1328
Langmuir 1999, 15, 1328-1336
Effect of the Reorientation of Oxyethylated Alcohol Molecules within the Surface Layer on Equilibrium and Dynamic Surface Pressure R. Miller,*,† E. V. Aksenenko,‡ L. Liggieri,§ F. Ravera,§ M. Ferrari,§ and V. B. Fainerman| MPI fu¨ r Kolloid- und Grenzfla¨ chenforschung, Rudower Chaussee 3, D-12489 Berlin, Germany, Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky Avenue, 252680 Kiev, Ukraine, ICFAM-CNR, Via De Marini 6, I-16149, Genoa, Italy, and Institute of Technical Ecology, Blvd. Shevchenko 25, 340017 Donetsk, Ukraine Received July 30, 1998. In Final Form: October 19, 1998 The diffusion-controlled adsorption kinetics of surfactant molecules at liquid interface is solved considering an additional rate constant for reorientation processes taking part within the surface layer. It is shown that a noninstantaneous reorientation can result in either acceleration or deceleration of surface tension changes as compared to a simple Langmuir model. The measured dynamic and equilibrium surface (interfacial) tensions for C10EO8 at the water/air and water/hexane interfaces agree well with the presented model derived for molecules which can exist in two (or more) orientation states. At low and medium surface pressures the reorientation model describes the experimental data well and predicts a more rapid decrease of surface tension as compared with the Langmuir model.
Introduction Equations of state of surface layers and adsorption isotherms describe the equilibrium behavior of adsorbed surfactant layers at liquid interfaces. These equations are the basis for calculations of adsorption kinetics and dynamic surface tension. The well-known equation proposed by Ward and Tordai1 represents a general relationship between the dynamic adsorption Γ(t) and the subsurface concentration c(0,t) for fresh nondeformed surfaces:
Γ)2
xDπ [c xt - ∫ 0
xt
0
c(0,t - t′)dxt′]
(1)
Here D is the diffusion coefficient, c0 is the bulk concentration, t is the time, and t′ is a dummy integration variable. Using eq 1 respective dependencies Γ(t) are obtained for different isotherms which serve as additional boundary conditions for the diffusion-controlled adsorption model. For example, in comparison to the Langmuir isotherm, the Frumkin isotherm predicts a lower initial value for the rate of surface tension changes.2,3 The adsorption isotherm which assumes aggregation within the surface layer also results in a slower surface tension decrease as compared with the Langmuir isotherm.4 This slower adsorption kinetics was often ascribed to the existence of adsorption barriers.5-11 The nature of the * Corresponding author. † MPI fu ¨ r Kolloid- und Grenzfla¨chenforschung. ‡ Institute of Colloid Chemistry and Chemistry of Water. § ICFAM-CNR. | Institute of Technical Ecology. (1) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 543. (2) Lin, S.-Y.; Lu, T.-L.; Hwang, W.-B. Langmuir 1994, 10, 3442. (3) Lin, S.-Y.; Hwang, W.-B.; Lu, T.-L. Colloids Surf. A 1996, 114, 143. (4) Aksenenko, E. V.; Fainerman, V. B.; Miller, R. J. Phys. Chem. B 1998, 102, 6025. (5) Baret, J. F. J. Chem. Phys. 1968, 65, 895. (6) Defay, R.; Prigogine, I.; Sanfeld, A. J. Colloid Interface Sci. 1977, 58, 498. (7) Bleys, G.; Joos, P. J. Phys. Chem. 1985, 89, 1027.
nondiffusional adsorption mechanism, energy barriers of adsorption or desorption for example, is not clear yet. It is possible that, at least for concentrated solutions, the nonequilibrium character of a surface layer can play a major role.8,9 The reorientation of surfactant molecules or their constituents (e.g. dipoles) is believed to be one of the explanations of the fact that the adsorption kinetics at liquid interfaces is faster than predicted by the diffusioncontrolled model. In the theoretical models developed so far,12-18 reorientation is treated as a first-order reaction, which results in a variation of the state of molecules within the surface layer. Experimental and theoretical results19-21 provide completely different estimates of the reorientation effect on dynamic surface tensions. In a number of experiments for Tritons containing different numbers of EO groups it was found that the surface tension decrease is faster than that expected from diffusion. This discrepancy can be understood by assuming reorientation processes of the Triton molecules at the surface. Two aspects of the reorientation (8) Fainerman, V. B. Usp. Khim. 1985, 54, 1613; Russ. Chem. Rev. 1985, 54, 948. (9) Liggieri, L.; Ravera, F.; Passerone, A. Colloids Surf. A 1996, 114, 351. (10) Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 1996, 178, 168. (11) Ravera, F.; Liggieri, L.; Passerone, A. J. Colloid Interface Sci. 1994, 169, 238. (12) Van den Tempel, M.; Lucassen-Reynders, E. H. Adv. Colloid Interface Sci. 1983, 18, 281. (13) Lucassen-Reynders, E. H. Colloids Surf. 1987, 25, 231. (14) Serrien, G.; Joos, P. J. Colloid Interface Sci. 1990, 139, 149. (15) Geeraerts, G.; Joos, P.; Ville´, F. Colloid Surf. A 1993, 75, 243. (16) Fainerman, V. B. Colloids Surf. 1991, 57, 249. (17) Joos, P.; Van Uffelen, M.; Serrien, G. J. Colloid Interface Sci. 1992, 152, 521. (18) Bois, A. G.; Panaiotov, I. I.; Baret, J. F. Chem. Phys. Lipids 1984, 34, 265. (19) Fainerman, V. B.; Makievski, A. V.; Joos, P. Colloids Surf. A 1994, 90, 213. (20) Fainerman, V. B.; Miller, R.; Makievski, A. V. Langmuir 1995, 11, 3054. (21) Fainerman, V. B.; Miller, R.; Wu¨stneck, R.; Makievski, A. V. J. Phys. Chem. 1996, 100, 3054.
10.1021/la980956b CCC: $18.00 © 1999 American Chemical Society Published on Web 01/21/1999
Reorientation of Oxyethylated Alcohol Molecules
mechanism are to be considered: the ability of the EO groups to be adsorbed at the water/air interface at low surface pressures; the adsorption relation derived by Joos for mixed adsorption layers,22 which says that the partial molar area of adsorbed molecules is larger at lower surface pressures and vice versa. Clearly, asymmetric surfactant molecules whose partial molar area depends on their orientation at the surface are capable of reorientation. Other surfactants, which do not possess any oxyethylene groups, can also demonstrate “super-diffusional” kinetics.23-25 Recently the dynamic surface tensions of aqueous solutions of alkyldimethylphosphine oxides with different alkyl chain lengths were studied.25 The experimental results obtained for the lower homologues (C8-C12) were found to correspond rather satisfactorily with the values predicted by the diffusion model for the Langmuir adsorption isotherm. For the higher homologues (C13-C15) however, the measured dynamic surface tensions are lower than the theoretically predicted. Fitting would lead to theoretical diffusion coefficients in e.g. C14DMPO which were 3-4 times larger than the physical values. A diffusion-controlled adsorption kinetics theory was proposed in ref 25 which considers two states of CnDMPO at the surface, each possessing its characteristic different partial molar area value.21,26 This theory agrees with the experimental data with typical diffusion coefficients. The theory proposed in ref 25 is a quasi-equilibrium model; i.e., it assumes that the composition of the surface layer under dynamic conditions and in equilibrium are essentially the same. However, the reorientation of molecules within the surface layer may require some time, which can affect both the adsorption rate and the rate of surface tension decrease. The reorientation process can account for new effects which were discussed earlier25 on a qualitative level. The present work aims at the derivation of a theoretical model which describes the kinetics of the transitions between various states of adsorbed molecules within the monolayer, based on the diffusion mechanism which governs the adsorption of reorienting surfactant molecules. The formulation of the present adsorption model has nothing in common with a barrier-controlled adsorption mechanism. It will be shown below that the kinetic effects arising from the transitions between the states of adsorbed molecules do not necessarily result in a deceleration of the surface tension decrease. On the contrary, under certain conditions an acceleration is predicted. To make comparison with other theoretical models valid for either dynamic or equilibrium conditions, the results of our measurements and dynamic and equilibrium surface tensions reported elsewhere for octaoxyethylated decyl alcohol C10EO8 at water/air and water/hexane interfaces will be used. The water/hexane interface has been selected due to the widespread use of alkane/water systems as models for the oil/water interface in the literature. Theory In line with refs 21, 25, and 26, we assume that adsorbed surfactant molecules can exist in two states 1 and 2, characterized by different values of the partial molar area (22) Joos, P.; Serrien, G. J. Colloid Interface Sci. 1991, 145, 291. (23) Miller, R.; Lunkenheimer, K. Colloid Polym. Sci. 1986, 264, 357. (24) Fang, J.-P.; Wantke, K.-D.; Lunkenheimer, K. J. Phys. Chem. 1995, 99, 4632. (25) Aksenenko, E. V.; Makievski, A. V.; Miller, R.; Fainerman, V. B. Colloids Surf. A 1998, 143, 311. (26) Fainerman, V. B.; Miller, R.; Wu¨stneck, R. J. Phys. Chem. 1997, 101, 6479.
Langmuir, Vol. 15, No. 4, 1999 1329
ω: ω1 > ω2. For either equilibrium or quasi-equilibrium surface layers, the adsorption in these states, Γ1 and Γ2, respectively, are related via the generalized adsorption equation proposed by Joos26
Γ1 Π (ω - ω2) ) β exp Γ2 RT 1
[
]
(2)
where R is the gas constant, T is the absolute temperature, Π ) σ0 - σ is the surface pressure, σ0 and σ are the surface tensions of the solvent and the solution, respectively, β ) (ω1/ω2)R, and R is a constant. The parameter β considers that the adsorption activity of surfactant molecules in state 1 is larger than in state 2. In fact, the nonideality of entropy of the surface layer gives rise to an extra exponential factor in the right-hand side of eq 2 which can additionally be incorporated into the constant β.21 It is seen that the ratio of adsorptions is determined both by the difference in molar areas and by the surface pressure. Under dynamic conditions, for example, when adsorption from the solution bulk takes place, the equilibrium condition (2) does not hold, if the transitions between the two states 1 and 2 are not instantaneous. This transition can be described by the kinetic scheme k12
} Γ2 Γ1 {\ k
(3)
21
where kij are the rate constants for the transition from state i into state j. The rate of this process can be expressed by a first-order equation
-
( ) ( )
dΓ1 dΓ2 ) ) k12Γ1 - k21Γ2 dt s dt s
(4)
The subscript “s” denotes that the transition between the two states takes place within the surface layer. For the quasi-equilibrium state, denoted by the superscript “(0)” it follows
[
k21 Γ1(0) Π(0) ) (0) ) β exp (ω - ω2) k12 Γ RT 1 2
]
(5)
Therefore, only one kinetic constant is required to describe the transition kinetics, because
[
]
Π(0)(ω1 - ω2) k21 ) k12β exp RT
(6a)
Equation 5 is also valid in the nonequilibrium state of the adsorption layer. Then k21 can be expressed by k12 via the relationship
[
k21 ) k12β exp -
]
Π(t)(ω1 - ω2) RT
(6b)
where the rate constant k21 is now a function of the actual value of Π. We will use this definition of the rate constants in the following calculations. Using definition (6a), slightly different results are obtained due to differences in the total values of k12 and k21. The variation in the adsorption states takes place both due to the diffusion-controlled flux of the surfactant from the bulk to the surface layer and the reorientation at the interface
1330 Langmuir, Vol. 15, No. 4, 1999
Miller et al.
( ) ( ) ( ) ( )
dΓ1 dΓ1 ) dt dt
b
dΓ2 dΓ2 ) dt dt
b
+ +
dΓ1 dt
s
dΓ2 dt
s
(7a) (7b)
c(0,t) )
where the subscript “b” stands for the flux of each “species” from the bulk phase. The total change of the surfactant amount at the interface is given by the sum of eqs 7a and 7b
( ) ( )
dΓ1 dΓ dΓ1 dΓ2 ) + ) dt dt dt dt
+
b
dΓ2 dt
(8)
b
where eq 4 was taken into account. Let us assume now that the diffusion-controlled fluxes from the solution bulk for the two states are equal to each other; that is, the two possible orientations of a surfactant molecule arriving at the surface are equally probable. Therefore, (dΓ1/dt)b ) (dΓ2/dt)b ) (1/2)(dΓ/dt), and the equations which describe the balance between the states 1 and 2 at a nondeformed surface layer follow from eqs 4, 6, and 7:
{ dΓ 1 dΓ ) + k{Γ dt 2 dt
[ ]} Π - βΓ exp[(ω - ω )]} RT
dΓ1 1 dΓ Π ) - k Γ1 - βΓ2 exp (ω - ω2) dt 2 dt RT 1 2
1
2
1
2
(9a)
bc2(0,t) )
ωΣΓ1 (1 - ωΣΓ)
ω1/ωΣ
ωΣΓ2 (1 - ωΣΓ)ω2/ωΣ
c1(0,t)Γ1/Γ1(0) + c2(0,t)Γ2/Γ2(0) Γ1/Γ1(0) + Γ2/Γ2(0)
(13)
The weighting is performed over the adsorption values according to the quasi-equilibrium state, eq 5. Clearly, for the equilibrium composition of the surface layer c1(0,t) ) c2(0,t) ) c(0,t). It is seen from eq 13 that the most significant contribution to c(0,t) results from that state the adsorption of which exceeds the equilibrium value. Thus,
c(0,t) )
{
ωΣ b
[
]
Γ22 Π(ω1 - ω2) Γ12 1 + β exp ω /ω ω /ω 1 Σ 2 Σ β(1 - Γω ) RT (1 - ΓωΣ) Σ
[
Γ1 + Γ2β exp -
]
Π(ω1 - ω2) RT
}
(14)
(9b)
Here k1 ) k12. More sophisticated models can also be used at this point, considering randomly oriented molecules moving toward the surface. This would result in somewhat different form of eqs 9. However, the surface effects (the structure of the subsurface water layer, electric potentials of oriented dipoles) and the conformational flexibility of the surfactant molecule chain would possibly reduce the asymmetry of the initial orientation distribution. In the following, we adopt the most simple and physically intelligible approach assuming that the two initial orientations are equally probable. The adsorption isotherms for the two states are26
βbc1(0,t) )
purpose we use the weighted average of the two concentrations to express the total subsurface concentration c(0,t):
Surface pressure can be calculated from the total adsorption and mean partial molar area as
Π)-
RT ln(1 - ΓωΣ) ωΣ
(15)
The numerical solution of the problem is rather similar to that of the quasi-stationary model given in ref 25. For this it is convenient to introduce values of all variables at equilibrium denoted by the superscript “(0)” and to transform all equations into dimensionless forms
γ ) Γ/Γ(0) γ1 ) Γ1/Γ(0) γ2 ) Γ2/Γ(0) w ) ωΣΓ(0) w1 ) ω1Γ(0) w2 ) ω2Γ(0)
(10a)
C ) c(0,t)/c0 B ) bc0 τ ) D (10b)
( ) c0
2
Γ(0)
t
(16)
Then, instead of eqs 1 and 9a, one obtains
where c1(0,t) and c2(0,t) are the subsurface concentrations of each state, b is a parameter representing the inverse concentration at about half surface coverage, and the mean partial molar area ωΣ and total adsorption Γ are defined as
ωΣΓ ) ω1Γ1 + ω2Γ2
(11)
Γ ) Γ1 + Γ2
(12)
It has to be noted that in equilibrium the adsorptions are related to each other by eq 2 only, and the subsurface concentrations in eqs 10 are equal to each other. A quasiequilibrium model was considered recently where the condition of an equilibrium surface layer composition was used to describe the diffusion-controlled adsorption process.25 In the general case of a nonequilibrium surface layer c1(0,t) * c2(0,t); therefore one has to postulate a relationship for the calculation of c(0,t) in eq 1. For this
γ) κ
(
2 [xτ xπ
)
∫0xτC(τ - τ′)d(xτ′)]
[
(17)
]
1 dγ dγ1 Π ) γ1 - βγ2 exp (ω1 - ω2) 2 dτ dτ RTΓ(0) (18)
where the “dimensionless characteristic transition time” is defined by κ ) D/k(c0/Γ(0))2. If the transition takes place instantly, then κ ) 0, and eq 18 reduces to eq 2. It can also be seen that at t f ∞ the derivatives on the left-hand side of eq 18 vanish; therefore, the value of Γ(∞) is identical to that in the quasi-equilibrium model. Also similarly to ref 25, the variable Ω ) w/w2 ) ωΣ/ω2 and the constant Ω1 ) w1/w2 ) ω1/ω2 are introduced, and the finite difference procedure for the numerical solution of the integral eq 17 is applied as described in detail in
Reorientation of Oxyethylated Alcohol Molecules
Langmuir, Vol. 15, No. 4, 1999 1331
Figure 1. Dynamic surface pressure calculated for different models: reorientation quasi-equilibrium model (bold solid); transfer kinetic model (dashed), and Langmuir model (thin solid), with ω ) ωmin ) 2.5 × 106 m2/mol, ωmax ) 7.5 × 105 m2/mol, R ) 2.0, k ) 0.01 s, and D ) 5 × 10-10 m2/s.
Figure 3. Dynamic surface tension calculated for different models: reorientation quasi-equilibrium model (1), transfer kinetic model (2-4), and Langmuir model (5), with ω ) ωmin ) 4.8 × 105 m2/mol, ωmax ) 1.1 × 106 m2/mol, R ) 7.5, D ) 5 × 10-10 m2/s, and k ) 0.0001 s (2), 0.01 s (3), 1 s (4).
Figure 2. Dynamic adsorption values in states 1 (solid) and 2 (dashed) for the kinetic (thin lines) and quasi-equilibrium (bold lines) models. The values of the parameters are the same as for Figure 1.
ref 27. At each successive step of the procedure, a first guess is made for the root of Ω ) Ω(γ) following from eq 18
κ
(
) {
[
Ω1 - Ω Ω1 - 1 1 dγ dγ1 Ω-1 β exp × )γ 2 dτ dt Ω1 - 1 Ω1 - 1 Ω ln(1 - γΩΓ(0)ω2)
]}
(19)
The derivatives on the left-hand side are calculated from a first-order finite differences. These values are introduced into the finite difference scheme of eq 17 via expressions (14) and (15) in dimensionless variables. The selfconsistency of the procedure is rather good. In general, it took not more than 7 iterations to obtain the solution accurate to within 14 digits at each time step of the numerical integration. Results of the calculated dynamic surface tensions and the composition of the surface layer are presented in Figures 1 and 2 for R ) 2. This value corresponds approximately to the contribution of nonideality of entropy discussed in ref 26 and is characteristic for non-oxyethylated nonionic surfactants.25 It is seen that the two-state quasi-stationary model predicts a more rapid kinetics of the surface tension decrease as compared to the Langmuir model. However, for low transition rate constants (k ) 0.01 s) the pressure increases less rapidly, and the peak value of Γ1 corresponds to higher time values (cf. Figure 2). The effect of slow transition kinetics becomes more significant for large R values characteristic to oxyethylated surfactants. In Figures 3 and 4 the surface pressure Π and Γ1 and Γ2 as a function of t, respectively, are shown for R ) 7.5. First, significant discrepancies between the reorientation and Langmuir models are observed. For (27) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces: Theory, Experiment, Application. In Studies of Interface Science; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1995; Vol. 1.
Figure 4. Dynamic adsorption values in state 1 (curves 1, 3, 5, 7) and state 2 (curves 2, 4, 6, 8) for the kinetic model at k ) 0.0001 s (curves 3 and 4), k ) 0.01 s (curves 5 and 6), and k ) 1 s (curves 6 and 7) and for the quasi-equilibrium model (curves 1 and 2). The parameters are the same as in Figure 3.
example, the quasi-stationary model for Π ) 15 mN/m predicts a surface lifetime seven times shorter than the Langmuir model (Figure 3). A decrease in k usually leads to a slower increase of the surface pressure not only as compared to the quasi-stationary reorientation model but for very low k and large times also with respect to the Langmuir model. However, for a slow transition between the states the dynamic pressure becomes even higher in certain points (shown by arrows in Figure 3 for k ) 10-2 s and k ) 10-4 s) than predicted by the quasi-stationary model. This phenomenon was discussed qualitatively in ref 25 and was ascribed to an oversaturation of the monolayer by molecules in state 1 possessing a maximum molar area. For k e 10-4 s, the model shows no transition between the states within surface layer; therefore, the adsorption values in both states are approximately equal to each other. At the same time, for the quasi-stationary process or for k ) 1 s, a sharp peak of the adsorption in state 1 occurs. To summarize the results of the model calculations, it can be concluded that slow transition kinetics of adsorbed
1332 Langmuir, Vol. 15, No. 4, 1999
Miller et al.
molecules produces relatively minor effects on dynamic surface tensions. This effect becomes most significant near the equilibrium, while at intermediate surface tensions the calculations performed for the quasi-equilibrium and transfer kinetic models lead to quite similar results. Thus the apparent ambiguity of some assumptions used in the formulation of the kinetic model do not affect significantly the results obtained. For example, when factors (2/3) and (1/3) are used in eqs 9 instead of (1/2), the differences between the quasi-equilibrium and kinetic models at low and medium Π values are slightly smaller. At the same time, the effect arising from the oversaturation of the monolayer by molecules in state 1 increases. Also, when the arithmetic mean value of c1(0,t) and c2(0,t) was used in eq 14 to estimate c(0,t) instead of the weighted average, the results show a slightly smaller difference between quasi-equilibrium and kinetic models near equilibrium. Materials and Methods The water used for the surfactant solutions was produced by a Millipore-MilliQ purifier fed with distilled water. C10EO8 by Sigma was used as received without further purification. All parts coming in contact with measurement liquids were cleaned with chromic acid and rinsed several times with distilled water and, at last, with MilliQ water. The hexane used was of spectrophotometric grade purity from Merk-Uvasol and had been purified according to a technique which consists of making impurities contained in the alkane to adsorb on solid basic alumna. This is obtained with several passages of the hexane through such an alumna column. After several cycles of passing hexane through this column and a final washing with water, any disturbing impurities had been removed and the interfacial tension had a value of 50.6 mN/m at 25 °C, showing no measurable time dependence. The dynamic interfacial tensions have been measured using a drop shape analysis technique based on ASTRA (automatic surface tension real-time acquisition) which has been described in detail elsewhere.28 This technique has an accuracy of (0.1 mN/m. Data for the water/air interface have been taken from the literature.29
Equilibrium. Let us consider first the equilibrium surface tension results obtained by Chang et al.29 for C10EO8 at the water/air interface using the pendent bubble method. The experimental surface tension isotherm29 is presented in Figure 5 along with the results of the calculations performed according to the reorientation, Langmuir, and Frumkin models. For an ideal surface layer, the basic equations of the reorientation model for two (or more) states within the monolayer were derived in refs 21 and 30. The equations of state for the surface layer and the adsorption isotherm for n > 2 are identical to eqs 16 and 10, respectively. For this case, however, the overall adsorption Γ ) Σig1Γi is the total adsorption of surfactant in all n states. The total adsorption in these equations can be expressed via the adsorption in state 1:
∑ i)1
Γ ) Γ1
R
i exp
ω1(i - 1) ωΣ
[
exp -
adsorption in the states with ωi > ωmin. In eq 20 the molar area of the ith state is defined as ωi ) ωmin + ∆ω(i - 1), while the total number of states is n ) 1 + (ωmax - ωmin)/ ∆ω. Here ∆ω is the molar area increment. The mean partial molar area for all states ω and the adsorption Γi in any ith state respectively can be expressed by n
ωΣ ) ω 1
]
(i - 1)Πω1 RT
The first exponential factor arises due to the nonideality of entropy of mixing. This factor (which was neglected here) and also iR correspond to a relative increase in (28) Liggieri, L.; Ravera, F. In Drops and Bubbles in Interfacial Research; Mo¨bius, D., Miller, R., Eds.; Elsevier: 1998; Vol. 6, p 239. (29) Chang, H. C.; Hsu, C. T.; Lin, S.-Y. Langmuir 1998, 14, 2476. (30) Fainerman, V. B.; Miller, R.; Wu¨stneck, R. J. Colloid Interface Sci. 1996, 183, 26.
(
n
[
iR exp n
[
)
iΠω1 RT
)
iΠω1 RT
]
(21)
(i - 1)Πω1
iR exp ∑ i)1
RT
]
(i - 1)Πω1 RT
(22)
The Frumkin equation of state and adsorption isotherm respectively are
Π)bc )
RT [ln(1 - θ) + aθ2] ω
(23)
θ exp(-2aθ) (1 - θ)
(24)
where θ ) ΓωΣ, and the parameter a is the intermolecular interaction constant. For a ) 0, these equations are simply the Szyszkowski-Langmuir equations:
Π)(20)
(
i(R+1) exp ∑ i)1 iR exp ∑ i)1
Γi ) Γ
Results and Discussion
n
Figure 5. Surface tension isotherm for C10EO8 at water/air interface. Experimental data are from Chang et al.29 ([) with the following theoretical isotherms: reorientation model for two states (ωmin ) 3.92 × 105 m2/mol, ωmax ) 1.06 × 106 m2/mol, R ) 3.8, bold solid line); reorientation model for 9 states (ωmin ) 2.41 × 105 m2/mol, ωmax ) 1.09 × 106 m2/mol, R ) 3.5, bold dotted line); Langmuir model (ω ) 5.65 × 105 m2/mol, thin solid line); Frumkin model (ω ) 3.08 × 105 m2/mol, a ) -5.1, thin dotted line).
RT RT ln(1 - Θ) ) ln(1 + bc) ωΣ ωΣ bc )
Θ 1-Θ
(25) (26)
It is seen from the results presented in Figure 5 that the Szyszkowski-Langmuir equation shows rather poor correspondence to the experimental data, while both the reorientation model and the Frumkin model agree with the experiment (not distinguishable in the graph). Note however that the Frumkin equation leads to a value of a ) -5.1, which is quite unrealistic and demonstrates that
Reorientation of Oxyethylated Alcohol Molecules
Figure 6. Dependencies of the monolayer coverage on surface pressure. The models used for the calculations are the same as in Figure 5.
the Frumkin isotherm is not suitable for the present system. The negative value of the constant corresponds to a repulsion between the adsorbed molecules, which is characteristic of solutions of ionic surfactants, where the parameter a compensates the Coulomb interaction.31 In such cases, however, the value of this parameter is usually much lower than that estimated here for C10EO8. We therefore conclude that for nonionic surfactants the constant a should be regarded as only a fitting parameter which has no physical meaning. For the data interpretation two reorientation models were further used, assuming 2 or 9 states of the adsorbed molecule in the monolayer, respectively. For the 9-state model of C10EO8 we assumed that the area per oxyethylene unit amounts to the increment of the molar surface area ∆ω. Calculations performed using a fitting program have resulted in the values ωmin ) 0.4 nm2 (per one molecule) and ωmax ) 1.8 nm2. Divided among eight EO groups, the difference between these areas amounts to ∆ω = 0.17 nm2, which agrees well with the actual area of a (CH2-CH2)O group. It follows from Figure 5 that the difference between the results of the two reorientation models is insignificant; therefore no definite conclusion can be drawn about the number of states the C10EO8 molecule possesses within the surface. Moreover, the maximum and minimum areas and the value of the constant R are very similar for the two models. It has to be noted that large positive R values are indicative of a high adsorption activity of the state which possesses maximum area ωmax; this agrees well with the results obtained in refs 19 and 20 showing that adsorption of oxyethylene chains at the water/air interface takes place. The fact that the value of R for C10EO8 is large supports the assumption that the contribution of nonideality of entropy, the first exponential factor in eq 20, can be neglected as compared with the preexponential factor. The dependence of the adsorption layer coverage Θ ) ΓωΣ on Π for all the models considered is shown in Figure 6. It is seen that for the reorientation models this value approaches 1 already at Π = 15 mN/m; further increase in surface pressure takes place at the expense of a decrease in the mean molar area of adsorbed C10EO8 molecules (Figure 7). Let us consider now the results obtained at the solution/ hexane interface using the pendent drop method and the automatic drop shape analysis software ASTRA. Figure 8 shows the experimental interfacial tension isotherm for C10EO8. Similarly to the water/air interface, the theoretical calculations were performed according to four models: the two reorientation, Langmuir, and Frumkin models. Again the experimental results are in perfect agreement with (31) Fainerman, V. B. Colloids Surf. 1991, 57, 249.
Langmuir, Vol. 15, No. 4, 1999 1333
Figure 7. Dependence of mean surface area per one C10EO8 molecule within the monolayer on surface pressure for the two reorientation models, two states (solid line) and 9 states (dotted line). Data used are as in Figure 5.
Figure 8. Surface tension isotherm for C10EO8 at water/hexane interface. Theoretical isotherms: reorientation model for two states (ωmin ) 4.82 × 105 m2/mol, ωmax ) 1.09 × 106 m2/mol, R ) 7.5, bold solid); reorientation model for 9 states (ωmin ) 2.41 × 105 m2/mol, ωmax ) 1.15 × 106 m2/mol, R ) 7.5, dashed solid); Langmuir model (ω ) 5.8 × 105 m2/mol, thin solid); Frumkin model (ω ) 3.8 × 105 m2/mol, a ) -10.8, thin dashed).
the two reorientation models and the Frumkin equation, while the Langmuir model yields worse agreement.32 However, for the Frumkin equation the value for a obtained from the fitting program is a ) -10.8, which is again quite unrealistic and lets us conclude that the model of interacting molecules is inapplicable to C10EO8 at the water/hexane interface. The geometric values of the parameters ωmax and ωmin for C10EO8 molecule at both interfaces are rather similar, while the R values are quite different: 3.5 at the water/air and 7.0 at the water/hexane interface. Thus the adsorption activity of oxyethylene groups at the water/hexane interface is significantly higher than that at the water/air interface (the ratio of the constants β for these two interfaces is about 10). This also agrees with the fact that the adsorption equilibrium constant b for C10EO8 at the water/hexane is 40 times higher than that at the water/air interface. For example, for the two-state model the value of b at the solution/air interface was found to be 2.9 × 108 cm3/mol, while at the water/hexane interface it is 1.2 × 1010 cm3/mol. The increase of the constant β results in an earlier (i.e., at lower Π) saturation of the surface layer (see Figure 9) and later (at higher Π values) beginning of a desorption of molecules in the state of maximum molar area (see Figure 10) as compared with the process at the solution/ air interface (cf. Figures 6 and 7). The adsorptions in states 1 and 2 (two-state model) as functions of the interfacial pressure for the two interfaces are shown in Figure 11. It is seen that the adsorption value in state 1 (ωΣ ) ωmax) (32) Ferrari, M.; Liggieri, L.; Ravera, F. Submitted for publication in J. Phys. Chem. B.
1334 Langmuir, Vol. 15, No. 4, 1999
Figure 9. Dependencies of the monolayer coverage on the interface pressure. Models and data used are the same as in Figure 8.
Miller et al.
Figure 12. Dynamic surface tension for aqueous solutions of C10EO8 at the air boundary for the concentrations: (4) c ) 4 × 10-9 mol/cm3, (0) c ) 6 × 10-9 mol/cm3, and (O) c ) 10-8 mol/cm3 from the data reported in ref 29. Theoretical calculations: Langmuir model (dotted lines); two-state model (solid lines). Table 1. Adsorption Parameters Determined for C10EO8 at the Water/Air and Water/Hexane Interfaces interface
Figure 10. Dependence of mean surface area per one C10EO8 molecule within the monolayer on the interface pressure for the two reorientation models: two states model (solid line) and 9 states model (dotted line) (data as in Figure 8).
Figure 11. Dependencies of the adsorption in the state 1 (1, L1) and state 2 (2, L2) on the interfacial pressure for the twostate model at water/air interface (1, 2) and water/hexane interface (L1, L2).
at both interfaces exhibits a maximum at Π = 8-10 mN/ m. The decrease of the adsorption of molecules in state 1 at the water/air interface becomes noticeable at Π = 10 mN/m, while for the water/hexane interface this decrease becomes evident at Π = 20 mN/m. On the contrary, the adsorption of molecules in state 2 (ωΣ ) ωmin) is more significant at the water/air interface. The values of adsorption in this state for the two interfaces become approximately equal at Π > 30 mN/m. To summarize, the experimental interfacial tension isotherms of C10EO8 at the solution/air and solution/hexane interfaces exhibit perfect agreement with the models that assume the capability of molecules to adsorb in two or more states within the surface layer. The adsorption characteristics of the C10EO8 molecule in both compact and unfolded states (including the area per oxyethylene unit) estimated from the experiment using the fitting program turn out to be roughly the same for the two interfaces and are close to the geometric dimensions of the molecule. The interface to hexane increases the
param
water/air
water/hexane
ωmin (m2/mol) ωmax (m2/mol) R b (cm3/mol)
(3.8-4.0) × 105 (1.0-1.2) × 106 3-4 2.9 × 108
(4.7-4.9) × 105 (1.0-1.4) × 106 5-8 1.2 × 1010
adsorption activity of C10EO8 molecule as a whole and even more significantly increases the adsorption activity of the oxyethylene chains. This effect is explained by an additional (as compared with the water/air interface) decrease of the free energy of the system due to the interaction of hydrophobic chains and the CH2-CH2 groups of C10EO8 molecule with the hydrocarbon molecules. The adsorption characteristics for C10EO8 at water/air and water/hexane interfaces presented in Figures 5 and 8 were calculated from the best fit of the experimental data by the models. The procedure used to minimize the deviation was described earlier.26 The optimum adsorption parameters for the two state model are summarized in Table 1 (ranges of the parameters are calculated with respect to the error in surface tensions). Dynamics. The experimental dynamic surface tensions for C10EO8 at the water/air interface, measured in ref 20 using the pendent bubble method, are compared with the calculations for the two models in Figure 12. The adsorption characteristics used were those estimated from the surface tension isotherm given in Figure 5. The diffusion coefficient D ) 5 × 10-10 m2/s was calculated from the equation proposed by Wilke and Chang.33 One can see that the Langmuir equation overestimates the dynamics, i.e., the value of D required to match the experimental data would be 1.5 × 10-9 m2/s, which is 3 times higher than the expected value. At the same time, the two-state model agrees well with the experimental data. Note that eq 17 describes the kinetics of adsorption at a plane interface with an infinite liquid volume, provided that no convection takes place. For the bubble immersed into the solution, sphericity and convection can lead to a more rapid adsorption than that calculated from eq 17. However, all the conclusions drawn above concerning the adsorption dynamics are relevant for the time interval t < 100 s (cf. Figure 12). For this case the thickness of the diffusion boundary layer (πDt)1/2 does not exceed 20% of the bubble (33) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264.
Reorientation of Oxyethylated Alcohol Molecules
Langmuir, Vol. 15, No. 4, 1999 1335
Figure 13. Dynamic surface tension for aqueous solutions of C10EO8 at the hexane boundary for the concentration c ) 4 × 10-8 mol/cm3 (O). Theoretical calculations: reorientation quasiequilibrium model (1-4); Langmuir model (dotted line). The parameters used were ω ) 5.8 × 105 m2/mol, ωmin ) 4.8 × 105 m2/mol, ωmax ) 1.4 × 106 m2/mol, D ) 5 × 10-10 m2/s, K* ) 1.5, and R ) 4 (1), 5 (2), 6 (3), and 7 (4).
diameter; therefore, both the sphericity and the convection transfer can be neglected. We thus conclude that the “super-diffusion” adsorption kinetics of C10EO8 at water/ air interface does really exist, being the result of the selfregulation of surfactant molecule states within the surface layer. In the analysis of adsorption kinetics at liquid/liquid interfaces, diffusion of C10EO8 into the oil phase has to be taken into account. In our experiments performed using the expanded drop method,28 no preliminary saturation of the oil phase with C10EO8 was made. For this case, instead of eq 1, the following expression should be used (cf. ref 34):
Γ)2
xDπ [cxt - (1 + Kx D )∫ Dm
xt
0
c(0,t - t′)dxt′
]
(27)
Here K is the equilibrium distribution coefficient of surfactant between the oil and water phases and Dm is the surfactant diffusion coefficient in the oil phase. To estimate the reduced distribution coefficient defined by K* ) K(Dm/D)1/2, the value of K was determined experimentally, and the ratio Dm/D was calculated using the equation proposed in ref 35. The values obtained in this way are K ) 0.85 and K* ) 1.8, which agree rather well with the results reported in refs 34 and 35 for Tritons X at the water/nonane interface: K* ) 1.5 for Triton X-45 possessing 4 to 5 oxyethylene groups, and K* ) 0.5 for Triton X-100 with 10 EO groups. It should be mentioned that the interfacial tension isotherm for C10EO8 (Figure 8) reasonably agrees with that obtained for Triton X-100 published in ref 34. For decyldimethylphosphine oxide (C10DMPO), the adsorption activity of which is close to that of C10EO8, a value of K* ) 1.3 was found.36,37 In Figure 13, experimental data for c0 ) 4 × 10-8 mol/ cm3 are compared with calculations performed for the Langmuir model and the quasi-stationary reorientation model using values of R within the range of 4-7. These values were found to be optimum for equilibrium conditions. It is seen that the behavior is similar to that characteristic for the water/air interface: the Langmuir (34) Fainerman, V. B.; Zholob, S. A.; Miller, R. Langmuir 1997, 13, 283. (35) Zholob, S. A.; Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 1997, 186, 149. (36) Ravera, F.; Ferrari, M.; Liggieri, L.; Miller, R. Prog. Colloid Polym. Sci. 1997, 105, 346. (37) Ferrari, M.; Liggieri, L.; Ravera, F.; Amodio, C.; Miller, R. J. Colloid Interface Sci. 1997, 186, 40.
Figure 14. Same experiment as in Figure 13. Theoretical calculations: reorientation quasi-equilibrium model with K* ) 4.5 (solid); kinetic model with k ) 0.5 s-1 (dashed). The parameters used were ωmin ) 4.8 × 105 m2/mol, ωmax ) 1.4 × 106 m2/mol, D ) 5 × 10-10 m2/s, and R ) 4 (bold lines) and 5 (thin lines).
model overestimates the dynamic surface tensions, while the reorientation model leads to rather good agreement with the experimental data for R ) 4-5 and realistic diffusion coefficient at short adsorption times t < 5 s. Similarly to the water/air interface, to achieve agreement with the Langmuir model within this time range, it was necessary to increase the diffusion coefficient by a factor of 3. Note that the dynamic interfacial tension data presented here are only measured for the short time range up to about 10 s. In contrast the isotherm data given in Figure 8 are equilibrium values and thus much lower than the dynamic γ(t) values of Figures 13 and 14. However, the isotherm data are the basis of the model calculations of dynamic interfacial tensions as the model requires the values of the isotherm parameters. In the region t > 5 s, larger differences between theory and experiment were found, as shown in Figure 13 (the quasi-stationary model predicts a faster interfacial tension decrease). To improve the agreement calculations for a slower transition rate between the two states in the surface layer have been performed. The results of the calculations for k ) 0.5 s-1 are shown in Figure 14. It is seen that the kinetic model improves the agreement in the long time range, but for t = 0.6 s, due to an oversaturation by molecules in state 1, the discrepancy between the theory and experiment becomes larger, as compared with the quasi-equilibrium model (Figure 13). The increase of the k value smoothes the distinctions between the models, while decrease leads to larger differences at t = 0.6 s and worsens the agreement with the experiment in the short time range. The discrepancies for t > 2 s can partly be ascribed to the fact that the theoretical model, eq 27, does not account for the actual geometry of the system. When adsorption from a drop takes place, the diffusion flux from the bulk to the drop surface is smaller and the flux from the drop to the oil phase is greater than those characteristic of a flat surface. This results in a decrease of the adsorption dynamics and increase of interfacial tensions. The effects caused by a possibly low transition rate between the states does not affect the adsorption of C10EO8 at the water/air interface (Figure 12). The deceleration of the surface tension decrease which results from these effects could possibly be observed at higher C10EO8 concentrations, when Π > 12 mN/m. However, such results were not presented in ref 29. It is interesting to compare the theoretical and experimental results with data reported in ref 25 for tetradecyldimethylphosphine oxide (C14DMPO). For the higher homologues (C14-C16) two states were shown to exist at
1336 Langmuir, Vol. 15, No. 4, 1999
the water/air surface layer.40 This results in a faster surface tension decrease for C14DMPO solutions, as compared with the Langmuir model. However, the dynamic surface pressure calculated from the quasi-equilibrium reorientation model for Π > 8 mN/m and a surface lifetime of ca. 100 s was found to be lower than the experimental data. This effect was presumably ascribed in ref 25 to the low transition rate between the adsorption states. It is seen now that this result agrees rather well with the kinetic reorientation model developed here. For C14DMPO, the slow transition rate results in an acceleration of the surface tension decrease at Π > 8 mN/m due to a decrease in σ for a slow transition Γ1 f Γ2 which leads to an oversaturation of the adsorption layer with molecules in state 1. This oversaturation results in an increase (within a certain range) of the total adsorption, surface layer coverage, and surface pressure, which was demonstrated theoretically (cf. Figures 3 and 14). Conclusions The problem of diffusion-controlled adsorption kinetics of reorientable surfactant molecules is formulated and solved for the case when the reorientation process within the surface layer requires some time. It is shown that this (38) Liggieri, L.; Ravera, F.; Ferrari, M.; Passerone, A.; Miller, R. J. Colloid Interface Sci. 1997, 186, 46. (39) Fainerman, V. B. Kolloidn. Zh. 1981, 43, 926. (40) Makievski, A. V.; Grigoriev, D. O. Colloids & Surf. A 1998, 143, 233.
Miller et al.
noninstantaneous reorientation can result in either acceleration or deceleration of the surface tension decrease, depending on the adsorption characteristics of the different molecular states and on the actual surface lifetime. Faster (for medium Π values) or slower (large Π values) decrease of σ is caused by an oversaturation of the surface layer by the state possessing maximum molar area. The values of both dynamic and equilibrium surface (interfacial) tensions of C10EO8 measured at the water/ air and water/hexane interfaces are in good agreement with the model which assumes two or more states of the adsorbed molecule, with the same values of adsorption parameters used to describe either the equilibrium or dynamic behavior. The nature of the interface gives rise to both an increase in the adsorption activity of the entire C10EO8 molecule at the water/hexane interface and of its oxyethylene chain. At low and intermediate surface pressures, the reorientation model predicts (in agreement with experimental data) a faster surface tension decrease, as compared with the Langmuir model. The transition kinetics of the C10EO8 molecules from the state of large molar area to the state of lower area produces only minor effects on the dynamic surface tensions. Acknowledgment. The work was financially supported by projects of the European Community (INCO ERB-IC15-CT96-0809), and the ESA (Topical Team and FAST projects). LA980956B
