Langmuir 2004, 20, 771-777
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Dynamic Surface Tension and Adsorption Kinetics of β-Casein at the Solution/Air Interface R. Miller,*,† V. B. Fainerman,‡ E. V. Aksenenko,§ M. E. Leser,| and M. Michel| Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Am Mu¨ hlenberg 1, 14424 Potsdam, Germany, Medical Physicochemical Centre, Donetsk Medical University, 16 Ilych Avenue, 83003 Donetsk, Ukraine, Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky Avenue, 03680 Kyiv (Kiev), Ukraine, and Nestec Limited, Nestle´ Research Centre, Vers-chez-les-Blanc, CH-1000 Lausanne, Switzerland Received August 11, 2003. In Final Form: November 10, 2003 A diffusion model is proposed to describe the adsorption kinetics of proteins at a liquid interface. The model is based on the simultaneous solution of the Ward-Tordai equation and a set of recently developed equations describing the equilibrium state of the adsorption layer: the adsorption isotherm, the surface layer equation of state, and the function of adsorption distribution over the states with different molar areas. The new kinetics model is compared with dynamic surface tensions of β-casein solutions measured with the drop/bubble profile and maximum bubble pressure methods. The adsorption process for low concentrations is governed by the diffusion mechanism, while at large protein concentrations this is only the case in the initial stage. The effective diffusion coefficients agree fairly well with literature data. The adsorption values calculated from the dynamic surface tension data agree very well with the used equilibrium adsorption model.
Introduction The direct measurement of protein adsorption kinetics at liquid interfaces is a quite complicated technical problem. The most common methods here are radiotracer techniques and ellipsometry.1-6 The comparison of β-casein adsorption dynamics data obtained from these methods with an approximation of the Ward and Tordai model for low subsurface concentration7 indicates that the protein adsorbs according to a diffusion mechanism, at least in the initial stage. The values of the apparent diffusion coefficient D, calculated from the dependence of adsorption Γ on the square root of time t, are similar to those in the solution bulk, D ) 0.6 × 10-10 m2/s at 20 °C.8 In particular, the radiotracer technique yields D ) 1.7 × 10-10 m2/s at a concentration of 6.3 × 10-7 mol/L,2 while from ellipsometric studies, values of D ) 2 × 10-10 m2/s for 3.3 × 10-8 mol/L and D ) 0.9 × 10-10 m2/s for a concentration of 6.3 × 10-7 mol/L were obtained.4 These data are somehow consistent and agree with the β-casein diffusion coefficient in the solution bulk. Measurements of dynamic surface tensions of protein solutions are technically simpler to perform. A number of results obtained by various tensiometer techniques for * Corresponding author. † Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. ‡ Donetsk Medical University. § Institute of Colloid Chemistry and Chemistry of Water. | Nestec Ltd. (1) Benjamins, J.; de Feijter, J. A.; Evans, M. T. A.; Graham, D. E.; Phillips, M. C. Discuss. Faraday Soc. 1975, 59, 218. (2) Feijter, J. A.; Benjamins, J.; Veer, F. A. Biopolymers 1978, 17, 1760. (3) Graham, D. E.; Phillips, M. C. J. Colloid Interface Sci. 1979, 70, 403. (4) Xu, S.; Damodaran, S. J. Colloid Interface Sci. 1993, 159, 124. (5) Grigoriev, D. O.; Fainerman, V. B.; Makievski, A. V.; Kra¨gel, J.; Wu¨stneck, R.; Miller, R. J. Colloid Interface Sci. 2002, 253, 257. (6) Russev, S. C.; Arguirov, T. Vl.; Gurkov, Th. D. Colloids Surf., B 2000, 19, 89. (7) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 543. (8) Handbook of Biochemistry; Sober, H. A., Ed.; Chemical Rubber Co.: Cleveland, OH, 1968.
protein solutions, and in particular β-casein, are reviewed in ref 9. From such measurements, however, the adsorption mechanism can only be revealed and the apparent diffusion coefficient estimated if a specific equation of state for the surface layer (a relation between surface tension and protein adsorption) is known for the system studied. Estimates based on the Szyszkowski-Langmuir equation of state lead to unrealistically high apparent diffusion coefficients for β-casein: the resulting values are in the range of 10-6 to 10-9 m2/s,10 which are several orders of magnitude higher than physically expected.2,4 This inconsistency should be ascribed to the particular features characteristic for the adsorption of flexible proteins at the solution/air interface: (i) surface denaturation of proteins leads to unfolding of adsorbed molecules, particularly at low surface pressure; (ii) in contrast to ordinary surfactants, the partial molar surface area of proteins is large and is subject to variation with surface pressure; (iii) for proteins, the nonideality of surface layer enthalpy and especially entropy is large. Only recently a theoretical model was proposed which accounts for all the above peculiarities of the adsorption of flexible proteins.11 In this model, a single set of parameters allows one to reproduce all known features for flexible protein systems: a sharp increase in surface pressure with concentration beyond a certain protein adsorption, an almost constant surface pressure at higher concentrations, and a significant increase in the adsorption layer thickness with increasing adsorption. This paper aims at the development of a theoretical model for the diffusional adsorption kinetics (quasiequilibrium model), which comprises the theories described elsewhere.7,11 Also, experimental dynamic surface tensions of β-casein solutions using the drop/bubble shape (9) 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. (10) Wu¨stneck, R.; Kra¨gel, J.; Miller, R.; Fainerman, V. B.; Wilde, P. J.; Sarker, D. K.; Clark, D. C. Food Hydrocolloids 1996, 10, 395. (11) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Adv. Colloid Interface Sci. 2003, 106, 237.
10.1021/la030332s CCC: $27.50 © 2004 American Chemical Society Published on Web 01/06/2004
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and maximum bubble pressure methods are reported, and a comparison between the experimental and theoretical results is made, which provides insight into the mechanism governing the adsorption process.
∑bi ) nbi.11 When compared with state 1, the distribution function for the adsorption in all states can be derived:
Γi ) Γ
Materials and Methods β-Casein was purchased from Sigma (Germany) and used without further purification. All 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 stock solutions were stored for not more than 3 days at 5 °C. The buffer solutions were prepared using Milli-Q or bidistilled water. No filtration of the prepared buffer and protein solutions was performed. The surface tension of the buffer solution was 72.0 mN/m. Chemical and physical degradations of proteins take place in aqueous solutions.12 Therefore, in some experiments 0.5 g/L of sodium azide (NaN3) was added to the phosphate buffer solution, to slow down degradation processes. The dynamic surface tensions of the protein solutions were measured in the time range from 10 to 30 000 s using the drop profile analysis tensiometer PAT1 (SINTERFACE Technologies, Germany).13 The solution drops were formed at the tip of a poly(tetrafluoroethylene) (PTFE) capillary immersed into a cuvette filled with a water-saturated atmosphere. The drop volume was kept constant at values between 16 and 17 mm3. The dynamic surface tensions of concentrated β-casein solutions in the time range of 10 ms to 50 s were measured by a maximum bubble pressure analyzer BPA1 (SINTERFACE Technologies), using special steel capillaries of 0.5 mm inner diameter.14
(1 - Γω)(ωi-ω1)/ω exp[2a(ωi - ω1)Γ] n
(1 - Γω) ∑ i)1
-
Πω0 ) ln(1 - ωΓ) + (ω - ω0)Γ + a(ωΓ)2 RT
(1)
where Π is the surface pressure, R is the gas law constant, T is the temperature, a is the intermolecular interaction n Γi is the total adsorption of protein parameter, Γ ) ∑i)1 n ωiΓi is the total surface in all n states, θ ) ωΓ ) ∑i)1 coverage, and ω is the average molecular area of adsorbed proteins. The equation of the adsorption isotherm for each state (j) of the protein is given by
bjc )
ωΓj (1 - ωΓ)ωj/ω
exp[-2a(ωj/ω)ωΓ]
(2)
Here c is the bulk concentration of the protein, and bj are the equilibrium adsorption constants of state j. It can be assumed that all constants bj have one and the same value for all states j from i ) 1 to i ) n, and therefore the adsorption constant for the protein molecule is given by (12) Wang, W. Int. J. Pharm. 1999, 185, 129. (13) Makievski, A. V.; Loglio, G.; Kra¨gel, J.; Miller, R.; Fainerman, V. B.; Neumann, A. W. J. Phys. Chem. B 1999, 103, 9557. (14) Fainerman, V. B.; Miller, R.; Makievski, A. V. Accurate analysis of the bubble formation process in maximum bubble pressure tensiometry. Rev. Sci. Instrum., in press.
(3)
exp[2a(ωi - ω1)Γ]
Equations 1-3 provide a good description of the behavior of the system at relatively low protein concentrations, where the increase in protein concentration results in an increase in surface pressure. However, it was shown experimentally that above a certain protein concentration c*, the surface pressure remains almost constant, while the adsorption often increases.1-3 Such a constant level of Π can be explained by a two-dimensional condensation (aggregation) of the protein layer which takes place beyond a certain critical adsorption Γ*. In the concentration range c > c*, the equation of state of the surface layer and the adsorption isotherm become11
-
Πω0 1 ) [ln(1 - Γω) + Γω - Γω0 + a(Γω)2] (4) RT Ψ
and
bjc )
Theory of Adsorption Kinetics The theoretical model developed recently is based on the experimental fact that protein molecules can exist in a number of states with different molar areas, varying from a maximum value (ωmax) at very low surface coverage to a minimum value (ωmin) at high surface coverage.11 It also assumes that molecules in different states are in equilibrium with each other but otherwise behave as independent components. Assuming that the molar area of the solvent (which, in turn, is equal to the molar area of the segment of the protein molecule), ω0, is much smaller than ωmin, the following equation of state for the surface layer was obtained:11
(ωi-ω1)/ω
Γjω ωj/ωΨ
(1 - Γω)
[
exp -
]
ωj (2aΓω) ωΨ
(5)
respectively, where
Ψ)
Π - Π* Γ exp ω Γ* RT
(
)
(6)
Π* is the critical pressure at c ) c*, and ) 0.0-0.2 is an adjustable parameter which accounts for the decrease of the area per protein molecule due to the surface layer condensation. Equations of state for surface layers and adsorption isotherms are the basis for calculations of adsorption kinetics and dynamic surface tensions. The equation proposed by Ward and Tordai7 represents a general relationship between the dynamic adsorption and the subsurface concentration c(0,t), and reads for fresh and nondeformed surfaces
Γ(t) ) 2
xDπ [c xt - ∫ 0
xt
0
c(0,t - t′) d(xt′)]
(7)
Here c0 is the protein bulk concentration, t is the time, and t′ is a dummy integration variable. Using eq 7, respective dependencies Γ(t) can be obtained for isotherms 2 and 5, which serve as an additional boundary condition for the diffusion-controlled adsorption model. When the adsorption from solution takes place at a spherical surface (bubble or drop), the effect of surface curvature can be approximately accounted for by introducing an additional term into eq 7:13,15
Γ(t) ) 2
x
D [c xt π 0
∫0xt c(0,t - t′) d(xt′)] (
c0D t (8) r
where r is the radius of curvature, and the signs “-” or “+” before the second term on the right-hand side correspond to a drop or bubble, respectively. Thus, if the (15) Liggieri, L.; Ravera, F.; Ferrari, M.; Passerone, A.; Miller, R. J. Colloid Interface Sci. 1997, 186, 46.
Adsorption Kinetics of β-Casein
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Figure 1. Dynamic surface tension of a 10-6 mol/L β-casein solution; the curves are labeled by the time in hours expired from the preparation of the solution.
molecules adsorb at the surface of a drop, the adsorption dynamics is slower than at a flat interface, while for the adsorption at a bubble surface the adsorption dynamics is faster than at a flat surface. When adsorption equilibrium is established, the adsorption values (and hence the corresponding surface pressure) can differ due to changes in the concentration of the solution in the drop or in the surrounding solution of the bubble.13 This allows us to calculate the protein adsorption from the difference between the initial concentration in a drop and in the surrounding solution of a bubble of the same size, cD and cB, respectively,
V Γ ) (cD - cB)Π)const S
(9)
V and S are the volume and surface area of the drop or bubble. The experiments are to be performed such that the same final surface pressure Π is reached.16 The mathematics of the adsorption kinetics model given by eq 8 and the numerical procedure used to obtain the solution for relevant parameters are summarized in the Appendix. Results and Discussion Prior to the tensiometry studies, the stability of β-casein solutions was investigated as follows. Fresh β-casein solutions with predefined concentration were prepared in the phosphate buffer and then stored in flasks at room temperature. First, measurements were made with a fresh solution, and subsequently new samples of the solution were taken in 2-4 h intervals. The dynamic surface tension was measured in the time range from 10 to 3000 s using the drop profile analysis technique. The dependencies obtained in this way for a 10-6 mol/L β-casein solution are shown in Figure 1. It is seen that during the initial period of 7-8 h the measured surface tensions are stable, while 10 h after preparation, the decrease in surface tension starts later and leads to higher values. This indicates a chemical degradation of the protein, which we ascribe to the presence of bacteria in the solution. To avoid this bacterial contamination (at least at the stage of the preparation of the solution), some authors make a microfiltration (pore size, 0.22 µm).17 It was shown in our (16) Fainerman, V. B.; Lylyk, S. V.; Makievski, A. V.; Miller, R. Interfacial tensiometry as a novel methodology for the determination of surfactant adsorption at a liquid surface. J. Colloid Interface Sci., submitted. (17) Suttiprasit, P.; Krisdhasima, V.; Mcguire, J. J. Colloid Interface Sci. 1992, 154, 316.
Figure 2. Dynamic surface tension of β-casein solutions at concentrations of 10-8 (2,4) and 10-7 mol/L (9,0), measured by drop profile (4,0) and bubble profile (2,9) analysis; theoretical curves calculated from eqs 1-8.
experiments that the degradation rate becomes lower with increasing concentration, while the degradation becomes more rapid at higher temperature. The addition of sodium azide lowers the degradation rate. This effect was more pronounced for diluted solutions. In particular, for solutions at concentrations lower than 10-7 mol/L, the addition of N3Na slows the degradation down several times. Note that quite similar results were obtained both with Milli-Q and bidistilled water: for Milli-Q the degradation rate was 10-20% lower than for the bidistilled water. Therefore, we concluded that dynamic experiments with β-casein solutions can be performed if the contact time of the solutions with ambient air does not exceed 8 h. The parameters in eqs 1-6 for β-casein are ω0 ) (2.02.5) × 105 m2/mol, ωmin ) 5 × 106 m2/mol, ωmax ) 4 × 107 m2/mol, a ) 1, Π* ) 22 mN/m, and ) 0.2.11 The area per water molecule ω0 for the dividing surface defined after Lucassen-Reynders, or the area per segment of the β-casein molecule, is similar to values typical for ordinary surfactants. At the same time, the value of ωmax is more than 2 orders of magnitude higher than ω0 and more than 1 order of magnitude higher than ωmin. The large difference between the values of ωmin and ωmax suggests that the adsorption layer thickness increases roughly 10 times with increasing β-casein concentration.11 We can assume that at the given conformational transition the density of the protein molecule remains constant. Figure 2 illustrates the dynamic surface tension of diluted β-casein solutions measured using drop/bubble profile analysis methods. The data using the bubble profile method were partly presented earlier.13 For a β-casein concentration of 10-8 mol/L, the data obtained from drop experiments do not show any surface tension decrease in the time range up to 20 000 s, while from the bubble profile method the surface tension starts to decrease after 10 000 s surface age, and at about 30 000 s this decrease amounts to 10 mN/m. This difference in the results obtained by the two methods is due to the decrease of the protein concentration in the drop bulk caused by the adsorption at the surface, in accordance with eq 9. The increase of the β-casein concentration to 10-7 mol/L leads to a significant surface tension decrease measured with the drop, while this decrease remains essentially less than that observed in the bubble experiments. The theoretical curves presented in Figure 2 were calculated from eq 8 for a diffusion coefficient of 2 × 10-10 m2/s, that is, assuming a finite radius of curvature of the surface. An average radius was taken for the whole experimental time range. For the bubble and drop experiments, the values for r were 2 and 1.5 mm, respectively. The quite satisfactory
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Figure 3. Dynamic surface tension of a 2 × 10-7 mol/L β-casein solution, measured by drop (4) and bubble profile analysis (2); theoretical curves calculated from eqs 1-8.
Figure 4. The same as in Figure 3 for a concentration of 10-6 mol/L.
Figure 5. Dynamic surface tension of a 2 × 10-6 mol/L β-casein solution, measured by the drop profile (4) and maximum bubble pressure method (],[); theoretical curve calculated from eqs 1-7.
agreement between experimental and theoretical values indicates the validity of the theoretical model given by eqs 1-6 not only for the equilibrium adsorption behavior but also during the adsorption dynamics. Therefore, the β-casein adsorption kinetics is controlled by diffusion from the solution bulk, at least at low protein concentrations, when the rate of reconformation of protein molecules in the surface layer is relatively high as compared with the adsorption rate. Figures 3-6 illustrate the experimental and theoretical dependencies for higher β-casein concentrations, in the range from 2 × 10-7 to 5 × 10-6 mol/L. While for 2 × 10-7 mol/L a small difference still exists between the data obtained by a drop or bubble, respectively, for higher concentrations this difference vanishes. This effect can be explained by the fact that when the adsorption
Miller et al.
Figure 6. The same as in Figure 5 for a concentration of 5 × 10-6 mol/L.
equilibrium is established, the diffusion layer thickness δ ) (Dπt)1/2 is remarkably lower than the drop radius. The behavior of the theoretical curves is quite similar: for concentrations above 10-6 mol/L the second term on the right-hand side of eq 8 becomes negligible as compared to the first one, and the ordinary Ward and Tordai eq 7 is applicable. Note that, to improve the agreement with the experimental results, the diffusion coefficients used in the calculations for Figures 3-6 were taken to be (0.50.7) × 10-10 m2/s, which is 3-4 times lower than those corresponding to the curves shown in Figure 2. Higher apparent diffusion coefficients are usually caused by a partial convective transport.18 A concentration increase leads to a decrease of the diffusion layer thickness δ. Hence the decrease of the apparent diffusion coefficient with increasing concentration can also be ascribed to convection effects. However, it is seen from our data that the convection plays only a minor role, because the values obtained for a drop (note that almost no convection can exist in such a small drop) and from a bubble surrounded by the solution (with a quite probable convection in the surrounding protein solution) are well described by the curves in Figures 2 and 3, calculated with the same diffusion coefficients. Figures 3-6 also show that the closer the system to the equilibrium state (i.e., the larger the time), the more significant the difference between the theoretical and experimental curves. The real process develops slower than the theoretical prediction. It can be supposed that in these conditions a secondary adsorption layer of β-casein forms (disregarded by the theoretical model of eqs 1-6 used here), and a low transition rate between the states (reconformation) of the adsorbed β-casein molecule in the adsorption layer exists.19 This process of conformational changes, disregarded in eqs 7 and 8, may play a significant role. It was shown recently that for globular proteins the consideration of the reconformation kinetics leads to a better agreement between theory and experiment.20 The formation of the second adsorption layer was taken into account in the general theoretical model for the equilibrium case. In our future studies, the presented adsorption dynamics model will have to be developed such that the finite transition rate between the molecular states in the adsorption layer is incorporated.11 Figure 7 illustrates the so-called induction time tInd (the time interval expired before a decrease in surface tension (18) Guzman, R. Z.; Carbonell, R. G.; Kilpatrick, P. K. J. Colloid Interface Sci. 1986, 114, 536. (19) Beverung, C. J.; Radke, C. J.; Blanch, H. W. Biophys. Chem. 1999, 81, 59. (20) Miller, R.; Aksenenko, E. V.; Fainerman, V. B.; Pison, U. Colloids Surf., A 2001, 183, 381.
Adsorption Kinetics of β-Casein
Figure 7. Induction time as a function of the β-casein concentration: 0, experimental values; solid line, calculated for D ) 0.5 × 10-10 m2/s; dotted line, calculated for D ) 2 × 10-10 m2/s. The arrow indicates the concentration range where the apparent diffusion coefficient changes.
starts to occur) as a function of the β-casein concentration. For β-casein concentrations lower than 2 × 10-7 mol/L, the experimental induction times were taken from bubble experiments, while for higher concentrations the induction times were either calculated from both methods or taken from drop experiments. Note that a dependence quite similar to the one in Figure 7 was obtained for bovine serum albumin (BSA).21 The theoretical induction times were calculated from eqs 1-8 for two diffusion coefficients, 2 × 10-10 and 0.5 × 10-10 m2/s. The calculated curves in Figures 3-6 show that the theoretical model provides quite a precise prediction for the induction time. Irrespective of the β-casein concentration, the onset of the surface tension decrease corresponds to a dynamic adsorption Γ of 0.6 mg/m2. This threshold adsorption value coincides exactly with that found in the experiment under equilibrium conditions.1 The comparison between experimental and theoretical induction times shows that for low β-casein concentrations the apparent diffusion coefficient is higher (2 × 10-10 m2/s), while D decreases to a value of 0.5 × 10-10 m2/s in the concentration range between 1 × 10-7 and 2 × 10-7 mol/L× and remains constant for higher concentrations. These diffusion coefficients coincide almost precisely with those obtained by ellipsometry in the same range of β-casein concentration.2 The decrease of the diffusion coefficient with increasing β-casein concentration can be explained by an aggregation (micelle formation) in the solution bulk. The formation of β-casein micelles was discussed for example in ref 23. In fact, a secondary adsorption layer can be formed by β-casein solution when the concentration exceeds 10-7 mol/L.5 In terms of thermodynamic quantities, these two processes (micelle formation and second layer formation) are roughly equivalent to each other and therefore should commence at similar bulk concentrations. Conclusions A diffusion kinetics model for protein adsorption at a liquid interface is proposed, which is based on the simultaneous solution of the Ward-Tordai equation and a set of recently developed equations (adsorption isotherm, surface layer equation of state, function of adsorption distribution over the states with different molar areas, (21) Ybert, Ch.; di Meglio, J.-M. Langmuir 1998, 14, 471. (22) Aksenenko, E. V. In Surfactants: Chemistry, Interfacial Properties, Applications; Fainerman, V. B., Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 2001; p 619. (23) de Kruif, C. G. In Progress and Trends in Rheology IV, Proceedings of the Fourth European Rheology Congress, Sevilla, 1994; Gallegos, C., Ed.; Steinkopff: Darmstadt, 1994; p 221.
Langmuir, Vol. 20, No. 3, 2004 775
etc.) which follows from the theory of equilibrium adsorption of proteins.11 The drop/bubble profile and maximum bubble pressure methods were used to study the dynamic surface tension of β-casein solutions. It was shown that, at least in the initial stage of the adsorption process for large protein concentrations and during all the adsorption process for low concentrations, the adsorption process is governed by the diffusion mechanism. The effective diffusion coefficients, obtained from best fit with experimental data, agree fairly well with bulk values and values obtained from direct studies of the adsorption dynamics using radiotracer techniques and ellipsometry. As the adsorption values in this study were calculated from the dynamic surface tension data, this fact supports the validity of the theoretical adsorption model proposed.11 Appendix: Solution of Model Equations To obtain the solution of the Ward-Tordai eq 8 by an iterative procedure explained in detail elsewhere,21 the dimensionless variables are introduced:
τ)D
()
c0 2 t Γ∞
γ(τ) )
Γ(τ) Γ∞
C(τ - τ′) ) c(0,t - t′) c0
ϑ ) xτ (A.1)
which reduces eq 8 to the form
γ(ϑ2) )
2 [ϑ xπ
Γ
∫0ϑ C(ϑ2 - ϑ′2) dϑ′] ( rc∞0ϑ2
(A.2)
Also, the set of equations which determine the isotherm and the state of the adsorbed monolayer should also be normalized by introducing the dimensionless equilibrium constant B:
B ) bc0
(A.3)
B introduced by eq A.3 is essential for the numerical integration of eq A.2 and should be determined for any given set of model parameters and given values of Π* and Π∞, where Π* is the critical surface pressure and Π∞ is the surface pressure which the system (as described by the mathematics) would attain at infinite time.21 This Π∞ value is a nonobservable quantity (because at large times the behavior of the system is governed by other physical factors, not accounted for in the present model). Therefore, this value should be calculated from the equilibrium constant b via the equilibrium solution of the model equations (the solution at t f ∞, with corresponding values denoted below by subscript ∞). If B is known, the normalized concentration C defined by eq A.1 and normalized adsorption γ defined via the adsorption at infinite time Γ∞
Γ ) γΓ∞
(A.4)
are determined, where the Γ∞ value should follow from the solution at infinite time. It is quite obvious from the definition of the model given above that the system can exhibit either a single-phase or two-phase regime (denoted below where appropriate by superscripts I and II), depending on the relation between the values Π* and Π∞. Distribution over the States of Adsorbed Molecules. Noting that the model parameters ω0, ωmin, ωmax, and n are interdependent via the relationship
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Miller et al.
ω0 ) (ωmax - ωmin)/(n - 1)
(A.5)
at the critical point. Then, the adsorption isotherm 5 becomes
(A.6)
1 θ1Ω exp - ΩΨ(2a(θ1Ω)} bc ) Ξ (1 - θ Ω)1/ΩΨ
and introducing the dimensionless variables
Ω ) ω/ωmin
θ1 ) Γωmin
p ) Πωmin/RT
Ω0 ) ω0/ωmin ) const
(A.7)
it is straightforward to reduce eq 3 to the form
(i - 1)ξi(θ1Ω,Ω) ∑ i)2
(A.8)
Ξ(θ1Ω,Ω)
where the auxiliary functions
ξi(θ1Ω,Ω) ) [1 + (i - 1)Ω0]R Ω0 exp (i - 1) [ln(1 - θ1Ω) + 2a(θ1Ω)] Ω
{
}
(A.9)
n
ξi(θ1Ω,Ω) ∑ i)2
were introduced. Precritical Phase. For c < c*, the behavior of the system is determined by eqs 1-3. The adsorption isotherm eq 2 for j ) 1, being transformed to the dimensionless quantities of eqs A.6 and A.7, becomes
1 θ1Ω exp - Ω(2a(θ1Ω)) bc ) Ξ (1 - θ1Ω)1/Ω
[
(A.15)
]
and the surface pressure is determined by the dimensionless equation of state
-(pΩ0) ) (1/Ψ)[ln(1 - θ1Ω) + θ1(Ω - Ω0) + a(θ1Ω)2] (A.16)
n
Ξ(θ1Ω,Ω) ) 1 +
]
1
and the dimensionless parameter
Ω ) 1 + Ω0
[
(A.10)
where b ≡ b1. The simultaneous solution of eqs A.8 and A.10 with eq A.9 for each c determines the dependencies of θ1 and Ω on c. The surface pressure can then be calculated from the equation of state (eq 1) which in dimensionless quantities reads
-(pΩ0) ) ln(1 - θ1Ω) + θ1(Ω - Ω0) + a(θ1Ω)2 (A.11)
The simultaneous solution of eqs A.8, A.9, and A.14-A16 for each c > c* gives the dependencies of p (and therefore Π), θ1, and Ω on c. Finally, the adsorption value is calculated from eq A.12. The Ward-Tordai procedure for the model can be implemented only if the equilibrium solution is known. Once the value Γ∞ is determined, the solution for any t can be obtained using the algorithm described elsewhere.21 In particular, the value of B introduced by eq A.3, which is essential for the numerical integration, has to be determined for the given set of model parameters and given values of Π* and Π∞. When B is known, the normalized concentration C and normalized adsorption γ are determined from eqs A.1 and A.4 via the adsorption at infinite time Γ∞ which follows from the solution at infinite time. This procedure is different for the two possible regimes of the system. The actual regime depends on the value of b as follows. For a given b, first the calculations are to be performed to determine the variables at the critical point as explained above. Next, it should be supposed that the system exhibits the two-phase regime. In this case, using the normalized variables and noting that γ(t ) ∞) ) 1, one can transform the equation of state in the limit t ) ∞ into the form
-p∞Ω0 exp[Ω(p∞ - p*)] ) θ/1 [ln(1 - θ1Ω) + θ1(Ω - Ω0) + a(θ1Ω)2] (A.17) θ1 Equation 8 becomes
and
Γ ) θ1/ωmin
(A.12)
Critical Point. The values at the critical point (denoted by *) are determined from the simultaneous solution of eq 1 at c ) c*,
-(p*Ω0) ) ln(1 - θ1Ω) + θ1(Ω - Ω0) + a(θ1Ω)2 (A.13) and eqs A.8 and A.9 with respect to θ1 ) θ/1 and Ω ) Ω*. Here p* ) Π*ω1/RT, cf. eq A.6. Then, the critical concentration c* follows from eq A.10, and the critical adsorption Γ* from eq A.12. Transcritical Phase. In the dimensionless form, eq 6 becomes
Ψ)
θ1 θ/1
Ψ∞ )
θ1∞ θ/1
exp[Ω∞(p∞ - p*)]
and the adsorption isotherm 5 takes the form
[
exp -
bc0 )
(A.14)
where θ/1 and p* are the values of θ1 and p, respectively,
]
2aθ1∞ Ψ∞
Ω∞θ1∞ Ξ∞ (1 - θ Ω )1/Ω∞Ψ∞ 1∞ ∞
(A.19)
where θ/1 and p* are the values of θ1 and p, respectively, at the critical point. The simultaneous solution of eqs A.17-A.20 with eqs A.8 and A.9 for θ1 ) θ1∞ and Ω ) Ω∞ enables one to calculate the value
Γ∞ ) θ1∞/ωmin exp[(p - p*)Ω]
(A.18)
(A.20)
(II) and p(II) ∞ can be determined. Then, if p∞ > p*, the system can undergo a phase transition to the transcritical branch, that is, a two-phase regime exists. Otherwise the system
Adsorption Kinetics of β-Casein
Langmuir, Vol. 20, No. 3, 2004 777
is unable to exist in a second phase for any model parameters, which is the single-phase regime of the system. For this regime, the set of equations to be solved is
1 θ1Ω exp - Ω(2a(θ1Ω)) bc0 ) Ξ (1 - θ1Ω)1/Ω
[
]
(A.21)
combined with eqs A.8, A.9, and A.11. These equations, being solved with respect to θ1 ) θ1∞ and Ω ) Ω∞, yield the values Γ∞ and p∞. Finally, with the normalizing coefficient Γ∞ determined for either of the two branches, the numerical integration of eq A.2 can be performed with the corresponding isotherm in the integrand. LA030332S
