Dynamic Elasticity of Adsorption Layers in the Presence of Internal

J. Phys. Chem. B 2001, 105, 195-203

195

Dynamic Elasticity of Adsorption Layers in the Presence of Internal Reorientation Processes Francesca Ravera,*,† Michele Ferrari,† Reinhard Miller,‡ and Libero Liggieri† Istituto di Chimica Fisica Applicata dei MaterialisCNR, Via De Marini 6, 16149 GenoVa, Italy, and Max Planck Institut fu¨ r Kolloid und Grenzfla¨ chenforschung, Max-Planck-Campus, Haus 2, Am Mu¨ hlenberg 2, D-14476 Golm, Germany ReceiVed: July 24, 2000; In Final Form: October 19, 2000

It has been shown that reorganization processes in the adsorbed layer, such as the orientation of the adsorbed molecules, can play an important role in determining the equilibrium and dynamic behavior of surfactant systems. In fact, a model allowing for two possible adsorption states of the molecules, with different molar surface areas, has been successfully applied to describe the equilibrium properties and the aging of liquid interfaces. In this work, the problem of the rheological behavior of an adsorption layer in the presence of reorientation processes is addressed, which represents a rather important need to increase the insight into the properties of these surfactant systems and to support the experimental investigations. Thus, the main goal of this paper is to calculate the frequency dependence of the dynamic dilational elasticity, , within the framework of the two-state model. According to the results, the phase of is no longer a monotonic function of the frequency, as predicted by the classical Langmuir model, but suggests the presence of a local maximum. Local extrema are found also in the modulus and phase of as a function of the equilibrium surface pressure. This behavior is qualitatively in agreement with experimental observations available so far. The dynamic elasticity of insoluble monolayers is also derived as a particular case of the general approach given here.

Introduction Dilational elasticity is an important property of adsorption layers at liquid interfaces, one that is correlated to the dynamic adsorption mechanism and has important consequences in many technologically relevant phenomena, such as liquid film physics, droplet or bubble coalescence, stability of foams, and emulsions.1,2 Usually, surfactant systems are characterized from the point of view of the surface adsorption isotherm and the dynamic interfacial tension during the aging of the surface. However, to understand the dynamic adsorption mechanisms in depth, it is useful to study them also from the viewpoint of rheological properties. In fact, an appropriate frequency study of the surfacetension response to controlled variations of the surface area can put in evidence all the processes involved in the adsorption dynamics. Dynamic surface elasticity and its relation to the transfer of surfactants from/to the surface has been widely theoretically studied,3-10 and some experimental methods have been applied11-21 to evaluate it. The expression for the dilational viscoelasticity modulus, found by Lucassen3,5 as a function of the frequency, is largely used22,23 also for liquid-liquid systems24 and for surfactant mixtures.25 This approach accounts for only the surface area changes and the consequent relaxation due to the diffusional exchange with the bulk, as the mechanisms responsible for the surface-tension variation. More recently, the effect of the micellization process also has been taken into account.26 However, in several experimental investigations,13,15,16,18 a non-monotonic behavior has been observed for the viscoelastic * Corresponding author. Fax, +039 0106475700; e-mail, ravera@ icfam.ge.cnr.it. † Istituto di Chimica Fisica Applicata dei Materiali. ‡ Max-Planck Institut fu ¨ r Kolloid und Grenzfla¨chenforschung.

modulus of the layer as a function of the perturbation frequency or of the surfactant concentration. This observation cannot be explained in the framework of these classical approaches, suggesting the existence of further relaxation mechanisms. In recent years, the idea that reorganization processes, such as reorientation or conformational modifications of adsorbed molecules, can occur within the surface layer for some surfactant systems has been theoretically and experimentally assessed. In fact, it has been shown27,28 that a model considering two possible states for the adsorbed molecules, characterized by different surface molar areas and surface activity, can remarkably improve the description of the equilibrium and dynamic properties of many surfactant systems. From the experimental point of view, the presence of such internal processes has been directly or indirectly assessed for the polyoxyethylated nonionic surfactants (CiEj)29-31 and alkyl dimethyl phosphine oxides (CnDMPO).32 Direct investigations33,34 of CiEj adsorption layers by neutron reflection have shown that molecular reorientations occur in the layer. Moreover, multistate models have been applied successfully to predict the adsorption behavior of some proteins.35 In most of these works, it has been shown that the utilization of a two-state isotherm, in the framework of the diffusioncontrolled adsorption, provides a good description of the aging of fresh interfaces. This means that, in these cases, processes internal to the adsorbed layer with changing surface area, such as orientation, should exist, but with characteristic times shorter than the diffusion relaxation time. Thus the dynamic aspect of these processes can be evidenced only when the adsorption dynamics is investigated in a comparable time scale. In fact, because of the very short characteristic time of the internal reorganization processes, a suitable way to access the dynamic properties of these internal processes is to study the system’s response to harmonic perturbations of the surface area.

10.1021/jp002614w CCC: $20.00 © 2001 American Chemical Society Published on Web 12/12/2000

196 J. Phys. Chem. B, Vol. 105, No. 1, 2001

Ravera et al.

Indeed, by tuning the perturbation frequency, it is possible to investigate the properties of the system in suitable time windows. The aim of this work is to provide a theoretical support to the study of the reorientation process at the interface by using the dilational properties of the adsorption layer. To reach this goal, the classical Lucassen approach must be extended to account for the dynamic exchange of molecules between two adsorption states which are characterized by different molar areas and surface activities. A first extension of this approach has been given in a recent work,36 which provides the dynamic elasticity behavior in the limit of high-frequency perturbations or an insoluble monolayer. In the present work, a different mathematical approach has been adopted which allows us to overcome these limitations. The theoretical treatment provides an expression for the dynamic elasticity modulus and the phase shift between the area variation and the surface tension as a function of the perturbation frequency. The description for insoluble surfactant monolayers results as a particular case of the theory, when the monolayer does not exchange molecules with the bulk. An approach that considers the presence of the timedependent processes internal to the adsorbed layer is also given in ref 7, which leads to a general expression of the surface elasticity. In that work surface chemical reactions are considered, but the surfactant reorientation has not been made explicit. Theory

where F-1 is the inverse Fourier operator and γ0 and A0 are the surface tension and the area corresponding to the reference state. This latter approach has been applied when nonharmonic variations of the area, such as trapezoidal or rectangular wave forms, for example, have been used.6 For both mentioned approaches, to obtain information about the adsorption mechanisms from the dilational experimental studies, it is mandatory to know the expression of (ν). In the following paragraphs, this expression is obtained assuming the two-state model proposed by Fainerman et al.27,37 Accordingly, a surfactant is considered which, once adsorbed to the surface, can assume two different states, 1 and 2, characterized by the surface molar areas ω1 and ω2 and by the parameters R and b2, related to the different surface activities (see the Appendix for details). These two states are expected to correspond to two different orientations with respect to the surface, so that the process internal to the adsorbed layer is in fact an orientation process. The surface area, harmonically oscillating at a given frequency ν, reads

A ) A0 + A ˜ e2πiνt

where A0 is the reference area, A ˜ is the oscillation amplitude, and ν is the frequency. This area variation provides an oscillating variation of the total adsorption Γ and of the partial adsorptions Γ1 and Γ2 of the two surface states, 1 and 2

The dilational properties of a surface system containing a soluble surfactant are described by the elasticity modulus, which is defined as

)

δγ δ ln A

(1)

and represents the response of the surface tension to compression-dilational perturbations of the surface area. Because of the presence of relaxation processes, such as transfer from/to the surface or internal processes, this elasticity modulus depends on the characteristic time of the input perturbation and can exhibit a viscoelastic behavior. Thus, it is convenient to express it as a frequency function, writing it as the surface-tension response to a harmonic perturbation of the surface area of a given frequency ν. In the frequency domain, (ν) is a complex quantity, and if small perturbations are considered, it can be written as

(ν) ) ||e

iφ

(2)

where φ is the phase difference between the surface tension and the applied area perturbation. Thus can be experimentally evaluated by measuring the amplitude and the phase shift of the surface-tension response of the system for different frequency values. As the frequency dependence of is linked to the adsorption properties, such measurements are appropriate to characterize the mechanism of the adsorption. In another approach widely used,6,9 assuming small relative area variation, the surface is considered to behave as a linear system. As any function describing the variable surface area can be written as a superposition of harmonics, the surfacetension response can be written using the Fourier transform formalism of linear systems

γ(t) - γ0 )

∫0tF-1((ν)) ln

(

)

A(t - τ) A0

dτ

(3)

(4)

Γj ) Γ0j + Γ ˜ je2πiνt

(5)

and the concentration is then

c(t) ) c0 + c˜ seaxe2πiνt

(6)

where x is the distance from the surface toward the solution and the apex 0 refers to the reference state that is considered at adsorption equilibrium. Γ ˜ j and c˜ s are complex quantities containing a phase shift with respect to the area oscillation, i.e.

˜ rj exp(iδj) Γ ˜j ) Γ

(7)

c˜ s ) c˜ rs exp(iδ)

(8)

where Γ ˜ rj and c˜ rs are the oscillation amplitudes. Because c(x, t) must be a convergent solution of the Fick equation

a ) -(i + 1)

xπνD

(9)

where D is the diffusion coefficient. In the framework of the two-state model, the surface tension is a function of Γ1 and Γ2; thus, from eq 1, the elasticity can be written as

) -01

d ln Γ1 d ln Γ2 - 02 d ln A d ln A

(10)

∂γ ∂ ln Γj

(11)

where

0j ) -

Dynamic Elasticity of Adsorption Layers

J. Phys. Chem. B, Vol. 105, No. 1, 2001 197

which can be expressed in terms of equilibrium quantities by using eqs A2 and A6 of the Appendix:

[

0j ) Γj0 Π0

Π0(ω0 - ωj) 0

Γω

0

+

]

RTωj

ω (1 - Γ0ω0) 0

(12)

reference equilibrium state, so that the oscillating quantities can be developed at a first order. Thus, because g is oscillating as a result of the oscillating Γ, at the first-order approximation we get

g ) g0 +

ω0

(dΓdg) (Γ - Γ ) 0

0

(22)

and are the equilibrium surface pressure and average surface area, respectively; R is the gas constant; and T is the absolute temperature. By taking into account the area dilation, the diffusional exchange, and the orientation process, the mass balance at the surface reads28

By comparing eq 15 with eq 22 and using eq 5, one obtains

dΓj d ln A ∂c + Φjor(Γ1,Γ2) + Γj ) qj D dt dt ∂x x)0

where g′ ) (dg/dΓ)0. A similar development can be done for the sublayer concentration, which is also oscillating due to the oscillation in Γ; this leads to

( )

(13)

g˜ ) g′Γ ˜

where qj is the probability that an adsorbing molecule takes up the state j in the adsorbed layer (obviously, q1 ) 1-q2) and Φjor are the fluxes of molecules between the two states

Φ2or ) -Φ1or ) Φor ) kor(Γ1 - gΓ2)

(14)

In eq 14, kor is the orientation coefficient and, by definition, g is the ratio between Γ1 and Γ2, which makes the orientation flux zero. In the case of a static surface area this ratio would be Γ10/Γ20, but when the surface area is oscillating, it is also an oscillating quantity characterized by a phase shift with respect to the surface-area change, which means it can be written as

g ) g0 + g˜ e2πiνt where

g0

)

Γ10/Γ20.

[

From eq 13, one finds

]

∂c qjD + Φorj d ln Γj ∂x x)0 )- 1d ln A dΓj dt

( )

(∂x∂c)

x)0

)Φ ˜ diffe2πiνt

∂cs 0 Γ ˜ ∂Γ

(24)

As reported in the Appendix, g′ and (∂cs/∂Γ)0 can be calculated from the equilibrium relationships supplied by the model.27,37 When we introduce the characteristic frequencies of the diffusion and orientation processes, ν0 ) (D/2π)((∂cs/∂Γ))2 and νor ) (kor/2π), respectively, the fluxes Φ ˜ diff and Φ ˜ or read

Φ ˜ diff ) -2π(i + 1)

x

ν0ν ˜ 2) (Γ ˜ +Γ 2 1

(25)

˜ 1 - (g0 + g′Γ20)Γ ˜2 Φ ˜ or ) 2πνor[(1 - g′Γ20)Γ

(16)

respectively. ˜ 1, we again use the mass To find an expression for Γ ˜ 2 and Γ balances (eq 13) for the two adsorption states. With some rearrangements, they lead to

-1

[

x [ x [ x [ x

(17)

q1(i + 1)

Moreover, when we consider only the first-order terms from eqs 14 and 15, the orientation flux reads

˜ or e2πiνt Φor ) Φ

(19)

Φ ˜ or ) kor(Γ1 - g0Γ ˜ 2 - g˜ Γ20)

(20)

with

Using eqs 16, 17, and 19, eq 10 becomes

]

The aim now is to write all quantities in this expression in terms of frequency, equilibrium values, and isotherm parameters. To do that, we assume the condition of small perturbation of the

]

]

where the second-order and higher terms have been neglected and K ˜ )A ˜ /A0. The solution of this set provides Γ ˜1 )

(

Γ0 q1(1 - i)

x

) (

ν0 νor + i (g0 + g′Γ20) - Γ10 1 + (1 - i) 2ν ν

(

)(

νor 1 - i (1 + g0) 1 + (1 - i) ν

2πiνΓ ˜1 + ) 01 2πiνΓ ˜ 1 - q1Φ ˜ diff + Φ ˜ or 2πiνΓ ˜2 (21) 02 2πiνΓ ˜ 2 - q2Φ ˜ diff - Φ ˜ or

]

ν0ν - νor(1 - g′Γ20) Γ ˜1 + 2 ν0ν + νor(g0 + g′Γ20) Γ iν + q2(i + 1) ˜ 2 + iνΓ20K ˜ )0 2 (28)

q2(i + 1)

(18)

(26)

ν0ν + νor(1 - g′Γ20) Γ ˜1 + 2 ν0ν ˜ 2 + iνΓ10K ˜ ) 0 (27) - νor(g0 + g′Γ20) Γ 2

iν + q1(i + 1)

where

Φ ˜ diff ) -(i + 1)xπDνc˜ s

( )

(15)

Using eq 6, the diffusion flux reads

D

c˜ s )

(23)

Γ ˜2 )

(

x

-Γ0 1 + q1(1 - i)

(

x)

x) ν0 2ν

) (

(29)

x)

νor ν0 - i (1 - g′Γ20) + Γ10 1 + (1 - i) 2ν ν

)(

x)

νor 1 - i (1 + g0) 1 + (1 - i) ν

K ˜

ν0 2ν

ν0 2ν

K ˜

ν0 2ν

(30)


Ravera et al.

Finally, using eqs 25, 26, 29, and 30, eq 10 provides

1 + (1 - i)ξ ) 01

1 + g0 [q1(1 - i)ξ + iλ(g0 + g′Γ20)] g0

(1 + (1 - i)ξ)(1 - iλ(1 + g0))

- 02g0 × 1 + (1 - i)ξ -

1 + g0 [1 + q1(1 - i)ξ - iλ(1 - g′Γ20)] g0

(1 + (1 - i)ξ)(1 - iλ(1 + g0)) (31) where, to simplify the discussion, ξ and λ have been defined as ξ ) xν0/2ν and λ ) νor/ν, and the expressions of 0i and g′ are provided by eqs 12 and A9, respectively, as a function of the thermodynamic parameters. Results and Discussion In the framework of this approach, it is important to emphasize that the whole dynamic process occurring during the oscillation of the surface is really the result of different processes, each with its own characteristic time, or, as is better in the present context, its own characteristic frequency. More precisely, the processes involved are the transfer of molecules from the bulk to the surface, which is described by a diffusion flux; the orientation changes inside the adsorbed layer; and the oscillation of the surface. The first two processes depend on the physical characteristics of the surfactant system, expressed by the isotherm parameters, the bulk concentration, the diffusion coefficient, and the orientation rate, whereas the last process can be considered as an imposed experimental condition. An important result of this work is that, as expressed by eq 31, the orientation process can be put in evidence by dilational studies with a suitable choice of the imposed frequency. It must be emphasized that eq 31 has been derived by assuming the existence of a process internal to the adsorbed layer, which results in a transition between two states characterized by different occupational areas without specifying the nature of the process. Thus, although the calculation has been referred to as an orientation process, which has been shown to exist for some surfactants, its validity is more general. For example, some systems undergoing changes in the conformational structure of the adsorbed molecules could be described by the two-state model from both the thermodynamic and the kinetic points of view. According to eq 31, is frequency-dependent through ξ and λ, which makes it easy to understand the influence of the characteristic times of orientation and diffusion. In many cases, considering ν0 , νor is physically realistic. In fact, the available investigations29-32 about the surfactant systems described by the two-state model show that the aging of fresh interfaces is controlled by diffusion. As already underscored, for these systems it is difficult to get information about the orientation process with experimental studies based on the aging of a fresh surface, whereas the rheological approach presented here offers an effective tool. To this aim, it is useful to investigate the behavior of the surface elasticity as a function of frequency, under different conditions, as predicted by eq 31. Figure 1 shows the modulus and the phase of , as a function of λ, for two sets of isotherm parameters corresponding to the systems C10E8 at the water-air interface (Figure 1a) and C10E8 at the water-hexane interface (Figure 1b), respectively, as reported previously.30 For the sake of

Figure 1. (a) Modulus (1, 2) and phase (3, 4) of the surface elasticity versus λ, by eq 31. Curves 1 and 3 are calculated for ω1 ) 1.06 × 1010 cm2/mol, ω2 ) 3.92 × 109 cm2/mol, R ) 3.8, corresponding to the C10E8 in water/air system, and for Π0 ) 10 mN/m and νor ) 100 ν0; curves 2 and 4 correspond to the Langmuir model with Γ∞ ) 1.77 × 10-10 mol/cm2 (obtained by the fitting on the equilibrium data). (b) Modulus (1, 2) and phase (3, 4) of the surface elasticity versus λ, by eq 31. Curves 1 and 3 are calculated for ω1 ) 1.09 × 1010 cm2/mol, ω2 ) 4.82 × 109 cm2/mol, R ) 7.5, corresponding to C10E8 in water/ hexane system, and for Π0 ) 10 mN/m and νor ) 100 ν0; curves 2 and 4 correspond to the Langmuir case with Γ∞ ) 1.72 × 10-10 mol/cm2 (obtained by the fitting on the equilibrium data).

comparison, the figures also show the curves calculated according to the Lucassen formula,3 using the Langmuir model with the value of the saturation adsorption (Γ∞) provided by fitting to the equilibrium adsorption data. The existence of the orientation process introduces a step rise in the modulus while increasing the frequency, which for λ , 1 reaches values larger than those predicted by the Langmuir model. For such high frequencies, the interface does not exchange surfactant molecules with the bulk, behaving like an insoluble monolayer. Moreover, the distribution of these molecules between the two states is about frozen in the equilibrium distribution. Thus the system loses its relaxation capability with a consequent increase in the interfacial rigidity. Considering the phase shift, or the phase, the most evident characteristic introduced by the two-state model is the presence of a relative maximum in the frequency dependence. It is useful to introduce two limiting cases, which allow us to study the separate effects of the relaxation due to the orientation process and to the diffusive exchange with the bulk. In the ν0 , νor context, it is interesting to consider the two



cases corresponding to imposed oscillation frequencies comparable to ν0 and νor, respectively. Limit Case A: ν = ν0 , νor. Under these hypotheses, the terms in λ2 are dominant with respect to the other frequencydependent terms. Thus, after some rearrangement, eq 31 becomes

)

(

)

g′Γ0(01 - g002) 1 + ξ + iξ + + 01 02 g0(1 + g0) 1 + 2ξ + 2ξ2

(32)

This particular situation corresponds to an oscillation period comparable with the characteristic time of diffusion, which is assumed to be much larger than the orientation time, so that the orientation process can be considered at equilibrium during the oscillation. Thus, this is the framework in which the classical calculation of the dynamic elasticity has been carried out, where only diffusive relaxation is considered. In fact, the dependence on ξ is the typical one3 for such relaxation processes, whereas the factor in brackets is specific to the surface model. It can be shown that

01 + 02 )

Γ0 RT 1 - ωΓ0

(33)

which corresponds to the Gibbs elasticity provided by the Langmuir model, assuming Γ∞ ) 1/ω. It can also be shown that

01 - g002 ∝ (ω1 - ω2)

(34)

Thus, if only one adsorption state exists, eq 32 reduces to the usual expression of the surface elasticity obtained by the classical Lucassen approach using the Langmuir isotherm. Limit Case B: ν = νor . ν0. This particular case corresponds to an oscillation period comparable with the characteristic time of the orientation process, which is much smaller than the characteristic diffusion time. In this case all the terms in ξ are vanishing, and eq 31 reads:

(01 - g002) g′Γ0 (35) g0 1 + λ (1 + g0)

Figure 2. (a) Surface elasticity modulus versus λ obtained with the same parameters as in Figure 1b. Solid line calculated from the exact eq 31; dashed line and dotted lines from the approximated expressions eqs 35 and 32, respectively. (b) Surface elasticity phase versus λ, obtained with the same parameters as in Figure 1b. Solid line calculated from the exact eq 31; dashed and dotted lines from the approximated expressions eq 35 and 32, respectively.

In this situation, the diffusion exchange is negligible and the only effective relaxation process is reorientation. If only one adsorption state exists (ω1 ) ω2) or for frequencies much larger than νor, eq 35 provides ) 01 + 02, which, as already underlined, corresponds to the Gibbs elasticity of the Langmuir model with Γ∞ ) 1/ω. Limit Case C: No Exchange of Surfactant. From the general eq 31, it is possible to derive the viscoelasticity modulus for a surface layer which behaves like an insoluble monolayer. In fact, for such systems, Φdiff is vanishing, as is ν0, which leads to neglecting the terms in ξ for all the values of the frequency, providing again eq 35. As expected, if ω1 ) ω2, which means also absence of internal processes, is constant and equal to the Gibbs elasticity, and this is also the limit at high frequency. Moreover, it is clear that a frequency dependence of for an insoluble monolayer exists only in the presence of internal processes. Otherwise is constant and equal to the Gibbs elasticity. The partial effects of the orientation and relaxation processes due to adsorption flux from the bulk can be observed in Figure 2a,b, where the modulus and the phase of the elasticity

calculated by the approximated eqs 32 and 35 are compared with the general result given by eq 31. The two approximated curves coincide with the general curve in two ranges of frequencies, (λ < 0.1) and (λ > 10). The remarkable separation into two frequency regions, corresponding to the different processes, depends on the fact that the two approximations have been carried out under the hypothesis νor , ν0. As far as this hypothesis is relaxed, the description requires the use of the general eq 31. Figure 3 shows the dependence of on the ratio between the characteristic frequencies. As far as νor approaches ν0, the two processes merge. In any case, it is clear that the introduction of the orientation process superposes its characteristic features on the behavior predicted by the diffusion relaxation models: a step rise in the modulus and a local maximum in the phase shift. From Figures 1, 2, and 3, it is evident that the frequency corresponding to the local maximum of the phase shift occurs close to the inflection point of the step rise in the modulus. In fact, this frequency provides the largest displacement from the equilibrium distribution: at smaller frequencies the system is closer to equilibrium because the relaxation process is more

) 01 + 02 +

λ2(1 + g0) - iλ 2

2


Ravera et al.

Figure 4. Surface elasticity phase versus λ, by eq 31, with R ) 0, Π0 ) 10 mN/m, νor ) 100 ν0, and for different values of ω1/ω2. ω1 ) ω2 corresponds to the Langmuir model.

considering the curves obtained by eq 35, plotted in Figure 2 (dashed curves). Its characteristic features are originated by the oscillation of the reference equilibrium state. While passing through a maximum at the position predicted by eq 36, the phase is zero at low and at high frequencies, where the adsorbed molecules are, for opposite reasons, at equilibrium distribution. In fact, although at high frequencies the distribution is frozen in the state corresponding to g0, at low frequencies it oscillates according to eq 15. Correspondingly, the modulus reaches the limit value 01 + 02 at high frequency, whereas at low frequencies it tends to

(01 - g002) || ) 01 + 02 + g′Γ0 g0(1 + g0) Figure 3. (a) Dependence of the surface elasticity modulus behavior on the νor/ν0 ratio. The isotherm parameters are the same as in Figure 1b. (b) Dependence of the surface elasticity phase behavior on the νor/ ν0 ratio. The isotherm parameters are the same as in Figure 1b.

effective, whereas at larger frequencies the possibility of displacing the system from equilibrium is reduced. The position of the relative maximum can be calculated from eq 35, and reads

νmax ) νor(1 + g0)

x

1+

g′Γ0 01 - g002 g0 01 + 02

(36)

Thus, νmax is linear in νor, by a coefficient which is linked only to the thermodynamic properties of the system, in particular to the relative variation of the reference distribution with Γ, i.e., g′/g0. Therefore, eq 36 provides an effective tool for the experimental evaluation of the rate coefficient of the orientation process. Note also that the factor outside the square root is equal to the characteristic time of the orientation process, τor ) 1/kor(1+g0).28 For a given R, the amplitude of the maximum changes with the ratio between the two-surface molar area. This is true even for R ) 0, i.e., for equal surface activities of the two states. As shown in Figure 4, for this value of R, such amplitude reaches a maximum around ω1/ω2 ) 2. The behavior of the system in the absence of surfactant exchange, as depicted in the above case C, can be studied by

(37)

Because it is always g′ < 0, owing to eq 34, this value is always smaller than 01 + 02, when ω1 > ω2. The dependence of the modulus and phase on the surface pressure, for different values of the frequency and for the isotherm parameters typical for C10E8 at water/air, are reported in parts a and b, respectively, of Figure 5. Analogous results have been obtained for the water/ hexane interface. The behavior of the modulus is rather different from the monotonic prediction of the Langmuir model, showing two local maxima which are smoothed as far as the frequency increases with respect to νor. At the limit of very high frequencies, a change of concavity remains related in some cases to a local maximum. It is interesting to note that the position of such maximum is close to the value of Π, which makes g0 ) 1, that is, Γ10 ) Γ20. This is true also in the case of no surfactant exchange (insoluble monolayer), as shown in Figure 6a. The phase also shows an interesting behavior as a function of the surface pressure (Figure 5b): a local maximum appears as the frequency increases, approaching νor. At very high frequencies a monotonic behavior is restored. A different behavior is observed for the phase of the insoluble monolayer (Figure 6b): it always passes through a maximum whose amplitude decreases with increasing frequency. At very high frequencies the phase tends to zero for all surface pressure values. The discontinuities in the phase plots are due to the utilization of the inverse trigonometric functions, the physical phase shift between the area changes, and the surface tension response being continuous between 0 and 2π.


Figure 5. (a) Surface elasticity modulus versus surface pressure for water/air (same isotherm parameters as in Figure 1a); two-state model with νor ) 10 Hz and λ ) 10 (1), λ ) 1 (2), limiting elasticity for high frequency (3), and Gibbs elasticity with Langmuir model (4). The vertical dotted line corresponds to Π ) 13.9 mN/m, for which g0 ) 1. (b) Surface elasticity phase versus surface pressure for water/air (same isotherm parameters as in Figure 1a); two-state model with νor ) 10 Hz and λ ) 10 (1), λ ) 1 (2).

At present, dynamic elasticity data of adsorbed monolayers are very scarce, in particular for frequencies larger than 1 Hz, which does not allow a satisfactory validation of the present theory. However, the predictions agree at least qualitatively with the few available experimental observations. In fact, a nonmonotonic behavior of the modulus of versus Π has often been observed,13,15,16,18 which cannot be explained on the basis of the classical models. The present theory is compared with dynamic elasticity data of the C10DMPO adsorption layer at water/air interface, collected by Wantke et al.13 with the use of an oscillating bubble technique, in Figures 7 and 8. C10DMPO is a soluble surfactant whose adsorption properties are fairly described by a two-state model. By using the two-state isotherm parameters reported by Aksenenko et al.,32 the present model gives a good description of the modulus versus frequency data in the range 0-100 Hz. Although the description of the modulus versus concentration is not as good, the two-state model provides a substantial improvement compared to the Langmuir model, and accounts for the appearance of a local maximum (Figure 7).


Figure 6. (a) Elasticity modulus versus surface pressure in the case of no exchange of surfactant with the bulk (insoluble monolayer) for λ ) 10 (1), λ ) 1 (2), λ ) 0.01 (3), and isotherm parameters as in Figure 1a. The vertical dotted line corresponds to Π ) 13.9 mN/m, for which g0 ) 1. (b) Elasticity phase versus surface pressure in the case of no exchange of surfactant with the bulk (insoluble monolayer) for λ ) 10 (1), λ ) 1 (2), λ ) 0.01 (3), and isotherm parameters as in Figure 1a.

It is interesting to compare the results given here with those obtained for a monolayer that does not exchange molecules with the bulk, reported in a previous work36 on the dynamic elasticity in the presence of reorientation. Qualitatively, in both works the modulus may show local extrema as a function of Π. The agreement is quite good on the whole surface-pressure range only for small values of λ, whereas in all the other cases a quantitative agreement is found only at low Π. These discrepancies possibly are due to the adoption here of an oscillating reference state, expressed by eq 15, an effect that was not taken into consideration in the previous treatment. Conclusion A model has been developed which yields a general expression of the dynamic surface elasticity as a function of frequency, to be written in terms of the characteristic diffusion and orientation frequency and of the isotherm parameters. The model predicts a deviation from the classical diffusioncontrolled behavior of the dynamic elasticity. In particular, the phase presents a characteristic maximum as a function of the


Ravera et al. Therefore, the oscillating bubble/drop technique,11,13,14 which allows the modulus and the phase of as a function of the frequency to be measured by interfacial tension response to sinusoidal area perturbations, is the most suitable one for these investigations. At present such a technique is being applied to study the dynamic elasticity of CiEj surfactant adsorbed layers. Acknowledgment. The work was partially supported by the European Space Agency Topical Team “Progress in Emulsion Science and Technology”, by the ESA projects FAST and FASES, and by the Italian Space Agency (contract ASI-CNR, ARS 9915). The authors thank Dr. V. I. Kovalchuk of the Institute of Biocolloid Chemistry - Kiev, for some helpful discussion. Appendix

Figure 7. Elasticity modulus versus frequency. Experimental data by Wantke et al.13 about C10DMPO at water-air, for c ) 5 × 10-7 mol/ cm3 (b) and c ) 5 × 10-8 mol/cm3 (9). Theoretical curves by the Langmuir model (dashed lines) obtained with Γ∞ ) 3.85 × 10-10 mol/ cm2 and aL ) 4.96 × 10-8 mol/cm3, and the two-state model (solid lines), with ω1 ) 5 × 109 cm2/mol and ω2 ) 2.7 × 109 cm2/mol, R ) 0, and νor ) 10 Hz.

Summary of the Equilibrium Relationships for the TwoState Model. In the framework of the model proposed by Fainerman et al.,27,37 the surfactant can adsorb in two different states, characterized respectively by the molar surface areas ω1 and ω2 and by the parameters b1 and b2, which are related to the different surface activities. These two states are expected to correspond to two different orientations with respect to the surface, so that the process internal to the adsorbed layer is in fact an orientation process. A specific relationship is also assumed between the surface activities and the molar areas:

b2 ) b1

() ω2 ω1

R

(A1)

thus, for R ) 0 the surface activity is independent of the molar surface area, and for R > 0 the molecules adsorbed in the state with the larger surface area have higher surface activity. The equilibrium distribution between the adsorptions in the two states, Γ1 and Γ2, is the expression of the Brown-Le Chatelier principle and depends on the surface pressure Π:

() [

ω1 Γ1 ) Γ2 ω2

Figure 8. Elasticity modulus versus C10DMPO bulk concentration for two values of the frequency: experimental data by Wantke et al.,13 theoretical curves by the Langmuir model (solid lines) and the twostate model (dashed lines) with the same isotherm parameters as in Figure 8 and for νor ) 10 Hz.

exp -

]

Π(ω1 - ω2) RT

(A2)

where R is the gas constant and T the absolute temperature. Thus, the adsorption in the state with smaller molar area is favored as the surface coverage increases. The surface isotherms and the Π-c isotherm are, respectively

c)

frequency. This means that experimental investigations of (ω) can be fruitfully used to investigate the orientation process in the adsorbed layer, if any, even if τD . τor. In principle, the results of this theoretical work can be employed in all the experimental approaches aimed at studying the adsorption properties through the surface rheology, i.e., evaluation of the response to harmonic and nonharmonic area perturbations. However, to gain insight into the orientation process, it is necessary to investigate the adsorption dynamics on a smaller time scale. To this aim, from the experimental point of view, a surface rheology approach can be more adequate, where the response of the interfacial tension to suitable perturbations of the surface area is studied: for example, the response to oscillations with frequency related to the characteristic time of the orientation process.

R

ωΓ1

()

ω1 R b (1 - ω1Γ1 - ω2Γ2)ω1/ω ω2 2

)

ωΓ2 b2(1 - ω1Γ1 - ω2Γ2)ω2/ω ) b2

ωΓ ω1 R (1 - ωΓ)ω1/ω + (1 - ωΓ)ω2/ω ω2

[( )

]

)

(A3)

and

(

1 - exp -

c)

Πω RT

)

[( ) ( ) ( )]

b2

ω1 ω2

R

Πω1 Πω2 exp + exp RT RT

(A4)



where ω is the average molar surface area

ω)

ω1Γ1 + ω2Γ2 Γ 1 + Γ2

(A5)

The surface equation of state is

Π)-

RT RT ln(1 - Γ1ω1 - Γ2ω2) ) - ln(1 - Γω) ω ω

(A6)

where Γ is the total adsorption Γ1 + Γ2. For the aims of the present paper, it is necessary to calculate the derivative of the ratio g ) Γ1/Γ2 and of the sublayer concentration in Γ at the equilibrium. From equation A4 it is

g)

ω - ω2 ω1 - ω

(A7)

and, using this latter equation together with eqs A2 and A6, one finds

1Γ)

[( ) ω1 ω2

-R

]

ω - ω2 ω1 - ω ω

(ω/(ω1-ω2))

(A8)

From these two latter equations, it is possible to calculate dg dg dΓ ) / ) dΓ dω dω

g′ )

1+g ) ω1 - ω (ω1 - ω) 1+g (1 - ωΓ) ln(1 ωΓ) + + Γ g(ω1 - ω2) ω2 ω2

[

)

ω1 - ω ω2

]

[

1+g

]

(ω1 - ω)Π ω1 - ω -Πω/RT 1+g + e ωRT g(ω1 - ω2) ω2

(A9)

By substituting eq A8 into eqs A3, a relationship between the equilibrium concentration c and the average area ω can be found, which, together with eq A8 and according to

dc dc dΓ ) / dΓ dω dω

(A10)

can be used to calculate an expression for (dc/dΓ) in terms of ω. References and Notes (1) Edwards, D.; Brenner, H.; Wasan, D. T. Interfacial Transport Process and Rheology; Butterworth-Heinemann: Boston, MA, 1991.

(2) Ivanov, I. B.; Kralchevsky, P. A. Colloids Surf., A 1997, 128, 155. (3) Lucassen J.; Hansen, R. S. J. Colloid Interface Sci. 1967, 23, 319. (4) Lucassen, J.; Van Den Tempel, M. Chem. Eng. Sci. 1972, 27, 1283. (5) Lucassen, J.; Van Den Tempel, M. J. Colloid Interface Sci. 1972, 41, 41. (6) Loglio, G.; Tesei, U.; Miller, R.; Cini, R. Colloids Surf. 1991, 61, 219. (7) Noskov, B. A.; Loglio, G. Colloid Surf., A 1998, 143, 167. (8) Miller, R.; Wu¨stneck, R.; Kra¨gel, J.; Kretzschmar, G. Colloids Surf., A 1996, 111, 75. (9) Miller, R.; Loglio, G.; Tesei, U.; Schano, K. H. AdV. Colloid Interface Sci. 1991, 37, 73. (10) Monroy, F.; Giermanska Kahn, J.; Langevin, D. Colloids Surf., A 1998, 143, 251. (11) Lunkenheimer, K.; Kretzschmar, G. Z. Phys. Chem. (Leipzig) 1975, 256, 593. (12) Loglio, G.; Tesei, U.; Cini, R. J. Colloid Interface Sci. 1979, 71, 316. (13) Wantke, K.-D.; Lunkenheimer, K.; Hempt, C. J. Colloid Interface Sci. 1993, 159, 28. (14) Fruhner, H.; Wantke, K.-D. Colloids Surf., A 1996, 114, 53. (15) Wantke, K.-D.; Fruhner, H.; Fang, J.; Lunkenheimer, K. J. Colloid Interface Sci. 1998, 208, 34. (16) Tian, Y.; Holt, R. G.; Apfel, R. E. J. Colloid Interface Sci. 1997, 187, 1. (17) Crone, A. H. M.; Snik, A. F. M.; Poulis, J. A.; Kruger, A. J.; Van Den Tempel, M. J. Colloid Interface Sci. 1980, 74, 1. (18) Jayalakshmi, Y.; Langevin, D. J. Colloid Interface Sci. 1997, 194, 22. (19) Loglio, G.; Tesei, U.; Cini, R. J. Colloid Interface Sci. 1984, 100, 393. (20) Fang, J.; Wantke, K.-D.; Lunkenheimer, K. J. Colloid Interface Sci. 1996, 182, 31. (21) Johnson, D. O.; Stebe, K. J. J. Colloid Interface Sci. 1994, 168, 21. (22) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1973, 42, 573. (23) Lucassen, J. Faraday Discuss. Chem. Soc. 1975, 59, 76. (24) Miller, R.; Loglio, G.; Tesei, U. Colloid Polym. Sci. 1992, 270, 598. (25) Garret, P. R.; Joos, P. J. Chem. Soc., Faraday Trans. 1976, 69, 2161. (26) Noskov, B. A.; Alexandrov, D. A.; Miller, R. J. Colloid Interface Sci. 1999, 219, 250. (27) Fainerman, V. B.; Miller, R.; Wu¨stneck, R.; Makievski, A. V. J. Phys. Chem. 1996, 100, 7669. (28) Ravera, F.; Liggieri, L.; Miller, R. Colloids Surf., A 2000, 175, 51. (29) Miller, R.; Aksenenko, E. V.; Liggieri, L.; Ravera, F.; Ferrari, M.; Fainerman, V. B. Langmuir 1999, 15, 1328. (30) Ferrari, M.; Liggieri, L.; Ravera, F. J. Phys. Chem. B 1998, 102, 10521. (31) Liggieri, L.; Ferrari, M.; Massa, A.; Ravera, F. Colloids Surf., A 1999, 156, 455. (32) Aksenenko, E. V.; Makievski, A.; Miller, R.; Fainerman, V. B. Colloids Surf., A 1998, 143, 311. (33) Lu, J. R.; Hromadova, M.; Thomas, R. K.; Penfold, J. Langmuir 1993, 9, 2417. (34) Lu, J. R.; Lee, Z. X.; Thomas, R. K.; Staples, E. J.; Thompson, L.; Tucker, I.; Penfold, J. J. Phys. Chem. 1994, 98, 6559. (35) Miller, R.; Fainerman, V. B.; Makievski, A.; Kra¨gel, J.; Grigoriev, D. O.; Kazakov, V. N.; Sinyachenko, O. V. AdV. Colloid Interface Sci. 2000, 86, 39. (36) Miller, R.; Aksenenko, E. V.; Fainerman, V. B. J. Phys. Chem., in press. (37) Fainerman, V. B.; Miller, R.; Aksenenko, E. V.; Makievski, A. V.; Kra¨gel, J.; Loglio, G.; Liggieri, L. AdV. Colloid Interface Sci. 2000, 86, 83.

Dynamic Elasticity of Adsorption Layers in the Presence of Internal

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