β-Lactoglobulin Adsorption Layers at the Water/Air Surface: 5. Adsorption Isotherm and Equation of State Revisited, Impact of pH

Abstract

The theoretical description of the adsorption of proteins at liquid/fluid interfaces suffers from the inapplicability of classical formalisms, which soundly calls for the development of more complicated adsorption models. A Frumkin-type thermodynamic 2-d solution model that accounts for nonidealities of interface enthalpy and entropy was proposed about two decades ago and has been continuously developed in the course of comparisons with experimental data. In a previous paper we investigated the adsorption of the globular protein β-lactoglobulin at the water/air interface and used such a model to analyze the experimental isotherms of the surface pressure, Π(c), and the frequency-, f-, dependent surface dilational viscoelasticity modulus, E(c)_f, in a wide range of protein concentrations, c, and at pH 7. However, the best fit between theory and experiment proposed in that paper appeared incompatible with new data on the surface excess, Γ, obtained from direct measurements with neutron reflectometry. Therefore, in this work, the same model is simultaneously applied to a larger set of experimental dependences, e.g., Π(c), Γ(c), E(Π)_f, etc., with E-values measured strictly in the linear viscoelasticity regime. Despite this ambitious complication, a best global fit was elaborated using a single set of parameter values, which well describes all experimental dependencies, thus corroborating the validity of the chosen thermodynamic model. Furthermore, we applied the model in the same manner to experimental results obtained at pH 3 and pH 5 in order to explain the well-pronounced effect of pH on the interfacial behavior of β-lactoglobulin. The results revealed that the propensity of β-lactoglobulin globules to unfold upon adsorption and stretch at the interface decreases in the order pH 3 > pH 7 > pH 5, i.e., with decreasing protein net charge. Finally, we discuss advantages and limitations in the current state of the model.

1. Introduction

Colloid stability is a large field of physical chemistry that has been developing for centuries [1,2]. The great body of experimental data obtained with various methodologies has been continuously bringing insights into important phenomena that can explain the mechanisms of colloid stabilization. The major identification feature of colloids as dispersed systems is their large interfacial area. Hence, interfacial phenomena inevitably play a central role in the behavior of colloids, and the key step toward colloid stabilization is the decrease of the free energy of liquid interfaces by adsorption of amphiphilic species (commonly known as surfactants). The latter can be low-molecular-weight surfactants, macromolecules (polymers, proteins) as well as supramolecular assemblies (molecular aggregates, nanogels, nanoparticles, etc.).

Understanding the process of adsorption at liquid interfaces and, importantly, its theoretical description, has probably started with the introduction of the Gibbs fundamental adsorption equation, dγ ~ Γdlnc, which relates the surface tension, γ, and the surface excess, Γ, via the surfactant concentration, c, in the liquid bulk phase. Later, several approaches for obtaining the adsorption isotherm and the equation of state of interfacial layers have been developed, among which perhaps the most familiar are those of Langmuir and Frumkin. These models have also been further modified by specific theoretical considerations or/and toward taking account for physicochemical features of the adsorbing species and the solvent, for instance, the proper localization of the Gibbs dividing surface and assuming penetrable or nonpenetrable interfaces in the former case and, in the latter case, accounting for charge effects, molecular structure, solvent conditions, etc. Even for the “simpler” case of small surfactants, the accounts for the nonionic or ionic nature of the molecules, as well as for counterion co-adsorption, already required modifications of the classical models [3,4,5]. On the other hand, description of the adsorption of polymers with various molecular structures (random coil, block- or graft copolymer, etc.) requires reconsiderations of the available theoretical formalisms and even creation of new ones, e.g., based on self-consistent field theories, scaling concepts and other approaches [6,7,8].

Proteins can be regarded as a special case of macromolecules because of their complicated chemical composition (heteropolymeric chain), which gives rise to well-defined secondary (α-helix, β-sheet, etc.) and tertiary (folding) molecular structures. Hence, the process of adsorption of proteins at liquid interfaces can exhibit unique features. Early studies showed that the classical approach of Gibbs is inapplicable to protein systems [9,10,11]. The Langmuir adsorption isotherm (or the Langmuir–Szyszkowski surface tension equation) also appears inadequate to describe protein adsorption [12]; nevertheless, some modifications have been proposed [13]. While, in the latter study, Damodaran and Razumovsky [13] concluded that the relative affinities of globular proteins to the water/air interface increases with increasing the molecular weight, M_w, (in a M_w^−2/3 manner), Ter-Minassian-Saraga [14] proposed the concept that the surface pressure, Π, of a protein monolayer is a function of the water activity at the surface. Joos [15] investigated spread layers of the globular protein β-lactoglobulin by treating the Π(θ) equation of state (θ is the fractional surface coverage) via the surface dilational modulus, E; he assumed a linear relation, E = uΠ, and claimed that the proportionality factor u is close to 8 for many proteins. In the same work, the author concluded that, at “low” Π, the protein globules are completely unfolded at the interface and the increase of Π leads to a decrease of the degree of unfolding. The concept of a Π-dependent degree of unfolding of globular proteins upon adsorption has been utilized also by Uraizee and Narsimhan [16], who proposed a lattice model for the treatment of the Π(Γ) equation of state. The model assumes a segmental molecular structure, as adsorbed segments are present only in the form of trains at the interface; it also accounts for segment–segment, segment–solvent and electrostatic interactions. In another statistical approach, Douillard et al. [17] described the equation of state for adsorption layers of the random coil protein β-casein by a simple relation: Π ≅ Γ^y, where the exponential factor y is characteristic of the regime of the interface and of the fractal dimension of the polypeptide chain. To our best knowledge, the applicability of this approach to globular proteins has not been affirmed so far.

The above considerations underline, on one hand, the conceptual difference between the interfacial behavior of low-molecular-weight surfactants and proteins and, on the other hand, the complexity of the problem of protein adsorption. However, in both cases, the heat of mixing at the interface, i.e., nonideality of enthalpy, is accounted for only in the classical Frumkin formalism [18] via an intermolecular interaction coefficient. Furthermore, it is obvious that the conformations of adsorbed proteins can be rather different from those in the bulk. Moreover, states of different molar areas can coexist, i.e., nonideality of entropy, that was considered by Joos and Serrien [19] and by Lucassen-Reynders [12].

In this paper, we focus on a thermodynamic model accounting for nonidealities of both enthalpy and entropy [20,21]. The basis of the model is the treatment of an interfacial protein layer as a 2-D solution, where the interfacial chemical composition is considered to be a mixture of the same components (solvent and solutes) as in the solution bulk and that the protein molecules can occupy a discreet spectrum of molar areas. We should stress here that previous theoretical treatments concern mostly the case of regimes at relatively low Π, which actually does not cover the whole range of experimentally measured surface pressures for protein adsorption layers. In the model considered here, such a “low” Π regime is defined as the precritical region in the surface pressure isotherm, Π(c), which is followed by a postcritical region, where θ and Π increase only slightly due to 2-D condensation (aggregation) of protein molecules adsorbed in a monolayer. At the same time, the surface excess, Γ, can significantly increase due to formation of additional layer(s) adjacent to the primary monolayer, a phenomenon experimentally observed, e.g., by ellipsometry and neutron reflectometry [22,23,24]. The two regions are divided by a critical surface pressure value, Π* (and a corresponding adsorption Γ*), achieved at a critical protein concentration, c* [20,21]. Aggregation in the postcritical region is accounted for by a dimensionless coefficient (ε = 0–0.1) that reflects the decrease of the area per protein molecule due to condensation [20]. Comparisons to experimental data for several proteins (β-casein, β-lactoglobulin, lysozyme, bovine and human serum albumins) showed quite satisfactory results [21,22,25,26] that credit adequacy to this approach. A refinement of the model for the postcritical region was introduced in [27], considering monomers and aggregates as independent kinetic units, and a semi-empirical linear relation between Π and Γ was proposed with a proportionality factor equal to the inverse aggregation number (see Equation (9), below). In this refined form, the model was compared to experimental data in several works [28,29,30,31,32,33]. We should mention here that, for the case of water/oil interfaces, the model was extended by the concept of a competitive adsorption of protein and oil molecules [34]. The general model was also extended to protein/surfactant mixtures [30,35]. In combination with surface dilational rheology theories [25,36], the model was successfully used to describe various dependencies of the surface dilational modulus E for proteins [25,27,28,29,30,31,32,36,37,38] and protein/surfactant mixtures [25,30,35].

The considered model is complicated, due to the number of model parameters. The state-of-the-art version (without accounting for co-adsorption in water/oil systems) used in the present work operates with eight free adjustable parameters for the description of a protein monolayer. To account for the formation of a secondary layer, an adsorption constant was additionally introduced; and finally, the protein diffusion coefficient is required for comparison with the available surface rheology data. Hence, to minimize speculations, fitting the model to experimental data should be done very carefully. One possibility is to constrain some parameters by a single value or to allow variation of the values only within a physically meaningful narrow range. A farther option is to apply the model simultaneously to as much experimental data as possible, ideally to the concentration dependencies of the key adsorption and rheologal parameters, i.e., Π(c), Γ(c) and E(c), which give access to the equation of state in terms of Π(Γ) and E(Π), and to other important dependencies. So far, the model was exclusively applied to experimental surface pressure isotherms, Π(c), and, only in some cases, experimental data on the surface dilational modulus, E [27,28,29,30,31,32,38] were considered. Only rarely, experimental data on both Π(c) and Γ(c) isotherms were simultaneously compared with the theory [22,26,39].

In the paper of Lucassen-Reynders, Fainerman and Miller [36], the model was applied for a detailed analysis of the dilational rheology behavior of protein layers, and an important advancement was achieved: it was demonstrated that the Gibbs (high-frequency limiting) elasticity of protein layers, where the protein molecules can exists in different conformational states, i.e., different partial molar areas, ω_i, is lower than the elasticity of a layer with uniform molar area, ω, by a factor of (1 + dlnω/dlnΓ). Actually, the latter study appears to be the only one where the model was compared to experimental data (for β-casein and bovine serum albumin) on the surface equation of state for both dependencies of Π(Γ) and E(Π). In the present work, we have the same aim, in order to theoretically describe the interfacial behavior of β-lactoglobulin (BLG) adsorption layers. Furthermore, we shall apply this approach, firstly, to reconsider our previous results on BLG adsorption layers at pH 7 [28] and, secondly, to analyze in this manner new experimental data obtained for BLG solutions at pH 3 and pH 5. The results are expected to well complement previous findings on the unique effect of pH on the properties of BLG adsorption layers at the water/air interface and on corresponding foam films and foams [24,40,41,42,43,44,45].

2. Materials and Methods

2.1. Chemicals and Solutions

Native β-lactoglobulin (molecular weight M_w ≈ 18.3 kg/mol) has been isolated and purified from whey protein isolate and supplied by the group of U. Kulozik at TU Munich, Germany [46]. The used sample contained total protein of ≈98.9% (of which BLG content >99%, BLG-A/BLG-B ≈ 1.22), salts of ≈0.7% and traces of lactose (<0.05%). Measurements were performed with solutions at pH 3, 5 and 7. While pH 5 is very close to the isoelectric point of BLG (pI ≈ 5.1 [40]), and the BLG net charge is negligible, the net charge at pH 7 is negative, and, at pH 3, it is positive. Aqueous stock solutions were prepared in 10 mM Na₂HPO₄/citric acid/Milli-Q buffers at pH 3 or pH 7 and at a BLG concentration of c = 2 × 10⁻⁴ M (≈3.65 mg/mL or ≈0.365 wt. %). To eliminate low-molecular-weight surface active contaminations, the initial stock solutions were further purified with activated charcoal (BLG/charcoal mass ratio 1/3, stirred for 20 min) [47] and then filtered through a 0.45 µm pore size protein nonbinding filter. The stock solutions at pH 3 or pH 7 were stored in a fridge for maximum 5 days, and the desired dilutions were freshly prepared before measurements. In the case of pH 5, all studied solutions were freshly prepared from a stock solution with pH 7 by dilution with 10 mM buffer of an appropriate pH to ensure a final value of pH 5. Solutions at pH 5 with concentration c > 10⁻⁵ M were not investigated, since they heavily precipitate during the measurements. All experiments were performed at room temperature of 22–23 °C.

2.2. Experimental Methods: Tensiometry and Surface Dilational Rheometry

Adsorption dynamics and dilational rheology experiments were performed using the drop/bubble Profile Analysis Tensiometer PAT-1 (SINTERFACE, Germany) [48] in the mode of a buoyant bubble in protein solution. The dynamic surface pressure, Π(t) = γ₀ − γ(t) (γ₀ = 72.3 ± 0.2 mN/m for the pure buffer/air interface, and γ(t) is the measured surface tension at the time moment t), was measured up to t = 80,000 s (≈22.2 h). The Π-values were then used to plot the surface pressure isotherms, Π(c). The surface dilational complex viscoelasticity modulus, E, and the phase angle, φ, were evaluated by the software integrated in the instrument through a Fourier transform of the surface tension response to harmonic area oscillations [48]. From these data, the corresponding real E’ and imaginary E” parts of the complex modulus were obtained. The dilational rheology data presented in this paper were measured at an oscillation frequency of f = 0.1 Hz and an amplitude of oscillations of g ≡ ΔA/A₀ × 100 = 2.7 ± 0.3%, where ΔA [mm²] is the amplitude of change of the undisturbed bubble area A₀ = 20 or 25 mm². In the following, we do not present the results for E’ and E”, since the values for E’ are very close to those for E, and the values for E” are much lower (with an order of magnitude) than those for E, which is typical for globular proteins [28,44,49,50,51]. The reported values for the experimental data for Π and E were averaged from 2–3 measurements, and the corresponding standard deviation is presented as error bars in the following figures (where applicable).

The used oscillating amplitude g ≈ 2.7% belongs to the linear viscoelasticity regime for BLG adsorption layers at the water/air interface at bulk pH 7 as found from measurements of the E(g)_f dependency [32]. In the present work we performed such measurements for all studied pH values and at different surface pressures. The obtained data in terms of E,φ,E”(g)_f dependencies (f = 0.1 Hz) are presented in Figure S1 in the Supporting Information.

Experimental data for the surface excess, Γ, and the molar area, ω, used in the present study were reported in a previous paper as evaluated from neutron reflectometry experiments with equivalent solution formulations [24], and the experimental data were gathered from single measurements for each sample.

2.3. Theoretical Model

The model assumes a discrete spectrum of n adsorbed states (1 ≤ j ≤ n) for a protein molecule, where the average molar area ω is distributed in the range between the boundary values: ω₁, which corresponds to state “1” with a minimum area, and ω_n, which corresponds to the n^th state with a maximum area. At an intermediate state j, the partial molar area is ω_j = ω₁ + (j − 1)ω₀, where ω₀ is an area increment taken to be in the order of the area per adsorbed water molecule. The average molar area, ω, is determined via the total coverage, θ, the partial adsorption in jth state, Γ_j, and the total adsorption, Γ:

$$\theta =\omega \Gamma ={\displaystyle \sum _{j=1}^n{\omega }_j}{\Gamma }_j, \Gamma ={\displaystyle \sum _{i=1}^n{\Gamma }_j}.$$

(1)

The adsorption isotherm equation for each j^th adsorbed state reads:

$$b_{j}c=\frac{\omega {\Gamma }_j}{{\left(1-\theta \right)}^{{\omega }_j/\omega }}exp\left[-2a \frac{{\omega }_j}{\omega }\theta \right] ,$$

(2)

where b_j is the adsorption equilibrium constant for the protein molecules in the jth state, and a is a Frumkin-type interaction parameter (a > 0 means intermolecular attraction), which accounts for the enthalpic nonideality. It was proposed earlier that the surface activity of adsorbed proteins increases with increasing the partial molar area, ω_j, according to a power law with a constant exponent, α, [20,21].

$$b_j={\left({\omega }_j/{\omega }_1\right)}^{\alpha }{b}_1.$$

(3)

Setting α > 0 means that the adsorption of molecules in states with larger molar areas is favored [27,28]. Then, combining Equations (2) and (3), one obtains:

$$b_{1}c=\frac{\omega {\Gamma }_j}{{({\omega }_j/ {\omega }_1)}^{\alpha }{\left(1-\theta \right)}^{{\omega }_j/\omega }}exp\left[-2a \frac{{\omega }_j}{\omega }\theta \right] .$$

(4)

Note b₁ in the left-hand side is the adsorption equilibrium constant for the protein molecules in the state “1.” Then, equating the right-hand sides of Equation (2) with j = 1 and Equation (3), it is straightforward to express the adsorption in any jth state via the adsorption in the state “1” and to eliminate all b_j except b₁, which, in what follows, is denoted by b:

$${\Gamma }_j={\Gamma }_1{\left(\frac{{\omega }_j}{{\omega }_1}\right)}^{\alpha }exp\left\{\frac{{\omega }_j-{\omega }_1}{\omega }\left[\ln \left(1-\theta \right)+2a\theta \right]\right\}.$$

(5)

Combining Equations (1) and (5) allows for eliminating the total adsorption, Γ, and the partial adsorptions, Γ_j, and obtaining an expression, which interrelates the model variables θ and ω via the model parameters ω_j:

$$\omega =\frac{{\displaystyle \sum _{j=1}^n{\omega }_j{\left(\frac{{\omega }_j}{{\omega }_1}\right)}^{\alpha }exp\left\{\frac{{\omega }_j-{\omega }_1}{\omega }\left[\ln \left(1-\theta \right)+2a\theta \right]\right\}}}{{\displaystyle \sum _{j=1}^n{\left(\frac{{\omega }_j}{{\omega }_1}\right)}^{\alpha }exp\left\{\frac{{\omega }_j-{\omega }_1}{\omega }\left[\ln \left(1-\theta \right)+2a\theta \right]\right\}}}.$$

(6)

In a similar way, using Equations (1) and (5), one transforms the adsorption isotherm Equation (2) to an expression that relates the model variables θ and ω with the protein concentration c:

$$\omega =\frac{{\displaystyle \sum _{j=1}^n{\omega }_j{\left(\frac{{\omega }_j}{{\omega }_1}\right)}^{\alpha }exp\left\{\frac{{\omega }_j-{\omega }_1}{\omega }\left[\ln \left(1-\theta \right)+2a\theta \right]\right\}}}{{\displaystyle \sum _{j=1}^n{\left(\frac{{\omega }_j}{{\omega }_1}\right)}^{\alpha }exp\left\{\frac{{\omega }_j-{\omega }_1}{\omega }\left[\ln \left(1-\theta \right)+2a\theta \right]\right\}}}.$$

(7)

For relatively low protein concentrations (precritical region), the equation of state reads:

$$-\frac{\Pi {\omega }_0}{RT}=\ln \left(1-\theta \right)+\theta \left(1-\frac{{\omega }_0}{\omega }\right)+a{\theta }^2,$$

(8)

where the first term on the right-hand side is relevant to the contribution of the ideal entropy, while the second and the third terms account for the nonideal entropy and enthalpy, respectively [20,21].

With Equations (1)–(8), a protein monolayer can be described by adjusting values for the six model parameters: a, α, b_j, ω₀, ω₁ and ω_n. A kink point, Π*, in the Π(c) isotherm is observed, corresponding to the critical bulk (c*) and surface (Γ*) protein concentrations, which divide the isotherms into a precritical region (Π < Π*, Γ < Γ*) and a postcritical region (Π > Π*, Γ > Γ*); the superscript “*” denotes the critical values of the different quantities. The precritical region, defined by Equation (8), is characterized by a steep increase of Π with increasing c, while, in the postcritical region, Π usually increases only slightly. Such behavior of the layer in the postcritical region is attributed to a 2-D condensation (surface aggregation), a compression of the layer [20,21,52] and the formation of a multilayer structure [20,21,53,54]. The protein molecules and aggregates are considered as independent kinetic units, and it is approximated that the increase of Π is proportional to the increase in Γ with a factor equal to the inverse value of the aggregation number, n_a, [27]:

$$\Pi ={\Pi }^{*}\left(1+\frac{1}{n_a}\frac{\Gamma -{\Gamma }^{*}}{{\Gamma }^{*}}\right),$$

(9)

where Γ(c) is computed via Equations (1), (5)–(7). Hence, two more parameters, Π* and n_a, should be included in the model calculations for the description of the primary protein monolayer in the postcritical region.

So far, Equations (1)–(9) describe the adsorption and the surface pressure isotherms as well as the surface equation of state for a protein monolayer within a 2-d solution model. Further accumulation of material onto the saturated monolayer gives rise to the formation of an adjacent protein layer; hence, the global interfacial structure tends to become heterogeneous. The secondary layer can be considered as adsorbed onto the primary monolayer and can be described by a Langmuir-type isotherm with the adsorption equilibrium constant, b₂. A previously derived approximation for the total adsorption, Γ_Σ, reads [20,55]:

$${\Gamma }_{\Sigma }\left(c\right)=\Gamma \left(c\right)\left(1+\frac{b_{2}c}{1+b_{2}c}\right),$$

(10)

and hence, for a bilayer, Γ_Σ instead of Γ should be used in Equation (9).

The surface dilational modulus E reads:

$$E=-\frac{\mathrm{d}\Pi }{\mathrm{d}\left(\ln A\right)},$$

(11)

where A is the surface area. Assuming harmonic surface area oscillations, the frequency-dependent dilational modulus can be expressed as a complex number [56]:

$$E\left(\varpi \right)=\left|E\right|e^{i\varphi }=E^{\prime }+i{E}^{″},$$

(12)

where $\varpi =2\pi f$ is the angular frequency (f is the frequency in [Hz]); $E^{\prime }\equiv \left|E\right|\cos \varphi$ is the real part accounting for the elastic contribution (conservation of energy) and $E^{″}\equiv \left|E\right|\sin \varphi$ is the imaginary part accounting for the viscous contribution (dissipation of energy). For a diffusion-controlled exchange of matter mechanism, the modulus E of surfactant and protein adsorption layers has been derived by Lucassen and van der Tempel [57,58] for a planar surface in oscillating barrier experiments. Joos [59] derived expressions for the case of a finite curvature of a drop or a bubble surface. For the particular case of diffusion from a reservoir onto the surface of a bubble, the modulus E reads [59,60]:

$$E\left(\varpi \right)=\frac{E_0}{1+\frac{\mathrm{d}c}{\mathrm{d}\Gamma }\frac{D}{i\varpi r}\left(1+r\sqrt{\frac{i\varpi }{D}}\right)},$$

(13)

where E₀ is the Gibbs elasticity, i.e., the high-frequency limiting elasticity under ideal elastic conditions; D is the diffusion coefficient of protein molecules in the bulk and r is the radius of the bubble. The limiting elasticity can be computed from the equation of state, Equation (8), taking into account the respective dependence ω(Γ) and assuming ω₀ << ω (which holds for proteins) [36]:

$$E_0={\left[\frac{\mathrm{d}\Pi }{\mathrm{d}\left(\ln \Gamma \right)}\right]}_{\mathrm{eq}}=\frac{RT}{{\omega }_0}\left(\frac{\theta }{1-\theta }-\theta -2a{\theta }^2\right)\left[1+\frac{\mathrm{d}\left(\ln \omega \right)}{\mathrm{d}\left(\ln \Gamma \right)}\right].$$

(14)

Note that such expression is valid for a protein adsorption layer in the precritical region, while, for the postcritical region, the protein layer should be considered as a composite surface [61] with a limiting elasticity [21,27]:

$$E_0=E_0^{*}\left({\Gamma }_{\Sigma }/\Gamma \right).$$

(15)

In some cases, it is convenient to introduce the reduced modulus, E/E₀ (0 < E/E₀ ≤ 1), which can be used as a measure for deviations from an ideal elastic behavior (E/E₀ = 1).

3. Results and Discussion

As mentioned in the introduction, one of the purposes of the present work is to reconsider our previous theoretical results obtained from comparison with experimental data for BLG adsorption layers at pH 7 [28] by taking into account new Γ(c) experimental data [24] and updated values of the dilational modulus, E, as measured in the linear viscoelasticity regime. In [28], the reported experimental E-data were measured at oscillation amplitudes (strains) of g ≈ 7%, which now appear to belong to the nonlinear viscoelasticity regime. Below a certain transition amplitude g_tr (yield strain), E is independent of g < g_tr, but beyond this threshold, E monotonically decreases with the increase of g > g_tr. For BLG, g_tr ≈ 4% was reported for solutions at pH 7 [32], as measured at relatively high surface pressures (Π > 20 mN/m). However, our data revealed that, in all studied cases at different pH, g_tr depends on the surface pressure, and it decreases with increasing Π (Figure S1 in the Supporting Information). Actually, Benjamins [62] reported earlier that the modulus E of interfacial layers of some globular proteins (bovine serum albumin and ovalbumin) do not depend on g (up to 15%) when E < 50 mN/m. We observed the same behavior for the studied BLG layers. For moduli E > 70 mN/m (that correspond to different surface pressures at different pH values, see the results below), g_tr was found to be of about 3–4% (independent of pH), which is in excellent agreement with literature [32]. Hence, in the present work, the theoretical analysis is performed on experimental E-data obtained at strains below g_tr, i.e., in the linear viscoelasticity regime.

The major purpose of this work is to examine the theoretical model as compared to several experimental dependencies of the surface pressure, Π, the surface excess, Γ, and the molar area, ω, as well as of the surface dilational modulus, E, and to compare the results for three different pH. We note, once again, that the Π- and E-data were measured by a single method (bubble shape analysis), while the Γ- and ω-data were obtained from independent neutron reflectometry experiments [24]. Hence, some of the dependencies presented further below were constructed with data from separate measurements. The model was simultaneously fitted to the following types of experimental data:

• the surface pressure isotherm, Π(c);
• the adsorption isotherm, Γ(c) [24]; and
• the dependency of the dilational modulus on the protein concentration, E(c)_f,g.

From these, one obtains:

4.

the equation of state, Π(Γ);

5.

the distribution of the average molar area over the protein concentration, ω(c) [24], the surface pressure, ω(Π) and the surface excess, ω(Γ); and

6.

the dependencies of the dilational modulus on the surface pressure, E(Π)_f,g, and on the surface excess, E(Γ)_f,g.

The simultaneous fit of the model to all experimental dependencies was facilitated by a calculation procedure integrated in a dedicated software application [http://www.thomascat.info/Scientific/adso/adso.htm (accessed on 1 March 2021)]. An extended description of the model and the calculation procedure implemented in the software is given in the Supporting Information. In this application, the plotted experimental data are compared with the model by means of an interactive tool, which displays the theoretical predictions for each dependency as computed on the basis of a set of values for the input parameters. Exemplary screenshots, illustrating the effects of different input parameters on the model predictions, are given in Figure S2 in the Supporting Information. The input parameter values used in the present work are listed in

Table 1. Values for input model parameters used for obtaining the best fits to the experimental data at different pH. Values for some output parameters obtained from the best fits are also listed.

Model Parameters	[unit]	pH 7		pH 3		pH 5
Input parameters
Π* (Critical surface pressure)	[mN/m]	15.1		15.2		15.0
ω1 (Minimum molar area) or (Area per molecule)	[m2/mol] [nm2/molecule]	9.7 × 106 16.1		8.4 × 106 14.0		4.5 × 106 7.5
ωn (Maximum molar area) or (Area per molecule)	[m2/mol] [nm2/molecule]	2.40 × 107 39.8		2.95 × 107 49.0		2.02 × 107 33.5
ω0 (Area increment) or (Area per increment)	[m2/mol] [nm2/increment]	2 × 105 0.33		2 × 105 0.33		2 × 105 0.33
bj (Local adsorption constant, monolayer)	[m3/mol]	2400		50		27
α (Exponential coefficient)	[-] [-] [-]	2.2		2.7		3.8
a (Frumkin interaction parameter)		0.70		0.66		0.92
na (Aggregation number)		11		16		7
m (Number of layers)	[-]	1	2	1	2	1	2
b2 (Adsorption constant, 2nd layer)	[m3/mol]	0	3	0	0.6	0	50
Output parameters a
n (Number of adsorbed states)	[-]	72		106		79
cΠ=0.1 c * cm2	[M] [M] [M]	7 × 10−10 7 × 10−9 5 × 10−6		7 × 10−9 8 × 10−8 3 × 10−5		3 × 10−9 2 × 10−8 3 × 10−7
ΓΠ=0.1 Γ* Γm2	[mg/m2] [mg/m2] [mg/m2]	0.42 0.97		0.31 0.80		0.69 1.09
		1.50 b		1.50 b		1.35 b
ωΠ=0.1 ω* ωm2	[nm2/molecule] [nm2/molecule] [nm2/molecule]	31.4 29.3		38.9 35.7		27.9 26.5
		20.4 ± 0.2 b		20.4 ± 0.2 b		22.4 ± 0.2 b
θΠ=0.1 θ*	[-] [-]	0.44		0.40		0.63
		0.94		0.94		0.95
E0*	[mN/m] [mN/m] [mN/m]	87		76		113
Πm2 E0,m2		20		20		19.5
		135		144		140
K* (Relative compression at ω*) c Km2 (Relative compression at ωm2) c	[%] [%]	7 35 d		8 48 d		5 20 d

^a Subscripts: “_Π=0.1” refers to values at the onset of measurable surface pressure, arbitrarily defined at Π = 0.1 mN/m [63]; “_m2” refers to values at the onset of secondary layer formation (defined in the text). ^b Values and error by definition; see above in the text. ^c K_x (defined in the text) is the monolayer compression at a given molar area ω_x relative to the origin ω_Π=0.1 at near-zero compression; ω_x ≡ ω* for K* and ω_x ≡ ω_m2 for K_m2. ^d Note that, in the postcritical region, compression is accompanied by surface aggregation.

.

Note that, in the model calculations, values for the surface excess, Γ, and the molar area, ω, are used in units [mol/m²] and [m²/mol], respectively, while for convenience, in most figures below, Γ is plotted in units of [mg/m²] as ${\Gamma }_{\left[\mathrm{mg}/{\mathrm{m}}^2\right]}=M_{\mathrm{w}\left[\mathrm{mg}/\mathrm{mol}\right]}{\Gamma }_{\left[\mathrm{mol}/{\mathrm{m}}^2\right]}$, and the area per molecule [nm²/molecule] is presented instead of the molar area as ${\omega }_{\left[{\mathrm{nm}}^2/\mathrm{molecule}\right]}={10}^{18}{\omega }_{\left[{\mathrm{m}}^2/\mathrm{mol}\right]}/N_{\mathrm{A}}$, where $N_{\mathrm{A}}$ is the Avogadro number.

As mentioned in the introduction, the model is complicated, due to the number of model parameters (there are nine input parameters for the description of the adsorption and rheological properties of a protein monolayer and, additionally, an adsorption constant, b₂, for the secondary layer); we used all of them as free adjustable parameters. Thereby, we attempted to facilitate the fitting procedure by following several steps and performing a number of subsequent iterations in order to achieve the final best global fit to all available experimental dependencies in each set for a given pH. In each fitting step in a given iteration, one or more fitting parameter(s) was/were optimized on the basis of the results from the previous iteration. In the following, we briefly explain the effects of each parameter on the model simulations. To visually support our explanations, we show original screenshots of the software tool in Figure S2 in the Supporting Information.

• At the very beginning, an approximate value for Π* should be set in order to divide the precritical and postcritical regions in the simulation curves. In the present case, the appropriate values for Π* are around 15 mN/m, where the three experimental Π(c) isotherms (for pH 3, pH 5 and pH 7) exhibit a kink;
• In accordance with Equations (1)–(7), the parameters ω₁, ω_n and b_j are essential for optimizing the model simulations in a way to fit the experimental Γ(c) isotherm and the corresponding dependency ω(c) and, at the same time, the experimental Π(c) isotherm. In the initial iteration, this step aimed at locating the precritical region of the Π(c) isotherm along the c-axis while maintaining a good fit to the Γ(c) isotherm. At other fixed parameters, increase of the local adsorption constant, b_j, shifts the precritical region of the Π(c) isotherm toward lower c; the individual effects of ω₁ or ω_n on the model simulations are illustrated in Figure S2b,c in the Supporting Information. In the final iterations, the values of these parameters were optimized to serve the best global fit to all processed dependencies. It must be stressed here that the input values of ω₁ and ω_n are boundary values. In the calculation procedure, the average molar area ω is allowed to vary between these minimal and maximal values, but the actual molar areas, computed as dependent on other parameters according to Equations (1)–(7), appear as output data (see Figure S3 in the Supporting Information). Those boundary values only guide the resulting best fit, and, therefore, a proper analysis should be based on the output data. Some characteristic values (ω_Π=0.1, ω* and ω_m2 (for definitions, see the legend in Table 1)) outputted from the best fits will be discussed further below;
• The molar area increment ω₀ was allowed to vary between 2.0 × 10⁵ and 4.5 × 10⁵ m²/mol [21,28,30,32,33], which corresponds to areas per increment between ≈0.33 nm² and ≈0.75 nm², respectively. We could mention here that Joos [15] used a limiting value of the solvent area per molecule of 0.10 nm² (≈6 × 10⁴ m²/mol); however, in our analysis we refrain from using such low values for ω₀. Assuming a weak effect of pH on this parameter, we aimed at setting a constant value in the processing of the data packs at the three different pH. The effect of ω₀ variation on the adsorption isotherm Γ(c) is less pronounced compared to that on the Π(c), Π(Γ) and E₀(Π) dependencies (Figure S2d in the Supporting Information). For the latter one, optimizing ω₀ was essential for description of the experimental E(Π) data. The best results were found using the lower values for ω₀ so that this parameter was fixed at 2.0 × 10⁵ m²/mol. It can be mentioned here that this value is very close to the molar area of a layer of adsorbed water molecules on mica surfaces (≈1.8 × 10⁵ m²/mol or ≈0.30 nm²/molecule) [64];
• For the exponential coefficient α, it was possible to use relatively close values in the cases of pH 3 (α = 2.7) and pH 7 (α = 2.2), but for pH 5, a substantially higher value (α = 3.8) was required. The effects of α on the model simulations are illustrated in Figure S2e in the Supporting Information;
• The effects of variation of the interaction parameter a on the model simulations are illustrated in Figure S2f in the Supporting Information. It should be noted that the simulations of the modulus E and those of the precritical region of the Π(Γ) equation of state are very sensitive to this parameter;
• The aggregation number n_a was optimized in respect to the postcritical region of the surface pressure isotherm Π(c) and to pin the local maximum of the E(Π) dependency observed at Π = 19–20 mN/m;
• Finally, the adsorption constant for the secondary layer b₂ was set to follow the experimental adsorption isotherm Γ(c), and then Π* and n_a were tuned in order to obtain best fits with the surface pressure isotherm Π(c) and the rheology data.

For each of the data packs at the three different pH, the same set of common input parameter values (listed in Table 1) were used to obtain fits by one-layer and two-layer models (in the following, denoted simply as “m1” and “m2,” respectively), with the only difference being the use of b₂ > 0 in the m2-fits. We should mention here that better m1- and m2-fits were obtained when using separate sets of parameter values, but these modified sets differ from each other only slightly, so that all the observed trends and their interpretation remain conceptually the same as those for the original results.

The aim of this approach is to determine the onset of the formation of a secondary layer by superimposing the results obtained by the m2-fit (describing a bilayer) and those obtained by the m1-fit (solely describing a monolayer). For this purpose, we analyzed in detail the theoretical ω(Γ) dependencies (Figure S3 in the Supporting Information). The onset of secondary layer formation was defined at a molar area of the saturated monolayer, denoted ω_m2, where the value of the surface excess obtained from the m2-fit (Γ_m2-fit) exceeds the one from the m1-fit (Γ_m1-fit) by about 1.5%; this corresponds to the situation where the difference between the ω-values at the average adsorption Γ_m2 = (Γ_m1-fit + Γ_m2-fit)/2 exceeds the area increment ω₀ ≈ 0.33 nm², and the error in ω_m2 was set as ±0.2 nm².

Despite the imposed restrictions, the obtained best global fits with the parameter values in Table 1 are quite satisfactory for the three cases of different pH values (Figure 1 and Figure 2) and our discussion and conclusions are based on these results. However, there is one conceptual discrepancy between theory and experiment: in the Π(c) isotherms computed by the m2-model, the model strongly overestimates Π at c > c_m2, i.e., beyond the onset of a secondary layer formation, while the experimental Π-values tend to level off.

Based on the obtained result, we can partition the run of the Π(c) and Γ(c) isotherms into three portions: (1) a precritical range, c_Π=0.1–c*, characterized by weak monolayer compression; (2) an initial part of the postcritical range, c*–c_m2, characterized by strong monolayer compression and surface aggregation; and (3) a range of growth of the secondary layer onto the saturated primary monolayer, c > c_m2. Definitions of the subscripts “_Π=0.1" and “_m2” are given in the legend of Table 1.

3.1. Precritical Region of the BLG Adsorption Layer

The monolayers formed in the precritical region are described by Equations (1)–(8), and, hence, the results from m1- and m2-fits based on the common parameter values listed in Table 1 are equivalent. Figure 1a,b show noticeable shifts along the c-axis of the Π(c) and Γ(c) isotherms as the values for both characteristic concentrations c_Π=0.1 and c* decrease in the order pH 3 < pH 5 < pH 7. Concerning the cases of pH 7 and pH 3, the lower surface activity at pH 3 is attributed to the higher protein net charge [40,43,65,66] within the concept of an electrostatic barrier of adsorption [65,66,67]. Effects of other pH-dependent factors, like exposed hydrophobicity, protein rigidity and degree of unfolding upon adsorption have also been considered in the literature [68,69,70]. Furthermore, the analysis by Sengupta et al. [71,72] of the potential energy profiles for a protein near the water/air interface, performed for sixteen proteins (which carry either positive or negative net charge under the experimental solvent conditions at pH 7), revealed the existence of an energy barrier to adsorption for the proteins with positive net charge. Such barrier appears as a local maximum in the profiles of net Van der Waals interactions (consisting of attractive Debye-Keesom interactions and London dispersion interactions) and is mainly attributable to the repulsive dispersion interactions between a protein and the water/air interface. It is worth to note that the dispersion interactions between a protein and a water/nonpolar oil interface have been found attractive; hence, such “Van der Waals force barrier” to adsorption has not been detected for this interface for any of the investigated proteins [71,72].

The lower surface activity of BLG at pH 3 than at pH 7 is well reflected by the model via the smaller adsorption constant, b_j, while the parameters α and a are kept comparatively close. In this line, the intermediate values of c_Π=0.1 and c* for pH 5 (negligible net charge) are surprising. Such “anomalous” behavior of BLG at pH 5 and at relatively low protein concentrations (c ≤ 10⁻⁷ M) was observed in previous adsorption kinetics studies [43]. Note that an analogous behavior was reported also for β-casein [27]. However, this intriguing behavior still remains unexplained.

Comparison of the Π(c) and Γ(c) isotherms shows that, for any of the studied pH values, the onset of Γ precedes the onset of Π, a typical situation for proteins [10,63]. This means that a certain minimum amount of adsorbed protein molecules is required to generate measurable Π-values. This phenomenon is the background of the so-called induction time in the dynamic surface pressure of protein solutions [43,55,73,74]. The origin of such behaviors is a first-order 2-D “gaseous” to “liquid expanded” phase transition, which occurs at extremely low Π [73,74]. The respective values of Γ_Π=0.1 for the studied BLG systems are listed in Table 1 and are well visible in the Π(Γ) data in Figure 2a.

The quantity Γ_Π=0.1 is most sensitive to the interaction parameter a and the maximum molar area ω_n (see Figure S2c,f in the Supporting Information) [20], and it increases in the order ${\Gamma }_{\Pi =0.1}^{\mathrm{pH}3}<{\Gamma }_{\Pi =0.1}^{\mathrm{pH}7}<{\Gamma }_{\Pi =0.1}^{\mathrm{pH}5}$; correspondingly, the molar area decreases in the same order, ${\omega }_{\Pi =0.1}^{\mathrm{pH}3}>{\omega }_{\Pi =0.1}^{\mathrm{pH}7}>{\omega }_{\Pi =0.1}^{\mathrm{pH}5}$ (Figure 1 and Figure 2). Such behavior of adsorbed proteins has been related to effects of the molecular net charge [16,66,75]. Indeed, vibrational sum-frequency-generation spectroscopy revealed that the strength of the electric field (independent of its sign) at water/air interfaces with adsorbed BLG decreases in the same pH-order [40]. This is in agreement with the absolute values, |Z|, of the BLG net charge, ±Z, in solution, that is, ca. +30 (pH 3), +2 (pH 5) and −11 (pH 7) electronic charges, e, as estimated from hydrogen ion titration experiments [76]. Interestingly, we found an excellent linear relation (R² > 0.99): ω_Π=0.1 = B|Z|+ 27.0, with B ≈ 0.4 [nm²/molecule.e]. For pH 5, ω_Π=0.1 = 27.9 nm²/molecule is very close to the area at zero net charge (27.0 nm²/molecule), which reveals very weak effect of the small |Z| at pH 5 on the molecular packing at the “gaseous” to “liquid expanded” phase transition.

At the very low c near the onset of Π, the induction times in the adsorption kinetics of BLG are very long (hours) [43], and, under these conditions, we consider the adsorbed protein molecules have approached the limit of unfolding upon adsorption [15,16], which apparently is pH-dependent. Note that the area “per” molecule ω is, in fact, slightly larger than the real area “of” a molecule at the interface, due to packing effects. Nevertheless, ω can be used to calculate arbitrary radii of adsorbed BLG molecules represented by oblate ellipsoids laterally packed in side-on configuration [24].

To illustrate the tendency of BLG to unfolding upon adsorption, one could also use the idea of de Feijter and Benjamins [77] about representing the adsorbed molecules as intrinsically “soft” particles that occupy an area per molecule with an “equivalent hard-core” radius, $R_{\mathrm{ehc}} \left[\mathrm{nm}\right]=\sqrt{\omega /\pi }$, (ω [nm²/molecule]). Later, a similar approach was used by Wierenga et al. [66], utilizing the same simple geometrical expression for estimating the “effective” radius of adsorbing globular proteins represented by hard-sphere particles. Thus, at the onset of the development of the “liquid expanded” phase, from ω_Π=0.1 we get $R_{\mathrm{ehc},\Pi =0.1}^{\mathrm{pH}3}$ ≈ 3.5 nm, $R_{\mathrm{ehc},\Pi =0.1}^{\mathrm{pH}7}$ ≈ 3.2 nm and $R_{\mathrm{ehc},\Pi =0.1}^{\mathrm{pH}5}$ ≈ 3.0 nm. The pH dependency of such “equivalent hard-core” radius can be explained by the pH-dependent molecular characteristics of the BLG globules and their net charge. The so-called acidic Q form (pH 2.5–4) of BLG has a less compact structure than the native N form (pH 4.5–6) and the so-called R form (pH 6.5–8) in the Tanford transition [78,79], which suggests the highest propensity to unfolding upon adsorption at pH 3. On the other hand, the largest net charge at pH 3 (compared to pH 5 nd pH 7) generates the strongest intermolecular electrostatic repulsion that counteracts protein–protein cohesive interactions [65] and also favors the protein–solvent interactions (wetting) at the interface. Having in mind constancy of the Debye length at the fixed ionic strength used, those prerequisites can be regarded as determining the highest values for ω_Π=0.1 and R_ehc at pH 3, as the latter is merely twice the radius (≈1.75 ±0.04 nm [80,81]) of the spherical native BLG monomeric unit in bulk. With this in line, the decrease of |Z| entails the decreasing values for ω_Π=0.1 and R_ehc at pH 7 and pH 5. Neglecting small pH-dependent variations in the molecular volume of BLG (ca. 1% [79]), the degree of unfolding in the primary monolayer can be monitored, for example, by the layer thickness, but we are not aware of data on the thickness of BLG layers at very low Π. Using the molecular volume of BLG (≈22.7 nm³ [24]) and the above estimated R_ehc,Π=0.1 values, for oblate ellipsoids adsorbed side-on at the interface, one estimates thicknesses (ellipsoids’ polar diameters) for the near-zero compressed monolayers of ≈0.9 nm (pH 3), ≈1.1 nm (pH 7) and ≈1.2 nm (pH 5), which are quite reasonable values [24,82,83,84].

In the concentration range c_Π=0.1–c*, Π and θ steeply increase, while the average molar area ω decreases only slightly (Figure 1a, Figure 2 and Figure 3a). For Γ* and ω*, Figure 2 reveals ${\Gamma }_{\mathrm{pH}3}^{*}<{\Gamma }_{\mathrm{pH}7}^{*}<{\Gamma }_{\mathrm{pH}5}^{*}$ and ${\omega }_{\mathrm{pH}3}^{*}>{\omega }_{\mathrm{pH}7}^{*}>{\omega }_{\mathrm{pH}5}^{*}$, which variations are equivalent to those of Γ_Π=0.1 and ω_Π=0.1. This behavior in the “liquid expanded” regime is consistent with the accumulation of adsorbed protein at the interface, accompanied by only a weak lateral compression of the monolayer. For a given surface pressure Π_x, the monolayer compression, K_x, relative to the origin ω_Π=0.1 (representing near-zero compression) can be estimated by a simple relation between ω_x and ω_Π=0.1: $K_{\mathrm{x}} [\%]=\left(1-\frac{{\omega }_{\mathrm{x}}}{{\omega }_{\Pi =0.1}}\right)\times 100$. At ω* we get $K_{\mathrm{pH}3}^{*}$ > $K_{\mathrm{pH}7}^{*}$ > $K_{\mathrm{pH}5}^{*}$ (for exact values, see Table 1). Simultaneous measurements of Π and Γ in adsorption kinetics experiments with solutions of succinylated variants of ovalbumin [1,66] or BLG [65] with varying |Z| have shown the same shift of the “dynamic” Π(Γ) curves toward smaller Γ with increasing |Z|. Song and Damodaran [65] concluded that electrostatic forces at the interface induce a partial increase of the surface pressure at constant Γ.

The current model used in our calculations does not provide a rigorous description of electrostatic effects in the adsorption isotherm and in the equation of state, but those are reflected by the enthalpic parameter a, which increases in the order a_pH3 < a_pH7 < a_pH5, meaning increasing intermolecular cohesive forces against decreasing electrostatic repulsion. From θ = ωΓ, the opposite trends in the Π(Γ) and Π(ω) data, respectively (Figure 2), cancel out to a great extent in the general equation of state Π(θ), as shown in Figure 3b. Indeed, the Π(θ) curves for BLG at pH 3 and pH 7 almost overlap, while the one for pH 5 appears below them, which illustrates the positive effect of significant electrostatic forces on the surface pressure [65]. However, the almost identical values at any pH for the characteristic quantities θ* (94–95%) and Π* (15.0–15.2 mN/m) reveal a pH-independent common behavior of the interfacial layers at the critical point (Π*,θ*), regardless of the pH-dependent adsorbed amount Γ* and degree of surface-induced unfolding (represented by ω*) of the adsorbed BLG globules. Again, we found an excellent linear relation (R² > 0.99): ω*= B|Z|+ 25.7, with B ≈ 0.33 [nm²/molecule.e]. Coincidently or not, this value of the prefactor B is virtually equal to the area increment ω₀, which means that the model settings allow detection of |Z|-induced variations in ω with the resolution of a single electronic charge.

The onset of surface pressure gives rise of the modulus E, which for the precritical region is computed via Equation (13). Figure 4a shows the dependencies lnω(lnΓ) at each pH required for determining the limiting elasticity E₀ via Equation (14). The prolongations of the curves above monolayer saturation correspond to increased values of Γ, due to development of the secondary layer, but the parallel decrease of the molar area is physically meaningless for θ > 1, and, therefore, the ω(c) data for c > c_m2 (that is, however, mathematically estimated by the model) are omitted in Figure 1c. Although the compression of the monolayer in the precritical region is weak, it gives a noticeable change of the derivative dlnω/dlnΓ (which is negative [36]) with increasing Π, as shown in Figure 4b. The steepest slope of this dependency at pH 3 agrees with the highest relative compressibility $K_{\mathrm{pH}3}^{*}$ ≈ 8%, as compared to the cases of pH 7 (≈7%) and pH 5 (≈5%).

The model computations of the dependencies E₀(Π) and E₀(Γ) are shown in full length in Figure 5; note that for Π > Π*, Equation (15) applies instead of Equation (14). The initial parts of the precritical regions of the E₀(Π) dependencies are linear [15,62] (R² > 0.99) almost up to Π* for pH 5 and pH 7 but only up to Π ≈ 10 mN/m for pH 3. Obviously, the model predictions for E₀ follow quite well the experimental data for the modulus E up to Π ≈ 18–19 mN/m. This means that the BLG monolayers exhibit a highly elastic behavior at the applied frequency of 0.1 Hz. The precritical regions of the E₀(Γ) dependencies for the different pH values are shifted in the same order as in the equation of state Π(Γ), as follows from the theory. The data in Figure 2a and Figure 5 show a strong effect of pH on E₀*, the values of which for each pH correspond to different Γ* and ω*, but to similar surface coverages of θ* = 0.94–0.95. Therefore, it follows from Equation (14) that this effect is accounted for by the interaction parameter a and the derivative dlnω/dlnΓ (Figure 4b). The observed pH-induced variations of the latter quantity reflect the intermolecular interactions and the resulting molecular packing as affected by the protein net charge [85,86].

Experimental and computed dependencies of the modulus E(c)_f,g are presented in Figure 6. The precritical regions of the data sets at different pH are localized along the c-axis in the same way as in the isotherms of the surface pressure Π(c) and the surface excess Γ(c) (Figure 1) that follows from the theory.

Experimental and computed dependencies E(Π)_f,g and E(Γ)_f,g are shown individually for each pH in Figure 7, where the theoretical E₀(Π) and E₀(Γ) results from Figure 5 are included, also, for comparison purposes. For the precritical region, the calculations for E by Equation (13) were performed using a diffusion coefficient of D_intr = 1 × 10⁻¹⁰ m²/s (denoted intrinsic), which is a typical value for the diffusivity of BLG in aqueous bulk media and is seemingly only weakly dependent on pH [87,88,89]. For pH 5 and pH 7, the data for E overlap quite well with the corresponding data for E₀ up to E₀*, while, for pH 3, a small but noticeable deviation appears beyond surface pressure of about 10 mN/m.

As mentioned above, quantification of the deviation from a purely elastic behavior of a surface layer is conveniently given by the reduced modulus E/E₀ (ideal elasticity at E/E₀ = 1). Calculations for f = 0.1 Hz and Π* (for each pH) yielded values for E*/E₀* of 0.99 (pH 7), 0.97 (pH 5) and 0.89 (pH 3); see also Figure S4a in the Supporting Information. The good agreement between theory and experiment with D_intr suggests that, apparently, protein diffusion dominates the surface pressure (stress) response of the weakly compressed monolayer to harmonic area oscillations (strain). The larger deviation from this behavior at pH 3 can be related to a high energy barrier to adsorption of positively charged proteins at the water/air interface, generated not solely by electrostatic interactions but also by repulsive dispersion interactions [71,72], as discussed in Section 3.1. Such high adsorption barrier seemingly interferes with the diffusion-controlled mechanism of the stress response, and the reduced modulus E*/E₀* for BLG layers at pH 3 increases only when lower diffusion coefficients are assumed. Just for demonstration, we computed that the reduced modulus at pH 3 at the critical point increases to a value E*/E₀* = 0.97 (as for pH 5) by using an “apparent” diffusion coefficient of D_app = 6 × 10⁻¹² m²/s.

3.2. Postcritical Region of Monolayer

Let us now discuss the behavior of BLG layers in the intermediate range of protein concentrations (c*–c_m2) up to the point of monolayer saturation (Π_m2,Γ_m2,ω_m2). This region spans over a larger c-range as compared to the precritical region but includes only a small range of surface coverages from θ* ≈ 0.95 to θ→1. Hence, it is characterized by strong compression of the monolayer accompanied by a comparatively weak increase of Π (≈5 mN/m) (see, for instance, Figure 2). The relative compression at monolayer saturation, K_m2, is much larger than K* (at θ*), also because of concomitant surface aggregation. Based on the values for ω_m2, we obtained $K_{\mathrm{m}2}^{\mathrm{pH}3}$ > $K_{\mathrm{m}2}^{\mathrm{pH}7}$ > $K_{\mathrm{m}2}^{\mathrm{pH}5}$ (for exact values, see Table 1). Since the values of ω_m2 at monolayer saturation are virtually the same for pH 3 and pH 7 and only slightly higher for pH 5, the variations of the relative compression K_m2 are determined by the origin ω_Π=0.1, i.e., by the pH-dependent degree of unfolding at near-zero compression.

As mentioned above, at pH 3, BLG attains its most flexible Q form, characterized by the lowest compressibility [78,79], whereas, at pH 5, the most compact molecular structure of BLG is attained (N form), characterized by the highest compressibility [78,79]. The latter is supposed to determine the lowest degree of unfolding at the interface, which subsequently results in the lowest values for ω_Π=0.1, ω*, K* and K_m2. At pH 7, BLG approaches the Tanford transition (centered at pH 7.5), which is characterized by a loosening of the interior packing of BLG [78,79]. Such molecular structure should render the protein globules moderate propensity to unfolding upon adsorption, which, in turn, results in intermediate values for ω_Π=0.1, ω*, K* and K_m2.

While, at pH 3, the highest net charge of BLG determines its lowest surface activity, the Π(c) and Γ(c) isotherms for pH 5 and pH 7 run through a crossover in the postcritical region. In the Γ(c) isotherms, this crossover occurs through a local overlap within the c-range between $c_{\mathrm{pH}7}^{*}$ (≈7 × 10⁻⁹ M) and ca. 1 × 10⁻⁷ M (including $c_{\mathrm{pH}5}^{*}$), whereas a sharp crossover of the Π(c) isotherms is observed. According to Equation (9), the observed steeper increase of Π > Π* for pH 5 at similar adsorptions is a result of the lower value for n_a (the effect of n_a on Π > Π* is illustrated in Figure S2g in the Supporting Information). Such behavior does not seem trivial, and, at the moment, we cannot provide a satisfactory explanation. A lower aggregation number for pH 5 is counterintuitive, because of the fact that proteins become more prone to aggregation when approaching pI. The most plausible reason behind this, at a first glance, paradox adsorption behavior, should be the entirely dimeric form of BLG in the bulk at pH 5 [90,91], while, for pH 7, the dimer fraction at 1 × 10⁻⁷ M can be estimated, as in the order of 10% [92] (at $c_{\mathrm{m}2}^{\mathrm{pH}3}$, the dimer fraction is < 5% [93]). Protein monomer/oligomer forms are not recognizable by the theory, and we stress here that the presented model computations are based on the molecular weight of a BLG monomeric unit. Further analysis is required to discover approaches for accounting bulk monomer/oligomer effects on the model’s performance, and such tests are currently running.

The increased monolayer compression in the considered region leads to a significant strengthening of the cohesive intermolecular interactions and to the formation of a strong protein network at the interface. This, in turn, enhances the elastic behavior of the monolayer, and the limiting elasticity E₀ increases linearly with Π and Γ (Figure 5) in accordance with Equations (9) and (15). The good quality of the fits for the E(Π)_f,g and E(Γ)_f,g dependencies at Π > Π* (see Figure 7) was achieved only by using much lower diffusion coefficients (see

Table 2. Values for the apparent diffusion coefficient, D_app, used in the model calculations.

	pH 7		pH 3		pH 5
	m1	m2	m1	m2	m1	m2
Dapp [m2/s]	8.0 × 10−15	1.2 × 10−14	1.3 × 10−15	1.8 × 10−15	1.0 × 10−11	3.8 × 10−11

) than the intrinsic bulk diffusivity of BLG. These “apparent” diffusion coefficients, D_app, reveal that the compressed BLG monolayers behave essentially as insoluble surface layers, where desorption is negligible [94,95]. However, the degree of reversibility of protein adsorption is dependent on Π [94,95], as also evidenced by the results and discussions in Section 3.1, where the diffusion-controlled mechanism seems to dominate the stress response of weakly compressed BLG monolayers (Π < Π*). At higher monolayer compressions, the strong protein network generates a high Gibbs free energy barrier to desorption [95]. However, it is evident in Figure 7 that the reduced modulus E/E₀ start to noticeably decrease before monolayer saturation at Π_m2 and Γ_m2, which suggests higher extent of energy dissipation for the strongly compressed monolayers, apparently due to in-plane (nondiffusional) relaxation processes that become detectable at the used frequency of 0.1 Hz. At this frequency, increase of the strain (oscillation) amplitude above g_tr (3–4%) leads to significant decrease of the modulus E (with about 40% at g = 20%), accompanied by increase of the viscous contribution E” (Figure S1 in the Supporting Information). Theoretical description of the effect of the dilational strain on the viscoelasticity of protein layers was attempted in [33].

3.3. Formation of a Secondary Layer

Once a BLG monolayer reaches the maximum compression K_m2 (θ→1), the observed further increase of the adsorption goes on through accumulation of protein material (presumably by hydrophobic interactions) as a discrete secondary sublayer adjacent to the primary monolayer [24]. The onset of this process is detected by the model as a splitting point between the m1- and m2-fits for the relevant dependencies. The coordinates of such splitting points in those dependencies are dictated by the characteristic molar area ω_m2 at monolayer saturation, as defined in the beginning of Section 3. For the different pH values studied here, we obtained ${\omega }_{\mathrm{m}2}^{\mathrm{pH}3}\approx {\omega }_{\mathrm{m}2}^{\mathrm{pH}7}<{\omega }_{\mathrm{m}2}^{\mathrm{pH}5}$ (see Table 1), which give the following “equivalent hard-core” radii: $R_{\mathrm{ehc},\Pi =0.1}^{\mathrm{pH}3}$≈ $R_{\mathrm{ehc},\Pi =0.1}^{\mathrm{pH}7}$ ≈ 2.6 nm and $R_{\mathrm{ehc},\Pi =0.1}^{\mathrm{pH}5}$ ≈ 2.7 nm. From these values, one gets the thicknesses of the monolayers saturated by oblate ellipsoids as ≈1.6 nm and ≈1.5 nm, respectively, the values of which are in very good agreement with previous neutron reflectometry results [24,82,83,84].

The slightly higher value for ${\omega }_{\mathrm{m}2}^{\mathrm{pH}5}$ is somehow surprising. However, the monolayers thicknesses suggest that adsorbing dimers either align in side-on stretched configurations at the interface or disintegrate into monomers (which also stretch) upon adsorption. The latter scenario seems probable, having in mind that the affinity of the BLG monomer to the water/air interface was found higher than that of the dimer [49,96]. Moreover, the pH-dependent dissociation constants of BLG dimers were found to decrease when approaching the isoelectric point [97], i.e., the dimer becomes more stable and eventually less prone to dissociate upon adsorption at an interface. It is possible that, under saturation conditions at $c_{\mathrm{m}2}^{\mathrm{pH}5}$ ≈ 3 × 10⁻⁷ M, the monolayer has gained some degree of heterogeneity near the onset of a secondary layer formation. The reasons for such disruption in the homogeneous 2-d monolayer’s architecture could be either nonperfect alignment of the adsorbed BLG entities (monomers, dimers or mixtures of them) or effects by newly arriving dimers (prolate ellipsoids [81]), which might be not able to get incorporated into the compressed monolayer but may only partially penetrate it, leading this way to a “pseudo saturation.” On the other hand, the monolayers at pH 3 and pH 7 exhibit similar adsorption behaviors, although the dimer fraction at $c_{\mathrm{m}2}^{\mathrm{pH}7}$ (ca. 65% [92]) is much higher than that at $c_{\mathrm{m}2}^{\mathrm{pH}3}$ (<5% [93]). There seems to be some small mismatch between theory and experiment, since neutron reflectometry measurements at c = 1 × 10⁻⁵ M (pH 7) revealed a monolayer surface structure [24], whereas the present computations revealed $c_{\mathrm{m}2}^{\mathrm{pH}7}$ ≈ 5 × 10⁻⁶ M, a concentration which, is after all, only twice lower. For pH 3, the model predictions revealed $c_{\mathrm{m}2}^{\mathrm{pH}3}$ ≈ 3 × 10⁻⁵ M, which agrees with the experimental results [24].

The propensity of BLG to build up a (heterogeneous) bilayer structure at the water/air interface is quantified by the adsorption constant b₂ of the secondary layer, which increases in the order pH 3 > pH 7 > pH 5 as it becomes extremely high at pH 5 (see Table 1). The physics behind this behavior can be explained in analogy to conventional adsorption to interfaces, but, in this case, the protein molecules adsorb on a proteinaceous surface instead of a liquid interface. At pH 3 and pH 7, the plane of the primary BLG monolayer is charged [40], which gives rise to an electrostatic barrier for further adsorption of molecules of the same type. Such adsorption barrier should be highest at pH 3, where the electric field at the interface is the strongest [40], and, accordingly, it is negligible at pH 5.

The experimental and the theoretical dependencies E(c)_f,g, E(Π)_f,g and E(Γ)_f,g for pH 3 and pH 7 in Figure 5, Figure 6 and Figure 7 show a relatively shallow but noticeable local maximum, which appears merely at c_m2, Π_m2 ≈ 20 mN/m and Γ_m2 ≈ 1.5 mg/m², respectively, and corresponds to the splitting point of the m1- and m2-fits. We should mention here that, for both pH values, slightly higher apparent diffusion coefficients D_app were used in the m2-fits than in the m1-fits (see Table 2). The experimental data beyond the maximum of E are well described only by m2-fits (Figure 5, Figure 6 and Figure 7). The slight decrease of E after the maximum suggests that the secondary layer somehow disrupts the elastic behavior of the primary monolayer, most probably because of loosening of the protein network due to disturbed lateral cohesion and/or of the appearance of relaxation processes originating from the looser structure of the secondary layer.

The situation at pH 5 is different, mostly due to the observed early splitting point of the m1- and m2-fits in Figure 5, Figure 6 and Figure 7, which occurs merely around c*, Π* and Γ*, respectively, i.e., well before the onset of secondary layer formation (c_m2,Π_m2,Γ_m2). For the m2-fit at pH 5, a broad plateau region is observed beyond the splitting point, as the model curves fall below the observed maximum of the experimental data (assigned to a saturated monolayer and well reflected by the m1-fit). For the following discussion, we recall previous data on the dynamic dilational modulus E(Π(t))_f,g [44]. Those data were measured in the nonlinear viscoelasticity regime (g ≈ 7% > g_tr) and, in Figure S5 in the Supporting Information, are shown new data measured in the linear viscoelasticity regime (g ≈ 2.7%). However, the results do not differ significantly, showing a local drop in the E(Π(t))_f,g curves at higher BLG concentrations (ca. c > 5 × 10⁻⁷ M) that deviates from the master curve observed at lower concentrations. This behavior can be attributed to the appearance of a transient step in the formation of the BLG bilayer due to the fast adsorption of dimers [24]. On the other hand, the Π(Γ) equations of state obtained from both m1- and m2-fits (Figure 2) are practically equal. This shows that BLG monolayers possess similar adsorption behaviors but can differ in their rheological characteristics in the postcritical region, depending on the formation conditions: monolayers formed at lower BLG concentrations c (slower adsorption kinetics) are more elastic than monolayers formed at higher c (faster adsorption kinetics). To explain this finding, at the moment, we could only speculate that, in the first case, the adsorbing dimers have enough time to eventually disintegrate into monomers and well arrange at the interface, whereas, in the latter case, dimers do not disintegrate into monomers but only arrange in side-on configuration at the interface. In both cases, it seems the monolayers’ thicknesses are comparable within the achievable neutron reflectometry measurement resolution of a few Å [24].

4. Conclusions

In this work, we applied an approach to compare a thermodynamic model simultaneously to the experimental isotherms of the surface pressure, Π(c), the surface excess, Γ(c), and the surface dilation viscoelasticity modulus, E(c)_f,g, and, in turn, to the equation of state in terms of various dependencies of Π, E₀ or E_f,g on Γ, ω or Π. Based on the obtained results we propose a scenario of the pH-dependent behavior of BLG adsorption layers at the water/air interface. Due to the fitting protocol of the model to all kinds of available experimental data sets, the resulting parameter values are most accurate.

The provided complex analysis provides a rich set of information, which is inaccessible by investigating only a limited number of protein bulk concentrations, as done in many studies. However, the lowest surface activity at pH 3 (compared to pH 5 and pH 7) is not a new finding [42,70]. We give here just a demonstrative example, by comparing the results for BLG at two arbitrary protein concentrations: c = 7 × 10⁻⁹ and 5 × 10⁻⁶ M. The first one is just the characteristic concentration c_{Π = 0.1} at pH 3, but, at the same time, it is already the critical concentration c* at pH 7 (${\Pi }_{\mathrm{pH}7}^{*}$ = 15.1 mN/m). For pH 5 and pH 7, the adsorption Γ (at c = 7 × 10⁻⁹ M) is comparable, but Π and E are much lower for pH 5, and the yield strain g_tr at the transition to a nonlinear viscoelasticity regime is apparently larger (see Figure S1 in the Supporting Information). The second concentration is assigned to the onset of double layer formation $c_{\mathrm{m}2}^{\mathrm{pH}7}$ at pH 7, which is almost one order of magnitude lower than the one $c_{\mathrm{m}2}^{\mathrm{pH}3}$ at pH 3; at the same time, a significant secondary layer is already accumulated at pH 5.

The theoretical results for the precritical region (c < c*) of a monolayer are in very good agreement with the experimental data. This made us confident to make adequate conclusions regarding the effect of pH on the behavior of weakly compressed BLG monolayers. The capacity of the model to account for the nonidealities of enthalpy and entropy via the interaction parameter a and a discrete spectrum of different protein adsorption states ω_j, respectively, allowed for obtaining precise computations of the variation of the molar area ω with the compression of the monolayer. This, in turn, yielded the characteristic values ω_Π=0.1 and ω* at the onset of measurable surface pressure values (near-zero compression at the “gaseous” to “liquid expanded” phase transition) and at the critical surface pressure Π* (weak compression), respectively. We interpret these values in terms of unfolding of BLG globules upon adsorption and their flattening at the water/air interface [15,16,24,84]. The results revealed a decrease of ω_Π=0.1 and ω*, i.e., reduced propensity to interfacial unfolding in the order pH 3 > pH 7 > pH 5, which excellently correlates with the decrease of the molecular net charge in a linear fashion [66]. Such sequence can be related to pH-dependent (charge-dependent) features of the tertiary structure of BLG globules in solution—the globular structure is most flexible at pH 3, most rigid at pH 5 and moderately flexible at pH 7 (near the Tanford transition) [78,79]. The results also revealed that a lower degree of interfacial unfolding of BLG globules leads to a higher limiting elasticity E₀ and, respectively, to a higher dilational viscoelasticity modulus E (see, for instance, Figure 5). At the same time, the reduced modulus E/E₀ is relatively high, as estimated at the oscillation frequency of f = 0.1 Hz and by using a diffusion coefficient equal to the intrinsic bulk diffusivity of BLG in aqueous solutions. This suggests that, at the given frequency, apparently, protein diffusion dominates the highly elastic stress response of the weakly compressed monolayer to dilational strains in the linear viscoelasticity regime.

The following regions of strong compression (accompanied by surface aggregation) of the monolayer (c* < c < c_m2), and of the development of a secondary layer (c > c _m2) are described by a different theory for the equation of state, which is based on a semi-empirical relation (Equation (9)) [27]. In addition, to describe the adsorption behavior of the heterogeneous surface layer (buildup by a 2-D monolayer and a discrete secondary sublayer), the adsorption of the secondary layer is expressed by a fairly crude (Langmuir-type) approximation (Equation (10)) [20,55]. The application of the latter equation results in a quite good overlap of the model predictions and the experimental adsorption isotherm Γ(c). However, our approach to detect the onset of the secondary layer formation is based on the detection of the characteristic value Γ_m2 at the splitting point of the m1- and m2-fits; hence, a more realistic performance of the model in respect to the m2-fits requires a rigorous derivation of the total adsorption Γ_Σ. Nevertheless, the obtained Γ_m2-values can be considered as relative, and they clearly reveal that the propensity of BLG to build up a heterogeneous (bilayer) surface layer structure increases with decrease of both the net charge and the surface unfolding ability of BLG. The need of further refinements of the model in respect to Equations (9) and (10) emerges from the observed significant discrepancy between theory and experiment in the bilayer region (c > c_m2) of the surface pressure isotherm and the equation of state (e.g., Figure 1a, Figure 2, Figure 5, and Figure 6).