Water Interface Studied by Shear

layers exhibit generalized Maxwell behavior in shear mode within a deformation frequency ... Dilational Properties of Sodium 2,5-Dialkyl Benzene S...
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Langmuir 2005, 21, 10555-10563

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Sorbitan Tristearate Layers at the Air/Water Interface Studied by Shear and Dilatational Interfacial Rheology Philipp Erni,* Peter Fischer, and Erich J. Windhab Laboratory of Food Process Engineering, ETH Zu¨ rich - Swiss Federal Institute of Technology, 8092 Zu¨ rich, Switzerland Received June 3, 2005. In Final Form: August 12, 2005 The shear and dilatational rheology of condensed interfacial layers of the water-insoluble surfactant sorbitan tristearate at the air/water interface is investigated. A new interfacial shear rheometer allows measurements in both stress- and strain-controlled modes, providing comprehensive interfacial rheological information such as the interfacial dynamic shear moduli, the creep response to a stress pulse, the stress relaxation response to a strain step, or steady shear curves. Our experiments show that the interfacial films are both viscoelastic and brittle in nature and subject to fracture at small deformations, as was supported by in-situ Brewster angle microscopy performed during the rheological experiments. Although any large-deformation test is destructive to the sample, it is still possible to study the linear viscoelastic regime if the deformations involved are controlled carefully. Complementary results for the dilatational rheology in area step compression/expansion experiments are reported. The dilatational behavior is predominantly elastic throughout the frequency spectrum measured, whereas the layers exhibit generalized Maxwell behavior in shear mode within a deformation frequency regime as narrow as two decades, indicating the presence of additional relaxation mechanisms in shear as opposed to expansion/compression. If the transient rheological response from stress relaxation experiments is considered, then the data can be described well with a stretched exponential model both in the shear and dilatational deformations.

Introduction Viscoelastic behavior in Langmuir layers is observed for various surface-active compounds, including (i) surfaceactive polymers and lipopolymers,1-4 (ii) single alkyl chain surfactants and fatty acids or fatty alcohols,1,5-7 and (iii) small-molecular surfactants with more than one hydrophobic side chain,2,8-11 for example, phospolipids and glycolipids. Nonpolymeric surfactant monolayers may exhibit interfacial viscous and elastic properties within their liquid-condensed or condensed phases,5,6 associated with intermolecular interaction between their alkyl side chains and, in some cases, between the headgroups. However, the interfacial rheology of small-molecular surfactants has not received as much attention as the related field of surface-active polymers primarily because of their low viscoelasticity compared to that of polymers. Strong viscoelastic effects have been found for monolayers of double-chain surfactants, such as synthetic glycolipids8,9 and phospholipids.2,10,11 For these molecules, the increased viscoelasticity can in many cases be attributed to side* Corresponding author. E-mail: [email protected]. (1) Brooks, C. F.; Fuller, G. G.; Frank, C. W.; Robertson, C. R. Langmuir 1999, 15, 2450. (2) Naumann, C. A.; Brooks, C. F.; Fuller, G. G.; Knoll, W.; Frank, C. W. Langmuir 1999, 15, 7752. (3) Fuller, G. G. Rheology of Mobile Interfaces. In Rheology Reviews; Binding, D. M., Walters, K., Eds.; The British Society of Rheology: Aberystwyth, U.K., 2003; p 77. (4) Brooks, C. F.; Thiele, J.; Frank, C. W.; O’Brien, D. F.; Knoll, W.; Fuller, G. G.; Robertson, C. R. Langmuir 2002, 18, 2166. (5) Gaub, H. E.; McConnell, H. M. J. Phys. Chem. 1986, 90, 6830. (6) Barentin, C.; Muller, P.; Ybert, C.; Joanny, J.-F.; di Meglio, J.-M. Eur. Phys. J. E 2000, 2, 153. (7) Garofalakis, G.; Murray, B. S. Langmuir 2002, 18, 4765. (8) Schneider, M. F.; Lim, K.; Fuller, G. G.; Tanaka, M. Phys. Chem. Chem. Phys. 2002, 4, 1949. (9) Tanaka, M.; Schiefer, S.; Gege, C.; Schmidt, R. R.; Fuller, G. G. J. Phys. Chem. B 2004, 108, 3211. (10) Kra¨gel, J.; Li, J. B.; Miller, R.; Bree, M.; Kretzschmar, G.; Mo¨hwald, H. Colloid Polym. Sci. 1996, 274, 1183. (11) Naumann, C. A.; Brooks, C. F.; Wiyatno, W.; Knoll, W.; Fuller, G. G.; Frank, C. W. Macromolecules 2001, 34, 3024.

chain condensation; therefore, it is expected that surfactants with similar headgroups but a larger number of hydrophilic alkyl chains will show stronger surface viscoelastic effects. Interfacial rheology is usually subdivided into the areas of shear rheology and dilatational rheology because in most applications (emulsion and foam stability, drop or bubble deformation in multiphase flow, etc.) both deformation modes exist.12 In recent years, progress in interfacial rheometrical instrumentation has led to an increase both in the number of materials studied (soluble and insoluble surfactants, surfactant polymers and proteins, model membranes, and surface-active particles) and in the quality and availability of rheological experiments. For example, whereas most of the older experimental literature on interfacial shear rheology exclusively discusses interfacial shear viscosities measured in the steady state, more recent instruments such as the stresscontrolled oscillating needle device1,3,4,13 or a modification of the classical rotating disk design with a new motor system14,15 allow a much broader selection of interfacial rheological tests, including oscillatory shear (with variable frequency or deformation amplitude), stress relaxation, creep, and so forth. Numerous methods for the measurement of interfacial shear and dilatational properties have been proposed over the past decades, all of them having individual advantages and limitations with respect to sensitivity, operating range, suitability for measurements at gas/liquid or liquid/ liquid interfaces, ease of operation, feasible tests (stress or strain rate control), and data acquisition (direct (12) Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heinemann: Boston, 1991. (13) Ding, J.; Warriner, H. E.; Zasadzinski, J. A.; Schwartz, D. K. Langmuir 2002, 18, 2800. (14) Erni, P.; Fischer, P.; Windhab, E. J.; Kusnezov, V.; Stettin, H.; La¨uger, J. Rev. Sci. Instrum. 2003, 74, 4916. (15) Erni, P.; Fischer, P.; Heyer, P.; Windhab, E. J.; Kusnezov, V.; La¨uger, J. Prog. Colloid Polym. Sci. 2004, 129, 16.

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measurements of torque, force, or displacement vs image analysis of tracer particles). Comprehensive reviews of different instrument designs are available.3,12,16-18 The material used for this study is sorbitan tristearate (trioctadecanyl ester of sorbic acid), a nonionic, waterinsoluble surfactant used in pharmaceutical and food applications. Compared to other sorbitan esters, most of which contain only one single alkyl chain and primarily act as surfactants by decreasing the interfacial tension, sorbitan tristearate exhibits strong viscoelastic effects when spread at the air/water interface. The interfacial rheology, morphology, and thermodynamics (surface pressure-area isotherms) of this surfactant on pure water subphases and at dodecane/water interfaces, its interfacial rheology on subphases containing biopolymers, and the physicochemical and rheological behavior of emulsion systems involving sorbitan surfactants have been described elsewhere.19-22 We present shear rheological data obtained with a new interfacial rheometer, as well as complementary results for the dilatational rheology obtained from compression/expansion experiments performed in a Langmuir film balance. The rheometer is briefly described, and its fluid mechanical background is summarized. We provide data obtained with different rheological operating modes, including creep recovery (i.e., the rheological answer to a stress pulse), stress relaxation after a strain step, steady shear experiments at variable shear rate, and oscillatory shear at variable deformations. An experimental setup combining interfacial rheometry with Brewster angle microscopy (BAM) was used for realtime imaging of the interface during rheological experiments. The static and transient dilatational moduli of the interfacial layers as well as the spectral dilatational moduli obtained by a fast Fourier transform method will be presented and discussed. Experimental Section Materials. Sorbitan tristearate (trioctadecanyl ester of sorbic acid, batch no. 407571/1) and chloroform (analysis grade) were purchased from Fluka and used as received. All water used was purified with a Milli-Q Biocel (Millipore) pure water system. The surfactant was characterized by surface pressure measurements using a Langmuir film balance (model 302LL, Nima Technology Ltd., Coventry, U.K., trough area 300 cm2 with two symmetrically driven barriers and a platinum Wilhelmy plate). Prior to every experiment, the trough and barriers were cleaned with chloroform. Before the surfactant was spread, the trough area was rapidly compressed to a minimum value, and possible impurities were removed from the water surface with a suction pump. Spreading and all experiments in the film balance were performed only if the surface pressure of the clean water surface did not change more than 0.1 mN/m upon expansion. Sorbitan tristearate was dissolved in chloroform and spread onto the clean subphase using a microsyringe. All experiments were performed with the Wilhelmy plate located halfway between the two barriers. For some of the shear experiments, the subphase viscosity was adjusted with nongelling sodium κ-carrageenan (κ-CN, CP Kelco, salt content 4.94 wt % Na, 0.77 wt % K, 0.016 wt % Ca, and 0.068 wt % Mg measured by atomic absorption spectroscopy). The (16) Miller, R.; Wu¨stneck, N.; Kra¨gel, J.; Kretzschmar, G. Colloids Surf. A 1996, 111, 75. (17) Murray, B. S. Curr. Opin. Colloid Interface Sci. 2002, 7, 426. (18) Slattery, J. C. Interfacial Transport Phenomena; SpringerVerlag: New York, 1990. (19) Rehage, H.; Achenbach, B.; Geest, M.; Wilhelm Siesler, H. Colloid Polym. Sci. 2001, 279, 597. (20) Rehage, H.; Achenbach, B.; Klaerner, F.-G.; Lee, J. Langmuir 2002, 18, 7115. (21) Peltonen, L.; Hirvonen, J.; Yliruusi, J. J. Colloid Interface Sci. 2001, 240, 272. (22) Peltonen, L.; Hirvonen, J.; Yliruusi, J. J. Colloid Interface Sci. 2001, 239, 134.

Erni et al. preparation and purification method23,24 as well as further physicochemical details25-27 are summarized elsewhere. The biopolymer was dissolved in water and first stirred at room temperature for 15 min and then heated at 95 °C for another 15 min. The biopolymer solutions are Newtonian in the range of concentrations used.28 Blank isotherms performed on freshly cleaned surfaces of the κ-carrageenan solutions during the time scale of the rheological experiments were flat with zero surface pressures. Surface tensions of the carrageenan subphases were measured by the drop volume method (TVT, Lauda). For the κ-carrageenan(1.5 wt %)/air system, the equilibrium surface tension was 71.6 mN/m (at 20 °C); therefore, the biopolymer exhibits very little interfacial activity on this time scale as compared to pure water (surface tension 72.8 mN/m), with some minor effects probably due to impurities from the production process. Interfacial Shear Rheology. The interfacial shear rheometer is based on the well-known rotating or oscillating biconical bob design, combined with the motor of a standard research rheometer with a brushless, electronically commutated (EC) motor system.14 This combination allows both stress- and straincontrolled measurements on the same sample without removing any parts of the instrument. The analysis of the steady-state interfacial rheological parameters is based on the BoussinesqScriven interfacial stress tensor29,30

Tσ ) [σ - (κ - η)divσvσ]P + 2ηDσ

(1)

which enters into the jump momentum balance at the interface. σ is the thermodynamic interfacial tension, κ is the dilatational viscosity, η is the shear viscosity of the interface, divσvσ is the divergence of the interfacial velocity field, P is a projection tensor,18 and Dσ is the interfacial strain rate tensor. A simplified numerical method was implemented to obtain the interfacial shear viscosity, stress, and dynamic moduli from the measured raw data (torque and angular deflection). A summary of the interfacial fluid mechanical analysis and numerical techniques can be found as Supporting Information for this article. The calculations are similar to previously published analyses of the interfacial shear flow in the biconical geometry,31-34 all of which yield implicit expressions for the interfacial velocity distribution needed for the calculation of the interfacial shear stess. All experiments are performed in either stress- or strain-controlled mode using a Physica MCR 300 rheometer adapted for interfacial rheometry. A 2D Searle-type measuring geometry with a rotating or oscillating biconical bob was used. The disk is positioned with its edge located at the surface of the aqueous phase (Figure 1). For the instrument used here, the biconical bob and its axis are connected to the top-driven motor and transducer unit of a standard rotational rheometer. The disk is rotating or oscillating at a controlled torque or rotational speed while the cup remains stationary, and depending on the kind of experiment either the (23) Walther, B.; Cramer, C.; Tiemeyer, A.; Hamberg, L.; Fischer, P.; Windhab, E. J.; Hermansson, A. M. J. Colloid Interface Sci. 2005, 286, 378. (24) Morris, V. J.; Chilvers, G. R. Carbohydr. Polym. 1983, 3, 129. (25) Ciancia, M.; Milas, M.; Rinaudo, M. Int. J. Biol. Macromol. 1997, 20, 35. (26) Harding, S. E.; Day, K.; Dhami, R.; Lowe, P. M. Carbohydr. Polym. 1997, 32, 81. (27) Slootmaekers, D.; Mandel, M.; Reynaers, H. Int. J. Biol. Macromol. 1991, 13, 17. (28) For the experiment with a κ-carrageenan subphase, the bulk rheology was measured in a double gap geometry. The bulk viscosity is 163 mPas (biopolymer concentration 1.5 wt %, temperature 20 °C). Both the elastic and loss moduli are strain-independent in the range of applied strains (up to γ ) 200%), and the phase angle had a value of π/2 for all angular frequencies measured (ω ) 10-1 to 102 s-1), indicating purely viscous (nonelastic) behavior. (29) Boussinesq, M. J. Ann. Chim. Phys. 1913, 29, 349. (30) Scriven, L. E. Chem. Eng. Sci. 1960, 12, 98. (31) Lee, H. O.; Jiang, T.-S.; Avramidis, K. S. J. Colloid Interface Sci. 1991, 146, 90. (32) Nagarajan, R.; Chung, S. I.; Wasan, D. T. J. Colloid Interface Sci. 1998, 204, 53. (33) Oh, S.-G.; Slattery, J. C. J. Colloid Interface Sci. 1978, 67, 516. (34) Ray, Y.-C.; Lee, H. O.; Jiang, T. L.; Jiang, T.-S. J. Colloid Interface Sci. 1987, 119, 81.

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Figure 1. Schematic of the combined setup for BAM and surface shear rheometry. (a) Laser, (b) polarizer, (c) CCD camera with microscope lens, (d) rotating biconical bob, (e) stationary cup, and (f) surface layer. (i) Front view and (ii) top view. angular velocity or the torque of the disk is measured. To account for bulk viscous contributions to the interfacial shear stress, a hydrodynamic analysis of the flow in the rheometer is necessary. Oh and Slattery first provided an exact solution of the velocity distribution in steady shear flow in the plane of the interface. The interfacial shear stress, τ, and interfacial shear viscosity, η, are calculated using an iterative procedure from the angular velocity and the disk torque. A dimensionless group, the Boussinesq number Bo, can be defined to describe to ratio of interfacial to bulk-phase stresses

Bo )

x xBR

(2)

where x is an interfacial parameter and xB is the corresponding bulk rheological parameter (for example, the interfacial and bulk shear viscosities η and ηB) and R is a characteristic experimental length scale (in our case, the radius of the rheometer cup). Small Boussinesq numbers indicate strong dissipation of interfacial stresses into the bulk whereas in the case of large Boussinesq numbers the interfacial flow dominates the bulk flow. In the latter case, a simplified method can be used by measuring a blank value for the rheology of the clean, surfactant-free interface, and the shear rheological properties may be obtained from the raw data using a simple proportionality relation with a geometry factor, as it is done in bulk rheometry. Dynamic experiments are performed to obtain information on the viscoelastic properties of the interfaces at different rheological time scales.35 In this case, the shear strain γ is varied sinusoidally with time at an angular frequency of ω: γ(t) ) γ0 sin(ωt), where t is time and γ0 is the oscillation amplitude. For linear viscoelastic materials, the response function is a sinusoidally changing shear stress, τ(t) ) -τ0 sin(ωt + φ) that is out of phase with the strain. A complex shear modulus G* ) G′ + iG′′ can be defined from the stress and strain waves as G* ) τ0eiφ/γ0, with φ being the phase (35) Macosko, C. W. Rheology: Principles, Measurements, and Applications; Wiley-VCH: New York, 1994.

angle. The storage modulus, G′, then describes the elastic properties of the sample, whereas the loss modulus, G′′, is proportional to the viscous resistance. Alternatively, viscoelasticity can be probed using transient experiments. In stress relaxation experiments, the time-dependent rheological response of the shear stress τ(t) to a given step function of the deformation γ-(t < 0) to γ+(t g 0) is measured. For this kind of test, deformation-controlled rheometers are used to provide sufficiently fast deformation changes. In creep experiments, a constant stress is imposed onto the sample during a defined time, and the deformation response function during this stress pulse (creep curve) is measured. Additionally, this experiment provides information about the recovery behavior of the strain upon removal of the stress. The material function defined from creep experiments is called the creep compliance J(t) ) γ(t)/τ0 and is defined as the ratio of the time-dependent deformation and the imposed stress pulse. Several authors have described and used biconical or disk rheometers based on torsion wire setups in rotation mode and in free and forced oscillation.33,36-40 Compared to these experiments, the method used here, in which the rotating disk is rigidly coupled to the top-driven motor and transducer unit of a standard rheometer, offers some advantages: measurements may be performed both in rotation and oscillation modes without exchanging any components (e.g., torsion wires) and without changing the sample. Torques (0.01 µNm) and deflection angles (0.1 µrad) can be attained in a range needed for interfacial rheometry,41 for example, with rigid, delicate interfacial layers. In summary, the interfacial shear rheological properties used in this article are the shear deformation (strain) γ, the interfacial shear viscosity η and shear stress τ, the transient and complex interfacial shear moduli G(t) and |G*|, the dynamic moduli G′ (storage) and G′′ (loss), and the creep compliance J(t). Interfacial Dilatational Rheology. For insoluble surfactant layers, a static dilatational modulus for extremely slow deformations can be obtained from the equilibrium surface pressure π versus the surface concentration Γ isotherm as

|

|

∂π ∂π ) -A Estatic ) Γ ∂Γ T ∂A T

(3)

(i.e., Estatic can be derived from the slope of the π-Γ isotherm12,42). To obtain meaningful data for Estatic, the compression or expansion of the surface must be performed at sufficiently slow deformation speeds to ensure that any relaxation process is absent from the recorded values. A dilatational relaxation modulus E(t) as a function of time is defined as

E(t) )

∆π(t) ∆A/A0

(4)

where ∆π ) π(t) - πeq is the transient surface pressure jump from the equilibrium value πeq after a small change in surface area ∆A/A0 at time t ) 0. In the absence of any relaxation processes, E is a purely elastic modulus with a constant value (i.e., the surface pressure adjusts itself to a new value instantaneously as soon as the area is changed; alternatively, it can be said that any possible relaxation process in this case happens on a time scale much shorter or much longer than the experimental time window). If the case of surface pressure relaxation after a decrease in area is considered, then two parameters can be defined from the experimental E versus t curve: (i) an instantaneous dilatational (36) Kra¨gel, J.; Siegel, S.; Miller, R.; Born, M.; Schano, K.-H. Colloids Surf., A 1994, 91, 169. (37) Briley, P. B.; Deemer, A. R.; Slattery, J. C. J. Colloid Interface Sci. 1976, 56, 1. (38) Goodrich, F. C.; Chatterchee, A. K. J. Colloid Interface Sci. 1970, 34, 36. (39) Shail, R. J. Eng. Math. 1978, 12, 59. (40) Jiang, T.-S.; Chen, J.-D.; Slattery, J. C. J. Colloid Interface Sci. 1983, 96, 7. (41) La¨uger, J.; Wollny, K.; Huck, S. Rheol. Acta 2002, 41, 356. (42) Monroy, F.; Ortega, F.; Rubio, R. G. Eur. Phys. J. B 2000, 13, 645.

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Figure 2. Example of a surface pressure relaxation experiment. The top curve shows the surface pressure jump ∆π ) π(t) - πeq, where πeq is the equilibrium surface pressure before the area change. The bottom curve shows the relative area change upon a compression experiment in the Langmuir film balance. The instantaneous value ∆π0 at t f 0 and the long-time asymptotic value ∆π∞ at t f ∞ are used to calculate the corresponding dilatational elastic parameters E0 and E∞. modulus E0 at t f 0 and (ii) an equilibrium dilatational modulus E∞ at t f ∞. The two parameters are presented graphically in Figure 2. For harmonic area changes, A(t) ) A0 exp(-iωt), the transient modulus E(t) is replaced by a complex dilatational modulus

E*(ω) ) E′ + iE′′

(5)

with a real part E′ (storage or elastic modulus) and an imaginary part E′′ (loss or viscous modulus). If E is measured in an experiment involving small sinusdoidal area perturbations, then the moduli are measured by analogy to bulk rheology from the stress response π(t) to a given strain function A(t). The loss modulus E′′ is experimentally reflected by a phase angle φ between the strain and stress function, with E′ ) |E| cos φ and E′′ ) |E| sin φ. With the methodology outlined so far, no assumptions are made about the physical nature of the relaxation processes (e.g., diffusion to/from the bulk solution, molecular rearrangements, exchange with secondary adsorption layers, etc.12,16,43-47). In the experiments presented here, we use a Langmuir film balance to acquire the π(t) versus A(t) raw data. The interfacial area is compressed or expanded using symmetric barriers with the surface pressure sensor located halfway between the two barriers either in oscillatory or in transient mode (step- or ramptype deformations). Figure 2 shows an example of the area versus time raw data (i.e., the controlled strain ∆A/A0(t) and the measured stress ∆π(t) of the experiment) from which the dilatational modulus and the phase angle are calculated. The transient dilatational modulus E(t) is related to the complex modulus E*(ω) via a Fourier transform:44,48

E*(ω) )

F{d∆π(t)/dt} F{d(A/A0)dt}

(6)

Software was written in our laboratory for the conversion of experimental data from the time domain to the frequency domain by a numerical Fourier transform (FFT) method. Brewster Angle Microscope. We use BAM49,50 as an in situ technique for real-time imaging and direct visualization of the structure, phase behavior, and orientation effects due to flow processes in Langmuir films. In the experiments performed here, (43) Benjamins, J.; Cagna, A.; Lucassen-Reynders, E. H. Colloids Surf., A 1996, 114, 245. (44) Loglio, G.; Tesei, U.; Cini, R. Rev. Sci. Instrum. 1988, 59, 2045. (45) Miller, R.; Li, J. B.; Bree, M.; Loglio, G.; Neuman, A. W.; Mo¨hwald, H. Thin Solid Films 1998, 327, 224. (46) van den Tempel, M.; Lucassen-Reynders, E. H. Adv. Colloid Interface Sci. 1983, 18, 281. (47) Monroy, F.; Ortega, F.; Rubio, R. G. Phys. Rev. E 1998, 58, 7629. (48) Miller, R.; Loglio, G.; Tesei, U.; Schano, K.-H. Adv. Colloid Interface Sci. 1991, 37, 73. (49) He´non, S.; Meunier, J. Rev. Sci. Instrum. 1991, 62, 936. (50) Ho¨nig, D.; Mo¨bius, D. J. Phys. Chem. 1991, 95, 4590.

Figure 3. Apparent interfacial shear viscosity of sorbitan tristearate (mean molecular area 0.6 nm2/molecule). The experiment was performed on a freshly spread interfacial film from low to high shear rates (up curve). The line is drawn to guide the eye. the technique is used for in situ imaging of the experiments performed in the interfacial rheometer in a setup built in our laboratory and depicted in Figure 1. A laser beam (He-Ne laser, 632.8 nm, Melles-Griot) is guided through a Glan-Thompson polarizer under the Brewster angle condition for the air/water interface, φ ) arctan(nwater/nair) ) 53.1°. For the empty surface, polarized light is annihilated, and no reflection is observed. In the presence of surface-active molecules or impurities, the Brewster angle condition is no longer met, and partial reflection occurs. A CCD camera equipped with a long working distance microscope lens (Mitutoyo) is used to capture the reflected laser beam. Images are acquired in real time during rheological experiments, therefore providing an opportunity to observe deformations, wall slip, and structural transformations in the interfacial films.

Results and Discussion Identification of the Linear Viscoelastic Regime and Onset of Film Fracture. A typical curve for the interfacial shear viscosity versus the applied shear rate is presented in Figure 3. At low shear rates, the interfacial shear viscosity is constant. As soon as a critical shear rate (or a critical cumulated strain) is exceeded, the film is apparently shear thinning. An interpretation of the data as exhibiting shear-thinning behavior of the surface is, however, not justified. Upon repeated runs of the identical experiment, we were not able to achieve repeatable results. A systematic shift of the entire flow curve toward lower interfacial viscosities was found when repeating the shear experiments on the same sample. For all newly prepared interfacial layers of the same concentration, the flow curves of the first run were in good agreement with each other for experiments from low to high shear rates (up curves). However, results from the corresponding down curves always differed from those of the up curves, with η values that sometimes were orders of magnitude lower than found in the up curves. This leads us to the conclusion that the steady shear experiments are destructive to the sample. Strain amplitude sweep experiments with sorbitan tristearate on either pure water or on the biopolymer subphases show that the linear viscoelastic regime of the interfacial films is indeed limited to very small strains (Figure 4). The dynamic interfacial shear moduli G′ and G′′ are strain-independent only up to deformations as small as approximately γ < 1.5% and decrease above this value. (Repeated runs of the identical experiment furnish much lower values of the shear moduli, as is the case for the shear viscosity in the experiment described above.) Brewster angle micrographs obtained during the rheological experiments are shown in Figure 4. The surface

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Figure 4. Strain amplitude sweep experiment (shear flow) with sorbitan tristearate (ω ) 1 rad/s). The insets are Brewster angle micrographs recorded during the experiment. (I) Linear viscoelastic regime and (II) nonlinear regime with observed fracture in the film, resulting in an inhomogeneous flow pattern with an intact film on the left-hand side and debris of the ruptured film flowing in the reduced shear gap on the righthand side. The scale bar corresponds to 100 µm.

morphology as recorded with BAM is shown, with typical images taken in the linear viscoelastic regime where neither shear modulus depends on the strain. All images taken at such subcritical deformations resemble the one shown in region I of Figure 4, with domains of high surface concentration (bright spots on the Figure) moving along with the rheometer disk in a homogeneous velocity distribution that is at its maximum and equal to the disk velocity on the right-hand side and zero at the wall of the cup. For deformations γ e γcrit, no changes in morphology are observed, and any movement on the surface due to the rheometric flow is steady and stable for hours, as long as the experimental conditions are kept constant. At all concentrations measured in the shear rheometer, the interfacial films were elastic at an angular frequency of ω ) 1 rad/s, with G′ almost an order of magnitude larger than G′′. If, however, the oscillation amplitude is increased and a critical deformation is exceeded, then the surface layer is disrupted, leaving behind a virtually unsheared portion of the surface with an intact film (left-hand side of Figure 4) and a suspension of film fragments moving in a reduced shear gap (right-hand side of Figure 4). In this second part of the strain sweep curves, both moduli decrease with increasing amplitude. In conclusion, the strain sweep experiments in combination with the BAM structure information provide more useful information than can be obtained from the steady shear experiments, which always involve large cumulated strains and therefore seem to be less suited to the study of fragile interfacial layers that are not purely viscous in nature. Interfacial Viscoelasticity: Shear Deformation. We will now discuss results from shear experiments performed in the linear viscoelastic regime that are nondestructive to the sample, with experimental errors usually within the margin of the data points. This means that in all cases of repeated transient experiments on the identical sample the tests were followed by an identical experiment in the negative strain direction to keep the overall cumulative deformation below the critical value. Frequency-dependent dynamic experiments were performed with small-amplitude oscillatory deformations to obtain information on the viscoelastic properties of the interfaces at different time scales. The interfacial storage and loss moduli G′ and G′′ are shown as a function of the deformation frequency in Figure 5. A range of angular frequencies from 10-2 to 5 rad/s was investigated. Three distinct regimes of the shear moduli are observed: (i) At

Figure 5. Dynamic interfacial shear moduli of sorbitan tristearate at the air/water interface (surface concentration mean molecular area 0.65 nm2/molecule, strain amplitude γ0 ) 0.2%. Lines are fits of a three-mode Maxwell model with parameters Gi ) 9.23, 11.88, and 13.21 mN/m and λi ) 24.1, 3.4, and 0.4 s.

low frequencies, the interfacial response to shear flow is predominantly viscous, with G′′ > G′ and the moduli scaling as approximately G′ ∝ ω1.5 and G′′ ∝ ω0.5. This indicates deviation from perfect Maxwell low-frequency behavior (a pure liquid where ideally G′ ∝ ω2 and G′′ ∝ ω). Scaling as seen in the viscous regime is encountered in concentrated solutions of 3D systems. It seems plausible that at lower frequencies than studied here the exponents would approach those of the pure liquid case. (ii) There is a transition region where a crossover of the storage and loss moduli is found at a frequency of approximately ω ) 0.08 rad/s, leading to an average relaxation time of λ ) 1/ω ) 62 s. (iii) At high frequencies ω > 0.2 rad/s, the elastic behavior dominates with G′ > G′′, and both moduli are much less frequency-dependent. This behavior is typical for temporary network liquids. Many interfacial networks show predominantly elastic behavior throughout the entire frequency spectrum either due to strong (covalent) bonds in the film or due to the confinement of macromolecules to a 2D geometry where entanglements (and hence reptation) are unlikely. Both effects can be excluded here because covalent bonds are not expected (i.e., the films are not chemically cross-linked) and the surfactant at hand is not a macromolecule. However, if the interfacial rheology of a surfactant such as sorbitan monostearate with the identical headgroup but only one stearyl side chain is considered, then none of the viscoelastic features described here is found (no elasticity or detectable interfacial viscosity, data not shown). This leads us to conclude that the three stearyl chains and the corresponding hydrophobic interactions of the surfactant might play a crucial role in its strong viscoelastic response. This argument will be confirmed below using the characteristic time scale found in creep experiments. An interesting aspect here is that the three regions shown in Figure 5 are observed within a very narrow range of frequencies (two decades) and without the use of any time-temperature or time-concentration superposition techniques. This is rather surprising because most interfacial films with comparably strong interfacial rheological response do not show such a pronounced frequency dependence; they are likely to possess a wide elastic regime

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(rubberlike), and it is rather unusual in interfacial rheology to find a crossover of G′ and G′′ and even a viscous regime in such a narrow range of experimental time scales. Even in bulk rheometry it is common that temperature shift factors must be utilized to experimentally access the different flow regimes seen in Figure 5. To describe the viscoelastic behavior in all three regimes shown in Figure 5, a phenomenological temporary network model may be used. The behavior of a generalized Maxwell material with n elements under harmonic oscillations can be represented by eqs 7 and 8, combining viscous (Newtonian) and elastic (Hookean) properties

ω2λi2 Gi i)1 1 + ω2λi2 n

G′(ω) )

∑ n

G′′(ω) )

Gi ∑ i)1

ωλi

1 + ω2λi2

(7)

(8)

where λi denotes the relaxation time and Gi denotes the plateau shear modulus of the ith Maxwell element. The solid line in Figure 5 shows a generalized Maxwell model fit to the shear moduli. Three modes (n ) 3 in eqs 7 and 8) are needed to obtain a reasonable description of the experimental data. The viscous regime (data up to the crossover point on the left-hand portion of the graph) could even be described with a single mode model (n ) 1). However, an inherent problem in describing interfaces as multimode Maxwell fluids is the fact that whereas the phenomenological description may be accurate and useful it is difficult to assign the individual model parameters (here, six parameters) to corresponding physical properties. Stress relaxation experiments in the interfacial shear rheometer were performed on the identical samples in strain-controlled mode. A step function in the deformation from γ- ) 0% to γ+ was performed at time t ) 0, and the resulting stresses were measured. The response of the interfacial shear relaxation modulus G(t g 0) to a step function of the strain to γ+ ) 0.8, 1.2, and 2% at time t ) 0 is plotted in Figure 7. Two classes of stress relaxation curves may be distinguished in this plot: (i) relaxation at subcritical strains γ < 2% and (ii) experiments at larger strains γ > 2% where the sample is disrupted. In the time domain, the Maxwell model for the transient relaxation modulus G(t) can be written as n

G(t) )

Gie-t/λ ∑ i)1

i

(9)

with Gi and λi being the same parameters as in the equations for the frequency domain. As for the frequency sweep experiments, a simple Maxwell model for the transient relaxation modulus G(t) gives a poor description of the data. Although a generalized Maxwell model performs much better, an interpretation of the model parameters is again difficult, and their physical meaning or relevance for comparison with other systems is limited. Therefore, we have attempted to describe our data with a stretched exponential (SE) law, sometimes also called the Kolrausch-Williams-Watts (KWW) law

G(t) ) G0 exp(-(t/λ)R)

(10)

where λ is a characteristic relaxation time and R is the SE

Figure 6. Creep recovery tests: shear deformation response to an interfacial stress pulse of τ0 ) 25 µN/m with sorbitan tristearate (mean molecular area 0.6 nm2/molecule). Experimental data: creep compliance J(t) ) γ(t)/τ0. The dashed line shows a Maxwell-Voigt model fit of the data (see inset; model parameters ηm ) 6.3 Ns/m, Jm ) 5.7 m/N, Jv ) 6.0 m/N, λv ) 9.6 s).

exponent. It describes the data remarkably well. The value of R implies a strong deviation from single Maxwell exponential relaxation (where R ) 1). This law offers a more compact description involving only three parameters to interpret the relaxation behavior and compare it to 3D systems. SE laws are popular for the description of the relaxation behavior of glasses. They are used to describe not only viscoelasticity but also a more general class of relaxation phenomena (i.e., dielectric relaxation). A derivation could be achieved in several different ways: as an average of exponential relaxation components with a relaxation time distribution52 or by relaxation rates depending on the length scales of the relaxing units.53 Creep recovery experiments were performed with intact surface layers with overall deformations in the linear viscoelastic regime (Figure 6). These experiments are stress-controlled because a pulse function of the torque is set at times t g 0 and the resulting deformation γ(t) is measured. The results obtained with clean and with surfactant-covered surfaces differ drastically: identical torque pulses applied to clean water surfaces lead to deflection angles of the disk that are several orders of magnitude higher than in the case of a surfactant layer present at the interface. This is true even for the smallest torque applied (M ) 0.01 µNm). Much smaller total deformations (γ < 0.4%) and typical viscoelastic creep response curves were observed for the surfactant-covered surfaces, including a phase of steady flow (constant shear rate) and partial relaxation of the strain upon removal of the stress. An attempt was made to describe the data with a simple phenomenological model, the MaxwellVoigt model (sometimes referred to as Burger’s model) for the creep compliance J(t),4,12

J(t) )

t + JM + JV(1 - e-t/λV) ηM

(11)

where indices M and V are used for the Maxwell and Voigt parameters, respectively. Figure 6 shows a nonlinear fit of the MV model to the creep portion of an experimental curve. A characteristic Maxwell time scale, which can be (51) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: Oxford, U.K., 1999. (52) Palmer, R. G.; Stein, D. L.; Abrahams, E.; Anderson, P. W. Phys. Rev. Lett. 1984, 53, 958. (53) Chamberlin, R. V. Phys. Rev. Lett. 1999, 82, 2520.

Sorbitan Tristearate Layers at Air/H2O Interface

Figure 7. Stress relaxation experiments: decay of the interfacial shear stress after deformation steps from γ ) 0 to γ+ at time t ) 0. The lines are fits of a stretched exponential (SE) model, with the model parameters summarized in the inset (surface coverage mean molecular area 0.6 nm2/molecule).

associated with an approximate lifetime of junction points in the interfacial network λm ) ηmJm ) 36 s, is obtained from the data in Figure 6. Published λm values for Langmuir monolayers are of the range of 103 s and higher for chemically cross-linked networks (e.g., polymerized lipopolymers) and are down to 102 s for physically crosslinked soft gels where hydrogen bonds are considered to be the relevant physical junctions. The value of λm ) 36 s for sorbitan tristearate confirms that the layers studied here are physically cross-linked and their description as temporary networks is justified. In this case, junction points due to chain condensation between the three alkyl side chains of sorbitan tristearate may play a dominant role in the formation of the network. This is supported by the somewhat weaker but still considerable viscoelasticity of comparable surfactants with only two side chains, such as phospholipids2,10,11 or those synthetic glycolipids where the viscoelasticity is assumed to be due to chain-chain correlation,8,9 and the much weaker surface rheological effects for single-chain sorbitan or sugar esters. The creep response tests also show the importance of a comprehensive analysis for the interfacial stress14 as outlined above: whereas for steady shear experiments it may often be appropriate to work with a simple analogy to 3D rheometric flows (i.e., to calculate the stress from the torque and rotational speed with a geometry factor and then subtract a blank value for the stresses measured on an uncovered interface) such blank values for the uncovered interface cannot be obtained in a creep test. Performing a stress (torque) pulse on a clean surface (i.e., a surface with negligible interfacial rheological effects) will result in a different flow situation where deformations of much higher magnitudes will be measured. The disk of the rheometer would visibly rotate several times if subjected to the same stress pulse on the surface of pure water, whereas the viscoelastic surfactant film is only slightly deformed. All interfacial shear stresses displayed in Figures 6 and 7 are transient and therefore cannot be calculated from the steady-state analysis outlined above. Although relaxation tests in bulk rheometry are subject to the same problem, the presence of coupled flow in both adjacent bulk fluids further complicates the interfacial stress relaxation problem. However, whereas no comprehensive analysis is available in relaxation mode, the Boussinesq numbers obtained from steady shear or oscillatory experiments may still be used as a measure of the coupling of the interfacial with the bulk-phase stresses. It should be noted that even with the analysis used here only the

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steady shear flow portion of the creep curve is based on a fully developed interfacial shear flow. In all other portions of the creep and relaxation experiments, the calculated stresses are transient and should therefore be treated as effective interfacial stresses. Although this is also true for similar tests in normal 3D rheology, the subphase flow adds a further degree of complexity to this problem. The steady shear flow and oscillatory flow problems have been solved,14,16,31-34,36 whereas methods to calculate the interfacial and bulk flow fields under startup and relaxation conditions are still to be developed. Because the data shown in Figure 4 were obtained on a subphase with increased viscosity, this work also shows that meaningful measurements of the interfacial shear moduli are still possible in systems where the subphase stresses play a larger role than on pure water. It should also be mentioned that the interaction between a polymer subphase and a insoluble surfactant layer can of course not be explained by hydrodynamics alone. Therefore, whereas the transport analysis used for the experiments at hand accounts for increased stress dissipation from the surface layer into the bulk, more work needs to be done to better characterize the influence of a biopolymer subphase on the thermodynamics of an insoluble monolayer. Biopolymer/surfactant interactions have been studied extensively for systems of ionic surfactants and oppositely charged polyelectrolytes. In the case of nonionic surfactants, polysaccharide/surfactant interactions are generally considered to be weak.54,55 Although we have found similar values of the interfacial moduli on water and on freshly cleaned carrageenan subphases, increased surface pressures and interfacial moduli were measured in the mixed system after extended waiting times. The data shown in Figure 4 were, however, acquired within a few minutes after clean-sucking the subphase, and on these time scales, no increases in surface pressure or in the moduli were observed. Although large deformation tests such as steady shear flow or large-amplitude oscillatory shear are not suited to the study of the intact surface, measurements in the linear viscoelastic regime are feasible with the motor and control system used in this study. Strain amplitude sweep, creep, stress relaxation, and small-amplitude oscillatory experiments were performed below a critical deformation of approximately 1.5% and showed both excellent repeatability and strain independence up to the critical strain. Interfacial Viscoelasticity: Dilatational Deformation. The spectral dilatational moduli E′ and E′′ are shown in Figure 8 (mean molecular area 0.58 nm2/ molecule). The data presented were obtained from the transient modulus E(t) using the FFT method; therefore, the upper frequency limit was set at ω ) 1 rad/s because the barriers of the Langmuir trough need a finite time (0.5 s) to perform the area compression. The following observations can be made: (i) the dilatational response is predominantly elastic throughout the entire frequency spectrum amenable with the method used (this is different from the shear response, where a crossover of G′ and G′′ and generalized Maxwell behavior are found); (ii) the magnitude of the complex dilatational modulus |E*| is always several orders of magnitude larger than the interfacial shear modulus; and (iii) the ratio E′/E′′ is several orders of magnitude greater than the ratio G′/G′′ at all (54) Kwak, J. C. T. Polymer-Surfactant Systems; Dekker: New York, 1998. (55) Goddard, E. D.; Ananthapadmanabhan, K. P. Interactions of Surfactants with Polymers and Proteins; CRC Press: Boca Raton, FL, 1993. (56) Martin, A.; Bos, M.; Cohen Stuart, M.; van Vliet, T. Langmuir 2002, 18, 1238.

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Figure 8. Dilatational storage (E′) and loss (E′′) moduli as a function of angular frequency. Data were obtained from the transient dilatational modulus E(t) using the FFT method (mean molecular area 0.6 nm2/molecule, ∆A/A0 ) -2%, original data acquisition rate 10 Hz).

frequencies tested. If the characteristic crossover frequency of the shear experiments is considered, where G′ ) G′′, then we still find a ratio E′/E′′ ) 12. Figure 9a shows the π-Γ isotherm at 20 °C along with the static elasticity modulus calculated from the isotherm data according to equation 3. The static modulus passes through a plateau with a local maximum at around 12 mN/m (i.e., the elasticity of the layer is only weakly dependent on the surface pressure in this region). A steep increase to large moduli at high surface pressures is observed, with values of several hundred mN/m. Step compression/expansion experiments were performed at different equilibrium surface pressures, labeled on the isotherm as I, II, and III. At each of these points, the surface pressure relaxation behavior after an area compression step ∆A/A0 ) -2% was investigated, and the area was then quickly reexpanded to the original value. In Figure 9b, the transient surface pressure responses π(t) in repeated compression/ expansion cycles performed at three different equilibrium surface pressures π ) 4, 10.1, and 37.1 mN/m are plotted. Experiment I in Figure 9 shows a pronounced difference in the relaxation behavior after compression as compared to the expansion case, indicated by the dashed boxes. Even though there are strong elastic effects in both compression and expansion, reflected by the instantaneous pressure response as soon as the area is changed, the relaxation behavior of the pressure following the change is markedly different. In the compression case (∆A/A0 < 0), the surface pressure slowly relaxes back from the short-time maximum pressure π0 to the equilibrium value π∞ on a time scale of the order of 102 s. In contrast, upon reversal of the area change (reexpansion with ∆A/A0 > 0) the surface response is almost purely elastic, and detectable relaxation processes are practically absent on the experimentally accessible time scale. From this difference, it is evident that the layer resists compression to a much higher degree than expansion, resulting in a nonsymmetric compression/ expansion response. It is a great advantage of the transient experiments performed here that such differences can be clearly distinguished, both visually from the π(t) or E(t) curves as well as after data analysis with a suitable model (see below). The transient dilatational moduli E(t) are shown in Figure 10. In all cases, E(t) decays to a nonzero value as t f ∞, indicating a permanent surface tension or surface pressure change upon compression of the area at a constant absolute amount of surfactant. Ideally, the asymptotic value E∞ of the transient modulus is identical to the static modulus calculated from the surface pressure isotherm under very slow compression according to eq 3. A comparison of the two quantities is given in Figure 10

Figure 9. (a) Surface pressure (π) vs area (mean molecular area, MMA) isotherm of sorbitan tristearate at 20 °C (solid line) and static dilatational elasticity Estatic calculated from the isotherm data (circles). (b) Step relaxation cycles of sorbitan tristearate at different surface pressures. The transient surface pressure responses to an area step function ∆A/A0 ) -2% are shown. Labels I-III indicate the pressures in the isotherm at which the relaxation experiments were performed. The dashed boxes in experiment I are drawn to point out the different responses upon compression and expansion, respectively.

where the static values are indicated by arrows. The instantaneous values of the transient dilatational modulus E0, as obtained from a stretched exponential model fit to the data, E(t) ) E0 exp(-(t/λ)R, are as high as 300 mN/m at an equilibrium surface pressure of 37 mN/m, a value far above the static modulus E∞. This indicates a stronger dilatational response for shorter time scales and clearly shows that the static elasticity obtained from the equilibrium π-Γ isotherm does not fully characterize the dilatational rheology of the surface layer. Additionally, the relaxation curves in Figure 10 point to qualitatively different relaxation mechanisms at different surface pressures, reflected in the values of the exponents R. In particular, we note that in the intermediate range of pressures around 10 mN/m the instantaneous and asymptotic values of E(t) are almost identical, indicating

Sorbitan Tristearate Layers at Air/H2O Interface

Figure 10. Step relaxation experiments of sorbitan tristearate performed at different equilibrium surface pressures πeq. The lines are stretched exponential (SE) model fits, with the fitting parameters summarized in the inset. On the right-hand axis, the corresponding static elasticity moduli Estatic, obtained from the surface pressure isotherm, are indicated by the arrows (area step for all experiments ∆A/A0 ) -2%).

the absence of relaxation processes detectable by our technique and, in consequence, purely elastic behavior. What prevents the surfactant layer from relaxing exactly in this concentration regime remains unclear. We note, however, that in the region where relaxation is absent the static modulus is only weakly dependent on the surface pressure. As one would expect from the behavior of the static modulus, both the instantaneous and the equilibrium portions of E(t) are large in the highly compressed region of the isotherm. Both at low (III) and high (I) surface pressures, we found pronounced differences between the instantaneous and equilibrium values of E(t). At π ) 37 mN/m, we find values for R comparable to those obtained in the shear relaxation experiments (R ≈ 0.25) whereas at lower pressures different exponents are found. The relaxation time scales upon compression are similar to those observed in shear experiments performed below a critical deformation at which the interfacial film is not destroyed. However, a correlation of the interfacial shear with the dilatational moduli is not straightforward, and it should also be mentioned that in dilatational deformation performed in the film balance there are shear components present (i.e., the deformation is not strictly dilatational). As is the case for numerous other interfacial films of surfactants or surface-active polymers, the rheological response in dilatational mode is much stronger than in shear mode, and the dilatational moduli exceed the shear ones by up to 2 orders of magnitude. Small changes in surface concentration induced by a dilatational deformation therefore generate higher surface stresses than small changes in the shape of an area element. Summary and Conclusions The rheology of condensed interfacial layers of the waterinsoluble surfactant sorbitan tristearate at the air/water

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interface was studied under interfacial shear and dilatational deformations. It is assumed that the large interfacial shear moduli are due to the specific structural feature of three stearyl side chains in the molecule and their condensation in the surface layer. Strain amplitude sweep, creep, stress relaxation, and small-amplitude oscillatory experiments were performed below a critical deformation of approximately 1.5% and showed both excellent repeatability and strain independence up to the critical strain. In shear flow, the surfactant layers exhibit rheological behavior similar to that of macromolecular temporary networks, characterized by generalized Maxwell behavior as well as a small linear viscoelastic regime limited by irreversible fracture events in the interface. A setup combining interfacial rheometry with BAM in real time provides useful information for interpretation of the measured interfacial rheological response of the gas/liquid interface during rheological experiments. The problem of fracture in interfacial films during shear flow even at small deformations can be adressed using a combination of interfacial rheometry and BAM images. Such fractures are not only relevant in systems such as the one at hand but they have been observed and deemed important in weakly structured gel systems (e.g., in protein interfacial layers). The dilatational behavior is predominantly elastic throughout the frequency spectrum measured, whereas the layers exhibit generalized Maxwell behavior in shear mode within a deformation frequency regime as narrow as two decades, indicating the presence of additional relaxation mechanisms in shear as opposed to expansion/ compression. If the transient rheological response from stress relaxation experiments is considered, then the data can be described well with a stretched exponential model, both in the shear and dilatational deformations. Condensed sorbitan tristearate interfacial layers may serve as a convenient 2D model for brittle materials, both bulk and interfacial. The frequency-dependent shear moduli (Figure 5) show a transition from viscous to elastic behavior within a very narrow range of deformation frequencies that is easily accessible by experiments. This is somewhat exceptional not only for interfacial films but even in normal 3D rheology where in most cases timetemperature superposition factors need to be used to have experimental access to such a variety of rheological regimes. Acknowledgment. Parts of this work were supported by the Swiss National Science Foundation (SNF project nos. 2100-065976 and 200020-108052). We thank Patrick Heyer, Jo¨rg La¨uger, and Victor Kusnezov at Anton Paar for providing technical assistance as well as Christoph Eschbach (ETH Zu¨rich) for contributions to the experiments. Supporting Information Available: Details of the transport analysis for the interfacial shear rheometer. This material is available free of charge via the Internet at http://pubs.acs.org. LA0514664