Accurately characterizing wetting interactions at liquid-solid interfaces is essential for optimizing processes in industries such as oil recovery, food production, pharmaceuticals, and coatings, where interfacial phenomena directly influence material performance and process efficiency [1,2]. Historically, determining the interfacial properties of materials has advanced in parallel with the development of thermodynamic principles, with foundational work by Gibbs [3] setting the stage for modern investigations. In particular, interfacial free energy (IFE) plays a key role in interface science [[4], [5], [6], [7]]. Understanding IFE is essential for the theoretical explanation of the interfacial phenomenon and for accurate predictions of material behavior in different industrial and engineering applications.

In the three-phase contact region of fluid-fluid-solid systems, IFEs are balanced as described by Young's equation [8]:γF1F2⋅cosθ=γSF2−γSF1,where θ is the wettability of fluid phase 1 (F1) on solid phase (S) surrounded by fluid phase 2 (F2), γF1F2, γSF2, and γSF1 are the IFEs between phases denoted in the subscript.

In experiments, although fluid-fluid IFE γF1F2 and wettability θ can be directly obtained [9,10], the measurement of fluid-solid IFE γSF is difficult and usually inaccurate [11]. Many techniques [[12], [13], [14]] exist for ascertaining the vacuum-solid IFE and a substantial body of experimental findings has been systematically correlated by Kumikov and Khokonov [15]. Meanwhile, due to the limitations of experimental techniques when approaching the nanoscale interfaces, indirect methods have been developed for the fluid-solid IFE of various systems, including cleavage test [16], solubility test [17], adhesion force measurement [18], deformation analysis on the solid film [19], contact angle/contact line curvature measurement [20], and contact angle experiment combined with Makkonen hypothesis [21]. A recent review comprehensively discusses both indirect and direct techniques for determining the IFE of fluid-solid interfaces in experiments, highlighting advancements and ongoing challenges [11]. However, precisely determining fluid-solid IFE experimentally remains challenging due to factors such as the size effects, contaminations, and irregularities of surfaces [11,22,23].

Semi-empirical theories have been extensively employed to estimate fluid-solid IFE using measured contact angle data. These theories encompass approaches like Neumann's equation of state approach [24,25], Zisman method [26], Fowkes method [27], geometric-mean approach [28], harmonic-mean approach [29], and van Oss-Good method [30]. A comprehensive summary of these semi-empirical theories, along with their assumptions, is provided by Żenkiewicz [31]. However, limited research has focused on providing a microscopic foundation for these semi-empirical theories, and significant controversies remain over the validity of several ad hoc assumptions inherent in their formulations [11,[32], [33], [34]].

First-principle theories have also been developed for determining the interfacial energies. In the statistical mechanical theory of Navascués and Berry [35], the Kirkwood-Buff method [36] was extended to the case where fluids are in contact with a rigid solid phase, and the fluid-solid IFE was split into one solid and two fluid-solid contributions. Note that this theory was later combined with molecular simulation [37] for estimating the fluid-solid relative IFE, which will be discussed in detail in the next section. The square gradient theory, which was first introduced by Rayleigh [38] and van der Waals [39] and later rediscovered by Cahn and Hilliard [40], was applied to study wetting problems [41,42]. In square gradient theory, the fluid-solid relative IFE is derived from the surface excess transverse stress, where principal stress profiles are computed based on density distribution across the interface [42,43]. Meanwhile, classical density functional theory (cDFT) has become a pivotal tool for analyzing wettability phenomena at the molecular level [32,44,45]. Within the cDFT framework, the fluid-solid relative IFE is typically calculated using excess grand potential: γSF∗=Ω+pV/A, where Ω, p, V, and A are the grand potential, bulk pressure, volume of the fluid, and interfacial area, respectively. The cDFT is a robust and versatile tool grounded in thermodynamic principles, offering exceptional accuracy and efficiency [46,47]. The cDFT has been used for computing the fluid-solid IFE for a broad range of systems including confined fluid [48], chemically patterned wall [49], and heterogeneous surface [50], while the application of other first-principle theories has been relatively rare nowadays. Despite the demonstrated precision and efficacy of cDFT, it remains underutilized across experimental, theoretical, and computational communities. Major obstacles to its wider adoption include the theoretical complexity and a lack of accessible, user-friendly software for fluid-solid IFE computations [[51], [52], [53]].

Molecular simulation is a powerful tool for investigating fluid-solid IFE due to several distinct advantages. Firstly, there is a wealth of open-source software available, offering a wide array of simulation tools to choose from [[54], [55], [56]]. Moreover, molecular simulations are based on precise atomic-level representations, offering a robust physical foundation for IFE calculations [57,58]. Furthermore, molecular simulations provide flexibility, enabling the study of diverse fluid-solid interfaces across a wide range of chemical systems. This versatility arises from the extensive library of force field parameters available in the literature [[59], [60], [61], [62], [63], [64]]. Although certain acceleration techniques, such as coarse-grained modeling [[65], [66], [67], [68], [69]], can reduce computational time, molecular simulation methods remain more computationally intensive than semi-empirical and first-principle approaches.

Despite the advancement of various molecular simulation methods for estimating fluid-solid IFE, there remains a scarcity of comprehensive reviews. Jiang and Patel [70] published a review paper focusing on molecular simulation techniques for estimating contact angles. Those methods are useful for calculating the differences of IFE. Nevertheless, several essential methodologies for the direct estimation of fluid-solid IFE remain uncovered in the previous review, and numerous innovative techniques have emerged over the past few years. This review aims to bridge these gaps by providing a comprehensive analysis of state-of-the-art molecular simulation methodologies for calculating fluid-solid IFE. The molecular simulation methods for estimating IFE can be broadly classified into two primary categories: mechanical and thermodynamic approaches. The mechanical approach encompasses methods such as the contact angle approach, the method using Bakker's equation, and the Wilhelmy simulation method, each of which will be discussed in detail. For the thermodynamic approach, key methods include the cleaving wall technique, the Frenkel-Ladd technique, and the test-volume/area methods.

Note that while there are various experimental [[71], [72], [73], [74], [75], [76]] and theoretical [[77], [78], [79]] methods available for studying fluid-solid IFE in the context of crystallization/nucleation, the focus of this review is on wetting problems and methods that are not suitable for studying common wetting problems were not included in the above discussion. Therefore, this review will not cover molecular simulation techniques such as the capillary fluctuation technique [80], umbrella sampling [81], extrapolation method [82,83], metadynamics [[84], [85], [86], [87]], the superheating and undercooling method [[88], [89], [90]], the seeding technique [[91], [92], [93]], tethered Monte Carlo [94], and the mold integration method [95].