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Dynamic Surface Elasticity of Adsorption Layers in the Presence of a Surface Phase Transition from Monomers to Large Aggregates Roberto Palazzolo, Francesca Ravera, Michele Ferrari, and Libero Liggieri* Istituto per l’Energetica e le Interfasi, Sezione di Genova-CNR, via De Marini 6, 16149 Genova, Italy Received November 21, 2001. In Final Form: February 1, 2002 It has been shown that reorganization processes in the adsorbed layer, like the phase transition of the adsorbed molecules from gaseous to liquid state, can play an important role in determining the equilibrium and dynamic behavior of surfactant systems. In fact, a model [Vollhardt D. et al. J. Phys. Chem. B 2000, 104, 5744] allowing for two possible adsorption states of the molecules, in the two different phases, has been successfully applied to describe the equilibrium properties and the aging of liquid interfaces. The problem of the rheological behavior of an adsorption layer in the presence of phase transition processes is here 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 the paper is the calculation of the frequency dependence of the dynamic surface elasticity (or dilational viscoelasticity), within the framework of the phase transition model. The dynamic surface elasticity of the insoluble monolayers, is also derived as a particular case of the general approach given here. According to the results, significant differences exist with respect to the predictions of the classical models. The specific features of the modulus and phase of the surface elasticity allow information about the kinetics of the aggregation process to be derived by experimental data.
1. Introduction The rheological properties of a liquid surface play an important role in many industrial and biophysical processes such as extension and contraction of lung alveolus, ink-jet printing, distillation, flotation, liquid-liquid extraction, emulsification, foaming, and so on. Despite the importance of the topic, only in the past three decades have efforts been made in investigating and modeling the surface tension response to area changes when relaxation phenomena occur. Because of such relaxation processes, the interface can be considered to behave as a linear viscoelastic medium. Under these conditions, the variations of the surface tension, γ, from its equilibrium value γ0, in response to a variation of the interfacial area from A0 to A, can be represented by the scalar Boltzman-Volterra constitutive equation,1 in terms of an integral nucleus H,
γ(t) - γ0 )
(
∫0tH(τ) ln
)
A(t - τ) A0
dτ
(1)
where τ is a dummy integral variable. As pointed out by Noskov and Loglio,1 using the linear system theory, the nucleus H can be expressed as a complex modulus that represents the transfer function between an input signal, which is the area deformation, and the output signal, i.e., the surface tension. Thus, H is representative of all the adsorption properties of the system. Accordingly, by applying Fourier transform to eq 1, one gets
(jν) ) J{H(t)} )
J{δγ(t)} J{δ ln A(t)}
(2)
where J is the Fourier operator, ν is the frequency, and * Corresponding author. Fax: +39 0106475700. E-mail: liggieri@ icfam.ge.cnr.it. (1) Noskov, B. A.; Loglio, G. Colloids Surf., A 1998, 143, 167.
j ) x-1. is a complex function of the frequency commonly called dynamic surface elasticity, or dilational viscoelasticity. The dependence of on the frequency expresses the fact that the relaxation processes, such as diffusion-adsorption, surface reorganization, and so on, affects the surface tension variation on a time scale comparable with their own characteristic time scale. The advantage of introducing is that, under the hypothesis of small oscillations at a given frequency ν, eq 2 can be recast as
(jν) )
dγ d ln A
(3)
This equation was first used by Lucassen-Reynders,2 and then by Lucassen,3 to derive for diffusion-controlled adsorption.
1 + ξ + jξ ) 0 1 + 2ξ + 2ξ2
(4)
where ξ ) xνD/2ν; νD is the characteristic frequency of the diffusion-controlled surfactant exchange
νD )
( )
D dcs 2π dΓ
0
2
(5)
D is the diffusion coefficient in the bulk, cs is the surfactant concentration just close to the surface (sublayer concentration), Γ is the adsorption, and 0 ) (∂Π/∂ ln Γ)0 is the Gibbs elasticity. The index 0 refers to the equilibrium state. As reported previously,4 the non-monotonic behavior of the modulus versus the surface pressure, often observed (2) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1973, 42, 573. (3) Lucassen, J. Faraday Discuss. Chem. Soc. 1975, 59, 76. (4) Ravera, F.; Ferrari, M.; Miller, R.; Liggieri, L. J. Phys. Chem. B 2001, 105, 195.
10.1021/la0156917 CCC: $22.00 © 2002 American Chemical Society Published on Web 03/28/2002
Dynamic Surface Elasticity of Adsoption Layers
experimentally, cannot be explained in the framework of such a classical approach, suggesting the existence of further relaxation mechanisms other than the surfactant diffusion. Van Uffelen and Joos5 show by measuring the dynamic surface tension that n-dodecanol adsorbed monolayers at water interface collapse above a critical surface pressure. A similar behavior was observed for n-decanol monolayers by Lin et al.,6 who suggested the existence of a phase transition for the surfactant adsorbed at the interface. Recently, by direct imaging of the adsorbed layer with a Brewster-angle microscope, Vollhardt and collaborators7,8 confirmed such a behavior for n-dodecanol and for some biological amphiphiles. They observed the formation of large aggregates on the surface. These authors have shown that for the different kinds of surfactant, the aggregation occurs above a critical surface pressure, i.e., above a critical adsorption. Hence, in analogy with the theoretical approach to bulk micellization, quasi-chemical methods have been developed9 to describe such systems. However, despite their attractive simplicity and their success in describing systems undergoing micellization,10 the quasi-chemical methods explicitly introduce the aggregation number, i.e., the number of molecules per aggregate, which cannot be determined experimentally. To overcome this problem, the quasi-chemical model has been extended to consider infinitely large aggregates, in layers of surfactants either insoluble8,11 or soluble.7 This is equivalent to considering the appearance of a surface phase transition. In fact, the same equations have been derived12 by assuming the existence of a first-order surface phase transition from monomers to large aggregates. As shown by the comparison with the experiment, these latter models seem to be more adherent to reality than the ones considering small aggregates. Moreover they have the advantage of involving less unknown parameters since they do not require the knowledge of the aggregation number. As already shown for surfactant molecular reorientation in the adsorbed layer,4,13 aggregation is expected to introduce a further relaxation mechanism in the adsorption layer. Thus, investigating the rheological properties of such monolayers can give access to the features of the aggregation process. To this aim, it is necessary to express the dynamic surface elasticity of the monolayer, as defined by eq 3, in the framework of such a monomer/aggregate phase transition model. 2. Theory To approach the problem, the following picture of the adsorption process is assumed. As the adsorption, Γm, of the free monomer surfactant progresses, the surface pressure, Π, increases until reaching a critical value, Πc. Above this point, the van der Waals attraction abruptly increases, triggering the nucleation of the new phase. Below the critical pressure, only free monomers are (5) Van Uffelen, M.; Joos, P. J. Colloid Interface Sci. 1993, 158, 452. (6) Lin, S.-Y.; Hwang, W.-B.; Lu, T.-L. Colloids Surf., A 1996, 114, 143. (7) Vollhardt, D.; Fainerman, V. B.; Emrich, G. J. Phys. Chem. B 2000, 104, 8536. (8) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145. (9) Fainerman, V. B.; Aksenenko, E. V.; Miller R. J. Phys. Chem. B 2000, 104, 5744. (10) Israelachvilli, J. Intermolecular and Surface Forces; Academic Press: New York, 1992. (11) Vollhardt, D.; Fainerman, V. B. Colloids Surf., A 2000, 176, 117. (12) Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 2000, 232, 1. (13) Ravera, F.; Liggieri, L.; Miller, R. Colloids Surf., A 2000, 175, 51.
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adsorbed. Above the critical pressure, the free monomer and the aggregate phase coexist at the interface, requiring a specific thermodynamic and kinetic model for the description of the adsorption layer behavior. 2.1. Thermodynamics. Below the critical pressure, the adsorption layer is suitably described by the Langmuir model.14 Thus, the equation of state reads
Π)-
RT ln(1 - ωmΓm) ωm
(6)
and the adsorption isotherm is
Γm )
bc ωm(1 + bc)
(7)
where c is the surfactant bulk concentration, ωm is the monomer molar area, and b is the isotherm parameter expressing the surface activity of the solute. The equilibrium relationships of the system above the critical pressure have already12 been worked out under the hypothesis that the surfactant molecules do not change their molar area when passing from the free to the aggregate state. In that case, above the critical pressure, the adsorption of monomers, Γm, remain fixed to the critical value, Γc, i.e.
Γm ) Γc
(8)
while only the adsorption of the molecules in the aggregate state, Γm*, increases with the surface pressure. The surface equation of state and the isotherm for a soluble layer are in that case given by the following set of equations7
Π)-
RT Γm ln(1 - ωmΓ) ωm Γ
bc )
ωmΓ
(9)
(10)
[1 - ωmΓ]θ
where θ ) Γm/Γ, c is the sublayer concentration, Γ ) Γm + Γm* is the total adsorption, and b is an isotherm parameter linked to the surface activity of the free monomer. R and T are the gas constant and the absolute temperature. Note that according to eq 8, Γ ) Γc + Γm* and θ ) Γc /(Γc + Γm*). However, as experimentally observed in evaluating the adsorption isotherm for insoluble monolayers,8 the hypothesis that the surfactant molecules do not change their molar area when passing from the free to the aggregate state seems in many cases unrealistic. Thus, the above model has been improved to account for a change of such molar area above the critical pressure, resulting8 in the dependence of Γm on Π according to
Γm ) Γc gΠ
(11)
where gΠ is a correction factor given by
(
gΠ ) exp -
)
Π - Πc ωmδ RT
(12)
The packing of the molecules in the aggregates is (14) Lyklema, J. Fundamental of Interface and Colloid Science; Academic Press: San Diego, 2000; Vol. 3, Chapter 4.
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change under the effect of this latter process and of the monomer exchange with the bulk. They are no longer linked by the equilibrium relationships reported above, and in particular eq 11 does not hold anymore. However, the mechanical properties of the interfacial layer and its composition are always at equilibrium. Hence, the surface equation of state, in the form of eq 13, can be used to link Π with Γm and Γm*. After these considerations, the interfacial mass balances read
1 d(ΓmA) ) ΦD - Φagg A dt
(15)
1 d(Γm*A) ) Φagg A dt
(16)
ΦD is the diffusion flux of the monomer Figure 1. Dependence of the adsorption of monomers, Γm (solid line), and of the aggregated molecules, Γm* (dashed line), on the surface pressure Π. Aggregation starts after the critical pressure Πc. The curves are calculated from eqs 12 and 13, for ωm ) 0.99 109 cm2/mol, Πc ) 15 mN/m, b ) 2.8 × 108 cm3/mol, and δ ) 0.2, which corresponds to n-dodecanol at the water/air system (see ref 7) at T ) 15 °C. The Langmuir model (eq 6) has been applied to describe the system below the critical surface pressure.
quantified by the factor δ ) 1 - ωm*/ωm, ωm* being the molar area of the molecules in the aggregates. The substitution of eq 11 into eq 9 provides the surface equation of state, in implicit form,
Π)-
Γc gΠ RT ln(1 - ωm(Γc gΠ + Γm*)) ωm Γc gΠ + Γm*
(13)
The substitution of eq 11 into eq 10 gives
bc )
ωm(Γc gΠ + Γm*) [1 - ωm(Γc gΠ + Γm*)]θ
(14)
where θ ) {Γc gΠ}/{(Γc gΠ + Γm*)}. This equation, depending also on the surface pressure, must be solved together with eq 13 to provide the adsorption isotherm. Figure 1 shows the dependence of the adsorptions Γm and Γm* on the surface pressure predicted from the above relationships. The critical values and isotherm parameters utilized in the calculation are close to those observed for n-dodecanol.7 The monomer behavior below the critical point is described according to the Langmuir model. Since δ > 0, the molar area of the aggregated surfactant is smaller than that of the free monomer, ωm* < ωm, causing the distribution of surfactant between the two phases to shift toward the aggregated phase. As a consequence, the adsorption of the free monomer decreases, while increasing the surface pressure. Kinetics. Surface area changes compress or dilute the surface area, displacing the system from the equilibrium. However, in this case only the monomers are exchanged with the bulk. In fact the large aggregates are nearly insoluble and remain at the interface. According to what is commonly observed experimentally, the exchange of monomer between the interface and the bulk can be assumed to be controlled by diffusion. The distribution of the surfactant between the monomer and aggregate states results from an aggregation/disaggregation process. In dynamic conditions, Γm and Γm*,
(∂z∂c)
ΦD ) D
z)0
(17)
Φagg is the aggregation flux, defined according to
Φagg ) kagg(Γm - Γc gΠ)
(18)
where kagg is the aggregation rate constant. This expression of the flux is similar to that introduced by Van Uffelen and Joos5 to describe the monolayer collapse for dodecanol. At the equilibrium, the aggregation flux is vanishing, providing relationship 11. kagg is a constant taking into account all processes internal to the interfacial layer which are involved in the aggregate formation. Among them, kagg also accounts for the monomer surface diffusion, at least in a first approximation. By summing the interfacial balances, it follows that the total adsorption increases only under the effect of the diffusion flux. Being the monomer exchange controlled by diffusion, it is possible to consider Γ locally at equilibrium with the sublayer concentration cs. Hence, they can be linked by the isotherm in the form of eq 10. According to eq 17 a normal diffusion flux is assumed. As a matter of fact, since only the monomers are exchanged with the bulk, close to the interface a deformation of the diffusion path occurs due to the presence of the aggregates. The hypothesis of normal diffusion flux is satisfied when the size of the aggregates is negligible with respect to the diffusion length,15 (DτD)1/2, where τD is the characteristic time for the diffusional bulk/interface exchange
τD )
1 dΓ D dc
2
( )
(19)
For n-dodecanol at the water/air interface, i.e., the case depicted in Figure 1, the typical size of the aggregates is of the order of some tens of micrometers; thus, the above assumption is satisfied up to surface pressures of the order of 35 mN/m, which corresponds to a significant portion of the isotherm. Calculation of the Dynamic Surface Elasticity. To calculate as a function of the frequency, it is useful to adopt a perturbative approach. An interface in the equilibrium state is perturbed by an harmonic oscillation of the surface area
˜ e2πjνt) A ) A0(1 + K
(20)
˜ is the area deformawhere A0 is the reference area, K tion amplitude, and ν is the frequency.
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If K ˜ is sufficiently small, the system behaves linearly and harmonic responses are expected for all the dynamic quantities. Thus, by indicating with the subscript 0 the equilibrium values, the adsorptions are
Γm ) Γm0 + Γ ˜ me2πjνt
(21)
˜ m*e2πjνt Γm* ) Γm0* + Γ
(22)
˜ e2πjνt Γ ) Γm + Γm* ) Γ0 + Γ
(23)
c(z,t) ) c0 + c˜seaze2πjνt
dgΠ dΠ
0
( ) (
ωmδ dΠ dΠ Γ ˜ )Γ ˜× dΓ 0 RT dΓ 0 Π - Πc ωmδ (31) exp RT
)
where (dΠ/dΓ)0 is calculated from eq 9. Thus, it is possible to progress by calculating the expressions of the amplitudes of the fluxes in terms of the adsorption amplitudes. By using eq 26, the diffusive flux can be recast as
Φ ˜ D ) -c˜ s(j + 1)xπDν
( )
Φ ˜D)-
(24)
where z is the distance from the surface toward the solution. The surfactant diffusion in the bulk is described by the Fick equation
∂c ∂c )D 2 ∂t ∂z
( )( )
(32)
which by carrying on the substitution of eq 30 provides
The bulk concentration is
2
g˜ Π )
dcs Γ ˜ (j + 1)xπDν dΓ 0
In a similar way, the amplitude of the aggregation flux can be expressed as a function of the adsorption amplitudes by substituting eqs 23, 21, 29, and 31 into eq 18
[
( )]
˜ m - Γc Φ ˜ agg ) kagg Γ z > 0; t > 0
(25)
and because c(z,t) must be a convergent solution of the Fick equation, it is
a ) -(j + 1)
x
πν D
(33)
dgΠ Γ ˜ dΓ 0
(34)
By using the diffusion characteristic frequency, eq 5, and by defining the aggregation characteristic frequency, νagg, as
νagg )
1 k 2π agg
(35)
(26) the flux amplitudes are
The fluxes defined by eqs 17 and 18 are as well oscillating quantities
x
Φ ˜ D ) -2π(j + 1)
ννD Γ ˜ 2
(36)
Φagg ) Φ ˜ agge2πjνt
(27)
Φ ˜ agg ) 2πνagg((1 - Ψ)Γ ˜ - ΨΓ ˜ m*)
˜ De2πjνt ΦD ) Φ
(28)
where also it has been defined Ψ ) Γc(dgΠ/dΓ)0. By substituting eqs 22, 21, 37, and 36 into the interface mass balances 15 and 16, neglecting the second-order terms, one obtains
Moreover, under the effect of the oscillating surface coverage, gΠ is also oscillating
gΠ ) gΠ0 + g˜ Πe2πjνt
(29)
˜ m*, c˜ s and g˜ Π are complex quanThe amplitudes Γ ˜, Γ ˜ m, Γ tities containing a phase shift with respect to the area oscillation. Because the kinetic exchange between subsurface and the surface is at equilibrium with respect to diffusion,13 Γ and cs are in phase during the oscillation. Thus, at the first order it is possible to link c˜ s with the amplitude of the total adsorption Γ ˜ by
( )
dcs Γ ˜ c˜ s ) dΓ 0
˜ + (1 - j)ξ(Γ ˜m + Γ ˜ m*) Γ ˜ m + Γm0K jλ((1 - Ψ)Γ ˜ m - ΨΓ ˜ m*) ) 0 (38) ˜ + jλ((1 - Ψ)Γ ˜ m - ΨΓ ˜ m*) ) 0 (39) Γ ˜ m* + Γm0*K where the dimensionless frequencies λ ) νagg/ν and ξ ) xνD/2ν have been defined. Solving such a set of equations provides expressions of ˜ m* amplitudes the Γ ˜ m and Γ
˜ Γ ˜ m ) -K (30)
where (dcs/dΓ)0 is calculated from eq 10. In a similar way, since gΠ is a function only of Π, because the equilibrium relationship between the dynamic surface pressure and the total adsorption can be assumed, it is possible to write at the first order (15) Jost, W. Diffusion in Solids, Liquids, and Gases; Academic Press: New York, 1952; Chapter 1.
(37)
(1 - jλΨ)Γm0 - (jλΨ - (j - 1)ξ)Γm0* (1 - jλ)(1 + ξ - jξ)
(40)
Γ ˜ m* ) K ˜
jλ(1 - Ψ)Γm0 + (jλ(1 - Ψ) + (j - 1)ξ - 1)Γm0* (1 - jλ)(1 + ξ - jξ) (41)
By using these expressions, all the amplitudes of the harmonic responses of the quantities characterizing the system are known as a function of their equilibrium values, of the area perturbation, and of the isotherm
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parameters. Thus, it is possible to calculate the dynamic surface elasticity. Due to the small harmonic perturbation approach, is expressed by eq 3. Γm and Γm* being independent variables, such a relationship can be written as
d ln Γm* d ln Γm -0m* (jν) ) -0m d ln A d ln A
(42)
where the equilibrium quantities
( ) ( )
0m ) Γm0
0m* ) Γm0*
∂Π ∂Γm
(43)
0
∂Π ∂Γm*
(44)
0
have been introduced. The explicit forms for 0m and 0m* can be found by using the eqs 8 and 9
0m )
[
]
RTΓm0 ωmΓm0 Γm0* ln(1 - ωmΓ0) ωmΓ0 1 - ωmΓ0 Γ0
0m* )
[
(45)
]
RTΓm0* ωmΓm0 Γm0 + ln(1 - ωmΓ0) ωmΓ0 1 - ωmΓ0 Γ0
(46)
The derivatives in eq 42 can be calculated from the balances 15 and 16, by substituting eqs 21, 22, 28, and 27. By this way, can be expressed as
2πjνΓ ˜m + (jν) ) 0m 2πjνΓ ˜m - Φ ˜D+Φ ˜ agg 0m*
2πjνΓ ˜ m* (47) 2πjνΓ ˜ m* - Φ ˜ agg
which after substitution of eqs 39 and 38 provides
(jν) )
[
0m
] [ [
] ]
+ Γ ˜ m* Γ ˜ m* 1 + (1 - j)ξ + 1 - jλ (1 - Ψ) - Ψ Γ ˜m Γ ˜m m0* (48) Γ ˜m -Ψ 1 + jλ (1 - Ψ) Γ ˜ m* ˜ m can be easily evaluated from the solutions of the Γ ˜ m*/Γ set of eqs 39 and 38
Γ ˜ m* 1 - jλ(1 - Ψ)(1 + q) - (j - 1)ξ ) Γ ˜m q - jλΨ(1 + q) + (j - 1)ξ
(49)
where q ) Γm0/Γm0* is the ratio between the equilibrium adsorptions. Eventually, the expression of is found by substituting eq 49 into eq 48
q - ξ - j(λΨ(1 + q) - ξ) + (jν) ) 0m q(1 + ξ - λξ - j(λ + λξ + ξ)) 1 + ξ - j(λ(1 - Ψ)(1 + q) + ξ) (50) 0m* (1 + ξ - λξ - j(λ + λξ + ξ)) 3. Results and Discussion The viscoelastic behavior results from the superposition of all the dynamic processes considered: the oscillatory
Figure 2. Modulus (a, top) and phase (b, bottom) of versus Π at various λ: 0.1 (solid line), 1 (dashed line), and 10 (dotted line). The curves are calculated from eq 50 for the same parameters as Figure 1 and for νagg ) 0.1 Hz. The Lucassen model (eq 4) with the Langmuir isotherm (eqs 6, 7) has been applied to describe the system below the critical surface pressure.
area dilation; the transfer of molecules from the bulk to the surface, which is described by a diffusion flux; and, above the critical surface pressure, the phase transition inside the adsorbed layer. The latter two processes depend on the physical characteristics of the surfactant system expressed by the isotherm parameters, on the bulk concentration, on the diffusion coefficient, and on the aggregation rate. The first process can be instead considered as an imposed experimental condition. In Figure 2a,b the modulus and phase of are plotted versus Π0. Each curve is calculated at given value of λ and for νagg ) 0.1 Hz. The precritical region has been calculated by using the classical Lucassen model with the Langmuir isotherm. As shown, at the critical concentration, a discontinuity is expected both in the modulus and phase of . Up to now, the phase transition process has been studied7 by experimentally investigating the aging of a fresh surface, which, for n-dodecanol, is allowed by the rather long characteristic time of the aggregation process. Indeed, these experiments have shown that such time is at least of the order of 10 s. As also demonstrated for the
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Figure 3. Modulus (1, 2) and phase (3, 4) of versus λ. Curves 1 and 3 are calculated from eq 50 for the same parameters as Figure 1 and for Π0 ) 35 mN/m and νagg ) 10νD; curves 2 and 4 corresponds to the Lucassen model with Langmuir isotherm (eqs 6, 7).
orientation process,4 the rheological approach described here, coupled with a suitable technique for the measurement of as a function of the frequency, is an effective tool for getting information on the aggregation process also in the case of smaller characteristic time. To this aim, it is useful to discuss the behavior of as a function of frequency, under different conditions. According to eq 50, is frequency dependent through ξ and λ, which makes it easy to understand the influence of the characteristic times of diffusion and aggregation. Figure 3 shows the behavior of the modulus and phase of as a function of λ, for a set of isotherm parameters characteristic of n-dodecanol at the water-air interface.7 For the sake of comparison, also the curves predicted by the diffusional relaxation approach of Lucassen2 (eq 4), using the Langmuir model (eqs 6, 7), are plotted. The curves show the same features observed for the reorientation model:4 a step rise in the modulus and a maximum in the phase. The maximum appears in the λ region of the order of the unit, i.e., at frequencies comparable with νagg. The value of the modulus at λ , 1 achieves values larger than those predicted by the diffusional relaxation approach. This corresponds to an increased interfacial rigidity, caused by the fact that, for such high oscillation frequencies, the interface is no longer able to exchange surfactant molecules with the bulk, behaving as an insoluble layer. Moreover, since the characteristic frequency of the perturbation is larger than that of the aggregation process, the surfactant distribution between free and aggregate phase is practically frozen in the equilibrium state. These two circumstances strongly limit any relaxation of the system. Parts a and b of Figure 4 show the modulus and the phase of as a function of ξ, for different values of the νaggr/νD ratio. The same set of isotherm parameters as those in Figure 3 has been used in the calculation. As far as the ratio νagg/νD increases, the maximum in the phase moves toward larger ξ, i.e., smaller frequencies, and becomes more pronounced. For a given surfactant, the characteristic frequency of the aggregation process is expected to be independent from the surfactant concentration. On the contrary, the char-
Figure 4. Modulus (a) and phase (b) of vs ξ, for different values of the νagg/νD ratio: 0.1 (solid line), 10 (dotted line), and 100 (dashed line). The dashed-dotted curve corresponds to diffusion-controlled adsorption calculated by eq 51. The isotherm parameters are the same as those in Figure 1, and the diffusion coefficient is D ) 4.0 × 10-6 cm2/s.
acteristic frequency of the diffusional exchange increases with the surfactant bulk concentration. In practice, the characteristic time of the diffusion process, as defined by eq 19, typically spans over several orders of magnitude, from milliseconds to hours. Thus, both situations, νagg , νD and vice versa, can be met during the investigation of the surface aggregation of soluble surfactants, so that it is useful to examine the behavior of under these two limiting cases. Limit Case A: νD , νagg. In such a framework, there are two subcases that have to be considered, corresponding to imposed oscillation frequencies comparable to νD or to νagg, respectively. Subcase 1: ν ≈ νD , νagg. This limit case allows one to check the consistency of this model with the classical Lucassen one valid under the hypothesis of diffusioncontrolled adsorption. That can be done by considering an imposed frequency of the same order of νD and a much larger aggregation frequency. Under these conditions, the terms in λ2 are dominant with respect to the other frequency dependent terms and, after some rearrangement, eq 50 becomes
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1+q 1 + ξ + jξ Ψ + 0m*(1 + q)(1 - Ψ) ) 0m q 1 + 2ξ + 2ξ2 (51)
[
]
Such a situation corresponds to an oscillation period comparable with the characteristic time of diffusion, which is assumed to be much larger than the aggregation time. As a consequence, the aggregation process can be considered at equilibrium during the oscillation and only diffusive relaxation exists. The modulus and phase of calculated according to eq 51 are plotted in Figure 4. It is evident that the features introduced by the aggregation process, specifically the local maximum in the phase shift, superpose to these curves. Equation 51 shows that can be written as the product between a function of ξ and a thermodynamic factor specific to the surface model. The same function of ξ can be observed in the classical Lucassen2 calculation and for layers showing reorientation4 when assuming the same particular case considered here. Thus, it is reasonable to hypothesise that, provided that only diffusional relaxation exists, such form of is kept. Equation 51 reduces to the usual expression of the dilational viscoelasticity obtained by the classical Lucassen approach using the Langmuir isotherm when aggregation is not considered, i.e., for Γm* ) 0. Subcase 2: ν ≈ νagg . νo. Under these conditions, the terms in ξ are vanishing, and eq 50 reads
)
(
1+q Ψ + 0m*(1 - jλ(1 + q)(1 - Ψ)) q 1 - jλ (52)
0m 1 - jλ
)
In this situation, the diffusion exchange is negligible and the only effective relaxation process is aggregation. Limit Case B: ν ≈ νagg , νD. Under these conditions, the terms in ξ2 are dominant with respect to the other frequency dependent terms. Thus, eq 50 becomes
1 0m - 0m* q )1 - jλ
(53)
which represents the expression of the dynamic surface elasticity for the aggregation controlled case. Limit Case C: νD ≈ 0 (Insoluble Monolayer). From the general eq 50 it is possible to derive the dilational viscoelasticity for a surface layer behaving as an insoluble monolayer. For such systems, Φdiff is vanishing, as well as νD, which leads to neglecting the terms in ξ for all the values of the frequency, providing again eq 53. The modulus and phase of calculated according to such equation are plotted in Figure 5a,b for different equilibrium surface pressures. It is interesting to note that the phase presents a narrow peak near λ ) 1, which is the range where the aggregation process is enhanced. Increasing Π0, the peak is smoothed, reflecting the circumstance that a more stable equilibrium state is expected increasing the fraction of aggregated molecules. 4. Conclusion In the present work a model for the dynamic surface elasticity has been worked out for adsorption layers in which the coexistence of free surfactant molecules and large aggregates is assumed. The thermodynamic bases of such a model have been already validated by the experimental study of equilibrium
Figure 5. Modulus (a, top) and phase (b, bottom) of versus λ at different Π0, for the insoluble monolayer. The curves are calculated from eq 53 for the same parameters as in Figure 1. The values of Π0 are 15.5, 20, 25, and 30 mN/m, from the bottom in a and from the top in b.
and kinetics adsorption,7,12 for several systems, among them n-dodecanol at the water/air interface. Here a general expression of the dynamic surface elasticity is found as a function of the frequency, in terms of the characteristic times of diffusion and phase transition processes, and of the isotherm parameters. The model predicts a deviation from the classical diffusion-controlled behavior. In particular the phase of presents a characteristic maximum as a function of the frequency. This means that measurements of (ν) can be fruitfully used to study the aggregation process in the adsorbed layer, if any, regardless of the value of the ratio of the characteristic times τD/τagg. A specific expression has been found for the insoluble monolayer that can find a large application. In fact, the formation of large aggregates is a typical phenomena observed in such systems. This work provides a theoretical tool for investigating the aggregation properties of soluble and insoluble adsorption layers through surface rheological studies. In fact, the obtained results can be exploited in all the experimental approaches based on the measurement of
Dynamic Surface Elasticity of Adsoption Layers
the surface elasticity as a response to harmonic area perturbations.16-18,19 The interpretation of these data according to the here presented model allows the parameters characterizing the (16) Fruehner, H.; Wantke, K.-D. Colloids Surf., A 1996, 114, 53. (17) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A.; Ravera, F.; Ferrari, M.; Liggieri, L. In Novel Methods to Study Interfacial Layers; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 2001; p 439. (18) Monroy, F.; Giermanska Kahn, J.; Langevin, D. Colloids Surf., A 1998, 143, 251. (19) Kovalchuk, V. I.; Kra¨gel, J.; Aksenenko, E.; Loglio, G.; Liggieri, L. In Novel Methods to Study Interfacial Layers; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 2001; p 485.
Langmuir, Vol. 18, No. 9, 2002 3599
aggregation process to be accessed. To this aim, the dynamic elasticity of n-dodecanol monolayers at the water interface is being experimentally investigated by the Oscillating Bubble19 and Drop Shape17 techniques. Acknowledgment. The work was partially supported by the European Space Agency under the MAP Project FASES (AO-99-052) and by the Italian Space Agency (ASICNR, ARS I/27/R/00). The authors thank Dr. Reinhard Miller of the Max-Planck Institute for Colloids and Interfaces (Golm, Germany) for many useful discussions. LA0156917
