J. Phys. Chem. B 2005, 109, 13327-13331
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Surface Dilational Rheology of Mixed β-Lactoglobulin/Surfactant Layers at the Air/Water Interface R. Miller,*,† M. E. Leser,‡ M. Michel,‡ and V. B. Fainerman§ Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung, Am Mu¨hlenberg 1, 14424 Potsdam, Germany, Nestec Ltd., Nestle´ Research Centre, Vers-chez-les-Blanc, CH-1000 Lausanne 26, Switzerland, and Medical Physicochemical Centre, Donetsk Medical UniVersity, 16 Ilych AVenue, 83003 Donetsk, Ukraine ReceiVed: March 2, 2005; In Final Form: May 16, 2005
The general theoretical model by Garrett and Joos proposed in 1976 for the estimation of the dilational elasticity of mixed surfactant solutions, and also the theoretical model proposed by Joos for the limiting elasticity of such mixtures, demonstrate quite satisfactory agreement with experimental results obtained from the oscillating bubble shape method for mixtures of a nonionic surfactant and a protein, that is, β-lactoglobuline and decyl dimethyl phosphine oxide, C10DMPO.
Introduction Addition of surfactants can modify adsorbed protein layers at liquid/fluid interfaces, which leads to changes of the adsorption and rheological characteristics.1-6 Although the dilational rheology of proteins or protein/surfactant mixtures is extremely important from a practical point of view, a corresponding theory is still not available. For much simpler systems, that is, surfactant mixtures, it appears possible to predict the rheological behavior of a mixture using data for the individual components. One of the first ever attempts to analyze theoretically the rheology of surfactant mixtures was made by Lucassen-Reynders.7 The theoretical analysis of the dilational rheology of surfactant mixtures was later performed by Garrett and Joos,8 who generalized the theory by Lucassen and van den Tempel.9,10 The theory of Garrett and Joos8 was further developed in refs 11-13. In particular, for the mixture of two surfactants the analytical expression for the complex dilational modulus was derived in ref 11. This expression involves four partial derivatives of the adsorptions with respect to concentrations and two partial derivatives of the surface tension with respect to the adsorptions. In ref 13, an irreversible thermodynamic approach had been applied and a general method for the derivation of equations for the surface elasticity of mixtures of an arbitrary number of surfactants and mixed adsorption kinetics was proposed. It was shown by Joos that by introducing the partial elasticities it becomes possible to express the limiting highfrequency elasticity for the mixture of surfactants in terms of the elasticities of the individual components.12 It can be expected that the results obtained in the theory for surfactant mixtures is applicable to protein/surfactant mixtures. Clearly, such application should account for the peculiarities of the adsorption and rheological behavior of the proteins. In the present publication, results are reported on experimental studies of the dilational rheology of a β-lactoglobulin (β-LG) mixture with the nonionic surfactant decyl dimethyl phosphine oxide (C10DMPO) at the solution/air interface. The theoretical * Corresponding author. E-mail:
[email protected]. † Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. ‡ Nestec Ltd. § Donetsk Medical University.
analysis is based on a model described in refs 8 and 12 and on the theoretical models developed for individual protein solutions and protein/nonionic surfactant mixtures, assuming multiple states of the protein molecule in the surface layer.14-16 Theory Surface Elasticity Modulus. For a diffusion-controlled adsorption, that is, in the absence of any other relaxation process, Lucassen and van den Tempel described the complex surface elasticity modulus ) r + ii as functions of angular frequency, ω ) 2πf, and surfactant concentration, c.10,11 This theory was generalized by Garrett and Joos8 for the case of surfactant mixtures, and the following expression for the complex dilational modulus was obtained
where ζj )
(
Γj 1 + ζj + iζj
∑Γ
) r + ii ) 0
1 + 2ζj + 2ζ2j
)
(1)
( )x ∂cj ∂Γj
Dj dγ , )2ω 0 d ln Γ
is the limiting (high frequency) elasticity, γ is the surface tension, Γ ) ∑Γj, and Γj and Dj are the adsorption and the diffusion coefficient for the jth component of the solution. Because ωDj ) Dj(∂cj/∂Γj)2 is the characteristic frequency of diffusional relaxation, the ζj parameters can be expressed by
ζj )
x
ωDj . 2ω
To determine the ∂cj/∂Γj derivatives, the equations
∂cj ∂Γj
)
[ ( ) ( ) ( )] ∑i
∂Γj
∂ci
∂cj
/
∂ci
c*ci
∂t
-1
(2)
∂t
were proposed in ref 8. Here the ∂c/∂t derivatives (with respect to time, t) refer to a dynamic subsurface layer. In some cases it becomes possible to neglect all components for i * j and assume that under dynamic conditions the inequality (∂ci/∂cj) , 1 holds.
10.1021/jp0510589 CCC: $30.25 © 2005 American Chemical Society Published on Web 06/18/2005
13328 J. Phys. Chem. B, Vol. 109, No. 27, 2005
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Then, from eq 2 the ∂cj/∂Γj derivatives can be expressed as ∂cj/ ∂Γj ) (∂cj/∂Γj)ci. The ∂cj/∂Γj derivatives were determined in ref 7 in this way. Similar results for two-component systems (with components 1 and 2) are obtained, for example, for component 1 if the condition (∂Γ1/∂c1)c2 . (∂Γ1/∂c2)c1 is valid. Here we also obtain ∂c1/∂Γ1 ) (∂c1/∂Γ1)c2. Let us consider the important particular case when eq 1 is applied to the solution of two surfactants. If the value of ζ2 for component 2 is quite high (ζ2 . 1, viscous monolayer behavior), then eq 1 for the elasticity modulus can be transformed into the form
|| )
(2r
+
2i )1/2
)
0(Γ1/Γ) (1 + 2ζ1 + 2ζ21)1/2
(3)
A general expression for the limiting (high-frequency) elasticity, 0, for the mixture of two surfactants was derived by Joos (see eq 2.179 in ref 12):
0 )
(Γ2 + Γ1Γ02/Γ01)0102 Γ201 + Γ1Γ02/Γ0102
(4)
In this equation, the 0 superscripts refer to the individual solutions with the same concentration as in the surfactant mixture, that is, 0j ) -(dγ/d ln Γ0j ) is the limiting elasticity of the jth surfactant individual solution, and Γ0j is the corresponding adsorption. The so-called parallel model for a solution that contains two proteins (with the mass fractions of the components in the mixture X1 and X2) leads to the following equation for the elasticity modulus:17
|1||2| || ) X1|2| + X2|1|
(5)
Equations 4 and 5 lead to correct limiting values for the individual solutions. It is interesting that recent experimental studies of the dilational rheology for mixtures of two proteins (β-lactoglobulin and β-casein)17 exhibit good agreement with eq 5. Possibly, eq 5 should be expressed via the mass portions of the components in the surface layer or the portions of occupied area in the surface layer, rather than via the mass portions of the components in the solution bulk. For the limiting (high-frequency) elasticity (|j| = 0j ), the more rigorous eq 4 differs from eq 5 only by the ratios of adsorptions in the individual solutions. Because eqs 1, 3, and 4 involve the adsorptions of the components both in the mixture, Γj, and in the individual solutions, Γ0j , we first consider briefly the results obtained for individual protein solutions. Individual Protein Solutions. Assuming that protein molecules can adsorb in a number of states with different molar area, varying from a maximum (ωmax) to a minimum value (ωmin), an equation of state for the surface layer was derived in ref 14
-
Πω0 ) ln(1 - θP) + θP(1 - ω0/ωΣ) + aPθ2P RT
ω0 is the molar area of the solvent, or the area occupied by one segment of the protein molecule n
is the total adsorption of proteins in all n states, and n
θ P ) ω ΣΓ P )
where Π is the surface pressure, R is the gas law constant, T is the temperature, aP is the intermolecular interaction parameter,
ωiΓPi ∑ i)1
is the total surface coverage by protein molecules. Here ωΣ is the average molar area of the adsorbed protein, ωi ) ω1 + (i 1)ω0 (1 e i e n) is the molar area in state i, assuming ω1 ) ωmin, ωmax ) ω1 + (n - 1)ω0. The equations for the adsorption isotherm for each state (j) of the adsorbed protein are
bPj cP )
ωΣΓPj (1 - θP)ωj/ωΣ
exp[-2aP(ωj/ωΣ)θP]
(7)
Here cP is the protein bulk concentration and bPj is the equilibrium adsorption constant for the protein in the jth state. When we assume that the bPi are equal to each other, bP ) ΣbPj ) nbPj, from eq 7 one can calculate the distribution function of adsorptions over various states of the protein molecules14
[
(1 - θP)(ωj-ω1)/ωΣ exp 2aPθP ΓPj ) ΓP
n
(1 - θP)(ω -ω )/ω ∑ i)1 j
1
Σ
[
]
ωj - ω1 ωΣ
]
ω i - ω1
exp 2aPθP
ωΣ
(8)
The model given by eqs 6-8 describes the evolution of the states of protein molecules during the increase of adsorption, which agrees in many details with known experimental results,14 also including the rheological behavior of protein adsorption layers.15 Mixture of Protein with a Nonionic Surfactant. With the approximation ω0 = ωS, the following equation of state for a protein/nonionic surfactant mixture was derived in ref 16
-
Πω0 ) ln(1 - θP - θS) + θP(1 - ω0/ωΣ) + aPθ2P + RT aSθ2S + 2aPSθPθS (9)
where aPS is a parameter that describes the interaction between the protein and surfactant molecules. A small difference between ω0 and ωS can be accounted for by introducing
ω*0 )
ω0θP + ωSθS θP + θ S
(10)
into eq 9 instead of ω0. For the protein in state j ) 1 and the surfactant, the adsorption isotherms read16
bP1cP ) (6)
ΓPi ∑ i)1
ΓP )
ωΣΓP1 (1 - θP - θS)ω1/ωΣ
bScS )
exp[-2aP(ω1/ωΣ)θP - 2aPSθS]
θS (1 - θP - θS)
(11) exp[-2aSθS - 2aPSθP]
(12)
β-Lactoglobulin/Surfactant Layers
J. Phys. Chem. B, Vol. 109, No. 27, 2005 13329
with θS ) ΓSωS. ΓS is the surfactant adsorption, and the subscript S refers to parameters characteristic for the pure surfactant. The distribution of protein adsorptions over the states, j, is given by the expression16
ΓPj ) ΓP
(1 - θP - θS)(ωj-ω1)/ωΣ exp[2aPθP(ωj - ω1)/ωΣ] n
(1 - θP - θS)(ω -ω )/ω ∑ i)1 j
1
Σ
exp[2aPθP(ωi - ω1)/ωΣ] (13)
Therefore, the problem of the theoretical description of a mixture can be formulated as follows: given the known values of T, ω0, ωmin, ωmax, aS, aP, aPS, bP1, cP, and bS for the individual components, the dependencies of parameters ωP, ΓP, ΓS, θP, θS, and Π as a function of the surfactant concentration, cS, can be calculated. Materials and Methods β-lactoglobulin was purchased from Sigma (Germany) and used without further purification. Decyl dimethyl phosphine oxide (C10DMPO) was synthesized and purified as described earlier.18 All of the measurements were performed at a temperature of 25 °C in phosphate buffer solutions (0.01 M, pH 7, prepared by mixing appropriate stock solutions of Na2HPO4 and NaH2PO4). The buffer solutions were prepared using Milli-Q water. The surface tension of the buffer solution was 72.0 mN/ m. In some long time experiments, 0.5 g/L of sodium azide (NaN3) was added to the phosphate buffer solution to slow the degradation processes of the protein. The surface tension of single and mixed β-LG solutions were measured with the profile analysis tensiometer PAT-1 (SINTERFACE Technologies, Germany). The solution drops were formed at the tip of a PTFE capillary immersed into a cuvette filled with a water-saturated atmosphere. After having reached the adsorption equilibrium, the solution drop was subjected to harmonic oscillations with frequencies f ) 0.01 to 0.2 Hz in order to study the dilational elasticity.
Figure 1. Dependence of the surface tension of β-LG solution (concentration 10-6 mol/L) on time. Figure 1b shows part of this dependence in larger scale.
Results and Discussion Figure 1a shows, as an example, the experimental curve of the dynamic surface tension for a 10-6 mol/L β-LG solution. After about 20.000 s, the adsorption equilibrium was achieved and harmonic oscillations of the drop area with a magnitude of ∆Ω ) ((7-8) % and frequencies in the range between 0.01 and 0.2 Hz were generated. In Figure 1b, the part of the dynamic curve is shown with harmonic area oscillations at 0.08 Hz. Figure 2 illustrates the experimental surface tension isotherm for C10DMPO (taken from ref 16) and β-LG/C10DMPO mixtures at different C10DMPO concentrations at a fixed concentration of 10-6 mol/L β-LG. The two experimental series refer to solutions with and without the addition of sodium azide. The data for the two series are in good agreement. The theoretical curve for C10DMPO shown in Figure 2 was calculated from the Frumkin model
-
ΠωS ) ln(1 - θS) + aSθ2S RT
bScS )
θS (1 - θS)
exp[-2aSθS]
(14) (15)
using the following parameters: ωS ) 3.0 × 105 m2/mol, aS )
Figure 2. Surface tension isotherms for individual C10DMPO (4, from ref 16) and the β-LG/C10DMPO mixtures ([, without sodium azide; ], with sodium azide) vs the C10DMPO concentration. The theoretical isotherms were calculated from eqs 9-12, 14, and 15 using the parameters listed in the text.
0.4, and bS ) 2.3 × 104 L/mol. The thermodynamic parameters for the individual β-LG solution are necessary in order to calculate the theoretical curves for the β-LG/C10DMPO mixtures. To estimate these values, we used the β-LG data measured by drop shape analysis in the presence of a phosphate buffer (0.01 M of Na2HPO4 and NaH2PO4, pH 7.0).5,19 These experimental dependencies are well described by eqs 6-8, leading to the following parameters:20 ω0 ) 3.5 × 105 m2/mol, ωmin ) 4.2 × 106 m2/mol, ωmax ) 1.5 × 107 m2/mol, aP ) 0.3, bP1 ) 1.7 × 106 L/mol (or bP ) 1.7 × 106‚31 ) 5.27 × 107 L/mol for the β-LG molecule as a whole).
13330 J. Phys. Chem. B, Vol. 109, No. 27, 2005
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Figure 3. Adsorption of β-LG (thin curve) and C10DMPO (thick curve) in the β-LG/C10DMPO mixture vs the C10DMPO concentration. Dotted line shows the C10DMPO adsorption in the individual solution.
The theoretical curves for the mixtures, shown in Figure 2, were calculated for the parameters given above, assuming aPS ) 0, because strong inhomogeneities in the mixed surface layer have to be expected. The protein and surfactant molecules practically do not mix at the surface but form domains containing essentially one of the components.21,22 Figure 2 shows quite satisfactory agreement between the experimental results and the theoretical curve. To calculate the limiting elasticity, 0, and the frequency dependent elasticity module, ||, using eqs 3 and 4, one should also know the adsorption values of β-LG and C10DMPO in the mixture and in the individual solutions. Figure 3 exhibits the dependence of the calculated values of C10DMPO adsorption in the individual solution and the adsorption of β-LG and C10DMPO in the mixture on the C10DMPO concentration at a fixed β-LG concentration of 10-6 mol/L. For C10DMPO concentrations above 10-4 mol/L, the protein is significantly displaced from the adsorption layer by the surfactant. The β-LG adsorptions at C10DMPO concentrations below 10-5 mol/L are equal to the individual β-LG solution in the absence of surfactant. The results of the experiments with harmonic oscillations of the surface area were analyzed using the Fourier transformation23,24
F[∆γ] (iω) ) Ω0 F[∆Ω]
(16)
where Ω0 is the initial area of the drop surface. The initial experimental data were first filtered to exclude the scattering errors. Also, the constant shift of the surface tension with time was eliminated, caused by the yet small deviation of the system from equilibrium when starting the harmonic oscillations. The dependencies of the dilational elasticity modulus on the oscillation frequency at various C10DMPO concentrations in the β-LG/C10DMPO mixtures are shown in Figure 4. Here the values of the phase angle, φ, determined as cos φ ) r/|| were in the range between 3 and 12°, that is, the mixed surface layers behave almost ideally elastic. With increasing oscillation frequency, the φ values decrease. Increase in C10DMPO concentration resulted at low concentrations in a decrease of angle φ as compared to the pure β-LG solution, and subsequently in an increase. This last effect is attributable to the increase of the fraction of the area covered by C10DMPO because for pure C10DMPO solutions in the frequency range studied, a viscous behavior was observed. It follows from Figure 4 that, with increasing C10DMPO concentration, the elasticity modulus of the β-LG/C10DMPO mixture decreases significantly.
Figure 4. Experimental dependence of surface dilational modulus magnitude on the harmonic oscillations frequency of the drop area for the β-LG/C10DMPO mixture at various C10DMPO concentrations: ×, 10-6 mol/L β-LG without C10DMPO; 2, with addition of 2 × 10-5 mol/L C10DMPO; ], 4 × 10-5 mol/L; [, 10-4 mol/L; 0, 2 × 10-4 mol/L; 9, 4 × 10-4 mol/L; 4, 7 × 10-4 mol/L.
For example, the modulus for β-LG mixed with 7 × 10-4 mol/L C10DMPO is 20 times lower than that for pure β-LG. The limiting elasticity, 0, and the diffusion relaxation frequency, ωD, for the β-LG solution of 10-6 mol/L were calculated from the corresponding frequency dependence shown in Figure 4 using the extrapolation procedure proposed in ref 25. The estimated values, 0 ) 82 mN/m and ωD ) 0.02 rad/s, agree well with the data obtained in ref 26. The values of 0 and ωD for the C10DMPO solutions were taken from ref 27 where these data were obtained from oscillating bubble experiments. For the C10DMPO concentrations studied in the present work, the 0 values vary in a narrow range, that is, between 30 and 35 mN/m.27 At the same time, it was found that the experimental dependence of ωD on cS is almost linear and can be described quite well by the linear relationship ωD ) 120cS (rad/s), where the C10DMPO concentration, cS, is expressed in mmol/L. The values of ωDj for the β-LG and C10DMPO components in the mixture are determined from the partial derivatives, ∂cj/ ∂Γj (ωDj ) Dj(∂cj/∂Γj)2), which were approximated by (∂cj/∂Γj)ci. This substitution is correct for β-LG (cf. eq 3), in view of the fact that the dependence of the β-LG adsorption on the C10DMPO concentration in the studied surfactant concentration range is very weak (cf. Figure 3). The ωDj values were estimated as follows. First, the dcj/dΓj derivatives were calculated for the individual β-LG and C10DMPO components from the theoretical models defined by eqs 6-8, 14, and 15, respectively. Then, the partial derivatives for these components in the mixture (∂cj/ ∂Γj)ci were calculated using eqs 9-13. The ωDj values for the components in the mixture were calculated from the experimental values of ωD for this component in the individual solution (ωD ) 120cs (rad/s) for C10DMPO (with cs in mmol/L) and ωD ) 0.02 rad/s for β-LG) as ωDj ) RωD, with the correction factor R ) ((∂cj/dΓj)ci/(dcj/dΓj))2 taken into account. It follows from Figure 3 that for C10DMPO at low concentrations the factor R . 1, but this value becomes lower with increasing concentration and approaches 1 for the highest C10DMPO concentration studied here. On the contrary, for β-LG in the mixture with C10DMPO at low C10DMPO concentrations R ) 1, which becomes lower with increasing C10DMPO concentration. We get R ) 0.3 for the highest C10DMPO concentrations studied here. Using the experimental values of ωDj measured in the individual solutions and applying the correction explained above
β-Lactoglobulin/Surfactant Layers
J. Phys. Chem. B, Vol. 109, No. 27, 2005 13331 with the experiments. Note that the parameters of the model used to describe the adsorption behavior of protein in the mixture account for the specific features characteristic of the protein molecule in the solution, namely, the capability of folding and unfolding of the molecule in the surface layer. In this context, the ability of the protein to decrease its molar area in a saturated monolayer was considered in the presence of a surfactant. Acknowledgment. The work was supported financially by projects of the European Space Agency (FASES MAP AO-99052), the DFG (Mi418/14), and the Ukrainian SFFR (Project no. 03.07/00227). References and Notes
Figure 5. Theoretical dependence of the surface dilational modulus magnitude on the frequency of bubble surface harmonic oscillations for the β-LG/C10DMPO mixture at various C10DMPO concentrations calculated from eqs 3 and 4. The notation is the same as that in Figure 4.
enabled us to determine the dilational characteristics of the mixture more exactly. Note, the simple theoretical models (e.g., Frumkin’s model) lead to quite unrealistic dependencies of dcj/ dΓj and j0 ) -(dγ/d ln Γj0) on the concentration for individual surfactant solutions. This deficiency could be overcome if the 2D compressibility of the molecules in the surface layer is taken into account. However, for mixed protein/surfactant layers these models are still to be developed. The values of ζ for the individual component C10DMPO in the frequency range studied are quite high (ζ . 1) and become even higher in mixtures with β-LG caused by the factor R . 1. Therefore, the influence of all of the derivatives in eq 2 on the value of ∂cj/∂Γj for C10DMPO is insignificant, and for the system studied eq 1 can be reduced to eq 3 where the subscript 1 refers to the protein. Figure 5 illustrates the calculations of the elasticity modulus for β-LG/C10DMPO mixtures according to eq 3. Here the 0 values for the mixtures calculated from eq 4 were used. Comparing the curves in Figure 4 with those in Figure 5 we find a quite satisfactory agreement between the experimental elasticity values and those predicted by the theoretical model. Calculations according to eq 5 show satisfactory agreement with the experimental data (although the calculated elasticity modulus values of the mixture are systematically lower by 3050% than the experimentally observed ones) when X1 and X2 used in the calculations are the fractions of the area occupied by the components in the surface layer. Conclusions The theoretical model proposed by Joos12 for the estimation of the limiting elasticity of mixed surfactant solutions, and also the general theoretical model proposed earlier by Garrett and Joos8 for the description of the dilational elasticity of such mixtures, demonstrate quite satisfactory agreement with experimental results obtained from the oscillating bubble shape method for mixtures of nonionic surfactant C10DMPO and protein β-LG. Also, the calculated limiting elasticities of the mixtures comply
(1) Miller, R.; Fainerman, V. B.; Makievski, A. V.; Kra¨gel, J.; Grigoriev, D. O.; Kazakov, V. N.; Sinyachenko, O. V. AdV. Colloid Interface Sci. 2000, 86, 39. (2) Dussaud, A.; Han, G. B.; Ter Minassian-Saraga, L.; Vignes-Adler, M. J. Colloid Interface Sci. 1994, 167, 247. (3) Turro, N. J.; Lei, X.-G.; Ananthapadmanabhan, K. P.; Aronson, M. Langmuir 1995, 11, 2525. (4) Wu¨stneck, R.; Kra¨gel, J.; Miller, R.; Wilde, P. J.; Clark, D. C. Colloids Surf., A 1996, 114, 255. (5) Kra¨gel, J.; O’Neill, M.; Makievski, A. V.; Michel, M.; Leser, M. E.; Miller, R. Colloids Surf., B 2003, 31, 107. (6) Dickinson, E. Colloids Surf., B 1996, 15, 161. (7) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1973, 42, 573. (8) Garrett, P. R.; Joos, P. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2161. (9) Lucassen, J.; van den Tempel, M. Chem. Eng. Sci. 1972, 27, 1283. (10) Lucassen, J.; van den Tempel, M. J. Colloid Interface Sci. 1972, 41, 491. (11) Jiang, Q.; Valentini, J. E.; Chiew, Y. C. J. Colloid Interface Sci. 1975, 174, 268. (12) Joos, P. Dynamic Surface Phenomena; VSP: Dordrecht, The Netherlands, 1999. (13) Noskov, B. A.; Loglio, G. Colloids Surf., A 1998, 141, 167. (14) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. AdV. Colloid Interface Sci. 2003, 106, 237. (15) Lucassen-Reynders, E. H.; Fainerman, V. B.; Miller, R. J. Phys. Chem. 2004, 108, 9173. (16) Fainerman, V. B.; Zholob, S. A.; Leser, M.; Michel, M.; Miller, R. J. Colloid Interface Sci. 2004, 274, 496. (17) Ridout, M. J.; Mackie, A. R.; Wilde, P. J. J. Agric. Food. Chem. 2004, 52, 3930. (18) Fainerman, V. B.; Makievski, A. V.; Vollhardt, D.; Siegel, S.; Miller, R. J. Phys. Chem. B 1999, 103, 330. (19) Kra¨gel, J.; Wu¨stneck, R.; Husband, F.; Wilde, P. J.; Makievski, A. V.; Grigoriev, D. O.; Li, J. B. Colloids Surf., B 1999, 12, 399. (20) Fainerman, V. B.; Zholob, S. A.; Leser, M. E.; Michel, M.; Miller, R. J. Phys. Chem. 2004, 108, 16780. (21) Mackie, A. R.; Gunning, A. P.; Ridout, M. J.; Wilde, P. J.; Morris, V. I. Langmuir 2001, 17, 6593. (22) Mackie, A. R.; Gunning, A. P.; Ridout, M. J.; Wilde, P. J.; Patino, J. R. Biomacromolecules 2001, 2, 1001. (23) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A. V.; Ravera, P.; Ferrari, M.; Liggieri, L. In NoVel Methods to Study Interfacial Layers; Mo¨bius, D., Miller, R., Eds.; Elsevier Science: Amsterdam, 2001. (24) Jenkins, G. M.; Watts, D. G. Spectral Analysis and its Applications; Holden Day: San Fransisco, CA, 1969. (25) Lucassen, J.; Giles, D. J. Chem. Soc., Faraday Trans. 1 1975, 71, 217. (26) Benjamins, J. Static and Dynamic Properties of Proteins Adsorbed at Liquid Interfaces. Thesis, Wageningen University, 2000. (27) Wantke, K.-D.; Fruhner, H. J. Colloid Interface Sci. 2001, 237, 185. (28) Fainerman, V. B.; Miller, R.; Kovalchuk, V. I. J. Phys. Chem. B. 2003, 107, 6119. (29) Fainerman, V. B.; Kovalchuk, V. I.; Aksenenko, E. V.; Michel, M.; Leser, M. E.; Miller, R. J. Phys. Chem. B. 2004, 108, 13700.
