New homogeneous nucleation experiments are presented at 240 K for water in carrier gas mixtures of nitrogen with carbon dioxide molar fractions of 5%, 15%, and 25%. The pulse expansion wave tube is used to test three different pressure conditions, namely, 0.1, 1, and 2 MPa. In addition, a restricted series of nucleation experiments is presented for 25% carbon dioxide mixtures at temperatures of 234 and 236 K at 0.1 MPa. As pressure and carbon dioxide content are increased, the nucleation rate increases accordingly. This behavior is attributed to the reduction in the water surface tension by the adsorption of carrier gas molecules. The new data are compared with theoretical predictions based on the classical nucleation theory and on extrapolations of empirical surface tension data to the supercooled conditions at 240 K. The extrapolation is carried out on the basis of a theoretical adsorption/surface tension model, extended to multi-component mixtures. The theoretical model appears to strongly overestimate the pressure and composition dependence. At relatively low pressures of 0.1 MPa, a reduction in the nucleation rates is found due to an incomplete thermalization of colliding clusters and carrier gas molecules. The observed decrease in the nucleation rate is supported by the theoretical model of Barrett, generalized here for water in multi-component carrier gas mixtures. The temperature dependence of the nucleation rate at 0.1 MPa follows the scaling model proposed by Hale [J. Chem. Phys. **122**, 204509 (2005)].

## I. INTRODUCTION

Dropwise condensation phenomena have been studied for over a century^{1} with important implications in many industrial, technological, and atmospheric applications.^{2,3} In climate science, for example, cloud formation is of paramount importance but extremely difficult to model. This is because clouds both shade the Earth and trap heat. A recent study of Zelinka *et al.*^{4} suggests that in a warmer world, clouds would become thinner on average, resulting in additional global warming. This can be characterized as an indirect effect (cloud feedback) of higher atmospheric carbon dioxide concentrations. This work investigates the effect of carbon dioxide concentration and pressure on water nucleation in the absence of foreign particles or surfaces (homogeneous nucleation) in a nitrogen carrier gas at low temperatures. Homogeneous nucleation is attractive because the number of stable droplets produced per unit of time and volume (nucleation rate) can directly be compared with the classical nucleation theory (CNT).^{5–11} However, it is not straightforward to realize experimentally as nucleation is almost always heterogeneous. As shown in previous publications,^{12–30} a Pulse Expansion Wave Tube (PEWT) is capable of eliminating contact of nuclei with foreign particles or surfaces and it enables experiments at different nucleation pressures, temperatures, and various gas compositions.

In our previous publication,^{30} we investigated pressure and carrier gas effects on nucleation, pointing out the central role played by adsorption. This work presents new homogeneous water nucleation experiments at 240 K and 0.1, 1, and 2 MPa in nitrogen mixtures with carbon dioxide molar fractions of 0%,^{30} 5%, 15%, and 25%. The enhancement of the saturated vapor pressure due to the carrier gas presence in the mixture (enhancement factor) is accounted for in the experimental results. The observed trends are explained in terms of two different phenomena: adsorption and incomplete thermalization. Adsorption of gas molecules on a liquid surface reduces the surface tension. Therefore, a theoretical analysis of the surface tension of water in the presence of mixed carrier gas formed by nitrogen and carbon dioxide is proposed. The CNT is used to predict the effect of the calculated surface tension decrease in the nucleation rates, and the results are compared with our experimental outcomes. At low pressure, the adsorption phenomena become negligible while the collision frequency of gas molecules with clusters is too low for full thermalization of the clusters.^{31–36} This effect is investigated theoretically according to the analytical model of Barrett.^{36} Predictions of this model, generalized for water in a mixed carrier gas formed by nitrogen and carbon dioxide, are used to interpret our experimental findings at 0.1 MPa and 240 K. Additional 0.1 MPa experiments are carried out at 236 and 234 K in nitrogen with 25% of carbon dioxide. The effect of temperature on the experimental data is investigated by means of Hale’s scaling model.^{37} Apart from the comparison of the novel nucleation rate data with theoretical predictions, the combined effect of adsorption and insufficient thermalization will also be shown by a comparison with reference nucleation rate data. These reference data are chosen such that both effects do not play a significant role.

## II. THEORETICAL ASPECTS

In this section, we shall discuss three phenomena with a common origin in the heteromolecular interaction taking place within the mixture: adsorption, incomplete thermalization, and enhancement effect.

### A. Adsorption effects

The surface tension decrease of water in the presence of a diluent gas is related to the adsorption of gas molecules on the liquid surface.^{38} As the pressure of the system increases or the temperature decreases, this effect becomes stronger.^{28} The pressure and temperature dependency of the surface tension is also influenced by the type of carrier gas in the mixture (almost negligible effect in helium and stronger effects in nitrogen and argon).^{28,30} We now extend this analysis to water–carbon dioxide–nitrogen mixtures. To this end, we first need to generalize the theoretical treatment of the adsorption effect to the case of *mixed* carrier gas. The complete derivation is given in the Appendix. As a first step, we consider the general expression of the water surface tension in the presence of one or more diluent gases,

where *p*_{g} = *p* − *p*_{s}(*T*), *p* is the total pressure of the mixture, *p*_{s}(*T*) and *σ*_{0} are the equilibrium (saturation) vapor pressure^{39} and the surface tension^{40} of pure water at the temperature *T*, respectively, *A*_{g} is the average surface area per adsorption site, and *k* is the Boltzmann constant. In the case of carbon dioxide and nitrogen as diluent gases (subscripts “2” and “3,” respectively), the Langmuir pressure *p*_{L} can be formulated as follows [see Eq. (A20)]:

where *y*_{2} and *y*_{3} are the molar fractions of carbon dioxide and nitrogen, respectively, and

The parameters $ug,iA$ and $viA$ are the interaction potential energy of the *i*th gas molecule with the liquid surface and the effective volume available for the translation of an adsorbed (*i*th gas) molecule, respectively. The two quantities $ug,2A$ and $v2A$ (water–carbon dioxide case) are derived as follows: The surface tension data available in the literature for water in carbon dioxide^{41–43} are used to calculate the Langmuir pressures *p*_{L,2} from Eq. (1) with *p*_{L} = *p*_{L,2} and *A*_{g} = 1.62 · 10^{−19} m^{2}.^{44} Finally, the *p*_{L,2}(*T*) values are fitted with a regression curve in the form of Eq. (3), leading to the following regression parameters: $v2A=1.37\u2009\u22c5\u200910\u221230\u2009m3$ and $ug,2A=\u22122.8\u2009\u22c5\u200910\u221220$ J. Figure 1 shows the experimental Langmuir pressures as a function of temperature (black squares, pentagons, and a triangle) and their fit (black dashed line). The same procedure can be applied to the water–nitrogen case (surface tension data taken from Refs. 45–49), leading to $v3A=6.94\u2009\u22c5\u200910\u221230\u2009m3$ and $ug,3A=\u22121.18\u2009\u22c5\u200910\u221220$ J. In this way, the water surface tension reduction in carbon dioxide and nitrogen can be calculated [Eqs. (1)–(3)] for the investigated experimental conditions. The final results are reported in Fig. 2.

For small ratios of *p*_{g}/*p*_{L}, the ln(1 + *p*_{g}/*p*_{L}) term in Eq. (1) can be approximated by *p*_{g}/*p*_{L} [see Eqs. (A21)–(A23) in the Appendix]. This approach was used in our previous publication^{30} for the water–nitrogen case, leading to similar values for $v3A$ and $ug,3A$ (8.85 · 10^{−30} m^{3} and −1.03 · 10^{−20} J, respectively). In the water–carbon dioxide case, the linear approximation is inaccurate, leading to a different *p*_{L,2}(*T*) curve (red dashed-dotted line in Fig. 1). For this reason, the general derivation is used in this work for water in carbon dioxide and, in order to keep a consistent approach, for water in nitrogen as well. Additional details can be found in the supplementary material.

### B. Incomplete thermalization effects

Nucleation in vapor–gas systems is generally affected by an incomplete thermalization of the growing clusters.^{31–36} This effect becomes more pronounced at lower pressures as the amount of the non-condensing component is insufficient to thermalize the clusters. The result is a nucleation rate reduction, with respect to the isothermal case, modeled by Barrett as (in the zeroth order approximation)^{34–36}

where *J*_{iso} is the isothermal condition. In Eq. (4), $\u0109bv=cv/k+1/2$, with *c*_{v} as the vapor monomer heat capacity at constant volume. The non-dimensional parameter *H* is defined as

with *M* as the molar mass of the vapor component, *L* as the latent heat of vaporization, *T* as temperature, *v*_{l} as the molecular liquid volume of the pure vapor component,^{50} $nGT*=32\pi vl2\sigma 03/[3(kT\u2009ln\u2009S)3]$ as the critical cluster size as defined by Gibbs and Thomson, Θ = *a*_{1}*σ*_{0}/*kT* as the non-dimensional surface tension of the pure vapor component, and $a1=(36\pi )1/3\u2002vl2/3$ as the surface area of the vapor monomer. In the Barrett model, the parameter *λ* of Eq. (4) is defined for a vapor–(single) carrier gas mixture as $\lambda =cbg\u2009ng\u2009\nu \xafg/(cbv\u2009n1\u2009\nu \xaf1)$, with *c*_{bg} = *c*_{g} + *k*/2 as the enhanced heat capacity of the gas at constant volume, *n*, and $\nu \xaf$ as the number densities and the mean thermal velocities of gas and vapor components (subscripts “g” and “1,” respectively). We propose the following generalization of this expression, accounting for the presence of two (or more) carrier gases (subscripts “2” and “3”):

We will use this approach to calculate $(J/Jiso)B$ at the investigated conditions and analyze the incomplete thermalization influence on the experimental data.

### C. Enhancement factor

Interactions between vapor and gas molecules cause an enhancement of the vapor molar fraction *y*_{eq} in the gas phase in equilibrium with the liquid phase.^{15,51–54} This effect can be quantified by the enhancement factor *f*_{e} as^{30}

with *p* and *T* as the total pressure and temperature of the mixture, respectively, and *x*_{1} = (1 − *x*_{2} − *x*_{3}) as the liquid fraction of the vapor component. In our analysis, we shall neglect the solubility of nitrogen in water as *x*_{3} ≤ 0.0015.^{55} It follows that *x*_{1} can be directly derived from the liquid fraction of carbon dioxide *x*_{2}.^{55} The parameter *ϕ*_{s} in Eq. (7) is the fugacity coefficient at equilibrium for pure water vapor, and it is obtained as *ϕ*_{s} = exp[*B*_{11}*p*_{s}/(*RT*)], with *B*_{11} standing for the second virial coefficient of pure water.^{55} The fugacity *ϕ*_{eq} for water vapor in the presence of carbon dioxide and nitrogen is computed with the following expression:

where *y*_{2} and *y*_{3} are the carbon dioxide and nitrogen molar fractions, respectively, and *B*_{12} and *B*_{13} are the second cross-virial coefficients for water–carbon dioxide^{56} and water–nitrogen^{57} mixtures, respectively. The compressibility factor *Z* = *pV*/*RT* can be expressed as *Z* = 1 + *B*/*V*, where *V* is the molar volume and *B* is the second virial coefficient of the mixture,

with *B*_{22} and *B*_{33} as the second virial coefficients of the pure carbon dioxide and nitrogen components, respectively, and *B*_{23} as the second cross-virial coefficients for carbon dioxide–nitrogen mixtures.^{55} The enhancement factor calculated according to the present analysis is reported in Fig. 3 for the investigated conditions.

The reader is referred to the supplementary material for more details on the carbon dioxide and the nitrogen solubility in water^{55,58–60} and on the virial and cross-virial coefficients^{55,56,61–66} used for this study.

## III. EXPERIMENTAL METHODOLOGY

The experimental setup used for this work is the Pulse Expansion Wave Tube (PEWT),^{30} a specially designed shock tube, implementing the so-called nucleation pulse method.^{67} This technique enables the nucleation and the droplet growth phenomena to be effectively decoupled, leading to a monodisperse cloud of condensing clusters.^{29} In this way, the nucleation rate *J* —the number of critical clusters generated per unit of time and volume—can be determined as *J* = *n*_{d}/Δ*t*, with Δ*t* as the nucleation pulse duration^{29} and *n*_{d} as the droplet number density. The Constant Angle Mie Scattering (CAMS)^{68–70} method is used to measure *n*_{d} by means of an optical setup placed at the observation section of the PEWT.^{29} A maximum standard uncertainty of 20% is calculated for *J*.^{30}

The other key parameter in homogeneous nucleation studies is supersaturation. It quantifies the deviation of the current state for the vapor component from its corresponding (same *p* and *T*) phase equilibrium and can be approximated as^{30} *S* ≈ *y*/*y*_{eq}, where *y*_{eq} [Eq. (7)] is the equilibrium vapor molar fraction for water in carbon dioxide and nitrogen at the nucleation conditions *p* and *T*. The pressure *p* is measured via two pressure transducers at the PEWT test section, while the temperature *T* is derived from the pressure signal by assuming isentropic conditions for the real mixture and measuring the initial temperature at the test section walls.^{29,30} A controlled vapor molar fraction *y* is obtained by means of the mixture preparation device (MPD) shown in Fig. 4. The pure nitrogen flowing from the gas bottle G_{I} (99.999% purity) is split into two controlled flows A and B (with normal volume flow rates Q_{A} and Q_{B}, respectively). Flow A is fully saturated with the pure water contained inside the saturators sat_{a} and sat_{b},^{29} while flow B remains dry. A third flow (with normal volume flow rate Q_{C}) of nitrogen and carbon dioxide is let into the MPD from the premixed gas bottle G_{II}. As a result, a well-defined total flow rate $Qtot=QA+QB+QC+QH2O$ is obtained at the MPD outlet, with $QH2O$ as the water flow rate added to the mixture after Q_{A} passes through sat_{a} and sat_{b}. Once the thermodynamic conditions *p*_{mpd} and *T*_{mpd} and the three flow rates Q_{A}, Q_{B}, and Q_{C} are known, the vapor molar fraction of water in the mixture can be determined as

where $yeq\u2032$ is the equilibrium vapor molar fraction calculated at the MPD conditions as

where *f*_{e,13} is the enhancement factor for water in nitrogen.^{30} By controlling the MPD thermodynamic conditions (*p*_{mpd} and *T*_{mpd}) and flow rates (Q_{A}, Q_{B}, and Q_{C}), the desired vapor molar fraction of carbon dioxide can be derived as

with $Q2=y2\u2009C\u2009QC$ as the carbon dioxide flow rate and $y2\u2009C=0.5$ as the molar fraction of carbon dioxide in G_{II}. This experimental methodology leads to a maximum standard uncertainty for *S* of 2.5%. A more extensive description can be found in Refs. 29, 30, and 58. The calculation of the described quantities at the investigated conditions is reported in the supplementary material.

## IV. RESULTS AND DISCUSSION

In this section, the experimental results will be presented and, afterward, interpreted with reference to adsorption (Sec. II A) and incomplete thermalization effects (Sec. II B). The enhancement of the equilibrium vapor pressure (enhancement factor), discussed in Sec. II C, is accounted for in the presented supersaturation values as described in Sec. III.

### A. Experiments at 240 K for various pressures and carbon dioxide contents

Homogeneous water nucleation experiments have been carried out at 240 K and 0.1, 1, and 2 MPa in nitrogen with 5%, 15%, and 25% of carbon dioxide.^{71,76} The resulting nucleation rate data (*J*) as a function of the supersaturation (*S*) are summarized in Fig. 5 for each pressure condition: 2 MPa [Fig. 5(a)], 1 MPa [Fig. 5(b)], and 0.1 MPa [Fig. 5(c)]. The *J*–*S* data are represented for variable carbon dioxide molar fractions: 5% (green circles), 15% (red squares), and 25% (blue triangles). The whole set of data is fitted according to the following regression curve *J*_{exp} = *C* exp[*D*/(ln *S*)^{2}], with the coefficients *C* and *D* calculated for each pressure and temperature condition and for each carbon dioxide vapor molar fraction. Different *J*_{exp}(*S*) line types are used to distinguish the pressure conditions: dashed lines for 0.1 MPa, dotted lines for 1 MPa, and solid lines for 2 MPa. The standard uncertainties for *J* (20% at the most—within the symbol sizes) and for *S* (2.5%) are shaded around the regression curves (Fig. 5). Literature experimental nucleation rates for water in nitrogen^{30} at the investigated pressures (0.1, 1, and 2 MPa) and temperature (240 K) are also included in the analysis, denoted as 0% carbon dioxide (Fig. 5). The *J*_{exp}(*S*) lines are colored according to the carbon dioxide content. Details on regression coefficients (*C* and *D*), fitting quality (adjusted R^{2}), and experimental *J*–*S* data can be found in the supplementary material. As pointed out previously,^{30} both surface tension reductions and incomplete thermalization effects can be considered negligible at 240 K for homogeneous water nucleation in helium at 1 MPa. Therefore, also for this work, this condition is used as an experimental reference at 240 K, and it is denoted as *J*_{id,exp} (dashed-dotted lines in Fig. 5).

The *J*–*S* data at 2 MPa and 240 K in Fig. 5(a) show that *J*_{exp} is higher than *J*_{id,exp} from about an order of magnitude at 0% of carbon dioxide to about three orders of magnitude at 25% of carbon dioxide. Consistently, intermediate results are found for *J*_{exp} with 5% and 15% of carbon dioxide in the mixture. In Table I, *J*_{exp}/*J*_{id,exp} is quantified at the average supersaturation value for each condition.

CO_{2} (%)
. | p (MPa) . | S^{a}
. | J_{exp}/J_{id,exp}^{b}
. | $(\u2212\Delta \sigma /\sigma 0)%$^{c}
. | J_{CNT}/J_{0,CNT}
. | $(J/Jiso)B$ . | References^{d}
. |
---|---|---|---|---|---|---|---|

0 | 0.101 72 | 12.6 | 0.33 | 0.25 | 1.4 | 0.6404 | Reference 30 |

5 | 0.100 95 | 12.3 | 0.39 | 0.48 | 1.8 | 0.6439 | c.w. |

15 | 0.100 67 | 12.2 | 0.38 | 0.97 | 3.0 | 0.6432 | c.w. |

25 | 0.100 72 | 12.0 | 0.48 | 1.42 | 4.8 | 0.6451 | c.w. |

0 | 1.001 8 | 11.9 | 3.5 | 2.35 | 27 | 0.9487 | Reference 30 |

5 | 1.002 0 | 11.5 | 5.3 | 4.41 | 208 | 0.9495 | c.w. |

15 | 0.996 0 | 11.2 | 11 | 8.11 | 7.4 · 10^{3} | 0.9500 | c.w. |

25 | 1.005 1 | 10.9 | 40 | 11.39 | 1.7 · 10^{5} | 0.9513 | c.w. |

0 | 2.000 4 | 11.6 | 14 | 4.46 | 410 | 0.9740 | Reference 30 |

5 | 1.995 7 | 11.3 | 40 | 8.10 | 1.2 · 10^{4} | 0.9745 | c.w. |

15 | 2.000 3 | 10.6 | 350 | 14.2 | 1.8 · 10^{6} | 0.9757 | c.w. |

25 | 2.000 2 | 9.9 | 4.6 · 10^{3} | 19.08 | 5.2 · 10^{8} | 0.9767 | c.w. |

CO_{2} (%)
. | p (MPa) . | S^{a}
. | J_{exp}/J_{id,exp}^{b}
. | $(\u2212\Delta \sigma /\sigma 0)%$^{c}
. | J_{CNT}/J_{0,CNT}
. | $(J/Jiso)B$ . | References^{d}
. |
---|---|---|---|---|---|---|---|

0 | 0.101 72 | 12.6 | 0.33 | 0.25 | 1.4 | 0.6404 | Reference 30 |

5 | 0.100 95 | 12.3 | 0.39 | 0.48 | 1.8 | 0.6439 | c.w. |

15 | 0.100 67 | 12.2 | 0.38 | 0.97 | 3.0 | 0.6432 | c.w. |

25 | 0.100 72 | 12.0 | 0.48 | 1.42 | 4.8 | 0.6451 | c.w. |

0 | 1.001 8 | 11.9 | 3.5 | 2.35 | 27 | 0.9487 | Reference 30 |

5 | 1.002 0 | 11.5 | 5.3 | 4.41 | 208 | 0.9495 | c.w. |

15 | 0.996 0 | 11.2 | 11 | 8.11 | 7.4 · 10^{3} | 0.9500 | c.w. |

25 | 1.005 1 | 10.9 | 40 | 11.39 | 1.7 · 10^{5} | 0.9513 | c.w. |

0 | 2.000 4 | 11.6 | 14 | 4.46 | 410 | 0.9740 | Reference 30 |

5 | 1.995 7 | 11.3 | 40 | 8.10 | 1.2 · 10^{4} | 0.9745 | c.w. |

15 | 2.000 3 | 10.6 | 350 | 14.2 | 1.8 · 10^{6} | 0.9757 | c.w. |

25 | 2.000 2 | 9.9 | 4.6 · 10^{3} | 19.08 | 5.2 · 10^{8} | 0.9767 | c.w. |

^{a}

Mean supersaturation at each experimental condition.

^{b}

Ratio of *J*_{exp}(*S*) and *J*_{id,exp}(*S*), both calculated at the same *S* value.^{a} As an example, consider the case with 25% of carbon dioxide at 2 MPa and *S* = 9.9 (last line in Table I): *J*_{exp}(*S* = 9.9) = 7.2 · 10^{15}, *J*_{id,exp}(*S* = 9.9) = 1.6 · 10^{12}, and *J*_{exp}/*J*_{id,exp} = 4.6 · 10^{3}. Note that *J*_{id,exp}(*S*) is the regression curve of the water–helium data at 1 MPa and 240 K (reference experimental case), with *J*_{id,exp}(*S*) = *C* exp[*D*/(ln *S*)^{2}], *C* = 5.5 · 10^{30}, and *D* = −224.7 (Ref. 30). The reader is referred to the supplementary material for the regression curves *J*_{exp}(*S*) of each condition in this work.

^{d}

Literature references, with *c*.*w*. (current work) denoting the new data.

The data at 1 MPa and 240 K with 5%, 15%, and 25% of carbon dioxide in Fig. 5(b) show a smaller nucleation rate increase than the 2 MPa case, as to be expected.

At 0.1 MPa, the increasing trend of *J*_{exp} with increasing carbon dioxide content, albeit present, is within the uncertainty bands. Moreover, the nucleation rates at 0.1 MPa and 240 K are found to be lower than the nucleation rates at the ideal experimental condition *J*_{id,exp}. Later, we will see that this can be attributed to the effect of incomplete thermalization.

#### 1. Comparison of the 240 K high pressure experiments with the theoretical predictions: Effect of adsorption of carrier gases on nucleation rates

The described experimental behavior can be qualitatively explained by taking into account the adsorption phenomena and incomplete thermalization effects. The surface tension of water in nitrogen and carbon dioxide decreases with respect to the surface tension of pure water *σ*_{0} as pressure and carbon dioxide content increase {see Sec. II A [Eqs. (1)–(3)]}. The values of the relative water surface tension decrease (−Δ*σ*/*σ*_{0}) for the investigated conditions, and their standard uncertainties are reported in Fig. 2 (mean values are given also in Table I). This reduction is caused by the adsorption of gas monomers on the cluster surface. A theoretical estimate of this effect on the nucleation rates can be made by employing the Classical Nucleation Theory (CNT),^{5–11}

The nucleation rates calculated according to the CNT will be indicated as *J*_{CNT} when it is assumed that *σ* = *σ*(*p*, *T*, *y*_{2}, *y*_{3}) [Eqs. (1)–(3)]. The nucleation rate increase predicted by the CNT can be estimated with the ratio *J*_{CNT}/*J*_{0,CNT}, where *J*_{0,CNT} is the value for *σ* = *σ*_{0}. These ratios are calculated for each experimental condition, and the results are reported in Table I.

At 2 and 1 MPa, incomplete thermalization phenomena do not play a role as $(J/Jiso)B$ is 0.98 and 0.95, respectively (Table I). On the contrary, *J*_{CNT}/*J*_{0,CNT} progressively increases with pressure and carbon dioxide content as the surface tension decreases. At 1 MPa, *J*_{CNT}/*J*_{0,CNT} goes from 27 for 0% of carbon dioxide up to 1.7 · 10^{5} for 25% of carbon dioxide. At 2 MPa, *J*_{CNT}/*J*_{0,CNT} increases from 410 for 0% of carbon dioxide up to 5.2 · 10^{8} for 25% of carbon dioxide.

We observe that the theoretical predictions of *J*_{CNT}/*J*_{0,CNT} appear to strongly overestimate the nucleation rate increase, due to gas adsorption, for all the analyzed conditions. This is not fully surprising considering that the CNT assumes bulk properties (capillarity approximation), which poorly describes the nucleation process in the case of clusters consisting of a few molecules only. This is the case for the experimental conditions presented in this work: the critical cluster size, as determined using the nucleation theorem,^{72,73} is about 29 ± 6 molecules. Hence, for this work, *J*_{CNT}/*J*_{0,CNT} can only be used as a qualitative reference to predict the trend of the nucleation rate increase as a function of pressure and carbon dioxide content. We can conclude that our experimental findings at 240 K and high pressures (2 and 1 MPa) seem to confirm the central role played by the adsorption phenomena in increasing the nucleation rate when pressure and carbon dioxide content are increased.

#### 2. Comparison of the 240 K low pressure experiments with the theoretical predictions: Effect of incomplete thermalization on nucleation rates

At 0.1 MPa and 240 K, *J*_{CNT}/*J*_{0,CNT} ranges from 1.4 with no carbon dioxide up to about 4 with 25% carbon dioxide. Considering that the theory strongly overpredicts the nucleation rate at high pressure, we postulate that *J*_{CNT}/*J*_{0,CNT} is overpredicted at low pressure as well and is close to unity in practice. On the other hand, incomplete thermalization (Barrett theory) at ambient pressure tends to decrease the nucleation rate with respect to the isothermal condition, with a $(J/Jiso)B$ of about 0.64 for any carbon dioxide content (Table I). Hence, the incomplete thermalization at 0.1 MPa explains the experimental data at this condition, within the uncertainty bands. It is clear that for a more detailed check of the Barrett model, experiments at much lower pressures would be indispensable.

### B. Experiments at 0.1 MPa and 25% of CO_{2}: Temperature effect on the nucleation rate and Hale scaling

Additional water nucleation experiments in nitrogen and 25% of carbon dioxide have been conducted at 0.1 MPa for 236 and 234 K. The corresponding *J*–*S* data, fitting curves, and uncertainty bands are represented in Fig. 6. The reader is referred to the supplementary material for more details. A nucleation rate decrease of about 2 orders of magnitude is observed when the temperature is reduced from 240 to 236 K. This result is in agreement with the nucleation rate reduction observed in our previous publication^{30} for water in argon. As pointed out,^{30} the *J*–*S* data can be successfully scaled to different temperatures by employing the Hale model:^{37} $Ssc=exp[C0\u2009ln\u2009S/(Tc/T\u22121)3/2]$, with $C0=(Tc/Tsc\u22121)3/2$, *T*_{sc} as the target scaling temperature, *S* as the actual supersaturation at the temperature *T*, and *T*_{c} as the critical temperature of water. We now verify the Hale scaling for our water–nitrogen–25% carbon dioxide mixture at 0.1 MPa. The 0.1 MPa *J*–*S* data obtained at 234 K (blue diamonds) and 236 K (blue pentagons) are shown in Fig. 6. These data are scaled^{37} to 240 K, and the regression curve for the scaled data is represented as a blue dashed line at 240 K in Fig. 6. The comparison between the scaled curve (blue dashed line at 240 K) and the actual *J*–*S* values at 240 K (blue triangles) shows a perfect agreement. Albeit over the relatively small temperature range analyzed in this work, this outcome demonstrates the validity of the Hale scaling model^{37} for the current water–nitrogen–carbon dioxide mixture as well as for the water–argon case discussed in our previous publication.^{30}

## V. CONCLUSIONS

Homogeneous water nucleation experiments were carried out with the pulse expansion wave tube in mixtures of nitrogen and 0%,^{30} 5%, 15%, and 25% carbon dioxide molar fractions. Most experiments were performed at a temperature of 240 K. Various pressures were tested: 0.1, 1, and 2 MPa. Nucleation rates were determined as functions of the supersaturation for a fixed pressure, temperature, and carrier gas composition. An experimental reference was identified in the literature for which adsorption and thermalization effects do not play a role (Sec. IV A).^{29,30}

As the pressure is increased from 1 to 2 MPa, the experimental nucleation rates are found to increase. This effect is stronger when the fraction of carbon dioxide increases (Sec. IV A and Fig. 5), and it can be attributed to the surface tension reduction (Sec. IV A 1). The adsorption of carrier gas molecules on a liquid surface reduces the surface tension. To compute this, use is made of an adsorption/surface tension theory, extended to multicomponents (Sec. II A and Appendix). Empirical literature data in the temperature range of 280–350 K for the Langmuir pressure were extrapolated to 240 K. The surface tension data are substituted in the Classical Nucleation Theory (CNT). The predicted effects of pressure and composition are in qualitative agreement with the experimental trend (Sec. IV A 1 and Table I), as extensively pointed out in the literature.^{19,26,28,30} A strong quantitative overestimation of the model is found, mainly due to the CNT (Sec. IV A 1).

At 0.1 MPa and 240 K, the increase in carbon dioxide from 0% to 25% does not appear to affect the nucleation rates. Additionally, they are found to be smaller than the experimental isothermal reference [Sec. IV A and Fig. 5(c)]. This can be explained by the incomplete thermalization of the colliding clusters and carrier gas molecules. These results were confirmed, within the experimental uncertainty, by the theoretical model of Barrett^{36} (Sec. IV A 1 and Table I). It is recognized that adsorption phenomena are negligible at 0.1 MPa and 240 K.

## SUPPLEMENTARY MATERIAL

The reader is referred to the supplementary material for more details on the data presented in this work.

## ACKNOWLEDGMENTS

J.H. acknowledges support from the Ministry of Education, Youth and Sports of the Czech Republic under OP RDE Grant No. CZ.02.1.01/0.0/0.0/16_019/0000753 “Research center for low carbon energy technologies” and institutional support RVO: 61388998.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.

### APPENDIX: GENERALIZED DERIVATION OF THE LANGMUIR ADSORPTION FOR MULTIPLE ADSORBATES

The derivation of the Langmuir adsorption was extensively discussed by Fransen *et al.*^{28} for a single type of gas molecules adsorbed at the condensing water cluster surfaces. This approach is now generalized to the case of *n* gases. Given *χ* available adsorption sites at the condensing cluster surfaces, one can consider that $Ng,iA$ sites are occupied by the adsorbed (superscript “A”) molecules of the *i*th gas. The number of possible configurations for the adsorbed layer can be calculated as

assuming no multiple occupation of the adsorption sites and mutual indistinguishability of the adsorbed molecules. The equilibrium condition between the adsorbed and the free molecules (superscript “G”) of the *i*th gas is given by the chemical potential equality

The chemical potentials in Eq. (A2) can be found as follows: For the adsorbed case, we first consider the Helmholtz free energy

with the internal energy

For the *i*th gas, $ug,iint$ and (3/2)*kT* are the internal molecular and the translation degrees of freedom, respectively, while $ug,iA$ is the potential energy of the interaction between the adsorbed gas molecules and the liquid surface. For our case, the potential energy of the interaction between adsorbed gas molecules is neglected, being weaker than the one between gas molecules and the liquid surface. In Eq. (A3), the quantity $SA$ denotes the entropy of the adsorbate,

where, for the *i*th gas, $sg,iint$ is the internal molecular entropy and *V*_{ref} and $viA$ are, respectively, the reference volume and the effective volume available for the translation of an adsorbed molecule. The configurational entropy $Sc$ on a per molecule basis is given by

where the probability of the *j*th configuration $Pj$ is considered equal for any configuration, i.e., $Pj=1/C$. Being the surface coverage of the *i*th gas component $\theta i=Ng,iA/\chi $ and applying Stirling’s approximation^{74} ln *x*! ≈ *x* ln *x* − *x*, Eq. (A6) becomes

We now consider the differential of the Helmholtz free energy of the adsorbed phase,

with *σ*_{g} as the contribution of the adsorbate to the surface tension and *A* as the surface area. Recalling Eq. (A3), the chemical potential for the adsorbed molecules of the *i*th gas can be expressed as

with the partial derivative of the configurational entropy that can be decomposed as $\u2202Sc/\u2202Ng,iA=(\u2202Sc/\u2202\theta g,i)\u2009(\u2202\theta g,i/\u2202Ng,iA)$ and then calculated by recalling Eq. (A7),

The chemical potential for the free gas molecules of the *i*th gas $\mu g,iG$ can be derived in a similar manner, assuming $viG=kT/pg,i$ (ideal gas law),

The equilibrium condition of Eq. (A2), combined with Eqs. (A11) and (A12), leads to the following expression:

The change in surface tension due to the chemical potential variation of the *n* gases can be derived from the Gibbs adsorption equation^{75}

at constant temperature and vapor chemical potential as

where *A* = *A*_{g}*χ* and *A*_{g} is the average surface area per adsorption site. It must be stressed that this derivation considers the same *A*_{g} for the *n* gases. The term d*μ*_{g,i} in Eq. (A15) can be calculated as

with the second equality obtained from the ideal gas law under the assumption of constant temperature, pressure, and *N*_{g,t≠i}. The combination of Eqs. (A15) and (A16) leads to

Finally, considering the derivation of *θ*_{i} from Eq. (A13),

the integration of Eq. (A17) leads to the following general expression for the surface tension:

where the Langmuir pressure is defined as

and *p*_{L,i} is the Langmuir pressure for the individual gas *i*, whose temperature dependence is given in Eq. (3). In Eq. (A20), $zg,i=yi/\u2211i=1nyi$, with *y*_{i} as the molar fraction of the *i*th gas. If the Langmuir pressure is much greater than the gas pressure, the general expression of the surface tension in Eq. (A19) becomes a linear function of pressure,

where

and

is the coefficient of the linearized dependence of the surface tension on pressure for the case of pure gas *i*.

## REFERENCES

_{2}+ N

_{2}+ H

_{2}O) system at temperatures of (298 to 448) K and pressures up to 40 MPa

_{2}dimer

_{2}-H

_{2}O mixtures

Note that the nucleation phenomena investigated in this work can be considered as essentially unary (water) nucleation.^{18,76}