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

: 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.


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 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].

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 Na 2 HPO 4 /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.

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) = γ 0 − γ(t) (γ 0 = 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 ≡ ∆A/A 0 × 100 = 2.7 ± 0.3%, where ∆A [mm 2 ] is the amplitude of change of the undisturbed bubble area A 0 = 20 or 25 mm 2 . 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.

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: ω 1 , 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 = ω 1 + (j − 1)ω 0 , where ω 0 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: The adsorption isotherm equation for each j th adsorbed state reads: 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].
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: Note b 1 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 b 1 , which, in what follows, is denoted by b: 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 : 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: For relatively low protein concentrations (precritical region), the equation of state reads: 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 , ω 0 , ω 1 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]: 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, b 2 . A previously derived approximation for the total adsorption, Γ Σ , reads [20,55]: and hence, for a bilayer, Γ Σ instead of Γ should be used in Equation (9). The surface dilational modulus E reads: where A is the surface area. Assuming harmonic surface area oscillations, the frequencydependent dilational modulus can be expressed as a complex number [56]: where = 2π f is the angular frequency (f is the frequency in [Hz]); E ≡ |E| cos φ is the real part accounting for the elastic contribution (conservation of energy) and E ≡ |E| sin φ 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]: where E 0 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 ω 0 << ω (which holds for proteins) [36]: 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]: In some cases, it is convenient to introduce the reduced modulus, E/E 0 (0 < E/E 0 ≤ 1), which can be used as a measure for deviations from an ideal elastic behavior (E/E 0 = 1).

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 Πand E-data were measured by a single method (bubble shape analysis), while the Γand 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: 1.
the dependency of the dilational modulus on the protein concentration, E(c) f,g .
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.  [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. 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, b 2 , 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 ω 1 , 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 ω 1 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 ω 1 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 ω 0 was allowed to vary between 2.0 × 10 5 and 4.5 × 10 5 m 2 /mol [21,28,30,32,33], which corresponds to areas per increment between ≈0.33 nm 2 and ≈0.75 nm 2 , respectively. We could mention here that Joos [15] used a limiting value of the solvent area per molecule of 0.10 nm 2 (≈6 × 10 4 m 2 /mol); however, in our analysis we refrain from using such low values for ω 0 . 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 ω 0 variation on the adsorption isotherm Γ(c) is less pronounced compared to that on the Π(c), Π(Γ) and E 0 (Π) dependencies ( Figure S2d in the Supporting Information). For the latter one, optimizing ω 0 was essential for description of the experimental E(Π) data. The best results were found using the lower values for ω 0 so that this parameter was fixed at 2.0 × 10 5 m 2 /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 × 10 5 m 2 /mol or ≈0.30 nm 2 /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  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 b 2 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 b 2 > 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 ω 0 ≈ 0.33 nm 2 , and the error in ω m2 was set as ±0.2 nm 2 .
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 (Figures 1 and 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.
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 ( Figures  1 and 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 > cm2, i.e., beyond the onset of a secondary layer formation, while the experimental Π-values tend to level off.  [24]). (c) Protein concentration dependencies of the area per molecule ω(c) (experimental data from [24]); note that an ω(c) dependency describes only the primary monolayer, and it is independent of the formation of further sublayers (see details in the text). Symbols are experimental data (exp), and lines are best fits for one-layer (m1) and two-layer (m2) models; asterisks indicate Π*, Γ* and ω* at c*, and arrows in (c) indicate the onset of double layer formation at cm2.  [24]). (c) Protein concentration dependencies of the area per molecule ω(c) (experimental data from [24]); note that an ω(c) dependency describes only the primary monolayer, and it is independent of the formation of further sublayers (see details in the text). Symbols are experimental data (exp), and lines are best fits for one-layer (m1) and two-layer (m2) models; asterisks indicate Π*, Γ* and ω* at c*, and arrows in (c) indicate the onset of double layer formation at c m2 .
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.
The lower surface activity of BLG at pH 3 than at pH 7 is well reflected by the model via the smaller adsorption constant, bj, while the parameters α and a are kept comparatively close. In this line, the intermediate values of cΠ=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.

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 pHdependent 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 Γ pH3 Π=0.1 ; correspondingly, the molar area decreases in the same order, ω (Figures 1 and 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 (R 2 > 0.99): ω Π=0.1 = B|Z|+ 27.0, with B ≈ 0.4 [nm 2 /molecule.e]. For pH 5, ω Π=0.1 = 27.9 nm 2 /molecule is very close to the area at zero net charge (27.0 nm 2 /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 hardcore" radius, R ehc [nm] = √ ω/π, (ω [nm 2 /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 pH3 ehc,Π=0.1 ≈ 3.5 nm, R pH7 ehc,Π=0.1 ≈ 3.2 nm and R pH5 ehc,Π=0.1 ≈ 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 nm 3 [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 (Figures 1a, 2 and 3a). For Γ* and ω*, Figure 2 reveals Γ * pH3 < Γ * pH7 < Γ * pH5 and ω * pH3 > ω * pH7 > ω * pH5 , 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.
In the concentration range cΠ=0.1-c*, Π and θ steeply increase, while the average molar area ω decreases only slightly (Figures 1a, 2 and 3a). For Γ* and ω*, Figure 2 reveals Γ * < Γ * < Γ * and * > * > * , which variations are equivalent to those of ΓΠ=0.1 and ωΠ=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 Πx, the monolayer compression, Kx, relative to the origin ωΠ=0.1 (representing near-zero compression) can be estimated by a simple relation between ωx and ωΠ=0.1: [%] = 1 .

100. At ω*
we get * > * > * (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 apH3 < apH7 < apH5, 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 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 pHdependent adsorbed amount Γ* and degree of surface-induced unfolding (represented by ω*) of the adsorbed BLG globules. Again, we found an excellent linear relation (R 2 > 0.99): ω*= B|Z|+ 25.7, with B ≈ 0.33 [nm 2 /molecule.e]. Coincidently or not, this value of the prefactor B is virtually equal to the area increment ω 0 , 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 E 0 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 * pH3 ≈ 8%, as compared to the cases of pH 7 (≈7%) and pH 5 (≈5%).
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 (R 2 > 0.99): ω*= B|Z|+ 25.7, with B ≈ 0.33 [nm 2 /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 > cm2 (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 * ≈ 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] (R 2 > 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 Figures 2a and 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 The model computations of the dependencies E 0 (Π) and E 0 (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 E 0 (Π) dependencies are linear [15,62] (R 2 > 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 E 0 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 E 0 (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 Figures 2a and 5 show a strong effect of pH on E 0 *, 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].
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 caxis 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.   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 Figure 5. BLG adsorption layers at the water/air interface at different bulk pH. Dependencies of the measured dilational complex viscoelasticity modulus E and the computed high-frequency limiting elasticity E 0 on (a) surface pressure, E 0 (Π), E(Π) f,g , and (b) surface excess, E 0 (Γ), E(Γ) f,g ; f = 0.1 Hz, g ≈ 2.7%. Symbols are experimental data for E (exp), lines are best fits for the two-layer (m2) model, asterisks indicate Π* and Γ*, arrows indicate the onset of secondary layer formation at Π m2 and Γ m2 .

Experimental and computed dependencies E(Π)f,g and E(Γ)f,g are shown individually for each pH in
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. [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 caxis 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 E 0 (Π) and E 0 (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 m 2 /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 E 0 up to E 0 *, while, for pH 3, a small but noticeable deviation appears beyond surface pressure of about 10 mN/m. by Equation (13) were performed using a diffusion coefficient of Dintr = 1×10 -10 m 2 /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. Figure 7. BLG adsorption layers at the water/air interface at (a1,a2) pH 3, (b1,b2) pH 7 and (c1,c2) pH 5. Dependencies of the dilational complex viscoelasticity modulus E and the high-frequency limiting elasticity E0 on (a1,b1,c1) surface pressure, E0(Π) and E(Π)f,g and (a2,b2,c2) surface excess, E0(Γ) and E(Γ)f,g; f = 0.1 Hz, g ≈ 2.7%. Symbols are experimental data for E (exp), lines are best fits for one-layer (m1) and two-layer (m2) models (note the common legend for each row), asterisks indicate Π* and Γ*, arrows indicate the onset of secondary layer formation at Πm2 and Γm2, Dintr = 1×10 −10 m 2 /s, Dapp is the apparent diffusion coefficient with values listed in Table 2.  Figure 7. BLG adsorption layers at the water/air interface at (a1,a2) pH 3, (b1,b2) pH 7 and (c1,c2) pH 5. Dependencies of the dilational complex viscoelasticity modulus E and the high-frequency limiting elasticity E 0 on (a1,b1,c1) surface pressure, E 0 (Π) and E(Π) f,g and (a2,b2,c2) surface excess, E 0 (Γ) and E(Γ) f,g ; f = 0.1 Hz, g ≈ 2.7%. Symbols are experimental data for E (exp), lines are best fits for one-layer (m1) and two-layer (m2) models (note the common legend for each row), asterisks indicate Π* and Γ*, arrows indicate the onset of secondary layer formation at Π m2 and Γ m2 , D intr = 1×10 −10 m 2 /s, D app is the apparent diffusion coefficient with values listed in Table 2. As mentioned above, quantification of the deviation from a purely elastic behavior of a surface layer is conveniently given by the reduced modulus E/E 0 (ideal elasticity at E/E 0 = 1). Calculations for f = 0.1 Hz and Π* (for each pH) yielded values for E*/E 0 * 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*/E 0 * 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*/E 0 * = 0.97 (as for pH 5) by using an "apparent" diffusion coefficient of D app = 6 × 10 −12 m 2 /s.

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 pH3 m2 > K pH7 m2 > K pH5 m2 (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 * pH7 (≈7 × 10 −9 M) and ca. 1 × 10 −7 M (including c * pH5 ), 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 pH3 m2 , 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 E 0 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) 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/E 0 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].

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 ω pH3 m2 ≈ ω pH7 m2 < ω pH5 m2 (see Table 1), which give the following "equivalent hard-core" radii: R pH3 ehc,Π=0.1 ≈ R pH7 ehc,Π=0.1 ≈ 2.6 nm and R pH5 ehc,Π=0.1 ≈ 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 ω pH5 m2 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 pH5 m2 ≈ 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 Colloids Interfaces 2021, 5, 14 20 of 26 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 pH7 m2 (ca. 65% [92]) is much higher than that at c pH3 m2 (<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 pH7 m2 ≈ 5 × 10 −6 M, a concentration which, is after all, only twice lower. For pH 3, the model predictions revealed c pH3 m2 ≈ 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 b 2 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 Figures 5-7 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/m 2 , 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 ( Figures 5-7). 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 Figures 5-7, 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].

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, E 0 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 (Π * pH7 = 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 pH7 m2 at pH 7, which is almost one order of magnitude lower than the one c pH3 m2 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 E 0 and, respectively, to a higher dilational viscoelasticity modulus E (see, for instance, Figure  5). At the same time, the reduced modulus E/E 0 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 semiempirical 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 m2fits; 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., Figures 1a, 2, 5, and 6).
Supplementary Materials: The following are available online at https://www.mdpi.com/2504-537 7/5/1/14/s1, Figure S1: Dilational rheology parameters: E (complex viscoelasticity modulus) and ϕ (phase angle), and the corresponding E" (imaginary part of E) as a function of the amplitude g of oscillating area deformation at various surface pressures Π and at constant oscillation frequency of f = 0.1 Hz; Figure S2: Exemplary screenshots of the interactive software for fitting the theoretical model to experimental data (pH 7). The symbols ( ) are experimental data and the lines are best m2-fits; where the model predictions are presented by green and red lines, the colors indicate the pre-critical (green) and the post-critical (red) ranges divided by the critical parameters Π*, ω*, Γ* and E_0ˆ*.; Figure S3. Model simulations of ω(Γ) dependencies by one-layer (m1) and two-layer (m2) fits. (top) The horizontal solid lines are the input (boundary) values for ω1 and ωn in the calculation procedure. The horizontal and vertical dotted lines indicate the critical points at coordinates (ω*,Γ*), and the splitting points at coordinates (ωm2,Γm2) at full saturation of the primary monolayer (θ→1). (bottom) a zoom-in portion of the ω(Γ) dependencies illustrating the determination of Γm2; ω0 ≈ 0.33 nm2 is the area increment used in the calculation procedures for all pH values; Figure S4. Funding: This work was financially supported by a DFG-AiF cluster project on "Protein Foams" Mi418/20-1 (DFG-199448917).
Data Availability Statement: Not applicable.