Anomalous Damping of the Capillary Waves at the Air− Water

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Anomalous Damping of the Capillary Waves at the Air-Water Interface of a Soluble Triblock Copolymer Mercedes G. Mun˜oz,† Francisco Monroy,† Paula Herna´ndez,† Francisco Ortega,† Ramo´n G. Rubio,*,† and Dominique Langevin‡ Departamento Quı´mica Fı´sica I, Facultad de Quı´mica, Universidad Complutense, 28040-Madrid, Spain, and Laboratoire Physique Solides, Universite´ Paris XI, Orsay, France Received July 1, 2002. In Final Form: November 5, 2002 The dilational viscoelasticity of Gibbs and Langmuir monolayers of pluronic F-68 (PEO76-PPO29PEO76) has been studied by capillary wave techniques over a broad frequency range. The results show that the behaviors of Gibbs and Langmuir monolayers are equivalent in the low surface pressure range. For concentrated monolayers, the dynamic elasticity  shows a noticeable frequency dependence, and the concentration dependence of  (at constant frequency) is not compatible with the classical Lucassen-Van den Tempel model. Moreover, in this concentration range, the dilational viscosity takes apparent negative values. The spectra of light scattered by thermal capillary waves have been analyzed in terms of two recently proposed dispersion equations. The equation of Buzza et al. (J. Chem. Phys. 1998, 109, 5008) is able to fit the experimental spectra only if one assumes that the thickness of the monolayer takes unrealistic values of the order of 0.1 µm. The equation of Hennenberg et al. (J. Colloid Interface Sci. 2000, 230, 216) is able to fit the experimental spectra, although the physical meaning of its new coupling parameter remains obscure.

* To whom correspondence should be addressed. E-mail: [email protected]. † Universidad Complutense. ‡ Universite ´ Paris XI.

the formation of polymolecular micelles, since the effect of T on the second break agrees with the effect of T on the cmc.6 This, however, is not a general behavior.7 Barentin et al.8,9 have given a theoretical description for the equilibrium properties of monolayers formed by polymers with a strong difference of the hydrophilichydrophobic character between blocks, grafted to the air/ water surface. The theory accounts rather well for the experimental surface tension data for the concentrated regime of the monolayers of two different PEO-PPOPEO copolymers.7 Alexandridis et al.5 calculated the area per molecule (from the initial slope of Gibbs isotherm) for pluronics with different PEO lengths and found it to be proportional to N1/2, where N is the number of ethylene oxide groups. However, for pluronics with different PPO block lengths, the limit area per molecule remains almost unchanged (it shows only a very slight decrease when the PPO block is increased). This suggests that the PPO blocks form coils out of the interface. This is in agreement with the conclusions of Yang and Sharma.10 For low values of c it has been found that the thermodynamic state of Gibbs (adsorbed) monolayers of a pluronic is equivalent to that of Langmuir (spread) monolayers at low surface concentration Γ. The results for these monolayers can be described by simple scaling laws and show at least three different concentration regimes: the dilute regime, which ends at the overlapping concentration Γ*; the semidilute regime, which extends between Γ* and Γ**; and the concentrated regime for Γ > Γ**. The behavior of the semidilute regime leads to a critical exponent that corresponds to good-solvent conditions.7

(1) Hamley, I. A. The Physics of Block Copolymers; Oxford University Press: Oxford, 1998. (2) Kok, H.-A.; Lecommandoux, S. Adv. Mater. 2001, 13, 1217. (3) Jo¨nsson, B.; Lindman, B.; Holmberg, K.; Kronberg, B. Surfactants and Polymers in Aqueous Solutions; John Wiley & Sons: Chichester, 1999. (4) Alexandridis, P. Curr. Opin. Colloid Interface Sci. 1997, 2, 478. (5) Alexandridis, P.; Athanassiou, V.; Fukuda, S.; Hatton, T. A. Langmuir 1994, 10, 2604.

(6) Alexandridis, P.; Holzwarth, J. F.; Hatton, T. A. Macromolecules 1994, 27, 2414. (7) Mun˜oz, M. G.; Monroy, F.; Ortega, F.; Rubio, R. G.; Langevin, D. Langmuir 2000, 16, 1083. (8) Barentin, C.; Muller, P.; Joanny, J. F. Macromolecules 1998, 31, 2198. (9) Barentin, C.; Joanny, J. F. Langmuir 1999, 15, 1802. (10) Yang, Z.; Sharma, R. Langmuir 2001, 17, 6254.

Introduction Block copolymers have attracted much attention in past years due to their rich structure and flow behavior.1,2 Their unique properties are also evident in solution when the solvent used has a different thermodynamic quality for the different blocks.3 Among the many block copolymers, the poly(oxyethylene)-b-poly(oxypropylene)-b-poly(oxyethylene) (PEO-PPO-PEO) copolymers (most frequently known as pluronic or synperionic) are thermoresponsive “intelligent” copolymers (the critical micelle concentration (cmc) may decrease 3 orders of magnitude with a 20 °C temperature increase) with a wide range of applications, i.e., emulsification, cleaning, rinsing, solubilization, foaming, water treatment, lubrication, fermentation, controlled drug delivery, thermal coating and printing, biomedical processing, and sensor development.4 Their solution behavior combines surfactant-like features with the polymeric behavior of the relatively long chains. The surface tension γ vs concentration c curves show at least two breaks. The first one occurs at c ≈ 0.001 wt % regardless of temperature T, even though the value of γ at the break may be different for different temperatures. It has been associated with a change in configuration or structural transition of the copolymer molecules at the air/water interface,5 with the PEO blocks protruding into the aqueous solution. It has been suggested that for some pluronics (i.e., for some relative values of the PEO and PPO molecular weights) the second break is due to

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We have recently presented a dynamic surface tension, γ(t), study of the adsorption-desorption process of two pluronics at the air/water interface.11 At low concentrations, γ(t) and time-resolved ellipsometry results point out that the adsorption kinetics is diffusion-controlled, while at higher concentrations, where the brush is formed, the adsorption process seems to be dominated for the existence of an adsorption barrier and a slow relaxation process in the interface plane. The studies mentioned above deal with surface tension data. However, it is well-known that γ may be insensitive to the state of the copolymer in the monolayer (e.g., to conformational or phase transitions); thus the use of other complementary techniques seems to be necessary for the study of these complex systems. Further insight into the dynamics of the pluronic monolayers can be gained from the study of the dilational viscoelasticity of the monolayers.12 Some studies already exist for related polymer monolayers. Noskov et al. have studied the viscoelastic moduli of the air/water interface of PEO solutions over a broad frequency (ω) range.13 Their results point out that the increase in the number of loops and tails at the transition from the dilute to the semidilute regimes has strong effects on the dynamic elasticity modulus, (ω), of the monolayer. However, they did not discuss any quantitative effects upon the dilational viscosity κ(ω). The groups of Richards14-17 and of Yu18,19 have carried out SQELS (light scattering by thermal capillary waves) experiments on monolayers of diblock copolymers that contain a PEO block. They also reported that the existence of subphase structures (loops, tails, micelles) has a profound influence on the dilational viscoelastic moduli of the monolayer. Similar conclusions were reached at by Lamp and Goedl in a study of the shear viscosity of end-tethered polymer monolayers.20 Monolayers of pluronic F-68 (PEO76-PPO29-PEO76) are well suited for this type of study. Indeed, F-68 solutions have a rather high value of cmc (between 1 and 10 mM),6,7,21 well beyond the concentration at which a brush is formed in a monolayer. Therefore, it is possible to study the viscoelasticity of the monolayers before micelles are formed in the bulk, which would add new contributions to the dynamics of the system.22 Also, the brush regime is reached at concentrations well below c ∼ 0.3 mM, where the bulk solutions start to depart from the Newtonian behavior, which might also give rise to new surface relaxation modes.23 In the present paper we have carried out a study of the viscoelastic moduli of F-68 monolayers in a broad range (11) Mun˜oz, M. G.; Monroy, F.; Ortega, F.; Rubio, R. G.; Langevin, D. Langmuir 2000, 16, 1094. (12) Langevin, D. Light Scattering by Liquid Surfaces and Complementary Techniques; Langevin, D., Ed.; Marcel Dekker: New York, 1992; Vol. 41. (13) Noskov, B. A.; Akentiev, A. V.; Loglio, G.; Miller, R. J. Phys. Chem. B 2000, 104, 7923. (14) Alexander, M.; Richards, R. W. J. Phys. Chem. B 2000, 104, 9179. (15) Richards, R. W.; Rochford, B. R.; Taylor, M. R. Macromolecules 1996, 29, 1980. (16) Peace, S. K.; Richards, R. W.; Williams, N. Langmuir 1998, 14, 667. (17) Milling, A. J.; Hutchings, L. R.; Richards, R. W. Langmuir 2001, 17, 5297. (18) Runge, F. E.; Kent, M. S.; Yu, H. Langmuir 1994, 10, 1962. (19) Esker, A. R.; Zhang, L.-H.; Sauer, B. B.; Lee, W.; Yu, H. Colloids Surf., A 2000, 171, 131. (20) Lamp, C.; Goedl, W. A. Macromolecules 2001, 34, 1343. (21) Lopes, J. R.; Loh, W. Langmuir 1998, 14, 750. (22) Noskov, B. A. Adv. Colloid Interface Sci. 2002, 95, 237. (23) Wang, Q. R.; Wang, C. H.; Deng, N. J. J. Chem. Phys. 1998, 108, 3827.

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of frequencies combining experiments with electrocapillary waves (ECW) (20 Hz-2 kHz) and light scattering by thermal capillary waves (SQELS) (10-200 kHz). As will be shown, the viscoelastic moduli show a complex dependence on c and ω, which are difficult to explain within the present theoretical description of the hydrodynamics of interfaces. Experimental Section Chemicals. The pluronic F-68 was purchased from Fluka (Germany), and was the same sample used in previous works.7,11 The polydispersity index, measured by GPC analysis, was 1.12. The copolymer was vacuum-dried at 80 °C for 24 h before use. Double distilled and deionized water from a Milli-Q-RG ultrafiltration system (resistivity > 18 MΩ) was used. The solutions were prepared by weighting the two components in an analytical balance (precision 10-5 g); the components were mixed at ∼15 °C since the solubility of the copolymer decreases with increasing T.4 Langmuir Trough. The spread monolayers of F-68 were formed on a Teflon trough (KSV Instruments Model 3000 with a minitrough, Finland) controlled by a computer. A Pt-Wilhelmy balance placed at the air-water (A/W) interface was used as a surface force sensor. Hydrophilic polyoxymethylene (Delrin) was used for the trough barriers. The whole setup was enclosed in a hermetically sealed box made of transparent methacrylate through which a small flow of filtered N2 was maintained. A Petry cell with subphase was placed inside the box to keep the humidity at saturation level. The temperature regulation of the trough was controlled by a flow of thermostated water passing through the jacket at the bottom of the trough. The temperature near the surface was measured with a 0.01 °C precision using a calibrated Pt-100 sensor. The temperature of the box was maintained at (298.1 ( 0.1) K. The box had two windows for the light scattering experiments. The surface of the subphase was cleaned by aspirating the impurities until the expected surface tension was obtained. To ensure that the spreading solvent (chloroform) did not add any significant impurities, blank experiments with the pure solvent were carried out. To form the monolayers, small amounts of the spreading solution (typically 10 µL) were carefully deposited by a Hamilton microsyringe at different places on the surface. The surface concentration G was increased by subsequent additions (10 mL) of the polymer solution. Times ranging from 10 min to more than 1 h were allowed for solvent evaporation and monolayer equilibration. The monolayer remained stable. The surface pressure Π was continuously monitored during the experiment, and the equilibrium value was taken when Π remained constant for at least 10 min. Electrocapillary Wave (ECW) Experiments. The technique used here is similar to the one described by Ito et al.24 and Jayalakshmi et al.25 In this technique, surface capillary waves at the A/W interface are excited in the frequency range from 20 Hz to 2 kHz by application of an electric ac field (∼500 V) through a blade positioned within 100 µm of the interface. The wavevector q and the corresponding spatial damping coefficient R of the electrocapillary waves excited at a given frequency are determined by optical reflectometry of a laser beam positioned at various distances from the blade. The measure of the phase difference and the amplitude ratio as a function of distance to the blade allows one to calculate q and R. These two parameters are then used to obtain the elasticity and dilational viscosity from the dispersion equation. The instrument was calibrated by determining the surface tension of several pure liquids. The measurements were done at 22.0 ( 0.2 °C. Surface Light Scattering Experiments (SQELS). The SQELS setup used here is the one described previously.26 A polarized 25 mW He-Ne Laser passes through a spatial filter that expands the beam and ensures a Gaussian intensity profile. (24) Ito, K.; Sauer, G. B.; Skarlupka, R. J.; Sano, M.; Yu, H. Langmuir 1990, 6, 1379. (25) Jayalakshmi, Y.; Ozanne, L.; Langevin, D. J. Colloid Interface Sci. 1995, 170, 358. (26) Monroy, F.; Ortega, F.; Rubio, R. G. Phys. Rev. E 1998, 58, 7629.

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Then the beam passes through a transmission diffraction grating from Align-Rite (UK). In all the experiments reported here we have used one 5-µm-wide dark line, each spaced 275 µm from each other. A 1:1 image of the grating is formed on the monolayer. The light scattered by the capillary waves is detected in the heterodyne mode, mixed with the light reflected by a diffraction mode. The wavevector q is selected by focusing each of the diffraction spots into the photomultiplier. The signal is collected and analyzed either by a NICOM Model 170 autocorrelator or by a Stanford Research (USA) Model 760 FFT spectrum analyzer. The spectrometer has been calibrated using simple liquids (nhexane, ethanol, and water) and can be used in the wavevector range 100 e q/cm-1 e 900. The temperature precision in these measurements was (0.1 °C in the whole measurement range of 10-40 °C.

Data Analysis For a monolayer-covered surface, the propagative parameters, the frequency ω and the wavevector q, are related by the dispersion equation D(ω) ) 0. For a surface in which the capillary and dilational modes are the only important ones12

D(ω) ) L(E˜ )‚T(γ) - [iωη(q - m)]2

(1)

L(E˜ ) ) E˜ q2 + iωη(q + m)

(2)

F T(γ) ) γq2 + iωη(q + m) - ω2 + Fg q

(3)

where

with i ) x-1; F, γ, and η are the liquid density, surface tension, and viscosity, respectively, and m-1 is the capillary penetration length

F m2 ) q2 + iω with Re(m) > 0 η

(4)

The elasticity modulus characteristic of the dilational mode is defined as E˜ (ω) ) E(ω) + iωK(ω), where (ω) is the dynamic elasticity and κ(ω) is the dilational viscosity. The corresponding modulus for the transverse mode is real and corresponds to the surface tension γ(ω). The last term in eq 1 takes into account the coupling between the capillary and the dilational modes. The spectrum of the scattered light is given by

Pq(ω) )

[

]

2kBT E˜ q2 + iωη(q + m) Im ω D(q,ω)

(5)

where kB is the Boltzmann constant, T is the temperature, and Im means that only the imaginary part of the expression in the square brackets is taken into account. However, it has to be considered that experimentally one obtains the convolution between the theoretical spectrum and the instrumental function G(ω). In our experimental technique G(ω) can be accurately described by a Gaussian function. As a result,in order to obtain the constitutive parameters γ(ω), (ω), and κ(ω) from the experimental data, one has to fit the experimental spectra to

P(ω) ) FT-1{FT[Pq(ω)] FT[G(ω)]}

(6)

where FT is the Fourier transform, and FT-1 denotes the inverse Fourier transform. Under certain conditions Pq(ω) may be approximated by a Lorentzian function, thus allowing a description of P(ω) by a Voigt function.

Figure 1. (a) Surface pressure vs concentration curve for the Gibbs monolayer of pluronic F-68 at 25 °C. The arrows mark the surface phase transitions discussed in ref 7. The inset shows the temperature dependence of the phase transition region at which a brush is formed in the subsurface. (b) Surface pressure vs area per molecule for the Langmuir monolayer of F-68 at 22 °C. The dashed line represents the best fit of the data in the semidilute regime to the scaling law indicated in the figure. The value of the exponent is close to that of good-solvent conditions.

Results Background of Equilibrium Results. Figure 1a gives the surface pressure Π vs c curves for the F-68 solutions at 22 °C. Π is defined as the difference between the surface tension of the solution and that of water. Figure 1b shows the corresponding data for the Langmuir monolayer at the same temperature. As can be observed, up to two plateau-like regions are found for the Gibbs monolayer; the values of Π for the first two transitions are similar for Gibbs and Langmuir monolayers. The first point to be noticed is that, contrary to what was suggested by Alexandridis,5,6 the second phase transition in the Π vs c curves takes place at a concentration well below the cmc (≈10 mM for F-68). The second point is that increasing T decreases the length of the plateau of the second transition; it has to be taken into account that increasing T decreases the cmc, which seems to confirm that this second transition is not directly linked to the formation of polymolecular micelles in the bulk. It must also be recalled that the ellipsometry experiments indicated a significant increase of the thickness of the monolayer (from 0.2 to 3.3 nm) at c ≈ 10-5, which was indicative of the formation of the brush.7,11 The inset of Figure 1a shows that the surface phase transition associated with the formation of the brush has

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Figure 2. (a) Concentration dependence of the wavelength (λ) and the spatial damping (R) of the electrocapillary waves for the Gibbs monolayers at 22 °C, and at a frequency of 800 Hz. (b) Same as (a) for a Langmuir monolayer.

a noticeable temperature dependence for the Gibbs monolayers. In general, for a given value of c, the surface pressure Π increases with T, and the c range of the phase transition shrinks. This behavior is reasonable if one takes into account that the solubility of F-68 in water decreases as T increases. The surface entropy (not shown) calculated from (∂γ/∂T) is negative for c e 100 mM, with a minimum ca. c ) 10 mM. Figure 1b shows the best fit of the Langmuir monolayer to a scaling law in the semidilute regime. The value of the critical exponent indicates that the water-air interface behaves as a good solvent for the polymer.28 Capillary Wave Experiments. Figure 2a shows the wavelenth λ and the spatial damping R for the Gibbs monolayers as a function of the copolymer concentration at 22 °C and for a capillary wave frequency of 800 Hz. Only solutions with c e 0.1 mM have been studied, to avoid any viscoelastic contribution from the subphase. Figure 2b shows the same magnitudes for the Langmuir monolayer as a function of the area per molecule. Although apparently the hydrodynamic behavior of the Gibbs and Langmuir monolayers is rather different, it will be shown below that this is not the case. Figure 3 shows the frequency dependence of λ and R for the Gibbs monolayers at 22 °C; as can be observed, the experimental data can be described by simple scaling laws ω ∼ qn and R ∼ ωm, where ω is the angular frequency and the wavevector q is calculated as q ) 2π/λ. However, the values of the exponents n and m are different from those of Kelvin’s law for λ (n ) 3/2) and Stokes’ law for R (m ) 1),12 and the differences slightly increase with c. This is indicative of the existence of viscoelastic loss in the monolayer. Figure 4 shows the wavevector dependence of the SQELS spectra at 25 °C for two well-separated values of c. It has been possible to fit all the experimental spectra within the experimental precision using a Voigt function: Figure 5 shows the concentration dependence of the capillary wave peak frequency of the SQELS spectra ωq and of their widths at half-height ∆Γ. As expected, there (27) Cook, J. T. E.; Richards, R. W. Eur. Phys. J. E 2002, 8, 111. (28) Kelarakis, A.; Castelletto, V.; Chaibundit, Ch.; Fundin, J.; Havredaki, V.; Hamley, I. W.; Booth, C. Langmuir 2001, 17, 4232.

Figure 3. Frequency dependence of wavelength and spatial damping of the electrocapillary waves of the Gibbs monolayers at different concentrations. Notice that the results are compatible with simple scaling laws: ω ∼ qn and R ∼ ωm, with the wavevector q calculated from λ and ω.

is a qualitative correlation between the concentration and temperature dependences of ωq and that of the surface tension, while the behavior of ∆Γ looks more complicated, and will be discussed below. Discussion The constitutive parameters γ(ω), (ω), and κ(ω) have been obtained from the analysis of the electrocapillary wave data and the SQELS spectra in terms of eqs 1-5. It has to be mentioned that the analysis of the data is

ECW Experiments on a Soluble Triblock Copolymer

Figure 4. Spectra of light scattered by thermal capillary waves as a function of wavevector q, for Gibbs monolayers at two different concentrations. Symbols: 0, q ) 101.2 cm-1; O, q ) 202.4 cm-1; 4, q ) 303.6 cm-1; 3, q ) 404.8 cm-1.

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Figure 6. Dynamic elasticity (ω) for Gibbs monolayers as a function of frequency. The values were calculated by fitting the electrocapillary wave data to eqs 1-5. Notice that for a given frequency, the concentration dependence of (ω) is not monotonic. The maximum of the (ω) vs ω curves for c g 10-3 mM is not compatible with the classical Lucassen-Van den Tempel model.

Figure 7. Dynamic elasticity for Gibbs monolayers of F-68 obtained from the spectra of light scattered by thermal capillary waves of wavector q ) 404.8 cm-1. Figure 5. Concentration and temperature dependence of frequency of the maximum (ωm) and of the width at half-height (∆Γ) of the spectra of scattered light for Gibbs monolayers at 25 °C. Both ωm and ∆Γ were obtained from fits of the experimental spectra to a Voigt function.

slightly different for ECW and SQELS data. While in the former the analysis of the frequency and the spatial damping leads to the elasticity (ω) and to the dilational loss modulus, in the latter (ω) and κ(ω) are obtained from the fits. As will be mentioned below, this leads to a significant difference. For this system, in the present c and T ranges, no noticeable differences were obtained when the SQELS spectra were analyzed in terms of ωq and ∆Γ.8 It also has to be noticed that we have found γ(ω) ≈ γ(ω)0) through the whole frequency range studied. It must be remarked that the frequencies of the dilational and capillary waves are equal only under resonance conditions, i.e., at the resonance wavevector qR.12,27 The value of qR depends on the constitutive parameters of the monolayer, and hence is different for the different Gibbs and Langmuir monolayers studied in this work. As a consequence, in what follows, we are going to discuss the constitutive parameters in terms of the capillary frequency, which is experimentally available,

and not in terms of the dilational frequency, which cannot be measured but is linked to the constitutive parameters through the dispersion equation. Moreover, when calculating the loss modulus from SQELS, one should use the frequency of the dilational mode; otherwise an apparent loss modulus is obtained. However, the use of this apparent loss modulus does not change the conclusions of the present work. Elasticity of the Monolayers. Figure 6 shows the values of (ω) obtained from the ECW experiments on Gibbs monolayers at 22 °C. As expected, for very low concentrations the elasticity is very small and almost frequency independent. For c > 10-5 mM (recall that cmc ≈ 10 mM!) (ω) increases markedly between 20 and 100 Hz, reaches a maximum between 100 and 200 Hz, and decreases again. It is important to remark that the dependence of  upon c is not monotonic, since it reaches a maximum near 0.01 mM. Figure 7 shows the values obtained from SQELS for q ) 386 cm-1 (capillary frequency ω ≈ 60 kHz), which, when compared with the data in Figure 6, confirms the decrease of (ω) at high frequencies. This figure also shows that the elasticity increases with decreasing T, a result which is qualitatively analogous to that of the bulk solutions.28

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Figure 8. Elasticity vs surface pressure for F-68 monolayers. Notice that, for the frequency range studied, the Langmuir and Gibbs monolayers show the same behavior. Below Π ≈ 10 mN‚m-1 the static and dynamic elasticities agree. In the dilute regime the simple relationship  ∼ Π holds; the slope of this simple equation indicates that the interface behaves as a good solvent for the polymer.

Figure 8 shows the same kind of results for the Langmuir monolayer; it also includes the compression modulus or static elasticity (ω)0) ) (∂Π/∂ ln A). As can be observed, the dynamic elasticity can be described by the scaling law  ) yΠ in the semidilute regime, with a critical exponent y ) 2.48 ( 0.04, close to that of the good-solvent conditions.29 In this concentration regime the agreement between static and dynamic elasticity is quite satisfactory. For surface concentrations corresponding to Π e 5 mN‚m-1, the polymer molecules have a flat two-dimensional conformation and do not form long tails and loops protruding into the water subphase. Then the spread film is elastic, and  increases with Π. At higher surface concentrations beyond the elasticity maximum, the number of loops and tails increases, which leads to the exchange of monomers between the monolayer plane and the subsurface and a reduction in the entanglement density in the surface plane, thus decreasing . For Π g 10 mN‚m-1, where the first phase transition is found, the elasticity increases again, which is probably due to the increase of repulsive interactions between the loops and tails. Finally, near Π ) 20 mN‚m-1, the dynamic elasticity reaches a second maximum and decreases to lower values. It must be recalled that the mushroom to brush transition was found for Π ≈ 20 mN‚m-1, and was accompanied by an large increase in the thickness of the monolayer.7 Finally, it has to be remarked that, in the frequency range of the ECW experiments shown in the figure, there are no significant differences between the Langmuir and the Gibbs monolayers. Figure 9 shows the frequency dependence of (ω) for the Langmuir monolayer at three different surface states: Π ≈ 0.5, 3, and 7 mN‚m-1, which correspond to the dilute regime, the transition to the semidilute regime (Γ*), and the transition to the concentrated regime (Γ**), respectively. It can be observed that even well below the formation of a mushroom or a brush in the subphase the elasticity shows a noticeable frequency dependence in the 100-200 Hz capillary frequency range, which coincides with the frequency at which the (ω) vs ω curve for a given value of c (Gibbs monolayers) showed a maximum. For these Langmuir monolayers the frequency dependence of (ω) has been well described by a simple Maxwell (29) Vilanove, R.; Poupinet, D.; Rondelez, F. Macromolecules 1988, 21, 2880.

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Figure 9. Frequency dependence of the elasticity of a Langmuir monolayer at three different surface pressures: 4, dilute regime, Π ) 0.5 mN‚m-1; O, Π* ) 3 mN‚m-1, crossover concentration from the dilute to the semidilute regimes; 0, Π** ) 7 mN‚m-1, crossover from the semidilute to the concentrated regimes. The curves are the fits of the data to a Maxwell relaxation model with the relaxation times indicated in the figure.

Figure 10. Frequency dependence of the dilational loss modulus for Gibbs monolayers of different concentrations at 25 °C. Notice that for c g 10-5 mM the dilational viscosity crosses over from positive to negative values at ω ≈ 500 Hz, irrespective of the solution concentration.

relaxation with relaxation times τ which increase with the surface concentration (see Figure 9). Dilational Viscosity. More complicated is the behavior of the dilational viscosity κ(ω). Figure 10 shows the results obtained from the ECW experiments for the Gibbs monolayers. As can be observed, κ(ω) ≈ 0 over the whole frequency range for c < 10-5 mM. Despite that the results are noisy, there is no doubt that for higher concentrations κ(ω) is positive at low frequencies and is apparently negative in the high-frequency range. Negative values are also obtained from the analysis of the SQELS spectra in the 2-70 kHz range (not shown). Moreover, the crossover from positive to negative values seems to happen near 500 Hz, independently of the concentration. Alexander and Richards have also reported a change from positive to negative values of κ(ω) for a soluble PS-PEO copolymer in the kilohertz range.14 It has been shown that the values of (ω) in the highfrequency range are essentially the same in Gibbs and Langmuir monolayers. Figure 11 shows the frequency dependence of κ(ω) for the Langmuir monolayer at three values of Γ. Taking into account that the values of Π corresponding to Γ* and Γ** are similar to those corresponding to Gibbs monolayers with concentrations c ) 5

ECW Experiments on a Soluble Triblock Copolymer

Figure 11. Frequency dependence of the loss modulus of the Langmuir monolayers at the same surface concentrations as Figure 9.

× 10-7 and 2 × 10-6 mM, respectively, the results of the Langmuir monolayers may be considered compatible with those of the Gibbs monolayers shown in Figure 10. Figures 7 and 10 show that the strong frequency dependence of the dilational viscoelastic moduli,  and κ, takes place at concentrations higher than c ) 10-5 mM. In ref 7 we found that this is the concentration for which the ellipsometry data indicate that the monolayer thickness increases from 0.2 to 3.3 nm. Also for c ≈ 10-5 mM the adsorption/desorption kinetics shows a crossover from a diffusion-mediated adsorption regime to a bimodal kinetics in which, in addition to a steric adsorption barrier, a slow reorganization process is dominant at long times.11 This process is outside the frequency range of the present study. Analysis of SQELS Spectra in Terms of Recent Dispersion Equations. Although apparent negative dilational viscosities have been interpreted as excess functions in some papers,30 and can be rationalized quite naturally for interfaces where there are macroscopic gradients (e.g., temperature or concentration gradients), there seems to be a wide consensus that, for thin equilibrium interfaces, this kind of results is a consequence of the inadequacy of the dispersion equation to describe the interfacial hydrodynamics. There are at least two aspects to be taken into account when trying to explain these awkward results. First, neutron reflectivity indicates that copolymer monolayers adsorbed at the air-water interface may have a thickness larger than that indicated by ellipsometry. Thick monolayers present dynamic modes not included in eq 1. Buzza et al. have proposed an extended dispersion equation that takes into account the bending and splaying modes and their coupling with the transversal and dilational ones.31 In a previous work we have shown that the use of some of these new contributions allows description of the experimental data with positive values of κ(ω).32 A second fact to consider is that the structure of the polymer interface is complex. The volume fraction profiles of loops and tails are different, as are their contributions to the rheology of the monolayers.33 Moreover, in some cases nonmonotonic profiles have been described. In general, the characteristic length of these interfacial structures is comparable to the penetration (30) Goodrich, F. C. Proc. R. Soc. London, Sect. A 1981, 374, 341. (31) Buzza, D. M. A.; Jones, J. L.; McLeish, T. C.; Richards, R. W. J. Chem. Phys. 1998, 109, 5008. (32) Mun˜oz, M. G.; Luna, L.; Monroy, F.; Rubio, R. G.; Ortega, F. Langmuir 2000, 16, 6657. (33) Jones, R. A. L.; Richards, R. W. Polymers at Surfaces and Interfaces; Cambridge University Press: Cambridge, 1999.

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Figure 12. Comparison of two experimental spectra of scattered light (O) with predictions of the classical dispersion equation with negative dilational viscosities (s), the equation of Buzza et al. with monolayer thickness of 0.1 µm (- -), and the equation of Hennenberg et al. (---).

length of the capillary waves. Thus, the wave might cross through regions with different monomer, loop, and tail concentrations. Hennenberg et al.34 have taken into account the contribution of concentration gradients in the subsurface. We have shown that this may also help in explaining negative apparent viscosities in an ionic surfactant solution.34 Therefore, it seems reasonable to test whether recently proposed dispersion equations allow one to interpret the results of the pluronic monolayers without leading to negative values of the dilational viscosity. Analysis in Terms of Buzza’s Dispersion Equation. When the bending mode is taken into account, Buzza’s dispersion equation for a liquid-air interface can be written as

D(ω) ) T(γ) L(E˜ ) - CB(F,η;2)

(7)

T(γ) ) (γ + B ˜ q2)q2 + iωη(q + m) - Fω2/q

(8)

CB(F,η;λ˜ ) ) [ηω(q - m) + iλ˜ q3]2

(9)

where B ˜ ) B + iB′ accounts for the contribution of the bending mode, and λ˜ ) λ + iλ′ accounts for the coupling between the bending and the longitudinal modes; the rest of the variables have the same meaning as in eqs 1-4. The values of λ and λ′ are linked to the elasticity and the thickness of the monolayer h0 by simple relations:

λ ∼ h0; λ′ ∼ h02

(10)

and similar relations hold for B and B′. We have found that for values of h0 below 100 nm the contribution of the bending mode to the SQELS spectra is almost negligible. Only for very thick interfaces does Buzza’s equation lead to noticeable differences with respect to eq 1. For the sake of example, Figure 12 compares the results of eqs 1 and 7 for two values of c and q. Notice that it is possible to obtain reasonable fits of the spectra with κ ) 0, although a value of h0 ∼ 0.1 µm has to be used. Although one may consider that the thickness of the interface is larger than the one measured by ellipsometry, (34) Hennenberg, M.; Slavtchev, S.; Weyssow, B.; Legros, J.-C. J. Colloid Interface Sci. 2000, 230, 216. (35) Mun˜oz, M. G.; Monroy, F.; J. L. F. Rubio; Ortega, F.; Rubio, R. G. J. Phys. Chem. B 2002, 106, 5636.

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there is no experimental support for such thick interfaces in pluronic systems. Analysis in Terms of Hennenberg’s Dispersion Equation. Hennenberg et al.34 have stated that surface renewal is due to both interfacial compressibility and to shape deformation, playing together. Thus, capillary deformation of the interface brings the monolayer into contact with bulk regions with different surfactant concentrations. Hence, the Marangoni force (surface tension gradients) caused mainly by surface dilation contains now additional terms which lead to the following dispersion equation

D(ω) ) T(γ) L(E˜ ) - CH(F,η;2)

(11)

where the longitudinal (L) and transversal (T) terms are the same as in eqs 1-3, respectively. The coupling term is given by

CH(F,η;2) ) [iωη(q - m) + 2q2]2

(12)

2 is the coupling constant between the capillary mode and the concentration gradient contribution. Notice that, if 2 ) 0, eq 12 reduces to eq 3. The effect of 2 is to effectively increase the coupling between the capillary and the dilational modes. As a consequence, when fitting SQELS spectra, the effect of 2 > 0 is qualitatively similar to that of negative effective dilational viscosities. Figure 12 shows that the fits obtained using κ ) 0 and 2 * 0 for two values of c and q are comparable to those obtained with eq 1 and κ < 0, and also to those obtained using Buzza’s equation. Conclusions The viscoelastic moduli of F-68 monolayers has been studied over a broad frequency range using a combination of electrocapillary wave technique and of light scattering by thermal capillary waves. For surface pressures below

that corresponding to the first surface phase transition (Π ≈ 10 mN‚m-1), there are no noticeable differences between the equilibrium and the dynamic elasticities. Moreover, for the frequency range studied, Gibbs and Langmuir monolayers show essentially the same dynamic elasticity (ω). In the case of Gibbs monolayers of very dilute solutions, (ω) is almost frequency independent. However, for c g 10-5 mM, the elasticity strongly increases with ω and then shows a shallow maximum. In this concentration range the thickness of the monolayer increases from 0.2 to 3.3 nm. The concentration dependence of  for a given frequency is not compatible with the classical LucassenVan den Tempel model. The dilational viscosity κ(ω) takes negative apparent values as the frequency and the concentration are increased. These apparent values indicate that the classical dispersion equation used for the analysis of the capillary wave experiments is not adequate. The use of the equation proposed by Buzza et al. allows one to fit the spectra of scattered light only if unrealistically high thicknesses (≈0.1 µm) are assumed for the monolayers. The use of the dispersion equation of Hennenberg et al., which assumes the existence of a concentration gradient in the subsurface, also allows one to fit the experimental spectra, although the meaning of the coupling parameters obtained remains obscure. Acknowledgment. This work was supported in part by MICyT under Grant BQU2000-786, and by a grant from the Del Amo program. M.G.M. is grateful to U. C. M. for a doctoral fellowship. We are grateful to the C. A. I. Espectroscopı´a for the use of some of its facilities. LA0206007