Langmuir 2007, 23, 12951-12958
12951
Nonlinear Viscoelasticity of Sorbitan Tristearate Monolayers at Liquid/Gas Interface Rema Krishnaswamy, Sayantan Majumdar, and A. K. Sood* Department of Physics, Indian Institute of Science, Bangalore 560012, India ReceiVed June 26, 2007. In Final Form: October 4, 2007 The interfacial rheology of sorbitan tristearate monolayers formed at the liquid/air interface reveal a distinct nonlinear viscoelastic behavior under oscillatory shear usually observed in many 3D metastable complex fluids with large structural relaxation times. At large strain amplitudes (γ), the storage modulus (G′) decreases monotonically whereas the loss modulus (G′′) exhibits a peak above a critical strain amplitude before it decreases at higher strain amplitudes. The power law decay exponents of G′ and G′′ are in the ratio 2:1. The peak in G′′ is absent at high temperatures and low concentration of sorbitan tristearate. Strain-rate frequency sweep measurements on the monolayers do indicate a strain-rate dependence on the structural relaxation time. The present study on sorbitan tristearate monolayers clearly indicates that the nonlinear viscoelastic behavior in 2D Langmuir monolayers is more general and exhibits many of the features observed in 3D complex fluids.
1. Introduction Low molecular weight surfactants are widely used as surfaceactive agents in drugs, cosmetics, and food industry to form stable emulsions and foams.1 Many industrial processes involving emulsions and foams rely on their stability under large deformation at the liquid/liquid or liquid/gas interfaces.2 Hence, the interfacial rheology of Langmuir monolayers of such systems at fluid interfaces is relevant as model systems to study the viscoelastic and flow behavior of 2D complex fluids as well as in many industrial and biomedical applications. The dynamics of many soft matter systems ranging from dense colloidal suspensions to gels have been well-characterized.3 However, very few analogous studies exist in 2D, particularly due to the fact that the instrumentation to probe the interfacial viscoelastic properties of such 2D systems is less well established. The pressure-area isotherms of Langmuir monolayers of insoluble surfactants have been extensively studied using filmbalance techniques to obtain the phase transformations, position, and shape of the isotherms. The surface rheology of Langmuir monolayers, however, is a step further, since it helps us to understand the correlation between the structure and the mechanical properties, which is crucial for many practical applications. The flow behavior of Langmuir monolayers is known to be strongly related to the surface morphology and composition of the monolayers. For example, a divergence of the viscosity due to close packing of solid lipid domains4 was observed recently for Langmuir monolayers of phospholipids: a behavior proposed for 3D hard-sphere suspensions.5 Monolayers of micrometersized polystyrene spheres as well as β-lactoglobulin proteins formed at fluid interfaces6 exhibit a jamming behavior similar to that seen in many 3D systems.7 Many of these sudies emphasize * Corresponding author.
[email protected]. (1) Bos, M. A.; Vliet, T. V. AdV. Colloid Interface Sci. 2001, 91, 437. (2) Edwards, D.A.; Brenner, H.; Wasan, D.T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Boston, 1991. (3) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press, New York, 1999. (4) Ding, J.; Warriner, H. E.; Zasadzinski, J. A. Phys. ReV. Lett. 2002, 88, 168102. (5) Brady, J. F. J. Chem. Phys. 1993, 99, 567. (6) Cicuta, P.; Stancik, E. J.; Fuller, G. G. Phys. ReV. Lett. 2003, 90, 236101. (7) Trappe, V.; Prasad, V.; Cipelletti, L.; Segre, P. N.; Weitz, D. A. Nature (London) 2001, 411, 772.
the relevance of Langmuir monolayers of surfactants as model systems to gain a theoretical understanding of the viscoelastic and flow behavior in 2D as well as to probe the generality of the flow behavior between 2D and 3D systems. The present study on sorbitan tristearate monolayers examines the nonlinear viscoelastic behavior of 2D films under oscillatory shear based on the insights obtained from similar studies carried out on 3D systems. The study of the viscoelastic properties of Langmuir monolayers at high surface concentrations also has some important practical applications as, for example, in understanding the role of pulmonary surfactant monolayers at the alveolar air-water interfaces during respiration.8 For a 2D film, the in-plane shear deformation at the interface arises due to a change in the shape of the surface, while keeping the area constant. For a viscoelastic surface, on applying a sinusoidal shear deformation γ ) γo exp(iωt) at an angular frequency ω, and strain amplitude γo, an out-of-phase response is obtained for the deviatoric part of the surface stress tensor defined as σ ) G*γ, where G* ) G′ + iG′′. Here, G′ and G′′ correspond to the interfacial storage and loss modulii which describes the elastic and viscous response of the film, respectively. In amplitude sweep experiments, an oscillatory shear of different strain amplitudes is applied at a constant angular frequency. At low strain amplitudes, G′ and G′′ remain constant and correspond to the linear viscoelastic regime. By applying an oscillatory shear of low strain amplitude at different angular frequencies, linear viscoelastic measurements probe the structural relaxation of the film. At large strain amplitudes, in amplitude sweep experiments, the viscoelastic response becomes nonlinear when G′ and G′′ are no longer constant, but decay monotonically. Recently, it was seen that an ultrathin film of metal nanoparticles, forming a 2D colloidal glass, exhibits a distinct nonlinear viscoelastic behavior characteristic of many 3D metastable complex fluids.9 Under oscillatory shear, at large strain amplitudes, G′ decreases monotonically, whereas G′′ exhibits a peak before it decays at larger strain amplitudes with the decay exponents of G′ and G′′ in the ratio 2:1. In 3D systems, this feature is found to be universal, exhibited by a wide variety of systems ranging from highly cross(8) Piknova, B.; Schram, V.; Hall, S. B. Curr. Opin. Struct. Biol. 2002, 12, 487. (9) Krishnaswamy, R.; Majumdar, S.; Ganapathy, R.; Agarwal, V. V.; Sood, A. K.; Rao, C. N. R. Langmuir 2007, 23, 3084.
10.1021/la701889w CCC: $37.00 © 2007 American Chemical Society Published on Web 11/22/2007
12952 Langmuir, Vol. 23, No. 26, 2007
linked associative polymers,10 dense colloidal suspensions,11 electrorheological fluids,12 colloidal glasses,13 emulsions,14 and gels.15 The nonlinear viscoelastic behavior oberved has been attributed to the flow behavior of these systems, where they exhibit a finite yield stress at low shear rates, followed by a viscous shear thinning beyond yield.11,12 A similar picture also emerges from the soft glassy rheology model where a peak in G′′ is present at the glass transition and absent above the glass transition.16 It was proposed recently that the nonlinear viscoelastic behavior seen in these systems is more general.17 A mode coupling theory proposed for the description of the dynamics of a supercooled colloidal suspension under time-dependent oscillatory strain reveals that the origin of the peak in G′′ arises from the strain-rate dependence of the structural relaxation time (τ). Moreover, a phenomenological Maxwell model, which takes into account the strain-rate dependence of τ, does reveal a peak in G′′ above a critical strain amplitude followed by a power law decay of the storage and loss modulii in the ratio 2:1. Further, the power law dependence of τ on the strain-rate amplitude was verified through a “strain-rate-frequency superposition” where the frequency sweep measurements are carried out at a fixed strain-rate amplitude.18 It is relevant in this context to examine whether a similar argument also holds for 2D systems. If so, it would also be a valuable tool for the extraction of structural relaxation time in 2D systems when they are experimentally inaccessible from the frequency sweep measurements. Our studies have been carried out on Langmuir monolayers of an insoluble surfactant film of sorbitan tristearate, at the air/ water interface. The phase behavior and interfacial rheology of sorbitan tristearate, commercially known as span 65 and commonly used as an emulsifier in the food and drug industry, has been well-characterized at liquid/gas and liquid/liquid interfaces19,20 through shear and dilational rheology.21 The viscoelastic properties of the film are known to arise from the physical cross-linking of the chains due to hydrogen bonding. These films exhibit additional relaxation mechanisms under shear as opposed to compression and dilation, and at low interfacial concentrations, the film fractures during shear flow. Many of these studies have been carried out using a highly sensitive stresscontrolled rheometer, capable of measuring very low torques, and by taking into account the contribution to the interfacial flow from the bulk phases in the measurements of the viscoelastic properties of the monolayers.22 This allows viscoelastic measurements to be carried out on even very fragile films with a viscoelastic behavior obtained at low surface concentrations. We have investigated the nonlinear viscoelastic behavior of sorbitan tristearate monolayers at different surface concentrations (10) Tirtaatmadja, V.; Tam, K. C.; Jenkins, D. R. Macromolecules 1997, 30, 1426. (11) Yziquel, F.; Carreau, P. J.; Tanguy, P. A. Rheol. Acta 1999, 38, 14. (12) Parthasarathy, M.; Klingenberg, D. J. J. Non-Newtonian Fluid Mech. 1999, 81, 83. (13) Mason, T. G.; Weitz, D. A. Phys. ReV. Lett. 1995, 75, 2770. (14) Mason, T. G.; Lacasse, M.; Grest, G. S.; Levine, D.; Bibette, J.; Weitz, D. A. Phys. ReV. E 1997, 56, 3150. (15) Panizza, P.; Roux, D.; Vuillaume, V.; Lu, C. Y. D.; Cates, M. E. Langmuir 1998, 58, 738. (16) Sollich, P.; Lequex, F.; Hebraud, P.; Cates, M. E. Phys. ReV. Lett. 1997, 78, 2020. (17) Miyazaki, K.; Wyss, H. M.; Weitz, D. A.; Reichman, D. R. Europhys. Lett. 2006, 75, 915. (18) Wyss, H. M.; Miyazaki, K.; Mattson, J.; Hu, Z.; Reichman, D. R.; Weitz, D. A. Phys. ReV. Lett. 2007, 98, 238303. (19) Rehage, H.; Achenbach, B.; Geest, M.; Siesler, H. W. Colloid Polym. Sci. 2001, 279, 597. (20) Erni, P.; Fischer, P.; Heyer, P.; Windhab, E. J.; Kusnezov, V.; Lauger, J. Progr. Colloid Polym. Sci. 2004, 129, 16. (21) Erni, P.; Fischer, P.; Windhab, E. J. Langmuir 2005, 21, 10555. (22) Erni, P.; Fischer, P.; Windhab, E. J.; Kusnezov, V.; Stettin, H.; Lauger, J. ReV. Sci. Instrum. 2003, 74, 4916.
Krishnaswamy et al.
Figure 1. Amplitude (γ) sweep measurements on sorbitan tristearate monolayers at the air-water interface at surface concentration Γm ) 0.54 µg cm-2 at 25 °C. The angular frequency was fixed at 5 rad/s.
and temperature under oscillatory shear. At low surface concentrations (Γm < 0.18 µg cm-2), under oscillatory shear, the amplitude sweep experiments on sorbitan tristearate monolayers reveal a monotonic decay of the storage and loss modulii at large strain amplitudes. With increase in surface concentration, e.g., at 0.54 µg cm-2, G′ decreases monotonically, whereas G′′ exhibits a peak before it decays above a critical strain amplitude (Figure 1). However, at these surface concentrations, on increasing the temperature to 40 °C, the peak in G′′ at large |γ| disappears, and the monotonic decay of G′ and G′′, observed at low surface concentrations, is recovered. At low temperatures ( 0.18 µg cm-2 also reveals an average structural relaxation time (τR) ≈ 100 s. With the increase in temperature (>30 °C), however, τR decreases by 2 orders of magnitude. At high surface concentrations and low temperatures, sorbitan tristearate monolayers exhibit a nonlinear viscoelastic behavior very similar to that seen recently in 2D monolayers of Ag nanoparticles. To probe the origin of the peak in G′′, strain-rate frequency sweep measurements were also carried out on the high surface concentration films at low temperatures. A power law dependence on the strain-rate amplitude is observed for the structural relaxation time. Such a behavior is remarkably similar to that observed in 3D and suggests a generality of this behavior for 2D and 3D systems. Our studies clearly reveal that many of the features observed in the nonlinear viscoelastic behavior of 3D systems are valid in 2D as well. 2. Experimental Details Sorbitan tristearate (trioctadecanyl ester of sorbic acid) obtained from Fluka was used as received. Sorbitan tristearate was dissolved in hexane (Merck) and ethanol (BDH) (4:1), and appropriate amounts of the solution were spread onto the aqueous surface (deionized water, Millipore) using a microsyringe (Hamilton) to obtain the desired surface concentration. The pressure-area isotherm of the surfactant used for the present study consisting of a mixture of different sugar esters has been well-characterized through filmbalance techniques combined with Brewster angle microscopy.19 The pressure-area isotherm obtained using hexane and ethanol (4: 1) as the spreading solvent was comparable to that obtained from the earlier studies, which were carried out using chloroform as the spreading solvent.19,21 Surface concentration was varied from 0.09 µg cm-2 (corresponding to a gas analogous phase) to 2.2 µg cm-2, which corresponds to a solid analogous phase obtained at high surface pressures.21 All rheological measurements were carried out using an interfacial rheology system (IRS) which consists of a rheometer (Physica MCR from Anton Paar) with an interfacial rheology cell based on bicone geometry.22 The raw data were numerically analyzed after each measurement to determine the interfacial modulii and the interfacial
Interface Rheology of Sorbitan Tristearate MLs
Figure 2. Frequency sweep measurements on sorbitan tristearate monolayers (surface concentration Γm ) 0.54 µg/cm2) formed at the air-water interface at different temperatures: (a) 25 °C and (b) 35 °C. The strain amplitude was fixed at 0.1%. The relaxation times τ as a function of 1/T are plotted in the inset. The solid line is the linear fit to the Arrhenius equation described in the text. steady shear viscosity and take into account the correction to the flow field from the bulk phases.20,22 To follow the film buildup at the interface, an oscillatory shear of strain amplitude 0.05% at an angular frequency of 5 rad/s was applied. All measurements were made on the film only after G′ and G′′ reached the saturation, which is after 100 s. Different measurements for a given concentration were made on a freshly spread film, and the preshear ensures that, at a given concentration and temperature, the initial conditions are identical.
3. Results and Discussion 3.1. Interfacial Oscillatory Shear on Sorbitan Tristearate Monolayers. 3.1.1. Frequency Sweep. To probe the structural relaxation of the film, constant strain frequency sweep measurements were carried out on the film at different surface concentrations and temperatures with the strain amplitude fixed at very low values of 0.1%. As seen from Figure 1, this corresponds to the linear viscoelastic regime. The angular frequencies were varied from 0.01 rad/s to 10 rad/s. As indicated by the frequency sweep curves (Figure 2) at low frequencies, a viscous behavior is observed with G′′ > G′, whereas at higher frequencies, a crossover from viscous to elastic behavior is observed with G′ > G′′ and G′ tending to a plateau. In the frequency sweeps, the loss modulus remains a constant over more than two decades in frequency, extending to the highest measured frequency. Such a loss spectrum has not been reported in many systems. The film is found to be viscoelastic at all the surface concentrations and temperatures probed including the gas analogous phase obtained at a surface concentration of 0.09 µg cm-2. The crossover frequency ωco is, however, temperaturedependent, and the average structural relaxation time τ ) 1/ωco
Langmuir, Vol. 23, No. 26, 2007 12953
decreases with increase in temperature (Figure 2). The average structural relaxation times, however, are found to be nearly insensitive to the interfacial concentration, consistent with the earlier reports.19,22 The low-frequency behavior of the film, below the crossover frequencies, exhibits a strong deviation from the Maxwell model3 corresponding to a single-exponential relaxation time for the stress relaxation with G′ ∼ ω2 and G′′ ∼ ω. In the present system, G′ ∼ ω1.2 and G′′ ∼ ω0.6 at an interfacial concentration of 0.54 µg cm-2 at 35 °C. On decreasing the temperature to 25 °C, interfacial modulii scales with angular frequency as G′ ∼ ω0.8 and G′′ ∼ ω0.34. In the absence of a theoretical model for the 2D Langmuir monolayers, the frequency sweep data at an interfacial concentration of 0.54 µg cm-2 were fitted to a Cole-Davidson model corresponding to a distribution of relaxation times,23 as well as to a generalized Maxwell model,24 consisting of n Maxwell elements. The values of the high-frequency plateau modulus Go obtained from the fits to the Cole-Davidson model were found to be at least 2 orders of magnitude higher than the extrapolation from the measured G′. Moreover, due to the flat nature of the loss spectra, the stretching exponent approaches zero. The fits to the generalized Maxwell model yield reasonable values for the plateau modulus. However, it is difficult to relate the different time scales obtained from the fits to physical properties of the system. Hence, these fits are not shown. On studying the effect of temperature on the average structural relaxation time, it is found that the crossover frequency shifts to higher values with increase in temperature. The variation of τ is given on a semilog plot vs 1/T (Figure 2). τ decreases exponentially with temperature T and can be represented by the Arrehenius relation,25 τ ) τo eEa/RT, where R is the gas constant. The activation energy Ea obtained from the linear fits to the Arrhenius plots for τ is 37 kJ/mol, which is consistent with the typical binding energies associated with the hydrogen bonds.26 The value of τo obtained from the fit is ∼10-13 s. 3.1.2. Amplitude Sweep. To probe the strength of the sorbitan tristearate monolayers, amplitude sweeps were carried out on the film at a fixed frequency of 5 rad/s at different surface concentrations (Γm) keeping the temperature fixed at 25 °C. At a low interfacial concentration of 0.09 µg cm-2, which corresponds to the gas analogous phase in the surface pressure-area isotherm of sorbitan tristearate monolayers,21 the film is elastic at the angular frequencies probed with G′ > G′′ (Figure 3). The elastic response of the film at interfacial concentrations corresponding to the gas analogous phase substantiates some of the earlier observations that the phase of the Langmuir monolayers cannot be determined from the pressure-area isotherms alone.27 The viscoelastic response in the gas phase for sorbitan tristearate monolayers can arise from the presence of solid domains of high surface concentrations coexisting with a gas analogous phase in the monolayer. The viscosities of Langmuir monolayers are known to be dependent on the area fraction of these domains, which is known to increase with surface concentration or corresponding surface pressure. The rigidity of the monolayers arises from the long-range repulsive interactions between the domains in the monolayer.4 At low values of γ, both G′ and G′′ are independent of strain amplitude, which corresponds to the (23) Menon, N.; Nagel, S. R.; Venerus, D. C. Phys. ReV. Lett. 1994, 73, 963. (24) Macosko, C. W. Rheology Principles, Measurements, and Applications; VCH Publishers, Inc.: New York, 1994. (25) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (26) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press Limited: London,1991. (27) Miyano, K.; Abraham, B. M. J. Chem. Phys. 1983, 78, 4776.
12954 Langmuir, Vol. 23, No. 26, 2007
Figure 3. Amplitude (γ) sweep measurements at 25 °C on sorbitan tristearate monolayers at the air-water interface at different surface concentrations Γm (µg cm-2). The angular frequency was fixed at 5 rad/s. (a) G′ (closed symbols) and (b) G′′ (open symbols) measured at different Γm (µg cm-2), (i) 0.09 (stars), (ii) 0.36 (triangles), (iii) 0.54 (squares), are shown. In the inset, amplitude (γ) sweep measurements at 25 °C on sorbitan tristearate monolayers at the air-water interface at a surface concentration Γm ) 2.2 µg cm-2 at different angular frequencies (i) 0.06 rad/s (filled circles), (ii) 1 rad/s (open squares), (iii) 10 rad/s (open traingles) are shown. The crossover frequency (ωCO) of the monolayers is 0.07 rad/s. Note that the peak in G′′ is absent below the crossover frequency.
linear viscoelastic regime. Above a critical strain amplitude (>1%), a monotonic decay of G′ and G′′ is observed. On increasing the surface concentration to 0.36 µg cm-2 corresponding to the liquid analogous phase, G′ decreases monotonically at high strain amplitudes, whereas G′′ exhibits a peak before it shear-thins at higher strain amplitudes. Both G′ and G′′ exhibit a power law decay where G′ ∼ γ-2ν and G′′ ∼ γ-ν. Such a behavior with the decay exponents in the ratio 2:1 persists up to the largest concentration studied (2.2 µg cm-2) corresponding to the highly condensed, solid analogous phase. At high surface concentrations, the peak in G′′ is present only when the amplitude sweeps are carried out at angular frequencies above the crossover frequency (G′ > G′′) and is absent otherwise (Figure 3). It was also seen that the peak position remains independent of the angular frequency at a given temperature and surface concentration. The influence of temperature on the strength of the film was also studied by keeping the surface concentration fixed at 0.54 µg cm-2 (Figure 4). The peak in G′′ observed at large strain amplitude persists up to 35 °C. However, the peak is absent at higher temperatures, and the monotonic decay of the storage and loss modulii, observed at low surface concentrations, is recovered. In addition to an increase in loss modulus at higher temperatures, the critical strain amplitude above which the nonlinearity sets in shifts to 0.02% at 40 °C. Thus, the nonlinear viscoelastic behavior observed so far in our studies on sorbitan tristearate monolayers may be summarized briefly as follows: (i) A peak
Krishnaswamy et al.
Figure 4. Amplitude (γ) sweep measurements on sorbitan tristearate monolayers at air-water interface at Γm ) 0.54 µg cm-2. The angular frequency was fixed at 5 rad/s. (a) G′ (closed symbols) and G′′ (open symbols) measured at different temperatures, (i) 30 °C (squares), (ii) 35 °C (triangles), (iii) 40 °C (stars) are shown.
in G′′ occurs above a critical strain amplitude at low temperatures and high surface concentrations of the monolayer and is absent at low surface concentrations and high temperatures. (ii) The peak is observed only for amplitude sweeps carried out at frequencies above the crossover frequency (G′ > G′′). (iii) The position of the peak always occurs for strain amplitude in the range 0.02 < |γ| < 0.1. (iv) The peak position remains independent of the angular frequency. (v) A power law decay is observed for the storage and loss modulii at large strain amplitudes with the decay exponents always in the ratio 2:1. Recently, a similar nonlinear viscoelastic behavior was also reported for a 2D monolayer of Ag nanoparticles, formed at the liquid/liquid interface.9 In the amplitude sweep experiments carried out on sorbitan tristearate monolayers, the distinct peak in G′′ is noteworthy, because such a behavior has not been observed for many 2D systems. All the features discussed above in the amplitude sweep experiments on monolayers are remarkably similar to that observed in many 3D soft glassy systems such as colloidal hard sphere suspensions, emulsions, and gels.13,14,29,30 Hence, we will examine the relevance of some of the mechanisms proposed for this behavior in 3D systems in the context of 2D monolayers. 3.1.3. Strain-Rate Frequency Sweep. It was proposed recently by Weitz and co-workers18 that, in 3D systems, a nonlinear viscoelastic behavior where the loss modulus exhibits a distinct shear thickening peak in G′′ at high strain amplitudes can arise in metastable systems with long structural relaxation time where the relaxation time is strain-rate dependent. It is relevant in this context to examine whether a strain-rate amplitude dependence exists for the structural relaxation times in 2D systems. (28) Adamson, W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons, Inc.: New York, 1990. (29) Hoffmann, H.; Rauscher, A. Colloid Polymer Sci. 1993, 271, 390. (30) Sollich, P. Phys. ReV. E 1998, 58, 738.
Interface Rheology of Sorbitan Tristearate MLs
Langmuir, Vol. 23, No. 26, 2007 12955
Figure 6. Interfacial shear viscosity vs shear rate curves for sorbitan tristearate monolayers at 25 °C at different surface concentrations Γm ) 0.09 µg cm-2 (a); 0.36 µg cm-2 (b); 0.54 µg cm-2 (c) are shown. The solid lines are the fit to the Carreau model described in the text. The corresponding interfacial shear stress vs shear rate curves for sorbitan tristearate monolayers at 25 °C at different surface concentrations Γm ) 0.09 µg cm-2 (open squares); 0.36 µg cm-2 (open triangles); 0.54 µg cm-2 (filled circles) are shown in the inset. A finite yield stress is absent at the lowest shear rates measured.
Figure 5. Frequency sweep measurements carried out on sorbitan tristearate monolayers at surface concentration Γm ) 0.54 µg cm-2 at 18 °C at constant strain-rate amplitudes |γ˘ | ) |γ|ω. G′ (open squares) and G′′ (open circles) at different strain-rate amplitudes (a) 0.000 05 s-1; (b) 0.0001 s-1; (c) 0.005 s-1; (d) 0.01 s-1; (e) 0.05 s-1; (f) 0.5 s-1; and (g) 1 s-1 are shown. The crossover frequency shifts to higher values with increase in |γ˘ |.
In the regular frequency sweep experiments, the strain amplitude is kept constant whereas the angular frequency is varied over a given range. Hence, the strain-rate amplitude |γ˘ | ) |γ| ω varies for each data point. The strain-rate dependence of the structural relaxation time can be obtained by varying both the strain amplitude and the angular frequency such that the product is constant during the frequency sweep measurements. The frequency sweep measurements carried out on sorbitan tristearate monolayers at surface concentration Γm ) 0.54 µg cm-2 at 18 °C, for different values of strain-rate amplitudes ranging from 0.000 05 s-1 to 1 s-1 (Figure 5) are shown. The crossover frequency shifts to higher frequencies with strain-rate amplitude, confirming the strain-rate dependence of τ. The lowfrequency region, where a liquid-like behavior is observed (G′ < G′′), shifts to higher frequencies on increasing the strain-rate amplitudes. Here, G′ ∼ ω1.6 and G′′ ∼ ω0.8. Unlike 3D systems,10,18 the frequency dependence of the storage and loss modulii at low frequencies, below the crossover, does not approach that of a Maxwellian liquid even at a high strain-rate amplitude of 5 s-1. 3.2. Interfacial Steady Shear on Sorbitan Tristearate Monolayers. From the flow curves obtained at different surface concentrations (Figure 6), it is seen that at very low shear rates, a finite yield stress is absent for the film at the surface concentrations studied. At higher shear rates, the flow curves reveal a power law dependence of the stress on shear rates with the stress-shear rate curves approaching a plateau. Further increase in shear rates lead to a sharp drop in stress indicating the rupture of the film due to the breakdown of the interfacial structure. Thus, a steady shear-thinning viscous behavior is observed for the film at all surface concentrations. The flow curves could not be fitted to a power law model, since it fails to describe the low shear rate region. Hence, the viscosity data (Figure 6) was fitted to a Carreau model24 given by η(γ˘ ) ) [ηo/(1 + γ˘ 2λ2)m]. The parameters ηo, λ, and m obtained from the fits are given in Table 1. As expected, the zero shear viscosity ηo is found to increase with the surface concentration, whereas m decreases with surface concentration. At a surface concentration of 0.09 µg cm-2, it is seen (Figure 6) that the stress vs shear rate curves reach a plateau even at a low shear rate of 0.005 s-1. Moreover, a gradual decrease in the interfacial stress is observed at higher shear rates, indicating the rupture of the film. Hence, the value of 0.5 for m at a low surface concentration is consistent
12956 Langmuir, Vol. 23, No. 26, 2007
Krishnaswamy et al.
Table 1. Parameters Obtained by Fitting the Interfacial Shear Viscosity vs Shear Rate Data at 25 °C for Different Interfacial Concentrations Γm (Figure 6) to the Carreau Model Described in the Text interfacial concentration (µg cm-2)
ηo (Pa-s)
λ (s)
m
0.09 0.36 0.54
0.214 0.275 1
250 282 347
0.5 0.4 0.3
with the breakdown in the structure of the film under steady shear at low surface concentrations. We have also examined whether the empirical Cox-Merz rule holds for the film under study. The Cox-Merz rule suggests that, for a shear-thinning fluid, the steady-shear viscosity η(γ˘ ) can be related to the dynamic viscosity ηf(ω) such that
η(γ˘ ) ) |ηf(ω)| with γ˘ ) ω Recall that
|ηf(ω)| )
x(G′)2 + (G′′)2 ω
Figure 7 shows the plot of η(γ˘ ) vs γ˘ and |ηf(ω)| vs ω at a surface concentration Γm ) 0.54 µg cm-2 at 18 °C. It can be seen that a significant deviation from the Cox-Merz rule is observed. This was found to be true for all surface concentrations studied, consistent with the earlier observations in these systems.19 For systems with a power law flow behavior under steady shear, an extended Cox-Merz rule is found to be valid at high strain amplitudes,33 where a correlation exists between steady shear-viscosity and the complex dynamic viscosity
η(γ˘ ) ) ηf(|γ|ω) with γ˘ ) |γ|ω where |γ|ω is the strain-rate amplitude. We examine the validity of this rule for monolayers of sorbitan tristearate at a surface concentration Γm ) 0.54 µg cm-2 at 18 °C (Figure 7). In the present system, at high shear rates, the superposition between steady and oscillatory shear is found to hold at different angular frequencies, consistent with the behavior of a shear-thinning fluid. 3.3. Nonlinear Viscoelastic Behavior of Sorbitan Tristearate Monolayers. 3.3.1. Strain Rate Dependence of Structural Relaxation Time. It was proposed recently that the peak in G′′ at high strain amplitudes arises from the dependence of the structural relaxation time τ, on the strain-rate amplitude17,18 given by
1 1 ) + A|γ˘ |ν τ τo
(1)
where A is a constant, and τo is the intrinsic relaxation time. Here, the strain-rate amplitude, |γ˘ | ) |γ|ω where |γ| is the strain amplitude and ω the angular frequency. From eq 1, the strain-rate amplitude dependence of the structural relaxation time τ(|γ˘ |) may be expressed as
τo ) b(|γ˘ |) ) 1 + Aτo|γ˘ |ν τ
(2)
Hence, the crossover frequency will shift to higher frequencies on increasing the strain-rate amplitude. The frequency scaling (31) Kurtz, R. E.; Lange, A.; Fuller, G. G. Langmuir 2006, 22, 5321. (32) Raghavan, S. R.; Khan, S. A. J. Rheol. 1995, 39, 1311. (33) Doraiswamy, D.; Mujumdar, A. N.; Tsao, I.; Beris, A. N.; Danforth, S. C.; Metzner, A. B. J. Rheol. 1991, 35, 647.
Figure 7. Dynamic and steady shear viscosities as a function of strain-rate amplitude |γ˘ | ) |γ|ω, angular frequency (ω), and shear rate (γ˘ ) measured on sorbitan tristearate monolayers at a surface concentration Γm ) 0.54 µg cm-2 at 18 °C are shown. A significant deviation from the empirical Cox-Merz rule (described in the text) is observed for dynamic (filled circles) and steady shear viscosities (open circles). However, the extended Cox-Merz rule is obeyed, as seen from the superposition of ηf (|γ|ω) (open symbols) with the steady shear viscosity (open circles) at an angular frequency of 1 rad/s (open triangles) and 10 rad/s (open stars).
factor b(|γ˘ |) can be obtained by scaling the frequency response curves onto a single master curve. It was also shown recently by Weitz and co-workers that this could be used successfully to extract the structural relaxation times in many 3D metastable systems when the crossover frequencies are not experimentally accessible.18 It can be seen from eq 2 that the scaling factor b(|γ˘ |) asymptotically approaches unity at low strain-rate amplitudes. Here, the viscoelastic behavior will remain independent of the strain rates imposed. The intrinsic relaxation time of the system is thus recovered by carrying out frequency sweep measurements with the strain-rate amplitude fixed at these values. On examining the frequency sweep curves at different strainrate amplitudes, it is seen that, at low frequencies, the shape of the frequency response curves remain independent of the strainrate imposed and can be scaled onto a single master curve (Figure 8 A). At high frequencies, the scaled curves for the loss modulus reveal a strain-rate dependence. The scaling factors for the viscoelastic moduli (a(|γ˘ |)) and the frequency (b(|γ˘ |)) are plotted in Figure 8B. The scaling factor for G*(ω) remains nearly independent of strain-rate amplitude. The frequency scaling factor b(|γ˘ |) could be fitted to eq 2 with A ) 9.5, τo ) 105 s, ν ) 0.88. The power law exponent ν ) 0.88 is the same as the decay exponent of G′′ obtained from the amplitude sweep experiments, the significance of which is discussed below. The asymptotic behavior of b(|γ˘ |) at low shear-rate amplitudes (|γ˘ | < 0.001 s-1) where it approaches a value of 1 is noteworthy. This indicates that, at these shear-rate amplitudes, the structural relaxation time remains nearly independent of the shear rates imposed and is found to be the same as the intrinsic relaxation time of the system obtained from the frequency sweep experiments (Figure 2). This is remarkably similar to the behavior for 3D systems.18 Our experiments reveal that the strain-rate frequency sweep can be successfuly employed to extract the intrinsic relaxation time for many 2D films such as protein monolayers, metal nanofilms, highly cross-linked polymers, and so on known to exhibit very long structural relaxation times of ∼1000 s or higher, which remain experimentally inaccessible in the regular frequency sweep measurements. It should be noted that the intrinsic relaxation time can be extracted from these measurements only if the frequency scaling factor b(|γ˘ |) exhibits the asymptotic behavior discussed above, where it approaches unity below a critical strain-rate amplitude.
Interface Rheology of Sorbitan Tristearate MLs
Figure 8. (A) Frequency sweep measurements carried out on sorbitan tristerate monolayers at surface concentration Γm ) 0.54 µg cm-2 at 18 °C at different strain rate amplitudes are scaled on to a single master curve. (B) The scaling factors for angular frequency (b) and G*(ω) (a) are plotted as a function of strain-rate amplitude. The dotted line corresponds to the fit to eq 2 described in the text. Amplitude sweep curves (G′ (filled squares) and G′′ (open circles) vs γ) at an angular frequency of 1 rad/s, obtained by incorporating the shear-rate dependence of τ (described in the text) into a Maxwell model are shown in the inset.
In the amplitude sweep experiments on sorbitan tristearate monolayers, the nonlinear viscoelastic behavior occurs at |γ| g 0.01. If the angular frequency is fixed at 1 rad/s, this would correspond to a strain-rate amplitude of 0.01 s-1, which is an order of magnitude higher than the critical strain-rate below which b(|γ˘ |) aymptotically approaches unity. To examine the origin of this discrepancy, we incorporate the strain-rate dependence of structural relaxation τ obtained from the fit to the frequency scaling factor (Figure 8B) into a Maxwell model.24 For an amplitude sweep carried out at an angular frequency of 1 rad/s, the peak in G′′ occurs at |γ| ) 0.1 (Figure 8B, inset), which agrees well with the experimentally observed behavior (Figure 1). Consistent with the asymptotic behavior of b(|γ˘ |), the linear regime extends only up to |γ| ) 0.001 with |γ˘ | ) |γ|ω ) 0.001 s-1, though experimentally it is observed at 0.01 s-1. Hence, at present we do not understand the origin of this discrepancy for sorbitan tristearate monolayers. It is likely that the discrepancy arises from a more complicated dependence of the storage and loss moduli on the structural relaxation time than that given by a Maxwell model. From eq 1, for large intrinsic relaxation time τo, the structural relaxation time τ ∼ |γ˘ |-ν. For a simple Maxwell model for Gf(ω), this would lead to a power law decay of G′ and G′′ at high strain amplitudes, with G′ ∼ |γ|-2ν and G′′ ∼ |γ|-ν. The value of the exponent is found to vary with the system. In the present system, the value of the exponent ν varies from 0.7 to 0.88 depending on temperature and interfacial concentration. However, it should be noted that the power law decay exponent of G′′ (ν
Langmuir, Vol. 23, No. 26, 2007 12957
) 0.88, for sorbitan tristearate monolayers at 0.54 µg cm-2 at 18 °C) is consistent with the large strain amplitude behavior expected in the amplitude sweep, if we take into account the dependence of structural relaxation time obtained from strainrate frequency sweep experiments (τ ∼ |γ˘ |-0.88) into a Maxwell model. Hence, our limitations in explaining the origin of the peak in G′′ at large strain amplitudes in the present system arises mainly from the absence of a rigorous model to describe the viscoelastic behavior for the Langmuir monolayers. 3.3.2. Origin of the Peak in the Loss Modulus for 2D Monolayers. The viscoelastic properties of 2D Langmuir monolayers are known to arise from the presence of long-range ordered solid-phase domains coexisting with a continuous liquid expanded phase in the monolayer.4,31 Recently, a divergence of the viscosity characteristic of a jamming behavior was observed for lipid monolayers above a critical area fraction of domains.4 The nature of divergence of the viscosity of the monolayer indicates that the solid lipid domains are stabilized by long-range repulsive interactions. The viscosity dependence on the area fraction of domains also suggests a strong analogy with the behavior of 3D colloidal suspensions where the viscosity diverges above a critical volume fraction determined by the range of repulsive interaction.5 Moreover, simulations of large-amplitude sweeps carried out on 2D suspensions of electrorheological fluids, consisting of percolating clusters of particles with repulsive dipole-dipole interactions, have shown that rearrangement of the clusters rather than a large-scale breakdown of the structure can give rise to the nonlinear viscoelastic behavior under study.12 Similarly, it is likely that the nonlinear viscoelastic behavior that we observe for the sorbitan tristearate monolayers at high surface concentrations arises from the rearrangement of the solid domains in the monolayer at large strain amplitudes. The decrease in storage modulus can be due to the modification of interdomain interactions during their structural rearrangement in the film, whereas the increase in loss modulus can arise from the viscous contribution, when the solid domain translates through the coexisting liquid phase. This is consistent with the observations from the realtime imaging of amplitude sweep experiments through Brewster angle microscopy (BAM). BAM on sorbitan tristearate monolayers does indicate that the film consists of a large number of solid domains of high surface concentration.21 At low strain amplitudes, BAM does not reveal any change in the morphology of the film under oscillatory shear. However, at higher strain amplitudes, where the peak in G′′ occurs, the morphology of the 2D film is modified due to the rearrangement of these domains, followed by a fragmentation of the film at large strain amplitudes. It is also likely that the nonlinear viscoelastic behavior described above for sorbitan tristearate monolayers can be a common feature of many 2D viscoelastic films of crosslinked polymers, membrane proteins, and so on, which forms micrometer-sized domains in the solid phase. However, at present, very few studies exist on the nonlinear viscoelastic behavior of such two-dimensional films. Further investigations are worthwhile to probe the viscoelastic behavior in these systems.
4. Conclusions In conclusion, we have studied the nonlinear viscoelastic behavior of sorbitan tristearate monolayers at the air-water interface at different surface concentrations and temperatures under oscillatory shear. At low surface concentrations corresponding to the gas analogous phase of the monolayer, the storage and loss moduli exhibit a monotonic decay at large strain amplitudes. On increasing the surface concentration to the highly condensed, solid analogous phase, at low temperatures, G′ decays
12958 Langmuir, Vol. 23, No. 26, 2007
monotonically, whereas G′′ reveals a distinct peak above a critical shear amplitude. At large γ, both the storage and loss moduli exhibit a power law decay with the decay exponents in the ratio 2:1. With increase in temperature, however, the nonlinear viscoelastic behavior observed at low surface concentrations is recovered. Strain-rate frequency sweep measurements were carried out on the Langmuir monolayers at high surface concentrations and at low temperatures. A shift in the crossover frequencies to higher values was observed on increasing the strain-rate amplitude. This indicates a strain-rate dependence of the structural relaxation time (τ) and the behavior is similar to that seen in 3D metastable complex fluids. We also show that
Krishnaswamy et al.
the nonlinear viscoelastic measurements can be used successfuly to extract the intrinsic structural relaxation time. Our studies clearly reveal the generality of the viscoelastic behavior in 2D and 3D systems. We hope that this work will motivate further theoretical studies on the viscoelastic behavior of 2D systems. Acknowledgment. We thank Rajesh Ganapathy for his help at the beginning of the experiments. We also thank Dr. S. Sampath for providing us with the surface pressure-area isotherm of sorbitan tristearate monolayers. LA701889W
