Surface Aggregation Adsorption of Binary Solutions Between Diiodomethane, Furfural, and N,N-Dimethylformamide

Abstract

The surface tensions (σ) of binary solutions of diiodomethane (DIM, 1)–furfural (FA, 2), DIM (1)–N,N-dimethylformamide (DMF, 2), and FA (1)–DMF (2) were determined at 25 °C over the entire bulk composition range, and the surface adsorption behavior was analyzed using the surface aggregation adsorption (SAA) model proposed recently. In particular, by combining the SAA model with the Gibbs adsorption equation, the changes in the Gibbs surface excess (Γ₂) and the adsorption layer thickness (τ) with the bulk composition (x_2,b) were investigated. The SAA model combined with the modified Eberhart model can well describe the σ-isotherms of the three binary solutions. The surface adsorption trends of component 2 in DIM–FA, DIM–DMF, and FA–DMF decrease in turn. The change trends of Γ₂ and τ with x_2,b are dependent on the SAA model parameters, namely, the adsorption equilibrium constant (K_x) and the average aggregation number (n). With an increase in x_2,b, Γ₂ continuously increases when K_x < 2v₁/[n(2n − 1)v₂] (where v₁ and v₂ are the partial molar volumes of components 1 and 2, respectively); otherwise (i.e., K_x ≥ 2v₁/[n(2n − 1)v₂]), Γ₂ initially increases and then decreases, showing a maximum on the Γ₂-isotherm. When n ≥ 1, τ gradually decreases with an increase in x_2,b; otherwise (i.e., n < 1), τ initially increases and then decreases, showing a maximum on the τ-isotherm. An increase in the adsorption trend leads to a decrease in both Γ₂ and τ. This work provides a better understanding of the surface adsorption behavior of liquid mixtures.

1. Introduction

Surface adsorption of liquid mixtures is an important interfacial phenomenon [1,2,3,4], which involves three fundamental physical quantities, namely, the surface tension, composition, and thickness of surface adsorption layers. Many techniques, such as ellipsometry [5,6], neutron reflection [7,8], mass spectrometry (combined with an indigenous cluster beam apparatus) [9], vibrational frequency spectroscopy [10,11], and metastable induced electron spectroscopy [12], have been used to determine the composition and thickness of surface layers, but it is currently difficult to accurately (or independently) obtain data over the entire composition range. Owing to the fact that the surface tension of liquid mixtures is easy to accurately determine over the entire composition range, thermodynamic models are typically used to predict (or estimate) the composition and thickness of surface layers from the surface tension data [8,13,14,15,16,17,18,19,20,21].

Many thermodynamic (or empirical or semi-empirical) models have been developed to relate the surface properties (tension, composition, and thickness) of liquid mixtures with their bulk compositions [22,23,24,25,26,27,28,29,30,31,32,33,34,35]. However, due to the complexity of surface phenomena, existing models still lack universality [33,34]. For instance, the surface tension isotherms of binary solutions can be divided into two main types, namely, the Langmuir-type (L-type) and the sigmoid-type (S-type) [33]. Most of the existing models can reasonably describe the L-type isotherms, but not the S-type ones [33]. This is because some assumptions in establishing the models deviate from the real state of solutions. In fact, the dependence of the surface composition and thickness on the bulk composition is still not fully understood.

Recently, we proposed a thermodynamic model, called the “surface aggregation adsorption” (SAA) model, for liquid mixtures, relating the surface composition with the bulk one [33,34]. By coupling it with the modified Eberhart model [25,28], an equation with two parameters (i.e., the adsorption equilibrium constant and the average aggregation number) was developed to relate the surface tension with the bulk composition, which can well describe the L-type and S-type isotherms [33]. In addition, the SAA model and the modified Eberhart model can be combined with the Gibbs adsorption equation to calculate the Gibbs excess and thickness of adsorption layers [36]. Previous reports on the change in surface layer thicknesses with bulk compositions mainly focused on the aqueous solutions of short-chain alcohols (especially ethanol) [8,9,10,13,17,18,36], but yet no reports on non-aqueous solutions. It is interesting to understand the dependence of the thickness of surface adsorption layers (or the Gibbs excess layers) on the bulk composition.

In the current work, the surface tensions (σ) of binary solutions between diiodomethane (DIM), furfural (FA), and N,N-dimethylformamide (DMF) were determined over the entire bulk composition range, and the surface adsorption behavior was analyzed using the SAA model from the σ data. In particular, by combining the SAA model and the modified Eberhart model with the Gibbs equation, the changes in the Gibbs excess and thickness of the adsorption layer with the bulk composition were investigated. This work provides a better understanding of the surface adsorption behavior of liquid mixtures.

2. Experimental Section and Theoretical Basis

2.1. Chemicals

Diiodomethane (DIM) was purchased from Aladdin (Shanghai, China). Furfural (FA) and N,N-dimethylformamide (DMF) were purchased from Macklin (Shanghai, China). All chemicals were of analytical grade and used as received. Ultrapure water (with a resistivity of 18.25 MΩ·cm at 25.0 °C) was obtained using a Hitech-Kflow water purification system (Hitech, Shanghai, China).

2.2. Surface Tension Determination

Surface tensions of pure liquids (σ₀) and their mixtures (σ) were measured at 25.0 ± 0.2 °C using a Sigma 700 automatic tensiometer (Biolin, Gothenburg, Sweden). Each test sample was equilibrated for at least 10 min before measurements. All of the tests were performed in triplicate, and their average values were reported here.

2.3. Theoretical Basis

For clarity, the theoretical basis of the SAA model [33] and the surface excess model are summarized in the following.

Let us consider a binary solution consisting of components 1 and 2, with a planar surface layer. The surface layer is considered as a homogeneous surface phase with a definite thickness (τ) [20,24,25,26,27,29,30,31,32,33,34].

The SAA model. The SAA model [33] assumes that: (1) the molecules of component 2 (commonly the surface-active component) adsorb into the surface layer, directly forming aggregates with an average aggregation number of n, and (2) the surface layer consists of surface sites, and each site is occupied by one molecule of component 1 or one aggregate formed by component 2. The adsorption equilibrium is schematically expressed as

$$\mathrm{Site}\text{-}(\mathrm{C}1)+{n\left(\mathrm{C}2\right)}_{bulk}\stackrel{K_x}{\rightleftharpoons }{\mathrm{Site}\text{-}\left(n\mathrm{C}2\right)}_{aggr}+\left(\mathrm{C}1\right)bulk$$

(1)

where Site represents the surface sites, (C1) and (C2) represent the molecules of components 1 and 2, respectively, (nC2)_aggr represents an aggregate formed by n molecules of component 2, the subscript “bulk” represents the bulk phase, and K_x is the adsorption equilibrium constant.

Further, assume that the activity coefficients of all of the species Site-(C1), Site-(nC2)_aggr, (C1)_bulk, and (C2)_bulk are unity. According to the adsorption equilibrium principle, the K_x can be expressed as,

$$K_x={\displaystyle \frac{x_{aggr}{x}_{1,\mathrm{b}}}{x_{1,\mathrm{s}}{x}_{2,\mathrm{b}}^n}}$$

(2)

where x_aggr and x_1,s are the mole fractions of aggregates and component 1 in the surface phase, respectively, and x_1,b and x_2,b are the mole fractions of components 1 and 2 in the bulk phase, respectively.

Assuming that the partial molar volume of aggregates (v_aggr) equals n times that of component 2 in the surface phase (v_2,s), i.e., v_aggr = nv_2,s, Equation (2) can be written as

$$K_x^{*}={\displaystyle \frac{{\varphi }_{2,\mathrm{s}}{x}_{1,\mathrm{b}}}{{\varphi }_{1,\mathrm{s}}{x}_{2,\mathrm{b}}^n}}$$

(3)

where ϕ_1,s and ϕ_2,s are the volume fractions of components 1 and 2 in the surface phase, and K_x^* ≡ nν_2,sK_x/ν_1,s, here v_1,s is the partial molar volume of component 1 in the surface phase.

Considering ϕ_1,s + ϕ_2,s = 1 and x_1,b + x_2,b = 1, from Equation (3) one has

$${\varphi }_{2,\mathrm{s}}={\displaystyle \frac{K_x^{*}{x}_{2,\mathrm{b}}^n}{1-x_{2,\mathrm{b}}+K_x^{*}{x}_{2,\mathrm{b}}^n}}$$

(4)

Equation (4) is the SAA model equation. If n = 1, Equation (4) becomes the Langmuir-like model [33,37].

Since the surface compositions are now difficult to determine, the estimation of the two parameters (K_x and n) requires another independent equation that connects the surface compositions with the surface tension. A suitable and widely used equation is the modified Eberhart mixing rule [25,28], which is represented as

$$\sigma ={\varphi }_{1,\mathrm{s}}{\sigma }_1^0+{\varphi }_{2,\mathrm{s}}{\sigma }_2^0$$

(5)

where σ is the surface tension of binary liquid mixtures, and σ₁⁰ and σ₂⁰ are the surface tensions of pure components 1 and 2, respectively. Equation (5) gives π_r = ϕ_2,s, where π_r is the reduced surface pressure (π_r) [27], defined as π_r = (σ₁⁰ − σ)/(σ₁⁰ − σ₂⁰). According to Equation (5), the surface composition can be obtained only from σ data.

Combining Equations (4) and (5) gives

$$\sigma ={\sigma }_1^0-{\displaystyle \frac{K_x^{*}{x}_{2,\mathrm{b}}^n}{x_{1,\mathrm{b}}+K_x^{*}{x}_{2,\mathrm{b}}^n}}\left({\sigma }_1^0-{\sigma }_2^0\right)$$

(6)

$${\pi }_r={\displaystyle \frac{K_x^{*}{x}_{2,\mathrm{b}}^n}{1-x_{2,\mathrm{b}}+K_x^{*}{x}_{2,\mathrm{b}}^n}}$$

(7)

Equation (7) can be written in a linear form as

$$\lg {\displaystyle \frac{{\pi }_r(1-x_{2,\mathrm{b}})}{1-{\pi }_r}}=\lg K_x^{*}+n\lg x_{2,\mathrm{b}}$$

(8)

Equation (8) indicates that the plot of lg[π_r(1 − x_2,b)/(1 − π_r)] vs. lgx_2,b should be a straight line, and the values of n and K_x^* (and thus K_x) can be estimated from its slope and intercept.

Adsorption free energy. Based on the adsorption equilibrium principle, the standard Gibbs free energy change in adsorption per mole of component 2 (∆G_ad⁰) can be represented as [33]

$$\Delta G_{\mathrm{ad}}^0=-{\displaystyle \frac{RT}{n}}\ln K_x$$

(9)

where R is the universal gas constant and T is the absolute temperature. Equation (9) indicates that the adsorption tendency is determined by K_x and n. For positive adsorption (K_x > 1 and ∆G_ad⁰ < 0), larger K_x and smaller n correspond to a stronger adsorption tendency (i.e., a larger |∆G_ad⁰|).

The surface excess. The surface excess (or adsorption amount) is a relative quantity characterizing the difference in compositions between the surface phase and the bulk phase. There exist different definitions for the surface excess [24], among which the most widely used is the Gibbs surface excess.

At constant T and pressure (p), and assuming the activity coefficients of all components being unity, the Gibbs adsorption equation can be expressed as [1]

$${\mathsf{\Gamma }}_2=-{\displaystyle \frac{x_{2,\mathrm{b}}}{RT}}{\displaystyle \frac{d\sigma }{d{x}_{2,\mathrm{b}}}}$$

(10)

where Γ₂ is the Gibbs surface excess of component 2, defined by the Gibbs dividing surface (a zero-volume dividing surface within the surface layer, located at the position with zero surface excess of component 1) [1]. Guggenheim and Adam [24] elucidated the physical meaning of the Gibbs excess, i.e., the difference between the amount of adsorbate (here component 2) contained in the adsorbed layer per unit area and that contained in the bulk solution with the same amount of solvent (here component 1) as the adsorbed layer [13,16,17].

Combining Equation (10) with Equation (6) gives,

$${\mathsf{\Gamma }}_2={\displaystyle \frac{K_x^{*}{x}_{2,\mathrm{b}}^n(n{x}_{1,\mathrm{b}}+x_{2,\mathrm{b}})}{RT{(x_{1,\mathrm{b}}+K_x^{*}{x}_{2,\mathrm{b}}^n)}^2}}\left({\sigma }_1^0-{\sigma }_2^0\right)$$

(11)

This is the algebraic expression for the Gibbs surface excess, which can be used to calculate Γ₂ over the entire concentration range (x_2,b = 0–1) with the parameters K_x^* and n obtained from the SAA model. For simplicity, Equation (11) may be called the Gibbs-SAA equation.

From Equation (11), the limiting values of Γ₂ at x_2,b → 0 and x_2,b → 1, denoted as Γ_2(x→0) and Γ_2(x→1), respectively, can be obtained as

$$\underset{x_{2,\mathrm{b}}\to 0}{\lim }{\mathsf{\Gamma }}_2\equiv {\mathsf{\Gamma }}_{2(x\to 0)}=0$$

(12)

$$\underset{x_{2,\mathrm{b}}\to 1}{\lim }{\mathsf{\Gamma }}_2\equiv {\mathsf{\Gamma }}_{2(x\to 1)}={\displaystyle \frac{1}{RT{K}_x^{*}}}\left({\sigma }_1^0-{\sigma }_2^0\right)$$

(13)

At x_2,b→0, Γ₂ = 0, an expected result; while at x_2,b→1, Γ₂ > 0 (i.e., Γ₂ ≠ 0), arising from the definition (or physical meaning) of the Gibbs surface excess [1,22,24]. A strong adsorption trend (i.e., large K_x^* or K_x) results in a low Γ_2(x→1). Over the entire concentration range (x_2,b = 0–1), Γ₂ ≥ 0, indicating positive adsorption of component 2 with lower surface tension (σ₂⁰ < σ₁⁰).

In addition, the surface excess can be expressed by the difference in the absolute concentrations between the surface and bulk phases, called “the absolute surface excess”:

$$\Delta {\mathsf{\Phi }}_2={\varphi }_{2,\mathrm{s}}-{\varphi }_{2,\mathrm{b}}$$

(14)

$$\Delta X_2=x_{2,\mathrm{s}}-x_{2,\mathrm{b}}$$

(15)

where ΔΦ₂ and ΔX₂ are the surface excess expressed by the mole and volume fractions, respectively, for component 2.

Adsorption layer thickness. Based on the Gibbs equation and the SAA model, an equation for predicting the adsorption layer thickness τ can be established [36]. According to the physical meaning of the Gibbs excess Γ₂ [1,24], one has,

$${\mathsf{\Gamma }}_2={\displaystyle \frac{\tau }{v_2}}\left({\varphi }_{2,\mathrm{s}}-{\displaystyle \frac{{\varphi }_{1,\mathrm{s}}}{{\varphi }_{1,\mathrm{b}}}}{\varphi }_{2,\mathrm{b}}\right)$$

(16)

Note that the term in the bracket is the Gibbs surface excess expressed in volume fraction (defined by the Gibbs dividing surface). Let ΔΦ₂⁽¹⁾ ≡ ϕ_2,s − ϕ_1,s ϕ_2,b/ϕ_1,b. The ΔΦ₂⁽¹⁾ values can be calculated using the modified Eberhart mixing rule Equation (5) from the σ data. At ϕ_2,b → 1 (or x_2,b → 1), one has

$$\underset{x_{2,\mathrm{b}}\to 1}{\lim }\Delta {\mathsf{\Phi }}_2^{(1)}\equiv \Delta {\mathsf{\Phi }}_{2(x\to 1)}^{(1)}=1-{\displaystyle \frac{1}{n{K}_x}}$$

(17)

Strong adsorption trends (large nK_x) lead to large ΔΦ_2(x→1)⁽¹⁾. If nK_x = 1 (or n = 1 and K_x = 1), ΔΦ_2(x→1)⁽¹⁾ = 0, corresponding to the ideal state.

Combining Equation (16) with Equations (3), (4), and (11) gives

$$\tau ={\displaystyle \frac{v_{1,\mathrm{s}}{v}_{2,\mathrm{s}}n{K}_x(n{x}_{1,\mathrm{b}}+x_{2,\mathrm{b}})}{RT(v_{1,\mathrm{s}}{x}_{1,\mathrm{b}}+v_{2,\mathrm{s}}n{K}_x{x}_{2,\mathrm{b}}^n)(n{K}_x-x_{2,\mathrm{b}}^{1-n})}}\left({\sigma }_1^0-{\sigma }_2^0\right)$$

(18)

Equation (18) is the adsorption layer thickness (ALT) equation, which can predict τ over the entire bulk concentration range. Note that the physical meaning of the so-obtained τ is the thickness of the surface excess layer, rather than that of the “real” surface layer of liquids. The thickness of the surface excess layer is determined only by its composition difference with the bulk phase, while that of the “real” surface layer is determined by all physical properties different from the bulk phase (such as density besides composition [6]).

If K_x = 1 and n = 1, Equation (18) gives τ → ∞, which is reasonable, because the condition (K_x = 1 and n = 1) corresponds to the “ideal” state, i.e., no surface adsorption occurs (with τ → ∞ or τ = 0). It should be noted that, for the binary systems with n ≤ 1, the ALT equation is applicable over the entire x_2,b range, but for those with n > 1, it gives unreasonable (negative or infinite) τ at extremely low x_2,b (i.e., when x_2,b ≤ (1/nK_x)^1/(n−1)). For example, for a binary system with n = 1.27 and K_x = 6.37 (corresponding to the DIM-FA system, see Section 3.1), Equation (18) gives negative or infinite τ at x_2,b ≤ 4.3 × 10⁻⁴. When K_x > 1 and n > 1, the σ-isotherm is S-type [33]. That is, Equation (18) is not fully applicable to systems with S-type isotherms. This is because the modified Eberhart mixing rule Equation (5) is an empirical equation [29], not a thermodynamic one. It can be seen from Equation (16) that if Equation (5) gives ΔΦ₂⁽¹⁾ ≤ 0 while the Gibbs equation (Equation (10)) gives Γ₂ > 0, an unreasonable negative or infinite τ will be obtained. In fact, a more reasonable scenario is that, at x_2,b → 0 (or ϕ_2,s → 0), no surface aggregation occurs, i.e., n = 1 (rather than n > 1). Therefore, for the systems with n > 1, the τ at extremely low x_2,b can be estimated by extrapolating τ at x_2,b > (1/nK_x)^1/(n−1) to x_2,b = 0.

From Equation (18), the limiting τ values at x_2,b → 0 (with n ≤ 1) and x_2,b→1, denoted as τ_x→0 and τ_x→1, respectively, can be obtained as

$$\underset{x\to 0}{\lim }\tau \equiv {\tau }_{x\to 0}={\displaystyle \frac{v_{2,\mathrm{s}}{K}_x}{RT(K_x-1)}}\left({\sigma }_1^0-{\sigma }_2^0\right)\hspace{1em}\hspace{1em}(\mathrm{for} n=1)$$

(19)

$$\underset{x\to 0}{\lim }\tau \equiv {\tau }_{x\to 0}={\displaystyle \frac{v_{2,\mathrm{s}}n}{RT}}\left({\sigma }_1^0-{\sigma }_2^0\right)\hspace{1em}\hspace{1em}(\mathrm{for} n<1)$$

(20)

$$\underset{x\to 1}{\lim }\tau \equiv {\tau }_{x\to 1}={\displaystyle \frac{v_{1,\mathrm{s}}}{RT(n{K}_x-1)}}\left({\sigma }_1^0-{\sigma }_2^0\right)$$

(21)

Equations (19)–(21) indicate that strong adsorption trend (i.e., large K_x and low n) results in small τ_x→0 and τ_x→1, similar to the case of Γ_2(x→1). In addition, τ_x→0 is related to v_2,s while τ_x→1 to v_1,s, which are consistent with the physical meaning of τ thus obtained.

2.4. Model Fitting

The model fitting for test data was performed using Origin2016 software with the Levenberg−Marquardt algorithm. The best-fit values of model parameters were automatically obtained from the software, which correspond to the minimum sum of squares of residuals. The average absolute deviation (AAD) and the average relative deviation (ARD) were used as accuracy criteria, which are represented as

$$\mathrm{AAD}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m\left|X_{\exp }-X_{\mathrm{cal}}\right|}$$

(22)

$$\mathrm{ARD}={\displaystyle \frac{100\%}{m}}{\displaystyle \sum _{i=1}^m\frac{\left|X_{\exp }-X_{\mathrm{cal}}\right|}{X_{\exp }}}$$

(23)

where X_exp and X_cal are the experimental and model-calculated values of the physical quantity X, and m represents the number of data points. Low AAD and ARD values indicate that the model fits the experimental data well.

3. Results and Discussion

Based on the SAA model, the Gibbs-SAA equation, and the ALT equation, we investigated the surface adsorption behavior of three binary solutions: DIM (1)–FA (2), DIM (1)–DMF (2), and FA (1)–DMF (2).

Table 1. Physical properties of pure liquids.

Liquid	Mr (g/mol)	ρ0 (g/cm3)	ρc0 (g/cm3)	vb0 (cm3/mol)	vc0 (cm3/mol)	vs0 (cm3/mol)	ls (nm)
DIM	267.84	3.33	~0.91	80.43	~297	176.1	0.66
FA	96.09	1.16	0.359	82.84	267.6	167.4	0.65
DMF	73.10	0.94	0.267	77.77	273.7	165.5	0.65

shows the physical properties of pure liquids (DIM, FA, and DMF) involved here, including their relative molar masses (M_r), densities (ρ⁰, at 25 °C) and critical densities (ρ_c⁰) [38]. Their bulk molar volumes (v_b⁰, at 25 °C) and critical molar volumes (v_c⁰) were calculated using M_r, ρ⁰, and ρ_c⁰, which are also listed in Table 1. Note that the v_c⁰ and ρ_c⁰ values of DIM were estimated using the Wetere method [38]. The surface tensions (σ⁰) of the pure liquids DIM, FA, and DMF were determined here at 25 °C to be 49.04, 43.10, and 36.40 mN/m, respectively, which are very close to the literature-reported values (ca. 50.0–50.55, 43.0–43.10, and 36.52–36.98 mN/m, respectively) [38,39,40,41,42,43,44,45,46], indicating that the σ data determined here are credible.

In the following model calculations, let v_1,s/v_2,s = v_1,b/v_2,b and assume the partial molar volume of component i is equal to its molar volume (v_i = v_i⁰) [22]. In addition, assume that the amount of matter of the surface phase is significantly lower than that of the bulk phase, that is, the surface adsorption does not induce a change in the bulk composition [21].

3.1. Surface Aggregation Adsorption

The surface tensions of DIM (1)–FA (2), DIM (1)–DMF (2), and FA (1)–DMF (2) binary solutions were determined at 25.0 °C over the entire x_2,b range. The σ and π_r data are shown in Tables S1–S3 in the Supporting Information (SI).

Figure 1 shows the isotherms of σ and π_r versus x_2,b for the three binary solutions. With an increase in x_2,b, σ gradually decreases and π_r gradually increases, showing the characteristics of conventional L-type isotherms [33]. Based on the modified Eberhart mixing rule [28], π_r = ϕ_2,s, thus the π_r–isotherms are the isotherms of ϕ_2,s versus x_2,b. For the three binary solutions, their π_r isotherms are above the “ideal” correlation line (as marked with the dotted line), indicating the positive adsorption of component 2 occurs. This is reasonable due to σ₂⁰ < σ₁⁰ for the three systems. The π_r values at given x_2,b increase in turn for DIM (1)–FA (2), DIM (1)–DMF (2), and FA (1)–DMF (2), indicating their adsorption trends increase in turn.

The π_r − x_2,b data of the three binary solutions were linearly fitted using Equation (8), as shown in Figure 2. All of the plots of lg[π_r(1 − x_2,b)/(1 − π_r)] versus lgx_2,b show good straight lines, with the adjusted coefficients of determination being 0.988, 0.995, and 0.993 for DIM–FA, DIM–DMF, and FA–DMF, respectively. From the slopes and intercepts of the plots, the n and K_x^* of the three solutions were estimated, and the K_x was then obtained, which are listed in

Table 2. Best-fitting parameter values for surface tension data of binary solutions. (AAD and ARD correspond to the σ − x_2,b data over the entire x_2,b range).

Binary Solution	n	Kx*	Kx	AAD (mN/m)	ARD (%)	∆Gad0 (kJ/mol)
DIM (1)–FA (2)	1.27	8.45	6.37	0.059	0.13	−3.62
DIM (1)–DMF (2)	0.92	2.21	2.47	0.042	0.10	−2.44
FA (1)–DMF (2)	1.00	1.28	1.38	0.040	0.10	−0.80

, where the standard deviations of the best-fitted n and lgK_x^* values were less than 0.05 and 0.04, respectively. For all of the three systems, K_x > 1, corresponding to positive adsorption.

The σ − x_2,b and π_r − x_2,b curves of the three binary solutions were then calculated using Equations (6) and (7), respectively, with the n and K_x^* values thus obtained (Table 2), which are also shown in Figure 1. All of the model curves coincide well with the experimental data. The AAD and ARD of the model-predicted σ values with the experimental ones are listed in Table 2, which are less than 0.06 mN/m and 0.15%, respectively. These results indicate that the SAA model is reasonable, which can well describe the σ or π_r–isotherms of the three binary solutions.

It should be noted that for DIM–DMF and FA–DMF systems, K_x > 1 and n ≤ 1, consistent with the L-type isotherms intuitively observed [33]. However, for the DIM–FA system, K_x > 1 and n > 1, which actually correspond to the S-type isotherms [33], different from the L-type isotherm intuitively observed. This is because the negative deviation of π_r from the ideal correlation line at extremely low x_2,b (x_2,b < 2.7 × 10⁻³) is too small to appear on the figure. The π_r − x_2,b data of the DIM–FA were also nonlinearly fitted using the Langmuir-like model (Equations (4) or (7) with n = 1), showing the best-fitted K_x^* = 5.22 (Figure S1). The AAD and ARD obtained by the Langmuir-like model for σ data were ca. 0.14 mN/m and 0.32%, respectively, which are larger than those obtained by the SAA model (0.059 mN/m and 0.13%, respectively). This indicates that the SAA model can better describe the σ or π_r–isotherms of the DIM–FA system than the Langmuir-like model (Figure S1).

In addition, the ∆G_ad⁰ of the three systems at 25.0 °C was calculated using Equation (9) with the K_x and n values obtained, listed in Table 2. The ∆G_ad⁰ values of the three systems are all negative, indicating that the surface adsorption is spontaneous. For DIM–FA, DIM–DMF, and FA–DMF, the absolute ∆G_ad⁰ (|∆G_ad⁰|) values decrease in turn, indicating the adsorption trends decrease in turn, consistent with their π_r–isotherms observed (Figure 1B).

3.2. The Surface Excess

The Gibbs excess Γ₂ values at different x_2,b were calculated using the Gibbs-SAA equation (Equation (11)), as shown in Figure 3A. The change of Γ₂ with x_2,b exhibits different trends for the three systems. For DIM–FA, with an increase in x_2,b, its Γ₂ initially increases and then decreases, showing a maximum at x_2,b ≈ 0.20. This result is similar to those of binary aqueous solutions of short-chain alcohols [24,36,47]. For DIM–DMF, Γ₂ initially rapidly increases and then gradually reach equilibrium; while for FA–DMF, its Γ₂ continuously (almost linearly) increases. The shape of Γ₂-isotherms is determined by K_x^* and n values. Owing to the fact that dΓ₂/dx_2,b ≥ 0 at x_2,b → 0, if dΓ₂/dx_2,b ≤ 0 at x_2,b → 1, a maximum Γ₂ will appear on the Γ₂-isotherm. Therefore, it can be derived that only when K_x^* ≥ 2/(2n − 1) (or K_x ≥ 2v_1,s/[n(2n − 1)v_2,s]), there exists a maximum on Γ₂-isotherm. In addition, the Γ_2(x→1) values of DIM–FA, DIM–DMF, and FA–DMF were obtained to be ca. 0.29, 2.23, and 2.09 μmol/m², which are listed in

Table 3. Adsorption parameters of binary solutions at 25 °C.

Binary Solution	Γ2(x→1) (μmol/m2)	$\mathbf{\Delta }{\mathbf{\Phi }}_{2(\mathit{x}\to 1)}^{(1)}$	τx→0 (nm)	τx→1 (nm)	N2(x→0)
DIM (1)–FA (2)	0.29	0.88	~0.8	0.05	1.2
DIM (1)–DMF (2)	2.23	0.56	0.78	0.69	1.2
FA (1)–DMF (2)	2.09	0.28	1.62	1.25	2.5

for clarity.

The Gibbs surface excess ΔΦ₂⁽¹⁾ and the absolute surface excess ΔΦ₂ are also shown in Figure 3 (B and C) as a function of x_2,b. With an increase in x_2,b, ΔΦ₂⁽¹⁾ gradually increases. The ΔΦ_2(x→1)⁽¹⁾ values of DIM–FA, DIM–DMF, and FA–DMF were obtained to be ca. 0.88, 0.56, and 0.28, respectively, which are listed in Table 3. Large ΔΦ₂⁽¹⁾ corresponds to the strong adsorption trend. In addition, with an increase in x_2,b, ΔΦ₂ initially increases and then decreases, showing a maximum. At x_2,b = 0 and x_2,b = 1, ΔΦ₂ = 0, which is an expected result. The maximum ΔΦ₂ values of DIM–FA, DIM–DMF, and FA–DMF increases sequentially, corresponding to their increasing adsorption trends. ΔΦ₂⁽¹⁾ and ΔΦ₂ can well reflect the adsorption trend of binary solutions.

3.3. Adsorption Layer Thickness

It has been demonstrated that the molar volumes of components in the surface phase (v_s) are larger than those in the bulk phase (v_b) [13,48,49,50,51]. Many models were suggested to estimate v_s [48,49,50], and the Paquette model was chosen here, which was commonly used in the literature [13,48,49,50,51]. The Paquette model [48] can be written as

$$v_{\mathrm{s}}=v_c^{3/5}{v}_{\mathrm{b}}^{2/5}$$

(24)

where v_c is the critical molar volume. The so-obtained v_s values of pure liquids (v_s⁰) involved here at 25 °C are shown in Table 1.

The change in τ with x_2,b was calculated using the ALT equation (Equation (18)) for DIM (1)–FA (2), DIM (1)–DMF (2), and FA (1)–DMF (2) systems, as shown in Figure 4, where the partial molar volumes of component 2 in the surface phase are assumed to be equal to their molar volumes (v_2.s = v_2,s⁰). Note that for the DIM–FA system with n > 1, negative τ will be obtained at x_2,b < 4.3 × 10⁻⁴, which is unreasonable, thus only the τ values at x_2,b > 6 × 10⁻³ are shown here. The three systems exhibit different trends. With an increase in x_2,b, for the DIM–FA system (with n > 1), its τ gradually decreases (or exponentially decays, with x_2,b from 6 × 10⁻³ to 1), similar to the literature-reported results for aqueous solutions of short-chain alcohols [7,8,14,36]. For the FA–DMF system (with n = 1), its τ almost linearly decreases. However, for the DIM–DMF system (with n < 1), its τ initially sharply increases and then gradually decreases, showing a maximum at x_2,b = 0.04. Similar results were reported in the literature [13,17,18] for the water-ethanol solution. These results indicate that the binary solutions with n < 1 exhibit a maximum on the τ–x_2,b isotherms.

In addition, except in the extremely low x_2,b (x_2,b → 0) range, the τ values of DIM–FA, DIM–DMF, and FA–DMF increase in turn, suggesting strong adsorption leads to a small τ. The τ values observed here for the three systems (at x_2,b = 0.5) are ca. 0.2–1.4 nm, which are comparable with those obtained by other models [13,14,15,18,52,53], molecular simulations [8,11,54,55], and experimental measurements [4,6,8,11,56,57,58] for various binary mixtures (0.25–1.7 nm).

The τ_x→0 and τ_x→1 of the three binary solutions were calculated using Equations (19)–(21), which are listed in Table 3. Note that the τ_x→0 of the DIM–FA was obtained by extrapolation (or at x_2,b ≈ 6 × 10⁻³). The so-obtained τ_x→0 values are ca. 0.78–1.62 nm and the τ_x→1 values (of DIM–DMF and FA–DMF) are ca. 0.69–1.25 nm, which are comparable with the thickness of surface layers reported for pure liquids (0.45–1.43 nm) [4,6,55,58]. Note that a small τ_x→1 value was observed for DIM–FA (only ~0.05 nm), arising from its small Γ_2(x→1) and large ΔΦ₂⁽¹⁾ (Table 3).

Taking the side-length of the equivalent cubes of molecules in the surface phase (l_s, calculated using v_s⁰, Table 1) as a measure of the molecular size, the number of molecular layers of component 2 in the surface phase at x_2,b → 0, denoted as N_2(x→0), was estimated using N_2(x→0) = τ_(x→0)/l_2,s to be 1.2, 1.2, and 2.5 for DIM–FA, DIM–DMF, and FA–DMF, respectively, which are listed in Table 3. The N_2(x→0) values obtained here (ca. 1.2–2.5) suggest that the surface phase contains one to three molecular layers, which is close to those reported in the literature [4,6,13,16,26,52].

4. Conclusions

The surface tensions (σ) of binary solutions of DIM (1)–FA (2), DIM (1)–DMF (2), and FA (1)–DMF (2) were determined at 25 °C over the entire bulk composition range, and their surface adsorption behaviors were analyzed using the SAA model, the modified Eberhart model, and the Gibbs adsorption equation.

The SAA model combined with the modified Eberhart model can well describe the σ-isotherms of the three binary solutions. The surface adsorption trends of component 2 in DIM–FA, DIM–DMF, and FA–DMF decrease in turn. The change trends of the Gibbs excess Γ₂ and adsorption layer thickness τ with x_2,b are dependent of the K_x^* or K_x and n values. With an increase in x_2,b, Γ₂ continuously increases when K_x^* < 2/(2n − 1), while when K_x^* ≥ 2/(2n − 1), Γ₂ initially increases and then gradually decreases, showing a maximum on the Γ₂-isotherm. When n ≥ 1, τ gradually decreases with an increase in x_2,b, while when n < 1, τ initially increases and then decreases, showing a maximum on the τ-isotherm. With an increase in the adsorption trend (or |∆G_ad⁰|), Γ₂ and τ decrease while ΔΦ₂⁽¹⁾ and ΔΦ₂ increase. This work provides a better understanding of the surface adsorption behavior of liquid mixtures.