Complex transition pathways in boiling and cavitation

Abstract

This work combines Navier-Stokes-Korteweg dynamics and rare event techniques to investigate the transition pathways and times of vapour bubble nucleation in metastable liquids under homogeneous and heterogeneous conditions. The nucleation pathways deviate from classical theory, showing that bubble volume alone is an inadequate reaction coordinate. The nucleation mechanism is driven by long-wavelength fluctuations with densities slightly different from the metastable liquid. We propose a new strategy to evaluate the typical nucleation times by inferring the diffusion coefficients from hydrodynamics. The methodology is validated against state-of-the-art nucleation theories in homogeneous conditions, revealing non-trivial, significant effects of surface wettability on heterogeneous nucleation. Notably, homogeneous nucleation is detected at moderate hydrophilic wettabilities despite the presence of a wall, an effect not captured by classical theories but consistent with atomistic simulations. Hydrophobic surfaces, instead, anticipate the spinodal.The proposed approach is fairly general and, despite the paper discussing results for a prototypical fluid, it can be easily extended, also in complex geometries, to any real fluid provided the equation of state is available, paving the way to model complex nucleation problems in real systems.

1. Introduction

The formation of vapour bubbles in a liquid is ubiquitous in natural phenomena. Fascinating examples are the spore release mechanism in the fern sporangium annulus (Noblin et al. 2012; Montagna et al. 2023), the hunting tactic adopted by the snapping shrimp (Versluis et al. 2000), and the loss of sap transport capacity due to xylem cavitation (Cochard 2006). Bubble nucleation is also central to several technological applications. Cavitation damage has been a critical issue for decades (Silberrad 1912; Reuter et al. 2022; Abbondanza et al. 2023a, 2024), while boiling is now actively exploited as an efficient mechanism of heat removal for thermal management applications (Bakir & Meindl 2008; Fan & Duan 2020; Chakraborty et al. 2024; Zhang et al. 2023). Cavitation and boiling are the two sides of the same coin: bubble formation is achieved equivalently by reducing the pressure below the saturation pressure at a given temperature, p_sat (T), or by increasing the temperature above the evaporation point at a given pressure, T_sat(p). Exceeding these threshold levels does not guarantee the bubble’s appearance. The liquid can indeed be held in metastable equilibrium in stretched (p < p_sat) or superheated (T > T_sat) conditions without any phase change (Debenedetti 1996; Azouzi et al. 2013; Magaletti et al. 2021). The origin of the metastability can be traced back to a finite free energy barrier between the liquid and the vapour states that renders the liquid/vapour phase change an activated process, where thermal fluctuations are the activating mechanism. This means that a nucleation event always occurs in metastable conditions after waiting a sufficiently long time. However, the time to be awaited exponentially grows with the energy barrier, resulting in an unlikely occurrence, a so-called rare event, except for conditions characterised by small barriers. The thermodynamic limit of metastability is reached at spinodal conditions where the spinodal decomposition, a barrier-less phase transition mechanism, occurs. The quantitative prediction of vapour bubble formation in liquids, and phase transition in general, is a long-lasting problem of fluid dynamics and statistical physics, see for example (Lutsko 2017) for an enlightening nucleation résumé. The practical nucleation limit, expressed through the cavitation pressure (Menzl et al. 2016), p_cav, or the superheat at boiling onset (Gallo et al. 2023), ΔT_ons = T_ons − T_sat, precedes the spinodal threshold. However, determining the actual nucleation limit remains challenging. For instance, the cavitation pressure of water has been widely debated due to significant discrepancies between measurement techniques (Caupin & Herbert 2006), with reported values ranging from −30, MPa to −120, MPa at ambient temperature. These discrepancies are largely attributed to sample purity (Lohse & Prosperetti 2016) and difficulty achieving homogeneous nucleation under laboratory conditions (Caupin & Herbert 2006). Lower measured cavitation pressures are considered more representative, as they likely correspond to purer samples. Specifically, since nucleation is an activated process, the cavitation pressure is defined as the pressure at which there is a 50% in probability of observing a bubble, given the experimental sample volume V_exp and observation time window τ_exp (Azouzi et al. 2013). The same arguments apply to the superheat limit, where extremely high measured superheats T_ons ~ 302°C (Skripov 1970) stand in stark contrast to boiling temperature measurements showing only a few degrees of superheat (Theofanous et al. 2002). Nucleation at solid surfaces introduces further complexity and an even stronger discrepancy in data due to the difficult control in the experimental measurements of dissolved gas content, surface roughness, and chemical impurities (Frenkel 1955).

Understanding the fluid dynamics of multiphase systems—and ultimately reconciling them with experimental observations—requires a deep grasp of nucleation and the incipient stages of phase change (Vincent & Marmottant 2017; Yatsyshin & Kalliadasis 2021; Gao et al. 2021; Alamé & Mahesh 2024; Chen et al. 2025). Nucleation is a key process in multiphase fluid dynamics, driving phase transitions such as bubble and droplet formation. However, its highly uncertain triggering conditions limit the predictive power of current models. Accurately capturing nucleation is therefore essential for developing realistic, multiscale simulations that move beyond empirical assumptions and better represent the complexity of multiphase behaviour.

The reference theory is the Classical Nucleation Theory (CNT) (Blander & Katz 1975), which allows the estimation of the energy barrier, critical cluster dimension and nucleation rate. Despite its physically grounded description of the phenomenon, the estimates for the nucleation rates obtained from CNT differ by orders of magnitude from experiments (Caupin & Herbert 2006; Menzl et al. 2016). More sophisticated theories like different extensions of CNT (Lutsko & Durán-Olivencia 2015; Menzl et al. 2016), density functional theory (DFT) (Oxtoby & Evans 1988; Talanquer & Oxtoby 1996; Lutsko 2008; Baidakov 2016; Yatsyshin & Kalliadasis 2021), and molecular dynamics (MD) simulations (Allen et al. 2009; Diemand et al. 2014) can provide better barrier and nucleation rate estimates to correct some of the CNT mispredictions. A common viewpoint in these approaches is considering a multi-parameter description of the nucleation process by enriching the list of reactive coordinates with thermodynamic cluster properties (Duran-Olivencia et al. 2018).

Here, we propose a mesoscale strategy that clarifies the onset of phase transitions and can be directly coupled one-to-one with hydrodynamic equations, bridging microscopic physics and macroscopic fluid dynamics. By combining Navier-Stokes-Korteweg dynamics with rare event techniques, we offer new insights into nucleation pathways and transition times of vapor bubbles in metastable liquids under both homogeneous and heterogeneous conditions.

The nucleation pathways are found to be significantly different from those predicted by classical nucleation theory, demonstrating that the bubble volume is an inadequate reaction coordinate. The nucleation mechanism arises from long-wavelength fluctuations at large radii, with densities only slightly different from the metastable liquid. The comparison with spherically averaged Fluctuating Hydrodynamic (FH) simulations of the homogeneous nucleation process, as proposed by these authors in (Gallo et al. 2020), supports this evidence. For the liquid/vapour thermodynamics, we exploit an approximate DFT approach, specifically, the van der Waals’ square gradient model for capillary fluids, also known as the Diffuse Interface (DI) model (Lutsko 2011; Anderson et al. 1998). The string method (E et al. 2002) is used to obtain the minimum energy path (MEP), which, in a system with simple gradient dynamics, is associated with the most likely transition path (MLP). Once the MEP has been estimated, we developed a simplified dynamical model that describes the thermodynamic system as a Brownian walker within a metastable basin. This approach allows us to analyse the system’s stochastic evolution in the presence of thermal fluctuations. Using Kramers’ theory, we then estimated the typical transition frequencies associated with bubble formation. In this framework, the formation of bubbles is interpreted as a rare event driven by noise-induced transitions across an energy barrier. Furthermore, within this metastable landscape, we estimated the effective diffusion of the bubble, which characterises how the bubble’s state fluctuates due to stochastic forces. The effective diffusion coefficient is estimated from hydrodynamics, providing a quantitative measure of the bubble’s mobility before it undergoes a phase transition. The approach we propose bears some resemblance to previous proposals (Menzl et al. 2016), but differs under several substantial respects. In our case, we extract the free energy profile from the MEP of a more realistic model of the liquid/vapour interfacial properties, accounting for the finite thickness of the interface and the intrinsic dependence of surface tension on bubble size, properties which are crucial given the breakdown of the sharp interface assumption at the scale of the nucleating embryo. Hence, the Brownian walker evolves over the appropriate landscape. Moreover, we managed to extract the friction coefficient from the appropriate NSK dynamics. The advantages are that the only input parameters needed by the model are experimentally measurable physical quantities, like planar surface tension and transport coefficients, and the model is naturally suited to deal with heterogeneous conditions and complex geometries.

In the present paper, the homogeneous case is also used as a validation of the proposed approach against state-of-the-art nucleation theories (Menzl et al. 2016). Applying this methodology to heterogeneous nucleation reveals distinct effects of surface wettability, particularly under low energy barrier conditions. For moderately hydrophilic surfaces, homogeneous nucleation occurs despite the presence of a wall, an effect overlooked by classical theories but supported by atomistic simulations (Zou et al. 2018; Sullivan et al. 2025) since the surface is unable to significantly reduce the nucleation barrier compared to the bulk. In contrast, hydrophobic surfaces anticipate the spinodal limit (Talanquer & Oxtoby 1996), triggering an earlier onset of nucleation via spinodal-like mechanisms. These results demonstrate how surface wettability can fundamentally influence nucleation pathways, with important implications for controlling phase transitions in confined or engineered systems.

2. Results

The two-phase system in the isothermal condition is described here by following the DI approach through the (Landau) free energy functional

$$\Omega [\rho ]={\displaystyle {\int }_V\left[f_b(\rho )+f_c(\nabla \rho )-{\mu }_{eq}\rho \right]dV+{\oint }_{\partial V}{f}_w(\rho )dS},$$ (2.1)

where the van der Waals’ square gradient approximation is used to express the non-local excess free energy contribution as f_c = (λ/2) |∇ρ|², while f_b is the classical Helmholtz bulk free energy density and μ_eq the equilibrium chemical potential, see the Supplementary Information (SI) for additonal information. The capillary coefficient λ controls the liquidvapour interface properties, namely the surface tension and the interface thickness. The temperature dependence of all the terms has been neglected for the ease of notation. Finally, the surface contribution f_w arises as a mean field approximation of the fluid-wall interactions and accounts for the wetting properties of the surface. Its explicit expression, derived in (Gallo et al. 2021),

$$f_w(\rho )=f_w\left({\rho }_V\right)-\cos \theta {\int }_{{\rho }_V}^{\rho }\sqrt{2\lambda \left[w_b(\tilde{\rho })-w_b\left({\rho }_V\right)\right]}d\tilde{\rho },$$ (2.2)

has been shown to recover the well-known Young condition for the equilibrium contact angle, θ. In the previous expression, f_w(ρ_V) = γ_SV is the surface energy at the solid-vapor interface, ρ_V is the vapor density, and w_b = f_b − μ_satρ, with μ_sat the chemical potential at the saturation condition (μ = ∂f_b/∂ρ). The starting point for deriving the form of the wall free energy is based on geometric considerations regarding the orientation of the interface normal, aligned with the density gradient, and its contact with the solid surface, along with the expression for surface tension from diffuse interface theory, see (Gallo et al. 2021) for details. In this work, the modified Benedict–Webb–Rubin Equation of State (Johnson et al. 1993), mimicking the behaviour of a Lennard-Jones fluid, has been used as f_b(ρ). f_b (ρ) is rendered dimensionless using the reference energy f_bref = ϵ/σ³ where σ = 3.4 × 10⁻¹⁰ m, ϵ = 1.65 × 10⁻²¹ J. The density field is non-dimensionalized with the reference density, ρ_ref = m/σ³, with m = 6.63 × 10⁻²⁶ Kg. The capillary coefficient is set to λ = 5.224 with its reference value λ_ref = σ⁵ϵ/m² ensuring consistency with surface tension values derived from Monte Carlo simulations (Gallo et al. 2018; Magaletti et al. 2022).

The minimization of Eq. (2.1) leads to the following Euler-Lagrange equation,

$$\frac{\delta \Omega }{\delta \rho }=\mu -\lambda {\nabla }^2\rho -{\mu }_{eq}=0,$$ (2.3)

expressing the chemical equilibrium in terms of a constant (generalised) chemical potential. This equilibrium condition is supplemented with the boundary condition ∂f_w/∂ρ + λ∂ρ/∂n = 0 at the solid surface. When the prescribed equilibrium chemical potential corresponds to a metastable condition, μ_spin < μ_eq < μ_sat, Eq. (2.3) has three solutions: i) the homogeneous liquid ρ(x) = ρ_L; ii) the homogeneous vapour ρ(x, t) = ρ_V ; iii) a two-phase solution with a bubble of a given radius surrounded by the metastable liquid, the critical nucleus representing the transition (or critical) state, i.e. the density configuration ρ_crit (x) that must be reached to trigger the complete transition from the metastable liquid to the more stable vapour state. Several previous studies have addressed nucleation within frameworks similar to the one employed in the present work (Talanquer & Oxtoby 1996; Baidakov 2016). However, these approaches have been limited to the calculation of critical nucleus profiles and the subsequent estimation of nucleation rates, without resolving the full transition pathway using a rare-event technique. We address here the problem of evaluating the non-trivial solution of case (iii) by exploiting the powerful string method (E et al. 2007; Magaletti et al. 2021), with the twofold objective of determining both the density field of the critical nucleus and the complete MEP, which describes the system configurations ρ(s, x) along a suitable reaction coordinate s advancing through the transition. As detailed in Appendix B, the MEP corresponds to the MLP for a gradient dynamics and thus provides a physically grounded description of the liquid–vapour phase transition. This path can be visualised as a curve (the string), parametrised with s, in the infinite configurational space of the density fields, with the configuration of maximum energy along the MEP representing the transition state. For the homogeneous case, the bubble radius, in each string image along the path, is extracted a-posteriori from the density field. In the spherical case, a natural radius definition is

$$R(s)={\int }_0^{\infty }r{(\partial \rho (r,s)/\partial r)}^2\text{d}r/{\int }_0^{\infty }{(\partial \rho (r,s)/\partial r)}^2\text{d}r,$$

which, as it will be discussed below, provides a good indicator for the bubble radius converging to the classical one used in sharp interface models. In classical theories, the bubble radius (or its volume, equivalently) is used as the reaction coordinate to describe the progress of the nucleation process, both in homogeneous and heterogeneous conditions. CNT is often used to quantitatively depict the energy landscape of the system in terms of the bubble size, with the energy maximum representing the barrier for nucleation and its corresponding critical radius. In panel a of Fig. 1, the landscapes are plotted in terms of the normalised grand potential ΔΩ / (k_BT) (Ω [ρ] − Ω [ρ_L]) /(k_BT) as a function of the bubble radius R. Both homogeneous and heterogeneous CNT predictions, at non-dimensional temperature T = 1.2 (T ≃ 1.31 being the critical temperature) and metastability level μ_lev = (μ_eq − μ_sat) / (μ_spin − μ_sat) = 0.2, are compared with those obtained by applying the string method to the DI modelling of the two-phase system, as previously described. For T = 1.2, we have μ_sat = −3.9506, μ_spin = −4.0491, ρ_Lsat = 0.5669 and ρ_Lspin = 0.4798. The heterogeneous landscape in the plot refers to the specific case of a neutrally wetting surface (θ = 90°). Despite the low metastability level – corresponding to a condition close to saturation –, the discrepancy between the different energy barrier predictions is apparent in both cases, with a difference on the order of 6%, in line with the results in (Shen & Debenedetti 2001). The differences in the nucleation barrier are minor, and the critical radii are accurate when approaching saturation conditions. However, discrepancies in the barrier become significantly more pronounced near the spinodal, where CNT predicts a finite barrier, whereas it should vanish. This aspect will be discussed in detail later in the paper. The major difference, however, is the behaviour at low energies where the DI model predicts a non-bijective correspondence between the radius and the energy, suggesting that the radius alone is insufficient to fully describe the nucleation process.

Figure 1.

Panel a: Comparison of energy landscapes for homogeneous and heterogeneous bubble nucleation predicted by CNT (dashed lines) and DI model (solid lines). The heterogeneous case corresponds to a neutrally wetting surface (θ = 90°). All curves are computed at T = 1.2 and μ_lev = 0.2. Panel b: Transition path of homogeneous nucleation projected onto the two-coordinate space {ρ_av, R} at T = 1.2, μ_lev 0.9, Arrows identify the phase change direction, red DI and blue CNT, respectively. Transition states from CNT and DI models are marked with full and empty circles, respectively. Red squares indicate ρ_av and R during a Fluctuating Hydrodynamics simulation. Inset: Transition paths at varying metastabilities, with circles marking the corresponding transition states.Panel c,d: Minimum Energy Paths (MEPs) of heterogeneous nucleation in {ρ_av, V} space for different surface wettabilities. Panel c: T = 1.2, μ_lev 0.2; Panel d: T = 1.2, μ_lev = 0.6.

New insights on the transition pathway are gained when the homogeneous nucleation MEP is plotted using a two-coordinates system, {ρ_av, R}, with the average density inside the bubble a-posteriori evaluated along the string as ${\rho }_{av}=3/\left(4\pi R^3\right){\int }_0^R\rho (r)4\pi r^2\text{d}r$, see Fig. 1 panel b. The DI model reveals a non-classical C-shape nucleation pathway. In line with the findings on crystal nucleation (Lutsko 2019; Lutsko & Lam 2020), the process starts with a spatially-extended density variation of small intensity, hence characterised by a large radius and ρ_av ≃ ρ_L. Successively, the embryo spatially localises, and its density variation increases. After the transition state is reached, the bubble further grows and expands, see the red arrows indicating the direction of the transition. The characteristic C-shaped form of the transition pathway is not sensitive to the specific definition of the radius and can also be observed when using the equimolar radius, as shown in the SI. The position of the transition state along the curve strongly depends on the degree of metastability μ_lev, as shown in the inset of panel b in Fig. 1: a large and low-density critical embryo characterises conditions close to saturation (small μ_lev), in agreement with CNT; when approaching spinodal conditions the critical embryo is instead characterised by small size and a high average density. The C-shape pathway is also confirmed by dynamic simulations with the spherically-averaged FH model proposed by these authors in (Gallo et al. 2020), see Appendix A for details. Due to thermal fluctuations, the system explores a pseudo-tube in the {ρ_av, R} space around the zero-temperature MEP provided by the string method. Fluctuations of the average density intensify as the vapour embryo localises in a small region and, consequently, triggers the formation of a low-density cluster that successively expands beyond the transition state. Please note that fluctuations increase when reducing the average (bubble) volume (Gallo 2022). We observe that the MEP is perfectly followed by brute force FH simulations, as demonstrated in panel b of Fig. 1. This shows unequivocally that, among all the possible thermally-induced fluctuations (always present also in stable liquids), the relevant ones for triggering the transition from the metastable liquid state are indeed those predicted by the MEP we have calculated. In addition, despite the nucleation path in FH being expected to differ from the MEP (Grafke et al. 2015; Yao & Ren 2022), the bubble nucleation mechanism appears to be well described by free-energy calculations.

Heterogeneous nucleation follows a similar pathway (with the volume V instead of the radius) with a small but spatially extended density variation as a precursor of the actual bubble formation. Results at different wettabilities and degrees of metastability are plotted in panels c and d of Fig. 1. As expected, the volume of the critical nuclei decreases as the contact angle increases, namely, hydrophilic surfaces require a larger critical bubble to initiate the phase transition compared to hydrophobic walls. In addition, the critical volume decreases when the metastability level is increased.

In the case of homogeneous nucleation, panel a of Fig.2 reports the nucleation barrier normalised by k_BT as a function of the degree of metastability. As shown, CNT captures well the qualitative trend of the barrier with increasing metastability. However, as the system approaches the spinodal limit, μ_lev → 1, CNT predicts a large limit barrier of approximately ${\Delta \Omega }_{CNT}^{\star }\simeq 18k_{B}T$, whereas the DI approach yields the expected vanishing barrier ${\Delta \Omega }_{DI}^{\star }\simeq 0$. This discrepancy arises from the gradual breakdown of the sharp interface assumption as the liquid-vapour interfacial thickness becomes comparable to the typical bubble radius. This effect is illustrated in the inset of panel a in Fig.2 (red curve right axis), which shows the ratio of the bubble radius to the interface thickness l₁₀₋₉₀, defined as the width of the region over which the density transitions from ρ₁₀ = 0.1ρ_L + 0.9ρ_V to ρ₉₀ = 0.9ρ_L + 0.1ρ_V (Caupin 2005). On the right axis, CNT (dashed line) and DI (solid line) radii are depicted. DI predicts a non-monotonic behaviour of the critical radius as the spinodal is approached, in contrast to the monotonic trend obtained from CNT. Nevertheless, the actual values of the critical radii remain in reasonably good agreement over the range of metastabilities considered. Panel b of Fig.2 shows the transition pathway for homogeneous nucleation in terms of the density profiles ρ(r) along the reaction coordinate s of the string. Precritical profiles are shown in blue, postcritical ones in red, and the critical (saddle-point) profile in black. Based on the structure of these profiles, we evaluate the quantity (∂ρ/∂r)² = ρ′(r)², normalised by its L₂ norm, $\left|\right|{\rho }^{\prime }(r)|{|}_{L_2}$. This normalised profile offers insight into the interfacial region and, in the sharp interface limit, serves as an indicator of the interface location. Specifically, as the interface thickness tends to zero, ${\rho }^{\prime 2}/\left|\right|{\rho }^{\prime }(r)|{|}_{L_2}^2\to \delta (r-R)$, converging to the sharp interface limit, where δ denotes the Dirac delta function. Within the diffuse interface framework, this provides a generalised representation of the interface. This normalised gradient is reported in panel c, where the main figure shows precritical profiles and the inset includes postcritical profiles and the saddle point. As indicated by the arrow representing the direction of the transition (increasing s), the initial profiles are broad, with maxima located at larger distances r and only modest density variations (red curves in panel b), corresponding to a slightly rarefied liquid. As the transition proceeds, the maxima shift toward smaller radii and eventually increase in amplitude near the saddle point and beyond (blue curves in panel b), consistent with the complex C-shaped transition pathway observed in the (R, ρ_av) plane. To better highlight the nucleation mechanism, panel d also reports the free energy landscape as a function of the two variables R and ρ_av, with the direction of the reaction coordinate s indicated.

Figure 2.

Panel a: Normalised free-energy barrier ΔΩ ^⋆/k_BT vs μ_lev. Dashed lines refer to CNT, while solid lines refer to the diffuse interface. Inset: the left axis reports R vs μ_lev, the right axis depicts R normalised with interface tickness l₁₀₋₉₀ vs μ_lev. Panel b: Density profiles along the transitions, pre/postcritical profiles are depicted in blue and red, respectively. The critical profile is reported in black (μ_lev = 0.8, T = 1.20). Panel c: (∂ρ/∂r)² normalised with its L₂ norm vs radial coordinate. The main panel refers to precritical states, while the inset refers to postcritical conditions. Panel d: Energy landscape as a function of the tuple (ρ_av, R), μ_lev = 0.2, T = 1.20. All cases refer to homogeneous nucleation, the reaction coordinate s increasing directions are also indicated.

A further comparison with heterogeneous CNT is shown in panel b of Fig. 3 where the ratio between the energy barriers of the heterogeneous and homogeneous cases, ${\Delta \Omega }_{het}^{\ast }/{\Delta \Omega }_{hom}^{\ast }$, is reported as a function of the contact angle. CNT again provides a theoretical prediction, with this ratio equal to the purely geometrical function Ψ(θ) = 1/4(2 − cos(ϕ))(1 + cos(ϕ))². The numerical results obtained with the string method applied to the DI model show, instead, a behaviour strongly dependent also on the degree of metastability. Close to saturation, μ_lev = 0.2, CNT predictions are well reproduced by the DI model. Mesoscale properties of the critical bubble start to be effective at larger metastability levels. When the interface thickness is comparable with the critical bubble dimension, at μ_lev = 0.5 and even more apparently at μ_lev = 0.8, the deviation from CNT is more pronounced. It is worth noticing that at high metastability, the hydrophobic surface can anticipate the spinodal condition. The energy barrier at μ_lev = 0.8 and θ = 110° is actually zero, suggesting a barrier-less mechanism of phase transition typical of the spinodal decomposition. This result is consistent with the findings of (Talanquer & Oxtoby 1996), where the so-called surface spinodal was identified using a different form of fluid-solid free-energy. Moreover, on the hydrophilic side of the plot, the case at μ_lev = 0.8 shows a heterogeneous energy barrier independent of the contact angle up to 60° and with a value slightly smaller than the homogeneous barrier. To understand this peculiar behaviour, it is instrumental to look at the full-density fields during the transition progress, shown in panel a of Fig. 3. The highly hydrophilic surface, even at a moderate metastability level, induces the localisation of the density variation in a region close to the wall but not directly in contact with it. As a consequence, the critical nuclei are almost spherical, resembling homogeneous nucleation. This behaviour has also been observed in full 3D FH simulations (Gallo et al. 2021), in MD simulations (Nagayama et al. 2006; Chen et al. 2018; Zou et al. 2018), and it could be ascribed to the mesoscale interaction between the vapour-liquid interface and the fluid-solid layering induced by the strong attraction of the hydrophilic surface. The configurations along the MEP show that, after the transition state, the bubble grows at the solid surface, recovering the expected behaviour always observed in experiments. Conversely, in the hydrophobic case, the embryos always sit at the solid wall due to the high affinity with the vapour.

Figure 3.

Panel a: Density fields along the Minimum Energy Path (MEP), showing nucleation progress from top to bottom. The second row corresponds to the transition states. Left: Hydrophilic case at T = 1.2, μ_lev = 0.5, θ = 30°; Right: Hydrophobic case at T = 1.2, μ_lev = 0.5, θ = 110°. Panel b: Energy barrier ratio between heterogeneous and homogeneous nucleation as a function of contact angle, obtained using the string method with the DI model. Symbols indicate different levels of metastability; the solid black curve shows the CNT prediction, based solely on the geometrical factor Ψ. Panel c: Mean first passage time for homogeneous nucleation versus metastability. The solid line is the DI model prediction, while red squares refer to Corrected CNT prediction (Menzl et al. 2016), and the blue triangle corresponds to brute force FH simulations. Panel d: DI model prediction of heterogeneous nucleation’s mean first passage time. Each curve corresponds to a different contact angle.

A key observable in stochastic processes such as nucleation is the mean waiting time for forming a critical nucleus. This timescale is usually prohibitively long for direct brute-force simulations (e.g., MD or FH), making theoretical approaches essential. Kramers’ theory provides a fundamental framework for estimating the characteristic nucleation rate, see (Hänggi et al. 1990) for a general discussion on barrier-crossing problems and (Gallo et al. 2020) for bubble nucleation application. The theory requires the free energy profile and the diffusion coefficient of the fluctuating vapour nucleus embedded in the liquid. Having identified the MEP, described through parameterisation with the curvilinear abscissa s, we can conjecture the simplest dynamics for the reaction coordinate to determine the nucleation times. Such dynamics sees the thermodynamic system as a Brownian walker in the energy landscape Ω = Ω(s)

$$\frac{\text{d}s}{\text{d}t}=-\alpha \frac{\text{d}\Omega (s)}{\text{d}s}+\sqrt{2D}\eta (t)$$ (2.4)

with α a friction coefficient, D = k_BTα the diffusion and η(t) a white noise with zero mean and ⟨η(t)η(q)⟩ = δ(t − q). The friction coefficient is estimated using the following procedure: the critical bubble ρ_crit (x, s^⋆) is identified and perturbed along the directions of the two stable basins, ρ_V and ρ_L. The perturbed configurations are then allowed to evolve according to the isothermal Navier-Stokes-Korteweg (NSK) equations

$$\begin{array}{c}\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \mathbf{\text{v}})=0,\\ \frac{\partial \rho \mathbf{\text{v}}}{\partial t}+\nabla \cdot (\rho \mathbf{\text{v}}\otimes \mathbf{\text{v}})=-\rho \nabla \left(\frac{\delta \Omega }{\delta \rho }\right)+\nabla \cdot \Sigma ,\end{array}$$ (2.5)

where ρ is the density, v is the velocity field and Σ = η(∇v + (∇v)^T) +(ζ − 2/3η) (∇⋅v) I is the viscous stress tensor, η, ζ the two viscosity coefficients. We adopt the viscosity expression proposed by (Rowley & Painter 1997) to ensure full consistency with the Lennard-Jones properties. By monitoring the energy variation δΩ occurring as the system transitions between two nearby configurations ρ(x, s^⋆) and ρ(x, s^⋆ ± δs), with $\delta s=\left|\right|\rho \left(\mathbf{\text{x}},s^{\star }\pm \delta s\right)-\rho \left(\mathbf{\text{x}},s^{\star }\right)|{|}_{L^2}$ and the time required for this transition δt (as measured in NSK equations), the friction α is determined as α = −(δs/δt)/(δΩ/δs). Specifically, as detailed in Appendix B, the string is a collection of fields ρ(x, s). We aim to estimate the friction coefficient at the saddle point. Starting from the saddle point, defined by the field ρ_crit (x) = ρ(x, s^⋆), we introduce perturbations in the directions of the basins of the liquid and vapor by considering the configurations just before and after the saddle point, ρ(x, s^⋆ ± δs). These are then evolved under the NSK equations toward the respective basins of liquid and vapour. By measuring the time taken to reach the subsequent configurations ρ(x, s^⋆ ± 2δs), we obtain two values of α, namely α_r = −(δs/δt_r)/(δΩ_r/δs) and α_l = −(δs/δt_l)/(δΩ_l/δs), corresponding to the right and left of the saddle point, with δΩ_r = Ω(s^⋆ + 2δs) − Ω(s^⋆ + δs) and δΩ_l = Ω(s^⋆ − 2δs) − Ω(s^⋆ − δs), and δt_r/s the time measured from NSK numerical simulations. We then define the effective diffusion coefficient at the saddle as α^⋆ = (α_r +α_l)/2. The parameter α has physical dimensions of energy times density squared per unit time, while s has the dimensions of a density. This is only an estimate, as the MEP does not represent the most likely path of the NSK dynamics, given that the two systems evolve in different spaces and their trajectories are not necessarily close (Grafke & Vanden-Eijnden 2019; Zakine & Vanden-Eijnden 2023). However, numerical results suggest that the forecast is remarkably accurate, as discussed subsequently. The procedure was repeated for both the homogeneous and heterogeneous cases. After estimating the friction coefficient, the mean first passage time τ was evaluated by using Kramers’ theory (Kramers 1940; Schulten et al. 1981),

$$\tau ={\displaystyle {\int }_{B_L}\exp \left(-\frac{\Delta \Omega }{k_{B}T}\right)}ds{\displaystyle {\int }_{B_{SP}}\frac{1}{D(s)}\exp }\left(\frac{\Delta \Omega }{k_{B}T}\right)ds,$$ (2.6)

where B_L and B_SP are the metastable and saddle-point basins. The two integrals can be evaluated using the classical saddle-point approximation, after noting that the free energy profiles’ curvatures (Ω″(s)) are positive and negative in the two basins, respectively. The second integral can be expanded as

$$\begin{array}{c}{\displaystyle {\int }_{B_{SP}}\frac{1}{D(s)}\exp \left(\frac{\Delta \Omega }{k_{B}T}\right)}ds~\exp \left(\frac{{\Delta \Omega }^{\star }}{k_{B}T}\right){\int }_{-\infty }^{\infty }\exp \left(-\frac{{\left(s-s^{\star }\right)}^2}{2{\nu }^2}\right)\times \\ \frac{1}{D^{\star }+D_{\star }^{\prime }\left(s-s^{\star }\right)+1/2D_{\star }^{″}{\left(s-s^{\star }\right)}^2+\dots }d\left(s-s^{\star }\right),\end{array}$$ (2.7)

with ν² = k_BT /∣Ω″(s^⋆)∣. Our data show that the diffusion coefficient is gently varying in the neighbourhood of the saddle point, leading to the conclusion that, within an order one prefactor,

$$\tau ~\frac{2\pi }{\alpha \left(s^{\star }\right)\sqrt{{\Omega }^{\prime \prime }(0)\left|{\Omega }^{\prime \prime }\left(s^{\star }\right)\right|}}\exp \left(\frac{{\Delta \Omega }^{\star }}{k_{B}T}\right).$$ (2.8)

τ is reported for homogeneous and heterogeneous nucleation in panels (c) and (d) of Fig. 3, respectively. Panel c shows the first passage time as a function of the metastability parameter μ_lev. The time is nondimensionalised using t_r = 2.15 ×10⁻¹² s as a reference. The solid black line depicts our prediction, while the red squares represent the mean first passage time evaluated with an adaptation of the nucleation theory proposed by Menzl et al. (Menzl et al. 2016) (see below), and the blue triangle depicts the FH brute force simulations. Kramers’ theory becomes increasingly accurate in the limit of large energy barriers (Hänggi et al. 1990), see (Gallo et al. 2020), where the agreement with brute-force simulations emerges for energy barriers ~6−7k_BT, at a metastability level of rougly μ_lev = 0.82, corresponding to a reduced density of ρ_L = 0.475 with T = 1.25. In the present case, for the highest metastability level considered (μ_lev = 0.8), the energy barrier (ΔΩ^⋆ ~ 10k_BT) is well in the range where Kramers’ theory is expected to be robust. This conclusion is supported by 200 FH simulations we performed using the equations reported in Appendix B. The resulting mean first passage for all cases is consistently in the range of τ ~ 10⁶ (τ_FH ~ 0.25 × 10⁶, τ_DI ~ 1.00 × 10⁶, τ_cCNT ~ 3.60 × 10⁶) confirming the robustness of Kramers’ theory even at moderate metastability. Obviously, also in the heterogeneous case, the accuracy of Kramers’ theory may deteriorate for low energy barriers. Such regimes represent only a minor portion of the results presented here, which can, however, be directly investigated via brute-force simulations.

The Menzl et al. theory – applied by the authors to the TIP4P/2005 water model – combines MD microscopic information (vapour embryo surface energy) with the stochastic overdamped Rayleigh Plesset equation to estimate the diffusion coefficient D. It is worth noting that the procedure proposed by Menzl et al. is completely general. It consists of correcting the CNT barrier height by either taking the surface energy from MD simulations or by applying a correction using the Tolman length, which is estimated by adjusting the CNT barrier relative to MD results. The diffusion coefficient of the nucleating bubble is estimated via the fluctuation-dissipation theorem applied to an overdamped Rayleigh–Plesset (RP) dynamics, requiring that it samples the equilibrium probability distribution function of the radius R, P(R) ~ exp(−ΔΩ(R)/k_BT). The diffusion coefficient is then computed in closed form as D_R = k_BT/(16πηR). In the present case, which deals with a Lennard-Jones fluid, the DFT barrier automatically incorporates the curvature correction (Blokhuis & Kuipers 2006; Wilhelmsen et al. 2015; Rehner et al. 2019; Magaletti et al. 2021) while the diffusion coefficient for reaction variable s is estimated using the hydrodynamic equations with the procedure mentioned above. With this approach, we can directly compare the results of the two theories. We found perfect agreement between the two approaches. The theoretical framework developed by Menzl et al. is not directly applicable to heterogeneous nucleation, as the Rayleigh-Plesset equations describe bubble dynamics in an unbounded medium. In contrast, the approach proposed here is more general, as it allows for the determination of the MEP and the estimation of diffusion coefficients under arbitrary conditions. In panel d, the heterogeneous case is presented, highlighting how the wall chemistry plays a crucial role in promoting nucleation by significantly reducing the characteristic transition times. It can be observed that when the contact angle is below approximately 50 degrees, the timescales become independent of wettability, approaching the homogeneous limit. This result is consistent with our previous findings of FH for boiling (Gallo et al. 2023) and with MD simulations (Zou et al. 2018; Sullivan et al. 2025) identifying the homogeneous nucleation as the main transition mechanism when hydrophilic chemistries are considered.

As a final analysis, we examine the velocity fields and the dissipation function of the vapour bubble as it evolves from the saddle point toward the two basins, corresponding to the metastable liquid and the stable vapour. To this end, we introduce the total free energy of the system, defined as the sum of the grand potential and the kinetic energy of the fluid

$$H[\rho ,\mathbf{\text{v}}]=\Omega [\rho ]+K[\rho ,\mathbf{\text{v}}]={{\displaystyle \int }}_V{f}_b(\rho )+\frac{\lambda }{2}|\nabla \rho {|}^2+\frac{1}{2}\rho |\mathbf{\text{v}}{|}^2-{\mu }_{eq}\rho dV+{\oint }_{\partial V}{f}_w(\rho )dS.$$ (2.9)

The dissipation reads

$$\begin{array}{l}\frac{\text{d}H}{\text{d}t}={\displaystyle {\int }_V\frac{\delta H}{\delta \mathbf{\text{v}}}\cdot \frac{\partial \mathbf{\text{v}}}{\partial t}+\frac{\delta H}{\delta \rho }\frac{\partial \rho }{\partial t}dV}=-{\displaystyle {\int }_V\mathbf{\text{v}}\cdot \left(\rho \mathbf{\text{v}}\cdot \nabla \mathbf{\text{v}}+\rho \nabla \left(\frac{\delta \Omega }{\delta \rho }\right)+\nabla \cdot \Sigma \right)+\left(\left(\frac{\delta \Omega }{\delta \rho }\right)+\frac{1}{2}|\mathbf{\text{v}}{|}^2\right)\nabla \cdot \rho \mathbf{\text{v}}}\\ =-{\displaystyle {\int }_{V}2\eta \tilde{E}:\tilde{E}+\frac{1}{3}\zeta {(\nabla \cdot \mathbf{\text{v}})}^{2}dV-{\oint }_{\partial V}\left(\left(\frac{\delta \Omega }{\delta \rho }\right)+\frac{1}{2}|\mathbf{\text{v}}{|}^2\right)\rho \mathbf{\text{v}}\cdot \widehat{\nu }dS}\end{array}$$ (2.10)

where Eq.s 2.5 have been invoked and $\tilde{E}$ is the deviatoric part of the symmetric part of the velocity gradient E = sym(∇ ⊗ v). In the numerical simulations, the system is large enough that the velocity at the boundary is always zero, so the boundary term in the above equation vanishes and dH / dt < 0. It is worth noting that, if the total chemical potential δΩ/δρ + 1/2|v|² is set to zero, then H remains monotonically decreasing in time as well. The simulations were performed in a domain of length L = 3000 (dimensionless units), which corresponds to approximately one micron in physical length. Since the present analysis is carried out for homogeneous nucleation, we exploit spherical symmetry to solve Eqs. 2.5 (Abbondanza et al. 2023b). We adopt the same grid space Δr, temporal integrator and integration timestep Δt of stochastic equations (see Appendix A for details). The thermodynamic conditions are T = 1.20 and μ_lev = 0.5. In Fig. 4, the time evolutions of H(t) (red curves) and dH / dt (blue curves) are shown during the hydrodynamic evolution of the bubble in the precritical and postcritical states, in panels a and c, respectively. Analogously, panels b and d display the corresponding velocity fields. In both configurations, H(t) decreases monotonically over time, with greater dissipation observed in the precritical phase, characterised by a more abrupt change in the velocity fields. It is particularly interesting to note (panel b) that the bubble initially shrinks (black curve t = 10 and blue curve t = 20), then slightly compresses the liquid and emits a wave that propagates through the liquid with the isotermal speed of sound velocity $c_T=\sqrt{\partial p/{\partial \rho |}_T}$ (purple and green curves at t = 75 and t = 100). In the postcritical phase, the bubble expands with lower velocities compared to its precritical counterpart.

Figure 4.

Panel a,c: Time evolution of the dissipation rate dH/dt (blue) and total free energy H(t) (red) computed from the NSK dynamics during relaxation towards (a) the metastable liquid basin and (c) the stable vapour basin. Panel b,d: Velocity profiles at selected time instants during the relaxation towards (b) the metastable liquid and (d) the stable vapour state.

3. Conclusions

In this work, the minimum-energy pathways for vapour nanobubble formation in metastable liquids are identified, revealing significant deviations from classical nucleation theory. Unlike CNT, which assumes a single reaction coordinate (e.g., bubble size), the study shows that nucleation involves both bubble size and average density. Nucleation is triggered by long-wavelength fluctuations with liquid-like density, forming a C-shaped trajectory in the {ρ_av, R} space. In both homogeneous and heterogeneous cases, nucleation begins with a large, slightly rarefied cluster that evolves into a stable vapour bubble.

The transition mechanism aligns with fluctuating hydrodynamics simulations, indicating that the MEP closely approximates the most likely path (MLP) under Navier-Stokes-Korteweg dynamics. Although identifying the MLP remains a complex task, the MEP provides a natural reaction coordinate to construct a simplified stochastic model for estimating transition times. Using Korteweg-Navier-Stokes dynamics and Kramers’ theory, first-passage times are calculated. For the homogeneous case, the predictions match those of (Menzl et al. 2016) without relying on molecular simulations. In the heterogeneous case, for surfaces with ϕ < 50°, nucleation follows a homogeneous-like pathway with transition times independent of contact angle — a behaviour inconsistent with CNT but supported by molecular dynamics and fluctuating hydrodynamics simulations.

A natural extension of this work involves computing the MLP of the fluctuating hydrodynamics via, for example, the Minimum Action Method, which minimises the dynamical action rather than the free energy Weinan et al. (2004). This approach, unlike MEP methods, accounts for non-equilibrium dynamics and full conservation laws (Yao & Ren 2022; Zakine & Vanden-Eijnden 2023; Grafke & Vanden-Eijnden 2019; Soons et al. 2025), and it will be reported elsewhere. An interesting example is the stochastic lubrication theory dynamics in film rupture rare events (Sprittles et al. 2023; Liu et al. 2024), where the importance of conservation laws shapes rare event kinetics and transition times.

As a final note, we stress that this procedure can be readily extended to real fluids by employing appropriate equations of state, enabling a quantitative prediction of nucleation in realistic systems.

4. Fluctuating Hydrodynamics Simulations

The FH equations are Navier-Stokes equations augmented with stochastic fluxes, i.e. the stress tensor and energy flux, to account for thermal fluctuations arising from the discrete nature of matter. These fluxes are derived from a fluctuation-dissipation theorem, ensuring that the equilibrium statistics of the stochastic system sample the Einstein-Boltzmann distribution. FH offers a powerful framework for quantifying the impact of thermal fluctuations on macroscopic fluid behaviour (Chaudhri et al. 2014; Bandak et al. 2022; Bell et al. 2022; Eyink & Jafari 2022; Barker et al. 2023; Eyink & Jafari 2024; Gallo et al. 2023; Gallo & Casciola 2024). In this work, we use our proposed FH extension to multiphase fluids (Gallo et al. 2020). This model represents a two-phase liquid–vapour system incorporating thermal fluctuations within the framework of Landau and Lifshitz’s fluctuating hydrodynamics. The inclusion of stochastic forcing enables the spontaneous nucleation of vapour clusters within the liquid, while the diffuse interface formulation captures the ensuing hydrodynamic processes of growth and transport. Our focus is on a coarse-grained variant of this model, derived by averaging the full three-dimensional equations over spherical shells. The resulting stochastic equations exhibit a spatial dependence solely on the radial distance from the center of the vapour cluster. The equations are reported here for the readers’ convenience.

$$\frac{\partial \rho }{\partial t}=-\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\rho v_r\right),$$ (4.1)

$$\rho \frac{D{v}_r}{Dt}=-\rho \frac{\partial }{\partial r}\left(\frac{\delta \Omega }{\delta \rho }\right)+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{\Sigma }_{rr}^v\right)-\frac{2}{r}{\Sigma }_{\phi \phi }^v+\frac{1}{r^2}\frac{\partial }{\partial r}(\beta r\xi (r,t))+\frac{\beta }{r^2}\xi (r,t),$$ (4.2)

with

$${\Sigma }_{rr}^v=\frac{4}{3}\eta \left(\frac{\partial v_r}{\partial r}-\frac{v_r}{r}\right),{\Sigma }_{rr}^v=\frac{2}{3}\eta \left(\frac{v_r}{r}-\frac{\partial v_r}{\partial r}\right),$$ (4.3)

and

$$\beta =\sqrt{\frac{2k_{B}T\eta }{3\pi }},⟨\xi (r,t)\xi \left(r^{\prime },t^{\prime }\right)⟩=\delta \left(r-r^{\prime }\right)\delta \left(t-t^{\prime }\right).$$ (4.4)

These equations represent the stochastic extension of Eqs. 2.5, under the assumption ζ = 0. Numerical simulations have been conducted starting from an initial state with homogeneous (metastable) density and zero velocity. At the boundary, we imposed a fixed density and a vanishing normal derivative of the velocity. Employing the same thermodynamic conditions used for the string computations and a 1000-point grid with dimensionless spacing Δr = 3. The temporal integrator used is a second-order Runge-Kutta explicit integrator, as it is well-suited for stochastic equations (Delong et al. 2013). The (non-dimensional) time step for integration in time was Δt = 10⁻³.

5. Appendix B: MLP for gradient systems

In this appendix, we aim to elucidate the relationship between the MLP and the MEP for rare trajectories that identify nucleation pathways. These discussions can be found in specialised literature, e.g. in non-equilibrium statistical mechanics of gradient flows (Bouchet 2020; Grafke 2019), or in stochastic process theory (Mielke et al. 2014). Here it is reported in its declination for bubble nucleation. As anticipated in the main text, when the system is subject to a chemical potential within the metastable range, that is

$${\mu }_{\text{spin}}<{\mu }_{\text{eq}}<{\mu }_{\text{sat}},$$ (5.1)

Eq. (2.3) admits three distinct steady-state solutions. These correspond to: (i) the uniform liquid phase, characterised by a constant density ρ(x) = ρ_L; (ii) the uniform vapour phase, with ρ(x, t) = ρ_V; (iii) a non-trivial two-phase configuration in which a vapour nucleus is embedded within a metastable liquid background. This third solution represents a critical state — the so-called critical nucleus — described by the spatially varying density profile ρ_crit (x). This configuration acts as a transition threshold: only once the system crosses this critical barrier can it proceed from the metastable liquid to the stable vapour phase, completing the phase transformation. If we assume that the thermodynamic system evolves according to a stochastic gradient dynamics

$$\frac{\partial \rho }{\partial t}=-M\left(\frac{\delta \Omega }{\delta \rho }\right)+\sqrt{2ϵ}\xi ,$$ (5.2)

with ϵ = Mk_BT the (small) intensity of the noise, M the mobility and ξ(x, t) a space-time with noise, i.e. ⟨ξ(x, t)ξ(y, q)⟩ = δ(x − y)δ(t − q). The probability of the phase transition from the liquid state to the vapour one in a certain time window T_w, $P_{{\rho }_L\to {\rho }_V}$, is the probability of observing a given path ρ(x, t) with −T_w < t < T_w with the end states ρ(x, −T_w) = ρ_L and ρ(x, T_w) = ρ_V. The Large Deviation theory provides (Freidlin et al. 1998) the probability of observing the path ρ(x, t)

$$P[\rho ]~\exp \left(-\frac{1}{2ϵ}{S}_{T_w}[\rho ]\right),$$ (5.3)

where

$$S_{T_w}[\rho ]=\frac{1}{2}{\int }_{-T_w}^{T_w}\left|\right|\frac{\partial \rho }{\partial t}+M\left(\frac{\delta \Omega }{\delta \rho }\right)|{|}^2\text{d}t,$$ (5.4)

with ‖⋅‖ the L₂ norm. Assuming that the trajectory ρ(x, t) crosses the separatrix between the attraction basins of the two minima ρ_L and ρ_V at $t=T_c,S_{T_w}[\rho ]$ reads

$$\begin{array}{c}{S}_{T_w}[\rho ]=\frac{1}{2}{\int }_{-T_w}^{T_c}\left|\right|\frac{\partial \rho }{\partial t}-M\left(\frac{\delta \Omega }{\delta \rho }\right)|{|}^2\text{d}t+\frac{1}{2}{\int }_{T_c}^{T_w}||\frac{\partial \rho }{\partial t}+M\left(\frac{\delta \Omega }{\delta \rho }\right)|{|}^2\text{d}t+\\ +2M\left(\Omega \left[\rho \left(\mathbf{\text{x}},T_c\right)\right]-\Omega \left[\rho \left(\mathbf{\text{x}},-T_w\right]\right)\right).\end{array}$$ (5.5)

The most likely path ρ^⋆(x, t) minimises the functional in Eq. 5.5 in both the window time (T_w) and density space. Concerning time, being ρ_L and ρ_V two minima of the free energy, the minimum of $S_{T_w}[\rho ]$ is achieved when T_w → ∞, $S_{T_w}[\rho ]\to S_{\infty }[\rho ]$. Therefore, the most likely path is given by the two (heteroclinic) orbits

$$\frac{\partial {\rho }^{\star }}{\partial t}=M\left(\frac{\delta \Omega }{\delta {\rho }^{\star }}\right)\text{for}-\infty <t<T_c,$$ (5.6)

$$\frac{\partial {\rho }^{\star }}{\partial t}=-M\left(\frac{\delta \Omega }{\delta {\rho }^{\star }}\right)\text{for}{T}_c<t<+\infty .$$ (5.7)

Eq. 5.7 represents the deterministic dynamics of the spontaneous expansion of the bubble, while Eq.5.6 is its time-reversed counterpart. The latter shows that the most likely nucleation event consists of a forward evolution in physical time starting from ρ_L at t = −∞ against the potential gradient up to the saddle point ρ_crit (x). The most likely path ρ^⋆ has the probability

$$P_{{\rho }_L\to {\rho }_V}^{\star }~\exp \left(-\frac{1}{2ϵ}{S}_{\infty }\left[{\rho }^{\star }\right]\right)=\exp \left(-\frac{{\Delta \Omega }^{\star }}{k_{B}T}\right),$$ (5.8)

where ΔΩ^⋆ = Ω [ρ_crit (x)] − ΔΩ [ρ_L] is the free energy barrier. This result highlights how the nucleation probability and the transition rate exponentially depend on the energy barrier. In principle, one can calculate ρ^⋆ by looking for the saddle point of ΔΩ[ρ], e.g., with the Gentlest Ascent Dynamics (Weinan & Zhou 2011), perturb the dynamics in the directions of maximum descent toward the two stable basins ρ_L and ρ_V and then follow the system’s evolution according to Eq.s 5.6 forward in time and the dynamics of Eq. 5.7 backwards in time, i.e. t → −t. This procedure identifies the most likely path with the minimum energy path (MEP). The MEP is a continuum sequence of fields ρ(s, x) satisfying the condition

$${\left(\frac{\delta \Omega }{\delta \rho }\right)}^{\perp }[\rho (s,\text{x})]=0,$$ (5.9)

with the symbol ⊥ referring to the projection of the functional derivative onto the space perpendicular to the path

$${\left(\frac{\delta \Omega }{\delta \rho }\right)}^{\perp }[\rho (s,\mathbf{\text{x}})]=\frac{\delta \Omega }{\delta \rho }-\frac{1}{\left|\right|\frac{\partial \rho (\mathbf{\text{x}},s)}{\partial s}|{|}^2}\frac{\partial \rho (\mathbf{\text{x}},s)}{\partial s}{{\displaystyle \int }}_V\frac{\delta \Omega }{\delta \rho }\frac{\partial \rho (\mathbf{\text{x}},s)}{\partial s}\text{d}V.$$ (5.10)

The determination of the MEP is achieved by discretising ρ(s, x) in a finite number of fields ρ^s(x) (images) with s = 1…N. The set of ρ^s forms the string. Starting from a guess string, e.g. linear interpolation between the initial and final states, the sequence is evolved over the pseudo-time $\tilde{t}$ according to the steepest descent algorithm

$$\frac{\partial {\rho }^s}{\partial \tilde{t}}={\mu }_{eq}-\mu \left({\rho }^s\right)+\lambda {\nabla }^2{\rho }^s.$$ (5.11)

To integrate this equation, we employ a staggered spatial scheme: spherical symmetry is used for homogeneous nucleation, while cylindrical symmetry is adopted for heterogeneous cases. Time integration is performed using a forward Euler scheme. After each evolution step over a pseudo-time interval $\Delta \tilde{t}$, the images are redistributed along the string through a reparameterization procedure that enforces equal arclength (E et al. 2007), see also (Bottacchiari et al. 2022, 2024) for application to other DI (Ginzburg-Landau) functionals,

$$\delta s=\left|\right|{\rho }^{s+1}(\mathbf{\text{x}})-{\rho }^s(\mathbf{\text{x}})\left|\right|=\text{constant}.$$ (5.12)

This two-step procedure is iterated up to the complete convergence of the whole string to the MEP. Concerning the initial string, we have that s = 0 corresponds to the homogeneous metastable liquid, which remains stationary in time. The final condition at s = 1 represents the vapour phase at the same chemical potential as the metastable liquid. To accelerate convergence, we initialise the string with a configuration that is qualitatively similar to the formation of a vapour bubble. In particular, the final instance of the initial string is chosen as a large supercritical bubble, with a radius approximately three-quarters of the domain size, which will eventually evolve into a fully developed vapour phase. This choice facilitates the generation of circular, rarefied regions in the liquid. At iteration 0, all intermediate instances are obtained through linear interpolation between the initial and final states. Upon convergence, we obtain the Minimum Energy Path (MEP), which we define operationally as the condition when the energy profile Ω(s) no longer changes during iterations. For all simulations, the MEP has been discretised using 200 instances along the path. The bubble free energy (Ω(s) − Ω(0))/(k_BT) as a function of the reaction coordinate s is shown in Fig. 5 for both homogeneous and heterogeneous nucleation. The maxima along these profiles correspond to the nucleation energy barriers.

Figure 5.

Free-energy profiles as a function of the reaction coordinate s for T = 1.20 and μ_lev = 0.2. The red and blue curves refer to heterogeneous and homogeneous cases.

Data Availability

The final data supporting the research will be made publicly available in an open repository after the completion of the review process.