Adsorption of proteins at the aqueous solution/alkane interface: Co-adsorption of protein and alkane R. Miller, E.V. Aksenenko, Igor I. Zinkovych, V.B. Fainerman PII: DOI: Reference:

S0001-8686(15)00015-9 doi: 10.1016/j.cis.2015.01.004 CIS 1510

To appear in:

Advances in Colloid and Interface Science

Please cite this article as: Miller R, Aksenenko EV, Zinkovych Igor I., Fainerman VB, Adsorption of proteins at the aqueous solution/alkane interface: Coadsorption of protein and alkane, Advances in Colloid and Interface Science (2015), doi: 10.1016/j.cis.2015.01.004

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ACCEPTED MANUSCRIPT Adsorption of proteins at the aqueous solution/alkane interface: co-adsorption of protein and alkane

Max-Planck-Institut für Kolloid- und Grenzflächenforschung, 14424 Potsdam, Germany 2

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R. Miller1, E.V. Aksenenko2, Igor I. Zinkovych3 and V.B. Fainerman3*

Institute of Colloid Chemistry and Chemistry of Water, 03680 Kiev, Ukraine Donetsk Medical University, 83003 Donetsk, Ukraine

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Abstract

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The equations of state, adsorption isotherms, functions of the distribution of protein molecules in liquid interfacial layers with respect to molar area and the equations for their

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viscoelastic behavior are presented. This theory was used to determine the adsorption characteristics of -casein and -lactoglobulin at water/oil interfaces. The experimental results are shown to be describable quite adequately by the proposed theory with consistent model

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parameters. The data analysis demonstrated that the -casein molecule adsorbed at equilibrium

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conditions is more unfolded as compared with dynamic conditions and this fact causes the significant increase of the adsorption equilibrium constant. The theory assumes the adsorption of

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protein molecules from the aqueous solution and a competitive adsorption of alkane molecules from the alkane phase. The comparison of the experimental equilibrium interfacial tension isotherms for -lactoglobulin at the solution/hexane interface with data calculated using the

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proposed theoretical model demonstrates that the assumption of a competitive adsorption is essential, and the influence of the hexane molecules on the shape of the adsorption isotherm does in fact exist.

Keywords: Whey protein adsorption, water/hexane interface, drop profile analysis tensiometry, equation of state, interfacial dilational viscoelasticity, competitive adsorption of protein and hexane molecules

* corresponding author

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ACCEPTED MANUSCRIPT 1.

Introduction The practical significance of protein adsorption at fluid interfaces stimulated the

development of new experiments and theoretical models to describe the equilibrium and

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dynamic behaviour of their adsorption layers. Various theoretical models for the description of

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the adsorption of macromolecules have been proposed so far [1-22], some of them used a phenomenological approach based on surface thermodynamics. For proteins solutions scaling

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theory and thermodynamic modelling, more simple as compared to statistical models, are to be explored as possibilities for formulating respective equations of state and adsorption isotherms [22]. Joos [14] showed that the degree of surface denaturation decreases with increasing surface

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pressure. Joos and Serrien [19] derived a relationship between the adsorption of proteins possessing two modifications with different partial molar area and showed that the fraction of

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adsorbed molecules with minimal surface area demand increases with increasing surface pressure. The proposed concept was further developed for an arbitrary number of different conformations (states) of protein molecules at the surface [20-22]. It should be noted that for

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some proteins the number of different states in the surface layer is rather high, up to several

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hundreds. The model proposed in [22] is the most rigorous one from the thermodynamic point of view, because it is based on the comparison of the chemical potentials of protein and solvent in

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the solution bulk and in the surface layer, and also assumes the non-ideality of enthalpy and entropy of the protein molecules in the surface layer. The thermodynamic model developed in [22] was further generalised in [23] onto the post-critical region, where the surface (or interfacial) tension is almost independent of the protein concentration. In [24,25] the model

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proposed in [22] was extended to describe the adsorption from mixed protein/surfactant solutions, for both non-ionic and ionic surfactant. The equations were derived to calculate the surface or interfacial tension, adsorption, average molecular area and dilation rheological characteristics of the protein/surfactant mixed layer (or protein+surfactant complex which is formed in the solution of ionic surfactant). In [26] a new physical model for a competitive adsorption of alkane and surfactant molecules at the water/alkane interface was proposed. Thus, the oil phase does not only provide a hydrophobic environment for the alkyl chains adsorbed at the interface, but the alkane molecules themselves compete for space in the interfacial layer. In the present article this model is refined and applied to describe the adsorption of proteins from its aqueous solution at the solution/alkane interface. Protein-stabilized food emulsions and foams constitute the most important class of food colloids [27,28]. The functional properties of proteins in food are related to their structural and 2

ACCEPTED MANUSCRIPT other

physicochemical

characteristics.

Thus,

a

fundamental

understanding

of

the

physicochemical properties of proteins is important. The adsorption behaviour and interfacial rheological properties of proteins at fluid interfaces have been extensively investigated. Foams are important in many modern technologies and are investigated for decades both experimentally

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and theoretically under various aspects. The methods and techniques supported by modern

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instruments help obtaining precise experimental results. The description of the dynamic and

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equilibrium behaviour of adsorbed protein layers requires finding the adsorption isotherm and the corresponding equation of state. There are attempts for the theoretical description of the adsorption dynamics of globular proteins, and particularly of -lactoglobulin (BLG) [29, 30]. In [31] experimental data are reported for the simultaneous adsorption isotherm and the dilational

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visco-elasticity of -сasein (BCS) under dynamic and equilibrium conditions. In the present study we will use the different results for BLG and BCS adsorption layers at water/oil interfaces

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published in literature [32-41]. In the present work these results are processed using most recently proposed thermodynamic models. Also, the analysis of these systems assuming the simultaneous (competitive) adsorption of protein and alkane at the protein solution/alkane

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2.

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interface is presented. Experimental results

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At first let us consider the experimental results obtained for two proteins: β-casein (BCS) and β-lactoglobulin (BLG) at the aqueous solution/alkane interface. Fig. 1 shows the dependence of equilibrium interface pressure for aqueous BCS solutions at pH 7 at the interfaces with

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tetradecane and hexane, as reported in [32-34]. In [32] the drop profile method was used, when the protein solution drop was formed in tetradecane. In [33] the Wilhelmy plate method was employed for the solution/tetradecane interface, and in [34] hexane drops were studied in aqueous protein solution by using the drop profile method. It is seen that the experimental results obtained by different methods show essential differences. The major inconsistence between the results obtained in [32] and [34] is attributable to the losses of BCS from the drop bulk due to the BCS adsorption at the drop surface. The difference between the initial concentration of protein in the drop and its concentration after the establishment of the adsorption equilibrium is Δc = ΓA/V, where Γ is the adsorbed amount of protein, A and V are the drop area and volume, respectively [35, 36]. It can be estimated that the decrease of the BCS bulk concentration due to the adsorption from the drop bulk is much more significant than the concentration decrease due to the BCS adsorption from the solution bulk around the drop. To obtain similar values of the interfacial tension for these two experimental protocols, the protein concentration inside an 3

ACCEPTED MANUSCRIPT aqueous solution drop in oil should be more than one order of magnitude higher than in the case when the proteins adsorb from the aqueous solution around an oil drop. This conclusion is in agreement with the results obtained in [32, 34] and shown in Fig. 1. An interesting result obtained for the first time in [34] should be noted: for low BCS concentrations (10−10 to

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10−9 mol/l) the equilibrium interfacial tension virtually does not depend on the concentration.

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The interfacial pressure within this concentration range depends on the surface lifetime. Fig. 1

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also illustrates the results reported in [34] for the adsorption time 15000 s. It is seen that in the dynamic conditions, in contrast to the equilibrium conditions, for low BCS concentrations the interfacial pressure is essentially low, while for high concentrations the influence of the lifetime

Fig. 1 are discussed in the next section.

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is much weaker, because the equilibrium is attained faster. The theoretical curves shown in

Fig. 2 illustrates the dependence of the BCS dilation viscoelasticity modulus

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E  Er2  Ei2 on the interfacial pressure  at the frequency of 0.1 Hz. The experimental points reproduced from [32] and [34] correspond to the notation in Fig. 1. Also shown in Fig. 2 are the

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experimental viscoelasticity modulus values at the same oscillation frequencies as reported in [37], which were obtained in [38] with the oscillating drop method for tetradecane drops in

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aqueous BCS solutions. It is important to emphasize that these experiments were performed under dynamical conditions, with a maximum surface lifetime less than 6000 s. The red

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theoretical curve in Fig. 2 is also reproduced from [37] while the other calculated dependencies correspond to the isotherms shown in Fig. 1 and are discussed in the next section. Fig. 3 illustrates the dependence of the interfacial pressure  on the BLG solution

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concentration. The results shown here are taken from [39, 40] and correspond to aqueous protein solution drops in hexane, as well as data from [41] corresponding to experiments with tetradecane drops formed in aqueous protein solutions. The surface pressure values at the interface to hexane and tetradecane are rather close to each other and agree with the data given in [26] for alkyl trimethylammonium bromide solutions. For the sake of comparison, this Figure also shows the dependence of the equilibrium interfacial pressure for BCS taken from [34] (shown also in Fig. 1). When we compare the interface pressure isotherms of these proteins we see that the concentration range of the BCS isotherms is almost 2 orders of magnitude lower than that for BLG for the same interfacial pressure values. This could be ascribed to the fact that the BLG adsorption activity is lower than that of BCS. It is seen also that the results obtained in [39] and [41] by different methods virtually coincide. It can be estimated that for a BLG concentration of 10−7 M, the protein loss caused by its adsorption from the drop bulk is about 30 %, while for a concentration of 10−6 M it is much lower and only about 4-5 %. It remains 4

ACCEPTED MANUSCRIPT unclear why the results obtained by the two methods are quite similar in the concentration range below 10−7 M, although a shift of the BLG isotherm towards higher concentrations measured with protein solution drops could be expected. A possible explanation for this fact is the lower adsorption activity of BLG at its interface with tetradecane as compared with that at the BLG

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solution/hexane interface. It was shown in [42] that for ordinary surfactants (sorbitan

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monoesters) at the aqueous surfactant solution/alkane (pentane to dodecane) interface, the

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adsorption activity of the surfactant becomes higher with the decrease of the alkane molecular weight. A similar dependence could also exist for BLG.

The dependence of the viscoelasticity modulus |E| of BLG solutions on the surface pressure  at a surface area oscillation frequency of 0.1 Hz is presented in Fig. 4. The data are

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similar to those in Fig. 2 and represent experimental values obtained by measurements of with oscillating tetradecane drops in aqueous protein solutions [37]. The red theoretical curve is also

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reproduced from [37]. In Fig. 4 also the results from [41] obtained by experiments similar to those in [37] are shown. The dependences of the viscoelasticity modulus on surface pressure

Discussion of results

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3.

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with a maximum at 20-23 mN/m are quite similar in both publications.

3.1. Equilibrium adsorption properties of protein solutions

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Taking into account the non-ideality of enthalpy and entropy, and assuming that the protein molecules can absorb in n states of different molar areas (with relative fractions of these adsorption layer states depending on the surface tension [19] or adsorption [22]) varying between



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a maximum max and minimum area min, the following equation of state was proposed in [22]:

0  ln(1   )   (1  0 / )  a 2 . RT

(1)

Here  is the surface pressure, R is the gas law constant, T is the temperature, a is the intermolecular interaction parameter, 0 is the molar area of the solvent or the area occupied by one segment of the protein molecule (area increment),   i1 i is the total adsorption of n

proteins in all n states (1  i  n),     i1 ii is the total surface coverage by protein n

molecules,  is the average molar area of the adsorbed protein, and i = 1 + (i − 1)0 is the molar area in state i; assuming 1 = min, max = 1 + (n  1)0. In [43] it was assumed that the surface activity of reorientable surfactant molecules increases with increasing molar area, according to a power law with the constant exponent α. For α = 0 and very small surface pressure (or adsorption) the fractions of states with different molar 5

ACCEPTED MANUSCRIPT area are almost equal to each other [18, 22]. For α > 0 the fraction of the surface covered by molecules with larger molar areas is larger, independent of Π. In [23] it was assumed for protein solutions that α > 0, which supports the adsorption of protein molecules in states with larger molar areas. In this case the increase of the equilibrium adsorption constant in the jth state (bj) as

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compared to that in the 1st state (b1) is determined by the relation: 

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j  b j    b1  1 

(2)

Therefore, the distribution of states of protein molecule with different molecular areas depends on two factors: the coefficient α and the surface pressure  (or total adsorption ); this fact is

j  /  exp  2a ( j / )  (1   )

bj c 

(3)

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j

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accounted for by the adsorption isotherm for any jth state:

where c is the protein bulk concentration. Combined with Eq. (2) we obtain [22, 23]:  j /

( j / 1 ) (1   )



exp  2a ( j / )

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j



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b1c 

(4)

The adsorption constant b for the protein molecule as a whole is: 

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  b  b j  b1  j  j 1 j 1  1  n

n

(5)

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Note that for α = 0 b = n×b1. For α > 0, to compare the parameters b obtained for different systems this value should be multiplied by the coefficient: 

1 n   k    j  n j 1  1 

(6)

Combining Eq. (4) with the above expressions for the total adsorption of proteins and total surface coverage by protein molecules one obtains a distribution function of adsorptions over all states of the protein molecules: 

 j 1   1  j     1     exp2a j     j    1   n i 1  i    1  .   exp 2a i     1         i 1  1 

(7) 6

ACCEPTED MANUSCRIPT With the set of Eqs. (1), (4) and (7) the evolution of the protein adsorption states with increasing total adsorption can be described, which reflects many details known from experiments. The model proposes that with increasing total adsorption , the number of adsorbed protein molecules occupying larger areas progressively decreases, while those requiring smaller

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areas at the interface increases.

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With increasing protein concentration c, the formation of a secondary (or multiple)

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adsorption layer has to be considered. The isotherm equation for such a multilayer adsorption can be derived assuming that the coverage of the second and subsequent layers is proportional to the adsorption equilibrium constant b2 and the coverage of the previous layers [22]. This includes also the assumption that the formation of these layers does not influence the surface pressure. On

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the basis of a Langmuir type isotherm for multiple adsorption layers, a rough approximation for the total adsorption P in the first, second and subsequent layers, in total m, can be obtained [22]: i 1

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 bc  P   2  i 1  1  b2 c  . m

(8)

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The Eq. (8) implies that the adsorption in the first layer Γ is identical to that given by Eqs. (1),

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(4) and (7). This approximation is rather crude, as it ignores the non-ideality of enthalpy and entropy of the mixed surface layer, however, the parameter b2 in Eq. (8) takes these effects

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partially into account.

Above a certain protein concentration c*, the surface tension only slightly increases, while the adsorption often exhibits a strong increase. This critical bulk concentration refers to a

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critical adsorption * and surface pressure Π*, which was explained in [22, 44] by a condensation (aggregation) of the protein molecules in the surface layer or by the formation of a secondary layer adjacent to the surface. Both effects change the average molar area of adsorbed molecules. Approximate equations of state and adsorption isotherm for such surface layers were presented in [22]. In [23] it was assumed that in the post-critical range P > *, monomers and aggregates are independent kinetic units. Hence, the following approximation for the surface pressure was used:

 1 P   *      * 1   na  *  ,

(9)

with na being the aggregation number. For na = 1, i.e. in absence of any aggregates, the model almost coincides with Eqs. (1), (4), (7) and (8). With increasing na the changes in surface pressure calculated from Eq. (9) become progressively less significant. 7

ACCEPTED MANUSCRIPT 3.2. Surface dilatational rheology of adsorbed layers The surface dilational visco-elasticity can be presented as a complex quantity and is defined as the increase in surface tension  as response to a small relative surface area

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increase A:

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E  d d ln A  Er  iEi

(10)

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where Er and Ei are its real and imaginary constituents, respectively. An expression for the diffusion controlled exchange of proteins caused by harmonic oscillations of the surface area was

the bubble or an oil drop was obtained: 1

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D dc  1  r  E  E0 1  i r d   ,

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derived by Joos [45]. In particular, for the adsorption of proteins from solution at the surface of

(11)

and for the adsorption from inside a drop on its surface, which is immersed in air or oil: 1

Here

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E0  d d ln  ,

(12)

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D dc  r cothr  1 , E  E0 1  i r d  

(13)

is the limiting elasticity, D is the diffusion coefficient of the protein in the solution,  = 2f is

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the angular frequency of the surface area oscillations, 2 = i/D, and r is the radius of curvature of the interface. For a planar interface (r  ) Eqs. (11) and (12) transform into the classical expression derived in [46, 47]. Eq. (13) applies for protein solution below the critical point; here the Π and Γ values are calculated using Eqs. (1), (4), (7) and (8). Adsorption layer in the postcritical concentration range should be regarded to as composite layers [48], for which the limiting elasticity E0 is given by:

E0  E0* P  *

(14)

Here E*0 refers to the values in the critical point calculated from Eq. (13), and P is the total adsorption (monolayer and multilayer) above the critical point. This E0 value in the post-critical range is used to calculate the elasticity and viscosity via Eqs. (12) and (13).

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ACCEPTED MANUSCRIPT 3.3. Comparison between theory and experiments Let us at first consider the theoretical isotherms for fitting experimental data for BCS as shown in Figs. 1 and 2. The dependencies of surface pressure, adsorbed amounts, average molar area and rheological characteristics on the protein concentration c were calculated via the model

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equations above with the model parameters summarized in Table 1. Here the columns which

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refer to BCS are labelled according to the curve numbers in Figs. 1 and 2, and the rightmost

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column lists the parameters used to calculate the curves for BLG as shown in Figs. 3 and 4.

Table 1. Values for model parameters used to calculate the theoretical curves 1−5 for BCS in

2a 0.0 0.5 3.5 3.5 12.0 85.2 1

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1b 0.6 1.7 1.8 3.5 7.0 35 31.5 9.9 90 2

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BCS 1a 0.8 1.6 4.0 5.2 12.0 2461 1

2b 0.7 1.0 1.5 3.9 8.0 30 31.2 3.6 33 2

BLG 3 0.2 0.5 3.8 4.0 12.0 30 31.5 643 12 3

4 0.56 1.7 1.8 4.0 8.0 30 31.5 48 2 2

5 0.6 0.0 1.4 3.8 7.0 20 21.7 2.04 1.0 2

0.0 1.8 3.5 5.8 1.4 15 27.5 5.82 0.61 2

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Parameters\curves a  105 m2/mol 1 / 106 m2/mol max / 107 m2/mol na * / mN/m b / 105 m3/mol b2 / 102 m3/mol m

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Figs. 1 and 2, and the theoretical curves for BLG in Figs. 3 and 4.

It turned out to be impossible to obtain a sufficiently good fit for the results shown by curves 1 and 2 using a single set of parameters. Therefore these results were fitted by two

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theoretical curves with a smooth transition between them: the parameters listed in the columns labelled by 1a and 2a refer to the values at low concentrations, while those labelled by 1b and 2b refer to the values at high concentrations. The transition between the curves is described by inflection points in the concentration range (2−6)×10−9 M It is seen from Fig. 1 that the theoretical curves 1−3 and 5 provide satisfactory fitting for all corresponding experimental data, and the values of molar areas ω1 and ωmax shown in the Table 1 agree with those published earlier in [22-25, 29-32, 39-41]. It should be noted that the ωmax values in Table 1 for BCS at the aqueous solution/alkane interface are by a factor of 2 to 2.5 higher than those for the solution/air interface, due to the larger degree of unfolding of the BCS molecules in the interfacial layer [32]. The ωmax values listed in Table 1 correspond to a layer thickness of ca. 0.5 nm.

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ACCEPTED MANUSCRIPT Essential differences can be seen between some model parameters calculated for the curves 1a and 1b and for 2a and 2b. For the ‘a’ curves the values of ωmax are by a factor of 1.5, and the values of ω0 by a factor of 2 larger than those for the ‘b’ curves. This can be attributed to the fact that at low concentrations the BCS molecules are more unfolded (i.e., the adsorbed

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segments of the molecule occupy a larger area at the interface) than at higher concentrations.

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This implies a significant increase of the adsorption equilibrium constant b of the BCS molecules. The main difference between the parameters obtained for the curves 1a and 2a is that

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for the curve 1a the α value is higher, which indicates that the fraction of unfolded states of BCS molecule in the equilibrium state is higher. Curve 3 shows a satisfactory fitting of the experimental data [32] in the entire concentrations range. The b value for curve 3 is higher than

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those for other curves, except for curve 1a. For curve 1a this value is by a factor of approximately 4 higher than that for curve 3. Curve 5 agrees satisfactorily with the experimental

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data reported in [33]. The theoretical isotherm 4 coincides with its form with the upper part of curve 1, and with its lower part with curve 5. Also, according to the experimental data from [34] this isotherm corresponds to relatively small adsorption times, ca. 3000−6000 s. These

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adsorption times correspond to measurements of the viscoelasticity modulus in [37].

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Using the parameters listed in the Table 1, the dependencies of the viscoelasticity modulus |E| on the interfacial pressure  at the surface oscillations frequency 0.1 Hz were

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calculated with a diffusion coefficient of 10−11 m2/s. These dependencies shown in Fig. 2 are in satisfactory agreement with the experimental data. The curves 1 and 2 (which, similarly to the dependencies shown in Fig. 1 were obtained by joining the curves a and b at Π = 15-17 and

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5-6 mN/m, respectively) exhibit two maxima. Also, in Fig. 2 the minimum values of the modulus in curve 1 (at Π = 15 mN/m) and curve 2 (at Π = 4 mN/m) correspond to the almost horizontal portions of the corresponding curves in Fig. 1. It is interesting to note that the values of viscoelasticity modulus calculated with the parameters for Fig. 4 are almost identical with the data obtained in [37] under dynamic conditions. As mentioned above, these data were measured at an adsorption time of less than 6000 s, and therefore the curve 4 in Fig. 1 does not exhibit a stretched portion at low concentrations. It should be noted also that the variation of the adsorption activity coefficient value b1 resulting in a shift of curve 4 in Fig. 1 along the concentration axis does not lead to any change in the shape of curve 4 in Fig. 2. This fact follows from the functional dependencies described by Eqs. (4), and (9)−(11), which means that the isotherms with different values b1 can correspond to curve 4 in Fig. 2. The theoretical curve for BLG calculated using the parameters listed in the rightmost column of Table 1 is shown in Fig. 3. The ωmax and ω0 values are almost equal to those reported 10

ACCEPTED MANUSCRIPT in [22, 29] for the solution/air interface. This fact can be attributed to the globular structure of BLG molecules. The theoretical dependence of the viscoelasticity modulus on the interfacial pressure at an oscillation frequency of 0.1 Hz (calculated for D = 10−11 m2/s) are shown in Fig. 4. The calculated curves agrees well with the experimental data reported in [41]. It can be seen that

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the theoretical results are also quite close to the data reported in [37], also obtained under

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dynamical conditions.

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Figs. 5 and 6 illustrate the calculated values of adsorption and average molar area for the BCS and BLG adsorption layers. For BCS the curves 1 and 3 correspond to the curves shown in Figs. 1 and 2. The curves 1 in Figs. 5 and 6, similarly to the dependences of the surface pressure and viscoelasticity modulus on bulk concentration, were obtained by joining the curves 1a and

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1b with the transition region in the concentration range between 10−9 and 10−8 M. In this range the adsorption and average molecular area for curves 1a and 1b are approximately equal;

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however, the values of the parameters in Table 1 indicate that in this range a change of the maximum area of the molecule and the area of its adsorbed segments occurs. It follows from the calculations that for curve 5 in Fig. 1 the adsorption value in the inflection point is ca.

D

2.0 mg/m2, and increases to 5 mg/m2 with increasing concentration. This fact agrees with the

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results obtained in [33].

The results discussed above were obtained without considering any effect of the co-

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adsorbing alkane molecules. In other words, the theory which disregards the influence of the adjacent phase on the adsorption of protein from its aqueous solution at the solution/fluid interface was used, and the influence of alkane as the neighbouring phase was accounted for only

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via the water/alkane interfacial tension. This approach is commonly used in the analysis of adsorption processes at water/oil interfaces [49-51]. However, in [26] a model was proposed which accounts for the specific influences of the adjacent phase. This new model is capable to analyse the effects of the oil phase exerts on the adsorption of ordinary surfactants at their interface with alkane due to the attraction interaction. In the following section, an extension of this model is proposed to describe the co-adsorption of protein and alkane molecules. 3.4. Co-adsorption of protein and alkane To analyse the co-adsorption data, the model for mixed protein-non-ionic surfactant adsorption layers proposed in [24] is used here. This model was employed previously for a number of systems in [52-55]. Here, we apply the model for protein molecules adsorbing in competition with alkane molecules from the oil bulk phase, i.e. the alkanes play the role of the surfactants molecules in the original model. 11

ACCEPTED MANUSCRIPT With the approximation 0  S where ωS is the molar area of the alkane the equation of state for protein/alkane mixtures reads:

0*  ln(1  P  S )  P (1  0 / P )  aPP2  aSS2  2aPSPS , RT

(15)

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

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The subcsripts S and P refer to the alkane and protein, respectively. Hence, S = SS is the n

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surface coverage by alkane molecules, S is the alkane adsorption,  P  PP  P,i P,i is the i1

partial surface coverage by protein molecules, bS is the alkane adsorption equilibrium constant, and aS is the corresponding interaction constant. The additional parameter aPS describes the

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interaction between protein and alkane molecules. Small differences between 0 and S can be accounted for by introducing the averaged molar area

0 P  S 0S . P  S

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0 

(16)

Therefore, it can be proposed that the alkane and water molecules and the segments of the

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proteins require approximately the same molar area.

P ,1 / P

    exp 2 aP P, j  P  aPSS  ,    P

S exp  2aSS  2aPSP  , 1 P S 

(17)

(18)

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bS cS 

PP, j (P, j / P,1 ) 1   P  S  P

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bP,1cP 

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The adsorption isotherms for the protein and the alkane, respectively, read:

where the parameter αP, similarly to the parameter α introduced for the individual protein above, accounts for the influence of the molecular area on the surface activity. The distribution over the states of protein molecules is determined by a relation similar to the above given Eq. (7): 

P, j

P , j P ,1  P, j  P   P,1     1   P  S  P exp2aP P P, j    P P ,1      P P n  P ,i P ,1  P,i  P,1  .  P ,i      1     exp 2 a  P  P S  P P    P i1  P,1   

(19)

The alkane molar area S and the corresponding adsorption S depend on the surface pressure  and the total surface coverage   P  S :

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ACCEPTED MANUSCRIPT S  S 0 1   ,

(20)

S  SS  SS 0 1    ,

(21)

The equations (20) and (21) take into account the intrinsic compressibility  of alkane molecules

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in surface layer [55-57]. For alkane molecules this parameter can be interpreted for example by

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changes in the tilt angle of the adsorbed molecules upon surface layer compression, accompanied

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by an increase in the surface layer thickness. For the alkanes we assume ε = 0. Usually for aqueous solutions with increasing protein adsorption the dividing surface is displaced towards the gas or oil phases, which leads to a decrease of the solvent adsorption [18, 21,22]. In the present case, two dividing surfaces should be considered: the first one which

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crosses the alkane, and the second one, which is displaced towards water or the aqueous solution bulk. The displacement of the dividing surface towards the oil or water phases is determined by

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the requirement that the total coverage of water (0), alkane (s) and protein (p) is equal to 1, i.e.

0   P   S  1. In absence of protein in the solution, the surface layer contains only water and alkane. The alkane adsorption is very fast and requires probably less than 10−4 s [58]. Therefore,

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the dividing surface moves toward the oil phase. With the increase in protein concentration the

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adosprtion of alkane decreases and the dividing surface moves towards the aqueous protein solution.

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In Fig. 7 the calculated interfacial pressure isotherms for BLG aqueous solutions at the interface with hexane is shown as example. The experimental results and the curve calculated using the model Eqs. (1), (4), (7)-(9) are reproduced from Fig. 3. The red solid line is calculated

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using the BLG and hexane co-adsorption model, Eqs. (15)-(21), with cS = 7.58 mol/dm3 (100 % hexane). The model parameter values for hexane are exactly the same as obtained in [26]: bS = 2.6 dm3/mol, ωS = 3.3·105 m2/mol and ε = 0. Also the model parameters for BLG used in the calculations were the same as those in Table 1. Only the parameter bP,1 was essentially increased (by a factor of 45) as compared with b1 that was used in the model Eqs. (1), (4), (7)-(9): from b1 = 104 m3/mol to bP,1 = 4.5∙105 m3/mol. This is required due to the displacement of hexane from the surface layer: according to [26] this should lead to an increase of b1 at least by a factor of BCSS. This factor is equal to 19.7, which is close to the actual ratio of the adsorption activity coefficients for the two models. The influence of displaced alkane on the interfacial tension caused by the contribution of the second adsorption layer (into which the alkane is displaced) was considered in [58]. This approach based on the correction to the equation of state of the mixture predicts a decrease of the adsorption equilibrium coefficient by almost one order of 13

ACCEPTED MANUSCRIPT magnitude, leading to the value which is close to that calculated from the usual model for a water/alkane interface. To improve the agreement between theory and experimental data, the protein-hexane attraction coefficient aPS was also taken into account, with the other parameters kept constant.

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The result is also shown in Fig. 7. The optimum fitting of the experimental data was obtained for

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aPS = 0.7. The curve calculated from the model Eqs. (15)-(21) agrees well with that calculated

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from the model Eqs. (1), (4), (7)-(9). In the critical point (at a concentration of ca. 6×10−7 M) the adsorption of hexane amounts to only ~0.25% when compared with the BLG adsorption. Therefore, the interfacial pressure in the post-critical region was calculated from Eq. (9). The dependences of BLG and hexane adsorptions calculated from the model Eqs. (15)-(21) are

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shown in Fig. 8. Note that at a BLG concentration of 10−12 M its adsorption is 0.0035 mg/m2, which is about 75 times lower than the hexane adsorption (0.26 mg/m2). In contrast, however, in

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the critical point the adsorbed amounts of BLG and hexane are 2.76 mg/m2 and 0.007 mg/m2, respectively. The introduction of the coefficient aPS into the proposed model results in an improved agreement between the calculated values and the experimental data, due to the account

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for the influence of alkane on the adsorption of protein. A similar conclusion was made in [26]

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for CnTAB solutions. In contrast, the dashed curve calculated from the model Eqs. (15)(21) with aPS = 0.7 shown in Fig. 7 yields a worse correspondence with the experimental data,

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especially in the low surface pressure range (less than 15 mN/m) where the hexane contributes significantly to the total adsorbed amount of proteins.

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Conclusions

The surface (interfacial) layer equations of state, adsorption isotherms and functions of the distribution of adsorbed protein (BLG and BCS) molecules in respect to their molar area (assuming that the protein molecules can adsorb in n states with different molar areas) are presented. The given thermodynamic analysis is based on a model for the non-ideality of surface layer enthalpy and entropy, and the number of formed layers with an additional adsorption constant for the second and subsequent layers. In the post-critical concentration range the aggregation of protein molecules in the surface layer is assumed by accounting for the number of molecules in aggregates formed during the adsorption process. Both adsorption and surface pressure isotherms (for the pre-critical and post-critical range) at liquid/fluid interfaces are described with a single a set of parameters. Also the equations for the dilational viscoelasticity behavior of the protein solutions at the interfaces are presented. A fitting programme was used to

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ACCEPTED MANUSCRIPT determine the adsorption characteristics of the most frequently studied proteins: -casein and -lactoglobulin at water/oil interfaces. To verify the applicability of the proposed approach, the experimental results on interfacial tension, adsorption and rheological characteristics of these proteins on the

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protein solution/hexane and protein solution/tetradecane interfaces obtained by drop profile

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analysis Tensiometry (with the protein solution either as drop formed in the ambient alkane, or

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as the media in which an alkane drop is formed) published earlier [32-34, 37-41], are employed. It is shown that these experimental results can be described quite adequately by the proposed theory with consistent model parameters. The processing of the experimental results using the model equations has revealed interesting details of the arrangement of the adsorbed molecules at

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the interface. In particular, it was shown that for low BCS concentrations at equilibrium the maximum area per protein molecule is by a factor of 1.5 larger, and the area of an adsorbed

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segment of the molecule is two times larger than the corresponding values under dynamic conditions. This indicates that the BCS molecules adsorbed in equilibrium conditions are more unfolded, and their adsorbed segments occupy larger interfacial areas, as compared with

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dynamic conditions. These factors cause a significant increase of the adsorption equilibrium

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constant of the BCS molecules.

The applicability of the theory assumes the adsorption of protein molecules from the

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aqueous solution, and in addition a competitive adsorption of alkane molecules from the alkane phase. The model was verified using experimental data for aqueous BCS and BLG solutions and alkanes (hexane and tetradecane). The comparison of the experimental equilibrium interfacial

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tension isotherms for BLG at the protein solution/hexane interface with data calculated using the proposed model demonstrates that the physical picture of a competitive adsorption of protein and alkane molecules is suitable. An essential influence of the hexane on the shape of the BLG adsorption isotherm is found, which is disregarded in all earlier classical approaches. Acknowledgements The work was supported by projects of the DFG (SPP 1273 and Mi418/20-1), and the COST actions CM1101 and MP1106.

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ACCEPTED MANUSCRIPT References S.J. Singer, J. Chem. Phys., 16 (1948) 872.

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19. P. Joos, G. Serrien, J. Colloid Interface Sci., 145 (1991) 291. 20. V.B. Fainerman, R. Miller, R. Wüstneck, J. Colloid Interface Sci., 183 (1996) 26. 21. V.B. Fainerman, E.H. Lucassen-Reynders, R. Miller, Colloids & Surfaces A, 143(1998) 141. 22. V.B. Fainerman, E.H. Lucassen-Reynders, R. Miller, Adv. Colloid Interface Sci., 106 (2003) 237-259. 23. R. Wüstneck, V.B. Fainerman, E. Aksenenko, Cs. Kotsmar, V. Pradines, J. Krägel, R. Miller, Colloids Surfaces A, 404 (2012) 17-24.. 24. V.B. Fainerman, S.A. Zholob, M. Leser, M. Michel and R. Miller, J. Colloid Interface Sci., 274 (2004) 496-501. 25. V.B. Fainerman, S.A. Zholob, M.E. Leser, M. Michel and R. Miller, J. Phys. Chem., 108 (2004) 16780-16785. 16

ACCEPTED MANUSCRIPT 26. V.B. Fainerman, N. Mucic, V. Pradines, E.V. Aksenenko, and R. Miller, Langmuir 2013, 29, 13783−13789. 27. E. Dickinson, and R. Miller (Eds), Food Colloids – Fundamentals of formulation. Special publication, Royal Society of Chemistry, No. 258 (2001).

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28. E. Dickinson, Colloids and Surfaces B., 20 (2001) 197-210.

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29. V.B. Fainerman, R. Miller, Colloid Journal, 67 (2005) 393–404.

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30. R. Wüstneck, J. Krägel, R. Miller, V.B. Fainerman, P.J. Wilde, D.K. Sarker, D.C. Klark, Food Hydrocol., 10 (1996) 395-405.

31. R. Wüstneck, V.B. Fainerman, E. Aksenenko, Cs. Kotsmar, V. Pradines, J. Krägel, R. Miller, Colloids Surfaces A, 404 (2012) 17-24.

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32. J. Maldonado-Valderrama, Valentin B. Fainerman, M.J. Gálvez-Ruiz, A. Marín-Rodriguez, M.A. Cabrerizo-Vílchez, and R. Miller, J. Phys. Chem. B, 109 (2005) 17608-17616.

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33. D.E. Graham, M.C. Phillips, J. Colloid Interface Sci., 70 (1979) 415. 34. A. Dan, R. Wüstneck, J. Krägel, E.V. Aksenenko, V.B. Fainerman and R. Miller, Food Hydrocolloids, 34 (2014) 193-201.

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35. A.V. Makievski, G. Loglio, J. Krägel, R. Miller, V.B. Fainerman and A.W. Neumann, J.

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Phys. Chem., 103 (1999) 9557-9561. 36. V.B. Fainerman, S.V. Lylyk, A.V. Makievski and R. Miller, J. Colloid Interface Sci., 275

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37. E.H. Lucassen-Reynders, J. Benjamins, V.B. Fainerman, Current Opinion Colloid & Interface Sci., 15 (2010) 264-270.

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38. J. Benjamins, A. Cagna, E. H. Lucassen-Reynders, Colloids & Surf. A, 114 (1996) 245. 39. V. Pradines, J. Krägel, V.B. Fainerman and R. Miller, J. Phys. Chem. B, 113 (2009) 745751.

40. V. Pradines, V.B. Fainerman, E.V. Aksenenko, J. Krägel, R. Wüstneck and R. Miller, Langmuir, 27 (2011) 965–971. 41. J. Maldonado-Valderrama, P.J. Wilde, V.J. Morris, V.B. Fainerman and R. Miller, Langmuir, 26 (2010) 15901-15908. 42. L. Peltonen, J. Hirvonen and J. Yliruusi, J. Colloid Interface Sci., 240 (2001) 272-276. 43. V.B. Fainerman, R. Miller, R. Wüstneck and A.V. Makievski, J. Phys. Chem., 100 (1996) 7669- 7675. 44. T. Sengupta, S. Damodaran, Langmuir, 14 (1998) 6457-6469. 45. P. Joos, Dynamic Surface Phenomena, Dordrecht, VSP, 1999. 46. J. Lucassen, M. van den Tempel, Chem. Eng. Sci., 1972 (27) 1283-1291. 17

ACCEPTED MANUSCRIPT 47. J. Lucassen, M. van den Tempel, J. Colloid Interface Sci., 1972 (41) 491-498. 48. J. Lucassen, Colloids Surfaces, 1992 (65) 139-149. 49. E. Kiss, R. Borba, Colloids and Surfaces B., 31 (2003) 169-176 50. R. Douillard, M. Daoud, V. Aguieґ-Beґghin, Current Opinion in Colloid and Interface

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Science, 8 (2003) 380–386.

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51. B.S. Murray, Rong Xu, E. Dickinson, Food Hydrocolloids, 23 (2009) 1190–1197.

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52. R. Miller, V.S. Alahverdjieva and V.B. Fainerman, Soft Matter, 4 (2008) 1141-1146. 53. V.S. Alahverdjieva, D.O. Grigoriev, V.B. Fainerman, E.V. Aksenenko, R. Miller and H. Möhwald, J. Phys. Chem. B, 112 (2008) 2136-2143.

54. Cs. Kotsmar, D.O. Grigoriev, F. Xu, E.V. Aksenenko, V.B. Fainerman, M.E. Leser and R.

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Miller, Langmuir, 24 (2008) 13977-13984.

55. V.B. Fainerman and R. Miller, Adsorption isotherms at liquid interfaces, Encyclopedia of

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Surface and Colloid Science, P. Somasundaran and A. Hubbard (Eds.), 2nd Edition, 1 (2009) 1-15.

56. V.B. Fainerman, R. Miller and V.I. Kovalchuk, Langmuir, 18 (2002) 7748-7752.

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57. V.B. Fainerman, R. Miller and V.I. Kovalchuk, J. Phys. Chem. 107 (2003) 6119-6121

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58. V.B. Fainerman, E.V. Aksenenko, N. Mucic, A. Javadi and R. Miller, Thermodynamics of adsorption of ionic surfactants at water/alkane interfaces, Soft Matter, 10 (2014) 6873-

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6887.

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Captions to Figures

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Fig. 1. Surface pressure dependence on the BCS concentration:  - data from [34], drop profile method, adsorption time 72.000 s, hexane drop in aqueous protein solution;  - data from

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[34], adsorption time 15000 s;  - data from [32], protein solution drop in tetradecane;  - data from [33], Wilhelmy plate; lines correspond to the theoretical calculations using the parameters

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listed in the Table 1; for detailed discussion see text.

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Fig. 2. Dependence of BCS solutions viscoelasticity modulus on surface pressure at the

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oscillations frequency 0.1 Hz. Red points and red lines are the experimental data and theoretical calculations from [37] using the model Eqs. (1)-(4), (11)-(13) for tetradecane drops in protein solutions; , ,  - data from [32,34], notation the same as in Fig. 1;  - data from [38] for adsorption time 25000 s; the lines correspond to the theoretical calculations using the parameters

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given in Table 1 (further details are given in the text.

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Fig. 3. Surface pressure dependence on the BLG concentration:  - data from [39] for the protein solution drop/hexane in the cell interface;  - data from [41] for the protein solution in

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the cell/tetradecane drop interface;  - data from [40] for the protein solution drop/hexane interface;  - results for BCS from Fig. 1. The solid line corresponds to the theoretical

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calculations with the parameters listed in the Table 1; for detailed discussion see text.

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Fig. 4. Dependence of BLG solutions viscoelasticity modulus on surface pressure at an

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oscillations frequency of 0.1 Hz. Red lines and red points, data from [37], tetradecane drop in protein solution;  - data from [41];  - data from [39]; the line corresponds to the theoretical

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calculations with the parameters listed in the Table 1; for detailed discussion see text.

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Fig. 5. Calculated adsorption dependencies of BCS (parameters for curves 1 and 3

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corresponding to those shown in Figs. 1 and 2) and BLG on the protein concentration.

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Fig. 6. Calculated dependencies of average molar area of BCS (curves 1 and 3 in Figs. 1

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and 2) and BLG as a function of the protein concentration.

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Fig. 7. Surface pressure dependence on the BLG concentration: The experimental points

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( - data from [39];  - data from [41];  - data from [40]) and theoretical curve (black) are reproduced from Fig. 3; red curves calculated using the protein – alkane co-adsorption model:

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solid line - aPS = 0.7; dotted line - aPS = 0.

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Fig. 8. Dependence of BLG adsorption (solid line) and hexane (dotted line) on the

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concentration in the pre-critical range, calculated using the protein – alkane co-adsorption model

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with the same parameter values as those taken for the solid red curve in Fig. 7.

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Graphical abstract

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ACCEPTED MANUSCRIPT Highlights Adsorption of proteins at the aqueous solution/alkane interface: co-adsorption of protein and alkane

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R. Miller1*, E.V. Aksenenko2, Igor I. Zinkovych3 and V.B. Fainerman3

Proteins adsorb stronger at the water/alkane than at the water/air interface. Using solution drop in alkane and alkane drop in solutions leads to different results.

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Depletion of protein molecules at low bulk concentrations is the reason for the differences. The interfacial tension isotherms can be well described by a new co-adsorption model.

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The co-adsorption model considers competitive adsorption of alkane and protein molecules.

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