-dimensional detector with a 17 × 17 cm² active area, which consists of four 1024 × 1024 pixel charge coupled device chips. The sample-to-detector distance was 2.262 m. One-dimensional scattering spectra were obtained by averaging the two-dimensional data using the APS data inversion program. To make the SAXS measurements as a function of position in the nozzle, we mounted the plenum, supersonic nozzle, and a section of vacuum tubing on a 3-axis motorized stage as shown in Fig. 2.

3. Analytical methods

3.1. Analysis of PTM results

The temperature, mass density, and velocity of the flowing gas mixture, and the heat released due to the phase changes and mass fractions of condensates (CO₂ and H₂O) were derived from the static pressure profile determined by PTM by integrating the one-dimensional adiabatic steady flow equations.¹⁹ In order to solve the flow equations for the binary condensation, we made the same assumption as in ref. 19, i.e.,where p_{v_i} (p_{v_j}) is the partial pressure of species i (j) in the vapor phase, and x^av_i (x^av_j) is the average mole fraction of species i (j) in the condensate. The variable in the right side, p^mix_i (p^mix_j) denotes the equilibrium vapor pressure of species i (j) over the bulk liquid at temperature T and mole fractions x^av_i and x^av_j. The value of p^mix_i is calculated as p^mix_i = p_ix^av_iγ_i, where p_i is the equilibrium vapor pressure of the pure condensed species i and γ_i is the activity coefficient of species i in the condensed phase.

For CO₂ and H₂O mixtures, p_CO2 ≫ p_H2O, and analysis of the PTM based on eqn (1) should match the process described in the Introduction (Fig. 1). That is, initially only H₂O condenses until the H₂O in the vapor phase is completely exhausted, and CO₂ only starts to condense after that. Here, we ignore the small amount of CO₂ that will adsorb to the surface of the particles prior to nucleation/condensation since the heat released by this process is too small to detect.

The assumption expressed by eqn (1) determines the compositions (ratio between the concentrations of CO₂ and H₂O) in the vapor and condensed phases, and hence, can affect the thermodynamic properties of the gas mixture. Under our conditions, however, this effect is insignificant, because the concentration of H₂O is so low (w_CO2 ≤ 0.346 and w_H2O ≤ 0.0032 as shown in Table 1) that the properties of the gas mixture are dominated by those of N₂ and CO₂.

Any effect of eqn (1) on the heat release determined for the phase change should be negligible, and we can investigate the process of the nucleation/condensation of CO₂/H₂O mixture in the supersonic flow based on the heat release derived from the PTM with eqn (1) even if this assumption is not entirely correct. The analytical result is insensitive to the accuracy of the determinations of p^mix_i and p^mix_j in eqn (1), because p_CO2 ≫ p_H2O, therefore, γ_i and γ_j were assumed to be unity for simplicity.

In our earlier work we have shown that the displacement thickness of the boundary layer on the nozzle surface is affected by condensation, and (A/A*)_wet deviates from (A/A*)_dry downstream of the onset point of condensation, where (A/A*)_wet and (A/A*)_dry denote the effective flow area ratios in condensing flow and non-condensing flow, respectively.²⁰ In order to determine (A/A*)_wet, an additional parameter must be measured beyond the static pressure. In this study, we first analyze the PTM results assuming (A/A*) = (A/A*)_dry, and, if necessary, use the results of SAXS or a reasonable alternative assumption to determine the (A/A*)_wet in order to improve the accuracy of the analysis.

The thermodynamic properties of the materials used in this work are summarized in Appendix 2.

3.2. Analysis of the SAXS measurements

In the current experiments we found that the SAXS spectra of the H₂O or H₂O/CO₂ aerosol could not, in most cases, be modelled as scattering from a collection of polydisperse spheres. For the H₂O particles, we expect that this arises because the low temperatures and rapid cooling rates characteristic of the current experiments prevent full scintering as particles coagulate. Futhermore, the H₂O/CO₂ particles may adopt a non-spherical shape as suggested by mid-infrared extinction spectroscopies.^11,13,14 The question of particle structure is the subject of ongoing research, and in this study we characterize particle size by using the model independent Guinier analysis²¹ to determine the radius of gyration of the particles.

According to the Guinier law, scattering intensity I(q_s) is expressed by eqn (2) in the limit of small q_s as,

where q_s is the scattering vector (momentum transfer vector), and r_G denotes the radius of gyration. The radius of gyration can, therefore, be determined from the slope of the Guinier plot, ln[I(q_s)] vs. q_s².

4. Experimental results and discussion

4.1. Phase changes detected by pressure trace measurements

Fig. 3(a) illustrates a typical pressure profile measured in Nozzle T1 together with the derived temperature profile. The isentropic pressure p_is and temperature T_is profiles are also shown, where these values correspond to the fictitious expansion of a gas mixture with the same physical properties as those of the condensing gas mixture but where neither condensation nor clustering occur. It is clear in Fig. 3(a) that T deviates from T_is in two distinct steps, suggesting two distinct phase transitions, where the first deviation is observed around z = 1 cm, and the start of the second one is around z = 8 cm. Applying our conventional criterion for the onset of condensation, T − T_is = 0.5 K, we find that the first deviation starts at z = 0.6 cm.

Fig. 3 Typical results of PTM. (a) Pressures and temperatures. (b) Partial pressures of CO₂ and H₂O compared with the corresponding onset pressures required for homogeneous nucleation of the pure species, respectively.

To identify the nature of these phase transitions we turn to Fig. 3(b). This figure shows the partial pressures of CO₂ and H₂O in the isentropic flow, as well as the pressures required for the onset of CO₂ and H₂O condensation initiated by homogeneous unary nucleation of each species in supersonic flow. The latter come from empirical equations for the onset pressure, p_on = a₀ exp (a₁T), that we derived by fitting to the experimental results in ref. 4 for p_{on_CO2} and in ref. 5 for p_{on_H2O}. The two equations for p_{on_CO2} correspond to the measurements made in Duff's Nozzle I (d(A/A*)/dz = ∼0.7 cm⁻¹) in the temperature range 160.8 ≤ T/K ≤ 182.6, and Nozzle II (d(A/A*)/dz = ∼0.07 cm⁻¹) at temperatures 167.2 ≤ T/K ≤ 193.1. The equation for p_{on_H2O} corresponds to measurements in Khan et al.'s nozzle (d(A/A*)/dz = 0.0477 cm⁻¹) at temperatures 191.7 ≤ T/K ≤ 232.4. The expansion rate of Nozzle T1 (d(A/A*)/dz = ∼0.17 cm⁻¹) is intermediate to those of Duff's nozzles and a factor of ∼3.6 times higher than Khan et al.'s nozzle. Although onset conditions depend on the expansion rate of the nozzle, this dependence is not strong and our equations should be accurate enough to determine whether homogeneous nucleation is possible and, if so, to estimate the onset points in Nozzle T1.

As shown in Fig. 3(b), the onset pressure of CO₂ lies 1–2 orders of magnitude above the isentropic partial pressure of CO₂ in the experiment, thereby confirming that homogeneous nucleation of pure CO₂ cannot occur in Nozzle T1 under these conditions. In fact, similar analysis precludes homogeneous nucleation of pure CO₂ under any conditions used in this study. In contrast, for H₂O the p_{is_H2O} line crosses the p_{on_H2O} line at z = 0.6 cm, the location where we first observe T deviate from T_is in Fig. 3(a). This observation strongly suggests the first deviation of T from T_is is caused by condensation of H₂O initiated by homogeneous nucleation of H₂O.

In order to confirm this finding, we investigated the latent heat, q, release profiles derived from the PTMs for a series of experiments with a constant H₂O mass fraction and CO₂ mass fractions varying from 0 to 0.309. Fig. 4 shows these profiles, labeled (1)–(5), for experiments conducted in Nozzle T1, where the thick black solid line corresponds to the experiment illustrated in Fig. 3. For z < 6 cm, all of the profiles align except for small fluctuations. In particular, q starts to increase at z = 0.6–0.8 cm and, after a steep increase, reaches a constant value near z = ∼1.5 cm that persists as long as z < 6 cm. Hence, it is clear that CO₂ does not contribute to the phase change detected by the first increase in q, and that the initial heat release is solely due to the homogeneous nucleation/condensation of H₂O.

Fig. 4 Latent heat released per unit mass of gas mixture, q.

For experiments (1)–(5) in Fig. 4, the temperature at z = 6 cm is less than 150 K, a value that is far below the lowest temperature, 202 K, at which Manka et al.²² observed the onset of H₂O droplet freezing in a supersonic nozzle. Hence, the H₂O droplets produced in Nozzle T1 should freeze well before reaching this point. Furthermore, since the vapor pressure of H₂O ice is only 6 × 10⁻⁶ Pa at T = 150 K,⁸ by this point in the expansion, almost all of the H₂O molecules in the core of the flow should be in the condensed phase.

Thus, the second heat release, observed when z > 6 cm in the flows (2)–(5) that contain both H₂O and CO₂, must be due to a CO₂ phase change. The fact that no heat is released in flow (6), where CO₂ is the only condensable species, confirms our earlier analysis that homogeneous nucleation of CO₂ cannot occur under the conditions in this study. We therefore attribute the second increase in q to the heterogeneous nucleation of CO₂ on the H₂O ice particles.

On the basis of PTM alone, we can confirm our intuitive picture that as the CO₂/H₂O mixture expands in the supersonic flow, H₂O ice particles are first produced by homogeneous nucleation and subsequent freezing. A further decrease in the temperature then drives the heterogeneous nucleation of CO₂ onto the H₂O ice particles.

4.2. Determining the onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particle in Nozzle T1

In our earlier homogeneous nucleation studies in supersonic nozzles, the onset point was defined as the position where the gas temperature derived from PTM is 0.5 K higher than the isentropic temperature.²³ At the onset point for the heterogeneous nucleation of CO₂ on H₂O ice particles, however, the temperature has already deviated from the isentropic value due to the condensation of H₂O.

In this study we therefore determined the onset point for heterogeneous nucleation of CO₂ from the latent heat, q, release curves. As illustrated in Fig. 5, we first fit a straight line (broken line) to q between the first and second onset points and used this as the base line. A quadratic curve (dotted line) was then fit to the q values downstream of the second onset point using the constraint that this curve is tangent to the straight line. The tangent point between the quadratic curve and the straight base line was defined as the onset point. The quadratic curve fit was restricted to values of q that were 0.52 to 1.56 kJ kg⁻¹ above the baseline, which corresponds to an increase in temperature of 0.5–1.5 K for pure nitrogen gas at 298.15 K.

Fig. 5 The method used to determine the onset point for heterogeneous nucleation of CO₂ on H₂O ice particle.

A subtle point to consider here is that heat release to a supersonic flow can affect boundary layer development, and hence the values of the variables (T, q, g_CO2, g_H2O, u, ρ) estimated from the PTM downstream of the onset of condensation, where g_CO2 and g_H2O denote mass fractions of condensates of CO₂ and H₂O, respectively, and u and ρ are the velocity and mass density of gas mixture, respectively. Fortunately, as discussed in Appendix 3, for the experiments conducted in Nozzle T1 the effect of H₂O condensation on the boundary layer was negligible and we could assume (A/A*)_wet = (A/A*)_dry prior to CO₂ condensation.

4.3. Analytical results of pressure trace measurements with Nozzle T3 for determining the onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particle

Although experiments in Nozzle T1 let us confirm the basic physics, nucleation of CO₂ on ice occurred fairly close to the end of the nozzle and we were not able to follow aerosol evolution to the point where CO₂ condensation was complete. In order to do so we designed a faster nozzle, T3, and Fig. 6(a) illustrates the effective flow area ratios, (A/A*)_dry, and Mach numbers for pure nitrogen flows at two inlet temperatures. The results in Nozzle T1 are also shown for comparison. The data in this figure confirm that such small changes in temperature do not affect A/A* and Mach number, and that the expansion rate, d(A/A*)/dz = ∼0.38 cm⁻¹, of Nozzle T3 is 2.2 times that of Nozzle T1, ∼0.17 cm⁻¹.

Fig. 6 Flow properties in Nozzle T3. (a) (A/A*)_dry and Mach number compared with those in Nozzle T1, respectively. (b) Typical temperature profiles corresponding to experiments in Nozzle T3 at the same CO₂ partial pressure and different H₂O partial pressures.

Typical temperature profiles in Nozzle T3 are illustrated in Fig. 6(b), and exhibit the same features as those observed in Nozzle T1. In particular, the first small deviation of T from T_is occurs at the onset point for the homogeneous nucleation of H₂O, and the deviation is almost constant until the second stronger deviation starts due to the heterogeneous nucleation of CO₂.

The onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particle in Nozzle T3 were determined from the q profiles shown in Fig. 7 using the method described above for Nozzle T1. In analyzing the nozzle T3 data, however, we found that we did need to correct for the deviation of (A/A*)_wet from (A/A*)_dry in the flows with higher water content illustrated in Fig. 7(b) and (c). The approach we used to do so is outlined in Appendix 3.

Fig. 7 The latent heat, q release profiles determined for experiments in Nozzle T3.

4.4. Onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particle in nozzles T1 and T3

The onset conditions for heterogeneous CO₂ nucleation measured in both Nozzles T1 and T3 are summarized in Table 2 and plotted in Fig. 8, together with the vapor pressures of CO₂ liquid and CO₂ solid, p_CO2(l) and p_CO2(s). Here the values p_CO2(l) are estimated by extrapolating the empirical equation²⁴ for 217 ≤ T/K ≤ 276. It is striking how the data corresponding to the same weight fraction of H₂O in a particular nozzle line up consistently.

Table 2 Onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particle in supersonic flow, partial pressure of CO₂, p_{on_CO2}, temperature, T_{on_CO2}, and radius of gyration of H₂O ice particle, r_{G_on}

Nozzle	Symbol	pon_CO2 (kPa)	Ton_CO2 (K)	rG_on (nm)
a rG_on for T3,10 was determined by linearly extrapolating the two rG_on for T3,11 and T3,13 to the onset point for T3,10.
Nozzle T1	T1,21	0.45	133.9	4.3
	T1,22	0.98	137.4	4.2
	T1,23	2.2	141.7	4.2
	T1,24	3.5	145.9	4.1
Nozzle T3	T3,10	0.24	123.7	2.3^a
	T3,11	0.38	126.9	2.2
	T3,13	2.0	137.4	2.1
	T3,21	0.31	130.2	3.2
	T3,22	0.70	135.0	3.1
	T3,23	1.5	137.4	3.0
	T3,24	2.8	141.7	3.1
	T3,31	0.30	131.0	3.6
	T3,32	0.68	136.3	3.6
	T3,33	1.5	139.4	3.7
	T3,34	2.8	143.8	3.7

---

Fig. 8 Onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particles, p_{on_CO2} vs. T_{on_CO2}. The radii of gyration of H₂O ice particles at the onset points, r_{G_on}, were determined by SAXS as described in Section 4.6. ^a Equilibrium vapor pressure of CO₂ liquid. Equation in ref. 24 for 217 ≤ T/K ≤ 276 is extrapolated towards supercooled temperature. (see Appendix 2). ^b Equilibrium vapor pressure of CO₂ solid. (see Appendix 2).

In the legend of Fig. 8, the radii of gyration of the H₂O ice particles at the onset points, r_{G_on}, are also indicated. These were determined by SAXS as we explain below in Sections 4.5 and 4.6. In Nozzle T3, the onset conditions move towards lower saturation (lower partial pressure and/or higher temperature) as w_H2O (r_{G_on}) increases. Finally, the data at the highest w_H2O (largest r_{G_on}) in Nozzle T3 (open circles) seem to align closely with the data measured in Nozzle T1 on slightly larger particles (closed circles). The dependence of the onset conditions on r_{G_on} will be discussed further in Section 4.6.

4.5. Flow properties in Nozzle T1_mica and Nozzle T3_mica used in SAXS measurements

In order to make the SAXS measurements we used Nozzles T1_mica and T3_mica. These nozzles have mica windows mounted on the sidewalls as illustrated in Fig. 15 of Appendix 1. Fig. 9 shows the effective flow area ratios and temperature profiles in those nozzles, derived from the PTMs of pure N₂ flow. Near the throat the profiles agree reasonably well with those in Nozzle T1 and Nozzle T3, respectively, but deviations between the matched nozzles increase further downstream. The large deviations near z = ∼8.5 cm, just downstream of the edge of the mica window, are attributed to the abrupt end of the window groove 8.38 cm downstream of the physical throat (see Fig. 15(a)). Since we cannot make SAXS measurements beyond the end of the window, these large deviations are irrelevant. As shown in Fig. 9(b), the temperature deviations between Nozzle T1_mica (T3_mica) and Nozzle T1 (T3) reflect the deviations in A/A*.

Fig. 9 Flow properties in Nozzle T1_mica and Nozzle T3_mica. The deviations between the nozzles are discussed in the text.

Although a better match between the two sets of nozzles would be ideal, the differences do not impact our results because in the region where H₂O nucleation/condensation occurs the expansion rates are actually quite close for matched nozzles. In particular, the average expansion rates are d(A/A*)_dry/dz = 0.21 cm⁻¹ in Nozzle T1 versus 0.22 cm⁻¹ in Nozzle T1_mica between z = 0.6 cm and 1.3 cm, and d(A/A*)_dry/dz = 0.49 cm⁻¹ in Nozzle T3 versus 0.50 cm⁻¹ in Nozzle T3_mica between z = 0.5 cm and 1.5 cm. Similar expansion rates in this critical region of the nozzle should ensure that the size distributions of the H₂O ice particle produced during the SAXS experiments are almost the same as those produced during the PTMs.

4.6. Particle size from SAXS

Typical SAXS data are displayed on an absolute intensity scale in a Guinier plot in Fig. 10. The thick black lines are the fits to the data used to determine the radius of gyration, r_G, of the CO₂/H₂O particles. The values of r_G are shown in Fig. 11 as a function of position in the nozzle, and the conditions corresponding to the experiments are summarized in Table 1. The rapid increase in particle size, 3–5 cm downstream of the throat, corresponds to the onset of CO₂ condensation. Upstream of the onset point, r_G is larger for higher values of w_H2O and/or when particles form in less rapidly expanding flows. These observations agree with our previous observations about H₂O and D₂O.^22,23,25 Furthermore, r_G does not show any systematic dependence on w_CO2, again emphasizing that CO₂ does not appear to play a role in the H₂O ice particle formation process.

Fig. 10 Typical Guinier plots of the SAXS data used to determine the radii of gyration of the particles.

Fig. 11 Radii of gyration determined from Guinier plots of SAXS spectra. Flow conditions are indicated by the symbols in Table 1.

In Nozzle T3_mica, the values of r_G at the onset point for heterogeneous nucleation of CO₂ on H₂O ice particle were determined by linearly extrapolating the values of r_G measured upstream of the onset point. In Nozzle T1_mica (T1,21 ∼ T1,24), CO₂ only starts to condense when z = 6.7–8.5 cm, and thus, the r_G values at z = 1.9 and 2.9 cm in Fig. 11 are measured far upstream of the onset points. In these cases we extrapolated the r_G values at z = 2.9 cm to the corresponding onset points, by assuming that the slope (dr_G/dz) is equal to that determined for pure H₂O (T1,20) between z = 2.9 and 7.9 cm.

As illustrated in Fig. 8, the plots at almost the same r_G seem to lie along a line, and these lines move systematically toward lower saturation (lower partial pressure and/or higher temperature) as r_G increases, until the lines corresponding to (3) and (4) almost coincide. According to Fletcher's heterogeneous nucleation theory,²⁶ the rate of heterogeneous nucleation on a spherical particle increases with the radius of particle. The fact that the onset conditions (3) and (4) in Fig. 8 agree suggests that these values should be close to those for the heterogeneous nucleation of CO₂ on the flat H₂O ice surface at these saturations. Thus, we first directly compare our results, (3) and (4) in Fig. 8, to those of Glandorf et al.¹⁰

In Glandorf et al.'s FTIR experiments,¹⁰ the onset of heterogeneous nucleation of CO₂ solid on a flat H₂O ice surface, i.e. the critical saturation for CO₂ solid S_CO2(s), was S_CO2(s) = 1.34 in the temperature range T = 130.2–140.0 K. Our experiments cover essentially the same temperature range but the lowest critical saturations with respect to the CO₂ solid, i.e. those for the results (3) and (4) in Fig. 8, are in the range S_CO2(s) = 6.4–8.2, values that are much higher than those reported by ref. 10.

One reason for the difference in S_CO2(s) is that the timescales in the nozzle experiments are much shorter than those accessed by ref. 10. In particular, in the FTIR studies, heterogeneous nucleation was observed within 10 s after the saturation of CO₂ reached the critical value (onset condition). In our study, the travel time from the point where S_CO2(s) = 1.34 to the onset point is ∼20 μs for results (3) and ∼50 μs for results (4). A difference in time scales of 5–6 orders of magnitude should not, however, require such a large difference of the critical saturations. In Fig. 6 in ref. 27, for example, the nucleation rate is estimated to increase by a factor of 10⁷ when S_CO2(s) is increased from 1.34 to 1.40 under conditions corresponding to the experiments in ref. 10.

A possible explanation for the large difference in the onset conditions between the current study and the work of ref. 10 is that heterogeneous nucleation of CO₂ onto the H₂O ice particles starts from the supercooled liquid phase rather than the solid phase on the ∼50 μs timescale. This, in turn, suggests that freezing of the supercooled CO₂ liquid adsorbed on the surface of H₂O ice takes more than 50 μs under the conditions investigated here.

For heterogeneous nucleation of CO₂ to start from the supercooled liquid phase, the saturation of CO₂ with respect to the supercooled liquid, S_CO2(l), must be equal to or greater than unity at the onset conditions determined here. Although the onset conditions corresponding to (3) and (4) in Fig. 8 are below the vapor–liquid equilibrium line (V–L line) extrapolated from the experimental equation for 217 ≤ T/K ≤ 276, there is no reason to assume that the extrapolated line accurately represents the V–L line in the supercooled state, and we expect the V–L line could lie on or below the plots of the onset conditions (3) and (4).

Our interpretation is consistent with the analysis of experimental results by Mensah,²⁸ where heterogeneous nucleation of Ar on H₂O ice particles (radius was expected to be larger than 20 nm) was observed in a cryogenic nucleation pulse chamber in the timescale of a few 10 ms. The onset pressures determined by Mensah at temperatures between 52 K and 72 K (below the triple point 83.3 K), were about 2–5 times higher than the vapor pressure of Ar solid, and strongly suggested the heterogeneous nucleation of Ar on H₂O ice particles occurs from the supercooled liquid phase.

On the timescale of about 10 s, heterogeneous nucleation of CO₂ on an H₂O ice surface occurs in the solid phase as indicated in ref. 10. At the beginning of the nucleation process, however, the CO₂ on the H₂O ice film may be liquid as found in this study, because Glandorf et al.'s experiments were done under conditions (temperature and saturation of CO₂) similar to our study. If this is true, then freezing of the supercooled CO₂ liquid deposited on the H₂O ice surface is required for heterogeneous nucleation to proceed at S_CO2(l) < 1 ≤ S_CO2(s). The time lag related to the freezing may be a reason why heterogeneous nucleation does not occur in the solid phase on the short timescale of our experiment. Further investigation is necessary to confirm this interpretation.

The dependence of the onset conditions on the radius for the smallest H₂O ice particles, (1) and (2) in Fig. 8, is also the subject for a future study.

5. Further discussion

5.1. Contact parameter between supercooled CO₂ liquid and H₂O ice

In order to predict the vapor–liquid equilibrium line of CO₂ from the onset conditions determined in this study, we first need to estimate the contact parameter of liquid CO₂ on H₂O ice. The contact parameter of solid CO₂ on H₂O ice, m_CO2(s) and that of liquid CO₂ on H₂O ice, m_CO2(l) are defined as,where σ_H2O, σ_CO2(s), and σ_CO2(l) are the surface free energies of H₂O ice, CO₂ solid, and CO₂ liquid, respectively, and σ_H2O/CO2(s) and σ_H2O/CO2(l) denote the interfacial free energies between H₂O ice and CO₂ solid, and between H₂O ice and CO₂ liquid, respectively. The value of m_CO2(s) = 0.952 is that estimated by Glandorf et al.¹⁰

To estimate the value of m_CO2(l) we substitute the values of the surface free energies,²⁷ σ_H2O = 0.106 J m⁻² and σ_CO2(s) = 0.080 J m⁻² into eqn (3) assuming m_CO2(s) = 0.952 to obtain σ_H2O/CO2(s) = 0.030 J m⁻². A Block equation,²⁹ using parameters obtained from experimental values of σ_CO2(l) in the range 221.0 ≤ T/K ≤ 293.2, was extrapolated to the relevant temperature range, 130 to 140 K, and gave an average value of σ_CO2(l) = 0.038 J m⁻². Although the value of σ_H2O/CO2(l) is not available in the literature, we can assume that σ_H2O/CO2(l) < σ_H2O/CO2(s), based on the Duprés equation,³⁰ σ_H2O/CO2(l) = σ_H2O + σ_CO2(l) − W_H2O/CO2(l) (for liquid CO₂), where W_H2O/CO2(l) denotes the work of adhesion at the interface. As noted above, σ_CO2(l) < σ_CO2(s) and W_H2O/CO2(l) should be larger than W_H2O/CO2(s) given that liquid CO₂ can move more freely and find a more stable configuration on the interface than solid CO₂. Using these vales we estimate m_CO2(l) as m_CO2(l) > (0.106 − 0.030)/0.038 = 2 > 1. Since the maximum possible value for the contact parameter is 1, this results suggests m_CO2(l) = 1 and the contact angle is zero, i.e. the CO₂ liquid fully wets the H₂O ice surface and the critical saturation of CO₂ on the flat H₂O ice surface is unity.

If these arguments are valid, the V–L line for supercooled CO₂ liquid should lie very close to curves (3) and (4) in Fig. 8. The fully wetting behavior of CO₂ molecules on H₂O ice surface at a temperature of 50 or 100 K observed in the molecular dynamics simulation¹⁵ may be attributed to the supercooled CO₂ liquid in a period prior to the freezing, because the simulation time was 3.5 ns, which is much shorter than the 50 μs travel time in the supersonic flow between the onset point expected for CO₂ solid and the onset point for CO₂ liquid observed in this study.

5.2. Condensation of CO₂ on H₂O ice particle downstream of onset point

Even if the initial critical nucleus of CO₂ is liquid like, at these low temperatures CO₂ is unlikely to remain in this state for long. To investigate the progress of CO₂ condensation/deposition onto the H₂O ice particles we therefore extended the PTM analysis downstream to the nozzle exit. Since heat release is high and will affect the boundary layers in this region, we first determined (A/A*)_wet using the r_G values in Fig. 11 as follows.

The volume of condensed droplet per unit mass of gas mixture, V′_c can be approximately derived from r_G neglecting the effect of size distribution and assuming spherical shape as,

V′_c = N′_c(4π/3)r_G³, (5)

where N′_c denotes the number of droplets per unit mass of gas mixture and is assumed to be constant downstream of the onset point. On the other hand, V′_c can be determined from the mass fractions of the condensed CO₂ and H₂O derived from PTM, g_CO2 and g_H2O, as,

V′_c = g_CO2/d_CO2 + g_H2O/d_H2O, (6)

where d_CO2 and d_H2O are the densities of the condensed CO₂ and H₂O, respectively.

The effect of the deviation of (A/A*)_wet from (A/A*)_dry on the determination of mass fraction of condensate is negligible just downstream of the onset point as shown in Fig. 6 in ref. 31. Hence we can determine N′_c so that V′_c in eqn (5) agrees with V′_c in eqn (6) just downstream of the onset point, where g_CO2 and g_H2O are derived assuming (A/A*)_wet = (A/A*)_dry. Using this value of N′_c, we determined (A/A*)_wet so that V′_c in eqn (6) agree with V′_c in eqn (5) downstream of the onset point. In eqn (6), d_H2O = 930 kg m⁻³ is an averaged value of the density of H₂O ice (Ih) in the temperature range T = 130–140 K, and d_CO2 = 1600 kg m⁻³ is the value used by ref. 27 and corresponds to the density of CO₂ solid at T = 168.2 K,³² and is very close to the density at the temperature in this analysis (T = ∼140 K), 1630 kg m⁻³. We used the density of CO₂ solid for analysis, though the heterogeneous nucleation of CO₂ was found to start in liquid phase as discussed above, because, the condensation/deposition of CO₂ after nucleation was found to proceed via the solid phase as explained below.

The values of (A/A*)_wet and V′_c determined using this analysis are shown in Fig. 12. As shown in Fig. 12(a), N′_c was determined to be 3.8 × 10¹⁸ kg⁻¹ so that V′_c in eqn (5) agree with V′_c from PTM with (A/A*)_dry just downstream of the onset point. And then (A/A*)_wet was determined as indicated in Fig. 12(b) so that V′_c from PTM using (A/A*)_wet reproduces V′_c in eqn (5) well in the whole region downstream of the onset point as shown in Fig. 12(a).

Fig. 12 SAXS and PTM data were combined to determine the deviation of (A/A*)_wet from (A/A*)_dry downstream of the onset point for heterogeneous nucleation of CO₂ as described in the text. (a) Volume of condensed droplet per unit mass of gas mixture, V′_c. (b) Flow area ratio, A/A*.

By using (A/A*)_wet determined above, the partial pressure of CO₂ and gas temperature in the condensing flow were derived and the partial pressure–temperature diagram, p_{v_CO2}–T line, are shown in Fig. 13. We restricted the p_{v_CO2}–T line to z < 8 cm, because the temperature is significantly underestimated for z ≥ ∼8 cm. The problem arises because the smaller expansion rate in Nozzle T3_mica relative to Nozzle T3 (Fig. 9(a)) leads to underestimated values of r_G, and thus V′_c {eqn (5)} in Fig. 12(a) when z > ∼6 cm.

Fig. 13 The expansion history is followed on a partial pressure–temperature diagram. The onset of CO₂ heterogeneous condensation is detected well before the lowest temperature reached in the expansion.

In Fig. 13 the onset points (3) and (4), that are expected to be very close to the V–L line of the supercooled CO₂ liquid, are also shown. The solid line is a tentative V–L line to guide the eye, where the slope of this line (that is the heat of vaporization) is constrained to be less than that of the solid. As shown in the figure, downstream of the onset point the p_{v_CO2}–T line crosses the V–L line and moves toward V–S line (vapor–solid equilibrium line), suggesting the supercooled CO₂ liquid adsorbed on the surface of H₂O ice freezes just after the nucleation/condensation start. The travel time between the onset point and the intersection of p_{v_CO2}–T line and V–L line is 28 μs, hence the freezing should occur within 28 μs at the most after the onset.

It is not yet known what triggers the freezing of the supercooled CO₂ liquid on H₂O ice particle. The decrease in temperature downstream of the onset point should not be the only trigger for freezing, because the nucleation is expected to start in liquid phase even at T = 124 K as shown in Fig. 8. Further investigation is necessary to elucidate the mechanism of the freezing of CO₂ adsorbed on the H₂O ice surface.

6. Conclusion

We have conducted the first systematic studies of heterogeneous nucleation in supersonic nozzles. Our findings contribute to elucidating the mechanism of heterogeneous nucleation, with particular reference to proposed innovative CO₂ capture technologies and cloud formation in the Martian atmosphere. We found that in the supersonic flow of gas mixture (CO₂ + H₂O + N₂) heterogeneous nucleation of CO₂ occurs on the H₂O ice particles produced by the homogeneous nucleation/condensation and freezing of H₂O. The data suggest that the heterogeneous nucleation of CO₂ may start from the supercooled liquid phase in the time scale of at least ∼50 μs, and that the supercooled CO₂ liquid on the H₂O ice freezes just after the onset of the nucleation within at most ∼30 μs and subsequent condensation proceeds in the solid phase.

Appendix 1

Supersonic Laval nozzles used in this study

The supersonic nozzle is assembled from the symmetrically-arranged same-shaped two aluminum blocks in Fig. 14(b) and two flat side walls as shown by the cross section in Fig. 14(a). The dimensions of the shaped block are shown in Fig. 14(c). The nominal opening angle of the nozzle made of these blocks is 6.38°. For PTM, we assembled two nozzles with different throat height, h*, from these blocks, namely, Nozzle T1 and Nozzle T3, where h* = 0.500 cm for T1 and h* = 0.211 cm for T3. The smaller h* gives larger nominal expansion rate, d(A/A*)/dz′, where A and A* denote the cross section of the nozzle (nominal flow area) and that at the physical throat, respectively, and z′ is the distance from the physical throat, where the nominal flow area has a minimum. When the Nozzle T1 is used, the third liquid N₂ Dewar was connected to the main flow, because the required N₂ flow was about 670 SLM.

Fig. 14 Supersonic Laval nozzle used in PTM. (a) Cross section of assembled nozzle with flat side walls for PTM. (b) Inner blocks determine the flow area of the nozzle. (c) Dimension of the inner block.

For SAXS experiments, we assembled Nozzle T1_mica and Nozzle T3_mica from the same shaped blocks as those for Nozzles T1 and T3, and from the sidewalls with mica window shown in Fig. 15(a). As shown in the figure, mica window starts at 0.51 cm upstream from the physical throat and ends at 8.38 cm downstream. The cross section of the assembled nozzle is shown in Fig. 15(b). As shown in the figure, the flow area in this nozzle includes the two of small rectangular (0.074 cm × 0.089 cm) in addition to the rectangular in the Nozzles T1 and T3 {see Fig. 14(a)}. Therefore, the throat height of Nozzle T1_mica (T3_mica) was set to 0.480 cm (0.191 cm) so that the nominal flow area of Nozzle T1_mica (T3_mica) is the same as that of Nozzle T1 (T3) for −0.51 ≤ z′/cm ≤ 8.38.

Fig. 15 Supersonic Laval nozzle used in SAXS measurement. (a) Side wall with a mica window. (b) Cross section of assembled nozzle with mica windows.

Appendix 2

Thermodynamic properties of materials

Molar isobaric heat capacity of N₂ gas, C_{p_N2(v)} = 29.124 J mol⁻¹ K⁻¹ at T = 298.15 K was used. Ideal gas molar isobaric heat capacity of H₂O for 160 ≤ T/K ≤ 340 in ref. 19 was extrapolated down to 110 K. The relative error caused by this extrapolation is less than 0.02%.³³ The ideal gas molar isobaric heat capacity of CO₂ was obtained by fitting a quadratic function to the data for 100 ≤ T/K ≤ 400 in ref. 34 as,

C⁰_{p_CO2(v)} = 25.92 + 2.930 × 10⁻²T + 2.38 × 10⁻⁵T² (J mol⁻¹ K⁻¹).

Vapor pressure of CO₂ solid is given in ref. 35 as,

log(p_CO2(s)/bar) = 6.81228 − 1301.679/(T − 3.494),

which well reproduces the data for 65 ≤ T/K ≤ 195 in ref. 36. The heat of sublimation of CO₂ solid was obtained by applying the Clausius–Clapeyron equation to the p_CO2(s) as,

Δh_{sub_CO2} = 2.303R × 1301.679T²/(T − 3.494)² (J mol⁻¹).

Vapor pressure of CO₂ liquid for 217 ≤ T/K ≤ 276 is given in ref. 24 as,

log(p_CO2(l)/atm) = −1353.202/T − 8.142537 log T + 6.259156 × 10⁻³T + 24.61930.

Vapor pressure and heat of vaporization of H₂O liquid for 123 ≤ T/K ≤ 350 in ref. 19 were used extrapolating down to 110 K in this study. The accurate value of vapor pressure is not necessary for the analysis in this study as described in the explanation about eqn (1). The H₂O droplet is expected to be frozen at the onset point for the heterogeneous nucleation of CO₂ as described in the main text. However, at the onset temperature or less, the heat of vaporization of H₂O liquid is smaller than the heat of sublimation of H₂O solid (hexagonal or cubic ice) only by 3%,⁸ therefore, we can safely use the empirical equation for the H₂O liquid without warring about where the H₂O droplet freeze in the supersonic nozzle.

Appendix 3

Correction of effective flow area ratio in condensing flow

As mentioned in the Section 3, (A/A*)_wet deviates from (A/A*)_dry downstream of the onset point, however, this deviation is small just downstream of the onset point and negligible for determination of the onset conditions for the typical homogeneous nucleations.¹⁹ For determination of the onset conditions for heterogeneous nucleation of CO₂ on H₂O ice particle, however, the deviation of (A/A*)_wet from (A/A*)_dry may not be negligible, because (A/A*)_wet should be affected by the condensation of H₂O.

Fig. 16 shows mass fraction ratio of H₂O in Nozzle T1, g_H2O/w_H2O, where g_H2O and w_H2O are the mass fractions of condensed H₂O and all H₂O, respectively. This value is expected to reach unity at z = 6 cm, where the condensation and freezing of H₂O should be completed as explained in Section 4.1. As shown in the figure, the values of g_H2O/w_H2O range 0.9 to 1.0 at z = 6 cm. These underestimations of g_H2O/w_H2O can be explained by the increase in (A/A*)_wet relative to (A/A*)_dry, and should be accompanied by the underestimations of the temperature.²⁰ We adjusted the values of (A/A*)_wet so that g_H2O/w_H2O reach unity at z = 6 cm, and found the increases in the temperature due to these corrections are 0.3 K or less, which are negligibly small. Therefore, we approximated (A/A*)_wet as (A/A*)_dry for the analyses of the flows in Nozzle T1.

Fig. 16 Mass fraction ratio of H₂O, g_H2O/w_H2O, where g_H2O and w_H2O are the mass fractions of condensed H₂O and all H₂O, respectively.

In Nozzle T3, however, the effect of the deviation of (A/A*)_wet from (A/A*)_dry on the determination of temperature was estimated to be about 1 K under the conditions in Fig. 7(b) and (c). Therefore, for the flows in Fig. 7(b) and (c), we determined the (A/A*)_wet so that the mass fraction ratio, g_H2O/w_H2O reaches unity as shown in Fig. 17 for a condensing flow as an example. As shown in Fig. 17(a), the g_H2O/w_H2O derived assuming A/A* = (A/A*)_dry peaks at only about 0.75, therefore (A/A*)_wet was determined as shown in Fig. 17(b) so that g_H2O/w_H2O reaches unity as indicated in Fig. 17(a). We assumed a simple function for the deviation of (A/A*)_wet from (A/A*)_dry as shown in Fig. 17(b), that is, increases linearly and reaches a constant. This shape is consistent with our experimental results {Fig. 8(a) and 10(b) in ref. 20}, where the change of the boundary-layer displacement thickness reaches a constant value downstream of the onset point of condensation. However, we neglected the decrease of (A/A*)_wet (that is the increase of the displacement thickness) just downstream of the onset point shown in those figures in ref. 20, because the change of (A/A*)_wet near the onset point of H₂O condensation (first onset point) should not affect the conditions at the onset point for the heterogeneous nucleation of CO₂, which is far downstream of the first onset point. The start point of the deviation of (A/A*)_wet was arbitrary set to the point where the slope of g_H2O/w_H2O has a maximum, and the endpoint of the increase was determined so that g_H2O/w_H2O peaks at unity except fluctuation by trial and error. The difference of the temperatures derived with (A/A*)_wet and with (A/A*)_dry, T_(A/A*)wet − T_(A/A*)dry is 1.2 K at most as shown in Fig. 17(c), and, under the conditions in this study, this difference was 1.4 K or less.

Fig. 17 The correction to the flow area ratio in a condensing flow is required because heat addition compresses the boundary layers relative to the flow of carrier gas alone. (a) Mass fraction ratio of H₂O, g_H2O/w_H2O. (b) The effective flow area ratio, A/A*. (c) The gas temperatures derived with (A/A*)_dry or (A/A*)_wet, T_(A/A*)dry and T_(A/A*)wet differ by less than 1.2 K.

Acknowledgements

We thank Anthony Castrogiovanni for his input and useful discussions during the course of the work. We thank Judith Wölk, S. Seifert and R. Winnans for their help in the SAXS experiments. The work was initially funded through the National Energy Technology Laboratory (NETL), DOE Carbon Capture Program, with Orbital ATK as a prime contractor (DE-FE0013122). Further support was provided by NSF grant CHE-1213959. This research used resources of the Advanced Photon Source, a U. S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.

References

1. N. Sipöcz, A. Hernandez-Nogales, M. A. Gonzalez-Salazar, R. Shisler and V. Lissianski, Energy Procedia, 2013, 37, 1228–1238.
2. P. M. Sforza, A. Castrogiovanni and R. Voland, Appl. Therm. Eng., 2012, 49, 154–160.
3. D. Berstad, R. Anantharaman and P. Nekså, Int. J. Refrig., 2013, 36, 1403–1416.
4. K. M. Duff, Non-equilibrium condensation of carbon dioxide in supersonic nozzles, Doctor thesis, Massachusetts Institute of Technology, 1966.
5. A. Khan, C. H. Heath, U. M. Dieregsweiler, B. E. Wyslouzil and R. Strey, J. Chem. Phys., 2003, 119, 3138–3147.
6. R. Span and W. Wagner, J. Phys. Chem. Ref. Data, 1996, 25, 1509–1596.
7. W. Wagner, A. Saul and A. Pruss, J. Phys. Chem. Ref. Data, 1994, 23, 515–527.
8. D. M. Murphy and T. Koop, Q. J. R. Meteorol. Soc., 2005, 131, 1539–1565.
9. M. T. Ho, G. W. Allinson and D. E. Wiley, Int. J. Greenhouse Gas Control, 2011, 5, 49–60.
10. D. L. Glandorf, A. Colaprete, M. A. Tolbert and O. B. Toon, Icarus, 2002, 160, 66–72.
11. M. Isenor, R. Escribano, T. C. Preston and R. Signorell, Icarus, 2013, 223, 591–601.
12. J. A. Noble, E. Congiu, F. Dulieu and H. J. Fraser, Mon. Not. R. Astron. Soc., 2012, 421, 768–779.
13. J. P. Devlin, C. A. Yinnon and V. Buch, Phys. Chem. Chem. Phys., 2009, 11, 7819–7825.
14. M. Taraschewski, H. K. Cammenga, R. Tuckermann and S. Bauerecker, J. Phys. Chem., 2005, 109, 3337–3343.
15. L. J. Karssemeijer, G. A. de Wijs and H. M. Cuppen, Phys. Chem. Chem. Phys., 2014, 16, 15630–15639.
16. H. Laksmono, S. Tanimura, H. Allen, G. Wilemski, M. Zahniser, J. Shorter, D. Nelson, B. McManus and B. E. Wyslouzil, Phys. Chem. Chem. Phys., 2011, 13, 5855–5871.
17. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, Ronald, New York, 1953, vol. 1, ch. 4.
18. S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson and J. B. McManus, J. Chem. Phys., 2007, 127, 034305.
19. S. Tanimura, U. M. Dieregsweiler and B. E. Wyslouzil, J. Chem. Phys., 2010, 133, 174305.
20. S. Tanimura, Y. Zvinevich, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson and B. McManus, J. Chem. Phys., 2005, 122, 194304.
21. R. J. Roe, Methods of X-Ray and Neutron Scattering in Polymer Science, Oxford University, New York, 2000.
22. A. Manka, H. Pathak, S. Tanimura, J. Wölk, R. Strey and B. E. Wyslouzil, Phys. Chem. Chem. Phys., 2012, 14, 4505–4516.
23. Y. J. Kim, B. E. Wyslouzil, G. Wilemski, J. Wölk and R. Strey, J. Phys. Chem., 2004, 108, 4365–4377.
24. A. Michels, T. Wassenaar, T. H. Zwietering and P. Smits, Physica, 1950, 16, 501–504.
25. B. E. Wysouzil, G. Wilemski, R. Strey, S. Seifert and R. E. Winans, Phys. Chem. Chem. Phys., 2007, 9, 5353–5358.
26. N. H. Fletcher, J. Chem. Phys., 1958, 29, 572–576; N. H. Fletcher, J. Chem. Phys., 1959, 31, 1136–1137.
27. A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi and M. Kulmala, J. Geophys. Res., 2005, 110, E02002.
28. A. Mensah, First Experimental Data on the Gas-Liquid Vapor Pressure Curve of Liquid Argon between 52 K and 72 K, Diploma thesis, Universität zu Köln, 2005.
29. E. L. Quinn, J. Am. Chem. Soc., 1927, 49, 2704–2711.
30. R. Defay, I. Prigogine, A. Bellemans, and D. H. Everett, Surface Tension and Adsorption, Longmans, Green, Bristol, 1966.
31. P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson and B. McManus, J. Chem. Phys., 2004, 121, 9964–9970.
32. D. M. Newitt, M. U. Pai, N. R. Kuloor and J. A. W. Huggill, Carbon Dioxide, in Thermodynamic Functions of Gases, ed. F. Din, Butterworths Scientific Publications, London, 1956, vol. 1.
33. A. S. Friedman and L. Haar, J. Chem. Phys., 1954, 22, 2051–2058.
34. M. W. Chase Jr, J. Phys. Chem. Ref. Data, 1998, 643 , monograph 9.
35. National Institute of Standards and technology, http://webbook.nist.gov/chemistry.
36. M. A. Aïnou, Monatshefte für chemie, 2005, 136, 2017–2027.

---