## 1 Introduction  
The nucleation of crystals from solution is a ubiquitous phenomenon that spans a broad spectrum of scientific disciplines. Gaining a profound insight into the process holds crucial implications for practical areas, including the production of pharmaceutical compounds [1; 2; 3], food and fertilizer manufacturing [4; 5], and the fabrication of high-quality optoelectrical devices [6; 7]. Of particular interest are the initial stages of crystallization in metastable states because of their potential uses for synthesizing materials with diverse structures and functionalities. However, achieving precise control over the crystallization requires an accurate understanding of the mechanism by which atoms and molecules aggregate during the pre-nucleation stage, ultimately culminating in crystal formation. In a stable undersaturated regime in an aqueous solution, the solute molecules are fully solvated, thereby minimizing the system's free energy and preventing the process of crystallization. As the solution becomes supersaturated, the possibility of crystal nucleation emerges because of a drastic increase in the thermodynamic driving force. During this process, the excess solute molecules aggregate and align, forming clusters or nuclei of the solute phase that disrupt the initial hydration shell structure [8; 9; 10; 11]. In practice, classical nucleation theory (CNT) is widely used to study such phase transitions in various metastable states because of its analytical simplicity [12; 13; 14]. It provides a theoretical framework to describe the initial steps of nucleation, where a small nucleus of the new phase forms in a metastable parent phase. In this model, nuclei must surpass a critical size to proceed with stable phase formation. While CNT has proven useful in metallic systems composed of alloys [15; 16], its applicability to complex systems or non-equilibrium conditions has been brought into question [17; 18; 19; 20; 21; 22]. In particular, the classical theory assumes that clusters evolve in size by adding individual molecules at a time, during which fluctuation between the distributions of embryo clusters is neglected. Since the theory only accounts for growth units of monomers, it fails to provide insights into the nature of pre-nucleation clusters formed in solutions [23; 24; 25]. Therefore, more sophisticated theories and experimental validations of pre-nucleating dynamics are highly desirable. Despite the extensive historical use and widespread application of crystallization, comprehending the intricate relationship between the degree of supersaturation and the kinetics and thermodynamics of crystallization remains a challenging task. This complexity is due to the underlying mechanisms operating at microscopic time and length scales and the difficulty in maintaining extreme levels of metastability. However, investigating a highly supersaturated aqueous solution in a contact-based experiment proves extremely difficult because of the interface between the aqueous solution and the container, which acts as an efficient heterogeneous nucleation site. Additionally, the characterization of sample properties can be contaminated by interactions between the sample and the container or by the addition of foreign substances that are used to control crystallization [1]. Such shortcomings have prompted the development of space-platform experiments to exploit microgravity conditions. The use of acoustic levitation in crystallization studies offers a notable advantage by providing a non-contact environment. This eliminates the influence of vessel walls, which can act as unwanted heterogeneous nucleation sites and prohibit manifesting highly supersaturated states, as well as affect crystal shape and purity. The absence of a container also minimizes the risk of impurities or foreign particles being introduced to interfere with the crystallization process.