Evaluation of a novel multimode interfacial rheometer

Fluid interfaces populated by surface-active molecules or particles are sometimes characterized by complex microstructures, which then lead to static and dynamic mechanical properties of the interface that are structure dependent and, in some cases, may also have memory [1]. Droplets, foams, vesicles, and surfactant- and particle-stabilized emulsions are examples of systems that are controlled by the properties of their liquid interfaces [2–4]. Determination of the rheological and structural properties of these complex interfaces is crucial for the understanding of physiological and biomedical systems [5,6], for the design and engineering of industrial processes [7,8], and for the development of environmental remediation technologies [9].

Interfacial rheology measurements may be considered as a two-dimensional analog to bulk rheology [10]. However, this analogy warrants careful consideration due to two principal factors: (i) Interfacial flows are inherently coupled to adjacent bulk phases, requiring minimization of bulk phase contributions to rheological measurements. This is achieved by proper design of the rheometer probes characterized by large values of the Boussinesq number (Bq, the measure of the ratio between the momentum transfer to the interface to that transferred to the bulk phase) and, if necessary, application of proper corrections for subphase contribution to the momentum transfer [11]. (ii) Interfaces are compressible, introducing increased complexity in the interplay between shear and dilatational rheological properties in comparison to bulk materials [10,12]. Thus, unlike bulk rheology, evaluation of the dilatational rheology of the interface is at least as important as the study of its shear rheology. Interfacial shear rheology involves subjecting an interfacial area element to shear deformation resulting in shape deformation while maintaining its area. On the other hand, interfacial dilatational rheology is concerned with the momentum transfer at an interfacial area element in response to changes in its area, while preserving its shape. Of particular concern for interfaces comprised of complex fluids and soft materials with memory is the critical role of flow deformation history on the measured material properties [1,12].

Different methods have been proposed to determine the rheological material functions for each deformation mode. Interfacial shear rheology measurements are usually carried out by employing one of the commercially available devices such as the bi-cone [13], the double wall ring (DWR) [14], or the magnetic rod interfacial stress rheometer (ISR) [15–17]. All these instruments are based on the drag on a probe positioned on the interface and capable of executing rotational or oscillatory motion through an applied external force. The force is either applied directly when the probe is a fixture mounted on a conventional rotational rheometer, e.g., the bicone and DWR [13,14], or remotely by, e.g., a magnetic field [15–17].

Despite the advancement in measuring the shear rheological properties of interfaces, performing accurate interfacial shear rheology measurements remains a challenge. Issues regarding the small probe size, resulting in limited contact area in comparison to bulk measurement instruments and the inherent low force/torque signals due to the thin interface drive operation to the edge of instrument resolution. Consequently, instrument and device inertia are more significant here than in the bulk counterparts due to the smaller forces involved. Finally, momentum transfer to the bulk phase, bulk fluid inertia, and effects of misalignment and imperfections in the measurement probes are also potentially detrimental factors. Recently, the operating windows for shear interfacial rheometers have been derived, accounting for torque/force limits and possible instrumental artifacts [11]. In the past, the lack of such clear operating windows had led to contradicting results when using different rheometers [18]. Substantial performance discrepancies in measuring the interfacial shear rheology between instruments incapable of maintaining surface pressure during measurement and those that can were highlighted with a clear indication of the advantage of the latter [18].

Dilatational interfacial rheometry is most frequently based on the implementation of the oscillating pendant drop technique analyzed by fitting the Young–Laplace (Y–L) equation to the time-dependent drop shape [19–22]. However, this analysis is only appropriate when the interfacial stresses are isotropic. In the case of complex interfaces and complex microstructure at the interface, anisotropy in the stress distribution and in surface composition, Marangoni stresses, and mixed-flow modes cause extra and deviatoric stresses to develop, which require careful analysis beyond the simple Y-L equation [23–25]. Variations on the pendant drop method include the capillary-pressure tensiometer [26,27] and the pulsating-bubble surfactometer [28]. Both methods integrate shape analysis with pressure measurements, utilizing small diameter capillaries to enhance the pressure signal. When capillary pressure is available, the shape analysis is notably simplified, focusing solely on determining curvature. Nevertheless, as already pointed out unequal strains in different directions may induce errors in the moduli values as discussed in depth by Nagel et al. [25].

The rectangular Langmuir–Pockels (LP) trough serves for over a century as the benchmark device for obtaining interfacial compression–expansion isotherms for surface active materials. In the case of complex and microstructured interfaces, the substantial contribution from the mixed flow field composed of superimposed normal (dilatation) and shear strain components generated by the unidirectional compression/expansion of the interface may not be ignored. Both the area and shape of the interface change. The subsequent surface pressure response of the interface can even be strongly anisotropic and is clearly manifested by the significant difference between the two isotherms obtained with the Wilhelmy plate oriented parallel and normal to the compression direction, respectively. The divergence between these two isotherms prompted attempts to use the LP trough as a means for the study of the steady and oscillatory shear and dilatational properties of interfaces despite the nontrivial analysis of the complex flow fields involved [29–32].

The recently developed circular trough [32–34] has been shown to generate a purely dilatational flow field (at high enough Bq) overcoming the mixed-flow issues encountered in the standard LP rectangular trough and capable of providing a better understanding of the dilatational characteristics of complex interfaces [33]. Similar features are also expected from the newly introduced Interfacial Dilational Rheometer [35].

Building on the concept of the circular trough mechanics [32,33], a novel interfacial rheometer—labeled the Quadrotrough (QT)—has been recently introduced [36]. It is designed to carry out pure shear and pure dilatational interfacial deformations at the air–liquid interface (or a combination thereof), either in steady or in oscillatory modes. It offers the advantage of enabling the entire rheological and thermodynamic characterization of the interface in one instrument without requiring sample exchange. Tein et al. [36] utilizing a small molecular weight surfactant (d-stearic acid) demonstrated the surface pressure response of the interface under a range of well-defined shear and dilatation interfacial deformations for both inviscid (at the air–water interface) and viscoelastic (over an aluminum nitrate solution subphase) interfaces. Moreover, the kinematics of the deformations were validated by particle tracking velocimetry. Furthermore, the QT is coupled with a Brewster angle microscope for in-plane interfacial microstructure and in situ neutron reflectometry for the out-of-plane interfacial nanostructure. This unique and valuable combination facilitates the development of robust structure–property relationships for complex fluid interfaces. Previous analysis of the QT [36] was limited to obtaining the surface pressure under different conditions, validating the flow fields generated and verifying the stability of shear and compression rates generated. No attempt was made to derive rheological material functions and no attempt to carry out oscillatory deformations. It is the purpose of the work reported here to validate the QT ability to serve as a reliable shear and dilatational interfacial rheometer by subjecting it to highly stringent tests comprised of deriving steady and complex shear and dilatation moduli on highly viscoelastic interfaces and comparing the results to data obtained by other established methods.

Interfacial rheological measurements of insoluble polymers deposited on the air/water interface are notoriously difficult [18]. In contrast to proteins and small molecular weight amphiphilic surfactants, their anchoring to the water subphase may be weak, and they may exhibit memory effects, while their conformation, flexibility, or rigidity may also play a role. Specific for the case of glassy polymers, the method by which the polymer layer has been obtained may lead to different ill-defined morphologies impacting the reproducibility of the rheological measurements and divergent results as detailed in Ref. [18]. For this reason, in this paper, we tested the ability of the QT to perform the challenging shear and dilatational rheology measurements using a well-studied polymeric system comprised of poly(tert-butyl methacrylate) (PtBMA) as an insoluble layer at the air–water interface and simultaneously extract in situ microstructure information using BAM. The interfacial shear and dilatational rheology of PtBMA, a glassy polymer, have been extensively studied [18,32,33]. As shown by Alexandris et al. [18], maintaining the surface pressure during shear rheology measurement of glassy polymers is of crucial importance for obtaining meaningful results. Therefore, oscillatory shear measurements carried out with the QT were compared to the results obtained by the DWRT, a modified DWR mounted onto an LP trough [37]. The oscillatory dilatational rheology measurements were compared to the values obtained by the radial trough as measured by Pepicelli et al. [32,33]. This work clearly demonstrates the ability of the QT to perform precise pure shear and dilatational, steady, and oscillatory rheological measurements, coupled with in situ extraction of microstructural information.

An acrylic polymer, PtBMA, with a molar mass of 135 kg/mol (polydispersity index D = 1.04), was obtained from Polymer Source (Canada) and used as received. PtBMA is hydrophobic and will not dissolve in the water subphase. However, the acrylate side groups provide anchoring to the water, allowing the formation of an insoluble polymer layer on the interface. The polymer is amorphous with a bulk Tg value of 105 °C. As all our measurements were performed at room temperature even when considering plasticization, the polymer at the interface is expected to be rigid or glassy.

A dilute solution of PtBMA was prepared by dissolution in chloroform at c ≈ 0.3 mg/ml (well below the overlap concentration c*). After gentle stirring to homogenize the solution, it was added dropwise to the water–air interface in the amount required to obtain the desired polymer concentration (reported as area per monomer). The nominal area per monomer upon compression/expansion is calculated from the interfacial area assuming that the number of molecules is constant. At low surface pressures, the interface contains clusters of glassy molecules that, upon increase in surface pressure, form a close-packed arrangement that we refer to as a glassy, soft-glassy, or elastic film, which by BAM is shown to be relatively uniform and effectively continuous, prior to buckling at very high compression ratios.

Barrier compression isotherm measurements were performed using both a standard rectangular Langmuir–Pockels (LP) trough and the newly developed Quadrotrough (QT). The mechanical design of the QT is described in detail elsewhere [36] and summarized here for convenience. An elastic barrier is held by four aluminum fingers that are controlled by stepper motors (Nanotec Electronics) and move along individual linear axes using another step motor (MOVTEC) capable of generating linear velocities in the range between 1 and 20 mm/min. Each aluminum finger can also move up and down by a servo motor (HITEC) to control the accurate placement of the elastic barrier at the air–water interface. These motors are programmed through LabVIEW (National Instruments™) for the elastic barrier movement. Each of the four linear motors can be addressed independently to provide different modes of deformation as discussed below.

Surface pressure isotherms are obtained by instructing all motors to move in tandem along the diagonals at the same speed leading to pure compression kinematics as has been demonstrated previously [36]. The surface tensiometer force balance is controlled by KSV NIMA software, and the surface tension is measured through a standard platinum Wilhelmy plate (wetted perimeter of 39.24 mm, height of 9 mm, KSV NIMA). The sample bath is maintained at the measurement temperature controlled by a thermal bath (all measurements were performed at 22 ± 2 °C).

The LP trough was the KSV-Nima (Finland) mini trough (30 × 7.5 cm²) equipped with a platinum Wilhelmy plate similar to the one above. In both troughs, the platinum Wilhelmy plate was positioned in the center of the trough in the desired orientation relative to the barriers.

Before each measurement using either the LP or the QT, the trough and barriers were thoroughly cleaned by several milli-Q water and ethanol rinsing cycles. The Wilhelmy plate was first cleaned with a propane burner and then rinsed repeatedly with ethanol and water. The trough was filled with milli-Q water. After positioning the barriers in the fully open position, the Wilhelmy plate was lowered slowly, hanging from a microbalance until submerged in the water, and the force was zeroed. For a Newtonian interface, changes in the force, measured by the Wilhelmy plate, are due to changes in surface tension, where the contact angle between the subphase and the plate is assumed to be close to zero. The cleanliness of the interface was verified by the measurement of the surface tension of water (72.8 ± 0.3 mN/m at room temperature) and a maximum increase in surface pressure π below 0.2 mN/m upon the complete compression of the pristine interface.

Drops of the dilute solution of the investigated polymer were evenly spread over the entire water surface by a 50 μl Hamilton microsyringe to achieve a homogeneous distribution. At least 30 min were allowed before the onset of compression to enable solvent evaporation, polymer diffusion on the interface, and system equilibration as evidenced by steady value of the surface pressure. At the onset of all compression isotherms, the barriers were completely open (barriers 30 cm apart for the LP trough and 11 cm apart for the QT). At the end of compression, the surface area of the LP trough was 56.5 cm² and that of the LP 40 cm², corresponding to compression ratios of 4 and 3 for the LP and QT, respectively. Experiments were carried out at room temperature at a constant compression speed of the barriers of 3 mm/min for the LP trough and 0.71 mm/min for the QT (corresponding to 1 mm/min finger speed along the diagonal—the lowest available controllable speed).

Oscillatory interfacial shear rheology was measured by means of a DWR mounted on an LP trough (DWRT), which sets the desired surface concentration by compression of an initially dilute solution and enables control of the surface pressure during the entire duration of the experiment [37]. The clear advantage of using this setup has been recently demonstrated especially for the rheology of interfacial films of glassy polymers [18]. A specially designed Teflon cup (inner radius of 14.25 mm, outer radius of 22 mm) equipped with two rectangular openings is placed on an LP trough (Ribbon trough, Biolin Scientific). The 3D-printed Ti6Al4V alloy ring (3D Systems, Belgium; inner radius of 17 mm, outer radius of 18 mm) also has three openings. The two openings in the cup allow a uniform compression inside and outside the cup and the openings in the ring ensure a uniform interface on both sides of the geometry and enable control over the surface pressure during a rheological experiment.

The trough and cup were thoroughly cleaned, aligned inside the trough, and filled with milli-Q water. The ring was rinsed with ethanol and milli-Q water and dried before attaching it to a stress-controlled rheometer (DHR-3, TA Instruments, USA). Calibrations and instrument inertia effects’ estimation were carried out as discussed in detail by Renggli et al. [11]. The ring was lowered to the interface until the interface appeared flat. Subsequently, the polymer solution was added dropwise to the interface. The system was compressed to the desired concentration/surface pressure and allowed to equilibrate for at least 30 min. The whole procedure was repeated with a fresh polymer sample for each measured concentration. The surface pressure was monitored throughout the experiment to prevent changes during the measurement.

The pure shear mode involves deformation that imparts directional properties, characterized by the primary axis of extension and the minor axis of compression. Consequently, the orientation of the Wilhelmy plate relative to the deformation axes will lead to distinct surface pressure responses during shear deformation. The specifics of strain directionality and the kinematics of the generated flow field are expounded elsewhere [36].

The experimental procedure involved spreading the polymer at the interface as described in Sec. II B followed by compression of the interface to achieve the target surface concentration spanning the range of 19–7 Ų/monomer. Surface equilibration and determination of the equilibrium surface pressure is followed by inducing the desired shear deformation either in steady or in oscillatory fashion. For the QT, the characteristic length scale L_C in Eq. (1) is approximated as the average length from the barrier to the center of the trough [36]. For the present setup and deformations employed L_C is varied between 150 mm (completely open trough, lowest compression) and 90 mm (largest compression). Accordingly, for PtBMA, the Bq values range between approximately 13 and 2000, indicating only minor contribution from the bulk; therefore, no subphase corrections were performed.

In steady shear experiments, the speed of the fingers was set at 1 mm/min resulting in a shear rate ranging from $γ ˙ s=4× 10 − 4 s − 1$ at completely open barriers to 8 × 10⁻⁴ s⁻¹ at the smallest area. Deformation ratios of up to λ = 0.22 were induced with position and surface pressure data collected every 1 s. The steady shear modulus for each surface concentration was obtained from Eq. (3).

The QT was used as a strain-controlled rheometer. The strain amplitude was kept at 1%–2%, and the frequency was varied by changing the velocity of the four fingers between 1 and 20 mm/min yielding frequencies between 0.1 and 0.7 rad/s. Due to design limitations related to the linear velocity executed by the step motors, SAOS measurements employed triangular rather than conventional sinusoidal oscillations. The complex modulus was calculated from the ratio between the amplitudes of the stress (surface pressure) and strain waves. The exact values of the frequency and the stress amplitude were determined by FFT analysis. The strain wave was obtained from the traces of the linear motion of the four fingers followed by the application of Eq. (2). The shape of the resulting strain wave as well as the surface pressure wave remained triangular and symmetric at the small strain values employed here. Typically, up to ten cycles were generated, the first two were neglected and between 5 and 7 were used for the analysis. Figure 1(a) provides a schematic description of the procedures employed for SAOS and steady shear measurement.

Steady elastic dilatational modulus experiments were carried out in a similar fashion to the procedure described for the compression isotherm measurements (Sec. II B). During compression, the fingers’ speed was maintained at a constant value of 1 mm/min, which results in area compression rates between 6 and 3 mm²/s and interfacial dilatational strain rate of $α ˙ s=2−3× 10 − 4 s − 1$⁠.

For the small amplitude oscillatory dilatational (SAOD) experiments, the polymer was initially spread at the interface as described in Sec. II B followed by compression of the interface to achieve the target surface concentration spanning the range of 19–7 Ų/monomer. Surface equilibration and determination of the equilibrium surface pressure are followed by applying oscillatory triangular wave dilatational deformation at strain amplitudes of 1%–2% and frequencies of 0.1–0.7 rad/s. The magnitude of the complex dilatational modulus was calculated from the ratio between the amplitudes of the stress (surface pressure) and strain waves. The exact values of the frequency and the stress amplitude were determined by FFT analysis. The strain was calculated as ln(A/A₀) using values obtained from the traces of the linear motion of the four fingers. The shape of the resulting strain wave as well as the surface pressure wave remained triangular and symmetric at the small strain values employed here. Figure 1(b) provides a schematic description of the procedures used during SAOD and steady dilatation measurements.

Two possible schemes are possible: (1) Discrete testing—For each surface concentration tested either in shear or compression, a fresh sample is used. The fresh polymer solution is spread on the interface with the barriers at the initial undeformed state, allowed to equilibrate, and then compressed to achieve the desired surface concentration as described above. (2) Stepwise testing—After each shear or dilatation experiment, the interface is further compressed to a new surface concentration; the surface pressure is allowed to relax to a state of equilibrium followed by the desired rheological test. Up to ten concentration steps have been performed on each sample, with concentrations spanning the range of 19–7 Ų/monomer.

The stepwise scheme offers a clear advantage in terms of experimental efficiency and reduction of reproducibility errors stemming from initial concentration errors and deposition variability. To ascertain the validity of the stepwise scheme, the complex shear and dilatation moduli as a function of frequency obtained by the two schemes at two different surface concentrations were compared. The results provided in  Appendix B demonstrate a reasonable agreement between the two procedures. Based on these results, all measurements reported here were obtained by the stepwise procedure.

A MicroBAM (KSV-Nima, Biolin Scientific) was used for Brewster angle microscopy (BAM) imaging of the polymer-laden interfaces. A P-polarized laser beam reaching the interface at the Brewster angle corresponding to the subphase liquid will not be reflected from the pristine liquid interface but will be reflected from the polymer film on top of it due to the differences in refractive indices between the subphase and the polymer. The reflected light goes through an analyzer to a CCD camera, which captures the reflected light and generates images with a spatial resolution of about 2 μm. With proper alignment, the nonreflecting water surface appears in BAM images as a dark surface. Objects protruding or elevated relative to the surface appear increasingly brighter in proportion to their relative elevation.

Motion video capturing the evolution of the structure of the interface during the compression in the LP trough and in the QT composed of images taken at a rate of one image per minute can be found in the supplementary material.

In this section, we aim to verify the isotropic nature of the compression in the QT and to compare the compression isotherms obtained by the QT to those obtained by either the LP trough and the radial trough. The impact of the Wilhelmy plate orientation on the compression isotherms obtained by the standard rectangular LP trough due to the anisotropy in the compression kinematics was described elsewhere [29,30,32]. This has been shown to be alleviated by the radial trough design [32,33]. The QT has been shown to generate isotropic interfacial deformations [36]; therefore, the compression isotherms and the stresses arising from the compression are independent of the Wilhelmy plate orientation. To test this claim for the PtBMA-laden interface, the compression isotherms obtained by means of the QT are compared to the isotherms obtained by the LP trough [Fig. 2(a)] and the radial trough [Fig. 2(b)].

The three compression isotherms depicted in Fig. 2(a), corresponding to the two Wilhelmy plate orientations in the LB trough and the one obtained by the QT, overlap for surface concentrations up to ∼14 Ų/monomer. At these low surface concentrations, the interface is associated with fluid-like behavior and is not expected to develop large anisotropic stresses. At higher concentrations above the bifurcation point between the two LP isotherms, the interface assumes elastic solid-like characteristics [29,30,32] and the surface pressure values obtained with the Wilhelmy plate perpendicular to the barriers are substantially lower than those measured in the parallel orientation. At these higher concentrations, the isotherm obtained by the QT follows more closely although not in perfect agreement, the isotherm for the LP|| orientation which indicates that in parallel orientation, the stresses that act on the plate are mostly isotropic compressional stresses. At concentrations above ∼10 Ų/monomer, these two isotherms diverge with the QT isotherm reaching higher surface pressures than the LP|| isotherm. At the highest compression achieved by the QT (7 Ų/monomer), the measured surface pressure is 70 mN/m, whereas the corresponding value measured by the LP trough is 65 mN/m. We observe that the QT isotherm does not trivially lie between the LP isotherms for the moderate to high compressions. Similar observations were reported for a mild viscoelastic interface [Fig. 7(b) in Ref. 36] formed by spreading d-stearic acid on the interface of aluminum nitrate solution as well as for PtBMA isotherms obtained by radial troughs using circular Wilhelmy rods [32–34]. This observation is a clear indication that the deformation history of the glassy polymer interface and probably other viscoelastic interfaces, resulting from the type of flow fields generated during compression, affects its material properties.

The good agreement between the isotherms obtained by the two orientations of the Wilhelmy plate in the QT [Fig. 2(b)] and between these two and the isotherm obtained by the radial trough up to relatively high compression values (<10 Ų/monomer) confirms the isotropic nature of the strain and the pure dilatational deformation generated in both devices (QT and radial trough). Moreover, the relatively small variations between the three isotherms are an indication of the good reproducibility obtained by these devices for a relatively challenging interfacial measurement.

At a compression of 8 Ų/monomer, the QT yields 10%–15% higher surface pressure value than those obtained by the LP trough. We speculate that the differences in the compression isotherms at these high compressions as well as in the rheological material functions reported below stem from the different complex deformation fields in the various trough geometries. Similar differences between different trough geometries at high compressions have been reported for other interfacial layers [40]. The exact structure of the interface at these high compressions in the different devices, the different buckling patterns discussed below, and the possible multilayering of the interface merit further investigation.

Scaling analysis of the surface pressure isotherms yields for low concentrations (below 20 Ų/monomer and π < 10 mN/m) a critical exponent value of 4/7, which corresponds to polymer chains in poor solvent [41]. At higher surface concentrations (between 14 and 10 Ų/monomer), a fractal dimension of 2 is obtained [42] in agreement with a 2d space filling film.

The ability of the QT to perform small SAOS measurements was evaluated by testing its sensitivity to surface concentration and by comparison to measurements carried out with the DWRT. Frequency sweeps between 0.1 and 0.7 rad/s with increasing surface concentrations (19–7 Ų/monomer) were performed with the QT at approximately 1% strain amplitude [Fig. 3(a)]. In addition, the complex shear modulus measured by the QT at the highest frequency (0.7 rad/s) was plotted against the molecular area and compared to the complex modulus measured by the DWRT (time sweep upon compression at 1% strain amplitude and 1 rad/s) as depicted in Fig. 3(b).

The magnitude of the complex shear modulus measured by the QT is shown to be highly sensitive to the surface concentration as depicted in Fig. 3(a), exhibiting two orders of magnitude increase over the concentration range explored. Similar variation for PtBMA has been reported by Alexandris et al. [18] for measurements carried out with the DWRT and ISR and deemed as a proper sensitivity test for the validation of an interfacial shear rheometer.

The frequency dependence of the shear modulus gradually decreases as surface concentration increases, and the interface shifts from liquid-like to elastic solid-like interface. At the lowest compression (17 Ų/monomer), the modulus increases by 300% over one decade of frequency, whereas at the highest compression tested (10 Ų/monomer), the increase is only by 25% over the same frequency range.

A comparison of the QT frequency sweep data to that obtained by the DWRT at the lowest and highest compression values (17 and 10 Ų/monomer, respectively) is provided in Fig. 3(a). Very good agreement between the data obtained by the two instruments is observed for the moduli values at 17 Ų/monomer. In contrast, for the highest compression, the moduli values measured by the DWRT data are at least three times lower than those measured by the QT. This is further demonstrated by the data presented in Fig. 3(b) in which results for the DWRT and QT for the highest frequency are compared over the entire concentration range. At low surface concentrations between 19 and 15 Ų/monomer, there is a good agreement between the complex shear moduli measured with the two instruments. At 13–12 Ų/monomer, the shear complex moduli measured by the two instruments start to diverge, with the QT yielding continuously increasing higher values than those measured by the DWRT reaching a fourfold difference at surface concentrations above 10 Ų/monomer. It should be noted that this coincides with the concentration at which the LP and QT isotherms diverge, suggesting that differences in flow histories, and in particular, the large amount of shear present during area changes in the LP trough, influence the interface microstructure and properties. That the values are lower in the LP trough and LP-mounted DWRT as compared to the pure compression of the QT is consistent with shear relaxation of the microstructure.

Interestingly, the exceedingly high values of the complex interfacial shear modulus obtained by the QT for PtBMA at large compressions (|G^s*| = 100−200 mN/m) are similar to those reported by Alicke et al. for colloidal particle-laden interfaces with surface coverage ϕs > 0.7 [Fig. 7 in Ref. 43]. The similarity between the interfacial shear moduli values for these two very different systems may be fortuitous. On the other hand, high surface concentrations result in a large increase in the interactions between closely packed fractal particle aggregates (in the colloidal system) and between closely packed clusters of glassy polymers (PtBMA). It is plausible that interfacial shear modulus values above 100 mN/m are the fingerprints of interfaces covered by highly packed, strongly interacting rigid clusters.

Excellent reproducibility was obtained between different measurements carried out with the QT as typically variations amounting to 10%–20% of the average complex shear modulus values were observed. For the DWRT that showed the best reproducibility among the different interfacial rheometers tested by Alexandris et al. [18], variations up to 100% in the measured modulus values were observed. Although these relatively large reproducibility errors in the DWRT may account for the difference between the two curves in Fig. 3(b) at small and moderate surface concentrations (up to approximately 12 Ų/monomer), it cannot be the source for the substantial differences observed between instruments at higher surface concentrations.

To further explore this discrepancy, we have resorted to BAM to determine whether the dissimilar shear moduli determined by the two instruments could be due to the formation of distinct interfaces at elevated surface concentrations as a result of the disparate flow regimes inherent to each instrument. Figure 4 is a comparison between BAM images of the QT interface [Fig. 4(a)] and the LP trough interface [Fig. 4(b)] taken at the maximum compression (i.e., 7 Ų/monomer). Figure 4(b) (LP trough) reveals luminous bands that we interpret as the appearance of interface buckling normal to the compression direction (parallel to the barriers). In contrast, no distinct interface patterns are evident at the maximum compression carried out in the QT, which further demonstrates the isotropic nature of the compression even at high compressions.

Motion video strips of the compression in the LP trough and the QT, comprised of BAM images taken one image per minute during the compression, are provided in the supplementary material. Based on the BAM images, we speculate that the lower shear modulus values measured at high compressions by the DWRT (mounted on a rectangular LP trough) as well as lower surface pressure values at high concentrations measured by the LP trough [Fig. 2(a)] are due to the alleviation of stresses by the quasi-2d buckled interface, while the higher values measured by the QT relate to the isotopically compressed interface similar to those generated in the radial trough for highly compressed particle-laden interfaces [43].

The excellent reproducibility in the complex shear modulus over the entire frequency range and the good agreement with the DWRT at low and moderate surface concentrations indicate that the QT is suitable for carrying out interfacial shear rheology measurements of glassy polymers-laden interfaces.

In this section, we aim to validate the dynamic dilatational data obtained by means of the QT. This is achieved by comparison of complex dilatational modulus values obtained by frequency sweeps at 1% strain amplitude with increasing interfacial concentration and those obtained by Pepicelli [33] utilizing the radial trough. Results are depicted in Fig. 5. The complex dilatational moduli as a function of the radial frequency at two surface concentrations, namely, 17 and 10 Ų/monomer, are presented in Fig. 5(a). While the trend observed for the very mild frequency dependence of the moduli is reproduced by both instruments, values obtained by the QT are consistently somewhat higher than those obtained by the radial trough. Yet, these differences are in line with those observed between other interfacial rheological instruments [18]. QT measurements were highly reproducible with variance below 5% from the complex moduli values. The data depicted in Fig. 3(b) show an order of magnitude increase upon compression of the liquid-like interface from 19 to ∼12 Ų/monomer. Little change is observed upon compression of the elastic solid-like interface (12–8 Ų/monomer). The values obtained here for the complex dilatational modulus and its mild dependence on frequency are similar to the ones obtained by the radial trough for the densely packed asphaltene layer at the air/water interface [40].

To determine the interfacial shear properties at fixed molecular areas, finite strain planar shear measurements were analyzed by means of the Hencky strain model adaptable for elastic interfaces under finite deformations as described by Eq. (3). Steady shear experiments were carried out at rates ranging from 4 × 10⁻⁴ to 8 × 10⁻⁴ s⁻¹ (fingers moving at a constant speed of 1 mm/min) in accordance with the procedure outlined in Sec. II C 2. The results are portrayed in Fig. 6.

In Fig. 6(a), the reduced surface pressure π–π₀ is plotted as a function of the extension ratio L/L₀ for a selected number of surface concentration values. Here, π₀ is the equilibrium surface pressure at a given concentration before the onset of shear [end of step 2 in Fig. 1(a)]. Verwijlen et al. [39] showed that the Hencky strain model is only valid as long as the shear strains are below 2. All the data reported here are limited to strains below 0.22. The linearity of the individual results (curves) indicates that the Hencky strain model is appropriate for extracting the steady shear modulus [Eq. (3)]. The increase in the slope with increasing surface concentration confirms the expected increase in the elastic shear modulus with compression.

In Fig. 6(b), the steady shear elastic modulus G^s derived from the slope of steady measurements is compared to the magnitude of the complex shear modulus |G^s*| obtained by the SAOS measurements at the lowest frequency tested (0.1 rad/s). At low surface concentrations up to approximately 14 Ų/monomer, the complex modulus values are higher than those of the steady shear moduli values. As compression further increases, rendering the interface more elastic, very good agreement is observed between the steady and the dynamic moduli values. This agreement between the moduli indicates that the Hencky strain model, at strain values below the limit set by Verwijlen et al. [39], and particularly under conditions of high compression where the interface is highly elastic, effectively characterizes the stress–strain relationship for shear deformation. It is important to note that due to instrument limitations (highest shear rate and lowest accessible frequency), the comparison between the moduli was conducted at very different shear deformation rates. The steady shear measurements were performed at shear rates one to two orders of magnitude lower than that for the lowest frequency measured (0.1 rad/s) in SAOS. This observation suggests that for elastic interfaces such as those observed at surface concentrations higher than ∼14 Ų/monomer, the complex moduli exhibit relatively little dependence on frequency at frequencies lower than 0.1 rad/s.

In Fig. 7(a), the apparent dilatational modulus and the Gibbs elasticity, derived from the slope of the compression (negative dilatation) isotherms [Eqs. (5) and (6)], are depicted. As pointed out above, only surface concentrations above the isotherm bifurcation point have been utilized. At high compressions (molecular areas smaller than ∼9 Ų/monomer), the surface pressure (SP) levels off probably due to departure from the two-dimensional interfacial film. For this reason, determination of K_app was limited to molecular areas above this value as indicated by the dashed line in Fig. 7(a).

From the apparent dilatational modulus and the Gibbs elasticity, the steady elastic dilatational modulus K^s is calculated over the relevant range of compressions. The results are presented in Fig. 7(b) and compared to the complex moduli values obtained by small amplitude oscillations at the lowest frequency tested (0.1 rad/s). The numerical values are summarized in Table I. A significant difference is observed between the dynamic and steady dilatational moduli values, unlike the good agreement found for the shear moduli [cf. Fig. 6(b)].

We are unclear about the reason for the substantial difference between the steady and oscillatory dilatational moduli values. One possible explanation rests on the different flow deformation histories prior to the two moduli evaluations [cf. Fig. 1(b)]. SAOD experiments at each concentration were carried out only after relaxation, whereas steady dilatation was obtained from continuous compression without allowing for relaxation. Thus, annealing either during or between the dilatational oscillation cycles is responsible for the higher oscillatory modulus values. In contrast, considerably smaller differences in deformation history exist in the shear measurements [Fig. 1(a)], both steady and SAOS were carried out following relaxation at each concentration tested, resulting in equally high shear modulus values.

Another interesting observation is that while the steady dilatational modulus remains constant throughout the compression over the range of compressions between 12 and 9 Ų/monomer, the steady shear modulus increases by an order of magnitude over the same range of compressions.

Thus, when $K s≫ G s$⁠, the 2D Poisson ratio ν_2D→ 1; when K^s ≅ G^s, it assumes values close to zero; and when $K s≪ G s$⁠, the Poisson ratio approaches −1. Since interfaces are highly compressible, the upper limit of the Poisson ratio unlike the case for 3D materials is not necessarily an indication of incompressibility but rather an indication of low shear resistance. Thus, the evaluation of the Poisson ratio provides insight into the fundamental elastic response of the interface.

The combined shear and dilatational interfacial measurements carried out with the QT enable us to determine the Poisson ratio for the interface utilizing the same instrument and sample. The Poisson ratio computed from the values of the complex moduli at 0.1 rad/s and the Poisson ratio calculated from the values of the steady moduli are compared in Fig. 8. For the latter, the value for the steady dilatational modulus has been used over the entire concentration range despite the fact it was determined only over a limited range between 12 and 9 Ų/monomer. In both cases, the Poisson ratio at low compressions assumes values close to 1, indicating that the shear modulus is significantly smaller than the dilatational modulus. At low polymer surface concentrations, the shear modulus is low since the spatially removed molecules or individual glassy clusters hardly interact, and shear stresses are poorly transmitted. The dilatation modulus, which represents mainly the compressibility of the clusters, is considerably larger than the shear component.

At surface concentrations around 14 Ų/monomer, which coincides with the bifurcation point in Fig. 2(b) (the assumed transition from liquid-like to solid-like interface), the values of ν_2D start decreasing sharply with increasing surface concentrations. This is the result of closer interaction of the molecular clusters and the formation of a more uniform interfacial film [18] manifested by increasing values of the shear moduli. Similar trends were observed for particle-laden interfaces [43]. At the maximum compression, the Poisson ratio obtained from the complex moduli reaches a value of approximately ν_2D-dynamic ∼ +0.4, whereas the Poisson ratio is calculated from the steady moduli ν_2D-steady ∼ −0.5. The latter value is similar to the Poisson ratio value reported by Alicke et al. for a close-packed 2D suspension [43]. It should be pointed out that the nature of the two systems is quite different.

Most negative Poisson's ratio materials, particularly isotropic ones, contain void space; however, void space is not a necessary condition. Materials or structures in the sheet or layer form can exhibit negative Poisson's ratio if reorientation of structural elements occurs [45]. The auxetic behavior of the 2D colloidal suspension is associated with the occurrence of rotating hinge structures (particle clusters), whereas the PtBMA-laden interface is characterized by tightly pressed glassy clusters. Thus, the decrease in the Poisson ratio values as surface concentration increases and the auxetic behavior at high compressions can be attributed to the closer interactions between the polymeric entities comprising the interface. In both systems, shear modulus stems from interaction between clusters or particulate domains while dilatational modulus is mostly due to intracluster compressibility. As a result, the two moduli are highly decoupled—a characteristic feature of auxetic materials.

As surface concentration increases the values of both the dynamic dilatational modulus [Fig. 5(b)] and the shear modulus [Fig. 3(b)] increase by two orders of magnitude in a similar fashion with |K^S*| > |G^S*| for all concentrations. As a result, the Poisson ratio is always positive. In contrast, for steady measurements while G^S increases with concentration by two orders of magnitude [Fig. 6(b)], K^S remains constant [Fig. 7(b)] leading to K^S< G^S at high concentrations, which leads to a negative Poisson's ratio. As already discussed above, we speculate that this substantial discrepancy between the Poisson values derived from the dynamic and steady moduli, respectively, is due to the different flow deformation histories experienced prior to the determination of the steady and the complex dilatational moduli. These history differences are expected to attain increasing importance as polymer concentrations at the interface increase or area per monomer decreases as actually observed in Fig. 8.

This study focused on the potential of the newly developed pure shear and compression trough—the Quadrotough (QT) to serve as a unified interfacial shear and dilatational rheometer for highly challenging complex interfaces. Compression isotherms obtained by utilizing the QT verified the purely isotropic nature of the compressions generated up to the very high compressions commensurate with those obtained by standard LP troughs. The QT demonstrated its utility in both shear and dilatational rheology assessments of hydrophobic glassy polymers-laden water interfaces.

Complex shear moduli measurements were characterized by a high degree of reproducibility and satisfactory agreement with measurements obtained by other instruments, particularly at lower surface concentrations. Discrepancies between results obtained by the QT and those obtained by the DWRT were attributed to the buckling phenomena observed at higher surface concentrations as evinced by BAM images in the LP trough. The QT displayed no such patterns, providing further support to the isotropic nature of the compression flow field in the QT.

The QT yielded very reproducible oscillatory dilatational data over the entire range of accessible compressions. The experiments yielded consistently higher complex dilatational modulus values relative to those obtained by the radial trough but not larger than those encountered between other interfacial rheometers [18]. For both shear and compression results, the transition from liquid-like to solid-like interface is clearly observed.

The Hencky strain model was employed to derive steady shear and dilatational interfacial moduli values and to compare those values to the ones obtained by SAOS and SAOD. Very good agreement was observed between the steady and complex shear moduli values over the entire interfacial concentration range. This was not the case for the dilatational modulus, for which the constant steady dilatational modulus value was markedly smaller than the values obtained for the dynamic dilatational modulus at the same molecular areas. This discrepancy is also manifested in the large difference between the values of the 2D Poisson ratio obtained by either the steady or the complex moduli at medium and high compressions. Further research is needed to elucidate the underlying phenomenon responsible for these observations. In both cases, the Poisson ratio decreases substantially with compression in a manner similar to recent observations on particle-laden interfaces [43] and more so when derived from the steady moduli values. This drop in interfacial Poisson ratio with decreasing molecular area is due to the greater sensitivity of the shear modulus as compared to the dilatational modulus to interfacial concentrations when the polymer interface is compressed beyond the point of strongly interacting interfacial components and possibly, interfacial buckling. Indications for possible auxetic behavior at high compressions are revealed when the Poisson ratio is based on the measured steady moduli values.

In summary, this work underscores the potential of the QT to serve as a reliable, convenient, and valuable tool for exploring interfacial rheology in various systems, providing insights into complex interfacial behaviors, and contributing to our understanding of interfacial science.

See the supplementary material for the following:A.Motion video strips capturing the evolution of the structure of the interface during the compression in the LP trough and in the QT comprised of BAM images taken one image per minute during the compression. In the LP, clear buckling patterns are formed at high compressions. For the QT, relatively uniform brightening is observed indicating a relatively homogeneous increase in film thickness with increased compression.B.Sample data illustrating the analysis of oscillatory deformations in SAOS and SAOD.

D.A. and M.G. acknowledge the financial support and travel grants from the Jim Blum fund for the BGU Chemical Engineering Water Technology Research Program and EUSMI Grant No. E190700306. N.J.W. and K.P. acknowledge funding from the National Institute of Technology Center for Neutron Research under cooperative Agreement Nos. 70NANB17H302 and 70NANB20H133.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Figure 9 provides a schematic illustration of the geometry of the trough upon deformation. The deformation is defined by means of λ = L/L₀, the extension ratio of the main diagonal ratio where L₀ and L are the initial and deformed lengths of the main diagonal axis as indicated in the illustration.

Two possible schemes are possible for carrying out experiments spanning a wide range of surface concentrations and frequencies:

1. Discrete testing—For each surface concentration tested in either shear or compression, a fresh sample is used. The sample is compressed to the desired surface concentration and allowed to equilibrate, and then either SAOS or SAOD frequency sweep is carried out.

2. Stepwise testing (ST)—The sample is compressed to an initial concentration and allowed to equilibrate; then, SAOS or SAOD frequency sweep is carried out. The interface is now further compressed to a new surface concentration; the surface pressure is allowed to relax to a state of equilibrium followed by the desired rheological test. Up to ten concentration steps have been performed on each sample, with concentrations spanning the range of 19–7 Ų/monomer.

To ascertain the validity of the ST scheme, the complex shear and dilatation moduli as a function of frequency obtained by the two schemes at two different surface concentrations were compared. The first concentration (17 Ų/monomer) was achieved in the ST scheme at the second compression step. The second concentration shown (10 Ų/monomer) was obtained after nine compression steps of the ST scheme. The results provided in Fig. 10 clearly indicate a good agreement between the two procedures.