Homogeneous nucleation of water is investigated in argon and in nitrogen at about 240 K and 0.1 MPa, 1 MPa, and 2 MPa by means of a pulse expansion wave tube. The surface tension reduction at high pressure qualitatively explains the observed enhancement of the nucleation rate of water in argon as well as in nitrogen. The differences in nucleation rates for the two mixtures at high pressure are consistent with the differences in adsorption behavior of the different carrier gas molecules. At low pressure, there is not enough carrier gas available to ensure the growing clusters are adequately thermalized by collisions with carrier gas molecules so that the nucleation rate is lower than under isothermal conditions. This reduction depends on the carrier gas, pressure, and temperature. A qualitative agreement between experiments and theory is found for argon and nitrogen as carrier gases. As expected, the reduction in the nucleation rates is more pronounced at higher temperatures. For helium as the carrier gas, non-isothermal effects appear to be substantially stronger than predicted by theory. The critical cluster sizes are determined experimentally and theoretically according to the Gibbs–Thomson equation, showing a reasonable agreement as documented in the literature. Finally, we propose an empirical correction of the classical nucleation theory for the nucleation rate calculation. The empirical expression is in agreement with the experimental data for the analyzed mixtures (water–helium, water–argon, and water–nitrogen) and thermodynamic conditions (0.06 MPa–2 MPa and 220 K–260 K).

## I. INTRODUCTION

Numerous studies have been reported on the dropwise vapor to liquid phase transition since the pioneering work of Wilson in 1879.^{1} An extensive overview of the past 100+ years of experimental and theoretical developments in condensation study has been published by Wyslouzil and Wölk (2016).^{2} One of the puzzling aspects is the effect of the carrier gas and pressure on nucleation.^{3–7} A successful facility to study homogeneous nucleation is the Pulse Expansion Wave Tube (PEWT). It offers the possibility to investigate dropwise condensation in a large variety of vapor–gas mixtures at pressures from 0.1 MPa to 4 MPa and temperatures between 200 K and 260 K.^{6–22} In this article, we present new data on homogeneous nucleation of water at 0.1 MPa, 1 MPa, and 2 MPa in two different carrier gases: nitrogen at 240 K and argon at 236 K and 240 K. The novel experiments have been carried out with a recently modified version of the PEWT.^{23} A comparison is made with the homogeneous water nucleation data by Wölk and Strey^{24} in argon between 220 K and 260 K, at about 0.06 MPa, and by Fransen *et al.* at 240 K and 1 MPa in nitrogen^{21} and at 0.1 MPa and 1 MPa in helium.^{22,23} We shall discuss several aspects of the role of the carrier gas in the nucleation process. It reduces warming of the growing clusters and causes the nucleation process to be isothermal at sufficiently high pressure. At high pressure, real gas effects become important as well as the reduction in surface tension due to adsorption of gas at the phase interface. Additionally, we shall compare experimental critical cluster sizes with predictions from Classical Nucleation Theory (CNT). Finally, an empirical correction of theoretical (CNT) nucleation rate is proposed, which takes into account the influence of the different carrier gases and the effect of the investigated pressure and temperature conditions.

## II. THEORETICAL BACKGROUND

Nucleation and droplet growth are the two steps in the dropwise condensation process. Nucleation is called homogeneous when aerosol particles and foreign surfaces do not affect the nucleation process. The carrier gas (non-condensing component) acts as a heat reservoir, which reduces the warming up of the clusters during nucleation. The key parameter in quantifying the nucleation process is the nucleation rate *J*, which is the number of droplets formed per unit of time and volume. The other important parameter in homogeneous nucleation studies is the supersaturation *S*. For a vapor–gas mixture, it quantifies the current state deviation of the vapor component from its corresponding (same *p* and *T*) phase equilibrium. More details on the *J* and *S* definitions will be given further on.

### A. Classical nucleation theory and nucleation theorem

The Classical Nucleation Theory (CNT)^{25–31} is the most used model to predict the nucleation rates,

with the work of cluster formation Δ*G*^{*} proposed by Becker and Döring,^{26}

As far as the kinetic prefactor *K* is concerned, we employ its corrected form suggested by Courtney,^{30,31}

In Eqs. (1)–(3), *k* is the Boltzmann constant and the quantities *v*_{l}, *p*_{s}, and *σ*_{0} are the molecular liquid volume,^{32} the saturation vapor pressure,^{33} and the surface tension of pure water^{34} at the temperature *T*. Of central importance in the nucleation theory is the so-called critical cluster, which is defined by equal probabilities of growth and decay. The size of the critical cluster (number of vapor molecules forming it) can be determined by the Gibbs–Thomson equation as

It should be noted that the CNT stipulates that bulk properties, such as surface tension, are also valid for clusters that may consist of a few molecules only (capillarity approximation). An independent way for the determination of the critical cluster size is provided by the first nucleation theorem^{35,36}

which enables a straightforward derivation from the experimental *J*–*S* curves.

### B. Pressure and carrier gas influence on the surface tension

The pressure dependency of the water surface tension, in the presence of helium, argon, and nitrogen, has been largely investigated experimentally^{37–41} and, recently, also numerically^{40} for different conditions. The surface tension decrease with increasing pressure is generally ascribed to the adsorption of carrier gas molecules on the condensing cluster surfaces.^{42} As already pointed out by Fransen *et al.*,^{22} adsorption of nitrogen is pronounced at high pressure and mild at ambient pressure.^{37–39,41} The analysis was based on the pressure dependency of the water surface tension in the presence of nitrogen, available in the literature between 273 K and 423 K. Such a dependency was linearly extrapolated into the supercooled liquid region of interest, namely, 240 K. For the present study, we extend the Fransen analysis^{22} with the water interfacial tension data of Chow *et al.* (2016)^{40} for nitrogen and argon as diluent gases. Following this approach,^{22} the surface tension of water in the presence of the diluent gas can be expressed as

with *σ*_{0}(*T*) as the surface tension of pure water, *A*_{g} as the average surface area per adsorption site (1.62 · 10^{−19} m^{2} for nitrogen and 1.38 · 10^{−19} m^{2} for argon),^{43} *v* as the effective volume available for the translation of an adsorbed molecule, and *u*_{g} as the interaction potential energy of a gas molecule with the liquid surface. The following values for *v* and *u*_{g} have been found by fitting the *σ*_{p0}(*T*) data available in the literature^{37–41} (see Fig. 1): for nitrogen *v* = 8.85 · 10^{−30} m^{3} and *u*_{g} = −1.03 · 10^{−20} J and for argon *v* = 7.13 · 10^{−30} m^{3} and *u*_{g} = −1.03 · 10^{−20} J.

The resulting water surface tension reduction at 240 K (*σ*_{0} = 79.95 · 10^{−3} N m^{−1}) is

1.45% ± 0.55% for argon at 1 MPa,

2.89% ± 0.98% for argon at 2 MPa,

1.53% ± 0.49% for nitrogen at 1 MPa, and

3.06% ± 0.87% for nitrogen at 2 MPa.

Thus, the pressure effect on the surface tension of water in argon and in nitrogen is expected to be similar. Additionally, values of surface tension reduction within 0.2% ± 0.1% confirm a negligible adsorption effect for both diluent gases at pressures up to 0.1 MPa and temperatures between 220 K and 260 K. A negligible effect has also been calculated for helium at pressures up to 1 MPa and 240 K. It should be stressed that this analysis is based on the extrapolation to the supercooled liquid region of the few literature data available only at temperatures above 298 K. This leads to a relatively high uncertainty, as confirmed also by Fransen *et al.*^{22}

### C. Kinetic model for non-isothermal nucleation

The cluster growth is characterized by two linked aspects: the trapping of vapor monomers by collision with the clusters (size) and the release of latent heat due to impinging and evaporating vapor monomers (energy). For mixtures of vapors and non-condensing carrier gases, the effect of latent heat release is further reduced by collisions of vapor clusters and gas molecules. The non-isothermal nucleation phenomena have been analyzed since 1966 by means of a 2D size-energy space.^{3–5,44–47} We follow here the analysis of Feder *et al.* (1966),^{3} later modified by Barrett and co-workers.^{4,46,47} The analytical expression of the non-isothermal nucleation rate, approximated to the first order, reads

where *J*_{iso} is the isothermal nucleation rate, $\u0109bv=cbv/k$, and *c*_{bv} = *c*_{v} + *k*/2 is the vapor monomer heat capacity at constant volume *c*_{v}, accounting also for the increased collision rate of the high energy monomers *k*/2. The non-dimensional parameter *H* of Eq. (7) is defined as

with *L* as the latent heat of vaporization, Θ = *a*_{1}*σ*_{0}/*kT* as the non-dimensional surface tension, and $a1=(36\pi )1/3\u2002vl2/3$ as the surface area of the vapor monomer. The parameter *λ* is given by

with *c*_{bg} = *c*_{g} + *k*/2 as the enhanced heat capacity of the gas at constant volume, *n*_{g} and *n*_{1} as the molecular gas and vapor number densities, $\nu \xafg$ and $\nu \xaf$ as the mean molecular (thermal) velocities of gas and vapor. Additionally, *x* is defined as

with *c*_{1} as the heat capacity per molecule in the liquid. Figure 2 shows *J*/J_{iso} as a function of (*λ* + 1) for a water-carrier gas system at 240 K and a supersaturation of 12. The parameter *λ* strongly increases with pressure and with the reciprocal of the gas molecular mass. As a result, negligible non-isothermal effects are predicted at high pressure (*J*/J_{iso} ≈ 1). On the other hand, at low pressures, thermalization is more pronounced for gases with larger molecular mass (*J*/J_{iso} < 1). Additionally, a larger thermalization effect is predicted for increasing temperature. The described model^{4} will be used for the analysis of the low pressure experiments presented in this work. Note that the zeroth order approximation of Eq. (7) (*x* = 0) makes a small difference (over the third digit for our conditions) in the calculation of *J*/J_{iso}, as also pointed out by Barrett.^{4}

### D. Enhancement factor and supersaturation definition

The presence of a carrier gas in a vapor-gas mixture in equilibrium with a liquid phase is responsible for the increase in the vapor content. In order to quantify this enhancement effect, we consider the equality of chemical potentials in the coexisting phases. This equilibrium condition leads to the following expression of the equilibrium vapor molar fraction:^{11,48–51}

with *p* as the total pressure of the mixture and *p*_{s} as the equilibrium (saturation) vapor pressure for the pure condensing component at the total (mixture) temperature *T* (Ref. 33). The term *x*_{g} is the carrier gas molar fraction dissolved in the liquid, which can be assumed negligibly small for the carrier gases investigated at temperatures well above 200 K. The quantities *ϕ*_{s} and *ϕ*_{eq} are the fugacity coefficients at equilibrium for pure water vapor and for water vapor in the presence of the carrier gas, respectively. Assuming negligible liquid compressibility, Eq. (11) can be reformulated as follows:

The enhancement factor *f*_{e} describes the increase in saturated vapor pressure due to the presence of the carrier (real) gas in the non-ideal mixture.^{10} It consists of two contributions, both related to the heteromolecular interactions taking place within the mixture. On the one hand, *ϕ*_{s}/*ϕ*_{eq} accounts for the non-ideality of the water vapor in the presence of the surrounding carrier gas molecules (real gas effect). The second contribution is the Poynting correction [exponent on the right-hand side of Eqs. (11) and (12)], which takes into consideration the increase in equilibrium vapor fraction due to the pressure difference between the mixture and the pure saturated vapor.

The fugacity coefficient *ϕ*_{s} can be expressed as a function of the second virial coefficient for pure water B_{11},^{51}

The fugacity coefficient *ϕ*_{eq} can be written as follows:^{51}

with the second virial coefficient of the mixture *B* defined as

*B*_{22} and *B*_{12} are the second virial coefficients for the pure carrier gas and for the interacting components of the mixture, respectively. The analytical expression of *B*_{12} in the form proposed by Hodges *et al.*^{52} is

with the temperature *T* in K. The coefficients *a*_{i} and *b*_{i} were obtained by fitting computational quantum chemistry results^{52–54} for the different water–gas mixtures analyzed in this work. In Fig. 3, the results of this analysis are presented in terms of pressure (0.05 MPa–5 MPa) and temperature (220 K–300 K) dependency of the real gas effect (*ϕ*_{s}/*ϕ*_{eq}), Poynting correction {exp[*M* (*p* − *p*_{s})/(*ρ*_{l}*RT*)]}, and total effect (*f*_{e}) for the three mixtures: water–nitrogen [Fig. 3(a)], water–argon [Fig. 3(b)], and water–helium [Fig. 3(c)]. It should be noted that a link exists between the increase in enhancement factor *f*_{e} and the reduction in water surface tension with the pressure, as pointed out by Luijten *et al.* (1997).^{11} Both aspects account for the heteromolecular interaction taking place in the vapor–gas mixture. Extending this qualitative consideration to different gases at fixed thermodynamic conditions, a larger *f*_{e} corresponds to a stronger reduction in the water surface tension. This outcome is confirmed by our calculations of *f*_{e} (Fig. 3) and *σ* (Sec. II B): at the same pressure and temperature conditions, *f*_{e} for the water–argon mixture is smaller than the one for the water–nitrogen mixture and the same applies for the *σ* reduction.

For a non-ideal vapor–gas mixture, the supersaturation of the condensing component can be defined as the ratio of its fugacities at the actual and at the corresponding—same total (mixture) pressure *p* and temperature *T*—equilibrium condition,^{55}

with *y* as the vapor molar fraction. The fugacities $F(p,T,y)$ and $Feq(p,T,yeq)$ can be expressed as a function of the chemical potential of the condensing component in the vapor state *μ*^{v}(*p*, *T*, *y*) and in the corresponding vapor–liquid equilibrium state $\mu eqv(p,T,yeq)$ as^{55}

with the subscript 0 denoting the (arbitrary reference) properties of the pure condensing component. Hence, substituting Eq. (18) into Eq. (17), the supersaturation definition can be reformulated as follows:^{56}

with the last two equalities derived from the expression $F=\varphi \u2009y\u2009p$ and by recalling Eq. (11). The validity of *S* ≈ *y*/*y*_{eq} = *y p*/(*f*_{e}*p*_{s}) is restricted to the case in which the vapor monomers interact mostly with the diluent gas molecules, and consequently, *ϕ* ≈ *ϕ*_{eq}. This condition is closely satisfied in the present application because of the small vapor molar fraction *y*. Note that the definition of supersaturation^{56} [Eq. (19)] differs from that by Wedekind *et al.*,^{5} who took the saturated state of the pure condensing component as reference.

## III. EXPERIMENTAL

The experimental facility is a wave tube in which a well defined pattern of pressure waves can be generated. Geometrical details of the Pulse Expansion Wave Tube (PEWT) can be found in Ref. 23. A simplified schematic of the tube is represented in Fig. 4 (bottom-right). The device consists of a high pressure section (HPS) (0.65 m) and a low pressure section (LPS) (9.2 m), separated by a polyester diaphragm D. The cross section areas of HPS and LPS are the same, except for a local widening W (0.15 m long) of the LPS, placed at about 0.18 m from the diaphragm location.

The test mixture of water vapor and carrier gas is brought into the high pressure section. A well-defined composition of the mixture is obtained by sending a controlled carrier gas flow through a saturator and by flushing the test mixture through the HPS until the tube walls are saturated. This technique leads to a vapor molar fraction *y* with a maximum estimated standard uncertainty of 1%. More details on the mixture preparation device can be found in Ref. 57.

The PEWT is designed to generate a specific wave pattern (see Fig. 4). The diaphragm rupture generates an expansion fan (E), moving toward the end wall of the HPS, and a shock wave (sh), in the opposite direction. The latter is reflected back as a weak expansion (sh_{1}) at the widening cross section enlargement W_{1} and as a small shock (sh_{2}) at the restriction W_{2}. In this way, at the observation point O, the test mixture undergoes a first large adiabatic expansion (E), immediately followed by an additional weak expansion (sh_{1}), and a small re-compression (sh_{2}), shortly after the weak expansion (sh_{1}). This pressure–time pattern generated at the observation point O is able to greatly reproduce the features prescribed by the nucleation pulse method, represented in the two left diagrams of Fig. 4. The combination of the two expansions E and sh_{1} increases the saturation level of the initially under-saturated mixture far above unity (*S* ≫ 1). The pressure drop is chosen such that the formation of a large number of droplets is triggered. This condition is maintained for a short period of time Δ*t*_{p}, the nucleation pulse. It is followed by a slight but sudden re-compression, which lowers the supersaturation level and inhibits the formation of new droplets. In this stage, the supersaturation is kept above unity (*S* > 1) for a long period of time, leading to a cloud of growing droplets. As a consequence, separation in time of nucleation and droplet growth is achieved, both at reasonably constant thermodynamic conditions. Additionally, if Δ*t*_{p} is much shorter than the growth time, the formed cloud of droplets can be assumed to be monodisperse to a large degree.^{58} The nucleation pulse method has been successfully implemented by means of several experimental facilities since 1965.^{58–61} Peters^{58} was the first to use a wave tube with this purpose. Looijmans and co-authors^{61} later built the first version of the PEWT. The nucleation experiments presented in this work have been carried out with an upgraded version^{23} of the Looijmans’s design.

The pressure signal is measured via two pressure transducers, placed at the bottom wall of the observation plane: piezo-resistive (Kistler 4073A50) and piezo-electric (Kistler 603B). The combination of the two transducers leads to a pressure standard uncertainty of 0.2%.^{19} With the pressure in time known, also Δ*t*_{p} can be measured with a maximum standard uncertainty of 3%. Before the test mixture undergoes expansion E, its temperature *T*_{0} is measured at the HPS wall via two platinum resistance thermometers (Tempcontrol PT-8316), placed at the walls of the observation plane. The time dependent temperature is calculated from the pressure signal as follows: Knowing the vapor molar fraction *y*, the entropy *s*_{0} for the actual mixture can be calculated at the initial conditions. Assuming the process to be isentropic *s*(*t*) = *s*_{0} = *const*, the temperature as a function of time *T*(*t*) can be finally derived for each known pressure condition *p*(*t*).^{62} A standard uncertainty of 0.1 K has been calculated for the pulse temperature. Once the thermodynamic conditions are known, the supersaturation can be computed from Eq. (19). The final standard uncertainty on the supersaturation has been calculated to be within 2%, depending on the experimental conditions.

The droplet monodispersity, obtained with the nucleation pulse method, leads to an accurate determination of the nucleation rate *J* by optical means. It is calculated as the ratio of droplet number density *n*_{d} and Δ*t*_{p},

with Δ*t*_{p} obtained from the pressure signal and *n*_{d} measured by means of a dedicated optical setup represented in Fig. 5. The latter consists of a linearly polarized 100:1 laser light (Lasos Lasnova GLK 3220 T01—wavelength of 532 nm), which probes the test section through the optical window *A*_{1}. A photomultiplier (PM) (Hamamatsu 1P28A) records the 90° scattered light at the optical window *A*_{2}, and a photodiode (PD) (Telefunken BPW 34) measures the light attenuation at the optical window *A*_{3} (*A*_{1}, *A*_{2}, and *A*_{3} are BK7 glass windows). The droplet radius in time is obtained by comparing the PM time signal with the theoretical (Mie theory) scattered light as a function of the droplet radius. The PD time signal is used in combination with the Lambert–Beer law and the calculated droplet growth rate to obtain *n*_{d}. This technique is known as the Constant Angle Mie Scattering (CAMS) method.^{60,63,64} The estimated standard uncertainty of *J* is within 20%. More details on the optical technique and the experimental setup can be found in Refs. 21 and 23.

## IV. RESULTS AND DISCUSSION

Novel homogeneous water nucleation experiments have been carried out

in nitrogen at 240 K and 0.1 MPa, 1 MPa, and 2 MPa;

in argon at 236 K and 0.1 MPa, 1 MPa, and 2 MPa; and

in argon at 240 K and 1 MPa and 2 MPa.

The obtained nucleation rates *J* as a function of the supersaturation *S* are presented in Figs. 6 and 7. The *J*_{CNT}(*S*) curves [Eqs. (1)–(3)] at the investigated temperatures are also reported for reference. For each mixture, the *J*–*S* data at fixed temperature and pressure conditions are fitted by means of the following regression curve:

with the coefficients *K*_{0} and *g*_{0} calculated for each condition analyzed in this work. The reader is referred to the supplementary material for more details on experimental data and regression coefficients. The estimated standard uncertainties of *S* and *J* (see Sec. III) are 2% and 20%, respectively. The *J* standard uncertainty is within the represented symbol sizes. Therefore, only the *S* standard uncertainty, combined with the regression error, is reported in the *J*–*S* plots.

In Fig. 6, the water–nitrogen experiments are shown in comparison with the Fransen results.^{21} The latter were obtained for the same mixture at 1 MPa and 240 K, by means of the previous version of the PEWT. Most of these literature data^{21} fall within the uncertainty bands identified in this study at the same conditions (water–nitrogen mixture at 240 K and 1 MPa). However, a relevant data scatter is noticeable (*R*^{2} = 0.96), which is not present for our data (*R*^{2} = 0.995, see the supplementary material). Hence, we can conclude that the new PEWT^{23} has significantly improved the quality of the experiments, with a consistent reduction of the data scatter.

Figure 7(a) shows the nucleation data obtained for water in argon at 1 MPa and 2 MPa. These experiments have been carried out at 236 K (yellow data points and lines) and 240 K (red data points and lines) in order to test the Hale scaling model^{65} of *S*,

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. Using Hale’s approach, the *J*_{exp}(*S*) fits of our data at 236 K for 1 MPa and 2 MPa are scaled to 240 K (red lines), as represented in Fig. 7(a): solid line for 1 MPa and dotted line for 2 MPa. The actual *J*–*S* data obtained at 240 K for 1 MPa (red circles) and 2 MPa (red stars) are shown in Fig. 7(a) to fall on top of the 240 K scaled curves for both high pressure conditions. It should be considered that Eq. (22) was developed and verified by Hale^{65} at low pressure (0.06 MPa).^{24} In view of our experimental findings, this scaling model^{65} appears to be also valid at high pressure conditions.

The water nucleation data in argon at 236 K and 0.1 MPa are compared with the Wölk and Strey *J*–*S* values^{24} measured for the same mixture at 0.06 MPa and two temperatures, 230 K and 240 K. This comparison is reported in Fig. 7(b). The intermediate *J*–*S* values of our experiments in comparison with the literature data^{24} are consistent with the investigated thermodynamic conditions (water–argon mixture at 236 K and 0.1 MPa). The Hale scaling model^{65} is also tested for the latter case. The *J*_{exp}(*S*) fit of our 236 K and 0.1 MPa data is scaled to 240 K and to 230 K, as shown in Fig. 7(b) with the red and purple dashed lines, respectively. These scaled curves are aligned with the 0.06 MPa water–argon data of Wölk and Strey at 240 K and 230 K^{24} (green triangular markers in Fig. 7(b). Such an outcome confirms the validity of the Hale scaling function^{65} at low pressure conditions, considering that, at 0.06 MPa and 0.1 MPa, negligible adsorption effects are present (see Sec. II B) and that the small pressure difference has a negligible influence on the thermalization.

### A. Critical cluster size

The nucleation theorem [Eq. (5)] leads to experimental values of the critical cluster size *n*^{*} from the *J*–*S* curves. These values can be compared with the theoretical ones$nGT*$, based on the Gibbs–Thomson formulation [Eq. (4)] and the capillarity approximation (CNT). Therefore, with the purpose of testing the Gibbs–Thomson equation, we calculate and compare *n*^{*} and $nGT*$. The results are presented in Fig. 8 for the conditions investigated in this work (more details in the supplementary material).

Reasonable agreement is found, within the error bars, between the experimental and theoretical values of the critical cluster sizes, as already pointed out by Wölk and Strey (2001).^{24} As for them, we confirm that using the macroscopic values for surface tension and density leads to a reasonable approximation of *J*_{exp}(*S*) slopes and experimental critical cluster sizes.

### B. Empirical correlation for the water nucleation rate

The CNT requires only bulk properties for the calculation of the nucleation rates *J*_{CNT} [Eqs. (1)–(3)]. This simplification makes possible to use the CNT relatively easily but, on the other hand, does not lead to a quantitative agreement with the experimental nucleation rates, as shown in Figs. 6 and 7(a). Therefore, we have investigated an empirical correction of *J*_{CNT}, leading to a one to one correspondence with the experimental values *J*_{exp}(*S*) [see Eq. (21)] discussed in this work. A similar approach has been proposed in the past by Wölk and Strey.^{24} Their empirical correlation took into account the temperature dependency (between 200 K and 260 K) at about 0.06 MPa for water in argon. We propose a possible empirical expression for the water nucleation rate, which includes not only the temperature but also the pressure and the carrier gas dependencies investigated in the present work. The proposed form of the empirical nucleation rate is

with *Z* as the compressibility factor of the mixture, approximated with the one of the carrier gas *Z*_{g} (small *y* for the fitted conditions), and Ψ = Ψ(*p*, *T*) as the empirical correction function, given by following equation:

The coefficients in Eq. (24) are *c*_{1} = −2.477 · 10^{−5} K^{−2}, *c*_{2} = 13.5 · 10^{−3} K^{−1}, *d*_{1} = −0.8324, *d*_{2} = 0.0274 MPa^{−1}, *d*_{3} = −0.0634 MPa^{−1/2}, with *T* in K and *p* in MPa. The pressure–temperature dependency of Ψ is shown in Fig. 9.

It must be stressed that the proposed empirical correlation in Eq. (23) is obtained by fitting the experimental conditions discussed in this work (including literature sources),^{22–24} and therefore, the domain of *J*_{emp} is strictly limited to water–helium, water–argon, and water–nitrogen mixtures with pressures between 0.05 MPa and 2 MPa and temperatures between 220 K and 260 K. For all the conditions analyzed, the *J*_{emp}(*S*) curves are shown in Fig. 10 to fall within the uncertainty bands of *J*_{exp}(*S*) [Eq. (21)]. It is worth mentioning that the empirical correlation of Wölk and Strey^{24} consists in a temperature correction of *J*_{CNT} [Eqs. (1) and (2) with kinetic prefactor *K* calculated according to Becker and Döring]^{26} such that a correspondence with their experimental data is achieved. Since the proposed function *J*_{emp} is shown in Fig. 10(a) to satisfactorily reproduce such data,^{24} the two temperature-dependent fits can be considered equivalent, albeit formally different.

Pressure, temperature, and carrier gas dependencies are explicitly considered in *J*_{emp} through *σ* [Eq. (6)], *f*_{e} [Eq. (12)], and *Z*, while Ψ only accounts for residual pressure effects and almost completely for the temperature corrections (partially considered also in *σ*, *f*_{e}, and *Z*). Additionally, the $(fe/Z)2$ term accounts for the presence of the non-ideal carrier gas in the mixture, taking into consideration its compressibility factor (*Z* ≈ *Z*_{g}) and the enhancement effects (*f*_{e}) of the non-ideal vapor–gas mixture discussed in Sec. II D. The parameters *σ*/*σ*_{0}, *f*_{e}, *Z*, and Ψ are all between 0.9 and 1.1, but because *σ* and Ψ appear cubed, in the exponent, they provide the major contribution to the pressure dependency of *J*_{emp}.

The pressure dependency of Ψ is obtained by fitting the experimental data at the four pressure conditions (0.06 MPa, 0.1 MPA, 1 MPa, and 2 MPa) analyzed in this work. Similarly, *σ* is derived, in Sec. II B, by fitting the data available in the literature. A slightly non-monotonic pressure dependency of Ψ between 1 MPa and 2 MPa (see Fig. 9) is obtained as a result. We might speculate that the reason of the non-monotonicity is the fact that the correction includes non-isothermal effects at low pressure, whereas, at high pressures, it represents deviations from the true magnitudes of the modeled gas effects on surface tension and enhancement factor. Currently, the empirical correlation proposed in this work [Eq. (23)] appears to be the solution that provides the most satisfactory reproducibility of the available data (see Fig. 10) with the minimum number of fitting coefficients possible.

### C. Effect of carrier gas adsorption on *J*_{exp} at high pressure

The water–helium mixture at 1 MPa and 240 K^{22,23} has been demonstrated to be rather insensitive to adsorption and thermalization effects, as mentioned in Secs. II B and II C (*J*/*J*_{iso} ≈ 1 and Δ*σ*/*σ*_{0} < −0.12%). Thus, its *J*_{exp}(*S*) curve is used as an experimental reference to analyze the water nucleation data at high and low pressure. The non-isothermal effect on nucleation can be reasonably excluded at high pressure (*J*/*J*_{iso} ≈ 1, see Sec. II C). In a previous publication, Fransen *et al.*^{22} compared the nucleation rates for water–nitrogen and water–helium mixtures at 1 MPa and 240 K. They argued that the larger *J* increase for the water–nitrogen mixture than for the water–helium mixture was due to the significant pressure effect on the reduction in water surface tension. We have shown this in Sec. II B as this effect is expected to be somewhat smaller for argon than for nitrogen at the same pressure conditions. In order to verify this theoretical outcome, we compare the high pressure nucleation experiments carried out with water–nitrogen and water–argon mixtures at 1 Ma and 2 MPa, both at 240 K. The results of these experiments are collected in Fig. 11 for a comparative analysis. The reference water–helium *J*_{exp}(*S*) curve at 1 MPa and 240 K^{22,23} is also represented.

The nucleation rates for the water–argon mixture at 240 K and 1 MPa are shown to overlap with the ones obtained with the water–nitrogen mixture at the same conditions. The differences are within the uncertainty bands. This is consistent with the predicted *σ* reduction at 1 MPa for the two mixtures (see Sec. II B): 1.45% ± 0.55% for argon and 1.53% ± 0.49% for nitrogen. At 1 MPa and 240 K, the adsorption effect for argon and nitrogen appears to be significantly larger than for helium. The nucleation rates of the water–nitrogen mixture at 240 K and 2 MPa show larger values than the water–argon mixture at the same conditions, albeit a significant overlap of the uncertainty bands is still present. Nevertheless, the *J* increase is shown to be more pronounced at 2 MPa than at 1 MPa for the two mixtures. This is in qualitative agreement with the larger *σ* reduction at 2 MPa (see Sec. II B): 2.89% ± 0.98% for argon and 3.06% ± 0.87% for nitrogen. The qualitative agreement between the experimental and the predicted *J* increase is further analyzed in Table I. The expected nucleation rate enhancement *J*/*J*_{0} is calculated according to the CNT [Eqs. (1)–(3)] at the upper and lower uncertainty bands of the surface tension reduction (see Sec. II B). The experimental nucleation rate increase $(J/J0)exp$ is calculated as the ratio of *J*_{exp} and reference$(J0)exp$, which is considered to be *J*_{exp} of the water–helium mixture at 240 K and 1 MPa (*J*/*J*_{0} ≈ 1). The experimental increase $(J/J0)exp$ is shown to be systematically close to the lower limit of $(J/J0)CNT$ for all pressure conditions and mixtures analyzed at 240 K.

. | . | . | . | $(\u2212\Delta \sigma /\sigma 0)%$ . | $(J/J0)CNT$ . | . | . | ||
---|---|---|---|---|---|---|---|---|---|

T (K) . | . | p (MPa) . | S^{a}
. | Min . | Max . | Min . | Max . | $JJ0exp$^{b}
. | References^{c}
. |

240 | He | 0.9968 | 12.7 | 0.10 | 0.14 | 1.11 | 1.15 | 1 | 22 and 23 |

Ar | 1.0002 | 11.9 | 0.89 | 2.00 | 2.6 | 8.2 | 2.9 | c.w. | |

1.9939 | 11.8 | 1.91 | 3.88 | 7.6 | 56.7 | 8 | c.w. | ||

N_{2} | 1.0018 | 11.8 | 1.03 | 2.02 | 3.0 | 8.5 | 3.5 | c.w. | |

2.0004 | 11.6 | 2.19 | 3.93 | 10.5 | 63.1 | 14.2 | c.w. | ||

236 | He | 0.9968 | 13.2 | 0.10 | 0.14 | 1.11 | 1.16 | 1 | Eq. (23) |

Ar | 1.0002 | 13.1 | 0.97 | 2.08 | 2.8 | 9.0 | 3.5 | c.w. | |

2.0004 | 13.0 | 2.07 | 4.03 | 9.1 | 68.2 | 9.4 | c.w. |

. | . | . | . | $(\u2212\Delta \sigma /\sigma 0)%$ . | $(J/J0)CNT$ . | . | . | ||
---|---|---|---|---|---|---|---|---|---|

T (K) . | . | p (MPa) . | S^{a}
. | Min . | Max . | Min . | Max . | $JJ0exp$^{b}
. | References^{c}
. |

240 | He | 0.9968 | 12.7 | 0.10 | 0.14 | 1.11 | 1.15 | 1 | 22 and 23 |

Ar | 1.0002 | 11.9 | 0.89 | 2.00 | 2.6 | 8.2 | 2.9 | c.w. | |

1.9939 | 11.8 | 1.91 | 3.88 | 7.6 | 56.7 | 8 | c.w. | ||

N_{2} | 1.0018 | 11.8 | 1.03 | 2.02 | 3.0 | 8.5 | 3.5 | c.w. | |

2.0004 | 11.6 | 2.19 | 3.93 | 10.5 | 63.1 | 14.2 | c.w. | ||

236 | He | 0.9968 | 13.2 | 0.10 | 0.14 | 1.11 | 1.16 | 1 | Eq. (23) |

Ar | 1.0002 | 13.1 | 0.97 | 2.08 | 2.8 | 9.0 | 3.5 | c.w. | |

2.0004 | 13.0 | 2.07 | 4.03 | 9.1 | 68.2 | 9.4 | c.w. |

^{a}

Mean supersaturation at each condition.

^{b}

Ratio of *J*_{exp} [Eq. (21)] and experimental reference$(J0)exp$, both calculated at the same *S*-value.^{a} The reference $(J0)exp$ is taken equal to *J*_{exp} [Eq. (21)] of the water–helium mixture at 1 MPa for 240 K (see Fig. 11) and to *J*_{emp} [Eq. (23)] of the water–helium mixture at 1 MPa for 236 K (see Fig. 12).

^{c}

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

Additionally, we analyze the 236 K experiments carried out in the water–argon mixture at 1 MPa and 2 MPa. In this case, the reference experimental condition (*J*/*J*_{0} ≈ 1) is $(J0)exp$ of the water–helium mixture at 1 MPa and 236 K. Since this condition is not available in the literature, we compute it as *J*_{emp} by employing the proposed empirical correlation [see Eq. (23)]. Equivalently, the reference condition can be obtained by scaling^{65} the water–helium data at 1 MPa and 240 K–236 K. The *J*–*S* plots of these conditions are shown in Fig. 12. Similar to the 240 K case, the experimental nucleation rate increase $(J/J0)exp$ is found to be consistently close to the lower limit of $(J/J0)CNT$ for all pressure conditions analyzed at 236 K, as reported in Table I.

Considering these experimental evidences, we can conclude that the adsorption phenomena and the corresponding water surface tension reduction seem to be the predominant causes of the nucleation rate increase at high pressure conditions. For a definitive confirmation of our findings, experiments at higher pressure conditions and with carrier gases that affect the surface tension more profoundly are needed. Further research studies should also be made in order to extend the few interfacial tension data, available above 298 K,^{37–41} to the supercooled liquid region.

### D. Thermalization effect on *J*_{exp} at low pressure

Adsorption effects are not important at low pressure, as discussed in Sec. II B (Δ*σ*/*σ*_{0} < −0.2%). So, we assume that only thermalization affects the nucleation rates at pressures up to the ambient condition. The experimental data at 0.1 MPa and 240 K for water–nitrogen and water–argon mixtures are analyzed in comparison with the 240 K water–helium data sets^{22,23} at 0.1 MPa and 1 MPa. The 1 MPa case is taken as experimental isothermal reference $(Jiso)exp$ at 240 K, being negligible thermalization effects (*J*/*J*_{iso} = 0.97 ≈ 1, see Table II) and adsorption phenomena (Δ*σ*/*σ*_{0} = −0.12%, see Table I). The *J*_{exp}(*S*) curves of these conditions are collected in Fig. 13. The nucleation rates of water–argon and water–nitrogen mixtures at 0.1 MPa and 240 K are shown to be almost identical. The water–helium data at 0.1 MPa and 240 K present smaller nucleation rates than water–argon and water–nitrogen mixtures at the same conditions. Non-isothermal effects are investigated in order to explain the nucleation rate variations for the different carrier gases at low pressure. A quantitative analysis is presented in Table II. According to the Barrett kinetic model,^{4} the theoretical (CNT-based) nucleation rate reduction [see Eq. (7)] is expected to be larger with increasing molecular mass of the carrier gas, at fixed conditions (0.1 MPa and 240 K): smaller for the water–helium mixture (*J*/*J*_{iso} = 0.757), intermediate for the water–nitrogen mixture (*J*/*J*_{iso} = 0.641), and larger for the water–argon mixture (*J*/*J*_{iso} = 0.514). At the same conditions, the experimental nucleation rate reduction is found to be in qualitative agreement with the predicted trend: smaller for the water–nitrogen mixture [$(J/Jiso)exp=0.328$] than for the water–argon mixture [$(J/Jiso)exp=0.258$]. However, the predicted trend is not respected for the water–helium mixture, which experimentally shows the most significant nucleation rate reduction [$(J/Jiso)exp=0.187$], while it was expected to be least significant [(*J*/*J*_{iso}) = 0.757].

. | p (MPa) . | S^{a}
. | λ
. | J/J_{iso}
. | $J/Jisoexp$^{b}
. | References^{c}
. |
---|---|---|---|---|---|---|

He | 0.1054 | 13.5 | 253 | 0.757 | 0.187 | 22 and 23 |

0.9968 | 12.7 | 2526 | 0.968 | 1 | 22 and 23 | |

Ar | 0.1028 | 12.0 | 87 | 0.514 | 0.258 | c.w. |

N_{2} | 0.1017 | 12.6 | 147 | 0.641 | 0.328 | c.w. |

. | p (MPa) . | S^{a}
. | λ
. | J/J_{iso}
. | $J/Jisoexp$^{b}
. | References^{c}
. |
---|---|---|---|---|---|---|

He | 0.1054 | 13.5 | 253 | 0.757 | 0.187 | 22 and 23 |

0.9968 | 12.7 | 2526 | 0.968 | 1 | 22 and 23 | |

Ar | 0.1028 | 12.0 | 87 | 0.514 | 0.258 | c.w. |

N_{2} | 0.1017 | 12.6 | 147 | 0.641 | 0.328 | c.w. |

Additionally, we investigate the temperature dependency of the thermalization effect. The Barrett model^{4} is employed to predict the nucleation rate reduction for the water–argon experiments carried out at 0.06 MPa and different temperatures (220 K–260 K) by Wölk and Strey.^{24} The results of this analysis are summarized in Table III. The theory^{4} predicts a larger nucleation rate reduction for increasing temperatures: from *J*/*J*_{iso} = 0.681 at 220 K to (*J*/*J*_{iso}) = 0.163 at 260 K.

p (MPa) . | T (K) . | S^{a}
. | λ
. | J/J_{iso}
. | $J/Jisoexp$^{b}
. |
---|---|---|---|---|---|

0.0482 | 220 | 19.5 | 217 | 0.681 | 0.446 |

0.0571 | 230 | 14.8 | 110 | 0.551 | 0.297 |

0.0611 | 240 | 11.4 | 55 | 0.397 | 0.205 |

0.0517 | 250 | 9.2 | 23 | 0.223 | 0.195 |

0.0631 | 260 | 7.6 | 14 | 0.163 | … |

p (MPa) . | T (K) . | S^{a}
. | λ
. | J/J_{iso}
. | $J/Jisoexp$^{b}
. |
---|---|---|---|---|---|

0.0482 | 220 | 19.5 | 217 | 0.681 | 0.446 |

0.0571 | 230 | 14.8 | 110 | 0.551 | 0.297 |

0.0611 | 240 | 11.4 | 55 | 0.397 | 0.205 |

0.0517 | 250 | 9.2 | 23 | 0.223 | 0.195 |

0.0631 | 260 | 7.6 | 14 | 0.163 | … |

^{a}

Mean supersaturation at each condition.

^{b}

Ratio of *J*_{exp} [Eq. (21)] and corresponding (same *S*-value)^{a} isothermal references $(Jiso)exp$ at different temperatures. The latter is taken equal to *J*_{emp} [Eq. (23)] of the water–helium mixture at 0.219 MPa for 220 K, at 0.458 MPa for 230 K, at 0.922 MPa for 240 K, at 1.834 MPa for 250 K, and at 3.339 MPa for 260 K (out of *J*_{emp} domain, see Sec. IV A). These conditions are chosen such as the theoretical *J*/*J*_{iso} = *const* ≈ 1 and adsorption effects can be considered negligible (Δ*σ*/*σ*_{0} < −0.2%).

In order to verify the predicted temperature dependency of the thermalization effect, the experimental isothermal references $(Jiso)exp$ at each temperature condition are needed. These references are chosen such that adsorption effects can be considered negligible and the theoretical *J*/*J*_{iso} = *const* ≈ 1. Exploiting Eqs. (7)–(10), we have found that such requirements are met by a water–helium mixture (Δ*σ*/*σ*_{0} < −0.2%, see Sec. II B) at 0.219 MPa for 220 K, at 0.458 MPa for 230 K, at 0.922 MPa for 240 K, at 1.834 MPa for 250 K, and at 3.339 MPa for 260 K. Since experimental data for these conditions are not available in the literature, we calculate them as *J*_{emp}(*S*) by employing the proposed empirical correlation [Eq. (23)]. Note that *J*_{emp} for the water–helium mixture at 3.69 MPa and 260 K is not calculated outside of the *J*_{emp} domain (see Sec. IV A). The comparison between *J*_{exp} and $(Jiso)exp$ is shown in Fig. 14 as a function of *S* at different temperature conditions. As a result, the experimental nucleation rate reduction is also found to be larger with increasing temperatures: from $(J/Jiso)exp=0.446$ at 220 K to $(J/Jiso)exp=0.195$ at 250 K. The experimental trend is quantitatively well predicted by the Barrett model^{4} at 250 K: $(J/Jiso)exp=0.195$ and *J*/*J*_{iso} = 0.223. With decreasing temperatures, the theoretical nucleation rate reductions deviate from the experimental trend: $(J/Jiso)exp=0.446$ and *J*/*J*_{iso} = 0.681 at 220 K. This outcome is not fully surprising since the Barrett model^{4} was validated for the water–argon case at 260 K and 0.06 MPa. However, at low and high temperatures, the experimental trend is in qualitative agreement with the Barrett model,^{4} albeit the theory quantitatively underestimates the thermalization effect on the nucleation rate reduction, as already pointed out by Hrubý and co-workers.^{66} This qualitative agreement is also confirmed when comparing the water–argon and water–nitrogen mixtures at fixed temperature conditions (240 K). We argue that other aspects poorly treated by the classical nucleation theory are responsible for the lack of quantitative agreement. It is a matter of on-going research to find a possible reason for such a discrepancy between theory and experiments.

## V. CONCLUSIONS

Homogeneous nucleation of water has been experimentally studied in argon and nitrogen at about 240 K with the PEWT facility. Three different pressure conditions have been analyzed for both mixtures: 0.1 MPa, 1 MPa, and 2 MPa. A substantial scatter reduction of the *J*–*S* data has been found with respect to the experiments carried out with the previous version of the PEWT,^{21} which confirms the improvements introduced with the new version of the setup.^{23} A good agreement of the *J*–*S* data has been found with the literature results, available at two of the investigated conditions (water–nitrogen^{21} mixture at 240 K and 1 MPa and water–argon^{24} mixture at 240 K and 0.06 MPa). The surface tension decrease due to adsorption effects appears to be the predominant cause of the nucleation rate enhancement at high pressure for both mixtures. Further research studies on the surface tension of water in diluent gases at supercooled conditions are the key for a more accurate quantitative analysis. Higher pressure conditions and carrier gases with a larger influence on the surface tension are needed to give further confirmation of our findings. At fixed low pressure conditions, the theory of Barrett^{4} on non-isothermal effects has been shown to qualitatively explain the experimental nucleation rate differences between water–argon and water–nitrogen mixtures at fixed temperature, as well as the temperature dependency of the thermalization phenomenon in the water–argon mixture. For helium as the carrier gas, the non-isothermal effects appear to be much stronger than those predicted by the Barrett model.^{4} The critical cluster size has been calculated from the experimental *J*–*S* curves by means of the nucleation theorem. The comparison with the Gibbs–Thomson equation has revealed an agreement within the experimental error bars, as already pointed out by Wölk and Strey.^{24} This confirms that, at the analyzed conditions, the capillarity assumption can be considered an acceptable approximation to predict the critical cluster size and slopes of the experimental *J*(*S*) curves. On the other hand, using the macroscopic values of surface tension and density in the *J*_{CNT}, even with corrections for real gas, adsorption, and imperfect thermalization, does not lead to quantitatively correct predictions of nucleation rates. Thus, we have introduced an empirical correction of *J*_{CNT}, based on the experimental values at the presented conditions. The proposed expression takes into account pressure, temperature, and the type of carrier gas, showing a one to one correspondence with the experimental nucleation rates within their uncertainty bands. This outcome suggests that the proposed empirical correlation can be used as a tool for the determination of actual nucleation rates within the following domain: water–argon, water–nitrogen, and water–helium mixtures at pressures between 0.06 MPa and 2 MPa and temperatures from 220 K to 260 K.

## SUPPLEMENTARY MATERIAL

Details on the new data presented in Sec. IV can be found in the supplementary material. A 2D representation at 240 K of Fig. 3 is also available as a supplementary figure.

## ACKNOWLEDGMENTS

J.H. acknowledges support from the Ministry of Education, Youth and Sports of the Czech Republic under OP RDE Grant No. CZ.02.1.01/0.0/0.0/16019/0000753 “Research center for low carbon energy technologies.”

## DATA AVAILABILITY

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

## REFERENCES

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_{2}O in comparison: The isotope effect