Physically Motivated Direct Non-Iterative Calculation of Surface Free Energy from Contact Angle Measurements with a Single Wetting Liquid: 1. Derivation and Demonstration of Utility

1. Introduction

The estimation of surface free energy (SFE) of polymers and other solid materials is of major interest in (bio)material and (bio)interfacial research [1,2,3,4]. The most commonly employed methods for this purpose are based on measurements of the contact angle formed between a probe liquid and the solid surface under investigation (Figure 1). However, SFE determination remains entangled with theoretical, computational and experimental challenges. Part of these challenges are related to the fact that SFE quantification involves parameters that are not directly experimentally accessible which greatly complicates its evaluation [5,6,7].

The solid-liquid, solid-vapor and liquid-vapor interfacial energies are commonly related via the Girifalco-Good equation [8,9,10,11]:

$${\gamma }_{\mathit{sl}}={\gamma }_{\mathit{sv}}+{\gamma }_{\mathit{lv}}-2{\Phi }_{\mathit{SL}}\sqrt{{\gamma }_{\mathit{sv}}{\gamma }_{\mathit{lv}}}$$

(1)

where: γ_lv - surface tension of the wetting liquid, γ_sv- solid-vapor surface tension (typically denoted as SFE), γ_sl- liquid-solid interfacial energy. Φ is the Girifalco–Good interaction parameter that quantifies the degree of similarity (in terms of surface energy and polarity matching) between the contacting entities. It decreases (0.5 < Φ < 1.15) away from unity with the attenuation of the interaction (i.e., increase in θ) between the liquid and the solid [8,9,10,11].

When combined with the Young equation

γ_sv = γ_sl + γ_lvcosθ

(2)

it yields:

$${\gamma }_{\mathit{lv}}\left(1+\cos \mathsf{\theta }\right)=2{\Phi }_{\mathit{SL}}\sqrt{{\gamma }_{\mathit{sv}}{\gamma }_{\mathit{lv}}}$$

(3)

where θ is the contact angle between the sessile droplet of wetting liquid and the solid surface.

Obviously if the value of Girifalco-Good parameter is known, SFE calculation from contact angle goniometry becomes routine. However, solving Equations (1) and (2) for Ф one obtains:

$$\Phi ={\displaystyle \frac{{\gamma }_{\mathit{sv}}+{\gamma }_{\mathit{lv}}-{\gamma }_{\mathit{sl}}}{2\sqrt{{\gamma }_{\mathit{sv}}{\gamma }_{\mathit{lv}}}}}={\displaystyle \frac{{\gamma }_{\mathit{lv}}\left(1+\cos \theta \right)}{2\sqrt{{\gamma }_{\mathit{sv}}{\gamma }_{\mathit{lv}}}}}={\displaystyle \frac{1+\cos \theta }{2\sqrt{\frac{{\gamma }_{\mathit{sv}}}{{\gamma }_{\mathit{lv}}}}}}$$

(4)

As solid-vapor and solid-liquid interfacial energies are not directly accessible, apparently Ф cannot be straightly calculated from the experimentally measurable surface tension of the wetting liquid and its contact angle with the solid surface. Hence, sophisticated approaches for γ_sv and Ф estimation were developed implementing single or multiple wetting liquids. Single wetting liquid methods relied on semi-empirical but theoretically grounded equations like Neumann (4) [12,13,14,15] and Gerson equation of state (5) [16,17,18,19,20,21]:

$$\cos {\theta }_Y=-1+2\sqrt{{\displaystyle \frac{{\gamma }_{\mathit{sv}}}{{\gamma }_{\mathit{lv}}}}}{e}^{-\beta {\left({\gamma }_{\mathit{lv}}-{\gamma }_{\mathit{sv}}\right)}^2}$$

(5)

where β = 0.0001247 (m²/mJ)² (or 0.0001247 (m/mN)²).

$${\displaystyle \frac{{\gamma }_{\mathit{lv}}\left(\cos \theta +1\right)}{2\sqrt{{\gamma }_{\mathit{lv}}{\gamma }_{\mathit{sv}}}}}-\exp \left[\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\cos \theta \right)\left({\mathit{a\gamma }}_{\mathit{sv}}+b\right)]=0\right.$$

(6)

where a = 6.5 × 10⁻⁵ (m/mN)²and b = −0.010 (m/mN).

The parameters a, b and β are best fit estimates (hence semi-empirical) from extended dataset of contact angles measured between numerous liquids and solids of various polarity and interfacial energies. All the same, the derivations of Equations (5) and (6) stem from theoretically substantiated attempts to amend Berthelot’s combining rule while accounting the complex nonlinear dependence of Ф on the interfacial properties of both participants in the solid-liquid pair [14,16]. As justly summarized by Neumann [12,14], although not entirely analytical (as is none of the currently existing frameworks for SFE evaluation) the equations of state (EoS) are undoubtedly “physically motivated” [12].

Both equations gained traction (especially Equation (5)), however the two share the same limitation: as they cannot be solved analytically, γ_sv and Ф are obtained iteratively. Although iterative solvers are broadly available, this is cumbersome and off-putting for practitioners as a transparent closed-form solution for γ_sv is always preferable, both for explicit understanding of the calculation itself and for its rapid implementation in practical settings. Moreover, any uncertainty in the measurement of θ will also have impact on the error in γ_sv and an explicit expression for γ_sv will allow analysis of the propagation of uncertainties in a simple and efficient way. To partially mitigate these issues equations of state are often fitted by high order polynomials (f(θ, γ_lv)) where each numerical coefficient is high order polynomial (f(γ_lv)) itself [22,23]. Apart from being tedious, such an approach has no physical justification. However, up to now no theoretically grounded way for closed-form assessment of SFE based on the contact angle measurements with a single wetting liquid is reported.

Here we show that by combining the Girifalco-Good, Neumann and Gerson equations it becomes possible to obtain physically motivated closed-form estimation of the interaction parameter Ф which allows for instant non-iterative calculation of the solid surface free energy directly from the surface tension of the wetting liquid and its contact angle with the solid surface. Apart from removing significant practical hurdles in SFE calculation such closed-form expression might be of broader interest when elucidating the interplay between the liquid and solid interfacial properties in wetting phenomena.

2. Materials and Methods

Contact angle data between solids with diverse SFE and polarity and liquids with broad range of surface tension are obtained from the classical studies of Neumann [14,15,16,17,18,19,20] and Zdziennicka and Janczuk [24]. Preference was given to these sources as the values reported in them are known for their precision and are commonly used as a reference standard in the field. Neumann used PD-ADSA and sessile drop techniques for his measurements, while Zdziennicka and Janczuk relied primarily on sessile drop technique that is also the most used methodology in contact angle measurements today.

3. Results

1. Derivation of physically motivated closed form solution for Girifalco-Good parameter and for surface free energy calculation

As can be seen in Figure 2 (upper panel) for all practical purposes γ_sv values obtained by Equations (5) and (6) can be assumed equal (mean absolute difference< 0.2 mN/m; mean discrepancy = 0.086 mN/m with 95% confidence interval: -0.015 ÷0.17 mN/m) [22]. As emphasized by Neumann [12,14,15], such differences are merely a result of the different functional form of the equations of state and have no influence on their capability to fit real world data (see also Supplementary file, point I). Then obviously from Equations (4), (5) and (6) one obtains:

$$\Phi =e^{-\beta {\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\right)}^2}=\exp \left[\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\cos \theta \right)\left(a{\gamma }_{\mathit{sv}}+b\right)\right]$$

(7)

Taking the natural logarithm of both sides eliminates the exponentials:

$$-\beta {\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\right)}^2=\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\cos \theta \right)\left(a{\gamma }_{\mathit{sv}}+b\right)$$

(8)

After some algebraic manipulations this equation can be rearranged to a quadratic equation:

$$A{\gamma }_{\mathit{sv}}^2+B{\gamma }_{\mathit{sv}}+C=0$$

(9)

where the coefficients are:

A = a + β

(9a)

B = b − aγ_lvcosθ − 2βγ_lv

(9b)

$$C=\beta {\gamma }_{\mathit{lv}}^2-b{\gamma }_{\mathit{lv}}\cos \theta$$

(9c)

The following physically meaningful root of the quadratic equation is obtained for the solid-vapor SFE that contains only the numerical coefficients a, b and β and the experimentally measurable quantities θ and γ_lv:

$${\gamma }_{\mathit{sv}}^{\mathit{quadratic}}={\displaystyle \frac{-B-\sqrt{B^2-4\mathit{AC}}}{2A}}$$

(10)

Equation (10) provides physically sound estimates across all relevant regimes (SFE ≥ 3 mN/m; see Figure S1 in the Supplementary Material). While it can yield negative values at extremely low surface energies (SFE < 2-3 mN/m as calculated via Equations (5) and (6))— where C < 0 or B² < 4AC may occur—this regime lies well below the physical lower bound for both solid and liquid surface free energies. At such extremely low energies, cohesive forces are insufficient to maintain a stable interface, rendering the thermodynamic concepts of contact angle and surface free energy undefined. For context, the lowest reported experimental SFE for a solid layer is 6.7 mN/m (measured from a hexagonally close-packed alignment of trifluoromethyl (-CF₃) groups on a vapor-deposited layer of n-perfluoroeicosane [25]), while the theoretical lower limit for a smooth, flat surface is approximately 11.5 mN/m (corresponding to a maximum water contact angle of 120°) [12,25]. Nevertheless, we do not recommend using Equation (10) directly for SFE calculations, as such an approximation is unnecessary. Instead, it is best utilized as a conversion factor to determine the interaction parameter as defined in the exponential form of Φ (see point II in the Supplementary material for further discussion).

Thus, the quadratic root ${\gamma }_{\mathit{sv}}^{\mathit{quadratic}}$ should be plugged in the expression for Gificalco-Good interaction parameter $\mathsf{\Phi }=e^{-\beta {\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\right)}^2}$ (see Equations (5) and (7)) provided by the Neumann equation of state when one obtains:

$$\Phi =\exp \left[-\beta {\left({\displaystyle \frac{-B-\sqrt{B^2-4\mathit{AC}}}{2A}}-{\gamma }_{\mathit{lv}}\right)}^2\right]$$

(11)

where coefficients A, B and C are defined in equations (9a-c). Because Equation (11) contains only experimentally measurable parameters it allows for direct analytical closed-form calculation of Ф once the contact angle of a wetting liquid with known surface tension is measured. As Ф is calculated as an exponent (as in Equation (7)) it remains strictly positive over the entire range of θ (0-180°) for any surface tension of the wetting liquid and its value remains in excellent agreement with the iteratively calculated Ф estimates obtained from the Neumann EoS (see point III in the Supplementary file).

Finally, after inserting Equation (11) in the Girifalco-Good equation the following hybrid Neumann-Gerson equation of state is obtained about γ_sv requiring only experimentally measurable parameters to calculate SFE:

$${\gamma }_{\mathit{sv}}={\displaystyle \frac{{\gamma }_{\mathit{lv}}{\left(1+\cos \theta \right)}^2}{4}}\exp \left[2\beta {\left({\displaystyle \frac{-B-\sqrt{B^2-4\mathit{AC}}}{2A}}-{\gamma }_{\mathit{lv}}\right)}^2\right]$$

(12)

As can be seen in Figure 2 (Bottom panel) the values of γ_sv calculated with Equation (12) are practically indistinguishable (mean absolute difference between the predictions of the two equations is less than 0.06 mN/m; mean discrepancy = -0.03 mN/m with 95% confidence interval: -0.044 ÷-0.019 mN/m) from the ones obtained with Equations (5) and (6). To put these numbers in perspective it should be considered that Neumann developed three equations of state based on the augmentation of the Berthelot’s combining rule (see point I in Supplementary file), that he considered to be virtually equivalent to each other. The mean discrepancy between Neumann’s own equations could reach 0.3 mN/m with 95% confidence interval: 0.47 and −0.30, i.e., the agreement of Equation (12) with Equation (5) was better than the one among Neumann’s own expressions.

However, in contrast to them, the estimates determined with Equation (12) are not computed iteratively but analytically directly from the experimentally measurable θ and γ_lv. Thus Equation (11) provides, to use the description coined by Neumann [12], physically motivated (as it is derived entirely from physically motivated predecessors) analytical closed-form solution of the Girifalco-Good equation.

2. Performance of Equation (12) on real world contact angle data compared to traditional surface energy estimation approaches

Table 1 summarizes the values of SFE calculated for variety of solid surfaces. As can be seen from practical point of view the values obtained iteratively with Neumann equation of state and calculated analytically via Equation (12) are essentially identical for all surfaces and for the entire range of probe liquid surface tensions and contact angles.

Table 2 is interesting that it presents data for a reasonably polar quartz surface with water contact angle of 26°. Once again there is excellent quantitative agreement in SFE values assessed iteratively with Equation (5) and computed in closed-form with Equation (12). For such hydrophilic surfaces apparent complication stems from the observation that Neumann equation of state (and respectively Equation (12)) yields different SFE for the same solid (i.e., γ_sv=f(γ_lv)) depending on the probe liquid that wetted the sample i.e., γ_sv is a function of γ_lv. Detailed discussion of the phenomenon is beyond the scope of the current study. Yet, it suffices to say that such result is in excellent agreement with the postulates of Frumkin–Derjaguin theory of wetting [26,27] assuming the universal formation of very thin (few nm), wetting liquid-specific, precursor film at the triple air-solid-liquid line of the sessile drop, i.e., that the contact angle based SFE estimates represent apparent SFE measuring the properties of the solid at the triple air/solid/liquid line, i.e., γ_sv =f(γ_lv), which is different from the SFE of the dry solid surface located away from the triple line. This apparent γ_sv =f(γ_lv) determined in contact angle studies, is nevertheless immensely useful as it reveals invaluable information about the chemical and structural dynamics of the solid surface. Furthermore, it is found that the average of the individual values of the solid surface free energy determined with distinct wetting liquids matches the value obtained with the model of Owens-Wendt (or of van Oss-Chaudhury-Good) [28,29,30,31,32,33,34]. The latter result is significant as it demonstrates that the controversies regarding SFE evaluations from single- and multiple wetting liquids are only apparent, and both approaches yield essentially identical estimates of γ_sv. Similar results were already demonstrated in number of studies [28,29,30,31,32,33,34,35,36].

4. Discussion

In the current study we combined Girifalco-Good, Neumann and Gerson equations to obtain a convenient new equation that provides closed-form solutions for the interaction parameter Ф and for the surface free energy γ_sv. Expectedly the newly derived Equation (12) shows excellent quantitative agreement with its predecessors Equations (5) and (6). In contrast to tedious high order polynomial fittings to the Neumann equation of state that have no physical justification [22,23], the newly derived equation inherits the “physical motivation”(a term originally coined by Neumann [12]) of its predecessors while providing possibility for immediate analytical non-iterative calculation of important physicochemical parameters.

This can be clearly seen for example when looking at the rigorous expression for Ф derived by Good and Girifalco [9,10,11]:

$${\Phi }_G={\displaystyle \frac{\frac{3}{4}{\alpha }_1{\alpha }_2\frac{2I_1{I}_2}{I_1+I_2}+\frac{{\alpha }_1{\mu }_2^2+{\alpha }_2{\mu }_1^2}{2}+\frac{{\mu }_1^2{\mu }_2^2}{3\mathit{\kappa T}}}{{\left(\frac{3}{4}{\alpha }_1^2{I}_1+{\alpha }_1{\mu }_1^2+\frac{{\mu }_1^4}{3\mathit{\kappa T}}\right)}^{0.5}{\left(\frac{3}{4}{\alpha }_2^2{I}_2+{\alpha }_2{\mu }_2^2+\frac{{\mu }_2^4}{3\mathit{\kappa T}}\right)}^{0.5}}}$$

(13)

where: α_1,2: Polarizability of molecules; μ_1,2: Permanent dipole moment of molecules; I_1,2: First ionization potential of molecules in phase 1 and 2 respectively and κT: Boltzmann constant multiplied by absolute temperature (k_BT)

The problem is that while information about the molecular properties is often available for pure liquids, these parameters are very challenging to obtain for most solids and attempts to achieve so have frequently resulted in misleading results. Therefore, although this expression is theoretically rigorous, it is with limited practical applicability and that is the reason Neumann [12,14] and Gerson [16,17] started to look for alternative ways to formulate Ф. However, as already shown these expressions contained unknown quantities like γ_sv and γ_ls and therefore Neumann and Gerson equations required iterative computations, while calculation of Ф directly from the surface tension of the wetting liquid and its contact angle with the studied surface remained elusive. Nevertheless, the values of Ф calculated from Equations (5) and (6) were found to be in good agreement with the values obtained by the rigorous expression based on molecular properties provided by Good and Girifalco [12,13,14,15,16].

From that point of view Equation (11) provides convenient opportunity as the interaction parameter values calculated from it match the Ф values computed iteratively with Equation (5) while doing so via direct calculation from the immediately available experimentally measurable γ_lv and θ. The existence of such opportunity is logical as the contact angle is an integral quantity, representing the manifestation of the individual molecular properties of both participants in the liquid-solid pair [9,10,11] as well as their interactions and combining Equation (5) and (6) allowed to untangle it in explicit form.

One of the reasons for implementing empirical high-order polynomial fittings to the equations of state is the fact that explicit expression for γ_sv allows transparent calculation of the propagation of uncertainty ( (dθ/dγ_sv )σ_θ ) in the value of the contact angle to the value of SFE [22,23]. This is possible here as well using Equation (14):

$${\displaystyle \frac{\partial {\gamma }_{\mathit{sv}}}{\partial \theta }}=\left[-{\displaystyle \frac{{\gamma }_{\mathit{lv}}\left(1+\cos \theta \right)\sin \theta }{2}}+4\beta {\gamma }_{\mathit{sv}}\left({\displaystyle \frac{\partial {\gamma }_{\mathit{sv}}^{\left(\mathrm{quad}\right)}}{\partial \theta }}\right)\left({\gamma }_{\mathit{sv}}^{\left(\mathrm{quad}\right)}-{\gamma }_{\mathit{lv}}\right)\right]\exp \left[2\beta {\left({\gamma }_{\mathit{sv}}^{\left(\mathrm{quad}\right)}-{\gamma }_{\mathit{lv}}\right)}^2\right]$$

(14)

Where

$${\displaystyle \frac{\partial {\gamma }_{\mathit{sv}}^{\left(\mathrm{quad}\right)}}{\partial \theta }}={\displaystyle \frac{-a{\gamma }_{\mathit{lv}}\sin \theta -\frac{B\left(a{\gamma }_{\mathit{lv}}\sin \theta \right)-2A\left(b{\gamma }_{\mathit{lv}}\sin \theta \right)}{\sqrt{B^2-4\mathit{AC}}}}{2A}}$$

(14a)

Such in depth analysis is beyond the scope of the current study, but it can be seen that propagation of uncertainty in θ as specified by Equation (14) will be easier to calculate compared to 5th order polynomial fitting of an equation of state [22] where uncertainty propagation will take the form of ${\displaystyle \frac{\partial \left(\mathit{PM}{\gamma }_{\mathit{sv}}\right)}{\partial \theta }}=5A\left({\gamma }_{\mathit{lv}}\right){\theta }^4+4B\left({\gamma }_{\mathit{lv}}\right){\theta }^3+3C\left({\gamma }_{\mathit{lv}}\right){\theta }^2+2D\left({\gamma }_{\mathit{lv}}\right)\theta +E\left({\gamma }_{\mathit{lv}}\right)$ (here A, B, C, D, E are high order polynomials themselves), i.e., it requires 24 fitted parameters and exhibits systematic edge amplification near the boundaries of the fitted range.

Equation (12) allows direct physically motivated closed-form calculation of SFE from the static contact angle of a sessile drop formed by liquid with known γ_lv over the surface of interest. Up to now no such opportunity was available as Equations (5) and (6) and other similar equations of state all required iterative computations.

The only option for analytical estimation from wetting with single liquid was provided by the equation of Chibowski [29,30,31,33] utilizing the advancing and receding contact angles (θ_a and θ_r):

$${\gamma }_s={\displaystyle \frac{{{\gamma }_l\left(1+\mathit{cos}{\theta }_a\right)}^2}{2+\cos {\theta }_r+\cos {\theta }_a}}$$

(15)

However, Equation (15) comes with its own limitations as it assumes that Ф remains the same for both the “dry” solid surface in front of the advancing edge of the liquid drop and for the wetted surface left behind the drop receding edge. As pointed out by Neumann [12], such assumption is not necessarily correct as the composition and structure of the hydrated solid surface might be significantly altered compared to its non-hydrated state. Nevertheless, it is shown on multiple occasions that for surfaces with moderate contact angle hysteresis ( θ_a - θ_r≤ 20°) SFE estimates from Equation (14) and Equation (5) are in very good agreement [28]. Another limitation of Equation (15) is that it requires two contact angles instead of only one and measurement of contact angle hysteresis has its own methodological challenges.

Until very recently there was a tendency among the adherents of single- (applying equations of state) and multiple-wetting liquids (applying mainly Owens-Wendt and van Oss-Chaudhury-Good models) to be mutually dismissive about the theoretical justification of the “competing” approach [37]. Nowadays it became clear that the trend observed when applying Neumann equation of state that SFE of the solid depends on the surface tension and polarity of the wetting liquid is actually not a bug but a feature as it completely agrees with the underlying (and experimentally backed) assumptions of Frumkin–Derjaguin theory of wetting [26,27] about the existence of nanometrically thin precursor film, which emanates from the three-phase contact line region of the sessile droplet and instantaneously spreads ahead of the latter. Therefore, contact angle-based estimates represent apparent SFE characterizing the properties of the solid at the triple air/solid/liquid line, i.e., γ_sv =f(γ_lv), and are distinct from the SFE of the dry surface faraway from the contact line [38]. Thus γ_sv=f(γ_lv) behavior of Equation (5) (and Equation (12)) provides information about the differential interaction of the solid surface with wetting liquids of distinct nature and thus for the intrinsic features of the solid surface itself [31,33]. Furthermore, it was demonstrated that the single SFE value calculated via Owens-Wendt or van Oss-Chaudhury-Good models is identical with the mean value of the individual SFE obtained via Equation (5) with different wetting liquids for the same solid, thus removing the apparent controversy between the two approaches. The result is unsurprising as from mathematical point of view it is obvious that in Owens-Wendt (and van Oss-Chaudhury-Good) model “interactions with several types of diagnostic liquids are averaged” and the computed SFE “describes the mean properties of the force field in terms of its force components as well as its total value” [39]. Furthermore, in contrast to the popular belief that equation of state approach and surface tension component theories are incompatible and mutually exclusive, as we recently pointed out knowing γ_sv and combining it with Owens-Wendt model allows to calculate the surface tension components of the SFE in analytical closed-form manner via the equations (γ^d and γ^p are the dispersive and polar components of the solid (s) and liquid (l) surface energy):

$$\sqrt{{\gamma }_s^d}=\sqrt{{\gamma }_l^d}{\displaystyle \frac{\left[1+\cos \theta \right]}{2}}+\sqrt{{\gamma }_l^p\left[{\displaystyle \frac{{\gamma }_s}{{\gamma }_l}}-{\left({\displaystyle \frac{\left[1+\cos \theta \right]}{2}}\right)}^2\right]}$$

(16)

$$\sqrt{{\gamma }_s^p}=\sqrt{{\gamma }_l^p}{\displaystyle \frac{\left[1+\cos \theta \right]}{2}}-\sqrt{{\gamma }_l^d\left[{\displaystyle \frac{{\gamma }_s}{{\gamma }_l}}-{\left({\displaystyle \frac{\left[1+\cos \theta \right]}{2}}\right)}^2\right]}$$

(17)

Curiously this opportunity that was demonstrated back in the 1970s by Kloubek [40,41] and by Neumann [42] remained almost neglected in practice, although it offers some interesting opportunities that will be explored in our subsequent publication. Interested readers may refer to our recent review [28] for examples of the combined application of the equations of state and Equations (16) and (17). It should be noted that implementing Equations (12), (15) and (16) it becomes possible to obtain the dispersive (${\gamma }_s^d$) and non-dispersive (${\gamma }_s^p$) components of SFE as well as its total value in completely transparent analytical fashion merely from the contact angle and the surface tension of the wetting liquid avoiding any iterative or multiple regression calculations [28], which is significant advantage compared to the currently used approaches.

Equations of state, like Equations (5) and (6), are sometimes criticized for not being entirely analytical and relying on best fit constants β, a and b [43]. It should be kept in mind that at the moment no approach for calculation of SFE is free from this limitation as in Owens-Wendt and van Oss-Chaudhury-Good model the surface tension component values of the wetting liquids cannot be obtained analytically as well and there are significant discrepancies about their best values (and how to measure them) in the literature [44]. At the same time, it should be noted that the constants β, a and b emerged in the derivation of the equations of state not in an arbitrary fashion but as a result of a physical necessity while amending the Berthelot’s combining rule to account for the nonlinear dependence of Ф on the interfacial properties of both participants in the solid-liquid pair. Regarding β an expression matching Equation (5) was derived by Neumann from molecular parameters and a density depletion theory at the interface. It provided a physical explanation for β as a measure for the molecular packing of the solid interface at the three-phase contact line [12].

$${\gamma }_{\mathit{lv}}\left(1+\cos \theta \right)\approx 2\sqrt{{\gamma }_{\mathit{l\nu }}{\gamma }_{\mathit{s\nu }}}{e}^{-{\displaystyle \frac{{\Delta }^2}{k_{B}T}}\left({\gamma }_{\mathit{sv}}-{\gamma }_{\mathit{lv}}\mathit{cos\theta }\right)}$$

(18)

where δ² is the area occupied by an atom in the liquid–solid interface, k_B is Boltzmann’s constant, and T is temperature.

As the Gerson and Neumann models describe the same system, their constants cannot be independent. The following analysis will allow us to access the physical sense behind the constants a and b in Equation (6) as well as their relationship with Neumann’s β. For this purpose, let us return to the expression for ${\gamma }_{\mathit{sv}}^{\mathit{quadratic}}$ (Equation (10)) and substitute the coefficients defined in Equations (9a-c) into it. Doing so one obtains:

$${\gamma }_{\mathit{sv}}^{\mathit{quadratic}}={\displaystyle \frac{{\gamma }_{\mathit{lv}}}{2A}}\left[\left(2\beta +a\cos \theta -{\displaystyle \frac{b}{{\gamma }_{\mathit{lv}}}}\right)-\sqrt{{\left(2\beta +a\cos \theta -{\displaystyle \frac{b}{{\gamma }_{\mathit{lv}}}}\right)}^2-4A\left(\beta -{\displaystyle \frac{b}{{\gamma }_{\mathit{lv}}}}\cos \theta \right)}\right]$$

(19)

Equation (19) allows for intriguing normalization and dimensionless reparameterization. Let us define:

- Fractional Interaction Parameter (α):

$$\alpha ={\displaystyle \frac{a}{A}}={\displaystyle \frac{a}{a+\beta }}$$

(19a)

Consequently, ${\displaystyle \frac{\beta }{A}}=1-\alpha$.

- Dimensionless Constant (q):

$$q={\displaystyle \frac{b}{A{\gamma }_{\mathit{lv}}}}=\left({\displaystyle \frac{b}{a}}\right){\displaystyle \frac{a}{A{\gamma }_{\mathit{lv}}}}=\left({\displaystyle \frac{b}{a}}\right){\displaystyle \frac{\alpha }{{\gamma }_{\mathit{lv}}}}$$

(19b)

- Universal Liquid Ratio r

To eliminate q as an independent parameter while remaining valid across liquids of varying surface tension γ_lv, let us define a universal system constant, relating interfacial state parameters:

$$C_{\mathit{ba}}=-{\displaystyle \frac{b}{a}}\approx 153.85\mathrm{mN}/\mathrm{m}$$

(19c)

Defining the dimensionless liquid parameter r as:

$$r={\displaystyle \frac{153.85}{{\gamma }_{\mathit{lv}}}}$$

(19d)

yields the exact relationship:

q = −rα

(19e)

Substituting expressions (19a-e) into Equation (19) yields a notably simple relationship for ${\gamma }_{\mathit{sv}}^{\mathit{quadratic}}$:

$${\gamma }_{\mathit{sv}}^{\mathit{quadratic}}={\displaystyle \frac{{\gamma }_{\mathit{lv}}}{2}}\left[\left(2+\left(r-2+\cos \theta \right)\alpha \right)-\sqrt{{\left(2+\left(r-2+\cos \theta \right)\alpha \right)}^2-4\left(1+\left(r\cos \theta -1\right)\alpha \right)}\right]$$

(20)

This new equation (20) is remarkable as it allows us to have a glimpse into the interplay among the characteristics of the system. In the original equation (19), a, b, and βappear as independent parameters. Re-parameterization reveals that the system’s behavior does not depend on a, b, and β individually, but rather on two tightly coupled dimensionless groupings:

$\alpha ={\displaystyle \frac{a}{a+\beta }}$: Captures the intrinsic balance between the linear and non-linear interfacial energy contributions.

$r={\displaystyle \frac{C_{\mathit{ba}}}{{\gamma }_{\mathit{lv}}}}$: Links the internal material constant C_ba = −b/a directly to the external probe liquid’s surface tension γ_lv.

This reveals a hidden link: changing the probe liquid (γ_lv) is mathematically equivalent to altering the system’s internal scale parameter (C_ba). They do not act independently; they only ever interact as the single ratio r.

It should be specifically kept in mind that Equation (5) and (6) were developed while working with different datasets, i.e., while Neumann [12,13,14,15] used mainly his own carefully measured data, Gerson [16] gathered a very broad dataset from range of literature sources. Nevertheless, although using different data and working independently, both reached remarkably similar conclusions as can be seen from the overlap in the SFE values obtained by Equation (5) and Equation (6). This complete agreement between the two equations, as well as the concordance of their estimates with the models of Chibowski, Owens-Wendt and van Oss-Chaudhury good regarding the total value of SFE, strengthens the conviction in the overall validity of the approach and emphasizes that the hypothesized exclusiveness of the different approaches is largely apparent and that actually all these models can mutually complement each other and provide valuable information about the structure and dynamics of solid surface upon wetting.

5. Conclusions

Combining the Girifalco-Good, Neumann and Gerson equations allowed us to obtain a convenient new expression that provides closed-form solutions for the interaction parameter Ф and for the surface free energy γ_sv. This newly derived Neumann-Gerson equation shows excellent quantitative agreement with its predecessors. In contrast to tedious and physically unjustified fitting of high order polynomials to the Neumann equation of state, the newly derived equation inherits the physical motivation of its predecessors while providing possibility for immediate analytical non-iterative calculation of important physicochemical parameters. The explicit expression for γ_sv also allows analysis of the propagation of uncertainties in a simple and efficient way, thus also delivering the practical utility sought after the polynomial fittings of the equations of state.