Conventional understanding of supercritical fluids often contradicts observations. While previously considered structureless, we now know supercritical liquid and gaseous states can be distinguished, separated by a pseudo-boiling phase transition across the Widom line. Observed droplets and sharp interfaces at supercritical pressures were attributed to surface tension from phase equilibria in mixtures. This paper proposes an alternative mechanism: thermal gradient-induced interfaces (TGIIF). This research questions the established understanding of droplets and phase interfaces, specifically focusing on the unexpected behavior of supercritical fluids under thermal gradients. The implications of this discovery are significant for technologies relying on supercritical fluid injection and heat transfer, such as diesel engines, jet engines, rocket engines, high-efficiency carbon-capture power cycles, and novel automotive combustion cycles. These applications demand a thorough understanding of supercritical fluid behavior to optimize efficiency and performance. The existence of a supercritical phase transition—pseudoboiling—has been established, differing from subcritical boiling in the absence of equilibrium phase coexistence. The transition occurs over a finite temperature interval. Droplet formation at supercritical pressures has been largely analyzed through mixture thermodynamics, where local mixtures might experience subcritical pressures enabling phase separation, despite the nominal supercritical pressure of pure components. However, the paper investigates whether pseudoboiling could induce droplet formation under truly supercritical conditions, devoid of phase equilibrium.

Existing research on droplet formation at supercritical pressures primarily focuses on mixtures, assuming phase equilibrium is necessary. Studies have explored supercritical phase separation in mixtures, linking it to interfacial thickness on a molecular scale. These studies, however, consistently focus on mixtures where phase equilibrium is a prerequisite for droplet formation. The relationship between the Widom line, defined as the locus of extrema in thermodynamic response functions, and pseudoboiling, a supercritical phase transition, is also relevant. The pseudoboiling line, based on the curvature of the Gibbs free enthalpy, marks the steepest isobaric thermal density gradient (∂ρ/∂T). Many injection processes involve a spatial temperature gradient (∂T/∂x), making the pseudoboiling line a potential candidate for inducing a maximum in the spatial density gradient (∂ρ/∂x).

The paper utilizes a combination of first-principles analysis and computational fluid dynamics (CFD) simulations to investigate TGIIF. The analytical analysis begins by examining a steady interface at constant pressure. An inflection point in the spatial density distribution is linked to the thermodynamic thermal density gradient via the spatial temperature gradient. The authors derive equations (1-9) to define the conditions for the existence of a density inflection point (interface), which is a necessary condition for an interface to form. These equations relate heat transfer characteristics (right-hand side) and fluid thermodynamic properties (left-hand side). The analysis assesses different fluid states (ideal gas, liquid, transcritical fluid) to determine their potential to form interfaces. A crucial condition derived is sgn(p_TT) = sgn(k_T) (Equation 9), where the concavity of the thermal density gradient and the slope of thermal conductivity must have opposite signs. This condition is fulfilled under transcritical conditions, not in liquids or gases. For the analytical model of a steady, 1D interface, a piece-wise linear approximation of the complex thermal conductivity distribution is used to solve for the temperature distribution analytically. The analytical solutions (equations 10-11) are then compared against CFD simulations using the open-source solver SU2, enhanced with tiny neural networks (TNNs) for accurate fluid property modeling. The transient interface is studied through 2D parametric studies of n-heptane droplets under various pressures and environmental temperatures, mirroring experimental Diesel injection studies. Dimensional analysis, specifically using Fourier's Law of heat conduction and transient heat conduction to derive equations (12-14), is employed to determine a characteristic vaporization timescale and a non-dimensional vaporization number (Va) to correlate droplet lifetime with other parameters.

The paper's key findings demonstrate the existence and characteristics of TGIIF. First-principles analysis and simulations show that under specific thermodynamic conditions, a temperature difference between a cold/dense region and a warmer/lighter environment creates self-steepening and self-stabilizing density gradients, even in pure fluids. This leads to the formation of stable droplets and bubbles without surface tension, a phenomenon confirmed through both analytical calculations and CFD simulations. The study found that the necessary condition for the existence of an interface, derived from first principles, is that the concavity of the thermal density gradient and the slope of the thermal conductivity must have opposite signs. This condition is fulfilled under transcritical conditions but not in liquids or gases. This was confirmed using SU2 simulations, enhanced with TNNs for accurate fluid property modeling. The agreement between analytical and numerical solutions was excellent for a wide range of conditions. Further, the transient behavior of droplets was investigated. Initially diffuse interfaces sharpen over time, and elliptical cross-sections regress towards circular shapes, similar to classical droplet evaporation. A key observation was that droplet vaporization follows a D² law, and that the interfacial temperature matches the pseudoboiling temperature (T_pb), analogous to the subcritical saturation temperature. A nondimensional vaporization number (Va) was identified, accurately predicting droplet lifetimes over five decades across various parameters. The results obtained using the linear approximation showed a very good agreement with the SU2 results. The results from both analytical and numerical solutions confirmed the existence of thermal gradient-induced interfaces in supercritical fluids.

The findings significantly advance our understanding of supercritical fluid behavior. The existence of TGIIF challenges the traditional definition of droplets and bubbles, showing that their formation doesn’t necessarily require phase equilibrium or surface tension. The striking similarity between TGIIF droplet vaporization and classical droplet evaporation, following a D² law and maintaining an interfacial temperature near T_pb, suggests potential simplifications in modeling supercritical heat transfer. The introduction of the vaporization number Va provides a valuable tool for predicting droplet lifetimes under a wide range of conditions. This new understanding could lead to improved designs and operation of high-pressure systems that involve supercritical fluids.

This research introduces thermal gradient-induced interfaces (TGIIF) as a novel mechanism for stabilizing liquid-gas interfaces in supercritical fluids without surface tension. The study demonstrates TGIIF's surprising similarity to classical droplet evaporation, suggesting that photographic evidence of stable spherical structures in high-pressure injection systems does not automatically imply the presence of phase equilibrium or surface tension. The development of a nondimensional vaporization number (Va) provides a practical tool for predicting droplet lifetimes. Future research could explore the effects of other factors like flow conditions and fluid properties on TGIIF and extend the analytical models for greater accuracy.

The study's analytical model utilizes a simplified linear approximation of the thermal conductivity. While the agreement with CFD simulations is good, more sophisticated models could improve accuracy, particularly for complex fluid behavior under extreme conditions. The 2D simulations, while insightful, do not fully capture the complexities of 3D droplet behavior. Future research using 3D simulations would provide more comprehensive insights. The current study focuses primarily on n-heptane and oxygen. Further investigation with a wider range of fluids is necessary to assess the generality of the findings.